LMFDB - Embedded newform 112.4.i.b.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,4,Mod(65,112)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("112.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.60821392064$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.4.i.b.65.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(-3.50000 - 6.06218i) q^{5} +(10.0000 + 15.5885i) q^{7} +(13.0000 + 22.5167i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(-3.50000 - 6.06218i) q^{5} +(10.0000 + 15.5885i) q^{7} +(13.0000 + 22.5167i) q^{9} +(17.5000 - 30.3109i) q^{11} +66.0000 q^{13} +7.00000 q^{15} +(-29.5000 + 51.0955i) q^{17} +(68.5000 + 118.645i) q^{19} +(-18.5000 + 0.866025i) q^{21} +(-3.50000 - 6.06218i) q^{23} +(38.0000 - 65.8179i) q^{25} -53.0000 q^{27} +106.000 q^{29} +(37.5000 - 64.9519i) q^{31} +(17.5000 + 30.3109i) q^{33} +(59.5000 - 115.181i) q^{35} +(-5.50000 - 9.52628i) q^{37} +(-33.0000 + 57.1577i) q^{39} -498.000 q^{41} -260.000 q^{43} +(91.0000 - 157.617i) q^{45} +(-85.5000 - 148.090i) q^{47} +(-143.000 + 311.769i) q^{49} +(-29.5000 - 51.0955i) q^{51} +(208.500 - 361.133i) q^{53} -245.000 q^{55} -137.000 q^{57} +(-8.50000 + 14.7224i) q^{59} +(-25.5000 - 44.1673i) q^{61} +(-221.000 + 427.817i) q^{63} +(-231.000 - 400.104i) q^{65} +(219.500 - 380.185i) q^{67} +7.00000 q^{69} +784.000 q^{71} +(-147.500 + 255.477i) q^{73} +(38.0000 + 65.8179i) q^{75} +(647.500 - 30.3109i) q^{77} +(-247.500 - 428.683i) q^{79} +(-324.500 + 562.050i) q^{81} -932.000 q^{83} +413.000 q^{85} +(-53.0000 + 91.7987i) q^{87} +(436.500 + 756.040i) q^{89} +(660.000 + 1028.84i) q^{91} +(37.5000 + 64.9519i) q^{93} +(479.500 - 830.518i) q^{95} -290.000 q^{97} +910.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9}+O(q^{10}) $ 2 * q - q^3 - 7 * q^5 + 20 * q^7 + 26 * q^9	$ 2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9} + 35 q^{11} + 132 q^{13} + 14 q^{15} - 59 q^{17} + 137 q^{19} - 37 q^{21} - 7 q^{23} + 76 q^{25} - 106 q^{27} + 212 q^{29} + 75 q^{31} + 35 q^{33} + 119 q^{35} - 11 q^{37} - 66 q^{39} - 996 q^{41} - 520 q^{43} + 182 q^{45} - 171 q^{47} - 286 q^{49} - 59 q^{51} + 417 q^{53} - 490 q^{55} - 274 q^{57} - 17 q^{59} - 51 q^{61} - 442 q^{63} - 462 q^{65} + 439 q^{67} + 14 q^{69} + 1568 q^{71} - 295 q^{73} + 76 q^{75} + 1295 q^{77} - 495 q^{79} - 649 q^{81} - 1864 q^{83} + 826 q^{85} - 106 q^{87} + 873 q^{89} + 1320 q^{91} + 75 q^{93} + 959 q^{95} - 580 q^{97} + 1820 q^{99}+O(q^{100}) $ 2 * q - q^3 - 7 * q^5 + 20 * q^7 + 26 * q^9 + 35 * q^11 + 132 * q^13 + 14 * q^15 - 59 * q^17 + 137 * q^19 - 37 * q^21 - 7 * q^23 + 76 * q^25 - 106 * q^27 + 212 * q^29 + 75 * q^31 + 35 * q^33 + 119 * q^35 - 11 * q^37 - 66 * q^39 - 996 * q^41 - 520 * q^43 + 182 * q^45 - 171 * q^47 - 286 * q^49 - 59 * q^51 + 417 * q^53 - 490 * q^55 - 274 * q^57 - 17 * q^59 - 51 * q^61 - 442 * q^63 - 462 * q^65 + 439 * q^67 + 14 * q^69 + 1568 * q^71 - 295 * q^73 + 76 * q^75 + 1295 * q^77 - 495 * q^79 - 649 * q^81 - 1864 * q^83 + 826 * q^85 - 106 * q^87 + 873 * q^89 + 1320 * q^91 + 75 * q^93 + 959 * q^95 - 580 * q^97 + 1820 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.0962250	+	0.166667i	−0.910119	−	0.414346i	$-0.864010\pi$
$3$	−0.500000	+	0.866025i	−0.0962250	+	0.166667i	0.813894	+	0.581013i	$0.197344\pi$
$4$		0			0
$5$	−3.50000	−	6.06218i	−0.313050	−	0.542218i	0.665971	−	0.745977i	$-0.268017\pi$
$5$	−3.50000	−	6.06218i	−0.313050	−	0.542218i	−0.979021	+	0.203760i	$0.934684\pi$
$6$		0			0
$7$	10.0000	+	15.5885i	0.539949	+	0.841698i
$7$	10.0000	+	15.5885i	0.539949	+	0.841698i
$8$		0			0
$9$	13.0000	+	22.5167i	0.481481	+	0.833950i
$10$		0			0
$11$	17.5000	−	30.3109i	0.479677	−	0.830825i	−0.520051	−	0.854135i	$-0.674087\pi$
$11$	17.5000	−	30.3109i	0.479677	−	0.830825i	0.999728	+	0.0233099i	$0.00742046\pi$
$12$		0			0
$13$	66.0000			1.40809			0.704043	−	0.710158i	$-0.251376\pi$
$13$	66.0000			1.40809			0.704043	+	0.710158i	$0.251376\pi$
$14$		0			0
$15$	7.00000			0.120493
$16$		0			0
$17$	−29.5000	+	51.0955i	−0.420871	+	0.728969i	−0.996025	−	0.0890757i	$-0.971609\pi$
$17$	−29.5000	+	51.0955i	−0.420871	+	0.728969i	0.575154	+	0.818045i	$0.304942\pi$
$18$		0			0
$19$	68.5000	+	118.645i	0.827104	+	1.43259i	0.900301	+	0.435269i	$0.143347\pi$
$19$	68.5000	+	118.645i	0.827104	+	1.43259i	−0.0731965	+	0.997318i	$0.523320\pi$
$20$		0			0
$21$	−18.5000	+	0.866025i	−0.192240	+	0.00899915i
$22$		0			0
$23$	−3.50000	−	6.06218i	−0.0317305	−	0.0549588i	0.849724	−	0.527228i	$-0.176768\pi$
$23$	−3.50000	−	6.06218i	−0.0317305	−	0.0549588i	−0.881455	+	0.472269i	$0.843435\pi$
$24$		0			0
$25$	38.0000	−	65.8179i	0.304000	−	0.526543i
$26$		0			0
$27$	−53.0000			−0.377772
$28$		0			0
$29$	106.000			0.678748			0.339374	−	0.940651i	$-0.389785\pi$
$29$	106.000			0.678748			0.339374	+	0.940651i	$0.389785\pi$
$30$		0			0
$31$	37.5000	−	64.9519i	0.217264	−	0.376313i	−0.736706	−	0.676213i	$-0.763620\pi$
$31$	37.5000	−	64.9519i	0.217264	−	0.376313i	0.953971	+	0.299900i	$0.0969533\pi$
$32$		0			0
$33$	17.5000	+	30.3109i	0.0923139	+	0.159892i
$34$		0			0
$35$	59.5000	−	115.181i	0.287352	−	0.556263i
$36$		0			0
$37$	−5.50000	−	9.52628i	−0.0244377	−	0.0423273i	0.853548	−	0.521014i	$-0.174446\pi$
$37$	−5.50000	−	9.52628i	−0.0244377	−	0.0423273i	−0.877986	+	0.478687i	$0.841113\pi$
$38$		0			0
$39$	−33.0000	+	57.1577i	−0.135493	+	0.234681i
$40$		0			0
$41$	−498.000			−1.89694			−0.948470	−	0.316867i	$-0.897369\pi$
$41$	−498.000			−1.89694			−0.948470	+	0.316867i	$0.897369\pi$
$42$		0			0
$43$	−260.000			−0.922084			−0.461042	−	0.887378i	$-0.652524\pi$
$43$	−260.000			−0.922084			−0.461042	+	0.887378i	$0.652524\pi$
$44$		0			0
$45$	91.0000	−	157.617i	0.301455	−	0.522136i
$46$		0			0
$47$	−85.5000	−	148.090i	−0.265350	−	0.459600i	0.702305	−	0.711876i	$-0.252154\pi$
$47$	−85.5000	−	148.090i	−0.265350	−	0.459600i	−0.967655	+	0.252276i	$0.918821\pi$
$48$		0			0
$49$	−143.000	+	311.769i	−0.416910	+	0.908948i
$50$		0			0
$51$	−29.5000	−	51.0955i	−0.0809966	−	0.140290i
$52$		0			0
$53$	208.500	−	361.133i	0.540371	−	0.935951i	−0.458511	−	0.888689i	$-0.651617\pi$
$53$	208.500	−	361.133i	0.540371	−	0.935951i	0.998883	−	0.0472619i	$-0.0150495\pi$
$54$		0			0
$55$	−245.000			−0.600651
$56$		0			0
$57$	−137.000			−0.318353
$58$		0			0
$59$	−8.50000	+	14.7224i	−0.0187560	+	0.0324864i	−0.875251	−	0.483669i	$-0.839304\pi$
$59$	−8.50000	+	14.7224i	−0.0187560	+	0.0324864i	0.856495	+	0.516155i	$0.172637\pi$
$60$		0			0
$61$	−25.5000	−	44.1673i	−0.0535236	−	0.0927056i	0.838022	−	0.545636i	$-0.183712\pi$
$61$	−25.5000	−	44.1673i	−0.0535236	−	0.0927056i	−0.891546	+	0.452930i	$0.850379\pi$
$62$		0			0
$63$	−221.000	+	427.817i	−0.441958	+	0.855553i
$64$		0			0
$65$	−231.000	−	400.104i	−0.440800	−	0.763489i
$66$		0			0
$67$	219.500	−	380.185i	0.400242	−	0.693239i	−0.593513	−	0.804824i	$-0.702260\pi$
$67$	219.500	−	380.185i	0.400242	−	0.693239i	0.993755	+	0.111585i	$0.0355928\pi$
$68$		0			0
$69$	7.00000			0.0122131
$70$		0			0
$71$	784.000			1.31047			0.655237	−	0.755423i	$-0.272569\pi$
$71$	784.000			1.31047			0.655237	+	0.755423i	$0.272569\pi$
$72$		0			0
$73$	−147.500	+	255.477i	−0.236487	+	0.409608i	−0.959704	−	0.281013i	$-0.909329\pi$
$73$	−147.500	+	255.477i	−0.236487	+	0.409608i	0.723217	+	0.690621i	$0.242663\pi$
$74$		0			0
$75$	38.0000	+	65.8179i	0.0585048	+	0.101333i
$76$		0			0
$77$	647.500	−	30.3109i	0.958305	−	0.0448603i
$78$		0			0
$79$	−247.500	−	428.683i	−0.352480	−	0.610513i	0.634203	−	0.773166i	$-0.281328\pi$
$79$	−247.500	−	428.683i	−0.352480	−	0.610513i	−0.986683	+	0.162653i	$0.947995\pi$
$80$		0			0
$81$	−324.500	+	562.050i	−0.445130	+	0.770988i
$82$		0			0
$83$	−932.000			−1.23253			−0.616267	−	0.787537i	$-0.711356\pi$
$83$	−932.000			−1.23253			−0.616267	+	0.787537i	$0.711356\pi$
$84$		0			0
$85$	413.000			0.527013
$86$		0			0
$87$	−53.0000	+	91.7987i	−0.0653126	+	0.113125i
$88$		0			0
$89$	436.500	+	756.040i	0.519875	+	0.900451i	0.999733	+	0.0231042i	$0.00735495\pi$
$89$	436.500	+	756.040i	0.519875	+	0.900451i	−0.479858	+	0.877346i	$0.659312\pi$
$90$		0			0
$91$	660.000	+	1028.84i	0.760294	+	1.18518i
$92$		0			0
$93$	37.5000	+	64.9519i	0.0418126	+	0.0724215i
$94$		0			0
$95$	479.500	−	830.518i	0.517849	−	0.896941i
$96$		0			0
$97$	−290.000			−0.303557			−0.151779	−	0.988415i	$-0.548500\pi$
$97$	−290.000			−0.303557			−0.151779	+	0.988415i	$0.548500\pi$
$98$		0			0
$99$	910.000			0.923823
$100$		0			0
$101$	542.500	−	939.638i	0.534463	−	0.925717i	−0.464726	−	0.885454i	$-0.653847\pi$
$101$	542.500	−	939.638i	0.534463	−	0.925717i	0.999189	−	0.0402627i	$-0.0128195\pi$
$102$		0			0
$103$	776.500	+	1344.94i	0.742823	+	1.28661i	0.951205	+	0.308560i	$0.0998472\pi$
$103$	776.500	+	1344.94i	0.742823	+	1.28661i	−0.208381	+	0.978048i	$0.566819\pi$
$104$		0			0
$105$	70.0000	+	109.119i	0.0650600	+	0.101419i
$106$		0			0
$107$	64.5000	+	111.717i	0.0582752	+	0.100936i	0.893691	−	0.448682i	$-0.148107\pi$
$107$	64.5000	+	111.717i	0.0582752	+	0.100936i	−0.835416	+	0.549618i	$0.814773\pi$
$108$		0			0
$109$	482.500	−	835.715i	0.423992	−	0.734376i	−0.572334	−	0.820021i	$-0.693962\pi$
$109$	482.500	−	835.715i	0.423992	−	0.734376i	0.996326	+	0.0856452i	$0.0272952\pi$
$110$		0			0
$111$	11.0000			0.00940607
$112$		0			0
$113$	−50.0000			−0.0416248			−0.0208124	−	0.999783i	$-0.506625\pi$
$113$	−50.0000			−0.0416248			−0.0208124	+	0.999783i	$0.506625\pi$
$114$		0			0
$115$	−24.5000	+	42.4352i	−0.0198664	+	0.0344096i
$116$		0			0
$117$	858.000	+	1486.10i	0.677967	+	1.17427i
$118$		0			0
$119$	−1091.50	+	51.0955i	−0.840821	+	0.0393606i
$120$		0			0
$121$	53.0000	+	91.7987i	0.0398197	+	0.0689697i
$122$		0			0
$123$	249.000	−	431.281i	0.182533	−	0.316157i
$124$		0			0
$125$	−1407.00			−1.00677
$126$		0			0
$127$	−936.000			−0.653989			−0.326994	−	0.945026i	$-0.606036\pi$
$127$	−936.000			−0.653989			−0.326994	+	0.945026i	$0.606036\pi$
$128$		0			0
$129$	130.000	−	225.167i	0.0887276	−	0.153681i
$130$		0			0
$131$	−377.500	−	653.849i	−0.251773	−	0.436084i	0.712241	−	0.701935i	$-0.247680\pi$
$131$	−377.500	−	653.849i	−0.251773	−	0.436084i	−0.964014	+	0.265851i	$0.914347\pi$
$132$		0			0
$133$	−1164.50	+	2254.26i	−0.759210	+	1.46970i
$134$		0			0
$135$	185.500	+	321.295i	0.118261	+	0.204835i
$136$		0			0
$137$	1178.50	−	2041.22i	0.734935	−	1.27294i	−0.219817	−	0.975541i	$-0.570546\pi$
$137$	1178.50	−	2041.22i	0.734935	−	1.27294i	0.954752	−	0.297403i	$-0.0961205\pi$
$138$		0			0
$139$	−28.0000			−0.0170858			−0.00854291	−	0.999964i	$-0.502719\pi$
$139$	−28.0000			−0.0170858			−0.00854291	+	0.999964i	$0.502719\pi$
$140$		0			0
$141$	171.000			0.102133
$142$		0			0
$143$	1155.00	−	2000.52i	0.675426	−	1.16987i
$144$		0			0
$145$	−371.000	−	642.591i	−0.212482	−	0.368029i
$146$		0			0
$147$	−198.500	−	279.726i	−0.111374	−	0.156948i
$148$		0			0
$149$	−1147.50	−	1987.53i	−0.630919	−	1.09278i	−0.987364	−	0.158467i	$-0.949345\pi$
$149$	−1147.50	−	1987.53i	−0.630919	−	1.09278i	0.356446	−	0.934316i	$-0.383988\pi$
$150$		0			0
$151$	−554.500	+	960.422i	−0.298838	+	0.517603i	−0.975870	−	0.218350i	$-0.929932\pi$
$151$	−554.500	+	960.422i	−0.298838	+	0.517603i	0.677032	+	0.735953i	$0.263266\pi$
$152$		0			0
$153$	−1534.00			−0.810566
$154$		0			0
$155$	−525.000			−0.272058
$156$		0			0
$157$	−779.500	+	1350.13i	−0.396248	+	0.686321i	−0.993260	−	0.115911i	$-0.963021\pi$
$157$	−779.500	+	1350.13i	−0.396248	+	0.686321i	0.597012	+	0.802232i	$0.296354\pi$
$158$		0			0
$159$	208.500	+	361.133i	0.103995	+	0.180124i
$160$		0			0
$161$	59.5000	−	115.181i	0.0291258	−	0.0563824i
$162$		0			0
$163$	−1125.50	−	1949.42i	−0.540834	−	0.936752i	−0.998856	−	0.0478115i	$-0.984775\pi$
$163$	−1125.50	−	1949.42i	−0.540834	−	0.936752i	0.458022	−	0.888941i	$-0.348558\pi$
$164$		0			0
$165$	122.500	−	212.176i	0.0577976	−	0.100108i
$166$		0			0
$167$	−2788.00			−1.29187			−0.645934	−	0.763393i	$-0.723532\pi$
$167$	−2788.00			−1.29187			−0.645934	+	0.763393i	$0.723532\pi$
$168$		0			0
$169$	2159.00			0.982704
$170$		0			0
$171$	−1781.00	+	3084.78i	−0.796471	+	1.37953i
$172$		0			0
$173$	−789.500	−	1367.45i	−0.346963	−	0.600957i	0.638746	−	0.769418i	$-0.279454\pi$
$173$	−789.500	−	1367.45i	−0.346963	−	0.600957i	−0.985708	+	0.168461i	$0.946120\pi$
$174$		0			0
$175$	1406.00	−	65.8179i	0.607335	−	0.0284307i
$176$		0			0
$177$	−8.50000	−	14.7224i	−0.00360960	−	0.00625201i
$178$		0			0
$179$	1225.50	−	2122.63i	0.511722	−	0.886328i	−0.488186	−	0.872740i	$-0.662341\pi$
$179$	1225.50	−	2122.63i	0.511722	−	0.886328i	0.999908	−	0.0135883i	$-0.00432541\pi$
$180$		0			0
$181$	−1170.00			−0.480472			−0.240236	−	0.970715i	$-0.577225\pi$
$181$	−1170.00			−0.480472			−0.240236	+	0.970715i	$0.577225\pi$
$182$		0			0
$183$	51.0000			0.0206012
$184$		0			0
$185$	−38.5000	+	66.6840i	−0.0153004	+	0.0265011i
$186$		0			0
$187$	1032.50	+	1788.34i	0.403764	+	0.699340i
$188$		0			0
$189$	−530.000	−	826.188i	−0.203978	−	0.317970i
$190$		0			0
$191$	−637.500	−	1104.18i	−0.241507	−	0.418303i	0.719637	−	0.694351i	$-0.244308\pi$
$191$	−637.500	−	1104.18i	−0.241507	−	0.418303i	−0.961144	+	0.276048i	$0.910975\pi$
$192$		0			0
$193$	−17.5000	+	30.3109i	−0.00652683	+	0.0113048i	−0.869270	−	0.494337i	$-0.835411\pi$
$193$	−17.5000	+	30.3109i	−0.00652683	+	0.0113048i	0.862744	+	0.505642i	$0.168744\pi$
$194$		0			0
$195$	462.000			0.169664
$196$		0			0
$197$	−2734.00			−0.988779			−0.494389	−	0.869241i	$-0.664608\pi$
$197$	−2734.00			−0.988779			−0.494389	+	0.869241i	$0.664608\pi$
$198$		0			0
$199$	1121.50	−	1942.49i	0.399503	−	0.691959i	−0.594162	−	0.804345i	$-0.702516\pi$
$199$	1121.50	−	1942.49i	0.399503	−	0.691959i	0.993665	+	0.112387i	$0.0358495\pi$
$200$		0			0
$201$	219.500	+	380.185i	0.0770265	+	0.133414i
$202$		0			0
$203$	1060.00	+	1652.38i	0.366490	+	0.571301i
$204$		0			0
$205$	1743.00	+	3018.96i	0.593836	+	1.02855i
$206$		0			0
$207$	91.0000	−	157.617i	0.0305553	−	0.0529232i
$208$		0			0
$209$	4795.00			1.58697
$210$		0			0
$211$	−1172.00			−0.382388			−0.191194	−	0.981552i	$-0.561236\pi$
$211$	−1172.00			−0.382388			−0.191194	+	0.981552i	$0.561236\pi$
$212$		0			0
$213$	−392.000	+	678.964i	−0.126100	+	0.218412i
$214$		0			0
$215$	910.000	+	1576.17i	0.288658	+	0.499970i
$216$		0			0
$217$	1387.50	−	64.9519i	0.434054	−	0.0203190i
$218$		0			0
$219$	−147.500	−	255.477i	−0.0455120	−	0.0788291i
$220$		0			0
$221$	−1947.00	+	3372.30i	−0.592622	+	1.02645i
$222$		0			0
$223$	−2024.00			−0.607790			−0.303895	−	0.952706i	$-0.598287\pi$
$223$	−2024.00			−0.607790			−0.303895	+	0.952706i	$0.598287\pi$
$224$		0			0
$225$	1976.00			0.585481
$226$		0			0
$227$	1285.50	−	2226.55i	0.375866	−	0.651019i	−0.614590	−	0.788847i	$-0.710679\pi$
$227$	1285.50	−	2226.55i	0.375866	−	0.651019i	0.990456	+	0.137827i	$0.0440119\pi$
$228$		0			0
$229$	−447.500	−	775.093i	−0.129134	−	0.223666i	0.794207	−	0.607647i	$-0.207886\pi$
$229$	−447.500	−	775.093i	−0.129134	−	0.223666i	−0.923341	+	0.383980i	$0.874553\pi$
$230$		0			0
$231$	−297.500	+	575.907i	−0.0847362	+	0.164034i
$232$		0			0
$233$	−893.500	−	1547.59i	−0.251224	−	0.435132i	0.712639	−	0.701531i	$-0.247500\pi$
$233$	−893.500	−	1547.59i	−0.251224	−	0.435132i	−0.963863	+	0.266398i	$0.914166\pi$
$234$		0			0
$235$	−598.500	+	1036.63i	−0.166135	+	0.287755i
$236$		0			0
$237$	495.000			0.135670
$238$		0			0
$239$	5100.00			1.38030			0.690150	−	0.723667i	$-0.257545\pi$
$239$	5100.00			1.38030			0.690150	+	0.723667i	$0.257545\pi$
$240$		0			0
$241$	2088.50	−	3617.39i	0.558225	−	0.966873i	−0.439420	−	0.898282i	$-0.644816\pi$
$241$	2088.50	−	3617.39i	0.558225	−	0.966873i	0.997645	−	0.0685917i	$-0.0218506\pi$
$242$		0			0
$243$	−1040.00	−	1801.33i	−0.274552	−	0.475537i
$244$		0			0
$245$	2390.50	−	224.301i	0.623361	−	0.0584900i
$246$		0			0
$247$	4521.00	+	7830.60i	1.16463	+	2.01720i
$248$		0			0
$249$	466.000	−	807.136i	0.118601	−	0.205422i
$250$		0			0
$251$	4680.00			1.17689			0.588444	−	0.808538i	$-0.299741\pi$
$251$	4680.00			1.17689			0.588444	+	0.808538i	$0.299741\pi$
$252$		0			0
$253$	−245.000			−0.0608815
$254$		0			0
$255$	−206.500	+	357.668i	−0.0507119	+	0.0878356i
$256$		0			0
$257$	874.500	+	1514.68i	0.212256	+	0.367638i	0.952420	−	0.304788i	$-0.0985856\pi$
$257$	874.500	+	1514.68i	0.212256	+	0.367638i	−0.740164	+	0.672426i	$0.765252\pi$
$258$		0			0
$259$	93.5000	−	180.999i	0.0224317	−	0.0434237i
$260$		0			0
$261$	1378.00	+	2386.77i	0.326805	+	0.566043i
$262$		0			0
$263$	−2236.50	+	3873.73i	−0.524367	+	0.908230i	0.475231	+	0.879861i	$0.342365\pi$
$263$	−2236.50	+	3873.73i	−0.524367	+	0.908230i	−0.999598	+	0.0283689i	$0.990969\pi$
$264$		0			0
$265$	−2919.00			−0.676652
$266$		0			0
$267$	−873.000			−0.200100
$268$		0			0
$269$	−987.500	+	1710.40i	−0.223825	+	0.387676i	−0.955966	−	0.293476i	$-0.905188\pi$
$269$	−987.500	+	1710.40i	−0.223825	+	0.387676i	0.732141	+	0.681153i	$0.238521\pi$
$270$		0			0
$271$	−4219.50	−	7308.39i	−0.945817	−	1.63820i	−0.754107	−	0.656751i	$-0.771930\pi$
$271$	−4219.50	−	7308.39i	−0.945817	−	1.63820i	−0.191710	−	0.981452i	$-0.561403\pi$
$272$		0			0
$273$	−1221.00	+	57.1577i	−0.270690	+	0.0126716i
$274$		0			0
$275$	−1330.00	−	2303.63i	−0.291644	−	0.505142i
$276$		0			0
$277$	−263.500	+	456.395i	−0.0571559	+	0.0989969i	−0.893188	−	0.449684i	$-0.851537\pi$
$277$	−263.500	+	456.395i	−0.0571559	+	0.0989969i	0.836032	+	0.548681i	$0.184870\pi$
$278$		0			0
$279$	1950.00			0.418435
$280$		0			0
$281$	−202.000			−0.0428837			−0.0214418	−	0.999770i	$-0.506826\pi$
$281$	−202.000			−0.0428837			−0.0214418	+	0.999770i	$0.506826\pi$
$282$		0			0
$283$	−3974.50	+	6884.04i	−0.834839	+	1.44598i	0.0593220	+	0.998239i	$0.481106\pi$
$283$	−3974.50	+	6884.04i	−0.834839	+	1.44598i	−0.894161	+	0.447745i	$0.852227\pi$
$284$		0			0
$285$	479.500	+	830.518i	0.0996601	+	0.172616i
$286$		0			0
$287$	−4980.00	−	7763.05i	−1.02425	−	1.59665i
$288$		0			0
$289$	716.000	+	1240.15i	0.145736	+	0.252422i
$290$		0			0
$291$	145.000	−	251.147i	0.0292098	−	0.0505929i
$292$		0			0
$293$	318.000			0.0634053			0.0317027	−	0.999497i	$-0.489907\pi$
$293$	318.000			0.0634053			0.0317027	+	0.999497i	$0.489907\pi$
$294$		0			0
$295$	119.000			0.0234863
$296$		0			0
$297$	−927.500	+	1606.48i	−0.181209	+	0.313863i
$298$		0			0
$299$	−231.000	−	400.104i	−0.0446792	−	0.0773866i
$300$		0			0
$301$	−2600.00	−	4053.00i	−0.497879	−	0.776116i
$302$		0			0
$303$	542.500	+	939.638i	0.102857	+	0.178154i
$304$		0			0
$305$	−178.500	+	309.171i	−0.0335111	+	0.0580429i
$306$		0			0
$307$	8132.00			1.51178			0.755892	−	0.654696i	$-0.227203\pi$
$307$	8132.00			1.51178			0.755892	+	0.654696i	$0.227203\pi$
$308$		0			0
$309$	−1553.00			−0.285913
$310$		0			0
$311$	−464.500	+	804.538i	−0.0846925	+	0.146692i	−0.905260	−	0.424858i	$-0.860324\pi$
$311$	−464.500	+	804.538i	−0.0846925	+	0.146692i	0.820568	+	0.571549i	$0.193657\pi$
$312$		0			0
$313$	104.500	+	180.999i	0.0188712	+	0.0326859i	0.875307	−	0.483568i	$-0.160659\pi$
$313$	104.500	+	180.999i	0.0188712	+	0.0326859i	−0.856436	+	0.516254i	$0.827326\pi$
$314$		0			0
$315$	3367.00	−	157.617i	0.602251	−	0.0281927i
$316$		0			0
$317$	−3565.50	−	6175.63i	−0.631730	−	1.09419i	−0.987198	−	0.159500i	$-0.949012\pi$
$317$	−3565.50	−	6175.63i	−0.631730	−	1.09419i	0.355468	−	0.934689i	$-0.384322\pi$
$318$		0			0
$319$	1855.00	−	3212.95i	0.325580	−	0.563921i
$320$		0			0
$321$	−129.000			−0.0224301
$322$		0			0
$323$	−8083.00			−1.39242
$324$		0			0
$325$	2508.00	−	4343.98i	0.428058	−	0.741418i
$326$		0			0
$327$	482.500	+	835.715i	0.0815973	+	0.141331i
$328$		0			0
$329$	1453.50	−	2813.72i	0.243569	−	0.471505i
$330$		0			0
$331$	−3285.50	−	5690.65i	−0.545581	−	0.944975i	−0.998570	−	0.0534583i	$-0.982976\pi$
$331$	−3285.50	−	5690.65i	−0.545581	−	0.944975i	0.452989	−	0.891516i	$-0.350358\pi$
$332$		0			0
$333$	143.000	−	247.683i	0.0235326	−	0.0407596i
$334$		0			0
$335$	−3073.00			−0.501182
$336$		0			0
$337$	−11466.0			−1.85339			−0.926696	−	0.375813i	$-0.877364\pi$
$337$	−11466.0			−1.85339			−0.926696	+	0.375813i	$0.877364\pi$
$338$		0			0
$339$	25.0000	−	43.3013i	0.00400535	−	0.00693747i
$340$		0			0
$341$	−1312.50	−	2273.32i	−0.208434	−	0.361018i
$342$		0			0
$343$	−6290.00	+	888.542i	−0.990169	+	0.139874i
$344$		0			0
$345$	−24.5000	−	42.4352i	−0.00382329	−	0.00662214i
$346$		0			0
$347$	−4888.50	+	8467.13i	−0.756278	+	1.30991i	0.188459	+	0.982081i	$0.439651\pi$
$347$	−4888.50	+	8467.13i	−0.756278	+	1.30991i	−0.944737	+	0.327831i	$0.893682\pi$
$348$		0			0
$349$	11914.0			1.82734			0.913670	−	0.406456i	$-0.133236\pi$
$349$	11914.0			1.82734			0.913670	+	0.406456i	$0.133236\pi$
$350$		0			0
$351$	−3498.00			−0.531936
$352$		0			0
$353$	−4561.50	+	7900.75i	−0.687774	+	1.19126i	0.284783	+	0.958592i	$0.408079\pi$
$353$	−4561.50	+	7900.75i	−0.687774	+	1.19126i	−0.972556	+	0.232667i	$0.925255\pi$
$354$		0			0
$355$	−2744.00	−	4752.75i	−0.410243	−	0.710562i
$356$		0			0
$357$	501.500	−	970.814i	0.0743479	−	0.143924i
$358$		0			0
$359$	4074.50	+	7057.24i	0.599008	+	1.03751i	0.992968	+	0.118385i	$0.0377716\pi$
$359$	4074.50	+	7057.24i	0.599008	+	1.03751i	−0.393960	+	0.919128i	$0.628895\pi$
$360$		0			0
$361$	−5955.00	+	10314.4i	−0.868202	+	1.50377i
$362$		0			0
$363$	−106.000			−0.0153266
$364$		0			0
$365$	2065.00			0.296129
$366$		0			0
$367$	4835.50	−	8375.33i	0.687769	−	1.19125i	−0.284790	−	0.958590i	$-0.591924\pi$
$367$	4835.50	−	8375.33i	0.687769	−	1.19125i	0.972558	−	0.232660i	$-0.0747429\pi$
$368$		0			0
$369$	−6474.00	−	11213.3i	−0.913341	−	1.58195i
$370$		0			0
$371$	7714.50	−	361.133i	1.07956	−	0.0505366i
$372$		0			0
$373$	2054.50	+	3558.50i	0.285196	+	0.493973i	0.972657	−	0.232248i	$-0.0746081\pi$
$373$	2054.50	+	3558.50i	0.285196	+	0.493973i	−0.687461	+	0.726221i	$0.741275\pi$
$374$		0			0
$375$	703.500	−	1218.50i	0.0968762	−	0.167795i
$376$		0			0
$377$	6996.00			0.955736
$378$		0			0
$379$	3488.00			0.472735			0.236367	−	0.971664i	$-0.424043\pi$
$379$	3488.00			0.472735			0.236367	+	0.971664i	$0.424043\pi$
$380$		0			0
$381$	468.000	−	810.600i	0.0629301	−	0.108998i
$382$		0			0
$383$	4358.50	+	7549.14i	0.581485	+	1.00716i	0.995304	+	0.0968028i	$0.0308616\pi$
$383$	4358.50	+	7549.14i	0.581485	+	1.00716i	−0.413818	+	0.910360i	$0.635805\pi$
$384$		0			0
$385$	−2450.00	−	3819.17i	−0.324321	−	0.505566i
$386$		0			0
$387$	−3380.00	−	5854.33i	−0.443967	−	0.768973i
$388$		0			0
$389$	−81.5000	+	141.162i	−0.0106227	+	0.0183990i	−0.871288	−	0.490772i	$-0.836715\pi$
$389$	−81.5000	+	141.162i	−0.0106227	+	0.0183990i	0.860665	+	0.509171i	$0.170048\pi$
$390$		0			0
$391$	413.000			0.0534177
$392$		0			0
$393$	755.000			0.0969077
$394$		0			0
$395$	−1732.50	+	3000.78i	−0.220687	+	0.382242i
$396$		0			0
$397$	−499.500	−	865.159i	−0.0631466	−	0.109373i	0.832724	−	0.553689i	$-0.186780\pi$
$397$	−499.500	−	865.159i	−0.0631466	−	0.109373i	−0.895870	+	0.444316i	$0.853447\pi$
$398$		0			0
$399$	−1370.00	−	2135.62i	−0.171894	−	0.267957i
$400$		0			0
$401$	7378.50	+	12779.9i	0.918865	+	1.59152i	0.801143	+	0.598474i	$0.204226\pi$
$401$	7378.50	+	12779.9i	0.918865	+	1.59152i	0.117722	+	0.993047i	$0.462441\pi$
$402$		0			0
$403$	2475.00	−	4286.83i	0.305927	−	0.529881i
$404$		0			0
$405$	4543.00			0.557391
$406$		0			0
$407$	−385.000			−0.0468888
$408$		0			0
$409$	66.5000	−	115.181i	0.00803964	−	0.0139251i	−0.861978	−	0.506946i	$-0.830774\pi$
$409$	66.5000	−	115.181i	0.00803964	−	0.0139251i	0.870017	+	0.493021i	$0.164108\pi$
$410$		0			0
$411$	1178.50	+	2041.22i	0.141438	+	0.244978i
$412$		0			0
$413$	−314.500	+	14.7224i	−0.0374710	+	0.00175410i
$414$		0			0
$415$	3262.00	+	5649.95i	0.385844	+	0.668302i
$416$		0			0
$417$	14.0000	−	24.2487i	0.00164408	−	0.00284764i
$418$		0			0
$419$	6420.00			0.748538			0.374269	−	0.927320i	$-0.377894\pi$
$419$	6420.00			0.748538			0.374269	+	0.927320i	$0.377894\pi$
$420$		0			0
$421$	10266.0			1.18844			0.594221	−	0.804302i	$-0.297460\pi$
$421$	10266.0			1.18844			0.594221	+	0.804302i	$0.297460\pi$
$422$		0			0
$423$	2223.00	−	3850.35i	0.255522	−	0.442578i
$424$		0			0
$425$	2242.00	+	3883.26i	0.255889	+	0.443213i
$426$		0			0
$427$	433.500	−	839.179i	0.0491301	−	0.0951070i
$428$		0			0
$429$	1155.00	+	2000.52i	0.129986	+	0.225142i
$430$		0			0
$431$	−7606.50	+	13174.8i	−0.850098	+	1.47241i	0.0310213	+	0.999519i	$0.490124\pi$
$431$	−7606.50	+	13174.8i	−0.850098	+	1.47241i	−0.881119	+	0.472894i	$0.843209\pi$
$432$		0			0
$433$	−1378.00			−0.152939			−0.0764693	−	0.997072i	$-0.524365\pi$
$433$	−1378.00			−0.152939			−0.0764693	+	0.997072i	$0.524365\pi$
$434$		0			0
$435$	742.000			0.0817843
$436$		0			0
$437$	479.500	−	830.518i	0.0524888	−	0.0909132i
$438$		0			0
$439$	−1381.50	−	2392.83i	−0.150195	−	0.260145i	0.781104	−	0.624401i	$-0.214657\pi$
$439$	−1381.50	−	2392.83i	−0.150195	−	0.260145i	−0.931299	+	0.364256i	$0.881323\pi$
$440$		0			0
$441$	−8879.00	+	833.116i	−0.958752	+	0.0899597i
$442$		0			0
$443$	2924.50	+	5065.38i	0.313651	+	0.543259i	0.979150	−	0.203140i	$-0.0651146\pi$
$443$	2924.50	+	5065.38i	0.313651	+	0.543259i	−0.665499	+	0.746399i	$0.731781\pi$
$444$		0			0
$445$	3055.50	−	5292.28i	0.325493	−	0.563771i
$446$		0			0
$447$	2295.00			0.242841
$448$		0			0
$449$	4582.00			0.481599			0.240799	−	0.970575i	$-0.422590\pi$
$449$	4582.00			0.481599			0.240799	+	0.970575i	$0.422590\pi$
$450$		0			0
$451$	−8715.00	+	15094.8i	−0.909919	+	1.57603i
$452$		0			0
$453$	−554.500	−	960.422i	−0.0575114	−	0.0996127i
$454$		0			0
$455$	3927.00	−	7601.97i	0.404617	−	0.783266i
$456$		0			0
$457$	−5775.50	−	10003.5i	−0.591174	−	1.02394i	−0.994075	−	0.108700i	$-0.965331\pi$
$457$	−5775.50	−	10003.5i	−0.591174	−	1.02394i	0.402901	−	0.915244i	$-0.368002\pi$
$458$		0			0
$459$	1563.50	−	2708.06i	0.158993	−	0.275384i
$460$		0			0
$461$	−9494.00			−0.959175			−0.479587	−	0.877494i	$-0.659214\pi$
$461$	−9494.00			−0.959175			−0.479587	+	0.877494i	$0.659214\pi$
$462$		0			0
$463$	10160.0			1.01982			0.509908	−	0.860229i	$-0.329679\pi$
$463$	10160.0			1.01982			0.509908	+	0.860229i	$0.329679\pi$
$464$		0			0
$465$	262.500	−	454.663i	0.0261788	−	0.0453430i
$466$		0			0
$467$	−653.500	−	1131.90i	−0.0647545	−	0.112158i	0.831831	−	0.555030i	$-0.187293\pi$
$467$	−653.500	−	1131.90i	−0.0647545	−	0.112158i	−0.896585	+	0.442872i	$0.853960\pi$
$468$		0			0
$469$	8121.50	−	380.185i	0.799608	−	0.0374314i
$470$		0			0
$471$	−779.500	−	1350.13i	−0.0762579	−	0.132083i
$472$		0			0
$473$	−4550.00	+	7880.83i	−0.442303	+	0.766091i
$474$		0			0
$475$	10412.0			1.00576
$476$		0			0
$477$	10842.0			1.04072
$478$		0			0
$479$	9143.50	−	15837.0i	0.872186	−	1.51067i	0.0124559	−	0.999922i	$-0.496035\pi$
$479$	9143.50	−	15837.0i	0.872186	−	1.51067i	0.859730	−	0.510748i	$-0.170632\pi$
$480$		0			0
$481$	−363.000	−	628.734i	−0.0344103	−	0.0596005i
$482$		0			0
$483$	70.0000	+	109.119i	0.00659443	+	0.0102797i
$484$		0			0
$485$	1015.00	+	1758.03i	0.0950284	+	0.164594i
$486$		0			0
$487$	−7476.50	+	12949.7i	−0.695673	+	1.20494i	0.274281	+	0.961650i	$0.411560\pi$
$487$	−7476.50	+	12949.7i	−0.695673	+	1.20494i	−0.969953	+	0.243291i	$0.921773\pi$
$488$		0			0
$489$	2251.00			0.208167
$490$		0			0
$491$	−14352.0			−1.31914			−0.659569	−	0.751644i	$-0.729261\pi$
$491$	−14352.0			−1.31914			−0.659569	+	0.751644i	$0.729261\pi$
$492$		0			0
$493$	−3127.00	+	5416.12i	−0.285665	+	0.494787i
$494$		0			0
$495$	−3185.00	−	5516.58i	−0.289202	−	0.500913i
$496$		0			0
$497$	7840.00	+	12221.4i	0.707590	+	1.10302i
$498$		0			0
$499$	−2765.50	−	4789.99i	−0.248098	−	0.429718i	0.714900	−	0.699226i	$-0.246472\pi$
$499$	−2765.50	−	4789.99i	−0.248098	−	0.429718i	−0.962998	+	0.269509i	$0.913139\pi$
$500$		0			0
$501$	1394.00	−	2414.48i	0.124310	−	0.215311i
$502$		0			0
$503$	−8400.00			−0.744607			−0.372304	−	0.928111i	$-0.621432\pi$
$503$	−8400.00			−0.744607			−0.372304	+	0.928111i	$0.621432\pi$
$504$		0			0
$505$	−7595.00			−0.669254
$506$		0			0
$507$	−1079.50	+	1869.75i	−0.0945607	+	0.163784i
$508$		0			0
$509$	1192.50	+	2065.47i	0.103844	+	0.179863i	0.913265	−	0.407365i	$-0.133552\pi$
$509$	1192.50	+	2065.47i	0.103844	+	0.179863i	−0.809421	+	0.587228i	$0.800219\pi$
$510$		0			0
$511$	−5457.50	+	255.477i	−0.472457	+	0.0221167i
$512$		0			0
$513$	−3630.50	−	6288.21i	−0.312457	−	0.541192i
$514$		0			0
$515$	5435.50	−	9414.56i	0.465081	−	0.805544i
$516$		0			0
$517$	−5985.00			−0.509130
$518$		0			0
$519$	1579.00			0.133546
$520$		0			0
$521$	4576.50	−	7926.73i	0.384837	−	0.666557i	−0.606910	−	0.794771i	$-0.707591\pi$
$521$	4576.50	−	7926.73i	0.384837	−	0.666557i	0.991747	+	0.128214i	$0.0409243\pi$
$522$		0			0
$523$	−6903.50	−	11957.2i	−0.577187	−	0.999718i	−0.995800	−	0.0915530i	$-0.970817\pi$
$523$	−6903.50	−	11957.2i	−0.577187	−	0.999718i	0.418613	−	0.908165i	$-0.362516\pi$
$524$		0			0
$525$	−646.000	+	1250.54i	−0.0537024	+	0.103958i
$526$		0			0
$527$	2212.50	+	3832.16i	0.182880	+	0.316758i
$528$		0			0
$529$	6059.00	−	10494.5i	0.497986	−	0.862538i
$530$		0			0
$531$	−442.000			−0.0361227
$532$		0			0
$533$	−32868.0			−2.67105
$534$		0			0
$535$	451.500	−	782.021i	0.0364861	−	0.0631957i
$536$		0			0
$537$	1225.50	+	2122.63i	0.0984809	+	0.170574i
$538$		0			0
$539$	6947.50	+	9790.42i	0.555195	+	0.782381i
$540$		0			0
$541$	−4087.50	−	7079.76i	−0.324834	−	0.562629i	0.656645	−	0.754200i	$-0.271975\pi$
$541$	−4087.50	−	7079.76i	−0.324834	−	0.562629i	−0.981479	+	0.191571i	$0.938642\pi$
$542$		0			0
$543$	585.000	−	1013.25i	0.0462334	−	0.0800787i
$544$		0			0
$545$	−6755.00			−0.530922
$546$		0			0
$547$	−4656.00			−0.363942			−0.181971	−	0.983304i	$-0.558248\pi$
$547$	−4656.00			−0.363942			−0.181971	+	0.983304i	$0.558248\pi$
$548$		0			0
$549$	663.000	−	1148.35i	0.0515413	−	0.0892721i
$550$		0			0
$551$	7261.00	+	12576.4i	0.561396	+	0.972366i
$552$		0			0
$553$	4207.50	−	8144.97i	0.323546	−	0.626328i
$554$		0			0
$555$	−38.5000	−	66.6840i	−0.00294457	−	0.00510014i
$556$		0			0
$557$	−3501.50	+	6064.78i	−0.266361	+	0.461352i	−0.967919	−	0.251261i	$-0.919155\pi$
$557$	−3501.50	+	6064.78i	−0.266361	+	0.461352i	0.701558	+	0.712612i	$0.252488\pi$
$558$		0			0
$559$	−17160.0			−1.29837
$560$		0			0
$561$	−2065.00			−0.155409
$562$		0			0
$563$	−9876.50	+	17106.6i	−0.739334	+	1.28056i	0.213462	+	0.976951i	$0.431526\pi$
$563$	−9876.50	+	17106.6i	−0.739334	+	1.28056i	−0.952796	+	0.303612i	$0.901807\pi$
$564$		0			0
$565$	175.000	+	303.109i	0.0130306	+	0.0225697i
$566$		0			0
$567$	−12006.5	+	562.050i	−0.889287	+	0.0416295i
$568$		0			0
$569$	3448.50	+	5972.98i	0.254075	+	0.440071i	0.964644	−	0.263557i	$-0.0848957\pi$
$569$	3448.50	+	5972.98i	0.254075	+	0.440071i	−0.710569	+	0.703628i	$0.751562\pi$
$570$		0			0
$571$	12457.5	−	21577.0i	0.913013	−	1.58138i	0.103227	−	0.994658i	$-0.467083\pi$
$571$	12457.5	−	21577.0i	0.913013	−	1.58138i	0.809785	−	0.586726i	$-0.199584\pi$
$572$		0			0
$573$	1275.00			0.0929562
$574$		0			0
$575$	−532.000			−0.0385842
$576$		0			0
$577$	−63.5000	+	109.985i	−0.00458152	+	0.00793543i	−0.868307	−	0.496027i	$-0.834792\pi$
$577$	−63.5000	+	109.985i	−0.00458152	+	0.00793543i	0.863726	+	0.503962i	$0.168125\pi$
$578$		0			0
$579$	−17.5000	−	30.3109i	−0.00125609	−	0.00217561i
$580$		0			0
$581$	−9320.00	−	14528.4i	−0.665506	−	1.03742i
$582$		0			0
$583$	−7297.50	−	12639.6i	−0.518407	−	0.897908i
$584$		0			0
$585$	6006.00	−	10402.7i	0.424474	−	0.735211i
$586$		0			0
$587$	−9044.00			−0.635921			−0.317961	−	0.948104i	$-0.602998\pi$
$587$	−9044.00			−0.635921			−0.317961	+	0.948104i	$0.602998\pi$
$588$		0			0
$589$	10275.0			0.718801
$590$		0			0
$591$	1367.00	−	2367.71i	0.0951453	−	0.164796i
$592$		0			0
$593$	5350.50	+	9267.34i	0.370521	+	0.641760i	0.989646	−	0.143532i	$-0.0458460\pi$
$593$	5350.50	+	9267.34i	0.370521	+	0.641760i	−0.619125	+	0.785292i	$0.712513\pi$
$594$		0			0
$595$	4130.00	+	6438.03i	0.284560	+	0.443586i
$596$		0			0
$597$	1121.50	+	1942.49i	0.0768843	+	0.133168i
$598$		0			0
$599$	10399.5	−	18012.5i	0.709369	−	1.22866i	−0.255722	−	0.966750i	$-0.582313\pi$
$599$	10399.5	−	18012.5i	0.709369	−	1.22866i	0.965091	−	0.261913i	$-0.0843533\pi$
$600$		0			0
$601$	−1402.00			−0.0951560			−0.0475780	−	0.998868i	$-0.515150\pi$
$601$	−1402.00			−0.0951560			−0.0475780	+	0.998868i	$0.515150\pi$
$602$		0			0
$603$	11414.0			0.770836
$604$		0			0
$605$	371.000	−	642.591i	0.0249311	−	0.0431819i
$606$		0			0
$607$	3262.50	+	5650.82i	0.218156	+	0.377858i	0.954244	−	0.299028i	$-0.0966625\pi$
$607$	3262.50	+	5650.82i	0.218156	+	0.377858i	−0.736088	+	0.676886i	$0.763329\pi$
$608$		0			0
$609$	−1961.00	+	91.7987i	−0.130482	+	0.00610816i
$610$		0			0
$611$	−5643.00	−	9773.96i	−0.373636	−	0.647156i
$612$		0			0
$613$	−7525.50	+	13034.5i	−0.495844	+	0.858826i	−0.999989	−	0.00479285i	$-0.998474\pi$
$613$	−7525.50	+	13034.5i	−0.495844	+	0.858826i	0.504145	+	0.863619i	$0.331808\pi$
$614$		0			0
$615$	−3486.00			−0.228568
$616$		0			0
$617$	11150.0			0.727524			0.363762	−	0.931492i	$-0.381492\pi$
$617$	11150.0			0.727524			0.363762	+	0.931492i	$0.381492\pi$
$618$		0			0
$619$	1707.50	−	2957.48i	0.110873	−	0.192037i	−0.805250	−	0.592936i	$-0.797969\pi$
$619$	1707.50	−	2957.48i	0.110873	−	0.192037i	0.916122	+	0.400899i	$0.131302\pi$
$620$		0			0
$621$	185.500	+	321.295i	0.0119869	+	0.0207619i
$622$		0			0
$623$	−7420.50	+	14364.8i	−0.477201	+	0.923775i
$624$		0			0
$625$	174.500	+	302.243i	0.0111680	+	0.0193435i
$626$		0			0
$627$	−2397.50	+	4152.59i	−0.152706	+	0.264495i
$628$		0			0
$629$	649.000			0.0411404
$630$		0			0
$631$	21184.0			1.33648			0.668242	−	0.743944i	$-0.267047\pi$
$631$	21184.0			1.33648			0.668242	+	0.743944i	$0.267047\pi$
$632$		0			0
$633$	586.000	−	1014.98i	0.0367953	−	0.0637313i
$634$		0			0
$635$	3276.00	+	5674.20i	0.204731	+	0.354604i
$636$		0			0
$637$	−9438.00	+	20576.8i	−0.587044	+	1.27988i
$638$		0			0
$639$	10192.0	+	17653.1i	0.630969	+	1.09287i
$640$		0			0
$641$	5352.50	−	9270.80i	0.329814	−	0.571255i	−0.652660	−	0.757651i	$-0.726347\pi$
$641$	5352.50	−	9270.80i	0.329814	−	0.571255i	0.982475	+	0.186395i	$0.0596805\pi$
$642$		0			0
$643$	−6860.00			−0.420734			−0.210367	−	0.977622i	$-0.567466\pi$
$643$	−6860.00			−0.420734			−0.210367	+	0.977622i	$0.567466\pi$
$644$		0			0
$645$	−1820.00			−0.111105
$646$		0			0
$647$	7231.50	−	12525.3i	0.439412	−	0.761084i	−0.558232	−	0.829685i	$-0.688520\pi$
$647$	7231.50	−	12525.3i	0.439412	−	0.761084i	0.997644	+	0.0686008i	$0.0218535\pi$
$648$		0			0
$649$	297.500	+	515.285i	0.0179937	+	0.0311660i
$650$		0			0
$651$	−637.500	+	1234.09i	−0.0383803	+	0.0742975i
$652$		0			0
$653$	−2989.50	−	5177.97i	−0.179155	−	0.310305i	0.762436	−	0.647063i	$-0.224003\pi$
$653$	−2989.50	−	5177.97i	−0.179155	−	0.310305i	−0.941591	+	0.336758i	$0.890670\pi$
$654$		0			0
$655$	−2642.50	+	4576.94i	−0.157635	+	0.273032i
$656$		0			0
$657$	−7670.00			−0.455457
$658$		0			0
$659$	6940.00			0.410234			0.205117	−	0.978737i	$-0.434243\pi$
$659$	6940.00			0.410234			0.205117	+	0.978737i	$0.434243\pi$
$660$		0			0
$661$	−6699.50	+	11603.9i	−0.394221	+	0.682812i	−0.993001	−	0.118102i	$-0.962319\pi$
$661$	−6699.50	+	11603.9i	−0.394221	+	0.682812i	0.598780	+	0.800914i	$0.295652\pi$
$662$		0			0
$663$	−1947.00	−	3372.30i	−0.114050	−	0.197541i
$664$		0			0
$665$	17741.5	−	830.518i	1.03457	−	0.0484303i
$666$		0			0
$667$	−371.000	−	642.591i	−0.0215370	−	0.0373032i
$668$		0			0
$669$	1012.00	−	1752.84i	0.0584846	−	0.101298i
$670$		0			0
$671$	−1785.00			−0.102696
$672$		0			0
$673$	29510.0			1.69023			0.845117	−	0.534582i	$-0.179531\pi$
$673$	29510.0			1.69023			0.845117	+	0.534582i	$0.179531\pi$
$674$		0			0
$675$	−2014.00	+	3488.35i	−0.114843	+	0.198914i
$676$		0			0
$677$	13000.5	+	22517.5i	0.738035	+	1.27831i	0.953379	+	0.301776i	$0.0975795\pi$
$677$	13000.5	+	22517.5i	0.738035	+	1.27831i	−0.215344	+	0.976538i	$0.569087\pi$
$678$		0			0
$679$	−2900.00	−	4520.65i	−0.163905	−	0.255503i
$680$		0			0
$681$	1285.50	+	2226.55i	0.0723355	+	0.125289i
$682$		0			0
$683$	−4402.50	+	7625.35i	−0.246643	+	0.427198i	−0.962592	−	0.270954i	$-0.912661\pi$
$683$	−4402.50	+	7625.35i	−0.246643	+	0.427198i	0.715949	+	0.698152i	$0.245994\pi$
$684$		0			0
$685$	−16499.0			−0.920284
$686$		0			0
$687$	895.000			0.0497036
$688$		0			0
$689$	13761.0	−	23834.8i	0.760889	−	1.31790i
$690$		0			0
$691$	14342.5	+	24841.9i	0.789601	+	1.36763i	0.926211	+	0.377004i	$0.123046\pi$
$691$	14342.5	+	24841.9i	0.789601	+	1.36763i	−0.136610	+	0.990625i	$0.543621\pi$
$692$		0			0
$693$	9100.00	+	14185.5i	0.498817	+	0.777579i
$694$		0			0
$695$	98.0000	+	169.741i	0.00534871	+	0.00926423i
$696$		0			0
$697$	14691.0	−	25445.6i	0.798366	−	1.38281i
$698$		0			0
$699$	1787.00			0.0966961
$700$		0			0
$701$	−3146.00			−0.169505			−0.0847523	−	0.996402i	$-0.527010\pi$
$701$	−3146.00			−0.169505			−0.0847523	+	0.996402i	$0.527010\pi$
$702$		0			0
$703$	753.500	−	1305.10i	0.0404250	−	0.0700182i
$704$		0			0
$705$	−598.500	−	1036.63i	−0.0319728	−	0.0553785i
$706$		0			0
$707$	20072.5	−	939.638i	1.06776	−	0.0499840i
$708$		0			0
$709$	−629.500	−	1090.33i	−0.0333447	−	0.0577547i	0.848871	−	0.528599i	$-0.177283\pi$
$709$	−629.500	−	1090.33i	−0.0333447	−	0.0577547i	−0.882216	+	0.470845i	$0.843949\pi$
$710$		0			0
$711$	6435.00	−	11145.7i	0.339425	−	0.587902i
$712$		0			0
$713$	−525.000			−0.0275756
$714$		0			0
$715$	−16170.0			−0.845767
$716$		0			0
$717$	−2550.00	+	4416.73i	−0.132819	+	0.230050i
$718$		0			0
$719$	8212.50	+	14224.5i	0.425973	+	0.737807i	0.996511	−	0.0834645i	$-0.0265985\pi$
$719$	8212.50	+	14224.5i	0.425973	+	0.737807i	−0.570538	+	0.821271i	$0.693265\pi$
$720$		0			0
$721$	−13200.5	+	25553.8i	−0.681848	+	1.31994i
$722$		0			0
$723$	2088.50	+	3617.39i	0.107430	+	0.186075i
$724$		0			0
$725$	4028.00	−	6976.70i	0.206340	−	0.357391i
$726$		0			0
$727$	6032.00			0.307723			0.153861	−	0.988092i	$-0.450829\pi$
$727$	6032.00			0.307723			0.153861	+	0.988092i	$0.450829\pi$
$728$		0			0
$729$	−15443.0			−0.784586
$730$		0			0
$731$	7670.00	−	13284.8i	0.388078	−	0.672171i
$732$		0			0
$733$	−7621.50	−	13200.8i	−0.384047	−	0.665189i	0.607589	−	0.794251i	$-0.292137\pi$
$733$	−7621.50	−	13200.8i	−0.384047	−	0.665189i	−0.991636	+	0.129062i	$0.958803\pi$
$734$		0			0
$735$	−1001.00	+	2182.38i	−0.0502346	+	0.109522i
$736$		0			0
$737$	−7682.50	−	13306.5i	−0.383974	−	0.665062i
$738$		0			0
$739$	−5026.50	+	8706.15i	−0.250207	+	0.433371i	−0.963583	−	0.267411i	$-0.913832\pi$
$739$	−5026.50	+	8706.15i	−0.250207	+	0.433371i	0.713376	+	0.700782i	$0.247165\pi$
$740$		0			0
$741$	−9042.00			−0.448267
$742$		0			0
$743$	−24384.0			−1.20399			−0.601993	−	0.798501i	$-0.705627\pi$
$743$	−24384.0			−1.20399			−0.601993	+	0.798501i	$0.705627\pi$
$744$		0			0
$745$	−8032.50	+	13912.7i	−0.395017	+	0.684190i
$746$		0			0
$747$	−12116.0	−	20985.5i	−0.593442	−	1.02787i
$748$		0			0
$749$	−1096.50	+	2122.63i	−0.0534916	+	0.103550i
$750$		0			0
$751$	5794.50	+	10036.4i	0.281550	+	0.487660i	0.971767	−	0.235943i	$-0.0758180\pi$
$751$	5794.50	+	10036.4i	0.281550	+	0.487660i	−0.690216	+	0.723603i	$0.742485\pi$
$752$		0			0
$753$	−2340.00	+	4053.00i	−0.113246	+	0.196148i
$754$		0			0
$755$	7763.00			0.374205
$756$		0			0
$757$	14562.0			0.699161			0.349581	−	0.936906i	$-0.386324\pi$
$757$	14562.0			0.699161			0.349581	+	0.936906i	$0.386324\pi$
$758$		0			0
$759$	122.500	−	212.176i	0.00585832	−	0.0101469i
$760$		0			0
$761$	11382.5	+	19715.1i	0.542201	+	0.939120i	0.998777	+	0.0494360i	$0.0157424\pi$
$761$	11382.5	+	19715.1i	0.542201	+	0.939120i	−0.456576	+	0.889684i	$0.650924\pi$
$762$		0			0
$763$	17852.5	−	835.715i	0.847056	−	0.0396526i
$764$		0			0
$765$	5369.00	+	9299.38i	0.253747	+	0.439503i
$766$		0			0
$767$	−561.000	+	971.681i	−0.0264101	+	0.0457436i
$768$		0			0
$769$	3766.00			0.176600			0.0883000	−	0.996094i	$-0.471857\pi$
$769$	3766.00			0.176600			0.0883000	+	0.996094i	$0.471857\pi$
$770$		0			0
$771$	−1749.00			−0.0816974
$772$		0			0
$773$	13430.5	−	23262.3i	0.624918	−	1.08239i	−0.363639	−	0.931540i	$-0.618466\pi$
$773$	13430.5	−	23262.3i	0.624918	−	1.08239i	0.988557	−	0.150849i	$-0.0482009\pi$
$774$		0			0
$775$	−2850.00	−	4936.34i	−0.132097	−	0.228798i
$776$		0			0
$777$	110.000	+	171.473i	0.00507880	+	0.00791707i
$778$		0			0
$779$	−34113.0	−	59085.4i	−1.56897	−	2.71753i
$780$		0			0
$781$	13720.0	−	23763.7i	0.628605	−	1.08878i
$782$		0			0
$783$	−5618.00			−0.256412
$784$		0			0
$785$	10913.0			0.496180
$786$		0			0
$787$	−1048.50	+	1816.06i	−0.0474905	+	0.0822559i	−0.888793	−	0.458308i	$-0.848456\pi$
$787$	−1048.50	+	1816.06i	−0.0474905	+	0.0822559i	0.841303	+	0.540564i	$0.181789\pi$
$788$		0			0
$789$	−2236.50	−	3873.73i	−0.100914	−	0.174789i
$790$		0			0
$791$	−500.000	−	779.423i	−0.0224753	−	0.0350355i
$792$		0			0
$793$	−1683.00	−	2915.04i	−0.0753658	−	0.130537i
$794$		0			0
$795$	1459.50	−	2527.93i	0.0651109	−	0.112775i
$796$		0			0
$797$	−35334.0			−1.57038			−0.785191	−	0.619254i	$-0.787435\pi$
$797$	−35334.0			−1.57038			−0.785191	+	0.619254i	$0.787435\pi$
$798$		0			0
$799$	10089.0			0.446712
$800$		0			0
$801$	−11349.0	+	19657.0i	−0.500621	+	0.867101i
$802$		0			0
$803$	5162.50	+	8941.71i	0.226875	+	0.392959i
$804$		0			0
$805$	−906.500	+	42.4352i	−0.0396894	+	0.00185795i
$806$		0			0
$807$	−987.500	−	1710.40i	−0.0430752	−	0.0746083i
$808$		0			0
$809$	−21267.5	+	36836.4i	−0.924259	+	1.60086i	−0.131510	+	0.991315i	$0.541983\pi$
$809$	−21267.5	+	36836.4i	−0.924259	+	1.60086i	−0.792749	+	0.609549i	$0.791351\pi$
$810$		0			0
$811$	−30676.0			−1.32821			−0.664106	−	0.747638i	$-0.731188\pi$
$811$	−30676.0			−1.32821			−0.664106	+	0.747638i	$0.731188\pi$
$812$		0			0
$813$	8439.00			0.364045
$814$		0			0
$815$	−7878.50	+	13646.0i	−0.338616	+	0.586500i
$816$		0			0
$817$	−17810.0	−	30847.8i	−0.762660	−	1.32097i
$818$		0			0
$819$	−14586.0	+	28235.9i	−0.622315	+	1.20469i
$820$		0			0
$821$	−18671.5	−	32340.0i	−0.793715	−	1.37475i	−0.923652	−	0.383232i	$-0.874811\pi$
$821$	−18671.5	−	32340.0i	−0.793715	−	1.37475i	0.129937	−	0.991522i	$-0.458522\pi$
$822$		0			0
$823$	1407.50	−	2437.86i	0.0596141	−	0.103255i	−0.834678	−	0.550738i	$-0.814346\pi$
$823$	1407.50	−	2437.86i	0.0596141	−	0.103255i	0.894292	+	0.447483i	$0.147680\pi$
$824$		0			0
$825$	2660.00			0.112254
$826$		0			0
$827$	9276.00			0.390034			0.195017	−	0.980800i	$-0.437524\pi$
$827$	9276.00			0.390034			0.195017	+	0.980800i	$0.437524\pi$
$828$		0			0
$829$	−9285.50	+	16083.0i	−0.389021	+	0.673805i	−0.992318	−	0.123712i	$-0.960520\pi$
$829$	−9285.50	+	16083.0i	−0.389021	+	0.673805i	0.603297	+	0.797517i	$0.293853\pi$
$830$		0			0
$831$	−263.500	−	456.395i	−0.0109997	−	0.0190520i
$832$		0			0
$833$	−11711.5	−	16503.8i	−0.487130	−	0.686464i
$834$		0			0
$835$	9758.00	+	16901.4i	0.404419	+	0.700474i
$836$		0			0
$837$	−1987.50	+	3442.45i	−0.0820765	+	0.142161i
$838$		0			0
$839$	−29048.0			−1.19529			−0.597645	−	0.801761i	$-0.703897\pi$
$839$	−29048.0			−1.19529			−0.597645	+	0.801761i	$0.703897\pi$
$840$		0			0
$841$	−13153.0			−0.539301
$842$		0			0
$843$	101.000	−	174.937i	0.00412648	−	0.00714728i
$844$		0			0
$845$	−7556.50	−	13088.2i	−0.307635	−	0.532839i
$846$		0			0
$847$	−901.000	+	1744.18i	−0.0365510	+	0.0707563i
$848$		0			0
$849$	−3974.50	−	6884.04i	−0.160665	−	0.278280i
$850$		0			0
$851$	−38.5000	+	66.6840i	−0.00155084	+	0.00268613i
$852$		0			0
$853$	32090.0			1.28809			0.644045	−	0.764988i	$-0.277255\pi$
$853$	32090.0			1.28809			0.644045	+	0.764988i	$0.277255\pi$
$854$		0			0
$855$	24934.0			0.997339
$856$		0			0
$857$	12268.5	−	21249.7i	0.489013	−	0.846995i	−0.510907	−	0.859636i	$-0.670690\pi$
$857$	12268.5	−	21249.7i	0.489013	−	0.846995i	0.999920	+	0.0126408i	$0.00402379\pi$
$858$		0			0
$859$	10412.5	+	18035.0i	0.413585	+	0.716351i	0.995279	−	0.0970571i	$-0.0309430\pi$
$859$	10412.5	+	18035.0i	0.413585	+	0.716351i	−0.581693	+	0.813408i	$0.697610\pi$
$860$		0			0
$861$	9213.00	−	431.281i	0.364667	−	0.0170709i
$862$		0			0
$863$	−11423.5	−	19786.1i	−0.450591	−	0.780447i	0.547831	−	0.836589i	$-0.315454\pi$
$863$	−11423.5	−	19786.1i	−0.450591	−	0.780447i	−0.998423	+	0.0561414i	$0.982120\pi$
$864$		0			0
$865$	−5526.50	+	9572.18i	−0.217233	+	0.376259i
$866$		0			0
$867$	−1432.00			−0.0560937
$868$		0			0
$869$	−17325.0			−0.676307
$870$		0			0
$871$	14487.0	−	25092.2i	0.563574	−	0.976139i
$872$		0			0
$873$	−3770.00	−	6529.83i	−0.146157	−	0.253152i
$874$		0			0
$875$	−14070.0	−	21933.0i	−0.543603	−	0.847394i
$876$		0			0
$877$	21368.5	+	37011.3i	0.822763	+	1.42507i	0.903617	+	0.428341i	$0.140902\pi$
$877$	21368.5	+	37011.3i	0.822763	+	1.42507i	−0.0808543	+	0.996726i	$0.525765\pi$
$878$		0			0
$879$	−159.000	+	275.396i	−0.00610118	+	0.0105676i
$880$		0			0
$881$	6162.00			0.235645			0.117822	−	0.993035i	$-0.462409\pi$
$881$	6162.00			0.235645			0.117822	+	0.993035i	$0.462409\pi$
$882$		0			0
$883$	−7748.00			−0.295290			−0.147645	−	0.989040i	$-0.547169\pi$
$883$	−7748.00			−0.295290			−0.147645	+	0.989040i	$0.547169\pi$
$884$		0			0
$885$	−59.5000	+	103.057i	−0.00225997	+	0.00391438i
$886$		0			0
$887$	−12961.5	−	22450.0i	−0.490648	−	0.849827i	0.509294	−	0.860592i	$-0.329906\pi$
$887$	−12961.5	−	22450.0i	−0.490648	−	0.849827i	−0.999942	+	0.0107656i	$0.996573\pi$
$888$		0			0
$889$	−9360.00	−	14590.8i	−0.353121	−	0.550461i
$890$		0			0
$891$	11357.5	+	19671.8i	0.427038	+	0.739651i
$892$		0			0
$893$	11713.5	−	20288.4i	0.438944	−	0.760274i
$894$		0			0
$895$	−17157.0			−0.640777
$896$		0			0
$897$	462.000			0.0171970
$898$		0			0
$899$	3975.00	−	6884.90i	0.147468	−	0.255422i
$900$		0			0
$901$	12301.5	+	21306.8i	0.454853	+	0.787828i
$902$		0			0
$903$	4810.00	−	225.167i	0.177261	−	0.00829798i
$904$		0			0
$905$	4095.00	+	7092.75i	0.150411	+	0.260520i
$906$		0			0
$907$	15967.5	−	27656.5i	0.584556	−	1.01248i	−0.410375	−	0.911917i	$-0.634602\pi$
$907$	15967.5	−	27656.5i	0.584556	−	1.01248i	0.994931	−	0.100563i	$-0.0320645\pi$
$908$		0			0
$909$	28210.0			1.02934
$910$		0			0
$911$	−3408.00			−0.123943			−0.0619715	−	0.998078i	$-0.519739\pi$
$911$	−3408.00			−0.123943			−0.0619715	+	0.998078i	$0.519739\pi$
$912$		0			0
$913$	−16310.0	+	28249.7i	−0.591218	+	1.02402i
$914$		0			0
$915$	−178.500	−	309.171i	−0.00644921	−	0.0111704i
$916$		0			0
$917$	6417.50	−	12423.1i	0.231106	−	0.447381i
$918$		0			0
$919$	6954.50	+	12045.5i	0.249628	+	0.432368i	0.963423	−	0.267987i	$-0.0863585\pi$
$919$	6954.50	+	12045.5i	0.249628	+	0.432368i	−0.713795	+	0.700355i	$0.753025\pi$
$920$		0			0
$921$	−4066.00	+	7042.52i	−0.145472	+	0.251964i
$922$		0			0
$923$	51744.0			1.84526
$924$		0			0
$925$	−836.000			−0.0297162
$926$		0			0
$927$	−20189.0	+	34968.4i	−0.715311	+	1.23896i
$928$		0			0
$929$	12268.5	+	21249.7i	0.433279	+	0.750462i	0.997153	−	0.0753990i	$-0.0240231\pi$
$929$	12268.5	+	21249.7i	0.433279	+	0.750462i	−0.563874	+	0.825861i	$0.690690\pi$
$930$		0			0
$931$	−46785.5	+	4389.88i	−1.64697	+	0.154536i
$932$		0			0
$933$	−464.500	−	804.538i	−0.0162991	−	0.0282308i
$934$		0			0
$935$	7227.50	−	12518.4i	0.252796	−	0.437856i
$936$		0			0
$937$	−32758.0			−1.14211			−0.571055	−	0.820912i	$-0.693466\pi$
$937$	−32758.0			−1.14211			−0.571055	+	0.820912i	$0.693466\pi$
$938$		0			0
$939$	−209.000			−0.00726353
$940$		0			0
$941$	19280.5	−	33394.8i	0.667934	−	1.15690i	−0.310546	−	0.950558i	$-0.600512\pi$
$941$	19280.5	−	33394.8i	0.667934	−	1.15690i	0.978481	−	0.206338i	$-0.0661547\pi$
$942$		0			0
$943$	1743.00	+	3018.96i	0.0601908	+	0.104253i
$944$		0			0
$945$	−3153.50	+	6104.61i	−0.108554	+	0.210141i
$946$		0			0
$947$	19830.5	+	34347.4i	0.680470	+	1.17861i	0.974838	+	0.222916i	$0.0715575\pi$
$947$	19830.5	+	34347.4i	0.680470	+	1.17861i	−0.294368	+	0.955692i	$0.595109\pi$
$948$		0			0
$949$	−9735.00	+	16861.5i	−0.332994	+	0.576763i
$950$		0			0
$951$	7131.00			0.243153
$952$		0			0
$953$	−46618.0			−1.58458			−0.792290	−	0.610144i	$-0.791111\pi$
$953$	−46618.0			−1.58458			−0.792290	+	0.610144i	$0.791111\pi$
$954$		0			0
$955$	−4462.50	+	7729.28i	−0.151207	+	0.261899i
$956$		0			0
$957$	1855.00	+	3212.95i	0.0626579	+	0.108527i
$958$		0			0
$959$	43604.5	−	2041.22i	1.46826	−	0.0687325i
$960$		0			0
$961$	12083.0	+	20928.4i	0.405592	+	0.702506i
$962$		0			0
$963$	−1677.00	+	2904.65i	−0.0561169	+	0.0971973i
$964$		0			0
$965$	245.000			0.00817288
$966$		0			0
$967$	−14816.0			−0.492710			−0.246355	−	0.969180i	$-0.579233\pi$
$967$	−14816.0			−0.492710			−0.246355	+	0.969180i	$0.579233\pi$
$968$		0			0
$969$	4041.50	−	7000.08i	0.133985	−	0.232069i
$970$		0			0
$971$	−8437.50	−	14614.2i	−0.278859	−	0.482998i	0.692242	−	0.721665i	$-0.256623\pi$
$971$	−8437.50	−	14614.2i	−0.278859	−	0.482998i	−0.971102	+	0.238667i	$0.923290\pi$
$972$		0			0
$973$	−280.000	−	436.477i	−0.00922548	−	0.0143811i
$974$		0			0
$975$	2508.00	+	4343.98i	0.0823798	+	0.142686i
$976$		0			0
$977$	7918.50	−	13715.2i	0.259299	−	0.449119i	−0.706755	−	0.707458i	$-0.749842\pi$
$977$	7918.50	−	13715.2i	0.259299	−	0.449119i	0.966054	+	0.258339i	$0.0831751\pi$
$978$		0			0
$979$	30555.0			0.997489
$980$		0			0
$981$	25090.0			0.816577
$982$		0			0
$983$	4957.50	−	8586.64i	0.160854	−	0.278608i	−0.774321	−	0.632793i	$-0.781908\pi$
$983$	4957.50	−	8586.64i	0.160854	−	0.278608i	0.935175	+	0.354185i	$0.115242\pi$
$984$		0			0
$985$	9569.00	+	16574.0i	0.309537	+	0.536133i
$986$		0			0
$987$	1710.00	+	2665.63i	0.0551468	+	0.0859654i
$988$		0			0
$989$	910.000	+	1576.17i	0.0292582	+	0.0506766i
$990$		0			0
$991$	−21840.5	+	37828.9i	−0.700087	+	1.21259i	0.268348	+	0.963322i	$0.413522\pi$
$991$	−21840.5	+	37828.9i	−0.700087	+	1.21259i	−0.968435	+	0.249265i	$0.919811\pi$
$992$		0			0
$993$	6571.00			0.209994
$994$		0			0
$995$	−15701.0			−0.500256
$996$		0			0
$997$	23556.5	−	40801.1i	0.748287	−	1.29607i	−0.200357	−	0.979723i	$-0.564210\pi$
$997$	23556.5	−	40801.1i	0.748287	−	1.29607i	0.948643	−	0.316348i	$-0.102457\pi$
$998$		0			0
$999$	291.500	+	504.893i	0.00923188	+	0.0159901i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.4.i.b.81.1		2
4.3	odd	2		14.4.c.b.11.1	yes	2
7.2	even	3	inner	112.4.i.b.65.1		2
7.3	odd	6		784.4.a.j.1.1		1
7.4	even	3		784.4.a.l.1.1		1
8.3	odd	2		448.4.i.c.193.1		2
8.5	even	2		448.4.i.d.193.1		2
12.11	even	2		126.4.g.c.109.1		2
20.3	even	4		350.4.j.d.249.2		4
20.7	even	4		350.4.j.d.249.1		4
20.19	odd	2		350.4.e.b.151.1		2
28.3	even	6		98.4.a.c.1.1		1
28.11	odd	6		98.4.a.b.1.1		1
28.19	even	6		98.4.c.e.79.1		2
28.23	odd	6		14.4.c.b.9.1	✓	2
28.27	even	2		98.4.c.e.67.1		2
56.37	even	6		448.4.i.d.65.1		2
56.51	odd	6		448.4.i.c.65.1		2
84.11	even	6		882.4.a.k.1.1		1
84.23	even	6		126.4.g.c.37.1		2
84.47	odd	6		882.4.g.d.667.1		2
84.59	odd	6		882.4.a.p.1.1		1
84.83	odd	2		882.4.g.d.361.1		2
140.23	even	12		350.4.j.d.149.1		4
140.39	odd	6		2450.4.a.bh.1.1		1
140.59	even	6		2450.4.a.bf.1.1		1
140.79	odd	6		350.4.e.b.51.1		2
140.107	even	12		350.4.j.d.149.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.c.b.9.1	✓	2	28.23	odd	6
14.4.c.b.11.1	yes	2	4.3	odd	2
98.4.a.b.1.1		1	28.11	odd	6
98.4.a.c.1.1		1	28.3	even	6
98.4.c.e.67.1		2	28.27	even	2
98.4.c.e.79.1		2	28.19	even	6
112.4.i.b.65.1		2	7.2	even	3	inner
112.4.i.b.81.1		2	1.1	even	1	trivial
126.4.g.c.37.1		2	84.23	even	6
126.4.g.c.109.1		2	12.11	even	2
350.4.e.b.51.1		2	140.79	odd	6
350.4.e.b.151.1		2	20.19	odd	2
350.4.j.d.149.1		4	140.23	even	12
350.4.j.d.149.2		4	140.107	even	12
350.4.j.d.249.1		4	20.7	even	4
350.4.j.d.249.2		4	20.3	even	4
448.4.i.c.65.1		2	56.51	odd	6
448.4.i.c.193.1		2	8.3	odd	2
448.4.i.d.65.1		2	56.37	even	6
448.4.i.d.193.1		2	8.5	even	2
784.4.a.j.1.1		1	7.3	odd	6
784.4.a.l.1.1		1	7.4	even	3
882.4.a.k.1.1		1	84.11	even	6
882.4.a.p.1.1		1	84.59	odd	6
882.4.g.d.361.1		2	84.83	odd	2
882.4.g.d.667.1		2	84.47	odd	6
2450.4.a.bf.1.1		1	140.59	even	6
2450.4.a.bh.1.1		1	140.39	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-3.50000 - 6.06218i) q^{5} +(10.0000 + 15.5885i) q^{7} +(13.0000 + 22.5167i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-3.50000 - 6.06218i) q^{5} +(10.0000 + 15.5885i) q^{7} +(13.0000 + 22.5167i) q^{9} +(17.5000 - 30.3109i) q^{11} +66.0000 q^{13} +7.00000 q^{15} +(-29.5000 + 51.0955i) q^{17} +(68.5000 + 118.645i) q^{19} +(-18.5000 + 0.866025i) q^{21} +(-3.50000 - 6.06218i) q^{23} +(38.0000 - 65.8179i) q^{25} -53.0000 q^{27} +106.000 q^{29} +(37.5000 - 64.9519i) q^{31} +(17.5000 + 30.3109i) q^{33} +(59.5000 - 115.181i) q^{35} +(-5.50000 - 9.52628i) q^{37} +(-33.0000 + 57.1577i) q^{39} -498.000 q^{41} -260.000 q^{43} +(91.0000 - 157.617i) q^{45} +(-85.5000 - 148.090i) q^{47} +(-143.000 + 311.769i) q^{49} +(-29.5000 - 51.0955i) q^{51} +(208.500 - 361.133i) q^{53} -245.000 q^{55} -137.000 q^{57} +(-8.50000 + 14.7224i) q^{59} +(-25.5000 - 44.1673i) q^{61} +(-221.000 + 427.817i) q^{63} +(-231.000 - 400.104i) q^{65} +(219.500 - 380.185i) q^{67} +7.00000 q^{69} +784.000 q^{71} +(-147.500 + 255.477i) q^{73} +(38.0000 + 65.8179i) q^{75} +(647.500 - 30.3109i) q^{77} +(-247.500 - 428.683i) q^{79} +(-324.500 + 562.050i) q^{81} -932.000 q^{83} +413.000 q^{85} +(-53.0000 + 91.7987i) q^{87} +(436.500 + 756.040i) q^{89} +(660.000 + 1028.84i) q^{91} +(37.5000 + 64.9519i) q^{93} +(479.500 - 830.518i) q^{95} -290.000 q^{97} +910.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9}+O(q^{10}) \) 2 * q - q^3 - 7 * q^5 + 20 * q^7 + 26 * q^9	\( 2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9} + 35 q^{11} + 132 q^{13} + 14 q^{15} - 59 q^{17} + 137 q^{19} - 37 q^{21} - 7 q^{23} + 76 q^{25} - 106 q^{27} + 212 q^{29} + 75 q^{31} + 35 q^{33} + 119 q^{35} - 11 q^{37} - 66 q^{39} - 996 q^{41} - 520 q^{43} + 182 q^{45} - 171 q^{47} - 286 q^{49} - 59 q^{51} + 417 q^{53} - 490 q^{55} - 274 q^{57} - 17 q^{59} - 51 q^{61} - 442 q^{63} - 462 q^{65} + 439 q^{67} + 14 q^{69} + 1568 q^{71} - 295 q^{73} + 76 q^{75} + 1295 q^{77} - 495 q^{79} - 649 q^{81} - 1864 q^{83} + 826 q^{85} - 106 q^{87} + 873 q^{89} + 1320 q^{91} + 75 q^{93} + 959 q^{95} - 580 q^{97} + 1820 q^{99}+O(q^{100}) \) 2 * q - q^3 - 7 * q^5 + 20 * q^7 + 26 * q^9 + 35 * q^11 + 132 * q^13 + 14 * q^15 - 59 * q^17 + 137 * q^19 - 37 * q^21 - 7 * q^23 + 76 * q^25 - 106 * q^27 + 212 * q^29 + 75 * q^31 + 35 * q^33 + 119 * q^35 - 11 * q^37 - 66 * q^39 - 996 * q^41 - 520 * q^43 + 182 * q^45 - 171 * q^47 - 286 * q^49 - 59 * q^51 + 417 * q^53 - 490 * q^55 - 274 * q^57 - 17 * q^59 - 51 * q^61 - 442 * q^63 - 462 * q^65 + 439 * q^67 + 14 * q^69 + 1568 * q^71 - 295 * q^73 + 76 * q^75 + 1295 * q^77 - 495 * q^79 - 649 * q^81 - 1864 * q^83 + 826 * q^85 - 106 * q^87 + 873 * q^89 + 1320 * q^91 + 75 * q^93 + 959 * q^95 - 580 * q^97 + 1820 * q^99

Introduction

L-functions

Modular forms

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Database

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