LMFDB - Embedded newform 1425.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.a (trivial)

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:	yes
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66908 q^{2} +1.00000 q^{3} +5.12398 q^{4} -2.66908 q^{6} +0.454904 q^{7} -8.33816 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.66908 q^{2} +1.00000 q^{3} +5.12398 q^{4} -2.66908 q^{6} +0.454904 q^{7} -8.33816 q^{8} +1.00000 q^{9} -1.45490 q^{11} +5.12398 q^{12} -3.33816 q^{13} -1.21417 q^{14} +12.0072 q^{16} -2.66908 q^{18} +1.00000 q^{19} +0.454904 q^{21} +3.88325 q^{22} -6.79306 q^{23} -8.33816 q^{24} +8.90981 q^{26} +1.00000 q^{27} +2.33092 q^{28} +3.00000 q^{29} +8.79306 q^{31} -15.3719 q^{32} -1.45490 q^{33} +5.12398 q^{36} +10.2480 q^{37} -2.66908 q^{38} -3.33816 q^{39} +3.97345 q^{41} -1.21417 q^{42} +8.00000 q^{43} -7.45490 q^{44} +18.1312 q^{46} +2.42835 q^{47} +12.0072 q^{48} -6.79306 q^{49} -17.1047 q^{52} +13.6763 q^{53} -2.66908 q^{54} -3.79306 q^{56} +1.00000 q^{57} -8.00724 q^{58} +7.54510 q^{59} -3.90981 q^{61} -23.4694 q^{62} +0.454904 q^{63} +17.0145 q^{64} +3.88325 q^{66} +14.1312 q^{67} -6.79306 q^{69} -9.13122 q^{71} -8.33816 q^{72} -11.6763 q^{73} -27.3526 q^{74} +5.12398 q^{76} -0.661842 q^{77} +8.90981 q^{78} +8.97345 q^{79} +1.00000 q^{81} -10.6054 q^{82} -4.54510 q^{83} +2.33092 q^{84} -21.3526 q^{86} +3.00000 q^{87} +12.1312 q^{88} -0.0901918 q^{89} -1.51854 q^{91} -34.8075 q^{92} +8.79306 q^{93} -6.48146 q^{94} -15.3719 q^{96} +12.0145 q^{97} +18.1312 q^{98} -1.45490 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9	$ 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 6 q^{13} + 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} - 3 q^{23} - 9 q^{24} + 24 q^{26} + 3 q^{27} + 15 q^{28} + 9 q^{29} + 9 q^{31} - 18 q^{32} - 3 q^{33} + 6 q^{36} + 12 q^{37} + 6 q^{39} + 3 q^{42} + 24 q^{43} - 21 q^{44} + 21 q^{46} - 6 q^{47} + 12 q^{48} - 3 q^{49} - 6 q^{52} + 9 q^{53} + 6 q^{56} + 3 q^{57} + 24 q^{59} - 9 q^{61} - 21 q^{62} + 3 q^{64} - 3 q^{66} + 9 q^{67} - 3 q^{69} + 6 q^{71} - 9 q^{72} - 3 q^{73} - 18 q^{74} + 6 q^{76} - 18 q^{77} + 24 q^{78} + 15 q^{79} + 3 q^{81} - 33 q^{82} - 15 q^{83} + 15 q^{84} + 9 q^{87} + 3 q^{88} - 3 q^{89} + 6 q^{91} - 39 q^{92} + 9 q^{93} - 30 q^{94} - 18 q^{96} - 12 q^{97} + 21 q^{98} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9 - 3 * q^11 + 6 * q^12 + 6 * q^13 + 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 - 3 * q^23 - 9 * q^24 + 24 * q^26 + 3 * q^27 + 15 * q^28 + 9 * q^29 + 9 * q^31 - 18 * q^32 - 3 * q^33 + 6 * q^36 + 12 * q^37 + 6 * q^39 + 3 * q^42 + 24 * q^43 - 21 * q^44 + 21 * q^46 - 6 * q^47 + 12 * q^48 - 3 * q^49 - 6 * q^52 + 9 * q^53 + 6 * q^56 + 3 * q^57 + 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 3 * q^66 + 9 * q^67 - 3 * q^69 + 6 * q^71 - 9 * q^72 - 3 * q^73 - 18 * q^74 + 6 * q^76 - 18 * q^77 + 24 * q^78 + 15 * q^79 + 3 * q^81 - 33 * q^82 - 15 * q^83 + 15 * q^84 + 9 * q^87 + 3 * q^88 - 3 * q^89 + 6 * q^91 - 39 * q^92 + 9 * q^93 - 30 * q^94 - 18 * q^96 - 12 * q^97 + 21 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.66908	−1.88732	−0.943662	−	0.330911i	$-0.892644\pi$
$2$	−2.66908	−1.88732	−0.943662	+	0.330911i	$0.892644\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.12398	2.56199
$5$	0	0
$5$	0	0
$6$	−2.66908	−1.08965
$7$	0.454904	0.171938	0.0859688	−	0.996298i	$-0.472601\pi$
$7$	0.454904	0.171938	0.0859688	+	0.996298i	$0.472601\pi$
$8$	−8.33816	−2.94798
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.45490	−0.438670	−0.219335	−	0.975650i	$-0.570389\pi$
$11$	−1.45490	−0.438670	−0.219335	+	0.975650i	$0.570389\pi$
$12$	5.12398	1.47917
$13$	−3.33816	−0.925838	−0.462919	−	0.886400i	$-0.653198\pi$
$13$	−3.33816	−0.925838	−0.462919	+	0.886400i	$0.653198\pi$
$14$	−1.21417	−0.324502
$15$	0	0
$16$	12.0072	3.00181
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−2.66908	−0.629108
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.454904	0.0992682
$22$	3.88325	0.827913
$23$	−6.79306	−1.41645	−0.708226	−	0.705986i	$-0.750504\pi$
$23$	−6.79306	−1.41645	−0.708226	+	0.705986i	$0.750504\pi$
$24$	−8.33816	−1.70202
$25$	0	0
$26$	8.90981	1.74736
$27$	1.00000	0.192450
$28$	2.33092	0.440503
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	8.79306	1.57928	0.789640	−	0.613570i	$-0.210267\pi$
$31$	8.79306	1.57928	0.789640	+	0.613570i	$0.210267\pi$
$32$	−15.3719	−2.71740
$33$	−1.45490	−0.253266
$34$	0	0
$35$	0	0
$36$	5.12398	0.853997
$37$	10.2480	1.68476	0.842378	−	0.538888i	$-0.181155\pi$
$37$	10.2480	1.68476	0.842378	+	0.538888i	$0.181155\pi$
$38$	−2.66908	−0.432982
$39$	−3.33816	−0.534533
$40$	0	0
$41$	3.97345	0.620548	0.310274	−	0.950647i	$-0.399579\pi$
$41$	3.97345	0.620548	0.310274	+	0.950647i	$0.399579\pi$
$42$	−1.21417	−0.187351
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−7.45490	−1.12387
$45$	0	0
$46$	18.1312	2.67330
$47$	2.42835	0.354211	0.177106	−	0.984192i	$-0.443327\pi$
$47$	2.42835	0.354211	0.177106	+	0.984192i	$0.443327\pi$
$48$	12.0072	1.73310
$49$	−6.79306	−0.970437
$50$	0	0
$51$	0	0
$52$	−17.1047	−2.37199
$53$	13.6763	1.87859	0.939293	−	0.343115i	$-0.111482\pi$
$53$	13.6763	1.87859	0.939293	+	0.343115i	$0.111482\pi$
$54$	−2.66908	−0.363216
$55$	0	0
$56$	−3.79306	−0.506869
$57$	1.00000	0.132453
$58$	−8.00724	−1.05140
$59$	7.54510	0.982288	0.491144	−	0.871078i	$-0.336579\pi$
$59$	7.54510	0.982288	0.491144	+	0.871078i	$0.336579\pi$
$60$	0	0
$61$	−3.90981	−0.500600	−0.250300	−	0.968168i	$-0.580529\pi$
$61$	−3.90981	−0.500600	−0.250300	+	0.968168i	$0.580529\pi$
$62$	−23.4694	−2.98061
$63$	0.454904	0.0573125
$64$	17.0145	2.12681
$65$	0	0
$66$	3.88325	0.477996
$67$	14.1312	1.72640	0.863202	−	0.504859i	$-0.168456\pi$
$67$	14.1312	1.72640	0.863202	+	0.504859i	$0.168456\pi$
$68$	0	0
$69$	−6.79306	−0.817789
$70$	0	0
$71$	−9.13122	−1.08368	−0.541838	−	0.840483i	$-0.682271\pi$
$71$	−9.13122	−1.08368	−0.541838	+	0.840483i	$0.682271\pi$
$72$	−8.33816	−0.982661
$73$	−11.6763	−1.36661	−0.683305	−	0.730133i	$-0.739458\pi$
$73$	−11.6763	−1.36661	−0.683305	+	0.730133i	$0.739458\pi$
$74$	−27.3526	−3.17968
$75$	0	0
$76$	5.12398	0.587761
$77$	−0.661842	−0.0754239
$78$	8.90981	1.00884
$79$	8.97345	1.00959	0.504796	−	0.863239i	$-0.331568\pi$
$79$	8.97345	1.00959	0.504796	+	0.863239i	$0.331568\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−10.6054	−1.17118
$83$	−4.54510	−0.498889	−0.249445	−	0.968389i	$-0.580248\pi$
$83$	−4.54510	−0.498889	−0.249445	+	0.968389i	$0.580248\pi$
$84$	2.33092	0.254324
$85$	0	0
$86$	−21.3526	−2.30251
$87$	3.00000	0.321634
$88$	12.1312	1.29319
$89$	−0.0901918	−0.00956031	−0.00478016	−	0.999989i	$-0.501522\pi$
$89$	−0.0901918	−0.00956031	−0.00478016	+	0.999989i	$0.501522\pi$
$90$	0	0
$91$	−1.51854	−0.159186
$92$	−34.8075	−3.62894
$93$	8.79306	0.911798
$94$	−6.48146	−0.668511
$95$	0	0
$96$	−15.3719	−1.56889
$97$	12.0145	1.21989	0.609943	−	0.792446i	$-0.291193\pi$
$97$	12.0145	1.21989	0.609943	+	0.792446i	$0.291193\pi$
$98$	18.1312	1.83153
$99$	−1.45490	−0.146223
$100$	0	0
$101$	5.51854	0.549115	0.274558	−	0.961571i	$-0.411469\pi$
$101$	5.51854	0.549115	0.274558	+	0.961571i	$0.411469\pi$
$102$	0	0
$103$	−2.54510	−0.250776	−0.125388	−	0.992108i	$-0.540018\pi$
$103$	−2.54510	−0.250776	−0.125388	+	0.992108i	$0.540018\pi$
$104$	27.8341	2.72936
$105$	0	0
$106$	−36.5032	−3.54550
$107$	−4.63529	−0.448110	−0.224055	−	0.974576i	$-0.571930\pi$
$107$	−4.63529	−0.448110	−0.224055	+	0.974576i	$0.571930\pi$
$108$	5.12398	0.493056
$109$	−6.24797	−0.598447	−0.299223	−	0.954183i	$-0.596728\pi$
$109$	−6.24797	−0.598447	−0.299223	+	0.954183i	$0.596728\pi$
$110$	0	0
$111$	10.2480	0.972694
$112$	5.46214	0.516124
$113$	−5.20694	−0.489827	−0.244914	−	0.969545i	$-0.578760\pi$
$113$	−5.20694	−0.489827	−0.244914	+	0.969545i	$0.578760\pi$
$114$	−2.66908	−0.249982
$115$	0	0
$116$	15.3719	1.42725
$117$	−3.33816	−0.308613
$118$	−20.1385	−1.85390
$119$	0	0
$120$	0	0
$121$	−8.88325	−0.807569
$122$	10.4356	0.944794
$123$	3.97345	0.358274
$124$	45.0555	4.04610
$125$	0	0
$126$	−1.21417	−0.108167
$127$	−10.6127	−0.941723	−0.470861	−	0.882207i	$-0.656057\pi$
$127$	−10.6127	−0.941723	−0.470861	+	0.882207i	$0.656057\pi$
$128$	−14.6691	−1.29658
$129$	8.00000	0.704361
$130$	0	0
$131$	21.7029	1.89619	0.948094	−	0.317989i	$-0.103008\pi$
$131$	21.7029	1.89619	0.948094	+	0.317989i	$0.103008\pi$
$132$	−7.45490	−0.648866
$133$	0.454904	0.0394452
$134$	−37.7173	−3.25828
$135$	0	0
$136$	0	0
$137$	22.3155	1.90655	0.953273	−	0.302110i	$-0.0976911\pi$
$137$	22.3155	1.90655	0.953273	+	0.302110i	$0.0976911\pi$
$138$	18.1312	1.54343
$139$	−10.4018	−0.882269	−0.441134	−	0.897441i	$-0.645424\pi$
$139$	−10.4018	−0.882269	−0.441134	+	0.897441i	$0.645424\pi$
$140$	0	0
$141$	2.42835	0.204504
$142$	24.3719	2.04525
$143$	4.85670	0.406138
$144$	12.0072	1.00060
$145$	0	0
$146$	31.1650	2.57923
$147$	−6.79306	−0.560282
$148$	52.5104	4.31633
$149$	5.15777	0.422541	0.211271	−	0.977428i	$-0.432240\pi$
$149$	5.15777	0.422541	0.211271	+	0.977428i	$0.432240\pi$
$150$	0	0
$151$	−8.49593	−0.691389	−0.345695	−	0.938347i	$-0.612357\pi$
$151$	−8.49593	−0.691389	−0.345695	+	0.938347i	$0.612357\pi$
$152$	−8.33816	−0.676314
$153$	0	0
$154$	1.76651	0.142349
$155$	0	0
$156$	−17.1047	−1.36947
$157$	22.6498	1.80765	0.903824	−	0.427905i	$-0.140748\pi$
$157$	22.6498	1.80765	0.903824	+	0.427905i	$0.140748\pi$
$158$	−23.9508	−1.90543
$159$	13.6763	1.08460
$160$	0	0
$161$	−3.09019	−0.243541
$162$	−2.66908	−0.209703
$163$	3.72548	0.291802	0.145901	−	0.989299i	$-0.453392\pi$
$163$	3.72548	0.291802	0.145901	+	0.989299i	$0.453392\pi$
$164$	20.3599	1.58984
$165$	0	0
$166$	12.1312	0.941565
$167$	12.2214	0.945721	0.472861	−	0.881137i	$-0.343221\pi$
$167$	12.2214	0.945721	0.472861	+	0.881137i	$0.343221\pi$
$168$	−3.79306	−0.292641
$169$	−1.85670	−0.142823
$170$	0	0
$171$	1.00000	0.0764719
$172$	40.9919	3.12560
$173$	24.1722	1.83778	0.918891	−	0.394511i	$-0.129086\pi$
$173$	24.1722	1.83778	0.918891	+	0.394511i	$0.129086\pi$
$174$	−8.00724	−0.607027
$175$	0	0
$176$	−17.4694	−1.31680
$177$	7.54510	0.567124
$178$	0.240729	0.0180434
$179$	−6.22141	−0.465010	−0.232505	−	0.972595i	$-0.574692\pi$
$179$	−6.22141	−0.465010	−0.232505	+	0.972595i	$0.574692\pi$
$180$	0	0
$181$	−20.4959	−1.52345	−0.761725	−	0.647900i	$-0.775647\pi$
$181$	−20.4959	−1.52345	−0.761725	+	0.647900i	$0.775647\pi$
$182$	4.05311	0.300436
$183$	−3.90981	−0.289021
$184$	56.6416	4.17568
$185$	0	0
$186$	−23.4694	−1.72086
$187$	0	0
$188$	12.4428	0.907486
$189$	0.454904	0.0330894
$190$	0	0
$191$	0.793062	0.0573840	0.0286920	−	0.999588i	$-0.490866\pi$
$191$	0.793062	0.0573840	0.0286920	+	0.999588i	$0.490866\pi$
$192$	17.0145	1.22791
$193$	−15.8196	−1.13872	−0.569360	−	0.822088i	$-0.692809\pi$
$193$	−15.8196	−1.13872	−0.569360	+	0.822088i	$0.692809\pi$
$194$	−32.0676	−2.30232
$195$	0	0
$196$	−34.8075	−2.48625
$197$	3.75203	0.267321	0.133661	−	0.991027i	$-0.457327\pi$
$197$	3.75203	0.267321	0.133661	+	0.991027i	$0.457327\pi$
$198$	3.88325	0.275971
$199$	6.45490	0.457576	0.228788	−	0.973476i	$-0.426524\pi$
$199$	6.45490	0.457576	0.228788	+	0.973476i	$0.426524\pi$
$200$	0	0
$201$	14.1312	0.996739
$202$	−14.7294	−1.03636
$203$	1.36471	0.0957840
$204$	0	0
$205$	0	0
$206$	6.79306	0.473295
$207$	−6.79306	−0.472150
$208$	−40.0821	−2.77919
$209$	−1.45490	−0.100638
$210$	0	0
$211$	23.9653	1.64984	0.824920	−	0.565249i	$-0.191220\pi$
$211$	23.9653	1.64984	0.824920	+	0.565249i	$0.191220\pi$
$212$	70.0772	4.81292
$213$	−9.13122	−0.625661
$214$	12.3719	0.845729
$215$	0	0
$216$	−8.33816	−0.567340
$217$	4.00000	0.271538
$218$	16.6763	1.12946
$219$	−11.6763	−0.789012
$220$	0	0
$221$	0	0
$222$	−27.3526	−1.83579
$223$	−3.20694	−0.214752	−0.107376	−	0.994218i	$-0.534245\pi$
$223$	−3.20694	−0.214752	−0.107376	+	0.994218i	$0.534245\pi$
$224$	−6.99276	−0.467224
$225$	0	0
$226$	13.8977	0.924463
$227$	−9.13122	−0.606060	−0.303030	−	0.952981i	$-0.597998\pi$
$227$	−9.13122	−0.606060	−0.303030	+	0.952981i	$0.597998\pi$
$228$	5.12398	0.339344
$229$	16.7400	1.10621	0.553104	−	0.833112i	$-0.313443\pi$
$229$	16.7400	1.10621	0.553104	+	0.833112i	$0.313443\pi$
$230$	0	0
$231$	−0.661842	−0.0435460
$232$	−25.0145	−1.64228
$233$	−9.57165	−0.627060	−0.313530	−	0.949578i	$-0.601512\pi$
$233$	−9.57165	−0.627060	−0.313530	+	0.949578i	$0.601512\pi$
$234$	8.90981	0.582452
$235$	0	0
$236$	38.6609	2.51661
$237$	8.97345	0.582888
$238$	0	0
$239$	17.5185	1.13318	0.566590	−	0.824000i	$-0.308262\pi$
$239$	17.5185	1.13318	0.566590	+	0.824000i	$0.308262\pi$
$240$	0	0
$241$	16.9098	1.08926	0.544628	−	0.838678i	$-0.316671\pi$
$241$	16.9098	1.08926	0.544628	+	0.838678i	$0.316671\pi$
$242$	23.7101	1.52414
$243$	1.00000	0.0641500
$244$	−20.0338	−1.28253
$245$	0	0
$246$	−10.6054	−0.676178
$247$	−3.33816	−0.212402
$248$	−73.3179	−4.65569
$249$	−4.54510	−0.288034
$250$	0	0
$251$	−11.6392	−0.734662	−0.367331	−	0.930090i	$-0.619728\pi$
$251$	−11.6392	−0.734662	−0.367331	+	0.930090i	$0.619728\pi$
$252$	2.33092	0.146834
$253$	9.88325	0.621355
$254$	28.3261	1.77734
$255$	0	0
$256$	5.12398	0.320249
$257$	−4.76651	−0.297327	−0.148663	−	0.988888i	$-0.547497\pi$
$257$	−4.76651	−0.297327	−0.148663	+	0.988888i	$0.547497\pi$
$258$	−21.3526	−1.32936
$259$	4.66184	0.289673
$260$	0	0
$261$	3.00000	0.185695
$262$	−57.9267	−3.57872
$263$	14.8606	0.916347	0.458173	−	0.888863i	$-0.348504\pi$
$263$	14.8606	0.916347	0.458173	+	0.888863i	$0.348504\pi$
$264$	12.1312	0.746625
$265$	0	0
$266$	−1.21417	−0.0744458
$267$	−0.0901918	−0.00551965
$268$	72.4081	4.42303
$269$	−21.5330	−1.31289	−0.656446	−	0.754373i	$-0.727941\pi$
$269$	−21.5330	−1.31289	−0.656446	+	0.754373i	$0.727941\pi$
$270$	0	0
$271$	−1.13122	−0.0687167	−0.0343584	−	0.999410i	$-0.510939\pi$
$271$	−1.13122	−0.0687167	−0.0343584	+	0.999410i	$0.510939\pi$
$272$	0	0
$273$	−1.51854	−0.0919063
$274$	−59.5620	−3.59827
$275$	0	0
$276$	−34.8075	−2.09517
$277$	−13.2624	−0.796863	−0.398431	−	0.917198i	$-0.630445\pi$
$277$	−13.2624	−0.796863	−0.398431	+	0.917198i	$0.630445\pi$
$278$	27.7632	1.66513
$279$	8.79306	0.526427
$280$	0	0
$281$	14.1988	0.847030	0.423515	−	0.905889i	$-0.360796\pi$
$281$	14.1988	0.847030	0.423515	+	0.905889i	$0.360796\pi$
$282$	−6.48146	−0.385965
$283$	−19.5330	−1.16112	−0.580559	−	0.814218i	$-0.697166\pi$
$283$	−19.5330	−1.16112	−0.580559	+	0.814218i	$0.697166\pi$
$284$	−46.7882	−2.77637
$285$	0	0
$286$	−12.9629	−0.766513
$287$	1.80754	0.106696
$288$	−15.3719	−0.905801
$289$	−17.0000	−1.00000
$290$	0	0
$291$	12.0145	0.704301
$292$	−59.8292	−3.50124
$293$	−24.9734	−1.45896	−0.729482	−	0.684000i	$-0.760239\pi$
$293$	−24.9734	−1.45896	−0.729482	+	0.684000i	$0.760239\pi$
$294$	18.1312	1.05743
$295$	0	0
$296$	−85.4492	−4.96663
$297$	−1.45490	−0.0844221
$298$	−13.7665	−0.797472
$299$	22.6763	1.31141
$300$	0	0
$301$	3.63923	0.209762
$302$	22.6763	1.30488
$303$	5.51854	0.317032
$304$	12.0072	0.688662
$305$	0	0
$306$	0	0
$307$	11.4018	0.650735	0.325367	−	0.945588i	$-0.394512\pi$
$307$	11.4018	0.650735	0.325367	+	0.945588i	$0.394512\pi$
$308$	−3.39127	−0.193235
$309$	−2.54510	−0.144785
$310$	0	0
$311$	1.32368	0.0750592	0.0375296	−	0.999296i	$-0.488051\pi$
$311$	1.32368	0.0750592	0.0375296	+	0.999296i	$0.488051\pi$
$312$	27.8341	1.57580
$313$	20.6127	1.16510	0.582549	−	0.812796i	$-0.302056\pi$
$313$	20.6127	1.16510	0.582549	+	0.812796i	$0.302056\pi$
$314$	−60.4540	−3.41162
$315$	0	0
$316$	45.9798	2.58657
$317$	−18.1722	−1.02066	−0.510328	−	0.859980i	$-0.670476\pi$
$317$	−18.1722	−1.02066	−0.510328	+	0.859980i	$0.670476\pi$
$318$	−36.5032	−2.04700
$319$	−4.36471	−0.244377
$320$	0	0
$321$	−4.63529	−0.258717
$322$	8.24797	0.459641
$323$	0	0
$324$	5.12398	0.284666
$325$	0	0
$326$	−9.94360	−0.550725
$327$	−6.24797	−0.345513
$328$	−33.1312	−1.82937
$329$	1.10467	0.0609022
$330$	0	0
$331$	−19.5225	−1.07305	−0.536526	−	0.843883i	$-0.680264\pi$
$331$	−19.5225	−1.07305	−0.536526	+	0.843883i	$0.680264\pi$
$332$	−23.2890	−1.27815
$333$	10.2480	0.561585
$334$	−32.6199	−1.78488
$335$	0	0
$336$	5.46214	0.297984
$337$	0.413875	0.0225452	0.0112726	−	0.999936i	$-0.496412\pi$
$337$	0.413875	0.0225452	0.0112726	+	0.999936i	$0.496412\pi$
$338$	4.95568	0.269553
$339$	−5.20694	−0.282802
$340$	0	0
$341$	−12.7931	−0.692783
$342$	−2.66908	−0.144327
$343$	−6.27452	−0.338792
$344$	−66.7053	−3.59651
$345$	0	0
$346$	−64.5176	−3.46849
$347$	−30.9243	−1.66010	−0.830051	−	0.557687i	$-0.811689\pi$
$347$	−30.9243	−1.66010	−0.830051	+	0.557687i	$0.811689\pi$
$348$	15.3719	0.824023
$349$	−14.7665	−0.790433	−0.395217	−	0.918588i	$-0.629330\pi$
$349$	−14.7665	−0.790433	−0.395217	+	0.918588i	$0.629330\pi$
$350$	0	0
$351$	−3.33816	−0.178178
$352$	22.3647	1.19204
$353$	−27.5330	−1.46543	−0.732717	−	0.680533i	$-0.761748\pi$
$353$	−27.5330	−1.46543	−0.732717	+	0.680533i	$0.761748\pi$
$354$	−20.1385	−1.07035
$355$	0	0
$356$	−0.462141	−0.0244934
$357$	0	0
$358$	16.6054	0.877625
$359$	−3.83409	−0.202356	−0.101178	−	0.994868i	$-0.532261\pi$
$359$	−3.83409	−0.202356	−0.101178	+	0.994868i	$0.532261\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	54.7053	2.87524
$363$	−8.88325	−0.466250
$364$	−7.78098	−0.407834
$365$	0	0
$366$	10.4356	0.545477
$367$	21.2254	1.10795	0.553977	−	0.832532i	$-0.313109\pi$
$367$	21.2254	1.10795	0.553977	+	0.832532i	$0.313109\pi$
$368$	−81.5659	−4.25192
$369$	3.97345	0.206849
$370$	0	0
$371$	6.22141	0.323000
$372$	45.0555	2.33602
$373$	−22.2624	−1.15271	−0.576353	−	0.817201i	$-0.695525\pi$
$373$	−22.2624	−1.15271	−0.576353	+	0.817201i	$0.695525\pi$
$374$	0	0
$375$	0	0
$376$	−20.2480	−1.04421
$377$	−10.0145	−0.515772
$378$	−1.21417	−0.0624504
$379$	−3.33816	−0.171470	−0.0857348	−	0.996318i	$-0.527324\pi$
$379$	−3.33816	−0.171470	−0.0857348	+	0.996318i	$0.527324\pi$
$380$	0	0
$381$	−10.6127	−0.543704
$382$	−2.11675	−0.108302
$383$	34.5370	1.76476	0.882378	−	0.470541i	$-0.155941\pi$
$383$	34.5370	1.76476	0.882378	+	0.470541i	$0.155941\pi$
$384$	−14.6691	−0.748578
$385$	0	0
$386$	42.2238	2.14914
$387$	8.00000	0.406663
$388$	61.5620	3.12534
$389$	19.1047	0.968645	0.484323	−	0.874890i	$-0.339066\pi$
$389$	19.1047	0.968645	0.484323	+	0.874890i	$0.339066\pi$
$390$	0	0
$391$	0	0
$392$	56.6416	2.86083
$393$	21.7029	1.09476
$394$	−10.0145	−0.504522
$395$	0	0
$396$	−7.45490	−0.374623
$397$	13.8301	0.694115	0.347058	−	0.937844i	$-0.387181\pi$
$397$	13.8301	0.694115	0.347058	+	0.937844i	$0.387181\pi$
$398$	−17.2286	−0.863594
$399$	0.454904	0.0227737
$400$	0	0
$401$	6.79306	0.339229	0.169615	−	0.985510i	$-0.445748\pi$
$401$	6.79306	0.339229	0.169615	+	0.985510i	$0.445748\pi$
$402$	−37.7173	−1.88117
$403$	−29.3526	−1.46216
$404$	28.2769	1.40683
$405$	0	0
$406$	−3.64252	−0.180775
$407$	−14.9098	−0.739052
$408$	0	0
$409$	−12.5104	−0.618600	−0.309300	−	0.950965i	$-0.600095\pi$
$409$	−12.5104	−0.618600	−0.309300	+	0.950965i	$0.600095\pi$
$410$	0	0
$411$	22.3155	1.10074
$412$	−13.0410	−0.642485
$413$	3.43229	0.168892
$414$	18.1312	0.891101
$415$	0	0
$416$	51.3140	2.51588
$417$	−10.4018	−0.509378
$418$	3.88325	0.189936
$419$	−2.06758	−0.101008	−0.0505040	−	0.998724i	$-0.516083\pi$
$419$	−2.06758	−0.101008	−0.0505040	+	0.998724i	$0.516083\pi$
$420$	0	0
$421$	−3.51854	−0.171483	−0.0857416	−	0.996317i	$-0.527326\pi$
$421$	−3.51854	−0.171483	−0.0857416	+	0.996317i	$0.527326\pi$
$422$	−63.9653	−3.11378
$423$	2.42835	0.118070
$424$	−114.035	−5.53804
$425$	0	0
$426$	24.3719	1.18082
$427$	−1.77859	−0.0860719
$428$	−23.7511	−1.14805
$429$	4.85670	0.234484
$430$	0	0
$431$	31.4468	1.51474	0.757369	−	0.652987i	$-0.226485\pi$
$431$	31.4468	1.51474	0.757369	+	0.652987i	$0.226485\pi$
$432$	12.0072	0.577698
$433$	−20.9774	−1.00811	−0.504055	−	0.863672i	$-0.668159\pi$
$433$	−20.9774	−1.00811	−0.504055	+	0.863672i	$0.668159\pi$
$434$	−10.6763	−0.512480
$435$	0	0
$436$	−32.0145	−1.53322
$437$	−6.79306	−0.324956
$438$	31.1650	1.48912
$439$	3.75598	0.179263	0.0896315	−	0.995975i	$-0.471431\pi$
$439$	3.75598	0.179263	0.0896315	+	0.995975i	$0.471431\pi$
$440$	0	0
$441$	−6.79306	−0.323479
$442$	0	0
$443$	−20.0184	−0.951104	−0.475552	−	0.879688i	$-0.657752\pi$
$443$	−20.0184	−0.951104	−0.475552	+	0.879688i	$0.657752\pi$
$444$	52.5104	2.49203
$445$	0	0
$446$	8.55957	0.405307
$447$	5.15777	0.243954
$448$	7.73995	0.365678
$449$	22.4057	1.05739	0.528696	−	0.848811i	$-0.322681\pi$
$449$	22.4057	1.05739	0.528696	+	0.848811i	$0.322681\pi$
$450$	0	0
$451$	−5.78098	−0.272216
$452$	−26.6803	−1.25493
$453$	−8.49593	−0.399174
$454$	24.3719	1.14383
$455$	0	0
$456$	−8.33816	−0.390470
$457$	34.2890	1.60397	0.801986	−	0.597343i	$-0.203777\pi$
$457$	34.2890	1.60397	0.801986	+	0.597343i	$0.203777\pi$
$458$	−44.6803	−2.08777
$459$	0	0
$460$	0	0
$461$	14.5104	0.675817	0.337909	−	0.941179i	$-0.390281\pi$
$461$	14.5104	0.675817	0.337909	+	0.941179i	$0.390281\pi$
$462$	1.76651	0.0821854
$463$	33.8486	1.57308	0.786538	−	0.617542i	$-0.211871\pi$
$463$	33.8486	1.57308	0.786538	+	0.617542i	$0.211871\pi$
$464$	36.0217	1.67227
$465$	0	0
$466$	25.5475	1.18346
$467$	−9.22141	−0.426716	−0.213358	−	0.976974i	$-0.568440\pi$
$467$	−9.22141	−0.426716	−0.213358	+	0.976974i	$0.568440\pi$
$468$	−17.1047	−0.790663
$469$	6.42835	0.296834
$470$	0	0
$471$	22.6498	1.04365
$472$	−62.9122	−2.89577
$473$	−11.6392	−0.535172
$474$	−23.9508	−1.10010
$475$	0	0
$476$	0	0
$477$	13.6763	0.626196
$478$	−46.7584	−2.13868
$479$	27.3035	1.24753	0.623764	−	0.781613i	$-0.285603\pi$
$479$	27.3035	1.24753	0.623764	+	0.781613i	$0.285603\pi$
$480$	0	0
$481$	−34.2093	−1.55981
$482$	−45.1336	−2.05578
$483$	−3.09019	−0.140609
$484$	−45.5176	−2.06898
$485$	0	0
$486$	−2.66908	−0.121072
$487$	14.4815	0.656218	0.328109	−	0.944640i	$-0.393589\pi$
$487$	14.4815	0.656218	0.328109	+	0.944640i	$0.393589\pi$
$488$	32.6006	1.47576
$489$	3.72548	0.168472
$490$	0	0
$491$	−38.2093	−1.72436	−0.862182	−	0.506599i	$-0.830902\pi$
$491$	−38.2093	−1.72436	−0.862182	+	0.506599i	$0.830902\pi$
$492$	20.3599	0.917894
$493$	0	0
$494$	8.90981	0.400871
$495$	0	0
$496$	105.580	4.74070
$497$	−4.15383	−0.186325
$498$	12.1312	0.543613
$499$	35.5740	1.59251	0.796256	−	0.604959i	$-0.206811\pi$
$499$	35.5740	1.59251	0.796256	+	0.604959i	$0.206811\pi$
$500$	0	0
$501$	12.2214	0.546012
$502$	31.0660	1.38654
$503$	−6.84223	−0.305080	−0.152540	−	0.988297i	$-0.548745\pi$
$503$	−6.84223	−0.305080	−0.152540	+	0.988297i	$0.548745\pi$
$504$	−3.79306	−0.168956
$505$	0	0
$506$	−26.3792	−1.17270
$507$	−1.85670	−0.0824589
$508$	−54.3792	−2.41269
$509$	25.1352	1.11410	0.557048	−	0.830480i	$-0.311934\pi$
$509$	25.1352	1.11410	0.557048	+	0.830480i	$0.311934\pi$
$510$	0	0
$511$	−5.31160	−0.234972
$512$	15.6618	0.692162
$513$	1.00000	0.0441511
$514$	12.7222	0.561152
$515$	0	0
$516$	40.9919	1.80457
$517$	−3.53302	−0.155382
$518$	−12.4428	−0.546706
$519$	24.1722	1.06104
$520$	0	0
$521$	10.5861	0.463787	0.231893	−	0.972741i	$-0.425508\pi$
$521$	10.5861	0.463787	0.231893	+	0.972741i	$0.425508\pi$
$522$	−8.00724	−0.350467
$523$	−33.9614	−1.48503	−0.742513	−	0.669831i	$-0.766366\pi$
$523$	−33.9614	−1.48503	−0.742513	+	0.669831i	$0.766366\pi$
$524$	111.205	4.85802
$525$	0	0
$526$	−39.6642	−1.72944
$527$	0	0
$528$	−17.4694	−0.760257
$529$	23.1457	1.00633
$530$	0	0
$531$	7.54510	0.327429
$532$	2.33092	0.101058
$533$	−13.2640	−0.574527
$534$	0.240729	0.0104174
$535$	0	0
$536$	−117.828	−5.08941
$537$	−6.22141	−0.268474
$538$	57.4733	2.47785
$539$	9.88325	0.425702
$540$	0	0
$541$	−33.7318	−1.45024	−0.725122	−	0.688620i	$-0.758217\pi$
$541$	−33.7318	−1.45024	−0.725122	+	0.688620i	$0.758217\pi$
$542$	3.01932	0.129691
$543$	−20.4959	−0.879565
$544$	0	0
$545$	0	0
$546$	4.05311	0.173457
$547$	−20.7255	−0.886158	−0.443079	−	0.896483i	$-0.646114\pi$
$547$	−20.7255	−0.886158	−0.443079	+	0.896483i	$0.646114\pi$
$548$	114.344	4.88455
$549$	−3.90981	−0.166867
$550$	0	0
$551$	3.00000	0.127804
$552$	56.6416	2.41083
$553$	4.08206	0.173587
$554$	35.3985	1.50394
$555$	0	0
$556$	−53.2986	−2.26037
$557$	−13.3237	−0.564543	−0.282271	−	0.959335i	$-0.591088\pi$
$557$	−13.3237	−0.564543	−0.282271	+	0.959335i	$0.591088\pi$
$558$	−23.4694	−0.993538
$559$	−26.7053	−1.12951
$560$	0	0
$561$	0	0
$562$	−37.8977	−1.59862
$563$	11.7786	0.496408	0.248204	−	0.968708i	$-0.420160\pi$
$563$	11.7786	0.496408	0.248204	+	0.968708i	$0.420160\pi$
$564$	12.4428	0.523937
$565$	0	0
$566$	52.1352	2.19140
$567$	0.454904	0.0191042
$568$	76.1376	3.19466
$569$	43.2278	1.81220	0.906101	−	0.423062i	$-0.139045\pi$
$569$	43.2278	1.81220	0.906101	+	0.423062i	$0.139045\pi$
$570$	0	0
$571$	−3.41782	−0.143031	−0.0715157	−	0.997439i	$-0.522784\pi$
$571$	−3.41782	−0.143031	−0.0715157	+	0.997439i	$0.522784\pi$
$572$	24.8856	1.04052
$573$	0.793062	0.0331307
$574$	−4.82446	−0.201369
$575$	0	0
$576$	17.0145	0.708936
$577$	−32.4081	−1.34917	−0.674584	−	0.738198i	$-0.735677\pi$
$577$	−32.4081	−1.34917	−0.674584	+	0.738198i	$0.735677\pi$
$578$	45.3743	1.88732
$579$	−15.8196	−0.657441
$580$	0	0
$581$	−2.06758	−0.0857778
$582$	−32.0676	−1.32924
$583$	−19.8977	−0.824080
$584$	97.3590	4.02874
$585$	0	0
$586$	66.6561	2.75354
$587$	−7.93636	−0.327569	−0.163784	−	0.986496i	$-0.552370\pi$
$587$	−7.93636	−0.327569	−0.163784	+	0.986496i	$0.552370\pi$
$588$	−34.8075	−1.43544
$589$	8.79306	0.362312
$590$	0	0
$591$	3.75203	0.154338
$592$	123.050	5.05731
$593$	1.98553	0.0815358	0.0407679	−	0.999169i	$-0.487020\pi$
$593$	1.98553	0.0815358	0.0407679	+	0.999169i	$0.487020\pi$
$594$	3.88325	0.159332
$595$	0	0
$596$	26.4283	1.08255
$597$	6.45490	0.264182
$598$	−60.5249	−2.47505
$599$	−21.3526	−0.872445	−0.436222	−	0.899839i	$-0.643684\pi$
$599$	−21.3526	−0.872445	−0.436222	+	0.899839i	$0.643684\pi$
$600$	0	0
$601$	−18.2480	−0.744350	−0.372175	−	0.928163i	$-0.621388\pi$
$601$	−18.2480	−0.744350	−0.372175	+	0.928163i	$0.621388\pi$
$602$	−9.71340	−0.395889
$603$	14.1312	0.575468
$604$	−43.5330	−1.77133
$605$	0	0
$606$	−14.7294	−0.598342
$607$	−23.6353	−0.959327	−0.479663	−	0.877453i	$-0.659241\pi$
$607$	−23.6353	−0.959327	−0.479663	+	0.877453i	$0.659241\pi$
$608$	−15.3719	−0.623415
$609$	1.36471	0.0553009
$610$	0	0
$611$	−8.10622	−0.327942
$612$	0	0
$613$	−36.9122	−1.49087	−0.745435	−	0.666578i	$-0.767758\pi$
$613$	−36.9122	−1.49087	−0.745435	+	0.666578i	$0.767758\pi$
$614$	−30.4323	−1.22815
$615$	0	0
$616$	5.51854	0.222348
$617$	14.6908	0.591429	0.295714	−	0.955276i	$-0.404442\pi$
$617$	14.6908	0.591429	0.295714	+	0.955276i	$0.404442\pi$
$618$	6.79306	0.273257
$619$	−24.3487	−0.978656	−0.489328	−	0.872100i	$-0.662758\pi$
$619$	−24.3487	−0.978656	−0.489328	+	0.872100i	$0.662758\pi$
$620$	0	0
$621$	−6.79306	−0.272596
$622$	−3.53302	−0.141661
$623$	−0.0410286	−0.00164378
$624$	−40.0821	−1.60457
$625$	0	0
$626$	−55.0169	−2.19892
$627$	−1.45490	−0.0581033
$628$	116.057	4.63118
$629$	0	0
$630$	0	0
$631$	21.3237	0.848882	0.424441	−	0.905455i	$-0.360471\pi$
$631$	21.3237	0.848882	0.424441	+	0.905455i	$0.360471\pi$
$632$	−74.8220	−2.97626
$633$	23.9653	0.952536
$634$	48.5032	1.92631
$635$	0	0
$636$	70.0772	2.77874
$637$	22.6763	0.898468
$638$	11.6498	0.461219
$639$	−9.13122	−0.361225
$640$	0	0
$641$	−26.9919	−1.06611	−0.533057	−	0.846079i	$-0.678957\pi$
$641$	−26.9919	−1.06611	−0.533057	+	0.846079i	$0.678957\pi$
$642$	12.3719	0.488282
$643$	40.1683	1.58408	0.792042	−	0.610467i	$-0.209018\pi$
$643$	40.1683	1.58408	0.792042	+	0.610467i	$0.209018\pi$
$644$	−15.8341	−0.623951
$645$	0	0
$646$	0	0
$647$	−17.2890	−0.679701	−0.339850	−	0.940480i	$-0.610376\pi$
$647$	−17.2890	−0.679701	−0.339850	+	0.940480i	$0.610376\pi$
$648$	−8.33816	−0.327554
$649$	−10.9774	−0.430900
$650$	0	0
$651$	4.00000	0.156772
$652$	19.0893	0.747594
$653$	−21.6537	−0.847375	−0.423688	−	0.905808i	$-0.639265\pi$
$653$	−21.6537	−0.847375	−0.423688	+	0.905808i	$0.639265\pi$
$654$	16.6763	0.652096
$655$	0	0
$656$	47.7101	1.86277
$657$	−11.6763	−0.455536
$658$	−2.94844	−0.114942
$659$	−11.6392	−0.453400	−0.226700	−	0.973965i	$-0.572794\pi$
$659$	−11.6392	−0.453400	−0.226700	+	0.973965i	$0.572794\pi$
$660$	0	0
$661$	−4.96292	−0.193035	−0.0965175	−	0.995331i	$-0.530770\pi$
$661$	−4.96292	−0.193035	−0.0965175	+	0.995331i	$0.530770\pi$
$662$	52.1071	2.02520
$663$	0	0
$664$	37.8977	1.47072
$665$	0	0
$666$	−27.3526	−1.05989
$667$	−20.3792	−0.789085
$668$	62.6223	2.42293
$669$	−3.20694	−0.123987
$670$	0	0
$671$	5.68840	0.219598
$672$	−6.99276	−0.269752
$673$	13.1578	0.507195	0.253597	−	0.967310i	$-0.418386\pi$
$673$	13.1578	0.507195	0.253597	+	0.967310i	$0.418386\pi$
$674$	−1.10467	−0.0425502
$675$	0	0
$676$	−9.51370	−0.365912
$677$	−13.1352	−0.504825	−0.252413	−	0.967620i	$-0.581224\pi$
$677$	−13.1352	−0.504825	−0.252413	+	0.967620i	$0.581224\pi$
$678$	13.8977	0.533739
$679$	5.46543	0.209744
$680$	0	0
$681$	−9.13122	−0.349909
$682$	34.1457	1.30751
$683$	46.5370	1.78069	0.890344	−	0.455289i	$-0.150464\pi$
$683$	46.5370	1.78069	0.890344	+	0.455289i	$0.150464\pi$
$684$	5.12398	0.195920
$685$	0	0
$686$	16.7472	0.639411
$687$	16.7400	0.638669
$688$	96.0579	3.66217
$689$	−45.6537	−1.73927
$690$	0	0
$691$	−4.26244	−0.162151	−0.0810754	−	0.996708i	$-0.525835\pi$
$691$	−4.26244	−0.162151	−0.0810754	+	0.996708i	$0.525835\pi$
$692$	123.858	4.70838
$693$	−0.661842	−0.0251413
$694$	82.5394	3.13315
$695$	0	0
$696$	−25.0145	−0.948171
$697$	0	0
$698$	39.4130	1.49180
$699$	−9.57165	−0.362033
$700$	0	0
$701$	−36.4428	−1.37643	−0.688213	−	0.725508i	$-0.741605\pi$
$701$	−36.4428	−1.37643	−0.688213	+	0.725508i	$0.741605\pi$
$702$	8.90981	0.336279
$703$	10.2480	0.386509
$704$	−24.7544	−0.932968
$705$	0	0
$706$	73.4878	2.76575
$707$	2.51041	0.0944136
$708$	38.6609	1.45297
$709$	−5.23349	−0.196548	−0.0982740	−	0.995159i	$-0.531332\pi$
$709$	−5.23349	−0.196548	−0.0982740	+	0.995159i	$0.531332\pi$
$710$	0	0
$711$	8.97345	0.336531
$712$	0.752034	0.0281837
$713$	−59.7318	−2.23697
$714$	0	0
$715$	0	0
$716$	−31.8784	−1.19135
$717$	17.5185	0.654242
$718$	10.2335	0.381911
$719$	50.2809	1.87516	0.937580	−	0.347770i	$-0.113061\pi$
$719$	50.2809	1.87516	0.937580	+	0.347770i	$0.113061\pi$
$720$	0	0
$721$	−1.15777	−0.0431178
$722$	−2.66908	−0.0993328
$723$	16.9098	0.628883
$724$	−105.021	−3.90307
$725$	0	0
$726$	23.7101	0.879965
$727$	−2.81567	−0.104427	−0.0522137	−	0.998636i	$-0.516628\pi$
$727$	−2.81567	−0.104427	−0.0522137	+	0.998636i	$0.516628\pi$
$728$	12.6618	0.469279
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−20.0338	−0.740470
$733$	9.23349	0.341047	0.170523	−	0.985354i	$-0.445454\pi$
$733$	9.23349	0.341047	0.170523	+	0.985354i	$0.445454\pi$
$734$	−56.6522	−2.09107
$735$	0	0
$736$	104.423	3.84907
$737$	−20.5596	−0.757322
$738$	−10.6054	−0.390392
$739$	−11.1843	−0.411422	−0.205711	−	0.978613i	$-0.565951\pi$
$739$	−11.1843	−0.411422	−0.205711	+	0.978613i	$0.565951\pi$
$740$	0	0
$741$	−3.33816	−0.122630
$742$	−16.6054	−0.609605
$743$	−23.1602	−0.849664	−0.424832	−	0.905272i	$-0.639667\pi$
$743$	−23.1602	−0.849664	−0.424832	+	0.905272i	$0.639667\pi$
$744$	−73.3179	−2.68797
$745$	0	0
$746$	59.4202	2.17553
$747$	−4.54510	−0.166296
$748$	0	0
$749$	−2.10861	−0.0770470
$750$	0	0
$751$	11.7520	0.428838	0.214419	−	0.976742i	$-0.431214\pi$
$751$	11.7520	0.428838	0.214419	+	0.976742i	$0.431214\pi$
$752$	29.1578	1.06327
$753$	−11.6392	−0.424157
$754$	26.7294	0.973428
$755$	0	0
$756$	2.33092	0.0847748
$757$	17.6232	0.640526	0.320263	−	0.947329i	$-0.396229\pi$
$757$	17.6232	0.640526	0.320263	+	0.947329i	$0.396229\pi$
$758$	8.90981	0.323619
$759$	9.88325	0.358739
$760$	0	0
$761$	−22.5556	−0.817641	−0.408820	−	0.912615i	$-0.634060\pi$
$761$	−22.5556	−0.817641	−0.408820	+	0.912615i	$0.634060\pi$
$762$	28.3261	1.02615
$763$	−2.84223	−0.102895
$764$	4.06364	0.147017
$765$	0	0
$766$	−92.1819	−3.33067
$767$	−25.1867	−0.909440
$768$	5.12398	0.184896
$769$	−26.8486	−0.968184	−0.484092	−	0.875017i	$-0.660850\pi$
$769$	−26.8486	−0.968184	−0.484092	+	0.875017i	$0.660850\pi$
$770$	0	0
$771$	−4.76651	−0.171662
$772$	−81.0594	−2.91739
$773$	−33.8751	−1.21840	−0.609202	−	0.793015i	$-0.708510\pi$
$773$	−33.8751	−1.21840	−0.609202	+	0.793015i	$0.708510\pi$
$774$	−21.3526	−0.767505
$775$	0	0
$776$	−100.179	−3.59620
$777$	4.66184	0.167243
$778$	−50.9919	−1.82815
$779$	3.97345	0.142363
$780$	0	0
$781$	13.2850	0.475376
$782$	0	0
$783$	3.00000	0.107211
$784$	−81.5659	−2.91307
$785$	0	0
$786$	−57.9267	−2.06618
$787$	−30.0781	−1.07217	−0.536084	−	0.844164i	$-0.680097\pi$
$787$	−30.0781	−1.07217	−0.536084	+	0.844164i	$0.680097\pi$
$788$	19.2254	0.684875
$789$	14.8606	0.529053
$790$	0	0
$791$	−2.36866	−0.0842198
$792$	12.1312	0.431064
$793$	13.0516	0.463474
$794$	−36.9138	−1.31002
$795$	0	0
$796$	33.0748	1.17231
$797$	−29.3792	−1.04066	−0.520332	−	0.853964i	$-0.674192\pi$
$797$	−29.3792	−1.04066	−0.520332	+	0.853964i	$0.674192\pi$
$798$	−1.21417	−0.0429813
$799$	0	0
$800$	0	0
$801$	−0.0901918	−0.00318677
$802$	−18.1312	−0.640236
$803$	16.9879	0.599491
$804$	72.4081	2.55364
$805$	0	0
$806$	78.3445	2.75957
$807$	−21.5330	−0.757998
$808$	−46.0145	−1.61878
$809$	−45.7358	−1.60798	−0.803992	−	0.594640i	$-0.797295\pi$
$809$	−45.7358	−1.60798	−0.803992	+	0.594640i	$0.797295\pi$
$810$	0	0
$811$	6.10227	0.214280	0.107140	−	0.994244i	$-0.465831\pi$
$811$	6.10227	0.214280	0.107140	+	0.994244i	$0.465831\pi$
$812$	6.99276	0.245398
$813$	−1.13122	−0.0396736
$814$	39.7955	1.39483
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	33.3913	1.16750
$819$	−1.51854	−0.0530621
$820$	0	0
$821$	−11.2175	−0.391492	−0.195746	−	0.980655i	$-0.562713\pi$
$821$	−11.2175	−0.391492	−0.195746	+	0.980655i	$0.562713\pi$
$822$	−59.5620	−2.07746
$823$	24.6353	0.858732	0.429366	−	0.903131i	$-0.358737\pi$
$823$	24.6353	0.858732	0.429366	+	0.903131i	$0.358737\pi$
$824$	21.2214	0.739283
$825$	0	0
$826$	−9.16107	−0.318754
$827$	−32.6311	−1.13469	−0.567347	−	0.823479i	$-0.692030\pi$
$827$	−32.6311	−1.13469	−0.567347	+	0.823479i	$0.692030\pi$
$828$	−34.8075	−1.20965
$829$	−33.3382	−1.15788	−0.578941	−	0.815369i	$-0.696534\pi$
$829$	−33.3382	−1.15788	−0.578941	+	0.815369i	$0.696534\pi$
$830$	0	0
$831$	−13.2624	−0.460069
$832$	−56.7970	−1.96908
$833$	0	0
$834$	27.7632	0.961362
$835$	0	0
$836$	−7.45490	−0.257833
$837$	8.79306	0.303933
$838$	5.51854	0.190635
$839$	51.1312	1.76525	0.882623	−	0.470082i	$-0.155776\pi$
$839$	51.1312	1.76525	0.882623	+	0.470082i	$0.155776\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	9.39127	0.323644
$843$	14.1988	0.489033
$844$	122.798	4.22688
$845$	0	0
$846$	−6.48146	−0.222837
$847$	−4.04103	−0.138851
$848$	164.215	5.63916
$849$	−19.5330	−0.670371
$850$	0	0
$851$	−69.6151	−2.38637
$852$	−46.7882	−1.60294
$853$	40.9122	1.40081	0.700404	−	0.713747i	$-0.253003\pi$
$853$	40.9122	1.40081	0.700404	+	0.713747i	$0.253003\pi$
$854$	4.74719	0.162446
$855$	0	0
$856$	38.6498	1.32102
$857$	17.8220	0.608788	0.304394	−	0.952546i	$-0.401546\pi$
$857$	17.8220	0.608788	0.304394	+	0.952546i	$0.401546\pi$
$858$	−12.9629	−0.442547
$859$	14.1231	0.481873	0.240937	−	0.970541i	$-0.422545\pi$
$859$	14.1231	0.481873	0.240937	+	0.970541i	$0.422545\pi$
$860$	0	0
$861$	1.80754	0.0616007
$862$	−83.9339	−2.85880
$863$	−32.4307	−1.10396	−0.551978	−	0.833859i	$-0.686127\pi$
$863$	−32.4307	−1.10396	−0.551978	+	0.833859i	$0.686127\pi$
$864$	−15.3719	−0.522964
$865$	0	0
$866$	55.9903	1.90263
$867$	−17.0000	−0.577350
$868$	20.4959	0.695677
$869$	−13.0555	−0.442878
$870$	0	0
$871$	−47.1722	−1.59837
$872$	52.0965	1.76421
$873$	12.0145	0.406628
$874$	18.1312	0.613298
$875$	0	0
$876$	−59.8292	−2.02144
$877$	20.9243	0.706563	0.353281	−	0.935517i	$-0.385066\pi$
$877$	20.9243	0.706563	0.353281	+	0.935517i	$0.385066\pi$
$878$	−10.0250	−0.338327
$879$	−24.9734	−0.842333
$880$	0	0
$881$	−31.3671	−1.05678	−0.528392	−	0.849000i	$-0.677205\pi$
$881$	−31.3671	−1.05678	−0.528392	+	0.849000i	$0.677205\pi$
$882$	18.1312	0.610510
$883$	0.635288	0.0213791	0.0106896	−	0.999943i	$-0.496597\pi$
$883$	0.635288	0.0213791	0.0106896	+	0.999943i	$0.496597\pi$
$884$	0	0
$885$	0	0
$886$	53.4307	1.79504
$887$	−55.0660	−1.84894	−0.924468	−	0.381259i	$-0.875491\pi$
$887$	−55.0660	−1.84894	−0.924468	+	0.381259i	$0.875491\pi$
$888$	−85.4492	−2.86749
$889$	−4.82775	−0.161918
$890$	0	0
$891$	−1.45490	−0.0487411
$892$	−16.4323	−0.550194
$893$	2.42835	0.0812616
$894$	−13.7665	−0.460421
$895$	0	0
$896$	−6.67302	−0.222930
$897$	22.6763	0.757140
$898$	−59.8027	−1.99564
$899$	26.3792	0.879795
$900$	0	0
$901$	0	0
$902$	15.4299	0.513759
$903$	3.63923	0.121106
$904$	43.4163	1.44400
$905$	0	0
$906$	22.6763	0.753370
$907$	−8.49593	−0.282103	−0.141051	−	0.990002i	$-0.545048\pi$
$907$	−8.49593	−0.282103	−0.141051	+	0.990002i	$0.545048\pi$
$908$	−46.7882	−1.55272
$909$	5.51854	0.183038
$910$	0	0
$911$	58.9798	1.95409	0.977044	−	0.213039i	$-0.0683361\pi$
$911$	58.9798	1.95409	0.977044	+	0.213039i	$0.0683361\pi$
$912$	12.0072	0.397599
$913$	6.61268	0.218848
$914$	−91.5200	−3.02721
$915$	0	0
$916$	85.7752	2.83409
$917$	9.87272	0.326026
$918$	0	0
$919$	8.22141	0.271199	0.135600	−	0.990764i	$-0.456704\pi$
$919$	8.22141	0.271199	0.135600	+	0.990764i	$0.456704\pi$
$920$	0	0
$921$	11.4018	0.375702
$922$	−38.7294	−1.27549
$923$	30.4815	1.00331
$924$	−3.39127	−0.111564
$925$	0	0
$926$	−90.3445	−2.96890
$927$	−2.54510	−0.0835919
$928$	−46.1158	−1.51383
$929$	1.76651	0.0579573	0.0289786	−	0.999580i	$-0.490775\pi$
$929$	1.76651	0.0579573	0.0289786	+	0.999580i	$0.490775\pi$
$930$	0	0
$931$	−6.79306	−0.222634
$932$	−49.0450	−1.60652
$933$	1.32368	0.0433355
$934$	24.6127	0.805351
$935$	0	0
$936$	27.8341	0.909786
$937$	−6.20694	−0.202772	−0.101386	−	0.994847i	$-0.532328\pi$
$937$	−6.20694	−0.202772	−0.101386	+	0.994847i	$0.532328\pi$
$938$	−17.1578	−0.560221
$939$	20.6127	0.672669
$940$	0	0
$941$	15.8833	0.517779	0.258890	−	0.965907i	$-0.416643\pi$
$941$	15.8833	0.517779	0.258890	+	0.965907i	$0.416643\pi$
$942$	−60.4540	−1.96970
$943$	−26.9919	−0.878976
$944$	90.5958	2.94864
$945$	0	0
$946$	31.0660	1.01004
$947$	16.1352	0.524322	0.262161	−	0.965024i	$-0.415565\pi$
$947$	16.1352	0.524322	0.262161	+	0.965024i	$0.415565\pi$
$948$	45.9798	1.49335
$949$	38.9774	1.26526
$950$	0	0
$951$	−18.1722	−0.589276
$952$	0	0
$953$	41.1110	1.33172	0.665858	−	0.746078i	$-0.268066\pi$
$953$	41.1110	1.33172	0.665858	+	0.746078i	$0.268066\pi$
$954$	−36.5032	−1.18183
$955$	0	0
$956$	89.7647	2.90320
$957$	−4.36471	−0.141091
$958$	−72.8751	−2.35449
$959$	10.1514	0.327807
$960$	0	0
$961$	46.3179	1.49413
$962$	91.3074	2.94387
$963$	−4.63529	−0.149370
$964$	86.6456	2.79067
$965$	0	0
$966$	8.24797	0.265374
$967$	30.9798	0.996243	0.498121	−	0.867107i	$-0.334023\pi$
$967$	30.9798	0.996243	0.498121	+	0.867107i	$0.334023\pi$
$968$	74.0700	2.38070
$969$	0	0
$970$	0	0
$971$	51.2133	1.64351	0.821756	−	0.569839i	$-0.192995\pi$
$971$	51.2133	1.64351	0.821756	+	0.569839i	$0.192995\pi$
$972$	5.12398	0.164352
$973$	−4.73182	−0.151695
$974$	−38.6522	−1.23850
$975$	0	0
$976$	−46.9460	−1.50270
$977$	20.2093	0.646554	0.323277	−	0.946304i	$-0.395215\pi$
$977$	20.2093	0.646554	0.323277	+	0.946304i	$0.395215\pi$
$978$	−9.94360	−0.317961
$979$	0.131220	0.00419382
$980$	0	0
$981$	−6.24797	−0.199482
$982$	101.984	3.25443
$983$	41.1191	1.31150	0.655748	−	0.754979i	$-0.272353\pi$
$983$	41.1191	1.31150	0.655748	+	0.754979i	$0.272353\pi$
$984$	−33.1312	−1.05618
$985$	0	0
$986$	0	0
$987$	1.10467	0.0351619
$988$	−17.1047	−0.544172
$989$	−54.3445	−1.72805
$990$	0	0
$991$	27.4549	0.872134	0.436067	−	0.899914i	$-0.356371\pi$
$991$	27.4549	0.872134	0.436067	+	0.899914i	$0.356371\pi$
$992$	−135.167	−4.29154
$993$	−19.5225	−0.619527
$994$	11.0869	0.351655
$995$	0	0
$996$	−23.2890	−0.737940
$997$	13.1086	0.415154	0.207577	−	0.978219i	$-0.433442\pi$
$997$	13.1086	0.415154	0.207577	+	0.978219i	$0.433442\pi$
$998$	−94.9499	−3.00559
$999$	10.2480	0.324231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.a.v.1.1	yes	3
3.2	odd	2		4275.2.a.bh.1.3		3
5.2	odd	4		1425.2.c.p.799.1		6
5.3	odd	4		1425.2.c.p.799.6		6
5.4	even	2		1425.2.a.u.1.3	✓	3
15.14	odd	2		4275.2.a.be.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1425.2.a.u.1.3	✓	3	5.4	even	2
1425.2.a.v.1.1	yes	3	1.1	even	1	trivial
1425.2.c.p.799.1		6	5.2	odd	4
1425.2.c.p.799.6		6	5.3	odd	4
4275.2.a.be.1.1		3	15.14	odd	2
4275.2.a.bh.1.3		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66908 q^{2} +1.00000 q^{3} +5.12398 q^{4} -2.66908 q^{6} +0.454904 q^{7} -8.33816 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.66908 q^{2} +1.00000 q^{3} +5.12398 q^{4} -2.66908 q^{6} +0.454904 q^{7} -8.33816 q^{8} +1.00000 q^{9} -1.45490 q^{11} +5.12398 q^{12} -3.33816 q^{13} -1.21417 q^{14} +12.0072 q^{16} -2.66908 q^{18} +1.00000 q^{19} +0.454904 q^{21} +3.88325 q^{22} -6.79306 q^{23} -8.33816 q^{24} +8.90981 q^{26} +1.00000 q^{27} +2.33092 q^{28} +3.00000 q^{29} +8.79306 q^{31} -15.3719 q^{32} -1.45490 q^{33} +5.12398 q^{36} +10.2480 q^{37} -2.66908 q^{38} -3.33816 q^{39} +3.97345 q^{41} -1.21417 q^{42} +8.00000 q^{43} -7.45490 q^{44} +18.1312 q^{46} +2.42835 q^{47} +12.0072 q^{48} -6.79306 q^{49} -17.1047 q^{52} +13.6763 q^{53} -2.66908 q^{54} -3.79306 q^{56} +1.00000 q^{57} -8.00724 q^{58} +7.54510 q^{59} -3.90981 q^{61} -23.4694 q^{62} +0.454904 q^{63} +17.0145 q^{64} +3.88325 q^{66} +14.1312 q^{67} -6.79306 q^{69} -9.13122 q^{71} -8.33816 q^{72} -11.6763 q^{73} -27.3526 q^{74} +5.12398 q^{76} -0.661842 q^{77} +8.90981 q^{78} +8.97345 q^{79} +1.00000 q^{81} -10.6054 q^{82} -4.54510 q^{83} +2.33092 q^{84} -21.3526 q^{86} +3.00000 q^{87} +12.1312 q^{88} -0.0901918 q^{89} -1.51854 q^{91} -34.8075 q^{92} +8.79306 q^{93} -6.48146 q^{94} -15.3719 q^{96} +12.0145 q^{97} +18.1312 q^{98} -1.45490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9	\( 3 q + 3 q^{3} + 6 q^{4} - 9 q^{8} + 3 q^{9} - 3 q^{11} + 6 q^{12} + 6 q^{13} + 3 q^{14} + 12 q^{16} + 3 q^{19} - 3 q^{22} - 3 q^{23} - 9 q^{24} + 24 q^{26} + 3 q^{27} + 15 q^{28} + 9 q^{29} + 9 q^{31} - 18 q^{32} - 3 q^{33} + 6 q^{36} + 12 q^{37} + 6 q^{39} + 3 q^{42} + 24 q^{43} - 21 q^{44} + 21 q^{46} - 6 q^{47} + 12 q^{48} - 3 q^{49} - 6 q^{52} + 9 q^{53} + 6 q^{56} + 3 q^{57} + 24 q^{59} - 9 q^{61} - 21 q^{62} + 3 q^{64} - 3 q^{66} + 9 q^{67} - 3 q^{69} + 6 q^{71} - 9 q^{72} - 3 q^{73} - 18 q^{74} + 6 q^{76} - 18 q^{77} + 24 q^{78} + 15 q^{79} + 3 q^{81} - 33 q^{82} - 15 q^{83} + 15 q^{84} + 9 q^{87} + 3 q^{88} - 3 q^{89} + 6 q^{91} - 39 q^{92} + 9 q^{93} - 30 q^{94} - 18 q^{96} - 12 q^{97} + 21 q^{98} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^4 - 9 * q^8 + 3 * q^9 - 3 * q^11 + 6 * q^12 + 6 * q^13 + 3 * q^14 + 12 * q^16 + 3 * q^19 - 3 * q^22 - 3 * q^23 - 9 * q^24 + 24 * q^26 + 3 * q^27 + 15 * q^28 + 9 * q^29 + 9 * q^31 - 18 * q^32 - 3 * q^33 + 6 * q^36 + 12 * q^37 + 6 * q^39 + 3 * q^42 + 24 * q^43 - 21 * q^44 + 21 * q^46 - 6 * q^47 + 12 * q^48 - 3 * q^49 - 6 * q^52 + 9 * q^53 + 6 * q^56 + 3 * q^57 + 24 * q^59 - 9 * q^61 - 21 * q^62 + 3 * q^64 - 3 * q^66 + 9 * q^67 - 3 * q^69 + 6 * q^71 - 9 * q^72 - 3 * q^73 - 18 * q^74 + 6 * q^76 - 18 * q^77 + 24 * q^78 + 15 * q^79 + 3 * q^81 - 33 * q^82 - 15 * q^83 + 15 * q^84 + 9 * q^87 + 3 * q^88 - 3 * q^89 + 6 * q^91 - 39 * q^92 + 9 * q^93 - 30 * q^94 - 18 * q^96 - 12 * q^97 + 21 * q^98 - 3 * q^99

Introduction

L-functions

Modular forms

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