LMFDB - Embedded newform 675.2.e.a.451.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(226,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("675.226");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.e (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:	$5.38990213644$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	675.451
Dual form			675.2.e.a.226.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{7} -3.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{7} -3.00000 q^{8} +(-1.00000 + 1.73205i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-1.50000 - 2.59808i) q^{14} +(0.500000 - 0.866025i) q^{16} +4.00000 q^{17} -8.00000 q^{19} +(-1.00000 - 1.73205i) q^{22} +(-1.50000 - 2.59808i) q^{23} +2.00000 q^{26} -3.00000 q^{28} +(-0.500000 + 0.866025i) q^{29} +(-2.50000 - 4.33013i) q^{32} +(-2.00000 + 3.46410i) q^{34} +4.00000 q^{37} +(4.00000 - 6.92820i) q^{38} +(2.50000 + 4.33013i) q^{41} +(-4.00000 + 6.92820i) q^{43} -2.00000 q^{44} +3.00000 q^{46} +(-3.50000 + 6.06218i) q^{47} +(-1.00000 - 1.73205i) q^{49} +(1.00000 - 1.73205i) q^{52} -2.00000 q^{53} +(4.50000 - 7.79423i) q^{56} +(-0.500000 - 0.866025i) q^{58} +(-7.00000 - 12.1244i) q^{59} +(-3.50000 + 6.06218i) q^{61} +7.00000 q^{64} +(-1.50000 - 2.59808i) q^{67} +(2.00000 + 3.46410i) q^{68} -2.00000 q^{71} -4.00000 q^{73} +(-2.00000 + 3.46410i) q^{74} +(-4.00000 - 6.92820i) q^{76} +(-3.00000 - 5.19615i) q^{77} +(3.00000 - 5.19615i) q^{79} -5.00000 q^{82} +(-4.50000 + 7.79423i) q^{83} +(-4.00000 - 6.92820i) q^{86} +(3.00000 - 5.19615i) q^{88} +15.0000 q^{89} +6.00000 q^{91} +(1.50000 - 2.59808i) q^{92} +(-3.50000 - 6.06218i) q^{94} +(1.00000 - 1.73205i) q^{97} +2.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + q^{4} - 3 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + q^4 - 3 * q^7 - 6 * q^8	$ 2 q - q^{2} + q^{4} - 3 q^{7} - 6 q^{8} - 2 q^{11} - 2 q^{13} - 3 q^{14} + q^{16} + 8 q^{17} - 16 q^{19} - 2 q^{22} - 3 q^{23} + 4 q^{26} - 6 q^{28} - q^{29} - 5 q^{32} - 4 q^{34} + 8 q^{37} + 8 q^{38} + 5 q^{41} - 8 q^{43} - 4 q^{44} + 6 q^{46} - 7 q^{47} - 2 q^{49} + 2 q^{52} - 4 q^{53} + 9 q^{56} - q^{58} - 14 q^{59} - 7 q^{61} + 14 q^{64} - 3 q^{67} + 4 q^{68} - 4 q^{71} - 8 q^{73} - 4 q^{74} - 8 q^{76} - 6 q^{77} + 6 q^{79} - 10 q^{82} - 9 q^{83} - 8 q^{86} + 6 q^{88} + 30 q^{89} + 12 q^{91} + 3 q^{92} - 7 q^{94} + 2 q^{97} + 4 q^{98}+O(q^{100}) $ 2 * q - q^2 + q^4 - 3 * q^7 - 6 * q^8 - 2 * q^11 - 2 * q^13 - 3 * q^14 + q^16 + 8 * q^17 - 16 * q^19 - 2 * q^22 - 3 * q^23 + 4 * q^26 - 6 * q^28 - q^29 - 5 * q^32 - 4 * q^34 + 8 * q^37 + 8 * q^38 + 5 * q^41 - 8 * q^43 - 4 * q^44 + 6 * q^46 - 7 * q^47 - 2 * q^49 + 2 * q^52 - 4 * q^53 + 9 * q^56 - q^58 - 14 * q^59 - 7 * q^61 + 14 * q^64 - 3 * q^67 + 4 * q^68 - 4 * q^71 - 8 * q^73 - 4 * q^74 - 8 * q^76 - 6 * q^77 + 6 * q^79 - 10 * q^82 - 9 * q^83 - 8 * q^86 + 6 * q^88 + 30 * q^89 + 12 * q^91 + 3 * q^92 - 7 * q^94 + 2 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

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 $

$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i	−0.986869	−	0.161521i	$-0.948360\pi$
$2$	−0.500000	+	0.866025i	−0.353553	+	0.612372i	0.633316	+	0.773893i	$0.281693\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	0.429919	+	0.902867i	$0.358542\pi$
$7$	−1.50000	+	2.59808i	−0.566947	+	0.981981i	−0.996866	+	0.0791130i	$0.974791\pi$
$8$	−3.00000			−1.06066
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	−1.50000	−	2.59808i	−0.400892	−	0.694365i
$15$		0			0
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$	4.00000			0.970143			0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000			0.970143			0.485071	+	0.874475i	$0.338794\pi$
$18$		0			0
$19$	−8.00000			−1.83533			−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000			−1.83533			−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		0			0
$22$	−1.00000	−	1.73205i	−0.213201	−	0.369274i
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	0.666190	−	0.745782i	$-0.267924\pi$
$23$	−1.50000	−	2.59808i	−0.312772	−	0.541736i	−0.978961	+	0.204046i	$0.934591\pi$
$24$		0			0
$25$		0			0
$26$	2.00000			0.392232
$27$		0			0
$28$	−3.00000			−0.566947
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	−0.908708	−	0.417432i	$-0.862930\pi$
$29$	−0.500000	+	0.866025i	−0.0928477	+	0.160817i	0.815861	+	0.578249i	$0.196264\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	−2.50000	−	4.33013i	−0.441942	−	0.765466i
$33$		0			0
$34$	−2.00000	+	3.46410i	−0.342997	+	0.594089i
$35$		0			0
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$	4.00000	−	6.92820i	0.648886	−	1.12390i
$39$		0			0
$40$		0			0
$41$	2.50000	+	4.33013i	0.390434	+	0.676252i	0.992507	−	0.122189i	$-0.0389915\pi$
$41$	2.50000	+	4.33013i	0.390434	+	0.676252i	−0.602072	+	0.798441i	$0.705658\pi$
$42$		0			0
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	0.381246	+	0.924473i	$0.375495\pi$
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	−0.991241	+	0.132068i	$0.957838\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$	3.00000			0.442326
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	0.489398	+	0.872060i	$0.337217\pi$
$47$	−3.50000	+	6.06218i	−0.510527	+	0.884260i	−0.999926	+	0.0121990i	$0.996117\pi$
$48$		0			0
$49$	−1.00000	−	1.73205i	−0.142857	−	0.247436i
$50$		0			0
$51$		0			0
$52$	1.00000	−	1.73205i	0.138675	−	0.240192i
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$		0			0
$56$	4.50000	−	7.79423i	0.601338	−	1.04155i
$57$		0			0
$58$	−0.500000	−	0.866025i	−0.0656532	−	0.113715i
$59$	−7.00000	−	12.1244i	−0.911322	−	1.57846i	−0.812198	−	0.583382i	$-0.801729\pi$
$59$	−7.00000	−	12.1244i	−0.911322	−	1.57846i	−0.0991242	−	0.995075i	$-0.531604\pi$
$60$		0			0
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	−0.998264	−	0.0588933i	$-0.981243\pi$
$61$	−3.50000	+	6.06218i	−0.448129	+	0.776182i	0.550135	+	0.835076i	$0.314576\pi$
$62$		0			0
$63$		0			0
$64$	7.00000			0.875000
$65$		0			0
$66$		0			0
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	0.759733	−	0.650236i	$-0.225330\pi$
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	−0.942987	+	0.332830i	$0.891996\pi$
$68$	2.00000	+	3.46410i	0.242536	+	0.420084i
$69$		0			0
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$	−2.00000	+	3.46410i	−0.232495	+	0.402694i
$75$		0			0
$76$	−4.00000	−	6.92820i	−0.458831	−	0.794719i
$77$	−3.00000	−	5.19615i	−0.341882	−	0.592157i
$78$		0			0
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	−0.646440	−	0.762964i	$-0.723743\pi$
$79$	3.00000	−	5.19615i	0.337526	−	0.584613i	0.983967	+	0.178352i	$0.0570765\pi$
$80$		0			0
$81$		0			0
$82$	−5.00000			−0.552158
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	−0.999976	−	0.00698436i	$-0.997777\pi$
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	0.506036	+	0.862512i	$0.331110\pi$
$84$		0			0
$85$		0			0
$86$	−4.00000	−	6.92820i	−0.431331	−	0.747087i
$87$		0			0
$88$	3.00000	−	5.19615i	0.319801	−	0.553912i
$89$	15.0000			1.59000			0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000			1.59000			0.794998	+	0.606612i	$0.207472\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$	1.50000	−	2.59808i	0.156386	−	0.270868i
$93$		0			0
$94$	−3.50000	−	6.06218i	−0.360997	−	0.625266i
$95$		0			0
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$	2.00000			0.202031
$99$		0			0
$100$		0			0
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.0623905	+	0.998052i	$0.519872\pi$
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.833143	+	0.553058i	$0.813461\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$	3.00000	+	5.19615i	0.294174	+	0.509525i
$105$		0			0
$106$	1.00000	−	1.73205i	0.0971286	−	0.168232i
$107$	3.00000			0.290021			0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000			0.290021			0.145010	+	0.989430i	$0.453678\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$		0			0
$112$	1.50000	+	2.59808i	0.141737	+	0.245495i
$113$	4.00000	+	6.92820i	0.376288	+	0.651751i	0.990519	−	0.137376i	$-0.0438669\pi$
$113$	4.00000	+	6.92820i	0.376288	+	0.651751i	−0.614231	+	0.789127i	$0.710534\pi$
$114$		0			0
$115$		0			0
$116$	−1.00000			−0.0928477
$117$		0			0
$118$	14.0000			1.28880
$119$	−6.00000	+	10.3923i	−0.550019	+	0.952661i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	−3.50000	−	6.06218i	−0.316875	−	0.548844i
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	5.00000			0.443678			0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000			0.443678			0.221839	+	0.975083i	$0.428794\pi$
$128$	1.50000	−	2.59808i	0.132583	−	0.229640i
$129$		0			0
$130$		0			0
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$	12.0000	−	20.7846i	1.04053	−	1.80225i
$134$	3.00000			0.259161
$135$		0			0
$136$	−12.0000			−1.02899
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	0.487278	+	0.873247i	$0.337990\pi$
$137$	−6.00000	+	10.3923i	−0.512615	+	0.887875i	−0.999893	+	0.0146279i	$0.995344\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$	1.00000	−	1.73205i	0.0839181	−	0.145350i
$143$	4.00000			0.334497
$144$		0			0
$145$		0			0
$146$	2.00000	−	3.46410i	0.165521	−	0.286691i
$147$		0			0
$148$	2.00000	+	3.46410i	0.164399	+	0.284747i
$149$	8.50000	+	14.7224i	0.696347	+	1.20611i	0.969724	+	0.244202i	$0.0785259\pi$
$149$	8.50000	+	14.7224i	0.696347	+	1.20611i	−0.273377	+	0.961907i	$0.588141\pi$
$150$		0			0
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	−0.822464	−	0.568818i	$-0.807401\pi$
$151$	1.00000	−	1.73205i	0.0813788	−	0.140952i	0.903842	+	0.427865i	$0.140734\pi$
$152$	24.0000			1.94666
$153$		0			0
$154$	6.00000			0.483494
$155$		0			0
$156$		0			0
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	−0.438948	+	0.898513i	$0.644649\pi$
$158$	3.00000	+	5.19615i	0.238667	+	0.413384i
$159$		0			0
$160$		0			0
$161$	9.00000			0.709299
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$	−2.50000	+	4.33013i	−0.195217	+	0.338126i
$165$		0			0
$166$	−4.50000	−	7.79423i	−0.349268	−	0.604949i
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	0.985933	−	0.167139i	$-0.0534527\pi$
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	−0.637713	+	0.770274i	$0.720119\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$		0			0
$171$		0			0
$172$	−8.00000			−0.609994
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	1.00000	+	1.73205i	0.0753778	+	0.130558i
$177$		0			0
$178$	−7.50000	+	12.9904i	−0.562149	+	0.973670i
$179$	2.00000			0.149487			0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000			0.149487			0.0747435	+	0.997203i	$0.476186\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	−3.00000	+	5.19615i	−0.222375	+	0.385164i
$183$		0			0
$184$	4.50000	+	7.79423i	0.331744	+	0.574598i
$185$		0			0
$186$		0			0
$187$	−4.00000	+	6.92820i	−0.292509	+	0.506640i
$188$	−7.00000			−0.510527
$189$		0			0
$190$		0			0
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	−0.684244	−	0.729253i	$-0.739868\pi$
$191$	4.00000	−	6.92820i	0.289430	−	0.501307i	0.973674	+	0.227946i	$0.0732010\pi$
$192$		0			0
$193$	−5.00000	−	8.66025i	−0.359908	−	0.623379i	0.628037	−	0.778183i	$-0.283859\pi$
$193$	−5.00000	−	8.66025i	−0.359908	−	0.623379i	−0.987945	+	0.154805i	$0.950525\pi$
$194$	1.00000	+	1.73205i	0.0717958	+	0.124354i
$195$		0			0
$196$	1.00000	−	1.73205i	0.0714286	−	0.123718i
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$		0			0
$202$	−9.00000	−	15.5885i	−0.633238	−	1.09680i
$203$	−1.50000	−	2.59808i	−0.105279	−	0.182349i
$204$		0			0
$205$		0			0
$206$	−8.00000			−0.557386
$207$		0			0
$208$	−2.00000			−0.138675
$209$	8.00000	−	13.8564i	0.553372	−	0.958468i
$210$		0			0
$211$	11.0000	+	19.0526i	0.757271	+	1.31163i	0.944237	+	0.329266i	$0.106801\pi$
$211$	11.0000	+	19.0526i	0.757271	+	1.31163i	−0.186966	+	0.982366i	$0.559865\pi$
$212$	−1.00000	−	1.73205i	−0.0686803	−	0.118958i
$213$		0			0
$214$	−1.50000	+	2.59808i	−0.102538	+	0.177601i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−2.50000	+	4.33013i	−0.169321	+	0.293273i
$219$		0			0
$220$		0			0
$221$	−4.00000	−	6.92820i	−0.269069	−	0.466041i
$222$		0			0
$223$	−9.50000	+	16.4545i	−0.636167	+	1.10187i	0.350100	+	0.936713i	$0.386148\pi$
$223$	−9.50000	+	16.4545i	−0.636167	+	1.10187i	−0.986267	+	0.165161i	$0.947186\pi$
$224$	15.0000			1.00223
$225$		0			0
$226$	−8.00000			−0.532152
$227$	2.00000	−	3.46410i	0.132745	−	0.229920i	−0.791989	−	0.610535i	$-0.790954\pi$
$227$	2.00000	−	3.46410i	0.132745	−	0.229920i	0.924734	+	0.380615i	$0.124288\pi$
$228$		0			0
$229$	−7.50000	−	12.9904i	−0.495614	−	0.858429i	0.504373	−	0.863486i	$-0.331724\pi$
$229$	−7.50000	−	12.9904i	−0.495614	−	0.858429i	−0.999987	+	0.00505719i	$0.998390\pi$
$230$		0			0
$231$		0			0
$232$	1.50000	−	2.59808i	0.0984798	−	0.170572i
$233$	24.0000			1.57229			0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000			1.57229			0.786146	+	0.618041i	$0.212073\pi$
$234$		0			0
$235$		0			0
$236$	7.00000	−	12.1244i	0.455661	−	0.789228i
$237$		0			0
$238$	−6.00000	−	10.3923i	−0.388922	−	0.673633i
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	0.707166	−	0.707048i	$-0.249973\pi$
$239$	−4.00000	−	6.92820i	−0.258738	−	0.448148i	−0.965904	+	0.258900i	$0.916640\pi$
$240$		0			0
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	−0.632709	−	0.774389i	$-0.718057\pi$
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	0.986996	+	0.160748i	$0.0513906\pi$
$242$	−7.00000			−0.449977
$243$		0			0
$244$	−7.00000			−0.448129
$245$		0			0
$246$		0			0
$247$	8.00000	+	13.8564i	0.509028	+	0.881662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	6.00000			0.377217
$254$	−2.50000	+	4.33013i	−0.156864	+	0.271696i
$255$		0			0
$256$	8.50000	+	14.7224i	0.531250	+	0.920152i
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	0.944294	−	0.329104i	$-0.106747\pi$
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	−0.757159	+	0.653231i	$0.773413\pi$
$258$		0			0
$259$	−6.00000	+	10.3923i	−0.372822	+	0.645746i
$260$		0			0
$261$		0			0
$262$	6.00000			0.370681
$263$	8.00000	−	13.8564i	0.493301	−	0.854423i	−0.506669	−	0.862141i	$-0.669123\pi$
$263$	8.00000	−	13.8564i	0.493301	−	0.854423i	0.999970	+	0.00771799i	$0.00245674\pi$
$264$		0			0
$265$		0			0
$266$	12.0000	+	20.7846i	0.735767	+	1.27439i
$267$		0			0
$268$	1.50000	−	2.59808i	0.0916271	−	0.158703i
$269$	−25.0000			−1.52428			−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000			−1.52428			−0.762138	+	0.647414i	$0.775850\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	2.00000	−	3.46410i	0.121268	−	0.210042i
$273$		0			0
$274$	−6.00000	−	10.3923i	−0.362473	−	0.627822i
$275$		0			0
$276$		0			0
$277$	6.00000	−	10.3923i	0.360505	−	0.624413i	−0.627539	−	0.778585i	$-0.715938\pi$
$277$	6.00000	−	10.3923i	0.360505	−	0.624413i	0.988044	+	0.154172i	$0.0492710\pi$
$278$	−16.0000			−0.959616
$279$		0			0
$280$		0			0
$281$	−7.50000	+	12.9904i	−0.447412	+	0.774941i	−0.998217	−	0.0596933i	$-0.980988\pi$
$281$	−7.50000	+	12.9904i	−0.447412	+	0.774941i	0.550804	+	0.834634i	$0.314321\pi$
$282$		0			0
$283$	10.5000	+	18.1865i	0.624160	+	1.08108i	0.988703	+	0.149890i	$0.0478921\pi$
$283$	10.5000	+	18.1865i	0.624160	+	1.08108i	−0.364542	+	0.931187i	$0.618775\pi$
$284$	−1.00000	−	1.73205i	−0.0593391	−	0.102778i
$285$		0			0
$286$	−2.00000	+	3.46410i	−0.118262	+	0.204837i
$287$	−15.0000			−0.885422
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$	−2.00000	−	3.46410i	−0.117041	−	0.202721i
$293$	−6.00000	−	10.3923i	−0.350524	−	0.607125i	0.635818	−	0.771839i	$-0.280663\pi$
$293$	−6.00000	−	10.3923i	−0.350524	−	0.607125i	−0.986341	+	0.164714i	$0.947330\pi$
$294$		0			0
$295$		0			0
$296$	−12.0000			−0.697486
$297$		0			0
$298$	−17.0000			−0.984784
$299$	−3.00000	+	5.19615i	−0.173494	+	0.300501i
$300$		0			0
$301$	−12.0000	−	20.7846i	−0.691669	−	1.19800i
$302$	1.00000	+	1.73205i	0.0575435	+	0.0996683i
$303$		0			0
$304$	−4.00000	+	6.92820i	−0.229416	+	0.397360i
$305$		0			0
$306$		0			0
$307$	−7.00000			−0.399511			−0.199756	−	0.979846i	$-0.564015\pi$
$307$	−7.00000			−0.399511			−0.199756	+	0.979846i	$0.564015\pi$
$308$	3.00000	−	5.19615i	0.170941	−	0.296078i
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	7.00000	−	12.1244i	0.395663	−	0.685309i	−0.597522	−	0.801852i	$-0.703848\pi$
$313$	7.00000	−	12.1244i	0.395663	−	0.685309i	0.993186	+	0.116543i	$0.0371814\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	6.00000			0.337526
$317$	17.0000	−	29.4449i	0.954815	−	1.65379i	0.220024	−	0.975494i	$-0.429386\pi$
$317$	17.0000	−	29.4449i	0.954815	−	1.65379i	0.734791	−	0.678294i	$-0.237280\pi$
$318$		0			0
$319$	−1.00000	−	1.73205i	−0.0559893	−	0.0969762i
$320$		0			0
$321$		0			0
$322$	−4.50000	+	7.79423i	−0.250775	+	0.434355i
$323$	−32.0000			−1.78053
$324$		0			0
$325$		0			0
$326$	−2.00000	+	3.46410i	−0.110770	+	0.191859i
$327$		0			0
$328$	−7.50000	−	12.9904i	−0.414118	−	0.717274i
$329$	−10.5000	−	18.1865i	−0.578884	−	1.00266i
$330$		0			0
$331$	3.00000	−	5.19615i	0.164895	−	0.285606i	−0.771723	−	0.635959i	$-0.780605\pi$
$331$	3.00000	−	5.19615i	0.164895	−	0.285606i	0.936618	+	0.350352i	$0.113938\pi$
$332$	−9.00000			−0.493939
$333$		0			0
$334$	−9.00000			−0.492458
$335$		0			0
$336$		0			0
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	0.736270	−	0.676688i	$-0.236585\pi$
$337$	−4.00000	−	6.92820i	−0.217894	−	0.377403i	−0.954164	+	0.299285i	$0.903252\pi$
$338$	4.50000	+	7.79423i	0.244768	+	0.423950i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−15.0000			−0.809924
$344$	12.0000	−	20.7846i	0.646997	−	1.12063i
$345$		0			0
$346$		0			0
$347$	−2.00000	−	3.46410i	−0.107366	−	0.185963i	0.807337	−	0.590091i	$-0.200908\pi$
$347$	−2.00000	−	3.46410i	−0.107366	−	0.185963i	−0.914702	+	0.404128i	$0.867575\pi$
$348$		0			0
$349$	2.50000	−	4.33013i	0.133822	−	0.231786i	−0.791325	−	0.611396i	$-0.790608\pi$
$349$	2.50000	−	4.33013i	0.133822	−	0.231786i	0.925147	+	0.379610i	$0.123942\pi$
$350$		0			0
$351$		0			0
$352$	10.0000			0.533002
$353$	−12.0000	+	20.7846i	−0.638696	+	1.10625i	0.347024	+	0.937856i	$0.387192\pi$
$353$	−12.0000	+	20.7846i	−0.638696	+	1.10625i	−0.985719	+	0.168397i	$0.946141\pi$
$354$		0			0
$355$		0			0
$356$	7.50000	+	12.9904i	0.397499	+	0.688489i
$357$		0			0
$358$	−1.00000	+	1.73205i	−0.0528516	+	0.0915417i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$	3.50000	−	6.06218i	0.183956	−	0.318621i
$363$		0			0
$364$	3.00000	+	5.19615i	0.157243	+	0.272352i
$365$		0			0
$366$		0			0
$367$	12.0000	−	20.7846i	0.626395	−	1.08495i	−0.361874	−	0.932227i	$-0.617863\pi$
$367$	12.0000	−	20.7846i	0.626395	−	1.08495i	0.988269	−	0.152721i	$-0.0488036\pi$
$368$	−3.00000			−0.156386
$369$		0			0
$370$		0			0
$371$	3.00000	−	5.19615i	0.155752	−	0.269771i
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$	−4.00000	−	6.92820i	−0.206835	−	0.358249i
$375$		0			0
$376$	10.5000	−	18.1865i	0.541496	−	0.937899i
$377$	2.00000			0.103005
$378$		0			0
$379$	−26.0000			−1.33553			−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000			−1.33553			−0.667765	+	0.744372i	$0.732749\pi$
$380$		0			0
$381$		0			0
$382$	4.00000	+	6.92820i	0.204658	+	0.354478i
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.799783	−	0.600289i	$-0.795052\pi$
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.119974	−	0.992777i	$-0.538281\pi$
$384$		0			0
$385$		0			0
$386$	10.0000			0.508987
$387$		0			0
$388$	2.00000			0.101535
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	−0.0561516	−	0.998422i	$-0.517883\pi$
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	0.892735	−	0.450582i	$-0.148784\pi$
$390$		0			0
$391$	−6.00000	−	10.3923i	−0.303433	−	0.525561i
$392$	3.00000	+	5.19615i	0.151523	+	0.262445i
$393$		0			0
$394$	6.00000	−	10.3923i	0.302276	−	0.523557i
$395$		0			0
$396$		0			0
$397$	−34.0000			−1.70641			−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000			−1.70641			−0.853206	+	0.521575i	$0.825345\pi$
$398$	−2.00000	+	3.46410i	−0.100251	+	0.173640i
$399$		0			0
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$		0			0
$404$	−18.0000			−0.895533
$405$		0			0
$406$	3.00000			0.148888
$407$	−4.00000	+	6.92820i	−0.198273	+	0.343418i
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$		0			0
$411$		0			0
$412$	−4.00000	+	6.92820i	−0.197066	+	0.341328i
$413$	42.0000			2.06668
$414$		0			0
$415$		0			0
$416$	−5.00000	+	8.66025i	−0.245145	+	0.424604i
$417$		0			0
$418$	8.00000	+	13.8564i	0.391293	+	0.677739i
$419$	13.0000	+	22.5167i	0.635092	+	1.10001i	0.986496	+	0.163787i	$0.0523710\pi$
$419$	13.0000	+	22.5167i	0.635092	+	1.10001i	−0.351404	+	0.936224i	$0.614296\pi$
$420$		0			0
$421$	−17.0000	+	29.4449i	−0.828529	+	1.43505i	0.0706626	+	0.997500i	$0.477489\pi$
$421$	−17.0000	+	29.4449i	−0.828529	+	1.43505i	−0.899192	+	0.437555i	$0.855845\pi$
$422$	−22.0000			−1.07094
$423$		0			0
$424$	6.00000			0.291386
$425$		0			0
$426$		0			0
$427$	−10.5000	−	18.1865i	−0.508131	−	0.880108i
$428$	1.50000	+	2.59808i	0.0725052	+	0.125583i
$429$		0			0
$430$		0			0
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$	−28.0000			−1.34559			−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000			−1.34559			−0.672797	+	0.739827i	$0.734907\pi$
$434$		0			0
$435$		0			0
$436$	2.50000	+	4.33013i	0.119728	+	0.207375i
$437$	12.0000	+	20.7846i	0.574038	+	0.994263i
$438$		0			0
$439$	−14.0000	+	24.2487i	−0.668184	+	1.15733i	0.310228	+	0.950662i	$0.399595\pi$
$439$	−14.0000	+	24.2487i	−0.668184	+	1.15733i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$		0			0
$442$	8.00000			0.380521
$443$	−7.50000	+	12.9904i	−0.356336	+	0.617192i	−0.987346	−	0.158583i	$-0.949307\pi$
$443$	−7.50000	+	12.9904i	−0.356336	+	0.617192i	0.631010	+	0.775775i	$0.282641\pi$
$444$		0			0
$445$		0			0
$446$	−9.50000	−	16.4545i	−0.449838	−	0.779142i
$447$		0			0
$448$	−10.5000	+	18.1865i	−0.496078	+	0.859233i
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$	−4.00000	+	6.92820i	−0.188144	+	0.325875i
$453$		0			0
$454$	2.00000	+	3.46410i	0.0938647	+	0.162578i
$455$		0			0
$456$		0			0
$457$	−10.0000	+	17.3205i	−0.467780	+	0.810219i	−0.999322	−	0.0368128i	$-0.988279\pi$
$457$	−10.0000	+	17.3205i	−0.467780	+	0.810219i	0.531542	+	0.847032i	$0.321613\pi$
$458$	15.0000			0.700904
$459$		0			0
$460$		0			0
$461$	4.50000	−	7.79423i	0.209586	−	0.363013i	−0.741998	−	0.670402i	$-0.766122\pi$
$461$	4.50000	−	7.79423i	0.209586	−	0.363013i	0.951584	+	0.307388i	$0.0994551\pi$
$462$		0			0
$463$	−18.0000	−	31.1769i	−0.836531	−	1.44891i	−0.892778	−	0.450497i	$-0.851247\pi$
$463$	−18.0000	−	31.1769i	−0.836531	−	1.44891i	0.0562469	−	0.998417i	$-0.482087\pi$
$464$	0.500000	+	0.866025i	0.0232119	+	0.0402042i
$465$		0			0
$466$	−12.0000	+	20.7846i	−0.555889	+	0.962828i
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$	9.00000			0.415581
$470$		0			0
$471$		0			0
$472$	21.0000	+	36.3731i	0.966603	+	1.67421i
$473$	−8.00000	−	13.8564i	−0.367840	−	0.637118i
$474$		0			0
$475$		0			0
$476$	−12.0000			−0.550019
$477$		0			0
$478$	8.00000			0.365911
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	−0.995023	−	0.0996406i	$-0.968231\pi$
$479$	−9.00000	+	15.5885i	−0.411220	+	0.712255i	0.583803	+	0.811895i	$0.301564\pi$
$480$		0			0
$481$	−4.00000	−	6.92820i	−0.182384	−	0.315899i
$482$	5.50000	+	9.52628i	0.250518	+	0.433910i
$483$		0			0
$484$	−3.50000	+	6.06218i	−0.159091	+	0.275554i
$485$		0			0
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$	10.5000	−	18.1865i	0.475313	−	0.823266i
$489$		0			0
$490$		0			0
$491$	10.0000	+	17.3205i	0.451294	+	0.781664i	0.998467	−	0.0553560i	$-0.0176294\pi$
$491$	10.0000	+	17.3205i	0.451294	+	0.781664i	−0.547173	+	0.837020i	$0.684296\pi$
$492$		0			0
$493$	−2.00000	+	3.46410i	−0.0900755	+	0.156015i
$494$	−16.0000			−0.719874
$495$		0			0
$496$		0			0
$497$	3.00000	−	5.19615i	0.134568	−	0.233079i
$498$		0			0
$499$	16.0000	+	27.7128i	0.716258	+	1.24060i	0.962472	+	0.271380i	$0.0874801\pi$
$499$	16.0000	+	27.7128i	0.716258	+	1.24060i	−0.246214	+	0.969216i	$0.579187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7.00000			−0.312115			−0.156057	−	0.987748i	$-0.549878\pi$
$503$	−7.00000			−0.312115			−0.156057	+	0.987748i	$0.549878\pi$
$504$		0			0
$505$		0			0
$506$	−3.00000	+	5.19615i	−0.133366	+	0.230997i
$507$		0			0
$508$	2.50000	+	4.33013i	0.110920	+	0.192118i
$509$	21.5000	+	37.2391i	0.952971	+	1.65059i	0.738945	+	0.673766i	$0.235324\pi$
$509$	21.5000	+	37.2391i	0.952971	+	1.65059i	0.214026	+	0.976828i	$0.431342\pi$
$510$		0			0
$511$	6.00000	−	10.3923i	0.265424	−	0.459728i
$512$	−11.0000			−0.486136
$513$		0			0
$514$	−6.00000			−0.264649
$515$		0			0
$516$		0			0
$517$	−7.00000	−	12.1244i	−0.307860	−	0.533229i
$518$	−6.00000	−	10.3923i	−0.263625	−	0.456612i
$519$		0			0
$520$		0			0
$521$	−11.0000			−0.481919			−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000			−0.481919			−0.240959	+	0.970535i	$0.577462\pi$
$522$		0			0
$523$	29.0000			1.26808			0.634041	−	0.773300i	$-0.281395\pi$
$523$	29.0000			1.26808			0.634041	+	0.773300i	$0.281395\pi$
$524$	3.00000	−	5.19615i	0.131056	−	0.226995i
$525$		0			0
$526$	8.00000	+	13.8564i	0.348817	+	0.604168i
$527$		0			0
$528$		0			0
$529$	7.00000	−	12.1244i	0.304348	−	0.527146i
$530$		0			0
$531$		0			0
$532$	24.0000			1.04053
$533$	5.00000	−	8.66025i	0.216574	−	0.375117i
$534$		0			0
$535$		0			0
$536$	4.50000	+	7.79423i	0.194370	+	0.336659i
$537$		0			0
$538$	12.5000	−	21.6506i	0.538913	−	0.933425i
$539$	4.00000			0.172292
$540$		0			0
$541$	−39.0000			−1.67674			−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000			−1.67674			−0.838370	+	0.545101i	$0.816491\pi$
$542$	4.00000	−	6.92820i	0.171815	−	0.297592i
$543$		0			0
$544$	−10.0000	−	17.3205i	−0.428746	−	0.742611i
$545$		0			0
$546$		0			0
$547$	−14.5000	+	25.1147i	−0.619975	+	1.07383i	0.369514	+	0.929225i	$0.379524\pi$
$547$	−14.5000	+	25.1147i	−0.619975	+	1.07383i	−0.989490	+	0.144604i	$0.953809\pi$
$548$	−12.0000			−0.512615
$549$		0			0
$550$		0			0
$551$	4.00000	−	6.92820i	0.170406	−	0.295151i
$552$		0			0
$553$	9.00000	+	15.5885i	0.382719	+	0.662889i
$554$	6.00000	+	10.3923i	0.254916	+	0.441527i
$555$		0			0
$556$	−8.00000	+	13.8564i	−0.339276	+	0.587643i
$557$	−30.0000			−1.27114			−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000			−1.27114			−0.635570	+	0.772043i	$0.719235\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$		0			0
$562$	−7.50000	−	12.9904i	−0.316368	−	0.547966i
$563$	10.5000	+	18.1865i	0.442522	+	0.766471i	0.997876	−	0.0651433i	$-0.0207504\pi$
$563$	10.5000	+	18.1865i	0.442522	+	0.766471i	−0.555354	+	0.831614i	$0.687417\pi$
$564$		0			0
$565$		0			0
$566$	−21.0000			−0.882696
$567$		0			0
$568$	6.00000			0.251754
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	−0.922032	−	0.387113i	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	−0.978022	−	0.208502i	$-0.933141\pi$
$571$	−16.0000	−	27.7128i	−0.669579	−	1.15975i	0.308443	−	0.951243i	$-0.400192\pi$
$572$	2.00000	+	3.46410i	0.0836242	+	0.144841i
$573$		0			0
$574$	7.50000	−	12.9904i	0.313044	−	0.542208i
$575$		0			0
$576$		0			0
$577$	10.0000			0.416305			0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000			0.416305			0.208153	+	0.978096i	$0.433255\pi$
$578$	0.500000	−	0.866025i	0.0207973	−	0.0360219i
$579$		0			0
$580$		0			0
$581$	−13.5000	−	23.3827i	−0.560074	−	0.970077i
$582$		0			0
$583$	2.00000	−	3.46410i	0.0828315	−	0.143468i
$584$	12.0000			0.496564
$585$		0			0
$586$	12.0000			0.495715
$587$	16.5000	−	28.5788i	0.681028	−	1.17957i	−0.293640	−	0.955916i	$-0.594867\pi$
$587$	16.5000	−	28.5788i	0.681028	−	1.17957i	0.974668	−	0.223659i	$-0.0718001\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	2.00000	−	3.46410i	0.0821995	−	0.142374i
$593$	20.0000			0.821302			0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000			0.821302			0.410651	+	0.911793i	$0.365302\pi$
$594$		0			0
$595$		0			0
$596$	−8.50000	+	14.7224i	−0.348174	+	0.603054i
$597$		0			0
$598$	−3.00000	−	5.19615i	−0.122679	−	0.212486i
$599$	−5.00000	−	8.66025i	−0.204294	−	0.353848i	0.745613	−	0.666379i	$-0.232157\pi$
$599$	−5.00000	−	8.66025i	−0.204294	−	0.353848i	−0.949908	+	0.312531i	$0.898823\pi$
$600$		0			0
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−1.00000	+	1.73205i	−0.0407909	+	0.0706518i	0.844909	+	0.534910i	$0.179654\pi$
$602$	24.0000			0.978167
$603$		0			0
$604$	2.00000			0.0813788
$605$		0			0
$606$		0			0
$607$	−20.5000	−	35.5070i	−0.832069	−	1.44119i	−0.896394	−	0.443257i	$-0.853823\pi$
$607$	−20.5000	−	35.5070i	−0.832069	−	1.44119i	0.0643251	−	0.997929i	$-0.479511\pi$
$608$	20.0000	+	34.6410i	0.811107	+	1.40488i
$609$		0			0
$610$		0			0
$611$	14.0000			0.566379
$612$		0			0
$613$	44.0000			1.77714			0.888572	−	0.458738i	$-0.151698\pi$
$613$	44.0000			1.77714			0.888572	+	0.458738i	$0.151698\pi$
$614$	3.50000	−	6.06218i	0.141249	−	0.244650i
$615$		0			0
$616$	9.00000	+	15.5885i	0.362620	+	0.628077i
$617$	18.0000	+	31.1769i	0.724653	+	1.25514i	0.959117	+	0.283011i	$0.0913331\pi$
$617$	18.0000	+	31.1769i	0.724653	+	1.25514i	−0.234464	+	0.972125i	$0.575334\pi$
$618$		0			0
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	−0.823029	−	0.567999i	$-0.807718\pi$
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	0.903416	+	0.428765i	$0.141051\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−22.5000	+	38.9711i	−0.901443	+	1.56135i
$624$		0			0
$625$		0			0
$626$	7.00000	+	12.1244i	0.279776	+	0.484587i
$627$		0			0
$628$	−7.00000	+	12.1244i	−0.279330	+	0.483814i
$629$	16.0000			0.637962
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	−9.00000	+	15.5885i	−0.358001	+	0.620076i
$633$		0			0
$634$	17.0000	+	29.4449i	0.675156	+	1.16940i
$635$		0			0
$636$		0			0
$637$	−2.00000	+	3.46410i	−0.0792429	+	0.137253i
$638$	2.00000			0.0791808
$639$		0			0
$640$		0			0
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	−0.330997	−	0.943632i	$-0.607385\pi$
$641$	16.5000	−	28.5788i	0.651711	−	1.12880i	0.982708	−	0.185164i	$-0.0592817\pi$
$642$		0			0
$643$	−4.50000	−	7.79423i	−0.177463	−	0.307374i	0.763548	−	0.645751i	$-0.223456\pi$
$643$	−4.50000	−	7.79423i	−0.177463	−	0.307374i	−0.941011	+	0.338377i	$0.890122\pi$
$644$	4.50000	+	7.79423i	0.177325	+	0.307136i
$645$		0			0
$646$	16.0000	−	27.7128i	0.629512	−	1.09035i
$647$	−17.0000			−0.668339			−0.334169	−	0.942513i	$-0.608456\pi$
$647$	−17.0000			−0.668339			−0.334169	+	0.942513i	$0.608456\pi$
$648$		0			0
$649$	28.0000			1.09910
$650$		0			0
$651$		0			0
$652$	2.00000	+	3.46410i	0.0783260	+	0.135665i
$653$	2.00000	+	3.46410i	0.0782660	+	0.135561i	0.902502	−	0.430686i	$-0.141728\pi$
$653$	2.00000	+	3.46410i	0.0782660	+	0.135561i	−0.824236	+	0.566247i	$0.808395\pi$
$654$		0			0
$655$		0			0
$656$	5.00000			0.195217
$657$		0			0
$658$	21.0000			0.818665
$659$	−4.00000	+	6.92820i	−0.155818	+	0.269884i	−0.933357	−	0.358951i	$-0.883135\pi$
$659$	−4.00000	+	6.92820i	−0.155818	+	0.269884i	0.777539	+	0.628835i	$0.216468\pi$
$660$		0			0
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	0.969442	−	0.245319i	$-0.0788928\pi$
$661$	7.00000	+	12.1244i	0.272268	+	0.471583i	−0.697174	+	0.716902i	$0.745559\pi$
$662$	3.00000	+	5.19615i	0.116598	+	0.201954i
$663$		0			0
$664$	13.5000	−	23.3827i	0.523902	−	0.907424i
$665$		0			0
$666$		0			0
$667$	3.00000			0.116160
$668$	−4.50000	+	7.79423i	−0.174110	+	0.301568i
$669$		0			0
$670$		0			0
$671$	−7.00000	−	12.1244i	−0.270232	−	0.468056i
$672$		0			0
$673$	3.00000	−	5.19615i	0.115642	−	0.200297i	−0.802395	−	0.596794i	$-0.796441\pi$
$673$	3.00000	−	5.19615i	0.115642	−	0.200297i	0.918036	+	0.396497i	$0.129774\pi$
$674$	8.00000			0.308148
$675$		0			0
$676$	9.00000			0.346154
$677$	21.0000	−	36.3731i	0.807096	−	1.39793i	−0.107772	−	0.994176i	$-0.534372\pi$
$677$	21.0000	−	36.3731i	0.807096	−	1.39793i	0.914867	−	0.403755i	$-0.132295\pi$
$678$		0			0
$679$	3.00000	+	5.19615i	0.115129	+	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$		0			0
$686$	7.50000	−	12.9904i	0.286351	−	0.495975i
$687$		0			0
$688$	4.00000	+	6.92820i	0.152499	+	0.264135i
$689$	2.00000	+	3.46410i	0.0761939	+	0.131972i
$690$		0			0
$691$	7.00000	−	12.1244i	0.266293	−	0.461232i	−0.701609	−	0.712562i	$-0.747535\pi$
$691$	7.00000	−	12.1244i	0.266293	−	0.461232i	0.967901	+	0.251330i	$0.0808679\pi$
$692$		0			0
$693$		0			0
$694$	4.00000			0.151838
$695$		0			0
$696$		0			0
$697$	10.0000	+	17.3205i	0.378777	+	0.656061i
$698$	2.50000	+	4.33013i	0.0946264	+	0.163898i
$699$		0			0
$700$		0			0
$701$	−23.0000			−0.868698			−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000			−0.868698			−0.434349	+	0.900745i	$0.643022\pi$
$702$		0			0
$703$	−32.0000			−1.20690
$704$	−7.00000	+	12.1244i	−0.263822	+	0.456954i
$705$		0			0
$706$	−12.0000	−	20.7846i	−0.451626	−	0.782239i
$707$	−27.0000	−	46.7654i	−1.01544	−	1.75879i
$708$		0			0
$709$	20.5000	−	35.5070i	0.769894	−	1.33349i	−0.167727	−	0.985834i	$-0.553643\pi$
$709$	20.5000	−	35.5070i	0.769894	−	1.33349i	0.937620	−	0.347661i	$-0.113024\pi$
$710$		0			0
$711$		0			0
$712$	−45.0000			−1.68645
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.00000	+	1.73205i	0.0373718	+	0.0647298i
$717$		0			0
$718$	12.0000	−	20.7846i	0.447836	−	0.775675i
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$	−22.5000	+	38.9711i	−0.837363	+	1.45036i
$723$		0			0
$724$	−3.50000	−	6.06218i	−0.130076	−	0.225299i
$725$		0			0
$726$		0			0
$727$	11.5000	−	19.9186i	0.426511	−	0.738739i	−0.570049	−	0.821611i	$-0.693076\pi$
$727$	11.5000	−	19.9186i	0.426511	−	0.738739i	0.996560	+	0.0828714i	$0.0264091\pi$
$728$	−18.0000			−0.667124
$729$		0			0
$730$		0			0
$731$	−16.0000	+	27.7128i	−0.591781	+	1.02500i
$732$		0			0
$733$	−17.0000	−	29.4449i	−0.627909	−	1.08757i	−0.987971	−	0.154642i	$-0.950578\pi$
$733$	−17.0000	−	29.4449i	−0.627909	−	1.08757i	0.360061	−	0.932929i	$-0.382756\pi$
$734$	12.0000	+	20.7846i	0.442928	+	0.767174i
$735$		0			0
$736$	−7.50000	+	12.9904i	−0.276454	+	0.478832i
$737$	6.00000			0.221013
$738$		0			0
$739$	−2.00000			−0.0735712			−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000			−0.0735712			−0.0367856	+	0.999323i	$0.511712\pi$
$740$		0			0
$741$		0			0
$742$	3.00000	+	5.19615i	0.110133	+	0.190757i
$743$	14.5000	+	25.1147i	0.531953	+	0.921370i	0.999304	+	0.0372984i	$0.0118752\pi$
$743$	14.5000	+	25.1147i	0.531953	+	0.921370i	−0.467351	+	0.884072i	$0.654791\pi$
$744$		0			0
$745$		0			0
$746$	−10.0000			−0.366126
$747$		0			0
$748$	−8.00000			−0.292509
$749$	−4.50000	+	7.79423i	−0.164426	+	0.284795i
$750$		0			0
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	0.760263	−	0.649616i	$-0.225070\pi$
$751$	−5.00000	−	8.66025i	−0.182453	−	0.316017i	−0.942715	+	0.333599i	$0.891737\pi$
$752$	3.50000	+	6.06218i	0.127632	+	0.221065i
$753$		0			0
$754$	−1.00000	+	1.73205i	−0.0364179	+	0.0630776i
$755$		0			0
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$	13.0000	−	22.5167i	0.472181	−	0.817842i
$759$		0			0
$760$		0			0
$761$	7.50000	+	12.9904i	0.271875	+	0.470901i	0.969342	−	0.245716i	$-0.0790230\pi$
$761$	7.50000	+	12.9904i	0.271875	+	0.470901i	−0.697467	+	0.716617i	$0.745690\pi$
$762$		0			0
$763$	−7.50000	+	12.9904i	−0.271518	+	0.470283i
$764$	8.00000			0.289430
$765$		0			0
$766$	36.0000			1.30073
$767$	−14.0000	+	24.2487i	−0.505511	+	0.875570i
$768$		0			0
$769$	−2.50000	−	4.33013i	−0.0901523	−	0.156148i	0.817423	−	0.576038i	$-0.195402\pi$
$769$	−2.50000	−	4.33013i	−0.0901523	−	0.156148i	−0.907575	+	0.419890i	$0.862069\pi$
$770$		0			0
$771$		0			0
$772$	5.00000	−	8.66025i	0.179954	−	0.311689i
$773$	−24.0000			−0.863220			−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000			−0.863220			−0.431610	+	0.902060i	$0.642054\pi$
$774$		0			0
$775$		0			0
$776$	−3.00000	+	5.19615i	−0.107694	+	0.186531i
$777$		0			0
$778$	16.5000	+	28.5788i	0.591554	+	1.02460i
$779$	−20.0000	−	34.6410i	−0.716574	−	1.24114i
$780$		0			0
$781$	2.00000	−	3.46410i	0.0715656	−	0.123955i
$782$	12.0000			0.429119
$783$		0			0
$784$	−2.00000			−0.0714286
$785$		0			0
$786$		0			0
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	0.999999	−	0.00110111i	$-0.000350496\pi$
$787$	14.0000	+	24.2487i	0.499046	+	0.864373i	−0.500953	+	0.865474i	$0.667017\pi$
$788$	−6.00000	−	10.3923i	−0.213741	−	0.370211i
$789$		0			0
$790$		0			0
$791$

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-1.50000 + 2.59808i) q^{7} -3.00000 q^{8} +(-1.00000 + 1.73205i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-1.50000 - 2.59808i) q^{14} +(0.500000 - 0.866025i) q^{16} +4.00000 q^{17} -8.00000 q^{19} +(-1.00000 - 1.73205i) q^{22} +(-1.50000 - 2.59808i) q^{23} +2.00000 q^{26} -3.00000 q^{28} +(-0.500000 + 0.866025i) q^{29} +(-2.50000 - 4.33013i) q^{32} +(-2.00000 + 3.46410i) q^{34} +4.00000 q^{37} +(4.00000 - 6.92820i) q^{38} +(2.50000 + 4.33013i) q^{41} +(-4.00000 + 6.92820i) q^{43} -2.00000 q^{44} +3.00000 q^{46} +(-3.50000 + 6.06218i) q^{47} +(-1.00000 - 1.73205i) q^{49} +(1.00000 - 1.73205i) q^{52} -2.00000 q^{53} +(4.50000 - 7.79423i) q^{56} +(-0.500000 - 0.866025i) q^{58} +(-7.00000 - 12.1244i) q^{59} +(-3.50000 + 6.06218i) q^{61} +7.00000 q^{64} +(-1.50000 - 2.59808i) q^{67} +(2.00000 + 3.46410i) q^{68} -2.00000 q^{71} -4.00000 q^{73} +(-2.00000 + 3.46410i) q^{74} +(-4.00000 - 6.92820i) q^{76} +(-3.00000 - 5.19615i) q^{77} +(3.00000 - 5.19615i) q^{79} -5.00000 q^{82} +(-4.50000 + 7.79423i) q^{83} +(-4.00000 - 6.92820i) q^{86} +(3.00000 - 5.19615i) q^{88} +15.0000 q^{89} +6.00000 q^{91} +(1.50000 - 2.59808i) q^{92} +(-3.50000 - 6.06218i) q^{94} +(1.00000 - 1.73205i) q^{97} +2.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{4} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + q^4 - 3 * q^7 - 6 * q^8	\( 2 q - q^{2} + q^{4} - 3 q^{7} - 6 q^{8} - 2 q^{11} - 2 q^{13} - 3 q^{14} + q^{16} + 8 q^{17} - 16 q^{19} - 2 q^{22} - 3 q^{23} + 4 q^{26} - 6 q^{28} - q^{29} - 5 q^{32} - 4 q^{34} + 8 q^{37} + 8 q^{38} + 5 q^{41} - 8 q^{43} - 4 q^{44} + 6 q^{46} - 7 q^{47} - 2 q^{49} + 2 q^{52} - 4 q^{53} + 9 q^{56} - q^{58} - 14 q^{59} - 7 q^{61} + 14 q^{64} - 3 q^{67} + 4 q^{68} - 4 q^{71} - 8 q^{73} - 4 q^{74} - 8 q^{76} - 6 q^{77} + 6 q^{79} - 10 q^{82} - 9 q^{83} - 8 q^{86} + 6 q^{88} + 30 q^{89} + 12 q^{91} + 3 q^{92} - 7 q^{94} + 2 q^{97} + 4 q^{98}+O(q^{100}) \) 2 * q - q^2 + q^4 - 3 * q^7 - 6 * q^8 - 2 * q^11 - 2 * q^13 - 3 * q^14 + q^16 + 8 * q^17 - 16 * q^19 - 2 * q^22 - 3 * q^23 + 4 * q^26 - 6 * q^28 - q^29 - 5 * q^32 - 4 * q^34 + 8 * q^37 + 8 * q^38 + 5 * q^41 - 8 * q^43 - 4 * q^44 + 6 * q^46 - 7 * q^47 - 2 * q^49 + 2 * q^52 - 4 * q^53 + 9 * q^56 - q^58 - 14 * q^59 - 7 * q^61 + 14 * q^64 - 3 * q^67 + 4 * q^68 - 4 * q^71 - 8 * q^73 - 4 * q^74 - 8 * q^76 - 6 * q^77 + 6 * q^79 - 10 * q^82 - 9 * q^83 - 8 * q^86 + 6 * q^88 + 30 * q^89 + 12 * q^91 + 3 * q^92 - 7 * q^94 + 2 * q^97 + 4 * q^98

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