LMFDB - Embedded newform 1503.2.c.a.1502.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1502,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("1503.1502");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.c (of order $2$, degree $1$, 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:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1502.19
Character	$\chi$	$=$	1503.1502
Dual form			1503.2.c.a.1502.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84274i q^{2} -1.39570 q^{4} -1.15735 q^{5} +0.906437 q^{7} -1.11357i q^{8} +O(q^{10})$	\(q-1.84274i q^{2} -1.39570 q^{4} -1.15735 q^{5} +0.906437 q^{7} -1.11357i q^{8} +2.13270i q^{10} +6.02781i q^{11} -5.63485i q^{13} -1.67033i q^{14} -4.84342 q^{16} -3.48670 q^{17} +0.634686 q^{19} +1.61532 q^{20} +11.1077 q^{22} -5.82348 q^{23} -3.66054 q^{25} -10.3836 q^{26} -1.26512 q^{28} -3.72741i q^{29} +1.33842 q^{31} +6.69805i q^{32} +6.42508i q^{34} -1.04907 q^{35} -8.17734i q^{37} -1.16956i q^{38} +1.28879i q^{40} -10.2231 q^{41} -6.92718i q^{43} -8.41302i q^{44} +10.7312i q^{46} +3.80855i q^{47} -6.17837 q^{49} +6.74543i q^{50} +7.86457i q^{52} -4.94220 q^{53} -6.97629i q^{55} -1.00938i q^{56} -6.86867 q^{58} +2.99830 q^{59} -1.28898 q^{61} -2.46637i q^{62} +2.65594 q^{64} +6.52150i q^{65} +14.3484i q^{67} +4.86639 q^{68} +1.93316i q^{70} +8.86785 q^{71} -6.52150i q^{73} -15.0687 q^{74} -0.885833 q^{76} +5.46383i q^{77} +10.5670i q^{79} +5.60554 q^{80} +18.8385i q^{82} -0.895126 q^{83} +4.03533 q^{85} -12.7650 q^{86} +6.71236 q^{88} -6.80816i q^{89} -5.10764i q^{91} +8.12784 q^{92} +7.01818 q^{94} -0.734555 q^{95} +10.1959 q^{97} +11.3852i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 48 q^{4}+O(q^{10}) $ 56 * q - 48 * q^4	$ 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) $ 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	1.84274i		−	1.30302i	−0.758642	−	0.651508i	$-0.774137\pi$
$2$		−	1.84274i		−	1.30302i	0.758642	−	0.651508i	$-0.225863\pi$
$3$		0			0
$3$		0			0
$4$	−1.39570			−0.697851
$5$	−1.15735			−0.517583			−0.258792	−	0.965933i	$-0.583324\pi$
$5$	−1.15735			−0.517583			−0.258792	+	0.965933i	$0.583324\pi$
$6$		0			0
$7$	0.906437			0.342601			0.171301	−	0.985219i	$-0.445203\pi$
$7$	0.906437			0.342601			0.171301	+	0.985219i	$0.445203\pi$
$8$		−	1.11357i		−	0.393705i
$9$		0			0
$10$			2.13270i			0.674419i
$11$			6.02781i			1.81745i	0.417393	+	0.908726i	$0.362944\pi$
$11$			6.02781i			1.81745i	−0.417393	+	0.908726i	$0.637056\pi$
$12$		0			0
$13$		−	5.63485i		−	1.56283i	−0.624014	−	0.781413i	$-0.714499\pi$
$13$		−	5.63485i		−	1.56283i	0.624014	−	0.781413i	$-0.285501\pi$
$14$		−	1.67033i		−	0.446415i
$15$		0			0
$16$	−4.84342			−1.21085
$17$	−3.48670			−0.845648			−0.422824	−	0.906212i	$-0.638961\pi$
$17$	−3.48670			−0.845648			−0.422824	+	0.906212i	$0.638961\pi$
$18$		0			0
$19$	0.634686			0.145607			0.0728035	−	0.997346i	$-0.476805\pi$
$19$	0.634686			0.145607			0.0728035	+	0.997346i	$0.476805\pi$
$20$	1.61532			0.361196
$21$		0			0
$22$	11.1077			2.36817
$23$	−5.82348			−1.21428			−0.607139	−	0.794595i	$-0.707683\pi$
$23$	−5.82348			−1.21428			−0.607139	+	0.794595i	$0.707683\pi$
$24$		0			0
$25$	−3.66054			−0.732108
$26$	−10.3836			−2.03639
$27$		0			0
$28$	−1.26512			−0.239085
$29$		−	3.72741i		−	0.692164i	−0.938204	−	0.346082i	$-0.887512\pi$
$29$		−	3.72741i		−	0.692164i	0.938204	−	0.346082i	$-0.112488\pi$
$30$		0			0
$31$	1.33842			0.240388			0.120194	−	0.992750i	$-0.461648\pi$
$31$	1.33842			0.240388			0.120194	+	0.992750i	$0.461648\pi$
$32$			6.69805i			1.18406i
$33$		0			0
$34$			6.42508i			1.10189i
$35$	−1.04907			−0.177325
$36$		0			0
$37$		−	8.17734i		−	1.34435i	−0.740394	−	0.672173i	$-0.765361\pi$
$37$		−	8.17734i		−	1.34435i	0.740394	−	0.672173i	$-0.234639\pi$
$38$		−	1.16956i		−	0.189728i
$39$		0			0
$40$			1.28879i			0.203775i
$41$	−10.2231			−1.59658			−0.798290	−	0.602273i	$-0.794262\pi$
$41$	−10.2231			−1.59658			−0.798290	+	0.602273i	$0.794262\pi$
$42$		0			0
$43$		−	6.92718i		−	1.05639i	−0.849125	−	0.528193i	$-0.822870\pi$
$43$		−	6.92718i		−	1.05639i	0.849125	−	0.528193i	$-0.177130\pi$
$44$		−	8.41302i		−	1.26831i
$45$		0			0
$46$			10.7312i			1.58222i
$47$			3.80855i			0.555534i	0.960648	+	0.277767i	$0.0895943\pi$
$47$			3.80855i			0.555534i	−0.960648	+	0.277767i	$0.910406\pi$
$48$		0			0
$49$	−6.17837			−0.882624
$50$			6.74543i			0.953948i
$51$		0			0
$52$			7.86457i			1.09062i
$53$	−4.94220			−0.678863			−0.339432	−	0.940631i	$-0.610235\pi$
$53$	−4.94220			−0.678863			−0.339432	+	0.940631i	$0.610235\pi$
$54$		0			0
$55$		−	6.97629i		−	0.940683i
$56$		−	1.00938i		−	0.134884i
$57$		0			0
$58$	−6.86867			−0.901900
$59$	2.99830			0.390346			0.195173	−	0.980769i	$-0.437473\pi$
$59$	2.99830			0.390346			0.195173	+	0.980769i	$0.437473\pi$
$60$		0			0
$61$	−1.28898			−0.165037			−0.0825185	−	0.996590i	$-0.526296\pi$
$61$	−1.28898			−0.165037			−0.0825185	+	0.996590i	$0.526296\pi$
$62$		−	2.46637i		−	0.313229i
$63$		0			0
$64$	2.65594			0.331992
$65$			6.52150i			0.808892i
$66$		0			0
$67$			14.3484i			1.75294i	0.481455	+	0.876471i	$0.340109\pi$
$67$			14.3484i			1.75294i	−0.481455	+	0.876471i	$0.659891\pi$
$68$	4.86639			0.590136
$69$		0			0
$70$			1.93316i			0.231057i
$71$	8.86785			1.05242			0.526210	−	0.850355i	$-0.323613\pi$
$71$	8.86785			1.05242			0.526210	+	0.850355i	$0.323613\pi$
$72$		0			0
$73$		−	6.52150i		−	0.763284i	−0.924310	−	0.381642i	$-0.875359\pi$
$73$		−	6.52150i		−	0.763284i	0.924310	−	0.381642i	$-0.124641\pi$
$74$	−15.0687			−1.75171
$75$		0			0
$76$	−0.885833			−0.101612
$77$			5.46383i			0.622661i
$78$		0			0
$79$			10.5670i			1.18889i	0.804138	+	0.594443i	$0.202627\pi$
$79$			10.5670i			1.18889i	−0.804138	+	0.594443i	$0.797373\pi$
$80$	5.60554			0.626718
$81$		0			0
$82$			18.8385i			2.08037i
$83$	−0.895126			−0.0982529			−0.0491264	−	0.998793i	$-0.515644\pi$
$83$	−0.895126			−0.0982529			−0.0491264	+	0.998793i	$0.515644\pi$
$84$		0			0
$85$	4.03533			0.437693
$86$	−12.7650			−1.37649
$87$		0			0
$88$	6.71236			0.715540
$89$		−	6.80816i		−	0.721664i	−0.932631	−	0.360832i	$-0.882493\pi$
$89$		−	6.80816i		−	0.721664i	0.932631	−	0.360832i	$-0.117507\pi$
$90$		0			0
$91$		−	5.10764i		−	0.535426i
$92$	8.12784			0.847385
$93$		0			0
$94$	7.01818			0.723870
$95$	−0.734555			−0.0753638
$96$		0			0
$97$	10.1959			1.03524			0.517621	−	0.855610i	$-0.326818\pi$
$97$	10.1959			1.03524			0.517621	+	0.855610i	$0.326818\pi$
$98$			11.3852i			1.15007i
$99$		0			0
$100$	5.10902			0.510902
$101$	−7.11532			−0.708001			−0.354001	−	0.935245i	$-0.615179\pi$
$101$	−7.11532			−0.708001			−0.354001	+	0.935245i	$0.615179\pi$
$102$		0			0
$103$			8.50074i			0.837603i	0.908078	+	0.418801i	$0.137550\pi$
$103$			8.50074i			0.837603i	−0.908078	+	0.418801i	$0.862450\pi$
$104$	−6.27477			−0.615292
$105$		0			0
$106$			9.10720i			0.884569i
$107$		−	9.19318i		−	0.888738i	−0.895844	−	0.444369i	$-0.853428\pi$
$107$		−	9.19318i		−	0.888738i	0.895844	−	0.444369i	$-0.146572\pi$
$108$		0			0
$109$			14.4762i			1.38657i	0.720664	+	0.693285i	$0.243837\pi$
$109$			14.4762i			1.38657i	−0.720664	+	0.693285i	$0.756163\pi$
$110$	−12.8555			−1.22572
$111$		0			0
$112$	−4.39026			−0.414840
$113$	0.758132			0.0713191			0.0356595	−	0.999364i	$-0.488647\pi$
$113$	0.758132			0.0713191			0.0356595	+	0.999364i	$0.488647\pi$
$114$		0			0
$115$	6.73981			0.628490
$116$			5.20236i			0.483027i
$117$		0			0
$118$		−	5.52510i		−	0.508627i
$119$	−3.16047			−0.289720
$120$		0			0
$121$	−25.3345			−2.30313
$122$			2.37526i			0.215046i
$123$		0			0
$124$	−1.86804			−0.167755
$125$	10.0233			0.896510
$126$		0			0
$127$	−1.55037			−0.137573			−0.0687864	−	0.997631i	$-0.521913\pi$
$127$	−1.55037			−0.137573			−0.0687864	+	0.997631i	$0.521913\pi$
$128$			8.50188i			0.751467i
$129$		0			0
$130$	12.0174			1.05400
$131$	−13.6253			−1.19045			−0.595224	−	0.803560i	$-0.702937\pi$
$131$	−13.6253			−1.19045			−0.595224	+	0.803560i	$0.702937\pi$
$132$		0			0
$133$	0.575303			0.0498851
$134$	26.4405			2.28411
$135$		0			0
$136$			3.88267i			0.332936i
$137$		−	18.6580i		−	1.59406i	−0.603942	−	0.797028i	$-0.706404\pi$
$137$		−	18.6580i		−	1.59406i	0.603942	−	0.797028i	$-0.293596\pi$
$138$		0			0
$139$		−	8.07323i		−	0.684763i	−0.939561	−	0.342381i	$-0.888767\pi$
$139$		−	8.07323i		−	0.684763i	0.939561	−	0.342381i	$-0.111233\pi$
$140$	1.46418			0.123746
$141$		0			0
$142$		−	16.3412i		−	1.37132i
$143$	33.9658			2.84036
$144$		0			0
$145$			4.31393i			0.358252i
$146$	−12.0175			−0.994572
$147$		0			0
$148$			11.4131i			0.938154i
$149$	−10.8495			−0.888829			−0.444414	−	0.895821i	$-0.646588\pi$
$149$	−10.8495			−0.888829			−0.444414	+	0.895821i	$0.646588\pi$
$150$		0			0
$151$		−	15.6094i		−	1.27028i	−0.772398	−	0.635139i	$-0.780943\pi$
$151$		−	15.6094i		−	1.27028i	0.772398	−	0.635139i	$-0.219057\pi$
$152$		−	0.706765i		−	0.0573262i
$153$		0			0
$154$	10.0684			0.811338
$155$	−1.54903			−0.124421
$156$		0			0
$157$	−6.16286			−0.491850			−0.245925	−	0.969289i	$-0.579092\pi$
$157$	−6.16286			−0.491850			−0.245925	+	0.969289i	$0.579092\pi$
$158$	19.4724			1.54914
$159$		0			0
$160$		−	7.75199i		−	0.612849i
$161$	−5.27862			−0.416013
$162$		0			0
$163$			2.24587i			0.175910i	0.996124	+	0.0879549i	$0.0280331\pi$
$163$			2.24587i			0.175910i	−0.996124	+	0.0879549i	$0.971967\pi$
$164$	14.2684			1.11418
$165$		0			0
$166$			1.64949i			0.128025i
$167$	12.7414	+	2.15770i	0.985962	+	0.166968i
$167$	12.7414	+	2.15770i	0.985962	+	0.166968i
$168$		0			0
$169$	−18.7515			−1.44242
$170$		−	7.43608i		−	0.570321i
$171$		0			0
$172$			9.66828i			0.737200i
$173$		−	18.1980i		−	1.38356i	−0.722106	−	0.691782i	$-0.756826\pi$
$173$		−	18.1980i		−	1.38356i	0.722106	−	0.691782i	$-0.243174\pi$
$174$		0			0
$175$	−3.31805			−0.250821
$176$		−	29.1952i		−	2.20067i
$177$		0			0
$178$	−12.5457			−0.940339
$179$			0.508304i			0.0379924i	0.999820	+	0.0189962i	$0.00604704\pi$
$179$			0.508304i			0.0379924i	−0.999820	+	0.0189962i	$0.993953\pi$
$180$		0			0
$181$	25.0235			1.85998			0.929990	−	0.367584i	$-0.119815\pi$
$181$	25.0235			1.85998			0.929990	+	0.367584i	$0.119815\pi$
$182$	−9.41206			−0.697668
$183$		0			0
$184$			6.48482i			0.478067i
$185$			9.46405i			0.695811i
$186$		0			0
$187$		−	21.0171i		−	1.53692i
$188$		−	5.31560i		−	0.387680i
$189$		0			0
$190$			1.35360i			0.0982002i
$191$			16.6469i			1.20453i	0.798297	+	0.602265i	$0.205735\pi$
$191$			16.6469i			1.20453i	−0.798297	+	0.602265i	$0.794265\pi$
$192$		0			0
$193$			5.61503i			0.404179i	0.979367	+	0.202090i	$0.0647732\pi$
$193$			5.61503i			0.404179i	−0.979367	+	0.202090i	$0.935227\pi$
$194$		−	18.7885i		−	1.34894i
$195$		0			0
$196$	8.62317			0.615940
$197$	21.7360			1.54863			0.774314	−	0.632801i	$-0.218095\pi$
$197$	21.7360			1.54863			0.774314	+	0.632801i	$0.218095\pi$
$198$		0			0
$199$	−13.0097			−0.922233			−0.461117	−	0.887340i	$-0.652551\pi$
$199$	−13.0097			−0.922233			−0.461117	+	0.887340i	$0.652551\pi$
$200$			4.07625i			0.288234i
$201$		0			0
$202$			13.1117i			0.922537i
$203$		−	3.37867i		−	0.237136i
$204$		0			0
$205$	11.8317			0.826363
$206$	15.6647			1.09141
$207$		0			0
$208$			27.2919i			1.89236i
$209$			3.82577i			0.264634i
$210$		0			0
$211$	5.05056			0.347695			0.173847	−	0.984773i	$-0.444380\pi$
$211$	5.05056			0.347695			0.173847	+	0.984773i	$0.444380\pi$
$212$	6.89783			0.473745
$213$		0			0
$214$	−16.9407			−1.15804
$215$			8.01718i			0.546767i
$216$		0			0
$217$	1.21320			0.0823571
$218$	26.6759			1.80672
$219$		0			0
$220$			9.73682i			0.656456i
$221$			19.6470i			1.32160i
$222$		0			0
$223$	−8.69554			−0.582296			−0.291148	−	0.956678i	$-0.594037\pi$
$223$	−8.69554			−0.582296			−0.291148	+	0.956678i	$0.594037\pi$
$224$			6.07136i			0.405660i
$225$		0			0
$226$		−	1.39704i		−	0.0929299i
$227$	14.3085			0.949690			0.474845	−	0.880069i	$-0.342504\pi$
$227$	14.3085			0.949690			0.474845	+	0.880069i	$0.342504\pi$
$228$		0			0
$229$	27.6613			1.82791			0.913956	−	0.405812i	$-0.133011\pi$
$229$	27.6613			1.82791			0.913956	+	0.405812i	$0.133011\pi$
$230$		−	12.4197i		−	0.818933i
$231$		0			0
$232$	−4.15072			−0.272508
$233$		−	5.19274i		−	0.340188i	−0.985428	−	0.170094i	$-0.945593\pi$
$233$		−	5.19274i		−	0.340188i	0.985428	−	0.170094i	$-0.0544071\pi$
$234$		0			0
$235$		−	4.40783i		−	0.287535i
$236$	−4.18474			−0.272403
$237$		0			0
$238$			5.82394i			0.377510i
$239$			26.6457i			1.72357i	0.507277	+	0.861783i	$0.330652\pi$
$239$			26.6457i			1.72357i	−0.507277	+	0.861783i	$0.669348\pi$
$240$		0			0
$241$		−	27.6024i		−	1.77803i	−0.457879	−	0.889014i	$-0.651391\pi$
$241$		−	27.6024i		−	1.77803i	0.457879	−	0.889014i	$-0.348609\pi$
$242$			46.6849i			3.00102i
$243$		0			0
$244$	1.79903			0.115171
$245$	7.15055			0.456832
$246$		0			0
$247$		−	3.57636i		−	0.227558i
$248$		−	1.49042i		−	0.0946419i
$249$		0			0
$250$		−	18.4703i		−	1.16817i
$251$		−	1.13725i		−	0.0717826i	−0.999356	−	0.0358913i	$-0.988573\pi$
$251$		−	1.13725i		−	0.0717826i	0.999356	−	0.0358913i	$-0.0114270\pi$
$252$		0			0
$253$		−	35.1028i		−	2.20689i
$254$			2.85693i			0.179260i
$255$		0			0
$256$	20.9787			1.31117
$257$	−9.01084			−0.562081			−0.281041	−	0.959696i	$-0.590680\pi$
$257$	−9.01084			−0.562081			−0.281041	+	0.959696i	$0.590680\pi$
$258$		0			0
$259$		−	7.41225i		−	0.460575i
$260$		−	9.10207i		−	0.564486i
$261$		0			0
$262$			25.1079i			1.55117i
$263$		−	22.8116i		−	1.40663i	−0.710881	−	0.703313i	$-0.751703\pi$
$263$		−	22.8116i		−	1.40663i	0.710881	−	0.703313i	$-0.248297\pi$
$264$		0			0
$265$	5.71986			0.351368
$266$		−	1.06014i		−	0.0650011i
$267$		0			0
$268$		−	20.0262i		−	1.22329i
$269$	−6.02885			−0.367585			−0.183793	−	0.982965i	$-0.558837\pi$
$269$	−6.02885			−0.367585			−0.183793	+	0.982965i	$0.558837\pi$
$270$		0			0
$271$		−	15.0624i		−	0.914974i	−0.889216	−	0.457487i	$-0.848750\pi$
$271$		−	15.0624i		−	0.914974i	0.889216	−	0.457487i	$-0.151250\pi$
$272$	16.8875			1.02396
$273$		0			0
$274$	−34.3818			−2.07708
$275$		−	22.0650i		−	1.33057i
$276$		0			0
$277$		−	21.4312i		−	1.28767i	−0.765162	−	0.643837i	$-0.777341\pi$
$277$		−	21.4312i		−	1.28767i	0.765162	−	0.643837i	$-0.222659\pi$
$278$	−14.8769			−0.892257
$279$		0			0
$280$			1.16820i			0.0698136i
$281$		−	20.7012i		−	1.23493i	−0.786598	−	0.617465i	$-0.788160\pi$
$281$		−	20.7012i		−	1.23493i	0.786598	−	0.617465i	$-0.211840\pi$
$282$		0			0
$283$	−20.0092			−1.18942			−0.594711	−	0.803939i	$-0.702734\pi$
$283$	−20.0092			−1.18942			−0.594711	+	0.803939i	$0.702734\pi$
$284$	−12.3769			−0.734432
$285$		0			0
$286$		−	62.5902i		−	3.70104i
$287$	−9.26660			−0.546990
$288$		0			0
$289$	−4.84295			−0.284880
$290$	7.94946			0.466808
$291$		0			0
$292$			9.10207i			0.532659i
$293$		−	17.5453i		−	1.02501i	−0.858686	−	0.512503i	$-0.828718\pi$
$293$		−	17.5453i		−	1.02501i	0.858686	−	0.512503i	$-0.171282\pi$
$294$		0			0
$295$	−3.47009			−0.202036
$296$	−9.10601			−0.529276
$297$		0			0
$298$			19.9929i			1.15816i
$299$			32.8144i			1.89771i
$300$		0			0
$301$		−	6.27906i		−	0.361919i
$302$	−28.7642			−1.65519
$303$		0			0
$304$	−3.07405			−0.176309
$305$	1.49180			0.0854204
$306$		0			0
$307$		−	21.6434i		−	1.23525i	−0.786472	−	0.617626i	$-0.788094\pi$
$307$		−	21.6434i		−	1.23525i	0.786472	−	0.617626i	$-0.211906\pi$
$308$		−	7.62588i		−	0.434525i
$309$		0			0
$310$			2.85446i			0.162122i
$311$		−	21.2353i		−	1.20414i	−0.798442	−	0.602072i	$-0.794342\pi$
$311$		−	21.2353i		−	1.20414i	0.798442	−	0.602072i	$-0.205658\pi$
$312$		0			0
$313$			16.7946i			0.949284i	0.880179	+	0.474642i	$0.157423\pi$
$313$			16.7946i			0.949284i	−0.880179	+	0.474642i	$0.842577\pi$
$314$			11.3566i			0.640889i
$315$		0			0
$316$		−	14.7485i		−	0.829665i
$317$			21.5938i			1.21283i	0.795148	+	0.606416i	$0.207393\pi$
$317$			21.5938i			1.21283i	−0.795148	+	0.606416i	$0.792607\pi$
$318$		0			0
$319$	22.4681			1.25797
$320$	−3.07385			−0.171834
$321$		0			0
$322$			9.72713i			0.542072i
$323$	−2.21296			−0.123132
$324$		0			0
$325$			20.6266i			1.14416i
$326$	4.13855			0.229213
$327$		0			0
$328$			11.3841i			0.628582i
$329$			3.45221i			0.190327i
$330$		0			0
$331$		−	12.0460i		−	0.662108i	−0.943612	−	0.331054i	$-0.892596\pi$
$331$		−	12.0460i		−	0.662108i	0.943612	−	0.331054i	$-0.107404\pi$
$332$	1.24933			0.0685659
$333$		0			0
$334$	3.97609	−	23.4792i	0.217562	−	1.28472i
$335$		−	16.6062i		−	0.907293i
$336$		0			0
$337$	27.2787			1.48596			0.742982	−	0.669311i	$-0.233411\pi$
$337$	27.2787			1.48596			0.742982	+	0.669311i	$0.233411\pi$
$338$			34.5542i			1.87950i
$339$		0			0
$340$	−5.63212			−0.305445
$341$			8.06776i			0.436893i
$342$		0			0
$343$	−11.9454			−0.644989
$344$	−7.71387			−0.415904
$345$		0			0
$346$	−33.5341			−1.80281
$347$	−34.8881			−1.87289			−0.936447	−	0.350809i	$-0.885907\pi$
$347$	−34.8881			−1.87289			−0.936447	+	0.350809i	$0.885907\pi$
$348$		0			0
$349$			33.8992i			1.81458i	0.420504	+	0.907291i	$0.361853\pi$
$349$			33.8992i			1.81458i	−0.420504	+	0.907291i	$0.638147\pi$
$350$			6.11431i			0.326824i
$351$		0			0
$352$	−40.3745			−2.15197
$353$			18.9580i			1.00903i	0.863402	+	0.504516i	$0.168329\pi$
$353$			18.9580i			1.00903i	−0.863402	+	0.504516i	$0.831671\pi$
$354$		0			0
$355$	−10.2632			−0.544715
$356$			9.50216i			0.503614i
$357$		0			0
$358$	0.936673			0.0495047
$359$		−	16.8967i		−	0.891772i	−0.895090	−	0.445886i	$-0.852889\pi$
$359$		−	16.8967i		−	0.891772i	0.895090	−	0.445886i	$-0.147111\pi$
$360$		0			0
$361$	−18.5972			−0.978799
$362$		−	46.1119i		−	2.42359i
$363$		0			0
$364$			7.12874i			0.373647i
$365$			7.54767i			0.395063i
$366$		0			0
$367$	−23.9554			−1.25046			−0.625230	−	0.780441i	$-0.714995\pi$
$367$	−23.9554			−1.25046			−0.625230	+	0.780441i	$0.714995\pi$
$368$	28.2055			1.47032
$369$		0			0
$370$	17.4398			0.906653
$371$	−4.47979			−0.232579
$372$		0			0
$373$		−	0.238882i		−	0.0123688i	−0.999981	−	0.00618442i	$-0.998031\pi$
$373$		−	0.238882i		−	0.0123688i	0.999981	−	0.00618442i	$-0.00196858\pi$
$374$	−38.7292			−2.00264
$375$		0			0
$376$	4.24107			0.218716
$377$	−21.0034			−1.08173
$378$		0			0
$379$			5.07242i			0.260552i	0.991478	+	0.130276i	$0.0415864\pi$
$379$			5.07242i			0.260552i	−0.991478	+	0.130276i	$0.958414\pi$
$380$	1.02522			0.0525927
$381$		0			0
$382$	30.6760			1.56952
$383$		−	9.08527i		−	0.464236i	−0.972688	−	0.232118i	$-0.925435\pi$
$383$		−	9.08527i		−	0.464236i	0.972688	−	0.232118i	$-0.0745655\pi$
$384$		0			0
$385$		−	6.32357i		−	0.322279i
$386$	10.3471			0.526652
$387$		0			0
$388$	−14.2305			−0.722444
$389$	−20.2208			−1.02523			−0.512616	−	0.858618i	$-0.671324\pi$
$389$	−20.2208			−1.02523			−0.512616	+	0.858618i	$0.671324\pi$
$390$		0			0
$391$	20.3047			1.02685
$392$			6.88002i			0.347494i
$393$		0			0
$394$		−	40.0539i		−	2.01789i
$395$		−	12.2298i		−	0.615347i
$396$		0			0
$397$	−0.766126			−0.0384508			−0.0192254	−	0.999815i	$-0.506120\pi$
$397$	−0.766126			−0.0384508			−0.0192254	+	0.999815i	$0.506120\pi$
$398$			23.9735i			1.20168i
$399$		0			0
$400$	17.7295			0.886476
$401$	9.41789			0.470307			0.235153	−	0.971958i	$-0.424441\pi$
$401$	9.41789			0.470307			0.235153	+	0.971958i	$0.424441\pi$
$402$		0			0
$403$		−	7.54181i		−	0.375684i
$404$	9.93087			0.494079
$405$		0			0
$406$	−6.22602			−0.308992
$407$	49.2914			2.44329
$408$		0			0
$409$	3.52132			0.174118			0.0870590	−	0.996203i	$-0.472253\pi$
$409$	3.52132			0.174118			0.0870590	+	0.996203i	$0.472253\pi$
$410$		−	21.8028i		−	1.07676i
$411$		0			0
$412$		−	11.8645i		−	0.584522i
$413$	2.71777			0.133733
$414$		0			0
$415$	1.03598			0.0508540
$416$	37.7425			1.85048
$417$		0			0
$418$	7.04991			0.344822
$419$			8.30172i			0.405565i	0.979224	+	0.202783i	$0.0649985\pi$
$419$			8.30172i			0.405565i	−0.979224	+	0.202783i	$0.935002\pi$
$420$		0			0
$421$	−14.2948			−0.696684			−0.348342	−	0.937368i	$-0.613255\pi$
$421$	−14.2948			−0.696684			−0.348342	+	0.937368i	$0.613255\pi$
$422$		−	9.30688i		−	0.453052i
$423$		0			0
$424$			5.50346i			0.267272i
$425$	12.7632			0.619105
$426$		0			0
$427$	−1.16838			−0.0565419
$428$			12.8309i			0.620207i
$429$		0			0
$430$	14.7736			0.712447
$431$			0.154164i			0.00742582i	0.999993	+	0.00371291i	$0.00118186\pi$
$431$			0.154164i			0.00742582i	−0.999993	+	0.00371291i	$0.998818\pi$
$432$		0			0
$433$	22.1080			1.06244			0.531220	−	0.847234i	$-0.321734\pi$
$433$	22.1080			1.06244			0.531220	+	0.847234i	$0.321734\pi$
$434$		−	2.23561i		−	0.107313i
$435$		0			0
$436$		−	20.2045i		−	0.967619i
$437$	−3.69608			−0.176807
$438$		0			0
$439$			4.38443i			0.209258i	0.994511	+	0.104629i	$0.0333654\pi$
$439$			4.38443i			0.209258i	−0.994511	+	0.104629i	$0.966635\pi$
$440$	−7.76856			−0.370352
$441$		0			0
$442$	36.2044			1.72207
$443$	−12.2689			−0.582914			−0.291457	−	0.956584i	$-0.594140\pi$
$443$	−12.2689			−0.582914			−0.291457	+	0.956584i	$0.594140\pi$
$444$		0			0
$445$			7.87943i			0.373521i
$446$			16.0236i			0.758741i
$447$		0			0
$448$	2.40744			0.113741
$449$		−	26.5286i		−	1.25196i	−0.779837	−	0.625982i	$-0.784698\pi$
$449$		−	26.5286i		−	1.25196i	0.779837	−	0.625982i	$-0.215302\pi$
$450$		0			0
$451$		−	61.6229i		−	2.90171i
$452$	−1.05813			−0.0497701
$453$		0			0
$454$		−	26.3669i		−	1.23746i
$455$			5.91133i			0.277127i
$456$		0			0
$457$			1.13945i			0.0533012i	0.999645	+	0.0266506i	$0.00848416\pi$
$457$			1.13945i			0.0533012i	−0.999645	+	0.0266506i	$0.991516\pi$
$458$		−	50.9727i		−	2.38180i
$459$		0			0
$460$	−9.40676			−0.438592
$461$		−	23.2160i		−	1.08128i	−0.841255	−	0.540639i	$-0.818183\pi$
$461$		−	23.2160i		−	1.08128i	0.841255	−	0.540639i	$-0.181817\pi$
$462$		0			0
$463$		−	10.7575i		−	0.499941i	−0.968253	−	0.249971i	$-0.919579\pi$
$463$		−	10.7575i		−	0.499941i	0.968253	−	0.249971i	$-0.0804209\pi$
$464$			18.0534i			0.838110i
$465$		0			0
$466$	−9.56889			−0.443270
$467$			40.7563i			1.88598i	0.332827	+	0.942988i	$0.391997\pi$
$467$			40.7563i			1.88598i	−0.332827	+	0.942988i	$0.608003\pi$
$468$		0			0
$469$			13.0060i			0.600560i
$470$	−8.12249			−0.374663
$471$		0			0
$472$		−	3.33881i		−	0.153681i
$473$	41.7557			1.91993
$474$		0			0
$475$	−2.32329			−0.106600
$476$	4.41108			0.202181
$477$		0			0
$478$	49.1012			2.24583
$479$	18.7118			0.854966			0.427483	−	0.904023i	$-0.359400\pi$
$479$	18.7118			0.854966			0.427483	+	0.904023i	$0.359400\pi$
$480$		0			0
$481$	−46.0781			−2.10098
$482$	−50.8642			−2.31680
$483$		0			0
$484$	35.3594			1.60724
$485$	−11.8003			−0.535823
$486$		0			0
$487$		−	15.8187i		−	0.716812i	−0.933566	−	0.358406i	$-0.883320\pi$
$487$		−	15.8187i		−	0.716812i	0.933566	−	0.358406i	$-0.116680\pi$
$488$			1.43536i			0.0649759i
$489$		0			0
$490$		−	13.1766i		−	0.595259i
$491$			5.11462i			0.230819i	0.993318	+	0.115410i	$0.0368181\pi$
$491$			5.11462i			0.230819i	−0.993318	+	0.115410i	$0.963182\pi$
$492$		0			0
$493$			12.9964i			0.585327i
$494$	−6.59031			−0.296512
$495$		0			0
$496$	−6.48254			−0.291075
$497$	8.03815			0.360560
$498$		0			0
$499$			40.3705i			1.80723i	0.428345	+	0.903615i	$0.359097\pi$
$499$			40.3705i			1.80723i	−0.428345	+	0.903615i	$0.640903\pi$
$500$	−13.9895			−0.625630
$501$		0			0
$502$	−2.09566			−0.0935339
$503$			14.6805i			0.654570i	0.944926	+	0.327285i	$0.106134\pi$
$503$			14.6805i			0.654570i	−0.944926	+	0.327285i	$0.893866\pi$
$504$		0			0
$505$	8.23493			0.366450
$506$	−64.6854			−2.87562
$507$		0			0
$508$	2.16385			0.0960054
$509$			31.9986i			1.41831i	0.705052	+	0.709156i	$0.250924\pi$
$509$			31.9986i			1.41831i	−0.705052	+	0.709156i	$0.749076\pi$
$510$		0			0
$511$		−	5.91133i		−	0.261502i
$512$		−	21.6545i		−	0.957004i
$513$		0			0
$514$			16.6047i			0.732401i
$515$		−	9.83834i		−	0.433529i
$516$		0			0
$517$	−22.9572			−1.00966
$518$	−13.6589			−0.600136
$519$		0			0
$520$	7.26212			0.318465
$521$	23.6538			1.03629			0.518146	−	0.855292i	$-0.326623\pi$
$521$	23.6538			1.03629			0.518146	+	0.855292i	$0.326623\pi$
$522$		0			0
$523$	14.6440			0.640336			0.320168	−	0.947361i	$-0.396261\pi$
$523$	14.6440			0.640336			0.320168	+	0.947361i	$0.396261\pi$
$524$	19.0169			0.830756
$525$		0			0
$526$	−42.0360			−1.83286
$527$	−4.66667			−0.203283
$528$		0			0
$529$	10.9129			0.474472
$530$		−	10.5402i		−	0.457838i
$531$		0			0
$532$	−0.802952			−0.0348124
$533$			57.6056i			2.49518i
$534$		0			0
$535$			10.6397i			0.459996i
$536$	15.9779			0.690142
$537$		0			0
$538$			11.1096i			0.478969i
$539$		−	37.2420i		−	1.60413i
$540$		0			0
$541$			5.61221i			0.241288i	0.992696	+	0.120644i	$0.0384959\pi$
$541$			5.61221i			0.241288i	−0.992696	+	0.120644i	$0.961504\pi$
$542$	−27.7561			−1.19223
$543$		0			0
$544$		−	23.3541i		−	1.00130i
$545$		−	16.7541i		−	0.717665i
$546$		0			0
$547$		−	21.1472i		−	0.904189i	−0.891970	−	0.452094i	$-0.850677\pi$
$547$		−	21.1472i		−	0.904189i	0.891970	−	0.452094i	$-0.149323\pi$
$548$			26.0409i			1.11241i
$549$		0			0
$550$	−40.6602			−1.73376
$551$		−	2.36574i		−	0.100784i
$552$		0			0
$553$			9.57837i			0.407314i
$554$	−39.4922			−1.67786
$555$		0			0
$556$			11.2678i			0.477862i
$557$		−	23.6824i		−	1.00345i	−0.865026	−	0.501727i	$-0.832698\pi$
$557$		−	23.6824i		−	1.00345i	0.865026	−	0.501727i	$-0.167302\pi$
$558$		0			0
$559$	−39.0336			−1.65095
$560$	5.08107			0.214714
$561$		0			0
$562$	−38.1470			−1.60913
$563$		−	39.6664i		−	1.67174i	−0.548928	−	0.835869i	$-0.684964\pi$
$563$		−	39.6664i		−	1.67174i	0.548928	−	0.835869i	$-0.315036\pi$
$564$		0			0
$565$	−0.877425			−0.0369136
$566$			36.8718i			1.54984i
$567$		0			0
$568$		−	9.87493i		−	0.414343i
$569$	20.6660			0.866363			0.433182	−	0.901307i	$-0.357391\pi$
$569$	20.6660			0.866363			0.433182	+	0.901307i	$0.357391\pi$
$570$		0			0
$571$		−	21.2349i		−	0.888654i	−0.895865	−	0.444327i	$-0.853443\pi$
$571$		−	21.2349i		−	0.888654i	0.895865	−	0.444327i	$-0.146557\pi$
$572$	−47.4061			−1.98215
$573$		0			0
$574$			17.0760i			0.712737i
$575$	21.3171			0.888983
$576$		0			0
$577$	−6.89715			−0.287132			−0.143566	−	0.989641i	$-0.545857\pi$
$577$	−6.89715			−0.287132			−0.143566	+	0.989641i	$0.545857\pi$
$578$			8.92432i			0.371203i
$579$		0			0
$580$		−	6.02096i		−	0.250007i
$581$	−0.811376			−0.0336616
$582$		0			0
$583$		−	29.7906i		−	1.23380i
$584$	−7.26212			−0.300509
$585$		0			0
$586$	−32.3314			−1.33560
$587$	17.4095			0.718567			0.359284	−	0.933228i	$-0.383021\pi$
$587$	17.4095			0.718567			0.359284	+	0.933228i	$0.383021\pi$
$588$		0			0
$589$	0.849479			0.0350022
$590$			6.39448i			0.263257i
$591$		0			0
$592$			39.6063i			1.62781i
$593$	13.4256			0.551323			0.275661	−	0.961255i	$-0.411103\pi$
$593$	13.4256			0.551323			0.275661	+	0.961255i	$0.411103\pi$
$594$		0			0
$595$	3.65778			0.149954
$596$	15.1427			0.620270
$597$		0			0
$598$	60.4685			2.47274
$599$		−	9.65922i		−	0.394665i	−0.980337	−	0.197333i	$-0.936772\pi$
$599$		−	9.65922i		−	0.394665i	0.980337	−	0.197333i	$-0.0632279\pi$
$600$		0			0
$601$	31.3715			1.27967			0.639836	−	0.768512i	$-0.279002\pi$
$601$	31.3715			1.27967			0.639836	+	0.768512i	$0.279002\pi$
$602$	−11.5707			−0.471586
$603$		0			0
$604$			21.7861i			0.886465i
$605$	29.3209			1.19206
$606$		0			0
$607$			36.0864i			1.46470i	0.680926	+	0.732352i	$0.261577\pi$
$607$			36.0864i			1.46470i	−0.680926	+	0.732352i	$0.738423\pi$
$608$			4.25116i			0.172407i
$609$		0			0
$610$		−	2.74901i		−	0.111304i
$611$	21.4606			0.868203
$612$		0			0
$613$	−16.0796			−0.649451			−0.324725	−	0.945808i	$-0.605272\pi$
$613$	−16.0796			−0.649451			−0.324725	+	0.945808i	$0.605272\pi$
$614$	−39.8832			−1.60955
$615$		0			0
$616$	6.08433			0.245145
$617$		−	2.90095i		−	0.116788i	−0.998294	−	0.0583940i	$-0.981402\pi$
$617$		−	2.90095i		−	0.116788i	0.998294	−	0.0583940i	$-0.0185980\pi$
$618$		0			0
$619$			2.51897i			0.101246i	0.998718	+	0.0506229i	$0.0161207\pi$
$619$			2.51897i			0.101246i	−0.998718	+	0.0506229i	$0.983879\pi$
$620$	2.16198			0.0868271
$621$		0			0
$622$	−39.1312			−1.56902
$623$		−	6.17117i		−	0.247243i
$624$		0			0
$625$	6.70223			0.268089
$626$	30.9481			1.23693
$627$		0			0
$628$	8.60152			0.343238
$629$			28.5119i			1.13684i
$630$		0			0
$631$	12.5708			0.500434			0.250217	−	0.968190i	$-0.419498\pi$
$631$	12.5708			0.500434			0.250217	+	0.968190i	$0.419498\pi$
$632$	11.7671			0.468070
$633$		0			0
$634$	39.7919			1.58034
$635$	1.79432			0.0712054
$636$		0			0
$637$			34.8142i			1.37939i
$638$		−	41.4030i		−	1.63916i
$639$		0			0
$640$		−	9.83966i		−	0.388947i
$641$	27.6221			1.09101			0.545503	−	0.838109i	$-0.316339\pi$
$641$	27.6221			1.09101			0.545503	+	0.838109i	$0.316339\pi$
$642$		0			0
$643$			7.09385i			0.279754i	0.990169	+	0.139877i	$0.0446707\pi$
$643$			7.09385i			0.279754i	−0.990169	+	0.139877i	$0.955329\pi$
$644$	7.36737			0.290315
$645$		0			0
$646$			4.07791i			0.160443i
$647$	0.420984			0.0165506			0.00827530	−	0.999966i	$-0.497366\pi$
$647$	0.420984			0.0165506			0.00827530	+	0.999966i	$0.497366\pi$
$648$		0			0
$649$			18.0732i			0.709435i
$650$	38.0095			1.49085
$651$		0			0
$652$		−	3.13456i		−	0.122759i
$653$		−	19.7133i		−	0.771441i	−0.922616	−	0.385720i	$-0.873953\pi$
$653$		−	19.7133i		−	0.771441i	0.922616	−	0.385720i	$-0.126047\pi$
$654$		0			0
$655$	15.7693			0.616156
$656$	49.5148			1.93323
$657$		0			0
$658$	6.36154			0.247999
$659$	−32.8638			−1.28019			−0.640096	−	0.768295i	$-0.721105\pi$
$659$	−32.8638			−1.28019			−0.640096	+	0.768295i	$0.721105\pi$
$660$		0			0
$661$			14.8421i			0.577289i	0.957436	+	0.288645i	$0.0932046\pi$
$661$			14.8421i			0.577289i	−0.957436	+	0.288645i	$0.906795\pi$
$662$	−22.1977			−0.862737
$663$		0			0
$664$			0.996782i			0.0386827i
$665$	−0.665828			−0.0258197
$666$		0			0
$667$			21.7065i			0.840479i
$668$	−17.7833	−	3.01151i	−0.688055	−	0.116519i
$669$		0			0
$670$	−30.6009			−1.18222
$671$		−	7.76973i		−	0.299947i
$672$		0			0
$673$			13.1410i			0.506548i	0.967395	+	0.253274i	$0.0815074\pi$
$673$			13.1410i			0.506548i	−0.967395	+	0.253274i	$0.918493\pi$
$674$		−	50.2676i		−	1.93624i
$675$		0			0
$676$	26.1715			1.00660
$677$		−	43.0156i		−	1.65322i	−0.562772	−	0.826612i	$-0.690265\pi$
$677$		−	43.0156i		−	1.65322i	0.562772	−	0.826612i	$-0.309735\pi$
$678$		0			0
$679$	9.24198			0.354675
$680$		−	4.49361i		−	0.172322i
$681$		0			0
$682$	14.8668			0.569279
$683$	33.5999			1.28567			0.642833	−	0.766007i	$-0.277759\pi$
$683$	33.5999			1.28567			0.642833	+	0.766007i	$0.277759\pi$
$684$		0			0
$685$			21.5938i			0.825057i
$686$			22.0122i			0.840431i
$687$		0			0
$688$			33.5513i			1.27913i
$689$			27.8485i			1.06094i
$690$		0			0
$691$			29.1001i			1.10702i	0.832843	+	0.553509i	$0.186712\pi$
$691$			29.1001i			1.10702i	−0.832843	+	0.553509i	$0.813288\pi$
$692$			25.3989i			0.965522i
$693$		0			0
$694$			64.2899i			2.44041i
$695$			9.34357i			0.354422i
$696$		0			0
$697$	35.6448			1.35014
$698$	62.4675			2.36443
$699$		0			0
$700$	4.63101			0.175036
$701$			33.2212i			1.25475i	0.778719	+	0.627373i	$0.215870\pi$
$701$			33.2212i			1.25475i	−0.778719	+	0.627373i	$0.784130\pi$
$702$		0			0
$703$		−	5.19005i		−	0.195746i
$704$			16.0095i			0.603380i
$705$		0			0
$706$	34.9347			1.31479
$707$	−6.44960			−0.242562
$708$		0			0
$709$			19.5179i			0.733010i	0.930416	+	0.366505i	$0.119446\pi$
$709$			19.5179i			0.733010i	−0.930416	+	0.366505i	$0.880554\pi$
$710$			18.9125i			0.709772i
$711$		0			0
$712$	−7.58134			−0.284123
$713$	−7.79427			−0.291898
$714$		0			0
$715$	−39.3103			−1.47012
$716$		−	0.709441i		−	0.0265130i
$717$		0			0
$718$	−31.1362			−1.16199
$719$	19.2276			0.717068			0.358534	−	0.933517i	$-0.383277\pi$
$719$	19.2276			0.717068			0.358534	+	0.933517i	$0.383277\pi$
$720$		0			0
$721$			7.70539i			0.286964i
$722$			34.2698i			1.27539i
$723$		0			0
$724$	−34.9253			−1.29799
$725$			13.6443i			0.506738i
$726$		0			0
$727$			21.8886i			0.811804i	0.913917	+	0.405902i	$0.133043\pi$
$727$			21.8886i			0.811804i	−0.913917	+	0.405902i	$0.866957\pi$
$728$	−5.68769			−0.210800
$729$		0			0
$730$	13.9084			0.514774
$731$			24.1530i			0.893330i
$732$		0			0
$733$	25.3028			0.934581			0.467291	−	0.884104i	$-0.345230\pi$
$733$	25.3028			0.934581			0.467291	+	0.884104i	$0.345230\pi$
$734$			44.1436i			1.62937i
$735$		0			0
$736$		−	39.0059i		−	1.43778i
$737$	−86.4897			−3.18589
$738$		0			0
$739$		−	49.0560i		−	1.80455i	−0.431157	−	0.902277i	$-0.641895\pi$
$739$		−	49.0560i		−	1.80455i	0.431157	−	0.902277i	$-0.358105\pi$
$740$		−	13.2090i		−	0.485572i
$741$		0			0
$742$			8.25511i			0.303054i
$743$			20.9713i			0.769364i	0.923049	+	0.384682i	$0.125689\pi$
$743$			20.9713i			0.769364i	−0.923049	+	0.384682i	$0.874311\pi$
$744$		0			0
$745$	12.5567			0.460043
$746$	−0.440198			−0.0161168
$747$		0			0
$748$			29.3337i			1.07254i
$749$		−	8.33304i		−	0.304483i
$750$		0			0
$751$		−	6.22334i		−	0.227093i	−0.993533	−	0.113546i	$-0.963779\pi$
$751$		−	6.22334i		−	0.227093i	0.993533	−	0.113546i	$-0.0362210\pi$
$752$		−	18.4464i		−	0.672671i
$753$		0			0
$754$			38.7039i			1.40951i
$755$			18.0656i			0.657475i
$756$		0			0
$757$	36.3413			1.32085			0.660424	−	0.750893i	$-0.270377\pi$
$757$	36.3413			1.32085			0.660424	+	0.750893i	$0.270377\pi$
$758$	9.34716			0.339504
$759$		0			0
$760$			0.817975i			0.0296711i
$761$		−	10.2535i		−	0.371688i	−0.982579	−	0.185844i	$-0.940498\pi$
$761$		−	10.2535i		−	0.371688i	0.982579	−	0.185844i	$-0.0595020\pi$
$762$		0			0
$763$			13.1218i			0.475040i
$764$		−	23.2341i		−	0.840582i
$765$		0			0
$766$	−16.7418			−0.604907
$767$		−	16.8950i		−	0.610042i
$768$		0			0
$769$		−	6.05922i		−	0.218501i	−0.994014	−	0.109251i	$-0.965155\pi$
$769$		−	6.05922i		−	0.218501i	0.994014	−	0.109251i	$-0.0348451\pi$
$770$	−11.6527			−0.419935
$771$		0			0
$772$		−	7.83692i		−	0.282057i
$773$	−20.3288			−0.731175			−0.365587	−	0.930777i	$-0.619132\pi$
$773$	−20.3288			−0.731175			−0.365587	+	0.930777i	$0.619132\pi$
$774$		0			0
$775$	−4.89935			−0.175990
$776$		−	11.3539i		−	0.407580i
$777$		0			0
$778$			37.2617i			1.33590i
$779$	−6.48846			−0.232473
$780$		0			0
$781$			53.4537i			1.91272i
$782$		−	37.4163i		−	1.33800i
$783$		0			0
$784$	29.9244			1.06873
$785$	7.13260			0.254573
$786$		0			0
$787$			25.4241i			0.906272i	0.891441	+	0.453136i	$0.149695\pi$
$787$			25.4241i			0.906272i	−0.891441	+	0.453136i	$0.850305\pi$
$788$	−30.3370			−1.08071
$789$		0			0
$790$	−22.5364			−0.801808
$791$	0.687199			0.0244340
$792$		0			0
$793$			7.26321i			0.257924i
$794$			1.41177i			0.0501020i
$795$		0			0
$796$	18.1577			0.643581
$797$	−1.07677			−0.0381412			−0.0190706	−	0.999818i	$-0.506071\pi$
$797$	−1.07677			−0.0381412			−0.0190706	+	0.999818i	$0.506071\pi$
$798$		0			0
$799$		−	13.2792i		−	0.469786i
$800$		−	24.5185i		−	0.866858i
$801$		0			0
$802$		−	17.3547i		−	0.612817i
$803$	39.3104			1.38723
$804$		0			0
$805$	6.10921			0.215321
$806$	−13.8976			−0.489523
$807$		0			0
$808$			7.92338i			0.278744i
$809$			4.79215i			0.168483i	0.996445	+	0.0842416i	$0.0268468\pi$
$809$			4.79215i			0.168483i	−0.996445	+	0.0842416i	$0.973153\pi$
$810$		0			0
$811$			13.8222i			0.485364i	0.970106	+	0.242682i	$0.0780271\pi$
$811$			13.8222i			0.485364i	−0.970106	+	0.242682i	$0.921973\pi$
$812$			4.71561i			0.165486i
$813$		0			0
$814$		−	90.8314i		−	3.18364i
$815$		−	2.59926i		−	0.0910479i
$816$		0			0
$817$		−	4.39659i		−	0.153817i
$818$		−	6.48888i		−	0.226878i
$819$		0			0
$820$	−16.5136			−0.576678
$821$	−50.1861			−1.75151			−0.875753	−	0.482760i	$-0.839634\pi$
$821$	−50.1861			−1.75151			−0.875753	+	0.482760i	$0.839634\pi$
$822$		0			0
$823$			21.8760i			0.762551i	0.924461	+	0.381275i	$0.124515\pi$
$823$			21.8760i			0.762551i	−0.924461	+	0.381275i	$0.875485\pi$
$824$	9.46614			0.329768
$825$		0			0
$826$		−	5.00816i		−	0.174256i
$827$	−15.7006			−0.545963			−0.272981	−	0.962019i	$-0.588010\pi$
$827$	−15.7006			−0.545963			−0.272981	+	0.962019i	$0.588010\pi$
$828$		0			0
$829$			39.4788i			1.37116i	0.727999	+	0.685578i	$0.240450\pi$
$829$			39.4788i			1.37116i	−0.727999	+	0.685578i	$0.759550\pi$
$830$		−	1.90904i		−	0.0662636i
$831$		0			0
$832$		−	14.9658i		−	0.518846i
$833$	21.5421			0.746390
$834$		0			0
$835$	−14.7463	−	2.49722i	−0.510318	−	0.0864199i
$836$		−	5.33963i		−	0.184675i
$837$		0			0
$838$	15.2979			0.528458
$839$			34.4411i			1.18904i	0.804082	+	0.594519i	$0.202657\pi$
$839$			34.4411i			1.18904i	−0.804082	+	0.594519i	$0.797343\pi$
$840$		0			0
$841$	15.1064			0.520910
$842$			26.3416i			0.907790i
$843$		0			0
$844$	−7.04907			−0.242639
$845$	21.7021			0.746574
$846$		0			0
$847$	−22.9641			−0.789056
$848$	23.9371			0.822005
$849$		0			0
$850$		−	23.5193i		−	0.806704i
$851$			47.6205i			1.63241i
$852$		0			0
$853$	−35.7953			−1.22561			−0.612804	−	0.790235i	$-0.709958\pi$
$853$	−35.7953			−1.22561			−0.612804	+	0.790235i	$0.709958\pi$
$854$			2.15302i			0.0736750i
$855$		0			0
$856$	−10.2372			−0.349901
$857$		−	35.5553i		−	1.21455i	−0.794493	−	0.607273i	$-0.792263\pi$
$857$		−	35.5553i		−	1.21455i	0.794493	−	0.607273i	$-0.207737\pi$
$858$		0			0
$859$	35.1778			1.20025			0.600126	−	0.799906i	$-0.295117\pi$
$859$	35.1778			1.20025			0.600126	+	0.799906i	$0.295117\pi$
$860$		−	11.1896i		−	0.381562i
$861$		0			0
$862$	0.284085			0.00967597
$863$		−	8.31229i		−	0.282954i	−0.989942	−	0.141477i	$-0.954815\pi$
$863$		−	8.31229i		−	0.282954i	0.989942	−	0.141477i	$-0.0451851\pi$
$864$		0			0
$865$			21.0614i			0.716110i
$866$		−	40.7393i		−	1.38438i
$867$		0			0
$868$	−1.69326			−0.0574730
$869$	−63.6961			−2.16074
$870$		0			0
$871$	80.8513			2.73954
$872$	16.1202			0.545899
$873$		0			0
$874$			6.81093i			0.230383i
$875$	9.08548			0.307145
$876$		0			0
$877$	−40.5456			−1.36913			−0.684564	−	0.728953i	$-0.740007\pi$
$877$	−40.5456			−1.36913			−0.684564	+	0.728953i	$0.740007\pi$
$878$	8.07938			0.272666
$879$		0			0
$880$			33.7891i			1.13903i
$881$	−49.8273			−1.67872			−0.839362	−	0.543573i	$-0.817071\pi$
$881$	−49.8273			−1.67872			−0.839362	+	0.543573i	$0.817071\pi$
$882$		0			0
$883$	−44.0175			−1.48131			−0.740654	−	0.671887i	$-0.765484\pi$
$883$	−44.0175			−1.48131			−0.740654	+	0.671887i	$0.765484\pi$
$884$		−	27.4214i		−	0.922280i
$885$		0			0
$886$			22.6085i			0.759546i
$887$	18.0370			0.605625			0.302812	−	0.953050i	$-0.402074\pi$
$887$	18.0370			0.605625			0.302812	+	0.953050i	$0.402074\pi$
$888$		0			0
$889$	−1.40531			−0.0471326
$890$	14.5198			0.486704
$891$		0			0
$892$	12.1364			0.406356
$893$			2.41723i			0.0808896i
$894$		0			0
$895$		−	0.588286i		−	0.0196642i
$896$			7.70642i			0.257453i
$897$		0			0
$898$	−48.8855			−1.63133
$899$		−	4.98886i		−	0.166388i
$900$		0			0
$901$	17.2319			0.574079
$902$	−113.555			−3.78097
$903$		0			0
$904$		−	0.844230i		−	0.0280787i
$905$	−28.9610			−0.962695
$906$		0			0
$907$	−29.2523			−0.971306			−0.485653	−	0.874152i	$-0.661418\pi$
$907$	−29.2523			−0.971306			−0.485653	+	0.874152i	$0.661418\pi$
$908$	−19.9704			−0.662742
$909$		0			0
$910$	10.8931			0.361101
$911$		−	45.5904i		−	1.51048i	−0.655451	−	0.755238i	$-0.727521\pi$
$911$		−	45.5904i		−	1.51048i	0.655451	−	0.755238i	$-0.272479\pi$
$912$		0			0
$913$		−	5.39565i		−	0.178570i
$914$	2.09971			0.0694524
$915$		0			0
$916$	−38.6070			−1.27561
$917$	−12.3505			−0.407849
$918$		0			0
$919$	8.93668			0.294794			0.147397	−	0.989077i	$-0.452911\pi$
$919$	8.93668			0.294794			0.147397	+	0.989077i	$0.452911\pi$
$920$		−	7.50522i		−	0.247440i
$921$		0			0
$922$	−42.7812			−1.40892
$923$		−	49.9690i		−	1.64475i
$924$		0			0
$925$			29.9335i			0.984206i
$926$	−19.8232			−0.651431
$927$		0			0
$928$	24.9664			0.819562
$929$		−	23.1276i		−	0.758794i	−0.925234	−	0.379397i	$-0.876132\pi$
$929$		−	23.1276i		−	0.758794i	0.925234	−	0.379397i	$-0.123868\pi$
$930$		0			0
$931$	−3.92133			−0.128516
$932$			7.24752i			0.237400i
$933$		0			0
$934$	75.1034			2.45746
$935$			24.3242i			0.795486i
$936$		0			0
$937$			31.4259i			1.02664i	0.858198	+	0.513319i	$0.171584\pi$
$937$			31.4259i			1.02664i	−0.858198	+	0.513319i	$0.828416\pi$
$938$	23.9667			0.782539
$939$		0			0
$940$			6.15201i			0.200657i
$941$	14.2097			0.463223			0.231611	−	0.972808i	$-0.425600\pi$
$941$	14.2097			0.463223			0.231611	+	0.972808i	$0.425600\pi$
$942$		0			0
$943$	59.5340			1.93869
$944$	−14.5220			−0.472652
$945$		0			0
$946$		−	76.9451i		−	2.50170i
$947$			28.8208i			0.936549i	0.883583	+	0.468275i	$0.155124\pi$
$947$			28.8208i			0.936549i	−0.883583	+	0.468275i	$0.844876\pi$
$948$		0			0
$949$	−36.7477			−1.19288
$950$			4.28123i			0.138902i
$951$		0			0
$952$			3.51939i			0.114064i
$953$	5.64016			0.182703			0.0913514	−	0.995819i	$-0.470881\pi$
$953$	5.64016			0.182703			0.0913514	+	0.995819i	$0.470881\pi$
$954$		0			0
$955$		−	19.2663i		−	0.623444i
$956$		−	37.1894i		−	1.20279i
$957$		0			0
$958$		−	34.4811i		−	1.11403i
$959$		−	16.9123i		−	0.546126i
$960$		0			0
$961$	−29.2086			−0.942214
$962$			84.9100i			2.73761i
$963$		0			0
$964$			38.5248i			1.24080i
$965$		−	6.49857i		−	0.209196i
$966$		0			0
$967$	−10.9649			−0.352608			−0.176304	−	0.984336i	$-0.556414\pi$
$967$	−10.9649			−0.352608			−0.176304	+	0.984336i	$0.556414\pi$
$968$			28.2116i			0.906755i
$969$		0			0
$970$			21.7449i			0.698187i
$971$	−55.2323			−1.77249			−0.886245	−	0.463218i	$-0.846695\pi$
$971$	−55.2323			−1.77249			−0.886245	+	0.463218i	$0.846695\pi$
$972$		0			0
$973$		−	7.31788i		−	0.234601i
$974$	−29.1497			−0.934017
$975$		0			0
$976$	6.24307			0.199836
$977$	−33.2212			−1.06284			−0.531421	−	0.847108i	$-0.678342\pi$
$977$	−33.2212			−1.06284			−0.531421	+	0.847108i	$0.678342\pi$
$978$		0			0
$979$	41.0383			1.31159
$980$	−9.98003			−0.318800
$981$		0			0
$982$	9.42493			0.300762
$983$	52.6421			1.67902			0.839512	−	0.543341i	$-0.182841\pi$
$983$	52.6421			1.67902			0.839512	+	0.543341i	$0.182841\pi$
$984$		0			0
$985$	−25.1562			−0.801544
$986$	23.9490			0.762690
$987$		0			0
$988$			4.99153i			0.158802i
$989$			40.3403i			1.28275i
$990$		0			0
$991$		−	34.6335i		−	1.10017i	−0.835109	−	0.550085i	$-0.814595\pi$
$991$		−	34.6335i		−	1.10017i	0.835109	−	0.550085i	$-0.185405\pi$
$992$			8.96482i			0.284633i
$993$		0			0
$994$		−	14.8122i		−	0.469816i
$995$	15.0568			0.477332
$996$		0			0
$997$	27.8498			0.882013			0.441006	−	0.897504i	$-0.354622\pi$
$997$	27.8498			0.882013			0.441006	+	0.897504i	$0.354622\pi$
$998$	74.3924			2.35485
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.c.a.1502.19	✓	56
3.2	odd	2	inner	1503.2.c.a.1502.38	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.37	yes	56
501.500	even	2	inner	1503.2.c.a.1502.20	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.2.c.a.1502.19	✓	56	1.1	even	1	trivial
1503.2.c.a.1502.20	yes	56	501.500	even	2	inner
1503.2.c.a.1502.37	yes	56	167.166	odd	2	inner
1503.2.c.a.1502.38	yes	56	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.84274i q^{2} -1.39570 q^{4} -1.15735 q^{5} +0.906437 q^{7} -1.11357i q^{8} +O(q^{10})\)	\(q-1.84274i q^{2} -1.39570 q^{4} -1.15735 q^{5} +0.906437 q^{7} -1.11357i q^{8} +2.13270i q^{10} +6.02781i q^{11} -5.63485i q^{13} -1.67033i q^{14} -4.84342 q^{16} -3.48670 q^{17} +0.634686 q^{19} +1.61532 q^{20} +11.1077 q^{22} -5.82348 q^{23} -3.66054 q^{25} -10.3836 q^{26} -1.26512 q^{28} -3.72741i q^{29} +1.33842 q^{31} +6.69805i q^{32} +6.42508i q^{34} -1.04907 q^{35} -8.17734i q^{37} -1.16956i q^{38} +1.28879i q^{40} -10.2231 q^{41} -6.92718i q^{43} -8.41302i q^{44} +10.7312i q^{46} +3.80855i q^{47} -6.17837 q^{49} +6.74543i q^{50} +7.86457i q^{52} -4.94220 q^{53} -6.97629i q^{55} -1.00938i q^{56} -6.86867 q^{58} +2.99830 q^{59} -1.28898 q^{61} -2.46637i q^{62} +2.65594 q^{64} +6.52150i q^{65} +14.3484i q^{67} +4.86639 q^{68} +1.93316i q^{70} +8.86785 q^{71} -6.52150i q^{73} -15.0687 q^{74} -0.885833 q^{76} +5.46383i q^{77} +10.5670i q^{79} +5.60554 q^{80} +18.8385i q^{82} -0.895126 q^{83} +4.03533 q^{85} -12.7650 q^{86} +6.71236 q^{88} -6.80816i q^{89} -5.10764i q^{91} +8.12784 q^{92} +7.01818 q^{94} -0.734555 q^{95} +10.1959 q^{97} +11.3852i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 48 q^{4}+O(q^{10}) \) 56 * q - 48 * q^4	\( 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) \) 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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