LMFDB - Embedded newform 8032.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.43537 q^{3} +2.52824 q^{5} -2.67250 q^{7} +8.80174 q^{9} +O(q^{10})$	$q-3.43537 q^{3} +2.52824 q^{5} -2.67250 q^{7} +8.80174 q^{9} -1.07977 q^{11} -0.0741393 q^{13} -8.68544 q^{15} +2.98601 q^{17} +5.23612 q^{19} +9.18102 q^{21} -0.357255 q^{23} +1.39201 q^{25} -19.9311 q^{27} -10.3306 q^{29} +0.191587 q^{31} +3.70939 q^{33} -6.75673 q^{35} +6.37070 q^{37} +0.254696 q^{39} -1.35606 q^{41} +3.54082 q^{43} +22.2529 q^{45} -1.08959 q^{47} +0.142258 q^{49} -10.2580 q^{51} -7.34832 q^{53} -2.72991 q^{55} -17.9880 q^{57} -1.39758 q^{59} +0.128667 q^{61} -23.5227 q^{63} -0.187442 q^{65} -5.46554 q^{67} +1.22730 q^{69} +8.48764 q^{71} -5.95742 q^{73} -4.78207 q^{75} +2.88567 q^{77} +13.8610 q^{79} +42.0655 q^{81} -8.71606 q^{83} +7.54936 q^{85} +35.4895 q^{87} -8.31495 q^{89} +0.198137 q^{91} -0.658173 q^{93} +13.2382 q^{95} +3.53227 q^{97} -9.50382 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	$ 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) $ 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * 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 $

$2$	0	0
$2$	0	0
$3$	−3.43537	−1.98341	−0.991705	−	0.128535i	$-0.958972\pi$
$3$	−3.43537	−1.98341	−0.991705	+	0.128535i	$0.958972\pi$
$4$	0	0
$5$	2.52824	1.13066	0.565332	−	0.824863i	$-0.308748\pi$
$5$	2.52824	1.13066	0.565332	+	0.824863i	$0.308748\pi$
$6$	0	0
$7$	−2.67250	−1.01011	−0.505055	−	0.863087i	$-0.668528\pi$
$7$	−2.67250	−1.01011	−0.505055	+	0.863087i	$0.668528\pi$
$8$	0	0
$9$	8.80174	2.93391
$10$	0	0
$11$	−1.07977	−0.325561	−0.162781	−	0.986662i	$-0.552046\pi$
$11$	−1.07977	−0.325561	−0.162781	+	0.986662i	$0.552046\pi$
$12$	0	0
$13$	−0.0741393	−0.0205625	−0.0102813	−	0.999947i	$-0.503273\pi$
$13$	−0.0741393	−0.0205625	−0.0102813	+	0.999947i	$0.503273\pi$
$14$	0	0
$15$	−8.68544	−2.24257
$16$	0	0
$17$	2.98601	0.724214	0.362107	−	0.932137i	$-0.382058\pi$
$17$	2.98601	0.724214	0.362107	+	0.932137i	$0.382058\pi$
$18$	0	0
$19$	5.23612	1.20125	0.600624	−	0.799531i	$-0.294919\pi$
$19$	5.23612	1.20125	0.600624	+	0.799531i	$0.294919\pi$
$20$	0	0
$21$	9.18102	2.00346
$22$	0	0
$23$	−0.357255	−0.0744927	−0.0372464	−	0.999306i	$-0.511859\pi$
$23$	−0.357255	−0.0744927	−0.0372464	+	0.999306i	$0.511859\pi$
$24$	0	0
$25$	1.39201	0.278402
$26$	0	0
$27$	−19.9311	−3.83575
$28$	0	0
$29$	−10.3306	−1.91835	−0.959175	−	0.282813i	$-0.908732\pi$
$29$	−10.3306	−1.91835	−0.959175	+	0.282813i	$0.908732\pi$
$30$	0	0
$31$	0.191587	0.0344101	0.0172051	−	0.999852i	$-0.494523\pi$
$31$	0.191587	0.0344101	0.0172051	+	0.999852i	$0.494523\pi$
$32$	0	0
$33$	3.70939	0.645722
$34$	0	0
$35$	−6.75673	−1.14210
$36$	0	0
$37$	6.37070	1.04734	0.523669	−	0.851922i	$-0.324563\pi$
$37$	6.37070	1.04734	0.523669	+	0.851922i	$0.324563\pi$
$38$	0	0
$39$	0.254696	0.0407839
$40$	0	0
$41$	−1.35606	−0.211781	−0.105891	−	0.994378i	$-0.533769\pi$
$41$	−1.35606	−0.211781	−0.105891	+	0.994378i	$0.533769\pi$
$42$	0	0
$43$	3.54082	0.539971	0.269985	−	0.962864i	$-0.412981\pi$
$43$	3.54082	0.539971	0.269985	+	0.962864i	$0.412981\pi$
$44$	0	0
$45$	22.2529	3.31727
$46$	0	0
$47$	−1.08959	−0.158933	−0.0794665	−	0.996838i	$-0.525322\pi$
$47$	−1.08959	−0.158933	−0.0794665	+	0.996838i	$0.525322\pi$
$48$	0	0
$49$	0.142258	0.0203225
$50$	0	0
$51$	−10.2580	−1.43641
$52$	0	0
$53$	−7.34832	−1.00937	−0.504684	−	0.863304i	$-0.668391\pi$
$53$	−7.34832	−1.00937	−0.504684	+	0.863304i	$0.668391\pi$
$54$	0	0
$55$	−2.72991	−0.368101
$56$	0	0
$57$	−17.9880	−2.38257
$58$	0	0
$59$	−1.39758	−0.181950	−0.0909748	−	0.995853i	$-0.528998\pi$
$59$	−1.39758	−0.181950	−0.0909748	+	0.995853i	$0.528998\pi$
$60$	0	0
$61$	0.128667	0.0164741	0.00823705	−	0.999966i	$-0.497378\pi$
$61$	0.128667	0.0164741	0.00823705	+	0.999966i	$0.497378\pi$
$62$	0	0
$63$	−23.5227	−2.96358
$64$	0	0
$65$	−0.187442	−0.0232493
$66$	0	0
$67$	−5.46554	−0.667721	−0.333861	−	0.942622i	$-0.608352\pi$
$67$	−5.46554	−0.667721	−0.333861	+	0.942622i	$0.608352\pi$
$68$	0	0
$69$	1.22730	0.147750
$70$	0	0
$71$	8.48764	1.00730	0.503649	−	0.863909i	$-0.331991\pi$
$71$	8.48764	1.00730	0.503649	+	0.863909i	$0.331991\pi$
$72$	0	0
$73$	−5.95742	−0.697264	−0.348632	−	0.937260i	$-0.613354\pi$
$73$	−5.95742	−0.697264	−0.348632	+	0.937260i	$0.613354\pi$
$74$	0	0
$75$	−4.78207	−0.552186
$76$	0	0
$77$	2.88567	0.328853
$78$	0	0
$79$	13.8610	1.55948	0.779740	−	0.626104i	$-0.215351\pi$
$79$	13.8610	1.55948	0.779740	+	0.626104i	$0.215351\pi$
$80$	0	0
$81$	42.0655	4.67394
$82$	0	0
$83$	−8.71606	−0.956711	−0.478356	−	0.878166i	$-0.658767\pi$
$83$	−8.71606	−0.956711	−0.478356	+	0.878166i	$0.658767\pi$
$84$	0	0
$85$	7.54936	0.818843
$86$	0	0
$87$	35.4895	3.80487
$88$	0	0
$89$	−8.31495	−0.881383	−0.440691	−	0.897659i	$-0.645267\pi$
$89$	−8.31495	−0.881383	−0.440691	+	0.897659i	$0.645267\pi$
$90$	0	0
$91$	0.198137	0.0207704
$92$	0	0
$93$	−0.658173	−0.0682494
$94$	0	0
$95$	13.2382	1.35821
$96$	0	0
$97$	3.53227	0.358648	0.179324	−	0.983790i	$-0.442609\pi$
$97$	3.53227	0.358648	0.179324	+	0.983790i	$0.442609\pi$
$98$	0	0
$99$	−9.50382	−0.955170
$100$	0	0
$101$	−3.65025	−0.363213	−0.181607	−	0.983371i	$-0.558130\pi$
$101$	−3.65025	−0.363213	−0.181607	+	0.983371i	$0.558130\pi$
$102$	0	0
$103$	−3.01956	−0.297526	−0.148763	−	0.988873i	$-0.547529\pi$
$103$	−3.01956	−0.297526	−0.148763	+	0.988873i	$0.547529\pi$
$104$	0	0
$105$	23.2118	2.26524
$106$	0	0
$107$	−15.3432	−1.48328	−0.741641	−	0.670797i	$-0.765952\pi$
$107$	−15.3432	−1.48328	−0.741641	+	0.670797i	$0.765952\pi$
$108$	0	0
$109$	9.51981	0.911833	0.455916	−	0.890023i	$-0.349312\pi$
$109$	9.51981	0.911833	0.455916	+	0.890023i	$0.349312\pi$
$110$	0	0
$111$	−21.8857	−2.07730
$112$	0	0
$113$	7.81448	0.735125	0.367562	−	0.929999i	$-0.380192\pi$
$113$	7.81448	0.735125	0.367562	+	0.929999i	$0.380192\pi$
$114$	0	0
$115$	−0.903226	−0.0842263
$116$	0	0
$117$	−0.652555	−0.0603287
$118$	0	0
$119$	−7.98011	−0.731536
$120$	0	0
$121$	−9.83411	−0.894010
$122$	0	0
$123$	4.65857	0.420049
$124$	0	0
$125$	−9.12187	−0.815885
$126$	0	0
$127$	3.47611	0.308455	0.154227	−	0.988035i	$-0.450711\pi$
$127$	3.47611	0.308455	0.154227	+	0.988035i	$0.450711\pi$
$128$	0	0
$129$	−12.1640	−1.07098
$130$	0	0
$131$	−3.44162	−0.300696	−0.150348	−	0.988633i	$-0.548039\pi$
$131$	−3.44162	−0.300696	−0.150348	+	0.988633i	$0.548039\pi$
$132$	0	0
$133$	−13.9935	−1.21339
$134$	0	0
$135$	−50.3907	−4.33694
$136$	0	0
$137$	−0.0555930	−0.00474963	−0.00237481	−	0.999997i	$-0.500756\pi$
$137$	−0.0555930	−0.00474963	−0.00237481	+	0.999997i	$0.500756\pi$
$138$	0	0
$139$	22.1356	1.87752	0.938759	−	0.344574i	$-0.111976\pi$
$139$	22.1356	1.87752	0.938759	+	0.344574i	$0.111976\pi$
$140$	0	0
$141$	3.74314	0.315229
$142$	0	0
$143$	0.0800530	0.00669437
$144$	0	0
$145$	−26.1183	−2.16901
$146$	0	0
$147$	−0.488707	−0.0403079
$148$	0	0
$149$	16.4537	1.34794	0.673968	−	0.738760i	$-0.264588\pi$
$149$	16.4537	1.34794	0.673968	+	0.738760i	$0.264588\pi$
$150$	0	0
$151$	8.94021	0.727544	0.363772	−	0.931488i	$-0.381489\pi$
$151$	8.94021	0.727544	0.363772	+	0.931488i	$0.381489\pi$
$152$	0	0
$153$	26.2821	2.12478
$154$	0	0
$155$	0.484379	0.0389063
$156$	0	0
$157$	−8.44926	−0.674325	−0.337162	−	0.941447i	$-0.609467\pi$
$157$	−8.44926	−0.674325	−0.337162	+	0.941447i	$0.609467\pi$
$158$	0	0
$159$	25.2442	2.00199
$160$	0	0
$161$	0.954763	0.0752459
$162$	0	0
$163$	−3.83200	−0.300145	−0.150073	−	0.988675i	$-0.547951\pi$
$163$	−3.83200	−0.300145	−0.150073	+	0.988675i	$0.547951\pi$
$164$	0	0
$165$	9.37824	0.730095
$166$	0	0
$167$	12.1609	0.941037	0.470518	−	0.882390i	$-0.344067\pi$
$167$	12.1609	0.941037	0.470518	+	0.882390i	$0.344067\pi$
$168$	0	0
$169$	−12.9945	−0.999577
$170$	0	0
$171$	46.0870	3.52436
$172$	0	0
$173$	−17.1141	−1.30116	−0.650579	−	0.759438i	$-0.725474\pi$
$173$	−17.1141	−1.30116	−0.650579	+	0.759438i	$0.725474\pi$
$174$	0	0
$175$	−3.72015	−0.281217
$176$	0	0
$177$	4.80120	0.360880
$178$	0	0
$179$	−12.4502	−0.930570	−0.465285	−	0.885161i	$-0.654048\pi$
$179$	−12.4502	−0.930570	−0.465285	+	0.885161i	$0.654048\pi$
$180$	0	0
$181$	20.1644	1.49881	0.749403	−	0.662114i	$-0.230340\pi$
$181$	20.1644	1.49881	0.749403	+	0.662114i	$0.230340\pi$
$182$	0	0
$183$	−0.442018	−0.0326749
$184$	0	0
$185$	16.1067	1.18419
$186$	0	0
$187$	−3.22419	−0.235776
$188$	0	0
$189$	53.2659	3.87453
$190$	0	0
$191$	−7.61906	−0.551296	−0.275648	−	0.961259i	$-0.588892\pi$
$191$	−7.61906	−0.551296	−0.275648	+	0.961259i	$0.588892\pi$
$192$	0	0
$193$	21.6627	1.55932	0.779658	−	0.626206i	$-0.215393\pi$
$193$	21.6627	1.55932	0.779658	+	0.626206i	$0.215393\pi$
$194$	0	0
$195$	0.643932	0.0461129
$196$	0	0
$197$	−17.8479	−1.27161	−0.635805	−	0.771850i	$-0.719332\pi$
$197$	−17.8479	−1.27161	−0.635805	+	0.771850i	$0.719332\pi$
$198$	0	0
$199$	5.87898	0.416750	0.208375	−	0.978049i	$-0.433183\pi$
$199$	5.87898	0.416750	0.208375	+	0.978049i	$0.433183\pi$
$200$	0	0
$201$	18.7761	1.32437
$202$	0	0
$203$	27.6086	1.93774
$204$	0	0
$205$	−3.42845	−0.239453
$206$	0	0
$207$	−3.14446	−0.218555
$208$	0	0
$209$	−5.65378	−0.391080
$210$	0	0
$211$	−23.1007	−1.59032	−0.795160	−	0.606399i	$-0.792613\pi$
$211$	−23.1007	−1.59032	−0.795160	+	0.606399i	$0.792613\pi$
$212$	0	0
$213$	−29.1581	−1.99788
$214$	0	0
$215$	8.95206	0.610525
$216$	0	0
$217$	−0.512017	−0.0347580
$218$	0	0
$219$	20.4659	1.38296
$220$	0	0
$221$	−0.221381	−0.0148917
$222$	0	0
$223$	5.01634	0.335919	0.167960	−	0.985794i	$-0.446282\pi$
$223$	5.01634	0.335919	0.167960	+	0.985794i	$0.446282\pi$
$224$	0	0
$225$	12.2521	0.816808
$226$	0	0
$227$	17.5364	1.16393	0.581967	−	0.813213i	$-0.302283\pi$
$227$	17.5364	1.16393	0.581967	+	0.813213i	$0.302283\pi$
$228$	0	0
$229$	−3.72572	−0.246202	−0.123101	−	0.992394i	$-0.539284\pi$
$229$	−3.72572	−0.246202	−0.123101	+	0.992394i	$0.539284\pi$
$230$	0	0
$231$	−9.91334	−0.652250
$232$	0	0
$233$	6.80924	0.446088	0.223044	−	0.974808i	$-0.428401\pi$
$233$	6.80924	0.446088	0.223044	+	0.974808i	$0.428401\pi$
$234$	0	0
$235$	−2.75475	−0.179700
$236$	0	0
$237$	−47.6175	−3.09309
$238$	0	0
$239$	28.4375	1.83947	0.919734	−	0.392543i	$-0.128405\pi$
$239$	28.4375	1.83947	0.919734	+	0.392543i	$0.128405\pi$
$240$	0	0
$241$	28.4474	1.83246	0.916228	−	0.400658i	$-0.131219\pi$
$241$	28.4474	1.83246	0.916228	+	0.400658i	$0.131219\pi$
$242$	0	0
$243$	−84.7169	−5.43459
$244$	0	0
$245$	0.359662	0.0229780
$246$	0	0
$247$	−0.388202	−0.0247007
$248$	0	0
$249$	29.9428	1.89755
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	0.385751	0.0242520
$254$	0	0
$255$	−25.9348	−1.62410
$256$	0	0
$257$	2.56067	0.159731	0.0798653	−	0.996806i	$-0.474551\pi$
$257$	2.56067	0.159731	0.0798653	+	0.996806i	$0.474551\pi$
$258$	0	0
$259$	−17.0257	−1.05793
$260$	0	0
$261$	−90.9276	−5.62828
$262$	0	0
$263$	−15.7114	−0.968808	−0.484404	−	0.874845i	$-0.660963\pi$
$263$	−15.7114	−0.968808	−0.484404	+	0.874845i	$0.660963\pi$
$264$	0	0
$265$	−18.5783	−1.14126
$266$	0	0
$267$	28.5649	1.74814
$268$	0	0
$269$	−1.23806	−0.0754861	−0.0377431	−	0.999287i	$-0.512017\pi$
$269$	−1.23806	−0.0754861	−0.0377431	+	0.999287i	$0.512017\pi$
$270$	0	0
$271$	−20.9812	−1.27451	−0.637257	−	0.770651i	$-0.719931\pi$
$271$	−20.9812	−1.27451	−0.637257	+	0.770651i	$0.719931\pi$
$272$	0	0
$273$	−0.680674	−0.0411963
$274$	0	0
$275$	−1.50305	−0.0906370
$276$	0	0
$277$	−13.0485	−0.784010	−0.392005	−	0.919963i	$-0.628218\pi$
$277$	−13.0485	−0.784010	−0.392005	+	0.919963i	$0.628218\pi$
$278$	0	0
$279$	1.68630	0.100956
$280$	0	0
$281$	−13.4173	−0.800412	−0.400206	−	0.916425i	$-0.631061\pi$
$281$	−13.4173	−0.800412	−0.400206	+	0.916425i	$0.631061\pi$
$282$	0	0
$283$	2.57715	0.153195	0.0765977	−	0.997062i	$-0.475594\pi$
$283$	2.57715	0.153195	0.0765977	+	0.997062i	$0.475594\pi$
$284$	0	0
$285$	−45.4780	−2.69389
$286$	0	0
$287$	3.62407	0.213922
$288$	0	0
$289$	−8.08375	−0.475514
$290$	0	0
$291$	−12.1347	−0.711346
$292$	0	0
$293$	−21.9627	−1.28307	−0.641537	−	0.767092i	$-0.721703\pi$
$293$	−21.9627	−1.28307	−0.641537	+	0.767092i	$0.721703\pi$
$294$	0	0
$295$	−3.53342	−0.205724
$296$	0	0
$297$	21.5209	1.24877
$298$	0	0
$299$	0.0264866	0.00153176
$300$	0	0
$301$	−9.46285	−0.545430
$302$	0	0
$303$	12.5399	0.720401
$304$	0	0
$305$	0.325301	0.0186267
$306$	0	0
$307$	4.84527	0.276534	0.138267	−	0.990395i	$-0.455847\pi$
$307$	4.84527	0.276534	0.138267	+	0.990395i	$0.455847\pi$
$308$	0	0
$309$	10.3733	0.590115
$310$	0	0
$311$	18.2079	1.03248	0.516238	−	0.856445i	$-0.327332\pi$
$311$	18.2079	1.03248	0.516238	+	0.856445i	$0.327332\pi$
$312$	0	0
$313$	24.6288	1.39210	0.696051	−	0.717992i	$-0.254939\pi$
$313$	24.6288	1.39210	0.696051	+	0.717992i	$0.254939\pi$
$314$	0	0
$315$	−59.4710	−3.35081
$316$	0	0
$317$	−12.1681	−0.683429	−0.341715	−	0.939804i	$-0.611008\pi$
$317$	−12.1681	−0.683429	−0.341715	+	0.939804i	$0.611008\pi$
$318$	0	0
$319$	11.1547	0.624541
$320$	0	0
$321$	52.7095	2.94196
$322$	0	0
$323$	15.6351	0.869961
$324$	0	0
$325$	−0.103203	−0.00572466
$326$	0	0
$327$	−32.7041	−1.80854
$328$	0	0
$329$	2.91193	0.160540
$330$	0	0
$331$	3.73968	0.205551	0.102776	−	0.994705i	$-0.467228\pi$
$331$	3.73968	0.205551	0.102776	+	0.994705i	$0.467228\pi$
$332$	0	0
$333$	56.0733	3.07280
$334$	0	0
$335$	−13.8182	−0.754969
$336$	0	0
$337$	−22.8095	−1.24251	−0.621257	−	0.783607i	$-0.713378\pi$
$337$	−22.8095	−1.24251	−0.621257	+	0.783607i	$0.713378\pi$
$338$	0	0
$339$	−26.8456	−1.45805
$340$	0	0
$341$	−0.206869	−0.0112026
$342$	0	0
$343$	18.3273	0.989582
$344$	0	0
$345$	3.10291	0.167055
$346$	0	0
$347$	−9.17584	−0.492585	−0.246293	−	0.969196i	$-0.579212\pi$
$347$	−9.17584	−0.492585	−0.246293	+	0.969196i	$0.579212\pi$
$348$	0	0
$349$	−14.3018	−0.765557	−0.382779	−	0.923840i	$-0.625033\pi$
$349$	−14.3018	−0.765557	−0.382779	+	0.923840i	$0.625033\pi$
$350$	0	0
$351$	1.47768	0.0788726
$352$	0	0
$353$	−19.8451	−1.05625	−0.528125	−	0.849167i	$-0.677105\pi$
$353$	−19.8451	−1.05625	−0.528125	+	0.849167i	$0.677105\pi$
$354$	0	0
$355$	21.4588	1.13891
$356$	0	0
$357$	27.4146	1.45094
$358$	0	0
$359$	−16.5556	−0.873772	−0.436886	−	0.899517i	$-0.643919\pi$
$359$	−16.5556	−0.873772	−0.436886	+	0.899517i	$0.643919\pi$
$360$	0	0
$361$	8.41698	0.442999
$362$	0	0
$363$	33.7838	1.77319
$364$	0	0
$365$	−15.0618	−0.788371
$366$	0	0
$367$	−35.0140	−1.82771	−0.913857	−	0.406036i	$-0.866911\pi$
$367$	−35.0140	−1.82771	−0.913857	+	0.406036i	$0.866911\pi$
$368$	0	0
$369$	−11.9357	−0.621348
$370$	0	0
$371$	19.6384	1.01957
$372$	0	0
$373$	5.04363	0.261149	0.130575	−	0.991438i	$-0.458318\pi$
$373$	5.04363	0.261149	0.130575	+	0.991438i	$0.458318\pi$
$374$	0	0
$375$	31.3370	1.61823
$376$	0	0
$377$	0.765906	0.0394461
$378$	0	0
$379$	−30.8051	−1.58235	−0.791176	−	0.611589i	$-0.790531\pi$
$379$	−30.8051	−1.58235	−0.791176	+	0.611589i	$0.790531\pi$
$380$	0	0
$381$	−11.9417	−0.611792
$382$	0	0
$383$	19.4063	0.991618	0.495809	−	0.868432i	$-0.334872\pi$
$383$	19.4063	0.991618	0.495809	+	0.868432i	$0.334872\pi$
$384$	0	0
$385$	7.29568	0.371822
$386$	0	0
$387$	31.1654	1.58423
$388$	0	0
$389$	18.1626	0.920880	0.460440	−	0.887691i	$-0.347692\pi$
$389$	18.1626	0.920880	0.460440	+	0.887691i	$0.347692\pi$
$390$	0	0
$391$	−1.06677	−0.0539487
$392$	0	0
$393$	11.8232	0.596404
$394$	0	0
$395$	35.0439	1.76325
$396$	0	0
$397$	−21.3211	−1.07007	−0.535037	−	0.844829i	$-0.679702\pi$
$397$	−21.3211	−1.07007	−0.535037	+	0.844829i	$0.679702\pi$
$398$	0	0
$399$	48.0729	2.40666
$400$	0	0
$401$	13.6392	0.681111	0.340555	−	0.940224i	$-0.389385\pi$
$401$	13.6392	0.681111	0.340555	+	0.940224i	$0.389385\pi$
$402$	0	0
$403$	−0.0142042	−0.000707559 0
$404$	0	0
$405$	106.352	5.28466
$406$	0	0
$407$	−6.87886	−0.340973
$408$	0	0
$409$	−34.5790	−1.70982	−0.854911	−	0.518776i	$-0.826388\pi$
$409$	−34.5790	−1.70982	−0.854911	+	0.518776i	$0.826388\pi$
$410$	0	0
$411$	0.190982	0.00942046
$412$	0	0
$413$	3.73504	0.183789
$414$	0	0
$415$	−22.0363	−1.08172
$416$	0	0
$417$	−76.0440	−3.72389
$418$	0	0
$419$	−18.2939	−0.893715	−0.446857	−	0.894605i	$-0.647457\pi$
$419$	−18.2939	−0.893715	−0.446857	+	0.894605i	$0.647457\pi$
$420$	0	0
$421$	31.7019	1.54506	0.772528	−	0.634981i	$-0.218992\pi$
$421$	31.7019	1.54506	0.772528	+	0.634981i	$0.218992\pi$
$422$	0	0
$423$	−9.59030	−0.466296
$424$	0	0
$425$	4.15656	0.201623
$426$	0	0
$427$	−0.343862	−0.0166407
$428$	0	0
$429$	−0.275011	−0.0132777
$430$	0	0
$431$	16.4025	0.790078	0.395039	−	0.918664i	$-0.370731\pi$
$431$	16.4025	0.790078	0.395039	+	0.918664i	$0.370731\pi$
$432$	0	0
$433$	6.66978	0.320529	0.160265	−	0.987074i	$-0.448765\pi$
$433$	6.66978	0.320529	0.160265	+	0.987074i	$0.448765\pi$
$434$	0	0
$435$	89.7261	4.30204
$436$	0	0
$437$	−1.87063	−0.0894843
$438$	0	0
$439$	−27.1049	−1.29365	−0.646824	−	0.762639i	$-0.723903\pi$
$439$	−27.1049	−1.29365	−0.646824	+	0.762639i	$0.723903\pi$
$440$	0	0
$441$	1.25212	0.0596246
$442$	0	0
$443$	−15.0669	−0.715851	−0.357926	−	0.933750i	$-0.616516\pi$
$443$	−15.0669	−0.715851	−0.357926	+	0.933750i	$0.616516\pi$
$444$	0	0
$445$	−21.0222	−0.996548
$446$	0	0
$447$	−56.5244	−2.67351
$448$	0	0
$449$	31.9354	1.50713	0.753563	−	0.657375i	$-0.228333\pi$
$449$	31.9354	1.50713	0.753563	+	0.657375i	$0.228333\pi$
$450$	0	0
$451$	1.46423	0.0689478
$452$	0	0
$453$	−30.7129	−1.44302
$454$	0	0
$455$	0.500939	0.0234844
$456$	0	0
$457$	−21.4393	−1.00289	−0.501444	−	0.865190i	$-0.667198\pi$
$457$	−21.4393	−1.00289	−0.501444	+	0.865190i	$0.667198\pi$
$458$	0	0
$459$	−59.5145	−2.77790
$460$	0	0
$461$	34.7856	1.62013	0.810064	−	0.586341i	$-0.199432\pi$
$461$	34.7856	1.62013	0.810064	+	0.586341i	$0.199432\pi$
$462$	0	0
$463$	3.72853	0.173280	0.0866398	−	0.996240i	$-0.472387\pi$
$463$	3.72853	0.173280	0.0866398	+	0.996240i	$0.472387\pi$
$464$	0	0
$465$	−1.66402	−0.0771671
$466$	0	0
$467$	−36.6483	−1.69588	−0.847941	−	0.530090i	$-0.822158\pi$
$467$	−36.6483	−1.69588	−0.847941	+	0.530090i	$0.822158\pi$
$468$	0	0
$469$	14.6066	0.674472
$470$	0	0
$471$	29.0263	1.33746
$472$	0	0
$473$	−3.82326	−0.175794
$474$	0	0
$475$	7.28874	0.334430
$476$	0	0
$477$	−64.6780	−2.96140
$478$	0	0
$479$	−41.2764	−1.88597	−0.942983	−	0.332842i	$-0.891993\pi$
$479$	−41.2764	−1.88597	−0.942983	+	0.332842i	$0.891993\pi$
$480$	0	0
$481$	−0.472319	−0.0215359
$482$	0	0
$483$	−3.27996	−0.149243
$484$	0	0
$485$	8.93045	0.405511
$486$	0	0
$487$	3.43645	0.155720	0.0778602	−	0.996964i	$-0.475191\pi$
$487$	3.43645	0.155720	0.0778602	+	0.996964i	$0.475191\pi$
$488$	0	0
$489$	13.1643	0.595311
$490$	0	0
$491$	−26.3038	−1.18707	−0.593536	−	0.804807i	$-0.702269\pi$
$491$	−26.3038	−1.18707	−0.593536	+	0.804807i	$0.702269\pi$
$492$	0	0
$493$	−30.8474	−1.38930
$494$	0	0
$495$	−24.0280	−1.07998
$496$	0	0
$497$	−22.6832	−1.01748
$498$	0	0
$499$	−19.1646	−0.857927	−0.428963	−	0.903322i	$-0.641121\pi$
$499$	−19.1646	−0.857927	−0.428963	+	0.903322i	$0.641121\pi$
$500$	0	0
$501$	−41.7771	−1.86646
$502$	0	0
$503$	−21.3388	−0.951448	−0.475724	−	0.879595i	$-0.657814\pi$
$503$	−21.3388	−0.951448	−0.475724	+	0.879595i	$0.657814\pi$
$504$	0	0
$505$	−9.22872	−0.410673
$506$	0	0
$507$	44.6409	1.98257
$508$	0	0
$509$	−34.4582	−1.52733	−0.763667	−	0.645611i	$-0.776603\pi$
$509$	−34.4582	−1.52733	−0.763667	+	0.645611i	$0.776603\pi$
$510$	0	0
$511$	15.9212	0.704313
$512$	0	0
$513$	−104.362	−4.60768
$514$	0	0
$515$	−7.63417	−0.336402
$516$	0	0
$517$	1.17650	0.0517425
$518$	0	0
$519$	58.7931	2.58073
$520$	0	0
$521$	21.1688	0.927422	0.463711	−	0.885986i	$-0.346518\pi$
$521$	21.1688	0.927422	0.463711	+	0.885986i	$0.346518\pi$
$522$	0	0
$523$	−41.6267	−1.82021	−0.910105	−	0.414378i	$-0.863999\pi$
$523$	−41.6267	−1.82021	−0.910105	+	0.414378i	$0.863999\pi$
$524$	0	0
$525$	12.7801	0.557768
$526$	0	0
$527$	0.572082	0.0249203
$528$	0	0
$529$	−22.8724	−0.994451
$530$	0	0
$531$	−12.3011	−0.533824
$532$	0	0
$533$	0.100537	0.00435476
$534$	0	0
$535$	−38.7913	−1.67709
$536$	0	0
$537$	42.7709	1.84570
$538$	0	0
$539$	−0.153605	−0.00661623
$540$	0	0
$541$	−38.6250	−1.66062	−0.830309	−	0.557303i	$-0.811836\pi$
$541$	−38.6250	−1.66062	−0.830309	+	0.557303i	$0.811836\pi$
$542$	0	0
$543$	−69.2720	−2.97275
$544$	0	0
$545$	24.0684	1.03098
$546$	0	0
$547$	−17.2538	−0.737718	−0.368859	−	0.929485i	$-0.620251\pi$
$547$	−17.2538	−0.737718	−0.368859	+	0.929485i	$0.620251\pi$
$548$	0	0
$549$	1.13249	0.0483336
$550$	0	0
$551$	−54.0925	−2.30442
$552$	0	0
$553$	−37.0434	−1.57525
$554$	0	0
$555$	−55.3324	−2.34873
$556$	0	0
$557$	−2.64059	−0.111885	−0.0559426	−	0.998434i	$-0.517816\pi$
$557$	−2.64059	−0.111885	−0.0559426	+	0.998434i	$0.517816\pi$
$558$	0	0
$559$	−0.262514	−0.0111032
$560$	0	0
$561$	11.0763	0.467641
$562$	0	0
$563$	4.08187	0.172030	0.0860151	−	0.996294i	$-0.472587\pi$
$563$	4.08187	0.172030	0.0860151	+	0.996294i	$0.472587\pi$
$564$	0	0
$565$	19.7569	0.831179
$566$	0	0
$567$	−112.420	−4.72119
$568$	0	0
$569$	34.3729	1.44099	0.720493	−	0.693462i	$-0.243915\pi$
$569$	34.3729	1.44099	0.720493	+	0.693462i	$0.243915\pi$
$570$	0	0
$571$	−17.3164	−0.724670	−0.362335	−	0.932048i	$-0.618020\pi$
$571$	−17.3164	−0.724670	−0.362335	+	0.932048i	$0.618020\pi$
$572$	0	0
$573$	26.1743	1.09345
$574$	0	0
$575$	−0.497302	−0.0207389
$576$	0	0
$577$	−9.38251	−0.390599	−0.195300	−	0.980744i	$-0.562568\pi$
$577$	−9.38251	−0.390599	−0.195300	+	0.980744i	$0.562568\pi$
$578$	0	0
$579$	−74.4193	−3.09276
$580$	0	0
$581$	23.2937	0.966384
$582$	0	0
$583$	7.93446	0.328612
$584$	0	0
$585$	−1.64982	−0.0682115
$586$	0	0
$587$	20.5311	0.847409	0.423704	−	0.905801i	$-0.360730\pi$
$587$	20.5311	0.847409	0.423704	+	0.905801i	$0.360730\pi$
$588$	0	0
$589$	1.00318	0.0413351
$590$	0	0
$591$	61.3141	2.52212
$592$	0	0
$593$	−18.9900	−0.779826	−0.389913	−	0.920852i	$-0.627495\pi$
$593$	−18.9900	−0.779826	−0.389913	+	0.920852i	$0.627495\pi$
$594$	0	0
$595$	−20.1757	−0.827121
$596$	0	0
$597$	−20.1965	−0.826586
$598$	0	0
$599$	29.1078	1.18931	0.594656	−	0.803981i	$-0.297288\pi$
$599$	29.1078	1.18931	0.594656	+	0.803981i	$0.297288\pi$
$600$	0	0
$601$	−31.6350	−1.29042	−0.645210	−	0.764006i	$-0.723230\pi$
$601$	−31.6350	−1.29042	−0.645210	+	0.764006i	$0.723230\pi$
$602$	0	0
$603$	−48.1062	−1.95904
$604$	0	0
$605$	−24.8630	−1.01083
$606$	0	0
$607$	25.3820	1.03022	0.515112	−	0.857123i	$-0.327750\pi$
$607$	25.3820	1.03022	0.515112	+	0.857123i	$0.327750\pi$
$608$	0	0
$609$	−94.8457	−3.84334
$610$	0	0
$611$	0.0807814	0.00326807
$612$	0	0
$613$	8.71186	0.351869	0.175934	−	0.984402i	$-0.443705\pi$
$613$	8.71186	0.351869	0.175934	+	0.984402i	$0.443705\pi$
$614$	0	0
$615$	11.7780	0.474934
$616$	0	0
$617$	−16.1031	−0.648287	−0.324143	−	0.946008i	$-0.605076\pi$
$617$	−16.1031	−0.648287	−0.324143	+	0.946008i	$0.605076\pi$
$618$	0	0
$619$	−4.67699	−0.187984	−0.0939920	−	0.995573i	$-0.529963\pi$
$619$	−4.67699	−0.187984	−0.0939920	+	0.995573i	$0.529963\pi$
$620$	0	0
$621$	7.12048	0.285735
$622$	0	0
$623$	22.2217	0.890294
$624$	0	0
$625$	−30.0224	−1.20089
$626$	0	0
$627$	19.4228	0.775673
$628$	0	0
$629$	19.0230	0.758496
$630$	0	0
$631$	1.80517	0.0718627	0.0359314	−	0.999354i	$-0.488560\pi$
$631$	1.80517	0.0718627	0.0359314	+	0.999354i	$0.488560\pi$
$632$	0	0
$633$	79.3595	3.15426
$634$	0	0
$635$	8.78844	0.348759
$636$	0	0
$637$	−0.0105469	−0.000417883 0
$638$	0	0
$639$	74.7060	2.95532
$640$	0	0
$641$	−10.3446	−0.408585	−0.204293	−	0.978910i	$-0.565489\pi$
$641$	−10.3446	−0.408585	−0.204293	+	0.978910i	$0.565489\pi$
$642$	0	0
$643$	24.8360	0.979437	0.489719	−	0.871881i	$-0.337100\pi$
$643$	24.8360	0.979437	0.489719	+	0.871881i	$0.337100\pi$
$644$	0	0
$645$	−30.7536	−1.21092
$646$	0	0
$647$	−19.4849	−0.766031	−0.383015	−	0.923742i	$-0.625114\pi$
$647$	−19.4849	−0.766031	−0.383015	+	0.923742i	$0.625114\pi$
$648$	0	0
$649$	1.50906	0.0592358
$650$	0	0
$651$	1.75897	0.0689394
$652$	0	0
$653$	−4.02381	−0.157464	−0.0787318	−	0.996896i	$-0.525087\pi$
$653$	−4.02381	−0.157464	−0.0787318	+	0.996896i	$0.525087\pi$
$654$	0	0
$655$	−8.70126	−0.339986
$656$	0	0
$657$	−52.4357	−2.04571
$658$	0	0
$659$	33.6892	1.31234	0.656172	−	0.754611i	$-0.272175\pi$
$659$	33.6892	1.31234	0.656172	+	0.754611i	$0.272175\pi$
$660$	0	0
$661$	43.9382	1.70900	0.854499	−	0.519453i	$-0.173864\pi$
$661$	43.9382	1.70900	0.854499	+	0.519453i	$0.173864\pi$
$662$	0	0
$663$	0.760524	0.0295363
$664$	0	0
$665$	−35.3791	−1.37194
$666$	0	0
$667$	3.69067	0.142903
$668$	0	0
$669$	−17.2330	−0.666266
$670$	0	0
$671$	−0.138930	−0.00536333
$672$	0	0
$673$	−26.1182	−1.00678	−0.503392	−	0.864058i	$-0.667915\pi$
$673$	−26.1182	−1.00678	−0.503392	+	0.864058i	$0.667915\pi$
$674$	0	0
$675$	−27.7443	−1.06788
$676$	0	0
$677$	−22.7181	−0.873129	−0.436564	−	0.899673i	$-0.643805\pi$
$677$	−22.7181	−0.873129	−0.436564	+	0.899673i	$0.643805\pi$
$678$	0	0
$679$	−9.44000	−0.362274
$680$	0	0
$681$	−60.2440	−2.30856
$682$	0	0
$683$	−18.2486	−0.698263	−0.349131	−	0.937074i	$-0.613523\pi$
$683$	−18.2486	−0.698263	−0.349131	+	0.937074i	$0.613523\pi$
$684$	0	0
$685$	−0.140553	−0.00537024
$686$	0	0
$687$	12.7992	0.488320
$688$	0	0
$689$	0.544799	0.0207552
$690$	0	0
$691$	20.3590	0.774493	0.387247	−	0.921976i	$-0.373426\pi$
$691$	20.3590	0.774493	0.387247	+	0.921976i	$0.373426\pi$
$692$	0	0
$693$	25.3990	0.964826
$694$	0	0
$695$	55.9642	2.12284
$696$	0	0
$697$	−4.04921	−0.153375
$698$	0	0
$699$	−23.3922	−0.884775
$700$	0	0
$701$	−17.8188	−0.673005	−0.336503	−	0.941683i	$-0.609244\pi$
$701$	−17.8188	−0.673005	−0.336503	+	0.941683i	$0.609244\pi$
$702$	0	0
$703$	33.3578	1.25811
$704$	0	0
$705$	9.46357	0.356419
$706$	0	0
$707$	9.75529	0.366886
$708$	0	0
$709$	−17.4801	−0.656478	−0.328239	−	0.944595i	$-0.606455\pi$
$709$	−17.4801	−0.656478	−0.328239	+	0.944595i	$0.606455\pi$
$710$	0	0
$711$	122.001	4.57538
$712$	0	0
$713$	−0.0684455	−0.00256330
$714$	0	0
$715$	0.202393	0.00756909
$716$	0	0
$717$	−97.6932	−3.64842
$718$	0	0
$719$	−23.2104	−0.865602	−0.432801	−	0.901490i	$-0.642475\pi$
$719$	−23.2104	−0.865602	−0.432801	+	0.901490i	$0.642475\pi$
$720$	0	0
$721$	8.06976	0.300534
$722$	0	0
$723$	−97.7271	−3.63451
$724$	0	0
$725$	−14.3804	−0.534073
$726$	0	0
$727$	18.4540	0.684423	0.342211	−	0.939623i	$-0.388824\pi$
$727$	18.4540	0.684423	0.342211	+	0.939623i	$0.388824\pi$
$728$	0	0
$729$	164.837	6.10509
$730$	0	0
$731$	10.5729	0.391054
$732$	0	0
$733$	18.7375	0.692086	0.346043	−	0.938219i	$-0.387525\pi$
$733$	18.7375	0.692086	0.346043	+	0.938219i	$0.387525\pi$
$734$	0	0
$735$	−1.23557	−0.0455747
$736$	0	0
$737$	5.90150	0.217384
$738$	0	0
$739$	−40.4600	−1.48835	−0.744173	−	0.667987i	$-0.767156\pi$
$739$	−40.4600	−1.48835	−0.744173	+	0.667987i	$0.767156\pi$
$740$	0	0
$741$	1.33362	0.0489917
$742$	0	0
$743$	−39.5617	−1.45138	−0.725690	−	0.688022i	$-0.758479\pi$
$743$	−39.5617	−1.45138	−0.725690	+	0.688022i	$0.758479\pi$
$744$	0	0
$745$	41.5989	1.52406
$746$	0	0
$747$	−76.7165	−2.80691
$748$	0	0
$749$	41.0047	1.49828
$750$	0	0
$751$	−49.2947	−1.79879	−0.899394	−	0.437139i	$-0.855992\pi$
$751$	−49.2947	−1.79879	−0.899394	+	0.437139i	$0.855992\pi$
$752$	0	0
$753$	−3.43537	−0.125192
$754$	0	0
$755$	22.6030	0.822609
$756$	0	0
$757$	−11.7893	−0.428489	−0.214244	−	0.976780i	$-0.568729\pi$
$757$	−11.7893	−0.428489	−0.214244	+	0.976780i	$0.568729\pi$
$758$	0	0
$759$	−1.32520	−0.0481016
$760$	0	0
$761$	−35.8370	−1.29909	−0.649545	−	0.760323i	$-0.725041\pi$
$761$	−35.8370	−1.29909	−0.649545	+	0.760323i	$0.725041\pi$
$762$	0	0
$763$	−25.4417	−0.921052
$764$	0	0
$765$	66.4475	2.40241
$766$	0	0
$767$	0.103616	0.00374134
$768$	0	0
$769$	−36.6683	−1.32229	−0.661146	−	0.750258i	$-0.729929\pi$
$769$	−36.6683	−1.32229	−0.661146	+	0.750258i	$0.729929\pi$
$770$	0	0
$771$	−8.79686	−0.316811
$772$	0	0
$773$	−27.5717	−0.991685	−0.495842	−	0.868413i	$-0.665141\pi$
$773$	−27.5717	−0.991685	−0.495842	+	0.868413i	$0.665141\pi$
$774$	0	0
$775$	0.266692	0.00957985
$776$	0	0
$777$	58.4895	2.09830
$778$	0	0
$779$	−7.10050	−0.254402
$780$	0	0
$781$	−9.16465	−0.327937
$782$	0	0
$783$	205.901	7.35830
$784$	0	0
$785$	−21.3618	−0.762435
$786$	0	0
$787$	−27.1094	−0.966347	−0.483174	−	0.875525i	$-0.660516\pi$
$787$	−27.1094	−0.966347	−0.483174	+	0.875525i	$0.660516\pi$
$788$	0	0
$789$	53.9745	1.92154
$790$	0	0
$791$	−20.8842	−0.742557
$792$	0	0
$793$	−0.00953927	−0.000338749 0
$794$	0	0
$795$	63.8234	2.26358
$796$	0	0
$797$	41.4290	1.46749	0.733745	−	0.679425i	$-0.237771\pi$
$797$	41.4290	1.46749	0.733745	+	0.679425i	$0.237771\pi$
$798$	0	0
$799$	−3.25353	−0.115102
$800$	0	0
$801$	−73.1861	−2.58590
$802$	0	0
$803$	6.43262	0.227002
$804$	0	0
$805$	2.41387	0.0850778
$806$	0	0
$807$	4.25321	0.149720
$808$	0	0
$809$	−3.78227	−0.132977	−0.0664887	−	0.997787i	$-0.521180\pi$
$809$	−3.78227	−0.132977	−0.0664887	+	0.997787i	$0.521180\pi$
$810$	0	0
$811$	−43.1888	−1.51656	−0.758281	−	0.651927i	$-0.773961\pi$
$811$	−43.1888	−1.51656	−0.758281	+	0.651927i	$0.773961\pi$
$812$	0	0
$813$	72.0780	2.52788
$814$	0	0
$815$	−9.68822	−0.339364
$816$	0	0
$817$	18.5402	0.648639
$818$	0	0
$819$	1.74395	0.0609387
$820$	0	0
$821$	3.69958	0.129116	0.0645581	−	0.997914i	$-0.479436\pi$
$821$	3.69958	0.129116	0.0645581	+	0.997914i	$0.479436\pi$
$822$	0	0
$823$	−2.97072	−0.103553	−0.0517763	−	0.998659i	$-0.516488\pi$
$823$	−2.97072	−0.103553	−0.0517763	+	0.998659i	$0.516488\pi$
$824$	0	0
$825$	5.16351	0.179770
$826$	0	0
$827$	1.53073	0.0532288	0.0266144	−	0.999646i	$-0.491527\pi$
$827$	1.53073	0.0532288	0.0266144	+	0.999646i	$0.491527\pi$
$828$	0	0
$829$	−9.79044	−0.340036	−0.170018	−	0.985441i	$-0.554383\pi$
$829$	−9.79044	−0.340036	−0.170018	+	0.985441i	$0.554383\pi$
$830$	0	0
$831$	44.8265	1.55501
$832$	0	0
$833$	0.424783	0.0147179
$834$	0	0
$835$	30.7456	1.06400
$836$	0	0
$837$	−3.81855	−0.131988
$838$	0	0
$839$	12.1216	0.418486	0.209243	−	0.977864i	$-0.432900\pi$
$839$	12.1216	0.418486	0.209243	+	0.977864i	$0.432900\pi$
$840$	0	0
$841$	77.7219	2.68007
$842$	0	0
$843$	46.0935	1.58754
$844$	0	0
$845$	−32.8533	−1.13019
$846$	0	0
$847$	26.2817	0.903048
$848$	0	0
$849$	−8.85344	−0.303849
$850$	0	0
$851$	−2.27596	−0.0780190
$852$	0	0
$853$	17.2971	0.592242	0.296121	−	0.955150i	$-0.404307\pi$
$853$	17.2971	0.592242	0.296121	+	0.955150i	$0.404307\pi$
$854$	0	0
$855$	116.519	3.98487
$856$	0	0
$857$	51.0962	1.74541	0.872706	−	0.488246i	$-0.162363\pi$
$857$	51.0962	1.74541	0.872706	+	0.488246i	$0.162363\pi$
$858$	0	0
$859$	−11.8729	−0.405097	−0.202548	−	0.979272i	$-0.564922\pi$
$859$	−11.8729	−0.405097	−0.202548	+	0.979272i	$0.564922\pi$
$860$	0	0
$861$	−12.4500	−0.424296
$862$	0	0
$863$	−16.5405	−0.563046	−0.281523	−	0.959554i	$-0.590840\pi$
$863$	−16.5405	−0.563046	−0.281523	+	0.959554i	$0.590840\pi$
$864$	0	0
$865$	−43.2685	−1.47117
$866$	0	0
$867$	27.7706	0.943140
$868$	0	0
$869$	−14.9666	−0.507706
$870$	0	0
$871$	0.405211	0.0137300
$872$	0	0
$873$	31.0902	1.05224
$874$	0	0
$875$	24.3782	0.824134
$876$	0	0
$877$	−14.6946	−0.496202	−0.248101	−	0.968734i	$-0.579807\pi$
$877$	−14.6946	−0.496202	−0.248101	+	0.968734i	$0.579807\pi$
$878$	0	0
$879$	75.4499	2.54486
$880$	0	0
$881$	29.3761	0.989706	0.494853	−	0.868977i	$-0.335222\pi$
$881$	29.3761	0.989706	0.494853	+	0.868977i	$0.335222\pi$
$882$	0	0
$883$	13.8415	0.465805	0.232902	−	0.972500i	$-0.425178\pi$
$883$	13.8415	0.465805	0.232902	+	0.972500i	$0.425178\pi$
$884$	0	0
$885$	12.1386	0.408035
$886$	0	0
$887$	−30.9969	−1.04077	−0.520386	−	0.853931i	$-0.674212\pi$
$887$	−30.9969	−1.04077	−0.520386	+	0.853931i	$0.674212\pi$
$888$	0	0
$889$	−9.28990	−0.311573
$890$	0	0
$891$	−45.4208	−1.52165
$892$	0	0
$893$	−5.70523	−0.190918
$894$	0	0
$895$	−31.4771	−1.05216
$896$	0	0
$897$	−0.0909912	−0.00303811
$898$	0	0
$899$	−1.97922	−0.0660107
$900$	0	0
$901$	−21.9421	−0.730999
$902$	0	0
$903$	32.5084	1.08181
$904$	0	0
$905$	50.9805	1.69465
$906$	0	0
$907$	19.9302	0.661771	0.330885	−	0.943671i	$-0.392653\pi$
$907$	19.9302	0.661771	0.330885	+	0.943671i	$0.392653\pi$
$908$	0	0
$909$	−32.1286	−1.06564
$910$	0	0
$911$	−31.0122	−1.02748	−0.513740	−	0.857946i	$-0.671740\pi$
$911$	−31.0122	−1.02748	−0.513740	+	0.857946i	$0.671740\pi$
$912$	0	0
$913$	9.41129	0.311468
$914$	0	0
$915$	−1.11753	−0.0369443
$916$	0	0
$917$	9.19774	0.303736
$918$	0	0
$919$	33.5593	1.10702	0.553510	−	0.832842i	$-0.313288\pi$
$919$	33.5593	1.10702	0.553510	+	0.832842i	$0.313288\pi$
$920$	0	0
$921$	−16.6453	−0.548480
$922$	0	0
$923$	−0.629267	−0.0207126
$924$	0	0
$925$	8.86809	0.291581
$926$	0	0
$927$	−26.5774	−0.872915
$928$	0	0
$929$	40.8161	1.33913	0.669566	−	0.742753i	$-0.266480\pi$
$929$	40.8161	1.33913	0.669566	+	0.742753i	$0.266480\pi$
$930$	0	0
$931$	0.744879	0.0244124
$932$	0	0
$933$	−62.5509	−2.04782
$934$	0	0
$935$	−8.15153	−0.266584
$936$	0	0
$937$	−30.2926	−0.989615	−0.494808	−	0.869003i	$-0.664761\pi$
$937$	−30.2926	−0.989615	−0.494808	+	0.869003i	$0.664761\pi$
$938$	0	0
$939$	−84.6089	−2.76111
$940$	0	0
$941$	11.9704	0.390223	0.195112	−	0.980781i	$-0.437493\pi$
$941$	11.9704	0.390223	0.195112	+	0.980781i	$0.437493\pi$
$942$	0	0
$943$	0.484459	0.0157762
$944$	0	0
$945$	134.669	4.38079
$946$	0	0
$947$	15.7971	0.513336	0.256668	−	0.966500i	$-0.417375\pi$
$947$	15.7971	0.513336	0.256668	+	0.966500i	$0.417375\pi$
$948$	0	0
$949$	0.441679	0.0143375
$950$	0	0
$951$	41.8019	1.35552
$952$	0	0
$953$	−44.2772	−1.43428	−0.717139	−	0.696930i	$-0.754549\pi$
$953$	−44.2772	−1.43428	−0.717139	+	0.696930i	$0.754549\pi$
$954$	0	0
$955$	−19.2628	−0.623331
$956$	0	0
$957$	−38.3203	−1.23872
$958$	0	0
$959$	0.148572	0.00479765
$960$	0	0
$961$	−30.9633	−0.998816
$962$	0	0
$963$	−135.047	−4.35182
$964$	0	0
$965$	54.7686	1.76306
$966$	0	0
$967$	56.9939	1.83280	0.916400	−	0.400265i	$-0.131082\pi$
$967$	56.9939	1.83280	0.916400	+	0.400265i	$0.131082\pi$
$968$	0	0
$969$	−53.7123	−1.72549
$970$	0	0
$971$	−14.9656	−0.480268	−0.240134	−	0.970740i	$-0.577191\pi$
$971$	−14.9656	−0.480268	−0.240134	+	0.970740i	$0.577191\pi$
$972$	0	0
$973$	−59.1574	−1.89650
$974$	0	0
$975$	0.354539	0.0113543
$976$	0	0
$977$	−28.5800	−0.914354	−0.457177	−	0.889376i	$-0.651139\pi$
$977$	−28.5800	−0.914354	−0.457177	+	0.889376i	$0.651139\pi$
$978$	0	0
$979$	8.97819	0.286944
$980$	0	0
$981$	83.7910	2.67524
$982$	0	0
$983$	21.1641	0.675029	0.337515	−	0.941320i	$-0.390414\pi$
$983$	21.1641	0.675029	0.337515	+	0.941320i	$0.390414\pi$
$984$	0	0
$985$	−45.1238	−1.43776
$986$	0	0
$987$	−10.0035	−0.318416
$988$	0	0
$989$	−1.26498	−0.0402239
$990$	0	0
$991$	47.3326	1.50357	0.751785	−	0.659409i	$-0.229193\pi$
$991$	47.3326	1.50357	0.751785	+	0.659409i	$0.229193\pi$
$992$	0	0
$993$	−12.8472	−0.407693
$994$	0	0
$995$	14.8635	0.471204
$996$	0	0
$997$	26.8937	0.851734	0.425867	−	0.904786i	$-0.359969\pi$
$997$	26.8937	0.851734	0.425867	+	0.904786i	$0.359969\pi$
$998$	0	0
$999$	−126.975	−4.01732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.g.1.1	✓	30
4.3	odd	2		8032.2.a.j.1.30	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.1	✓	30	1.1	even	1	trivial
8032.2.a.j.1.30	yes	30	4.3	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-3.43537 q^{3} +2.52824 q^{5} -2.67250 q^{7} +8.80174 q^{9} +O(q^{10})\)	\(q-3.43537 q^{3} +2.52824 q^{5} -2.67250 q^{7} +8.80174 q^{9} -1.07977 q^{11} -0.0741393 q^{13} -8.68544 q^{15} +2.98601 q^{17} +5.23612 q^{19} +9.18102 q^{21} -0.357255 q^{23} +1.39201 q^{25} -19.9311 q^{27} -10.3306 q^{29} +0.191587 q^{31} +3.70939 q^{33} -6.75673 q^{35} +6.37070 q^{37} +0.254696 q^{39} -1.35606 q^{41} +3.54082 q^{43} +22.2529 q^{45} -1.08959 q^{47} +0.142258 q^{49} -10.2580 q^{51} -7.34832 q^{53} -2.72991 q^{55} -17.9880 q^{57} -1.39758 q^{59} +0.128667 q^{61} -23.5227 q^{63} -0.187442 q^{65} -5.46554 q^{67} +1.22730 q^{69} +8.48764 q^{71} -5.95742 q^{73} -4.78207 q^{75} +2.88567 q^{77} +13.8610 q^{79} +42.0655 q^{81} -8.71606 q^{83} +7.54936 q^{85} +35.4895 q^{87} -8.31495 q^{89} +0.198137 q^{91} -0.658173 q^{93} +13.2382 q^{95} +3.53227 q^{97} -9.50382 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9} - 31 q^{11} + 5 q^{13} - 8 q^{15} - q^{17} - 29 q^{19} + 6 q^{21} - 27 q^{23} + 34 q^{25} - 36 q^{27} + q^{29} + 9 q^{31} - 8 q^{33} - 51 q^{35} + 5 q^{37} + 8 q^{39} - 3 q^{41} - 43 q^{43} - 42 q^{47} + 41 q^{49} - 54 q^{51} - 11 q^{53} + 10 q^{55} + 2 q^{57} - 49 q^{59} + 21 q^{61} - 15 q^{63} - 14 q^{65} - 68 q^{67} - 10 q^{69} - 44 q^{71} + 25 q^{73} - 51 q^{75} - 20 q^{77} + 49 q^{79} + 6 q^{81} - 88 q^{83} - 5 q^{85} - 16 q^{87} - 16 q^{89} - 46 q^{91} - 4 q^{93} - 28 q^{95} - 4 q^{97} - 71 q^{99}+O(q^{100}) \) 30 * q - 9 * q^3 - 5 * q^7 + 35 * q^9 - 31 * q^11 + 5 * q^13 - 8 * q^15 - q^17 - 29 * q^19 + 6 * q^21 - 27 * q^23 + 34 * q^25 - 36 * q^27 + q^29 + 9 * q^31 - 8 * q^33 - 51 * q^35 + 5 * q^37 + 8 * q^39 - 3 * q^41 - 43 * q^43 - 42 * q^47 + 41 * q^49 - 54 * q^51 - 11 * q^53 + 10 * q^55 + 2 * q^57 - 49 * q^59 + 21 * q^61 - 15 * q^63 - 14 * q^65 - 68 * q^67 - 10 * q^69 - 44 * q^71 + 25 * q^73 - 51 * q^75 - 20 * q^77 + 49 * q^79 + 6 * q^81 - 88 * q^83 - 5 * q^85 - 16 * q^87 - 16 * q^89 - 46 * q^91 - 4 * q^93 - 28 * q^95 - 4 * q^97 - 71 * q^99

Introduction

L-functions

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