LMFDB - Embedded newform 4732.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(1,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, 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("4732.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25492.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} - 2x + 8 $ x^4 - 8x^2 - 2x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.77129$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.77129 q^{3} +2.32791 q^{5} +1.00000 q^{7} +4.68002 q^{9} -0.908738 q^{11} +6.45131 q^{15} -1.93294 q^{17} +4.19046 q^{19} +2.77129 q^{21} +1.41917 q^{23} +0.419174 q^{25} +4.65582 q^{27} -7.75502 q^{29} -3.87048 q^{31} -2.51837 q^{33} +2.32791 q^{35} +11.9939 q^{37} +6.19839 q^{41} +6.91667 q^{43} +10.8947 q^{45} +3.16959 q^{47} +1.00000 q^{49} -5.35672 q^{51} +10.7792 q^{53} -2.11546 q^{55} +11.6130 q^{57} +4.97580 q^{59} -0.0483985 q^{61} +4.68002 q^{63} -3.76908 q^{67} +3.93294 q^{69} +15.3381 q^{71} -7.16959 q^{73} +1.16165 q^{75} -0.908738 q^{77} -17.3460 q^{79} -1.13745 q^{81} -12.1173 q^{83} -4.49971 q^{85} -21.4914 q^{87} +11.7814 q^{89} -10.7262 q^{93} +9.75502 q^{95} +6.37298 q^{97} -4.25291 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{17} + 3 q^{19} + 3 q^{23} - q^{25} + 6 q^{27} - q^{29} + 13 q^{31} + 10 q^{33} + 3 q^{35} - 10 q^{41} + 3 q^{43} + 13 q^{45} - 3 q^{47} + 4 q^{49} + 4 q^{51} + 11 q^{53}+ \cdots - 26 q^{99}+O(q^{100}) $ 4 * q + 3 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^17 + 3 * q^19 + 3 * q^23 - q^25 + 6 * q^27 - q^29 + 13 * q^31 + 10 * q^33 + 3 * q^35 - 10 * q^41 + 3 * q^43 + 13 * q^45 - 3 * q^47 + 4 * q^49 + 4 * q^51 + 11 * q^53 - 10 * q^55 + 26 * q^57 + 22 * q^59 + 4 * q^61 + 4 * q^63 - 12 * q^67 + 10 * q^69 + 26 * q^71 - 13 * q^73 + 10 * q^75 - 13 * q^79 - 12 * q^81 + 19 * q^83 + 12 * q^85 - 16 * q^87 + 7 * q^89 - 32 * q^93 + 9 * q^95 + 19 * q^97 - 26 * q^99

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
$3$	2.77129	1.60000	0.800001	−	0.599998i	$-0.204832\pi$
$3$	2.77129	1.60000	0.800001	+	0.599998i	$0.204832\pi$
$4$	0	0
$5$	2.32791	1.04107	0.520537	−	0.853839i	$-0.325732\pi$
$5$	2.32791	1.04107	0.520537	+	0.853839i	$0.325732\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.68002	1.56001
$10$	0	0
$11$	−0.908738	−0.273995	−0.136997	−	0.990571i	$-0.543745\pi$
$11$	−0.908738	−0.273995	−0.136997	+	0.990571i	$0.543745\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	6.45131	1.66572
$16$	0	0
$17$	−1.93294	−0.468806	−0.234403	−	0.972140i	$-0.575314\pi$
$17$	−1.93294	−0.468806	−0.234403	+	0.972140i	$0.575314\pi$
$18$	0	0
$19$	4.19046	0.961357	0.480679	−	0.876897i	$-0.340390\pi$
$19$	4.19046	0.961357	0.480679	+	0.876897i	$0.340390\pi$
$20$	0	0
$21$	2.77129	0.604744
$22$	0	0
$23$	1.41917	0.295918	0.147959	−	0.988993i	$-0.452730\pi$
$23$	1.41917	0.295918	0.147959	+	0.988993i	$0.452730\pi$
$24$	0	0
$25$	0.419174	0.0838348
$26$	0	0
$27$	4.65582	0.896014
$28$	0	0
$29$	−7.75502	−1.44007	−0.720036	−	0.693937i	$-0.755875\pi$
$29$	−7.75502	−1.44007	−0.720036	+	0.693937i	$0.755875\pi$
$30$	0	0
$31$	−3.87048	−0.695159	−0.347580	−	0.937650i	$-0.612996\pi$
$31$	−3.87048	−0.695159	−0.347580	+	0.937650i	$0.612996\pi$
$32$	0	0
$33$	−2.51837	−0.438392
$34$	0	0
$35$	2.32791	0.393489
$36$	0	0
$37$	11.9939	1.97178	0.985891	−	0.167390i	$-0.0535338\pi$
$37$	11.9939	1.97178	0.985891	+	0.167390i	$0.0535338\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.19839	0.968027	0.484013	−	0.875061i	$-0.339179\pi$
$41$	6.19839	0.968027	0.484013	+	0.875061i	$0.339179\pi$
$42$	0	0
$43$	6.91667	1.05478	0.527391	−	0.849622i	$-0.323170\pi$
$43$	6.91667	1.05478	0.527391	+	0.849622i	$0.323170\pi$
$44$	0	0
$45$	10.8947	1.62408
$46$	0	0
$47$	3.16959	0.462332	0.231166	−	0.972914i	$-0.425746\pi$
$47$	3.16959	0.462332	0.231166	+	0.972914i	$0.425746\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.35672	−0.750091
$52$	0	0
$53$	10.7792	1.48064	0.740320	−	0.672255i	$-0.234674\pi$
$53$	10.7792	1.48064	0.740320	+	0.672255i	$0.234674\pi$
$54$	0	0
$55$	−2.11546	−0.285249
$56$	0	0
$57$	11.6130	1.53817
$58$	0	0
$59$	4.97580	0.647794	0.323897	−	0.946092i	$-0.395007\pi$
$59$	4.97580	0.647794	0.323897	+	0.946092i	$0.395007\pi$
$60$	0	0
$61$	−0.0483985	−0.00619679	−0.00309840	−	0.999995i	$-0.500986\pi$
$61$	−0.0483985	−0.00619679	−0.00309840	+	0.999995i	$0.500986\pi$
$62$	0	0
$63$	4.68002	0.589628
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.76908	−0.460466	−0.230233	−	0.973136i	$-0.573949\pi$
$67$	−3.76908	−0.460466	−0.230233	+	0.973136i	$0.573949\pi$
$68$	0	0
$69$	3.93294	0.473470
$70$	0	0
$71$	15.3381	1.82029	0.910146	−	0.414287i	$-0.135969\pi$
$71$	15.3381	1.82029	0.910146	+	0.414287i	$0.135969\pi$
$72$	0	0
$73$	−7.16959	−0.839137	−0.419568	−	0.907724i	$-0.637819\pi$
$73$	−7.16959	−0.839137	−0.419568	+	0.907724i	$0.637819\pi$
$74$	0	0
$75$	1.16165	0.134136
$76$	0	0
$77$	−0.908738	−0.103560
$78$	0	0
$79$	−17.3460	−1.95158	−0.975788	−	0.218717i	$-0.929813\pi$
$79$	−17.3460	−1.95158	−0.975788	+	0.218717i	$0.929813\pi$
$80$	0	0
$81$	−1.13745	−0.126384
$82$	0	0
$83$	−12.1173	−1.33004	−0.665022	−	0.746824i	$-0.731578\pi$
$83$	−12.1173	−1.33004	−0.665022	+	0.746824i	$0.731578\pi$
$84$	0	0
$85$	−4.49971	−0.488062
$86$	0	0
$87$	−21.4914	−2.30412
$88$	0	0
$89$	11.7814	1.24883	0.624414	−	0.781093i	$-0.285338\pi$
$89$	11.7814	1.24883	0.624414	+	0.781093i	$0.285338\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−10.7262	−1.11226
$94$	0	0
$95$	9.75502	1.00084
$96$	0	0
$97$	6.37298	0.647079	0.323539	−	0.946215i	$-0.395127\pi$
$97$	6.37298	0.647079	0.323539	+	0.946215i	$0.395127\pi$
$98$	0	0
$99$	−4.25291	−0.427434
$100$	0	0
$101$	−7.60963	−0.757187	−0.378593	−	0.925563i	$-0.623592\pi$
$101$	−7.60963	−0.757187	−0.378593	+	0.925563i	$0.623592\pi$
$102$	0	0
$103$	3.72510	0.367045	0.183522	−	0.983016i	$-0.441250\pi$
$103$	3.72510	0.367045	0.183522	+	0.983016i	$0.441250\pi$
$104$	0	0
$105$	6.45131	0.629583
$106$	0	0
$107$	−8.94769	−0.865006	−0.432503	−	0.901633i	$-0.642369\pi$
$107$	−8.94769	−0.865006	−0.432503	+	0.901633i	$0.642369\pi$
$108$	0	0
$109$	−0.268784	−0.0257449	−0.0128724	−	0.999917i	$-0.504098\pi$
$109$	−0.268784	−0.0257449	−0.0128724	+	0.999917i	$0.504098\pi$
$110$	0	0
$111$	33.2385	3.15486
$112$	0	0
$113$	12.1718	1.14503	0.572513	−	0.819896i	$-0.305969\pi$
$113$	12.1718	1.14503	0.572513	+	0.819896i	$0.305969\pi$
$114$	0	0
$115$	3.30371	0.308073
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.93294	−0.177192
$120$	0	0
$121$	−10.1742	−0.924927
$122$	0	0
$123$	17.1775	1.54884
$124$	0	0
$125$	−10.6638	−0.953796
$126$	0	0
$127$	11.1742	0.991550	0.495775	−	0.868451i	$-0.334884\pi$
$127$	11.1742	0.991550	0.495775	+	0.868451i	$0.334884\pi$
$128$	0	0
$129$	19.1681	1.68765
$130$	0	0
$131$	1.54257	0.134775	0.0673875	−	0.997727i	$-0.478534\pi$
$131$	1.54257	0.134775	0.0673875	+	0.997727i	$0.478534\pi$
$132$	0	0
$133$	4.19046	0.363359
$134$	0	0
$135$	10.8383	0.932817
$136$	0	0
$137$	−15.2571	−1.30350	−0.651752	−	0.758432i	$-0.725966\pi$
$137$	−15.2571	−1.30350	−0.651752	+	0.758432i	$0.725966\pi$
$138$	0	0
$139$	9.49138	0.805048	0.402524	−	0.915409i	$-0.368133\pi$
$139$	9.49138	0.805048	0.402524	+	0.915409i	$0.368133\pi$
$140$	0	0
$141$	8.78383	0.739732
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−18.0530	−1.49922
$146$	0	0
$147$	2.77129	0.228572
$148$	0	0
$149$	19.1071	1.56532	0.782659	−	0.622451i	$-0.213863\pi$
$149$	19.1071	1.56532	0.782659	+	0.622451i	$0.213863\pi$
$150$	0	0
$151$	−18.0423	−1.46826	−0.734130	−	0.679009i	$-0.762410\pi$
$151$	−18.0423	−1.46826	−0.734130	+	0.679009i	$0.762410\pi$
$152$	0	0
$153$	−9.04619	−0.731341
$154$	0	0
$155$	−9.01014	−0.723712
$156$	0	0
$157$	−17.7179	−1.41404	−0.707021	−	0.707193i	$-0.749961\pi$
$157$	−17.7179	−1.41404	−0.707021	+	0.707193i	$0.749961\pi$
$158$	0	0
$159$	29.8723	2.36903
$160$	0	0
$161$	1.41917	0.111847
$162$	0	0
$163$	1.14578	0.0897445	0.0448722	−	0.998993i	$-0.485712\pi$
$163$	1.14578	0.0897445	0.0448722	+	0.998993i	$0.485712\pi$
$164$	0	0
$165$	−5.86255	−0.456399
$166$	0	0
$167$	1.10532	0.0855321	0.0427660	−	0.999085i	$-0.486383\pi$
$167$	1.10532	0.0855321	0.0427660	+	0.999085i	$0.486383\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	19.6114	1.49972
$172$	0	0
$173$	−18.6046	−1.41448	−0.707242	−	0.706972i	$-0.750061\pi$
$173$	−18.6046	−1.41448	−0.707242	+	0.706972i	$0.750061\pi$
$174$	0	0
$175$	0.419174	0.0316866
$176$	0	0
$177$	13.7894	1.03647
$178$	0	0
$179$	−8.44337	−0.631087	−0.315544	−	0.948911i	$-0.602187\pi$
$179$	−8.44337	−0.631087	−0.315544	+	0.948911i	$0.602187\pi$
$180$	0	0
$181$	−19.3506	−1.43832	−0.719159	−	0.694845i	$-0.755473\pi$
$181$	−19.3506	−1.43832	−0.719159	+	0.694845i	$0.755473\pi$
$182$	0	0
$183$	−0.134126	−0.00991488
$184$	0	0
$185$	27.9207	2.05277
$186$	0	0
$187$	1.75653	0.128450
$188$	0	0
$189$	4.65582	0.338661
$190$	0	0
$191$	11.6360	0.841954	0.420977	−	0.907071i	$-0.361687\pi$
$191$	11.6360	0.841954	0.420977	+	0.907071i	$0.361687\pi$
$192$	0	0
$193$	−16.8058	−1.20971	−0.604855	−	0.796336i	$-0.706769\pi$
$193$	−16.8058	−1.20971	−0.604855	+	0.796336i	$0.706769\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.56344	−0.467626	−0.233813	−	0.972282i	$-0.575120\pi$
$197$	−6.56344	−0.467626	−0.233813	+	0.972282i	$0.575120\pi$
$198$	0	0
$199$	−26.1472	−1.85353	−0.926763	−	0.375647i	$-0.877421\pi$
$199$	−26.1472	−1.85353	−0.926763	+	0.375647i	$0.877421\pi$
$200$	0	0
$201$	−10.4452	−0.736747
$202$	0	0
$203$	−7.75502	−0.544296
$204$	0	0
$205$	14.4293	1.00779
$206$	0	0
$207$	6.64177	0.461635
$208$	0	0
$209$	−3.80803	−0.263407
$210$	0	0
$211$	1.82820	0.125859	0.0629294	−	0.998018i	$-0.479956\pi$
$211$	1.82820	0.125859	0.0629294	+	0.998018i	$0.479956\pi$
$212$	0	0
$213$	42.5061	2.91247
$214$	0	0
$215$	16.1014	1.09811
$216$	0	0
$217$	−3.87048	−0.262746
$218$	0	0
$219$	−19.8690	−1.34262
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.98815	−0.601891	−0.300946	−	0.953641i	$-0.597302\pi$
$223$	−8.98815	−0.601891	−0.300946	+	0.953641i	$0.597302\pi$
$224$	0	0
$225$	1.96175	0.130783
$226$	0	0
$227$	−10.4452	−0.693272	−0.346636	−	0.938000i	$-0.612676\pi$
$227$	−10.4452	−0.693272	−0.346636	+	0.938000i	$0.612676\pi$
$228$	0	0
$229$	20.3517	1.34488	0.672440	−	0.740152i	$-0.265246\pi$
$229$	20.3517	1.34488	0.672440	+	0.740152i	$0.265246\pi$
$230$	0	0
$231$	−2.51837	−0.165697
$232$	0	0
$233$	−22.6621	−1.48464	−0.742320	−	0.670045i	$-0.766275\pi$
$233$	−22.6621	−1.48464	−0.742320	+	0.670045i	$0.766275\pi$
$234$	0	0
$235$	7.37852	0.481322
$236$	0	0
$237$	−48.0707	−3.12253
$238$	0	0
$239$	20.0159	1.29472	0.647360	−	0.762185i	$-0.275873\pi$
$239$	20.0159	1.29472	0.647360	+	0.762185i	$0.275873\pi$
$240$	0	0
$241$	21.8380	1.40671	0.703353	−	0.710841i	$-0.251685\pi$
$241$	21.8380	1.40671	0.703353	+	0.710841i	$0.251685\pi$
$242$	0	0
$243$	−17.1197	−1.09823
$244$	0	0
$245$	2.32791	0.148725
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−33.5804	−2.12807
$250$	0	0
$251$	10.7713	0.679878	0.339939	−	0.940448i	$-0.389594\pi$
$251$	10.7713	0.679878	0.339939	+	0.940448i	$0.389594\pi$
$252$	0	0
$253$	−1.28966	−0.0810801
$254$	0	0
$255$	−12.4700	−0.780900
$256$	0	0
$257$	−17.3089	−1.07970	−0.539848	−	0.841762i	$-0.681518\pi$
$257$	−17.3089	−1.07970	−0.539848	+	0.841762i	$0.681518\pi$
$258$	0	0
$259$	11.9939	0.745263
$260$	0	0
$261$	−36.2937	−2.24652
$262$	0	0
$263$	−5.37410	−0.331381	−0.165691	−	0.986178i	$-0.552985\pi$
$263$	−5.37410	−0.331381	−0.165691	+	0.986178i	$0.552985\pi$
$264$	0	0
$265$	25.0931	1.54146
$266$	0	0
$267$	32.6497	1.99813
$268$	0	0
$269$	−17.0851	−1.04170	−0.520850	−	0.853648i	$-0.674385\pi$
$269$	−17.0851	−1.04170	−0.520850	+	0.853648i	$0.674385\pi$
$270$	0	0
$271$	−18.4452	−1.12047	−0.560233	−	0.828335i	$-0.689288\pi$
$271$	−18.4452	−1.12047	−0.560233	+	0.828335i	$0.689288\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.380919	−0.0229703
$276$	0	0
$277$	8.35765	0.502162	0.251081	−	0.967966i	$-0.419214\pi$
$277$	8.35765	0.502162	0.251081	+	0.967966i	$0.419214\pi$
$278$	0	0
$279$	−18.1140	−1.08445
$280$	0	0
$281$	−9.28966	−0.554174	−0.277087	−	0.960845i	$-0.589369\pi$
$281$	−9.28966	−0.554174	−0.277087	+	0.960845i	$0.589369\pi$
$282$	0	0
$283$	−1.91427	−0.113792	−0.0568958	−	0.998380i	$-0.518120\pi$
$283$	−1.91427	−0.113792	−0.0568958	+	0.998380i	$0.518120\pi$
$284$	0	0
$285$	27.0339	1.60135
$286$	0	0
$287$	6.19839	0.365880
$288$	0	0
$289$	−13.2638	−0.780221
$290$	0	0
$291$	17.6614	1.03533
$292$	0	0
$293$	−11.6473	−0.680443	−0.340221	−	0.940345i	$-0.610502\pi$
$293$	−11.6473	−0.680443	−0.340221	+	0.940345i	$0.610502\pi$
$294$	0	0
$295$	11.5832	0.674402
$296$	0	0
$297$	−4.23092	−0.245503
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.91667	0.398670
$302$	0	0
$303$	−21.0885	−1.21150
$304$	0	0
$305$	−0.112667	−0.00645132
$306$	0	0
$307$	−5.53131	−0.315689	−0.157844	−	0.987464i	$-0.550454\pi$
$307$	−5.53131	−0.315689	−0.157844	+	0.987464i	$0.550454\pi$
$308$	0	0
$309$	10.3233	0.587272
$310$	0	0
$311$	19.2490	1.09151	0.545755	−	0.837945i	$-0.316243\pi$
$311$	19.2490	1.09151	0.545755	+	0.837945i	$0.316243\pi$
$312$	0	0
$313$	20.8932	1.18095	0.590476	−	0.807055i	$-0.298940\pi$
$313$	20.8932	1.18095	0.590476	+	0.807055i	$0.298940\pi$
$314$	0	0
$315$	10.8947	0.613846
$316$	0	0
$317$	6.93515	0.389517	0.194758	−	0.980851i	$-0.437608\pi$
$317$	6.93515	0.389517	0.194758	+	0.980851i	$0.437608\pi$
$318$	0	0
$319$	7.04728	0.394572
$320$	0	0
$321$	−24.7966	−1.38401
$322$	0	0
$323$	−8.09989	−0.450690
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.744878	−0.0411918
$328$	0	0
$329$	3.16959	0.174745
$330$	0	0
$331$	−28.6436	−1.57439	−0.787197	−	0.616702i	$-0.788468\pi$
$331$	−28.6436	−1.57439	−0.787197	+	0.616702i	$0.788468\pi$
$332$	0	0
$333$	56.1316	3.07599
$334$	0	0
$335$	−8.77408	−0.479379
$336$	0	0
$337$	10.1234	0.551457	0.275728	−	0.961236i	$-0.411081\pi$
$337$	10.1234	0.551457	0.275728	+	0.961236i	$0.411081\pi$
$338$	0	0
$339$	33.7315	1.83204
$340$	0	0
$341$	3.51725	0.190470
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	9.15553	0.492917
$346$	0	0
$347$	−6.18252	−0.331895	−0.165948	−	0.986135i	$-0.553068\pi$
$347$	−6.18252	−0.331895	−0.165948	+	0.986135i	$0.553068\pi$
$348$	0	0
$349$	18.1738	0.972821	0.486411	−	0.873730i	$-0.338306\pi$
$349$	18.1738	0.972821	0.486411	+	0.873730i	$0.338306\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28.2638	−1.50433	−0.752164	−	0.658976i	$-0.770990\pi$
$353$	−28.2638	−1.50433	−0.752164	+	0.658976i	$0.770990\pi$
$354$	0	0
$355$	35.7056	1.89506
$356$	0	0
$357$	−5.35672	−0.283508
$358$	0	0
$359$	−26.7949	−1.41418	−0.707090	−	0.707123i	$-0.749993\pi$
$359$	−26.7949	−1.41418	−0.707090	+	0.707123i	$0.749993\pi$
$360$	0	0
$361$	−1.44005	−0.0757920
$362$	0	0
$363$	−28.1956	−1.47989
$364$	0	0
$365$	−16.6902	−0.873603
$366$	0	0
$367$	14.8932	0.777417	0.388709	−	0.921361i	$-0.372921\pi$
$367$	14.8932	0.777417	0.388709	+	0.921361i	$0.372921\pi$
$368$	0	0
$369$	29.0086	1.51013
$370$	0	0
$371$	10.7792	0.559629
$372$	0	0
$373$	26.5760	1.37605	0.688026	−	0.725686i	$-0.258477\pi$
$373$	26.5760	1.37605	0.688026	+	0.725686i	$0.258477\pi$
$374$	0	0
$375$	−29.5523	−1.52608
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−0.349711	−0.0179634	−0.00898172	−	0.999960i	$-0.502859\pi$
$379$	−0.349711	−0.0179634	−0.00898172	+	0.999960i	$0.502859\pi$
$380$	0	0
$381$	30.9669	1.58648
$382$	0	0
$383$	28.3968	1.45101	0.725504	−	0.688218i	$-0.241607\pi$
$383$	28.3968	1.45101	0.725504	+	0.688218i	$0.241607\pi$
$384$	0	0
$385$	−2.11546	−0.107814
$386$	0	0
$387$	32.3702	1.64547
$388$	0	0
$389$	6.14187	0.311405	0.155703	−	0.987804i	$-0.450236\pi$
$389$	6.14187	0.311405	0.155703	+	0.987804i	$0.450236\pi$
$390$	0	0
$391$	−2.74317	−0.138728
$392$	0	0
$393$	4.27490	0.215640
$394$	0	0
$395$	−40.3799	−2.03174
$396$	0	0
$397$	−7.30371	−0.366563	−0.183281	−	0.983060i	$-0.558672\pi$
$397$	−7.30371	−0.366563	−0.183281	+	0.983060i	$0.558672\pi$
$398$	0	0
$399$	11.6130	0.581375
$400$	0	0
$401$	−6.84500	−0.341823	−0.170912	−	0.985286i	$-0.554671\pi$
$401$	−6.84500	−0.341823	−0.170912	+	0.985286i	$0.554671\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.64789	−0.131575
$406$	0	0
$407$	−10.8993	−0.540258
$408$	0	0
$409$	−3.16293	−0.156397	−0.0781985	−	0.996938i	$-0.524917\pi$
$409$	−3.16293	−0.156397	−0.0781985	+	0.996938i	$0.524917\pi$
$410$	0	0
$411$	−42.2819	−2.08561
$412$	0	0
$413$	4.97580	0.244843
$414$	0	0
$415$	−28.2080	−1.38467
$416$	0	0
$417$	26.3033	1.28808
$418$	0	0
$419$	−27.1522	−1.32647	−0.663236	−	0.748410i	$-0.730817\pi$
$419$	−27.1522	−1.32647	−0.663236	+	0.748410i	$0.730817\pi$
$420$	0	0
$421$	−26.6238	−1.29757	−0.648783	−	0.760973i	$-0.724722\pi$
$421$	−26.6238	−1.29757	−0.648783	+	0.760973i	$0.724722\pi$
$422$	0	0
$423$	14.8337	0.721241
$424$	0	0
$425$	−0.810237	−0.0393023
$426$	0	0
$427$	−0.0483985	−0.00234217
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.5468	−0.797031	−0.398515	−	0.917162i	$-0.630474\pi$
$431$	−16.5468	−0.797031	−0.398515	+	0.917162i	$0.630474\pi$
$432$	0	0
$433$	−10.9994	−0.528598	−0.264299	−	0.964441i	$-0.585141\pi$
$433$	−10.9994	−0.528598	−0.264299	+	0.964441i	$0.585141\pi$
$434$	0	0
$435$	−50.0300	−2.39876
$436$	0	0
$437$	5.94699	0.284483
$438$	0	0
$439$	25.6024	1.22194	0.610968	−	0.791655i	$-0.290780\pi$
$439$	25.6024	1.22194	0.610968	+	0.791655i	$0.290780\pi$
$440$	0	0
$441$	4.68002	0.222858
$442$	0	0
$443$	−22.4278	−1.06558	−0.532789	−	0.846248i	$-0.678856\pi$
$443$	−22.4278	−1.06558	−0.532789	+	0.846248i	$0.678856\pi$
$444$	0	0
$445$	27.4261	1.30012
$446$	0	0
$447$	52.9513	2.50451
$448$	0	0
$449$	−4.49918	−0.212329	−0.106165	−	0.994349i	$-0.533857\pi$
$449$	−4.49918	−0.212329	−0.106165	+	0.994349i	$0.533857\pi$
$450$	0	0
$451$	−5.63272	−0.265234
$452$	0	0
$453$	−50.0003	−2.34922
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−40.1439	−1.87785	−0.938926	−	0.344120i	$-0.888177\pi$
$457$	−40.1439	−1.87785	−0.938926	+	0.344120i	$0.888177\pi$
$458$	0	0
$459$	−8.99941	−0.420057
$460$	0	0
$461$	9.66748	0.450259	0.225130	−	0.974329i	$-0.427719\pi$
$461$	9.66748	0.450259	0.225130	+	0.974329i	$0.427719\pi$
$462$	0	0
$463$	−22.7513	−1.05734	−0.528671	−	0.848827i	$-0.677310\pi$
$463$	−22.7513	−1.05734	−0.528671	+	0.848827i	$0.677310\pi$
$464$	0	0
$465$	−24.9697	−1.15794
$466$	0	0
$467$	24.8074	1.14795	0.573976	−	0.818872i	$-0.305400\pi$
$467$	24.8074	1.14795	0.573976	+	0.818872i	$0.305400\pi$
$468$	0	0
$469$	−3.76908	−0.174040
$470$	0	0
$471$	−49.1013	−2.26247
$472$	0	0
$473$	−6.28544	−0.289005
$474$	0	0
$475$	1.75653	0.0805953
$476$	0	0
$477$	50.4470	2.30981
$478$	0	0
$479$	40.3922	1.84557	0.922783	−	0.385320i	$-0.125909\pi$
$479$	40.3922	1.84557	0.922783	+	0.385320i	$0.125909\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.93294	0.178955
$484$	0	0
$485$	14.8357	0.673657
$486$	0	0
$487$	10.5793	0.479394	0.239697	−	0.970848i	$-0.422952\pi$
$487$	10.5793	0.479394	0.239697	+	0.970848i	$0.422952\pi$
$488$	0	0
$489$	3.17529	0.143591
$490$	0	0
$491$	13.8186	0.623623	0.311812	−	0.950144i	$-0.399064\pi$
$491$	13.8186	0.623623	0.311812	+	0.950144i	$0.399064\pi$
$492$	0	0
$493$	14.9900	0.675114
$494$	0	0
$495$	−9.90041	−0.444990
$496$	0	0
$497$	15.3381	0.688006
$498$	0	0
$499$	−12.8366	−0.574647	−0.287324	−	0.957834i	$-0.592766\pi$
$499$	−12.8366	−0.574647	−0.287324	+	0.957834i	$0.592766\pi$
$500$	0	0
$501$	3.06315	0.136852
$502$	0	0
$503$	−19.3882	−0.864475	−0.432238	−	0.901760i	$-0.642276\pi$
$503$	−19.3882	−0.864475	−0.432238	+	0.901760i	$0.642276\pi$
$504$	0	0
$505$	−17.7146	−0.788287
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.23386	−0.409283	−0.204642	−	0.978837i	$-0.565603\pi$
$509$	−9.23386	−0.409283	−0.204642	+	0.978837i	$0.565603\pi$
$510$	0	0
$511$	−7.16959	−0.317164
$512$	0	0
$513$	19.5100	0.861389
$514$	0	0
$515$	8.67169	0.382121
$516$	0	0
$517$	−2.88032	−0.126676
$518$	0	0
$519$	−51.5587	−2.26318
$520$	0	0
$521$	39.5902	1.73448	0.867239	−	0.497893i	$-0.165893\pi$
$521$	39.5902	1.73448	0.867239	+	0.497893i	$0.165893\pi$
$522$	0	0
$523$	30.3920	1.32895	0.664474	−	0.747311i	$-0.268655\pi$
$523$	30.3920	1.32895	0.664474	+	0.747311i	$0.268655\pi$
$524$	0	0
$525$	1.16165	0.0506986
$526$	0	0
$527$	7.48140	0.325895
$528$	0	0
$529$	−20.9859	−0.912432
$530$	0	0
$531$	23.2869	1.01056
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−20.8294	−0.900535
$536$	0	0
$537$	−23.3990	−1.00974
$538$	0	0
$539$	−0.908738	−0.0391421
$540$	0	0
$541$	−19.2290	−0.826720	−0.413360	−	0.910568i	$-0.635645\pi$
$541$	−19.2290	−0.826720	−0.413360	+	0.910568i	$0.635645\pi$
$542$	0	0
$543$	−53.6260	−2.30131
$544$	0	0
$545$	−0.625706	−0.0268023
$546$	0	0
$547$	−8.33845	−0.356526	−0.178263	−	0.983983i	$-0.557048\pi$
$547$	−8.33845	−0.356526	−0.178263	+	0.983983i	$0.557048\pi$
$548$	0	0
$549$	−0.226506	−0.00966704
$550$	0	0
$551$	−32.4971	−1.38442
$552$	0	0
$553$	−17.3460	−0.737627
$554$	0	0
$555$	77.3762	3.28444
$556$	0	0
$557$	37.2621	1.57884	0.789422	−	0.613850i	$-0.210380\pi$
$557$	37.2621	1.57884	0.789422	+	0.613850i	$0.210380\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	4.86785	0.205521
$562$	0	0
$563$	9.02753	0.380465	0.190232	−	0.981739i	$-0.439076\pi$
$563$	9.02753	0.380465	0.190232	+	0.981739i	$0.439076\pi$
$564$	0	0
$565$	28.3349	1.19206
$566$	0	0
$567$	−1.13745	−0.0477685
$568$	0	0
$569$	38.8560	1.62893	0.814465	−	0.580213i	$-0.197031\pi$
$569$	38.8560	1.62893	0.814465	+	0.580213i	$0.197031\pi$
$570$	0	0
$571$	13.4509	0.562903	0.281452	−	0.959575i	$-0.409184\pi$
$571$	13.4509	0.562903	0.281452	+	0.959575i	$0.409184\pi$
$572$	0	0
$573$	32.2468	1.34713
$574$	0	0
$575$	0.594881	0.0248083
$576$	0	0
$577$	11.5167	0.479446	0.239723	−	0.970841i	$-0.422943\pi$
$577$	11.5167	0.479446	0.239723	+	0.970841i	$0.422943\pi$
$578$	0	0
$579$	−46.5737	−1.93554
$580$	0	0
$581$	−12.1173	−0.502709
$582$	0	0
$583$	−9.79548	−0.405688
$584$	0	0
$585$	0	0
$586$	0	0
$587$	46.4498	1.91719	0.958594	−	0.284776i	$-0.0919191\pi$
$587$	46.4498	1.91719	0.958594	+	0.284776i	$0.0919191\pi$
$588$	0	0
$589$	−16.2191	−0.668296
$590$	0	0
$591$	−18.1892	−0.748203
$592$	0	0
$593$	8.51044	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$593$	8.51044	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$594$	0	0
$595$	−4.49971	−0.184470
$596$	0	0
$597$	−72.4614	−2.96565
$598$	0	0
$599$	17.9952	0.735262	0.367631	−	0.929972i	$-0.380169\pi$
$599$	17.9952	0.735262	0.367631	+	0.929972i	$0.380169\pi$
$600$	0	0
$601$	−10.0809	−0.411210	−0.205605	−	0.978635i	$-0.565916\pi$
$601$	−10.0809	−0.411210	−0.205605	+	0.978635i	$0.565916\pi$
$602$	0	0
$603$	−17.6394	−0.718331
$604$	0	0
$605$	−23.6846	−0.962917
$606$	0	0
$607$	39.8239	1.61640	0.808201	−	0.588907i	$-0.200441\pi$
$607$	39.8239	1.61640	0.808201	+	0.588907i	$0.200441\pi$
$608$	0	0
$609$	−21.4914	−0.870875
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−6.13854	−0.247933	−0.123967	−	0.992286i	$-0.539562\pi$
$613$	−6.13854	−0.247933	−0.123967	+	0.992286i	$0.539562\pi$
$614$	0	0
$615$	39.9878	1.61246
$616$	0	0
$617$	28.3968	1.14321	0.571606	−	0.820528i	$-0.306321\pi$
$617$	28.3968	1.14321	0.571606	+	0.820528i	$0.306321\pi$
$618$	0	0
$619$	−9.95160	−0.399989	−0.199994	−	0.979797i	$-0.564092\pi$
$619$	−9.95160	−0.399989	−0.199994	+	0.979797i	$0.564092\pi$
$620$	0	0
$621$	6.60743	0.265147
$622$	0	0
$623$	11.7814	0.472013
$624$	0	0
$625$	−26.9202	−1.07681
$626$	0	0
$627$	−10.5531	−0.421452
$628$	0	0
$629$	−23.1834	−0.924383
$630$	0	0
$631$	46.4171	1.84783	0.923917	−	0.382592i	$-0.124969\pi$
$631$	46.4171	1.84783	0.923917	+	0.382592i	$0.124969\pi$
$632$	0	0
$633$	5.06648	0.201374
$634$	0	0
$635$	26.0125	1.03228
$636$	0	0
$637$	0	0
$638$	0	0
$639$	71.7825	2.83967
$640$	0	0
$641$	19.2052	0.758560	0.379280	−	0.925282i	$-0.376172\pi$
$641$	19.2052	0.758560	0.379280	+	0.925282i	$0.376172\pi$
$642$	0	0
$643$	7.98776	0.315006	0.157503	−	0.987518i	$-0.449656\pi$
$643$	7.98776	0.315006	0.157503	+	0.987518i	$0.449656\pi$
$644$	0	0
$645$	44.6216	1.75697
$646$	0	0
$647$	−22.5655	−0.887139	−0.443570	−	0.896240i	$-0.646288\pi$
$647$	−22.5655	−0.887139	−0.443570	+	0.896240i	$0.646288\pi$
$648$	0	0
$649$	−4.52170	−0.177492
$650$	0	0
$651$	−10.7262	−0.420393
$652$	0	0
$653$	−25.7446	−1.00746	−0.503732	−	0.863860i	$-0.668040\pi$
$653$	−25.7446	−1.00746	−0.503732	+	0.863860i	$0.668040\pi$
$654$	0	0
$655$	3.59097	0.140311
$656$	0	0
$657$	−33.5538	−1.30906
$658$	0	0
$659$	−43.3529	−1.68879	−0.844396	−	0.535720i	$-0.820040\pi$
$659$	−43.3529	−1.68879	−0.844396	+	0.535720i	$0.820040\pi$
$660$	0	0
$661$	−5.46928	−0.212730	−0.106365	−	0.994327i	$-0.533921\pi$
$661$	−5.46928	−0.212730	−0.106365	+	0.994327i	$0.533921\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.75502	0.378284
$666$	0	0
$667$	−11.0057	−0.426143
$668$	0	0
$669$	−24.9087	−0.963027
$670$	0	0
$671$	0.0439815	0.00169789
$672$	0	0
$673$	−7.17601	−0.276615	−0.138307	−	0.990389i	$-0.544166\pi$
$673$	−7.17601	−0.276615	−0.138307	+	0.990389i	$0.544166\pi$
$674$	0	0
$675$	1.95160	0.0751172
$676$	0	0
$677$	0.609048	0.0234076	0.0117038	−	0.999932i	$-0.496274\pi$
$677$	0.609048	0.0234076	0.0117038	+	0.999932i	$0.496274\pi$
$678$	0	0
$679$	6.37298	0.244573
$680$	0	0
$681$	−28.9466	−1.10924
$682$	0	0
$683$	−34.2763	−1.31155	−0.655773	−	0.754958i	$-0.727657\pi$
$683$	−34.2763	−1.31155	−0.655773	+	0.754958i	$0.727657\pi$
$684$	0	0
$685$	−35.5173	−1.35704
$686$	0	0
$687$	56.4004	2.15181
$688$	0	0
$689$	0	0
$690$	0	0
$691$	21.4049	0.814282	0.407141	−	0.913365i	$-0.366526\pi$
$691$	21.4049	0.814282	0.407141	+	0.913365i	$0.366526\pi$
$692$	0	0
$693$	−4.25291	−0.161555
$694$	0	0
$695$	22.0951	0.838115
$696$	0	0
$697$	−11.9811	−0.453817
$698$	0	0
$699$	−62.8030	−2.37543
$700$	0	0
$701$	34.1562	1.29006	0.645031	−	0.764156i	$-0.276844\pi$
$701$	34.1562	1.29006	0.645031	+	0.764156i	$0.276844\pi$
$702$	0	0
$703$	50.2599	1.89559
$704$	0	0
$705$	20.4480	0.770116
$706$	0	0
$707$	−7.60963	−0.286190
$708$	0	0
$709$	−24.6272	−0.924893	−0.462447	−	0.886647i	$-0.653028\pi$
$709$	−24.6272	−0.924893	−0.462447	+	0.886647i	$0.653028\pi$
$710$	0	0
$711$	−81.1796	−3.04447
$712$	0	0
$713$	−5.49289	−0.205710
$714$	0	0
$715$	0	0
$716$	0	0
$717$	55.4697	2.07155
$718$	0	0
$719$	18.6090	0.694000	0.347000	−	0.937865i	$-0.387200\pi$
$719$	18.6090	0.694000	0.347000	+	0.937865i	$0.387200\pi$
$720$	0	0
$721$	3.72510	0.138730
$722$	0	0
$723$	60.5192	2.25073
$724$	0	0
$725$	−3.25071	−0.120728
$726$	0	0
$727$	29.3061	1.08690	0.543451	−	0.839441i	$-0.317117\pi$
$727$	29.3061	1.08690	0.543451	+	0.839441i	$0.317117\pi$
$728$	0	0
$729$	−44.0312	−1.63078
$730$	0	0
$731$	−13.3695	−0.494489
$732$	0	0
$733$	−15.1865	−0.560928	−0.280464	−	0.959865i	$-0.590488\pi$
$733$	−15.1865	−0.560928	−0.280464	+	0.959865i	$0.590488\pi$
$734$	0	0
$735$	6.45131	0.237960
$736$	0	0
$737$	3.42510	0.126165
$738$	0	0
$739$	15.8531	0.583166	0.291583	−	0.956546i	$-0.405818\pi$
$739$	15.8531	0.583166	0.291583	+	0.956546i	$0.405818\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.0404	−0.625151	−0.312575	−	0.949893i	$-0.601192\pi$
$743$	−17.0404	−0.625151	−0.312575	+	0.949893i	$0.601192\pi$
$744$	0	0
$745$	44.4797	1.62961
$746$	0	0
$747$	−56.7091	−2.07488
$748$	0	0
$749$	−8.94769	−0.326941
$750$	0	0
$751$	12.1966	0.445060	0.222530	−	0.974926i	$-0.428569\pi$
$751$	12.1966	0.445060	0.222530	+	0.974926i	$0.428569\pi$
$752$	0	0
$753$	29.8503	1.08781
$754$	0	0
$755$	−42.0008	−1.52857
$756$	0	0
$757$	8.90080	0.323505	0.161753	−	0.986831i	$-0.448285\pi$
$757$	8.90080	0.323505	0.161753	+	0.986831i	$0.448285\pi$
$758$	0	0
$759$	−3.57401	−0.129728
$760$	0	0
$761$	24.5299	0.889210	0.444605	−	0.895727i	$-0.353344\pi$
$761$	24.5299	0.889210	0.444605	+	0.895727i	$0.353344\pi$
$762$	0	0
$763$	−0.268784	−0.00973064
$764$	0	0
$765$	−21.0587	−0.761380
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−31.7896	−1.14636	−0.573180	−	0.819429i	$-0.694291\pi$
$769$	−31.7896	−1.14636	−0.573180	+	0.819429i	$0.694291\pi$
$770$	0	0
$771$	−47.9678	−1.72752
$772$	0	0
$773$	3.29354	0.118460	0.0592302	−	0.998244i	$-0.481135\pi$
$773$	3.29354	0.118460	0.0592302	+	0.998244i	$0.481135\pi$
$774$	0	0
$775$	−1.62241	−0.0582786
$776$	0	0
$777$	33.2385	1.19242
$778$	0	0
$779$	25.9741	0.930620
$780$	0	0
$781$	−13.9383	−0.498751
$782$	0	0
$783$	−36.1060	−1.29032
$784$	0	0
$785$	−41.2457	−1.47212
$786$	0	0
$787$	−11.8265	−0.421569	−0.210785	−	0.977533i	$-0.567602\pi$
$787$	−11.8265	−0.421569	−0.210785	+	0.977533i	$0.567602\pi$
$788$	0	0
$789$	−14.8932	−0.530211
$790$	0	0
$791$	12.1718	0.432779
$792$	0	0
$793$	0	0
$794$	0	0
$795$	69.5401	2.46633
$796$	0	0
$797$	−24.5512	−0.869648	−0.434824	−	0.900515i	$-0.643189\pi$
$797$	−24.5512	−0.869648	−0.434824	+	0.900515i	$0.643189\pi$
$798$	0	0
$799$	−6.12661	−0.216744
$800$	0	0
$801$	55.1374	1.94818
$802$	0	0
$803$	6.51527	0.229919
$804$	0	0
$805$	3.30371	0.116441
$806$	0	0
$807$	−47.3478	−1.66672
$808$	0	0
$809$	20.1958	0.710046	0.355023	−	0.934858i	$-0.384473\pi$
$809$	20.1958	0.710046	0.355023	+	0.934858i	$0.384473\pi$
$810$	0	0
$811$	−42.1330	−1.47949	−0.739744	−	0.672888i	$-0.765053\pi$
$811$	−42.1330	−1.47949	−0.739744	+	0.672888i	$0.765053\pi$
$812$	0	0
$813$	−51.1169	−1.79275
$814$	0	0
$815$	2.66728	0.0934307
$816$	0	0
$817$	28.9840	1.01402
$818$	0	0
$819$	0	0
$820$	0	0
$821$	41.8862	1.46184	0.730919	−	0.682465i	$-0.239092\pi$
$821$	41.8862	1.46184	0.730919	+	0.682465i	$0.239092\pi$
$822$	0	0
$823$	−21.9316	−0.764488	−0.382244	−	0.924061i	$-0.624849\pi$
$823$	−21.9316	−0.764488	−0.382244	+	0.924061i	$0.624849\pi$
$824$	0	0
$825$	−1.05564	−0.0367525
$826$	0	0
$827$	35.5193	1.23513	0.617563	−	0.786522i	$-0.288120\pi$
$827$	35.5193	1.23513	0.617563	+	0.786522i	$0.288120\pi$
$828$	0	0
$829$	−13.8731	−0.481832	−0.240916	−	0.970546i	$-0.577448\pi$
$829$	−13.8731	−0.481832	−0.240916	+	0.970546i	$0.577448\pi$
$830$	0	0
$831$	23.1614	0.803461
$832$	0	0
$833$	−1.93294	−0.0669723
$834$	0	0
$835$	2.57308	0.0890452
$836$	0	0
$837$	−18.0203	−0.622872
$838$	0	0
$839$	3.89096	0.134331	0.0671655	−	0.997742i	$-0.478604\pi$
$839$	3.89096	0.134331	0.0671655	+	0.997742i	$0.478604\pi$
$840$	0	0
$841$	31.1404	1.07381
$842$	0	0
$843$	−25.7443	−0.886681
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.1742	−0.349590
$848$	0	0
$849$	−5.30500	−0.182067
$850$	0	0
$851$	17.0214	0.583486
$852$	0	0
$853$	22.8271	0.781585	0.390792	−	0.920479i	$-0.372201\pi$
$853$	22.8271	0.781585	0.390792	+	0.920479i	$0.372201\pi$
$854$	0	0
$855$	45.6537	1.56132
$856$	0	0
$857$	−24.7641	−0.845925	−0.422962	−	0.906147i	$-0.639010\pi$
$857$	−24.7641	−0.845925	−0.422962	+	0.906147i	$0.639010\pi$
$858$	0	0
$859$	37.0312	1.26349	0.631743	−	0.775178i	$-0.282340\pi$
$859$	37.0312	1.26349	0.631743	+	0.775178i	$0.282340\pi$
$860$	0	0
$861$	17.1775	0.585408
$862$	0	0
$863$	26.8486	0.913938	0.456969	−	0.889483i	$-0.348935\pi$
$863$	26.8486	0.913938	0.456969	+	0.889483i	$0.348935\pi$
$864$	0	0
$865$	−43.3099	−1.47258
$866$	0	0
$867$	−36.7577	−1.24836
$868$	0	0
$869$	15.7630	0.534722
$870$	0	0
$871$	0	0
$872$	0	0
$873$	29.8257	1.00945
$874$	0	0
$875$	−10.6638	−0.360501
$876$	0	0
$877$	−8.60874	−0.290697	−0.145348	−	0.989381i	$-0.546430\pi$
$877$	−8.60874	−0.290697	−0.145348	+	0.989381i	$0.546430\pi$
$878$	0	0
$879$	−32.2780	−1.08871
$880$	0	0
$881$	−0.0606391	−0.00204298	−0.00102149	−	0.999999i	$-0.500325\pi$
$881$	−0.0606391	−0.00204298	−0.00102149	+	0.999999i	$0.500325\pi$
$882$	0	0
$883$	−27.3929	−0.921844	−0.460922	−	0.887441i	$-0.652481\pi$
$883$	−27.3929	−0.921844	−0.460922	+	0.887441i	$0.652481\pi$
$884$	0	0
$885$	32.1004	1.07904
$886$	0	0
$887$	15.5259	0.521309	0.260655	−	0.965432i	$-0.416062\pi$
$887$	15.5259	0.521309	0.260655	+	0.965432i	$0.416062\pi$
$888$	0	0
$889$	11.1742	0.374770
$890$	0	0
$891$	1.03365	0.0346284
$892$	0	0
$893$	13.2820	0.444466
$894$	0	0
$895$	−19.6554	−0.657009
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.0157	1.00108
$900$	0	0
$901$	−20.8356	−0.694133
$902$	0	0
$903$	19.1681	0.637874
$904$	0	0
$905$	−45.0465	−1.49740
$906$	0	0
$907$	20.5427	0.682109	0.341055	−	0.940043i	$-0.389216\pi$
$907$	20.5427	0.682109	0.341055	+	0.940043i	$0.389216\pi$
$908$	0	0
$909$	−35.6133	−1.18122
$910$	0	0
$911$	−26.9607	−0.893246	−0.446623	−	0.894722i	$-0.647374\pi$
$911$	−26.9607	−0.893246	−0.446623	+	0.894722i	$0.647374\pi$
$912$	0	0
$913$	11.0114	0.364425
$914$	0	0
$915$	−0.312234	−0.0103221
$916$	0	0
$917$	1.54257	0.0509402
$918$	0	0
$919$	50.4079	1.66280	0.831401	−	0.555673i	$-0.187539\pi$
$919$	50.4079	1.66280	0.831401	+	0.555673i	$0.187539\pi$
$920$	0	0
$921$	−15.3288	−0.505102
$922$	0	0
$923$	0	0
$924$	0	0
$925$	5.02753	0.165304
$926$	0	0
$927$	17.4335	0.572592
$928$	0	0
$929$	−1.44117	−0.0472831	−0.0236415	−	0.999720i	$-0.507526\pi$
$929$	−1.44117	−0.0472831	−0.0236415	+	0.999720i	$0.507526\pi$
$930$	0	0
$931$	4.19046	0.137337
$932$	0	0
$933$	53.3445	1.74642
$934$	0	0
$935$	4.08905	0.133726
$936$	0	0
$937$	−5.55565	−0.181495	−0.0907475	−	0.995874i	$-0.528926\pi$
$937$	−5.55565	−0.181495	−0.0907475	+	0.995874i	$0.528926\pi$
$938$	0	0
$939$	57.9009	1.88953
$940$	0	0
$941$	−21.1131	−0.688266	−0.344133	−	0.938921i	$-0.611827\pi$
$941$	−21.1131	−0.688266	−0.344133	+	0.938921i	$0.611827\pi$
$942$	0	0
$943$	8.79660	0.286457
$944$	0	0
$945$	10.8383	0.352572
$946$	0	0
$947$	54.5512	1.77268	0.886338	−	0.463039i	$-0.153241\pi$
$947$	54.5512	1.77268	0.886338	+	0.463039i	$0.153241\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	19.2193	0.623227
$952$	0	0
$953$	−4.78814	−0.155103	−0.0775515	−	0.996988i	$-0.524710\pi$
$953$	−4.78814	−0.155103	−0.0775515	+	0.996988i	$0.524710\pi$
$954$	0	0
$955$	27.0877	0.876537
$956$	0	0
$957$	19.5300	0.631316
$958$	0	0
$959$	−15.2571	−0.492678
$960$	0	0
$961$	−16.0194	−0.516754
$962$	0	0
$963$	−41.8754	−1.34942
$964$	0	0
$965$	−39.1225	−1.25940
$966$	0	0
$967$	−15.9411	−0.512630	−0.256315	−	0.966593i	$-0.582508\pi$
$967$	−15.9411	−0.512630	−0.256315	+	0.966593i	$0.582508\pi$
$968$	0	0
$969$	−22.4471	−0.721105
$970$	0	0
$971$	7.43432	0.238579	0.119289	−	0.992860i	$-0.461938\pi$
$971$	7.43432	0.238579	0.119289	+	0.992860i	$0.461938\pi$
$972$	0	0
$973$	9.49138	0.304280
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.8212	0.410185	0.205093	−	0.978743i	$-0.434250\pi$
$977$	12.8212	0.410185	0.205093	+	0.978743i	$0.434250\pi$
$978$	0	0
$979$	−10.7062	−0.342173
$980$	0	0
$981$	−1.25792	−0.0401622
$982$	0	0
$983$	24.2873	0.774643	0.387322	−	0.921945i	$-0.373400\pi$
$983$	24.2873	0.774643	0.387322	+	0.921945i	$0.373400\pi$
$984$	0	0
$985$	−15.2791	−0.486833
$986$	0	0
$987$	8.78383	0.279592
$988$	0	0
$989$	9.81596	0.312130
$990$	0	0
$991$	−23.9734	−0.761539	−0.380769	−	0.924670i	$-0.624341\pi$
$991$	−23.9734	−0.761539	−0.380769	+	0.924670i	$0.624341\pi$
$992$	0	0
$993$	−79.3795	−2.51903
$994$	0	0
$995$	−60.8684	−1.92966
$996$	0	0
$997$	−23.3275	−0.738790	−0.369395	−	0.929272i	$-0.620435\pi$
$997$	−23.3275	−0.738790	−0.369395	+	0.929272i	$0.620435\pi$
$998$	0	0
$999$	55.8414	1.76674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.p.1.4		4
13.5	odd	4		364.2.g.a.337.7	✓	8
13.8	odd	4		364.2.g.a.337.8	yes	8
13.12	even	2		4732.2.a.o.1.4		4
39.5	even	4		3276.2.e.f.2521.6		8
39.8	even	4		3276.2.e.f.2521.3		8
52.31	even	4		1456.2.k.d.337.1		8
52.47	even	4		1456.2.k.d.337.2		8
91.5	even	12		2548.2.y.e.753.8		16
91.18	odd	12		2548.2.y.f.961.2		16
91.31	even	12		2548.2.y.e.961.7		16
91.34	even	4		2548.2.g.g.2157.1		8
91.44	odd	12		2548.2.y.f.753.1		16
91.47	even	12		2548.2.y.e.753.7		16
91.60	odd	12		2548.2.y.f.961.1		16
91.73	even	12		2548.2.y.e.961.8		16
91.83	even	4		2548.2.g.g.2157.2		8
91.86	odd	12		2548.2.y.f.753.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.g.a.337.7	✓	8	13.5	odd	4
364.2.g.a.337.8	yes	8	13.8	odd	4
1456.2.k.d.337.1		8	52.31	even	4
1456.2.k.d.337.2		8	52.47	even	4
2548.2.g.g.2157.1		8	91.34	even	4
2548.2.g.g.2157.2		8	91.83	even	4
2548.2.y.e.753.7		16	91.47	even	12
2548.2.y.e.753.8		16	91.5	even	12
2548.2.y.e.961.7		16	91.31	even	12
2548.2.y.e.961.8		16	91.73	even	12
2548.2.y.f.753.1		16	91.44	odd	12
2548.2.y.f.753.2		16	91.86	odd	12
2548.2.y.f.961.1		16	91.60	odd	12
2548.2.y.f.961.2		16	91.18	odd	12
3276.2.e.f.2521.3		8	39.8	even	4
3276.2.e.f.2521.6		8	39.5	even	4
4732.2.a.o.1.4		4	13.12	even	2
4732.2.a.p.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)

\(f(q)\)	\(=\)	\(q+2.77129 q^{3} +2.32791 q^{5} +1.00000 q^{7} +4.68002 q^{9} -0.908738 q^{11} +6.45131 q^{15} -1.93294 q^{17} +4.19046 q^{19} +2.77129 q^{21} +1.41917 q^{23} +0.419174 q^{25} +4.65582 q^{27} -7.75502 q^{29} -3.87048 q^{31} -2.51837 q^{33} +2.32791 q^{35} +11.9939 q^{37} +6.19839 q^{41} +6.91667 q^{43} +10.8947 q^{45} +3.16959 q^{47} +1.00000 q^{49} -5.35672 q^{51} +10.7792 q^{53} -2.11546 q^{55} +11.6130 q^{57} +4.97580 q^{59} -0.0483985 q^{61} +4.68002 q^{63} -3.76908 q^{67} +3.93294 q^{69} +15.3381 q^{71} -7.16959 q^{73} +1.16165 q^{75} -0.908738 q^{77} -17.3460 q^{79} -1.13745 q^{81} -12.1173 q^{83} -4.49971 q^{85} -21.4914 q^{87} +11.7814 q^{89} -10.7262 q^{93} +9.75502 q^{95} +6.37298 q^{97} -4.25291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{17} + 3 q^{19} + 3 q^{23} - q^{25} + 6 q^{27} - q^{29} + 13 q^{31} + 10 q^{33} + 3 q^{35} - 10 q^{41} + 3 q^{43} + 13 q^{45} - 3 q^{47} + 4 q^{49} + 4 q^{51} + 11 q^{53}+ \cdots - 26 q^{99}+O(q^{100}) \) 4 * q + 3 * q^5 + 4 * q^7 + 4 * q^9 - 2 * q^17 + 3 * q^19 + 3 * q^23 - q^25 + 6 * q^27 - q^29 + 13 * q^31 + 10 * q^33 + 3 * q^35 - 10 * q^41 + 3 * q^43 + 13 * q^45 - 3 * q^47 + 4 * q^49 + 4 * q^51 + 11 * q^53 - 10 * q^55 + 26 * q^57 + 22 * q^59 + 4 * q^61 + 4 * q^63 - 12 * q^67 + 10 * q^69 + 26 * q^71 - 13 * q^73 + 10 * q^75 - 13 * q^79 - 12 * q^81 + 19 * q^83 + 12 * q^85 - 16 * q^87 + 7 * q^89 - 32 * q^93 + 9 * q^95 + 19 * q^97 - 26 * q^99

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