LMFDB - Embedded newform 396.4.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [396,4,Mod(1,396)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("396.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(396, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 396 = 2^{2} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	396.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-12,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.3647563623$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.56776$ of defining polynomial
Character	$\chi$	$=$	396.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-17.1355 q^{5} -23.1355 q^{7} +11.0000 q^{11} -59.6776 q^{13} +80.8132 q^{17} -11.7289 q^{19} +56.0513 q^{23} +168.626 q^{25} +85.4579 q^{29} -99.8974 q^{31} +396.440 q^{35} +402.982 q^{37} -27.0842 q^{41} -74.0000 q^{43} +408.491 q^{47} +192.253 q^{49} -463.099 q^{53} -188.491 q^{55} +498.813 q^{59} -635.912 q^{61} +1022.61 q^{65} -701.421 q^{67} -27.7435 q^{71} +619.626 q^{73} -254.491 q^{77} -208.322 q^{79} +1300.98 q^{83} -1384.78 q^{85} +1391.18 q^{89} +1380.67 q^{91} +200.982 q^{95} +1051.76 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} - 24 q^{7} + 22 q^{11} - 8 q^{13} + 28 q^{17} - 68 q^{19} + 268 q^{23} + 70 q^{25} + 260 q^{29} + 112 q^{31} + 392 q^{35} + 316 q^{37} + 124 q^{41} - 148 q^{43} + 572 q^{47} - 150 q^{49}+ \cdots + 500 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 - 24 * q^7 + 22 * q^11 - 8 * q^13 + 28 * q^17 - 68 * q^19 + 268 * q^23 + 70 * q^25 + 260 * q^29 + 112 * q^31 + 392 * q^35 + 316 * q^37 + 124 * q^41 - 148 * q^43 + 572 * q^47 - 150 * q^49 + 76 * q^53 - 132 * q^55 + 864 * q^59 - 136 * q^61 + 1288 * q^65 - 512 * q^67 + 724 * q^71 + 972 * q^73 - 264 * q^77 - 528 * q^79 + 2112 * q^83 - 1656 * q^85 + 288 * q^89 + 1336 * q^91 - 88 * q^95 + 500 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−17.1355	−1.53265	−0.766324	−	0.642454i	$-0.777916\pi$
$5$	−17.1355	−1.53265	−0.766324	+	0.642454i	$0.777916\pi$
$6$	0	0
$7$	−23.1355	−1.24920	−0.624601	−	0.780944i	$-0.714738\pi$
$7$	−23.1355	−1.24920	−0.624601	+	0.780944i	$0.714738\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−59.6776	−1.27320	−0.636600	−	0.771194i	$-0.719660\pi$
$13$	−59.6776	−1.27320	−0.636600	+	0.771194i	$0.719660\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	80.8132	1.15295	0.576473	−	0.817116i	$-0.304429\pi$
$17$	80.8132	1.15295	0.576473	+	0.817116i	$0.304429\pi$
$18$	0	0
$19$	−11.7289	−0.141621	−0.0708106	−	0.997490i	$-0.522559\pi$
$19$	−11.7289	−0.141621	−0.0708106	+	0.997490i	$0.522559\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	56.0513	0.508152	0.254076	−	0.967184i	$-0.418229\pi$
$23$	56.0513	0.508152	0.254076	+	0.967184i	$0.418229\pi$
$24$	0	0
$25$	168.626	1.34901
$26$	0	0
$27$	0	0
$28$	0	0
$29$	85.4579	0.547211	0.273606	−	0.961842i	$-0.411784\pi$
$29$	85.4579	0.547211	0.273606	+	0.961842i	$0.411784\pi$
$30$	0	0
$31$	−99.8974	−0.578778	−0.289389	−	0.957212i	$-0.593452\pi$
$31$	−99.8974	−0.578778	−0.289389	+	0.957212i	$0.593452\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	396.440	1.91459
$36$	0	0
$37$	402.982	1.79053	0.895267	−	0.445530i	$-0.146985\pi$
$37$	402.982	1.79053	0.895267	+	0.445530i	$0.146985\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−27.0842	−0.103167	−0.0515835	−	0.998669i	$-0.516427\pi$
$41$	−27.0842	−0.103167	−0.0515835	+	0.998669i	$0.516427\pi$
$42$	0	0
$43$	−74.0000	−0.262439	−0.131220	−	0.991353i	$-0.541889\pi$
$43$	−74.0000	−0.262439	−0.131220	+	0.991353i	$0.541889\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	408.491	1.26776	0.633878	−	0.773433i	$-0.281462\pi$
$47$	408.491	1.26776	0.633878	+	0.773433i	$0.281462\pi$
$48$	0	0
$49$	192.253	0.560503
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−463.099	−1.20022	−0.600109	−	0.799919i	$-0.704876\pi$
$53$	−463.099	−1.20022	−0.600109	+	0.799919i	$0.704876\pi$
$54$	0	0
$55$	−188.491	−0.462111
$56$	0	0
$57$	0	0
$58$	0	0
$59$	498.813	1.10068	0.550339	−	0.834942i	$-0.314499\pi$
$59$	498.813	1.10068	0.550339	+	0.834942i	$0.314499\pi$
$60$	0	0
$61$	−635.912	−1.33476	−0.667379	−	0.744719i	$-0.732584\pi$
$61$	−635.912	−1.33476	−0.667379	+	0.744719i	$0.732584\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1022.61	1.95137
$66$	0	0
$67$	−701.421	−1.27899	−0.639494	−	0.768796i	$-0.720856\pi$
$67$	−701.421	−1.27899	−0.639494	+	0.768796i	$0.720856\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−27.7435	−0.0463739	−0.0231870	−	0.999731i	$-0.507381\pi$
$71$	−27.7435	−0.0463739	−0.0231870	+	0.999731i	$0.507381\pi$
$72$	0	0
$73$	619.626	0.993449	0.496725	−	0.867908i	$-0.334536\pi$
$73$	619.626	0.993449	0.496725	+	0.867908i	$0.334536\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−254.491	−0.376648
$78$	0	0
$79$	−208.322	−0.296685	−0.148342	−	0.988936i	$-0.547394\pi$
$79$	−208.322	−0.296685	−0.148342	+	0.988936i	$0.547394\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1300.98	1.72050	0.860249	−	0.509875i	$-0.170308\pi$
$83$	1300.98	1.72050	0.860249	+	0.509875i	$0.170308\pi$
$84$	0	0
$85$	−1384.78	−1.76706
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1391.18	1.65691	0.828453	−	0.560058i	$-0.189221\pi$
$89$	1391.18	1.65691	0.828453	+	0.560058i	$0.189221\pi$
$90$	0	0
$91$	1380.67	1.59048
$92$	0	0
$93$	0	0
$94$	0	0
$95$	200.982	0.217056
$96$	0	0
$97$	1051.76	1.10093	0.550463	−	0.834859i	$-0.314451\pi$
$97$	1051.76	1.10093	0.550463	+	0.834859i	$0.314451\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	775.524	0.764035	0.382017	−	0.924155i	$-0.375229\pi$
$101$	775.524	0.764035	0.382017	+	0.924155i	$0.375229\pi$
$102$	0	0
$103$	618.168	0.591359	0.295679	−	0.955287i	$-0.404454\pi$
$103$	618.168	0.591359	0.295679	+	0.955287i	$0.404454\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2169.68	−1.96029	−0.980146	−	0.198275i	$-0.936466\pi$
$107$	−2169.68	−1.96029	−0.980146	+	0.198275i	$0.936466\pi$
$108$	0	0
$109$	54.4908	0.0478832	0.0239416	−	0.999713i	$-0.492378\pi$
$109$	54.4908	0.0478832	0.0239416	+	0.999713i	$0.492378\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2090.09	−1.74000	−0.869998	−	0.493055i	$-0.835880\pi$
$113$	−2090.09	−1.74000	−0.869998	+	0.493055i	$0.835880\pi$
$114$	0	0
$115$	−960.469	−0.778819
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1869.66	−1.44026
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−747.560	−0.534911
$126$	0	0
$127$	1801.70	1.25886	0.629429	−	0.777058i	$-0.283289\pi$
$127$	1801.70	1.25886	0.629429	+	0.777058i	$0.283289\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2987.55	1.99254	0.996271	−	0.0862774i	$-0.0274971\pi$
$131$	2987.55	1.99254	0.996271	+	0.0862774i	$0.0274971\pi$
$132$	0	0
$133$	271.355	0.176913
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1848.44	−1.15272	−0.576361	−	0.817195i	$-0.695528\pi$
$137$	−1848.44	−1.15272	−0.576361	+	0.817195i	$0.695528\pi$
$138$	0	0
$139$	1857.49	1.13345	0.566727	−	0.823906i	$-0.308210\pi$
$139$	1857.49	1.13345	0.566727	+	0.823906i	$0.308210\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−656.454	−0.383884
$144$	0	0
$145$	−1464.37	−0.838683
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−923.905	−0.507982	−0.253991	−	0.967207i	$-0.581743\pi$
$149$	−923.905	−0.507982	−0.253991	+	0.967207i	$0.581743\pi$
$150$	0	0
$151$	−1945.88	−1.04870	−0.524348	−	0.851504i	$-0.675691\pi$
$151$	−1945.88	−1.04870	−0.524348	+	0.851504i	$0.675691\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1711.79	0.887062
$156$	0	0
$157$	−3034.42	−1.54251	−0.771253	−	0.636529i	$-0.780370\pi$
$157$	−3034.42	−1.54251	−0.771253	+	0.636529i	$0.780370\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1296.78	−0.634784
$162$	0	0
$163$	116.879	0.0561636	0.0280818	−	0.999606i	$-0.491060\pi$
$163$	116.879	0.0561636	0.0280818	+	0.999606i	$0.491060\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−233.495	−0.108194	−0.0540969	−	0.998536i	$-0.517228\pi$
$167$	−233.495	−0.108194	−0.0540969	+	0.998536i	$0.517228\pi$
$168$	0	0
$169$	1364.42	0.621038
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2534.21	−1.11371	−0.556856	−	0.830609i	$-0.687992\pi$
$173$	−2534.21	−1.11371	−0.556856	+	0.830609i	$0.687992\pi$
$174$	0	0
$175$	−3901.26	−1.68519
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2051.62	0.856677	0.428339	−	0.903618i	$-0.359099\pi$
$179$	2051.62	0.856677	0.428339	+	0.903618i	$0.359099\pi$
$180$	0	0
$181$	−1648.58	−0.677005	−0.338502	−	0.940965i	$-0.609920\pi$
$181$	−1648.58	−0.677005	−0.338502	+	0.940965i	$0.609920\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6905.30	−2.74426
$186$	0	0
$187$	888.945	0.347626
$188$	0	0
$189$	0	0
$190$	0	0
$191$	878.117	0.332661	0.166331	−	0.986070i	$-0.446808\pi$
$191$	878.117	0.332661	0.166331	+	0.986070i	$0.446808\pi$
$192$	0	0
$193$	−1877.90	−0.700383	−0.350192	−	0.936678i	$-0.613884\pi$
$193$	−1877.90	−0.700383	−0.350192	+	0.936678i	$0.613884\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3244.61	−1.17345	−0.586723	−	0.809788i	$-0.699582\pi$
$197$	−3244.61	−1.17345	−0.586723	+	0.809788i	$0.699582\pi$
$198$	0	0
$199$	3887.56	1.38483	0.692417	−	0.721498i	$-0.256546\pi$
$199$	3887.56	1.38483	0.692417	+	0.721498i	$0.256546\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1977.11	−0.683577
$204$	0	0
$205$	464.103	0.158119
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−129.018	−0.0427004
$210$	0	0
$211$	4567.95	1.49038	0.745191	−	0.666851i	$-0.232358\pi$
$211$	4567.95	1.49038	0.745191	+	0.666851i	$0.232358\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1268.03	0.402227
$216$	0	0
$217$	2311.18	0.723010
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4822.74	−1.46793
$222$	0	0
$223$	3885.48	1.16678	0.583388	−	0.812194i	$-0.301727\pi$
$223$	3885.48	1.16678	0.583388	+	0.812194i	$0.301727\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−985.011	−0.288006	−0.144003	−	0.989577i	$-0.545998\pi$
$227$	−985.011	−0.288006	−0.144003	+	0.989577i	$0.545998\pi$
$228$	0	0
$229$	−795.861	−0.229659	−0.114830	−	0.993385i	$-0.536632\pi$
$229$	−795.861	−0.229659	−0.114830	+	0.993385i	$0.536632\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3453.50	0.971014	0.485507	−	0.874233i	$-0.338635\pi$
$233$	3453.50	0.971014	0.485507	+	0.874233i	$0.338635\pi$
$234$	0	0
$235$	−6999.71	−1.94302
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1235.52	−0.334389	−0.167194	−	0.985924i	$-0.553471\pi$
$239$	−1235.52	−0.334389	−0.167194	+	0.985924i	$0.553471\pi$
$240$	0	0
$241$	5914.20	1.58078	0.790389	−	0.612606i	$-0.209879\pi$
$241$	5914.20	1.58078	0.790389	+	0.612606i	$0.209879\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3294.35	−0.859055
$246$	0	0
$247$	699.956	0.180312
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7353.71	1.84925	0.924626	−	0.380876i	$-0.124378\pi$
$251$	7353.71	1.84925	0.924626	+	0.380876i	$0.124378\pi$
$252$	0	0
$253$	616.564	0.153214
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1838.71	−0.446286	−0.223143	−	0.974786i	$-0.571632\pi$
$257$	−1838.71	−0.446286	−0.223143	+	0.974786i	$0.571632\pi$
$258$	0	0
$259$	−9323.19	−2.23674
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2923.71	0.685489	0.342744	−	0.939429i	$-0.388644\pi$
$263$	2923.71	0.685489	0.342744	+	0.939429i	$0.388644\pi$
$264$	0	0
$265$	7935.44	1.83951
$266$	0	0
$267$	0	0
$268$	0	0
$269$	209.517	0.0474887	0.0237444	−	0.999718i	$-0.492441\pi$
$269$	209.517	0.0474887	0.0237444	+	0.999718i	$0.492441\pi$
$270$	0	0
$271$	−1239.79	−0.277905	−0.138952	−	0.990299i	$-0.544373\pi$
$271$	−1239.79	−0.277905	−0.138952	+	0.990299i	$0.544373\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1854.89	0.406742
$276$	0	0
$277$	−1947.43	−0.422417	−0.211209	−	0.977441i	$-0.567740\pi$
$277$	−1947.43	−0.422417	−0.211209	+	0.977441i	$0.567740\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2202.79	0.467642	0.233821	−	0.972280i	$-0.424877\pi$
$281$	2202.79	0.467642	0.233821	+	0.972280i	$0.424877\pi$
$282$	0	0
$283$	3251.80	0.683037	0.341519	−	0.939875i	$-0.389059\pi$
$283$	3251.80	0.683037	0.341519	+	0.939875i	$0.389059\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	626.608	0.128876
$288$	0	0
$289$	1617.77	0.329283
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4009.12	0.799370	0.399685	−	0.916653i	$-0.369120\pi$
$293$	4009.12	0.799370	0.399685	+	0.916653i	$0.369120\pi$
$294$	0	0
$295$	−8547.43	−1.68695
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3345.01	−0.646980
$300$	0	0
$301$	1712.03	0.327840
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10896.7	2.04571
$306$	0	0
$307$	−10113.3	−1.88012	−0.940062	−	0.341004i	$-0.889233\pi$
$307$	−10113.3	−1.88012	−0.940062	+	0.341004i	$0.889233\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7289.25	1.32905	0.664527	−	0.747265i	$-0.268633\pi$
$311$	7289.25	1.32905	0.664527	+	0.747265i	$0.268633\pi$
$312$	0	0
$313$	−6692.49	−1.20857	−0.604284	−	0.796769i	$-0.706541\pi$
$313$	−6692.49	−1.20857	−0.604284	+	0.796769i	$0.706541\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5903.35	1.04595	0.522973	−	0.852349i	$-0.324823\pi$
$317$	5903.35	1.04595	0.522973	+	0.852349i	$0.324823\pi$
$318$	0	0
$319$	940.037	0.164990
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−947.853	−0.163282
$324$	0	0
$325$	−10063.2	−1.71756
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9450.65	−1.58368
$330$	0	0
$331$	10051.4	1.66911	0.834553	−	0.550928i	$-0.185726\pi$
$331$	10051.4	1.66911	0.834553	+	0.550928i	$0.185726\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12019.2	1.96024
$336$	0	0
$337$	7157.33	1.15693	0.578464	−	0.815708i	$-0.303653\pi$
$337$	7157.33	1.15693	0.578464	+	0.815708i	$0.303653\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1098.87	−0.174508
$342$	0	0
$343$	3487.62	0.549020
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7625.44	−1.17970	−0.589848	−	0.807514i	$-0.700813\pi$
$347$	−7625.44	−1.17970	−0.589848	+	0.807514i	$0.700813\pi$
$348$	0	0
$349$	6075.13	0.931789	0.465895	−	0.884840i	$-0.345733\pi$
$349$	6075.13	0.931789	0.465895	+	0.884840i	$0.345733\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3776.97	0.569484	0.284742	−	0.958604i	$-0.408092\pi$
$353$	3776.97	0.569484	0.284742	+	0.958604i	$0.408092\pi$
$354$	0	0
$355$	475.400	0.0710749
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2975.31	−0.437412	−0.218706	−	0.975791i	$-0.570184\pi$
$359$	−2975.31	−0.437412	−0.218706	+	0.975791i	$0.570184\pi$
$360$	0	0
$361$	−6721.43	−0.979943
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10617.6	−1.52261
$366$	0	0
$367$	4485.19	0.637942	0.318971	−	0.947764i	$-0.396663\pi$
$367$	4485.19	0.637942	0.318971	+	0.947764i	$0.396663\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10714.0	1.49931
$372$	0	0
$373$	−4989.98	−0.692684	−0.346342	−	0.938108i	$-0.612576\pi$
$373$	−4989.98	−0.692684	−0.346342	+	0.938108i	$0.612576\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5099.93	−0.696710
$378$	0	0
$379$	6105.92	0.827546	0.413773	−	0.910380i	$-0.364211\pi$
$379$	6105.92	0.827546	0.413773	+	0.910380i	$0.364211\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13409.8	1.78906	0.894531	−	0.447006i	$-0.147510\pi$
$383$	13409.8	1.78906	0.894531	+	0.447006i	$0.147510\pi$
$384$	0	0
$385$	4360.83	0.577269
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14748.7	1.92233	0.961166	−	0.275971i	$-0.0889995\pi$
$389$	14748.7	1.92233	0.961166	+	0.275971i	$0.0889995\pi$
$390$	0	0
$391$	4529.68	0.585872
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3569.71	0.454713
$396$	0	0
$397$	11148.0	1.40932	0.704660	−	0.709545i	$-0.251099\pi$
$397$	11148.0	1.40932	0.704660	+	0.709545i	$0.251099\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8105.61	1.00941	0.504707	−	0.863291i	$-0.331601\pi$
$401$	8105.61	1.00941	0.504707	+	0.863291i	$0.331601\pi$
$402$	0	0
$403$	5961.64	0.736900
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4432.80	0.539866
$408$	0	0
$409$	−3940.27	−0.476366	−0.238183	−	0.971220i	$-0.576552\pi$
$409$	−3940.27	−0.476366	−0.238183	+	0.971220i	$0.576552\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11540.3	−1.37497
$414$	0	0
$415$	−22293.0	−2.63692
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3979.21	−0.463955	−0.231977	−	0.972721i	$-0.574520\pi$
$419$	−3979.21	−0.463955	−0.231977	+	0.972721i	$0.574520\pi$
$420$	0	0
$421$	4480.43	0.518677	0.259338	−	0.965787i	$-0.416495\pi$
$421$	4480.43	0.518677	0.259338	+	0.965787i	$0.416495\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13627.2	1.55534
$426$	0	0
$427$	14712.2	1.66738
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−269.524	−0.0301218	−0.0150609	−	0.999887i	$-0.504794\pi$
$431$	−269.524	−0.0301218	−0.0150609	+	0.999887i	$0.504794\pi$
$432$	0	0
$433$	4813.88	0.534274	0.267137	−	0.963659i	$-0.413922\pi$
$433$	4813.88	0.534274	0.267137	+	0.963659i	$0.413922\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−657.422	−0.0719652
$438$	0	0
$439$	−1642.33	−0.178552	−0.0892758	−	0.996007i	$-0.528455\pi$
$439$	−1642.33	−0.178552	−0.0892758	+	0.996007i	$0.528455\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11791.1	1.26458	0.632291	−	0.774731i	$-0.282115\pi$
$443$	11791.1	1.26458	0.632291	+	0.774731i	$0.282115\pi$
$444$	0	0
$445$	−23838.6	−2.53946
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16120.5	1.69437	0.847184	−	0.531300i	$-0.178296\pi$
$449$	16120.5	1.69437	0.847184	+	0.531300i	$0.178296\pi$
$450$	0	0
$451$	−297.927	−0.0311060
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−23658.6	−2.43765
$456$	0	0
$457$	−16023.9	−1.64019	−0.820096	−	0.572226i	$-0.806080\pi$
$457$	−16023.9	−1.64019	−0.820096	+	0.572226i	$0.806080\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−18815.8	−1.90095	−0.950476	−	0.310799i	$-0.899403\pi$
$461$	−18815.8	−1.90095	−0.950476	+	0.310799i	$0.899403\pi$
$462$	0	0
$463$	10838.9	1.08796	0.543982	−	0.839097i	$-0.316916\pi$
$463$	10838.9	1.08796	0.543982	+	0.839097i	$0.316916\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12978.1	−1.28599	−0.642993	−	0.765872i	$-0.722308\pi$
$467$	−12978.1	−1.28599	−0.642993	+	0.765872i	$0.722308\pi$
$468$	0	0
$469$	16227.7	1.59771
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−814.000	−0.0791285
$474$	0	0
$475$	−1977.81	−0.191049
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13221.3	1.26117	0.630583	−	0.776122i	$-0.282816\pi$
$479$	13221.3	1.26117	0.630583	+	0.776122i	$0.282816\pi$
$480$	0	0
$481$	−24049.0	−2.27971
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18022.4	−1.68733
$486$	0	0
$487$	10876.2	1.01200	0.506002	−	0.862532i	$-0.331123\pi$
$487$	10876.2	1.01200	0.506002	+	0.862532i	$0.331123\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16259.9	−1.49449	−0.747247	−	0.664546i	$-0.768625\pi$
$491$	−16259.9	−1.49449	−0.747247	+	0.664546i	$0.768625\pi$
$492$	0	0
$493$	6906.12	0.630905
$494$	0	0
$495$	0	0
$496$	0	0
$497$	641.861	0.0579304
$498$	0	0
$499$	−15871.4	−1.42385	−0.711927	−	0.702253i	$-0.752177\pi$
$499$	−15871.4	−1.42385	−0.711927	+	0.702253i	$0.752177\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2755.87	−0.244290	−0.122145	−	0.992512i	$-0.538977\pi$
$503$	−2755.87	−0.244290	−0.122145	+	0.992512i	$0.538977\pi$
$504$	0	0
$505$	−13289.0	−1.17100
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−17150.8	−1.49351	−0.746756	−	0.665098i	$-0.768390\pi$
$509$	−17150.8	−1.49351	−0.746756	+	0.665098i	$0.768390\pi$
$510$	0	0
$511$	−14335.4	−1.24102
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10592.6	−0.906345
$516$	0	0
$517$	4493.40	0.382243
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3971.87	0.333994	0.166997	−	0.985957i	$-0.446593\pi$
$521$	3971.87	0.333994	0.166997	+	0.985957i	$0.446593\pi$
$522$	0	0
$523$	−10449.8	−0.873688	−0.436844	−	0.899537i	$-0.643904\pi$
$523$	−10449.8	−0.873688	−0.436844	+	0.899537i	$0.643904\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8073.03	−0.667299
$528$	0	0
$529$	−9025.25	−0.741781
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1616.32	0.131352
$534$	0	0
$535$	37178.7	3.00444
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2114.78	0.168998
$540$	0	0
$541$	10902.6	0.866430	0.433215	−	0.901291i	$-0.357379\pi$
$541$	10902.6	0.866430	0.433215	+	0.901291i	$0.357379\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−933.729	−0.0733882
$546$	0	0
$547$	−4206.15	−0.328779	−0.164389	−	0.986396i	$-0.552565\pi$
$547$	−4206.15	−0.328779	−0.164389	+	0.986396i	$0.552565\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1002.33	−0.0774968
$552$	0	0
$553$	4819.65	0.370619
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4765.22	0.362494	0.181247	−	0.983438i	$-0.441987\pi$
$557$	4765.22	0.362494	0.181247	+	0.983438i	$0.441987\pi$
$558$	0	0
$559$	4416.15	0.334138
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8664.98	−0.648642	−0.324321	−	0.945947i	$-0.605136\pi$
$563$	−8664.98	−0.648642	−0.324321	+	0.945947i	$0.605136\pi$
$564$	0	0
$565$	35814.9	2.66680
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11067.6	0.815429	0.407714	−	0.913109i	$-0.366326\pi$
$569$	11067.6	0.815429	0.407714	+	0.913109i	$0.366326\pi$
$570$	0	0
$571$	−15956.5	−1.16946	−0.584729	−	0.811229i	$-0.698799\pi$
$571$	−15956.5	−1.16946	−0.584729	+	0.811229i	$0.698799\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	9451.73	0.685503
$576$	0	0
$577$	1299.52	0.0937605	0.0468803	−	0.998901i	$-0.485072\pi$
$577$	1299.52	0.0937605	0.0468803	+	0.998901i	$0.485072\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30098.9	−2.14925
$582$	0	0
$583$	−5094.09	−0.361879
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6714.43	0.472120	0.236060	−	0.971739i	$-0.424144\pi$
$587$	6714.43	0.472120	0.236060	+	0.971739i	$0.424144\pi$
$588$	0	0
$589$	1171.69	0.0819672
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10130.3	0.701522	0.350761	−	0.936465i	$-0.385923\pi$
$593$	10130.3	0.701522	0.350761	+	0.936465i	$0.385923\pi$
$594$	0	0
$595$	32037.5	2.20741
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7732.64	−0.527457	−0.263729	−	0.964597i	$-0.584952\pi$
$599$	−7732.64	−0.527457	−0.263729	+	0.964597i	$0.584952\pi$
$600$	0	0
$601$	13318.2	0.903925	0.451963	−	0.892037i	$-0.350724\pi$
$601$	13318.2	0.903925	0.451963	+	0.892037i	$0.350724\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2073.40	−0.139332
$606$	0	0
$607$	−18128.9	−1.21224	−0.606120	−	0.795373i	$-0.707275\pi$
$607$	−18128.9	−1.21224	−0.606120	+	0.795373i	$0.707275\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24377.8	−1.61411
$612$	0	0
$613$	−15524.8	−1.02291	−0.511454	−	0.859311i	$-0.670893\pi$
$613$	−15524.8	−1.02291	−0.511454	+	0.859311i	$0.670893\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3365.91	−0.219621	−0.109811	−	0.993953i	$-0.535024\pi$
$617$	−3365.91	−0.219621	−0.109811	+	0.993953i	$0.535024\pi$
$618$	0	0
$619$	13998.9	0.908990	0.454495	−	0.890749i	$-0.349820\pi$
$619$	13998.9	0.908990	0.454495	+	0.890749i	$0.349820\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−32185.7	−2.06981
$624$	0	0
$625$	−8268.45	−0.529181
$626$	0	0
$627$	0	0
$628$	0	0
$629$	32566.2	2.06439
$630$	0	0
$631$	13632.3	0.860052	0.430026	−	0.902816i	$-0.358504\pi$
$631$	13632.3	0.860052	0.430026	+	0.902816i	$0.358504\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−30873.1	−1.92939
$636$	0	0
$637$	−11473.2	−0.713633
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2510.82	0.154713	0.0773567	−	0.997003i	$-0.475352\pi$
$641$	2510.82	0.154713	0.0773567	+	0.997003i	$0.475352\pi$
$642$	0	0
$643$	−11707.5	−0.718037	−0.359018	−	0.933330i	$-0.616888\pi$
$643$	−11707.5	−0.718037	−0.359018	+	0.933330i	$0.616888\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7293.14	−0.443157	−0.221579	−	0.975142i	$-0.571121\pi$
$647$	−7293.14	−0.443157	−0.221579	+	0.975142i	$0.571121\pi$
$648$	0	0
$649$	5486.94	0.331867
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3663.23	−0.219530	−0.109765	−	0.993958i	$-0.535010\pi$
$653$	−3663.23	−0.219530	−0.109765	+	0.993958i	$0.535010\pi$
$654$	0	0
$655$	−51193.2	−3.05387
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18384.4	1.08673	0.543365	−	0.839496i	$-0.317150\pi$
$659$	18384.4	1.08673	0.543365	+	0.839496i	$0.317150\pi$
$660$	0	0
$661$	18470.0	1.08684	0.543420	−	0.839461i	$-0.317129\pi$
$661$	18470.0	1.08684	0.543420	+	0.839461i	$0.317129\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4649.82	−0.271146
$666$	0	0
$667$	4790.03	0.278067
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6995.03	−0.402444
$672$	0	0
$673$	−6242.62	−0.357556	−0.178778	−	0.983889i	$-0.557214\pi$
$673$	−6242.62	−0.357556	−0.178778	+	0.983889i	$0.557214\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26567.6	1.50823	0.754117	−	0.656740i	$-0.228065\pi$
$677$	26567.6	1.50823	0.754117	+	0.656740i	$0.228065\pi$
$678$	0	0
$679$	−24333.0	−1.37528
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21274.9	1.19189	0.595946	−	0.803025i	$-0.296777\pi$
$683$	21274.9	1.19189	0.595946	+	0.803025i	$0.296777\pi$
$684$	0	0
$685$	31674.0	1.76672
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27636.6	1.52812
$690$	0	0
$691$	−28567.4	−1.57273	−0.786363	−	0.617765i	$-0.788038\pi$
$691$	−28567.4	−1.57273	−0.786363	+	0.617765i	$0.788038\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−31829.0	−1.73719
$696$	0	0
$697$	−2188.76	−0.118946
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12457.4	−0.671199	−0.335600	−	0.942005i	$-0.608939\pi$
$701$	−12457.4	−0.671199	−0.335600	+	0.942005i	$0.608939\pi$
$702$	0	0
$703$	−4726.55	−0.253578
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17942.2	−0.954433
$708$	0	0
$709$	3973.22	0.210462	0.105231	−	0.994448i	$-0.466442\pi$
$709$	3973.22	0.210462	0.105231	+	0.994448i	$0.466442\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5599.38	−0.294107
$714$	0	0
$715$	11248.7	0.588360
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21019.6	−1.09026	−0.545131	−	0.838351i	$-0.683520\pi$
$719$	−21019.6	−1.09026	−0.545131	+	0.838351i	$0.683520\pi$
$720$	0	0
$721$	−14301.7	−0.738726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14410.5	0.738194
$726$	0	0
$727$	20360.3	1.03868	0.519342	−	0.854567i	$-0.326177\pi$
$727$	20360.3	1.03868	0.519342	+	0.854567i	$0.326177\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5980.17	−0.302578
$732$	0	0
$733$	−19038.5	−0.959350	−0.479675	−	0.877446i	$-0.659245\pi$
$733$	−19038.5	−0.959350	−0.479675	+	0.877446i	$0.659245\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7715.63	−0.385630
$738$	0	0
$739$	−1317.95	−0.0656045	−0.0328023	−	0.999462i	$-0.510443\pi$
$739$	−1317.95	−0.0656045	−0.0328023	+	0.999462i	$0.510443\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16501.6	0.814784	0.407392	−	0.913253i	$-0.366438\pi$
$743$	16501.6	0.814784	0.407392	+	0.913253i	$0.366438\pi$
$744$	0	0
$745$	15831.6	0.778557
$746$	0	0
$747$	0	0
$748$	0	0
$749$	50196.8	2.44880
$750$	0	0
$751$	7514.31	0.365115	0.182557	−	0.983195i	$-0.441562\pi$
$751$	7514.31	0.365115	0.182557	+	0.983195i	$0.441562\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	33343.6	1.60728
$756$	0	0
$757$	−18274.4	−0.877402	−0.438701	−	0.898633i	$-0.644561\pi$
$757$	−18274.4	−0.877402	−0.438701	+	0.898633i	$0.644561\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30769.7	1.46570	0.732852	−	0.680388i	$-0.238189\pi$
$761$	30769.7	1.46570	0.732852	+	0.680388i	$0.238189\pi$
$762$	0	0
$763$	−1260.67	−0.0598158
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−29768.0	−1.40138
$768$	0	0
$769$	−9841.74	−0.461511	−0.230756	−	0.973012i	$-0.574120\pi$
$769$	−9841.74	−0.461511	−0.230756	+	0.973012i	$0.574120\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−10693.8	−0.497581	−0.248791	−	0.968557i	$-0.580033\pi$
$773$	−10693.8	−0.497581	−0.248791	+	0.968557i	$0.580033\pi$
$774$	0	0
$775$	−16845.3	−0.780777
$776$	0	0
$777$	0	0
$778$	0	0
$779$	317.669	0.0146106
$780$	0	0
$781$	−305.179	−0.0139823
$782$	0	0
$783$	0	0
$784$	0	0
$785$	51996.5	2.36412
$786$	0	0
$787$	−24419.1	−1.10603	−0.553015	−	0.833171i	$-0.686523\pi$
$787$	−24419.1	−1.10603	−0.553015	+	0.833171i	$0.686523\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	48355.5	2.17361
$792$	0	0
$793$	37949.7	1.69941
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14848.3	−0.659919	−0.329959	−	0.943995i	$-0.607035\pi$
$797$	−14848.3	−0.659919	−0.329959	+	0.943995i	$0.607035\pi$
$798$	0	0
$799$	33011.4	1.46165
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6815.89	0.299536
$804$	0	0
$805$	22220.9	0.972901
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20269.0	0.880864	0.440432	−	0.897786i	$-0.354825\pi$
$809$	20269.0	0.880864	0.440432	+	0.897786i	$0.354825\pi$
$810$	0	0
$811$	31691.6	1.37219	0.686094	−	0.727513i	$-0.259324\pi$
$811$	31691.6	1.37219	0.686094	+	0.727513i	$0.259324\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2002.78	−0.0860791
$816$	0	0
$817$	867.942	0.0371670
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−16106.6	−0.684680	−0.342340	−	0.939576i	$-0.611220\pi$
$821$	−16106.6	−0.684680	−0.342340	+	0.939576i	$0.611220\pi$
$822$	0	0
$823$	43327.1	1.83510	0.917550	−	0.397620i	$-0.130164\pi$
$823$	43327.1	1.83510	0.917550	+	0.397620i	$0.130164\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	22443.8	0.943711	0.471855	−	0.881676i	$-0.343584\pi$
$827$	22443.8	0.943711	0.471855	+	0.881676i	$0.343584\pi$
$828$	0	0
$829$	−23235.7	−0.973474	−0.486737	−	0.873549i	$-0.661813\pi$
$829$	−23235.7	−0.973474	−0.486737	+	0.873549i	$0.661813\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	15536.5	0.646230
$834$	0	0
$835$	4001.05	0.165823
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6100.81	0.251041	0.125521	−	0.992091i	$-0.459940\pi$
$839$	6100.81	0.251041	0.125521	+	0.992091i	$0.459940\pi$
$840$	0	0
$841$	−17085.9	−0.700560
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−23380.1	−0.951833
$846$	0	0
$847$	−2799.40	−0.113564
$848$	0	0
$849$	0	0
$850$	0	0
$851$	22587.6	0.909864
$852$	0	0
$853$	−24426.3	−0.980471	−0.490235	−	0.871590i	$-0.663089\pi$
$853$	−24426.3	−0.980471	−0.490235	+	0.871590i	$0.663089\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8143.04	0.324575	0.162288	−	0.986743i	$-0.448113\pi$
$857$	8143.04	0.324575	0.162288	+	0.986743i	$0.448113\pi$
$858$	0	0
$859$	24479.5	0.972329	0.486164	−	0.873867i	$-0.338396\pi$
$859$	24479.5	0.972329	0.486164	+	0.873867i	$0.338396\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24905.7	0.982387	0.491194	−	0.871050i	$-0.336561\pi$
$863$	24905.7	0.982387	0.491194	+	0.871050i	$0.336561\pi$
$864$	0	0
$865$	43424.9	1.70693
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2291.55	−0.0894538
$870$	0	0
$871$	41859.2	1.62841
$872$	0	0
$873$	0	0
$874$	0	0
$875$	17295.2	0.668211
$876$	0	0
$877$	−9738.75	−0.374976	−0.187488	−	0.982267i	$-0.560035\pi$
$877$	−9738.75	−0.374976	−0.187488	+	0.982267i	$0.560035\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	8663.22	0.331296	0.165648	−	0.986185i	$-0.447029\pi$
$881$	8663.22	0.331296	0.165648	+	0.986185i	$0.447029\pi$
$882$	0	0
$883$	−34151.4	−1.30157	−0.650785	−	0.759262i	$-0.725560\pi$
$883$	−34151.4	−1.30157	−0.650785	+	0.759262i	$0.725560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32057.4	−1.21351	−0.606754	−	0.794890i	$-0.707529\pi$
$887$	−32057.4	−1.21351	−0.606754	+	0.794890i	$0.707529\pi$
$888$	0	0
$889$	−41683.3	−1.57257
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4791.17	−0.179541
$894$	0	0
$895$	−35155.6	−1.31298
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8537.02	−0.316714
$900$	0	0
$901$	−37424.5	−1.38378
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28249.3	1.03761
$906$	0	0
$907$	−8298.83	−0.303813	−0.151906	−	0.988395i	$-0.548541\pi$
$907$	−8298.83	−0.303813	−0.151906	+	0.988395i	$0.548541\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32365.3	−1.17707	−0.588535	−	0.808472i	$-0.700295\pi$
$911$	−32365.3	−1.17707	−0.588535	+	0.808472i	$0.700295\pi$
$912$	0	0
$913$	14310.8	0.518749
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−69118.4	−2.48909
$918$	0	0
$919$	−15578.2	−0.559170	−0.279585	−	0.960121i	$-0.590197\pi$
$919$	−15578.2	−0.559170	−0.279585	+	0.960121i	$0.590197\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1655.67	0.0590433
$924$	0	0
$925$	67953.3	2.41545
$926$	0	0
$927$	0	0
$928$	0	0
$929$	28956.6	1.02264	0.511321	−	0.859390i	$-0.329156\pi$
$929$	28956.6	1.02264	0.511321	+	0.859390i	$0.329156\pi$
$930$	0	0
$931$	−2254.92	−0.0793792
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15232.5	−0.532789
$936$	0	0
$937$	−33515.3	−1.16851	−0.584256	−	0.811569i	$-0.698614\pi$
$937$	−33515.3	−1.16851	−0.584256	+	0.811569i	$0.698614\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−32394.3	−1.12224	−0.561119	−	0.827735i	$-0.689629\pi$
$941$	−32394.3	−1.12224	−0.561119	+	0.827735i	$0.689629\pi$
$942$	0	0
$943$	−1518.11	−0.0524245
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10750.3	−0.368889	−0.184445	−	0.982843i	$-0.559049\pi$
$947$	−10750.3	−0.368889	−0.184445	+	0.982843i	$0.559049\pi$
$948$	0	0
$949$	−36977.8	−1.26486
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22205.9	0.754794	0.377397	−	0.926052i	$-0.376819\pi$
$953$	22205.9	0.754794	0.377397	+	0.926052i	$0.376819\pi$
$954$	0	0
$955$	−15047.0	−0.509853
$956$	0	0
$957$	0	0
$958$	0	0
$959$	42764.6	1.43998
$960$	0	0
$961$	−19811.5	−0.665017
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32178.8	1.07344
$966$	0	0
$967$	14348.5	0.477161	0.238581	−	0.971123i	$-0.423318\pi$
$967$	14348.5	0.477161	0.238581	+	0.971123i	$0.423318\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41006.2	1.35525	0.677627	−	0.735406i	$-0.263008\pi$
$971$	41006.2	1.35525	0.677627	+	0.735406i	$0.263008\pi$
$972$	0	0
$973$	−42973.9	−1.41591
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17160.2	0.561928	0.280964	−	0.959718i	$-0.409346\pi$
$977$	17160.2	0.561928	0.280964	+	0.959718i	$0.409346\pi$
$978$	0	0
$979$	15303.0	0.499576
$980$	0	0
$981$	0	0
$982$	0	0
$983$	43576.2	1.41390	0.706950	−	0.707264i	$-0.250071\pi$
$983$	43576.2	1.41390	0.706950	+	0.707264i	$0.250071\pi$
$984$	0	0
$985$	55598.1	1.79848
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4147.80	−0.133359
$990$	0	0
$991$	−38954.3	−1.24866	−0.624332	−	0.781159i	$-0.714629\pi$
$991$	−38954.3	−1.24866	−0.624332	+	0.781159i	$0.714629\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−66615.4	−2.12246
$996$	0	0
$997$	−38907.9	−1.23593	−0.617967	−	0.786204i	$-0.712043\pi$
$997$	−38907.9	−1.23593	−0.617967	+	0.786204i	$0.712043\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	396.4.a.h.1.1	✓	2
3.2	odd	2		396.4.a.j.1.2	yes	2
4.3	odd	2		1584.4.a.y.1.1		2
12.11	even	2		1584.4.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
396.4.a.h.1.1	✓	2	1.1	even	1	trivial
396.4.a.j.1.2	yes	2	3.2	odd	2
1584.4.a.y.1.1		2	4.3	odd	2
1584.4.a.bi.1.2		2	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	396.a (trivial)

\(f(q)\)	\(=\)	\(q-17.1355 q^{5} -23.1355 q^{7} +11.0000 q^{11} -59.6776 q^{13} +80.8132 q^{17} -11.7289 q^{19} +56.0513 q^{23} +168.626 q^{25} +85.4579 q^{29} -99.8974 q^{31} +396.440 q^{35} +402.982 q^{37} -27.0842 q^{41} -74.0000 q^{43} +408.491 q^{47} +192.253 q^{49} -463.099 q^{53} -188.491 q^{55} +498.813 q^{59} -635.912 q^{61} +1022.61 q^{65} -701.421 q^{67} -27.7435 q^{71} +619.626 q^{73} -254.491 q^{77} -208.322 q^{79} +1300.98 q^{83} -1384.78 q^{85} +1391.18 q^{89} +1380.67 q^{91} +200.982 q^{95} +1051.76 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} - 24 q^{7} + 22 q^{11} - 8 q^{13} + 28 q^{17} - 68 q^{19} + 268 q^{23} + 70 q^{25} + 260 q^{29} + 112 q^{31} + 392 q^{35} + 316 q^{37} + 124 q^{41} - 148 q^{43} + 572 q^{47} - 150 q^{49}+ \cdots + 500 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 - 24 * q^7 + 22 * q^11 - 8 * q^13 + 28 * q^17 - 68 * q^19 + 268 * q^23 + 70 * q^25 + 260 * q^29 + 112 * q^31 + 392 * q^35 + 316 * q^37 + 124 * q^41 - 148 * q^43 + 572 * q^47 - 150 * q^49 + 76 * q^53 - 132 * q^55 + 864 * q^59 - 136 * q^61 + 1288 * q^65 - 512 * q^67 + 724 * q^71 + 972 * q^73 - 264 * q^77 - 528 * q^79 + 2112 * q^83 - 1656 * q^85 + 288 * q^89 + 1336 * q^91 - 88 * q^95 + 500 * q^97

Introduction

L-functions

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