LMFDB - Embedded newform 1932.2.k.a.1609.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1609,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("1932.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.27
Character	$\chi$	$=$	1932.1609
Dual form			1932.2.k.a.1609.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.18466 q^{5} +(-2.46306 + 0.966093i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -4.18466 q^{5} +(-2.46306 + 0.966093i) q^{7} -1.00000 q^{9} -1.82537i q^{11} -5.63628i q^{13} -4.18466i q^{15} +1.34961 q^{17} -4.27595 q^{19} +(-0.966093 - 2.46306i) q^{21} +(1.26310 + 4.62651i) q^{23} +12.5113 q^{25} -1.00000i q^{27} -5.54797 q^{29} +2.52845i q^{31} +1.82537 q^{33} +(10.3071 - 4.04276i) q^{35} -0.592620i q^{37} +5.63628 q^{39} +0.941186i q^{41} -2.90862i q^{43} +4.18466 q^{45} -2.40543i q^{47} +(5.13333 - 4.75909i) q^{49} +1.34961i q^{51} +7.80446i q^{53} +7.63854i q^{55} -4.27595i q^{57} +11.6435i q^{59} +11.5488 q^{61} +(2.46306 - 0.966093i) q^{63} +23.5859i q^{65} +9.55560i q^{67} +(-4.62651 + 1.26310i) q^{69} -5.05917 q^{71} +0.229691i q^{73} +12.5113i q^{75} +(1.76347 + 4.49599i) q^{77} -9.67783i q^{79} +1.00000 q^{81} -6.18486 q^{83} -5.64765 q^{85} -5.54797i q^{87} +18.7078 q^{89} +(5.44517 + 13.8825i) q^{91} -2.52845 q^{93} +17.8934 q^{95} +13.1577 q^{97} +1.82537i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{9}+O(q^{10}) $ 32 * q - 32 * q^9	$ 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) $ 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times$.

$n$	$829$	$925$	$967$	$1289$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−4.18466			−1.87143			−0.935717	−	0.352750i	$-0.885246\pi$
$5$	−4.18466			−1.87143			−0.935717	+	0.352750i	$0.885246\pi$
$6$		0			0
$7$	−2.46306	+	0.966093i	−0.930949	+	0.365149i
$7$	−2.46306	+	0.966093i	−0.930949	+	0.365149i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	1.82537i		−	0.550369i	−0.961391	−	0.275185i	$-0.911261\pi$
$11$		−	1.82537i		−	0.550369i	0.961391	−	0.275185i	$-0.0887390\pi$
$12$		0			0
$13$		−	5.63628i		−	1.56322i	−0.623765	−	0.781612i	$-0.714398\pi$
$13$		−	5.63628i		−	1.56322i	0.623765	−	0.781612i	$-0.285602\pi$
$14$		0			0
$15$		−	4.18466i		−	1.08047i
$16$		0			0
$17$	1.34961			0.327328			0.163664	−	0.986516i	$-0.447669\pi$
$17$	1.34961			0.327328			0.163664	+	0.986516i	$0.447669\pi$
$18$		0			0
$19$	−4.27595			−0.980970			−0.490485	−	0.871450i	$-0.663180\pi$
$19$	−4.27595			−0.980970			−0.490485	+	0.871450i	$0.663180\pi$
$20$		0			0
$21$	−0.966093	−	2.46306i	−0.210819	−	0.537484i
$22$		0			0
$23$	1.26310	+	4.62651i	0.263374	+	0.964694i
$23$	1.26310	+	4.62651i	0.263374	+	0.964694i
$24$		0			0
$25$	12.5113			2.50227
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−5.54797			−1.03023			−0.515116	−	0.857120i	$-0.672251\pi$
$29$	−5.54797			−1.03023			−0.515116	+	0.857120i	$0.672251\pi$
$30$		0			0
$31$			2.52845i			0.454123i	0.973880	+	0.227061i	$0.0729118\pi$
$31$			2.52845i			0.454123i	−0.973880	+	0.227061i	$0.927088\pi$
$32$		0			0
$33$	1.82537			0.317756
$34$		0			0
$35$	10.3071	−	4.04276i	1.74221	−	0.683352i
$36$		0			0
$37$		−	0.592620i		−	0.0974261i	−0.998813	−	0.0487131i	$-0.984488\pi$
$37$		−	0.592620i		−	0.0974261i	0.998813	−	0.0487131i	$-0.0155120\pi$
$38$		0			0
$39$	5.63628			0.902527
$40$		0			0
$41$			0.941186i			0.146989i	0.997296	+	0.0734943i	$0.0234151\pi$
$41$			0.941186i			0.146989i	−0.997296	+	0.0734943i	$0.976585\pi$
$42$		0			0
$43$		−	2.90862i		−	0.443560i	−0.975097	−	0.221780i	$-0.928813\pi$
$43$		−	2.90862i		−	0.443560i	0.975097	−	0.221780i	$-0.0711867\pi$
$44$		0			0
$45$	4.18466			0.623812
$46$		0			0
$47$		−	2.40543i		−	0.350867i	−0.984491	−	0.175434i	$-0.943867\pi$
$47$		−	2.40543i		−	0.350867i	0.984491	−	0.175434i	$-0.0561328\pi$
$48$		0			0
$49$	5.13333	−	4.75909i	0.733333	−	0.679870i
$50$		0			0
$51$			1.34961i			0.188983i
$52$		0			0
$53$			7.80446i			1.07202i	0.844210	+	0.536012i	$0.180070\pi$
$53$			7.80446i			1.07202i	−0.844210	+	0.536012i	$0.819930\pi$
$54$		0			0
$55$			7.63854i			1.02998i
$56$		0			0
$57$		−	4.27595i		−	0.566363i
$58$		0			0
$59$			11.6435i			1.51585i	0.652341	+	0.757926i	$0.273787\pi$
$59$			11.6435i			1.51585i	−0.652341	+	0.757926i	$0.726213\pi$
$60$		0			0
$61$	11.5488			1.47867			0.739335	−	0.673338i	$-0.235140\pi$
$61$	11.5488			1.47867			0.739335	+	0.673338i	$0.235140\pi$
$62$		0			0
$63$	2.46306	−	0.966093i	0.310316	−	0.121716i
$64$		0			0
$65$			23.5859i			2.92547i
$66$		0			0
$67$			9.55560i			1.16740i	0.811968	+	0.583701i	$0.198396\pi$
$67$			9.55560i			1.16740i	−0.811968	+	0.583701i	$0.801604\pi$
$68$		0			0
$69$	−4.62651	+	1.26310i	−0.556966	+	0.152059i
$70$		0			0
$71$	−5.05917			−0.600414			−0.300207	−	0.953874i	$-0.597056\pi$
$71$	−5.05917			−0.600414			−0.300207	+	0.953874i	$0.597056\pi$
$72$		0			0
$73$			0.229691i			0.0268833i	0.999910	+	0.0134417i	$0.00427874\pi$
$73$			0.229691i			0.0268833i	−0.999910	+	0.0134417i	$0.995721\pi$
$74$		0			0
$75$			12.5113i			1.44469i
$76$		0			0
$77$	1.76347	+	4.49599i	0.200967	+	0.512366i
$78$		0			0
$79$		−	9.67783i		−	1.08884i	−0.838812	−	0.544420i	$-0.816750\pi$
$79$		−	9.67783i		−	1.08884i	0.838812	−	0.544420i	$-0.183250\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−6.18486			−0.678876			−0.339438	−	0.940628i	$-0.610237\pi$
$83$	−6.18486			−0.678876			−0.339438	+	0.940628i	$0.610237\pi$
$84$		0			0
$85$	−5.64765			−0.612573
$86$		0			0
$87$		−	5.54797i		−	0.594805i
$88$		0			0
$89$	18.7078			1.98302			0.991510	−	0.130028i	$-0.0415068\pi$
$89$	18.7078			1.98302			0.991510	+	0.130028i	$0.0415068\pi$
$90$		0			0
$91$	5.44517	+	13.8825i	0.570809	+	1.45528i
$92$		0			0
$93$	−2.52845			−0.262188
$94$		0			0
$95$	17.8934			1.83582
$96$		0			0
$97$	13.1577			1.33596			0.667980	−	0.744179i	$-0.267159\pi$
$97$	13.1577			1.33596			0.667980	+	0.744179i	$0.267159\pi$
$98$		0			0
$99$			1.82537i			0.183456i
$100$		0			0
$101$			14.2453i			1.41746i	0.705479	+	0.708731i	$0.250732\pi$
$101$			14.2453i			1.41746i	−0.705479	+	0.708731i	$0.749268\pi$
$102$		0			0
$103$	−1.67757			−0.165296			−0.0826481	−	0.996579i	$-0.526338\pi$
$103$	−1.67757			−0.165296			−0.0826481	+	0.996579i	$0.526338\pi$
$104$		0			0
$105$	4.04276	+	10.3071i	0.394533	+	1.00587i
$106$		0			0
$107$		−	2.36527i		−	0.228659i	−0.993443	−	0.114329i	$-0.963528\pi$
$107$		−	2.36527i		−	0.228659i	0.993443	−	0.114329i	$-0.0364719\pi$
$108$		0			0
$109$		−	13.9734i		−	1.33841i	−0.743080	−	0.669203i	$-0.766636\pi$
$109$		−	13.9734i		−	1.33841i	0.743080	−	0.669203i	$-0.233364\pi$
$110$		0			0
$111$	0.592620			0.0562490
$112$		0			0
$113$			2.50110i			0.235284i	0.993056	+	0.117642i	$0.0375336\pi$
$113$			2.50110i			0.235284i	−0.993056	+	0.117642i	$0.962466\pi$
$114$		0			0
$115$	−5.28563	−	19.3603i	−0.492887	−	1.80536i
$116$		0			0
$117$			5.63628i			0.521074i
$118$		0			0
$119$	−3.32417	+	1.30385i	−0.304726	+	0.119523i
$120$		0			0
$121$	7.66803			0.697094
$122$		0			0
$123$	−0.941186			−0.0848639
$124$		0			0
$125$	−31.4324			−2.81140
$126$		0			0
$127$	4.82083			0.427779			0.213890	−	0.976858i	$-0.431387\pi$
$127$	4.82083			0.427779			0.213890	+	0.976858i	$0.431387\pi$
$128$		0			0
$129$	2.90862			0.256090
$130$		0			0
$131$			13.3351i			1.16510i	0.812796	+	0.582548i	$0.197944\pi$
$131$			13.3351i			1.16510i	−0.812796	+	0.582548i	$0.802056\pi$
$132$		0			0
$133$	10.5319	−	4.13096i	0.913233	−	0.358200i
$134$		0			0
$135$			4.18466i			0.360158i
$136$		0			0
$137$		−	12.3538i		−	1.05545i	−0.849414	−	0.527727i	$-0.823044\pi$
$137$		−	12.3538i		−	1.05545i	0.849414	−	0.527727i	$-0.176956\pi$
$138$		0			0
$139$		−	13.5018i		−	1.14521i	−0.819833	−	0.572603i	$-0.805934\pi$
$139$		−	13.5018i		−	1.14521i	0.819833	−	0.572603i	$-0.194066\pi$
$140$		0			0
$141$	2.40543			0.202573
$142$		0			0
$143$	−10.2883			−0.860350
$144$		0			0
$145$	23.2163			1.92801
$146$		0			0
$147$	4.75909	+	5.13333i	0.392523	+	0.423390i
$148$		0			0
$149$		−	12.7461i		−	1.04420i	−0.852884	−	0.522101i	$-0.825149\pi$
$149$		−	12.7461i		−	1.04420i	0.852884	−	0.522101i	$-0.174851\pi$
$150$		0			0
$151$	16.9527			1.37959			0.689794	−	0.724005i	$-0.257701\pi$
$151$	16.9527			1.37959			0.689794	+	0.724005i	$0.257701\pi$
$152$		0			0
$153$	−1.34961			−0.109109
$154$		0			0
$155$		−	10.5807i		−	0.849862i
$156$		0			0
$157$	−11.6179			−0.927206			−0.463603	−	0.886043i	$-0.653444\pi$
$157$	−11.6179			−0.927206			−0.463603	+	0.886043i	$0.653444\pi$
$158$		0			0
$159$	−7.80446			−0.618934
$160$		0			0
$161$	−7.58072	−	10.1751i	−0.597445	−	0.801910i
$162$		0			0
$163$	5.74976			0.450356			0.225178	−	0.974318i	$-0.427704\pi$
$163$	5.74976			0.450356			0.225178	+	0.974318i	$0.427704\pi$
$164$		0			0
$165$	−7.63854			−0.594659
$166$		0			0
$167$			19.5553i			1.51323i	0.653858	+	0.756617i	$0.273149\pi$
$167$			19.5553i			1.51323i	−0.653858	+	0.756617i	$0.726851\pi$
$168$		0			0
$169$	−18.7677			−1.44367
$170$		0			0
$171$	4.27595			0.326990
$172$		0			0
$173$		−	22.7092i		−	1.72655i	−0.504734	−	0.863275i	$-0.668410\pi$
$173$		−	22.7092i		−	1.72655i	0.504734	−	0.863275i	$-0.331590\pi$
$174$		0			0
$175$	−30.8162	+	12.0871i	−2.32948	+	0.913700i
$176$		0			0
$177$	−11.6435			−0.875177
$178$		0			0
$179$	12.0764			0.902636			0.451318	−	0.892363i	$-0.350954\pi$
$179$	12.0764			0.902636			0.451318	+	0.892363i	$0.350954\pi$
$180$		0			0
$181$	−3.54990			−0.263862			−0.131931	−	0.991259i	$-0.542118\pi$
$181$	−3.54990			−0.263862			−0.131931	+	0.991259i	$0.542118\pi$
$182$		0			0
$183$			11.5488i			0.853710i
$184$		0			0
$185$			2.47991i			0.182327i
$186$		0			0
$187$		−	2.46353i		−	0.180151i
$188$		0			0
$189$	0.966093	+	2.46306i	0.0702729	+	0.179161i
$190$		0			0
$191$		−	9.51786i		−	0.688688i	−0.938843	−	0.344344i	$-0.888101\pi$
$191$		−	9.51786i		−	0.688688i	0.938843	−	0.344344i	$-0.111899\pi$
$192$		0			0
$193$	−2.10667			−0.151641			−0.0758206	−	0.997121i	$-0.524158\pi$
$193$	−2.10667			−0.151641			−0.0758206	+	0.997121i	$0.524158\pi$
$194$		0			0
$195$	−23.5859			−1.68902
$196$		0			0
$197$	−17.7016			−1.26119			−0.630593	−	0.776114i	$-0.717188\pi$
$197$	−17.7016			−1.26119			−0.630593	+	0.776114i	$0.717188\pi$
$198$		0			0
$199$	10.0241			0.710590			0.355295	−	0.934754i	$-0.384380\pi$
$199$	10.0241			0.710590			0.355295	+	0.934754i	$0.384380\pi$
$200$		0			0
$201$	−9.55560			−0.674000
$202$		0			0
$203$	13.6650	−	5.35985i	0.959094	−	0.376188i
$204$		0			0
$205$		−	3.93854i		−	0.275080i
$206$		0			0
$207$	−1.26310	−	4.62651i	−0.0877914	−	0.321565i
$208$		0			0
$209$			7.80518i			0.539896i
$210$		0			0
$211$	15.8426			1.09065			0.545323	−	0.838226i	$-0.316407\pi$
$211$	15.8426			1.09065			0.545323	+	0.838226i	$0.316407\pi$
$212$		0			0
$213$		−	5.05917i		−	0.346649i
$214$		0			0
$215$			12.1716i			0.830094i
$216$		0			0
$217$	−2.44272	−	6.22772i	−0.165822	−	0.422765i
$218$		0			0
$219$	−0.229691			−0.0155211
$220$		0			0
$221$		−	7.60677i		−	0.511687i
$222$		0			0
$223$			24.1995i			1.62052i	0.586071	+	0.810259i	$0.300674\pi$
$223$			24.1995i			1.62052i	−0.586071	+	0.810259i	$0.699326\pi$
$224$		0			0
$225$	−12.5113			−0.834089
$226$		0			0
$227$	17.7083			1.17534			0.587671	−	0.809100i	$-0.300045\pi$
$227$	17.7083			1.17534			0.587671	+	0.809100i	$0.300045\pi$
$228$		0			0
$229$	−10.9498			−0.723584			−0.361792	−	0.932259i	$-0.617835\pi$
$229$	−10.9498			−0.723584			−0.361792	+	0.932259i	$0.617835\pi$
$230$		0			0
$231$	−4.49599	+	1.76347i	−0.295814	+	0.116028i
$232$		0			0
$233$	14.7878			0.968784			0.484392	−	0.874851i	$-0.339041\pi$
$233$	14.7878			0.968784			0.484392	+	0.874851i	$0.339041\pi$
$234$		0			0
$235$			10.0659i			0.656625i
$236$		0			0
$237$	9.67783			0.628643
$238$		0			0
$239$	−3.80996			−0.246446			−0.123223	−	0.992379i	$-0.539323\pi$
$239$	−3.80996			−0.246446			−0.123223	+	0.992379i	$0.539323\pi$
$240$		0			0
$241$	1.90190			0.122512			0.0612562	−	0.998122i	$-0.480489\pi$
$241$	1.90190			0.122512			0.0612562	+	0.998122i	$0.480489\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−21.4812	+	19.9151i	−1.37238	+	1.27233i
$246$		0			0
$247$			24.1005i			1.53348i
$248$		0			0
$249$		−	6.18486i		−	0.391950i
$250$		0			0
$251$	5.54304			0.349874			0.174937	−	0.984580i	$-0.444028\pi$
$251$	5.54304			0.349874			0.174937	+	0.984580i	$0.444028\pi$
$252$		0			0
$253$	8.44508	−	2.30562i	0.530938	−	0.144953i
$254$		0			0
$255$		−	5.64765i		−	0.353669i
$256$		0			0
$257$		−	18.4310i		−	1.14969i	−0.818261	−	0.574846i	$-0.805062\pi$
$257$		−	18.4310i		−	1.14969i	0.818261	−	0.574846i	$-0.194938\pi$
$258$		0			0
$259$	0.572526	+	1.45966i	0.0355750	+	0.0906988i
$260$		0			0
$261$	5.54797			0.343411
$262$		0			0
$263$		−	18.2412i		−	1.12480i	−0.826865	−	0.562400i	$-0.809878\pi$
$263$		−	18.2412i		−	1.12480i	0.826865	−	0.562400i	$-0.190122\pi$
$264$		0			0
$265$		−	32.6590i		−	2.00622i
$266$		0			0
$267$			18.7078i			1.14490i
$268$		0			0
$269$			11.7654i			0.717350i	0.933463	+	0.358675i	$0.116771\pi$
$269$			11.7654i			0.717350i	−0.933463	+	0.358675i	$0.883229\pi$
$270$		0			0
$271$		−	10.6020i		−	0.644024i	−0.946736	−	0.322012i	$-0.895641\pi$
$271$		−	10.6020i		−	0.644024i	0.946736	−	0.322012i	$-0.104359\pi$
$272$		0			0
$273$	−13.8825	+	5.44517i	−0.840207	+	0.329557i
$274$		0			0
$275$		−	22.8378i		−	1.37717i
$276$		0			0
$277$	1.65673			0.0995434			0.0497717	−	0.998761i	$-0.484151\pi$
$277$	1.65673			0.0995434			0.0497717	+	0.998761i	$0.484151\pi$
$278$		0			0
$279$		−	2.52845i		−	0.151374i
$280$		0			0
$281$			4.69240i			0.279925i	0.990157	+	0.139963i	$0.0446982\pi$
$281$			4.69240i			0.279925i	−0.990157	+	0.139963i	$0.955302\pi$
$282$		0			0
$283$	17.5213			1.04153			0.520766	−	0.853699i	$-0.325646\pi$
$283$	17.5213			1.04153			0.520766	+	0.853699i	$0.325646\pi$
$284$		0			0
$285$			17.8934i			1.05991i
$286$		0			0
$287$	−0.909273	−	2.31820i	−0.0536727	−	0.136839i
$288$		0			0
$289$	−15.1786			−0.892856
$290$		0			0
$291$			13.1577i			0.771317i
$292$		0			0
$293$	17.1995			1.00481			0.502404	−	0.864633i	$-0.332449\pi$
$293$	17.1995			1.00481			0.502404	+	0.864633i	$0.332449\pi$
$294$		0			0
$295$		−	48.7239i		−	2.83682i
$296$		0			0
$297$	−1.82537			−0.105919
$298$		0			0
$299$	26.0763	−	7.11917i	1.50803	−	0.411712i
$300$		0			0
$301$	2.80999	+	7.16410i	0.161965	+	0.412932i
$302$		0			0
$303$	−14.2453			−0.818372
$304$		0			0
$305$	−48.3277			−2.76723
$306$		0			0
$307$		−	18.2741i		−	1.04296i	−0.853265	−	0.521478i	$-0.825381\pi$
$307$		−	18.2741i		−	1.04296i	0.853265	−	0.521478i	$-0.174619\pi$
$308$		0			0
$309$		−	1.67757i		−	0.0954338i
$310$		0			0
$311$			21.5738i			1.22334i	0.791115	+	0.611668i	$0.209501\pi$
$311$			21.5738i			1.22334i	−0.791115	+	0.611668i	$0.790499\pi$
$312$		0			0
$313$	21.2454			1.20086			0.600429	−	0.799678i	$-0.294996\pi$
$313$	21.2454			1.20086			0.600429	+	0.799678i	$0.294996\pi$
$314$		0			0
$315$	−10.3071	+	4.04276i	−0.580737	+	0.227784i
$316$		0			0
$317$	−14.9417			−0.839211			−0.419606	−	0.907706i	$-0.637832\pi$
$317$	−14.9417			−0.839211			−0.419606	+	0.907706i	$0.637832\pi$
$318$		0			0
$319$			10.1271i			0.567008i
$320$		0			0
$321$	2.36527			0.132016
$322$		0			0
$323$	−5.77086			−0.321099
$324$		0			0
$325$		−	70.5174i		−	3.91160i
$326$		0			0
$327$	13.9734			0.772729
$328$		0			0
$329$	2.32386	+	5.92471i	0.128119	+	0.326640i
$330$		0			0
$331$	−27.6313			−1.51875			−0.759376	−	0.650652i	$-0.774496\pi$
$331$	−27.6313			−1.51875			−0.759376	+	0.650652i	$0.774496\pi$
$332$		0			0
$333$			0.592620i			0.0324754i
$334$		0			0
$335$		−	39.9869i		−	2.18472i
$336$		0			0
$337$			34.9895i			1.90600i	0.302972	+	0.952999i	$0.402021\pi$
$337$			34.9895i			1.90600i	−0.302972	+	0.952999i	$0.597979\pi$
$338$		0			0
$339$	−2.50110			−0.135841
$340$		0			0
$341$	4.61535			0.249935
$342$		0			0
$343$	−8.04598	+	16.6812i	−0.434442	+	0.900700i
$344$		0			0
$345$	19.3603	−	5.28563i	1.04233	−	0.284569i
$346$		0			0
$347$	23.9364			1.28497			0.642487	−	0.766296i	$-0.277903\pi$
$347$	23.9364			1.28497			0.642487	+	0.766296i	$0.277903\pi$
$348$		0			0
$349$		−	35.3685i		−	1.89323i	−0.322359	−	0.946617i	$-0.604476\pi$
$349$		−	35.3685i		−	1.89323i	0.322359	−	0.946617i	$-0.395524\pi$
$350$		0			0
$351$	−5.63628			−0.300842
$352$		0			0
$353$			30.4548i			1.62095i	0.585775	+	0.810473i	$0.300790\pi$
$353$			30.4548i			1.62095i	−0.585775	+	0.810473i	$0.699210\pi$
$354$		0			0
$355$	21.1709			1.12363
$356$		0			0
$357$	−1.30385	−	3.32417i	−0.0690069	−	0.175934i
$358$		0			0
$359$		−	3.39530i		−	0.179197i	−0.995978	−	0.0895986i	$-0.971442\pi$
$359$		−	3.39530i		−	0.179197i	0.995978	−	0.0895986i	$-0.0285584\pi$
$360$		0			0
$361$	−0.716256			−0.0376977
$362$		0			0
$363$			7.66803i			0.402467i
$364$		0			0
$365$		−	0.961179i		−	0.0503104i
$366$		0			0
$367$	27.1391			1.41665			0.708324	−	0.705887i	$-0.249452\pi$
$367$	27.1391			1.41665			0.708324	+	0.705887i	$0.249452\pi$
$368$		0			0
$369$		−	0.941186i		−	0.0489962i
$370$		0			0
$371$	−7.53983	−	19.2228i	−0.391448	−	0.998001i
$372$		0			0
$373$			20.1619i			1.04394i	0.852963	+	0.521972i	$0.174804\pi$
$373$			20.1619i			1.04394i	−0.852963	+	0.521972i	$0.825196\pi$
$374$		0			0
$375$		−	31.4324i		−	1.62316i
$376$		0			0
$377$			31.2699i			1.61048i
$378$		0			0
$379$			12.8262i			0.658840i	0.944184	+	0.329420i	$0.106853\pi$
$379$			12.8262i			0.658840i	−0.944184	+	0.329420i	$0.893147\pi$
$380$		0			0
$381$			4.82083i			0.246978i
$382$		0			0
$383$	7.15646			0.365678			0.182839	−	0.983143i	$-0.441471\pi$
$383$	7.15646			0.365678			0.182839	+	0.983143i	$0.441471\pi$
$384$		0			0
$385$	−7.37953	−	18.8142i	−0.376096	−	0.958859i
$386$		0			0
$387$			2.90862i			0.147853i
$388$		0			0
$389$		−	25.9304i		−	1.31472i	−0.753575	−	0.657362i	$-0.771672\pi$
$389$		−	25.9304i		−	1.31472i	0.753575	−	0.657362i	$-0.228328\pi$
$390$		0			0
$391$	1.70469	+	6.24397i	0.0862097	+	0.315771i
$392$		0			0
$393$	−13.3351			−0.672668
$394$		0			0
$395$			40.4984i			2.03769i
$396$		0			0
$397$			25.3619i			1.27288i	0.771327	+	0.636439i	$0.219593\pi$
$397$			25.3619i			1.27288i	−0.771327	+	0.636439i	$0.780407\pi$
$398$		0			0
$399$	4.13096	+	10.5319i	0.206807	+	0.527256i
$400$		0			0
$401$			11.5114i			0.574852i	0.957803	+	0.287426i	$0.0927996\pi$
$401$			11.5114i			0.574852i	−0.957803	+	0.287426i	$0.907200\pi$
$402$		0			0
$403$	14.2511			0.709896
$404$		0			0
$405$	−4.18466			−0.207937
$406$		0			0
$407$	−1.08175			−0.0536203
$408$		0			0
$409$		−	32.9472i		−	1.62913i	−0.580070	−	0.814566i	$-0.696975\pi$
$409$		−	32.9472i		−	1.62913i	0.580070	−	0.814566i	$-0.303025\pi$
$410$		0			0
$411$	12.3538			0.609367
$412$		0			0
$413$	−11.2487	−	28.6786i	−0.553511	−	1.41118i
$414$		0			0
$415$	25.8815			1.27047
$416$		0			0
$417$	13.5018			0.661185
$418$		0			0
$419$	−19.1690			−0.936469			−0.468234	−	0.883604i	$-0.655110\pi$
$419$	−19.1690			−0.936469			−0.468234	+	0.883604i	$0.655110\pi$
$420$		0			0
$421$		−	26.8573i		−	1.30894i	−0.756087	−	0.654471i	$-0.772891\pi$
$421$		−	26.8573i		−	1.30894i	0.756087	−	0.654471i	$-0.227109\pi$
$422$		0			0
$423$			2.40543i			0.116956i
$424$		0			0
$425$	16.8854			0.819063
$426$		0			0
$427$	−28.4453	+	11.1572i	−1.37657	+	0.539934i
$428$		0			0
$429$		−	10.2883i		−	0.496723i
$430$		0			0
$431$			31.8793i			1.53557i	0.640705	+	0.767787i	$0.278642\pi$
$431$			31.8793i			1.53557i	−0.640705	+	0.767787i	$0.721358\pi$
$432$		0			0
$433$	−9.21633			−0.442909			−0.221454	−	0.975171i	$-0.571080\pi$
$433$	−9.21633			−0.442909			−0.221454	+	0.975171i	$0.571080\pi$
$434$		0			0
$435$			23.2163i			1.11314i
$436$		0			0
$437$	−5.40094	−	19.7827i	−0.258362	−	0.946336i
$438$		0			0
$439$			31.8254i			1.51894i	0.650539	+	0.759472i	$0.274543\pi$
$439$			31.8254i			1.51894i	−0.650539	+	0.759472i	$0.725457\pi$
$440$		0			0
$441$	−5.13333	+	4.75909i	−0.244444	+	0.226623i
$442$		0			0
$443$	8.72364			0.414473			0.207236	−	0.978291i	$-0.433553\pi$
$443$	8.72364			0.414473			0.207236	+	0.978291i	$0.433553\pi$
$444$		0			0
$445$	−78.2856			−3.71109
$446$		0			0
$447$	12.7461			0.602870
$448$		0			0
$449$	9.14168			0.431422			0.215711	−	0.976457i	$-0.430793\pi$
$449$	9.14168			0.431422			0.215711	+	0.976457i	$0.430793\pi$
$450$		0			0
$451$	1.71801			0.0808980
$452$		0			0
$453$			16.9527i			0.796506i
$454$		0			0
$455$	−22.7862	−	58.0935i	−1.06823	−	2.72346i
$456$		0			0
$457$			8.78417i			0.410906i	0.978667	+	0.205453i	$0.0658668\pi$
$457$			8.78417i			0.410906i	−0.978667	+	0.205453i	$0.934133\pi$
$458$		0			0
$459$		−	1.34961i		−	0.0629943i
$460$		0			0
$461$		−	12.7282i		−	0.592810i	−0.955062	−	0.296405i	$-0.904212\pi$
$461$		−	12.7282i		−	0.592810i	0.955062	−	0.296405i	$-0.0957878\pi$
$462$		0			0
$463$	0.736780			0.0342411			0.0171205	−	0.999853i	$-0.494550\pi$
$463$	0.736780			0.0342411			0.0171205	+	0.999853i	$0.494550\pi$
$464$		0			0
$465$	10.5807			0.490668
$466$		0			0
$467$	17.5289			0.811140			0.405570	−	0.914064i	$-0.367073\pi$
$467$	17.5289			0.811140			0.405570	+	0.914064i	$0.367073\pi$
$468$		0			0
$469$	−9.23160	−	23.5360i	−0.426275	−	1.08679i
$470$		0			0
$471$		−	11.6179i		−	0.535323i
$472$		0			0
$473$	−5.30930			−0.244122
$474$		0			0
$475$	−53.4979			−2.45465
$476$		0			0
$477$		−	7.80446i		−	0.357342i
$478$		0			0
$479$	−15.2212			−0.695475			−0.347737	−	0.937592i	$-0.613050\pi$
$479$	−15.2212			−0.695475			−0.347737	+	0.937592i	$0.613050\pi$
$480$		0			0
$481$	−3.34017			−0.152299
$482$		0			0
$483$	10.1751	−	7.58072i	0.462983	−	0.344935i
$484$		0			0
$485$	−55.0604			−2.50016
$486$		0			0
$487$	−2.00583			−0.0908928			−0.0454464	−	0.998967i	$-0.514471\pi$
$487$	−2.00583			−0.0908928			−0.0454464	+	0.998967i	$0.514471\pi$
$488$		0			0
$489$			5.74976i			0.260013i
$490$		0			0
$491$	22.9197			1.03435			0.517175	−	0.855880i	$-0.326984\pi$
$491$	22.9197			1.03435			0.517175	+	0.855880i	$0.326984\pi$
$492$		0			0
$493$	−7.48759			−0.337224
$494$		0			0
$495$		−	7.63854i		−	0.343327i
$496$		0			0
$497$	12.4611	−	4.88763i	0.558955	−	0.219240i
$498$		0			0
$499$	−34.1508			−1.52880			−0.764400	−	0.644743i	$-0.776964\pi$
$499$	−34.1508			−1.52880			−0.764400	+	0.644743i	$0.776964\pi$
$500$		0			0
$501$	−19.5553			−0.873666
$502$		0			0
$503$	−32.3500			−1.44241			−0.721207	−	0.692720i	$-0.756412\pi$
$503$	−32.3500			−1.44241			−0.721207	+	0.692720i	$0.756412\pi$
$504$		0			0
$505$		−	59.6118i		−	2.65269i
$506$		0			0
$507$		−	18.7677i		−	0.833501i
$508$		0			0
$509$		−	15.3113i		−	0.678660i	−0.940667	−	0.339330i	$-0.889800\pi$
$509$		−	15.3113i		−	0.678660i	0.940667	−	0.339330i	$-0.110200\pi$
$510$		0			0
$511$	−0.221903	−	0.565743i	−0.00981641	−	0.0250270i
$512$		0			0
$513$			4.27595i			0.188788i
$514$		0			0
$515$	7.02007			0.309341
$516$		0			0
$517$	−4.39079			−0.193107
$518$		0			0
$519$	22.7092			0.996824
$520$		0			0
$521$	−36.5112			−1.59958			−0.799792	−	0.600278i	$-0.795057\pi$
$521$	−36.5112			−1.59958			−0.799792	+	0.600278i	$0.795057\pi$
$522$		0			0
$523$	25.0196			1.09403			0.547015	−	0.837123i	$-0.315764\pi$
$523$	25.0196			1.09403			0.547015	+	0.837123i	$0.315764\pi$
$524$		0			0
$525$	−12.0871	−	30.8162i	−0.527525	−	1.34493i
$526$		0			0
$527$			3.41242i			0.148647i
$528$		0			0
$529$	−19.8092	+	11.6875i	−0.861268	+	0.508151i
$530$		0			0
$531$		−	11.6435i		−	0.505284i
$532$		0			0
$533$	5.30479			0.229776
$534$		0			0
$535$			9.89782i			0.427920i
$536$		0			0
$537$			12.0764i			0.521137i
$538$		0			0
$539$	−8.68709	−	9.37022i	−0.374179	−	0.403604i
$540$		0			0
$541$	14.5238			0.624426			0.312213	−	0.950012i	$-0.398930\pi$
$541$	14.5238			0.624426			0.312213	+	0.950012i	$0.398930\pi$
$542$		0			0
$543$		−	3.54990i		−	0.152341i
$544$		0			0
$545$			58.4737i			2.50474i
$546$		0			0
$547$	38.0343			1.62623			0.813116	−	0.582102i	$-0.197770\pi$
$547$	38.0343			1.62623			0.813116	+	0.582102i	$0.197770\pi$
$548$		0			0
$549$	−11.5488			−0.492890
$550$		0			0
$551$	23.7228			1.01063
$552$		0			0
$553$	9.34968	+	23.8371i	0.397589	+	1.01366i
$554$		0			0
$555$	−2.47991			−0.105266
$556$		0			0
$557$			15.1605i			0.642372i	0.947016	+	0.321186i	$0.104082\pi$
$557$			15.1605i			0.642372i	−0.947016	+	0.321186i	$0.895918\pi$
$558$		0			0
$559$	−16.3938			−0.693383
$560$		0			0
$561$	2.46353			0.104010
$562$		0			0
$563$	12.0126			0.506270			0.253135	−	0.967431i	$-0.418538\pi$
$563$	12.0126			0.506270			0.253135	+	0.967431i	$0.418538\pi$
$564$		0			0
$565$		−	10.4663i		−	0.440319i
$566$		0			0
$567$	−2.46306	+	0.966093i	−0.103439	+	0.0405721i
$568$		0			0
$569$			14.2498i			0.597385i	0.954349	+	0.298692i	$0.0965504\pi$
$569$			14.2498i			0.597385i	−0.954349	+	0.298692i	$0.903450\pi$
$570$		0			0
$571$			43.8426i			1.83476i	0.398017	+	0.917378i	$0.369698\pi$
$571$			43.8426i			1.83476i	−0.398017	+	0.917378i	$0.630302\pi$
$572$		0			0
$573$	9.51786			0.397614
$574$		0			0
$575$	15.8030	+	57.8838i	0.659033	+	2.41392i
$576$		0			0
$577$			26.3434i			1.09669i	0.836252	+	0.548345i	$0.184742\pi$
$577$			26.3434i			1.09669i	−0.836252	+	0.548345i	$0.815258\pi$
$578$		0			0
$579$		−	2.10667i		−	0.0875501i
$580$		0			0
$581$	15.2337	−	5.97515i	0.632000	−	0.247891i
$582$		0			0
$583$	14.2460			0.590009
$584$		0			0
$585$		−	23.5859i		−	0.975157i
$586$		0			0
$587$		−	41.1320i		−	1.69770i	−0.528635	−	0.848849i	$-0.677296\pi$
$587$		−	41.1320i		−	1.69770i	0.528635	−	0.848849i	$-0.322704\pi$
$588$		0			0
$589$		−	10.8115i		−	0.445481i
$590$		0			0
$591$		−	17.7016i		−	0.728146i
$592$		0			0
$593$			19.1990i			0.788409i	0.919023	+	0.394205i	$0.128980\pi$
$593$			19.1990i			0.788409i	−0.919023	+	0.394205i	$0.871020\pi$
$594$		0			0
$595$	13.9105	−	5.45615i	0.570274	−	0.223680i
$596$		0			0
$597$			10.0241i			0.410259i
$598$		0			0
$599$	−34.2099			−1.39778			−0.698889	−	0.715230i	$-0.746322\pi$
$599$	−34.2099			−1.39778			−0.698889	+	0.715230i	$0.746322\pi$
$600$		0			0
$601$		−	44.4374i		−	1.81264i	−0.422594	−	0.906319i	$-0.638880\pi$
$601$		−	44.4374i		−	1.81264i	0.422594	−	0.906319i	$-0.361120\pi$
$602$		0			0
$603$		−	9.55560i		−	0.389134i
$604$		0			0
$605$	−32.0881			−1.30457
$606$		0			0
$607$		−	6.08184i		−	0.246854i	−0.992354	−	0.123427i	$-0.960611\pi$
$607$		−	6.08184i		−	0.246854i	0.992354	−	0.123427i	$-0.0393885\pi$
$608$		0			0
$609$	5.35985	+	13.6650i	0.217192	+	0.553733i
$610$		0			0
$611$	−13.5577			−0.548484
$612$		0			0
$613$		−	9.98116i		−	0.403135i	−0.979475	−	0.201568i	$-0.935396\pi$
$613$		−	9.98116i		−	0.403135i	0.979475	−	0.201568i	$-0.0646036\pi$
$614$		0			0
$615$	3.93854			0.158817
$616$		0			0
$617$			13.6405i			0.549147i	0.961566	+	0.274574i	$0.0885367\pi$
$617$			13.6405i			0.549147i	−0.961566	+	0.274574i	$0.911463\pi$
$618$		0			0
$619$	11.3459			0.456030			0.228015	−	0.973658i	$-0.426776\pi$
$619$	11.3459			0.456030			0.228015	+	0.973658i	$0.426776\pi$
$620$		0			0
$621$	4.62651	−	1.26310i	0.185655	−	0.0506864i
$622$		0			0
$623$	−46.0784	+	18.0734i	−1.84609	+	0.724097i
$624$		0			0
$625$	68.9770			2.75908
$626$		0			0
$627$	−7.80518			−0.311709
$628$		0			0
$629$		−	0.799805i		−	0.0318903i
$630$		0			0
$631$		−	12.0968i		−	0.481565i	−0.970579	−	0.240782i	$-0.922596\pi$
$631$		−	12.0968i		−	0.481565i	0.970579	−	0.240782i	$-0.0774040\pi$
$632$		0			0
$633$			15.8426i			0.629685i
$634$		0			0
$635$	−20.1735			−0.800561
$636$		0			0
$637$	−26.8236	−	28.9329i	−1.06279	−	1.14636i
$638$		0			0
$639$	5.05917			0.200138
$640$		0			0
$641$		−	21.9652i		−	0.867571i	−0.901016	−	0.433786i	$-0.857177\pi$
$641$		−	21.9652i		−	0.867571i	0.901016	−	0.433786i	$-0.142823\pi$
$642$		0			0
$643$	−34.5336			−1.36187			−0.680936	−	0.732343i	$-0.738427\pi$
$643$	−34.5336			−1.36187			−0.680936	+	0.732343i	$0.738427\pi$
$644$		0			0
$645$	−12.1716			−0.479255
$646$		0			0
$647$			24.4249i			0.960240i	0.877203	+	0.480120i	$0.159407\pi$
$647$			24.4249i			0.960240i	−0.877203	+	0.480120i	$0.840593\pi$
$648$		0			0
$649$	21.2536			0.834278
$650$		0			0
$651$	6.22772	−	2.44272i	0.244084	−	0.0957376i
$652$		0			0
$653$	−7.28377			−0.285036			−0.142518	−	0.989792i	$-0.545520\pi$
$653$	−7.28377			−0.285036			−0.142518	+	0.989792i	$0.545520\pi$
$654$		0			0
$655$		−	55.8029i		−	2.18040i
$656$		0			0
$657$		−	0.229691i		−	0.00896111i
$658$		0			0
$659$			5.05252i			0.196818i	0.995146	+	0.0984092i	$0.0313754\pi$
$659$			5.05252i			0.196818i	−0.995146	+	0.0984092i	$0.968625\pi$
$660$		0			0
$661$	16.8335			0.654747			0.327374	−	0.944895i	$-0.393836\pi$
$661$	16.8335			0.654747			0.327374	+	0.944895i	$0.393836\pi$
$662$		0			0
$663$	7.60677			0.295422
$664$		0			0
$665$	−44.0725	+	17.2867i	−1.70906	+	0.670348i
$666$		0			0
$667$	−7.00763	−	25.6677i	−0.271336	−	0.993859i
$668$		0			0
$669$	−24.1995			−0.935607
$670$		0			0
$671$		−	21.0808i		−	0.813814i
$672$		0			0
$673$	36.9364			1.42379			0.711896	−	0.702284i	$-0.247837\pi$
$673$	36.9364			1.42379			0.711896	+	0.702284i	$0.247837\pi$
$674$		0			0
$675$		−	12.5113i		−	0.481562i
$676$		0			0
$677$	7.40366			0.284546			0.142273	−	0.989827i	$-0.454559\pi$
$677$	7.40366			0.284546			0.142273	+	0.989827i	$0.454559\pi$
$678$		0			0
$679$	−32.4082	+	12.7115i	−1.24371	+	0.487824i
$680$		0			0
$681$			17.7083i			0.678583i
$682$		0			0
$683$	−13.3099			−0.509288			−0.254644	−	0.967035i	$-0.581958\pi$
$683$	−13.3099			−0.509288			−0.254644	+	0.967035i	$0.581958\pi$
$684$		0			0
$685$			51.6963i			1.97521i
$686$		0			0
$687$		−	10.9498i		−	0.417762i
$688$		0			0
$689$	43.9881			1.67581
$690$		0			0
$691$		−	20.1229i		−	0.765511i	−0.923850	−	0.382755i	$-0.874975\pi$
$691$		−	20.1229i		−	0.765511i	0.923850	−	0.382755i	$-0.125025\pi$
$692$		0			0
$693$	−1.76347	−	4.49599i	−0.0669888	−	0.170789i
$694$		0			0
$695$			56.5003i			2.14318i
$696$		0			0
$697$			1.27023i			0.0481135i
$698$		0			0
$699$			14.7878i			0.559328i
$700$		0			0
$701$		−	12.6451i		−	0.477598i	−0.971069	−	0.238799i	$-0.923246\pi$
$701$		−	12.6451i		−	0.477598i	0.971069	−	0.238799i	$-0.0767536\pi$
$702$		0			0
$703$			2.53401i			0.0955721i
$704$		0			0
$705$	−10.0659			−0.379103
$706$		0			0
$707$	−13.7623	−	35.0871i	−0.517584	−	1.31959i
$708$		0			0
$709$			23.4317i			0.879996i	0.897999	+	0.439998i	$0.145021\pi$
$709$			23.4317i			0.879996i	−0.897999	+	0.439998i	$0.854979\pi$
$710$		0			0
$711$			9.67783i			0.362947i
$712$		0			0
$713$	−11.6979	+	3.19368i	−0.438090	+	0.119604i
$714$		0			0
$715$	43.0529			1.61009
$716$		0			0
$717$		−	3.80996i		−	0.142286i
$718$		0			0
$719$			11.5922i			0.432317i	0.976358	+	0.216159i	$0.0693528\pi$
$719$			11.5922i			0.432317i	−0.976358	+	0.216159i	$0.930647\pi$
$720$		0			0
$721$	4.13196	−	1.62069i	0.153882	−	0.0603577i
$722$		0			0
$723$			1.90190i			0.0707326i
$724$		0			0
$725$	−69.4125			−2.57792
$726$		0			0
$727$	48.2282			1.78869			0.894343	−	0.447382i	$-0.147644\pi$
$727$	48.2282			1.78869			0.894343	+	0.447382i	$0.147644\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	3.92549i		−	0.145190i
$732$		0			0
$733$	−21.4926			−0.793849			−0.396924	−	0.917851i	$-0.629923\pi$
$733$	−21.4926			−0.793849			−0.396924	+	0.917851i	$0.629923\pi$
$734$		0			0
$735$	−19.9151	−	21.4812i	−0.734581	−	0.792347i
$736$		0			0
$737$	17.4425			0.642502
$738$		0			0
$739$	−24.5260			−0.902203			−0.451101	−	0.892473i	$-0.648969\pi$
$739$	−24.5260			−0.902203			−0.451101	+	0.892473i	$0.648969\pi$
$740$		0			0
$741$	−24.1005			−0.885352
$742$		0			0
$743$			34.1958i			1.25452i	0.778809	+	0.627262i	$0.215824\pi$
$743$			34.1958i			1.25452i	−0.778809	+	0.627262i	$0.784176\pi$
$744$		0			0
$745$			53.3380i			1.95415i
$746$		0			0
$747$	6.18486			0.226292
$748$		0			0
$749$	2.28507	+	5.82579i	0.0834945	+	0.212870i
$750$		0			0
$751$		−	24.5018i		−	0.894082i	−0.894513	−	0.447041i	$-0.852478\pi$
$751$		−	24.5018i		−	0.894082i	0.894513	−	0.447041i	$-0.147522\pi$
$752$		0			0
$753$			5.54304i			0.202000i
$754$		0			0
$755$	−70.9411			−2.58181
$756$		0			0
$757$			30.9296i			1.12416i	0.827084	+	0.562078i	$0.189998\pi$
$757$			30.9296i			1.12416i	−0.827084	+	0.562078i	$0.810002\pi$
$758$		0			0
$759$	2.30562	+	8.44508i	0.0836886	+	0.306537i
$760$		0			0
$761$		−	26.6994i		−	0.967851i	−0.875109	−	0.483925i	$-0.839211\pi$
$761$		−	26.6994i		−	0.967851i	0.875109	−	0.483925i	$-0.160789\pi$
$762$		0			0
$763$	13.4996	+	34.4172i	0.488717	+	1.24599i
$764$		0			0
$765$	5.64765			0.204191
$766$		0			0
$767$	65.6259			2.36961
$768$		0			0
$769$	−3.64634			−0.131490			−0.0657451	−	0.997836i	$-0.520942\pi$
$769$	−3.64634			−0.131490			−0.0657451	+	0.997836i	$0.520942\pi$
$770$		0			0
$771$	18.4310			0.663776
$772$		0			0
$773$	55.0457			1.97986			0.989928	−	0.141571i	$-0.0452154\pi$
$773$	55.0457			1.97986			0.989928	+	0.141571i	$0.0452154\pi$
$774$		0			0
$775$			31.6343i			1.13634i
$776$		0			0
$777$	−1.45966	+	0.572526i	−0.0523650	+	0.0205392i
$778$		0			0
$779$		−	4.02446i		−	0.144191i
$780$		0			0
$781$			9.23485i			0.330449i
$782$		0			0
$783$			5.54797i			0.198268i
$784$		0			0
$785$	48.6167			1.73521
$786$		0			0
$787$	13.4562			0.479661			0.239831	−	0.970815i	$-0.422908\pi$
$787$	13.4562			0.479661			0.239831	+	0.970815i	$0.422908\pi$
$788$		0			0
$789$	18.2412			0.649404
$790$		0			0
$791$	−2.41630	−	6.16037i	−0.0859137	−	0.219038i
$792$		0			0
$793$		−	65.0922i		−	2.31149i
$794$		0			0
$795$	32.6590			1.15829
$796$		0			0
$797$	36.1027			1.27882			0.639411	−	0.768865i	$-0.279178\pi$
$797$	36.1027			1.27882			0.639411	+	0.768865i	$0.279178\pi$
$798$		0			0
$799$		−	3.24638i		−	0.114849i
$800$		0			0
$801$	−18.7078			−0.661007
$802$		0			0
$803$	0.419271			0.0147957
$804$		0			0
$805$	31.7227	+	42.5793i	1.11808	+	1.50072i
$806$		0			0
$807$	−11.7654			−0.414162
$808$		0			0
$809$	−12.8480			−0.451713			−0.225857	−	0.974161i	$-0.572518\pi$
$809$	−12.8480			−0.451713			−0.225857	+	0.974161i	$0.572518\pi$
$810$		0			0
$811$			34.5522i			1.21329i	0.794973	+	0.606645i	$0.207485\pi$
$811$			34.5522i			1.21329i	−0.794973	+	0.606645i	$0.792515\pi$
$812$		0			0
$813$	10.6020			0.371828
$814$		0			0
$815$	−24.0608			−0.842812
$816$		0			0
$817$			12.4371i			0.435119i
$818$		0			0
$819$	−5.44517	−	13.8825i	−0.190270	−	0.485094i
$820$		0			0
$821$	52.2393			1.82317			0.911583	−	0.411116i	$-0.134861\pi$
$821$	52.2393			1.82317			0.911583	+	0.411116i	$0.134861\pi$
$822$		0			0
$823$	44.6554			1.55659			0.778295	−	0.627899i	$-0.216085\pi$
$823$	44.6554			1.55659			0.778295	+	0.627899i	$0.216085\pi$
$824$		0			0
$825$	22.8378			0.795110
$826$		0			0
$827$			27.0087i			0.939185i	0.882883	+	0.469593i	$0.155599\pi$
$827$			27.0087i			0.939185i	−0.882883	+	0.469593i	$0.844401\pi$
$828$		0			0
$829$		−	32.9928i		−	1.14589i	−0.819595	−	0.572943i	$-0.805802\pi$
$829$		−	32.9928i		−	1.14589i	0.819595	−	0.572943i	$-0.194198\pi$
$830$		0			0
$831$			1.65673i			0.0574714i
$832$		0			0
$833$	6.92798	−	6.42290i	0.240040	−	0.222540i
$834$		0			0
$835$		−	81.8322i		−	2.83192i
$836$		0			0
$837$	2.52845			0.0873960
$838$		0			0
$839$	21.8396			0.753987			0.376994	−	0.926216i	$-0.376958\pi$
$839$	21.8396			0.753987			0.376994	+	0.926216i	$0.376958\pi$
$840$		0			0
$841$	1.77997			0.0613783
$842$		0			0
$843$	−4.69240			−0.161615
$844$		0			0
$845$	78.5362			2.70173
$846$		0			0
$847$	−18.8868	+	7.40803i	−0.648959	+	0.254543i
$848$		0			0
$849$			17.5213i			0.601329i
$850$		0			0
$851$	2.74176	−	0.748537i	0.0939864	−	0.0256595i
$852$		0			0
$853$			10.9620i			0.375332i	0.982233	+	0.187666i	$0.0600922\pi$
$853$			10.9620i			0.375332i	−0.982233	+	0.187666i	$0.939908\pi$
$854$		0			0
$855$	−17.8934			−0.611941
$856$		0			0
$857$		−	12.6558i		−	0.432314i	−0.976359	−	0.216157i	$-0.930648\pi$
$857$		−	12.6558i		−	0.432314i	0.976359	−	0.216157i	$-0.0693523\pi$
$858$		0			0
$859$		−	9.16755i		−	0.312793i	−0.987694	−	0.156396i	$-0.950012\pi$
$859$		−	9.16755i		−	0.312793i	0.987694	−	0.156396i	$-0.0499877\pi$
$860$		0			0
$861$	2.31820	−	0.909273i	0.0790040	−	0.0309879i
$862$		0			0
$863$	42.3252			1.44077			0.720384	−	0.693576i	$-0.243966\pi$
$863$	42.3252			1.44077			0.720384	+	0.693576i	$0.243966\pi$
$864$		0			0
$865$			95.0302i			3.23112i
$866$		0			0
$867$		−	15.1786i		−	0.515491i
$868$		0			0
$869$	−17.6656			−0.599264
$870$		0			0
$871$	53.8581			1.82491
$872$		0			0
$873$	−13.1577			−0.445320
$874$		0			0
$875$	77.4198	−	30.3666i	2.61727	−	1.02658i
$876$		0			0
$877$	−49.7990			−1.68159			−0.840796	−	0.541351i	$-0.817913\pi$
$877$	−49.7990			−1.68159			−0.840796	+	0.541351i	$0.817913\pi$
$878$		0			0
$879$			17.1995i			0.580126i
$880$		0			0
$881$	22.2859			0.750830			0.375415	−	0.926857i	$-0.377500\pi$
$881$	22.2859			0.750830			0.375415	+	0.926857i	$0.377500\pi$
$882$		0			0
$883$	−45.8981			−1.54459			−0.772297	−	0.635262i	$-0.780892\pi$
$883$	−45.8981			−1.54459			−0.772297	+	0.635262i	$0.780892\pi$
$884$		0			0
$885$	48.7239			1.63784
$886$		0			0
$887$			2.69739i			0.0905696i	0.998974	+	0.0452848i	$0.0144195\pi$
$887$			2.69739i			0.0905696i	−0.998974	+	0.0452848i	$0.985580\pi$
$888$		0			0
$889$	−11.8740	+	4.65737i	−0.398241	+	0.156203i
$890$		0			0
$891$		−	1.82537i		−	0.0611521i
$892$		0			0
$893$			10.2855i			0.344190i
$894$		0			0
$895$	−50.5357			−1.68922
$896$		0			0
$897$	7.11917	+	26.0763i	0.237702	+	0.870662i
$898$		0			0
$899$		−	14.0278i		−	0.467852i
$900$		0			0
$901$			10.5330i			0.350904i
$902$		0			0
$903$	−7.16410	+	2.80999i	−0.238406	+	0.0935108i
$904$		0			0
$905$	14.8551			0.493800
$906$		0			0
$907$			17.3172i			0.575009i	0.957779	+	0.287504i	$0.0928255\pi$
$907$			17.3172i			0.575009i	−0.957779	+	0.287504i	$0.907174\pi$
$908$		0			0
$909$		−	14.2453i		−	0.472487i
$910$		0			0
$911$		−	11.1520i		−	0.369483i	−0.982787	−	0.184742i	$-0.940855\pi$
$911$		−	11.1520i		−	0.369483i	0.982787	−	0.184742i	$-0.0591448\pi$
$912$		0			0
$913$			11.2896i			0.373633i
$914$		0			0
$915$		−	48.3277i		−	1.59766i
$916$		0			0
$917$	−12.8830	−	32.8452i	−0.425433	−	1.08465i
$918$		0			0
$919$			2.90313i			0.0957655i	0.998853	+	0.0478828i	$0.0152474\pi$
$919$			2.90313i			0.0957655i	−0.998853	+	0.0478828i	$0.984753\pi$
$920$		0			0
$921$	18.2741			0.602151
$922$		0			0
$923$			28.5149i			0.938580i
$924$		0			0
$925$		−	7.41447i		−	0.243786i
$926$		0			0
$927$	1.67757			0.0550987
$928$		0			0
$929$		−	42.4157i		−	1.39161i	−0.718229	−	0.695807i	$-0.755047\pi$
$929$		−	42.4157i		−	1.39161i	0.718229	−	0.695807i	$-0.244953\pi$
$930$		0			0
$931$	−21.9499	+	20.3496i	−0.719378	+	0.666932i
$932$		0			0
$933$	−21.5738			−0.706293
$934$		0			0
$935$			10.3090i			0.337141i
$936$		0			0
$937$	−21.6954			−0.708757			−0.354378	−	0.935102i	$-0.615308\pi$
$937$	−21.6954			−0.708757			−0.354378	+	0.935102i	$0.615308\pi$
$938$		0			0
$939$			21.2454i			0.693316i
$940$		0			0
$941$	−31.2006			−1.01711			−0.508556	−	0.861029i	$-0.669820\pi$
$941$	−31.2006			−1.01711			−0.508556	+	0.861029i	$0.669820\pi$
$942$		0			0
$943$	−4.35441	+	1.18881i	−0.141799	+	0.0387130i
$944$		0			0
$945$	−4.04276	−	10.3071i	−0.131511	−	0.335289i
$946$		0			0
$947$	−23.3015			−0.757197			−0.378598	−	0.925561i	$-0.623594\pi$
$947$	−23.3015			−0.757197			−0.378598	+	0.925561i	$0.623594\pi$
$948$		0			0
$949$	1.29460			0.0420246
$950$		0			0
$951$		−	14.9417i		−	0.484519i
$952$		0			0
$953$		−	6.59921i		−	0.213769i	−0.994271	−	0.106885i	$-0.965912\pi$
$953$		−	6.59921i		−	0.213769i	0.994271	−	0.106885i	$-0.0340875\pi$
$954$		0			0
$955$			39.8290i			1.28884i
$956$		0			0
$957$	−10.1271			−0.327362
$958$		0			0
$959$	11.9349	+	30.4281i	0.385398	+	0.982574i
$960$		0			0
$961$	24.6069			0.793772
$962$		0			0
$963$			2.36527i			0.0762196i
$964$		0			0
$965$	8.81567			0.283787
$966$		0			0
$967$	−17.6781			−0.568489			−0.284245	−	0.958752i	$-0.591743\pi$
$967$	−17.6781			−0.568489			−0.284245	+	0.958752i	$0.591743\pi$
$968$		0			0
$969$		−	5.77086i		−	0.185387i
$970$		0			0
$971$	40.4369			1.29768			0.648841	−	0.760924i	$-0.275254\pi$
$971$	40.4369			1.29768			0.648841	+	0.760924i	$0.275254\pi$
$972$		0			0
$973$	13.0440	+	33.2557i	0.418170	+	1.06613i
$974$		0			0
$975$	70.5174			2.25837
$976$		0			0
$977$		−	23.5277i		−	0.752719i	−0.926474	−	0.376360i	$-0.877176\pi$
$977$		−	23.5277i		−	0.752719i	0.926474	−	0.376360i	$-0.122824\pi$
$978$		0			0
$979$		−	34.1486i		−	1.09139i
$980$		0			0
$981$			13.9734i			0.446135i
$982$		0			0
$983$	20.6661			0.659147			0.329573	−	0.944130i	$-0.393095\pi$
$983$	20.6661			0.659147			0.329573	+	0.944130i	$0.393095\pi$
$984$		0			0
$985$	74.0750			2.36023
$986$		0			0
$987$	−5.92471	+	2.32386i	−0.188585	+	0.0739694i
$988$		0			0
$989$	13.4567	−	3.67387i	0.427900	−	0.116822i
$990$		0			0
$991$	13.8115			0.438737			0.219368	−	0.975642i	$-0.429600\pi$
$991$	13.8115			0.438737			0.219368	+	0.975642i	$0.429600\pi$
$992$		0			0
$993$		−	27.6313i		−	0.876852i
$994$		0			0
$995$	−41.9474			−1.32982
$996$		0			0
$997$			5.36830i			0.170016i	0.996380	+	0.0850079i	$0.0270915\pi$
$997$			5.36830i			0.170016i	−0.996380	+	0.0850079i	$0.972908\pi$
$998$		0			0
$999$	−0.592620			−0.0187497

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.k.a.1609.27	yes	32
3.2	odd	2		5796.2.k.d.5473.32		32
7.6	odd	2	inner	1932.2.k.a.1609.26	yes	32
21.20	even	2		5796.2.k.d.5473.2		32
23.22	odd	2	inner	1932.2.k.a.1609.28	yes	32
69.68	even	2		5796.2.k.d.5473.1		32
161.160	even	2	inner	1932.2.k.a.1609.25	✓	32
483.482	odd	2		5796.2.k.d.5473.31		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.25	✓	32	161.160	even	2	inner
1932.2.k.a.1609.26	yes	32	7.6	odd	2	inner
1932.2.k.a.1609.27	yes	32	1.1	even	1	trivial
1932.2.k.a.1609.28	yes	32	23.22	odd	2	inner
5796.2.k.d.5473.1		32	69.68	even	2
5796.2.k.d.5473.2		32	21.20	even	2
5796.2.k.d.5473.31		32	483.482	odd	2
5796.2.k.d.5473.32		32	3.2	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.18466 q^{5} +(-2.46306 + 0.966093i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -4.18466 q^{5} +(-2.46306 + 0.966093i) q^{7} -1.00000 q^{9} -1.82537i q^{11} -5.63628i q^{13} -4.18466i q^{15} +1.34961 q^{17} -4.27595 q^{19} +(-0.966093 - 2.46306i) q^{21} +(1.26310 + 4.62651i) q^{23} +12.5113 q^{25} -1.00000i q^{27} -5.54797 q^{29} +2.52845i q^{31} +1.82537 q^{33} +(10.3071 - 4.04276i) q^{35} -0.592620i q^{37} +5.63628 q^{39} +0.941186i q^{41} -2.90862i q^{43} +4.18466 q^{45} -2.40543i q^{47} +(5.13333 - 4.75909i) q^{49} +1.34961i q^{51} +7.80446i q^{53} +7.63854i q^{55} -4.27595i q^{57} +11.6435i q^{59} +11.5488 q^{61} +(2.46306 - 0.966093i) q^{63} +23.5859i q^{65} +9.55560i q^{67} +(-4.62651 + 1.26310i) q^{69} -5.05917 q^{71} +0.229691i q^{73} +12.5113i q^{75} +(1.76347 + 4.49599i) q^{77} -9.67783i q^{79} +1.00000 q^{81} -6.18486 q^{83} -5.64765 q^{85} -5.54797i q^{87} +18.7078 q^{89} +(5.44517 + 13.8825i) q^{91} -2.52845 q^{93} +17.8934 q^{95} +13.1577 q^{97} +1.82537i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9}+O(q^{10}) \) 32 * q - 32 * q^9	\( 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) \) 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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