LMFDB - Embedded newform 1904.2.j.b.783.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1904,2,Mod(783,1904)] 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("1904.783"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1904, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1904 = 2^{4} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1904.j (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$15.2035165449$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.3
Character	$\chi$	$=$	1904.783
Dual form			1904.2.j.b.783.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69181 q^{3} +1.35274i q^{5} +(-1.38205 + 2.25609i) q^{7} +4.24582 q^{9} +3.05189i q^{11} -1.89309i q^{13} -3.64131i q^{15} +1.00000i q^{17} +6.45459 q^{19} +(3.72022 - 6.07295i) q^{21} +9.21870i q^{23} +3.17010 q^{25} -3.35352 q^{27} +10.1082 q^{29} -8.51357 q^{31} -8.21511i q^{33} +(-3.05189 - 1.86955i) q^{35} -1.85537 q^{37} +5.09582i q^{39} +4.02548i q^{41} -6.23137i q^{43} +5.74348i q^{45} -1.46028 q^{47} +(-3.17987 - 6.23606i) q^{49} -2.69181i q^{51} -0.520824 q^{53} -4.12841 q^{55} -17.3745 q^{57} +11.0102 q^{59} +10.5171i q^{61} +(-5.86795 + 9.57895i) q^{63} +2.56085 q^{65} +10.0173i q^{67} -24.8149i q^{69} -12.7096i q^{71} +1.60926i q^{73} -8.53331 q^{75} +(-6.88534 - 4.21787i) q^{77} -10.2819i q^{79} -3.71045 q^{81} +1.20784 q^{83} -1.35274 q^{85} -27.2093 q^{87} +16.6194i q^{89} +(4.27097 + 2.61634i) q^{91} +22.9169 q^{93} +8.73136i q^{95} +10.8127i q^{97} +12.9578i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{9} - 32 q^{25} - 8 q^{29} + 8 q^{49} + 40 q^{53} + 8 q^{57} - 48 q^{65} - 16 q^{77} + 24 q^{81} + 32 q^{93}+O(q^{100}) $ 24 * q + 32 * q^9 - 32 * q^25 - 8 * q^29 + 8 * q^49 + 40 * q^53 + 8 * q^57 - 48 * q^65 - 16 * q^77 + 24 * q^81 + 32 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1904\mathbb{Z}\right)^\times$.

$n$	$239$	$785$	$1361$	$1429$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−2.69181			−1.55412			−0.777058	−	0.629429i	$-0.783289\pi$
$3$	−2.69181			−1.55412			−0.777058	+	0.629429i	$0.783289\pi$
$4$		0			0
$5$			1.35274i			0.604962i	0.953155	+	0.302481i	$0.0978149\pi$
$5$			1.35274i			0.604962i	−0.953155	+	0.302481i	$0.902185\pi$
$6$		0			0
$7$	−1.38205	+	2.25609i	−0.522366	+	0.852721i
$7$	−1.38205	+	2.25609i	−0.522366	+	0.852721i
$8$		0			0
$9$	4.24582			1.41527
$10$		0			0
$11$			3.05189i			0.920180i	0.887872	+	0.460090i	$0.152183\pi$
$11$			3.05189i			0.920180i	−0.887872	+	0.460090i	$0.847817\pi$
$12$		0			0
$13$		−	1.89309i		−	0.525048i	−0.964925	−	0.262524i	$-0.915445\pi$
$13$		−	1.89309i		−	0.525048i	0.964925	−	0.262524i	$-0.0845549\pi$
$14$		0			0
$15$		−	3.64131i		−	0.940181i
$16$		0			0
$17$			1.00000i			0.242536i
$17$			1.00000i			0.242536i
$18$		0			0
$19$	6.45459			1.48079			0.740393	−	0.672175i	$-0.234640\pi$
$19$	6.45459			1.48079			0.740393	+	0.672175i	$0.234640\pi$
$20$		0			0
$21$	3.72022	−	6.07295i	0.811818	−	1.32523i
$22$		0			0
$23$			9.21870i			1.92223i	0.276147	+	0.961115i	$0.410942\pi$
$23$			9.21870i			1.92223i	−0.276147	+	0.961115i	$0.589058\pi$
$24$		0			0
$25$	3.17010			0.634021
$26$		0			0
$27$	−3.35352			−0.645384
$28$		0			0
$29$	10.1082			1.87704			0.938522	−	0.345219i	$-0.112195\pi$
$29$	10.1082			1.87704			0.938522	+	0.345219i	$0.112195\pi$
$30$		0			0
$31$	−8.51357			−1.52908			−0.764541	−	0.644575i	$-0.777034\pi$
$31$	−8.51357			−1.52908			−0.764541	+	0.644575i	$0.777034\pi$
$32$		0			0
$33$		−	8.21511i		−	1.43007i
$34$		0			0
$35$	−3.05189	−	1.86955i	−0.515864	−	0.316012i
$36$		0			0
$37$	−1.85537			−0.305022			−0.152511	−	0.988302i	$-0.548736\pi$
$37$	−1.85537			−0.305022			−0.152511	+	0.988302i	$0.548736\pi$
$38$		0			0
$39$			5.09582i			0.815985i
$40$		0			0
$41$			4.02548i			0.628674i	0.949311	+	0.314337i	$0.101782\pi$
$41$			4.02548i			0.628674i	−0.949311	+	0.314337i	$0.898218\pi$
$42$		0			0
$43$		−	6.23137i		−	0.950275i	−0.879912	−	0.475137i	$-0.842398\pi$
$43$		−	6.23137i		−	0.950275i	0.879912	−	0.475137i	$-0.157602\pi$
$44$		0			0
$45$			5.74348i			0.856188i
$46$		0			0
$47$	−1.46028			−0.213004			−0.106502	−	0.994312i	$-0.533965\pi$
$47$	−1.46028			−0.213004			−0.106502	+	0.994312i	$0.533965\pi$
$48$		0			0
$49$	−3.17987	−	6.23606i	−0.454267	−	0.890866i
$50$		0			0
$51$		−	2.69181i		−	0.376928i
$52$		0			0
$53$	−0.520824			−0.0715407			−0.0357703	−	0.999360i	$-0.511388\pi$
$53$	−0.520824			−0.0715407			−0.0357703	+	0.999360i	$0.511388\pi$
$54$		0			0
$55$	−4.12841			−0.556674
$56$		0			0
$57$	−17.3745			−2.30131
$58$		0			0
$59$	11.0102			1.43340			0.716702	−	0.697379i	$-0.245651\pi$
$59$	11.0102			1.43340			0.716702	+	0.697379i	$0.245651\pi$
$60$		0			0
$61$			10.5171i			1.34658i	0.739379	+	0.673290i	$0.235119\pi$
$61$			10.5171i			1.34658i	−0.739379	+	0.673290i	$0.764881\pi$
$62$		0			0
$63$	−5.86795	+	9.57895i	−0.739292	+	1.20683i
$64$		0			0
$65$	2.56085			0.317634
$66$		0			0
$67$			10.0173i			1.22380i	0.790934	+	0.611902i	$0.209595\pi$
$67$			10.0173i			1.22380i	−0.790934	+	0.611902i	$0.790405\pi$
$68$		0			0
$69$		−	24.8149i		−	2.98737i
$70$		0			0
$71$		−	12.7096i		−	1.50836i	−0.656669	−	0.754179i	$-0.728035\pi$
$71$		−	12.7096i		−	1.50836i	0.656669	−	0.754179i	$-0.271965\pi$
$72$		0			0
$73$			1.60926i			0.188349i	0.995556	+	0.0941745i	$0.0300212\pi$
$73$			1.60926i			0.188349i	−0.995556	+	0.0941745i	$0.969979\pi$
$74$		0			0
$75$	−8.53331			−0.985341
$76$		0			0
$77$	−6.88534	−	4.21787i	−0.784657	−	0.480671i
$78$		0			0
$79$		−	10.2819i		−	1.15681i	−0.815750	−	0.578405i	$-0.803676\pi$
$79$		−	10.2819i		−	1.15681i	0.815750	−	0.578405i	$-0.196324\pi$
$80$		0			0
$81$	−3.71045			−0.412273
$82$		0			0
$83$	1.20784			0.132578			0.0662890	−	0.997800i	$-0.478884\pi$
$83$	1.20784			0.132578			0.0662890	+	0.997800i	$0.478884\pi$
$84$		0			0
$85$	−1.35274			−0.146725
$86$		0			0
$87$	−27.2093			−2.91714
$88$		0			0
$89$			16.6194i			1.76165i	0.473443	+	0.880825i	$0.343011\pi$
$89$			16.6194i			1.76165i	−0.473443	+	0.880825i	$0.656989\pi$
$90$		0			0
$91$	4.27097	+	2.61634i	0.447719	+	0.274267i
$92$		0			0
$93$	22.9169			2.37637
$94$		0			0
$95$			8.73136i			0.895819i
$96$		0			0
$97$			10.8127i			1.09786i	0.835867	+	0.548931i	$0.184965\pi$
$97$			10.8127i			1.09786i	−0.835867	+	0.548931i	$0.815035\pi$
$98$		0			0
$99$			12.9578i			1.30231i
$100$		0			0
$101$		−	2.21511i		−	0.220411i	−0.993909	−	0.110206i	$-0.964849\pi$
$101$		−	2.21511i		−	0.220411i	0.993909	−	0.110206i	$-0.0351510\pi$
$102$		0			0
$103$	−16.9392			−1.66907			−0.834536	−	0.550954i	$-0.814264\pi$
$103$	−16.9392			−1.66907			−0.834536	+	0.550954i	$0.814264\pi$
$104$		0			0
$105$	8.21511	+	5.03247i	0.801712	+	0.491119i
$106$		0			0
$107$			9.35556i			0.904436i	0.891907	+	0.452218i	$0.149367\pi$
$107$			9.35556i			0.904436i	−0.891907	+	0.452218i	$0.850633\pi$
$108$		0			0
$109$	−6.37926			−0.611022			−0.305511	−	0.952189i	$-0.598827\pi$
$109$	−6.37926			−0.611022			−0.305511	+	0.952189i	$0.598827\pi$
$110$		0			0
$111$	4.99431			0.474039
$112$		0			0
$113$	−16.8891			−1.58879			−0.794396	−	0.607400i	$-0.792212\pi$
$113$	−16.8891			−1.58879			−0.794396	+	0.607400i	$0.792212\pi$
$114$		0			0
$115$	−12.4705			−1.16288
$116$		0			0
$117$		−	8.03771i		−	0.743087i
$118$		0			0
$119$	−2.25609	−	1.38205i	−0.206815	−	0.126692i
$120$		0			0
$121$	1.68595			0.153268
$122$		0			0
$123$		−	10.8358i		−	0.977032i
$124$		0			0
$125$			11.0520i			0.988521i
$126$		0			0
$127$			8.56472i			0.759996i	0.924987	+	0.379998i	$0.124075\pi$
$127$			8.56472i			0.759996i	−0.924987	+	0.379998i	$0.875925\pi$
$128$		0			0
$129$			16.7736i			1.47684i
$130$		0			0
$131$	−10.1861			−0.889965			−0.444982	−	0.895539i	$-0.646790\pi$
$131$	−10.1861			−0.889965			−0.444982	+	0.895539i	$0.646790\pi$
$132$		0			0
$133$	−8.92058	+	14.5621i	−0.773512	+	1.26270i
$134$		0			0
$135$		−	4.53642i		−	0.390433i
$136$		0			0
$137$	−5.27033			−0.450275			−0.225137	−	0.974327i	$-0.572283\pi$
$137$	−5.27033			−0.450275			−0.225137	+	0.974327i	$0.572283\pi$
$138$		0			0
$139$	−8.46101			−0.717653			−0.358827	−	0.933404i	$-0.616823\pi$
$139$	−8.46101			−0.717653			−0.358827	+	0.933404i	$0.616823\pi$
$140$		0			0
$141$	3.93080			0.331033
$142$		0			0
$143$	5.77750			0.483139
$144$		0			0
$145$			13.6737i			1.13554i
$146$		0			0
$147$	8.55959	+	16.7863i	0.705983	+	1.38451i
$148$		0			0
$149$	−16.0463			−1.31456			−0.657281	−	0.753645i	$-0.728294\pi$
$149$	−16.0463			−1.31456			−0.657281	+	0.753645i	$0.728294\pi$
$150$		0			0
$151$			20.9297i			1.70323i	0.524167	+	0.851616i	$0.324377\pi$
$151$			20.9297i			1.70323i	−0.524167	+	0.851616i	$0.675623\pi$
$152$		0			0
$153$			4.24582i			0.343254i
$154$		0			0
$155$		−	11.5166i		−	0.925037i
$156$		0			0
$157$			4.98650i			0.397966i	0.980003	+	0.198983i	$0.0637638\pi$
$157$			4.98650i			0.397966i	−0.980003	+	0.198983i	$0.936236\pi$
$158$		0			0
$159$	1.40196			0.111182
$160$		0			0
$161$	−20.7982	−	12.7407i	−1.63913	−	1.00411i
$162$		0			0
$163$		−	14.3260i		−	1.12210i	−0.827781	−	0.561051i	$-0.810397\pi$
$163$		−	14.3260i		−	1.12210i	0.827781	−	0.561051i	$-0.189603\pi$
$164$		0			0
$165$	11.1129			0.865136
$166$		0			0
$167$	10.6697			0.825646			0.412823	−	0.910811i	$-0.364543\pi$
$167$	10.6697			0.825646			0.412823	+	0.910811i	$0.364543\pi$
$168$		0			0
$169$	9.41622			0.724325
$170$		0			0
$171$	27.4051			2.09572
$172$		0			0
$173$		−	21.1472i		−	1.60779i	−0.594771	−	0.803895i	$-0.702757\pi$
$173$		−	21.1472i		−	1.60779i	0.594771	−	0.803895i	$-0.297243\pi$
$174$		0			0
$175$	−4.38125	+	7.15203i	−0.331191	+	0.540643i
$176$		0			0
$177$	−29.6373			−2.22768
$178$		0			0
$179$		−	1.59560i		−	0.119260i	−0.998221	−	0.0596302i	$-0.981008\pi$
$179$		−	1.59560i		−	0.119260i	0.998221	−	0.0596302i	$-0.0189921\pi$
$180$		0			0
$181$		−	17.2111i		−	1.27929i	−0.768670	−	0.639646i	$-0.779081\pi$
$181$		−	17.2111i		−	1.27929i	0.768670	−	0.639646i	$-0.220919\pi$
$182$		0			0
$183$		−	28.3101i		−	2.09274i
$184$		0			0
$185$		−	2.50983i		−	0.184527i
$186$		0			0
$187$	−3.05189			−0.223177
$188$		0			0
$189$	4.63473	−	7.56583i	0.337127	−	0.550333i
$190$		0			0
$191$		−	7.49646i		−	0.542425i	−0.962519	−	0.271213i	$-0.912575\pi$
$191$		−	7.49646i		−	0.542425i	0.962519	−	0.271213i	$-0.0874246\pi$
$192$		0			0
$193$	12.1150			0.872057			0.436029	−	0.899933i	$-0.356385\pi$
$193$	12.1150			0.872057			0.436029	+	0.899933i	$0.356385\pi$
$194$		0			0
$195$	−6.89331			−0.493640
$196$		0			0
$197$	−3.76117			−0.267972			−0.133986	−	0.990983i	$-0.542778\pi$
$197$	−3.76117			−0.267972			−0.133986	+	0.990983i	$0.542778\pi$
$198$		0			0
$199$	8.17379			0.579424			0.289712	−	0.957114i	$-0.406440\pi$
$199$	8.17379			0.579424			0.289712	+	0.957114i	$0.406440\pi$
$200$		0			0
$201$		−	26.9646i		−	1.90193i
$202$		0			0
$203$	−13.9700	+	22.8050i	−0.980505	+	1.60060i
$204$		0			0
$205$	−5.44541			−0.380324
$206$		0			0
$207$			39.1410i			2.72048i
$208$		0			0
$209$			19.6987i			1.36259i
$210$		0			0
$211$		−	15.4188i		−	1.06147i	−0.847537	−	0.530737i	$-0.821915\pi$
$211$		−	15.4188i		−	1.06147i	0.847537	−	0.530737i	$-0.178085\pi$
$212$		0			0
$213$			34.2119i			2.34416i
$214$		0			0
$215$	8.42940			0.574881
$216$		0			0
$217$	11.7662	−	19.2074i	0.798741	−	1.30388i
$218$		0			0
$219$		−	4.33180i		−	0.292716i
$220$		0			0
$221$	1.89309			0.127343
$222$		0			0
$223$	1.02475			0.0686224			0.0343112	−	0.999411i	$-0.489076\pi$
$223$	1.02475			0.0686224			0.0343112	+	0.999411i	$0.489076\pi$
$224$		0			0
$225$	13.4597			0.897313
$226$		0			0
$227$	8.51877			0.565410			0.282705	−	0.959207i	$-0.408768\pi$
$227$	8.51877			0.565410			0.282705	+	0.959207i	$0.408768\pi$
$228$		0			0
$229$		−	20.2401i		−	1.33750i	−0.743486	−	0.668752i	$-0.766829\pi$
$229$		−	20.2401i		−	1.33750i	0.743486	−	0.668752i	$-0.233171\pi$
$230$		0			0
$231$	18.5340	+	11.3537i	1.21945	+	0.747019i
$232$		0			0
$233$	−1.80636			−0.118339			−0.0591694	−	0.998248i	$-0.518845\pi$
$233$	−1.80636			−0.118339			−0.0591694	+	0.998248i	$0.518845\pi$
$234$		0			0
$235$		−	1.97538i		−	0.128859i
$236$		0			0
$237$			27.6770i			1.79781i
$238$		0			0
$239$		−	15.2557i		−	0.986811i	−0.869800	−	0.493405i	$-0.835752\pi$
$239$		−	15.2557i		−	0.986811i	0.869800	−	0.493405i	$-0.164248\pi$
$240$		0			0
$241$			24.1482i			1.55552i	0.628559	+	0.777762i	$0.283645\pi$
$241$			24.1482i			1.55552i	−0.628559	+	0.777762i	$0.716355\pi$
$242$		0			0
$243$	20.0484			1.28610
$244$		0			0
$245$	8.43575	−	4.30152i	0.538940	−	0.274814i
$246$		0			0
$247$		−	12.2191i		−	0.777483i
$248$		0			0
$249$	−3.25128			−0.206041
$250$		0			0
$251$	11.2742			0.711618			0.355809	−	0.934559i	$-0.384205\pi$
$251$	11.2742			0.711618			0.355809	+	0.934559i	$0.384205\pi$
$252$		0			0
$253$	−28.1345			−1.76880
$254$		0			0
$255$	3.64131			0.228027
$256$		0			0
$257$		−	8.86963i		−	0.553272i	−0.960975	−	0.276636i	$-0.910780\pi$
$257$		−	8.86963i		−	0.553272i	0.960975	−	0.276636i	$-0.0892196\pi$
$258$		0			0
$259$	2.56422	−	4.18589i	0.159333	−	0.260098i
$260$		0			0
$261$	42.9176			2.65653
$262$		0			0
$263$			3.81724i			0.235381i	0.993050	+	0.117690i	$0.0375490\pi$
$263$			3.81724i			0.235381i	−0.993050	+	0.117690i	$0.962451\pi$
$264$		0			0
$265$		−	0.704538i		−	0.0432794i
$266$		0			0
$267$		−	44.7361i		−	2.73781i
$268$		0			0
$269$			8.92926i			0.544427i	0.962237	+	0.272213i	$0.0877556\pi$
$269$			8.92926i			0.544427i	−0.962237	+	0.272213i	$0.912244\pi$
$270$		0			0
$271$	−20.1147			−1.22188			−0.610941	−	0.791676i	$-0.709209\pi$
$271$	−20.1147			−1.22188			−0.610941	+	0.791676i	$0.709209\pi$
$272$		0			0
$273$	−11.4966	−	7.04269i	−0.695807	−	0.426243i
$274$		0			0
$275$			9.67482i			0.583413i
$276$		0			0
$277$	3.52916			0.212047			0.106023	−	0.994364i	$-0.466188\pi$
$277$	3.52916			0.212047			0.106023	+	0.994364i	$0.466188\pi$
$278$		0			0
$279$	−36.1471			−2.16407
$280$		0			0
$281$	4.27003			0.254729			0.127364	−	0.991856i	$-0.459348\pi$
$281$	4.27003			0.254729			0.127364	+	0.991856i	$0.459348\pi$
$282$		0			0
$283$	−6.29981			−0.374485			−0.187242	−	0.982314i	$-0.559955\pi$
$283$	−6.29981			−0.374485			−0.187242	+	0.982314i	$0.559955\pi$
$284$		0			0
$285$		−	23.5031i		−	1.39221i
$286$		0			0
$287$	−9.08183	−	5.56342i	−0.536084	−	0.328398i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		−	29.1057i		−	1.70621i
$292$		0			0
$293$			19.2803i			1.12637i	0.826332	+	0.563184i	$0.190424\pi$
$293$			19.2803i			1.12637i	−0.826332	+	0.563184i	$0.809576\pi$
$294$		0			0
$295$			14.8939i			0.867155i
$296$		0			0
$297$		−	10.2346i		−	0.593870i
$298$		0			0
$299$	17.4518			1.00926
$300$		0			0
$301$	14.0585	+	8.61208i	0.810320	+	0.496392i
$302$		0			0
$303$			5.96264i			0.342545i
$304$		0			0
$305$	−14.2269			−0.814630
$306$		0			0
$307$	4.60562			0.262857			0.131428	−	0.991326i	$-0.458044\pi$
$307$	4.60562			0.262857			0.131428	+	0.991326i	$0.458044\pi$
$308$		0			0
$309$	45.5971			2.59393
$310$		0			0
$311$	3.60084			0.204185			0.102092	−	0.994775i	$-0.467446\pi$
$311$	3.60084			0.204185			0.102092	+	0.994775i	$0.467446\pi$
$312$		0			0
$313$		−	22.9391i		−	1.29659i	−0.761388	−	0.648297i	$-0.775482\pi$
$313$		−	22.9391i		−	1.29659i	0.761388	−	0.648297i	$-0.224518\pi$
$314$		0			0
$315$	−12.9578	−	7.93779i	−0.730089	−	0.447244i
$316$		0			0
$317$	−10.0962			−0.567061			−0.283530	−	0.958963i	$-0.591506\pi$
$317$	−10.0962			−0.567061			−0.283530	+	0.958963i	$0.591506\pi$
$318$		0			0
$319$			30.8491i			1.72722i
$320$		0			0
$321$		−	25.1834i		−	1.40560i
$322$		0			0
$323$			6.45459i			0.359143i
$324$		0			0
$325$		−	6.00128i		−	0.332891i
$326$		0			0
$327$	17.1717			0.949599
$328$		0			0
$329$	2.01819	−	3.29453i	0.111266	−	0.181633i
$330$		0			0
$331$			31.0860i			1.70864i	0.519745	+	0.854322i	$0.326027\pi$
$331$			31.0860i			1.70864i	−0.519745	+	0.854322i	$0.673973\pi$
$332$		0			0
$333$	−7.87759			−0.431689
$334$		0			0
$335$	−13.5507			−0.740355
$336$		0			0
$337$	−31.1997			−1.69955			−0.849777	−	0.527142i	$-0.823264\pi$
$337$	−31.1997			−1.69955			−0.849777	+	0.527142i	$0.823264\pi$
$338$		0			0
$339$	45.4622			2.46917
$340$		0			0
$341$		−	25.9825i		−	1.40703i
$342$		0			0
$343$	18.4638	+	1.44450i	0.996954	+	0.0779957i
$344$		0			0
$345$	33.5681			1.80725
$346$		0			0
$347$		−	17.8870i		−	0.960227i	−0.877206	−	0.480113i	$-0.840595\pi$
$347$		−	17.8870i		−	0.960227i	0.877206	−	0.480113i	$-0.159405\pi$
$348$		0			0
$349$			23.3335i			1.24901i	0.781019	+	0.624507i	$0.214700\pi$
$349$			23.3335i			1.24901i	−0.781019	+	0.624507i	$0.785300\pi$
$350$		0			0
$351$			6.34850i			0.338858i
$352$		0			0
$353$		−	18.0248i		−	0.959362i	−0.877443	−	0.479681i	$-0.840752\pi$
$353$		−	18.0248i		−	0.959362i	0.877443	−	0.479681i	$-0.159248\pi$
$354$		0			0
$355$	17.1928			0.912499
$356$		0			0
$357$	6.07295	+	3.72022i	0.321415	+	0.196895i
$358$		0			0
$359$		−	4.43692i		−	0.234172i	−0.993122	−	0.117086i	$-0.962645\pi$
$359$		−	4.43692i		−	0.234172i	0.993122	−	0.117086i	$-0.0373553\pi$
$360$		0			0
$361$	22.6618			1.19272
$362$		0			0
$363$	−4.53825			−0.238196
$364$		0			0
$365$	−2.17690			−0.113944
$366$		0			0
$367$	−17.8725			−0.932937			−0.466469	−	0.884538i	$-0.654474\pi$
$367$	−17.8725			−0.932937			−0.466469	+	0.884538i	$0.654474\pi$
$368$		0			0
$369$			17.0915i			0.889746i
$370$		0			0
$371$	0.719806	−	1.17502i	0.0373704	−	0.0610042i
$372$		0			0
$373$	−19.2353			−0.995967			−0.497983	−	0.867187i	$-0.665926\pi$
$373$	−19.2353			−0.995967			−0.497983	+	0.867187i	$0.665926\pi$
$374$		0			0
$375$		−	29.7498i		−	1.53628i
$376$		0			0
$377$		−	19.1357i		−	0.985538i
$378$		0			0
$379$		−	10.2704i		−	0.527556i	−0.964583	−	0.263778i	$-0.915031\pi$
$379$		−	10.2704i		−	0.527556i	0.964583	−	0.263778i	$-0.0849686\pi$
$380$		0			0
$381$		−	23.0546i		−	1.18112i
$382$		0			0
$383$	−0.00772947			−0.000394958			−0.000197479	−	1.00000i	$-0.500063\pi$
$383$	−0.00772947			−0.000394958			−0.000197479		1.00000i	$0.500063\pi$
$384$		0			0
$385$	5.70567	−	9.31405i	0.290788	−	0.474688i
$386$		0			0
$387$		−	26.4573i		−	1.34490i
$388$		0			0
$389$	25.4132			1.28850			0.644250	−	0.764815i	$-0.277170\pi$
$389$	25.4132			1.28850			0.644250	+	0.764815i	$0.277170\pi$
$390$		0			0
$391$	−9.21870			−0.466209
$392$		0			0
$393$	27.4190			1.38311
$394$		0			0
$395$	13.9088			0.699826
$396$		0			0
$397$			6.38504i			0.320456i	0.987080	+	0.160228i	$0.0512230\pi$
$397$			6.38504i			0.320456i	−0.987080	+	0.160228i	$0.948777\pi$
$398$		0			0
$399$	24.0125	−	39.1984i	1.20213	−	1.96238i
$400$		0			0
$401$	27.4417			1.37037			0.685187	−	0.728367i	$-0.259721\pi$
$401$	27.4417			1.37037			0.685187	+	0.728367i	$0.259721\pi$
$402$		0			0
$403$			16.1169i			0.802841i
$404$		0			0
$405$		−	5.01927i		−	0.249409i
$406$		0			0
$407$		−	5.66240i		−	0.280675i
$408$		0			0
$409$			3.77313i			0.186570i	0.995639	+	0.0932848i	$0.0297367\pi$
$409$			3.77313i			0.186570i	−0.995639	+	0.0932848i	$0.970263\pi$
$410$		0			0
$411$	14.1867			0.699779
$412$		0			0
$413$	−15.2166	+	24.8399i	−0.748762	+	1.22229i
$414$		0			0
$415$			1.63389i			0.0802047i
$416$		0			0
$417$	22.7754			1.11532
$418$		0			0
$419$	33.3561			1.62955			0.814777	−	0.579774i	$-0.196859\pi$
$419$	33.3561			1.62955			0.814777	+	0.579774i	$0.196859\pi$
$420$		0			0
$421$	18.7223			0.912468			0.456234	−	0.889860i	$-0.349198\pi$
$421$	18.7223			0.912468			0.456234	+	0.889860i	$0.349198\pi$
$422$		0			0
$423$	−6.20010			−0.301459
$424$		0			0
$425$			3.17010i			0.153773i
$426$		0			0
$427$	−23.7276	−	14.5352i	−1.14826	−	0.703408i
$428$		0			0
$429$	−15.5519			−0.750853
$430$		0			0
$431$		−	7.90301i		−	0.380674i	−0.981719	−	0.190337i	$-0.939042\pi$
$431$		−	7.90301i		−	0.380674i	0.981719	−	0.190337i	$-0.0609581\pi$
$432$		0			0
$433$			20.8841i			1.00363i	0.864976	+	0.501814i	$0.167334\pi$
$433$			20.8841i			1.00363i	−0.864976	+	0.501814i	$0.832666\pi$
$434$		0			0
$435$		−	36.8070i		−	1.76476i
$436$		0			0
$437$			59.5029i			2.84641i
$438$		0			0
$439$	−7.65077			−0.365151			−0.182576	−	0.983192i	$-0.558443\pi$
$439$	−7.65077			−0.365151			−0.182576	+	0.983192i	$0.558443\pi$
$440$		0			0
$441$	−13.5012	−	26.4772i	−0.642912	−	1.26082i
$442$		0			0
$443$		−	0.700477i		−	0.0332807i	−0.999862	−	0.0166403i	$-0.994703\pi$
$443$		−	0.700477i		−	0.0332807i	0.999862	−	0.0166403i	$-0.00529703\pi$
$444$		0			0
$445$	−22.4816			−1.06573
$446$		0			0
$447$	43.1935			2.04298
$448$		0			0
$449$	−6.77949			−0.319944			−0.159972	−	0.987122i	$-0.551140\pi$
$449$	−6.77949			−0.319944			−0.159972	+	0.987122i	$0.551140\pi$
$450$		0			0
$451$	−12.2853			−0.578494
$452$		0			0
$453$		−	56.3386i		−	2.64702i
$454$		0			0
$455$	−3.53922	+	5.77750i	−0.165921	+	0.270853i
$456$		0			0
$457$	3.05801			0.143048			0.0715238	−	0.997439i	$-0.477214\pi$
$457$	3.05801			0.143048			0.0715238	+	0.997439i	$0.477214\pi$
$458$		0			0
$459$		−	3.35352i		−	0.156529i
$460$		0			0
$461$		−	16.3370i		−	0.760890i	−0.924804	−	0.380445i	$-0.875771\pi$
$461$		−	16.3370i		−	0.760890i	0.924804	−	0.380445i	$-0.124229\pi$
$462$		0			0
$463$			5.70202i			0.264995i	0.991183	+	0.132498i	$0.0422997\pi$
$463$			5.70202i			0.264995i	−0.991183	+	0.132498i	$0.957700\pi$
$464$		0			0
$465$			31.0005i			1.43761i
$466$		0			0
$467$	13.1500			0.608511			0.304255	−	0.952590i	$-0.401592\pi$
$467$	13.1500			0.608511			0.304255	+	0.952590i	$0.401592\pi$
$468$		0			0
$469$	−22.5998	−	13.8444i	−1.04356	−	0.639274i
$470$		0			0
$471$		−	13.4227i		−	0.618485i
$472$		0			0
$473$	19.0175			0.874424
$474$		0			0
$475$	20.4617			0.938848
$476$		0			0
$477$	−2.21133			−0.101250
$478$		0			0
$479$	−3.69837			−0.168983			−0.0844915	−	0.996424i	$-0.526927\pi$
$479$	−3.69837			−0.168983			−0.0844915	+	0.996424i	$0.526927\pi$
$480$		0			0
$481$			3.51238i			0.160151i
$482$		0			0
$483$	55.9847	+	34.2955i	2.54739	+	1.56050i
$484$		0			0
$485$	−14.6267			−0.664166
$486$		0			0
$487$			10.6055i			0.480581i	0.970701	+	0.240291i	$0.0772428\pi$
$487$			10.6055i			0.480581i	−0.970701	+	0.240291i	$0.922757\pi$
$488$		0			0
$489$			38.5630i			1.74388i
$490$		0			0
$491$		−	41.1786i		−	1.85837i	−0.369620	−	0.929183i	$-0.620512\pi$
$491$		−	41.1786i		−	1.85837i	0.369620	−	0.929183i	$-0.379488\pi$
$492$		0			0
$493$			10.1082i			0.455250i
$494$		0			0
$495$	−17.5285			−0.787847
$496$		0			0
$497$	28.6741	+	17.5654i	1.28621	+	0.787915i
$498$		0			0
$499$		−	5.38220i		−	0.240940i	−0.992717	−	0.120470i	$-0.961560\pi$
$499$		−	5.38220i		−	0.240940i	0.992717	−	0.120470i	$-0.0384402\pi$
$500$		0			0
$501$	−28.7208			−1.28315
$502$		0			0
$503$	−21.5148			−0.959297			−0.479648	−	0.877461i	$-0.659236\pi$
$503$	−21.5148			−0.959297			−0.479648	+	0.877461i	$0.659236\pi$
$504$		0			0
$505$	2.99646			0.133341
$506$		0			0
$507$	−25.3467			−1.12568
$508$		0			0
$509$			23.8277i			1.05615i	0.849199	+	0.528073i	$0.177085\pi$
$509$			23.8277i			1.05615i	−0.849199	+	0.528073i	$0.822915\pi$
$510$		0			0
$511$	−3.63062	−	2.22407i	−0.160609	−	0.0983872i
$512$		0			0
$513$	−21.6456			−0.955676
$514$		0			0
$515$		−	22.9143i		−	1.00973i
$516$		0			0
$517$		−	4.45663i		−	0.196002i
$518$		0			0
$519$			56.9241i			2.49869i
$520$		0			0
$521$			1.98790i			0.0870914i	0.999051	+	0.0435457i	$0.0138654\pi$
$521$			1.98790i			0.0870914i	−0.999051	+	0.0435457i	$0.986135\pi$
$522$		0			0
$523$	−3.01620			−0.131889			−0.0659447	−	0.997823i	$-0.521006\pi$
$523$	−3.01620			−0.131889			−0.0659447	+	0.997823i	$0.521006\pi$
$524$		0			0
$525$	11.7935	−	19.2519i	0.514709	−	0.840221i
$526$		0			0
$527$		−	8.51357i		−	0.370857i
$528$		0			0
$529$	−61.9844			−2.69497
$530$		0			0
$531$	46.7473			2.02866
$532$		0			0
$533$	7.62058			0.330084
$534$		0			0
$535$	−12.6556			−0.547150
$536$		0			0
$537$			4.29503i			0.185344i
$538$		0			0
$539$	19.0318	−	9.70461i	0.819757	−	0.418007i
$540$		0			0
$541$	−27.6065			−1.18690			−0.593449	−	0.804872i	$-0.702234\pi$
$541$	−27.6065			−1.18690			−0.593449	+	0.804872i	$0.702234\pi$
$542$		0			0
$543$			46.3290i			1.98817i
$544$		0			0
$545$		−	8.62946i		−	0.369645i
$546$		0			0
$547$			8.71443i			0.372602i	0.982493	+	0.186301i	$0.0596500\pi$
$547$			8.71443i			0.372602i	−0.982493	+	0.186301i	$0.940350\pi$
$548$		0			0
$549$			44.6539i			1.90578i
$550$		0			0
$551$	65.2443			2.77950
$552$		0			0
$553$	23.1970	+	14.2102i	0.986436	+	0.604278i
$554$		0			0
$555$			6.75599i			0.286776i
$556$		0			0
$557$	−38.7779			−1.64307			−0.821537	−	0.570156i	$-0.806883\pi$
$557$	−38.7779			−1.64307			−0.821537	+	0.570156i	$0.806883\pi$
$558$		0			0
$559$	−11.7965			−0.498940
$560$		0			0
$561$	8.21511			0.346842
$562$		0			0
$563$	10.2349			0.431351			0.215675	−	0.976465i	$-0.430805\pi$
$563$	10.2349			0.431351			0.215675	+	0.976465i	$0.430805\pi$
$564$		0			0
$565$		−	22.8465i		−	0.961159i
$566$		0			0
$567$	5.12804	−	8.37111i	0.215357	−	0.351554i
$568$		0			0
$569$	−13.5851			−0.569516			−0.284758	−	0.958599i	$-0.591913\pi$
$569$	−13.5851			−0.569516			−0.284758	+	0.958599i	$0.591913\pi$
$570$		0			0
$571$		−	22.2164i		−	0.929728i	−0.885382	−	0.464864i	$-0.846103\pi$
$571$		−	22.2164i		−	0.929728i	0.885382	−	0.464864i	$-0.153897\pi$
$572$		0			0
$573$			20.1790i			0.842991i
$574$		0			0
$575$			29.2242i			1.21873i
$576$		0			0
$577$			10.8984i			0.453705i	0.973929	+	0.226852i	$0.0728435\pi$
$577$			10.8984i			0.453705i	−0.973929	+	0.226852i	$0.927157\pi$
$578$		0			0
$579$	−32.6113			−1.35528
$580$		0			0
$581$	−1.66930	+	2.72500i	−0.0692543	+	0.113052i
$582$		0			0
$583$		−	1.58950i		−	0.0658303i
$584$		0			0
$585$	10.8729			0.449539
$586$		0			0
$587$	27.1918			1.12233			0.561163	−	0.827705i	$-0.310354\pi$
$587$	27.1918			1.12233			0.561163	+	0.827705i	$0.310354\pi$
$588$		0			0
$589$	−54.9516			−2.26424
$590$		0			0
$591$	10.1243			0.416460
$592$		0			0
$593$		−	24.9911i		−	1.02626i	−0.858311	−	0.513130i	$-0.828486\pi$
$593$		−	24.9911i		−	1.02626i	0.858311	−	0.513130i	$-0.171514\pi$
$594$		0			0
$595$	1.86955	−	3.05189i	0.0766442	−	0.125115i
$596$		0			0
$597$	−22.0023			−0.900492
$598$		0			0
$599$			22.9381i			0.937225i	0.883404	+	0.468613i	$0.155246\pi$
$599$			22.9381i			0.937225i	−0.883404	+	0.468613i	$0.844754\pi$
$600$		0			0
$601$		−	4.31687i		−	0.176089i	−0.996117	−	0.0880445i	$-0.971938\pi$
$601$		−	4.31687i		−	0.176089i	0.996117	−	0.0880445i	$-0.0280618\pi$
$602$		0			0
$603$			42.5316i			1.73202i
$604$		0			0
$605$			2.28064i			0.0927213i
$606$		0			0
$607$	−18.5595			−0.753306			−0.376653	−	0.926354i	$-0.622925\pi$
$607$	−18.5595			−0.753306			−0.376653	+	0.926354i	$0.622925\pi$
$608$		0			0
$609$	37.6047	−	61.3866i	1.52382	−	2.48751i
$610$		0			0
$611$			2.76444i			0.111837i
$612$		0			0
$613$	−43.8859			−1.77254			−0.886268	−	0.463173i	$-0.846711\pi$
$613$	−43.8859			−1.77254			−0.886268	+	0.463173i	$0.846711\pi$
$614$		0			0
$615$	14.6580			0.591067
$616$		0			0
$617$	21.2429			0.855208			0.427604	−	0.903966i	$-0.359358\pi$
$617$	21.2429			0.855208			0.427604	+	0.903966i	$0.359358\pi$
$618$		0			0
$619$	−40.5663			−1.63050			−0.815249	−	0.579110i	$-0.803400\pi$
$619$	−40.5663			−1.63050			−0.815249	+	0.579110i	$0.803400\pi$
$620$		0			0
$621$		−	30.9150i		−	1.24058i
$622$		0			0
$623$	−37.4948	−	22.9688i	−1.50220	−	0.920227i
$624$		0			0
$625$	0.900070			0.0360028
$626$		0			0
$627$		−	53.0252i		−	2.11762i
$628$		0			0
$629$		−	1.85537i		−	0.0739786i
$630$		0			0
$631$		−	17.0673i		−	0.679439i	−0.940527	−	0.339720i	$-0.889668\pi$
$631$		−	17.0673i		−	0.679439i	0.940527	−	0.339720i	$-0.110332\pi$
$632$		0			0
$633$			41.5044i			1.64965i
$634$		0			0
$635$	−11.5858			−0.459769
$636$		0			0
$637$	−11.8054	+	6.01976i	−0.467747	+	0.238512i
$638$		0			0
$639$		−	53.9629i		−	2.13474i
$640$		0			0
$641$	43.4525			1.71627			0.858135	−	0.513424i	$-0.171623\pi$
$641$	43.4525			1.71627			0.858135	+	0.513424i	$0.171623\pi$
$642$		0			0
$643$	0.217686			0.00858472			0.00429236	−	0.999991i	$-0.498634\pi$
$643$	0.217686			0.00858472			0.00429236	+	0.999991i	$0.498634\pi$
$644$		0			0
$645$	−22.6903			−0.893431
$646$		0			0
$647$	16.1546			0.635104			0.317552	−	0.948241i	$-0.397139\pi$
$647$	16.1546			0.635104			0.317552	+	0.948241i	$0.397139\pi$
$648$		0			0
$649$			33.6019i			1.31899i
$650$		0			0
$651$	−31.6723	+	51.7025i	−1.24134	+	2.02638i
$652$		0			0
$653$	−12.5307			−0.490364			−0.245182	−	0.969477i	$-0.578848\pi$
$653$	−12.5307			−0.490364			−0.245182	+	0.969477i	$0.578848\pi$
$654$		0			0
$655$		−	13.7791i		−	0.538395i
$656$		0			0
$657$			6.83261i			0.266566i
$658$		0			0
$659$			11.7139i			0.456309i	0.973625	+	0.228155i	$0.0732692\pi$
$659$			11.7139i			0.456309i	−0.973625	+	0.228155i	$0.926731\pi$
$660$		0			0
$661$		−	0.827718i		−	0.0321945i	−0.999870	−	0.0160972i	$-0.994876\pi$
$661$		−	0.827718i		−	0.0321945i	0.999870	−	0.0160972i	$-0.00512413\pi$
$662$		0			0
$663$	−5.09582			−0.197905
$664$		0			0
$665$	−19.6987	−	12.0672i	−0.763884	−	0.467946i
$666$		0			0
$667$			93.1844i			3.60811i
$668$		0			0
$669$	−2.75843			−0.106647
$670$		0			0
$671$	−32.0971			−1.23910
$672$		0			0
$673$	3.76740			0.145222			0.0726112	−	0.997360i	$-0.476867\pi$
$673$	3.76740			0.145222			0.0726112	+	0.997360i	$0.476867\pi$
$674$		0			0
$675$	−10.6310			−0.409187
$676$		0			0
$677$			40.2594i			1.54729i	0.633618	+	0.773646i	$0.281569\pi$
$677$			40.2594i			1.54729i	−0.633618	+	0.773646i	$0.718431\pi$
$678$		0			0
$679$	−24.3944	−	14.9437i	−0.936171	−	0.573487i
$680$		0			0
$681$	−22.9309			−0.878713
$682$		0			0
$683$			14.6864i			0.561958i	0.959714	+	0.280979i	$0.0906591\pi$
$683$			14.6864i			0.561958i	−0.959714	+	0.280979i	$0.909341\pi$
$684$		0			0
$685$		−	7.12937i		−	0.272399i
$686$		0			0
$687$			54.4825i			2.07864i
$688$		0			0
$689$			0.985965i			0.0375623i
$690$		0			0
$691$	−10.2903			−0.391462			−0.195731	−	0.980658i	$-0.562708\pi$
$691$	−10.2903			−0.391462			−0.195731	+	0.980658i	$0.562708\pi$
$692$		0			0
$693$	−29.2339	−	17.9083i	−1.11051	−	0.680282i
$694$		0			0
$695$		−	11.4455i		−	0.434153i
$696$		0			0
$697$	−4.02548			−0.152476
$698$		0			0
$699$	4.86238			0.183912
$700$		0			0
$701$	−19.9181			−0.752296			−0.376148	−	0.926560i	$-0.622752\pi$
$701$	−19.9181			−0.752296			−0.376148	+	0.926560i	$0.622752\pi$
$702$		0			0
$703$	−11.9757			−0.451672
$704$		0			0
$705$			5.31734i			0.200262i
$706$		0			0
$707$	4.99748	+	3.06139i	0.187949	+	0.115136i
$708$		0			0
$709$	45.8787			1.72301			0.861505	−	0.507749i	$-0.169522\pi$
$709$	45.8787			1.72301			0.861505	+	0.507749i	$0.169522\pi$
$710$		0			0
$711$		−	43.6553i		−	1.63720i
$712$		0			0
$713$		−	78.4840i		−	2.93925i
$714$		0			0
$715$			7.81543i			0.292281i
$716$		0			0
$717$			41.0655i			1.53362i
$718$		0			0
$719$	−13.5152			−0.504033			−0.252017	−	0.967723i	$-0.581094\pi$
$719$	−13.5152			−0.504033			−0.252017	+	0.967723i	$0.581094\pi$
$720$		0			0
$721$	23.4109	−	38.2164i	0.871867	−	1.42325i
$722$		0			0
$723$		−	65.0023i		−	2.41746i
$724$		0			0
$725$	32.0440			1.19008
$726$		0			0
$727$	29.4777			1.09327			0.546634	−	0.837371i	$-0.315909\pi$
$727$	29.4777			1.09327			0.546634	+	0.837371i	$0.315909\pi$
$728$		0			0
$729$	−42.8350			−1.58648
$730$		0			0
$731$	6.23137			0.230476
$732$		0			0
$733$			23.3329i			0.861822i	0.902395	+	0.430911i	$0.141808\pi$
$733$			23.3329i			0.861822i	−0.902395	+	0.430911i	$0.858192\pi$
$734$		0			0
$735$	−22.7074	+	11.5789i	−0.837575	+	0.427093i
$736$		0			0
$737$	−30.5716			−1.12612
$738$		0			0
$739$		−	17.8344i		−	0.656049i	−0.944669	−	0.328025i	$-0.893617\pi$
$739$		−	17.8344i		−	0.656049i	0.944669	−	0.328025i	$-0.106383\pi$
$740$		0			0
$741$			32.8915i			1.20830i
$742$		0			0
$743$		−	42.9431i		−	1.57543i	−0.616041	−	0.787714i	$-0.711264\pi$
$743$		−	42.9431i		−	1.57543i	0.616041	−	0.787714i	$-0.288736\pi$
$744$		0			0
$745$		−	21.7064i		−	0.795261i
$746$		0			0
$747$	5.12829			0.187634
$748$		0			0
$749$	−21.1070	−	12.9299i	−0.771232	−	0.472447i
$750$		0			0
$751$			41.4068i			1.51096i	0.655174	+	0.755478i	$0.272596\pi$
$751$			41.4068i			1.51096i	−0.655174	+	0.755478i	$0.727404\pi$
$752$		0			0
$753$	−30.3479			−1.10594
$754$		0			0
$755$	−28.3123			−1.03039
$756$		0			0
$757$	35.7665			1.29995			0.649977	−	0.759954i	$-0.274778\pi$
$757$	35.7665			1.29995			0.649977	+	0.759954i	$0.274778\pi$
$758$		0			0
$759$	75.7326			2.74892
$760$		0			0
$761$		−	13.6918i		−	0.496329i	−0.968718	−	0.248164i	$-0.920173\pi$
$761$		−	13.6918i		−	0.496329i	0.968718	−	0.248164i	$-0.0798273\pi$
$762$		0			0
$763$	8.81646	−	14.3922i	0.319177	−	0.521031i
$764$		0			0
$765$	−5.74348			−0.207656
$766$		0			0
$767$		−	20.8432i		−	0.752606i
$768$		0			0
$769$			39.0000i			1.40638i	0.711004	+	0.703188i	$0.248241\pi$
$769$			39.0000i			1.40638i	−0.711004	+	0.703188i	$0.751759\pi$
$770$		0			0
$771$			23.8753i			0.859849i
$772$		0			0
$773$			31.2203i			1.12292i	0.827505	+	0.561458i	$0.189760\pi$
$773$			31.2203i			1.12292i	−0.827505	+	0.561458i	$0.810240\pi$
$774$		0			0
$775$	−26.9889			−0.969470
$776$		0			0
$777$	−6.90239	+	11.2676i	−0.247622	+	0.404223i
$778$		0			0
$779$			25.9828i			0.930931i
$780$		0			0
$781$	38.7885			1.38796
$782$		0			0
$783$	−33.8980			−1.21142
$784$		0			0
$785$	−6.74542			−0.240754
$786$		0			0
$787$	3.80226			0.135536			0.0677680	−	0.997701i	$-0.478412\pi$
$787$	3.80226			0.135536			0.0677680	+	0.997701i	$0.478412\pi$
$788$		0			0
$789$		−	10.2753i		−	0.365809i
$790$		0			0
$791$	23.3416	−	38.1033i	0.829932	−	1.35480i
$792$		0			0
$793$	19.9098			0.707019
$794$		0			0
$795$			1.89648i			0.0672612i
$796$		0			0
$797$			11.6361i			0.412173i	0.978534	+	0.206087i	$0.0660729\pi$
$797$			11.6361i			0.412173i	−0.978534	+	0.206087i	$0.933927\pi$
$798$		0			0
$799$		−	1.46028i		−	0.0516611i
$800$		0			0
$801$			70.5629i			2.49322i
$802$		0			0
$803$	−4.91127			−0.173315
$804$		0			0
$805$	17.2348	−	28.1345i	0.607448	−	0.991610i
$806$		0			0
$807$		−	24.0358i		−	0.846102i
$808$		0			0
$809$	15.4899			0.544597			0.272299	−	0.962213i	$-0.412216\pi$
$809$	15.4899			0.544597			0.272299	+	0.962213i	$0.412216\pi$
$810$		0			0
$811$	−33.2013			−1.16586			−0.582928	−	0.812524i	$-0.698093\pi$
$811$	−33.2013			−1.16586			−0.582928	+	0.812524i	$0.698093\pi$
$812$		0			0
$813$	54.1449			1.89895
$814$		0			0
$815$	19.3794			0.678830
$816$		0			0
$817$		−	40.2210i		−	1.40715i
$818$		0			0
$819$	18.1338	+	11.1085i	0.633646	+	0.388164i
$820$		0			0
$821$	3.18895			0.111295			0.0556476	−	0.998450i	$-0.482278\pi$
$821$	3.18895			0.111295			0.0556476	+	0.998450i	$0.482278\pi$
$822$		0			0
$823$			47.1640i			1.64403i	0.569463	+	0.822017i	$0.307151\pi$
$823$			47.1640i			1.64403i	−0.569463	+	0.822017i	$0.692849\pi$
$824$		0			0
$825$		−	26.0427i		−	0.906692i
$826$		0			0
$827$			21.0568i			0.732218i	0.930572	+	0.366109i	$0.119310\pi$
$827$			21.0568i			0.732218i	−0.930572	+	0.366109i	$0.880690\pi$
$828$		0			0
$829$		−	43.4789i		−	1.51009i	−0.655676	−	0.755043i	$-0.727616\pi$
$829$		−	43.4789i		−	1.51009i	0.655676	−	0.755043i	$-0.272384\pi$
$830$		0			0
$831$	−9.49981			−0.329545
$832$		0			0
$833$	6.23606	−	3.17987i	0.216067	−	0.110176i
$834$		0			0
$835$			14.4333i			0.499485i
$836$		0			0
$837$	28.5504			0.986846
$838$		0			0
$839$	−12.4982			−0.431487			−0.215743	−	0.976450i	$-0.569217\pi$
$839$	−12.4982			−0.431487			−0.215743	+	0.976450i	$0.569217\pi$
$840$		0			0
$841$	73.1756			2.52330
$842$		0			0
$843$	−11.4941			−0.395878
$844$		0			0
$845$			12.7377i			0.438189i
$846$		0			0
$847$	−2.33007	+	3.80365i	−0.0800620	+	0.130695i
$848$		0			0
$849$	16.9579			0.581993
$850$		0			0
$851$		−	17.1041i		−	0.586322i
$852$		0			0
$853$			30.6440i			1.04923i	0.851339	+	0.524616i	$0.175791\pi$
$853$			30.6440i			1.04923i	−0.851339	+	0.524616i	$0.824209\pi$
$854$		0			0
$855$			37.0718i			1.26783i
$856$		0			0
$857$			40.3156i			1.37715i	0.725163	+	0.688577i	$0.241764\pi$
$857$			40.3156i			1.37715i	−0.725163	+	0.688577i	$0.758236\pi$
$858$		0			0
$859$	20.4629			0.698184			0.349092	−	0.937089i	$-0.386490\pi$
$859$	20.4629			0.698184			0.349092	+	0.937089i	$0.386490\pi$
$860$		0			0
$861$	24.4465	+	14.9756i	0.833136	+	0.510369i
$862$		0			0
$863$			2.80969i			0.0956428i	0.998856	+	0.0478214i	$0.0152278\pi$
$863$			2.80969i			0.0956428i	−0.998856	+	0.0478214i	$0.984772\pi$
$864$		0			0
$865$	28.6066			0.972652
$866$		0			0
$867$	2.69181			0.0914185
$868$		0			0
$869$	31.3794			1.06447
$870$		0			0
$871$	18.9636			0.642556
$872$		0			0
$873$			45.9088i			1.55378i
$874$		0			0
$875$	−24.9343	−	15.2744i	−0.842933	−	0.516370i
$876$		0			0
$877$	21.4369			0.723873			0.361936	−	0.932203i	$-0.382116\pi$
$877$	21.4369			0.723873			0.361936	+	0.932203i	$0.382116\pi$
$878$		0			0
$879$		−	51.8989i		−	1.75051i
$880$		0			0
$881$		−	8.49388i		−	0.286166i	−0.989711	−	0.143083i	$-0.954298\pi$
$881$		−	8.49388i		−	0.286166i	0.989711	−	0.143083i	$-0.0457016\pi$
$882$		0			0
$883$			22.9524i			0.772411i	0.922413	+	0.386206i	$0.126214\pi$
$883$			22.9524i			0.772411i	−0.922413	+	0.386206i	$0.873786\pi$
$884$		0			0
$885$		−	40.0915i		−	1.34766i
$886$		0			0
$887$	49.2403			1.65333			0.826663	−	0.562697i	$-0.190236\pi$
$887$	49.2403			1.65333			0.826663	+	0.562697i	$0.190236\pi$
$888$		0			0
$889$	−19.3228	−	11.8369i	−0.648064	−	0.396996i
$890$		0			0
$891$		−	11.3239i		−	0.379365i
$892$		0			0
$893$	−9.42553			−0.315413
$894$		0			0
$895$	2.15842			0.0721480
$896$		0			0
$897$	−46.9768			−1.56851
$898$		0			0
$899$	−86.0568			−2.87015
$900$		0			0
$901$		−	0.520824i		−	0.0173512i
$902$		0			0
$903$	−37.8428	−	23.1820i	−1.25933	−	0.771450i
$904$		0			0
$905$	23.2821			0.773923
$906$		0			0
$907$		−	26.8663i		−	0.892081i	−0.895013	−	0.446040i	$-0.852834\pi$
$907$		−	26.8663i		−	0.892081i	0.895013	−	0.446040i	$-0.147166\pi$
$908$		0			0
$909$		−	9.40495i		−	0.311943i
$910$		0			0
$911$			37.7912i			1.25208i	0.779791	+	0.626040i	$0.215325\pi$
$911$			37.7912i			1.25208i	−0.779791	+	0.626040i	$0.784675\pi$
$912$		0			0
$913$			3.68621i			0.121996i
$914$		0			0
$915$	38.2961			1.26603
$916$		0			0
$917$	14.0777	−	22.9808i	0.464888	−	0.758892i
$918$		0			0
$919$			41.5090i			1.36926i	0.728893	+	0.684628i	$0.240035\pi$
$919$			41.5090i			1.36926i	−0.728893	+	0.684628i	$0.759965\pi$
$920$		0			0
$921$	−12.3974			−0.408510
$922$		0			0
$923$	−24.0605			−0.791960
$924$		0			0
$925$	−5.88173			−0.193390
$926$		0			0
$927$	−71.9210			−2.36219
$928$		0			0
$929$		−	0.950505i		−	0.0311851i	−0.999878	−	0.0155925i	$-0.995037\pi$
$929$		−	0.950505i		−	0.0311851i	0.999878	−	0.0155925i	$-0.00496346\pi$
$930$		0			0
$931$	−20.5247	−	40.2512i	−0.672671	−	1.31918i
$932$		0			0
$933$	−9.69276			−0.317327
$934$		0			0
$935$		−	4.12841i		−	0.135013i
$936$		0			0
$937$		−	30.6491i		−	1.00126i	−0.865661	−	0.500631i	$-0.833101\pi$
$937$		−	30.6491i		−	1.00126i	0.865661	−	0.500631i	$-0.166899\pi$
$938$		0			0
$939$			61.7476i			2.01506i
$940$		0			0
$941$			3.75169i			0.122302i	0.998129	+	0.0611508i	$0.0194771\pi$
$941$			3.75169i			0.122302i	−0.998129	+	0.0611508i	$0.980523\pi$
$942$		0			0
$943$	−37.1097			−1.20846
$944$		0			0
$945$	10.2346	+	6.26957i	0.332931	+	0.203949i
$946$		0			0
$947$			15.0583i			0.489327i	0.969608	+	0.244664i	$0.0786776\pi$
$947$			15.0583i			0.489327i	−0.969608	+	0.244664i	$0.921322\pi$
$948$		0			0
$949$	3.04646			0.0988923
$950$		0			0
$951$	27.1771			0.881278
$952$		0			0
$953$	22.3945			0.725431			0.362715	−	0.931900i	$-0.381850\pi$
$953$	22.3945			0.725431			0.362715	+	0.931900i	$0.381850\pi$
$954$		0			0
$955$	10.1407			0.328147
$956$		0			0
$957$		−	83.0399i		−	2.68430i
$958$		0			0
$959$	7.28387	−	11.8903i	0.235208	−	0.383959i
$960$		0			0
$961$	41.4808			1.33809
$962$		0			0
$963$			39.7221i			1.28003i
$964$		0			0
$965$			16.3884i			0.527562i
$966$		0			0
$967$		−	4.50070i		−	0.144733i	−0.997378	−	0.0723664i	$-0.976945\pi$
$967$		−	4.50070i		−	0.144733i	0.997378	−	0.0723664i	$-0.0230551\pi$
$968$		0			0
$969$		−	17.3745i		−	0.558150i
$970$		0			0
$971$	−37.8954			−1.21612			−0.608060	−	0.793891i	$-0.708052\pi$
$971$	−37.8954			−1.21612			−0.608060	+	0.793891i	$0.708052\pi$
$972$		0			0
$973$	11.6936	−	19.0888i	0.374878	−	0.611958i
$974$		0			0
$975$			16.1543i			0.517351i
$976$		0			0
$977$	−39.6445			−1.26834			−0.634170	−	0.773194i	$-0.718658\pi$
$977$	−39.6445			−1.26834			−0.634170	+	0.773194i	$0.718658\pi$
$978$		0			0
$979$	−50.7205			−1.62104
$980$		0			0
$981$	−27.0852			−0.864764
$982$		0			0
$983$	−13.6688			−0.435967			−0.217984	−	0.975952i	$-0.569948\pi$
$983$	−13.6688			−0.435967			−0.217984	+	0.975952i	$0.569948\pi$
$984$		0			0
$985$		−	5.08788i		−	0.162113i
$986$		0			0
$987$	−5.43257	+	8.86823i	−0.172921	+	0.282279i
$988$		0			0
$989$	57.4451			1.82665
$990$		0			0
$991$			16.5321i			0.525160i	0.964910	+	0.262580i	$0.0845734\pi$
$991$			16.5321i			0.525160i	−0.964910	+	0.262580i	$0.915427\pi$
$992$		0			0
$993$		−	83.6776i		−	2.65543i
$994$		0			0
$995$			11.0570i			0.350530i
$996$		0			0
$997$		−	35.9526i		−	1.13863i	−0.822119	−	0.569315i	$-0.807208\pi$
$997$		−	35.9526i		−	1.13863i	0.822119	−	0.569315i	$-0.192792\pi$
$998$		0			0
$999$	6.22203			0.196856

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1904.2.j.b.783.3	✓	24
4.3	odd	2	inner	1904.2.j.b.783.21	yes	24
7.6	odd	2	inner	1904.2.j.b.783.22	yes	24
28.27	even	2	inner	1904.2.j.b.783.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1904.2.j.b.783.3	✓	24	1.1	even	1	trivial
1904.2.j.b.783.4	yes	24	28.27	even	2	inner
1904.2.j.b.783.21	yes	24	4.3	odd	2	inner
1904.2.j.b.783.22	yes	24	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1904.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.69181 q^{3} +1.35274i q^{5} +(-1.38205 + 2.25609i) q^{7} +4.24582 q^{9} +3.05189i q^{11} -1.89309i q^{13} -3.64131i q^{15} +1.00000i q^{17} +6.45459 q^{19} +(3.72022 - 6.07295i) q^{21} +9.21870i q^{23} +3.17010 q^{25} -3.35352 q^{27} +10.1082 q^{29} -8.51357 q^{31} -8.21511i q^{33} +(-3.05189 - 1.86955i) q^{35} -1.85537 q^{37} +5.09582i q^{39} +4.02548i q^{41} -6.23137i q^{43} +5.74348i q^{45} -1.46028 q^{47} +(-3.17987 - 6.23606i) q^{49} -2.69181i q^{51} -0.520824 q^{53} -4.12841 q^{55} -17.3745 q^{57} +11.0102 q^{59} +10.5171i q^{61} +(-5.86795 + 9.57895i) q^{63} +2.56085 q^{65} +10.0173i q^{67} -24.8149i q^{69} -12.7096i q^{71} +1.60926i q^{73} -8.53331 q^{75} +(-6.88534 - 4.21787i) q^{77} -10.2819i q^{79} -3.71045 q^{81} +1.20784 q^{83} -1.35274 q^{85} -27.2093 q^{87} +16.6194i q^{89} +(4.27097 + 2.61634i) q^{91} +22.9169 q^{93} +8.73136i q^{95} +10.8127i q^{97} +12.9578i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{9} - 32 q^{25} - 8 q^{29} + 8 q^{49} + 40 q^{53} + 8 q^{57} - 48 q^{65} - 16 q^{77} + 24 q^{81} + 32 q^{93}+O(q^{100}) \) 24 * q + 32 * q^9 - 32 * q^25 - 8 * q^29 + 8 * q^49 + 40 * q^53 + 8 * q^57 - 48 * q^65 - 16 * q^77 + 24 * q^81 + 32 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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