LMFDB - Embedded newform 1904.2.j.b.783.22

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			783.22
Character	$\chi$	$=$	1904.783
Dual form			1904.2.j.b.783.21

$q$-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	$-0.216711\pi$
$3$	2.69181			1.55412			0.777058	+	0.629429i	$0.216711\pi$
$4$		0			0
$5$		−	1.35274i		−	0.604962i	−0.953155	−	0.302481i	$-0.902185\pi$
$5$		−	1.35274i		−	0.604962i	0.953155	−	0.302481i	$-0.0978149\pi$
$6$		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	$0.0845549\pi$
$13$			1.89309i			0.525048i	−0.964925	+	0.262524i	$0.915445\pi$
$14$		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	$-0.765360\pi$
$19$	−6.45459			−1.48079			−0.740393	+	0.672175i	$0.765360\pi$
$20$		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	$-0.222966\pi$
$31$	8.51357			1.52908			0.764541	+	0.644575i	$0.222966\pi$
$32$		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	$-0.898218\pi$
$41$		−	4.02548i		−	0.628674i	0.949311	−	0.314337i	$-0.101782\pi$
$42$		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	$-0.466035\pi$
$47$	1.46028			0.213004			0.106502	+	0.994312i	$0.466035\pi$
$48$		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	$-0.754349\pi$
$59$	−11.0102			−1.43340			−0.716702	+	0.697379i	$0.754349\pi$
$60$		0			0
$61$		−	10.5171i		−	1.34658i	−0.739379	−	0.673290i	$-0.764881\pi$
$61$		−	10.5171i		−	1.34658i	0.739379	−	0.673290i	$-0.235119\pi$
$62$		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	$-0.969979\pi$
$73$		−	1.60926i		−	0.188349i	0.995556	−	0.0941745i	$-0.0300212\pi$
$74$		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	$-0.521116\pi$
$83$	−1.20784			−0.132578			−0.0662890	+	0.997800i	$0.521116\pi$
$84$		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	$-0.656989\pi$
$89$		−	16.6194i		−	1.76165i	0.473443	−	0.880825i	$-0.343011\pi$
$90$		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	$-0.815035\pi$
$97$		−	10.8127i		−	1.09786i	0.835867	−	0.548931i	$-0.184965\pi$
$98$		0			0
$99$			12.9578i			1.30231i
$100$		0			0
$101$			2.21511i			0.220411i	0.993909	+	0.110206i	$0.0351510\pi$
$101$			2.21511i			0.220411i	−0.993909	+	0.110206i	$0.964849\pi$
$102$		0			0
$103$	16.9392			1.66907			0.834536	−	0.550954i	$-0.185736\pi$
$103$	16.9392			1.66907			0.834536	+	0.550954i	$0.185736\pi$
$104$		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	$-0.353210\pi$
$131$	10.1861			0.889965			0.444982	+	0.895539i	$0.353210\pi$
$132$		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	$-0.383177\pi$
$139$	8.46101			0.717653			0.358827	+	0.933404i	$0.383177\pi$
$140$		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	$-0.936236\pi$
$157$		−	4.98650i		−	0.397966i	0.980003	−	0.198983i	$-0.0637638\pi$
$158$		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	$-0.635457\pi$
$167$	−10.6697			−0.825646			−0.412823	+	0.910811i	$0.635457\pi$
$168$		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	$0.297243\pi$
$173$			21.1472i			1.60779i	−0.594771	+	0.803895i	$0.702757\pi$
$174$		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	$0.220919\pi$
$181$			17.2111i			1.27929i	−0.768670	+	0.639646i	$0.779081\pi$
$182$		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	$-0.593560\pi$
$199$	−8.17379			−0.579424			−0.289712	+	0.957114i	$0.593560\pi$
$200$		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	$-0.510924\pi$
$223$	−1.02475			−0.0686224			−0.0343112	+	0.999411i	$0.510924\pi$
$224$		0			0
$225$	13.4597			0.897313
$226$		0			0
$227$	−8.51877			−0.565410			−0.282705	−	0.959207i	$-0.591232\pi$
$227$	−8.51877			−0.565410			−0.282705	+	0.959207i	$0.591232\pi$
$228$		0			0
$229$			20.2401i			1.33750i	0.743486	+	0.668752i	$0.233171\pi$
$229$			20.2401i			1.33750i	−0.743486	+	0.668752i	$0.766829\pi$
$230$		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	$-0.716355\pi$
$241$		−	24.1482i		−	1.55552i	0.628559	−	0.777762i	$-0.283645\pi$
$242$		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	$-0.615795\pi$
$251$	−11.2742			−0.711618			−0.355809	+	0.934559i	$0.615795\pi$
$252$		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	$0.0892196\pi$
$257$			8.86963i			0.553272i	−0.960975	+	0.276636i	$0.910780\pi$
$258$		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	$-0.912244\pi$
$269$		−	8.92926i		−	0.544427i	0.962237	−	0.272213i	$-0.0877556\pi$
$270$		0			0
$271$	20.1147			1.22188			0.610941	−	0.791676i	$-0.290791\pi$
$271$	20.1147			1.22188			0.610941	+	0.791676i	$0.290791\pi$
$272$		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	$-0.440045\pi$
$283$	6.29981			0.374485			0.187242	+	0.982314i	$0.440045\pi$
$284$		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	$-0.809576\pi$
$293$		−	19.2803i		−	1.12637i	0.826332	−	0.563184i	$-0.190424\pi$
$294$		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	$-0.541956\pi$
$307$	−4.60562			−0.262857			−0.131428	+	0.991326i	$0.541956\pi$
$308$		0			0
$309$	45.5971			2.59393
$310$		0			0
$311$	−3.60084			−0.204185			−0.102092	−	0.994775i	$-0.532554\pi$
$311$	−3.60084			−0.204185			−0.102092	+	0.994775i	$0.532554\pi$
$312$		0			0
$313$			22.9391i			1.29659i	0.761388	+	0.648297i	$0.224518\pi$
$313$			22.9391i			1.29659i	−0.761388	+	0.648297i	$0.775482\pi$
$314$		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	$-0.785300\pi$
$349$		−	23.3335i		−	1.24901i	0.781019	−	0.624507i	$-0.214700\pi$
$350$		0			0
$351$			6.34850i			0.338858i
$352$		0			0
$353$			18.0248i			0.959362i	0.877443	+	0.479681i	$0.159248\pi$
$353$			18.0248i			0.959362i	−0.877443	+	0.479681i	$0.840752\pi$
$354$		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	$-0.345526\pi$
$367$	17.8725			0.932937			0.466469	+	0.884538i	$0.345526\pi$
$368$		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	$-0.499937\pi$
$383$	0.00772947			0.000394958			0.000197479		1.00000i	$0.499937\pi$
$384$		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	$-0.948777\pi$
$397$		−	6.38504i		−	0.320456i	0.987080	−	0.160228i	$-0.0512230\pi$
$398$		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	$-0.970263\pi$
$409$		−	3.77313i		−	0.186570i	0.995639	−	0.0932848i	$-0.0297367\pi$
$410$		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	$-0.803141\pi$
$419$	−33.3561			−1.62955			−0.814777	+	0.579774i	$0.803141\pi$
$420$		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	$-0.832666\pi$
$433$		−	20.8841i		−	1.00363i	0.864976	−	0.501814i	$-0.167334\pi$
$434$		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	$-0.441557\pi$
$439$	7.65077			0.365151			0.182576	+	0.983192i	$0.441557\pi$
$440$		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	$0.124229\pi$
$461$			16.3370i			0.760890i	−0.924804	+	0.380445i	$0.875771\pi$
$462$		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	$-0.598408\pi$
$467$	−13.1500			−0.608511			−0.304255	+	0.952590i	$0.598408\pi$
$468$		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	$-0.473073\pi$
$479$	3.69837			0.168983			0.0844915	+	0.996424i	$0.473073\pi$
$480$		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	$-0.340764\pi$
$503$	21.5148			0.959297			0.479648	+	0.877461i	$0.340764\pi$
$504$		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	$-0.822915\pi$
$509$		−	23.8277i		−	1.05615i	0.849199	−	0.528073i	$-0.177085\pi$
$510$		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	$-0.986135\pi$
$521$		−	1.98790i		−	0.0870914i	0.999051	−	0.0435457i	$-0.0138654\pi$
$522$		0			0
$523$	3.01620			0.131889			0.0659447	−	0.997823i	$-0.478994\pi$
$523$	3.01620			0.131889			0.0659447	+	0.997823i	$0.478994\pi$
$524$		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	$-0.569195\pi$
$563$	−10.2349			−0.431351			−0.215675	+	0.976465i	$0.569195\pi$
$564$		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	$-0.927157\pi$
$577$		−	10.8984i		−	0.453705i	0.973929	−	0.226852i	$-0.0728435\pi$
$578$		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	$-0.689646\pi$
$587$	−27.1918			−1.12233			−0.561163	+	0.827705i	$0.689646\pi$
$588$		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	$0.171514\pi$
$593$			24.9911i			1.02626i	−0.858311	+	0.513130i	$0.828486\pi$
$594$		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	$0.0280618\pi$
$601$			4.31687i			0.176089i	−0.996117	+	0.0880445i	$0.971938\pi$
$602$		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	$-0.377075\pi$
$607$	18.5595			0.753306			0.376653	+	0.926354i	$0.377075\pi$
$608$		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	$-0.196600\pi$
$619$	40.5663			1.63050			0.815249	+	0.579110i	$0.196600\pi$
$620$		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	$-0.501366\pi$
$643$	−0.217686			−0.00858472			−0.00429236	+	0.999991i	$0.501366\pi$
$644$		0			0
$645$	−22.6903			−0.893431
$646$		0			0
$647$	−16.1546			−0.635104			−0.317552	−	0.948241i	$-0.602861\pi$
$647$	−16.1546			−0.635104			−0.317552	+	0.948241i	$0.602861\pi$
$648$		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	$0.00512413\pi$
$661$			0.827718i			0.0321945i	−0.999870	+	0.0160972i	$0.994876\pi$
$662$		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	$-0.718431\pi$
$677$		−	40.2594i		−	1.54729i	0.633618	−	0.773646i	$-0.281569\pi$
$678$		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	$-0.437292\pi$
$691$	10.2903			0.391462			0.195731	+	0.980658i	$0.437292\pi$
$692$		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	$-0.418906\pi$
$719$	13.5152			0.504033			0.252017	+	0.967723i	$0.418906\pi$
$720$		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	$-0.684091\pi$
$727$	−29.4777			−1.09327			−0.546634	+	0.837371i	$0.684091\pi$
$728$		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	$-0.858192\pi$
$733$		−	23.3329i		−	0.861822i	0.902395	−	0.430911i	$-0.141808\pi$
$734$		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	$0.0798273\pi$
$761$			13.6918i			0.496329i	−0.968718	+	0.248164i	$0.920173\pi$
$762$		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	$-0.751759\pi$
$769$		−	39.0000i		−	1.40638i	0.711004	−	0.703188i	$-0.248241\pi$
$770$		0			0
$771$			23.8753i			0.859849i
$772$		0			0
$773$		−	31.2203i		−	1.12292i	−0.827505	−	0.561458i	$-0.810240\pi$
$773$		−	31.2203i		−	1.12292i	0.827505	−	0.561458i	$-0.189760\pi$
$774$		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	$-0.521588\pi$
$787$	−3.80226			−0.135536			−0.0677680	+	0.997701i	$0.521588\pi$
$788$		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	$-0.933927\pi$
$797$		−	11.6361i		−	0.412173i	0.978534	−	0.206087i	$-0.0660729\pi$
$798$		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	$-0.301907\pi$
$811$	33.2013			1.16586			0.582928	+	0.812524i	$0.301907\pi$
$812$		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	$0.272384\pi$
$829$			43.4789i			1.51009i	−0.655676	+	0.755043i	$0.727616\pi$
$830$		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	$-0.430783\pi$
$839$	12.4982			0.431487			0.215743	+	0.976450i	$0.430783\pi$
$840$		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	$-0.824209\pi$
$853$		−	30.6440i		−	1.04923i	0.851339	−	0.524616i	$-0.175791\pi$
$854$		0			0
$855$			37.0718i			1.26783i
$856$		0			0
$857$		−	40.3156i		−	1.37715i	−0.725163	−	0.688577i	$-0.758236\pi$
$857$		−	40.3156i		−	1.37715i	0.725163	−	0.688577i	$-0.241764\pi$
$858$		0			0
$859$	−20.4629			−0.698184			−0.349092	−	0.937089i	$-0.613510\pi$
$859$	−20.4629			−0.698184			−0.349092	+	0.937089i	$0.613510\pi$
$860$		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	$0.0457016\pi$
$881$			8.49388i			0.286166i	−0.989711	+	0.143083i	$0.954298\pi$
$882$		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	$-0.809764\pi$
$887$	−49.2403			−1.65333			−0.826663	+	0.562697i	$0.809764\pi$
$888$		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	$0.00496346\pi$
$929$			0.950505i			0.0311851i	−0.999878	+	0.0155925i	$0.995037\pi$
$930$		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	$0.166899\pi$
$937$			30.6491i			1.00126i	−0.865661	+	0.500631i	$0.833101\pi$
$938$		0			0
$939$			61.7476i			2.01506i
$940$		0			0
$941$		−	3.75169i		−	0.122302i	−0.998129	−	0.0611508i	$-0.980523\pi$
$941$		−	3.75169i		−	0.122302i	0.998129	−	0.0611508i	$-0.0194771\pi$
$942$		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	$-0.291948\pi$
$971$	37.8954			1.21612			0.608060	+	0.793891i	$0.291948\pi$
$972$		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	$-0.430052\pi$
$983$	13.6688			0.435967			0.217984	+	0.975952i	$0.430052\pi$
$984$		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	$0.192792\pi$
$997$			35.9526i			1.13863i	−0.822119	+	0.569315i	$0.807208\pi$
$998$		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.783.22	yes	24
4.3	odd	2	inner	1904.2.j.b.783.4	yes	24
7.6	odd	2	inner	1904.2.j.b.783.3	✓	24
28.27	even	2	inner	1904.2.j.b.783.21	yes	24

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Related objects

Downloads

Learn more