LMFDB - Embedded newform 315.4.g.a.314.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(314,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("315.314");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.g (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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			314.18
Character	$\chi$	$=$	315.314
Dual form			315.4.g.a.314.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41590 q^{2} -5.99524 q^{4} +(-2.48890 + 10.8998i) q^{5} +(-5.69164 - 17.6240i) q^{7} +19.8158 q^{8} +(3.52402 - 15.4330i) q^{10} -30.5871i q^{11} -16.5994 q^{13} +(8.05877 + 24.9538i) q^{14} +19.9047 q^{16} -81.3047i q^{17} +87.9721i q^{19} +(14.9215 - 65.3468i) q^{20} +43.3082i q^{22} +51.9402 q^{23} +(-112.611 - 54.2569i) q^{25} +23.5030 q^{26} +(34.1227 + 105.660i) q^{28} +160.205i q^{29} +306.111i q^{31} -186.710 q^{32} +115.119i q^{34} +(206.264 - 18.1734i) q^{35} +104.929i q^{37} -124.559i q^{38} +(-49.3195 + 215.988i) q^{40} +477.920 q^{41} -265.819i q^{43} +183.377i q^{44} -73.5419 q^{46} +140.739i q^{47} +(-278.210 + 200.619i) q^{49} +(159.445 + 76.8222i) q^{50} +99.5173 q^{52} +305.963 q^{53} +(333.393 + 76.1282i) q^{55} +(-112.784 - 349.234i) q^{56} -226.833i q^{58} +134.064 q^{59} +303.805i q^{61} -433.422i q^{62} +105.123 q^{64} +(41.3142 - 180.930i) q^{65} +671.592i q^{67} +487.441i q^{68} +(-292.048 + 25.7316i) q^{70} +616.486i q^{71} +1021.48 q^{73} -148.568i q^{74} -527.413i q^{76} +(-539.067 + 174.091i) q^{77} -558.731 q^{79} +(-49.5409 + 216.958i) q^{80} -676.686 q^{82} +335.471i q^{83} +(886.204 + 202.359i) q^{85} +376.372i q^{86} -606.108i q^{88} +931.364 q^{89} +(94.4778 + 292.548i) q^{91} -311.394 q^{92} -199.272i q^{94} +(-958.877 - 218.953i) q^{95} +772.248 q^{97} +(393.917 - 284.056i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 192 q^{4} + 768 q^{16} - 432 q^{25} + 816 q^{46} + 456 q^{49} + 7968 q^{64} + 1464 q^{70} + 4368 q^{79} - 1440 q^{85} - 4392 q^{91}+O(q^{100}) $ 48 * q + 192 * q^4 + 768 * q^16 - 432 * q^25 + 816 * q^46 + 456 * q^49 + 7968 * q^64 + 1464 * q^70 + 4368 * q^79 - 1440 * q^85 - 4392 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−1.41590			−0.500595			−0.250298	−	0.968169i	$-0.580528\pi$
$2$	−1.41590			−0.500595			−0.250298	+	0.968169i	$0.580528\pi$
$3$		0			0
$3$		0			0
$4$	−5.99524			−0.749405
$5$	−2.48890	+	10.8998i	−0.222614	+	0.974907i
$5$	−2.48890	+	10.8998i	−0.222614	+	0.974907i
$6$		0			0
$7$	−5.69164	−	17.6240i	−0.307320	−	0.951606i
$7$	−5.69164	−	17.6240i	−0.307320	−	0.951606i
$8$	19.8158			0.875743
$9$		0			0
$10$	3.52402	−	15.4330i	0.111439	−	0.488034i
$11$		−	30.5871i		−	0.838397i	−0.907895	−	0.419198i	$-0.862311\pi$
$11$		−	30.5871i		−	0.838397i	0.907895	−	0.419198i	$-0.137689\pi$
$12$		0			0
$13$	−16.5994			−0.354142			−0.177071	−	0.984198i	$-0.556662\pi$
$13$	−16.5994			−0.354142			−0.177071	+	0.984198i	$0.556662\pi$
$14$	8.05877	+	24.9538i	0.153843	+	0.476369i
$15$		0			0
$16$	19.9047			0.311012
$17$		−	81.3047i		−	1.15996i	−0.814632	−	0.579979i	$-0.803061\pi$
$17$		−	81.3047i		−	1.15996i	0.814632	−	0.579979i	$-0.196939\pi$
$18$		0			0
$19$			87.9721i			1.06222i	0.847303	+	0.531110i	$0.178225\pi$
$19$			87.9721i			1.06222i	−0.847303	+	0.531110i	$0.821775\pi$
$20$	14.9215	−	65.3468i	0.166828	−	0.730600i
$21$		0			0
$22$			43.3082i			0.419697i
$23$	51.9402			0.470882			0.235441	−	0.971889i	$-0.424347\pi$
$23$	51.9402			0.470882			0.235441	+	0.971889i	$0.424347\pi$
$24$		0			0
$25$	−112.611	−	54.2569i	−0.900886	−	0.434055i
$26$	23.5030			0.177282
$27$		0			0
$28$	34.1227	+	105.660i	0.230307	+	0.713138i
$29$			160.205i			1.02584i	0.858437	+	0.512919i	$0.171436\pi$
$29$			160.205i			1.02584i	−0.858437	+	0.512919i	$0.828564\pi$
$30$		0			0
$31$			306.111i			1.77352i	0.462226	+	0.886762i	$0.347051\pi$
$31$			306.111i			1.77352i	−0.462226	+	0.886762i	$0.652949\pi$
$32$	−186.710			−1.03143
$33$		0			0
$34$			115.119i			0.580669i
$35$	206.264	−	18.1734i	0.996141	−	0.0877674i
$36$		0			0
$37$			104.929i			0.466221i	0.972450	+	0.233111i	$0.0748904\pi$
$37$			104.929i			0.466221i	−0.972450	+	0.233111i	$0.925110\pi$
$38$		−	124.559i		−	0.531742i
$39$		0			0
$40$	−49.3195	+	215.988i	−0.194952	+	0.853768i
$41$	477.920			1.82045			0.910227	−	0.414110i	$-0.135907\pi$
$41$	477.920			1.82045			0.910227	+	0.414110i	$0.135907\pi$
$42$		0			0
$43$		−	265.819i		−	0.942720i	−0.881941	−	0.471360i	$-0.843763\pi$
$43$		−	265.819i		−	0.942720i	0.881941	−	0.471360i	$-0.156237\pi$
$44$			183.377i			0.628298i
$45$		0			0
$46$	−73.5419			−0.235721
$47$			140.739i			0.436785i	0.975861	+	0.218393i	$0.0700813\pi$
$47$			140.739i			0.436785i	−0.975861	+	0.218393i	$0.929919\pi$
$48$		0			0
$49$	−278.210	+	200.619i	−0.811109	+	0.584895i
$50$	159.445	+	76.8222i	0.450979	+	0.217286i
$51$		0			0
$52$	99.5173			0.265396
$53$	305.963			0.792966			0.396483	−	0.918042i	$-0.370231\pi$
$53$	305.963			0.792966			0.396483	+	0.918042i	$0.370231\pi$
$54$		0			0
$55$	333.393	+	76.1282i	0.817358	+	0.186639i
$56$	−112.784	−	349.234i	−0.269133	−	0.833363i
$57$		0			0
$58$		−	226.833i		−	0.513529i
$59$	134.064			0.295825			0.147912	−	0.989000i	$-0.452745\pi$
$59$	134.064			0.295825			0.147912	+	0.989000i	$0.452745\pi$
$60$		0			0
$61$			303.805i			0.637676i	0.947809	+	0.318838i	$0.103293\pi$
$61$			303.805i			0.637676i	−0.947809	+	0.318838i	$0.896707\pi$
$62$		−	433.422i		−	0.887817i
$63$		0			0
$64$	105.123			0.205319
$65$	41.3142	−	180.930i	0.0788368	−	0.345255i
$66$		0			0
$67$			671.592i			1.22460i	0.790626	+	0.612299i	$0.209755\pi$
$67$			671.592i			1.22460i	−0.790626	+	0.612299i	$0.790245\pi$
$68$			487.441i			0.869277i
$69$		0			0
$70$	−292.048	+	25.7316i	−0.498663	+	0.0439359i
$71$			616.486i			1.03047i	0.857049	+	0.515236i	$0.172296\pi$
$71$			616.486i			1.03047i	−0.857049	+	0.515236i	$0.827704\pi$
$72$		0			0
$73$	1021.48			1.63774			0.818871	−	0.573977i	$-0.194600\pi$
$73$	1021.48			1.63774			0.818871	+	0.573977i	$0.194600\pi$
$74$		−	148.568i		−	0.233388i
$75$		0			0
$76$		−	527.413i		−	0.796033i
$77$	−539.067	+	174.091i	−0.797823	+	0.257656i
$78$		0			0
$79$	−558.731			−0.795723			−0.397862	−	0.917445i	$-0.630248\pi$
$79$	−558.731			−0.795723			−0.397862	+	0.917445i	$0.630248\pi$
$80$	−49.5409	+	216.958i	−0.0692355	+	0.303207i
$81$		0			0
$82$	−676.686			−0.911310
$83$			335.471i			0.443648i	0.975087	+	0.221824i	$0.0712010\pi$
$83$			335.471i			0.443648i	−0.975087	+	0.221824i	$0.928799\pi$
$84$		0			0
$85$	886.204	+	202.359i	1.13085	+	0.258222i
$86$			376.372i			0.471921i
$87$		0			0
$88$		−	606.108i		−	0.734220i
$89$	931.364			1.10926			0.554632	−	0.832096i	$-0.312859\pi$
$89$	931.364			1.10926			0.554632	+	0.832096i	$0.312859\pi$
$90$		0			0
$91$	94.4778	+	292.548i	0.108835	+	0.337004i
$92$	−311.394			−0.352881
$93$		0			0
$94$		−	199.272i		−	0.218653i
$95$	−958.877	−	218.953i	−1.03557	−	0.236465i
$96$		0			0
$97$	772.248			0.808350			0.404175	−	0.914682i	$-0.367559\pi$
$97$	772.248			0.808350			0.404175	+	0.914682i	$0.367559\pi$
$98$	393.917	−	284.056i	0.406037	−	0.292795i
$99$		0			0
$100$	675.128	+	325.283i	0.675128	+	0.325283i
$101$	−889.653			−0.876473			−0.438237	−	0.898860i	$-0.644397\pi$
$101$	−889.653			−0.876473			−0.438237	+	0.898860i	$0.644397\pi$
$102$		0			0
$103$	−1586.47			−1.51766			−0.758831	−	0.651288i	$-0.774229\pi$
$103$	−1586.47			−1.51766			−0.758831	+	0.651288i	$0.774229\pi$
$104$	−328.931			−0.310137
$105$		0			0
$106$	−433.211			−0.396955
$107$	−767.328			−0.693275			−0.346638	−	0.937999i	$-0.612677\pi$
$107$	−767.328			−0.693275			−0.346638	+	0.937999i	$0.612677\pi$
$108$		0			0
$109$	−814.720			−0.715927			−0.357963	−	0.933736i	$-0.616529\pi$
$109$	−814.720			−0.715927			−0.357963	+	0.933736i	$0.616529\pi$
$110$	−472.050	−	107.790i	−0.409166	−	0.0934303i
$111$		0			0
$112$	−113.291	−	350.801i	−0.0955800	−	0.295961i
$113$	1649.68			1.37335			0.686676	−	0.726964i	$-0.259069\pi$
$113$	1649.68			1.37335			0.686676	+	0.726964i	$0.259069\pi$
$114$		0			0
$115$	−129.274	+	566.137i	−0.104825	+	0.459066i
$116$		−	960.466i		−	0.768767i
$117$		0			0
$118$	−189.821			−0.148088
$119$	−1432.91	+	462.757i	−1.10382	+	0.356478i
$120$		0			0
$121$	395.428			0.297091
$122$		−	430.156i		−	0.319217i
$123$		0			0
$124$		−	1835.21i		−	1.32909i
$125$	871.665	−	1092.39i	0.623713	−	0.781654i
$126$		0			0
$127$			173.846i			0.121467i	0.998154	+	0.0607337i	$0.0193440\pi$
$127$			173.846i			0.121467i	−0.998154	+	0.0607337i	$0.980656\pi$
$128$	1344.83			0.928652
$129$		0			0
$130$	−58.4966	+	256.178i	−0.0394653	+	0.172833i
$131$	468.800			0.312666			0.156333	−	0.987704i	$-0.450033\pi$
$131$	468.800			0.312666			0.156333	+	0.987704i	$0.450033\pi$
$132$		0			0
$133$	1550.42	−	500.705i	1.01082	−	0.326441i
$134$		−	950.906i		−	0.613028i
$135$		0			0
$136$		−	1611.12i		−	1.01583i
$137$	1374.98			0.857463			0.428731	−	0.903432i	$-0.358961\pi$
$137$	1374.98			0.857463			0.428731	+	0.903432i	$0.358961\pi$
$138$		0			0
$139$		−	1723.01i		−	1.05140i	−0.850671	−	0.525698i	$-0.823804\pi$
$139$		−	1723.01i		−	1.05140i	0.850671	−	0.525698i	$-0.176196\pi$
$140$	−1236.60	+	108.954i	−0.746513	+	0.0657733i
$141$		0			0
$142$		−	872.881i		−	0.515849i
$143$			507.728i			0.296911i
$144$		0			0
$145$	−1746.20	−	398.733i	−1.00010	−	0.228365i
$146$	−1446.31			−0.819846
$147$		0			0
$148$		−	629.073i		−	0.349388i
$149$		−	621.643i		−	0.341792i	−0.985289	−	0.170896i	$-0.945334\pi$
$149$		−	621.643i		−	0.341792i	0.985289	−	0.170896i	$-0.0546662\pi$
$150$		0			0
$151$	2791.64			1.50451			0.752253	−	0.658874i	$-0.228967\pi$
$151$	2791.64			1.50451			0.752253	+	0.658874i	$0.228967\pi$
$152$			1743.24i			0.930232i
$153$		0			0
$154$	763.263	−	246.495i	0.399387	−	0.128981i
$155$	−3336.55	−	761.880i	−1.72902	−	0.394811i
$156$		0			0
$157$	−1735.14			−0.882034			−0.441017	−	0.897499i	$-0.645382\pi$
$157$	−1735.14			−0.882034			−0.441017	+	0.897499i	$0.645382\pi$
$158$	791.105			0.398335
$159$		0			0
$160$	464.701	−	2035.09i	0.229611	−	1.00555i
$161$	−295.625	−	915.393i	−0.144711	−	0.448094i
$162$		0			0
$163$		−	995.750i		−	0.478486i	−0.970960	−	0.239243i	$-0.923101\pi$
$163$		−	995.750i		−	0.478486i	0.970960	−	0.239243i	$-0.0768992\pi$
$164$	−2865.25			−1.36426
$165$		0			0
$166$		−	474.993i		−	0.222088i
$167$		−	1877.32i		−	0.869888i	−0.900457	−	0.434944i	$-0.856768\pi$
$167$		−	1877.32i		−	0.869888i	0.900457	−	0.434944i	$-0.143232\pi$
$168$		0			0
$169$	−1921.46			−0.874584
$170$	−1254.77	−	286.519i	−0.566098	−	0.129265i
$171$		0			0
$172$			1593.64i			0.706478i
$173$			2654.34i			1.16651i	0.812291	+	0.583253i	$0.198220\pi$
$173$			2654.34i			1.16651i	−0.812291	+	0.583253i	$0.801780\pi$
$174$		0			0
$175$	−315.283	+	2293.46i	−0.136190	+	0.990683i
$176$		−	608.829i		−	0.260751i
$177$		0			0
$178$	−1318.72			−0.555292
$179$			4472.92i			1.86772i	0.357642	+	0.933859i	$0.383581\pi$
$179$			4472.92i			1.86772i	−0.357642	+	0.933859i	$0.616419\pi$
$180$		0			0
$181$		−	396.033i		−	0.162635i	−0.996688	−	0.0813173i	$-0.974087\pi$
$181$		−	396.033i		−	0.162635i	0.996688	−	0.0813173i	$-0.0259127\pi$
$182$	−133.771	−	414.217i	−0.0544821	−	0.168702i
$183$		0			0
$184$	1029.24			0.412371
$185$	−1143.70	−	261.157i	−0.454522	−	0.103787i
$186$		0			0
$187$	−2486.87			−0.972504
$188$		−	843.764i		−	0.327329i
$189$		0			0
$190$	1357.67	+	310.015i	0.518399	+	0.118373i
$191$		−	1297.93i		−	0.491702i	−0.969308	−	0.245851i	$-0.920933\pi$
$191$		−	1297.93i		−	0.491702i	0.969308	−	0.245851i	$-0.0790673\pi$
$192$		0			0
$193$		−	4471.94i		−	1.66786i	−0.551868	−	0.833931i	$-0.686085\pi$
$193$		−	4471.94i		−	1.66786i	0.551868	−	0.833931i	$-0.313915\pi$
$194$	−1093.42			−0.404656
$195$		0			0
$196$	1667.94	−	1202.76i	0.607849	−	0.438323i
$197$	21.6839			0.00784220			0.00392110	−	0.999992i	$-0.498752\pi$
$197$	21.6839			0.00784220			0.00392110	+	0.999992i	$0.498752\pi$
$198$		0			0
$199$			1699.27i			0.605316i	0.953099	+	0.302658i	$0.0978740\pi$
$199$			1699.27i			0.605316i	−0.953099	+	0.302658i	$0.902126\pi$
$200$	−2231.47	−	1075.14i	−0.788945	−	0.380121i
$201$		0			0
$202$	1259.66			0.438758
$203$	2823.45	−	911.828i	0.976193	−	0.315260i
$204$		0			0
$205$	−1189.49	+	5209.23i	−0.405258	+	1.77477i
$206$	2246.27			0.759734
$207$		0			0
$208$	−330.407			−0.110142
$209$	2690.81			0.890562
$210$		0			0
$211$	−1882.71			−0.614269			−0.307135	−	0.951666i	$-0.599370\pi$
$211$	−1882.71			−0.614269			−0.307135	+	0.951666i	$0.599370\pi$
$212$	−1834.32			−0.594252
$213$		0			0
$214$	1086.46			0.347050
$215$	2897.37	+	661.595i	0.919064	+	0.209862i
$216$		0			0
$217$	5394.91	−	1742.28i	1.68770	−	0.545039i
$218$	1153.56			0.358389
$219$		0			0
$220$	−1998.77	−	456.406i	−0.612532	−	0.139868i
$221$			1349.61i			0.410790i
$222$		0			0
$223$	5818.27			1.74717			0.873587	−	0.486668i	$-0.161788\pi$
$223$	5818.27			1.74717			0.873587	+	0.486668i	$0.161788\pi$
$224$	1062.68	+	3290.57i	0.316980	+	0.981519i
$225$		0			0
$226$	−2335.77			−0.687493
$227$			2706.94i			0.791480i	0.918363	+	0.395740i	$0.129512\pi$
$227$			2706.94i			0.791480i	−0.918363	+	0.395740i	$0.870488\pi$
$228$		0			0
$229$			6228.53i			1.79735i	0.438616	+	0.898674i	$0.355469\pi$
$229$			6228.53i			1.79735i	−0.438616	+	0.898674i	$0.644531\pi$
$230$	183.038	−	801.592i	0.0524747	−	0.229806i
$231$		0			0
$232$			3174.59i			0.898370i
$233$	4128.59			1.16083			0.580414	−	0.814322i	$-0.302891\pi$
$233$	4128.59			1.16083			0.580414	+	0.814322i	$0.302891\pi$
$234$		0			0
$235$	−1534.03	−	350.285i	−0.425825	−	0.0972344i
$236$	−803.746			−0.221692
$237$		0			0
$238$	2028.86	−	655.216i	0.552568	−	0.178451i
$239$		−	4645.24i		−	1.25722i	−0.777721	−	0.628610i	$-0.783624\pi$
$239$		−	4645.24i		−	1.25722i	0.777721	−	0.628610i	$-0.216376\pi$
$240$		0			0
$241$			6926.50i			1.85135i	0.378320	+	0.925675i	$0.376502\pi$
$241$			6926.50i			1.85135i	−0.378320	+	0.925675i	$0.623498\pi$
$242$	−559.886			−0.148722
$243$		0			0
$244$		−	1821.38i		−	0.477877i
$245$	−1494.27	−	3531.76i	−0.389654	−	0.920961i
$246$		0			0
$247$		−	1460.28i		−	0.376177i
$248$			6065.85i			1.55315i
$249$		0			0
$250$	−1234.19	+	1546.72i	−0.312228	+	0.391292i
$251$	5572.63			1.40136			0.700680	−	0.713476i	$-0.252880\pi$
$251$	5572.63			1.40136			0.700680	+	0.713476i	$0.252880\pi$
$252$		0			0
$253$		−	1588.70i		−	0.394785i
$254$		−	246.148i		−	0.0608060i
$255$		0			0
$256$	−2745.13			−0.670198
$257$			2220.94i			0.539060i	0.962992	+	0.269530i	$0.0868683\pi$
$257$			2220.94i			0.539060i	−0.962992	+	0.269530i	$0.913132\pi$
$258$		0			0
$259$	1849.26	−	597.217i	0.443659	−	0.143279i
$260$	−247.688	+	1084.72i	−0.0590807	+	0.258736i
$261$		0			0
$262$	−663.772			−0.156519
$263$	4665.04			1.09376			0.546880	−	0.837211i	$-0.315816\pi$
$263$	4665.04			1.09376			0.546880	+	0.837211i	$0.315816\pi$
$264$		0			0
$265$	−761.509	+	3334.93i	−0.176525	+	0.773068i
$266$	−2195.23	+	708.947i	−0.506009	+	0.163415i
$267$		0			0
$268$		−	4026.36i		−	0.917719i
$269$	4373.91			0.991384			0.495692	−	0.868498i	$-0.334915\pi$
$269$	4373.91			0.991384			0.495692	+	0.868498i	$0.334915\pi$
$270$		0			0
$271$			1610.82i			0.361071i	0.983568	+	0.180535i	$0.0577830\pi$
$271$			1610.82i			0.361071i	−0.983568	+	0.180535i	$0.942217\pi$
$272$		−	1618.35i		−	0.360760i
$273$		0			0
$274$	−1946.83			−0.429242
$275$	−1659.56	+	3444.44i	−0.363910	+	0.755300i
$276$		0			0
$277$			4785.35i			1.03799i	0.854777	+	0.518996i	$0.173694\pi$
$277$			4785.35i			1.03799i	−0.854777	+	0.518996i	$0.826306\pi$
$278$			2439.61i			0.526324i
$279$		0			0
$280$	4087.28	−	360.120i	0.872364	−	0.0768617i
$281$			4559.34i			0.967927i	0.875088	+	0.483964i	$0.160803\pi$
$281$			4559.34i			0.967927i	−0.875088	+	0.483964i	$0.839197\pi$
$282$		0			0
$283$	−3503.70			−0.735949			−0.367975	−	0.929836i	$-0.619949\pi$
$283$	−3503.70			−0.735949			−0.367975	+	0.929836i	$0.619949\pi$
$284$		−	3695.98i		−	0.772240i
$285$		0			0
$286$		−	718.890i		−	0.148632i
$287$	−2720.15	−	8422.87i	−0.559461	−	1.73236i
$288$		0			0
$289$	−1697.45			−0.345501
$290$	2472.44	+	564.565i	0.500643	+	0.114319i
$291$		0			0
$292$	−6124.02			−1.22733
$293$		−	5727.17i		−	1.14193i	−0.820975	−	0.570964i	$-0.806569\pi$
$293$		−	5727.17i		−	1.14193i	0.820975	−	0.570964i	$-0.193431\pi$
$294$		0			0
$295$	−333.672	+	1461.27i	−0.0658546	+	0.288401i
$296$			2079.25i			0.408290i
$297$		0			0
$298$			880.182i			0.171099i
$299$	−862.176			−0.166759
$300$		0			0
$301$	−4684.78	+	1512.94i	−0.897098	+	0.289716i
$302$	−3952.67			−0.753149
$303$		0			0
$304$			1751.06i			0.330363i
$305$	−3311.41	−	756.139i	−0.621674	−	0.141955i
$306$		0			0
$307$	1876.50			0.348852			0.174426	−	0.984670i	$-0.444193\pi$
$307$	1876.50			0.348852			0.174426	+	0.984670i	$0.444193\pi$
$308$	3231.84	−	1043.72i	0.597893	−	0.193088i
$309$		0			0
$310$	4724.21	+	1078.74i	0.865539	+	0.197640i
$311$	−4517.34			−0.823650			−0.411825	−	0.911263i	$-0.635108\pi$
$311$	−4517.34			−0.823650			−0.411825	+	0.911263i	$0.635108\pi$
$312$		0			0
$313$	−7150.40			−1.29126			−0.645630	−	0.763650i	$-0.723405\pi$
$313$	−7150.40			−1.29126			−0.645630	+	0.763650i	$0.723405\pi$
$314$	2456.78			0.441542
$315$		0			0
$316$	3349.72			0.596318
$317$	−4062.32			−0.719756			−0.359878	−	0.932999i	$-0.617182\pi$
$317$	−4062.32			−0.719756			−0.359878	+	0.932999i	$0.617182\pi$
$318$		0			0
$319$	4900.20			0.860059
$320$	−261.641	+	1145.82i	−0.0457069	+	0.200167i
$321$		0			0
$322$	418.574	+	1296.10i	0.0724417	+	0.224314i
$323$	7152.54			1.23213
$324$		0			0
$325$	1869.27	+	900.632i	0.319042	+	0.153717i
$326$			1409.88i			0.239528i
$327$		0			0
$328$	9470.38			1.59425
$329$	2480.39	−	801.036i	0.415648	−	0.134233i
$330$		0			0
$331$	−868.986			−0.144301			−0.0721507	−	0.997394i	$-0.522986\pi$
$331$	−868.986			−0.144301			−0.0721507	+	0.997394i	$0.522986\pi$
$332$		−	2011.23i		−	0.332472i
$333$		0			0
$334$			2658.09i			0.435462i
$335$	−7320.22	−	1671.52i	−1.19387	−	0.272612i
$336$		0			0
$337$		−	1287.31i		−	0.208083i	−0.994573	−	0.104042i	$-0.966823\pi$
$337$		−	1287.31i		−	0.208083i	0.994573	−	0.104042i	$-0.0331775\pi$
$338$	2720.59			0.437812
$339$		0			0
$340$	−5313.00	−	1213.19i	−0.847464	−	0.193513i
$341$	9363.07			1.48692
$342$		0			0
$343$	5119.18	+	3761.33i	0.805859	+	0.592107i
$344$		−	5267.41i		−	0.825580i
$345$		0			0
$346$		−	3758.27i		−	0.583947i
$347$	422.265			0.0653268			0.0326634	−	0.999466i	$-0.489601\pi$
$347$	422.265			0.0653268			0.0326634	+	0.999466i	$0.489601\pi$
$348$		0			0
$349$			4640.81i			0.711797i	0.934525	+	0.355898i	$0.115825\pi$
$349$			4640.81i			0.711797i	−0.934525	+	0.355898i	$0.884175\pi$
$350$	446.409	−	3247.31i	0.0681758	−	0.495931i
$351$		0			0
$352$			5710.91i			0.864751i
$353$			1539.01i			0.232048i	0.993246	+	0.116024i	$0.0370150\pi$
$353$			1539.01i			0.232048i	−0.993246	+	0.116024i	$0.962985\pi$
$354$		0			0
$355$	−6719.57	−	1534.37i	−1.00461	−	0.229397i
$356$	−5583.75			−0.831287
$357$		0			0
$358$		−	6333.19i		−	0.934970i
$359$		−	1035.99i		−	0.152305i	−0.997096	−	0.0761525i	$-0.975736\pi$
$359$		−	1035.99i		−	0.152305i	0.997096	−	0.0761525i	$-0.0242636\pi$
$360$		0			0
$361$	−880.088			−0.128311
$362$			560.741i			0.0814141i
$363$		0			0
$364$	−566.417	−	1753.89i	−0.0815613	−	0.252552i
$365$	−2542.36	+	11133.9i	−0.364584	+	1.59665i
$366$		0			0
$367$	−2278.61			−0.324094			−0.162047	−	0.986783i	$-0.551810\pi$
$367$	−2278.61			−0.324094			−0.162047	+	0.986783i	$0.551810\pi$
$368$	1033.86			0.146450
$369$		0			0
$370$	1619.36	+	369.771i	0.227532	+	0.0519554i
$371$	−1741.43	−	5392.28i	−0.243694	−	0.754591i
$372$		0			0
$373$		−	8610.23i		−	1.19523i	−0.801783	−	0.597615i	$-0.796115\pi$
$373$		−	8610.23i		−	1.19523i	0.801783	−	0.597615i	$-0.203885\pi$
$374$	3521.16			0.486831
$375$		0			0
$376$			2788.86i			0.382512i
$377$		−	2659.30i		−	0.363292i
$378$		0			0
$379$	−16.5038			−0.00223679			−0.00111840	−	0.999999i	$-0.500356\pi$
$379$	−16.5038			−0.00223679			−0.00111840	+	0.999999i	$0.500356\pi$
$380$	5748.70	+	1312.68i	0.776057	+	0.177208i
$381$		0			0
$382$			1837.74i			0.246143i
$383$			9887.59i			1.31914i	0.751641	+	0.659572i	$0.229263\pi$
$383$			9887.59i			1.31914i	−0.751641	+	0.659572i	$0.770737\pi$
$384$		0			0
$385$	−555.871	−	6309.01i	−0.0735839	−	0.835161i
$386$			6331.81i			0.834924i
$387$		0			0
$388$	−4629.81			−0.605781
$389$		−	279.292i		−	0.0364028i	−0.999834	−	0.0182014i	$-0.994206\pi$
$389$		−	279.292i		−	0.0364028i	0.999834	−	0.0182014i	$-0.00579400\pi$
$390$		0			0
$391$		−	4222.98i		−	0.546203i
$392$	−5512.97	+	3975.43i	−0.710324	+	0.512218i
$393$		0			0
$394$	−30.7021			−0.00392577
$395$	1390.62	−	6090.05i	0.177139	−	0.775756i
$396$		0			0
$397$	−10894.0			−1.37721			−0.688607	−	0.725135i	$-0.741777\pi$
$397$	−10894.0			−1.37721			−0.688607	+	0.725135i	$0.741777\pi$
$398$		−	2405.99i		−	0.303018i
$399$		0			0
$400$	−2241.49	−	1079.97i	−0.280186	−	0.134996i
$401$		−	9631.75i		−	1.19947i	−0.800200	−	0.599734i	$-0.795273\pi$
$401$		−	9631.75i		−	1.19947i	0.800200	−	0.599734i	$-0.204727\pi$
$402$		0			0
$403$		−	5081.27i		−	0.628079i
$404$	5333.68			0.656833
$405$		0			0
$406$	−3997.71	+	1291.05i	−0.488678	+	0.157818i
$407$	3209.47			0.390878
$408$		0			0
$409$		−	4082.30i		−	0.493538i	−0.969074	−	0.246769i	$-0.920631\pi$
$409$		−	4082.30i		−	0.493538i	0.969074	−	0.246769i	$-0.0793689\pi$
$410$	1684.20	−	7375.73i	0.202870	−	0.888443i
$411$		0			0
$412$	9511.24			1.13734
$413$	−763.044	−	2362.74i	−0.0909127	−	0.281509i
$414$		0			0
$415$	−3656.57	−	834.953i	−0.432515	−	0.0987620i
$416$	3099.27			0.365274
$417$		0			0
$418$	−3809.91			−0.445811
$419$	−15952.5			−1.85998			−0.929988	−	0.367589i	$-0.880183\pi$
$419$	−15952.5			−1.85998			−0.929988	+	0.367589i	$0.880183\pi$
$420$		0			0
$421$	16707.5			1.93415			0.967073	−	0.254499i	$-0.0819104\pi$
$421$	16707.5			1.93415			0.967073	+	0.254499i	$0.0819104\pi$
$422$	2665.72			0.307500
$423$		0			0
$424$	6062.90			0.694435
$425$	−4411.34	+	9155.78i	−0.503486	+	1.04499i
$426$		0			0
$427$	5354.26	−	1729.15i	0.606816	−	0.195970i
$428$	4600.31			0.519543
$429$		0			0
$430$	−4102.37	−	936.750i	−0.460079	−	0.105056i
$431$			1021.69i			0.114183i	0.998369	+	0.0570916i	$0.0181827\pi$
$431$			1021.69i			0.114183i	−0.998369	+	0.0570916i	$0.981817\pi$
$432$		0			0
$433$	1052.40			0.116802			0.0584008	−	0.998293i	$-0.481400\pi$
$433$	1052.40			0.116802			0.0584008	+	0.998293i	$0.481400\pi$
$434$	−7638.63	+	2466.88i	−0.844853	+	0.272844i
$435$		0			0
$436$	4884.44			0.536519
$437$			4569.29i			0.500180i
$438$		0			0
$439$		−	13540.8i		−	1.47213i	−0.676909	−	0.736066i	$-0.736681\pi$
$439$		−	13540.8i		−	1.47213i	0.676909	−	0.736066i	$-0.263319\pi$
$440$	6606.45	+	1508.54i	0.715796	+	0.163447i
$441$		0			0
$442$		−	1910.91i		−	0.205639i
$443$	−12799.7			−1.37276			−0.686380	−	0.727243i	$-0.740801\pi$
$443$	−12799.7			−1.37276			−0.686380	+	0.727243i	$0.740801\pi$
$444$		0			0
$445$	−2318.07	+	10151.7i	−0.246937	+	1.08143i
$446$	−8238.06			−0.874627
$447$		0			0
$448$	−598.325	−	1852.70i	−0.0630986	−	0.195383i
$449$		−	523.887i		−	0.0550641i	−0.999621	−	0.0275320i	$-0.991235\pi$
$449$		−	523.887i		−	0.0550641i	0.999621	−	0.0275320i	$-0.00876482\pi$
$450$		0			0
$451$		−	14618.2i		−	1.52626i
$452$	−9890.21			−1.02920
$453$		0			0
$454$		−	3832.75i		−	0.396211i
$455$	−3423.85	+	301.667i	−0.352775	+	0.0310821i
$456$		0			0
$457$			5749.64i			0.588527i	0.955724	+	0.294263i	$0.0950743\pi$
$457$			5749.64i			0.588527i	−0.955724	+	0.294263i	$0.904926\pi$
$458$		−	8818.96i		−	0.899744i
$459$		0			0
$460$	775.027	−	3394.12i	0.0785561	−	0.344026i
$461$	−14278.7			−1.44257			−0.721284	−	0.692640i	$-0.756448\pi$
$461$	−14278.7			−1.44257			−0.721284	+	0.692640i	$0.756448\pi$
$462$		0			0
$463$			16272.2i			1.63333i	0.577113	+	0.816664i	$0.304179\pi$
$463$			16272.2i			1.63333i	−0.577113	+	0.816664i	$0.695821\pi$
$464$			3188.84i			0.319047i
$465$		0			0
$466$	−5845.66			−0.581105
$467$			15119.6i			1.49819i	0.662464	+	0.749094i	$0.269511\pi$
$467$			15119.6i			1.49819i	−0.662464	+	0.749094i	$0.730489\pi$
$468$		0			0
$469$	11836.1	−	3822.46i	1.16534	−	0.376343i
$470$	2172.02	+	495.967i	0.213166	+	0.0486751i
$471$		0			0
$472$	2656.59			0.259066
$473$	−8130.62			−0.790373
$474$		0			0
$475$	4773.09	−	9906.61i	0.461062	−	0.956940i
$476$	8590.65	−	2774.34i	0.827210	−	0.267146i
$477$		0			0
$478$			6577.18i			0.629358i
$479$	−13040.7			−1.24394			−0.621970	−	0.783041i	$-0.713667\pi$
$479$	−13040.7			−1.24394			−0.621970	+	0.783041i	$0.713667\pi$
$480$		0			0
$481$		−	1741.75i		−	0.165108i
$482$		−	9807.21i		−	0.926777i
$483$		0			0
$484$	−2370.69			−0.222642
$485$	−1922.05	+	8417.34i	−0.179950	+	0.788066i
$486$		0			0
$487$		−	12919.7i		−	1.20215i	−0.799192	−	0.601075i	$-0.794739\pi$
$487$		−	12919.7i		−	1.20215i	0.799192	−	0.601075i	$-0.205261\pi$
$488$			6020.14i			0.558440i
$489$		0			0
$490$	2115.73	+	5000.60i	0.195059	+	0.461029i
$491$		−	10382.4i		−	0.954275i	−0.878829	−	0.477138i	$-0.841674\pi$
$491$		−	10382.4i		−	0.954275i	0.878829	−	0.477138i	$-0.158326\pi$
$492$		0			0
$493$	13025.4			1.18993
$494$			2067.61i			0.188312i
$495$		0			0
$496$			6093.07i			0.551587i
$497$	10865.0	−	3508.82i	0.980603	−	0.316684i
$498$		0			0
$499$	13113.5			1.17643			0.588217	−	0.808703i	$-0.299830\pi$
$499$	13113.5			1.17643			0.588217	+	0.808703i	$0.299830\pi$
$500$	−5225.84	+	6549.16i	−0.467413	+	0.585775i
$501$		0			0
$502$	−7890.27			−0.701514
$503$			9544.92i			0.846097i	0.906107	+	0.423049i	$0.139040\pi$
$503$			9544.92i			0.846097i	−0.906107	+	0.423049i	$0.860960\pi$
$504$		0			0
$505$	2214.25	−	9697.03i	0.195115	−	0.854480i
$506$			2249.44i			0.197628i
$507$		0			0
$508$		−	1042.25i		−	0.0910282i
$509$	3806.73			0.331493			0.165747	−	0.986168i	$-0.446997\pi$
$509$	3806.73			0.331493			0.165747	+	0.986168i	$0.446997\pi$
$510$		0			0
$511$	−5813.90	−	18002.6i	−0.503311	−	1.55849i
$512$	−6871.84			−0.593155
$513$		0			0
$514$		−	3144.62i		−	0.269851i
$515$	3948.55	−	17292.2i	0.337852	−	1.47958i
$516$		0			0
$517$	4304.80			0.366199
$518$	−2618.37	+	845.597i	−0.222094	+	0.0717247i
$519$		0			0
$520$	818.674	−	3585.27i	0.0690408	−	0.302355i
$521$	7738.68			0.650744			0.325372	−	0.945586i	$-0.394510\pi$
$521$	7738.68			0.650744			0.325372	+	0.945586i	$0.394510\pi$
$522$		0			0
$523$	−19441.5			−1.62547			−0.812734	−	0.582635i	$-0.802022\pi$
$523$	−19441.5			−1.62547			−0.812734	+	0.582635i	$0.802022\pi$
$524$	−2810.57			−0.234313
$525$		0			0
$526$	−6605.21			−0.547530
$527$	24888.3			2.05721
$528$		0			0
$529$	−9469.22			−0.778271
$530$	1078.22	−	4721.91i	0.0883676	−	0.386994i
$531$		0			0
$532$	−9295.13	+	3001.85i	−0.757510	+	0.244636i
$533$	−7933.19			−0.644699
$534$		0			0
$535$	1909.80	−	8363.72i	0.154333	−	0.675879i
$536$			13308.1i			1.07243i
$537$		0			0
$538$	−6193.01			−0.496282
$539$	6136.35	+	8509.66i	0.490374	+	0.680031i
$540$		0			0
$541$	−10379.0			−0.824817			−0.412409	−	0.910999i	$-0.635312\pi$
$541$	−10379.0			−0.824817			−0.412409	+	0.910999i	$0.635312\pi$
$542$		−	2280.75i		−	0.180750i
$543$		0			0
$544$			15180.4i			1.19642i
$545$	2027.75	−	8880.28i	0.159375	−	0.697962i
$546$		0			0
$547$		−	16009.6i		−	1.25141i	−0.780061	−	0.625703i	$-0.784812\pi$
$547$		−	16009.6i		−	1.25141i	0.780061	−	0.625703i	$-0.215188\pi$
$548$	−8243.33			−0.642587
$549$		0			0
$550$	2349.77	−	4876.97i	0.182172	−	0.378099i
$551$	−14093.6			−1.08967
$552$		0			0
$553$	3180.09	+	9847.07i	0.244541	+	0.757215i
$554$		−	6775.56i		−	0.519614i
$555$		0			0
$556$			10329.9i			0.787921i
$557$	11746.0			0.893526			0.446763	−	0.894652i	$-0.352577\pi$
$557$	11746.0			0.893526			0.446763	+	0.894652i	$0.352577\pi$
$558$		0			0
$559$			4412.43i			0.333857i
$560$	4105.63	−	361.736i	0.309812	−	0.0272967i
$561$		0			0
$562$		−	6455.56i		−	0.484540i
$563$		−	9594.88i		−	0.718252i	−0.933289	−	0.359126i	$-0.883075\pi$
$563$		−	9594.88i		−	0.718252i	0.933289	−	0.359126i	$-0.116925\pi$
$564$		0			0
$565$	−4105.88	+	17981.1i	−0.305727	+	1.33889i
$566$	4960.88			0.368412
$567$		0			0
$568$			12216.2i			0.902428i
$569$		−	8438.18i		−	0.621700i	−0.950459	−	0.310850i	$-0.899386\pi$
$569$		−	8438.18i		−	0.621700i	0.950459	−	0.310850i	$-0.100614\pi$
$570$		0			0
$571$	18415.6			1.34968			0.674841	−	0.737963i	$-0.264212\pi$
$571$	18415.6			1.34968			0.674841	+	0.737963i	$0.264212\pi$
$572$		−	3043.95i		−	0.222507i
$573$		0			0
$574$	3851.45	+	11925.9i	0.280064	+	0.867209i
$575$	−5849.02	−	2818.11i	−0.424211	−	0.204389i
$576$		0			0
$577$	5724.61			0.413030			0.206515	−	0.978443i	$-0.433788\pi$
$577$	5724.61			0.413030			0.206515	+	0.978443i	$0.433788\pi$
$578$	2403.41			0.172956
$579$		0			0
$580$	10468.9	+	2390.50i	0.749476	+	0.171138i
$581$	5912.35	−	1909.38i	0.422178	−	0.136342i
$582$		0			0
$583$		−	9358.51i		−	0.664820i
$584$	20241.5			1.43424
$585$		0			0
$586$			8109.09i			0.571644i
$587$			20033.2i			1.40862i	0.709894	+	0.704309i	$0.248743\pi$
$587$			20033.2i			1.40862i	−0.709894	+	0.704309i	$0.751257\pi$
$588$		0			0
$589$	−26929.3			−1.88387
$590$	472.444	−	2069.01i	0.0329665	−	0.144372i
$591$		0			0
$592$			2088.58i			0.145000i
$593$		−	10763.1i		−	0.745341i	−0.927964	−	0.372670i	$-0.878442\pi$
$593$		−	10763.1i		−	0.745341i	0.927964	−	0.372670i	$-0.121558\pi$
$594$		0			0
$595$	−1477.58	−	16770.2i	−0.101806	−	1.15548i
$596$			3726.90i			0.256140i
$597$		0			0
$598$	1220.75			0.0834787
$599$			12751.4i			0.869799i	0.900479	+	0.434899i	$0.143216\pi$
$599$			12751.4i			0.869799i	−0.900479	+	0.434899i	$0.856784\pi$
$600$		0			0
$601$			8319.06i			0.564628i	0.959322	+	0.282314i	$0.0911020\pi$
$601$			8319.06i			0.564628i	−0.959322	+	0.282314i	$0.908898\pi$
$602$	6633.17	−	2142.17i	0.449083	−	0.145031i
$603$		0			0
$604$	−16736.5			−1.12748
$605$	−984.181	+	4310.09i	−0.0661366	+	0.289636i
$606$		0			0
$607$	12259.8			0.819786			0.409893	−	0.912134i	$-0.365566\pi$
$607$	12259.8			0.819786			0.409893	+	0.912134i	$0.365566\pi$
$608$		−	16425.2i		−	1.09561i
$609$		0			0
$610$	4688.61	+	1070.61i	0.311207	+	0.0710622i
$611$		−	2336.18i		−	0.154684i
$612$		0			0
$613$			23691.3i			1.56098i	0.625167	+	0.780491i	$0.285031\pi$
$613$			23691.3i			1.56098i	−0.625167	+	0.780491i	$0.714969\pi$
$614$	−2656.93			−0.174634
$615$		0			0
$616$	−10682.1	+	3449.75i	−0.698689	+	0.225640i
$617$	8805.60			0.574554			0.287277	−	0.957848i	$-0.407250\pi$
$617$	8805.60			0.574554			0.287277	+	0.957848i	$0.407250\pi$
$618$		0			0
$619$		−	1540.85i		−	0.100051i	−0.998748	−	0.0500257i	$-0.984070\pi$
$619$		−	1540.85i		−	0.100051i	0.998748	−	0.0500257i	$-0.0159303\pi$
$620$	20003.4	+	4567.65i	1.29574	+	0.295873i
$621$		0			0
$622$	6396.09			0.412315
$623$	−5300.99	−	16414.4i	−0.340898	−	1.05558i
$624$		0			0
$625$	9737.38	+	12219.8i	0.623192	+	0.782069i
$626$	10124.2			0.646399
$627$		0			0
$628$	10402.6			0.661000
$629$	8531.20			0.540797
$630$		0			0
$631$	4818.83			0.304017			0.152008	−	0.988379i	$-0.451426\pi$
$631$	4818.83			0.304017			0.152008	+	0.988379i	$0.451426\pi$
$632$	−11071.7			−0.696849
$633$		0			0
$634$	5751.83			0.360306
$635$	−1894.89	−	432.685i	−0.118419	−	0.0270403i
$636$		0			0
$637$	4618.13	−	3330.15i	0.287248	−	0.207136i
$638$	−6938.18			−0.430541
$639$		0			0
$640$	−3347.15	+	14658.4i	−0.206731	+	0.905350i
$641$			18206.7i			1.12187i	0.827858	+	0.560937i	$0.189559\pi$
$641$			18206.7i			1.12187i	−0.827858	+	0.560937i	$0.810441\pi$
$642$		0			0
$643$	−18717.3			−1.14796			−0.573981	−	0.818868i	$-0.694602\pi$
$643$	−18717.3			−1.14796			−0.573981	+	0.818868i	$0.694602\pi$
$644$	1772.34	+	5488.00i	0.108447	+	0.335804i
$645$		0			0
$646$	−10127.3			−0.616798
$647$		−	1556.10i		−	0.0945542i	−0.998882	−	0.0472771i	$-0.984946\pi$
$647$		−	1556.10i		−	0.0945542i	0.998882	−	0.0472771i	$-0.0150544\pi$
$648$		0			0
$649$		−	4100.63i		−	0.248018i
$650$	−2646.70	−	1275.20i	−0.159711	−	0.0769500i
$651$		0			0
$652$			5969.76i			0.358579i
$653$	−29408.2			−1.76238			−0.881189	−	0.472765i	$-0.843256\pi$
$653$	−29408.2			−1.76238			−0.881189	+	0.472765i	$0.843256\pi$
$654$		0			0
$655$	−1166.79	+	5109.82i	−0.0696037	+	0.304820i
$656$	9512.88			0.566182
$657$		0			0
$658$	−3511.97	+	1134.18i	−0.208071	+	0.0671962i
$659$		−	15993.9i		−	0.945425i	−0.881217	−	0.472713i	$-0.843275\pi$
$659$		−	15993.9i		−	0.945425i	0.881217	−	0.472713i	$-0.156725\pi$
$660$		0			0
$661$			17299.8i			1.01798i	0.860773	+	0.508988i	$0.169980\pi$
$661$			17299.8i			1.01798i	−0.860773	+	0.508988i	$0.830020\pi$
$662$	1230.39			0.0722366
$663$		0			0
$664$			6647.64i			0.388522i
$665$	1598.75	+	18145.5i	0.0932283	+	1.05812i
$666$		0			0
$667$			8321.07i			0.483048i
$668$			11255.0i			0.651898i
$669$		0			0
$670$	10364.7	+	2366.71i	0.597645	+	0.136468i
$671$	9292.51			0.534625
$672$		0			0
$673$			17783.4i			1.01857i	0.860597	+	0.509287i	$0.170091\pi$
$673$			17783.4i			1.01857i	−0.860597	+	0.509287i	$0.829909\pi$
$674$			1822.69i			0.104165i
$675$		0			0
$676$	11519.6			0.655417
$677$		−	12701.0i		−	0.721032i	−0.932753	−	0.360516i	$-0.882601\pi$
$677$		−	12701.0i		−	0.721032i	0.932753	−	0.360516i	$-0.117399\pi$
$678$		0			0
$679$	−4395.36	−	13610.1i	−0.248422	−	0.769231i
$680$	17560.8	+	4009.91i	0.990335	+	0.226137i
$681$		0			0
$682$	−13257.1			−0.744343
$683$	−27112.0			−1.51891			−0.759453	−	0.650562i	$-0.774533\pi$
$683$	−27112.0			−1.51891			−0.759453	+	0.650562i	$0.774533\pi$
$684$		0			0
$685$	−3422.18	+	14987.0i	−0.190883	+	0.835946i
$686$	−7248.23	−	5325.66i	−0.403409	−	0.296406i
$687$		0			0
$688$		−	5291.05i		−	0.293197i
$689$	−5078.80			−0.280822
$690$		0			0
$691$		−	521.160i		−	0.0286916i	−0.999897	−	0.0143458i	$-0.995433\pi$
$691$		−	521.160i		−	0.0286916i	0.999897	−	0.0143458i	$-0.00456656\pi$
$692$		−	15913.4i		−	0.874185i
$693$		0			0
$694$	−597.884			−0.0327023
$695$	18780.5	+	4288.40i	1.02501	+	0.234055i
$696$		0			0
$697$		−	38857.1i		−	2.11165i
$698$		−	6570.91i		−	0.356322i
$699$		0			0
$700$	1890.20	−	13749.9i	0.102061	−	0.742422i
$701$			3411.27i			0.183797i	0.995768	+	0.0918987i	$0.0292936\pi$
$701$			3411.27i			0.183797i	−0.995768	+	0.0918987i	$0.970706\pi$
$702$		0			0
$703$	−9230.80			−0.495230
$704$		−	3215.42i		−	0.172139i
$705$		0			0
$706$		−	2179.07i		−	0.116162i
$707$	5063.58	+	15679.2i	0.269357	+	0.834057i
$708$		0			0
$709$	−6386.93			−0.338316			−0.169158	−	0.985589i	$-0.554105\pi$
$709$	−6386.93			−0.338316			−0.169158	+	0.985589i	$0.554105\pi$
$710$	9514.22	+	2172.51i	0.502905	+	0.114835i
$711$		0			0
$712$	18455.7			0.971430
$713$			15899.5i			0.835120i
$714$		0			0
$715$	−5534.12	−	1263.68i	−0.289461	−	0.0660965i
$716$		−	26816.2i		−	1.39968i
$717$		0			0
$718$			1466.86i			0.0762431i
$719$	1659.01			0.0860510			0.0430255	−	0.999074i	$-0.486300\pi$
$719$	1659.01			0.0860510			0.0430255	+	0.999074i	$0.486300\pi$
$720$		0			0
$721$	9029.60	+	27959.9i	0.466407	+	1.44422i
$722$	1246.11			0.0642321
$723$		0			0
$724$			2374.31i			0.121879i
$725$	8692.21	−	18040.8i	0.445270	−	0.924163i
$726$		0			0
$727$	−25833.8			−1.31791			−0.658955	−	0.752182i	$-0.729001\pi$
$727$	−25833.8			−1.31791			−0.658955	+	0.752182i	$0.729001\pi$
$728$	1872.15	+	5797.07i	0.0953113	+	0.295129i
$729$		0			0
$730$	3599.72	−	15764.5i	0.182509	−	0.799273i
$731$	−21612.3			−1.09351
$732$		0			0
$733$	17410.3			0.877304			0.438652	−	0.898657i	$-0.355456\pi$
$733$	17410.3			0.877304			0.438652	+	0.898657i	$0.355456\pi$
$734$	3226.28			0.162240
$735$		0			0
$736$	−9697.73			−0.485683
$737$	20542.1			1.02670
$738$		0			0
$739$	−26348.6			−1.31157			−0.655784	−	0.754949i	$-0.727662\pi$
$739$	−26348.6			−1.31157			−0.655784	+	0.754949i	$0.727662\pi$
$740$	6856.76	+	1565.70i	0.340621	+	0.0777786i
$741$		0			0
$742$	2465.68	+	7634.92i	0.121992	+	0.377745i
$743$	−27257.0			−1.34585			−0.672923	−	0.739713i	$-0.734961\pi$
$743$	−27257.0			−1.34585			−0.672923	+	0.739713i	$0.734961\pi$
$744$		0			0
$745$	6775.78	+	1547.20i	0.333215	+	0.0760875i
$746$			12191.2i			0.598326i
$747$		0			0
$748$	14909.4			0.728799
$749$	4367.36	+	13523.4i	0.213057	+	0.659725i
$750$		0			0
$751$	12085.2			0.587210			0.293605	−	0.955927i	$-0.405145\pi$
$751$	12085.2			0.587210			0.293605	+	0.955927i	$0.405145\pi$
$752$			2801.38i			0.135845i
$753$		0			0
$754$			3765.30i			0.181862i
$755$	−6948.10	+	30428.3i	−0.334924	+	1.46675i
$756$		0			0
$757$		−	1931.31i		−	0.0927275i	−0.998925	−	0.0463638i	$-0.985237\pi$
$757$		−	1931.31i		−	0.0927275i	0.998925	−	0.0463638i	$-0.0147633\pi$
$758$	23.3677			0.00111973
$759$		0			0
$760$	−19000.9	−	4338.74i	−0.906890	−	0.207082i
$761$	26856.6			1.27931			0.639653	−	0.768664i	$-0.279078\pi$
$761$	26856.6			1.27931			0.639653	+	0.768664i	$0.279078\pi$
$762$		0			0
$763$	4637.09	+	14358.6i	0.220018	+	0.681280i
$764$			7781.40i			0.368483i
$765$		0			0
$766$		−	13999.8i		−	0.660357i
$767$	−2225.38			−0.104764
$768$		0			0
$769$		−	2772.05i		−	0.129990i	−0.997886	−	0.0649952i	$-0.979297\pi$
$769$		−	2772.05i		−	0.129990i	0.997886	−	0.0649952i	$-0.0207032\pi$
$770$	787.056	+	8932.91i	0.0368357	+	0.418078i
$771$		0			0
$772$			26810.3i			1.24990i
$773$		−	18591.8i		−	0.865071i	−0.901617	−	0.432535i	$-0.857619\pi$
$773$		−	18591.8i		−	0.865071i	0.901617	−	0.432535i	$-0.142381\pi$
$774$		0			0
$775$	16608.7	−	34471.5i	0.769807	−	1.59774i
$776$	15302.7			0.707907
$777$		0			0
$778$			395.449i			0.0182230i
$779$			42043.6i			1.93372i
$780$		0			0
$781$	18856.5			0.863944
$782$			5979.30i			0.273426i
$783$		0			0
$784$	−5537.71	+	3993.27i	−0.252264	+	0.181909i
$785$	4318.59	−	18912.7i	0.196353	−	0.859901i
$786$		0			0
$787$	10211.2			0.462504			0.231252	−	0.972894i	$-0.425718\pi$
$787$	10211.2			0.462504			0.231252	+	0.972894i	$0.425718\pi$
$788$	−130.000			−0.00587698
$789$		0			0
$790$	−1968.98	+	8622.88i	−0.0886748	+	0.388340i
$791$	−9389.37	−	29073.9i	−0.422058	−	1.30689i
$792$		0			0
$793$		−	5042.98i		−	0.225828i
$794$	15424.8			0.689427
$795$		0			0
$796$		−	10187.5i		−	0.453627i
$797$			15129.4i			0.672412i	0.941788	+	0.336206i	$0.109144\pi$
$797$			15129.4i			0.672412i	−0.941788	+	0.336206i	$0.890856\pi$
$798$		0			0
$799$	11442.7			0.506652
$800$	21025.5	+	10130.3i	0.929205	+	0.447699i
$801$		0			0
$802$			13637.6i			0.600448i
$803$		−	31244.1i		−	1.37308i
$804$		0			0
$805$	10713.4	−	943.928i	0.469064	−	0.0413281i
$806$			7194.55i			0.314413i
$807$		0			0
$808$	−17629.2			−0.767565
$809$		−	27312.2i		−	1.18695i	−0.804851	−	0.593477i	$-0.797755\pi$
$809$		−	27312.2i		−	1.18695i	0.804851	−	0.593477i	$-0.202245\pi$
$810$		0			0
$811$		−	955.054i		−	0.0413520i	−0.999786	−	0.0206760i	$-0.993418\pi$
$811$		−	955.054i		−	0.0413520i	0.999786	−	0.0206760i	$-0.00658184\pi$
$812$	−16927.2	+	5466.62i	−0.731564	+	0.236257i
$813$		0			0
$814$	−4544.28			−0.195672
$815$	10853.5	+	2478.32i	0.466479	+	0.106517i
$816$		0			0
$817$	23384.6			1.00138
$818$			5780.12i			0.247063i
$819$		0			0
$820$	7131.30	−	31230.6i	0.303702	−	1.33002i
$821$		−	4863.55i		−	0.206747i	−0.994643	−	0.103373i	$-0.967036\pi$
$821$		−	4863.55i		−	0.206747i	0.994643	−	0.103373i	$-0.0329636\pi$
$822$		0			0
$823$			24899.9i			1.05463i	0.849671	+	0.527313i	$0.176800\pi$
$823$			24899.9i			1.05463i	−0.849671	+	0.527313i	$0.823200\pi$
$824$	−31437.1			−1.32908
$825$		0			0
$826$	1080.39	+	3345.40i	0.0455105	+	0.140922i
$827$	4217.13			0.177321			0.0886603	−	0.996062i	$-0.471741\pi$
$827$	4217.13			0.177321			0.0886603	+	0.996062i	$0.471741\pi$
$828$		0			0
$829$			15003.0i			0.628561i	0.949330	+	0.314280i	$0.101763\pi$
$829$			15003.0i			0.628561i	−0.949330	+	0.314280i	$0.898237\pi$
$830$	5177.32	+	1182.21i	0.216515	+	0.0494398i
$831$		0			0
$832$	−1744.99			−0.0727122
$833$	16311.2	+	22619.8i	0.678453	+	0.940852i
$834$		0			0
$835$	20462.4	+	4672.45i	0.848060	+	0.193649i
$836$	−16132.1			−0.667391
$837$		0			0
$838$	22587.1			0.931095
$839$	2205.65			0.0907597			0.0453798	−	0.998970i	$-0.485550\pi$
$839$	2205.65			0.0907597			0.0453798	+	0.998970i	$0.485550\pi$
$840$		0			0
$841$	−1276.58			−0.0523425
$842$	−23656.1			−0.968224
$843$		0			0
$844$	11287.3			0.460336
$845$	4782.32	−	20943.5i	0.194694	−	0.852637i
$846$		0			0
$847$	−2250.64	−	6969.03i	−0.0913020	−	0.282714i
$848$	6090.11			0.246622
$849$		0			0
$850$	6246.00	−	12963.6i	0.252042	−	0.523117i
$851$			5450.02i			0.219535i
$852$		0			0
$853$	−41922.9			−1.68278			−0.841391	−	0.540426i	$-0.818263\pi$
$853$	−41922.9			−1.68278			−0.841391	+	0.540426i	$0.818263\pi$
$854$	−7581.07	+	2448.29i	−0.303769	+	0.0981018i
$855$		0			0
$856$	−15205.2			−0.607131
$857$		−	5480.18i		−	0.218436i	−0.994018	−	0.109218i	$-0.965165\pi$
$857$		−	5480.18i		−	0.218436i	0.994018	−	0.109218i	$-0.0348346\pi$
$858$		0			0
$859$		−	13807.4i		−	0.548431i	−0.961668	−	0.274216i	$-0.911582\pi$
$859$		−	13807.4i		−	0.548431i	0.961668	−	0.274216i	$-0.0884182\pi$
$860$	−17370.4	−	3966.42i	−0.688751	−	0.157272i
$861$		0			0
$862$		−	1446.60i		−	0.0571595i
$863$	−36042.1			−1.42165			−0.710827	−	0.703367i	$-0.751679\pi$
$863$	−36042.1			−1.42165			−0.710827	+	0.703367i	$0.751679\pi$
$864$		0			0
$865$	−28931.7	−	6606.37i	−1.13723	−	0.259680i
$866$	−1490.09			−0.0584703
$867$		0			0
$868$	−32343.7	+	10445.4i	−1.26477	+	0.408455i
$869$			17090.0i			0.667131i
$870$		0			0
$871$		−	11148.0i		−	0.433681i
$872$	−16144.3			−0.626968
$873$		0			0
$874$		−	6469.64i		−	0.250388i
$875$	−24213.6	−	9144.71i	−0.935506	−	0.353312i
$876$		0			0
$877$		−	25249.1i		−	0.972180i	−0.873909	−	0.486090i	$-0.838423\pi$
$877$		−	25249.1i		−	0.972180i	0.873909	−	0.486090i	$-0.161577\pi$
$878$			19172.4i			0.736943i
$879$		0			0
$880$	6636.11	+	1515.31i	0.254208	+	0.0580468i
$881$	−7038.44			−0.269161			−0.134581	−	0.990903i	$-0.542969\pi$
$881$	−7038.44			−0.269161			−0.134581	+	0.990903i	$0.542969\pi$
$882$		0			0
$883$			17328.3i			0.660413i	0.943909	+	0.330207i	$0.107118\pi$
$883$			17328.3i			0.660413i	−0.943909	+	0.330207i	$0.892882\pi$
$884$		−	8091.22i		−	0.307848i
$885$		0			0
$886$	18123.1			0.687197
$887$		−	23764.0i		−	0.899570i	−0.893137	−	0.449785i	$-0.851501\pi$
$887$		−	23764.0i		−	0.899570i	0.893137	−	0.449785i	$-0.148499\pi$
$888$		0			0
$889$	3063.86	−	989.470i	0.115589	−	0.0373293i
$890$	3282.15	−	14373.7i	0.123616	−	0.541358i
$891$		0			0
$892$	−34881.9			−1.30934
$893$	−12381.1			−0.463962
$894$		0			0
$895$	−48753.8	−	11132.6i	−1.82085	−	0.415779i
$896$	−7654.30	−	23701.3i	−0.285393	−	0.883712i
$897$		0			0
$898$			741.770i			0.0275648i
$899$	−49040.5			−1.81935
$900$		0			0
$901$		−	24876.2i		−	0.919807i
$902$			20697.9i			0.764039i
$903$		0			0
$904$	32689.7			1.20270
$905$	4316.67	+	985.684i	0.158554	+	0.0362047i
$906$		0			0
$907$			16998.5i			0.622301i	0.950361	+	0.311151i	$0.100714\pi$
$907$			16998.5i			0.622301i	−0.950361	+	0.311151i	$0.899286\pi$
$908$		−	16228.8i		−	0.593139i
$909$		0			0
$910$	4847.82	−	427.129i	0.176598	−	0.0155596i
$911$			40955.7i			1.48949i	0.667351	+	0.744743i	$0.267428\pi$
$911$			40955.7i			1.48949i	−0.667351	+	0.744743i	$0.732572\pi$
$912$		0			0
$913$	10261.1			0.371953
$914$		−	8140.90i		−	0.294614i
$915$		0			0
$916$		−	37341.5i		−	1.34694i
$917$	−2668.24	−	8262.13i	−0.0960884	−	0.297535i
$918$		0			0
$919$	42060.9			1.50975			0.754876	−	0.655867i	$-0.227697\pi$
$919$	42060.9			1.50975			0.754876	+	0.655867i	$0.227697\pi$
$920$	−2561.66	+	11218.5i	−0.0917995	+	0.402024i
$921$		0			0
$922$	20217.1			0.722142
$923$		−	10233.3i		−	0.364933i
$924$		0			0
$925$	5693.11	−	11816.1i	0.202366	−	0.420012i
$926$		−	23039.7i		−	0.817636i
$927$		0			0
$928$		−	29911.8i		−	1.05808i
$929$	−1206.74			−0.0426177			−0.0213088	−	0.999773i	$-0.506783\pi$
$929$	−1206.74			−0.0426177			−0.0213088	+	0.999773i	$0.506783\pi$
$930$		0			0
$931$	−17648.9	−	24474.8i	−0.621287	−	0.861577i
$932$	−24751.9			−0.869930
$933$		0			0
$934$		−	21407.9i		−	0.749985i
$935$	6189.57	−	27106.4i	0.216493	−	0.948101i
$936$		0			0
$937$	−40024.3			−1.39545			−0.697725	−	0.716366i	$-0.745804\pi$
$937$	−40024.3			−1.39545			−0.697725	+	0.716366i	$0.745804\pi$
$938$	−16758.8	+	5412.21i	−0.583361	+	0.188395i
$939$		0			0
$940$	9196.85	+	2100.04i	0.319115	+	0.0728679i
$941$	7112.58			0.246401			0.123201	−	0.992382i	$-0.460684\pi$
$941$	7112.58			0.246401			0.123201	+	0.992382i	$0.460684\pi$
$942$		0			0
$943$	24823.3			0.857218
$944$	2668.51			0.0920049
$945$		0			0
$946$	11512.1			0.395657
$947$	52031.5			1.78543			0.892713	−	0.450626i	$-0.148799\pi$
$947$	52031.5			1.78543			0.892713	+	0.450626i	$0.148799\pi$
$948$		0			0
$949$	−16956.0			−0.579993
$950$	−6758.21	+	14026.7i	−0.230805	+	0.479039i
$951$		0			0
$952$	−28394.3	+	9169.90i	−0.966666	+	0.312183i
$953$	50850.3			1.72844			0.864220	−	0.503114i	$-0.167812\pi$
$953$	50850.3			1.72844			0.864220	+	0.503114i	$0.167812\pi$
$954$		0			0
$955$	14147.2	+	3230.42i	0.479363	+	0.109459i
$956$			27849.3i			0.942166i
$957$		0			0
$958$	18464.3			0.622710
$959$	−7825.89	−	24232.6i	−0.263515	−	0.815967i
$960$		0			0
$961$	−63913.2			−2.14539
$962$			2466.14i			0.0826525i
$963$		0			0
$964$		−	41526.0i		−	1.38741i
$965$	48743.2	+	11130.2i	1.62601	+	0.371289i
$966$		0			0
$967$		−	35434.8i		−	1.17839i	−0.807990	−	0.589197i	$-0.799444\pi$
$967$		−	35434.8i		−	1.17839i	0.807990	−	0.589197i	$-0.200556\pi$
$968$	7835.74			0.260176
$969$		0			0
$970$	2721.42	−	11918.1i	0.0900820	−	0.394502i
$971$	−46207.8			−1.52717			−0.763584	−	0.645709i	$-0.776562\pi$
$971$	−46207.8			−1.52717			−0.763584	+	0.645709i	$0.776562\pi$
$972$		0			0
$973$	−30366.4	+	9806.77i	−1.00052	+	0.323115i
$974$			18293.0i			0.601791i
$975$		0			0
$976$			6047.16i			0.198325i
$977$	14154.5			0.463503			0.231752	−	0.972775i	$-0.425554\pi$
$977$	14154.5			0.463503			0.231752	+	0.972775i	$0.425554\pi$
$978$		0			0
$979$		−	28487.7i		−	0.930002i
$980$	8958.48	+	21173.7i	0.292008	+	0.690173i
$981$		0			0
$982$			14700.3i			0.477705i
$983$			12680.8i			0.411451i	0.978610	+	0.205725i	$0.0659554\pi$
$983$			12680.8i			0.411451i	−0.978610	+	0.205725i	$0.934045\pi$
$984$		0			0
$985$	−53.9689	+	236.350i	−0.00174578	+	0.00764541i
$986$	−18442.6			−0.595672
$987$		0			0
$988$			8754.75i			0.281908i
$989$		−	13806.7i		−	0.443909i
$990$		0			0
$991$	36020.9			1.15463			0.577317	−	0.816520i	$-0.304100\pi$
$991$	36020.9			1.15463			0.577317	+	0.816520i	$0.304100\pi$
$992$		−	57153.9i		−	1.82927i
$993$		0			0
$994$	−15383.7	+	4968.12i	−0.490885	+	0.158531i
$995$	−18521.7	−	4229.30i	−0.590127	−	0.134752i
$996$		0			0
$997$	61180.4			1.94343			0.971716	−	0.236152i	$-0.0758862\pi$
$997$	61180.4			1.94343			0.971716	+	0.236152i	$0.0758862\pi$
$998$	−18567.3			−0.588917
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.g.a.314.18	yes	48
3.2	odd	2	inner	315.4.g.a.314.32	yes	48
5.4	even	2	inner	315.4.g.a.314.30	yes	48
7.6	odd	2	inner	315.4.g.a.314.17	✓	48
15.14	odd	2	inner	315.4.g.a.314.20	yes	48
21.20	even	2	inner	315.4.g.a.314.31	yes	48
35.34	odd	2	inner	315.4.g.a.314.29	yes	48
105.104	even	2	inner	315.4.g.a.314.19	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.4.g.a.314.17	✓	48	7.6	odd	2	inner
315.4.g.a.314.18	yes	48	1.1	even	1	trivial
315.4.g.a.314.19	yes	48	105.104	even	2	inner
315.4.g.a.314.20	yes	48	15.14	odd	2	inner
315.4.g.a.314.29	yes	48	35.34	odd	2	inner
315.4.g.a.314.30	yes	48	5.4	even	2	inner
315.4.g.a.314.31	yes	48	21.20	even	2	inner
315.4.g.a.314.32	yes	48	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41590 q^{2} -5.99524 q^{4} +(-2.48890 + 10.8998i) q^{5} +(-5.69164 - 17.6240i) q^{7} +19.8158 q^{8} +(3.52402 - 15.4330i) q^{10} -30.5871i q^{11} -16.5994 q^{13} +(8.05877 + 24.9538i) q^{14} +19.9047 q^{16} -81.3047i q^{17} +87.9721i q^{19} +(14.9215 - 65.3468i) q^{20} +43.3082i q^{22} +51.9402 q^{23} +(-112.611 - 54.2569i) q^{25} +23.5030 q^{26} +(34.1227 + 105.660i) q^{28} +160.205i q^{29} +306.111i q^{31} -186.710 q^{32} +115.119i q^{34} +(206.264 - 18.1734i) q^{35} +104.929i q^{37} -124.559i q^{38} +(-49.3195 + 215.988i) q^{40} +477.920 q^{41} -265.819i q^{43} +183.377i q^{44} -73.5419 q^{46} +140.739i q^{47} +(-278.210 + 200.619i) q^{49} +(159.445 + 76.8222i) q^{50} +99.5173 q^{52} +305.963 q^{53} +(333.393 + 76.1282i) q^{55} +(-112.784 - 349.234i) q^{56} -226.833i q^{58} +134.064 q^{59} +303.805i q^{61} -433.422i q^{62} +105.123 q^{64} +(41.3142 - 180.930i) q^{65} +671.592i q^{67} +487.441i q^{68} +(-292.048 + 25.7316i) q^{70} +616.486i q^{71} +1021.48 q^{73} -148.568i q^{74} -527.413i q^{76} +(-539.067 + 174.091i) q^{77} -558.731 q^{79} +(-49.5409 + 216.958i) q^{80} -676.686 q^{82} +335.471i q^{83} +(886.204 + 202.359i) q^{85} +376.372i q^{86} -606.108i q^{88} +931.364 q^{89} +(94.4778 + 292.548i) q^{91} -311.394 q^{92} -199.272i q^{94} +(-958.877 - 218.953i) q^{95} +772.248 q^{97} +(393.917 - 284.056i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 192 q^{4} + 768 q^{16} - 432 q^{25} + 816 q^{46} + 456 q^{49} + 7968 q^{64} + 1464 q^{70} + 4368 q^{79} - 1440 q^{85} - 4392 q^{91}+O(q^{100}) \) 48 * q + 192 * q^4 + 768 * q^16 - 432 * q^25 + 816 * q^46 + 456 * q^49 + 7968 * q^64 + 1464 * q^70 + 4368 * q^79 - 1440 * q^85 - 4392 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more