LMFDB - Embedded newform 2912.2.i.a.337.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2912,2,Mod(337,2912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2912, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2912.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.i (of order $2$, degree $1$, not 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:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.20
Character	$\chi$	$=$	2912.337
Dual form			2912.2.i.a.337.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58827i q^{3} +1.51724 q^{5} -1.00000i q^{7} -3.69913 q^{9} -1.15039 q^{11} +(-3.39762 - 1.20671i) q^{13} +3.92702i q^{15} +0.0878473 q^{17} -8.25567 q^{19} +2.58827 q^{21} +6.48403 q^{23} -2.69799 q^{25} -1.80953i q^{27} +2.75060i q^{29} -5.46650i q^{31} -2.97752i q^{33} -1.51724i q^{35} -8.75613 q^{37} +(3.12330 - 8.79396i) q^{39} -0.221622i q^{41} -11.3062i q^{43} -5.61245 q^{45} +4.58078i q^{47} -1.00000 q^{49} +0.227372i q^{51} +10.8067i q^{53} -1.74542 q^{55} -21.3679i q^{57} -3.96379 q^{59} -11.0547i q^{61} +3.69913i q^{63} +(-5.15500 - 1.83087i) q^{65} -4.63551 q^{67} +16.7824i q^{69} -9.36899i q^{71} -10.4110i q^{73} -6.98312i q^{75} +1.15039i q^{77} -7.68869 q^{79} -6.41384 q^{81} -5.19464 q^{83} +0.133285 q^{85} -7.11928 q^{87} -1.08552i q^{89} +(-1.20671 + 3.39762i) q^{91} +14.1488 q^{93} -12.5258 q^{95} +10.4826i q^{97} +4.25545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) $ 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$			2.58827i			1.49434i	0.664635	+	0.747168i	$0.268587\pi$
$3$			2.58827i			1.49434i	−0.664635	+	0.747168i	$0.731413\pi$
$4$		0			0
$5$	1.51724			0.678529			0.339265	−	0.940691i	$-0.389822\pi$
$5$	1.51724			0.678529			0.339265	+	0.940691i	$0.389822\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−3.69913			−1.23304
$10$		0			0
$11$	−1.15039			−0.346857			−0.173428	−	0.984847i	$-0.555484\pi$
$11$	−1.15039			−0.346857			−0.173428	+	0.984847i	$0.555484\pi$
$12$		0			0
$13$	−3.39762	−	1.20671i	−0.942331	−	0.334682i
$13$	−3.39762	−	1.20671i	−0.942331	−	0.334682i
$14$		0			0
$15$			3.92702i			1.01395i
$16$		0			0
$17$	0.0878473			0.0213061			0.0106531	−	0.999943i	$-0.496609\pi$
$17$	0.0878473			0.0213061			0.0106531	+	0.999943i	$0.496609\pi$
$18$		0			0
$19$	−8.25567			−1.89398			−0.946990	−	0.321262i	$-0.895893\pi$
$19$	−8.25567			−1.89398			−0.946990	+	0.321262i	$0.895893\pi$
$20$		0			0
$21$	2.58827			0.564806
$22$		0			0
$23$	6.48403			1.35201			0.676007	−	0.736895i	$-0.263709\pi$
$23$	6.48403			1.35201			0.676007	+	0.736895i	$0.263709\pi$
$24$		0			0
$25$	−2.69799			−0.539598
$26$		0			0
$27$		−	1.80953i		−	0.348243i
$28$		0			0
$29$			2.75060i			0.510773i	0.966839	+	0.255387i	$0.0822027\pi$
$29$			2.75060i			0.510773i	−0.966839	+	0.255387i	$0.917797\pi$
$30$		0			0
$31$		−	5.46650i		−	0.981813i	−0.871212	−	0.490907i	$-0.836666\pi$
$31$		−	5.46650i		−	0.981813i	0.871212	−	0.490907i	$-0.163334\pi$
$32$		0			0
$33$		−	2.97752i		−	0.518321i
$34$		0			0
$35$		−	1.51724i		−	0.256460i
$36$		0			0
$37$	−8.75613			−1.43950			−0.719750	−	0.694234i	$-0.755744\pi$
$37$	−8.75613			−1.43950			−0.719750	+	0.694234i	$0.755744\pi$
$38$		0			0
$39$	3.12330	−	8.79396i	0.500128	−	1.40816i
$40$		0			0
$41$		−	0.221622i		−	0.0346115i	−0.999850	−	0.0173057i	$-0.994491\pi$
$41$		−	0.221622i		−	0.0346115i	0.999850	−	0.0173057i	$-0.00550886\pi$
$42$		0			0
$43$		−	11.3062i		−	1.72417i	−0.506760	−	0.862087i	$-0.669157\pi$
$43$		−	11.3062i		−	1.72417i	0.506760	−	0.862087i	$-0.330843\pi$
$44$		0			0
$45$	−5.61245			−0.836655
$46$		0			0
$47$			4.58078i			0.668175i	0.942542	+	0.334088i	$0.108428\pi$
$47$			4.58078i			0.668175i	−0.942542	+	0.334088i	$0.891572\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			0.227372i			0.0318385i
$52$		0			0
$53$			10.8067i			1.48441i	0.670171	+	0.742207i	$0.266221\pi$
$53$			10.8067i			1.48441i	−0.670171	+	0.742207i	$0.733779\pi$
$54$		0			0
$55$	−1.74542			−0.235352
$56$		0			0
$57$		−	21.3679i		−	2.83025i
$58$		0			0
$59$	−3.96379			−0.516041			−0.258021	−	0.966139i	$-0.583070\pi$
$59$	−3.96379			−0.516041			−0.258021	+	0.966139i	$0.583070\pi$
$60$		0			0
$61$		−	11.0547i		−	1.41541i	−0.706509	−	0.707704i	$-0.749731\pi$
$61$		−	11.0547i		−	1.41541i	0.706509	−	0.707704i	$-0.250269\pi$
$62$		0			0
$63$			3.69913i			0.466046i
$64$		0			0
$65$	−5.15500	−	1.83087i	−0.639399	−	0.227092i
$66$		0			0
$67$	−4.63551			−0.566317			−0.283159	−	0.959073i	$-0.591382\pi$
$67$	−4.63551			−0.566317			−0.283159	+	0.959073i	$0.591382\pi$
$68$		0			0
$69$			16.7824i			2.02037i
$70$		0			0
$71$		−	9.36899i		−	1.11190i	−0.831217	−	0.555948i	$-0.812356\pi$
$71$		−	9.36899i		−	1.11190i	0.831217	−	0.555948i	$-0.187644\pi$
$72$		0			0
$73$		−	10.4110i		−	1.21851i	−0.792974	−	0.609255i	$-0.791468\pi$
$73$		−	10.4110i		−	1.21851i	0.792974	−	0.609255i	$-0.208532\pi$
$74$		0			0
$75$		−	6.98312i		−	0.806341i
$76$		0			0
$77$			1.15039i			0.131099i
$78$		0			0
$79$	−7.68869			−0.865045			−0.432523	−	0.901623i	$-0.642376\pi$
$79$	−7.68869			−0.865045			−0.432523	+	0.901623i	$0.642376\pi$
$80$		0			0
$81$	−6.41384			−0.712649
$82$		0			0
$83$	−5.19464			−0.570186			−0.285093	−	0.958500i	$-0.592024\pi$
$83$	−5.19464			−0.570186			−0.285093	+	0.958500i	$0.592024\pi$
$84$		0			0
$85$	0.133285			0.0144568
$86$		0			0
$87$	−7.11928			−0.763267
$88$		0			0
$89$		−	1.08552i		−	0.115064i	−0.998344	−	0.0575322i	$-0.981677\pi$
$89$		−	1.08552i		−	0.115064i	0.998344	−	0.0575322i	$-0.0183232\pi$
$90$		0			0
$91$	−1.20671	+	3.39762i	−0.126498	+	0.356168i
$92$		0			0
$93$	14.1488			1.46716
$94$		0			0
$95$	−12.5258			−1.28512
$96$		0			0
$97$			10.4826i			1.06434i	0.846636	+	0.532172i	$0.178624\pi$
$97$			10.4826i			1.06434i	−0.846636	+	0.532172i	$0.821376\pi$
$98$		0			0
$99$	4.25545			0.427689
$100$		0			0
$101$			3.46200i			0.344482i	0.985055	+	0.172241i	$0.0551007\pi$
$101$			3.46200i			0.344482i	−0.985055	+	0.172241i	$0.944899\pi$
$102$		0			0
$103$	−5.66902			−0.558586			−0.279293	−	0.960206i	$-0.590100\pi$
$103$	−5.66902			−0.558586			−0.279293	+	0.960206i	$0.590100\pi$
$104$		0			0
$105$	3.92702			0.383237
$106$		0			0
$107$			17.3294i			1.67530i	0.546208	+	0.837650i	$0.316071\pi$
$107$			17.3294i			1.67530i	−0.546208	+	0.837650i	$0.683929\pi$
$108$		0			0
$109$	11.0393			1.05737			0.528685	−	0.848818i	$-0.322685\pi$
$109$	11.0393			1.05737			0.528685	+	0.848818i	$0.322685\pi$
$110$		0			0
$111$		−	22.6632i		−	2.15110i
$112$		0			0
$113$	−7.22371			−0.679550			−0.339775	−	0.940507i	$-0.610351\pi$
$113$	−7.22371			−0.679550			−0.339775	+	0.940507i	$0.610351\pi$
$114$		0			0
$115$	9.83782			0.917381
$116$		0			0
$117$	12.5682	+	4.46379i	1.16193	+	0.412677i
$118$		0			0
$119$		−	0.0878473i		−	0.00805295i
$120$		0			0
$121$	−9.67660			−0.879691
$122$		0			0
$123$	0.573616			0.0517212
$124$		0			0
$125$	−11.6797			−1.04466
$126$		0			0
$127$	10.8980			0.967043			0.483521	−	0.875333i	$-0.339358\pi$
$127$	10.8980			0.967043			0.483521	+	0.875333i	$0.339358\pi$
$128$		0			0
$129$	29.2634			2.57650
$130$		0			0
$131$		−	11.3300i		−	0.989908i	−0.868919	−	0.494954i	$-0.835185\pi$
$131$		−	11.3300i		−	0.989908i	0.868919	−	0.494954i	$-0.164815\pi$
$132$		0			0
$133$			8.25567i			0.715857i
$134$		0			0
$135$		−	2.74548i		−	0.236293i
$136$		0			0
$137$			5.25241i			0.448744i	0.974504	+	0.224372i	$0.0720331\pi$
$137$			5.25241i			0.448744i	−0.974504	+	0.224372i	$0.927967\pi$
$138$		0			0
$139$		−	6.66621i		−	0.565421i	−0.959205	−	0.282710i	$-0.908766\pi$
$139$		−	6.66621i		−	0.565421i	0.959205	−	0.282710i	$-0.0912335\pi$
$140$		0			0
$141$	−11.8563			−0.998479
$142$		0			0
$143$	3.90860	+	1.38820i	0.326854	+	0.116087i
$144$		0			0
$145$			4.17331i			0.346574i
$146$		0			0
$147$		−	2.58827i		−	0.213477i
$148$		0			0
$149$	17.9670			1.47192			0.735959	−	0.677027i	$-0.236732\pi$
$149$	17.9670			1.47192			0.735959	+	0.677027i	$0.236732\pi$
$150$		0			0
$151$		−	0.785249i		−	0.0639027i	−0.999489	−	0.0319513i	$-0.989828\pi$
$151$		−	0.785249i		−	0.0639027i	0.999489	−	0.0319513i	$-0.0101722\pi$
$152$		0			0
$153$	−0.324958			−0.0262713
$154$		0			0
$155$		−	8.29398i		−	0.666189i
$156$		0			0
$157$			11.0014i			0.878004i	0.898486	+	0.439002i	$0.144668\pi$
$157$			11.0014i			0.878004i	−0.898486	+	0.439002i	$0.855332\pi$
$158$		0			0
$159$	−27.9706			−2.21821
$160$		0			0
$161$		−	6.48403i		−	0.511014i
$162$		0			0
$163$	−9.28282			−0.727087			−0.363543	−	0.931577i	$-0.618433\pi$
$163$	−9.28282			−0.727087			−0.363543	+	0.931577i	$0.618433\pi$
$164$		0			0
$165$		−	4.51761i		−	0.351696i
$166$		0			0
$167$		−	9.91186i		−	0.767002i	−0.923540	−	0.383501i	$-0.874718\pi$
$167$		−	9.91186i		−	0.767002i	0.923540	−	0.383501i	$-0.125282\pi$
$168$		0			0
$169$	10.0877	+	8.19992i	0.775976	+	0.630763i
$170$		0			0
$171$	30.5388			2.33536
$172$		0			0
$173$		−	22.0729i		−	1.67817i	−0.543999	−	0.839086i	$-0.683090\pi$
$173$		−	22.0729i		−	1.67817i	0.543999	−	0.839086i	$-0.316910\pi$
$174$		0			0
$175$			2.69799i			0.203949i
$176$		0			0
$177$		−	10.2593i		−	0.771139i
$178$		0			0
$179$			20.2249i			1.51168i	0.654756	+	0.755841i	$0.272772\pi$
$179$			20.2249i			1.51168i	−0.654756	+	0.755841i	$0.727228\pi$
$180$		0			0
$181$		−	19.1349i		−	1.42229i	−0.703048	−	0.711143i	$-0.748178\pi$
$181$		−	19.1349i		−	1.42229i	0.703048	−	0.711143i	$-0.251822\pi$
$182$		0			0
$183$	28.6125			2.11510
$184$		0			0
$185$	−13.2851			−0.976743
$186$		0			0
$187$	−0.101059			−0.00739016
$188$		0			0
$189$	−1.80953			−0.131624
$190$		0			0
$191$	1.20317			0.0870584			0.0435292	−	0.999052i	$-0.486140\pi$
$191$	1.20317			0.0870584			0.0435292	+	0.999052i	$0.486140\pi$
$192$		0			0
$193$			16.6104i			1.19564i	0.801630	+	0.597820i	$0.203966\pi$
$193$			16.6104i			1.19564i	−0.801630	+	0.597820i	$0.796034\pi$
$194$		0			0
$195$	4.73878	−	13.3425i	0.339351	−	0.955478i
$196$		0			0
$197$	−12.3136			−0.877306			−0.438653	−	0.898656i	$-0.644544\pi$
$197$	−12.3136			−0.877306			−0.438653	+	0.898656i	$0.644544\pi$
$198$		0			0
$199$	3.59829			0.255076			0.127538	−	0.991834i	$-0.459293\pi$
$199$	3.59829			0.255076			0.127538	+	0.991834i	$0.459293\pi$
$200$		0			0
$201$		−	11.9979i		−	0.846269i
$202$		0			0
$203$	2.75060			0.193054
$204$		0			0
$205$		−	0.336252i		−	0.0234849i
$206$		0			0
$207$	−23.9853			−1.66709
$208$		0			0
$209$	9.49727			0.656940
$210$		0			0
$211$			13.3979i			0.922352i	0.887309	+	0.461176i	$0.152572\pi$
$211$			13.3979i			0.922352i	−0.887309	+	0.461176i	$0.847428\pi$
$212$		0			0
$213$	24.2495			1.66155
$214$		0			0
$215$		−	17.1541i		−	1.16990i
$216$		0			0
$217$	−5.46650			−0.371091
$218$		0			0
$219$	26.9463			1.82087
$220$		0			0
$221$	−0.298472	−	0.106007i	−0.0200774	−	0.00713078i
$222$		0			0
$223$		−	15.4009i		−	1.03132i	−0.856793	−	0.515660i	$-0.827547\pi$
$223$		−	15.4009i		−	1.03132i	0.856793	−	0.515660i	$-0.172453\pi$
$224$		0			0
$225$	9.98021			0.665347
$226$		0			0
$227$	18.4841			1.22683			0.613417	−	0.789760i	$-0.289795\pi$
$227$	18.4841			1.22683			0.613417	+	0.789760i	$0.289795\pi$
$228$		0			0
$229$	−9.79269			−0.647119			−0.323560	−	0.946208i	$-0.604880\pi$
$229$	−9.79269			−0.647119			−0.323560	+	0.946208i	$0.604880\pi$
$230$		0			0
$231$	−2.97752			−0.195907
$232$		0			0
$233$	−13.8891			−0.909903			−0.454952	−	0.890516i	$-0.650343\pi$
$233$	−13.8891			−0.909903			−0.454952	+	0.890516i	$0.650343\pi$
$234$		0			0
$235$			6.95013i			0.453376i
$236$		0			0
$237$		−	19.9004i		−	1.29267i
$238$		0			0
$239$			3.54335i			0.229200i	0.993412	+	0.114600i	$0.0365587\pi$
$239$			3.54335i			0.229200i	−0.993412	+	0.114600i	$0.963441\pi$
$240$		0			0
$241$		−	2.53135i		−	0.163059i	−0.996671	−	0.0815293i	$-0.974020\pi$
$241$		−	2.53135i		−	0.163059i	0.996671	−	0.0815293i	$-0.0259804\pi$
$242$		0			0
$243$		−	22.0293i		−	1.41318i
$244$		0			0
$245$	−1.51724			−0.0969327
$246$		0			0
$247$	28.0497	+	9.96223i	1.78476	+	0.633882i
$248$		0			0
$249$		−	13.4451i		−	0.852049i
$250$		0			0
$251$			28.4225i			1.79401i	0.442019	+	0.897006i	$0.354262\pi$
$251$			28.4225i			1.79401i	−0.442019	+	0.897006i	$0.645738\pi$
$252$		0			0
$253$	−7.45919			−0.468955
$254$		0			0
$255$			0.344978i			0.0216034i
$256$		0			0
$257$	1.65015			0.102934			0.0514668	−	0.998675i	$-0.483610\pi$
$257$	1.65015			0.102934			0.0514668	+	0.998675i	$0.483610\pi$
$258$		0			0
$259$			8.75613i			0.544080i
$260$		0			0
$261$		−	10.1748i		−	0.629805i
$262$		0			0
$263$	13.6929			0.844338			0.422169	−	0.906517i	$-0.361269\pi$
$263$	13.6929			0.844338			0.422169	+	0.906517i	$0.361269\pi$
$264$		0			0
$265$			16.3963i			1.00722i
$266$		0			0
$267$	2.80960			0.171945
$268$		0			0
$269$		−	12.0078i		−	0.732128i	−0.930590	−	0.366064i	$-0.880705\pi$
$269$		−	12.0078i		−	0.732128i	0.930590	−	0.366064i	$-0.119295\pi$
$270$		0			0
$271$		−	3.02364i		−	0.183673i	−0.995774	−	0.0918365i	$-0.970726\pi$
$271$		−	3.02364i		−	0.183673i	0.995774	−	0.0918365i	$-0.0292737\pi$
$272$		0			0
$273$	−8.79396	−	3.12330i	−0.532234	−	0.189031i
$274$		0			0
$275$	3.10375			0.187163
$276$		0			0
$277$			22.1000i			1.32786i	0.747794	+	0.663931i	$0.231113\pi$
$277$			22.1000i			1.32786i	−0.747794	+	0.663931i	$0.768887\pi$
$278$		0			0
$279$			20.2213i			1.21062i
$280$		0			0
$281$		−	25.7594i		−	1.53668i	−0.640044	−	0.768339i	$-0.721084\pi$
$281$		−	25.7594i		−	1.53668i	0.640044	−	0.768339i	$-0.278916\pi$
$282$		0			0
$283$		−	1.31917i		−	0.0784165i	−0.999231	−	0.0392083i	$-0.987516\pi$
$283$		−	1.31917i		−	0.0784165i	0.999231	−	0.0392083i	$-0.0124836\pi$
$284$		0			0
$285$		−	32.4201i		−	1.92040i
$286$		0			0
$287$	−0.221622			−0.0130819
$288$		0			0
$289$	−16.9923			−0.999546
$290$		0			0
$291$	−27.1317			−1.59049
$292$		0			0
$293$	3.67811			0.214878			0.107439	−	0.994212i	$-0.465735\pi$
$293$	3.67811			0.214878			0.107439	+	0.994212i	$0.465735\pi$
$294$		0			0
$295$	−6.01401			−0.350149
$296$		0			0
$297$			2.08167i			0.120791i
$298$		0			0
$299$	−22.0303	−	7.82437i	−1.27405	−	0.452495i
$300$		0			0
$301$	−11.3062			−0.651677
$302$		0			0
$303$	−8.96057			−0.514772
$304$		0			0
$305$		−	16.7726i		−	0.960396i
$306$		0			0
$307$	27.2095			1.55293			0.776463	−	0.630163i	$-0.217012\pi$
$307$	27.2095			1.55293			0.776463	+	0.630163i	$0.217012\pi$
$308$		0			0
$309$		−	14.6729i		−	0.834715i
$310$		0			0
$311$	14.8834			0.843960			0.421980	−	0.906605i	$-0.361335\pi$
$311$	14.8834			0.843960			0.421980	+	0.906605i	$0.361335\pi$
$312$		0			0
$313$	−11.4874			−0.649308			−0.324654	−	0.945833i	$-0.605248\pi$
$313$	−11.4874			−0.649308			−0.324654	+	0.945833i	$0.605248\pi$
$314$		0			0
$315$			5.61245i			0.316226i
$316$		0			0
$317$	13.0995			0.735743			0.367872	−	0.929877i	$-0.380087\pi$
$317$	13.0995			0.735743			0.367872	+	0.929877i	$0.380087\pi$
$318$		0			0
$319$		−	3.16427i		−	0.177165i
$320$		0			0
$321$	−44.8532			−2.50346
$322$		0			0
$323$	−0.725239			−0.0403534
$324$		0			0
$325$	9.16676	+	3.25570i	0.508480	+	0.180594i
$326$		0			0
$327$			28.5726i			1.58007i
$328$		0			0
$329$	4.58078			0.252546
$330$		0			0
$331$	−21.5841			−1.18637			−0.593183	−	0.805067i	$-0.702129\pi$
$331$	−21.5841			−1.18637			−0.593183	+	0.805067i	$0.702129\pi$
$332$		0			0
$333$	32.3900			1.77496
$334$		0			0
$335$	−7.03317			−0.384263
$336$		0			0
$337$	3.36280			0.183183			0.0915917	−	0.995797i	$-0.470805\pi$
$337$	3.36280			0.183183			0.0915917	+	0.995797i	$0.470805\pi$
$338$		0			0
$339$		−	18.6969i		−	1.01548i
$340$		0			0
$341$			6.28863i			0.340548i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			25.4629i			1.37088i
$346$		0			0
$347$			10.4226i			0.559513i	0.960071	+	0.279756i	$0.0902537\pi$
$347$			10.4226i			0.559513i	−0.960071	+	0.279756i	$0.909746\pi$
$348$		0			0
$349$	−5.88538			−0.315037			−0.157519	−	0.987516i	$-0.550349\pi$
$349$	−5.88538			−0.315037			−0.157519	+	0.987516i	$0.550349\pi$
$350$		0			0
$351$	−2.18358	+	6.14809i	−0.116551	+	0.328161i
$352$		0			0
$353$		−	18.2921i		−	0.973592i	−0.873516	−	0.486796i	$-0.838166\pi$
$353$		−	18.2921i		−	0.973592i	0.873516	−	0.486796i	$-0.161834\pi$
$354$		0			0
$355$		−	14.2150i		−	0.754453i
$356$		0			0
$357$	0.227372			0.0120338
$358$		0			0
$359$		−	10.4332i		−	0.550644i	−0.961352	−	0.275322i	$-0.911216\pi$
$359$		−	10.4332i		−	0.550644i	0.961352	−	0.275322i	$-0.0887845\pi$
$360$		0			0
$361$	49.1561			2.58716
$362$		0			0
$363$		−	25.0456i		−	1.31455i
$364$		0			0
$365$		−	15.7959i		−	0.826795i
$366$		0			0
$367$	−10.6000			−0.553316			−0.276658	−	0.960968i	$-0.589227\pi$
$367$	−10.6000			−0.553316			−0.276658	+	0.960968i	$0.589227\pi$
$368$		0			0
$369$			0.819806i			0.0426774i
$370$		0			0
$371$	10.8067			0.561056
$372$		0			0
$373$			21.9283i			1.13540i	0.823235	+	0.567701i	$0.192167\pi$
$373$			21.9283i			1.13540i	−0.823235	+	0.567701i	$0.807833\pi$
$374$		0			0
$375$		−	30.2301i		−	1.56108i
$376$		0			0
$377$	3.31918	−	9.34549i	0.170947	−	0.481317i
$378$		0			0
$379$	−10.9073			−0.560268			−0.280134	−	0.959961i	$-0.590379\pi$
$379$	−10.9073			−0.560268			−0.280134	+	0.959961i	$0.590379\pi$
$380$		0			0
$381$			28.2070i			1.44509i
$382$		0			0
$383$			11.6452i			0.595040i	0.954716	+	0.297520i	$0.0961594\pi$
$383$			11.6452i			0.595040i	−0.954716	+	0.297520i	$0.903841\pi$
$384$		0			0
$385$			1.74542i			0.0889548i
$386$		0			0
$387$			41.8229i			2.12598i
$388$		0			0
$389$		−	13.1587i		−	0.667172i	−0.942720	−	0.333586i	$-0.891741\pi$
$389$		−	13.1587i		−	0.667172i	0.942720	−	0.333586i	$-0.108259\pi$
$390$		0			0
$391$	0.569605			0.0288062
$392$		0			0
$393$	29.3251			1.47926
$394$		0			0
$395$	−11.6656			−0.586958
$396$		0			0
$397$	−2.01211			−0.100985			−0.0504925	−	0.998724i	$-0.516079\pi$
$397$	−2.01211			−0.100985			−0.0504925	+	0.998724i	$0.516079\pi$
$398$		0			0
$399$	−21.3679			−1.06973
$400$		0			0
$401$		−	3.32731i		−	0.166158i	−0.996543	−	0.0830789i	$-0.973525\pi$
$401$		−	3.32731i		−	0.166158i	0.996543	−	0.0830789i	$-0.0264753\pi$
$402$		0			0
$403$	−6.59651	+	18.5731i	−0.328595	+	0.925193i
$404$		0			0
$405$	−9.73132			−0.483553
$406$		0			0
$407$	10.0730			0.499300
$408$		0			0
$409$			35.5960i			1.76011i	0.474872	+	0.880055i	$0.342494\pi$
$409$			35.5960i			1.76011i	−0.474872	+	0.880055i	$0.657506\pi$
$410$		0			0
$411$	−13.5947			−0.670575
$412$		0			0
$413$			3.96379i			0.195045i
$414$		0			0
$415$	−7.88150			−0.386888
$416$		0			0
$417$	17.2539			0.844929
$418$		0			0
$419$		−	1.09920i		−	0.0536992i	−0.999639	−	0.0268496i	$-0.991452\pi$
$419$		−	1.09920i		−	0.0536992i	0.999639	−	0.0268496i	$-0.00854753\pi$
$420$		0			0
$421$	−9.75651			−0.475503			−0.237752	−	0.971326i	$-0.576410\pi$
$421$	−9.75651			−0.475503			−0.237752	+	0.971326i	$0.576410\pi$
$422$		0			0
$423$		−	16.9449i		−	0.823888i
$424$		0			0
$425$	−0.237011			−0.0114967
$426$		0			0
$427$	−11.0547			−0.534974
$428$		0			0
$429$	−3.59302	+	10.1165i	−0.173473	+	0.488430i
$430$		0			0
$431$			6.44866i			0.310621i	0.987866	+	0.155310i	$0.0496378\pi$
$431$			6.44866i			0.310621i	−0.987866	+	0.155310i	$0.950362\pi$
$432$		0			0
$433$	−25.7879			−1.23929			−0.619643	−	0.784884i	$-0.712723\pi$
$433$	−25.7879			−1.23929			−0.619643	+	0.784884i	$0.712723\pi$
$434$		0			0
$435$	−10.8016			−0.517899
$436$		0			0
$437$	−53.5301			−2.56069
$438$		0			0
$439$	20.8825			0.996668			0.498334	−	0.866985i	$-0.333945\pi$
$439$	20.8825			0.996668			0.498334	+	0.866985i	$0.333945\pi$
$440$		0			0
$441$	3.69913			0.176149
$442$		0			0
$443$			1.97890i			0.0940205i	0.998894	+	0.0470103i	$0.0149694\pi$
$443$			1.97890i			0.0940205i	−0.998894	+	0.0470103i	$0.985031\pi$
$444$		0			0
$445$		−	1.64698i		−	0.0780745i
$446$		0			0
$447$			46.5035i			2.19954i
$448$		0			0
$449$			23.3403i			1.10150i	0.834671	+	0.550749i	$0.185658\pi$
$449$			23.3403i			1.10150i	−0.834671	+	0.550749i	$0.814342\pi$
$450$		0			0
$451$			0.254952i			0.0120052i
$452$		0			0
$453$	2.03243			0.0954921
$454$		0			0
$455$	−1.83087	+	5.15500i	−0.0858326	+	0.241670i
$456$		0			0
$457$		−	12.8179i		−	0.599595i	−0.954003	−	0.299798i	$-0.903081\pi$
$457$		−	12.8179i		−	0.599595i	0.954003	−	0.299798i	$-0.0969191\pi$
$458$		0			0
$459$		−	0.158962i		−	0.00741971i
$460$		0			0
$461$	−30.2775			−1.41016			−0.705081	−	0.709127i	$-0.749089\pi$
$461$	−30.2775			−1.41016			−0.705081	+	0.709127i	$0.749089\pi$
$462$		0			0
$463$			0.839207i			0.0390013i	0.999810	+	0.0195006i	$0.00620764\pi$
$463$			0.839207i			0.0390013i	−0.999810	+	0.0195006i	$0.993792\pi$
$464$		0			0
$465$	21.4670			0.995510
$466$		0			0
$467$			31.5327i			1.45916i	0.683895	+	0.729581i	$0.260285\pi$
$467$			31.5327i			1.45916i	−0.683895	+	0.729581i	$0.739715\pi$
$468$		0			0
$469$			4.63551i			0.214048i
$470$		0			0
$471$	−28.4744			−1.31203
$472$		0			0
$473$			13.0065i			0.598041i
$474$		0			0
$475$	22.2737			1.02199
$476$		0			0
$477$		−	39.9753i		−	1.83035i
$478$		0			0
$479$			37.3603i			1.70704i	0.521062	+	0.853519i	$0.325536\pi$
$479$			37.3603i			1.70704i	−0.521062	+	0.853519i	$0.674464\pi$
$480$		0			0
$481$	29.7500	+	10.5661i	1.35649	+	0.481775i
$482$		0			0
$483$	16.7824			0.763626
$484$		0			0
$485$			15.9046i			0.722189i
$486$		0			0
$487$			38.6285i			1.75042i	0.483741	+	0.875211i	$0.339278\pi$
$487$			38.6285i			1.75042i	−0.483741	+	0.875211i	$0.660722\pi$
$488$		0			0
$489$		−	24.0264i		−	1.08651i
$490$		0			0
$491$			34.4399i			1.55425i	0.629346	+	0.777125i	$0.283323\pi$
$491$			34.4399i			1.55425i	−0.629346	+	0.777125i	$0.716677\pi$
$492$		0			0
$493$			0.241633i			0.0108826i
$494$		0			0
$495$	6.45653			0.290199
$496$		0			0
$497$	−9.36899			−0.420257
$498$		0			0
$499$	−4.62862			−0.207205			−0.103603	−	0.994619i	$-0.533037\pi$
$499$	−4.62862			−0.207205			−0.103603	+	0.994619i	$0.533037\pi$
$500$		0			0
$501$	25.6545			1.14616
$502$		0			0
$503$	−32.9666			−1.46991			−0.734954	−	0.678117i	$-0.762796\pi$
$503$	−32.9666			−1.46991			−0.734954	+	0.678117i	$0.762796\pi$
$504$		0			0
$505$			5.25267i			0.233741i
$506$		0			0
$507$	−21.2236	+	26.1096i	−0.942572	+	1.15957i
$508$		0			0
$509$	−4.90267			−0.217307			−0.108654	−	0.994080i	$-0.534654\pi$
$509$	−4.90267			−0.217307			−0.108654	+	0.994080i	$0.534654\pi$
$510$		0			0
$511$	−10.4110			−0.460554
$512$		0			0
$513$			14.9389i			0.659566i
$514$		0			0
$515$	−8.60126			−0.379017
$516$		0			0
$517$		−	5.26970i		−	0.231761i
$518$		0			0
$519$	57.1306			2.50775
$520$		0			0
$521$	6.67667			0.292510			0.146255	−	0.989247i	$-0.453278\pi$
$521$	6.67667			0.292510			0.146255	+	0.989247i	$0.453278\pi$
$522$		0			0
$523$		−	19.5635i		−	0.855451i	−0.903909	−	0.427726i	$-0.859315\pi$
$523$		−	19.5635i		−	0.855451i	0.903909	−	0.427726i	$-0.140685\pi$
$524$		0			0
$525$	−6.98312			−0.304768
$526$		0			0
$527$		−	0.480218i		−	0.0209186i
$528$		0			0
$529$	19.0427			0.827944
$530$		0			0
$531$	14.6626			0.636301
$532$		0			0
$533$	−0.267434	+	0.752986i	−0.0115838	+	0.0326155i
$534$		0			0
$535$			26.2929i			1.13674i
$536$		0			0
$537$	−52.3475			−2.25896
$538$		0			0
$539$	1.15039			0.0495509
$540$		0			0
$541$	−37.0198			−1.59160			−0.795802	−	0.605557i	$-0.792950\pi$
$541$	−37.0198			−1.59160			−0.795802	+	0.605557i	$0.792950\pi$
$542$		0			0
$543$	49.5262			2.12537
$544$		0			0
$545$	16.7492			0.717457
$546$		0			0
$547$		−	44.1142i		−	1.88619i	−0.332526	−	0.943094i	$-0.607901\pi$
$547$		−	44.1142i		−	1.88619i	0.332526	−	0.943094i	$-0.392099\pi$
$548$		0			0
$549$			40.8927i			1.74526i
$550$		0			0
$551$		−	22.7080i		−	0.967394i
$552$		0			0
$553$			7.68869i			0.326956i
$554$		0			0
$555$		−	34.3855i		−	1.45958i
$556$		0			0
$557$	−12.6022			−0.533972			−0.266986	−	0.963700i	$-0.586028\pi$
$557$	−12.6022			−0.533972			−0.266986	+	0.963700i	$0.586028\pi$
$558$		0			0
$559$	−13.6433	+	38.4141i	−0.577051	+	1.62474i
$560$		0			0
$561$		−	0.261568i		−	0.0110434i
$562$		0			0
$563$			4.78407i			0.201625i	0.994905	+	0.100812i	$0.0321442\pi$
$563$			4.78407i			0.201625i	−0.994905	+	0.100812i	$0.967856\pi$
$564$		0			0
$565$	−10.9601			−0.461094
$566$		0			0
$567$			6.41384i			0.269356i
$568$		0			0
$569$	28.1257			1.17909			0.589545	−	0.807736i	$-0.299307\pi$
$569$	28.1257			1.17909			0.589545	+	0.807736i	$0.299307\pi$
$570$		0			0
$571$			9.67429i			0.404856i	0.979297	+	0.202428i	$0.0648833\pi$
$571$			9.67429i			0.404856i	−0.979297	+	0.202428i	$0.935117\pi$
$572$		0			0
$573$			3.11413i			0.130095i
$574$		0			0
$575$	−17.4939			−0.729545
$576$		0			0
$577$		−	5.85466i		−	0.243732i	−0.992547	−	0.121866i	$-0.961112\pi$
$577$		−	5.85466i		−	0.243732i	0.992547	−	0.121866i	$-0.0388879\pi$
$578$		0			0
$579$	−42.9921			−1.78669
$580$		0			0
$581$			5.19464i			0.215510i
$582$		0			0
$583$		−	12.4320i		−	0.514879i
$584$		0			0
$585$	19.0690	+	6.77262i	0.788406	+	0.280014i
$586$		0			0
$587$	−42.1940			−1.74153			−0.870766	−	0.491698i	$-0.836376\pi$
$587$	−42.1940			−1.74153			−0.870766	+	0.491698i	$0.836376\pi$
$588$		0			0
$589$			45.1297i			1.85954i
$590$		0			0
$591$		−	31.8708i		−	1.31099i
$592$		0			0
$593$			19.0752i			0.783325i	0.920109	+	0.391663i	$0.128100\pi$
$593$			19.0752i			0.783325i	−0.920109	+	0.391663i	$0.871900\pi$
$594$		0			0
$595$		−	0.133285i		−	0.00546416i
$596$		0			0
$597$			9.31334i			0.381169i
$598$		0			0
$599$	3.43267			0.140255			0.0701275	−	0.997538i	$-0.477659\pi$
$599$	3.43267			0.140255			0.0701275	+	0.997538i	$0.477659\pi$
$600$		0			0
$601$	−38.6990			−1.57857			−0.789283	−	0.614029i	$-0.789548\pi$
$601$	−38.6990			−1.57857			−0.789283	+	0.614029i	$0.789548\pi$
$602$		0			0
$603$	17.1473			0.698293
$604$		0			0
$605$	−14.6817			−0.596896
$606$		0			0
$607$	−17.5335			−0.711664			−0.355832	−	0.934550i	$-0.615803\pi$
$607$	−17.5335			−0.711664			−0.355832	+	0.934550i	$0.615803\pi$
$608$		0			0
$609$			7.11928i			0.288488i
$610$		0			0
$611$	5.52769	−	15.5638i	0.223626	−	0.629642i
$612$		0			0
$613$	17.9396			0.724574			0.362287	−	0.932067i	$-0.381996\pi$
$613$	17.9396			0.724574			0.362287	+	0.932067i	$0.381996\pi$
$614$		0			0
$615$	0.870311			0.0350943
$616$		0			0
$617$			11.0038i			0.442995i	0.975161	+	0.221498i	$0.0710945\pi$
$617$			11.0038i			0.442995i	−0.975161	+	0.221498i	$0.928906\pi$
$618$		0			0
$619$	−43.6123			−1.75293			−0.876464	−	0.481467i	$-0.840104\pi$
$619$	−43.6123			−1.75293			−0.876464	+	0.481467i	$0.840104\pi$
$620$		0			0
$621$		−	11.7330i		−	0.470830i
$622$		0			0
$623$	−1.08552			−0.0434902
$624$		0			0
$625$	−4.23089			−0.169236
$626$		0			0
$627$			24.5815i			0.981689i
$628$		0			0
$629$	−0.769203			−0.0306701
$630$		0			0
$631$		−	31.3565i		−	1.24828i	−0.781312	−	0.624141i	$-0.785449\pi$
$631$		−	31.3565i		−	1.24828i	0.781312	−	0.624141i	$-0.214551\pi$
$632$		0			0
$633$	−34.6774			−1.37830
$634$		0			0
$635$	16.5349			0.656167
$636$		0			0
$637$	3.39762	+	1.20671i	0.134619	+	0.0478117i
$638$		0			0
$639$			34.6571i			1.37101i
$640$		0			0
$641$	−3.14378			−0.124172			−0.0620859	−	0.998071i	$-0.519775\pi$
$641$	−3.14378			−0.124172			−0.0620859	+	0.998071i	$0.519775\pi$
$642$		0			0
$643$	23.9503			0.944508			0.472254	−	0.881462i	$-0.343440\pi$
$643$	23.9503			0.944508			0.472254	+	0.881462i	$0.343440\pi$
$644$		0			0
$645$	44.3995			1.74823
$646$		0			0
$647$	−34.8065			−1.36838			−0.684192	−	0.729302i	$-0.739845\pi$
$647$	−34.8065			−1.36838			−0.684192	+	0.729302i	$0.739845\pi$
$648$		0			0
$649$	4.55991			0.178992
$650$		0			0
$651$		−	14.1488i		−	0.554534i
$652$		0			0
$653$		−	23.9093i		−	0.935643i	−0.883823	−	0.467822i	$-0.845039\pi$
$653$		−	23.9093i		−	0.935643i	0.883823	−	0.467822i	$-0.154961\pi$
$654$		0			0
$655$		−	17.1903i		−	0.671682i
$656$		0			0
$657$			38.5115i			1.50248i
$658$		0			0
$659$		−	9.71979i		−	0.378629i	−0.981916	−	0.189315i	$-0.939373\pi$
$659$		−	9.71979i		−	0.378629i	0.981916	−	0.189315i	$-0.0606266\pi$
$660$		0			0
$661$	3.10662			0.120834			0.0604168	−	0.998173i	$-0.480757\pi$
$661$	3.10662			0.120834			0.0604168	+	0.998173i	$0.480757\pi$
$662$		0			0
$663$	0.274373	−	0.772526i	0.0106558	−	0.0300024i
$664$		0			0
$665$			12.5258i			0.485730i
$666$		0			0
$667$			17.8350i			0.690573i
$668$		0			0
$669$	39.8616			1.54114
$670$		0			0
$671$			12.7172i			0.490944i
$672$		0			0
$673$	29.1531			1.12377			0.561884	−	0.827216i	$-0.310077\pi$
$673$	29.1531			1.12377			0.561884	+	0.827216i	$0.310077\pi$
$674$		0			0
$675$			4.88208i			0.187912i
$676$		0			0
$677$		−	33.3416i		−	1.28142i	−0.767782	−	0.640711i	$-0.778640\pi$
$677$		−	33.3416i		−	1.28142i	0.767782	−	0.640711i	$-0.221360\pi$
$678$		0			0
$679$	10.4826			0.402284
$680$		0			0
$681$			47.8418i			1.83330i
$682$		0			0
$683$	41.2913			1.57997			0.789984	−	0.613128i	$-0.210089\pi$
$683$	41.2913			1.57997			0.789984	+	0.613128i	$0.210089\pi$
$684$		0			0
$685$			7.96916i			0.304486i
$686$		0			0
$687$		−	25.3461i		−	0.967014i
$688$		0			0
$689$	13.0406	−	36.7171i	0.496807	−	1.39881i
$690$		0			0
$691$	−6.17715			−0.234990			−0.117495	−	0.993073i	$-0.537486\pi$
$691$	−6.17715			−0.234990			−0.117495	+	0.993073i	$0.537486\pi$
$692$		0			0
$693$		−	4.25545i		−	0.161651i
$694$		0			0
$695$		−	10.1142i		−	0.383654i
$696$		0			0
$697$		−	0.0194689i		−	0.000737436i
$698$		0			0
$699$		−	35.9486i		−	1.35970i
$700$		0			0
$701$			37.6124i			1.42060i	0.703898	+	0.710301i	$0.251441\pi$
$701$			37.6124i			1.42060i	−0.703898	+	0.710301i	$0.748559\pi$
$702$		0			0
$703$	72.2878			2.72638
$704$		0			0
$705$	−17.9888			−0.677497
$706$		0			0
$707$	3.46200			0.130202
$708$		0			0
$709$	−16.2893			−0.611758			−0.305879	−	0.952070i	$-0.598950\pi$
$709$	−16.2893			−0.611758			−0.305879	+	0.952070i	$0.598950\pi$
$710$		0			0
$711$	28.4414			1.06664
$712$		0			0
$713$		−	35.4450i		−	1.32743i
$714$		0			0
$715$	5.93028	+	2.10622i	0.221780	+	0.0787682i
$716$		0			0
$717$	−9.17113			−0.342502
$718$		0			0
$719$	12.6463			0.471627			0.235813	−	0.971798i	$-0.424225\pi$
$719$	12.6463			0.471627			0.235813	+	0.971798i	$0.424225\pi$
$720$		0			0
$721$			5.66902i			0.211126i
$722$		0			0
$723$	6.55181			0.243665
$724$		0			0
$725$		−	7.42109i		−	0.275612i
$726$		0			0
$727$	31.4098			1.16492			0.582462	−	0.812858i	$-0.302089\pi$
$727$	31.4098			1.16492			0.582462	+	0.812858i	$0.302089\pi$
$728$		0			0
$729$	37.7762			1.39912
$730$		0			0
$731$		−	0.993217i		−	0.0367355i
$732$		0			0
$733$	−7.05081			−0.260428			−0.130214	−	0.991486i	$-0.541566\pi$
$733$	−7.05081			−0.260428			−0.130214	+	0.991486i	$0.541566\pi$
$734$		0			0
$735$		−	3.92702i		−	0.144850i
$736$		0			0
$737$	5.33266			0.196431
$738$		0			0
$739$	−23.2829			−0.856475			−0.428237	−	0.903666i	$-0.640865\pi$
$739$	−23.2829			−0.856475			−0.428237	+	0.903666i	$0.640865\pi$
$740$		0			0
$741$	−25.7849	+	72.6000i	−0.947233	+	2.66703i
$742$		0			0
$743$		−	41.5984i		−	1.52610i	−0.646341	−	0.763049i	$-0.723702\pi$
$743$		−	41.5984i		−	1.52610i	0.646341	−	0.763049i	$-0.276298\pi$
$744$		0			0
$745$	27.2603			0.998739
$746$		0			0
$747$	19.2156			0.703063
$748$		0			0
$749$	17.3294			0.633204
$750$		0			0
$751$	−10.7904			−0.393748			−0.196874	−	0.980429i	$-0.563079\pi$
$751$	−10.7904			−0.393748			−0.196874	+	0.980429i	$0.563079\pi$
$752$		0			0
$753$	−73.5650			−2.68086
$754$		0			0
$755$		−	1.19141i		−	0.0433598i
$756$		0			0
$757$			6.37976i			0.231876i	0.993256	+	0.115938i	$0.0369874\pi$
$757$			6.37976i			0.231876i	−0.993256	+	0.115938i	$0.963013\pi$
$758$		0			0
$759$		−	19.3064i		−	0.700777i
$760$		0			0
$761$			21.0373i			0.762603i	0.924451	+	0.381301i	$0.124524\pi$
$761$			21.0373i			0.762603i	−0.924451	+	0.381301i	$0.875476\pi$
$762$		0			0
$763$		−	11.0393i		−	0.399649i
$764$		0			0
$765$	−0.493039			−0.0178259
$766$		0			0
$767$	13.4675	+	4.78316i	0.486282	+	0.172710i
$768$		0			0
$769$			7.90032i			0.284893i	0.989803	+	0.142446i	$0.0454969\pi$
$769$			7.90032i			0.284893i	−0.989803	+	0.142446i	$0.954503\pi$
$770$		0			0
$771$			4.27103i			0.153817i
$772$		0			0
$773$	26.2770			0.945119			0.472560	−	0.881299i	$-0.343330\pi$
$773$	26.2770			0.945119			0.472560	+	0.881299i	$0.343330\pi$
$774$		0			0
$775$			14.7486i			0.529785i
$776$		0			0
$777$	−22.6632			−0.813038
$778$		0			0
$779$			1.82963i			0.0655535i
$780$		0			0
$781$			10.7780i			0.385668i
$782$		0			0
$783$	4.97728			0.177873
$784$		0			0
$785$			16.6917i			0.595751i
$786$		0			0
$787$	−11.3359			−0.404080			−0.202040	−	0.979377i	$-0.564757\pi$
$787$	−11.3359			−0.404080			−0.202040	+	0.979377i	$0.564757\pi$
$788$		0			0
$789$			35.4408i			1.26173i
$790$		0			0
$791$			7.22371i			0.256846i
$792$		0			0
$793$	−13.3399	+	37.5597i	−0.473712	+	1.33378i
$794$		0			0
$795$	−42.4381			−1.50512
$796$		0			0
$797$		−	33.0929i		−	1.17221i	−0.810235	−	0.586105i	$-0.800661\pi$
$797$		−	33.0929i		−	1.17221i	0.810235	−	0.586105i	$-0.199339\pi$
$798$		0			0
$799$			0.402409i			0.0142362i
$800$		0			0
$801$			4.01546i			0.141879i
$802$		0			0
$803$			11.9767i			0.422648i
$804$		0			0
$805$		−	9.83782i		−	0.346738i
$806$		0			0
$807$	31.0794			1.09405
$808$		0			0
$809$	−36.5501			−1.28503			−0.642516	−	0.766272i	$-0.722109\pi$
$809$	−36.5501			−1.28503			−0.642516	+	0.766272i	$0.722109\pi$
$810$		0			0
$811$	−2.27406			−0.0798529			−0.0399265	−	0.999203i	$-0.512712\pi$
$811$	−2.27406			−0.0798529			−0.0399265	+	0.999203i	$0.512712\pi$
$812$		0			0
$813$	7.82599			0.274469
$814$		0			0
$815$	−14.0842			−0.493350
$816$		0			0
$817$			93.3400i			3.26555i
$818$		0			0
$819$	4.46379	−	12.5682i	0.155977	−	0.439170i
$820$		0			0
$821$	28.9611			1.01075			0.505375	−	0.862900i	$-0.331354\pi$
$821$	28.9611			1.01075			0.505375	+	0.862900i	$0.331354\pi$
$822$		0			0
$823$	−18.0668			−0.629768			−0.314884	−	0.949130i	$-0.601966\pi$
$823$	−18.0668			−0.629768			−0.314884	+	0.949130i	$0.601966\pi$
$824$		0			0
$825$			8.03333i			0.279685i
$826$		0			0
$827$	50.9766			1.77263			0.886315	−	0.463083i	$-0.153257\pi$
$827$	50.9766			1.77263			0.886315	+	0.463083i	$0.153257\pi$
$828$		0			0
$829$		−	42.8967i		−	1.48987i	−0.667140	−	0.744933i	$-0.732482\pi$
$829$		−	42.8967i		−	1.48987i	0.667140	−	0.744933i	$-0.267518\pi$
$830$		0			0
$831$	−57.2008			−1.98427
$832$		0			0
$833$	−0.0878473			−0.00304373
$834$		0			0
$835$		−	15.0386i		−	0.520434i
$836$		0			0
$837$	−9.89178			−0.341910
$838$		0			0
$839$			46.5281i			1.60633i	0.595758	+	0.803164i	$0.296852\pi$
$839$			46.5281i			1.60633i	−0.595758	+	0.803164i	$0.703148\pi$
$840$		0			0
$841$	21.4342			0.739111
$842$		0			0
$843$	66.6722			2.29631
$844$		0			0
$845$	15.3054	+	12.4412i	0.526522	+	0.427991i
$846$		0			0
$847$			9.67660i			0.332492i
$848$		0			0
$849$	3.41436			0.117181
$850$		0			0
$851$	−56.7751			−1.94622
$852$		0			0
$853$	34.8932			1.19472			0.597360	−	0.801973i	$-0.296216\pi$
$853$	34.8932			1.19472			0.597360	+	0.801973i	$0.296216\pi$
$854$		0			0
$855$	46.3346			1.58461
$856$		0			0
$857$	3.09683			0.105786			0.0528928	−	0.998600i	$-0.483156\pi$
$857$	3.09683			0.105786			0.0528928	+	0.998600i	$0.483156\pi$
$858$		0			0
$859$		−	32.0784i		−	1.09450i	−0.836969	−	0.547251i	$-0.815674\pi$
$859$		−	32.0784i		−	1.09450i	0.836969	−	0.547251i	$-0.184326\pi$
$860$		0			0
$861$		−	0.573616i		−	0.0195488i
$862$		0			0
$863$			4.04153i			0.137575i	0.997631	+	0.0687876i	$0.0219131\pi$
$863$			4.04153i			0.137575i	−0.997631	+	0.0687876i	$0.978087\pi$
$864$		0			0
$865$		−	33.4898i		−	1.13869i
$866$		0			0
$867$		−	43.9806i		−	1.49366i
$868$		0			0
$869$	8.84502			0.300047
$870$		0			0
$871$	15.7497	+	5.59373i	0.533658	+	0.189536i
$872$		0			0
$873$		−	38.7764i		−	1.31238i
$874$		0			0
$875$			11.6797i			0.394845i
$876$		0			0
$877$	6.27681			0.211953			0.105976	−	0.994369i	$-0.466203\pi$
$877$	6.27681			0.211953			0.105976	+	0.994369i	$0.466203\pi$
$878$		0			0
$879$			9.51994i			0.321100i
$880$		0			0
$881$	5.41652			0.182487			0.0912437	−	0.995829i	$-0.470916\pi$
$881$	5.41652			0.182487			0.0912437	+	0.995829i	$0.470916\pi$
$882$		0			0
$883$		−	16.4099i		−	0.552236i	−0.961124	−	0.276118i	$-0.910952\pi$
$883$		−	16.4099i		−	0.552236i	0.961124	−	0.276118i	$-0.0890480\pi$
$884$		0			0
$885$		−	15.5659i		−	0.523241i
$886$		0			0
$887$	−39.5735			−1.32875			−0.664374	−	0.747400i	$-0.731302\pi$
$887$	−39.5735			−1.32875			−0.664374	+	0.747400i	$0.731302\pi$
$888$		0			0
$889$		−	10.8980i		−	0.365508i
$890$		0			0
$891$	7.37844			0.247187
$892$		0			0
$893$		−	37.8174i		−	1.26551i
$894$		0			0
$895$			30.6860i			1.02572i
$896$		0			0
$897$	20.2516	−	57.0203i	0.676180	−	1.90385i
$898$		0			0
$899$	15.0362			0.501484
$900$		0			0
$901$			0.949340i			0.0316271i
$902$		0			0
$903$		−	29.2634i		−	0.973824i
$904$		0			0
$905$		−	29.0322i		−	0.965062i
$906$		0			0
$907$		−	54.8552i		−	1.82144i	−0.413026	−	0.910719i	$-0.635528\pi$
$907$		−	54.8552i		−	1.82144i	0.413026	−	0.910719i	$-0.364472\pi$
$908$		0			0
$909$		−	12.8064i		−	0.424760i
$910$		0			0
$911$	19.6423			0.650778			0.325389	−	0.945580i	$-0.394505\pi$
$911$	19.6423			0.650778			0.325389	+	0.945580i	$0.394505\pi$
$912$		0			0
$913$	5.97588			0.197773
$914$		0			0
$915$	43.4120			1.43515
$916$		0			0
$917$	−11.3300			−0.374150
$918$		0			0
$919$	−50.8099			−1.67606			−0.838032	−	0.545621i	$-0.816294\pi$
$919$	−50.8099			−1.67606			−0.838032	+	0.545621i	$0.816294\pi$
$920$		0			0
$921$			70.4253i			2.32059i
$922$		0			0
$923$	−11.3057	+	31.8323i	−0.372132	+	1.04777i
$924$		0			0
$925$	23.6240			0.776751
$926$		0			0
$927$	20.9704			0.688760
$928$		0			0
$929$			22.2352i			0.729515i	0.931103	+	0.364757i	$0.118848\pi$
$929$			22.2352i			0.729515i	−0.931103	+	0.364757i	$0.881152\pi$
$930$		0			0
$931$	8.25567			0.270569
$932$		0			0
$933$			38.5222i			1.26116i
$934$		0			0
$935$	−0.153330			−0.00501444
$936$		0			0
$937$	32.6708			1.06731			0.533653	−	0.845703i	$-0.320819\pi$
$937$	32.6708			1.06731			0.533653	+	0.845703i	$0.320819\pi$
$938$		0			0
$939$		−	29.7325i		−	0.970285i
$940$		0			0
$941$	−29.1551			−0.950428			−0.475214	−	0.879870i	$-0.657629\pi$
$941$	−29.1551			−0.950428			−0.475214	+	0.879870i	$0.657629\pi$
$942$		0			0
$943$		−	1.43700i		−	0.0467952i
$944$		0			0
$945$	−2.74548			−0.0893105
$946$		0			0
$947$	−46.5705			−1.51334			−0.756668	−	0.653799i	$-0.773174\pi$
$947$	−46.5705			−1.51334			−0.756668	+	0.653799i	$0.773174\pi$
$948$		0			0
$949$	−12.5631	+	35.3725i	−0.407814	+	1.14824i
$950$		0			0
$951$			33.9051i			1.09945i
$952$		0			0
$953$	2.59239			0.0839758			0.0419879	−	0.999118i	$-0.486631\pi$
$953$	2.59239			0.0839758			0.0419879	+	0.999118i	$0.486631\pi$
$954$		0			0
$955$	1.82550			0.0590716
$956$		0			0
$957$	8.18997			0.264744
$958$		0			0
$959$	5.25241			0.169609
$960$		0			0
$961$	1.11733			0.0360429
$962$		0			0
$963$		−	64.1038i		−	2.06571i
$964$		0			0
$965$			25.2019i			0.811277i
$966$		0			0
$967$			39.6322i			1.27449i	0.770663	+	0.637243i	$0.219925\pi$
$967$			39.6322i			1.27449i	−0.770663	+	0.637243i	$0.780075\pi$
$968$		0			0
$969$		−	1.87711i		−	0.0603015i
$970$		0			0
$971$		−	28.8005i		−	0.924252i	−0.886814	−	0.462126i	$-0.847087\pi$
$971$		−	28.8005i		−	0.924252i	0.886814	−	0.462126i	$-0.152913\pi$
$972$		0			0
$973$	−6.66621			−0.213709
$974$		0			0
$975$	−8.42663	+	23.7260i	−0.269868	+	0.759840i
$976$		0			0
$977$		−	46.4373i		−	1.48566i	−0.669480	−	0.742830i	$-0.733483\pi$
$977$		−	46.4373i		−	1.48566i	0.669480	−	0.742830i	$-0.266517\pi$
$978$		0			0
$979$			1.24877i			0.0399108i
$980$		0			0
$981$	−40.8357			−1.30378
$982$		0			0
$983$			46.2252i			1.47436i	0.675699	+	0.737178i	$0.263842\pi$
$983$			46.2252i			1.47436i	−0.675699	+	0.737178i	$0.736158\pi$
$984$		0			0
$985$	−18.6826			−0.595278
$986$		0			0
$987$			11.8563i			0.377389i
$988$		0			0
$989$		−	73.3096i		−	2.33111i
$990$		0			0
$991$	−10.6490			−0.338277			−0.169139	−	0.985592i	$-0.554099\pi$
$991$	−10.6490			−0.338277			−0.169139	+	0.985592i	$0.554099\pi$
$992$		0			0
$993$		−	55.8653i		−	1.77283i
$994$		0			0
$995$	5.45946			0.173077
$996$		0			0
$997$		−	50.3193i		−	1.59363i	−0.604225	−	0.796814i	$-0.706517\pi$
$997$		−	50.3193i		−	1.59363i	0.604225	−	0.796814i	$-0.293483\pi$
$998$		0			0
$999$			15.8445i			0.501296i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.i.a.337.20		84
4.3	odd	2		728.2.i.a.701.83	yes	84
8.3	odd	2		728.2.i.a.701.1	✓	84
8.5	even	2	inner	2912.2.i.a.337.65		84
13.12	even	2	inner	2912.2.i.a.337.66		84
52.51	odd	2		728.2.i.a.701.2	yes	84
104.51	odd	2		728.2.i.a.701.84	yes	84
104.77	even	2	inner	2912.2.i.a.337.19		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.i.a.701.1	✓	84	8.3	odd	2
728.2.i.a.701.2	yes	84	52.51	odd	2
728.2.i.a.701.83	yes	84	4.3	odd	2
728.2.i.a.701.84	yes	84	104.51	odd	2
2912.2.i.a.337.19		84	104.77	even	2	inner
2912.2.i.a.337.20		84	1.1	even	1	trivial
2912.2.i.a.337.65		84	8.5	even	2	inner
2912.2.i.a.337.66		84	13.12	even	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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.58827i q^{3} +1.51724 q^{5} -1.00000i q^{7} -3.69913 q^{9} -1.15039 q^{11} +(-3.39762 - 1.20671i) q^{13} +3.92702i q^{15} +0.0878473 q^{17} -8.25567 q^{19} +2.58827 q^{21} +6.48403 q^{23} -2.69799 q^{25} -1.80953i q^{27} +2.75060i q^{29} -5.46650i q^{31} -2.97752i q^{33} -1.51724i q^{35} -8.75613 q^{37} +(3.12330 - 8.79396i) q^{39} -0.221622i q^{41} -11.3062i q^{43} -5.61245 q^{45} +4.58078i q^{47} -1.00000 q^{49} +0.227372i q^{51} +10.8067i q^{53} -1.74542 q^{55} -21.3679i q^{57} -3.96379 q^{59} -11.0547i q^{61} +3.69913i q^{63} +(-5.15500 - 1.83087i) q^{65} -4.63551 q^{67} +16.7824i q^{69} -9.36899i q^{71} -10.4110i q^{73} -6.98312i q^{75} +1.15039i q^{77} -7.68869 q^{79} -6.41384 q^{81} -5.19464 q^{83} +0.133285 q^{85} -7.11928 q^{87} -1.08552i q^{89} +(-1.20671 + 3.39762i) q^{91} +14.1488 q^{93} -12.5258 q^{95} +10.4826i q^{97} +4.25545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) \) 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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