LMFDB - Embedded newform 1848.2.v.c.881.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1848,2,Mod(881,1848)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1848.881"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1848, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1848.v (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$14.7563542935$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 40x^{12} + 388x^{8} + 436x^{4} + 16 $ x^16 + 40x^12 + 388x^8 + 436*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.10
Root			$-0.312148 + 0.312148i$ of defining polynomial
Character	$\chi$	$=$	1848.881
Dual form			1848.2.v.c.881.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.312148 + 1.70369i) q^{3} +(-0.589216 - 2.57931i) q^{7} +(-2.80513 + 1.06361i) q^{9} +1.00000i q^{11} +2.74337i q^{13} -2.99983 q^{17} +1.65615i q^{19} +(4.21042 - 1.80897i) q^{21} -0.431822i q^{23} -5.00000 q^{25} +(-2.68768 - 4.44707i) q^{27} -5.61025i q^{29} +5.52646i q^{31} +(-1.70369 + 0.312148i) q^{33} -7.96712 q^{37} +(-4.67386 + 0.856340i) q^{39} +11.0632 q^{41} -10.0943 q^{43} +0.624297 q^{47} +(-6.30565 + 3.03954i) q^{49} +(-0.936391 - 5.11078i) q^{51} -6.55904i q^{53} +(-2.82157 + 0.516965i) q^{57} -12.7030 q^{59} -3.25628i q^{61} +(4.39620 + 6.60859i) q^{63} -6.35686 q^{67} +(0.735691 - 0.134793i) q^{69} +16.6863i q^{71} +11.5424i q^{73} +(-1.56074 - 8.51846i) q^{75} +(2.57931 - 0.589216i) q^{77} -12.1693 q^{79} +(6.73747 - 5.96712i) q^{81} -16.2615 q^{83} +(9.55814 - 1.75123i) q^{87} +9.50212 q^{89} +(7.07600 - 1.61644i) q^{91} +(-9.41538 + 1.72508i) q^{93} -7.32767i q^{97} +(-1.06361 - 2.80513i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{7} - 8 q^{9} + 20 q^{21} - 80 q^{25} + 24 q^{39} + 16 q^{43} - 24 q^{49} - 40 q^{51} - 72 q^{57} - 12 q^{63} - 48 q^{67} - 24 q^{79} - 16 q^{81} + 40 q^{91} - 40 q^{93} + 8 q^{99}+O(q^{100}) $ 16 * q + 4 * q^7 - 8 * q^9 + 20 * q^21 - 80 * q^25 + 24 * q^39 + 16 * q^43 - 24 * q^49 - 40 * q^51 - 72 * q^57 - 12 * q^63 - 48 * q^67 - 24 * q^79 - 16 * q^81 + 40 * q^91 - 40 * q^93 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times$.

$n$	$463$	$617$	$673$	$925$	$1585$
$\chi(n)$	$1$	$-1$	$1$	$1$	$-1$

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
$3$	0.312148	+	1.70369i	0.180219	+	0.983627i
$3$	0.312148	+	1.70369i	0.180219	+	0.983627i
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−0.589216	−	2.57931i	−0.222703	−	0.974886i
$7$	−0.589216	−	2.57931i	−0.222703	−	0.974886i
$8$		0			0
$9$	−2.80513	+	1.06361i	−0.935042	+	0.354536i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			2.74337i			0.760875i	0.924807	+	0.380438i	$0.124227\pi$
$13$			2.74337i			0.760875i	−0.924807	+	0.380438i	$0.875773\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.99983			−0.727565			−0.363782	−	0.931484i	$-0.618515\pi$
$17$	−2.99983			−0.727565			−0.363782	+	0.931484i	$0.618515\pi$
$18$		0			0
$19$			1.65615i			0.379947i	0.981789	+	0.189973i	$0.0608402\pi$
$19$			1.65615i			0.379947i	−0.981789	+	0.189973i	$0.939160\pi$
$20$		0			0
$21$	4.21042	−	1.80897i	0.918789	−	0.394749i
$22$		0			0
$23$		−	0.431822i		−	0.0900411i	−0.998986	−	0.0450205i	$-0.985665\pi$
$23$		−	0.431822i		−	0.0900411i	0.998986	−	0.0450205i	$-0.0143353\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$	−2.68768	−	4.44707i	−0.517244	−	0.855838i
$28$		0			0
$29$		−	5.61025i		−	1.04180i	−0.853618	−	0.520899i	$-0.825597\pi$
$29$		−	5.61025i		−	1.04180i	0.853618	−	0.520899i	$-0.174403\pi$
$30$		0			0
$31$			5.52646i			0.992581i	0.868156	+	0.496291i	$0.165305\pi$
$31$			5.52646i			0.992581i	−0.868156	+	0.496291i	$0.834695\pi$
$32$		0			0
$33$	−1.70369	+	0.312148i	−0.296575	+	0.0543381i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−7.96712			−1.30979			−0.654893	−	0.755722i	$-0.727286\pi$
$37$	−7.96712			−1.30979			−0.654893	+	0.755722i	$0.727286\pi$
$38$		0			0
$39$	−4.67386	+	0.856340i	−0.748417	+	0.137124i
$40$		0			0
$41$	11.0632			1.72778			0.863890	−	0.503681i	$-0.168021\pi$
$41$	11.0632			1.72778			0.863890	+	0.503681i	$0.168021\pi$
$42$		0			0
$43$	−10.0943			−1.53937			−0.769686	−	0.638423i	$-0.779587\pi$
$43$	−10.0943			−1.53937			−0.769686	+	0.638423i	$0.779587\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.624297			0.0910631			0.0455315	−	0.998963i	$-0.485502\pi$
$47$	0.624297			0.0910631			0.0455315	+	0.998963i	$0.485502\pi$
$48$		0			0
$49$	−6.30565	+	3.03954i	−0.900807	+	0.434220i
$50$		0			0
$51$	−0.936391	−	5.11078i	−0.131121	−	0.715652i
$52$		0			0
$53$		−	6.55904i		−	0.900953i	−0.892788	−	0.450477i	$-0.851254\pi$
$53$		−	6.55904i		−	0.900953i	0.892788	−	0.450477i	$-0.148746\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.82157	+	0.516965i	−0.373726	+	0.0684737i
$58$		0			0
$59$	−12.7030			−1.65379			−0.826896	−	0.562354i	$-0.809896\pi$
$59$	−12.7030			−1.65379			−0.826896	+	0.562354i	$0.809896\pi$
$60$		0			0
$61$		−	3.25628i		−	0.416924i	−0.978031	−	0.208462i	$-0.933154\pi$
$61$		−	3.25628i		−	0.416924i	0.978031	−	0.208462i	$-0.0668457\pi$
$62$		0			0
$63$	4.39620	+	6.60859i	0.553869	+	0.832604i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.35686			−0.776614			−0.388307	−	0.921530i	$-0.626940\pi$
$67$	−6.35686			−0.776614			−0.388307	+	0.921530i	$0.626940\pi$
$68$		0			0
$69$	0.735691	−	0.134793i	0.0885668	−	0.0162271i
$70$		0			0
$71$			16.6863i			1.98029i	0.140030	+	0.990147i	$0.455280\pi$
$71$			16.6863i			1.98029i	−0.140030	+	0.990147i	$0.544720\pi$
$72$		0			0
$73$			11.5424i			1.35094i	0.737388	+	0.675469i	$0.236059\pi$
$73$			11.5424i			1.35094i	−0.737388	+	0.675469i	$0.763941\pi$
$74$		0			0
$75$	−1.56074	−	8.51846i	−0.180219	−	0.983627i
$76$		0			0
$77$	2.57931	−	0.589216i	0.293939	−	0.0671474i
$78$		0			0
$79$	−12.1693			−1.36915			−0.684576	−	0.728941i	$-0.740013\pi$
$79$	−12.1693			−1.36915			−0.684576	+	0.728941i	$0.740013\pi$
$80$		0			0
$81$	6.73747	−	5.96712i	0.748608	−	0.663013i
$82$		0			0
$83$	−16.2615			−1.78493			−0.892466	−	0.451114i	$-0.851027\pi$
$83$	−16.2615			−1.78493			−0.892466	+	0.451114i	$0.851027\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	9.55814	−	1.75123i	1.02474	−	0.187752i
$88$		0			0
$89$	9.50212			1.00722			0.503611	−	0.863930i	$-0.332004\pi$
$89$	9.50212			1.00722			0.503611	+	0.863930i	$0.332004\pi$
$90$		0			0
$91$	7.07600	−	1.61644i	0.741767	−	0.169449i
$92$		0			0
$93$	−9.41538	+	1.72508i	−0.976329	+	0.178882i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	7.32767i		−	0.744012i	−0.928230	−	0.372006i	$-0.878670\pi$
$97$		−	7.32767i		−	0.744012i	0.928230	−	0.372006i	$-0.121330\pi$
$98$		0			0
$99$	−1.06361	−	2.80513i	−0.106897	−	0.281926i
$100$		0			0
$101$	11.1980			1.11424			0.557120	−	0.830432i	$-0.311906\pi$
$101$	11.1980			1.11424			0.557120	+	0.830432i	$0.311906\pi$
$102$		0			0
$103$		−	7.51074i		−	0.740056i	−0.929021	−	0.370028i	$-0.879348\pi$
$103$		−	7.51074i		−	0.740056i	0.929021	−	0.370028i	$-0.120652\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.254436i		−	0.0245973i	−0.999924	−	0.0122986i	$-0.996085\pi$
$107$		−	0.254436i		−	0.0245973i	0.999924	−	0.0122986i	$-0.00391488\pi$
$108$		0			0
$109$	1.82261			0.174575			0.0872874	−	0.996183i	$-0.472180\pi$
$109$	1.82261			0.174575			0.0872874	+	0.996183i	$0.472180\pi$
$110$		0			0
$111$	−2.48692	−	13.5735i	−0.236048	−	1.28834i
$112$		0			0
$113$		−	17.5774i		−	1.65354i	−0.562540	−	0.826770i	$-0.690176\pi$
$113$		−	17.5774i		−	1.65354i	0.562540	−	0.826770i	$-0.309824\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.91788	−	7.69551i	−0.269758	−	0.711450i
$118$		0			0
$119$	1.76754	+	7.73747i	0.162031	+	0.709293i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	3.45336	+	18.8482i	0.311379	+	1.69949i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.07600			0.627894			0.313947	−	0.949441i	$-0.398349\pi$
$127$	7.07600			0.627894			0.313947	+	0.949441i	$0.398349\pi$
$128$		0			0
$129$	−3.15093	−	17.1976i	−0.277424	−	1.51417i
$130$		0			0
$131$	−4.15016			−0.362601			−0.181301	−	0.983428i	$-0.558031\pi$
$131$	−4.15016			−0.362601			−0.181301	+	0.983428i	$0.558031\pi$
$132$		0			0
$133$	4.27172	−	0.975830i	0.370405	−	0.0846152i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.09329i		−	0.0934060i	−0.998909	−	0.0467030i	$-0.985129\pi$
$137$		−	1.09329i		−	0.0934060i	0.998909	−	0.0467030i	$-0.0148714\pi$
$138$		0			0
$139$		−	4.15334i		−	0.352282i	−0.984365	−	0.176141i	$-0.943639\pi$
$139$		−	4.15334i		−	0.352282i	0.984365	−	0.176141i	$-0.0563614\pi$
$140$		0			0
$141$	0.194873	+	1.06361i	0.0164113	+	0.0895720i
$142$		0			0
$143$	−2.74337			−0.229412
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.14673	−	9.79409i	−0.589452	−	0.807803i
$148$		0			0
$149$		−	4.22155i		−	0.345843i	−0.984936	−	0.172922i	$-0.944679\pi$
$149$		−	4.22155i		−	0.345843i	0.984936	−	0.172922i	$-0.0553207\pi$
$150$		0			0
$151$	−18.7806			−1.52834			−0.764171	−	0.645013i	$-0.776852\pi$
$151$	−18.7806			−1.52834			−0.764171	+	0.645013i	$0.776852\pi$
$152$		0			0
$153$	8.41489	−	3.19064i	0.680304	−	0.257948i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			6.43665i			0.513701i	0.966451	+	0.256850i	$0.0826848\pi$
$157$			6.43665i			0.513701i	−0.966451	+	0.256850i	$0.917315\pi$
$158$		0			0
$159$	11.1746	−	2.04739i	0.886202	−	0.162369i
$160$		0			0
$161$	−1.11380	+	0.254436i	−0.0877798	+	0.0200524i
$162$		0			0
$163$	−5.32293			−0.416924			−0.208462	−	0.978030i	$-0.566846\pi$
$163$	−5.32293			−0.416924			−0.208462	+	0.978030i	$0.566846\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	22.8621			1.76912			0.884560	−	0.466427i	$-0.154459\pi$
$167$	22.8621			1.76912			0.884560	+	0.466427i	$0.154459\pi$
$168$		0			0
$169$	5.47390			0.421069
$170$		0			0
$171$	−1.76150	−	4.64571i	−0.134705	−	0.355266i
$172$		0			0
$173$	0.367846			0.0279668			0.0139834	−	0.999902i	$-0.495549\pi$
$173$	0.367846			0.0279668			0.0139834	+	0.999902i	$0.495549\pi$
$174$		0			0
$175$	2.94608	+	12.8965i	0.222703	+	0.974886i
$176$		0			0
$177$	−3.96523	−	21.6420i	−0.298045	−	1.62671i
$178$		0			0
$179$		−	10.6113i		−	0.793126i	−0.918008	−	0.396563i	$-0.870203\pi$
$179$		−	10.6113i		−	0.793126i	0.918008	−	0.396563i	$-0.129797\pi$
$180$		0			0
$181$		−	4.91848i		−	0.365587i	−0.983151	−	0.182794i	$-0.941486\pi$
$181$		−	4.91848i		−	0.365587i	0.983151	−	0.182794i	$-0.0585140\pi$
$182$		0			0
$183$	5.54769	−	1.01644i	0.410097	−	0.0751375i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	2.99983i		−	0.219369i
$188$		0			0
$189$	−9.88673	+	9.55263i	−0.719153	+	0.694851i
$190$		0			0
$191$		−	7.82261i		−	0.566025i	−0.959116	−	0.283012i	$-0.908666\pi$
$191$		−	7.82261i		−	0.566025i	0.959116	−	0.283012i	$-0.0913337\pi$
$192$		0			0
$193$	−6.71373			−0.483265			−0.241632	−	0.970368i	$-0.577683\pi$
$193$	−6.71373			−0.483265			−0.241632	+	0.970368i	$0.577683\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			23.4421i			1.67018i	0.550115	+	0.835089i	$0.314584\pi$
$197$			23.4421i			1.67018i	−0.550115	+	0.835089i	$0.685416\pi$
$198$		0			0
$199$			20.4046i			1.44644i	0.690616	+	0.723221i	$0.257339\pi$
$199$			20.4046i			1.44644i	−0.690616	+	0.723221i	$0.742661\pi$
$200$		0			0
$201$	−1.98428	−	10.8301i	−0.139961	−	0.763899i
$202$		0			0
$203$	−14.4706	+	3.30565i	−1.01563	+	0.232011i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.459290	+	1.21132i	0.0319228	+	0.0841922i
$208$		0			0
$209$	−1.65615			−0.114558
$210$		0			0
$211$	19.1876			1.32093			0.660465	−	0.750857i	$-0.270359\pi$
$211$	19.1876			1.32093			0.660465	+	0.750857i	$0.270359\pi$
$212$		0			0
$213$	−28.4282	+	5.20859i	−1.94787	+	0.356887i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	14.2544	−	3.25628i	0.967654	−	0.221050i
$218$		0			0
$219$	−19.6647	+	3.60295i	−1.32882	+	0.243465i
$220$		0			0
$221$		−	8.22964i		−	0.553586i
$222$		0			0
$223$		−	0.854824i		−	0.0572432i	−0.999590	−	0.0286216i	$-0.990888\pi$
$223$		−	0.854824i		−	0.0572432i	0.999590	−	0.0286216i	$-0.00911179\pi$
$224$		0			0
$225$	14.0256	−	5.31805i	0.935042	−	0.354536i
$226$		0			0
$227$	21.7483			1.44348			0.721741	−	0.692163i	$-0.243342\pi$
$227$	21.7483			1.44348			0.721741	+	0.692163i	$0.243342\pi$
$228$		0			0
$229$			21.5255i			1.42244i	0.702968	+	0.711222i	$0.251858\pi$
$229$			21.5255i			1.42244i	−0.702968	+	0.711222i	$0.748142\pi$
$230$		0			0
$231$	1.80897	+	4.21042i	0.119021	+	0.277025i
$232$		0			0
$233$		−	11.5078i		−	0.753903i	−0.926233	−	0.376951i	$-0.876972\pi$
$233$		−	11.5078i		−	0.753903i	0.926233	−	0.376951i	$-0.123028\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−3.79863	−	20.7327i	−0.246747	−	1.34673i
$238$		0			0
$239$			18.2911i			1.18315i	0.806249	+	0.591576i	$0.201494\pi$
$239$			18.2911i			1.18315i	−0.806249	+	0.591576i	$0.798506\pi$
$240$		0			0
$241$			12.3249i			0.793917i	0.917836	+	0.396959i	$0.129934\pi$
$241$			12.3249i			0.793917i	−0.917836	+	0.396959i	$0.870066\pi$
$242$		0			0
$243$	12.2692	+	9.61594i	0.787070	+	0.616863i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.54344			−0.289092
$248$		0			0
$249$	−5.07600	−	27.7046i	−0.321679	−	1.75571i
$250$		0			0
$251$	−12.7619			−0.805525			−0.402762	−	0.915305i	$-0.631950\pi$
$251$	−12.7619			−0.805525			−0.402762	+	0.915305i	$0.631950\pi$
$252$		0			0
$253$	0.431822			0.0271484
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.8621			−1.42610			−0.713048	−	0.701115i	$-0.752686\pi$
$257$	−22.8621			−1.42610			−0.713048	+	0.701115i	$0.752686\pi$
$258$		0			0
$259$	4.69435	+	20.5496i	0.291693	+	1.27689i
$260$		0			0
$261$	5.96712	+	15.7375i	0.369355	+	0.974125i
$262$		0			0
$263$			25.8318i			1.59286i	0.604732	+	0.796429i	$0.293280\pi$
$263$			25.8318i			1.59286i	−0.604732	+	0.796429i	$0.706720\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.96607	+	16.1887i	0.181521	+	0.990730i
$268$		0			0
$269$	−1.76150			−0.107400			−0.0537002	−	0.998557i	$-0.517102\pi$
$269$	−1.76150			−0.107400			−0.0537002	+	0.998557i	$0.517102\pi$
$270$		0			0
$271$			12.6296i			0.767196i	0.923500	+	0.383598i	$0.125315\pi$
$271$			12.6296i			0.767196i	−0.923500	+	0.383598i	$0.874685\pi$
$272$		0			0
$273$	4.96268	+	11.5508i	0.300355	+	0.699084i
$274$		0			0
$275$		−	5.00000i		−	0.301511i
$276$		0			0
$277$	−14.1116			−0.847885			−0.423942	−	0.905689i	$-0.639354\pi$
$277$	−14.1116			−0.847885			−0.423942	+	0.905689i	$0.639354\pi$
$278$		0			0
$279$	−5.87799	−	15.5024i	−0.351906	−	0.928106i
$280$		0			0
$281$		−	7.25339i		−	0.432701i	−0.976316	−	0.216350i	$-0.930585\pi$
$281$		−	7.25339i		−	0.432701i	0.976316	−	0.216350i	$-0.0694154\pi$
$282$		0			0
$283$			29.3161i			1.74266i	0.490698	+	0.871330i	$0.336742\pi$
$283$			29.3161i			1.74266i	−0.490698	+	0.871330i	$0.663258\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.51860	−	28.5353i	−0.384781	−	1.68439i
$288$		0			0
$289$	−8.00105			−0.470650
$290$		0			0
$291$	12.4841	−	2.28732i	0.731830	−	0.134085i
$292$		0			0
$293$	17.7900			1.03930			0.519650	−	0.854379i	$-0.326062\pi$
$293$	17.7900			1.03930			0.519650	+	0.854379i	$0.326062\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.44707	−	2.68768i	0.258045	−	0.155955i
$298$		0			0
$299$	1.18465			0.0685100
$300$		0			0
$301$	5.94774	+	26.0364i	0.342822	+	1.50071i
$302$		0			0
$303$	3.49543	+	19.0779i	0.200807	+	1.09600i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	17.9730i		−	1.02577i	−0.858456	−	0.512887i	$-0.828576\pi$
$307$		−	17.9730i		−	1.02577i	0.858456	−	0.512887i	$-0.171424\pi$
$308$		0			0
$309$	12.7960	−	2.34447i	0.727938	−	0.133372i
$310$		0			0
$311$	−10.1590			−0.576066			−0.288033	−	0.957620i	$-0.593001\pi$
$311$	−10.1590			−0.576066			−0.288033	+	0.957620i	$0.593001\pi$
$312$		0			0
$313$			14.5554i			0.822719i	0.911473	+	0.411359i	$0.134946\pi$
$313$			14.5554i			0.822719i	−0.911473	+	0.411359i	$0.865054\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			14.0092i			0.786835i	0.919360	+	0.393417i	$0.128707\pi$
$317$			14.0092i			0.786835i	−0.919360	+	0.393417i	$0.871293\pi$
$318$		0			0
$319$	5.61025			0.314114
$320$		0			0
$321$	0.433481	−	0.0794219i	0.0241945	−	0.00443290i
$322$		0			0
$323$		−	4.96816i		−	0.276436i
$324$		0			0
$325$		−	13.7169i		−	0.760875i
$326$		0			0
$327$	0.568926	+	3.10517i	0.0314617	+	0.171716i
$328$		0			0
$329$	−0.367846	−	1.61025i	−0.0202800	−	0.0887761i
$330$		0			0
$331$	18.4410			1.01361			0.506805	−	0.862061i	$-0.330826\pi$
$331$	18.4410			1.01361			0.506805	+	0.862061i	$0.330826\pi$
$332$		0			0
$333$	22.3488	−	8.47390i	1.22471	−	0.464367i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	23.3067			1.26960			0.634800	−	0.772677i	$-0.281083\pi$
$337$	23.3067			1.26960			0.634800	+	0.772677i	$0.281083\pi$
$338$		0			0
$339$	29.9464	−	5.48675i	1.62647	−	0.297999i
$340$		0			0
$341$	−5.52646			−0.299275
$342$		0			0
$343$	11.5553	+	14.4733i	0.623927	+	0.781483i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			34.9149i			1.87433i	0.348886	+	0.937165i	$0.386560\pi$
$347$			34.9149i			1.87433i	−0.348886	+	0.937165i	$0.613440\pi$
$348$		0			0
$349$		−	10.3733i		−	0.555267i	−0.960687	−	0.277634i	$-0.910450\pi$
$349$		−	10.3733i		−	0.555267i	0.960687	−	0.277634i	$-0.0895502\pi$
$350$		0			0
$351$	12.2000	−	7.37330i	0.651186	−	0.393558i
$352$		0			0
$353$	−10.8628			−0.578166			−0.289083	−	0.957304i	$-0.593350\pi$
$353$	−10.8628			−0.578166			−0.289083	+	0.957304i	$0.593350\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−12.6305	+	5.42659i	−0.668478	+	0.287206i
$358$		0			0
$359$			0.898619i			0.0474273i	0.999719	+	0.0237137i	$0.00754900\pi$
$359$			0.898619i			0.0474273i	−0.999719	+	0.0237137i	$0.992451\pi$
$360$		0			0
$361$	16.2572			0.855640
$362$		0			0
$363$	−0.312148	−	1.70369i	−0.0163835	−	0.0894206i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	16.5205i		−	0.862362i	−0.902265	−	0.431181i	$-0.858097\pi$
$367$		−	16.5205i		−	0.862362i	0.902265	−	0.431181i	$-0.141903\pi$
$368$		0			0
$369$	−31.0336	+	11.7669i	−1.61555	+	0.612560i
$370$		0			0
$371$	−16.9178	+	3.86469i	−0.878327	+	0.200645i
$372$		0			0
$373$	16.5342			0.856111			0.428055	−	0.903752i	$-0.359199\pi$
$373$	16.5342			0.856111			0.428055	+	0.903752i	$0.359199\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15.3910			0.792678
$378$		0			0
$379$	−13.3725			−0.686900			−0.343450	−	0.939171i	$-0.611596\pi$
$379$	−13.3725			−0.686900			−0.343450	+	0.939171i	$0.611596\pi$
$380$		0			0
$381$	2.20876	+	12.0553i	0.113158	+	0.617613i
$382$		0			0
$383$	32.0421			1.63727			0.818637	−	0.574311i	$-0.194730\pi$
$383$	32.0421			1.63727			0.818637	+	0.574311i	$0.194730\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	28.3159	−	10.7364i	1.43938	−	0.545763i
$388$		0			0
$389$		−	20.2270i		−	1.02555i	−0.858524	−	0.512774i	$-0.828618\pi$
$389$		−	20.2270i		−	1.02555i	0.858524	−	0.512774i	$-0.171382\pi$
$390$		0			0
$391$			1.29539i			0.0655107i
$392$		0			0
$393$	−1.29547	−	7.07059i	−0.0653476	−	0.356664i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	21.6375i		−	1.08596i	−0.839747	−	0.542978i	$-0.817297\pi$
$397$		−	21.6375i		−	1.08596i	0.839747	−	0.542978i	$-0.182703\pi$
$398$		0			0
$399$	2.99592	+	6.97309i	0.149984	+	0.349091i
$400$		0			0
$401$			15.1547i			0.756792i	0.925644	+	0.378396i	$0.123524\pi$
$401$			15.1547i			0.756792i	−0.925644	+	0.378396i	$0.876476\pi$
$402$		0			0
$403$	−15.1611			−0.755231
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.96712i		−	0.394915i
$408$		0			0
$409$			22.6748i			1.12119i	0.828088	+	0.560597i	$0.189428\pi$
$409$			22.6748i			1.12119i	−0.828088	+	0.560597i	$0.810572\pi$
$410$		0			0
$411$	1.86263	−	0.341268i	0.0918766	−	0.0168335i
$412$		0			0
$413$	7.48482	+	32.7650i	0.368304	+	1.61226i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	7.07600	−	1.29646i	0.346513	−	0.0634878i
$418$		0			0
$419$	−29.6243			−1.44724			−0.723621	−	0.690197i	$-0.757524\pi$
$419$	−29.6243			−1.44724			−0.723621	+	0.690197i	$0.757524\pi$
$420$		0			0
$421$	−23.9509			−1.16730			−0.583648	−	0.812006i	$-0.698375\pi$
$421$	−23.9509			−1.16730			−0.583648	+	0.812006i	$0.698375\pi$
$422$		0			0
$423$	−1.75123	+	0.664008i	−0.0851478	+	0.0322852i
$424$		0			0
$425$	14.9991			0.727565
$426$		0			0
$427$	−8.39894	+	1.91865i	−0.406453	+	0.0928500i
$428$		0			0
$429$	−0.856340	−	4.67386i	−0.0413445	−	0.225656i
$430$		0			0
$431$		−	9.16924i		−	0.441667i	−0.975312	−	0.220833i	$-0.929122\pi$
$431$		−	9.16924i		−	0.441667i	0.975312	−	0.220833i	$-0.0708777\pi$
$432$		0			0
$433$		−	9.06863i		−	0.435811i	−0.975970	−	0.217905i	$-0.930078\pi$
$433$		−	9.06863i		−	0.435811i	0.975970	−	0.217905i	$-0.0699224\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.715162			0.0342108
$438$		0			0
$439$			17.0785i			0.815112i	0.913180	+	0.407556i	$0.133619\pi$
$439$			17.0785i			0.815112i	−0.913180	+	0.407556i	$0.866381\pi$
$440$		0			0
$441$	14.4553	−	15.2330i	0.688346	−	0.725382i
$442$		0			0
$443$		−	7.07059i		−	0.335934i	−0.985793	−	0.167967i	$-0.946280\pi$
$443$		−	7.07059i		−	0.335934i	0.985793	−	0.167967i	$-0.0537202\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.19222	−	1.31775i	0.340180	−	0.0623275i
$448$		0			0
$449$		−	16.0594i		−	0.757888i	−0.925420	−	0.378944i	$-0.876287\pi$
$449$		−	16.0594i		−	0.757888i	0.925420	−	0.378944i	$-0.123713\pi$
$450$		0			0
$451$			11.0632i			0.520945i
$452$		0			0
$453$	−5.86233	−	31.9963i	−0.275436	−	1.50332i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.7294			−0.642233			−0.321117	−	0.947040i	$-0.604058\pi$
$457$	−13.7294			−0.642233			−0.321117	+	0.947040i	$0.604058\pi$
$458$		0			0
$459$	8.06256	+	13.3404i	0.376328	+	0.622678i
$460$		0			0
$461$	−4.57338			−0.213003			−0.106502	−	0.994313i	$-0.533965\pi$
$461$	−4.57338			−0.213003			−0.106502	+	0.994313i	$0.533965\pi$
$462$		0			0
$463$	−20.7137			−0.962648			−0.481324	−	0.876543i	$-0.659844\pi$
$463$	−20.7137			−0.962648			−0.481324	+	0.876543i	$0.659844\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−11.2111			−0.518788			−0.259394	−	0.965772i	$-0.583523\pi$
$467$	−11.2111			−0.518788			−0.259394	+	0.965772i	$0.583523\pi$
$468$		0			0
$469$	3.74556	+	16.3963i	0.172954	+	0.757111i
$470$		0			0
$471$	−10.9661	+	2.00919i	−0.505290	+	0.0925787i
$472$		0			0
$473$		−	10.0943i		−	0.464138i
$474$		0			0
$475$		−	8.28075i		−	0.379947i
$476$		0			0
$477$	6.97625	+	18.3989i	0.319421	+	0.842430i
$478$		0			0
$479$	−14.9889			−0.684859			−0.342429	−	0.939544i	$-0.611250\pi$
$479$	−14.9889			−0.684859			−0.342429	+	0.939544i	$0.611250\pi$
$480$		0			0
$481$		−	21.8568i		−	0.996584i
$482$		0			0
$483$	−0.781152	−	1.81815i	−0.0355437	−	0.0827288i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	25.7660			1.16757			0.583785	−	0.811908i	$-0.301571\pi$
$487$	25.7660			1.16757			0.583785	+	0.811908i	$0.301571\pi$
$488$		0			0
$489$	−1.66155	−	9.06863i	−0.0751377	−	0.410098i
$490$		0			0
$491$		−	29.1219i		−	1.31425i	−0.753781	−	0.657125i	$-0.771772\pi$
$491$		−	29.1219i		−	1.31425i	0.753781	−	0.657125i	$-0.228228\pi$
$492$		0			0
$493$			16.8298i			0.757975i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	43.0390	−	9.83181i	1.93056	−	0.441017i
$498$		0			0
$499$	19.4749			0.871818			0.435909	−	0.899991i	$-0.356427\pi$
$499$	19.4749			0.871818			0.435909	+	0.899991i	$0.356427\pi$
$500$		0			0
$501$	7.13636	+	38.9499i	0.318829	+	1.74015i
$502$		0			0
$503$	−10.7507			−0.479350			−0.239675	−	0.970853i	$-0.577041\pi$
$503$	−10.7507			−0.479350			−0.239675	+	0.970853i	$0.577041\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.70867	+	9.32583i	0.0758846	+	0.414175i
$508$		0			0
$509$	−22.8415			−1.01243			−0.506217	−	0.862406i	$-0.668956\pi$
$509$	−22.8415			−1.01243			−0.506217	+	0.862406i	$0.668956\pi$
$510$		0			0
$511$	29.7715	−	6.80098i	1.31701	−	0.300858i
$512$		0			0
$513$	7.36501	−	4.45120i	0.325173	−	0.196525i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			0.624297i			0.0274565i
$518$		0			0
$519$	0.114822	+	0.626695i	0.00504014	+	0.0275089i
$520$		0			0
$521$	27.6599			1.21180			0.605901	−	0.795540i	$-0.292813\pi$
$521$	27.6599			1.21180			0.605901	+	0.795540i	$0.292813\pi$
$522$		0			0
$523$		−	24.2281i		−	1.05942i	−0.848179	−	0.529710i	$-0.822301\pi$
$523$		−	24.2281i		−	1.05942i	0.848179	−	0.529710i	$-0.177699\pi$
$524$		0			0
$525$	−21.0521	+	9.04484i	−0.918789	+	0.394749i
$526$		0			0
$527$		−	16.5784i		−	0.722167i
$528$		0			0
$529$	22.8135			0.991893
$530$		0			0
$531$	35.6336	−	13.5110i	1.54637	−	0.586330i
$532$		0			0
$533$			30.3505i			1.31462i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	18.0784	−	3.31230i	0.780140	−	0.142936i
$538$		0			0
$539$	−3.03954	−	6.30565i	−0.130922	−	0.271604i
$540$		0			0
$541$	10.5363			0.452993			0.226496	−	0.974012i	$-0.427273\pi$
$541$	10.5363			0.452993			0.226496	+	0.974012i	$0.427273\pi$
$542$		0			0
$543$	8.37956	−	1.53529i	0.359601	−	0.0658858i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.6103			0.581932			0.290966	−	0.956733i	$-0.406023\pi$
$547$	13.6103			0.581932			0.290966	+	0.956733i	$0.406023\pi$
$548$		0			0
$549$	3.46341	+	9.13427i	0.147815	+	0.389841i
$550$		0			0
$551$	9.29142			0.395828
$552$		0			0
$553$	7.17034	+	31.3883i	0.304914	+	1.33477i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			21.2555i			0.900624i	0.892871	+	0.450312i	$0.148687\pi$
$557$			21.2555i			0.900624i	−0.892871	+	0.450312i	$0.851313\pi$
$558$		0			0
$559$		−	27.6925i		−	1.17127i
$560$		0			0
$561$	5.11078	−	0.936391i	0.215777	−	0.0395345i
$562$		0			0
$563$	−17.7329			−0.747352			−0.373676	−	0.927559i	$-0.621903\pi$
$563$	−17.7329			−0.747352			−0.373676	+	0.927559i	$0.621903\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−19.3608	−	13.8621i	−0.813079	−	0.582153i
$568$		0			0
$569$			30.2033i			1.26619i	0.774075	+	0.633094i	$0.218215\pi$
$569$			30.2033i			1.26619i	−0.774075	+	0.633094i	$0.781785\pi$
$570$		0			0
$571$	−16.5106			−0.690945			−0.345473	−	0.938429i	$-0.612281\pi$
$571$	−16.5106			−0.690945			−0.345473	+	0.938429i	$0.612281\pi$
$572$		0			0
$573$	13.3273	−	2.44182i	0.556757	−	0.102008i
$574$		0			0
$575$			2.15911i			0.0900411i
$576$		0			0
$577$			15.9039i			0.662089i	0.943615	+	0.331045i	$0.107401\pi$
$577$			15.9039i			0.662089i	−0.943615	+	0.331045i	$0.892599\pi$
$578$		0			0
$579$	−2.09568	−	11.4381i	−0.0870935	−	0.475352i
$580$		0			0
$581$	9.58154	+	41.9434i	0.397509	+	1.74011i
$582$		0			0
$583$	6.55904			0.271648
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.32867			−0.0548400			−0.0274200	−	0.999624i	$-0.508729\pi$
$587$	−1.32867			−0.0548400			−0.0274200	+	0.999624i	$0.508729\pi$
$588$		0			0
$589$	−9.15265			−0.377128
$590$		0			0
$591$	−39.9380	+	7.31740i	−1.64283	+	0.300998i
$592$		0			0
$593$	8.40715			0.345240			0.172620	−	0.984988i	$-0.444777\pi$
$593$	8.40715			0.345240			0.172620	+	0.984988i	$0.444777\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−34.7631	+	6.36926i	−1.42276	+	0.260676i
$598$		0			0
$599$			4.63668i			0.189449i	0.995504	+	0.0947247i	$0.0301971\pi$
$599$			4.63668i			0.189449i	−0.995504	+	0.0947247i	$0.969803\pi$
$600$		0			0
$601$		−	17.9705i		−	0.733032i	−0.930412	−	0.366516i	$-0.880550\pi$
$601$		−	17.9705i		−	0.733032i	0.930412	−	0.366516i	$-0.119450\pi$
$602$		0			0
$603$	17.8318	−	6.76122i	0.726167	−	0.275338i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			17.7034i			0.718561i	0.933230	+	0.359280i	$0.116978\pi$
$607$			17.7034i			0.718561i	−0.933230	+	0.359280i	$0.883022\pi$
$608$		0			0
$609$	−10.1488	−	23.6215i	−0.411249	−	0.957192i
$610$		0			0
$611$			1.71268i			0.0692876i
$612$		0			0
$613$	0.179477			0.00724902			0.00362451	−	0.999993i	$-0.498846\pi$
$613$	0.179477			0.00724902			0.00362451	+	0.999993i	$0.498846\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	37.8070i		−	1.52205i	−0.648721	−	0.761027i	$-0.724696\pi$
$617$		−	37.8070i		−	1.52205i	0.648721	−	0.761027i	$-0.275304\pi$
$618$		0			0
$619$		−	0.364667i		−	0.0146572i	−0.999973	−	0.00732860i	$-0.997667\pi$
$619$		−	0.364667i		−	0.0146572i	0.999973	−	0.00732860i	$-0.00233279\pi$
$620$		0			0
$621$	−1.92034	+	1.16060i	−0.0770606	+	0.0465732i
$622$		0			0
$623$	−5.59880	−	24.5089i	−0.224311	−	0.981927i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	−0.516965	−	2.82157i	−0.0206456	−	0.112683i
$628$		0			0
$629$	23.9000			0.952954
$630$		0			0
$631$	16.7979			0.668713			0.334356	−	0.942447i	$-0.391481\pi$
$631$	16.7979			0.668713			0.334356	+	0.942447i	$0.391481\pi$
$632$		0			0
$633$	5.98939	+	32.6898i	0.238057	+	1.29930i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−8.33859	−	17.2988i	−0.330387	−	0.685402i
$638$		0			0
$639$	−17.7477	−	46.8071i	−0.702086	−	1.85166i
$640$		0			0
$641$		−	17.8912i		−	0.706659i	−0.935499	−	0.353329i	$-0.885050\pi$
$641$		−	17.8912i		−	0.706659i	0.935499	−	0.353329i	$-0.114950\pi$
$642$		0			0
$643$		−	20.2372i		−	0.798076i	−0.916934	−	0.399038i	$-0.869344\pi$
$643$		−	20.2372i		−	0.798076i	0.916934	−	0.399038i	$-0.130656\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−41.9464			−1.64908			−0.824541	−	0.565802i	$-0.808566\pi$
$647$	−41.9464			−1.64908			−0.824541	+	0.565802i	$0.808566\pi$
$648$		0			0
$649$		−	12.7030i		−	0.498637i
$650$		0			0
$651$	9.99719	+	23.2687i	0.391821	+	0.911973i
$652$		0			0
$653$			11.2134i			0.438815i	0.975633	+	0.219407i	$0.0704123\pi$
$653$			11.2134i			0.438815i	−0.975633	+	0.219407i	$0.929588\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−12.2766	−	32.3780i	−0.478957	−	1.26318i
$658$		0			0
$659$			27.2555i			1.06172i	0.847459	+	0.530861i	$0.178132\pi$
$659$			27.2555i			1.06172i	−0.847459	+	0.530861i	$0.821868\pi$
$660$		0			0
$661$		−	16.4254i		−	0.638874i	−0.947607	−	0.319437i	$-0.896506\pi$
$661$		−	16.4254i		−	0.638874i	0.947607	−	0.319437i	$-0.103494\pi$
$662$		0			0
$663$	14.0208	−	2.56887i	0.544522	−	0.0997667i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.42263			−0.0938046
$668$		0			0
$669$	1.45636	−	0.266832i	0.0563060	−	0.0103163i
$670$		0			0
$671$	3.25628			0.125707
$672$		0			0
$673$	−22.2890			−0.859178			−0.429589	−	0.903025i	$-0.641342\pi$
$673$	−22.2890			−0.859178			−0.429589	+	0.903025i	$0.641342\pi$
$674$		0			0
$675$	13.4384	+	22.2353i	0.517244	+	0.855838i
$676$		0			0
$677$	−31.1173			−1.19593			−0.597967	−	0.801521i	$-0.704025\pi$
$677$	−31.1173			−1.19593			−0.597967	+	0.801521i	$0.704025\pi$
$678$		0			0
$679$	−18.9003	+	4.31758i	−0.725327	+	0.165693i
$680$		0			0
$681$	6.78868	+	37.0523i	0.260143	+	1.41985i
$682$		0			0
$683$		−	37.2572i		−	1.42561i	−0.701364	−	0.712803i	$-0.747425\pi$
$683$		−	37.2572i		−	1.42561i	0.701364	−	0.712803i	$-0.252575\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−36.6728	+	6.71914i	−1.39915	+	0.256351i
$688$		0			0
$689$	17.9939			0.685513
$690$		0			0
$691$		−	2.75111i		−	0.104657i	−0.998630	−	0.0523286i	$-0.983336\pi$
$691$		−	2.75111i		−	0.104657i	0.998630	−	0.0523286i	$-0.0166643\pi$
$692$		0			0
$693$	−6.60859	+	4.39620i	−0.251039	+	0.166998i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−33.1876			−1.25707
$698$		0			0
$699$	19.6058	−	3.59215i	0.741559	−	0.135868i
$700$		0			0
$701$			36.5106i			1.37898i	0.724293	+	0.689492i	$0.242166\pi$
$701$			36.5106i			1.37898i	−0.724293	+	0.689492i	$0.757834\pi$
$702$		0			0
$703$		−	13.1947i		−	0.497649i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.59802	−	28.8830i	−0.248144	−	1.08626i
$708$		0			0
$709$	40.0862			1.50547			0.752735	−	0.658323i	$-0.228734\pi$
$709$	40.0862			1.50547			0.752735	+	0.658323i	$0.228734\pi$
$710$		0			0
$711$	34.1364	−	12.9434i	1.28022	−	0.485414i
$712$		0			0
$713$	2.38645			0.0893731
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−31.1624	+	5.70954i	−1.16378	+	0.213227i
$718$		0			0
$719$	43.1623			1.60968			0.804842	−	0.593490i	$-0.202250\pi$
$719$	43.1623			1.60968			0.804842	+	0.593490i	$0.202250\pi$
$720$		0			0
$721$	−19.3725	+	4.42545i	−0.721470	+	0.164812i
$722$		0			0
$723$	−20.9978	+	3.84720i	−0.780918	+	0.143079i
$724$		0			0
$725$			28.0513i			1.04180i
$726$		0			0
$727$		−	14.6624i		−	0.543799i	−0.962326	−	0.271900i	$-0.912348\pi$
$727$		−	14.6624i		−	0.543799i	0.962326	−	0.271900i	$-0.0876518\pi$
$728$		0			0
$729$	−12.5528	+	23.9046i	−0.464918	+	0.885354i
$730$		0			0
$731$	30.2812			1.11999
$732$		0			0
$733$		−	5.75346i		−	0.212509i	−0.994339	−	0.106255i	$-0.966114\pi$
$733$		−	5.75346i		−	0.212509i	0.994339	−	0.106255i	$-0.0338858\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.35686i		−	0.234158i
$738$		0			0
$739$	−8.28732			−0.304854			−0.152427	−	0.988315i	$-0.548709\pi$
$739$	−8.28732			−0.304854			−0.152427	+	0.988315i	$0.548709\pi$
$740$		0			0
$741$	−1.41823	−	7.74062i	−0.0520999	−	0.284359i
$742$		0			0
$743$			12.7137i			0.466421i	0.972426	+	0.233211i	$0.0749232\pi$
$743$			12.7137i			0.466421i	−0.972426	+	0.233211i	$0.925077\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	45.6156	−	17.2959i	1.66899	−	0.632823i
$748$		0			0
$749$	−0.656269	+	0.149918i	−0.0239796	+	0.00547788i
$750$		0			0
$751$	50.9845			1.86045			0.930225	−	0.366991i	$-0.119612\pi$
$751$	50.9845			1.86045			0.930225	+	0.366991i	$0.119612\pi$
$752$		0			0
$753$	−3.98361	−	21.7424i	−0.145171	−	0.792336i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−13.8297			−0.502650			−0.251325	−	0.967903i	$-0.580866\pi$
$757$	−13.8297			−0.502650			−0.251325	+	0.967903i	$0.580866\pi$
$758$		0			0
$759$	0.134793	+	0.735691i	0.00489266	+	0.0267039i
$760$		0			0
$761$	−7.97367			−0.289045			−0.144523	−	0.989501i	$-0.546165\pi$
$761$	−7.97367			−0.289045			−0.144523	+	0.989501i	$0.546165\pi$
$762$		0			0
$763$	−1.07391	−	4.70108i	−0.0388783	−	0.170191i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	34.8491i		−	1.25833i
$768$		0			0
$769$		−	5.27319i		−	0.190156i	−0.995470	−	0.0950780i	$-0.969690\pi$
$769$		−	5.27319i		−	0.190156i	0.995470	−	0.0950780i	$-0.0303101\pi$
$770$		0			0
$771$	−7.13636	−	38.9499i	−0.257010	−	1.40275i
$772$		0			0
$773$	2.20707			0.0793829			0.0396915	−	0.999212i	$-0.487362\pi$
$773$	2.20707			0.0793829			0.0396915	+	0.999212i	$0.487362\pi$
$774$		0			0
$775$		−	27.6323i		−	0.992581i
$776$		0			0
$777$	−33.5449	+	14.4123i	−1.20342	+	0.517037i
$778$		0			0
$779$			18.3223i			0.656464i
$780$		0			0
$781$	−16.6863			−0.597081
$782$		0			0
$783$	−24.9492	+	15.0786i	−0.891610	+	0.538863i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			34.0792i			1.21479i	0.794399	+	0.607396i	$0.207786\pi$
$787$			34.0792i			1.21479i	−0.794399	+	0.607396i	$0.792214\pi$
$788$		0			0
$789$	−44.0094	+	8.06336i	−1.56678	+	0.287063i
$790$		0			0
$791$	−45.3374	+	10.3569i	−1.61201	+	0.368248i
$792$		0			0
$793$	8.93319			0.317227
$794$		0			0
$795$		0			0
$796$		0			0
$797$	28.5716			1.01206			0.506029	−	0.862517i	$-0.331113\pi$
$797$	28.5716			1.01206			0.506029	+	0.862517i	$0.331113\pi$
$798$		0			0
$799$	−1.87278			−0.0662543
$800$		0			0
$801$	−26.6546	+	10.1065i	−0.941795	+	0.357097i
$802$		0			0
$803$	−11.5424			−0.407323
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.549848	−	3.00105i	−0.0193556	−	0.105642i
$808$		0			0
$809$			28.4264i			0.999419i	0.866193	+	0.499710i	$0.166560\pi$
$809$			28.4264i			0.999419i	−0.866193	+	0.499710i	$0.833440\pi$
$810$		0			0
$811$			2.87212i			0.100854i	0.998728	+	0.0504269i	$0.0160582\pi$
$811$			2.87212i			0.100854i	−0.998728	+	0.0504269i	$0.983942\pi$
$812$		0			0
$813$	−21.5170	+	3.94232i	−0.754635	+	0.138263i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	16.7177i		−	0.584880i
$818$		0			0
$819$	−18.1298	+	12.0604i	−0.633508	+	0.421425i
$820$		0			0
$821$		−	48.8803i		−	1.70594i	−0.521963	−	0.852968i	$-0.674800\pi$
$821$		−	48.8803i		−	1.70594i	0.521963	−	0.852968i	$-0.325200\pi$
$822$		0			0
$823$	−52.1725			−1.81862			−0.909309	−	0.416121i	$-0.863389\pi$
$823$	−52.1725			−1.81862			−0.909309	+	0.416121i	$0.863389\pi$
$824$		0			0
$825$	8.51846	−	1.56074i	0.296575	−	0.0543381i
$826$		0			0
$827$			0.678751i			0.0236025i	0.999930	+	0.0118012i	$0.00375654\pi$
$827$			0.678751i			0.0236025i	−0.999930	+	0.0118012i	$0.996243\pi$
$828$		0			0
$829$			20.6117i			0.715874i	0.933746	+	0.357937i	$0.116520\pi$
$829$			20.6117i			0.715874i	−0.933746	+	0.357937i	$0.883480\pi$
$830$		0			0
$831$	−4.40492	−	24.0418i	−0.152805	−	0.834002i
$832$		0			0
$833$	18.9158	−	9.11808i	0.655395	−	0.315923i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	24.5765	−	14.8533i	0.849489	−	0.513407i
$838$		0			0
$839$	2.54969			0.0880250			0.0440125	−	0.999031i	$-0.485986\pi$
$839$	2.54969			0.0880250			0.0440125	+	0.999031i	$0.485986\pi$
$840$		0			0
$841$	−2.47494			−0.0853429
$842$		0			0
$843$	12.3575	−	2.26413i	0.425616	−	0.0779809i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	0.589216	+	2.57931i	0.0202457	+	0.0886260i
$848$		0			0
$849$	−49.9455	+	9.15096i	−1.71413	+	0.314060i
$850$		0			0
$851$			3.44038i			0.117935i
$852$		0			0
$853$		−	23.5367i		−	0.805880i	−0.915226	−	0.402940i	$-0.867988\pi$
$853$		−	23.5367i		−	0.805880i	0.915226	−	0.402940i	$-0.132012\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−26.4216			−0.902544			−0.451272	−	0.892386i	$-0.649030\pi$
$857$	−26.4216			−0.902544			−0.451272	+	0.892386i	$0.649030\pi$
$858$		0			0
$859$			41.6137i			1.41984i	0.704282	+	0.709920i	$0.251269\pi$
$859$			41.6137i			1.41984i	−0.704282	+	0.709920i	$0.748731\pi$
$860$		0			0
$861$	46.5806	−	20.0129i	1.58746	−	0.682039i
$862$		0			0
$863$			8.34767i			0.284158i	0.989855	+	0.142079i	$0.0453787\pi$
$863$			8.34767i			0.284158i	−0.989855	+	0.142079i	$0.954621\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−2.49751	−	13.6313i	−0.0848200	−	0.462944i
$868$		0			0
$869$		−	12.1693i		−	0.412815i
$870$		0			0
$871$		−	17.4393i		−	0.590907i
$872$		0			0
$873$	7.79377	+	20.5550i	0.263779	+	0.695683i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−20.4027			−0.688951			−0.344475	−	0.938795i	$-0.611943\pi$
$877$	−20.4027			−0.688951			−0.344475	+	0.938795i	$0.611943\pi$
$878$		0			0
$879$	5.55311	+	30.3086i	0.187302	+	1.02228i
$880$		0			0
$881$	3.32647			0.112072			0.0560358	−	0.998429i	$-0.482154\pi$
$881$	3.32647			0.112072			0.0560358	+	0.998429i	$0.482154\pi$
$882$		0			0
$883$	−22.7612			−0.765976			−0.382988	−	0.923753i	$-0.625105\pi$
$883$	−22.7612			−0.765976			−0.382988	+	0.923753i	$0.625105\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	35.6233			1.19611			0.598057	−	0.801454i	$-0.295940\pi$
$887$	35.6233			1.19611			0.598057	+	0.801454i	$0.295940\pi$
$888$		0			0
$889$	−4.16929	−	18.2512i	−0.139834	−	0.612125i
$890$		0			0
$891$	5.96712	+	6.73747i	0.199906	+	0.225714i
$892$		0			0
$893$			1.03393i			0.0345991i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	0.369786	+	2.01828i	0.0123468	+	0.0673883i
$898$		0			0
$899$	31.0048			1.03407
$900$		0			0
$901$			19.6760i			0.655502i
$902$		0			0
$903$	−42.5014	+	18.2603i	−1.41436	+	0.607666i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−49.5774			−1.64619			−0.823095	−	0.567904i	$-0.807754\pi$
$907$	−49.5774			−1.64619			−0.823095	+	0.567904i	$0.807754\pi$
$908$		0			0
$909$	−31.4117	+	11.9103i	−1.04186	+	0.395039i
$910$		0			0
$911$		−	42.7067i		−	1.41494i	−0.706745	−	0.707469i	$-0.749837\pi$
$911$		−	42.7067i		−	1.41494i	0.706745	−	0.707469i	$-0.250163\pi$
$912$		0			0
$913$		−	16.2615i		−	0.538177i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.44534	+	10.7045i	0.0807522	+	0.353495i
$918$		0			0
$919$	13.3487			0.440334			0.220167	−	0.975462i	$-0.429340\pi$
$919$	13.3487			0.440334			0.220167	+	0.975462i	$0.429340\pi$
$920$		0			0
$921$	30.6205	−	5.61025i	1.00898	−	0.184864i
$922$		0			0
$923$	−45.7767			−1.50676
$924$		0			0
$925$	39.8356			1.30979
$926$		0			0
$927$	7.98850	+	21.0686i	0.262377	+	0.691983i
$928$		0			0
$929$	−49.4962			−1.62392			−0.811958	−	0.583715i	$-0.801598\pi$
$929$	−49.4962			−1.62392			−0.811958	+	0.583715i	$0.801598\pi$
$930$		0			0
$931$	−5.03393	−	10.4431i	−0.164980	−	0.342259i
$932$		0			0
$933$	−3.17113	−	17.3079i	−0.103818	−	0.566634i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	46.5889i		−	1.52199i	−0.648756	−	0.760996i	$-0.724710\pi$
$937$		−	46.5889i		−	1.52199i	0.648756	−	0.760996i	$-0.275290\pi$
$938$		0			0
$939$	−24.7979	+	4.54344i	−0.809248	+	0.148270i
$940$		0			0
$941$	0.833952			0.0271860			0.0135930	−	0.999908i	$-0.495673\pi$
$941$	0.833952			0.0271860			0.0135930	+	0.999908i	$0.495673\pi$
$942$		0			0
$943$		−	4.77733i		−	0.155571i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	21.6119i		−	0.702294i	−0.936320	−	0.351147i	$-0.885792\pi$
$947$		−	21.6119i		−	0.702294i	0.936320	−	0.351147i	$-0.114208\pi$
$948$		0			0
$949$	−31.6652			−1.02790
$950$		0			0
$951$	−23.8673	+	4.37295i	−0.773951	+	0.141803i
$952$		0			0
$953$		−	52.7101i		−	1.70745i	−0.520727	−	0.853723i	$-0.674339\pi$
$953$		−	52.7101i		−	1.70745i	0.520727	−	0.853723i	$-0.325661\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	1.75123	+	9.55814i	0.0566093	+	0.308971i
$958$		0			0
$959$	−2.81993	+	0.644183i	−0.0910602	+	0.0208018i
$960$		0			0
$961$	0.458244			0.0147821
$962$		0			0
$963$	0.270621	+	0.713726i	0.00872063	+	0.0229995i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−45.5967			−1.46629			−0.733146	−	0.680071i	$-0.761949\pi$
$967$	−45.5967			−1.46629			−0.733146	+	0.680071i	$0.761949\pi$
$968$		0			0
$969$	8.46421	−	1.55080i	0.271910	−	0.0498190i
$970$		0			0
$971$	−18.1692			−0.583079			−0.291539	−	0.956559i	$-0.594167\pi$
$971$	−18.1692			−0.583079			−0.291539	+	0.956559i	$0.594167\pi$
$972$		0			0
$973$	−10.7127	+	2.44721i	−0.343434	+	0.0784540i
$974$		0			0
$975$	23.3693	−	4.28170i	0.748417	−	0.137124i
$976$		0			0
$977$			32.0184i			1.02436i	0.858878	+	0.512179i	$0.171162\pi$
$977$			32.0184i			1.02436i	−0.858878	+	0.512179i	$0.828838\pi$
$978$		0			0
$979$			9.50212i			0.303689i
$980$		0			0
$981$	−5.11266	+	1.93855i	−0.163235	+	0.0618931i
$982$		0			0
$983$	−1.05778			−0.0337379			−0.0168689	−	0.999858i	$-0.505370\pi$
$983$	−1.05778			−0.0337379			−0.0168689	+	0.999858i	$0.505370\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	2.62855	−	1.12933i	0.0836677	−	0.0359471i
$988$		0			0
$989$			4.35895i			0.138607i
$990$		0			0
$991$	−46.6113			−1.48066			−0.740328	−	0.672245i	$-0.765330\pi$
$991$	−46.6113			−1.48066			−0.740328	+	0.672245i	$0.765330\pi$
$992$		0			0
$993$	5.75633	+	31.4178i	0.182672	+	0.997013i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			19.1134i			0.605328i	0.953097	+	0.302664i	$0.0978760\pi$
$997$			19.1134i			0.605328i	−0.953097	+	0.302664i	$0.902124\pi$
$998$		0			0
$999$	21.4130	+	35.4303i	0.677479	+	1.12096i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1848.2.v.c.881.10	yes	16
3.2	odd	2	inner	1848.2.v.c.881.8	yes	16
7.6	odd	2	inner	1848.2.v.c.881.7	✓	16
21.20	even	2	inner	1848.2.v.c.881.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1848.2.v.c.881.7	✓	16	7.6	odd	2	inner
1848.2.v.c.881.8	yes	16	3.2	odd	2	inner
1848.2.v.c.881.9	yes	16	21.20	even	2	inner
1848.2.v.c.881.10	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1848.v (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.312148 + 1.70369i) q^{3} +(-0.589216 - 2.57931i) q^{7} +(-2.80513 + 1.06361i) q^{9} +1.00000i q^{11} +2.74337i q^{13} -2.99983 q^{17} +1.65615i q^{19} +(4.21042 - 1.80897i) q^{21} -0.431822i q^{23} -5.00000 q^{25} +(-2.68768 - 4.44707i) q^{27} -5.61025i q^{29} +5.52646i q^{31} +(-1.70369 + 0.312148i) q^{33} -7.96712 q^{37} +(-4.67386 + 0.856340i) q^{39} +11.0632 q^{41} -10.0943 q^{43} +0.624297 q^{47} +(-6.30565 + 3.03954i) q^{49} +(-0.936391 - 5.11078i) q^{51} -6.55904i q^{53} +(-2.82157 + 0.516965i) q^{57} -12.7030 q^{59} -3.25628i q^{61} +(4.39620 + 6.60859i) q^{63} -6.35686 q^{67} +(0.735691 - 0.134793i) q^{69} +16.6863i q^{71} +11.5424i q^{73} +(-1.56074 - 8.51846i) q^{75} +(2.57931 - 0.589216i) q^{77} -12.1693 q^{79} +(6.73747 - 5.96712i) q^{81} -16.2615 q^{83} +(9.55814 - 1.75123i) q^{87} +9.50212 q^{89} +(7.07600 - 1.61644i) q^{91} +(-9.41538 + 1.72508i) q^{93} -7.32767i q^{97} +(-1.06361 - 2.80513i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7} - 8 q^{9} + 20 q^{21} - 80 q^{25} + 24 q^{39} + 16 q^{43} - 24 q^{49} - 40 q^{51} - 72 q^{57} - 12 q^{63} - 48 q^{67} - 24 q^{79} - 16 q^{81} + 40 q^{91} - 40 q^{93} + 8 q^{99}+O(q^{100}) \) 16 * q + 4 * q^7 - 8 * q^9 + 20 * q^21 - 80 * q^25 + 24 * q^39 + 16 * q^43 - 24 * q^49 - 40 * q^51 - 72 * q^57 - 12 * q^63 - 48 * q^67 - 24 * q^79 - 16 * q^81 + 40 * q^91 - 40 * q^93 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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