LMFDB - Embedded newform 4100.2.d.g.1149.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4100,2,Mod(1149,4100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4100.1149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4100 = 2^{2} \cdot 5^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4100.d (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:	$32.7386648287$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 261x^{10} + 1141x^{8} + 2289x^{6} + 1896x^{4} + 549x^{2} + 9 $ x^14 + 27x^12 + 261x^10 + 1141x^8 + 2289x^6 + 1896x^4 + 549x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.6
Root			$0.806545i$ of defining polynomial
Character	$\chi$	$=$	4100.1149
Dual form			4100.2.d.g.1149.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.806545i q^{3} +0.0998195i q^{7} +2.34949 q^{9} +O(q^{10})$	$q-0.806545i q^{3} +0.0998195i q^{7} +2.34949 q^{9} +5.37250 q^{11} +1.73922i q^{13} -4.30316i q^{17} +3.44930 q^{19} +0.0805089 q^{21} +1.32144i q^{23} -4.31460i q^{27} +1.62622 q^{29} -1.63408 q^{31} -4.33316i q^{33} +6.20042i q^{37} +1.40276 q^{39} +1.00000 q^{41} +7.87893i q^{43} -9.86748i q^{47} +6.99004 q^{49} -3.47069 q^{51} -0.388118i q^{53} -2.78202i q^{57} -1.30680 q^{59} +4.02584 q^{61} +0.234524i q^{63} +9.53602i q^{67} +1.06580 q^{69} -14.7282 q^{71} +3.26192i q^{73} +0.536280i q^{77} +7.68351 q^{79} +3.56854 q^{81} +6.27886i q^{83} -1.31162i q^{87} -10.3722 q^{89} -0.173608 q^{91} +1.31796i q^{93} -5.08866i q^{97} +12.6226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 12 q^{9}+O(q^{10}) $ 14 * q - 12 * q^9	$ 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) $ 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$.

$n$	$1477$	$2051$	$3901$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 0.806545i	− 0.465659i	−0.972518	−	0.232829i	$-0.925202\pi$
$3$	− 0.806545i	− 0.465659i	0.972518	−	0.232829i	$-0.0747984\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.0998195i	0.0377282i	0.999822	+	0.0188641i	$0.00600499\pi$
$7$	0.0998195i	0.0377282i	−0.999822	+	0.0188641i	$0.993995\pi$
$8$	0	0
$9$	2.34949	0.783162
$10$	0	0
$11$	5.37250	1.61987	0.809934	−	0.586521i	$-0.199503\pi$
$11$	5.37250	1.61987	0.809934	+	0.586521i	$0.199503\pi$
$12$	0	0
$13$	1.73922i	0.482373i	0.970479	+	0.241187i	$0.0775366\pi$
$13$	1.73922i	0.482373i	−0.970479	+	0.241187i	$0.922463\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.30316i	− 1.04367i	−0.853047	−	0.521835i	$-0.825248\pi$
$17$	− 4.30316i	− 1.04367i	0.853047	−	0.521835i	$-0.174752\pi$
$18$	0	0
$19$	3.44930	0.791325	0.395662	−	0.918396i	$-0.370515\pi$
$19$	3.44930	0.791325	0.395662	+	0.918396i	$0.370515\pi$
$20$	0	0
$21$	0.0805089	0.0175685
$22$	0	0
$23$	1.32144i	0.275539i	0.990464	+	0.137770i	$0.0439934\pi$
$23$	1.32144i	0.275539i	−0.990464	+	0.137770i	$0.956007\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.31460i	− 0.830345i
$28$	0	0
$29$	1.62622	0.301981	0.150990	−	0.988535i	$-0.451754\pi$
$29$	1.62622	0.301981	0.150990	+	0.988535i	$0.451754\pi$
$30$	0	0
$31$	−1.63408	−0.293490	−0.146745	−	0.989174i	$-0.546880\pi$
$31$	−1.63408	−0.293490	−0.146745	+	0.989174i	$0.546880\pi$
$32$	0	0
$33$	− 4.33316i	− 0.754306i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.20042i	1.01934i	0.860369	+	0.509672i	$0.170233\pi$
$37$	6.20042i	1.01934i	−0.860369	+	0.509672i	$0.829767\pi$
$38$	0	0
$39$	1.40276	0.224621
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	7.87893i	1.20152i	0.799428	+	0.600762i	$0.205136\pi$
$43$	7.87893i	1.20152i	−0.799428	+	0.600762i	$0.794864\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.86748i	− 1.43932i	−0.694326	−	0.719660i	$-0.744298\pi$
$47$	− 9.86748i	− 1.43932i	0.694326	−	0.719660i	$-0.255702\pi$
$48$	0	0
$49$	6.99004	0.998577
$50$	0	0
$51$	−3.47069	−0.485994
$52$	0	0
$53$	− 0.388118i	− 0.0533121i	−0.999645	−	0.0266560i	$-0.991514\pi$
$53$	− 0.388118i	− 0.0533121i	0.999645	−	0.0266560i	$-0.00848589\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.78202i	− 0.368487i
$58$	0	0
$59$	−1.30680	−0.170131	−0.0850653	−	0.996375i	$-0.527110\pi$
$59$	−1.30680	−0.170131	−0.0850653	+	0.996375i	$0.527110\pi$
$60$	0	0
$61$	4.02584	0.515455	0.257728	−	0.966218i	$-0.417026\pi$
$61$	4.02584	0.515455	0.257728	+	0.966218i	$0.417026\pi$
$62$	0	0
$63$	0.234524i	0.0295473i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.53602i	1.16501i	0.812827	+	0.582505i	$0.197927\pi$
$67$	9.53602i	1.16501i	−0.812827	+	0.582505i	$0.802073\pi$
$68$	0	0
$69$	1.06580	0.128307
$70$	0	0
$71$	−14.7282	−1.74792	−0.873961	−	0.485997i	$-0.838457\pi$
$71$	−14.7282	−1.74792	−0.873961	+	0.485997i	$0.838457\pi$
$72$	0	0
$73$	3.26192i	0.381779i	0.981612	+	0.190889i	$0.0611372\pi$
$73$	3.26192i	0.381779i	−0.981612	+	0.190889i	$0.938863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.536280i	0.0611148i
$78$	0	0
$79$	7.68351	0.864463	0.432231	−	0.901763i	$-0.357726\pi$
$79$	7.68351	0.864463	0.432231	+	0.901763i	$0.357726\pi$
$80$	0	0
$81$	3.56854	0.396504
$82$	0	0
$83$	6.27886i	0.689194i	0.938751	+	0.344597i	$0.111984\pi$
$83$	6.27886i	0.689194i	−0.938751	+	0.344597i	$0.888016\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.31162i	− 0.140620i
$88$	0	0
$89$	−10.3722	−1.09945	−0.549727	−	0.835344i	$-0.685268\pi$
$89$	−10.3722	−1.09945	−0.549727	+	0.835344i	$0.685268\pi$
$90$	0	0
$91$	−0.173608	−0.0181991
$92$	0	0
$93$	1.31796i	0.136666i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.08866i	− 0.516675i	−0.966055	−	0.258338i	$-0.916825\pi$
$97$	− 5.08866i	− 0.516675i	0.966055	−	0.258338i	$-0.0831747\pi$
$98$	0	0
$99$	12.6226	1.26862
$100$	0	0
$101$	5.32768	0.530124	0.265062	−	0.964231i	$-0.414608\pi$
$101$	5.32768	0.530124	0.265062	+	0.964231i	$0.414608\pi$
$102$	0	0
$103$	− 4.85956i	− 0.478826i	−0.970918	−	0.239413i	$-0.923045\pi$
$103$	− 4.85956i	− 0.478826i	0.970918	−	0.239413i	$-0.0769550\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.257735i	− 0.0249162i	−0.999922	−	0.0124581i	$-0.996034\pi$
$107$	− 0.257735i	− 0.0249162i	0.999922	−	0.0124581i	$-0.00396564\pi$
$108$	0	0
$109$	−8.01559	−0.767754	−0.383877	−	0.923384i	$-0.625411\pi$
$109$	−8.01559	−0.767754	−0.383877	+	0.923384i	$0.625411\pi$
$110$	0	0
$111$	5.00092	0.474666
$112$	0	0
$113$	15.7033i	1.47725i	0.674119	+	0.738623i	$0.264524\pi$
$113$	15.7033i	1.47725i	−0.674119	+	0.738623i	$0.735476\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.08628i	0.377776i
$118$	0	0
$119$	0.429539	0.0393758
$120$	0	0
$121$	17.8637	1.62397
$122$	0	0
$123$	− 0.806545i	− 0.0727237i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13.6041i	− 1.20717i	−0.797298	−	0.603587i	$-0.793738\pi$
$127$	− 13.6041i	− 1.20717i	0.797298	−	0.603587i	$-0.206262\pi$
$128$	0	0
$129$	6.35471	0.559501
$130$	0	0
$131$	10.7735	0.941287	0.470643	−	0.882324i	$-0.344022\pi$
$131$	10.7735	0.941287	0.470643	+	0.882324i	$0.344022\pi$
$132$	0	0
$133$	0.344308i	0.0298553i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.16448i	− 0.441231i	−0.975361	−	0.220616i	$-0.929193\pi$
$137$	− 5.16448i	− 0.441231i	0.975361	−	0.220616i	$-0.0708066\pi$
$138$	0	0
$139$	2.02609	0.171851	0.0859254	−	0.996302i	$-0.472615\pi$
$139$	2.02609	0.171851	0.0859254	+	0.996302i	$0.472615\pi$
$140$	0	0
$141$	−7.95857	−0.670232
$142$	0	0
$143$	9.34396i	0.781381i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 5.63778i	− 0.464996i
$148$	0	0
$149$	−4.83123	−0.395790	−0.197895	−	0.980223i	$-0.563410\pi$
$149$	−4.83123	−0.395790	−0.197895	+	0.980223i	$0.563410\pi$
$150$	0	0
$151$	11.2302	0.913897	0.456948	−	0.889493i	$-0.348942\pi$
$151$	11.2302	0.913897	0.456948	+	0.889493i	$0.348942\pi$
$152$	0	0
$153$	− 10.1102i	− 0.817362i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 15.8583i	− 1.26563i	−0.774304	−	0.632814i	$-0.781900\pi$
$157$	− 15.8583i	− 1.26563i	0.774304	−	0.632814i	$-0.218100\pi$
$158$	0	0
$159$	−0.313034	−0.0248252
$160$	0	0
$161$	−0.131905	−0.0103956
$162$	0	0
$163$	− 5.05063i	− 0.395596i	−0.980243	−	0.197798i	$-0.936621\pi$
$163$	− 5.05063i	− 0.395596i	0.980243	−	0.197798i	$-0.0633790\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.9343i	− 1.31042i	−0.755448	−	0.655209i	$-0.772581\pi$
$167$	− 16.9343i	− 1.31042i	0.755448	−	0.655209i	$-0.227419\pi$
$168$	0	0
$169$	9.97511	0.767316
$170$	0	0
$171$	8.10409	0.619735
$172$	0	0
$173$	2.33978i	0.177890i	0.996037	+	0.0889452i	$0.0283496\pi$
$173$	2.33978i	0.177890i	−0.996037	+	0.0889452i	$0.971650\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.05399i	0.0792228i
$178$	0	0
$179$	23.6363	1.76666	0.883331	−	0.468749i	$-0.155295\pi$
$179$	23.6363	1.76666	0.883331	+	0.468749i	$0.155295\pi$
$180$	0	0
$181$	−24.3993	−1.81358	−0.906792	−	0.421579i	$-0.861476\pi$
$181$	−24.3993	−1.81358	−0.906792	+	0.421579i	$0.861476\pi$
$182$	0	0
$183$	− 3.24702i	− 0.240026i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 23.1187i	− 1.69061i
$188$	0	0
$189$	0.430681	0.0313275
$190$	0	0
$191$	4.35958	0.315448	0.157724	−	0.987483i	$-0.449584\pi$
$191$	4.35958	0.315448	0.157724	+	0.987483i	$0.449584\pi$
$192$	0	0
$193$	17.5758i	1.26514i	0.774505	+	0.632568i	$0.217999\pi$
$193$	17.5758i	1.26514i	−0.774505	+	0.632568i	$0.782001\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.67895i	− 0.404608i	−0.979323	−	0.202304i	$-0.935157\pi$
$197$	− 5.67895i	− 0.404608i	0.979323	−	0.202304i	$-0.0648430\pi$
$198$	0	0
$199$	−4.92781	−0.349323	−0.174662	−	0.984629i	$-0.555883\pi$
$199$	−4.92781	−0.349323	−0.174662	+	0.984629i	$0.555883\pi$
$200$	0	0
$201$	7.69123	0.542497
$202$	0	0
$203$	0.162328i	0.0113932i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.10470i	0.215792i
$208$	0	0
$209$	18.5314	1.28184
$210$	0	0
$211$	13.1592	0.905920	0.452960	−	0.891531i	$-0.350368\pi$
$211$	13.1592	0.905920	0.452960	+	0.891531i	$0.350368\pi$
$212$	0	0
$213$	11.8790i	0.813935i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.163114i	− 0.0110729i
$218$	0	0
$219$	2.63088	0.177779
$220$	0	0
$221$	7.48415	0.503438
$222$	0	0
$223$	− 15.1477i	− 1.01436i	−0.861839	−	0.507182i	$-0.830687\pi$
$223$	− 15.1477i	− 1.01436i	0.861839	−	0.507182i	$-0.169313\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 15.5445i	− 1.03172i	−0.856672	−	0.515861i	$-0.827472\pi$
$227$	− 15.5445i	− 1.03172i	0.856672	−	0.515861i	$-0.172528\pi$
$228$	0	0
$229$	−0.850327	−0.0561912	−0.0280956	−	0.999605i	$-0.508944\pi$
$229$	−0.850327	−0.0561912	−0.0280956	+	0.999605i	$0.508944\pi$
$230$	0	0
$231$	0.432534	0.0284586
$232$	0	0
$233$	15.9948i	1.04786i	0.851763	+	0.523928i	$0.175534\pi$
$233$	15.9948i	1.04786i	−0.851763	+	0.523928i	$0.824466\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6.19710i	− 0.402545i
$238$	0	0
$239$	−8.93976	−0.578265	−0.289132	−	0.957289i	$-0.593367\pi$
$239$	−8.93976	−0.578265	−0.289132	+	0.957289i	$0.593367\pi$
$240$	0	0
$241$	13.0713	0.841998	0.420999	−	0.907061i	$-0.361680\pi$
$241$	13.0713	0.841998	0.420999	+	0.907061i	$0.361680\pi$
$242$	0	0
$243$	− 15.8220i	− 1.01498i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.99911i	0.381714i
$248$	0	0
$249$	5.06418	0.320929
$250$	0	0
$251$	29.0659	1.83462	0.917311	−	0.398171i	$-0.130355\pi$
$251$	29.0659	1.83462	0.917311	+	0.398171i	$0.130355\pi$
$252$	0	0
$253$	7.09943i	0.446337i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.30240i	0.143620i	0.997418	+	0.0718100i	$0.0228775\pi$
$257$	2.30240i	0.143620i	−0.997418	+	0.0718100i	$0.977122\pi$
$258$	0	0
$259$	−0.618923	−0.0384580
$260$	0	0
$261$	3.82077	0.236500
$262$	0	0
$263$	− 17.6940i	− 1.09106i	−0.838092	−	0.545528i	$-0.816329\pi$
$263$	− 17.6940i	− 1.09106i	0.838092	−	0.545528i	$-0.183671\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	8.36567i	0.511971i
$268$	0	0
$269$	−6.10169	−0.372026	−0.186013	−	0.982547i	$-0.559557\pi$
$269$	−6.10169	−0.372026	−0.186013	+	0.982547i	$0.559557\pi$
$270$	0	0
$271$	−0.445834	−0.0270825	−0.0135412	−	0.999908i	$-0.504310\pi$
$271$	−0.445834	−0.0270825	−0.0135412	+	0.999908i	$0.504310\pi$
$272$	0	0
$273$	0.140023i	0.00847457i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 27.6395i	− 1.66070i	−0.557245	−	0.830348i	$-0.688142\pi$
$277$	− 27.6395i	− 1.66070i	0.557245	−	0.830348i	$-0.311858\pi$
$278$	0	0
$279$	−3.83926	−0.229850
$280$	0	0
$281$	−19.1815	−1.14428	−0.572138	−	0.820158i	$-0.693886\pi$
$281$	−19.1815	−1.14428	−0.572138	+	0.820158i	$0.693886\pi$
$282$	0	0
$283$	− 0.279266i	− 0.0166006i	−0.999966	−	0.00830031i	$-0.997358\pi$
$283$	− 0.279266i	− 0.0166006i	0.999966	−	0.00830031i	$-0.00264210\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.0998195i	0.00589216i
$288$	0	0
$289$	−1.51717	−0.0892450
$290$	0	0
$291$	−4.10423	−0.240594
$292$	0	0
$293$	− 24.6673i	− 1.44108i	−0.693415	−	0.720539i	$-0.743895\pi$
$293$	− 24.6673i	− 1.44108i	0.693415	−	0.720539i	$-0.256105\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 23.1802i	− 1.34505i
$298$	0	0
$299$	−2.29828	−0.132913
$300$	0	0
$301$	−0.786470	−0.0453314
$302$	0	0
$303$	− 4.29702i	− 0.246857i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	31.1894i	1.78007i	0.455889	+	0.890036i	$0.349321\pi$
$307$	31.1894i	1.78007i	−0.455889	+	0.890036i	$0.650679\pi$
$308$	0	0
$309$	−3.91945	−0.222970
$310$	0	0
$311$	−3.75111	−0.212706	−0.106353	−	0.994328i	$-0.533917\pi$
$311$	−3.75111	−0.212706	−0.106353	+	0.994328i	$0.533917\pi$
$312$	0	0
$313$	12.8514i	0.726405i	0.931710	+	0.363202i	$0.118317\pi$
$313$	12.8514i	0.726405i	−0.931710	+	0.363202i	$0.881683\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.0562i	0.957971i	0.877823	+	0.478986i	$0.158995\pi$
$317$	17.0562i	0.957971i	−0.877823	+	0.478986i	$0.841005\pi$
$318$	0	0
$319$	8.73684	0.489169
$320$	0	0
$321$	−0.207875	−0.0116024
$322$	0	0
$323$	− 14.8429i	− 0.825881i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.46493i	0.357511i
$328$	0	0
$329$	0.984967	0.0543030
$330$	0	0
$331$	−5.20557	−0.286124	−0.143062	−	0.989714i	$-0.545695\pi$
$331$	−5.20557	−0.286124	−0.143062	+	0.989714i	$0.545695\pi$
$332$	0	0
$333$	14.5678i	0.798311i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 13.7503i	− 0.749024i	−0.927222	−	0.374512i	$-0.877810\pi$
$337$	− 13.7503i	− 0.749024i	0.927222	−	0.374512i	$-0.122190\pi$
$338$	0	0
$339$	12.6654	0.687893
$340$	0	0
$341$	−8.77911	−0.475416
$342$	0	0
$343$	1.39648i	0.0754028i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.8372i	− 1.33333i	−0.745357	−	0.666665i	$-0.767721\pi$
$347$	− 24.8372i	− 1.33333i	0.745357	−	0.666665i	$-0.232279\pi$
$348$	0	0
$349$	−21.0102	−1.12465	−0.562325	−	0.826916i	$-0.690093\pi$
$349$	−21.0102	−1.12465	−0.562325	+	0.826916i	$0.690093\pi$
$350$	0	0
$351$	7.50405	0.400536
$352$	0	0
$353$	14.0070i	0.745515i	0.927929	+	0.372758i	$0.121588\pi$
$353$	14.0070i	0.745515i	−0.927929	+	0.372758i	$0.878412\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 0.346443i	− 0.0183357i
$358$	0	0
$359$	27.8401	1.46934	0.734671	−	0.678423i	$-0.237336\pi$
$359$	27.8401	1.46934	0.734671	+	0.678423i	$0.237336\pi$
$360$	0	0
$361$	−7.10230	−0.373805
$362$	0	0
$363$	− 14.4079i	− 0.756217i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9.44809i	0.493186i	0.969119	+	0.246593i	$0.0793111\pi$
$367$	9.44809i	0.493186i	−0.969119	+	0.246593i	$0.920689\pi$
$368$	0	0
$369$	2.34949	0.122309
$370$	0	0
$371$	0.0387417	0.00201137
$372$	0	0
$373$	31.0031i	1.60528i	0.596465	+	0.802639i	$0.296572\pi$
$373$	31.0031i	1.60528i	−0.596465	+	0.802639i	$0.703428\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.82835i	0.145668i
$378$	0	0
$379$	−17.0823	−0.877457	−0.438729	−	0.898620i	$-0.644571\pi$
$379$	−17.0823	−0.877457	−0.438729	+	0.898620i	$0.644571\pi$
$380$	0	0
$381$	−10.9724	−0.562131
$382$	0	0
$383$	0.936370i	0.0478463i	0.999714	+	0.0239231i	$0.00761569\pi$
$383$	0.936370i	0.0478463i	−0.999714	+	0.0239231i	$0.992384\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.5114i	0.940988i
$388$	0	0
$389$	−35.9995	−1.82525	−0.912623	−	0.408803i	$-0.865946\pi$
$389$	−35.9995	−1.82525	−0.912623	+	0.408803i	$0.865946\pi$
$390$	0	0
$391$	5.68636	0.287572
$392$	0	0
$393$	− 8.68932i	− 0.438318i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.8251i	0.994996i	0.867465	+	0.497498i	$0.165748\pi$
$397$	19.8251i	0.994996i	−0.867465	+	0.497498i	$0.834252\pi$
$398$	0	0
$399$	0.277700	0.0139024
$400$	0	0
$401$	18.3613	0.916921	0.458461	−	0.888715i	$-0.348401\pi$
$401$	18.3613	0.916921	0.458461	+	0.888715i	$0.348401\pi$
$402$	0	0
$403$	− 2.84204i	− 0.141572i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.3118i	1.65120i
$408$	0	0
$409$	−24.2850	−1.20082	−0.600408	−	0.799694i	$-0.704995\pi$
$409$	−24.2850	−1.20082	−0.600408	+	0.799694i	$0.704995\pi$
$410$	0	0
$411$	−4.16538	−0.205463
$412$	0	0
$413$	− 0.130444i	− 0.00641872i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.63413i	− 0.0800239i
$418$	0	0
$419$	20.0326	0.978656	0.489328	−	0.872100i	$-0.337242\pi$
$419$	20.0326	0.978656	0.489328	+	0.872100i	$0.337242\pi$
$420$	0	0
$421$	−8.42063	−0.410396	−0.205198	−	0.978720i	$-0.565784\pi$
$421$	−8.42063	−0.410396	−0.205198	+	0.978720i	$0.565784\pi$
$422$	0	0
$423$	− 23.1835i	− 1.12722i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.401857i	0.0194472i
$428$	0	0
$429$	7.53632	0.363857
$430$	0	0
$431$	−6.69404	−0.322440	−0.161220	−	0.986918i	$-0.551543\pi$
$431$	−6.69404	−0.322440	−0.161220	+	0.986918i	$0.551543\pi$
$432$	0	0
$433$	16.8815i	0.811272i	0.914035	+	0.405636i	$0.132950\pi$
$433$	16.8815i	0.811272i	−0.914035	+	0.405636i	$0.867050\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.55805i	0.218041i
$438$	0	0
$439$	4.16902	0.198977	0.0994883	−	0.995039i	$-0.468279\pi$
$439$	4.16902	0.198977	0.0994883	+	0.995039i	$0.468279\pi$
$440$	0	0
$441$	16.4230	0.782047
$442$	0	0
$443$	− 6.52720i	− 0.310117i	−0.987905	−	0.155058i	$-0.950443\pi$
$443$	− 6.52720i	− 0.310117i	0.987905	−	0.155058i	$-0.0495566\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.89660i	0.184303i
$448$	0	0
$449$	−11.5290	−0.544086	−0.272043	−	0.962285i	$-0.587699\pi$
$449$	−11.5290	−0.544086	−0.272043	+	0.962285i	$0.587699\pi$
$450$	0	0
$451$	5.37250	0.252981
$452$	0	0
$453$	− 9.05762i	− 0.425564i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 32.1138i	− 1.50222i	−0.660178	−	0.751109i	$-0.729519\pi$
$457$	− 32.1138i	− 1.50222i	0.660178	−	0.751109i	$-0.270481\pi$
$458$	0	0
$459$	−18.5664	−0.866605
$460$	0	0
$461$	22.9409	1.06846	0.534232	−	0.845338i	$-0.320601\pi$
$461$	22.9409	1.06846	0.534232	+	0.845338i	$0.320601\pi$
$462$	0	0
$463$	33.6555i	1.56410i	0.623213	+	0.782052i	$0.285827\pi$
$463$	33.6555i	1.56410i	−0.623213	+	0.782052i	$0.714173\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.9102i	− 1.06016i	−0.847948	−	0.530080i	$-0.822162\pi$
$467$	− 22.9102i	− 1.06016i	0.847948	−	0.530080i	$-0.177838\pi$
$468$	0	0
$469$	−0.951881	−0.0439538
$470$	0	0
$471$	−12.7904	−0.589351
$472$	0	0
$473$	42.3295i	1.94631i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.911877i	− 0.0417520i
$478$	0	0
$479$	−29.0016	−1.32512	−0.662558	−	0.749011i	$-0.730529\pi$
$479$	−29.0016	−1.32512	−0.662558	+	0.749011i	$0.730529\pi$
$480$	0	0
$481$	−10.7839	−0.491704
$482$	0	0
$483$	0.106388i	0.00484081i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17.7421i	− 0.803971i	−0.915646	−	0.401986i	$-0.868320\pi$
$487$	− 17.7421i	− 0.803971i	0.915646	−	0.401986i	$-0.131680\pi$
$488$	0	0
$489$	−4.07356	−0.184213
$490$	0	0
$491$	−15.4970	−0.699368	−0.349684	−	0.936868i	$-0.613711\pi$
$491$	−15.4970	−0.699368	−0.349684	+	0.936868i	$0.613711\pi$
$492$	0	0
$493$	− 6.99787i	− 0.315168i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.47017i	− 0.0659460i
$498$	0	0
$499$	6.86077	0.307130	0.153565	−	0.988139i	$-0.450924\pi$
$499$	6.86077	0.307130	0.153565	+	0.988139i	$0.450924\pi$
$500$	0	0
$501$	−13.6583	−0.610208
$502$	0	0
$503$	− 30.1690i	− 1.34517i	−0.740021	−	0.672584i	$-0.765184\pi$
$503$	− 30.1690i	− 1.34517i	0.740021	−	0.672584i	$-0.234816\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8.04537i	− 0.357307i
$508$	0	0
$509$	−10.8676	−0.481697	−0.240849	−	0.970563i	$-0.577426\pi$
$509$	−10.8676	−0.481697	−0.240849	+	0.970563i	$0.577426\pi$
$510$	0	0
$511$	−0.325603	−0.0144038
$512$	0	0
$513$	− 14.8824i	− 0.657073i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 53.0130i	− 2.33151i
$518$	0	0
$519$	1.88714	0.0828362
$520$	0	0
$521$	−6.07283	−0.266056	−0.133028	−	0.991112i	$-0.542470\pi$
$521$	−6.07283	−0.266056	−0.133028	+	0.991112i	$0.542470\pi$
$522$	0	0
$523$	− 4.47515i	− 0.195685i	−0.995202	−	0.0978424i	$-0.968806\pi$
$523$	− 4.47515i	− 0.195685i	0.995202	−	0.0978424i	$-0.0311941\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.03172i	0.306307i
$528$	0	0
$529$	21.2538	0.924078
$530$	0	0
$531$	−3.07030	−0.133240
$532$	0	0
$533$	1.73922i	0.0753341i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 19.0638i	− 0.822662i
$538$	0	0
$539$	37.5539	1.61756
$540$	0	0
$541$	4.15133	0.178480	0.0892398	−	0.996010i	$-0.471556\pi$
$541$	4.15133	0.178480	0.0892398	+	0.996010i	$0.471556\pi$
$542$	0	0
$543$	19.6791i	0.844511i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17.1599i	0.733704i	0.930279	+	0.366852i	$0.119565\pi$
$547$	17.1599i	0.733704i	−0.930279	+	0.366852i	$0.880435\pi$
$548$	0	0
$549$	9.45864	0.403685
$550$	0	0
$551$	5.60932	0.238965
$552$	0	0
$553$	0.766965i	0.0326147i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.8445i	1.22218i	0.791561	+	0.611090i	$0.209268\pi$
$557$	28.8445i	1.22218i	−0.791561	+	0.611090i	$0.790732\pi$
$558$	0	0
$559$	−13.7032	−0.579584
$560$	0	0
$561$	−18.6463	−0.787246
$562$	0	0
$563$	− 16.6121i	− 0.700117i	−0.936728	−	0.350058i	$-0.886162\pi$
$563$	− 16.6121i	− 0.700117i	0.936728	−	0.350058i	$-0.113838\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.356210i	0.0149594i
$568$	0	0
$569$	18.0242	0.755615	0.377808	−	0.925884i	$-0.376678\pi$
$569$	18.0242	0.755615	0.377808	+	0.925884i	$0.376678\pi$
$570$	0	0
$571$	9.23605	0.386517	0.193258	−	0.981148i	$-0.438094\pi$
$571$	9.23605	0.386517	0.193258	+	0.981148i	$0.438094\pi$
$572$	0	0
$573$	− 3.51620i	− 0.146891i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10.1925i	0.424318i	0.977235	+	0.212159i	$0.0680495\pi$
$577$	10.1925i	0.424318i	−0.977235	+	0.212159i	$0.931951\pi$
$578$	0	0
$579$	14.1757	0.589122
$580$	0	0
$581$	−0.626753	−0.0260021
$582$	0	0
$583$	− 2.08516i	− 0.0863585i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.8213i	0.776838i	0.921483	+	0.388419i	$0.126979\pi$
$587$	18.8213i	0.776838i	−0.921483	+	0.388419i	$0.873021\pi$
$588$	0	0
$589$	−5.63646	−0.232246
$590$	0	0
$591$	−4.58033	−0.188410
$592$	0	0
$593$	− 27.2144i	− 1.11756i	−0.829315	−	0.558781i	$-0.811269\pi$
$593$	− 27.2144i	− 1.11756i	0.829315	−	0.558781i	$-0.188731\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.97450i	0.162665i
$598$	0	0
$599$	20.9408	0.855620	0.427810	−	0.903869i	$-0.359285\pi$
$599$	20.9408	0.855620	0.427810	+	0.903869i	$0.359285\pi$
$600$	0	0
$601$	−30.5470	−1.24604	−0.623020	−	0.782206i	$-0.714094\pi$
$601$	−30.5470	−1.24604	−0.623020	+	0.782206i	$0.714094\pi$
$602$	0	0
$603$	22.4047i	0.912391i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	39.0343i	1.58435i	0.610292	+	0.792176i	$0.291052\pi$
$607$	39.0343i	1.58435i	−0.610292	+	0.792176i	$0.708948\pi$
$608$	0	0
$609$	0.130925	0.00530535
$610$	0	0
$611$	17.1617	0.694290
$612$	0	0
$613$	8.99273i	0.363213i	0.983371	+	0.181607i	$0.0581297\pi$
$613$	8.99273i	0.363213i	−0.983371	+	0.181607i	$0.941870\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.2499i	0.734715i	0.930080	+	0.367358i	$0.119737\pi$
$617$	18.2499i	0.734715i	−0.930080	+	0.367358i	$0.880263\pi$
$618$	0	0
$619$	8.57874	0.344809	0.172404	−	0.985026i	$-0.444846\pi$
$619$	8.57874	0.344809	0.172404	+	0.985026i	$0.444846\pi$
$620$	0	0
$621$	5.70148	0.228793
$622$	0	0
$623$	− 1.03535i	− 0.0414805i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 14.9464i	− 0.596901i
$628$	0	0
$629$	26.6814	1.06386
$630$	0	0
$631$	−21.9165	−0.872483	−0.436242	−	0.899830i	$-0.643691\pi$
$631$	−21.9165	−0.872483	−0.436242	+	0.899830i	$0.643691\pi$
$632$	0	0
$633$	− 10.6135i	− 0.421850i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.1572i	0.481687i
$638$	0	0
$639$	−34.6038	−1.36891
$640$	0	0
$641$	−29.3594	−1.15963	−0.579814	−	0.814749i	$-0.696875\pi$
$641$	−29.3594	−1.15963	−0.579814	+	0.814749i	$0.696875\pi$
$642$	0	0
$643$	− 46.9899i	− 1.85310i	−0.376172	−	0.926550i	$-0.622760\pi$
$643$	− 46.9899i	− 1.85310i	0.376172	−	0.926550i	$-0.377240\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.51849i	0.295582i	0.989019	+	0.147791i	$0.0472164\pi$
$647$	7.51849i	0.295582i	−0.989019	+	0.147791i	$0.952784\pi$
$648$	0	0
$649$	−7.02076	−0.275589
$650$	0	0
$651$	−0.131558	−0.00515618
$652$	0	0
$653$	3.18138i	0.124497i	0.998061	+	0.0622486i	$0.0198272\pi$
$653$	3.18138i	0.124497i	−0.998061	+	0.0622486i	$0.980173\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.66383i	0.298995i
$658$	0	0
$659$	−8.61166	−0.335463	−0.167731	−	0.985833i	$-0.553644\pi$
$659$	−8.61166	−0.335463	−0.167731	+	0.985833i	$0.553644\pi$
$660$	0	0
$661$	47.7721	1.85812	0.929060	−	0.369928i	$-0.120618\pi$
$661$	47.7721	1.85812	0.929060	+	0.369928i	$0.120618\pi$
$662$	0	0
$663$	− 6.03630i	− 0.234430i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.14895i	0.0832076i
$668$	0	0
$669$	−12.2173	−0.472348
$670$	0	0
$671$	21.6288	0.834970
$672$	0	0
$673$	42.7516i	1.64795i	0.566623	+	0.823977i	$0.308250\pi$
$673$	42.7516i	1.64795i	−0.566623	+	0.823977i	$0.691750\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.5120i	1.82604i	0.407919	+	0.913018i	$0.366254\pi$
$677$	47.5120i	1.82604i	−0.407919	+	0.913018i	$0.633746\pi$
$678$	0	0
$679$	0.507947	0.0194932
$680$	0	0
$681$	−12.5373	−0.480431
$682$	0	0
$683$	− 5.15859i	− 0.197388i	−0.995118	−	0.0986940i	$-0.968533\pi$
$683$	− 5.15859i	− 0.197388i	0.995118	−	0.0986940i	$-0.0314665\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.685827i	0.0261659i
$688$	0	0
$689$	0.675023	0.0257163
$690$	0	0
$691$	−17.7485	−0.675184	−0.337592	−	0.941293i	$-0.609612\pi$
$691$	−17.7485	−0.675184	−0.337592	+	0.941293i	$0.609612\pi$
$692$	0	0
$693$	1.25998i	0.0478627i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.30316i	− 0.162994i
$698$	0	0
$699$	12.9005	0.487943
$700$	0	0
$701$	26.5043	1.00105	0.500527	−	0.865721i	$-0.333139\pi$
$701$	26.5043	1.00105	0.500527	+	0.865721i	$0.333139\pi$
$702$	0	0
$703$	21.3872i	0.806632i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.531807i	0.0200007i
$708$	0	0
$709$	−26.6306	−1.00013	−0.500066	−	0.865987i	$-0.666691\pi$
$709$	−26.6306	−1.00013	−0.500066	+	0.865987i	$0.666691\pi$
$710$	0	0
$711$	18.0523	0.677014
$712$	0	0
$713$	− 2.15934i	− 0.0808681i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.21031i	0.269274i
$718$	0	0
$719$	33.0626	1.23303	0.616513	−	0.787344i	$-0.288545\pi$
$719$	33.0626	1.23303	0.616513	+	0.787344i	$0.288545\pi$
$720$	0	0
$721$	0.485078	0.0180653
$722$	0	0
$723$	− 10.5426i	− 0.392084i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	48.4354i	1.79637i	0.439617	+	0.898185i	$0.355114\pi$
$727$	48.4354i	1.79637i	−0.439617	+	0.898185i	$0.644886\pi$
$728$	0	0
$729$	−2.05553	−0.0761307
$730$	0	0
$731$	33.9043	1.25399
$732$	0	0
$733$	16.2519i	0.600276i	0.953896	+	0.300138i	$0.0970327\pi$
$733$	16.2519i	0.600276i	−0.953896	+	0.300138i	$0.902967\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	51.2322i	1.88716i
$738$	0	0
$739$	−15.2083	−0.559447	−0.279723	−	0.960081i	$-0.590243\pi$
$739$	−15.2083	−0.559447	−0.279723	+	0.960081i	$0.590243\pi$
$740$	0	0
$741$	4.83855	0.177749
$742$	0	0
$743$	− 21.8396i	− 0.801216i	−0.916250	−	0.400608i	$-0.868799\pi$
$743$	− 21.8396i	− 0.801216i	0.916250	−	0.400608i	$-0.131201\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.7521i	0.539751i
$748$	0	0
$749$	0.0257270	0.000940044 0
$750$	0	0
$751$	−44.6547	−1.62947	−0.814736	−	0.579832i	$-0.803118\pi$
$751$	−44.6547	−1.62947	−0.814736	+	0.579832i	$0.803118\pi$
$752$	0	0
$753$	− 23.4429i	− 0.854308i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 15.8719i	− 0.576876i	−0.957499	−	0.288438i	$-0.906864\pi$
$757$	− 15.8719i	− 0.576876i	0.957499	−	0.288438i	$-0.0931359\pi$
$758$	0	0
$759$	5.72601	0.207841
$760$	0	0
$761$	4.08944	0.148242	0.0741211	−	0.997249i	$-0.476385\pi$
$761$	4.08944	0.148242	0.0741211	+	0.997249i	$0.476385\pi$
$762$	0	0
$763$	− 0.800112i	− 0.0289660i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.27281i	− 0.0820664i
$768$	0	0
$769$	−23.6100	−0.851399	−0.425699	−	0.904865i	$-0.639972\pi$
$769$	−23.6100	−0.851399	−0.425699	+	0.904865i	$0.639972\pi$
$770$	0	0
$771$	1.85699	0.0668779
$772$	0	0
$773$	− 34.8263i	− 1.25261i	−0.779576	−	0.626307i	$-0.784565\pi$
$773$	− 34.8263i	− 1.25261i	0.779576	−	0.626307i	$-0.215435\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.499189i	0.0179083i
$778$	0	0
$779$	3.44930	0.123584
$780$	0	0
$781$	−79.1274	−2.83140
$782$	0	0
$783$	− 7.01647i	− 0.250748i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.66596i	0.201970i	0.994888	+	0.100985i	$0.0321994\pi$
$787$	5.66596i	0.201970i	−0.994888	+	0.100985i	$0.967801\pi$
$788$	0	0
$789$	−14.2710	−0.508060
$790$	0	0
$791$	−1.56750	−0.0557339
$792$	0	0
$793$	7.00182i	0.248642i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.4500i	1.07859i	0.842116	+	0.539297i	$0.181310\pi$
$797$	30.4500i	1.07859i	−0.842116	+	0.539297i	$0.818690\pi$
$798$	0	0
$799$	−42.4613	−1.50217
$800$	0	0
$801$	−24.3694	−0.861051
$802$	0	0
$803$	17.5246i	0.618431i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.92128i	0.173237i
$808$	0	0
$809$	−53.3607	−1.87606	−0.938031	−	0.346552i	$-0.887352\pi$
$809$	−53.3607	−1.87606	−0.938031	+	0.346552i	$0.887352\pi$
$810$	0	0
$811$	−34.2828	−1.20383	−0.601916	−	0.798559i	$-0.705596\pi$
$811$	−34.2828	−1.20383	−0.601916	+	0.798559i	$0.705596\pi$
$812$	0	0
$813$	0.359585i	0.0126112i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	27.1768i	0.950796i
$818$	0	0
$819$	−0.407890	−0.0142528
$820$	0	0
$821$	1.02193	0.0356656	0.0178328	−	0.999841i	$-0.494323\pi$
$821$	1.02193	0.0356656	0.0178328	+	0.999841i	$0.494323\pi$
$822$	0	0
$823$	9.93953i	0.346470i	0.984880	+	0.173235i	$0.0554221\pi$
$823$	9.93953i	0.346470i	−0.984880	+	0.173235i	$0.944578\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.1416i	0.491751i	0.969301	+	0.245876i	$0.0790755\pi$
$827$	14.1416i	0.491751i	−0.969301	+	0.245876i	$0.920924\pi$
$828$	0	0
$829$	−11.2633	−0.391190	−0.195595	−	0.980685i	$-0.562664\pi$
$829$	−11.2633	−0.391190	−0.195595	+	0.980685i	$0.562664\pi$
$830$	0	0
$831$	−22.2925	−0.773318
$832$	0	0
$833$	− 30.0792i	− 1.04218i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.05042i	0.243698i
$838$	0	0
$839$	−12.2937	−0.424425	−0.212212	−	0.977224i	$-0.568067\pi$
$839$	−12.2937	−0.424425	−0.212212	+	0.977224i	$0.568067\pi$
$840$	0	0
$841$	−26.3554	−0.908808
$842$	0	0
$843$	15.4708i	0.532842i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.78315i	0.0612696i
$848$	0	0
$849$	−0.225240	−0.00773022
$850$	0	0
$851$	−8.19349	−0.280869
$852$	0	0
$853$	20.7417i	0.710181i	0.934832	+	0.355090i	$0.115550\pi$
$853$	20.7417i	0.710181i	−0.934832	+	0.355090i	$0.884450\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.8975i	0.372253i	0.982526	+	0.186126i	$0.0595934\pi$
$857$	10.8975i	0.372253i	−0.982526	+	0.186126i	$0.940407\pi$
$858$	0	0
$859$	31.6809	1.08094	0.540469	−	0.841364i	$-0.318247\pi$
$859$	31.6809	1.08094	0.540469	+	0.841364i	$0.318247\pi$
$860$	0	0
$861$	0.0805089	0.00274374
$862$	0	0
$863$	− 11.5138i	− 0.391934i	−0.980610	−	0.195967i	$-0.937215\pi$
$863$	− 11.5138i	− 0.391934i	0.980610	−	0.195967i	$-0.0627846\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.22366i	0.0415577i
$868$	0	0
$869$	41.2796	1.40032
$870$	0	0
$871$	−16.5853	−0.561970
$872$	0	0
$873$	− 11.9557i	− 0.404640i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.54515i	− 0.254782i	−0.991853	−	0.127391i	$-0.959340\pi$
$877$	− 7.54515i	− 0.254782i	0.991853	−	0.127391i	$-0.0406602\pi$
$878$	0	0
$879$	−19.8953	−0.671050
$880$	0	0
$881$	−33.8753	−1.14129	−0.570644	−	0.821197i	$-0.693306\pi$
$881$	−33.8753	−1.14129	−0.570644	+	0.821197i	$0.693306\pi$
$882$	0	0
$883$	2.90766i	0.0978505i	0.998802	+	0.0489253i	$0.0155796\pi$
$883$	2.90766i	0.0978505i	−0.998802	+	0.0489253i	$0.984420\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44.5070i	1.49440i	0.664600	+	0.747199i	$0.268602\pi$
$887$	44.5070i	1.49440i	−0.664600	+	0.747199i	$0.731398\pi$
$888$	0	0
$889$	1.35796	0.0455445
$890$	0	0
$891$	19.1720	0.642285
$892$	0	0
$893$	− 34.0360i	− 1.13897i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.85366i	0.0618920i
$898$	0	0
$899$	−2.65738	−0.0886284
$900$	0	0
$901$	−1.67013	−0.0556402
$902$	0	0
$903$	0.634324i	0.0211090i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 52.6463i	− 1.74809i	−0.485843	−	0.874046i	$-0.661487\pi$
$907$	− 52.6463i	− 1.74809i	0.485843	−	0.874046i	$-0.338513\pi$
$908$	0	0
$909$	12.5173	0.415173
$910$	0	0
$911$	−6.67724	−0.221227	−0.110613	−	0.993863i	$-0.535282\pi$
$911$	−6.67724	−0.221227	−0.110613	+	0.993863i	$0.535282\pi$
$912$	0	0
$913$	33.7331i	1.11640i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.07541i	0.0355131i
$918$	0	0
$919$	−44.0994	−1.45470	−0.727352	−	0.686264i	$-0.759249\pi$
$919$	−44.0994	−1.45470	−0.727352	+	0.686264i	$0.759249\pi$
$920$	0	0
$921$	25.1556	0.828907
$922$	0	0
$923$	− 25.6157i	− 0.843151i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 11.4175i	− 0.374998i
$928$	0	0
$929$	8.73003	0.286423	0.143211	−	0.989692i	$-0.454257\pi$
$929$	8.73003	0.286423	0.143211	+	0.989692i	$0.454257\pi$
$930$	0	0
$931$	24.1108	0.790198
$932$	0	0
$933$	3.02544i	0.0990485i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	49.3068i	1.61078i	0.592743	+	0.805392i	$0.298045\pi$
$937$	49.3068i	1.61078i	−0.592743	+	0.805392i	$0.701955\pi$
$938$	0	0
$939$	10.3652	0.338257
$940$	0	0
$941$	−33.3420	−1.08692	−0.543460	−	0.839435i	$-0.682886\pi$
$941$	−33.3420	−1.08692	−0.543460	+	0.839435i	$0.682886\pi$
$942$	0	0
$943$	1.32144i	0.0430320i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.3665i	− 0.661823i	−0.943662	−	0.330911i	$-0.892644\pi$
$947$	− 20.3665i	− 0.661823i	0.943662	−	0.330911i	$-0.107356\pi$
$948$	0	0
$949$	−5.67320	−0.184160
$950$	0	0
$951$	13.7566	0.446088
$952$	0	0
$953$	3.86028i	0.125047i	0.998044	+	0.0625233i	$0.0199148\pi$
$953$	3.86028i	0.125047i	−0.998044	+	0.0625233i	$0.980085\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 7.04665i	− 0.227786i
$958$	0	0
$959$	0.515516	0.0166469
$960$	0	0
$961$	−28.3298	−0.913863
$962$	0	0
$963$	− 0.605545i	− 0.0195134i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 10.0234i	− 0.322330i	−0.986927	−	0.161165i	$-0.948475\pi$
$967$	− 10.0234i	− 0.322330i	0.986927	−	0.161165i	$-0.0515251\pi$
$968$	0	0
$969$	−11.9715	−0.384579
$970$	0	0
$971$	32.0757	1.02936	0.514679	−	0.857383i	$-0.327911\pi$
$971$	32.0757	1.02936	0.514679	+	0.857383i	$0.327911\pi$
$972$	0	0
$973$	0.202243i	0.00648363i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.0038i	0.512007i	0.966676	+	0.256004i	$0.0824059\pi$
$977$	16.0038i	0.512007i	−0.966676	+	0.256004i	$0.917594\pi$
$978$	0	0
$979$	−55.7248	−1.78097
$980$	0	0
$981$	−18.8325	−0.601275
$982$	0	0
$983$	− 26.3906i	− 0.841731i	−0.907123	−	0.420865i	$-0.861726\pi$
$983$	− 26.3906i	− 0.841731i	0.907123	−	0.420865i	$-0.138274\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.794420i	− 0.0252867i
$988$	0	0
$989$	−10.4115	−0.331067
$990$	0	0
$991$	−1.60656	−0.0510341	−0.0255170	−	0.999674i	$-0.508123\pi$
$991$	−1.60656	−0.0510341	−0.0255170	+	0.999674i	$0.508123\pi$
$992$	0	0
$993$	4.19853i	0.133236i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 33.9705i	− 1.07586i	−0.842990	−	0.537929i	$-0.819207\pi$
$997$	− 33.9705i	− 1.07586i	0.842990	−	0.537929i	$-0.180793\pi$
$998$	0	0
$999$	26.7524	0.846407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4100.2.d.g.1149.6		14
5.2	odd	4		4100.2.a.j.1.3	yes	7
5.3	odd	4		4100.2.a.g.1.5	✓	7
5.4	even	2	inner	4100.2.d.g.1149.9		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4100.2.a.g.1.5	✓	7	5.3	odd	4
4100.2.a.j.1.3	yes	7	5.2	odd	4
4100.2.d.g.1149.6		14	1.1	even	1	trivial
4100.2.d.g.1149.9		14	5.4	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:	\( N \)	\(=\)	\( 4100 = 2^{2} \cdot 5^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.806545i q^{3} +0.0998195i q^{7} +2.34949 q^{9} +O(q^{10})\)	\(q-0.806545i q^{3} +0.0998195i q^{7} +2.34949 q^{9} +5.37250 q^{11} +1.73922i q^{13} -4.30316i q^{17} +3.44930 q^{19} +0.0805089 q^{21} +1.32144i q^{23} -4.31460i q^{27} +1.62622 q^{29} -1.63408 q^{31} -4.33316i q^{33} +6.20042i q^{37} +1.40276 q^{39} +1.00000 q^{41} +7.87893i q^{43} -9.86748i q^{47} +6.99004 q^{49} -3.47069 q^{51} -0.388118i q^{53} -2.78202i q^{57} -1.30680 q^{59} +4.02584 q^{61} +0.234524i q^{63} +9.53602i q^{67} +1.06580 q^{69} -14.7282 q^{71} +3.26192i q^{73} +0.536280i q^{77} +7.68351 q^{79} +3.56854 q^{81} +6.27886i q^{83} -1.31162i q^{87} -10.3722 q^{89} -0.173608 q^{91} +1.31796i q^{93} -5.08866i q^{97} +12.6226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 12 q^{9}+O(q^{10}) \) 14 * q - 12 * q^9	\( 14 q - 12 q^{9} - 2 q^{11} + 2 q^{19} - 2 q^{21} + 10 q^{29} + 8 q^{31} + 22 q^{39} + 14 q^{41} + 6 q^{49} + 54 q^{51} + 32 q^{59} + 20 q^{61} + 38 q^{69} + 54 q^{71} + 2 q^{79} + 54 q^{81} + 32 q^{89} + 18 q^{91} + 80 q^{99}+O(q^{100}) \) 14 * q - 12 * q^9 - 2 * q^11 + 2 * q^19 - 2 * q^21 + 10 * q^29 + 8 * q^31 + 22 * q^39 + 14 * q^41 + 6 * q^49 + 54 * q^51 + 32 * q^59 + 20 * q^61 + 38 * q^69 + 54 * q^71 + 2 * q^79 + 54 * q^81 + 32 * q^89 + 18 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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