LMFDB - Embedded newform 2205.2.b.c.881.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2205,2,Mod(881,2205)] 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("2205.881"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-16,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$17.6070136457$
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} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 790 x^{12} - 2192 x^{11} + 5036 x^{10} - 9472 x^{9} + \cdots + 17 $ x^16 - 8x^15 + 52x^14 - 224x^13 + 790x^12 - 2192x^11 + 5036x^10 - 9472x^9 + 14669x^8 - 18520x^7 + 18744x^6 - 14872x^5 + 8956x^4 - 3912x^3 + 1160x^2 - 208*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.16
Root			$0.500000 - 0.229342i$ of defining polynomial
Character	$\chi$	$=$	2205.881
Dual form			2205.2.b.c.881.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.65421i q^{2} -5.04484 q^{4} -1.00000 q^{5} -8.08165i q^{8} -2.65421i q^{10} +4.28711i q^{11} +3.72027i q^{13} +11.3607 q^{16} -5.64024 q^{17} +6.24192i q^{19} +5.04484 q^{20} -11.3789 q^{22} +2.66983i q^{23} +1.00000 q^{25} -9.87438 q^{26} -10.7034i q^{29} +6.21489i q^{31} +13.9905i q^{32} -14.9704i q^{34} +1.68281 q^{37} -16.5674 q^{38} +8.08165i q^{40} -4.50679 q^{41} +4.89765 q^{43} -21.6278i q^{44} -7.08629 q^{46} -7.85801 q^{47} +2.65421i q^{50} -18.7682i q^{52} -9.01748i q^{53} -4.28711i q^{55} +28.4090 q^{58} +7.79598 q^{59} -13.7521i q^{61} -16.4956 q^{62} -14.4123 q^{64} -3.72027i q^{65} +0.434026 q^{67} +28.4541 q^{68} +6.48172i q^{71} -7.40739i q^{73} +4.46652i q^{74} -31.4895i q^{76} -10.6369 q^{79} -11.3607 q^{80} -11.9620i q^{82} +8.90228 q^{83} +5.64024 q^{85} +12.9994i q^{86} +34.6469 q^{88} -9.92599 q^{89} -13.4689i q^{92} -20.8568i q^{94} -6.24192i q^{95} +3.04079i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4} - 16 q^{5} + 16 q^{20} - 16 q^{22} + 16 q^{25} - 32 q^{26} + 16 q^{38} + 32 q^{41} + 32 q^{43} - 16 q^{46} - 32 q^{47} + 48 q^{58} + 32 q^{59} + 32 q^{62} + 16 q^{64} + 16 q^{68} + 32 q^{79}+ \cdots - 64 q^{89}+O(q^{100}) $ 16 * q - 16 * q^4 - 16 * q^5 + 16 * q^20 - 16 * q^22 + 16 * q^25 - 32 * q^26 + 16 * q^38 + 32 * q^41 + 32 * q^43 - 16 * q^46 - 32 * q^47 + 48 * q^58 + 32 * q^59 + 32 * q^62 + 16 * q^64 + 16 * q^68 + 32 * q^79 + 16 * q^88 - 64 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$.

$n$	$442$	$1081$	$1226$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	2.65421i	1.87681i	0.345536	+	0.938406i	$0.387697\pi$
$2$	2.65421i	1.87681i	−0.345536	+	0.938406i	$0.612303\pi$
$3$	0	0
$3$	0	0
$4$	−5.04484	−2.52242
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 8.08165i	− 2.85730i
$9$	0	0
$10$	− 2.65421i	− 0.839335i
$11$	4.28711i	1.29261i	0.763078	+	0.646306i	$0.223687\pi$
$11$	4.28711i	1.29261i	−0.763078	+	0.646306i	$0.776313\pi$
$12$	0	0
$13$	3.72027i	1.03182i	0.856644	+	0.515908i	$0.172546\pi$
$13$	3.72027i	1.03182i	−0.856644	+	0.515908i	$0.827454\pi$
$14$	0	0
$15$	0	0
$16$	11.3607	2.84018
$17$	−5.64024	−1.36796	−0.683980	−	0.729501i	$-0.739753\pi$
$17$	−5.64024	−1.36796	−0.683980	+	0.729501i	$0.739753\pi$
$18$	0	0
$19$	6.24192i	1.43200i	0.698103	+	0.715998i	$0.254028\pi$
$19$	6.24192i	1.43200i	−0.698103	+	0.715998i	$0.745972\pi$
$20$	5.04484	1.12806
$21$	0	0
$22$	−11.3789	−2.42599
$23$	2.66983i	0.556698i	0.960480	+	0.278349i	$0.0897871\pi$
$23$	2.66983i	0.556698i	−0.960480	+	0.278349i	$0.910213\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−9.87438	−1.93653
$27$	0	0
$28$	0	0
$29$	− 10.7034i	− 1.98756i	−0.111346	−	0.993782i	$-0.535516\pi$
$29$	− 10.7034i	− 1.98756i	0.111346	−	0.993782i	$-0.464484\pi$
$30$	0	0
$31$	6.21489i	1.11623i	0.829764	+	0.558114i	$0.188475\pi$
$31$	6.21489i	1.11623i	−0.829764	+	0.558114i	$0.811525\pi$
$32$	13.9905i	2.47319i
$33$	0	0
$34$	− 14.9704i	− 2.56740i
$35$	0	0
$36$	0	0
$37$	1.68281	0.276652	0.138326	−	0.990387i	$-0.455828\pi$
$37$	1.68281	0.276652	0.138326	+	0.990387i	$0.455828\pi$
$38$	−16.5674	−2.68758
$39$	0	0
$40$	8.08165i	1.27782i
$41$	−4.50679	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$41$	−4.50679	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$42$	0	0
$43$	4.89765	0.746884	0.373442	−	0.927653i	$-0.378177\pi$
$43$	4.89765	0.746884	0.373442	+	0.927653i	$0.378177\pi$
$44$	− 21.6278i	− 3.26051i
$45$	0	0
$46$	−7.08629	−1.04482
$47$	−7.85801	−1.14621	−0.573104	−	0.819483i	$-0.694261\pi$
$47$	−7.85801	−1.14621	−0.573104	+	0.819483i	$0.694261\pi$
$48$	0	0
$49$	0	0
$50$	2.65421i	0.375362i
$51$	0	0
$52$	− 18.7682i	− 2.60268i
$53$	− 9.01748i	− 1.23865i	−0.785136	−	0.619323i	$-0.787407\pi$
$53$	− 9.01748i	− 1.23865i	0.785136	−	0.619323i	$-0.212593\pi$
$54$	0	0
$55$	− 4.28711i	− 0.578074i
$56$	0	0
$57$	0	0
$58$	28.4090	3.73028
$59$	7.79598	1.01495	0.507475	−	0.861667i	$-0.330579\pi$
$59$	7.79598	1.01495	0.507475	+	0.861667i	$0.330579\pi$
$60$	0	0
$61$	− 13.7521i	− 1.76078i	−0.474248	−	0.880391i	$-0.657280\pi$
$61$	− 13.7521i	− 1.76078i	0.474248	−	0.880391i	$-0.342720\pi$
$62$	−16.4956	−2.09495
$63$	0	0
$64$	−14.4123	−1.80153
$65$	− 3.72027i	− 0.461442i
$66$	0	0
$67$	0.434026	0.0530247	0.0265124	−	0.999648i	$-0.491560\pi$
$67$	0.434026	0.0530247	0.0265124	+	0.999648i	$0.491560\pi$
$68$	28.4541	3.45057
$69$	0	0
$70$	0	0
$71$	6.48172i	0.769239i	0.923075	+	0.384620i	$0.125667\pi$
$71$	6.48172i	0.769239i	−0.923075	+	0.384620i	$0.874333\pi$
$72$	0	0
$73$	− 7.40739i	− 0.866970i	−0.901161	−	0.433485i	$-0.857284\pi$
$73$	− 7.40739i	− 0.866970i	0.901161	−	0.433485i	$-0.142716\pi$
$74$	4.46652i	0.519223i
$75$	0	0
$76$	− 31.4895i	− 3.61209i
$77$	0	0
$78$	0	0
$79$	−10.6369	−1.19674	−0.598371	−	0.801219i	$-0.704185\pi$
$79$	−10.6369	−1.19674	−0.598371	+	0.801219i	$0.704185\pi$
$80$	−11.3607	−1.27017
$81$	0	0
$82$	− 11.9620i	− 1.32098i
$83$	8.90228	0.977152	0.488576	−	0.872521i	$-0.337517\pi$
$83$	8.90228	0.977152	0.488576	+	0.872521i	$0.337517\pi$
$84$	0	0
$85$	5.64024	0.611770
$86$	12.9994i	1.40176i
$87$	0	0
$88$	34.6469	3.69338
$89$	−9.92599	−1.05215	−0.526077	−	0.850437i	$-0.676338\pi$
$89$	−9.92599	−1.05215	−0.526077	+	0.850437i	$0.676338\pi$
$90$	0	0
$91$	0	0
$92$	− 13.4689i	− 1.40423i
$93$	0	0
$94$	− 20.8568i	− 2.15122i
$95$	− 6.24192i	− 0.640408i
$96$	0	0
$97$	3.04079i	0.308746i	0.988013	+	0.154373i	$0.0493357\pi$
$97$	3.04079i	0.308746i	−0.988013	+	0.154373i	$0.950664\pi$
$98$	0	0
$99$	0	0
$100$	−5.04484	−0.504484
$101$	−11.9159	−1.18567	−0.592836	−	0.805323i	$-0.701992\pi$
$101$	−11.9159	−1.18567	−0.592836	+	0.805323i	$0.701992\pi$
$102$	0	0
$103$	− 11.8981i	− 1.17236i	−0.810183	−	0.586178i	$-0.800632\pi$
$103$	− 11.8981i	− 1.17236i	0.810183	−	0.586178i	$-0.199368\pi$
$104$	30.0659	2.94820
$105$	0	0
$106$	23.9343	2.32471
$107$	− 3.69509i	− 0.357218i	−0.983920	−	0.178609i	$-0.942840\pi$
$107$	− 3.69509i	− 0.357218i	0.983920	−	0.178609i	$-0.0571598\pi$
$108$	0	0
$109$	5.08689	0.487236	0.243618	−	0.969871i	$-0.421666\pi$
$109$	5.08689	0.487236	0.243618	+	0.969871i	$0.421666\pi$
$110$	11.3789	1.08494
$111$	0	0
$112$	0	0
$113$	− 6.30402i	− 0.593032i	−0.955028	−	0.296516i	$-0.904175\pi$
$113$	− 6.30402i	− 0.593032i	0.955028	−	0.296516i	$-0.0958248\pi$
$114$	0	0
$115$	− 2.66983i	− 0.248963i
$116$	53.9967i	5.01347i
$117$	0	0
$118$	20.6922i	1.90487i
$119$	0	0
$120$	0	0
$121$	−7.37933	−0.670848
$122$	36.5011	3.30466
$123$	0	0
$124$	− 31.3531i	− 2.81560i
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.75023	0.244043	0.122022	−	0.992527i	$-0.461062\pi$
$127$	2.75023	0.244043	0.122022	+	0.992527i	$0.461062\pi$
$128$	− 10.2722i	− 0.907943i
$129$	0	0
$130$	9.87438	0.866040
$131$	−12.4355	−1.08650	−0.543249	−	0.839572i	$-0.682806\pi$
$131$	−12.4355	−1.08650	−0.543249	+	0.839572i	$0.682806\pi$
$132$	0	0
$133$	0	0
$134$	1.15200i	0.0995174i
$135$	0	0
$136$	45.5825i	3.90867i
$137$	8.50306i	0.726466i	0.931698	+	0.363233i	$0.118327\pi$
$137$	8.50306i	0.726466i	−0.931698	+	0.363233i	$0.881673\pi$
$138$	0	0
$139$	10.4963i	0.890287i	0.895459	+	0.445143i	$0.146847\pi$
$139$	10.4963i	0.890287i	−0.895459	+	0.445143i	$0.853153\pi$
$140$	0	0
$141$	0	0
$142$	−17.2039	−1.44372
$143$	−15.9492	−1.33374
$144$	0	0
$145$	10.7034i	0.888865i
$146$	19.6608	1.62714
$147$	0	0
$148$	−8.48949	−0.697832
$149$	10.5193i	0.861778i	0.902405	+	0.430889i	$0.141800\pi$
$149$	10.5193i	0.861778i	−0.902405	+	0.430889i	$0.858200\pi$
$150$	0	0
$151$	9.29005	0.756014	0.378007	−	0.925803i	$-0.376610\pi$
$151$	9.29005	0.756014	0.378007	+	0.925803i	$0.376610\pi$
$152$	50.4450	4.09163
$153$	0	0
$154$	0	0
$155$	− 6.21489i	− 0.499192i
$156$	0	0
$157$	8.87768i	0.708516i	0.935148	+	0.354258i	$0.115266\pi$
$157$	8.87768i	0.708516i	−0.935148	+	0.354258i	$0.884734\pi$
$158$	− 28.2325i	− 2.24606i
$159$	0	0
$160$	− 13.9905i	− 1.10605i
$161$	0	0
$162$	0	0
$163$	−24.8753	−1.94838	−0.974191	−	0.225724i	$-0.927525\pi$
$163$	−24.8753	−1.94838	−0.974191	+	0.225724i	$0.927525\pi$
$164$	22.7360	1.77538
$165$	0	0
$166$	23.6285i	1.83393i
$167$	−9.07164	−0.701984	−0.350992	−	0.936378i	$-0.614156\pi$
$167$	−9.07164	−0.701984	−0.350992	+	0.936378i	$0.614156\pi$
$168$	0	0
$169$	−0.840395	−0.0646457
$170$	14.9704i	1.14818i
$171$	0	0
$172$	−24.7079	−1.88396
$173$	3.95823	0.300938	0.150469	−	0.988615i	$-0.451922\pi$
$173$	3.95823	0.300938	0.150469	+	0.988615i	$0.451922\pi$
$174$	0	0
$175$	0	0
$176$	48.7047i	3.67126i
$177$	0	0
$178$	− 26.3457i	− 1.97469i
$179$	0.330331i	0.0246901i	0.999924	+	0.0123451i	$0.00392965\pi$
$179$	0.330331i	0.0246901i	−0.999924	+	0.0123451i	$0.996070\pi$
$180$	0	0
$181$	− 2.09148i	− 0.155459i	−0.996974	−	0.0777294i	$-0.975233\pi$
$181$	− 2.09148i	− 0.155459i	0.996974	−	0.0777294i	$-0.0247670\pi$
$182$	0	0
$183$	0	0
$184$	21.5766	1.59065
$185$	−1.68281	−0.123722
$186$	0	0
$187$	− 24.1804i	− 1.76824i
$188$	39.6424	2.89122
$189$	0	0
$190$	16.5674	1.20192
$191$	7.72494i	0.558957i	0.960152	+	0.279479i	$0.0901616\pi$
$191$	7.72494i	0.558957i	−0.960152	+	0.279479i	$0.909838\pi$
$192$	0	0
$193$	8.30309	0.597669	0.298835	−	0.954305i	$-0.403402\pi$
$193$	8.30309	0.597669	0.298835	+	0.954305i	$0.403402\pi$
$194$	−8.07090	−0.579457
$195$	0	0
$196$	0	0
$197$	− 25.4881i	− 1.81595i	−0.419019	−	0.907977i	$-0.637626\pi$
$197$	− 25.4881i	− 1.81595i	0.419019	−	0.907977i	$-0.362374\pi$
$198$	0	0
$199$	2.06372i	0.146293i	0.997321	+	0.0731465i	$0.0233041\pi$
$199$	2.06372i	0.146293i	−0.997321	+	0.0731465i	$0.976696\pi$
$200$	− 8.08165i	− 0.571459i
$201$	0	0
$202$	− 31.6272i	− 2.22528i
$203$	0	0
$204$	0	0
$205$	4.50679	0.314768
$206$	31.5801	2.20029
$207$	0	0
$208$	42.2650i	2.93055i
$209$	−26.7598	−1.85102
$210$	0	0
$211$	−6.21339	−0.427747	−0.213874	−	0.976861i	$-0.568608\pi$
$211$	−6.21339	−0.427747	−0.213874	+	0.976861i	$0.568608\pi$
$212$	45.4918i	3.12439i
$213$	0	0
$214$	9.80756	0.670431
$215$	−4.89765	−0.334017
$216$	0	0
$217$	0	0
$218$	13.5017i	0.914450i
$219$	0	0
$220$	21.6278i	1.45815i
$221$	− 20.9832i	− 1.41148i
$222$	0	0
$223$	− 24.0589i	− 1.61110i	−0.592526	−	0.805552i	$-0.701869\pi$
$223$	− 24.0589i	− 1.61110i	0.592526	−	0.805552i	$-0.298131\pi$
$224$	0	0
$225$	0	0
$226$	16.7322	1.11301
$227$	8.19698	0.544053	0.272026	−	0.962290i	$-0.412306\pi$
$227$	8.19698	0.544053	0.272026	+	0.962290i	$0.412306\pi$
$228$	0	0
$229$	− 3.06400i	− 0.202475i	−0.994862	−	0.101237i	$-0.967720\pi$
$229$	− 3.06400i	− 0.202475i	0.994862	−	0.101237i	$-0.0322802\pi$
$230$	7.08629	0.467256
$231$	0	0
$232$	−86.5008	−5.67906
$233$	− 13.3600i	− 0.875245i	−0.899159	−	0.437623i	$-0.855821\pi$
$233$	− 13.3600i	− 0.875245i	0.899159	−	0.437623i	$-0.144179\pi$
$234$	0	0
$235$	7.85801	0.512600
$236$	−39.3295	−2.56013
$237$	0	0
$238$	0	0
$239$	6.31296i	0.408352i	0.978934	+	0.204176i	$0.0654514\pi$
$239$	6.31296i	0.408352i	−0.978934	+	0.204176i	$0.934549\pi$
$240$	0	0
$241$	19.9082i	1.28240i	0.767374	+	0.641200i	$0.221563\pi$
$241$	19.9082i	1.28240i	−0.767374	+	0.641200i	$0.778437\pi$
$242$	− 19.5863i	− 1.25905i
$243$	0	0
$244$	69.3774i	4.44143i
$245$	0	0
$246$	0	0
$247$	−23.2216	−1.47756
$248$	50.2266	3.18939
$249$	0	0
$250$	− 2.65421i	− 0.167867i
$251$	1.12767	0.0711778	0.0355889	−	0.999367i	$-0.488669\pi$
$251$	1.12767	0.0711778	0.0355889	+	0.999367i	$0.488669\pi$
$252$	0	0
$253$	−11.4459	−0.719595
$254$	7.29969i	0.458023i
$255$	0	0
$256$	−1.55991	−0.0974942
$257$	−6.58199	−0.410573	−0.205287	−	0.978702i	$-0.565813\pi$
$257$	−6.58199	−0.410573	−0.205287	+	0.978702i	$0.565813\pi$
$258$	0	0
$259$	0	0
$260$	18.7682i	1.16395i
$261$	0	0
$262$	− 33.0065i	− 2.03915i
$263$	− 20.5273i	− 1.26577i	−0.774246	−	0.632885i	$-0.781871\pi$
$263$	− 20.5273i	− 1.26577i	0.774246	−	0.632885i	$-0.218129\pi$
$264$	0	0
$265$	9.01748i	0.553939i
$266$	0	0
$267$	0	0
$268$	−2.18959	−0.133751
$269$	−11.2229	−0.684271	−0.342135	−	0.939651i	$-0.611150\pi$
$269$	−11.2229	−0.684271	−0.342135	+	0.939651i	$0.611150\pi$
$270$	0	0
$271$	− 1.33130i	− 0.0808705i	−0.999182	−	0.0404352i	$-0.987126\pi$
$271$	− 1.33130i	− 0.0808705i	0.999182	−	0.0404352i	$-0.0128744\pi$
$272$	−64.0773	−3.88526
$273$	0	0
$274$	−22.5689	−1.36344
$275$	4.28711i	0.258523i
$276$	0	0
$277$	4.49357	0.269992	0.134996	−	0.990846i	$-0.456898\pi$
$277$	4.49357	0.269992	0.134996	+	0.990846i	$0.456898\pi$
$278$	−27.8595	−1.67090
$279$	0	0
$280$	0	0
$281$	− 7.64141i	− 0.455848i	−0.973679	−	0.227924i	$-0.926806\pi$
$281$	− 7.64141i	− 0.455848i	0.973679	−	0.227924i	$-0.0731938\pi$
$282$	0	0
$283$	26.6304i	1.58301i	0.611160	+	0.791507i	$0.290703\pi$
$283$	26.6304i	1.58301i	−0.611160	+	0.791507i	$0.709297\pi$
$284$	− 32.6993i	− 1.94034i
$285$	0	0
$286$	− 42.3326i	− 2.50318i
$287$	0	0
$288$	0	0
$289$	14.8124	0.871316
$290$	−28.4090	−1.66823
$291$	0	0
$292$	37.3691i	2.18686i
$293$	−8.53715	−0.498746	−0.249373	−	0.968408i	$-0.580224\pi$
$293$	−8.53715	−0.498746	−0.249373	+	0.968408i	$0.580224\pi$
$294$	0	0
$295$	−7.79598	−0.453899
$296$	− 13.5999i	− 0.790475i
$297$	0	0
$298$	−27.9206	−1.61739
$299$	−9.93248	−0.574410
$300$	0	0
$301$	0	0
$302$	24.6578i	1.41889i
$303$	0	0
$304$	70.9128i	4.06713i
$305$	13.7521i	0.787446i
$306$	0	0
$307$	34.1889i	1.95126i	0.219418	+	0.975631i	$0.429584\pi$
$307$	34.1889i	1.95126i	−0.219418	+	0.975631i	$0.570416\pi$
$308$	0	0
$309$	0	0
$310$	16.4956	0.936890
$311$	−16.2975	−0.924148	−0.462074	−	0.886841i	$-0.652894\pi$
$311$	−16.2975	−0.924148	−0.462074	+	0.886841i	$0.652894\pi$
$312$	0	0
$313$	27.1468i	1.53443i	0.641392	+	0.767214i	$0.278357\pi$
$313$	27.1468i	1.53443i	−0.641392	+	0.767214i	$0.721643\pi$
$314$	−23.5632	−1.32975
$315$	0	0
$316$	53.6614	3.01869
$317$	16.0827i	0.903295i	0.892196	+	0.451648i	$0.149164\pi$
$317$	16.0827i	0.903295i	−0.892196	+	0.451648i	$0.850836\pi$
$318$	0	0
$319$	45.8865	2.56915
$320$	14.4123	0.805670
$321$	0	0
$322$	0	0
$323$	− 35.2060i	− 1.95891i
$324$	0	0
$325$	3.72027i	0.206363i
$326$	− 66.0243i	− 3.65675i
$327$	0	0
$328$	36.4223i	2.01108i
$329$	0	0
$330$	0	0
$331$	7.39789	0.406625	0.203312	−	0.979114i	$-0.434829\pi$
$331$	7.39789	0.406625	0.203312	+	0.979114i	$0.434829\pi$
$332$	−44.9106	−2.46479
$333$	0	0
$334$	− 24.0780i	− 1.31749i
$335$	−0.434026	−0.0237134
$336$	0	0
$337$	−5.88030	−0.320320	−0.160160	−	0.987091i	$-0.551201\pi$
$337$	−5.88030	−0.320320	−0.160160	+	0.987091i	$0.551201\pi$
$338$	− 2.23059i	− 0.121328i
$339$	0	0
$340$	−28.4541	−1.54314
$341$	−26.6439	−1.44285
$342$	0	0
$343$	0	0
$344$	− 39.5811i	− 2.13407i
$345$	0	0
$346$	10.5060i	0.564805i
$347$	− 6.26608i	− 0.336381i	−0.985755	−	0.168190i	$-0.946208\pi$
$347$	− 6.26608i	− 0.336381i	0.985755	−	0.168190i	$-0.0537923\pi$
$348$	0	0
$349$	5.67407i	0.303726i	0.988402	+	0.151863i	$0.0485273\pi$
$349$	5.67407i	0.303726i	−0.988402	+	0.151863i	$0.951473\pi$
$350$	0	0
$351$	0	0
$352$	−59.9788	−3.19688
$353$	18.4457	0.981768	0.490884	−	0.871225i	$-0.336674\pi$
$353$	18.4457	0.981768	0.490884	+	0.871225i	$0.336674\pi$
$354$	0	0
$355$	− 6.48172i	− 0.344014i
$356$	50.0751	2.65397
$357$	0	0
$358$	−0.876769	−0.0463387
$359$	17.8403i	0.941574i	0.882247	+	0.470787i	$0.156030\pi$
$359$	17.8403i	0.941574i	−0.882247	+	0.470787i	$0.843970\pi$
$360$	0	0
$361$	−19.9616	−1.05061
$362$	5.55124	0.291767
$363$	0	0
$364$	0	0
$365$	7.40739i	0.387721i
$366$	0	0
$367$	− 23.3572i	− 1.21923i	−0.792696	−	0.609617i	$-0.791323\pi$
$367$	− 23.3572i	− 1.21923i	0.792696	−	0.609617i	$-0.208677\pi$
$368$	30.3312i	1.58112i
$369$	0	0
$370$	− 4.46652i	− 0.232203i
$371$	0	0
$372$	0	0
$373$	−21.2967	−1.10270	−0.551350	−	0.834274i	$-0.685887\pi$
$373$	−21.2967	−1.10270	−0.551350	+	0.834274i	$0.685887\pi$
$374$	64.1798	3.31866
$375$	0	0
$376$	63.5057i	3.27506i
$377$	39.8194	2.05080
$378$	0	0
$379$	19.2933	0.991028	0.495514	−	0.868600i	$-0.334980\pi$
$379$	19.2933	0.991028	0.495514	+	0.868600i	$0.334980\pi$
$380$	31.4895i	1.61538i
$381$	0	0
$382$	−20.5036	−1.04906
$383$	5.66172	0.289300	0.144650	−	0.989483i	$-0.453794\pi$
$383$	5.66172	0.289300	0.144650	+	0.989483i	$0.453794\pi$
$384$	0	0
$385$	0	0
$386$	22.0382i	1.12171i
$387$	0	0
$388$	− 15.3403i	− 0.778786i
$389$	34.7086i	1.75980i	0.475162	+	0.879898i	$0.342389\pi$
$389$	34.7086i	1.75980i	−0.475162	+	0.879898i	$0.657611\pi$
$390$	0	0
$391$	− 15.0585i	− 0.761540i
$392$	0	0
$393$	0	0
$394$	67.6509	3.40820
$395$	10.6369	0.535200
$396$	0	0
$397$	− 0.535451i	− 0.0268735i	−0.999910	−	0.0134367i	$-0.995723\pi$
$397$	− 0.535451i	− 0.0268735i	0.999910	−	0.0134367i	$-0.00427718\pi$
$398$	−5.47754	−0.274564
$399$	0	0
$400$	11.3607	0.568037
$401$	15.2904i	0.763568i	0.924252	+	0.381784i	$0.124690\pi$
$401$	15.2904i	0.763568i	−0.924252	+	0.381784i	$0.875310\pi$
$402$	0	0
$403$	−23.1211	−1.15174
$404$	60.1136	2.99077
$405$	0	0
$406$	0	0
$407$	7.21438i	0.357603i
$408$	0	0
$409$	− 28.5693i	− 1.41266i	−0.707881	−	0.706331i	$-0.750349\pi$
$409$	− 28.5693i	− 1.41266i	0.707881	−	0.706331i	$-0.249651\pi$
$410$	11.9620i	0.590759i
$411$	0	0
$412$	60.0241i	2.95717i
$413$	0	0
$414$	0	0
$415$	−8.90228	−0.436996
$416$	−52.0484	−2.55188
$417$	0	0
$418$	− 71.0262i	− 3.47401i
$419$	12.3199	0.601869	0.300934	−	0.953645i	$-0.402702\pi$
$419$	12.3199	0.601869	0.300934	+	0.953645i	$0.402702\pi$
$420$	0	0
$421$	13.5311	0.659466	0.329733	−	0.944074i	$-0.393041\pi$
$421$	13.5311	0.659466	0.329733	+	0.944074i	$0.393041\pi$
$422$	− 16.4916i	− 0.802801i
$423$	0	0
$424$	−72.8761	−3.53918
$425$	−5.64024	−0.273592
$426$	0	0
$427$	0	0
$428$	18.6412i	0.901055i
$429$	0	0
$430$	− 12.9994i	− 0.626887i
$431$	29.7477i	1.43289i	0.697641	+	0.716447i	$0.254233\pi$
$431$	29.7477i	1.43289i	−0.697641	+	0.716447i	$0.745767\pi$
$432$	0	0
$433$	− 26.2953i	− 1.26367i	−0.775102	−	0.631836i	$-0.782301\pi$
$433$	− 26.2953i	− 1.26367i	0.775102	−	0.631836i	$-0.217699\pi$
$434$	0	0
$435$	0	0
$436$	−25.6626	−1.22901
$437$	−16.6649	−0.797188
$438$	0	0
$439$	16.4055i	0.782993i	0.920180	+	0.391497i	$0.128043\pi$
$439$	16.4055i	0.782993i	−0.920180	+	0.391497i	$0.871957\pi$
$440$	−34.6469	−1.65173
$441$	0	0
$442$	55.6939	2.64909
$443$	− 19.8413i	− 0.942689i	−0.881949	−	0.471345i	$-0.843769\pi$
$443$	− 19.8413i	− 0.942689i	0.881949	−	0.471345i	$-0.156231\pi$
$444$	0	0
$445$	9.92599	0.470537
$446$	63.8574	3.02374
$447$	0	0
$448$	0	0
$449$	3.55114i	0.167589i	0.996483	+	0.0837944i	$0.0267039\pi$
$449$	3.55114i	0.167589i	−0.996483	+	0.0837944i	$0.973296\pi$
$450$	0	0
$451$	− 19.3211i	− 0.909795i
$452$	31.8028i	1.49588i
$453$	0	0
$454$	21.7565i	1.02108i
$455$	0	0
$456$	0	0
$457$	−0.513005	−0.0239973	−0.0119987	−	0.999928i	$-0.503819\pi$
$457$	−0.513005	−0.0239973	−0.0119987	+	0.999928i	$0.503819\pi$
$458$	8.13251	0.380007
$459$	0	0
$460$	13.4689i	0.627989i
$461$	4.51083	0.210091	0.105045	−	0.994467i	$-0.466501\pi$
$461$	4.51083	0.210091	0.105045	+	0.994467i	$0.466501\pi$
$462$	0	0
$463$	18.2002	0.845834	0.422917	−	0.906168i	$-0.361006\pi$
$463$	18.2002	0.845834	0.422917	+	0.906168i	$0.361006\pi$
$464$	− 121.598i	− 5.64504i
$465$	0	0
$466$	35.4604	1.64267
$467$	20.1747	0.933574	0.466787	−	0.884370i	$-0.345412\pi$
$467$	20.1747	0.933574	0.466787	+	0.884370i	$0.345412\pi$
$468$	0	0
$469$	0	0
$470$	20.8568i	0.962053i
$471$	0	0
$472$	− 63.0044i	− 2.90001i
$473$	20.9968i	0.965432i
$474$	0	0
$475$	6.24192i	0.286399i
$476$	0	0
$477$	0	0
$478$	−16.7559	−0.766399
$479$	−29.5954	−1.35225	−0.676125	−	0.736787i	$-0.736342\pi$
$479$	−29.5954	−1.35225	−0.676125	+	0.736787i	$0.736342\pi$
$480$	0	0
$481$	6.26049i	0.285454i
$482$	−52.8406	−2.40682
$483$	0	0
$484$	37.2275	1.69216
$485$	− 3.04079i	− 0.138075i
$486$	0	0
$487$	2.74195	0.124249	0.0621247	−	0.998068i	$-0.480212\pi$
$487$	2.74195	0.124249	0.0621247	+	0.998068i	$0.480212\pi$
$488$	−111.140	−5.03107
$489$	0	0
$490$	0	0
$491$	5.66445i	0.255633i	0.991798	+	0.127816i	$0.0407968\pi$
$491$	5.66445i	0.255633i	−0.991798	+	0.127816i	$0.959203\pi$
$492$	0	0
$493$	60.3696i	2.71891i
$494$	− 61.6351i	− 2.77309i
$495$	0	0
$496$	70.6058i	3.17029i
$497$	0	0
$498$	0	0
$499$	−32.9701	−1.47595	−0.737973	−	0.674830i	$-0.764217\pi$
$499$	−32.9701	−1.47595	−0.737973	+	0.674830i	$0.764217\pi$
$500$	5.04484	0.225612
$501$	0	0
$502$	2.99307i	0.133587i
$503$	−20.1522	−0.898543	−0.449271	−	0.893395i	$-0.648316\pi$
$503$	−20.1522	−0.898543	−0.449271	+	0.893395i	$0.648316\pi$
$504$	0	0
$505$	11.9159	0.530249
$506$	− 30.3797i	− 1.35054i
$507$	0	0
$508$	−13.8745	−0.615580
$509$	−20.3914	−0.903834	−0.451917	−	0.892060i	$-0.649260\pi$
$509$	−20.3914	−0.903834	−0.451917	+	0.892060i	$0.649260\pi$
$510$	0	0
$511$	0	0
$512$	− 24.6847i	− 1.09092i
$513$	0	0
$514$	− 17.4700i	− 0.770569i
$515$	11.8981i	0.524293i
$516$	0	0
$517$	− 33.6882i	− 1.48160i
$518$	0	0
$519$	0	0
$520$	−30.0659	−1.31848
$521$	−15.1913	−0.665545	−0.332772	−	0.943007i	$-0.607984\pi$
$521$	−15.1913	−0.665545	−0.332772	+	0.943007i	$0.607984\pi$
$522$	0	0
$523$	− 45.5824i	− 1.99318i	−0.0825054	−	0.996591i	$-0.526292\pi$
$523$	− 45.5824i	− 1.99318i	0.0825054	−	0.996591i	$-0.473708\pi$
$524$	62.7352	2.74060
$525$	0	0
$526$	54.4839	2.37561
$527$	− 35.0535i	− 1.52696i
$528$	0	0
$529$	15.8720	0.690088
$530$	−23.9343	−1.03964
$531$	0	0
$532$	0	0
$533$	− 16.7665i	− 0.726236i
$534$	0	0
$535$	3.69509i	0.159753i
$536$	− 3.50765i	− 0.151507i
$537$	0	0
$538$	− 29.7879i	− 1.28425i
$539$	0	0
$540$	0	0
$541$	−22.9445	−0.986460	−0.493230	−	0.869899i	$-0.664184\pi$
$541$	−22.9445	−0.986460	−0.493230	+	0.869899i	$0.664184\pi$
$542$	3.53354	0.151779
$543$	0	0
$544$	− 78.9098i	− 3.38323i
$545$	−5.08689	−0.217899
$546$	0	0
$547$	−42.2306	−1.80565	−0.902825	−	0.430008i	$-0.858511\pi$
$547$	−42.2306	−1.80565	−0.902825	+	0.430008i	$0.858511\pi$
$548$	− 42.8966i	− 1.83245i
$549$	0	0
$550$	−11.3789	−0.485198
$551$	66.8095	2.84618
$552$	0	0
$553$	0	0
$554$	11.9269i	0.506724i
$555$	0	0
$556$	− 52.9523i	− 2.24568i
$557$	− 39.9842i	− 1.69419i	−0.531444	−	0.847093i	$-0.678351\pi$
$557$	− 39.9842i	− 1.69419i	0.531444	−	0.847093i	$-0.321649\pi$
$558$	0	0
$559$	18.2206i	0.770648i
$560$	0	0
$561$	0	0
$562$	20.2819	0.855541
$563$	−31.0191	−1.30730	−0.653649	−	0.756798i	$-0.726763\pi$
$563$	−31.0191	−1.30730	−0.653649	+	0.756798i	$0.726763\pi$
$564$	0	0
$565$	6.30402i	0.265212i
$566$	−70.6828	−2.97102
$567$	0	0
$568$	52.3830	2.19794
$569$	− 28.0105i	− 1.17426i	−0.809492	−	0.587131i	$-0.800257\pi$
$569$	− 28.0105i	− 1.17426i	0.809492	−	0.587131i	$-0.199743\pi$
$570$	0	0
$571$	27.3784	1.14575	0.572875	−	0.819642i	$-0.305828\pi$
$571$	27.3784	1.14575	0.572875	+	0.819642i	$0.305828\pi$
$572$	80.4612	3.36425
$573$	0	0
$574$	0	0
$575$	2.66983i	0.111340i
$576$	0	0
$577$	42.4728i	1.76817i	0.467329	+	0.884084i	$0.345217\pi$
$577$	42.4728i	1.76817i	−0.467329	+	0.884084i	$0.654783\pi$
$578$	39.3152i	1.63529i
$579$	0	0
$580$	− 53.9967i	− 2.24209i
$581$	0	0
$582$	0	0
$583$	38.6589	1.60109
$584$	−59.8640	−2.47719
$585$	0	0
$586$	− 22.6594i	− 0.936052i
$587$	−29.2099	−1.20562	−0.602811	−	0.797884i	$-0.705953\pi$
$587$	−29.2099	−1.20562	−0.602811	+	0.797884i	$0.705953\pi$
$588$	0	0
$589$	−38.7929	−1.59843
$590$	− 20.6922i	− 0.851883i
$591$	0	0
$592$	19.1179	0.785741
$593$	18.3488	0.753496	0.376748	−	0.926316i	$-0.377042\pi$
$593$	18.3488	0.753496	0.376748	+	0.926316i	$0.377042\pi$
$594$	0	0
$595$	0	0
$596$	− 53.0684i	− 2.17377i
$597$	0	0
$598$	− 26.3629i	− 1.07806i
$599$	39.2269i	1.60277i	0.598149	+	0.801385i	$0.295903\pi$
$599$	39.2269i	1.60277i	−0.598149	+	0.801385i	$0.704097\pi$
$600$	0	0
$601$	20.7149i	0.844977i	0.906368	+	0.422488i	$0.138843\pi$
$601$	20.7149i	0.844977i	−0.906368	+	0.422488i	$0.861157\pi$
$602$	0	0
$603$	0	0
$604$	−46.8668	−1.90698
$605$	7.37933	0.300012
$606$	0	0
$607$	11.2157i	0.455229i	0.973751	+	0.227615i	$0.0730927\pi$
$607$	11.2157i	0.455229i	−0.973751	+	0.227615i	$0.926907\pi$
$608$	−87.3275	−3.54160
$609$	0	0
$610$	−36.5011	−1.47789
$611$	− 29.2339i	− 1.18268i
$612$	0	0
$613$	−44.6233	−1.80232	−0.901158	−	0.433490i	$-0.857282\pi$
$613$	−44.6233	−1.80232	−0.901158	+	0.433490i	$0.857282\pi$
$614$	−90.7445	−3.66215
$615$	0	0
$616$	0	0
$617$	9.26765i	0.373102i	0.982445	+	0.186551i	$0.0597309\pi$
$617$	9.26765i	0.373102i	−0.982445	+	0.186551i	$0.940269\pi$
$618$	0	0
$619$	− 27.0338i	− 1.08658i	−0.839546	−	0.543289i	$-0.817179\pi$
$619$	− 27.0338i	− 1.08658i	0.839546	−	0.543289i	$-0.182821\pi$
$620$	31.3531i	1.25917i
$621$	0	0
$622$	− 43.2571i	− 1.73445i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−72.0533	−2.87983
$627$	0	0
$628$	− 44.7865i	− 1.78717i
$629$	−9.49144	−0.378448
$630$	0	0
$631$	−10.6725	−0.424864	−0.212432	−	0.977176i	$-0.568138\pi$
$631$	−10.6725	−0.424864	−0.212432	+	0.977176i	$0.568138\pi$
$632$	85.9636i	3.41945i
$633$	0	0
$634$	−42.6869	−1.69531
$635$	−2.75023	−0.109140
$636$	0	0
$637$	0	0
$638$	121.792i	4.82181i
$639$	0	0
$640$	10.2722i	0.406045i
$641$	− 36.2449i	− 1.43159i	−0.698311	−	0.715794i	$-0.746065\pi$
$641$	− 36.2449i	− 1.43159i	0.698311	−	0.715794i	$-0.253935\pi$
$642$	0	0
$643$	5.28720i	0.208507i	0.994551	+	0.104253i	$0.0332453\pi$
$643$	5.28720i	0.208507i	−0.994551	+	0.104253i	$0.966755\pi$
$644$	0	0
$645$	0	0
$646$	93.4441	3.67651
$647$	6.28826	0.247217	0.123608	−	0.992331i	$-0.460553\pi$
$647$	6.28826	0.247217	0.123608	+	0.992331i	$0.460553\pi$
$648$	0	0
$649$	33.4222i	1.31194i
$650$	−9.87438	−0.387305
$651$	0	0
$652$	125.492	4.91464
$653$	26.7840i	1.04814i	0.851675	+	0.524070i	$0.175587\pi$
$653$	26.7840i	1.04814i	−0.851675	+	0.524070i	$0.824413\pi$
$654$	0	0
$655$	12.4355	0.485896
$656$	−51.2004	−1.99904
$657$	0	0
$658$	0	0
$659$	46.3143i	1.80415i	0.431579	+	0.902075i	$0.357957\pi$
$659$	46.3143i	1.80415i	−0.431579	+	0.902075i	$0.642043\pi$
$660$	0	0
$661$	14.5037i	0.564127i	0.959396	+	0.282063i	$0.0910189\pi$
$661$	14.5037i	0.564127i	−0.959396	+	0.282063i	$0.908981\pi$
$662$	19.6356i	0.763158i
$663$	0	0
$664$	− 71.9451i	− 2.79201i
$665$	0	0
$666$	0	0
$667$	28.5761	1.10647
$668$	45.7650	1.77070
$669$	0	0
$670$	− 1.15200i	− 0.0445055i
$671$	58.9570	2.27601
$672$	0	0
$673$	−42.3105	−1.63095	−0.815475	−	0.578792i	$-0.803524\pi$
$673$	−42.3105	−1.63095	−0.815475	+	0.578792i	$0.803524\pi$
$674$	− 15.6076i	− 0.601181i
$675$	0	0
$676$	4.23966	0.163064
$677$	−29.9586	−1.15140	−0.575702	−	0.817660i	$-0.695271\pi$
$677$	−29.9586	−1.15140	−0.575702	+	0.817660i	$0.695271\pi$
$678$	0	0
$679$	0	0
$680$	− 45.5825i	− 1.74801i
$681$	0	0
$682$	− 70.7187i	− 2.70796i
$683$	13.0827i	0.500597i	0.968169	+	0.250298i	$0.0805287\pi$
$683$	13.0827i	0.500597i	−0.968169	+	0.250298i	$0.919471\pi$
$684$	0	0
$685$	− 8.50306i	− 0.324885i
$686$	0	0
$687$	0	0
$688$	55.6409	2.12129
$689$	33.5474	1.27806
$690$	0	0
$691$	− 22.9993i	− 0.874934i	−0.899234	−	0.437467i	$-0.855876\pi$
$691$	− 22.9993i	− 0.874934i	0.899234	−	0.437467i	$-0.144124\pi$
$692$	−19.9686	−0.759093
$693$	0	0
$694$	16.6315	0.631323
$695$	− 10.4963i	− 0.398148i
$696$	0	0
$697$	25.4194	0.962827
$698$	−15.0602	−0.570037
$699$	0	0
$700$	0	0
$701$	− 4.26800i	− 0.161200i	−0.996747	−	0.0806000i	$-0.974316\pi$
$701$	− 4.26800i	− 0.161200i	0.996747	−	0.0806000i	$-0.0256836\pi$
$702$	0	0
$703$	10.5039i	0.396164i
$704$	− 61.7870i	− 2.32868i
$705$	0	0
$706$	48.9589i	1.84259i
$707$	0	0
$708$	0	0
$709$	−9.39557	−0.352858	−0.176429	−	0.984313i	$-0.556455\pi$
$709$	−9.39557	−0.352858	−0.176429	+	0.984313i	$0.556455\pi$
$710$	17.2039	0.645650
$711$	0	0
$712$	80.2184i	3.00631i
$713$	−16.5927	−0.621402
$714$	0	0
$715$	15.9492	0.596466
$716$	− 1.66647i	− 0.0622788i
$717$	0	0
$718$	−47.3519	−1.76716
$719$	5.99462	0.223561	0.111781	−	0.993733i	$-0.464345\pi$
$719$	5.99462	0.223561	0.111781	+	0.993733i	$0.464345\pi$
$720$	0	0
$721$	0	0
$722$	− 52.9823i	− 1.97180i
$723$	0	0
$724$	10.5512i	0.392132i
$725$	− 10.7034i	− 0.397513i
$726$	0	0
$727$	− 5.62979i	− 0.208797i	−0.994536	−	0.104399i	$-0.966708\pi$
$727$	− 5.62979i	− 0.208797i	0.994536	−	0.104399i	$-0.0332918\pi$
$728$	0	0
$729$	0	0
$730$	−19.6608	−0.727678
$731$	−27.6239	−1.02171
$732$	0	0
$733$	− 18.5974i	− 0.686911i	−0.939169	−	0.343455i	$-0.888403\pi$
$733$	− 18.5974i	− 0.686911i	0.939169	−	0.343455i	$-0.111597\pi$
$734$	61.9948	2.28827
$735$	0	0
$736$	−37.3522	−1.37682
$737$	1.86072i	0.0685404i
$738$	0	0
$739$	2.50186	0.0920326	0.0460163	−	0.998941i	$-0.485347\pi$
$739$	2.50186	0.0920326	0.0460163	+	0.998941i	$0.485347\pi$
$740$	8.48949	0.312080
$741$	0	0
$742$	0	0
$743$	− 12.2890i	− 0.450840i	−0.974262	−	0.225420i	$-0.927624\pi$
$743$	− 12.2890i	− 0.450840i	0.974262	−	0.225420i	$-0.0723755\pi$
$744$	0	0
$745$	− 10.5193i	− 0.385399i
$746$	− 56.5259i	− 2.06956i
$747$	0	0
$748$	121.986i	4.46025i
$749$	0	0
$750$	0	0
$751$	−3.91861	−0.142992	−0.0714960	−	0.997441i	$-0.522777\pi$
$751$	−3.91861	−0.142992	−0.0714960	+	0.997441i	$0.522777\pi$
$752$	−89.2727	−3.25544
$753$	0	0
$754$	105.689i	3.84897i
$755$	−9.29005	−0.338100
$756$	0	0
$757$	2.47039	0.0897878	0.0448939	−	0.998992i	$-0.485705\pi$
$757$	2.47039	0.0897878	0.0448939	+	0.998992i	$0.485705\pi$
$758$	51.2084i	1.85997i
$759$	0	0
$760$	−50.4450	−1.82983
$761$	13.1684	0.477356	0.238678	−	0.971099i	$-0.423286\pi$
$761$	13.1684	0.477356	0.238678	+	0.971099i	$0.423286\pi$
$762$	0	0
$763$	0	0
$764$	− 38.9711i	− 1.40993i
$765$	0	0
$766$	15.0274i	0.542962i
$767$	29.0031i	1.04724i
$768$	0	0
$769$	28.6058i	1.03155i	0.856723	+	0.515776i	$0.172496\pi$
$769$	28.6058i	1.03155i	−0.856723	+	0.515776i	$0.827504\pi$
$770$	0	0
$771$	0	0
$772$	−41.8878	−1.50757
$773$	27.3916	0.985207	0.492604	−	0.870254i	$-0.336045\pi$
$773$	27.3916	0.985207	0.492604	+	0.870254i	$0.336045\pi$
$774$	0	0
$775$	6.21489i	0.223246i
$776$	24.5746	0.882177
$777$	0	0
$778$	−92.1240	−3.30281
$779$	− 28.1310i	− 1.00790i
$780$	0	0
$781$	−27.7879	−0.994328
$782$	39.9684	1.42927
$783$	0	0
$784$	0	0
$785$	− 8.87768i	− 0.316858i
$786$	0	0
$787$	37.7772i	1.34661i	0.739363	+	0.673307i	$0.235127\pi$
$787$	37.7772i	1.34661i	−0.739363	+	0.673307i	$0.764873\pi$
$788$	128.584i	4.58060i
$789$	0	0
$790$	28.2325i	1.00447i
$791$	0	0
$792$	0	0
$793$	51.1617	1.81680
$794$	1.42120	0.0504365
$795$	0	0
$796$	− 10.4111i	− 0.369013i
$797$	−7.98820	−0.282956	−0.141478	−	0.989941i	$-0.545186\pi$
$797$	−7.98820	−0.282956	−0.141478	+	0.989941i	$0.545186\pi$
$798$	0	0
$799$	44.3211	1.56797
$800$	13.9905i	0.494639i
$801$	0	0
$802$	−40.5840	−1.43307
$803$	31.7563	1.12066
$804$	0	0
$805$	0	0
$806$	− 61.3682i	− 2.16160i
$807$	0	0
$808$	96.2999i	3.38782i
$809$	23.0803i	0.811460i	0.913993	+	0.405730i	$0.132983\pi$
$809$	23.0803i	0.811460i	−0.913993	+	0.405730i	$0.867017\pi$
$810$	0	0
$811$	6.69330i	0.235033i	0.993071	+	0.117517i	$0.0374934\pi$
$811$	6.69330i	0.235033i	−0.993071	+	0.117517i	$0.962507\pi$
$812$	0	0
$813$	0	0
$814$	−19.1485	−0.671154
$815$	24.8753	0.871343
$816$	0	0
$817$	30.5707i	1.06953i
$818$	75.8291	2.65130
$819$	0	0
$820$	−22.7360	−0.793976
$821$	− 17.1276i	− 0.597759i	−0.954291	−	0.298880i	$-0.903387\pi$
$821$	− 17.1276i	− 0.597759i	0.954291	−	0.298880i	$-0.0966129\pi$
$822$	0	0
$823$	−9.93194	−0.346206	−0.173103	−	0.984904i	$-0.555379\pi$
$823$	−9.93194	−0.346206	−0.173103	+	0.984904i	$0.555379\pi$
$824$	−96.1564	−3.34977
$825$	0	0
$826$	0	0
$827$	35.1507i	1.22231i	0.791511	+	0.611155i	$0.209295\pi$
$827$	35.1507i	1.22231i	−0.791511	+	0.611155i	$0.790705\pi$
$828$	0	0
$829$	− 47.9501i	− 1.66537i	−0.553743	−	0.832687i	$-0.686801\pi$
$829$	− 47.9501i	− 1.66537i	0.553743	−	0.832687i	$-0.313199\pi$
$830$	− 23.6285i	− 0.820158i
$831$	0	0
$832$	− 53.6175i	− 1.85885i
$833$	0	0
$834$	0	0
$835$	9.07164	0.313937
$836$	134.999	4.66904
$837$	0	0
$838$	32.6997i	1.12959i
$839$	−10.2976	−0.355512	−0.177756	−	0.984075i	$-0.556884\pi$
$839$	−10.2976	−0.355512	−0.177756	+	0.984075i	$0.556884\pi$
$840$	0	0
$841$	−85.5619	−2.95041
$842$	35.9144i	1.23769i
$843$	0	0
$844$	31.3456	1.07896
$845$	0.840395	0.0289105
$846$	0	0
$847$	0	0
$848$	− 102.445i	− 3.51798i
$849$	0	0
$850$	− 14.9704i	− 0.513481i
$851$	4.49280i	0.154011i
$852$	0	0
$853$	− 2.97705i	− 0.101932i	−0.998700	−	0.0509662i	$-0.983770\pi$
$853$	− 2.97705i	− 0.101932i	0.998700	−	0.0509662i	$-0.0162301\pi$
$854$	0	0
$855$	0	0
$856$	−29.8625	−1.02068
$857$	−20.3724	−0.695907	−0.347954	−	0.937512i	$-0.613123\pi$
$857$	−20.3724	−0.695907	−0.347954	+	0.937512i	$0.613123\pi$
$858$	0	0
$859$	46.0709i	1.57192i	0.618279	+	0.785959i	$0.287830\pi$
$859$	46.0709i	1.57192i	−0.618279	+	0.785959i	$0.712170\pi$
$860$	24.7079	0.842531
$861$	0	0
$862$	−78.9566	−2.68927
$863$	6.24165i	0.212468i	0.994341	+	0.106234i	$0.0338793\pi$
$863$	6.24165i	0.212468i	−0.994341	+	0.106234i	$0.966121\pi$
$864$	0	0
$865$	−3.95823	−0.134584
$866$	69.7934	2.37167
$867$	0	0
$868$	0	0
$869$	− 45.6015i	− 1.54693i
$870$	0	0
$871$	1.61469i	0.0547118i
$872$	− 41.1105i	− 1.39218i
$873$	0	0
$874$	− 44.2321i	− 1.49617i
$875$	0	0
$876$	0	0
$877$	24.1876	0.816756	0.408378	−	0.912813i	$-0.366094\pi$
$877$	24.1876	0.816756	0.408378	+	0.912813i	$0.366094\pi$
$878$	−43.5438	−1.46953
$879$	0	0
$880$	− 48.7047i	− 1.64184i
$881$	22.4548	0.756522	0.378261	−	0.925699i	$-0.376522\pi$
$881$	22.4548	0.756522	0.378261	+	0.925699i	$0.376522\pi$
$882$	0	0
$883$	32.8640	1.10596	0.552981	−	0.833194i	$-0.313490\pi$
$883$	32.8640	1.10596	0.552981	+	0.833194i	$0.313490\pi$
$884$	105.857i	3.56036i
$885$	0	0
$886$	52.6631	1.76925
$887$	−2.51188	−0.0843407	−0.0421703	−	0.999110i	$-0.513427\pi$
$887$	−2.51188	−0.0843407	−0.0421703	+	0.999110i	$0.513427\pi$
$888$	0	0
$889$	0	0
$890$	26.3457i	0.883110i
$891$	0	0
$892$	121.373i	4.06388i
$893$	− 49.0491i	− 1.64136i
$894$	0	0
$895$	− 0.330331i	− 0.0110418i
$896$	0	0
$897$	0	0
$898$	−9.42549	−0.314533
$899$	66.5202	2.21857
$900$	0	0
$901$	50.8608i	1.69442i
$902$	51.2823	1.70751
$903$	0	0
$904$	−50.9469	−1.69447
$905$	2.09148i	0.0695233i
$906$	0	0
$907$	−34.9873	−1.16174	−0.580868	−	0.813998i	$-0.697287\pi$
$907$	−34.9873	−1.16174	−0.580868	+	0.813998i	$0.697287\pi$
$908$	−41.3525	−1.37233
$909$	0	0
$910$	0	0
$911$	19.0468i	0.631048i	0.948918	+	0.315524i	$0.102180\pi$
$911$	19.0468i	0.631048i	−0.948918	+	0.315524i	$0.897820\pi$
$912$	0	0
$913$	38.1651i	1.26308i
$914$	− 1.36162i	− 0.0450385i
$915$	0	0
$916$	15.4574i	0.510727i
$917$	0	0
$918$	0	0
$919$	32.3628	1.06755	0.533775	−	0.845626i	$-0.320773\pi$
$919$	32.3628	1.06755	0.533775	+	0.845626i	$0.320773\pi$
$920$	−21.5766	−0.711360
$921$	0	0
$922$	11.9727i	0.394300i
$923$	−24.1138	−0.793714
$924$	0	0
$925$	1.68281	0.0553303
$926$	48.3071i	1.58747i
$927$	0	0
$928$	149.745	4.91563
$929$	11.5607	0.379295	0.189647	−	0.981852i	$-0.439265\pi$
$929$	11.5607	0.379295	0.189647	+	0.981852i	$0.439265\pi$
$930$	0	0
$931$	0	0
$932$	67.3993i	2.20774i
$933$	0	0
$934$	53.5479i	1.75214i
$935$	24.1804i	0.790782i
$936$	0	0
$937$	11.5562i	0.377524i	0.982023	+	0.188762i	$0.0604475\pi$
$937$	11.5562i	0.377524i	−0.982023	+	0.188762i	$0.939553\pi$
$938$	0	0
$939$	0	0
$940$	−39.6424	−1.29299
$941$	−7.69522	−0.250857	−0.125428	−	0.992103i	$-0.540031\pi$
$941$	−7.69522	−0.250857	−0.125428	+	0.992103i	$0.540031\pi$
$942$	0	0
$943$	− 12.0323i	− 0.391827i
$944$	88.5680	2.88264
$945$	0	0
$946$	−55.7299	−1.81193
$947$	− 29.6330i	− 0.962944i	−0.876462	−	0.481472i	$-0.840102\pi$
$947$	− 29.6330i	− 0.962944i	0.876462	−	0.481472i	$-0.159898\pi$
$948$	0	0
$949$	27.5575	0.894554
$950$	−16.5674	−0.537517
$951$	0	0
$952$	0	0
$953$	3.73027i	0.120835i	0.998173	+	0.0604177i	$0.0192433\pi$
$953$	3.73027i	0.120835i	−0.998173	+	0.0604177i	$0.980757\pi$
$954$	0	0
$955$	− 7.72494i	− 0.249973i
$956$	− 31.8479i	− 1.03003i
$957$	0	0
$958$	− 78.5525i	− 2.53792i
$959$	0	0
$960$	0	0
$961$	−7.62491	−0.245965
$962$	−16.6167	−0.535743
$963$	0	0
$964$	− 100.434i	− 3.23475i
$965$	−8.30309	−0.267286
$966$	0	0
$967$	7.55618	0.242990	0.121495	−	0.992592i	$-0.461231\pi$
$967$	7.55618	0.242990	0.121495	+	0.992592i	$0.461231\pi$
$968$	59.6371i	1.91681i
$969$	0	0
$970$	8.07090	0.259141
$971$	−23.8452	−0.765229	−0.382614	−	0.923908i	$-0.624976\pi$
$971$	−23.8452	−0.765229	−0.382614	+	0.923908i	$0.624976\pi$
$972$	0	0
$973$	0	0
$974$	7.27771i	0.233193i
$975$	0	0
$976$	− 156.235i	− 5.00094i
$977$	− 6.93993i	− 0.222028i	−0.993819	−	0.111014i	$-0.964590\pi$
$977$	− 6.93993i	− 0.222028i	0.993819	−	0.111014i	$-0.0354099\pi$
$978$	0	0
$979$	− 42.5538i	− 1.36003i
$980$	0	0
$981$	0	0
$982$	−15.0346	−0.479775
$983$	9.73963	0.310646	0.155323	−	0.987864i	$-0.450358\pi$
$983$	9.73963	0.310646	0.155323	+	0.987864i	$0.450358\pi$
$984$	0	0
$985$	25.4881i	0.812120i
$986$	−160.234	−5.10288
$987$	0	0
$988$	117.149	3.72702
$989$	13.0759i	0.415789i
$990$	0	0
$991$	−56.3031	−1.78853	−0.894263	−	0.447541i	$-0.852300\pi$
$991$	−56.3031	−1.78853	−0.894263	+	0.447541i	$0.852300\pi$
$992$	−86.9494	−2.76065
$993$	0	0
$994$	0	0
$995$	− 2.06372i	− 0.0654242i
$996$	0	0
$997$	− 52.5782i	− 1.66517i	−0.553899	−	0.832584i	$-0.686861\pi$
$997$	− 52.5782i	− 1.66517i	0.553899	−	0.832584i	$-0.313139\pi$
$998$	− 87.5097i	− 2.77007i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.b.c.881.16	yes	16
3.2	odd	2		2205.2.b.d.881.1	yes	16
7.6	odd	2		2205.2.b.d.881.16	yes	16
21.20	even	2	inner	2205.2.b.c.881.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.2.b.c.881.1	✓	16	21.20	even	2	inner
2205.2.b.c.881.16	yes	16	1.1	even	1	trivial
2205.2.b.d.881.1	yes	16	3.2	odd	2
2205.2.b.d.881.16	yes	16	7.6	odd	2

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.65421i q^{2} -5.04484 q^{4} -1.00000 q^{5} -8.08165i q^{8} -2.65421i q^{10} +4.28711i q^{11} +3.72027i q^{13} +11.3607 q^{16} -5.64024 q^{17} +6.24192i q^{19} +5.04484 q^{20} -11.3789 q^{22} +2.66983i q^{23} +1.00000 q^{25} -9.87438 q^{26} -10.7034i q^{29} +6.21489i q^{31} +13.9905i q^{32} -14.9704i q^{34} +1.68281 q^{37} -16.5674 q^{38} +8.08165i q^{40} -4.50679 q^{41} +4.89765 q^{43} -21.6278i q^{44} -7.08629 q^{46} -7.85801 q^{47} +2.65421i q^{50} -18.7682i q^{52} -9.01748i q^{53} -4.28711i q^{55} +28.4090 q^{58} +7.79598 q^{59} -13.7521i q^{61} -16.4956 q^{62} -14.4123 q^{64} -3.72027i q^{65} +0.434026 q^{67} +28.4541 q^{68} +6.48172i q^{71} -7.40739i q^{73} +4.46652i q^{74} -31.4895i q^{76} -10.6369 q^{79} -11.3607 q^{80} -11.9620i q^{82} +8.90228 q^{83} +5.64024 q^{85} +12.9994i q^{86} +34.6469 q^{88} -9.92599 q^{89} -13.4689i q^{92} -20.8568i q^{94} -6.24192i q^{95} +3.04079i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} - 16 q^{5} + 16 q^{20} - 16 q^{22} + 16 q^{25} - 32 q^{26} + 16 q^{38} + 32 q^{41} + 32 q^{43} - 16 q^{46} - 32 q^{47} + 48 q^{58} + 32 q^{59} + 32 q^{62} + 16 q^{64} + 16 q^{68} + 32 q^{79}+ \cdots - 64 q^{89}+O(q^{100}) \) 16 * q - 16 * q^4 - 16 * q^5 + 16 * q^20 - 16 * q^22 + 16 * q^25 - 32 * q^26 + 16 * q^38 + 32 * q^41 + 32 * q^43 - 16 * q^46 - 32 * q^47 + 48 * q^58 + 32 * q^59 + 32 * q^62 + 16 * q^64 + 16 * q^68 + 32 * q^79 + 16 * q^88 - 64 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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