LMFDB - Embedded newform 2205.2.b.d.881.2

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)

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")

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);

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			881.2
Root			$0.500000 - 3.00459i$ of defining polynomial
Character	$\chi$	$=$	2205.881
Dual form			2205.2.b.d.881.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14110i q^{2} -2.58431 q^{4} +1.00000 q^{5} +1.25106i q^{8} -2.14110i q^{10} +4.18075i q^{11} +4.24055i q^{13} -2.48998 q^{16} -3.08329 q^{17} -3.93398i q^{19} -2.58431 q^{20} +8.95140 q^{22} +3.52184i q^{23} +1.00000 q^{25} +9.07943 q^{26} +7.16190i q^{29} -4.25812i q^{31} +7.83340i q^{32} +6.60162i q^{34} +1.22460 q^{37} -8.42304 q^{38} +1.25106i q^{40} +7.62731 q^{41} +4.88515 q^{43} -10.8043i q^{44} +7.54061 q^{46} +2.24777 q^{47} -2.14110i q^{50} -10.9589i q^{52} +7.99310i q^{53} +4.18075i q^{55} +15.3343 q^{58} -6.03279 q^{59} +14.0213i q^{61} -9.11706 q^{62} +11.7921 q^{64} +4.24055i q^{65} +1.87745 q^{67} +7.96815 q^{68} +16.3448i q^{71} +11.1606i q^{73} -2.62199i q^{74} +10.1666i q^{76} +15.6282 q^{79} -2.48998 q^{80} -16.3308i q^{82} -17.0540 q^{83} -3.08329 q^{85} -10.4596i q^{86} -5.23035 q^{88} +3.04499 q^{89} -9.10151i q^{92} -4.81271i q^{94} -3.93398i q^{95} -19.3686i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 2.14110i	− 1.51399i	−0.653423	−	0.756993i	$-0.726668\pi$
$2$	− 2.14110i	− 1.51399i	0.653423	−	0.756993i	$-0.273332\pi$
$3$	0	0
$3$	0	0
$4$	−2.58431	−1.29215
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.25106i	0.442315i
$9$	0	0
$10$	− 2.14110i	− 0.677075i
$11$	4.18075i	1.26054i	0.776375	+	0.630272i	$0.217056\pi$
$11$	4.18075i	1.26054i	−0.776375	+	0.630272i	$0.782944\pi$
$12$	0	0
$13$	4.24055i	1.17612i	0.808819	+	0.588058i	$0.200107\pi$
$13$	4.24055i	1.17612i	−0.808819	+	0.588058i	$0.799893\pi$
$14$	0	0
$15$	0	0
$16$	−2.48998	−0.622494
$17$	−3.08329	−0.747807	−0.373903	−	0.927468i	$-0.621981\pi$
$17$	−3.08329	−0.747807	−0.373903	+	0.927468i	$0.621981\pi$
$18$	0	0
$19$	− 3.93398i	− 0.902516i	−0.892393	−	0.451258i	$-0.850975\pi$
$19$	− 3.93398i	− 0.902516i	0.892393	−	0.451258i	$-0.149025\pi$
$20$	−2.58431	−0.577868
$21$	0	0
$22$	8.95140	1.90844
$23$	3.52184i	0.734355i	0.930151	+	0.367177i	$0.119676\pi$
$23$	3.52184i	0.734355i	−0.930151	+	0.367177i	$0.880324\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.07943	1.78062
$27$	0	0
$28$	0	0
$29$	7.16190i	1.32993i	0.746874	+	0.664966i	$0.231554\pi$
$29$	7.16190i	1.32993i	−0.746874	+	0.664966i	$0.768446\pi$
$30$	0	0
$31$	− 4.25812i	− 0.764781i	−0.924001	−	0.382391i	$-0.875101\pi$
$31$	− 4.25812i	− 0.764781i	0.924001	−	0.382391i	$-0.124899\pi$
$32$	7.83340i	1.38476i
$33$	0	0
$34$	6.60162i	1.13217i
$35$	0	0
$36$	0	0
$37$	1.22460	0.201323	0.100662	−	0.994921i	$-0.467904\pi$
$37$	1.22460	0.201323	0.100662	+	0.994921i	$0.467904\pi$
$38$	−8.42304	−1.36640
$39$	0	0
$40$	1.25106i	0.197809i
$41$	7.62731	1.19119	0.595593	−	0.803286i	$-0.296917\pi$
$41$	7.62731	1.19119	0.595593	+	0.803286i	$0.296917\pi$
$42$	0	0
$43$	4.88515	0.744978	0.372489	−	0.928037i	$-0.378504\pi$
$43$	4.88515	0.744978	0.372489	+	0.928037i	$0.378504\pi$
$44$	− 10.8043i	− 1.62881i
$45$	0	0
$46$	7.54061	1.11180
$47$	2.24777	0.327872	0.163936	−	0.986471i	$-0.447581\pi$
$47$	2.24777	0.327872	0.163936	+	0.986471i	$0.447581\pi$
$48$	0	0
$49$	0	0
$50$	− 2.14110i	− 0.302797i
$51$	0	0
$52$	− 10.9589i	− 1.51972i
$53$	7.99310i	1.09794i	0.835843	+	0.548968i	$0.184979\pi$
$53$	7.99310i	1.09794i	−0.835843	+	0.548968i	$0.815021\pi$
$54$	0	0
$55$	4.18075i	0.563732i
$56$	0	0
$57$	0	0
$58$	15.3343	2.01350
$59$	−6.03279	−0.785402	−0.392701	−	0.919666i	$-0.628459\pi$
$59$	−6.03279	−0.785402	−0.392701	+	0.919666i	$0.628459\pi$
$60$	0	0
$61$	14.0213i	1.79524i	0.440767	+	0.897622i	$0.354707\pi$
$61$	14.0213i	1.79524i	−0.440767	+	0.897622i	$0.645293\pi$
$62$	−9.11706	−1.15787
$63$	0	0
$64$	11.7921	1.47402
$65$	4.24055i	0.525975i
$66$	0	0
$67$	1.87745	0.229367	0.114684	−	0.993402i	$-0.463415\pi$
$67$	1.87745	0.229367	0.114684	+	0.993402i	$0.463415\pi$
$68$	7.96815	0.966281
$69$	0	0
$70$	0	0
$71$	16.3448i	1.93977i	0.243561	+	0.969886i	$0.421684\pi$
$71$	16.3448i	1.93977i	−0.243561	+	0.969886i	$0.578316\pi$
$72$	0	0
$73$	11.1606i	1.30625i	0.757249	+	0.653126i	$0.226543\pi$
$73$	11.1606i	1.30625i	−0.757249	+	0.653126i	$0.773457\pi$
$74$	− 2.62199i	− 0.304800i
$75$	0	0
$76$	10.1666i	1.16619i
$77$	0	0
$78$	0	0
$79$	15.6282	1.75831	0.879156	−	0.476533i	$-0.158107\pi$
$79$	15.6282	1.75831	0.879156	+	0.476533i	$0.158107\pi$
$80$	−2.48998	−0.278388
$81$	0	0
$82$	− 16.3308i	− 1.80344i
$83$	−17.0540	−1.87192	−0.935962	−	0.352102i	$-0.885467\pi$
$83$	−17.0540	−1.87192	−0.935962	+	0.352102i	$0.885467\pi$
$84$	0	0
$85$	−3.08329	−0.334429
$86$	− 10.4596i	− 1.12789i
$87$	0	0
$88$	−5.23035	−0.557557
$89$	3.04499	0.322769	0.161384	−	0.986892i	$-0.448404\pi$
$89$	3.04499	0.322769	0.161384	+	0.986892i	$0.448404\pi$
$90$	0	0
$91$	0	0
$92$	− 9.10151i	− 0.948898i
$93$	0	0
$94$	− 4.81271i	− 0.496393i
$95$	− 3.93398i	− 0.403618i
$96$	0	0
$97$	− 19.3686i	− 1.96658i	−0.182036	−	0.983292i	$-0.558269\pi$
$97$	− 19.3686i	− 1.96658i	0.182036	−	0.983292i	$-0.441731\pi$
$98$	0	0
$99$	0	0
$100$	−2.58431	−0.258431
$101$	−3.06784	−0.305261	−0.152631	−	0.988283i	$-0.548774\pi$
$101$	−3.06784	−0.305261	−0.152631	+	0.988283i	$0.548774\pi$
$102$	0	0
$103$	7.31378i	0.720648i	0.932827	+	0.360324i	$0.117334\pi$
$103$	7.31378i	0.720648i	−0.932827	+	0.360324i	$0.882666\pi$
$104$	−5.30516	−0.520214
$105$	0	0
$106$	17.1140	1.66226
$107$	− 10.0286i	− 0.969505i	−0.874651	−	0.484752i	$-0.838910\pi$
$107$	− 10.0286i	− 0.969505i	0.874651	−	0.484752i	$-0.161090\pi$
$108$	0	0
$109$	−0.119632	−0.0114587	−0.00572933	−	0.999984i	$-0.501824\pi$
$109$	−0.119632	−0.0114587	−0.00572933	+	0.999984i	$0.501824\pi$
$110$	8.95140	0.853482
$111$	0	0
$112$	0	0
$113$	− 18.5189i	− 1.74211i	−0.491188	−	0.871054i	$-0.663437\pi$
$113$	− 18.5189i	− 1.74211i	0.491188	−	0.871054i	$-0.336563\pi$
$114$	0	0
$115$	3.52184i	0.328413i
$116$	− 18.5085i	− 1.71847i
$117$	0	0
$118$	12.9168i	1.18909i
$119$	0	0
$120$	0	0
$121$	−6.47866	−0.588969
$122$	30.0210	2.71797
$123$	0	0
$124$	11.0043i	0.988214i
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.99148	−0.442923	−0.221461	−	0.975169i	$-0.571083\pi$
$127$	−4.99148	−0.442923	−0.221461	+	0.975169i	$0.571083\pi$
$128$	− 9.58132i	− 0.846877i
$129$	0	0
$130$	9.07943	0.796319
$131$	6.63297	0.579525	0.289763	−	0.957099i	$-0.406424\pi$
$131$	6.63297	0.579525	0.289763	+	0.957099i	$0.406424\pi$
$132$	0	0
$133$	0	0
$134$	− 4.01981i	− 0.347259i
$135$	0	0
$136$	− 3.85736i	− 0.330766i
$137$	19.7452i	1.68694i	0.537173	+	0.843472i	$0.319492\pi$
$137$	19.7452i	1.68694i	−0.537173	+	0.843472i	$0.680508\pi$
$138$	0	0
$139$	− 15.8170i	− 1.34158i	−0.741647	−	0.670790i	$-0.765955\pi$
$139$	− 15.8170i	− 1.34158i	0.741647	−	0.670790i	$-0.234045\pi$
$140$	0	0
$141$	0	0
$142$	34.9958	2.93679
$143$	−17.7287	−1.48254
$144$	0	0
$145$	7.16190i	0.594763i
$146$	23.8960	1.97765
$147$	0	0
$148$	−3.16474	−0.260140
$149$	− 7.99054i	− 0.654611i	−0.944919	−	0.327305i	$-0.893859\pi$
$149$	− 7.99054i	− 0.654611i	0.944919	−	0.327305i	$-0.106141\pi$
$150$	0	0
$151$	20.0680	1.63311	0.816555	−	0.577268i	$-0.195881\pi$
$151$	20.0680	1.63311	0.816555	+	0.577268i	$0.195881\pi$
$152$	4.92163	0.399197
$153$	0	0
$154$	0	0
$155$	− 4.25812i	− 0.342020i
$156$	0	0
$157$	− 7.25448i	− 0.578970i	−0.957182	−	0.289485i	$-0.906516\pi$
$157$	− 7.25448i	− 0.578970i	0.957182	−	0.289485i	$-0.0934841\pi$
$158$	− 33.4616i	− 2.66206i
$159$	0	0
$160$	7.83340i	0.619284i
$161$	0	0
$162$	0	0
$163$	1.04273	0.0816726	0.0408363	−	0.999166i	$-0.486998\pi$
$163$	1.04273	0.0816726	0.0408363	+	0.999166i	$0.486998\pi$
$164$	−19.7113	−1.53919
$165$	0	0
$166$	36.5144i	2.83407i
$167$	10.2720	0.794874	0.397437	−	0.917629i	$-0.369900\pi$
$167$	10.2720	0.794874	0.397437	+	0.917629i	$0.369900\pi$
$168$	0	0
$169$	−4.98223	−0.383249
$170$	6.60162i	0.506321i
$171$	0	0
$172$	−12.6247	−0.962626
$173$	−9.21652	−0.700719	−0.350359	−	0.936615i	$-0.613941\pi$
$173$	−9.21652	−0.700719	−0.350359	+	0.936615i	$0.613941\pi$
$174$	0	0
$175$	0	0
$176$	− 10.4100i	− 0.784680i
$177$	0	0
$178$	− 6.51963i	− 0.488667i
$179$	11.0782i	0.828024i	0.910271	+	0.414012i	$0.135873\pi$
$179$	11.0782i	0.828024i	−0.910271	+	0.414012i	$0.864127\pi$
$180$	0	0
$181$	− 7.18944i	− 0.534387i	−0.963643	−	0.267193i	$-0.913904\pi$
$181$	− 7.18944i	− 0.534387i	0.963643	−	0.267193i	$-0.0860962\pi$
$182$	0	0
$183$	0	0
$184$	−4.40602	−0.324816
$185$	1.22460	0.0900344
$186$	0	0
$187$	− 12.8904i	− 0.942642i
$188$	−5.80894	−0.423660
$189$	0	0
$190$	−8.42304	−0.611071
$191$	18.0815i	1.30833i	0.756351	+	0.654166i	$0.226980\pi$
$191$	18.0815i	1.30833i	−0.756351	+	0.654166i	$0.773020\pi$
$192$	0	0
$193$	−10.2481	−0.737677	−0.368839	−	0.929493i	$-0.620245\pi$
$193$	−10.2481	−0.737677	−0.368839	+	0.929493i	$0.620245\pi$
$194$	−41.4701	−2.97738
$195$	0	0
$196$	0	0
$197$	− 8.85439i	− 0.630849i	−0.948951	−	0.315425i	$-0.897853\pi$
$197$	− 8.85439i	− 0.630849i	0.948951	−	0.315425i	$-0.102147\pi$
$198$	0	0
$199$	19.6938i	1.39605i	0.716071	+	0.698027i	$0.245939\pi$
$199$	19.6938i	1.39605i	−0.716071	+	0.698027i	$0.754061\pi$
$200$	1.25106i	0.0884630i
$201$	0	0
$202$	6.56855i	0.462161i
$203$	0	0
$204$	0	0
$205$	7.62731	0.532714
$206$	15.6595	1.09105
$207$	0	0
$208$	− 10.5589i	− 0.732125i
$209$	16.4470	1.13766
$210$	0	0
$211$	−15.7769	−1.08613	−0.543064	−	0.839691i	$-0.682736\pi$
$211$	−15.7769	−1.08613	−0.543064	+	0.839691i	$0.682736\pi$
$212$	− 20.6566i	− 1.41870i
$213$	0	0
$214$	−21.4723	−1.46782
$215$	4.88515	0.333164
$216$	0	0
$217$	0	0
$218$	0.256144i	0.0173483i
$219$	0	0
$220$	− 10.8043i	− 0.728428i
$221$	− 13.0748i	− 0.879507i
$222$	0	0
$223$	− 16.5834i	− 1.11050i	−0.831682	−	0.555252i	$-0.812622\pi$
$223$	− 16.5834i	− 1.11050i	0.831682	−	0.555252i	$-0.187378\pi$
$224$	0	0
$225$	0	0
$226$	−39.6507	−2.63753
$227$	−0.367489	−0.0243911	−0.0121956	−	0.999926i	$-0.503882\pi$
$227$	−0.367489	−0.0243911	−0.0121956	+	0.999926i	$0.503882\pi$
$228$	0	0
$229$	1.99141i	0.131596i	0.997833	+	0.0657979i	$0.0209593\pi$
$229$	1.99141i	0.131596i	−0.997833	+	0.0657979i	$0.979041\pi$
$230$	7.54061	0.497213
$231$	0	0
$232$	−8.95994	−0.588249
$233$	11.8456i	0.776029i	0.921653	+	0.388014i	$0.126839\pi$
$233$	11.8456i	0.776029i	−0.921653	+	0.388014i	$0.873161\pi$
$234$	0	0
$235$	2.24777	0.146629
$236$	15.5906	1.01486
$237$	0	0
$238$	0	0
$239$	− 0.815591i	− 0.0527562i	−0.999652	−	0.0263781i	$-0.991603\pi$
$239$	− 0.815591i	− 0.0527562i	0.999652	−	0.0263781i	$-0.00839738\pi$
$240$	0	0
$241$	− 24.8363i	− 1.59985i	−0.600103	−	0.799923i	$-0.704874\pi$
$241$	− 24.8363i	− 1.59985i	0.600103	−	0.799923i	$-0.295126\pi$
$242$	13.8714i	0.891690i
$243$	0	0
$244$	− 36.2353i	− 2.31973i
$245$	0	0
$246$	0	0
$247$	16.6822	1.06146
$248$	5.32715	0.338274
$249$	0	0
$250$	− 2.14110i	− 0.135415i
$251$	22.3983	1.41377	0.706884	−	0.707330i	$-0.250100\pi$
$251$	22.3983	1.41377	0.706884	+	0.707330i	$0.250100\pi$
$252$	0	0
$253$	−14.7239	−0.925686
$254$	10.6873i	0.670578i
$255$	0	0
$256$	3.06970	0.191856
$257$	−21.6129	−1.34818	−0.674088	−	0.738651i	$-0.735463\pi$
$257$	−21.6129	−1.34818	−0.674088	+	0.738651i	$0.735463\pi$
$258$	0	0
$259$	0	0
$260$	− 10.9589i	− 0.679640i
$261$	0	0
$262$	− 14.2018i	− 0.877393i
$263$	8.94959i	0.551856i	0.961178	+	0.275928i	$0.0889851\pi$
$263$	8.94959i	0.551856i	−0.961178	+	0.275928i	$0.911015\pi$
$264$	0	0
$265$	7.99310i	0.491012i
$266$	0	0
$267$	0	0
$268$	−4.85191	−0.296378
$269$	−4.12371	−0.251427	−0.125714	−	0.992067i	$-0.540122\pi$
$269$	−4.12371	−0.251427	−0.125714	+	0.992067i	$0.540122\pi$
$270$	0	0
$271$	12.1187i	0.736157i	0.929795	+	0.368078i	$0.119984\pi$
$271$	12.1187i	0.736157i	−0.929795	+	0.368078i	$0.880016\pi$
$272$	7.67731	0.465505
$273$	0	0
$274$	42.2764	2.55401
$275$	4.18075i	0.252109i
$276$	0	0
$277$	−29.9437	−1.79914	−0.899572	−	0.436773i	$-0.856121\pi$
$277$	−29.9437	−1.79914	−0.899572	+	0.436773i	$0.856121\pi$
$278$	−33.8658	−2.03113
$279$	0	0
$280$	0	0
$281$	− 0.257513i	− 0.0153619i	−0.999971	−	0.00768097i	$-0.997555\pi$
$281$	− 0.257513i	− 0.0153619i	0.999971	−	0.00768097i	$-0.00244495\pi$
$282$	0	0
$283$	− 12.1600i	− 0.722835i	−0.932404	−	0.361418i	$-0.882293\pi$
$283$	− 12.1600i	− 0.722835i	0.932404	−	0.361418i	$-0.117707\pi$
$284$	− 42.2400i	− 2.50648i
$285$	0	0
$286$	37.9588i	2.24455i
$287$	0	0
$288$	0	0
$289$	−7.49335	−0.440785
$290$	15.3343	0.900463
$291$	0	0
$292$	− 28.8425i	− 1.68788i
$293$	4.83106	0.282234	0.141117	−	0.989993i	$-0.454931\pi$
$293$	4.83106	0.282234	0.141117	+	0.989993i	$0.454931\pi$
$294$	0	0
$295$	−6.03279	−0.351242
$296$	1.53204i	0.0890483i
$297$	0	0
$298$	−17.1085	−0.991071
$299$	−14.9345	−0.863686
$300$	0	0
$301$	0	0
$302$	− 42.9676i	− 2.47250i
$303$	0	0
$304$	9.79551i	0.561811i
$305$	14.0213i	0.802857i
$306$	0	0
$307$	− 14.9603i	− 0.853831i	−0.904292	−	0.426915i	$-0.859600\pi$
$307$	− 14.9603i	− 0.853831i	0.904292	−	0.426915i	$-0.140400\pi$
$308$	0	0
$309$	0	0
$310$	−9.11706	−0.517814
$311$	−5.44050	−0.308503	−0.154251	−	0.988032i	$-0.549297\pi$
$311$	−5.44050	−0.308503	−0.154251	+	0.988032i	$0.549297\pi$
$312$	0	0
$313$	28.4987i	1.61084i	0.592703	+	0.805421i	$0.298061\pi$
$313$	28.4987i	1.61084i	−0.592703	+	0.805421i	$0.701939\pi$
$314$	−15.5326	−0.876553
$315$	0	0
$316$	−40.3881	−2.27201
$317$	10.0937i	0.566917i	0.958985	+	0.283458i	$0.0914818\pi$
$317$	10.0937i	0.566917i	−0.958985	+	0.283458i	$0.908518\pi$
$318$	0	0
$319$	−29.9421	−1.67644
$320$	11.7921	0.659200
$321$	0	0
$322$	0	0
$323$	12.1296i	0.674908i
$324$	0	0
$325$	4.24055i	0.235223i
$326$	− 2.23258i	− 0.123651i
$327$	0	0
$328$	9.54220i	0.526880i
$329$	0	0
$330$	0	0
$331$	−2.83527	−0.155841	−0.0779204	−	0.996960i	$-0.524828\pi$
$331$	−2.83527	−0.155841	−0.0779204	+	0.996960i	$0.524828\pi$
$332$	44.0728	2.41881
$333$	0	0
$334$	− 21.9935i	− 1.20343i
$335$	1.87745	0.102576
$336$	0	0
$337$	−15.7316	−0.856954	−0.428477	−	0.903553i	$-0.640950\pi$
$337$	−15.7316	−0.856954	−0.428477	+	0.903553i	$0.640950\pi$
$338$	10.6675i	0.580233i
$339$	0	0
$340$	7.96815	0.432134
$341$	17.8021	0.964039
$342$	0	0
$343$	0	0
$344$	6.11160i	0.329515i
$345$	0	0
$346$	19.7335i	1.06088i
$347$	10.9920i	0.590083i	0.955484	+	0.295042i	$0.0953335\pi$
$347$	10.9920i	0.590083i	−0.955484	+	0.295042i	$0.904667\pi$
$348$	0	0
$349$	28.4095i	1.52073i	0.649498	+	0.760363i	$0.274979\pi$
$349$	28.4095i	1.52073i	−0.649498	+	0.760363i	$0.725021\pi$
$350$	0	0
$351$	0	0
$352$	−32.7495	−1.74555
$353$	−14.0852	−0.749681	−0.374840	−	0.927089i	$-0.622302\pi$
$353$	−14.0852	−0.749681	−0.374840	+	0.927089i	$0.622302\pi$
$354$	0	0
$355$	16.3448i	0.867492i
$356$	−7.86919	−0.417066
$357$	0	0
$358$	23.7195	1.25362
$359$	− 9.56173i	− 0.504649i	−0.967643	−	0.252324i	$-0.918805\pi$
$359$	− 9.56173i	− 0.504649i	0.967643	−	0.252324i	$-0.0811950\pi$
$360$	0	0
$361$	3.52382	0.185464
$362$	−15.3933	−0.809054
$363$	0	0
$364$	0	0
$365$	11.1606i	0.584173i
$366$	0	0
$367$	0.293915i	0.0153423i	0.999971	+	0.00767113i	$0.00244182\pi$
$367$	0.293915i	0.0153423i	−0.999971	+	0.00767113i	$0.997558\pi$
$368$	− 8.76930i	− 0.457131i
$369$	0	0
$370$	− 2.62199i	− 0.136311i
$371$	0	0
$372$	0	0
$373$	−12.4392	−0.644075	−0.322038	−	0.946727i	$-0.604368\pi$
$373$	−12.4392	−0.644075	−0.322038	+	0.946727i	$0.604368\pi$
$374$	−27.5997	−1.42715
$375$	0	0
$376$	2.81209i	0.145023i
$377$	−30.3704	−1.56415
$378$	0	0
$379$	−29.9052	−1.53613	−0.768063	−	0.640375i	$-0.778779\pi$
$379$	−29.9052	−1.53613	−0.768063	+	0.640375i	$0.778779\pi$
$380$	10.1666i	0.521536i
$381$	0	0
$382$	38.7143	1.98080
$383$	14.4938	0.740599	0.370300	−	0.928912i	$-0.379255\pi$
$383$	14.4938	0.740599	0.370300	+	0.928912i	$0.379255\pi$
$384$	0	0
$385$	0	0
$386$	21.9423i	1.11683i
$387$	0	0
$388$	50.0544i	2.54113i
$389$	21.4184i	1.08595i	0.839748	+	0.542977i	$0.182703\pi$
$389$	21.4184i	1.08595i	−0.839748	+	0.542977i	$0.817297\pi$
$390$	0	0
$391$	− 10.8588i	− 0.549155i
$392$	0	0
$393$	0	0
$394$	−18.9581	−0.955097
$395$	15.6282	0.786341
$396$	0	0
$397$	− 39.0198i	− 1.95835i	−0.203024	−	0.979174i	$-0.565077\pi$
$397$	− 39.0198i	− 1.95835i	0.203024	−	0.979174i	$-0.434923\pi$
$398$	42.1663	2.11361
$399$	0	0
$400$	−2.48998	−0.124499
$401$	26.9225i	1.34444i	0.740350	+	0.672222i	$0.234660\pi$
$401$	26.9225i	1.34444i	−0.740350	+	0.672222i	$0.765340\pi$
$402$	0	0
$403$	18.0568	0.899471
$404$	7.92823	0.394444
$405$	0	0
$406$	0	0
$407$	5.11975i	0.253776i
$408$	0	0
$409$	13.5166i	0.668353i	0.942511	+	0.334176i	$0.108458\pi$
$409$	13.5166i	0.668353i	−0.942511	+	0.334176i	$0.891542\pi$
$410$	− 16.3308i	− 0.806522i
$411$	0	0
$412$	− 18.9010i	− 0.931188i
$413$	0	0
$414$	0	0
$415$	−17.0540	−0.837150
$416$	−33.2179	−1.62864
$417$	0	0
$418$	− 35.2146i	− 1.72240i
$419$	−1.52935	−0.0747138	−0.0373569	−	0.999302i	$-0.511894\pi$
$419$	−1.52935	−0.0747138	−0.0373569	+	0.999302i	$0.511894\pi$
$420$	0	0
$421$	5.22312	0.254559	0.127280	−	0.991867i	$-0.459375\pi$
$421$	5.22312	0.254559	0.127280	+	0.991867i	$0.459375\pi$
$422$	33.7800i	1.64438i
$423$	0	0
$424$	−9.99982	−0.485634
$425$	−3.08329	−0.149561
$426$	0	0
$427$	0	0
$428$	25.9171i	1.25275i
$429$	0	0
$430$	− 10.4596i	− 0.504406i
$431$	− 23.7489i	− 1.14394i	−0.820273	−	0.571972i	$-0.806179\pi$
$431$	− 23.7489i	− 1.14394i	0.820273	−	0.571972i	$-0.193821\pi$
$432$	0	0
$433$	− 37.7340i	− 1.81338i	−0.421799	−	0.906689i	$-0.638601\pi$
$433$	− 37.7340i	− 1.81338i	0.421799	−	0.906689i	$-0.361399\pi$
$434$	0	0
$435$	0	0
$436$	0.309166	0.0148064
$437$	13.8548	0.662767
$438$	0	0
$439$	26.1830i	1.24965i	0.780767	+	0.624823i	$0.214829\pi$
$439$	26.1830i	1.24965i	−0.780767	+	0.624823i	$0.785171\pi$
$440$	−5.23035	−0.249347
$441$	0	0
$442$	−27.9945	−1.33156
$443$	− 11.6259i	− 0.552361i	−0.961106	−	0.276181i	$-0.910931\pi$
$443$	− 11.6259i	− 0.552361i	0.961106	−	0.276181i	$-0.0890688\pi$
$444$	0	0
$445$	3.04499	0.144346
$446$	−35.5066	−1.68129
$447$	0	0
$448$	0	0
$449$	27.8649i	1.31502i	0.753444	+	0.657512i	$0.228391\pi$
$449$	27.8649i	1.31502i	−0.753444	+	0.657512i	$0.771609\pi$
$450$	0	0
$451$	31.8879i	1.50154i
$452$	47.8584i	2.25107i
$453$	0	0
$454$	0.786831i	0.0369278i
$455$	0	0
$456$	0	0
$457$	26.5423	1.24160	0.620799	−	0.783970i	$-0.286808\pi$
$457$	26.5423	1.24160	0.620799	+	0.783970i	$0.286808\pi$
$458$	4.26380	0.199234
$459$	0	0
$460$	− 9.10151i	− 0.424360i
$461$	−20.0220	−0.932516	−0.466258	−	0.884649i	$-0.654398\pi$
$461$	−20.0220	−0.932516	−0.466258	+	0.884649i	$0.654398\pi$
$462$	0	0
$463$	10.1206	0.470342	0.235171	−	0.971954i	$-0.424435\pi$
$463$	10.1206	0.470342	0.235171	+	0.971954i	$0.424435\pi$
$464$	− 17.8330i	− 0.827874i
$465$	0	0
$466$	25.3625	1.17490
$467$	35.9145	1.66193	0.830963	−	0.556327i	$-0.187790\pi$
$467$	35.9145	1.66193	0.830963	+	0.556327i	$0.187790\pi$
$468$	0	0
$469$	0	0
$470$	− 4.81271i	− 0.221994i
$471$	0	0
$472$	− 7.54736i	− 0.347395i
$473$	20.4236i	0.939077i
$474$	0	0
$475$	− 3.93398i	− 0.180503i
$476$	0	0
$477$	0	0
$478$	−1.74626	−0.0798721
$479$	28.0405	1.28120	0.640602	−	0.767873i	$-0.278685\pi$
$479$	28.0405	1.28120	0.640602	+	0.767873i	$0.278685\pi$
$480$	0	0
$481$	5.19297i	0.236779i
$482$	−53.1770	−2.42214
$483$	0	0
$484$	16.7428	0.761038
$485$	− 19.3686i	− 0.879483i
$486$	0	0
$487$	35.2987	1.59954	0.799769	−	0.600308i	$-0.204955\pi$
$487$	35.2987	1.59954	0.799769	+	0.600308i	$0.204955\pi$
$488$	−17.5414	−0.794063
$489$	0	0
$490$	0	0
$491$	− 0.959904i	− 0.0433199i	−0.999765	−	0.0216599i	$-0.993105\pi$
$491$	− 0.959904i	− 0.0433199i	0.999765	−	0.0216599i	$-0.00689511\pi$
$492$	0	0
$493$	− 22.0822i	− 0.994532i
$494$	− 35.7183i	− 1.60704i
$495$	0	0
$496$	10.6026i	0.476072i
$497$	0	0
$498$	0	0
$499$	37.7831	1.69141	0.845703	−	0.533654i	$-0.179182\pi$
$499$	37.7831	1.69141	0.845703	+	0.533654i	$0.179182\pi$
$500$	−2.58431	−0.115574
$501$	0	0
$502$	− 47.9570i	− 2.14042i
$503$	−29.1078	−1.29785	−0.648927	−	0.760850i	$-0.724782\pi$
$503$	−29.1078	−1.29785	−0.648927	+	0.760850i	$0.724782\pi$
$504$	0	0
$505$	−3.06784	−0.136517
$506$	31.5254i	1.40147i
$507$	0	0
$508$	12.8995	0.572324
$509$	−6.39121	−0.283285	−0.141643	−	0.989918i	$-0.545238\pi$
$509$	−6.39121	−0.283285	−0.141643	+	0.989918i	$0.545238\pi$
$510$	0	0
$511$	0	0
$512$	− 25.7352i	− 1.13734i
$513$	0	0
$514$	46.2754i	2.04112i
$515$	7.31378i	0.322284i
$516$	0	0
$517$	9.39738i	0.413296i
$518$	0	0
$519$	0	0
$520$	−5.30516	−0.232647
$521$	−27.1037	−1.18743	−0.593716	−	0.804674i	$-0.702340\pi$
$521$	−27.1037	−1.18743	−0.593716	+	0.804674i	$0.702340\pi$
$522$	0	0
$523$	− 20.0524i	− 0.876829i	−0.898773	−	0.438415i	$-0.855540\pi$
$523$	− 20.0524i	− 0.876829i	0.898773	−	0.438415i	$-0.144460\pi$
$524$	−17.1416	−0.748835
$525$	0	0
$526$	19.1620	0.835501
$527$	13.1290i	0.571908i
$528$	0	0
$529$	10.5966	0.460723
$530$	17.1140	0.743385
$531$	0	0
$532$	0	0
$533$	32.3440i	1.40097i
$534$	0	0
$535$	− 10.0286i	− 0.433576i
$536$	2.34880i	0.101453i
$537$	0	0
$538$	8.82927i	0.380657i
$539$	0	0
$540$	0	0
$541$	−18.0339	−0.775338	−0.387669	−	0.921799i	$-0.626720\pi$
$541$	−18.0339	−0.775338	−0.387669	+	0.921799i	$0.626720\pi$
$542$	25.9473	1.11453
$543$	0	0
$544$	− 24.1526i	− 1.03553i
$545$	−0.119632	−0.00512447
$546$	0	0
$547$	15.3974	0.658345	0.329173	−	0.944270i	$-0.393230\pi$
$547$	15.3974	0.658345	0.329173	+	0.944270i	$0.393230\pi$
$548$	− 51.0276i	− 2.17979i
$549$	0	0
$550$	8.95140	0.381689
$551$	28.1748	1.20029
$552$	0	0
$553$	0	0
$554$	64.1125i	2.72388i
$555$	0	0
$556$	40.8759i	1.73353i
$557$	2.76314i	0.117078i	0.998285	+	0.0585389i	$0.0186442\pi$
$557$	2.76314i	0.117078i	−0.998285	+	0.0585389i	$0.981356\pi$
$558$	0	0
$559$	20.7157i	0.876181i
$560$	0	0
$561$	0	0
$562$	−0.551361	−0.0232578
$563$	41.4623	1.74743	0.873713	−	0.486442i	$-0.161705\pi$
$563$	41.4623	1.74743	0.873713	+	0.486442i	$0.161705\pi$
$564$	0	0
$565$	− 18.5189i	− 0.779094i
$566$	−26.0357	−1.09436
$567$	0	0
$568$	−20.4483	−0.857990
$569$	− 4.09457i	− 0.171653i	−0.996310	−	0.0858266i	$-0.972647\pi$
$569$	− 4.09457i	− 0.171653i	0.996310	−	0.0858266i	$-0.0273531\pi$
$570$	0	0
$571$	−11.1852	−0.468087	−0.234043	−	0.972226i	$-0.575196\pi$
$571$	−11.1852	−0.468087	−0.234043	+	0.972226i	$0.575196\pi$
$572$	45.8163	1.91567
$573$	0	0
$574$	0	0
$575$	3.52184i	0.146871i
$576$	0	0
$577$	− 2.24718i	− 0.0935513i	−0.998905	−	0.0467756i	$-0.985105\pi$
$577$	− 2.24718i	− 0.0935513i	0.998905	−	0.0467756i	$-0.0148946\pi$
$578$	16.0440i	0.667342i
$579$	0	0
$580$	− 18.5085i	− 0.768525i
$581$	0	0
$582$	0	0
$583$	−33.4171	−1.38400
$584$	−13.9626	−0.577775
$585$	0	0
$586$	− 10.3438i	− 0.427298i
$587$	−8.04727	−0.332147	−0.166073	−	0.986113i	$-0.553109\pi$
$587$	−8.04727	−0.332147	−0.166073	+	0.986113i	$0.553109\pi$
$588$	0	0
$589$	−16.7514	−0.690228
$590$	12.9168i	0.531776i
$591$	0	0
$592$	−3.04923	−0.125322
$593$	−18.7307	−0.769179	−0.384589	−	0.923088i	$-0.625657\pi$
$593$	−18.7307	−0.769179	−0.384589	+	0.923088i	$0.625657\pi$
$594$	0	0
$595$	0	0
$596$	20.6500i	0.845857i
$597$	0	0
$598$	31.9763i	1.30761i
$599$	− 28.1486i	− 1.15012i	−0.818112	−	0.575060i	$-0.804979\pi$
$599$	− 28.1486i	− 1.15012i	0.818112	−	0.575060i	$-0.195021\pi$
$600$	0	0
$601$	21.1621i	0.863221i	0.902060	+	0.431610i	$0.142054\pi$
$601$	21.1621i	0.863221i	−0.902060	+	0.431610i	$0.857946\pi$
$602$	0	0
$603$	0	0
$604$	−51.8618	−2.11023
$605$	−6.47866	−0.263395
$606$	0	0
$607$	46.4470i	1.88522i	0.333889	+	0.942612i	$0.391639\pi$
$607$	46.4470i	1.88522i	−0.333889	+	0.942612i	$0.608361\pi$
$608$	30.8164	1.24977
$609$	0	0
$610$	30.0210	1.21551
$611$	9.53179i	0.385615i
$612$	0	0
$613$	12.5001	0.504874	0.252437	−	0.967613i	$-0.418768\pi$
$613$	12.5001	0.504874	0.252437	+	0.967613i	$0.418768\pi$
$614$	−32.0315	−1.29269
$615$	0	0
$616$	0	0
$617$	14.0495i	0.565610i	0.959177	+	0.282805i	$0.0912649\pi$
$617$	14.0495i	0.565610i	−0.959177	+	0.282805i	$0.908735\pi$
$618$	0	0
$619$	12.6776i	0.509556i	0.967000	+	0.254778i	$0.0820024\pi$
$619$	12.6776i	0.509556i	−0.967000	+	0.254778i	$0.917998\pi$
$620$	11.0043i	0.441943i
$621$	0	0
$622$	11.6487i	0.467069i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	61.0186	2.43879
$627$	0	0
$628$	18.7478i	0.748118i
$629$	−3.77579	−0.150551
$630$	0	0
$631$	11.8905	0.473354	0.236677	−	0.971588i	$-0.423942\pi$
$631$	11.8905	0.473354	0.236677	+	0.971588i	$0.423942\pi$
$632$	19.5518i	0.777728i
$633$	0	0
$634$	21.6115	0.858304
$635$	−4.99148	−0.198081
$636$	0	0
$637$	0	0
$638$	64.1090i	2.53810i
$639$	0	0
$640$	− 9.58132i	− 0.378735i
$641$	10.3455i	0.408624i	0.978906	+	0.204312i	$0.0654957\pi$
$641$	10.3455i	0.408624i	−0.978906	+	0.204312i	$0.934504\pi$
$642$	0	0
$643$	− 39.5611i	− 1.56014i	−0.625693	−	0.780070i	$-0.715184\pi$
$643$	− 39.5611i	− 1.56014i	0.625693	−	0.780070i	$-0.284816\pi$
$644$	0	0
$645$	0	0
$646$	25.9706	1.02180
$647$	−5.33850	−0.209878	−0.104939	−	0.994479i	$-0.533465\pi$
$647$	−5.33850	−0.209878	−0.104939	+	0.994479i	$0.533465\pi$
$648$	0	0
$649$	− 25.2216i	− 0.990033i
$650$	9.07943	0.356125
$651$	0	0
$652$	−2.69472	−0.105533
$653$	− 14.0866i	− 0.551251i	−0.961265	−	0.275625i	$-0.911115\pi$
$653$	− 14.0866i	− 0.551251i	0.961265	−	0.275625i	$-0.0888849\pi$
$654$	0	0
$655$	6.63297	0.259172
$656$	−18.9918	−0.741506
$657$	0	0
$658$	0	0
$659$	13.6532i	0.531853i	0.963993	+	0.265926i	$0.0856778\pi$
$659$	13.6532i	0.531853i	−0.963993	+	0.265926i	$0.914322\pi$
$660$	0	0
$661$	− 5.23274i	− 0.203530i	−0.994808	−	0.101765i	$-0.967551\pi$
$661$	− 5.23274i	− 0.203530i	0.994808	−	0.101765i	$-0.0324490\pi$
$662$	6.07060i	0.235941i
$663$	0	0
$664$	− 21.3356i	− 0.827980i
$665$	0	0
$666$	0	0
$667$	−25.2231	−0.976641
$668$	−26.5461	−1.02710
$669$	0	0
$670$	− 4.01981i	− 0.155299i
$671$	−58.6195	−2.26298
$672$	0	0
$673$	26.9352	1.03827	0.519137	−	0.854691i	$-0.326253\pi$
$673$	26.9352	1.03827	0.519137	+	0.854691i	$0.326253\pi$
$674$	33.6829i	1.29742i
$675$	0	0
$676$	12.8756	0.495216
$677$	29.8797	1.14837	0.574184	−	0.818726i	$-0.305319\pi$
$677$	29.8797	1.14837	0.574184	+	0.818726i	$0.305319\pi$
$678$	0	0
$679$	0	0
$680$	− 3.85736i	− 0.147923i
$681$	0	0
$682$	− 38.1161i	− 1.45954i
$683$	− 29.8268i	− 1.14129i	−0.821196	−	0.570646i	$-0.806693\pi$
$683$	− 29.8268i	− 1.14129i	0.821196	−	0.570646i	$-0.193307\pi$
$684$	0	0
$685$	19.7452i	0.754424i
$686$	0	0
$687$	0	0
$688$	−12.1639	−0.463744
$689$	−33.8951	−1.29130
$690$	0	0
$691$	− 33.5305i	− 1.27556i	−0.770218	−	0.637781i	$-0.779852\pi$
$691$	− 33.5305i	− 1.27556i	0.770218	−	0.637781i	$-0.220148\pi$
$692$	23.8183	0.905436
$693$	0	0
$694$	23.5350	0.893377
$695$	− 15.8170i	− 0.599973i
$696$	0	0
$697$	−23.5172	−0.890777
$698$	60.8276	2.30236
$699$	0	0
$700$	0	0
$701$	− 25.5057i	− 0.963338i	−0.876353	−	0.481669i	$-0.840031\pi$
$701$	− 25.5057i	− 0.963338i	0.876353	−	0.481669i	$-0.159969\pi$
$702$	0	0
$703$	− 4.81755i	− 0.181697i
$704$	49.2999i	1.85806i
$705$	0	0
$706$	30.1578i	1.13501i
$707$	0	0
$708$	0	0
$709$	−9.29640	−0.349134	−0.174567	−	0.984645i	$-0.555852\pi$
$709$	−9.29640	−0.349134	−0.174567	+	0.984645i	$0.555852\pi$
$710$	34.9958	1.31337
$711$	0	0
$712$	3.80946i	0.142765i
$713$	14.9964	0.561621
$714$	0	0
$715$	−17.7287	−0.663014
$716$	− 28.6295i	− 1.06993i
$717$	0	0
$718$	−20.4726	−0.764031
$719$	−11.3186	−0.422113	−0.211057	−	0.977474i	$-0.567690\pi$
$719$	−11.3186	−0.422113	−0.211057	+	0.977474i	$0.567690\pi$
$720$	0	0
$721$	0	0
$722$	− 7.54484i	− 0.280790i
$723$	0	0
$724$	18.5797i	0.690509i
$725$	7.16190i	0.265986i
$726$	0	0
$727$	− 11.3507i	− 0.420976i	−0.977596	−	0.210488i	$-0.932495\pi$
$727$	− 11.3507i	− 0.420976i	0.977596	−	0.210488i	$-0.0675053\pi$
$728$	0	0
$729$	0	0
$730$	23.8960	0.884430
$731$	−15.0623	−0.557100
$732$	0	0
$733$	27.4434i	1.01364i	0.862051	+	0.506822i	$0.169180\pi$
$733$	27.4434i	1.01364i	−0.862051	+	0.506822i	$0.830820\pi$
$734$	0.629302	0.0232280
$735$	0	0
$736$	−27.5880	−1.01691
$737$	7.84916i	0.289127i
$738$	0	0
$739$	−30.4714	−1.12091	−0.560454	−	0.828185i	$-0.689373\pi$
$739$	−30.4714	−1.12091	−0.560454	+	0.828185i	$0.689373\pi$
$740$	−3.16474	−0.116338
$741$	0	0
$742$	0	0
$743$	− 14.7965i	− 0.542830i	−0.962462	−	0.271415i	$-0.912508\pi$
$743$	− 14.7965i	− 0.542830i	0.962462	−	0.271415i	$-0.0874916\pi$
$744$	0	0
$745$	− 7.99054i	− 0.292751i
$746$	26.6335i	0.975120i
$747$	0	0
$748$	33.3128i	1.21804i
$749$	0	0
$750$	0	0
$751$	24.6806	0.900610	0.450305	−	0.892875i	$-0.351315\pi$
$751$	24.6806	0.900610	0.450305	+	0.892875i	$0.351315\pi$
$752$	−5.59690	−0.204098
$753$	0	0
$754$	65.0260i	2.36811i
$755$	20.0680	0.730349
$756$	0	0
$757$	−0.747953	−0.0271848	−0.0135924	−	0.999908i	$-0.504327\pi$
$757$	−0.747953	−0.0271848	−0.0135924	+	0.999908i	$0.504327\pi$
$758$	64.0299i	2.32567i
$759$	0	0
$760$	4.92163	0.178526
$761$	35.5771	1.28967	0.644834	−	0.764322i	$-0.276926\pi$
$761$	35.5771	1.28967	0.644834	+	0.764322i	$0.276926\pi$
$762$	0	0
$763$	0	0
$764$	− 46.7281i	− 1.69056i
$765$	0	0
$766$	− 31.0327i	− 1.12126i
$767$	− 25.5823i	− 0.923724i
$768$	0	0
$769$	24.6978i	0.890625i	0.895375	+	0.445312i	$0.146907\pi$
$769$	24.6978i	0.890625i	−0.895375	+	0.445312i	$0.853093\pi$
$770$	0	0
$771$	0	0
$772$	26.4843	0.953192
$773$	−4.35654	−0.156694	−0.0783469	−	0.996926i	$-0.524964\pi$
$773$	−4.35654	−0.156694	−0.0783469	+	0.996926i	$0.524964\pi$
$774$	0	0
$775$	− 4.25812i	− 0.152956i
$776$	24.2312	0.869850
$777$	0	0
$778$	45.8588	1.64412
$779$	− 30.0057i	− 1.07506i
$780$	0	0
$781$	−68.3335	−2.44516
$782$	−23.2499	−0.831413
$783$	0	0
$784$	0	0
$785$	− 7.25448i	− 0.258923i
$786$	0	0
$787$	− 4.96114i	− 0.176845i	−0.996083	−	0.0884227i	$-0.971817\pi$
$787$	− 4.96114i	− 0.176845i	0.996083	−	0.0884227i	$-0.0281826\pi$
$788$	22.8825i	0.815154i
$789$	0	0
$790$	− 33.4616i	− 1.19051i
$791$	0	0
$792$	0	0
$793$	−59.4580	−2.11141
$794$	−83.5453	−2.96491
$795$	0	0
$796$	− 50.8947i	− 1.80392i
$797$	−12.1509	−0.430406	−0.215203	−	0.976569i	$-0.569041\pi$
$797$	−12.1509	−0.430406	−0.215203	+	0.976569i	$0.569041\pi$
$798$	0	0
$799$	−6.93053	−0.245185
$800$	7.83340i	0.276952i
$801$	0	0
$802$	57.6436	2.03547
$803$	−46.6597	−1.64659
$804$	0	0
$805$	0	0
$806$	− 38.6613i	− 1.36179i
$807$	0	0
$808$	− 3.83804i	− 0.135022i
$809$	13.6668i	0.480498i	0.970711	+	0.240249i	$0.0772290\pi$
$809$	13.6668i	0.480498i	−0.970711	+	0.240249i	$0.922771\pi$
$810$	0	0
$811$	− 49.4766i	− 1.73736i	−0.495376	−	0.868679i	$-0.664970\pi$
$811$	− 49.4766i	− 1.73736i	0.495376	−	0.868679i	$-0.335030\pi$
$812$	0	0
$813$	0	0
$814$	10.9619	0.384214
$815$	1.04273	0.0365251
$816$	0	0
$817$	− 19.2181i	− 0.672355i
$818$	28.9404	1.01188
$819$	0	0
$820$	−19.7113	−0.688349
$821$	− 8.63972i	− 0.301528i	−0.988570	−	0.150764i	$-0.951827\pi$
$821$	− 8.63972i	− 0.301528i	0.988570	−	0.150764i	$-0.0481734\pi$
$822$	0	0
$823$	−8.53861	−0.297637	−0.148819	−	0.988865i	$-0.547547\pi$
$823$	−8.53861	−0.297637	−0.148819	+	0.988865i	$0.547547\pi$
$824$	−9.14995	−0.318754
$825$	0	0
$826$	0	0
$827$	− 0.166257i	− 0.00578131i	−0.999996	−	0.00289066i	$-0.999080\pi$
$827$	− 0.166257i	− 0.00578131i	0.999996	−	0.00289066i	$-0.000920126\pi$
$828$	0	0
$829$	− 26.9687i	− 0.936663i	−0.883553	−	0.468331i	$-0.844855\pi$
$829$	− 26.9687i	− 0.936663i	0.883553	−	0.468331i	$-0.155145\pi$
$830$	36.5144i	1.26743i
$831$	0	0
$832$	50.0051i	1.73361i
$833$	0	0
$834$	0	0
$835$	10.2720	0.355479
$836$	−42.5040	−1.47003
$837$	0	0
$838$	3.27450i	0.113116i
$839$	33.9620	1.17250	0.586249	−	0.810131i	$-0.300604\pi$
$839$	33.9620	1.17250	0.586249	+	0.810131i	$0.300604\pi$
$840$	0	0
$841$	−22.2928	−0.768718
$842$	− 11.1832i	− 0.385399i
$843$	0	0
$844$	40.7724	1.40344
$845$	−4.98223	−0.171394
$846$	0	0
$847$	0	0
$848$	− 19.9026i	− 0.683459i
$849$	0	0
$850$	6.60162i	0.226434i
$851$	4.31285i	0.147843i
$852$	0	0
$853$	28.4906i	0.975500i	0.872983	+	0.487750i	$0.162182\pi$
$853$	28.4906i	0.975500i	−0.872983	+	0.487750i	$0.837818\pi$
$854$	0	0
$855$	0	0
$856$	12.5464	0.428827
$857$	−1.65476	−0.0565255	−0.0282628	−	0.999601i	$-0.508998\pi$
$857$	−1.65476	−0.0565255	−0.0282628	+	0.999601i	$0.508998\pi$
$858$	0	0
$859$	− 17.2579i	− 0.588834i	−0.955677	−	0.294417i	$-0.904875\pi$
$859$	− 17.2579i	− 0.588834i	0.955677	−	0.294417i	$-0.0951254\pi$
$860$	−12.6247	−0.430499
$861$	0	0
$862$	−50.8487	−1.73191
$863$	− 52.7745i	− 1.79647i	−0.439519	−	0.898233i	$-0.644851\pi$
$863$	− 52.7745i	− 1.79647i	0.439519	−	0.898233i	$-0.355149\pi$
$864$	0	0
$865$	−9.21652	−0.313371
$866$	−80.7921	−2.74543
$867$	0	0
$868$	0	0
$869$	65.3377i	2.21643i
$870$	0	0
$871$	7.96142i	0.269763i
$872$	− 0.149666i	− 0.00506834i
$873$	0	0
$874$	− 29.6646i	− 1.00342i
$875$	0	0
$876$	0	0
$877$	−44.8989	−1.51613	−0.758064	−	0.652180i	$-0.773855\pi$
$877$	−44.8989	−1.51613	−0.758064	+	0.652180i	$0.773855\pi$
$878$	56.0603	1.89194
$879$	0	0
$880$	− 10.4100i	− 0.350920i
$881$	24.1868	0.814875	0.407437	−	0.913233i	$-0.366422\pi$
$881$	24.1868	0.814875	0.407437	+	0.913233i	$0.366422\pi$
$882$	0	0
$883$	20.3843	0.685988	0.342994	−	0.939338i	$-0.388559\pi$
$883$	20.3843	0.685988	0.342994	+	0.939338i	$0.388559\pi$
$884$	33.7893i	1.13646i
$885$	0	0
$886$	−24.8921	−0.836267
$887$	17.5964	0.590831	0.295415	−	0.955369i	$-0.404542\pi$
$887$	17.5964	0.590831	0.295415	+	0.955369i	$0.404542\pi$
$888$	0	0
$889$	0	0
$890$	− 6.51963i	− 0.218539i
$891$	0	0
$892$	42.8565i	1.43494i
$893$	− 8.84269i	− 0.295909i
$894$	0	0
$895$	11.0782i	0.370304i
$896$	0	0
$897$	0	0
$898$	59.6614	1.99093
$899$	30.4962	1.01711
$900$	0	0
$901$	− 24.6450i	− 0.821044i
$902$	68.2751	2.27331
$903$	0	0
$904$	23.1681	0.770561
$905$	− 7.18944i	− 0.238985i
$906$	0	0
$907$	−19.0178	−0.631475	−0.315738	−	0.948846i	$-0.602252\pi$
$907$	−19.0178	−0.631475	−0.315738	+	0.948846i	$0.602252\pi$
$908$	0.949705	0.0315171
$909$	0	0
$910$	0	0
$911$	− 24.1354i	− 0.799640i	−0.916594	−	0.399820i	$-0.869073\pi$
$911$	− 24.1354i	− 0.799640i	0.916594	−	0.399820i	$-0.130927\pi$
$912$	0	0
$913$	− 71.2986i	− 2.35964i
$914$	− 56.8298i	− 1.87976i
$915$	0	0
$916$	− 5.14640i	− 0.170042i
$917$	0	0
$918$	0	0
$919$	37.0034	1.22063	0.610315	−	0.792159i	$-0.291043\pi$
$919$	37.0034	1.22063	0.610315	+	0.792159i	$0.291043\pi$
$920$	−4.40602	−0.145262
$921$	0	0
$922$	42.8690i	1.41182i
$923$	−69.3109	−2.28140
$924$	0	0
$925$	1.22460	0.0402646
$926$	− 21.6691i	− 0.712091i
$927$	0	0
$928$	−56.1020	−1.84164
$929$	37.6178	1.23420	0.617100	−	0.786884i	$-0.288307\pi$
$929$	37.6178	1.23420	0.617100	+	0.786884i	$0.288307\pi$
$930$	0	0
$931$	0	0
$932$	− 30.6126i	− 1.00275i
$933$	0	0
$934$	− 76.8966i	− 2.51613i
$935$	− 12.8904i	− 0.421563i
$936$	0	0
$937$	48.5109i	1.58478i	0.610013	+	0.792391i	$0.291164\pi$
$937$	48.5109i	1.58478i	−0.610013	+	0.792391i	$0.708836\pi$
$938$	0	0
$939$	0	0
$940$	−5.80894	−0.189467
$941$	18.1829	0.592745	0.296373	−	0.955072i	$-0.404223\pi$
$941$	18.1829	0.592745	0.296373	+	0.955072i	$0.404223\pi$
$942$	0	0
$943$	26.8622i	0.874753i
$944$	15.0215	0.488908
$945$	0	0
$946$	43.7289	1.42175
$947$	28.3545i	0.921398i	0.887556	+	0.460699i	$0.152401\pi$
$947$	28.3545i	0.921398i	−0.887556	+	0.460699i	$0.847599\pi$
$948$	0	0
$949$	−47.3271	−1.53630
$950$	−8.42304	−0.273279
$951$	0	0
$952$	0	0
$953$	− 44.3923i	− 1.43801i	−0.695005	−	0.719004i	$-0.744598\pi$
$953$	− 44.3923i	− 1.43801i	0.695005	−	0.719004i	$-0.255402\pi$
$954$	0	0
$955$	18.0815i	0.585104i
$956$	2.10774i	0.0681691i
$957$	0	0
$958$	− 60.0375i	− 1.93972i
$959$	0	0
$960$	0	0
$961$	12.8684	0.415110
$962$	11.1187	0.358480
$963$	0	0
$964$	64.1846i	2.06725i
$965$	−10.2481	−0.329899
$966$	0	0
$967$	−44.9331	−1.44495	−0.722475	−	0.691397i	$-0.756996\pi$
$967$	−44.9331	−1.44495	−0.722475	+	0.691397i	$0.756996\pi$
$968$	− 8.10516i	− 0.260510i
$969$	0	0
$970$	−41.4701	−1.33152
$971$	−14.3993	−0.462097	−0.231048	−	0.972942i	$-0.574216\pi$
$971$	−14.3993	−0.462097	−0.231048	+	0.972942i	$0.574216\pi$
$972$	0	0
$973$	0	0
$974$	− 75.5781i	− 2.42168i
$975$	0	0
$976$	− 34.9127i	− 1.11753i
$977$	− 22.4408i	− 0.717946i	−0.933348	−	0.358973i	$-0.883127\pi$
$977$	− 22.4408i	− 0.717946i	0.933348	−	0.358973i	$-0.116873\pi$
$978$	0	0
$979$	12.7303i	0.406864i
$980$	0	0
$981$	0	0
$982$	−2.05525	−0.0655857
$983$	−41.3894	−1.32012	−0.660058	−	0.751214i	$-0.729468\pi$
$983$	−41.3894	−1.32012	−0.660058	+	0.751214i	$0.729468\pi$
$984$	0	0
$985$	− 8.85439i	− 0.282124i
$986$	−47.2801	−1.50571
$987$	0	0
$988$	−43.1119	−1.37157
$989$	17.2047i	0.547078i
$990$	0	0
$991$	−26.9764	−0.856934	−0.428467	−	0.903557i	$-0.640946\pi$
$991$	−26.9764	−0.856934	−0.428467	+	0.903557i	$0.640946\pi$
$992$	33.3556	1.05904
$993$	0	0
$994$	0	0
$995$	19.6938i	0.624335i
$996$	0	0
$997$	− 9.95252i	− 0.315200i	−0.987503	−	0.157600i	$-0.949624\pi$
$997$	− 9.95252i	− 0.315200i	0.987503	−	0.157600i	$-0.0503756\pi$
$998$	− 80.8975i	− 2.56076i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.b.d.881.2	yes	16
3.2	odd	2		2205.2.b.c.881.15	yes	16
7.6	odd	2		2205.2.b.c.881.2	✓	16
21.20	even	2	inner	2205.2.b.d.881.15	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.2.b.c.881.2	✓	16	7.6	odd	2
2205.2.b.c.881.15	yes	16	3.2	odd	2
2205.2.b.d.881.2	yes	16	1.1	even	1	trivial
2205.2.b.d.881.15	yes	16	21.20	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-2.14110i q^{2} -2.58431 q^{4} +1.00000 q^{5} +1.25106i q^{8} -2.14110i q^{10} +4.18075i q^{11} +4.24055i q^{13} -2.48998 q^{16} -3.08329 q^{17} -3.93398i q^{19} -2.58431 q^{20} +8.95140 q^{22} +3.52184i q^{23} +1.00000 q^{25} +9.07943 q^{26} +7.16190i q^{29} -4.25812i q^{31} +7.83340i q^{32} +6.60162i q^{34} +1.22460 q^{37} -8.42304 q^{38} +1.25106i q^{40} +7.62731 q^{41} +4.88515 q^{43} -10.8043i q^{44} +7.54061 q^{46} +2.24777 q^{47} -2.14110i q^{50} -10.9589i q^{52} +7.99310i q^{53} +4.18075i q^{55} +15.3343 q^{58} -6.03279 q^{59} +14.0213i q^{61} -9.11706 q^{62} +11.7921 q^{64} +4.24055i q^{65} +1.87745 q^{67} +7.96815 q^{68} +16.3448i q^{71} +11.1606i q^{73} -2.62199i q^{74} +10.1666i q^{76} +15.6282 q^{79} -2.48998 q^{80} -16.3308i q^{82} -17.0540 q^{83} -3.08329 q^{85} -10.4596i q^{86} -5.23035 q^{88} +3.04499 q^{89} -9.10151i q^{92} -4.81271i q^{94} -3.93398i q^{95} -19.3686i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{5} - 16 q^{20} - 16 q^{22} + 16 q^{25} + 32 q^{26} - 16 q^{38} - 32 q^{41} + 32 q^{43} - 16 q^{46} + 32 q^{47} + 48 q^{58} - 32 q^{59} - 32 q^{62} + 16 q^{64} - 16 q^{68} + 32 q^{79}+ \cdots + 64 q^{89}+O(q^{100}) \) 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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