LMFDB - Embedded newform 2205.2.b.d.881.6

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			881.6
Root			$0.500000 + 1.71572i$ of defining polynomial
Character	$\chi$	$=$	2205.881
Dual form			2205.2.b.d.881.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.15464i q^{2} +0.666795 q^{4} +1.00000 q^{5} -3.07920i q^{8} -1.15464i q^{10} -1.60302i q^{11} +0.357205i q^{13} -2.22179 q^{16} -1.71805 q^{17} -4.09896i q^{19} +0.666795 q^{20} -1.85092 q^{22} -2.70889i q^{23} +1.00000 q^{25} +0.412444 q^{26} -7.84601i q^{29} +6.44364i q^{31} -3.59302i q^{32} +1.98374i q^{34} -7.74723 q^{37} -4.73284 q^{38} -3.07920i q^{40} +4.07114 q^{41} +3.84370 q^{43} -1.06889i q^{44} -3.12781 q^{46} -5.12477 q^{47} -1.15464i q^{50} +0.238182i q^{52} -4.10804i q^{53} -1.60302i q^{55} -9.05935 q^{58} +7.51051 q^{59} -13.3611i q^{61} +7.44012 q^{62} -8.59225 q^{64} +0.357205i q^{65} +11.5119 q^{67} -1.14559 q^{68} -3.19867i q^{71} +16.6521i q^{73} +8.94530i q^{74} -2.73317i q^{76} -9.48008 q^{79} -2.22179 q^{80} -4.70072i q^{82} -0.627576 q^{83} -1.71805 q^{85} -4.43811i q^{86} -4.93602 q^{88} +9.40198 q^{89} -1.80628i q^{92} +5.91729i q^{94} -4.09896i q^{95} +5.93276i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 1.15464i	− 0.816457i	−0.912880	−	0.408229i	$-0.866147\pi$
$2$	− 1.15464i	− 0.816457i	0.912880	−	0.408229i	$-0.133853\pi$
$3$	0	0
$3$	0	0
$4$	0.666795	0.333398
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 3.07920i	− 1.08866i
$9$	0	0
$10$	− 1.15464i	− 0.365131i
$11$	− 1.60302i	− 0.483328i	−0.970360	−	0.241664i	$-0.922307\pi$
$11$	− 1.60302i	− 0.483328i	0.970360	−	0.241664i	$-0.0776932\pi$
$12$	0	0
$13$	0.357205i	0.0990707i	0.998772	+	0.0495354i	$0.0157740\pi$
$13$	0.357205i	0.0990707i	−0.998772	+	0.0495354i	$0.984226\pi$
$14$	0	0
$15$	0	0
$16$	−2.22179	−0.555448
$17$	−1.71805	−0.416688	−0.208344	−	0.978056i	$-0.566807\pi$
$17$	−1.71805	−0.416688	−0.208344	+	0.978056i	$0.566807\pi$
$18$	0	0
$19$	− 4.09896i	− 0.940365i	−0.882569	−	0.470183i	$-0.844188\pi$
$19$	− 4.09896i	− 0.940365i	0.882569	−	0.470183i	$-0.155812\pi$
$20$	0.666795	0.149100
$21$	0	0
$22$	−1.85092	−0.394617
$23$	− 2.70889i	− 0.564844i	−0.959290	−	0.282422i	$-0.908862\pi$
$23$	− 2.70889i	− 0.564844i	0.959290	−	0.282422i	$-0.0911378\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.412444	0.0808870
$27$	0	0
$28$	0	0
$29$	− 7.84601i	− 1.45697i	−0.685063	−	0.728484i	$-0.740225\pi$
$29$	− 7.84601i	− 1.45697i	0.685063	−	0.728484i	$-0.259775\pi$
$30$	0	0
$31$	6.44364i	1.15731i	0.815572	+	0.578656i	$0.196423\pi$
$31$	6.44364i	1.15731i	−0.815572	+	0.578656i	$0.803577\pi$
$32$	− 3.59302i	− 0.635162i
$33$	0	0
$34$	1.98374i	0.340208i
$35$	0	0
$36$	0	0
$37$	−7.74723	−1.27364	−0.636818	−	0.771014i	$-0.719750\pi$
$37$	−7.74723	−1.27364	−0.636818	+	0.771014i	$0.719750\pi$
$38$	−4.73284	−0.767768
$39$	0	0
$40$	− 3.07920i	− 0.486864i
$41$	4.07114	0.635805	0.317902	−	0.948123i	$-0.397022\pi$
$41$	4.07114	0.635805	0.317902	+	0.948123i	$0.397022\pi$
$42$	0	0
$43$	3.84370	0.586159	0.293080	−	0.956088i	$-0.405320\pi$
$43$	3.84370	0.586159	0.293080	+	0.956088i	$0.405320\pi$
$44$	− 1.06889i	− 0.161141i
$45$	0	0
$46$	−3.12781	−0.461171
$47$	−5.12477	−0.747524	−0.373762	−	0.927525i	$-0.621932\pi$
$47$	−5.12477	−0.747524	−0.373762	+	0.927525i	$0.621932\pi$
$48$	0	0
$49$	0	0
$50$	− 1.15464i	− 0.163291i
$51$	0	0
$52$	0.238182i	0.0330299i
$53$	− 4.10804i	− 0.564283i	−0.959373	−	0.282141i	$-0.908955\pi$
$53$	− 4.10804i	− 0.564283i	0.959373	−	0.282141i	$-0.0910447\pi$
$54$	0	0
$55$	− 1.60302i	− 0.216151i
$56$	0	0
$57$	0	0
$58$	−9.05935	−1.18955
$59$	7.51051	0.977785	0.488892	−	0.872344i	$-0.337401\pi$
$59$	7.51051	0.977785	0.488892	+	0.872344i	$0.337401\pi$
$60$	0	0
$61$	− 13.3611i	− 1.71071i	−0.518040	−	0.855356i	$-0.673338\pi$
$61$	− 13.3611i	− 1.71071i	0.518040	−	0.855356i	$-0.326662\pi$
$62$	7.44012	0.944896
$63$	0	0
$64$	−8.59225	−1.07403
$65$	0.357205i	0.0443058i
$66$	0	0
$67$	11.5119	1.40640	0.703202	−	0.710990i	$-0.251753\pi$
$67$	11.5119	1.40640	0.703202	+	0.710990i	$0.251753\pi$
$68$	−1.14559	−0.138923
$69$	0	0
$70$	0	0
$71$	− 3.19867i	− 0.379613i	−0.981822	−	0.189806i	$-0.939214\pi$
$71$	− 3.19867i	− 0.379613i	0.981822	−	0.189806i	$-0.0607860\pi$
$72$	0	0
$73$	16.6521i	1.94899i	0.224418	+	0.974493i	$0.427952\pi$
$73$	16.6521i	1.94899i	−0.224418	+	0.974493i	$0.572048\pi$
$74$	8.94530i	1.03987i
$75$	0	0
$76$	− 2.73317i	− 0.313516i
$77$	0	0
$78$	0	0
$79$	−9.48008	−1.06659	−0.533296	−	0.845929i	$-0.679047\pi$
$79$	−9.48008	−1.06659	−0.533296	+	0.845929i	$0.679047\pi$
$80$	−2.22179	−0.248404
$81$	0	0
$82$	− 4.70072i	− 0.519107i
$83$	−0.627576	−0.0688854	−0.0344427	−	0.999407i	$-0.510966\pi$
$83$	−0.627576	−0.0688854	−0.0344427	+	0.999407i	$0.510966\pi$
$84$	0	0
$85$	−1.71805	−0.186349
$86$	− 4.43811i	− 0.478574i
$87$	0	0
$88$	−4.93602	−0.526181
$89$	9.40198	0.996607	0.498304	−	0.867003i	$-0.333956\pi$
$89$	9.40198	0.996607	0.498304	+	0.867003i	$0.333956\pi$
$90$	0	0
$91$	0	0
$92$	− 1.80628i	− 0.188317i
$93$	0	0
$94$	5.91729i	0.610322i
$95$	− 4.09896i	− 0.420544i
$96$	0	0
$97$	5.93276i	0.602381i	0.953564	+	0.301190i	$0.0973840\pi$
$97$	5.93276i	0.602381i	−0.953564	+	0.301190i	$0.902616\pi$
$98$	0	0
$99$	0	0
$100$	0.666795	0.0666795
$101$	−4.36228	−0.434063	−0.217032	−	0.976165i	$-0.569637\pi$
$101$	−4.36228	−0.434063	−0.217032	+	0.976165i	$0.569637\pi$
$102$	0	0
$103$	− 0.506737i	− 0.0499303i	−0.999688	−	0.0249651i	$-0.992053\pi$
$103$	− 0.506737i	− 0.0499303i	0.999688	−	0.0249651i	$-0.00794748\pi$
$104$	1.09990	0.107855
$105$	0	0
$106$	−4.74333	−0.460713
$107$	− 19.5304i	− 1.88808i	−0.329836	−	0.944038i	$-0.606993\pi$
$107$	− 19.5304i	− 1.88808i	0.329836	−	0.944038i	$-0.393007\pi$
$108$	0	0
$109$	9.16489	0.877837	0.438918	−	0.898527i	$-0.355362\pi$
$109$	9.16489	0.877837	0.438918	+	0.898527i	$0.355362\pi$
$110$	−1.85092	−0.176478
$111$	0	0
$112$	0	0
$113$	− 4.81733i	− 0.453176i	−0.973991	−	0.226588i	$-0.927243\pi$
$113$	− 4.81733i	− 0.453176i	0.973991	−	0.226588i	$-0.0727571\pi$
$114$	0	0
$115$	− 2.70889i	− 0.252606i
$116$	− 5.23168i	− 0.485749i
$117$	0	0
$118$	− 8.67197i	− 0.798319i
$119$	0	0
$120$	0	0
$121$	8.43033	0.766394
$122$	−15.4273	−1.39672
$123$	0	0
$124$	4.29659i	0.385845i
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.2009	−1.17139	−0.585697	−	0.810530i	$-0.699179\pi$
$127$	−13.2009	−1.17139	−0.585697	+	0.810530i	$0.699179\pi$
$128$	2.73496i	0.241738i
$129$	0	0
$130$	0.412444	0.0361738
$131$	18.8531	1.64721	0.823603	−	0.567166i	$-0.191960\pi$
$131$	18.8531	1.64721	0.823603	+	0.567166i	$0.191960\pi$
$132$	0	0
$133$	0	0
$134$	− 13.2922i	− 1.14827i
$135$	0	0
$136$	5.29022i	0.453633i
$137$	4.35312i	0.371912i	0.982558	+	0.185956i	$0.0595382\pi$
$137$	4.35312i	0.371912i	−0.982558	+	0.185956i	$0.940462\pi$
$138$	0	0
$139$	1.62070i	0.137466i	0.997635	+	0.0687330i	$0.0218956\pi$
$139$	1.62070i	0.137466i	−0.997635	+	0.0687330i	$0.978104\pi$
$140$	0	0
$141$	0	0
$142$	−3.69333	−0.309938
$143$	0.572606	0.0478837
$144$	0	0
$145$	− 7.84601i	− 0.651576i
$146$	19.2273	1.59126
$147$	0	0
$148$	−5.16582	−0.424627
$149$	− 9.32158i	− 0.763653i	−0.924234	−	0.381827i	$-0.875295\pi$
$149$	− 9.32158i	− 0.763653i	0.924234	−	0.381827i	$-0.124705\pi$
$150$	0	0
$151$	−11.3489	−0.923560	−0.461780	−	0.886995i	$-0.652789\pi$
$151$	−11.3489	−0.923560	−0.461780	+	0.886995i	$0.652789\pi$
$152$	−12.6215	−1.02374
$153$	0	0
$154$	0	0
$155$	6.44364i	0.517566i
$156$	0	0
$157$	− 3.68158i	− 0.293822i	−0.989150	−	0.146911i	$-0.953067\pi$
$157$	− 3.68158i	− 0.293822i	0.989150	−	0.146911i	$-0.0469330\pi$
$158$	10.9461i	0.870827i
$159$	0	0
$160$	− 3.59302i	− 0.284053i
$161$	0	0
$162$	0	0
$163$	−10.8165	−0.847212	−0.423606	−	0.905846i	$-0.639236\pi$
$163$	−10.8165	−0.847212	−0.423606	+	0.905846i	$0.639236\pi$
$164$	2.71462	0.211976
$165$	0	0
$166$	0.724628i	0.0562420i
$167$	−20.5609	−1.59105	−0.795523	−	0.605923i	$-0.792804\pi$
$167$	−20.5609	−1.59105	−0.795523	+	0.605923i	$0.792804\pi$
$168$	0	0
$169$	12.8724	0.990185
$170$	1.98374i	0.152146i
$171$	0	0
$172$	2.56296	0.195424
$173$	14.6900	1.11686	0.558430	−	0.829552i	$-0.311404\pi$
$173$	14.6900	1.11686	0.558430	+	0.829552i	$0.311404\pi$
$174$	0	0
$175$	0	0
$176$	3.56158i	0.268464i
$177$	0	0
$178$	− 10.8559i	− 0.813687i
$179$	11.9475i	0.893000i	0.894784	+	0.446500i	$0.147330\pi$
$179$	11.9475i	0.893000i	−0.894784	+	0.446500i	$0.852670\pi$
$180$	0	0
$181$	− 17.3559i	− 1.29005i	−0.764161	−	0.645026i	$-0.776846\pi$
$181$	− 17.3559i	− 1.29005i	0.764161	−	0.645026i	$-0.223154\pi$
$182$	0	0
$183$	0	0
$184$	−8.34123	−0.614924
$185$	−7.74723	−0.569588
$186$	0	0
$187$	2.75407i	0.201397i
$188$	−3.41717	−0.249223
$189$	0	0
$190$	−4.73284	−0.343356
$191$	2.00877i	0.145350i	0.997356	+	0.0726748i	$0.0231535\pi$
$191$	2.00877i	0.145350i	−0.997356	+	0.0726748i	$0.976846\pi$
$192$	0	0
$193$	10.2964	0.741148	0.370574	−	0.928803i	$-0.379161\pi$
$193$	10.2964	0.741148	0.370574	+	0.928803i	$0.379161\pi$
$194$	6.85023	0.491818
$195$	0	0
$196$	0	0
$197$	21.9056i	1.56071i	0.625339	+	0.780353i	$0.284961\pi$
$197$	21.9056i	1.56071i	−0.625339	+	0.780353i	$0.715039\pi$
$198$	0	0
$199$	18.7030i	1.32582i	0.748697	+	0.662912i	$0.230680\pi$
$199$	18.7030i	1.32582i	−0.748697	+	0.662912i	$0.769320\pi$
$200$	− 3.07920i	− 0.217732i
$201$	0	0
$202$	5.03689i	0.354394i
$203$	0	0
$204$	0	0
$205$	4.07114	0.284341
$206$	−0.585101	−0.0407660
$207$	0	0
$208$	− 0.793635i	− 0.0550287i
$209$	−6.57071	−0.454505
$210$	0	0
$211$	5.32850	0.366829	0.183415	−	0.983036i	$-0.441285\pi$
$211$	5.32850	0.366829	0.183415	+	0.983036i	$0.441285\pi$
$212$	− 2.73922i	− 0.188130i
$213$	0	0
$214$	−22.5507	−1.54153
$215$	3.84370	0.262138
$216$	0	0
$217$	0	0
$218$	− 10.5822i	− 0.716716i
$219$	0	0
$220$	− 1.06889i	− 0.0720642i
$221$	− 0.613695i	− 0.0412816i
$222$	0	0
$223$	6.84113i	0.458116i	0.973413	+	0.229058i	$0.0735646\pi$
$223$	6.84113i	0.458116i	−0.973413	+	0.229058i	$0.926435\pi$
$224$	0	0
$225$	0	0
$226$	−5.56231	−0.369999
$227$	−26.7957	−1.77850	−0.889248	−	0.457426i	$-0.848772\pi$
$227$	−26.7957	−1.77850	−0.889248	+	0.457426i	$0.848772\pi$
$228$	0	0
$229$	23.8141i	1.57368i	0.617158	+	0.786839i	$0.288284\pi$
$229$	23.8141i	1.57368i	−0.617158	+	0.786839i	$0.711716\pi$
$230$	−3.12781	−0.206242
$231$	0	0
$232$	−24.1594	−1.58615
$233$	4.57554i	0.299753i	0.988705	+	0.149877i	$0.0478877\pi$
$233$	4.57554i	0.299753i	−0.988705	+	0.149877i	$0.952112\pi$
$234$	0	0
$235$	−5.12477	−0.334303
$236$	5.00797	0.325991
$237$	0	0
$238$	0	0
$239$	12.5361i	0.810890i	0.914119	+	0.405445i	$0.132883\pi$
$239$	12.5361i	0.810890i	−0.914119	+	0.405445i	$0.867117\pi$
$240$	0	0
$241$	2.09928i	0.135227i	0.997712	+	0.0676133i	$0.0215384\pi$
$241$	2.09928i	0.135227i	−0.997712	+	0.0676133i	$0.978462\pi$
$242$	− 9.73404i	− 0.625728i
$243$	0	0
$244$	− 8.90912i	− 0.570348i
$245$	0	0
$246$	0	0
$247$	1.46417	0.0931627
$248$	19.8413	1.25992
$249$	0	0
$250$	− 1.15464i	− 0.0730262i
$251$	−20.8036	−1.31311	−0.656555	−	0.754278i	$-0.727987\pi$
$251$	−20.8036	−1.31311	−0.656555	+	0.754278i	$0.727987\pi$
$252$	0	0
$253$	−4.34241	−0.273005
$254$	15.2424i	0.956393i
$255$	0	0
$256$	−14.0266	−0.876662
$257$	−17.7924	−1.10986	−0.554931	−	0.831896i	$-0.687255\pi$
$257$	−17.7924	−1.10986	−0.554931	+	0.831896i	$0.687255\pi$
$258$	0	0
$259$	0	0
$260$	0.238182i	0.0147714i
$261$	0	0
$262$	− 21.7687i	− 1.34487i
$263$	14.5539i	0.897431i	0.893675	+	0.448715i	$0.148118\pi$
$263$	14.5539i	0.897431i	−0.893675	+	0.448715i	$0.851882\pi$
$264$	0	0
$265$	− 4.10804i	− 0.252355i
$266$	0	0
$267$	0	0
$268$	7.67609	0.468892
$269$	21.9662	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$269$	21.9662	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$270$	0	0
$271$	− 31.3407i	− 1.90381i	−0.306390	−	0.951906i	$-0.599121\pi$
$271$	− 31.3407i	− 1.90381i	0.306390	−	0.951906i	$-0.400879\pi$
$272$	3.81715	0.231449
$273$	0	0
$274$	5.02631	0.303650
$275$	− 1.60302i	− 0.0966657i
$276$	0	0
$277$	30.9680	1.86069	0.930344	−	0.366688i	$-0.119508\pi$
$277$	30.9680	1.86069	0.930344	+	0.366688i	$0.119508\pi$
$278$	1.87133	0.112235
$279$	0	0
$280$	0	0
$281$	14.4971i	0.864822i	0.901677	+	0.432411i	$0.142337\pi$
$281$	14.4971i	0.864822i	−0.901677	+	0.432411i	$0.857663\pi$
$282$	0	0
$283$	− 9.96129i	− 0.592137i	−0.955167	−	0.296069i	$-0.904324\pi$
$283$	− 9.96129i	− 0.592137i	0.955167	−	0.296069i	$-0.0956757\pi$
$284$	− 2.13286i	− 0.126562i
$285$	0	0
$286$	− 0.661156i	− 0.0390950i
$287$	0	0
$288$	0	0
$289$	−14.0483	−0.826371
$290$	−9.05935	−0.531984
$291$	0	0
$292$	11.1036i	0.649787i
$293$	−19.0315	−1.11183	−0.555917	−	0.831238i	$-0.687633\pi$
$293$	−19.0315	−1.11183	−0.555917	+	0.831238i	$0.687633\pi$
$294$	0	0
$295$	7.51051	0.437279
$296$	23.8553i	1.38656i
$297$	0	0
$298$	−10.7631	−0.623490
$299$	0.967629	0.0559594
$300$	0	0
$301$	0	0
$302$	13.1039i	0.754047i
$303$	0	0
$304$	9.10704i	0.522324i
$305$	− 13.3611i	− 0.765054i
$306$	0	0
$307$	15.8498i	0.904596i	0.891867	+	0.452298i	$0.149396\pi$
$307$	15.8498i	0.904596i	−0.891867	+	0.452298i	$0.850604\pi$
$308$	0	0
$309$	0	0
$310$	7.44012	0.422570
$311$	23.7234	1.34523	0.672614	−	0.739994i	$-0.265171\pi$
$311$	23.7234	1.34523	0.672614	+	0.739994i	$0.265171\pi$
$312$	0	0
$313$	17.5892i	0.994198i	0.867694	+	0.497099i	$0.165601\pi$
$313$	17.5892i	0.994198i	−0.867694	+	0.497099i	$0.834399\pi$
$314$	−4.25091	−0.239893
$315$	0	0
$316$	−6.32127	−0.355599
$317$	2.74428i	0.154134i	0.997026	+	0.0770669i	$0.0245555\pi$
$317$	2.74428i	0.154134i	−0.997026	+	0.0770669i	$0.975444\pi$
$318$	0	0
$319$	−12.5773	−0.704194
$320$	−8.59225	−0.480321
$321$	0	0
$322$	0	0
$323$	7.04221i	0.391839i
$324$	0	0
$325$	0.357205i	0.0198141i
$326$	12.4892i	0.691713i
$327$	0	0
$328$	− 12.5359i	− 0.692177i
$329$	0	0
$330$	0	0
$331$	34.1835	1.87889	0.939447	−	0.342693i	$-0.111339\pi$
$331$	34.1835	1.87889	0.939447	+	0.342693i	$0.111339\pi$
$332$	−0.418465	−0.0229662
$333$	0	0
$334$	23.7405i	1.29902i
$335$	11.5119	0.628963
$336$	0	0
$337$	−4.72928	−0.257620	−0.128810	−	0.991669i	$-0.541116\pi$
$337$	−4.72928	−0.257620	−0.128810	+	0.991669i	$0.541116\pi$
$338$	− 14.8631i	− 0.808444i
$339$	0	0
$340$	−1.14559	−0.0621282
$341$	10.3293	0.559362
$342$	0	0
$343$	0	0
$344$	− 11.8355i	− 0.638129i
$345$	0	0
$346$	− 16.9617i	− 0.911868i
$347$	− 23.5145i	− 1.26232i	−0.775651	−	0.631162i	$-0.782578\pi$
$347$	− 23.5145i	− 1.26232i	0.775651	−	0.631162i	$-0.217422\pi$
$348$	0	0
$349$	− 3.64800i	− 0.195273i	−0.995222	−	0.0976364i	$-0.968872\pi$
$349$	− 3.64800i	− 0.195273i	0.995222	−	0.0976364i	$-0.0311282\pi$
$350$	0	0
$351$	0	0
$352$	−5.75968	−0.306992
$353$	2.09123	0.111305	0.0556523	−	0.998450i	$-0.482276\pi$
$353$	2.09123	0.111305	0.0556523	+	0.998450i	$0.482276\pi$
$354$	0	0
$355$	− 3.19867i	− 0.169768i
$356$	6.26919	0.332267
$357$	0	0
$358$	13.7952	0.729097
$359$	− 11.0492i	− 0.583155i	−0.956547	−	0.291578i	$-0.905820\pi$
$359$	− 11.0492i	− 0.583155i	0.956547	−	0.291578i	$-0.0941801\pi$
$360$	0	0
$361$	2.19855	0.115713
$362$	−20.0399	−1.05327
$363$	0	0
$364$	0	0
$365$	16.6521i	0.871613i
$366$	0	0
$367$	− 30.7668i	− 1.60601i	−0.595969	−	0.803007i	$-0.703232\pi$
$367$	− 30.7668i	− 1.60601i	0.595969	−	0.803007i	$-0.296768\pi$
$368$	6.01860i	0.313741i
$369$	0	0
$370$	8.94530i	0.465044i
$371$	0	0
$372$	0	0
$373$	18.0140	0.932730	0.466365	−	0.884592i	$-0.345563\pi$
$373$	18.0140	0.932730	0.466365	+	0.884592i	$0.345563\pi$
$374$	3.17997	0.164432
$375$	0	0
$376$	15.7802i	0.813801i
$377$	2.80263	0.144343
$378$	0	0
$379$	30.5269	1.56806	0.784031	−	0.620721i	$-0.213160\pi$
$379$	30.5269	1.56806	0.784031	+	0.620721i	$0.213160\pi$
$380$	− 2.73317i	− 0.140208i
$381$	0	0
$382$	2.31942	0.118672
$383$	19.0308	0.972430	0.486215	−	0.873839i	$-0.338377\pi$
$383$	19.0308	0.972430	0.486215	+	0.873839i	$0.338377\pi$
$384$	0	0
$385$	0	0
$386$	− 11.8886i	− 0.605116i
$387$	0	0
$388$	3.95594i	0.200832i
$389$	37.0522i	1.87862i	0.343071	+	0.939310i	$0.388533\pi$
$389$	37.0522i	1.87862i	−0.343071	+	0.939310i	$0.611467\pi$
$390$	0	0
$391$	4.65402i	0.235364i
$392$	0	0
$393$	0	0
$394$	25.2931	1.27425
$395$	−9.48008	−0.476995
$396$	0	0
$397$	− 12.8335i	− 0.644094i	−0.946724	−	0.322047i	$-0.895629\pi$
$397$	− 12.8335i	− 0.644094i	0.946724	−	0.322047i	$-0.104371\pi$
$398$	21.5954	1.08248
$399$	0	0
$400$	−2.22179	−0.111090
$401$	8.75444i	0.437176i	0.975817	+	0.218588i	$0.0701450\pi$
$401$	8.75444i	0.437176i	−0.975817	+	0.218588i	$0.929855\pi$
$402$	0	0
$403$	−2.30170	−0.114656
$404$	−2.90875	−0.144716
$405$	0	0
$406$	0	0
$407$	12.4190i	0.615585i
$408$	0	0
$409$	7.19948i	0.355991i	0.984031	+	0.177996i	$0.0569613\pi$
$409$	7.19948i	0.355991i	−0.984031	+	0.177996i	$0.943039\pi$
$410$	− 4.70072i	− 0.232152i
$411$	0	0
$412$	− 0.337890i	− 0.0166466i
$413$	0	0
$414$	0	0
$415$	−0.627576	−0.0308065
$416$	1.28344	0.0629260
$417$	0	0
$418$	7.58683i	0.371084i
$419$	−15.6058	−0.762395	−0.381198	−	0.924494i	$-0.624488\pi$
$419$	−15.6058	−0.762395	−0.381198	+	0.924494i	$0.624488\pi$
$420$	0	0
$421$	23.3453	1.13778	0.568891	−	0.822413i	$-0.307372\pi$
$421$	23.3453	1.13778	0.568891	+	0.822413i	$0.307372\pi$
$422$	− 6.15253i	− 0.299500i
$423$	0	0
$424$	−12.6495	−0.614313
$425$	−1.71805	−0.0833377
$426$	0	0
$427$	0	0
$428$	− 13.0228i	− 0.629480i
$429$	0	0
$430$	− 4.43811i	− 0.214025i
$431$	− 14.1788i	− 0.682967i	−0.939888	−	0.341484i	$-0.889071\pi$
$431$	− 14.1788i	− 0.682967i	0.939888	−	0.341484i	$-0.110929\pi$
$432$	0	0
$433$	35.2665i	1.69480i	0.530957	+	0.847399i	$0.321833\pi$
$433$	35.2665i	1.69480i	−0.530957	+	0.847399i	$0.678167\pi$
$434$	0	0
$435$	0	0
$436$	6.11110	0.292669
$437$	−11.1036	−0.531159
$438$	0	0
$439$	14.9937i	0.715611i	0.933796	+	0.357806i	$0.116475\pi$
$439$	14.9937i	0.715611i	−0.933796	+	0.357806i	$0.883525\pi$
$440$	−4.93602	−0.235315
$441$	0	0
$442$	−0.708600	−0.0337047
$443$	32.4617i	1.54230i	0.636652	+	0.771151i	$0.280319\pi$
$443$	32.4617i	1.54230i	−0.636652	+	0.771151i	$0.719681\pi$
$444$	0	0
$445$	9.40198	0.445696
$446$	7.89908	0.374032
$447$	0	0
$448$	0	0
$449$	− 12.1678i	− 0.574234i	−0.957896	−	0.287117i	$-0.907303\pi$
$449$	− 12.1678i	− 0.574234i	0.957896	−	0.287117i	$-0.0926968\pi$
$450$	0	0
$451$	− 6.52611i	− 0.307302i
$452$	− 3.21217i	− 0.151088i
$453$	0	0
$454$	30.9396i	1.45207i
$455$	0	0
$456$	0	0
$457$	29.7369	1.39103	0.695517	−	0.718510i	$-0.255175\pi$
$457$	29.7369	1.39103	0.695517	+	0.718510i	$0.255175\pi$
$458$	27.4968	1.28484
$459$	0	0
$460$	− 1.80628i	− 0.0842181i
$461$	17.5104	0.815539	0.407770	−	0.913085i	$-0.366307\pi$
$461$	17.5104	0.815539	0.407770	+	0.913085i	$0.366307\pi$
$462$	0	0
$463$	16.3138	0.758168	0.379084	−	0.925362i	$-0.376239\pi$
$463$	16.3138	0.758168	0.379084	+	0.925362i	$0.376239\pi$
$464$	17.4322i	0.809270i
$465$	0	0
$466$	5.28312	0.244736
$467$	−35.0551	−1.62216	−0.811079	−	0.584937i	$-0.801119\pi$
$467$	−35.0551	−1.62216	−0.811079	+	0.584937i	$0.801119\pi$
$468$	0	0
$469$	0	0
$470$	5.91729i	0.272944i
$471$	0	0
$472$	− 23.1264i	− 1.06448i
$473$	− 6.16153i	− 0.283307i
$474$	0	0
$475$	− 4.09896i	− 0.188073i
$476$	0	0
$477$	0	0
$478$	14.4747	0.662057
$479$	28.4078	1.29798	0.648992	−	0.760795i	$-0.275191\pi$
$479$	28.4078	1.29798	0.648992	+	0.760795i	$0.275191\pi$
$480$	0	0
$481$	− 2.76734i	− 0.126180i
$482$	2.42393	0.110407
$483$	0	0
$484$	5.62130	0.255514
$485$	5.93276i	0.269393i
$486$	0	0
$487$	17.6184	0.798368	0.399184	−	0.916871i	$-0.369293\pi$
$487$	17.6184	0.798368	0.399184	+	0.916871i	$0.369293\pi$
$488$	−41.1415	−1.86239
$489$	0	0
$490$	0	0
$491$	2.81103i	0.126860i	0.997986	+	0.0634300i	$0.0202040\pi$
$491$	2.81103i	0.126860i	−0.997986	+	0.0634300i	$0.979796\pi$
$492$	0	0
$493$	13.4798i	0.607101i
$494$	− 1.69059i	− 0.0760633i
$495$	0	0
$496$	− 14.3164i	− 0.642827i
$497$	0	0
$498$	0	0
$499$	5.40475	0.241950	0.120975	−	0.992656i	$-0.461398\pi$
$499$	5.40475	0.241950	0.120975	+	0.992656i	$0.461398\pi$
$500$	0.666795	0.0298200
$501$	0	0
$502$	24.0207i	1.07210i
$503$	6.96616	0.310606	0.155303	−	0.987867i	$-0.450365\pi$
$503$	6.96616	0.310606	0.155303	+	0.987867i	$0.450365\pi$
$504$	0	0
$505$	−4.36228	−0.194119
$506$	5.01394i	0.222897i
$507$	0	0
$508$	−8.80232	−0.390540
$509$	−4.29636	−0.190433	−0.0952163	−	0.995457i	$-0.530354\pi$
$509$	−4.29636	−0.190433	−0.0952163	+	0.995457i	$0.530354\pi$
$510$	0	0
$511$	0	0
$512$	21.6656i	0.957495i
$513$	0	0
$514$	20.5439i	0.906155i
$515$	− 0.506737i	− 0.0223295i
$516$	0	0
$517$	8.21510i	0.361300i
$518$	0	0
$519$	0	0
$520$	1.09990	0.0482340
$521$	−20.1993	−0.884946	−0.442473	−	0.896782i	$-0.645899\pi$
$521$	−20.1993	−0.884946	−0.442473	+	0.896782i	$0.645899\pi$
$522$	0	0
$523$	− 3.94123i	− 0.172338i	−0.996281	−	0.0861690i	$-0.972537\pi$
$523$	− 3.94123i	− 0.172338i	0.996281	−	0.0861690i	$-0.0274625\pi$
$524$	12.5712	0.549175
$525$	0	0
$526$	16.8046	0.732714
$527$	− 11.0705i	− 0.482238i
$528$	0	0
$529$	15.6619	0.680952
$530$	−4.74333	−0.206037
$531$	0	0
$532$	0	0
$533$	1.45423i	0.0629896i
$534$	0	0
$535$	− 19.5304i	− 0.844373i
$536$	− 35.4475i	− 1.53110i
$537$	0	0
$538$	− 25.3631i	− 1.09348i
$539$	0	0
$540$	0	0
$541$	−0.395732	−0.0170138	−0.00850692	−	0.999964i	$-0.502708\pi$
$541$	−0.395732	−0.0170138	−0.00850692	+	0.999964i	$0.502708\pi$
$542$	−36.1874	−1.55438
$543$	0	0
$544$	6.17299i	0.264665i
$545$	9.16489	0.392581
$546$	0	0
$547$	22.1746	0.948117	0.474058	−	0.880493i	$-0.342789\pi$
$547$	22.1746	0.948117	0.474058	+	0.880493i	$0.342789\pi$
$548$	2.90264i	0.123995i
$549$	0	0
$550$	−1.85092	−0.0789234
$551$	−32.1605	−1.37008
$552$	0	0
$553$	0	0
$554$	− 35.7571i	− 1.51917i
$555$	0	0
$556$	1.08067i	0.0458308i
$557$	− 11.2280i	− 0.475744i	−0.971297	−	0.237872i	$-0.923550\pi$
$557$	− 11.2280i	− 0.475744i	0.971297	−	0.237872i	$-0.0764499\pi$
$558$	0	0
$559$	1.37299i	0.0580712i
$560$	0	0
$561$	0	0
$562$	16.7390	0.706090
$563$	34.7365	1.46397	0.731984	−	0.681322i	$-0.238595\pi$
$563$	34.7365	1.46397	0.731984	+	0.681322i	$0.238595\pi$
$564$	0	0
$565$	− 4.81733i	− 0.202667i
$566$	−11.5018	−0.483455
$567$	0	0
$568$	−9.84936	−0.413270
$569$	39.8074i	1.66881i	0.551149	+	0.834407i	$0.314190\pi$
$569$	39.8074i	1.66881i	−0.551149	+	0.834407i	$0.685810\pi$
$570$	0	0
$571$	−15.4622	−0.647074	−0.323537	−	0.946216i	$-0.604872\pi$
$571$	−15.4622	−0.647074	−0.323537	+	0.946216i	$0.604872\pi$
$572$	0.381811	0.0159643
$573$	0	0
$574$	0	0
$575$	− 2.70889i	− 0.112969i
$576$	0	0
$577$	− 7.30058i	− 0.303927i	−0.988386	−	0.151964i	$-0.951440\pi$
$577$	− 7.30058i	− 0.303927i	0.988386	−	0.151964i	$-0.0485596\pi$
$578$	16.2208i	0.674696i
$579$	0	0
$580$	− 5.23168i	− 0.217234i
$581$	0	0
$582$	0	0
$583$	−6.58526	−0.272734
$584$	51.2753	2.12179
$585$	0	0
$586$	21.9747i	0.907765i
$587$	−6.99734	−0.288811	−0.144405	−	0.989519i	$-0.546127\pi$
$587$	−6.99734	−0.288811	−0.144405	+	0.989519i	$0.546127\pi$
$588$	0	0
$589$	26.4122	1.08830
$590$	− 8.67197i	− 0.357019i
$591$	0	0
$592$	17.2127	0.707439
$593$	8.58674	0.352615	0.176307	−	0.984335i	$-0.443585\pi$
$593$	8.58674	0.352615	0.176307	+	0.984335i	$0.443585\pi$
$594$	0	0
$595$	0	0
$596$	− 6.21558i	− 0.254600i
$597$	0	0
$598$	− 1.11727i	− 0.0456885i
$599$	− 8.94210i	− 0.365364i	−0.983172	−	0.182682i	$-0.941522\pi$
$599$	− 8.94210i	− 0.365364i	0.983172	−	0.182682i	$-0.0584779\pi$
$600$	0	0
$601$	13.7656i	0.561509i	0.959780	+	0.280754i	$0.0905846\pi$
$601$	13.7656i	0.561509i	−0.959780	+	0.280754i	$0.909415\pi$
$602$	0	0
$603$	0	0
$604$	−7.56739	−0.307913
$605$	8.43033	0.342742
$606$	0	0
$607$	− 30.9117i	− 1.25467i	−0.778751	−	0.627334i	$-0.784146\pi$
$607$	− 30.9117i	− 1.25467i	0.778751	−	0.627334i	$-0.215854\pi$
$608$	−14.7276	−0.597285
$609$	0	0
$610$	−15.4273	−0.624634
$611$	− 1.83059i	− 0.0740578i
$612$	0	0
$613$	14.0313	0.566718	0.283359	−	0.959014i	$-0.408551\pi$
$613$	14.0313	0.566718	0.283359	+	0.959014i	$0.408551\pi$
$614$	18.3009	0.738564
$615$	0	0
$616$	0	0
$617$	15.4280i	0.621107i	0.950556	+	0.310553i	$0.100514\pi$
$617$	15.4280i	0.621107i	−0.950556	+	0.310553i	$0.899486\pi$
$618$	0	0
$619$	− 22.4630i	− 0.902863i	−0.892306	−	0.451431i	$-0.850914\pi$
$619$	− 22.4630i	− 0.902863i	0.892306	−	0.451431i	$-0.149086\pi$
$620$	4.29659i	0.172555i
$621$	0	0
$622$	− 27.3920i	− 1.09832i
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	20.3092	0.811720
$627$	0	0
$628$	− 2.45486i	− 0.0979595i
$629$	13.3101	0.530709
$630$	0	0
$631$	22.5219	0.896581	0.448291	−	0.893888i	$-0.352033\pi$
$631$	22.5219	0.896581	0.448291	+	0.893888i	$0.352033\pi$
$632$	29.1911i	1.16116i
$633$	0	0
$634$	3.16866	0.125844
$635$	−13.2009	−0.523863
$636$	0	0
$637$	0	0
$638$	14.5223i	0.574944i
$639$	0	0
$640$	2.73496i	0.108109i
$641$	45.2571i	1.78755i	0.448516	+	0.893775i	$0.351953\pi$
$641$	45.2571i	1.78755i	−0.448516	+	0.893775i	$0.648047\pi$
$642$	0	0
$643$	35.7477i	1.40975i	0.709330	+	0.704877i	$0.248998\pi$
$643$	35.7477i	1.40975i	−0.709330	+	0.704877i	$0.751002\pi$
$644$	0	0
$645$	0	0
$646$	8.13125	0.319920
$647$	−13.9336	−0.547786	−0.273893	−	0.961760i	$-0.588311\pi$
$647$	−13.9336	−0.547786	−0.273893	+	0.961760i	$0.588311\pi$
$648$	0	0
$649$	− 12.0395i	− 0.472591i
$650$	0.412444	0.0161774
$651$	0	0
$652$	−7.21238	−0.282459
$653$	21.8033i	0.853230i	0.904433	+	0.426615i	$0.140294\pi$
$653$	21.8033i	0.853230i	−0.904433	+	0.426615i	$0.859706\pi$
$654$	0	0
$655$	18.8531	0.736653
$656$	−9.04523	−0.353157
$657$	0	0
$658$	0	0
$659$	− 16.8309i	− 0.655638i	−0.944741	−	0.327819i	$-0.893686\pi$
$659$	− 16.8309i	− 0.655638i	0.944741	−	0.327819i	$-0.106314\pi$
$660$	0	0
$661$	21.7907i	0.847560i	0.905765	+	0.423780i	$0.139297\pi$
$661$	21.7907i	0.847560i	−0.905765	+	0.423780i	$0.860703\pi$
$662$	− 39.4698i	− 1.53404i
$663$	0	0
$664$	1.93243i	0.0749930i
$665$	0	0
$666$	0	0
$667$	−21.2540	−0.822959
$668$	−13.7099	−0.530451
$669$	0	0
$670$	− 13.2922i	− 0.513521i
$671$	−21.4181	−0.826836
$672$	0	0
$673$	24.0769	0.928096	0.464048	−	0.885810i	$-0.346396\pi$
$673$	24.0769	0.928096	0.464048	+	0.885810i	$0.346396\pi$
$674$	5.46064i	0.210336i
$675$	0	0
$676$	8.58326	0.330125
$677$	−35.9713	−1.38249	−0.691244	−	0.722622i	$-0.742937\pi$
$677$	−35.9713	−1.38249	−0.691244	+	0.722622i	$0.742937\pi$
$678$	0	0
$679$	0	0
$680$	5.29022i	0.202871i
$681$	0	0
$682$	− 11.9266i	− 0.456695i
$683$	− 32.9651i	− 1.26137i	−0.776037	−	0.630687i	$-0.782773\pi$
$683$	− 32.9651i	− 1.26137i	0.776037	−	0.630687i	$-0.217227\pi$
$684$	0	0
$685$	4.35312i	0.166324i
$686$	0	0
$687$	0	0
$688$	−8.53991	−0.325581
$689$	1.46741	0.0559039
$690$	0	0
$691$	12.8724i	0.489689i	0.969562	+	0.244845i	$0.0787369\pi$
$691$	12.8724i	0.489689i	−0.969562	+	0.244845i	$0.921263\pi$
$692$	9.79522	0.372358
$693$	0	0
$694$	−27.1509	−1.03063
$695$	1.62070i	0.0614766i
$696$	0	0
$697$	−6.99442	−0.264932
$698$	−4.21214	−0.159432
$699$	0	0
$700$	0	0
$701$	− 26.9582i	− 1.01820i	−0.860708	−	0.509099i	$-0.829979\pi$
$701$	− 26.9582i	− 1.01820i	0.860708	−	0.509099i	$-0.170021\pi$
$702$	0	0
$703$	31.7556i	1.19768i
$704$	13.7735i	0.519110i
$705$	0	0
$706$	− 2.41462i	− 0.0908755i
$707$	0	0
$708$	0	0
$709$	11.4300	0.429264	0.214632	−	0.976695i	$-0.431145\pi$
$709$	11.4300	0.429264	0.214632	+	0.976695i	$0.431145\pi$
$710$	−3.69333	−0.138608
$711$	0	0
$712$	− 28.9506i	− 1.08497i
$713$	17.4551	0.653700
$714$	0	0
$715$	0.572606	0.0214142
$716$	7.96656i	0.297724i
$717$	0	0
$718$	−12.7579	−0.476121
$719$	48.2915	1.80097	0.900484	−	0.434889i	$-0.143213\pi$
$719$	48.2915	1.80097	0.900484	+	0.434889i	$0.143213\pi$
$720$	0	0
$721$	0	0
$722$	− 2.53854i	− 0.0944747i
$723$	0	0
$724$	− 11.5728i	− 0.430100i
$725$	− 7.84601i	− 0.291393i
$726$	0	0
$727$	25.8282i	0.957913i	0.877839	+	0.478957i	$0.158985\pi$
$727$	25.8282i	0.957913i	−0.877839	+	0.478957i	$0.841015\pi$
$728$	0	0
$729$	0	0
$730$	19.2273	0.711635
$731$	−6.60367	−0.244246
$732$	0	0
$733$	37.2569i	1.37612i	0.725656	+	0.688058i	$0.241537\pi$
$733$	37.2569i	1.37612i	−0.725656	+	0.688058i	$0.758463\pi$
$734$	−35.5247	−1.31124
$735$	0	0
$736$	−9.73311	−0.358767
$737$	− 18.4538i	− 0.679755i
$738$	0	0
$739$	−15.1832	−0.558524	−0.279262	−	0.960215i	$-0.590090\pi$
$739$	−15.1832	−0.558524	−0.279262	+	0.960215i	$0.590090\pi$
$740$	−5.16582	−0.189899
$741$	0	0
$742$	0	0
$743$	− 14.9120i	− 0.547069i	−0.961862	−	0.273534i	$-0.911807\pi$
$743$	− 14.9120i	− 0.547069i	0.961862	−	0.273534i	$-0.0881927\pi$
$744$	0	0
$745$	− 9.32158i	− 0.341516i
$746$	− 20.7998i	− 0.761534i
$747$	0	0
$748$	1.83640i	0.0671454i
$749$	0	0
$750$	0	0
$751$	−14.3840	−0.524881	−0.262440	−	0.964948i	$-0.584527\pi$
$751$	−14.3840	−0.524881	−0.262440	+	0.964948i	$0.584527\pi$
$752$	11.3862	0.415211
$753$	0	0
$754$	− 3.23604i	− 0.117850i
$755$	−11.3489	−0.413029
$756$	0	0
$757$	−3.90656	−0.141986	−0.0709931	−	0.997477i	$-0.522617\pi$
$757$	−3.90656	−0.141986	−0.0709931	+	0.997477i	$0.522617\pi$
$758$	− 35.2477i	− 1.28026i
$759$	0	0
$760$	−12.6215	−0.457830
$761$	−12.8066	−0.464240	−0.232120	−	0.972687i	$-0.574566\pi$
$761$	−12.8066	−0.464240	−0.232120	+	0.972687i	$0.574566\pi$
$762$	0	0
$763$	0	0
$764$	1.33944i	0.0484592i
$765$	0	0
$766$	− 21.9738i	− 0.793947i
$767$	2.68279i	0.0968698i
$768$	0	0
$769$	− 36.3639i	− 1.31132i	−0.755058	−	0.655658i	$-0.772391\pi$
$769$	− 36.3639i	− 1.31132i	0.755058	−	0.655658i	$-0.227609\pi$
$770$	0	0
$771$	0	0
$772$	6.86556	0.247097
$773$	−32.5499	−1.17074	−0.585370	−	0.810766i	$-0.699051\pi$
$773$	−32.5499	−1.17074	−0.585370	+	0.810766i	$0.699051\pi$
$774$	0	0
$775$	6.44364i	0.231462i
$776$	18.2682	0.655789
$777$	0	0
$778$	42.7821	1.53381
$779$	− 16.6874i	− 0.597889i
$780$	0	0
$781$	−5.12754	−0.183478
$782$	5.37373	0.192164
$783$	0	0
$784$	0	0
$785$	− 3.68158i	− 0.131401i
$786$	0	0
$787$	− 18.0485i	− 0.643360i	−0.946849	−	0.321680i	$-0.895753\pi$
$787$	− 18.0485i	− 0.643360i	0.946849	−	0.321680i	$-0.104247\pi$
$788$	14.6065i	0.520336i
$789$	0	0
$790$	10.9461i	0.389446i
$791$	0	0
$792$	0	0
$793$	4.77264	0.169482
$794$	−14.8181	−0.525875
$795$	0	0
$796$	12.4711i	0.442027i
$797$	17.0005	0.602190	0.301095	−	0.953594i	$-0.402648\pi$
$797$	17.0005	0.602190	0.301095	+	0.953594i	$0.402648\pi$
$798$	0	0
$799$	8.80461	0.311485
$800$	− 3.59302i	− 0.127032i
$801$	0	0
$802$	10.1083	0.356935
$803$	26.6937	0.942000
$804$	0	0
$805$	0	0
$806$	2.65764i	0.0936115i
$807$	0	0
$808$	13.4323i	0.472548i
$809$	44.8419i	1.57656i	0.615318	+	0.788279i	$0.289028\pi$
$809$	44.8419i	1.57656i	−0.615318	+	0.788279i	$0.710972\pi$
$810$	0	0
$811$	12.6240i	0.443289i	0.975128	+	0.221645i	$0.0711425\pi$
$811$	12.6240i	0.443289i	−0.975128	+	0.221645i	$0.928858\pi$
$812$	0	0
$813$	0	0
$814$	14.3395	0.502598
$815$	−10.8165	−0.378885
$816$	0	0
$817$	− 15.7552i	− 0.551204i
$818$	8.31284	0.290652
$819$	0	0
$820$	2.71462	0.0947985
$821$	− 2.19737i	− 0.0766888i	−0.999265	−	0.0383444i	$-0.987792\pi$
$821$	− 2.19737i	− 0.0766888i	0.999265	−	0.0383444i	$-0.0122084\pi$
$822$	0	0
$823$	−39.0579	−1.36147	−0.680736	−	0.732529i	$-0.738340\pi$
$823$	−39.0579	−1.36147	−0.680736	+	0.732529i	$0.738340\pi$
$824$	−1.56035	−0.0543572
$825$	0	0
$826$	0	0
$827$	5.09620i	0.177212i	0.996067	+	0.0886061i	$0.0282412\pi$
$827$	5.09620i	0.177212i	−0.996067	+	0.0886061i	$0.971759\pi$
$828$	0	0
$829$	− 6.08051i	− 0.211185i	−0.994409	−	0.105592i	$-0.966326\pi$
$829$	− 6.08051i	− 0.211185i	0.994409	−	0.105592i	$-0.0336739\pi$
$830$	0.724628i	0.0251522i
$831$	0	0
$832$	− 3.06919i	− 0.106405i
$833$	0	0
$834$	0	0
$835$	−20.5609	−0.711538
$836$	−4.38132	−0.151531
$837$	0	0
$838$	18.0192i	0.622463i
$839$	34.6534	1.19637	0.598183	−	0.801359i	$-0.295889\pi$
$839$	34.6534	1.19637	0.598183	+	0.801359i	$0.295889\pi$
$840$	0	0
$841$	−32.5599	−1.12275
$842$	− 26.9556i	− 0.928951i
$843$	0	0
$844$	3.55302	0.122300
$845$	12.8724	0.442824
$846$	0	0
$847$	0	0
$848$	9.12721i	0.313430i
$849$	0	0
$850$	1.98374i	0.0680416i
$851$	20.9864i	0.719405i
$852$	0	0
$853$	− 31.2245i	− 1.06911i	−0.845135	−	0.534554i	$-0.820480\pi$
$853$	− 31.2245i	− 1.06911i	0.845135	−	0.534554i	$-0.179520\pi$
$854$	0	0
$855$	0	0
$856$	−60.1381	−2.05548
$857$	−51.7185	−1.76667	−0.883336	−	0.468740i	$-0.844708\pi$
$857$	−51.7185	−1.76667	−0.883336	+	0.468740i	$0.844708\pi$
$858$	0	0
$859$	− 24.4200i	− 0.833199i	−0.909090	−	0.416600i	$-0.863222\pi$
$859$	− 24.4200i	− 0.833199i	0.909090	−	0.416600i	$-0.136778\pi$
$860$	2.56296	0.0873963
$861$	0	0
$862$	−16.3714	−0.557614
$863$	− 26.7371i	− 0.910141i	−0.890455	−	0.455071i	$-0.849614\pi$
$863$	− 26.7371i	− 0.910141i	0.890455	−	0.455071i	$-0.150386\pi$
$864$	0	0
$865$	14.6900	0.499475
$866$	40.7202	1.38373
$867$	0	0
$868$	0	0
$869$	15.1967i	0.515514i
$870$	0	0
$871$	4.11211i	0.139333i
$872$	− 28.2205i	− 0.955668i
$873$	0	0
$874$	12.8208i	0.433669i
$875$	0	0
$876$	0	0
$877$	32.7528	1.10598	0.552992	−	0.833186i	$-0.313486\pi$
$877$	32.7528	1.10598	0.552992	+	0.833186i	$0.313486\pi$
$878$	17.3124	0.584266
$879$	0	0
$880$	3.56158i	0.120061i
$881$	−44.2838	−1.49196	−0.745980	−	0.665969i	$-0.768018\pi$
$881$	−44.2838	−1.49196	−0.745980	+	0.665969i	$0.768018\pi$
$882$	0	0
$883$	−45.3627	−1.52658	−0.763288	−	0.646058i	$-0.776416\pi$
$883$	−45.3627	−1.52658	−0.763288	+	0.646058i	$0.776416\pi$
$884$	− 0.409209i	− 0.0137632i
$885$	0	0
$886$	37.4818	1.25922
$887$	19.4350	0.652563	0.326281	−	0.945273i	$-0.394204\pi$
$887$	19.4350	0.652563	0.326281	+	0.945273i	$0.394204\pi$
$888$	0	0
$889$	0	0
$890$	− 10.8559i	− 0.363892i
$891$	0	0
$892$	4.56163i	0.152735i
$893$	21.0062i	0.702946i
$894$	0	0
$895$	11.9475i	0.399362i
$896$	0	0
$897$	0	0
$898$	−14.0495	−0.468837
$899$	50.5569	1.68617
$900$	0	0
$901$	7.05782i	0.235130i
$902$	−7.53534	−0.250899
$903$	0	0
$904$	−14.8335	−0.493356
$905$	− 17.3559i	− 0.576929i
$906$	0	0
$907$	−41.2795	−1.37066	−0.685332	−	0.728230i	$-0.740343\pi$
$907$	−41.2795	−1.37066	−0.685332	+	0.728230i	$0.740343\pi$
$908$	−17.8673	−0.592946
$909$	0	0
$910$	0	0
$911$	11.3055i	0.374567i	0.982306	+	0.187283i	$0.0599683\pi$
$911$	11.3055i	0.374567i	−0.982306	+	0.187283i	$0.940032\pi$
$912$	0	0
$913$	1.00602i	0.0332943i
$914$	− 34.3356i	− 1.13572i
$915$	0	0
$916$	15.8791i	0.524661i
$917$	0	0
$918$	0	0
$919$	−38.9164	−1.28373	−0.641866	−	0.766817i	$-0.721839\pi$
$919$	−38.9164	−1.28373	−0.641866	+	0.766817i	$0.721839\pi$
$920$	−8.34123	−0.275002
$921$	0	0
$922$	− 20.2183i	− 0.665853i
$923$	1.14258	0.0376085
$924$	0	0
$925$	−7.74723	−0.254727
$926$	− 18.8367i	− 0.619012i
$927$	0	0
$928$	−28.1909	−0.925411
$929$	14.4205	0.473120	0.236560	−	0.971617i	$-0.423980\pi$
$929$	14.4205	0.473120	0.236560	+	0.971617i	$0.423980\pi$
$930$	0	0
$931$	0	0
$932$	3.05095i	0.0999370i
$933$	0	0
$934$	40.4762i	1.32442i
$935$	2.75407i	0.0900676i
$936$	0	0
$937$	− 45.7697i	− 1.49523i	−0.664133	−	0.747615i	$-0.731199\pi$
$937$	− 45.7697i	− 1.49523i	0.664133	−	0.747615i	$-0.268801\pi$
$938$	0	0
$939$	0	0
$940$	−3.41717	−0.111456
$941$	11.7592	0.383340	0.191670	−	0.981459i	$-0.438610\pi$
$941$	11.7592	0.383340	0.191670	+	0.981459i	$0.438610\pi$
$942$	0	0
$943$	− 11.0283i	− 0.359130i
$944$	−16.6868	−0.543109
$945$	0	0
$946$	−7.11438	−0.231308
$947$	35.6980i	1.16003i	0.814606	+	0.580015i	$0.196953\pi$
$947$	35.6980i	1.16003i	−0.814606	+	0.580015i	$0.803047\pi$
$948$	0	0
$949$	−5.94822	−0.193087
$950$	−4.73284	−0.153554
$951$	0	0
$952$	0	0
$953$	3.09755i	0.100339i	0.998741	+	0.0501697i	$0.0159762\pi$
$953$	3.09755i	0.100339i	−0.998741	+	0.0501697i	$0.984024\pi$
$954$	0	0
$955$	2.00877i	0.0650023i
$956$	8.35898i	0.270349i
$957$	0	0
$958$	− 32.8009i	− 1.05975i
$959$	0	0
$960$	0	0
$961$	−10.5205	−0.339371
$962$	−3.19530	−0.103021
$963$	0	0
$964$	1.39979i	0.0450842i
$965$	10.2964	0.331452
$966$	0	0
$967$	−30.4410	−0.978917	−0.489458	−	0.872027i	$-0.662806\pi$
$967$	−30.4410	−0.978917	−0.489458	+	0.872027i	$0.662806\pi$
$968$	− 25.9587i	− 0.834344i
$969$	0	0
$970$	6.85023	0.219948
$971$	−30.1997	−0.969154	−0.484577	−	0.874749i	$-0.661026\pi$
$971$	−30.1997	−0.969154	−0.484577	+	0.874749i	$0.661026\pi$
$972$	0	0
$973$	0	0
$974$	− 20.3430i	− 0.651833i
$975$	0	0
$976$	29.6856i	0.950213i
$977$	11.4654i	0.366809i	0.983038	+	0.183405i	$0.0587119\pi$
$977$	11.4654i	0.366809i	−0.983038	+	0.183405i	$0.941288\pi$
$978$	0	0
$979$	− 15.0715i	− 0.481689i
$980$	0	0
$981$	0	0
$982$	3.24574	0.103576
$983$	−31.3686	−1.00050	−0.500251	−	0.865881i	$-0.666759\pi$
$983$	−31.3686	−1.00050	−0.500251	+	0.865881i	$0.666759\pi$
$984$	0	0
$985$	21.9056i	0.697969i
$986$	15.5644	0.495672
$987$	0	0
$988$	0.976299	0.0310602
$989$	− 10.4122i	− 0.331088i
$990$	0	0
$991$	−0.520712	−0.0165410	−0.00827048	−	0.999966i	$-0.502633\pi$
$991$	−0.520712	−0.0165410	−0.00827048	+	0.999966i	$0.502633\pi$
$992$	23.1521	0.735081
$993$	0	0
$994$	0	0
$995$	18.7030i	0.592926i
$996$	0	0
$997$	29.1905i	0.924472i	0.886757	+	0.462236i	$0.152953\pi$
$997$	29.1905i	0.924472i	−0.886757	+	0.462236i	$0.847047\pi$
$998$	− 6.24056i	− 0.197542i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.b.d.881.6	yes	16
3.2	odd	2		2205.2.b.c.881.11	yes	16
7.6	odd	2		2205.2.b.c.881.6	✓	16
21.20	even	2	inner	2205.2.b.d.881.11	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2205.2.b.c.881.6	✓	16	7.6	odd	2
2205.2.b.c.881.11	yes	16	3.2	odd	2
2205.2.b.d.881.6	yes	16	1.1	even	1	trivial
2205.2.b.d.881.11	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)

\(f(q)\)	\(=\)	\(q-1.15464i q^{2} +0.666795 q^{4} +1.00000 q^{5} -3.07920i q^{8} -1.15464i q^{10} -1.60302i q^{11} +0.357205i q^{13} -2.22179 q^{16} -1.71805 q^{17} -4.09896i q^{19} +0.666795 q^{20} -1.85092 q^{22} -2.70889i q^{23} +1.00000 q^{25} +0.412444 q^{26} -7.84601i q^{29} +6.44364i q^{31} -3.59302i q^{32} +1.98374i q^{34} -7.74723 q^{37} -4.73284 q^{38} -3.07920i q^{40} +4.07114 q^{41} +3.84370 q^{43} -1.06889i q^{44} -3.12781 q^{46} -5.12477 q^{47} -1.15464i q^{50} +0.238182i q^{52} -4.10804i q^{53} -1.60302i q^{55} -9.05935 q^{58} +7.51051 q^{59} -13.3611i q^{61} +7.44012 q^{62} -8.59225 q^{64} +0.357205i q^{65} +11.5119 q^{67} -1.14559 q^{68} -3.19867i q^{71} +16.6521i q^{73} +8.94530i q^{74} -2.73317i q^{76} -9.48008 q^{79} -2.22179 q^{80} -4.70072i q^{82} -0.627576 q^{83} -1.71805 q^{85} -4.43811i q^{86} -4.93602 q^{88} +9.40198 q^{89} -1.80628i q^{92} +5.91729i q^{94} -4.09896i q^{95} +5.93276i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4} + 16 q^{5} - 16 q^{20} - 16 q^{22} + 16 q^{25} + 32 q^{26} - 16 q^{38} - 32 q^{41} + 32 q^{43} - 16 q^{46} + 32 q^{47} + 48 q^{58} - 32 q^{59} - 32 q^{62} + 16 q^{64} - 16 q^{68} + 32 q^{79}+ \cdots + 64 q^{89}+O(q^{100}) \) 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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