LMFDB - Embedded newform 261.2.c.b.28.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,2,Mod(28,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("261.28");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	261.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$2.08409549276$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			28.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	261.28
Dual form			261.2.c.b.28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.61803i q^{2} -0.618034 q^{4} -3.23607 q^{5} -4.23607 q^{7} +2.23607i q^{8} -5.23607i q^{10} -2.23607i q^{11} -1.00000 q^{13} -6.85410i q^{14} -4.85410 q^{16} +3.00000i q^{17} +7.70820i q^{19} +2.00000 q^{20} +3.61803 q^{22} -1.23607 q^{23} +5.47214 q^{25} -1.61803i q^{26} +2.61803 q^{28} +(4.47214 - 3.00000i) q^{29} -7.23607i q^{31} -3.38197i q^{32} -4.85410 q^{34} +13.7082 q^{35} +4.76393i q^{37} -12.4721 q^{38} -7.23607i q^{40} +4.47214i q^{41} -0.472136i q^{43} +1.38197i q^{44} -2.00000i q^{46} +10.2361i q^{47} +10.9443 q^{49} +8.85410i q^{50} +0.618034 q^{52} -6.76393 q^{53} +7.23607i q^{55} -9.47214i q^{56} +(4.85410 + 7.23607i) q^{58} -8.94427 q^{59} +2.76393i q^{61} +11.7082 q^{62} -4.23607 q^{64} +3.23607 q^{65} +4.70820 q^{67} -1.85410i q^{68} +22.1803i q^{70} -4.76393 q^{71} -7.70820i q^{73} -7.70820 q^{74} -4.76393i q^{76} +9.47214i q^{77} -2.29180i q^{79} +15.7082 q^{80} -7.23607 q^{82} +6.00000 q^{83} -9.70820i q^{85} +0.763932 q^{86} +5.00000 q^{88} +0.0557281i q^{89} +4.23607 q^{91} +0.763932 q^{92} -16.5623 q^{94} -24.9443i q^{95} +18.1803i q^{97} +17.7082i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 4 q^{5} - 8 q^{7} - 4 q^{13} - 6 q^{16} + 8 q^{20} + 10 q^{22} + 4 q^{23} + 4 q^{25} + 6 q^{28} - 6 q^{34} + 28 q^{35} - 32 q^{38} + 8 q^{49} - 2 q^{52} - 36 q^{53} + 6 q^{58} + 20 q^{62}+ \cdots - 26 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 - 4 * q^5 - 8 * q^7 - 4 * q^13 - 6 * q^16 + 8 * q^20 + 10 * q^22 + 4 * q^23 + 4 * q^25 + 6 * q^28 - 6 * q^34 + 28 * q^35 - 32 * q^38 + 8 * q^49 - 2 * q^52 - 36 * q^53 + 6 * q^58 + 20 * q^62 - 8 * q^64 + 4 * q^65 - 8 * q^67 - 28 * q^71 - 4 * q^74 + 36 * q^80 - 20 * q^82 + 24 * q^83 + 12 * q^86 + 20 * q^88 + 8 * q^91 + 12 * q^92 - 26 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/261\mathbb{Z}\right)^\times$.

$n$	$118$	$146$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.61803i			1.14412i	0.820211	+	0.572061i	$0.193856\pi$
$2$			1.61803i			1.14412i	−0.820211	+	0.572061i	$0.806144\pi$
$3$		0			0
$3$		0			0
$4$	−0.618034			−0.309017
$5$	−3.23607			−1.44721			−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607			−1.44721			−0.723607	+	0.690212i	$0.757517\pi$
$6$		0			0
$7$	−4.23607			−1.60108			−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607			−1.60108			−0.800542	+	0.599277i	$0.795455\pi$
$8$			2.23607i			0.790569i
$9$		0			0
$10$		−	5.23607i		−	1.65579i
$11$		−	2.23607i		−	0.674200i	−0.941469	−	0.337100i	$-0.890554\pi$
$11$		−	2.23607i		−	0.674200i	0.941469	−	0.337100i	$-0.109446\pi$
$12$		0			0
$13$	−1.00000			−0.277350			−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000			−0.277350			−0.138675	+	0.990338i	$0.544284\pi$
$14$		−	6.85410i		−	1.83184i
$15$		0			0
$16$	−4.85410			−1.21353
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$			7.70820i			1.76838i	0.467124	+	0.884192i	$0.345290\pi$
$19$			7.70820i			1.76838i	−0.467124	+	0.884192i	$0.654710\pi$
$20$	2.00000			0.447214
$21$		0			0
$22$	3.61803			0.771367
$23$	−1.23607			−0.257738			−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607			−0.257738			−0.128869	+	0.991662i	$0.541135\pi$
$24$		0			0
$25$	5.47214			1.09443
$26$		−	1.61803i		−	0.317323i
$27$		0			0
$28$	2.61803			0.494762
$29$	4.47214	−	3.00000i	0.830455	−	0.557086i
$29$	4.47214	−	3.00000i	0.830455	−	0.557086i
$30$		0			0
$31$		−	7.23607i		−	1.29964i	−0.760090	−	0.649818i	$-0.774845\pi$
$31$		−	7.23607i		−	1.29964i	0.760090	−	0.649818i	$-0.225155\pi$
$32$		−	3.38197i		−	0.597853i
$33$		0			0
$34$	−4.85410			−0.832472
$35$	13.7082			2.31711
$36$		0			0
$37$			4.76393i			0.783186i	0.920139	+	0.391593i	$0.128076\pi$
$37$			4.76393i			0.783186i	−0.920139	+	0.391593i	$0.871924\pi$
$38$	−12.4721			−2.02325
$39$		0			0
$40$		−	7.23607i		−	1.14412i
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$		−	0.472136i		−	0.0720001i	−0.999352	−	0.0360000i	$-0.988538\pi$
$43$		−	0.472136i		−	0.0720001i	0.999352	−	0.0360000i	$-0.0114616\pi$
$44$			1.38197i			0.208339i
$45$		0			0
$46$		−	2.00000i		−	0.294884i
$47$			10.2361i			1.49308i	0.665338	+	0.746542i	$0.268287\pi$
$47$			10.2361i			1.49308i	−0.665338	+	0.746542i	$0.731713\pi$
$48$		0			0
$49$	10.9443			1.56347
$50$			8.85410i			1.25216i
$51$		0			0
$52$	0.618034			0.0857059
$53$	−6.76393			−0.929098			−0.464549	−	0.885548i	$-0.653783\pi$
$53$	−6.76393			−0.929098			−0.464549	+	0.885548i	$0.653783\pi$
$54$		0			0
$55$			7.23607i			0.975711i
$56$		−	9.47214i		−	1.26577i
$57$		0			0
$58$	4.85410	+	7.23607i	0.637375	+	0.950142i
$59$	−8.94427			−1.16445			−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427			−1.16445			−0.582223	+	0.813029i	$0.697817\pi$
$60$		0			0
$61$			2.76393i			0.353885i	0.984221	+	0.176943i	$0.0566207\pi$
$61$			2.76393i			0.353885i	−0.984221	+	0.176943i	$0.943379\pi$
$62$	11.7082			1.48694
$63$		0			0
$64$	−4.23607			−0.529508
$65$	3.23607			0.401385
$66$		0			0
$67$	4.70820			0.575199			0.287599	−	0.957751i	$-0.407143\pi$
$67$	4.70820			0.575199			0.287599	+	0.957751i	$0.407143\pi$
$68$		−	1.85410i		−	0.224843i
$69$		0			0
$70$			22.1803i			2.65106i
$71$	−4.76393			−0.565375			−0.282687	−	0.959212i	$-0.591226\pi$
$71$	−4.76393			−0.565375			−0.282687	+	0.959212i	$0.591226\pi$
$72$		0			0
$73$		−	7.70820i		−	0.902177i	−0.892479	−	0.451089i	$-0.851036\pi$
$73$		−	7.70820i		−	0.902177i	0.892479	−	0.451089i	$-0.148964\pi$
$74$	−7.70820			−0.896061
$75$		0			0
$76$		−	4.76393i		−	0.546460i
$77$			9.47214i			1.07945i
$78$		0			0
$79$		−	2.29180i		−	0.257847i	−0.991655	−	0.128924i	$-0.958848\pi$
$79$		−	2.29180i		−	0.257847i	0.991655	−	0.128924i	$-0.0411522\pi$
$80$	15.7082			1.75623
$81$		0			0
$82$	−7.23607			−0.799090
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		−	9.70820i		−	1.05300i
$86$	0.763932			0.0823769
$87$		0			0
$88$	5.00000			0.533002
$89$			0.0557281i			0.00590717i	0.999996	+	0.00295358i	$0.000940156\pi$
$89$			0.0557281i			0.00590717i	−0.999996	+	0.00295358i	$0.999060\pi$
$90$		0			0
$91$	4.23607			0.444061
$92$	0.763932			0.0796454
$93$		0			0
$94$	−16.5623			−1.70827
$95$		−	24.9443i		−	2.55923i
$96$		0			0
$97$			18.1803i			1.84593i	0.384879	+	0.922967i	$0.374243\pi$
$97$			18.1803i			1.84593i	−0.384879	+	0.922967i	$0.625757\pi$
$98$			17.7082i			1.78880i
$99$		0			0
$100$	−3.38197			−0.338197
$101$			3.94427i			0.392470i	0.980557	+	0.196235i	$0.0628715\pi$
$101$			3.94427i			0.392470i	−0.980557	+	0.196235i	$0.937129\pi$
$102$		0			0
$103$	−19.4164			−1.91316			−0.956578	−	0.291477i	$-0.905853\pi$
$103$	−19.4164			−1.91316			−0.956578	+	0.291477i	$0.905853\pi$
$104$		−	2.23607i		−	0.219265i
$105$		0			0
$106$		−	10.9443i		−	1.06300i
$107$	−0.763932			−0.0738521			−0.0369260	−	0.999318i	$-0.511757\pi$
$107$	−0.763932			−0.0738521			−0.0369260	+	0.999318i	$0.511757\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	−11.7082			−1.11633
$111$		0			0
$112$	20.5623			1.94296
$113$		−	4.52786i		−	0.425946i	−0.977058	−	0.212973i	$-0.931685\pi$
$113$		−	4.52786i		−	0.425946i	0.977058	−	0.212973i	$-0.0683146\pi$
$114$		0			0
$115$	4.00000			0.373002
$116$	−2.76393	+	1.85410i	−0.256625	+	0.172149i
$117$		0			0
$118$		−	14.4721i		−	1.33227i
$119$		−	12.7082i		−	1.16496i
$120$		0			0
$121$	6.00000			0.545455
$122$	−4.47214			−0.404888
$123$		0			0
$124$			4.47214i			0.401610i
$125$	−1.52786			−0.136656
$126$		0			0
$127$		−	12.4721i		−	1.10672i	−0.832941	−	0.553362i	$-0.813345\pi$
$127$		−	12.4721i		−	1.10672i	0.832941	−	0.553362i	$-0.186655\pi$
$128$		−	13.6180i		−	1.20368i
$129$		0			0
$130$			5.23607i			0.459234i
$131$		−	12.2361i		−	1.06907i	−0.845146	−	0.534535i	$-0.820487\pi$
$131$		−	12.2361i		−	1.06907i	0.845146	−	0.534535i	$-0.179513\pi$
$132$		0			0
$133$		−	32.6525i		−	2.83133i
$134$			7.61803i			0.658098i
$135$		0			0
$136$	−6.70820			−0.575224
$137$			3.52786i			0.301406i	0.988579	+	0.150703i	$0.0481537\pi$
$137$			3.52786i			0.301406i	−0.988579	+	0.150703i	$0.951846\pi$
$138$		0			0
$139$	−6.70820			−0.568982			−0.284491	−	0.958679i	$-0.591825\pi$
$139$	−6.70820			−0.568982			−0.284491	+	0.958679i	$0.591825\pi$
$140$	−8.47214			−0.716026
$141$		0			0
$142$		−	7.70820i		−	0.646858i
$143$			2.23607i			0.186989i
$144$		0			0
$145$	−14.4721	+	9.70820i	−1.20185	+	0.806222i
$146$	12.4721			1.03220
$147$		0			0
$148$		−	2.94427i		−	0.242018i
$149$	5.52786			0.452860			0.226430	−	0.974027i	$-0.427295\pi$
$149$	5.52786			0.452860			0.226430	+	0.974027i	$0.427295\pi$
$150$		0			0
$151$	12.0000			0.976546			0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000			0.976546			0.488273	+	0.872691i	$0.337627\pi$
$152$	−17.2361			−1.39803
$153$		0			0
$154$	−15.3262			−1.23502
$155$			23.4164i			1.88085i
$156$		0			0
$157$			2.00000i			0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$			2.00000i			0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	3.70820			0.295009
$159$		0			0
$160$			10.9443i			0.865221i
$161$	5.23607			0.412660
$162$		0			0
$163$			7.41641i			0.580898i	0.956890	+	0.290449i	$0.0938046\pi$
$163$			7.41641i			0.580898i	−0.956890	+	0.290449i	$0.906195\pi$
$164$		−	2.76393i		−	0.215827i
$165$		0			0
$166$			9.70820i			0.753503i
$167$	−15.8885			−1.22949			−0.614746	−	0.788725i	$-0.710742\pi$
$167$	−15.8885			−1.22949			−0.614746	+	0.788725i	$0.710742\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$	15.7082			1.20476
$171$		0			0
$172$			0.291796i			0.0222492i
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	−23.1803			−1.75227
$176$			10.8541i			0.818159i
$177$		0			0
$178$	−0.0901699			−0.00675852
$179$	17.2361			1.28828			0.644142	−	0.764906i	$-0.277215\pi$
$179$	17.2361			1.28828			0.644142	+	0.764906i	$0.277215\pi$
$180$		0			0
$181$	20.4164			1.51754			0.758770	−	0.651359i	$-0.225801\pi$
$181$	20.4164			1.51754			0.758770	+	0.651359i	$0.225801\pi$
$182$			6.85410i			0.508060i
$183$		0			0
$184$		−	2.76393i		−	0.203760i
$185$		−	15.4164i		−	1.13344i
$186$		0			0
$187$	6.70820			0.490552
$188$		−	6.32624i		−	0.461388i
$189$		0			0
$190$	40.3607			2.92807
$191$		−	8.94427i		−	0.647185i	−0.946197	−	0.323592i	$-0.895109\pi$
$191$		−	8.94427i		−	0.647185i	0.946197	−	0.323592i	$-0.104891\pi$
$192$		0			0
$193$			17.4164i			1.25366i	0.779156	+	0.626830i	$0.215648\pi$
$193$			17.4164i			1.25366i	−0.779156	+	0.626830i	$0.784352\pi$
$194$	−29.4164			−2.11198
$195$		0			0
$196$	−6.76393			−0.483138
$197$	3.70820			0.264199			0.132099	−	0.991236i	$-0.457828\pi$
$197$	3.70820			0.264199			0.132099	+	0.991236i	$0.457828\pi$
$198$		0			0
$199$	−20.1246			−1.42660			−0.713298	−	0.700861i	$-0.752799\pi$
$199$	−20.1246			−1.42660			−0.713298	+	0.700861i	$0.752799\pi$
$200$			12.2361i			0.865221i
$201$		0			0
$202$	−6.38197			−0.449034
$203$	−18.9443	+	12.7082i	−1.32963	+	0.891941i
$204$		0			0
$205$		−	14.4721i		−	1.01078i
$206$		−	31.4164i		−	2.18888i
$207$		0			0
$208$	4.85410			0.336571
$209$	17.2361			1.19224
$210$		0			0
$211$			10.0000i			0.688428i	0.938891	+	0.344214i	$0.111855\pi$
$211$			10.0000i			0.688428i	−0.938891	+	0.344214i	$0.888145\pi$
$212$	4.18034			0.287107
$213$		0			0
$214$		−	1.23607i		−	0.0844959i
$215$			1.52786i			0.104199i
$216$		0			0
$217$			30.6525i			2.08083i
$218$			8.09017i			0.547935i
$219$		0			0
$220$		−	4.47214i		−	0.301511i
$221$		−	3.00000i		−	0.201802i
$222$		0			0
$223$	−12.7082			−0.851004			−0.425502	−	0.904957i	$-0.639903\pi$
$223$	−12.7082			−0.851004			−0.425502	+	0.904957i	$0.639903\pi$
$224$			14.3262i			0.957212i
$225$		0			0
$226$	7.32624			0.487334
$227$	−9.05573			−0.601050			−0.300525	−	0.953774i	$-0.597162\pi$
$227$	−9.05573			−0.601050			−0.300525	+	0.953774i	$0.597162\pi$
$228$		0			0
$229$			22.1803i			1.46572i	0.680380	+	0.732859i	$0.261815\pi$
$229$			22.1803i			1.46572i	−0.680380	+	0.732859i	$0.738185\pi$
$230$			6.47214i			0.426760i
$231$		0			0
$232$	6.70820	+	10.0000i	0.440415	+	0.656532i
$233$	10.4721			0.686052			0.343026	−	0.939326i	$-0.388548\pi$
$233$	10.4721			0.686052			0.343026	+	0.939326i	$0.388548\pi$
$234$		0			0
$235$		−	33.1246i		−	2.16081i
$236$	5.52786			0.359833
$237$		0			0
$238$	20.5623			1.33286
$239$	−26.1803			−1.69347			−0.846733	−	0.532019i	$-0.821434\pi$
$239$	−26.1803			−1.69347			−0.846733	+	0.532019i	$0.821434\pi$
$240$		0			0
$241$	−3.00000			−0.193247			−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000			−0.193247			−0.0966235	+	0.995321i	$0.530804\pi$
$242$			9.70820i			0.624067i
$243$		0			0
$244$		−	1.70820i		−	0.109357i
$245$	−35.4164			−2.26267
$246$		0			0
$247$		−	7.70820i		−	0.490461i
$248$	16.1803			1.02745
$249$		0			0
$250$		−	2.47214i		−	0.156352i
$251$			12.2361i			0.772334i	0.922429	+	0.386167i	$0.126201\pi$
$251$			12.2361i			0.772334i	−0.922429	+	0.386167i	$0.873799\pi$
$252$		0			0
$253$			2.76393i			0.173767i
$254$	20.1803			1.26623
$255$		0			0
$256$	13.5623			0.847644
$257$	26.4721			1.65129			0.825643	−	0.564193i	$-0.190812\pi$
$257$	26.4721			1.65129			0.825643	+	0.564193i	$0.190812\pi$
$258$		0			0
$259$		−	20.1803i		−	1.25395i
$260$	−2.00000			−0.124035
$261$		0			0
$262$	19.7984			1.22315
$263$			4.94427i			0.304877i	0.988313	+	0.152438i	$0.0487126\pi$
$263$			4.94427i			0.304877i	−0.988313	+	0.152438i	$0.951287\pi$
$264$		0			0
$265$	21.8885			1.34460
$266$	52.8328			3.23939
$267$		0			0
$268$	−2.90983			−0.177746
$269$			21.3607i			1.30238i	0.758913	+	0.651192i	$0.225731\pi$
$269$			21.3607i			1.30238i	−0.758913	+	0.651192i	$0.774269\pi$
$270$		0			0
$271$		−	23.4164i		−	1.42245i	−0.702967	−	0.711223i	$-0.748142\pi$
$271$		−	23.4164i		−	1.42245i	0.702967	−	0.711223i	$-0.251858\pi$
$272$		−	14.5623i		−	0.882969i
$273$		0			0
$274$	−5.70820			−0.344845
$275$		−	12.2361i		−	0.737863i
$276$		0			0
$277$	6.41641			0.385525			0.192762	−	0.981245i	$-0.438255\pi$
$277$	6.41641			0.385525			0.192762	+	0.981245i	$0.438255\pi$
$278$		−	10.8541i		−	0.650986i
$279$		0			0
$280$			30.6525i			1.83184i
$281$	1.41641			0.0844958			0.0422479	−	0.999107i	$-0.486548\pi$
$281$	1.41641			0.0844958			0.0422479	+	0.999107i	$0.486548\pi$
$282$		0			0
$283$	−13.8885			−0.825588			−0.412794	−	0.910824i	$-0.635447\pi$
$283$	−13.8885			−0.825588			−0.412794	+	0.910824i	$0.635447\pi$
$284$	2.94427			0.174710
$285$		0			0
$286$	−3.61803			−0.213939
$287$		−	18.9443i		−	1.11825i
$288$		0			0
$289$	8.00000			0.470588
$290$	−15.7082	−	23.4164i	−0.922417	−	1.37506i
$291$		0			0
$292$			4.76393i			0.278788i
$293$			29.9443i			1.74936i	0.484698	+	0.874682i	$0.338929\pi$
$293$			29.9443i			1.74936i	−0.484698	+	0.874682i	$0.661071\pi$
$294$		0			0
$295$	28.9443			1.68520
$296$	−10.6525			−0.619163
$297$		0			0
$298$			8.94427i			0.518128i
$299$	1.23607			0.0714837
$300$		0			0
$301$			2.00000i			0.115278i
$302$			19.4164i			1.11729i
$303$		0			0
$304$		−	37.4164i		−	2.14598i
$305$		−	8.94427i		−	0.512148i
$306$		0			0
$307$			28.1803i			1.60834i	0.594401	+	0.804168i	$0.297389\pi$
$307$			28.1803i			1.60834i	−0.594401	+	0.804168i	$0.702611\pi$
$308$		−	5.85410i		−	0.333568i
$309$		0			0
$310$	−37.8885			−2.15192
$311$			15.6525i			0.887570i	0.896133	+	0.443785i	$0.146365\pi$
$311$			15.6525i			0.887570i	−0.896133	+	0.443785i	$0.853635\pi$
$312$		0			0
$313$	−5.47214			−0.309303			−0.154652	−	0.987969i	$-0.549426\pi$
$313$	−5.47214			−0.309303			−0.154652	+	0.987969i	$0.549426\pi$
$314$	−3.23607			−0.182622
$315$		0			0
$316$			1.41641i			0.0796792i
$317$		−	23.8328i		−	1.33858i	−0.742999	−	0.669292i	$-0.766597\pi$
$317$		−	23.8328i		−	1.33858i	0.742999	−	0.669292i	$-0.233403\pi$
$318$		0			0
$319$	−6.70820	−	10.0000i	−0.375587	−	0.559893i
$320$	13.7082			0.766312
$321$		0			0
$322$			8.47214i			0.472134i
$323$	−23.1246			−1.28669
$324$		0			0
$325$	−5.47214			−0.303539
$326$	−12.0000			−0.664619
$327$		0			0
$328$	−10.0000			−0.552158
$329$		−	43.3607i		−	2.39055i
$330$		0			0
$331$		−	8.29180i		−	0.455758i	−0.973689	−	0.227879i	$-0.926821\pi$
$331$		−	8.29180i		−	0.455758i	0.973689	−	0.227879i	$-0.0731791\pi$
$332$	−3.70820			−0.203514
$333$		0			0
$334$		−	25.7082i		−	1.40669i
$335$	−15.2361			−0.832435
$336$		0			0
$337$			19.2361i			1.04786i	0.851763	+	0.523928i	$0.175534\pi$
$337$			19.2361i			1.04786i	−0.851763	+	0.523928i	$0.824466\pi$
$338$		−	19.4164i		−	1.05611i
$339$		0			0
$340$			6.00000i			0.325396i
$341$	−16.1803			−0.876215
$342$		0			0
$343$	−16.7082			−0.902158
$344$	1.05573			0.0569210
$345$		0			0
$346$			9.70820i			0.521916i
$347$	22.6525			1.21605			0.608024	−	0.793918i	$-0.291962\pi$
$347$	22.6525			1.21605			0.608024	+	0.793918i	$0.291962\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		−	37.5066i		−	2.00481i
$351$		0			0
$352$	−7.56231			−0.403072
$353$	−4.65248			−0.247626			−0.123813	−	0.992306i	$-0.539512\pi$
$353$	−4.65248			−0.247626			−0.123813	+	0.992306i	$0.539512\pi$
$354$		0			0
$355$	15.4164			0.818218
$356$		−	0.0344419i		−	0.00182541i
$357$		0			0
$358$			27.8885i			1.47396i
$359$			25.3050i			1.33554i	0.744366	+	0.667772i	$0.232752\pi$
$359$			25.3050i			1.33554i	−0.744366	+	0.667772i	$0.767248\pi$
$360$		0			0
$361$	−40.4164			−2.12718
$362$			33.0344i			1.73625i
$363$		0			0
$364$	−2.61803			−0.137222
$365$			24.9443i			1.30564i
$366$		0			0
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	6.00000			0.312772
$369$		0			0
$370$	24.9443			1.29679
$371$	28.6525			1.48756
$372$		0			0
$373$	−3.88854			−0.201341			−0.100671	−	0.994920i	$-0.532099\pi$
$373$	−3.88854			−0.201341			−0.100671	+	0.994920i	$0.532099\pi$
$374$			10.8541i			0.561252i
$375$		0			0
$376$	−22.8885			−1.18039
$377$	−4.47214	+	3.00000i	−0.230327	+	0.154508i
$378$		0			0
$379$			14.2918i			0.734120i	0.930197	+	0.367060i	$0.119636\pi$
$379$			14.2918i			0.734120i	−0.930197	+	0.367060i	$0.880364\pi$
$380$			15.4164i			0.790845i
$381$		0			0
$382$	14.4721			0.740459
$383$	5.34752			0.273246			0.136623	−	0.990623i	$-0.456375\pi$
$383$	5.34752			0.273246			0.136623	+	0.990623i	$0.456375\pi$
$384$		0			0
$385$		−	30.6525i		−	1.56219i
$386$	−28.1803			−1.43434
$387$		0			0
$388$		−	11.2361i		−	0.570425i
$389$			13.4721i			0.683064i	0.939870	+	0.341532i	$0.110946\pi$
$389$			13.4721i			0.683064i	−0.939870	+	0.341532i	$0.889054\pi$
$390$		0			0
$391$		−	3.70820i		−	0.187532i
$392$			24.4721i			1.23603i
$393$		0			0
$394$			6.00000i			0.302276i
$395$			7.41641i			0.373160i
$396$		0			0
$397$	6.94427			0.348523			0.174262	−	0.984699i	$-0.444246\pi$
$397$	6.94427			0.348523			0.174262	+	0.984699i	$0.444246\pi$
$398$		−	32.5623i		−	1.63220i
$399$		0			0
$400$	−26.5623			−1.32812
$401$	−33.7082			−1.68331			−0.841654	−	0.540018i	$-0.818418\pi$
$401$	−33.7082			−1.68331			−0.841654	+	0.540018i	$0.818418\pi$
$402$		0			0
$403$			7.23607i			0.360454i
$404$		−	2.43769i		−	0.121280i
$405$		0			0
$406$	−20.5623	−	30.6525i	−1.02049	−	1.52126i
$407$	10.6525			0.528024
$408$		0			0
$409$		−	10.5836i		−	0.523325i	−0.965159	−	0.261662i	$-0.915729\pi$
$409$		−	10.5836i		−	0.523325i	0.965159	−	0.261662i	$-0.0842707\pi$
$410$	23.4164			1.15645
$411$		0			0
$412$	12.0000			0.591198
$413$	37.8885			1.86437
$414$		0			0
$415$	−19.4164			−0.953114
$416$			3.38197i			0.165815i
$417$		0			0
$418$			27.8885i			1.36407i
$419$	−3.81966			−0.186603			−0.0933013	−	0.995638i	$-0.529742\pi$
$419$	−3.81966			−0.186603			−0.0933013	+	0.995638i	$0.529742\pi$
$420$		0			0
$421$			20.0000i			0.974740i	0.873195	+	0.487370i	$0.162044\pi$
$421$			20.0000i			0.974740i	−0.873195	+	0.487370i	$0.837956\pi$
$422$	−16.1803			−0.787647
$423$		0			0
$424$		−	15.1246i		−	0.734516i
$425$			16.4164i			0.796313i
$426$		0			0
$427$		−	11.7082i		−	0.566600i
$428$	0.472136			0.0228216
$429$		0			0
$430$	−2.47214			−0.119217
$431$	−14.3607			−0.691730			−0.345865	−	0.938284i	$-0.612414\pi$
$431$	−14.3607			−0.691730			−0.345865	+	0.938284i	$0.612414\pi$
$432$		0			0
$433$		−	14.2918i		−	0.686820i	−0.939186	−	0.343410i	$-0.888418\pi$
$433$		−	14.2918i		−	0.686820i	0.939186	−	0.343410i	$-0.111582\pi$
$434$	−49.5967			−2.38072
$435$		0			0
$436$	−3.09017			−0.147992
$437$		−	9.52786i		−	0.455780i
$438$		0			0
$439$	13.2918			0.634383			0.317191	−	0.948362i	$-0.397260\pi$
$439$	13.2918			0.634383			0.317191	+	0.948362i	$0.397260\pi$
$440$	−16.1803			−0.771367
$441$		0			0
$442$	4.85410			0.230886
$443$		−	9.65248i		−	0.458603i	−0.973355	−	0.229301i	$-0.926356\pi$
$443$		−	9.65248i		−	0.458603i	0.973355	−	0.229301i	$-0.0736442\pi$
$444$		0			0
$445$		−	0.180340i		−	0.00854893i
$446$		−	20.5623i		−	0.973653i
$447$		0			0
$448$	17.9443			0.847787
$449$		−	18.8885i		−	0.891405i	−0.895181	−	0.445703i	$-0.852954\pi$
$449$		−	18.8885i		−	0.891405i	0.895181	−	0.445703i	$-0.147046\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$			2.79837i			0.131624i
$453$		0			0
$454$		−	14.6525i		−	0.687675i
$455$	−13.7082			−0.642651
$456$		0			0
$457$	6.41641			0.300147			0.150073	−	0.988675i	$-0.452049\pi$
$457$	6.41641			0.300147			0.150073	+	0.988675i	$0.452049\pi$
$458$	−35.8885			−1.67696
$459$		0			0
$460$	−2.47214			−0.115264
$461$			18.9443i			0.882323i	0.897428	+	0.441161i	$0.145433\pi$
$461$			18.9443i			0.882323i	−0.897428	+	0.441161i	$0.854567\pi$
$462$		0			0
$463$	0.708204			0.0329130			0.0164565	−	0.999865i	$-0.494761\pi$
$463$	0.708204			0.0329130			0.0164565	+	0.999865i	$0.494761\pi$
$464$	−21.7082	+	14.5623i	−1.00778	+	0.676038i
$465$		0			0
$466$			16.9443i			0.784928i
$467$			13.5279i			0.625995i	0.949754	+	0.312997i	$0.101333\pi$
$467$			13.5279i			0.625995i	−0.949754	+	0.312997i	$0.898667\pi$
$468$		0			0
$469$	−19.9443			−0.920941
$470$	53.5967			2.47223
$471$		0			0
$472$		−	20.0000i		−	0.920575i
$473$	−1.05573			−0.0485424
$474$		0			0
$475$			42.1803i			1.93537i
$476$			7.85410i			0.359992i
$477$		0			0
$478$		−	42.3607i		−	1.93753i
$479$		−	10.4721i		−	0.478484i	−0.970960	−	0.239242i	$-0.923101\pi$
$479$		−	10.4721i		−	0.478484i	0.970960	−	0.239242i	$-0.0768989\pi$
$480$		0			0
$481$		−	4.76393i		−	0.217217i
$482$		−	4.85410i		−	0.221098i
$483$		0			0
$484$	−3.70820			−0.168555
$485$		−	58.8328i		−	2.67146i
$486$		0			0
$487$	20.3607			0.922630			0.461315	−	0.887236i	$-0.347378\pi$
$487$	20.3607			0.922630			0.461315	+	0.887236i	$0.347378\pi$
$488$	−6.18034			−0.279771
$489$		0			0
$490$		−	57.3050i		−	2.58877i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	9.00000	+	13.4164i	0.405340	+	0.604245i
$494$	12.4721			0.561148
$495$		0			0
$496$			35.1246i			1.57714i
$497$	20.1803			0.905212
$498$		0			0
$499$	21.1803			0.948162			0.474081	−	0.880481i	$-0.342780\pi$
$499$	21.1803			0.948162			0.474081	+	0.880481i	$0.342780\pi$
$500$	0.944272			0.0422291
$501$		0			0
$502$	−19.7984			−0.883645
$503$		−	25.1803i		−	1.12274i	−0.827566	−	0.561368i	$-0.810275\pi$
$503$		−	25.1803i		−	1.12274i	0.827566	−	0.561368i	$-0.189725\pi$
$504$		0			0
$505$		−	12.7639i		−	0.567988i
$506$	−4.47214			−0.198811
$507$		0			0
$508$			7.70820i			0.341996i
$509$	26.8328			1.18934			0.594672	−	0.803969i	$-0.297282\pi$
$509$	26.8328			1.18934			0.594672	+	0.803969i	$0.297282\pi$
$510$		0			0
$511$			32.6525i			1.44446i
$512$		−	5.29180i		−	0.233867i
$513$		0			0
$514$			42.8328i			1.88927i
$515$	62.8328			2.76874
$516$		0			0
$517$	22.8885			1.00664
$518$	32.6525			1.43467
$519$		0			0
$520$			7.23607i			0.317323i
$521$	3.52786			0.154559			0.0772793	−	0.997009i	$-0.475377\pi$
$521$	3.52786			0.154559			0.0772793	+	0.997009i	$0.475377\pi$
$522$		0			0
$523$	33.0689			1.44600			0.723001	−	0.690847i	$-0.242762\pi$
$523$	33.0689			1.44600			0.723001	+	0.690847i	$0.242762\pi$
$524$			7.56231i			0.330361i
$525$		0			0
$526$	−8.00000			−0.348817
$527$	21.7082			0.945624
$528$		0			0
$529$	−21.4721			−0.933571
$530$			35.4164i			1.53839i
$531$		0			0
$532$			20.1803i			0.874929i
$533$		−	4.47214i		−	0.193710i
$534$		0			0
$535$	2.47214			0.106880
$536$			10.5279i			0.454734i
$537$		0			0
$538$	−34.5623			−1.49009
$539$		−	24.4721i		−	1.05409i
$540$		0			0
$541$		−	10.6525i		−	0.457986i	−0.973428	−	0.228993i	$-0.926457\pi$
$541$		−	10.6525i		−	0.457986i	0.973428	−	0.228993i	$-0.0735432\pi$
$542$	37.8885			1.62745
$543$		0			0
$544$	10.1459			0.435002
$545$	−16.1803			−0.693090
$546$		0			0
$547$	−6.34752			−0.271401			−0.135700	−	0.990750i	$-0.543328\pi$
$547$	−6.34752			−0.271401			−0.135700	+	0.990750i	$0.543328\pi$
$548$		−	2.18034i		−	0.0931395i
$549$		0			0
$550$	19.7984			0.844205
$551$	23.1246	+	34.4721i	0.985142	+	1.46856i
$552$		0			0
$553$			9.70820i			0.412835i
$554$			10.3820i			0.441087i
$555$		0			0
$556$	4.14590			0.175825
$557$	16.4721			0.697947			0.348973	−	0.937133i	$-0.386530\pi$
$557$	16.4721			0.697947			0.348973	+	0.937133i	$0.386530\pi$
$558$		0			0
$559$			0.472136i			0.0199692i
$560$	−66.5410			−2.81187
$561$		0			0
$562$			2.29180i			0.0966736i
$563$		−	13.0689i		−	0.550788i	−0.961331	−	0.275394i	$-0.911192\pi$
$563$		−	13.0689i		−	0.550788i	0.961331	−	0.275394i	$-0.0888083\pi$
$564$		0			0
$565$			14.6525i			0.616434i
$566$		−	22.4721i		−	0.944574i
$567$		0			0
$568$		−	10.6525i		−	0.446968i
$569$		−	21.0000i		−	0.880366i	−0.897908	−	0.440183i	$-0.854914\pi$
$569$		−	21.0000i		−	0.880366i	0.897908	−	0.440183i	$-0.145086\pi$
$570$		0			0
$571$	18.8328			0.788129			0.394064	−	0.919083i	$-0.371069\pi$
$571$	18.8328			0.788129			0.394064	+	0.919083i	$0.371069\pi$
$572$		−	1.38197i		−	0.0577829i
$573$		0			0
$574$	30.6525			1.27941
$575$	−6.76393			−0.282075
$576$		0			0
$577$		−	24.8328i		−	1.03380i	−0.856045	−	0.516902i	$-0.827085\pi$
$577$		−	24.8328i		−	1.03380i	0.856045	−	0.516902i	$-0.172915\pi$
$578$			12.9443i			0.538411i
$579$		0			0
$580$	8.94427	−	6.00000i	0.371391	−	0.249136i
$581$	−25.4164			−1.05445
$582$		0			0
$583$			15.1246i			0.626397i
$584$	17.2361			0.713234
$585$		0			0
$586$	−48.4508			−2.00149
$587$	0.944272			0.0389743			0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272			0.0389743			0.0194871	+	0.999810i	$0.493797\pi$
$588$		0			0
$589$	55.7771			2.29825
$590$			46.8328i			1.92808i
$591$		0			0
$592$		−	23.1246i		−	0.950416i
$593$	−40.1803			−1.65001			−0.825004	−	0.565126i	$-0.808827\pi$
$593$	−40.1803			−1.65001			−0.825004	+	0.565126i	$0.808827\pi$
$594$		0			0
$595$			41.1246i			1.68594i
$596$	−3.41641			−0.139942
$597$		0			0
$598$			2.00000i			0.0817861i
$599$			1.76393i			0.0720723i	0.999350	+	0.0360362i	$0.0114731\pi$
$599$			1.76393i			0.0720723i	−0.999350	+	0.0360362i	$0.988527\pi$
$600$		0			0
$601$		−	18.5410i		−	0.756304i	−0.925744	−	0.378152i	$-0.876560\pi$
$601$		−	18.5410i		−	0.756304i	0.925744	−	0.378152i	$-0.123440\pi$
$602$	−3.23607			−0.131892
$603$		0			0
$604$	−7.41641			−0.301769
$605$	−19.4164			−0.789389
$606$		0			0
$607$			25.4164i			1.03162i	0.856703	+	0.515810i	$0.172509\pi$
$607$			25.4164i			1.03162i	−0.856703	+	0.515810i	$0.827491\pi$
$608$	26.0689			1.05723
$609$		0			0
$610$	14.4721			0.585960
$611$		−	10.2361i		−	0.414107i
$612$		0			0
$613$	−24.4164			−0.986169			−0.493085	−	0.869981i	$-0.664131\pi$
$613$	−24.4164			−0.986169			−0.493085	+	0.869981i	$0.664131\pi$
$614$	−45.5967			−1.84013
$615$		0			0
$616$	−21.1803			−0.853380
$617$		−	19.8885i		−	0.800683i	−0.916366	−	0.400341i	$-0.868892\pi$
$617$		−	19.8885i		−	0.800683i	0.916366	−	0.400341i	$-0.131108\pi$
$618$		0			0
$619$		−	32.5410i		−	1.30793i	−0.756523	−	0.653967i	$-0.773104\pi$
$619$		−	32.5410i		−	1.30793i	0.756523	−	0.653967i	$-0.226896\pi$
$620$		−	14.4721i		−	0.581215i
$621$		0			0
$622$	−25.3262			−1.01549
$623$		−	0.236068i		−	0.00945786i
$624$		0			0
$625$	−22.4164			−0.896656
$626$		−	8.85410i		−	0.353881i
$627$		0			0
$628$		−	1.23607i		−	0.0493245i
$629$	−14.2918			−0.569851
$630$		0			0
$631$	28.7082			1.14286			0.571428	−	0.820652i	$-0.306390\pi$
$631$	28.7082			1.14286			0.571428	+	0.820652i	$0.306390\pi$
$632$	5.12461			0.203846
$633$		0			0
$634$	38.5623			1.53150
$635$			40.3607i			1.60166i
$636$		0			0
$637$	−10.9443			−0.433628
$638$	16.1803	−	10.8541i	0.640586	−	0.429718i
$639$		0			0
$640$			44.0689i			1.74198i
$641$			19.4721i			0.769103i	0.923104	+	0.384552i	$0.125644\pi$
$641$			19.4721i			0.769103i	−0.923104	+	0.384552i	$0.874356\pi$
$642$		0			0
$643$	0.708204			0.0279288			0.0139644	−	0.999902i	$-0.495555\pi$
$643$	0.708204			0.0279288			0.0139644	+	0.999902i	$0.495555\pi$
$644$	−3.23607			−0.127519
$645$		0			0
$646$		−	37.4164i		−	1.47213i
$647$	−17.5967			−0.691800			−0.345900	−	0.938271i	$-0.612426\pi$
$647$	−17.5967			−0.691800			−0.345900	+	0.938271i	$0.612426\pi$
$648$		0			0
$649$			20.0000i			0.785069i
$650$		−	8.85410i		−	0.347286i
$651$		0			0
$652$		−	4.58359i		−	0.179507i
$653$			32.3050i			1.26419i	0.774891	+	0.632095i	$0.217805\pi$
$653$			32.3050i			1.26419i	−0.774891	+	0.632095i	$0.782195\pi$
$654$		0			0
$655$			39.5967i			1.54717i
$656$		−	21.7082i		−	0.847563i
$657$		0			0
$658$	70.1591			2.73508
$659$			26.2361i			1.02201i	0.859577	+	0.511006i	$0.170727\pi$
$659$			26.2361i			1.02201i	−0.859577	+	0.511006i	$0.829273\pi$
$660$		0			0
$661$	−29.8328			−1.16036			−0.580181	−	0.814488i	$-0.697018\pi$
$661$	−29.8328			−1.16036			−0.580181	+	0.814488i	$0.697018\pi$
$662$	13.4164			0.521443
$663$		0			0
$664$			13.4164i			0.520658i
$665$			105.666i			4.09754i
$666$		0			0
$667$	−5.52786	+	3.70820i	−0.214040	+	0.143582i
$668$	9.81966			0.379934
$669$		0			0
$670$		−	24.6525i		−	0.952408i
$671$	6.18034			0.238589
$672$		0			0
$673$	−4.41641			−0.170240			−0.0851200	−	0.996371i	$-0.527127\pi$
$673$	−4.41641			−0.170240			−0.0851200	+	0.996371i	$0.527127\pi$
$674$	−31.1246			−1.19888
$675$		0			0
$676$	7.41641			0.285246
$677$		−	18.0557i		−	0.693938i	−0.937877	−	0.346969i	$-0.887211\pi$
$677$		−	18.0557i		−	0.693938i	0.937877	−	0.346969i	$-0.112789\pi$
$678$		0			0
$679$		−	77.0132i		−	2.95549i
$680$	21.7082			0.832472
$681$		0			0
$682$		−	26.1803i		−	1.00250i
$683$	−32.9443			−1.26058			−0.630289	−	0.776361i	$-0.717064\pi$
$683$	−32.9443			−1.26058			−0.630289	+	0.776361i	$0.717064\pi$
$684$		0			0
$685$		−	11.4164i		−	0.436199i
$686$		−	27.0344i		−	1.03218i
$687$		0			0
$688$			2.29180i			0.0873739i
$689$	6.76393			0.257685
$690$		0			0
$691$	−30.2361			−1.15023			−0.575117	−	0.818071i	$-0.695044\pi$
$691$	−30.2361			−1.15023			−0.575117	+	0.818071i	$0.695044\pi$
$692$	−3.70820			−0.140965
$693$		0			0
$694$			36.6525i			1.39131i
$695$	21.7082			0.823439
$696$		0			0
$697$	−13.4164			−0.508183
$698$			16.1803i			0.612435i
$699$		0			0
$700$	14.3262			0.541481
$701$	−8.18034			−0.308967			−0.154484	−	0.987995i	$-0.549371\pi$
$701$	−8.18034			−0.308967			−0.154484	+	0.987995i	$0.549371\pi$
$702$		0			0
$703$	−36.7214			−1.38497
$704$			9.47214i			0.356995i
$705$		0			0
$706$		−	7.52786i		−	0.283315i
$707$		−	16.7082i		−	0.628377i
$708$		0			0
$709$	−13.4164			−0.503864			−0.251932	−	0.967745i	$-0.581066\pi$
$709$	−13.4164			−0.503864			−0.251932	+	0.967745i	$0.581066\pi$
$710$			24.9443i			0.936142i
$711$		0			0
$712$	−0.124612			−0.00467002
$713$			8.94427i			0.334966i
$714$		0			0
$715$		−	7.23607i		−	0.270614i
$716$	−10.6525			−0.398102
$717$		0			0
$718$	−40.9443			−1.52803
$719$	30.0000			1.11881			0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000			1.11881			0.559406	+	0.828894i	$0.311029\pi$
$720$		0			0
$721$	82.2492			3.06312
$722$		−	65.3951i		−	2.43375i
$723$		0			0
$724$	−12.6180			−0.468946
$725$	24.4721	−	16.4164i	0.908872	−	0.609690i
$726$		0			0
$727$			5.81966i			0.215839i	0.994160	+	0.107920i	$0.0344189\pi$
$727$			5.81966i			0.215839i	−0.994160	+	0.107920i	$0.965581\pi$
$728$			9.47214i			0.351061i
$729$		0			0
$730$	−40.3607			−1.49382
$731$	1.41641			0.0523877
$732$		0			0
$733$			8.47214i			0.312925i	0.987684	+	0.156463i	$0.0500091\pi$
$733$			8.47214i			0.312925i	−0.987684	+	0.156463i	$0.949991\pi$
$734$	29.1246			1.07501
$735$		0			0
$736$			4.18034i			0.154089i
$737$		−	10.5279i		−	0.387799i
$738$		0			0
$739$			12.1803i			0.448061i	0.974582	+	0.224031i	$0.0719215\pi$
$739$			12.1803i			0.448061i	−0.974582	+	0.224031i	$0.928078\pi$
$740$			9.52786i			0.350251i
$741$		0			0
$742$			46.3607i			1.70195i
$743$		−	43.0689i		−	1.58004i	−0.613078	−	0.790022i	$-0.710069\pi$
$743$		−	43.0689i		−	1.58004i	0.613078	−	0.790022i	$-0.289931\pi$
$744$		0			0
$745$	−17.8885			−0.655386
$746$		−	6.29180i		−	0.230359i
$747$		0			0
$748$	−4.14590			−0.151589
$749$	3.23607			0.118243
$750$		0			0
$751$			26.8328i			0.979143i	0.871963	+	0.489572i	$0.162847\pi$
$751$			26.8328i			0.979143i	−0.871963	+	0.489572i	$0.837153\pi$
$752$		−	49.6869i		−	1.81190i
$753$		0			0
$754$	−4.85410	−	7.23607i	−0.176776	−	0.263522i
$755$	−38.8328			−1.41327
$756$		0			0
$757$		−	38.0000i		−	1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$		−	38.0000i		−	1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	−23.1246			−0.839924
$759$		0			0
$760$	55.7771			2.02325
$761$	−2.65248			−0.0961522			−0.0480761	−	0.998844i	$-0.515309\pi$
$761$	−2.65248			−0.0961522			−0.0480761	+	0.998844i	$0.515309\pi$
$762$		0			0
$763$	−21.1803			−0.766780
$764$			5.52786i			0.199991i
$765$		0			0
$766$			8.65248i			0.312627i
$767$	8.94427			0.322959
$768$		0			0
$769$		−	40.1803i		−	1.44894i	−0.689306	−	0.724470i	$-0.742085\pi$
$769$		−	40.1803i		−	1.44894i	0.689306	−	0.724470i	$-0.257915\pi$
$770$	49.5967			1.78734
$771$		0			0
$772$		−	10.7639i		−	0.387402i
$773$		−	31.8885i		−	1.14695i	−0.819223	−	0.573476i	$-0.805595\pi$
$773$		−	31.8885i		−	1.14695i	0.819223	−	0.573476i	$-0.194405\pi$
$774$		0			0
$775$		−	39.5967i		−	1.42236i
$776$	−40.6525			−1.45934
$777$		0			0
$778$	−21.7984			−0.781510
$779$	−34.4721			−1.23509
$780$		0			0
$781$			10.6525i			0.381176i
$782$	6.00000			0.214560
$783$		0			0
$784$	−53.1246			−1.89731
$785$		−	6.47214i		−	0.231000i
$786$		0			0
$787$	−12.0000			−0.427754			−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000			−0.427754			−0.213877	+	0.976861i	$0.568609\pi$
$788$	−2.29180			−0.0816419
$789$		0			0
$790$	−12.0000			−0.426941
$791$			19.1803i			0.681974i
$792$		0			0
$793$		−	2.76393i		−	0.0981501i
$794$			11.2361i			0.398753i
$795$		0			0
$796$	12.4377			0.440842
$797$			44.8328i			1.58806i	0.607879	+	0.794030i	$0.292021\pi$
$797$			44.8328i			1.58806i	−0.607879	+	0.794030i	$0.707979\pi$
$798$		0			0
$799$	−30.7082			−1.08638
$800$		−	18.5066i		−	0.654306i
$801$		0			0
$802$		−	54.5410i		−	1.92591i
$803$	−17.2361			−0.608248
$804$		0			0
$805$	−16.9443			−0.597207
$806$	−11.7082			−0.412404
$807$		0			0
$808$	−8.81966			−0.310275
$809$		−	43.3607i		−	1.52448i	−0.647294	−	0.762240i	$-0.724100\pi$
$809$		−	43.3607i		−	1.52448i	0.647294	−	0.762240i	$-0.275900\pi$
$810$		0			0
$811$	46.5967			1.63623			0.818117	−	0.575052i	$-0.195018\pi$
$811$	46.5967			1.63623			0.818117	+	0.575052i	$0.195018\pi$
$812$	11.7082	−	7.85410i	0.410877	−	0.275625i
$813$		0			0
$814$			17.2361i			0.604124i
$815$		−	24.0000i		−	0.840683i
$816$		0			0
$817$	3.63932			0.127324
$818$	17.1246			0.598748
$819$		0			0
$820$			8.94427i			0.312348i
$821$	12.4721			0.435281			0.217640	−	0.976029i	$-0.430164\pi$
$821$	12.4721			0.435281			0.217640	+	0.976029i	$0.430164\pi$
$822$		0			0
$823$			26.3607i			0.918876i	0.888210	+	0.459438i	$0.151949\pi$
$823$			26.3607i			0.918876i	−0.888210	+	0.459438i	$0.848051\pi$
$824$		−	43.4164i		−	1.51248i
$825$		0			0
$826$			61.3050i			2.13307i
$827$		−	29.8885i		−	1.03933i	−0.854371	−	0.519663i	$-0.826057\pi$
$827$		−	29.8885i		−	1.03933i	0.854371	−	0.519663i	$-0.173943\pi$
$828$		0			0
$829$			27.3050i			0.948340i	0.880433	+	0.474170i	$0.157252\pi$
$829$			27.3050i			0.948340i	−0.880433	+	0.474170i	$0.842748\pi$
$830$		−	31.4164i		−	1.09048i
$831$		0			0
$832$	4.23607			0.146859
$833$			32.8328i			1.13759i
$834$		0			0
$835$	51.4164			1.77934
$836$	−10.6525			−0.368424
$837$		0			0
$838$		−	6.18034i		−	0.213496i
$839$			0.708204i			0.0244499i	0.999925	+	0.0122250i	$0.00389142\pi$
$839$			0.708204i			0.0244499i	−0.999925	+	0.0122250i	$0.996109\pi$
$840$		0			0
$841$	11.0000	−	26.8328i	0.379310	−	0.925270i
$842$	−32.3607			−1.11522
$843$		0			0
$844$		−	6.18034i		−	0.212736i
$845$	38.8328			1.33589
$846$		0			0
$847$	−25.4164			−0.873318
$848$	32.8328			1.12748
$849$		0			0
$850$	−26.5623			−0.911080
$851$		−	5.88854i		−	0.201857i
$852$		0			0
$853$		−	45.1935i		−	1.54740i	−0.633555	−	0.773698i	$-0.718405\pi$
$853$		−	45.1935i		−	1.54740i	0.633555	−	0.773698i	$-0.281595\pi$
$854$	18.9443			0.648260
$855$		0			0
$856$		−	1.70820i		−	0.0583852i
$857$	−37.5967			−1.28428			−0.642140	−	0.766587i	$-0.721953\pi$
$857$	−37.5967			−1.28428			−0.642140	+	0.766587i	$0.721953\pi$
$858$		0			0
$859$		−	36.3607i		−	1.24061i	−0.784361	−	0.620305i	$-0.787009\pi$
$859$		−	36.3607i		−	1.24061i	0.784361	−	0.620305i	$-0.212991\pi$
$860$		−	0.944272i		−	0.0321994i
$861$		0			0
$862$		−	23.2361i		−	0.791424i
$863$	46.6525			1.58807			0.794034	−	0.607873i	$-0.207977\pi$
$863$	46.6525			1.58807			0.794034	+	0.607873i	$0.207977\pi$
$864$		0			0
$865$	−19.4164			−0.660178
$866$	23.1246			0.785806
$867$		0			0
$868$		−	18.9443i		−	0.643010i
$869$	−5.12461			−0.173841
$870$		0			0
$871$	−4.70820			−0.159531
$872$			11.1803i			0.378614i
$873$		0			0
$874$	15.4164			0.521468
$875$	6.47214			0.218798
$876$		0			0
$877$	32.4721			1.09651			0.548253	−	0.836312i	$-0.315293\pi$
$877$	32.4721			1.09651			0.548253	+	0.836312i	$0.315293\pi$
$878$			21.5066i			0.725812i
$879$		0			0
$880$		−	35.1246i		−	1.18405i
$881$			55.2492i			1.86139i	0.365792	+	0.930697i	$0.380798\pi$
$881$			55.2492i			1.86139i	−0.365792	+	0.930697i	$0.619202\pi$
$882$		0			0
$883$	34.2492			1.15258			0.576289	−	0.817246i	$-0.304500\pi$
$883$	34.2492			1.15258			0.576289	+	0.817246i	$0.304500\pi$
$884$			1.85410i			0.0623602i
$885$		0			0
$886$	15.6180			0.524698
$887$			17.0689i			0.573117i	0.958063	+	0.286559i	$0.0925113\pi$
$887$			17.0689i			0.573117i	−0.958063	+	0.286559i	$0.907489\pi$
$888$		0			0
$889$			52.8328i			1.77196i
$890$	0.291796			0.00978103
$891$		0			0
$892$	7.85410			0.262975
$893$	−78.9017			−2.64034
$894$		0			0
$895$	−55.7771			−1.86442
$896$			57.6869i			1.92718i
$897$		0			0
$898$	30.5623			1.01988
$899$	−21.7082	−	32.3607i	−0.724009	−	1.07929i
$900$		0			0
$901$		−	20.2918i		−	0.676018i
$902$			16.1803i			0.538746i
$903$		0			0
$904$	10.1246			0.336740
$905$	−66.0689			−2.19620
$906$		0			0
$907$			26.4721i			0.878993i	0.898244	+	0.439496i	$0.144843\pi$
$907$			26.4721i			0.878993i	−0.898244	+	0.439496i	$0.855157\pi$
$908$	5.59675			0.185735
$909$		0			0
$910$		−	22.1803i		−	0.735271i
$911$			51.1803i			1.69568i	0.530252	+	0.847840i	$0.322097\pi$
$911$			51.1803i			1.69568i	−0.530252	+	0.847840i	$0.677903\pi$
$912$		0			0
$913$		−	13.4164i		−	0.444018i
$914$			10.3820i			0.343405i
$915$		0			0
$916$		−	13.7082i		−	0.452932i
$917$			51.8328i			1.71167i
$918$		0			0
$919$	−25.6525			−0.846197			−0.423099	−	0.906084i	$-0.639058\pi$
$919$	−25.6525			−0.846197			−0.423099	+	0.906084i	$0.639058\pi$
$920$			8.94427i			0.294884i
$921$		0			0
$922$	−30.6525			−1.00949
$923$	4.76393			0.156807
$924$		0			0
$925$			26.0689i			0.857140i
$926$			1.14590i			0.0376565i
$927$		0			0
$928$	−10.1459	−	15.1246i	−0.333055	−	0.496490i
$929$	32.3607			1.06172			0.530860	−	0.847460i	$-0.321869\pi$
$929$	32.3607			1.06172			0.530860	+	0.847460i	$0.321869\pi$
$930$		0			0
$931$			84.3607i			2.76481i
$932$	−6.47214			−0.212002
$933$		0			0
$934$	−21.8885			−0.716215
$935$	−21.7082			−0.709934
$936$		0			0
$937$	33.0000			1.07806			0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000			1.07806			0.539032	+	0.842286i	$0.318790\pi$
$938$		−	32.2705i		−	1.05367i
$939$		0			0
$940$			20.4721i			0.667727i
$941$	44.8328			1.46151			0.730754	−	0.682641i	$-0.239169\pi$
$941$	44.8328			1.46151			0.730754	+	0.682641i	$0.239169\pi$
$942$		0			0
$943$		−	5.52786i		−	0.180012i
$944$	43.4164			1.41308
$945$		0			0
$946$		−	1.70820i		−	0.0555385i
$947$			16.8197i			0.546566i	0.961934	+	0.273283i	$0.0881095\pi$
$947$			16.8197i			0.546566i	−0.961934	+	0.273283i	$0.911891\pi$
$948$		0			0
$949$			7.70820i			0.250219i
$950$	−68.2492			−2.21430
$951$		0			0
$952$	28.4164			0.920981
$953$	59.6656			1.93276			0.966380	−	0.257119i	$-0.0827733\pi$
$953$	59.6656			1.93276			0.966380	+	0.257119i	$0.0827733\pi$
$954$		0			0
$955$			28.9443i			0.936615i
$956$	16.1803			0.523310
$957$		0			0
$958$	16.9443			0.547445
$959$		−	14.9443i		−	0.482576i
$960$		0			0
$961$	−21.3607			−0.689054
$962$	7.70820			0.248522
$963$		0			0
$964$	1.85410			0.0597166
$965$		−	56.3607i		−	1.81431i
$966$		0			0
$967$			27.1246i			0.872269i	0.899882	+	0.436134i	$0.143653\pi$
$967$			27.1246i			0.872269i	−0.899882	+	0.436134i	$0.856347\pi$
$968$			13.4164i			0.431220i
$969$		0			0
$970$	95.1935			3.05648
$971$		−	16.5836i		−	0.532193i	−0.963946	−	0.266096i	$-0.914266\pi$
$971$		−	16.5836i		−	0.532193i	0.963946	−	0.266096i	$-0.0857340\pi$
$972$		0			0
$973$	28.4164			0.910988
$974$			32.9443i			1.05560i
$975$		0			0
$976$		−	13.4164i		−	0.429449i
$977$	−51.0132			−1.63206			−0.816028	−	0.578013i	$-0.803828\pi$
$977$	−51.0132			−1.63206			−0.816028	+	0.578013i	$0.803828\pi$
$978$		0			0
$979$	0.124612			0.00398261
$980$	21.8885			0.699204
$981$		0			0
$982$		0			0
$983$			13.8885i			0.442976i	0.975163	+	0.221488i	$0.0710913\pi$
$983$			13.8885i			0.442976i	−0.975163	+	0.221488i	$0.928909\pi$
$984$		0			0
$985$	−12.0000			−0.382352
$986$	−21.7082	+	14.5623i	−0.691330	+	0.463758i
$987$		0			0
$988$			4.76393i			0.151561i
$989$			0.583592i			0.0185572i
$990$		0			0
$991$	−21.2918			−0.676356			−0.338178	−	0.941082i	$-0.609811\pi$
$991$	−21.2918			−0.676356			−0.338178	+	0.941082i	$0.609811\pi$
$992$	−24.4721			−0.776991
$993$		0			0
$994$			32.6525i			1.03567i
$995$	65.1246			2.06459
$996$		0			0
$997$		−	24.5836i		−	0.778570i	−0.921117	−	0.389285i	$-0.872722\pi$
$997$		−	24.5836i		−	0.778570i	0.921117	−	0.389285i	$-0.127278\pi$
$998$			34.2705i			1.08481i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.2.c.b.28.4		4
3.2	odd	2		87.2.c.a.28.1	✓	4
4.3	odd	2		4176.2.o.l.289.2		4
12.11	even	2		1392.2.o.i.289.2		4
15.2	even	4		2175.2.f.b.724.3		4
15.8	even	4		2175.2.f.a.724.2		4
15.14	odd	2		2175.2.d.e.376.4		4
29.12	odd	4		7569.2.a.f.1.1		2
29.17	odd	4		7569.2.a.n.1.2		2
29.28	even	2	inner	261.2.c.b.28.1		4
87.17	even	4		2523.2.a.d.1.1		2
87.41	even	4		2523.2.a.e.1.2		2
87.86	odd	2		87.2.c.a.28.4	yes	4
116.115	odd	2		4176.2.o.l.289.1		4
348.347	even	2		1392.2.o.i.289.4		4
435.173	even	4		2175.2.f.b.724.4		4
435.347	even	4		2175.2.f.a.724.1		4
435.434	odd	2		2175.2.d.e.376.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.c.a.28.1	✓	4	3.2	odd	2
87.2.c.a.28.4	yes	4	87.86	odd	2
261.2.c.b.28.1		4	29.28	even	2	inner
261.2.c.b.28.4		4	1.1	even	1	trivial
1392.2.o.i.289.2		4	12.11	even	2
1392.2.o.i.289.4		4	348.347	even	2
2175.2.d.e.376.1		4	435.434	odd	2
2175.2.d.e.376.4		4	15.14	odd	2
2175.2.f.a.724.1		4	435.347	even	4
2175.2.f.a.724.2		4	15.8	even	4
2175.2.f.b.724.3		4	15.2	even	4
2175.2.f.b.724.4		4	435.173	even	4
2523.2.a.d.1.1		2	87.17	even	4
2523.2.a.e.1.2		2	87.41	even	4
4176.2.o.l.289.1		4	116.115	odd	2
4176.2.o.l.289.2		4	4.3	odd	2
7569.2.a.f.1.1		2	29.12	odd	4
7569.2.a.n.1.2		2	29.17	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	261.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.61803i q^{2} -0.618034 q^{4} -3.23607 q^{5} -4.23607 q^{7} +2.23607i q^{8} -5.23607i q^{10} -2.23607i q^{11} -1.00000 q^{13} -6.85410i q^{14} -4.85410 q^{16} +3.00000i q^{17} +7.70820i q^{19} +2.00000 q^{20} +3.61803 q^{22} -1.23607 q^{23} +5.47214 q^{25} -1.61803i q^{26} +2.61803 q^{28} +(4.47214 - 3.00000i) q^{29} -7.23607i q^{31} -3.38197i q^{32} -4.85410 q^{34} +13.7082 q^{35} +4.76393i q^{37} -12.4721 q^{38} -7.23607i q^{40} +4.47214i q^{41} -0.472136i q^{43} +1.38197i q^{44} -2.00000i q^{46} +10.2361i q^{47} +10.9443 q^{49} +8.85410i q^{50} +0.618034 q^{52} -6.76393 q^{53} +7.23607i q^{55} -9.47214i q^{56} +(4.85410 + 7.23607i) q^{58} -8.94427 q^{59} +2.76393i q^{61} +11.7082 q^{62} -4.23607 q^{64} +3.23607 q^{65} +4.70820 q^{67} -1.85410i q^{68} +22.1803i q^{70} -4.76393 q^{71} -7.70820i q^{73} -7.70820 q^{74} -4.76393i q^{76} +9.47214i q^{77} -2.29180i q^{79} +15.7082 q^{80} -7.23607 q^{82} +6.00000 q^{83} -9.70820i q^{85} +0.763932 q^{86} +5.00000 q^{88} +0.0557281i q^{89} +4.23607 q^{91} +0.763932 q^{92} -16.5623 q^{94} -24.9443i q^{95} +18.1803i q^{97} +17.7082i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 4 q^{5} - 8 q^{7} - 4 q^{13} - 6 q^{16} + 8 q^{20} + 10 q^{22} + 4 q^{23} + 4 q^{25} + 6 q^{28} - 6 q^{34} + 28 q^{35} - 32 q^{38} + 8 q^{49} - 2 q^{52} - 36 q^{53} + 6 q^{58} + 20 q^{62}+ \cdots - 26 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 - 4 * q^5 - 8 * q^7 - 4 * q^13 - 6 * q^16 + 8 * q^20 + 10 * q^22 + 4 * q^23 + 4 * q^25 + 6 * q^28 - 6 * q^34 + 28 * q^35 - 32 * q^38 + 8 * q^49 - 2 * q^52 - 36 * q^53 + 6 * q^58 + 20 * q^62 - 8 * q^64 + 4 * q^65 - 8 * q^67 - 28 * q^71 - 4 * q^74 + 36 * q^80 - 20 * q^82 + 24 * q^83 + 12 * q^86 + 20 * q^88 + 8 * q^91 + 12 * q^92 - 26 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more