LMFDB - Embedded newform 88.3.h.a.65.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,3,Mod(65,88)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("88.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	88.h (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:	$2.39782632637$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1750426112.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 4x^{3} + 228x^{2} + 368x + 548 $ x^6 - 2x^5 - 21x^4 + 4x^3 + 228x^2 + 368*x + 548
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.6
Root			$-0.615072 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	88.65
Dual form			88.3.h.a.65.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.50331 q^{3} +2.27316 q^{5} +1.73969i q^{7} +3.27316 q^{9} +(-0.230143 - 10.9976i) q^{11} +12.4212i q^{13} +7.96359 q^{15} +13.0534i q^{17} -27.2142i q^{19} +6.09465i q^{21} -4.49669 q^{23} -19.8327 q^{25} -20.0629 q^{27} +25.4746i q^{29} -27.6026 q^{31} +(-0.806263 - 38.5279i) q^{33} +3.95459i q^{35} -46.6739 q^{37} +43.5151i q^{39} -57.0438i q^{41} +23.2596i q^{43} +7.44044 q^{45} +37.7515 q^{47} +45.9735 q^{49} +45.7301i q^{51} +77.1323 q^{53} +(-0.523153 - 24.9993i) q^{55} -95.3399i q^{57} +102.636 q^{59} +89.8774i q^{61} +5.69428i q^{63} +28.2353i q^{65} -26.3114 q^{67} -15.7533 q^{69} +5.63570 q^{71} -44.8545i q^{73} -69.4801 q^{75} +(19.1323 - 0.400377i) q^{77} -140.194i q^{79} -99.7449 q^{81} +115.984i q^{83} +29.6725i q^{85} +89.2452i q^{87} +107.220 q^{89} -21.6089 q^{91} -96.7003 q^{93} -61.8624i q^{95} -15.9651 q^{97} +(-0.753297 - 35.9969i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 6 q^{9} + 10 q^{11} - 52 q^{23} + 22 q^{25} + 32 q^{27} - 36 q^{31} - 64 q^{33} - 48 q^{37} + 172 q^{45} - 60 q^{47} - 170 q^{49} + 108 q^{53} + 172 q^{55} + 236 q^{59} - 292 q^{67} + 92 q^{69}+ \cdots + 182 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 6 * q^9 + 10 * q^11 - 52 * q^23 + 22 * q^25 + 32 * q^27 - 36 * q^31 - 64 * q^33 - 48 * q^37 + 172 * q^45 - 60 * q^47 - 170 * q^49 + 108 * q^53 + 172 * q^55 + 236 * q^59 - 292 * q^67 + 92 * q^69 + 300 * q^71 - 244 * q^75 - 240 * q^77 - 362 * q^81 + 384 * q^89 - 48 * q^91 - 148 * q^93 + 400 * q^97 + 182 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$.

$n$	$23$	$45$	$57$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	3.50331			1.16777			0.583885	−	0.811837i	$-0.301532\pi$
$3$	3.50331			1.16777			0.583885	+	0.811837i	$0.301532\pi$
$4$		0			0
$5$	2.27316			0.454633			0.227316	−	0.973821i	$-0.427005\pi$
$5$	2.27316			0.454633			0.227316	+	0.973821i	$0.427005\pi$
$6$		0			0
$7$			1.73969i			0.248526i	0.992249	+	0.124263i	$0.0396567\pi$
$7$			1.73969i			0.248526i	−0.992249	+	0.124263i	$0.960343\pi$
$8$		0			0
$9$	3.27316			0.363685
$10$		0			0
$11$	−0.230143	−	10.9976i	−0.0209221	−	0.999781i
$11$	−0.230143	−	10.9976i	−0.0209221	−	0.999781i
$12$		0			0
$13$			12.4212i			0.955474i	0.878503	+	0.477737i	$0.158543\pi$
$13$			12.4212i			0.955474i	−0.878503	+	0.477737i	$0.841457\pi$
$14$		0			0
$15$	7.96359			0.530906
$16$		0			0
$17$			13.0534i			0.767847i	0.923365	+	0.383923i	$0.125427\pi$
$17$			13.0534i			0.767847i	−0.923365	+	0.383923i	$0.874573\pi$
$18$		0			0
$19$		−	27.2142i		−	1.43233i	−0.697932	−	0.716164i	$-0.745896\pi$
$19$		−	27.2142i		−	1.43233i	0.697932	−	0.716164i	$-0.254104\pi$
$20$		0			0
$21$			6.09465i			0.290222i
$22$		0			0
$23$	−4.49669			−0.195508			−0.0977542	−	0.995211i	$-0.531166\pi$
$23$	−4.49669			−0.195508			−0.0977542	+	0.995211i	$0.531166\pi$
$24$		0			0
$25$	−19.8327			−0.793309
$26$		0			0
$27$	−20.0629			−0.743069
$28$		0			0
$29$			25.4746i			0.878433i	0.898381	+	0.439216i	$0.144744\pi$
$29$			25.4746i			0.878433i	−0.898381	+	0.439216i	$0.855256\pi$
$30$		0			0
$31$	−27.6026			−0.890406			−0.445203	−	0.895430i	$-0.646868\pi$
$31$	−27.6026			−0.890406			−0.445203	+	0.895430i	$0.646868\pi$
$32$		0			0
$33$	−0.806263	−	38.5279i	−0.0244322	−	1.16751i
$34$		0			0
$35$			3.95459i			0.112988i
$36$		0			0
$37$	−46.6739			−1.26146			−0.630728	−	0.776004i	$-0.717244\pi$
$37$	−46.6739			−1.26146			−0.630728	+	0.776004i	$0.717244\pi$
$38$		0			0
$39$			43.5151i			1.11577i
$40$		0			0
$41$		−	57.0438i		−	1.39131i	−0.718375	−	0.695656i	$-0.755114\pi$
$41$		−	57.0438i		−	1.39131i	0.718375	−	0.695656i	$-0.244886\pi$
$42$		0			0
$43$			23.2596i			0.540922i	0.962731	+	0.270461i	$0.0871761\pi$
$43$			23.2596i			0.540922i	−0.962731	+	0.270461i	$0.912824\pi$
$44$		0			0
$45$	7.44044			0.165343
$46$		0			0
$47$	37.7515			0.803223			0.401612	−	0.915810i	$-0.368450\pi$
$47$	37.7515			0.803223			0.401612	+	0.915810i	$0.368450\pi$
$48$		0			0
$49$	45.9735			0.938235
$50$		0			0
$51$			45.7301i			0.896668i
$52$		0			0
$53$	77.1323			1.45533			0.727664	−	0.685934i	$-0.240606\pi$
$53$	77.1323			1.45533			0.727664	+	0.685934i	$0.240606\pi$
$54$		0			0
$55$	−0.523153	−	24.9993i	−0.00951188	−	0.454533i
$56$		0			0
$57$		−	95.3399i		−	1.67263i
$58$		0			0
$59$	102.636			1.73959			0.869794	−	0.493416i	$-0.164252\pi$
$59$	102.636			1.73959			0.869794	+	0.493416i	$0.164252\pi$
$60$		0			0
$61$			89.8774i			1.47340i	0.676219	+	0.736700i	$0.263617\pi$
$61$			89.8774i			1.47340i	−0.676219	+	0.736700i	$0.736383\pi$
$62$		0			0
$63$			5.69428i			0.0903853i
$64$		0			0
$65$			28.2353i			0.434390i
$66$		0			0
$67$	−26.3114			−0.392707			−0.196354	−	0.980533i	$-0.562910\pi$
$67$	−26.3114			−0.392707			−0.196354	+	0.980533i	$0.562910\pi$
$68$		0			0
$69$	−15.7533			−0.228309
$70$		0			0
$71$	5.63570			0.0793761			0.0396880	−	0.999212i	$-0.487364\pi$
$71$	5.63570			0.0793761			0.0396880	+	0.999212i	$0.487364\pi$
$72$		0			0
$73$		−	44.8545i		−	0.614445i	−0.951638	−	0.307222i	$-0.900600\pi$
$73$		−	44.8545i		−	0.614445i	0.951638	−	0.307222i	$-0.0993995\pi$
$74$		0			0
$75$	−69.4801			−0.926402
$76$		0			0
$77$	19.1323	−	0.400377i	0.248472	−	0.00519970i
$78$		0			0
$79$		−	140.194i		−	1.77461i	−0.461182	−	0.887306i	$-0.652574\pi$
$79$		−	140.194i		−	1.77461i	0.461182	−	0.887306i	$-0.347426\pi$
$80$		0			0
$81$	−99.7449			−1.23142
$82$		0			0
$83$			115.984i			1.39740i	0.715415	+	0.698700i	$0.246238\pi$
$83$			115.984i			1.39740i	−0.715415	+	0.698700i	$0.753762\pi$
$84$		0			0
$85$			29.6725i			0.349088i
$86$		0			0
$87$			89.2452i			1.02581i
$88$		0			0
$89$	107.220			1.20472			0.602361	−	0.798224i	$-0.294227\pi$
$89$	107.220			1.20472			0.602361	+	0.798224i	$0.294227\pi$
$90$		0			0
$91$	−21.6089			−0.237461
$92$		0			0
$93$	−96.7003			−1.03979
$94$		0			0
$95$		−	61.8624i		−	0.651184i
$96$		0			0
$97$	−15.9651			−0.164588			−0.0822942	−	0.996608i	$-0.526225\pi$
$97$	−15.9651			−0.164588			−0.0822942	+	0.996608i	$0.526225\pi$
$98$		0			0
$99$	−0.753297	−	35.9969i	−0.00760906	−	0.363605i
$100$		0			0
$101$		−	130.766i		−	1.29471i	−0.762188	−	0.647355i	$-0.775875\pi$
$101$		−	130.766i		−	1.29471i	0.762188	−	0.647355i	$-0.224125\pi$
$102$		0			0
$103$	54.3014			0.527198			0.263599	−	0.964632i	$-0.415090\pi$
$103$	54.3014			0.527198			0.263599	+	0.964632i	$0.415090\pi$
$104$		0			0
$105$			13.8541i			0.131944i
$106$		0			0
$107$		−	0.643741i		−	0.00601627i	−0.999995	−	0.00300814i	$-0.999042\pi$
$107$		−	0.643741i		−	0.00601627i	0.999995	−	0.00300814i	$-0.000957521\pi$
$108$		0			0
$109$			170.899i			1.56788i	0.620834	+	0.783942i	$0.286794\pi$
$109$			170.899i			1.56788i	−0.620834	+	0.783942i	$0.713206\pi$
$110$		0			0
$111$	−163.513			−1.47309
$112$		0			0
$113$	0.432018			0.00382317			0.00191159	−	0.999998i	$-0.499392\pi$
$113$	0.432018			0.00382317			0.00191159	+	0.999998i	$0.499392\pi$
$114$		0			0
$115$	−10.2217			−0.0888845
$116$		0			0
$117$			40.6565i			0.347491i
$118$		0			0
$119$	−22.7088			−0.190830
$120$		0			0
$121$	−120.894	+	5.06204i	−0.999125	+	0.0418351i
$122$		0			0
$123$		−	199.842i		−	1.62473i
$124$		0			0
$125$	−101.912			−0.815297
$126$		0			0
$127$			13.4423i			0.105845i	0.998599	+	0.0529223i	$0.0168536\pi$
$127$			13.4423i			0.105845i	−0.998599	+	0.0529223i	$0.983146\pi$
$128$		0			0
$129$			81.4857i			0.631672i
$130$		0			0
$131$			60.7665i			0.463866i	0.972732	+	0.231933i	$0.0745051\pi$
$131$			60.7665i			0.463866i	−0.972732	+	0.231933i	$0.925495\pi$
$132$		0			0
$133$	47.3442			0.355972
$134$		0			0
$135$	−45.6062			−0.337824
$136$		0			0
$137$	69.4584			0.506996			0.253498	−	0.967336i	$-0.418419\pi$
$137$	69.4584			0.506996			0.253498	+	0.967336i	$0.418419\pi$
$138$		0			0
$139$		−	152.227i		−	1.09516i	−0.836755	−	0.547578i	$-0.815550\pi$
$139$		−	152.227i		−	1.09516i	0.836755	−	0.547578i	$-0.184450\pi$
$140$		0			0
$141$	132.255			0.937979
$142$		0			0
$143$	136.603	−	2.85865i	0.955265	−	0.0199905i
$144$		0			0
$145$			57.9079i			0.399364i
$146$		0			0
$147$	161.059			1.09564
$148$		0			0
$149$			110.353i			0.740626i	0.928907	+	0.370313i	$0.120750\pi$
$149$			110.353i			0.740626i	−0.928907	+	0.370313i	$0.879250\pi$
$150$		0			0
$151$		−	61.3872i		−	0.406538i	−0.979123	−	0.203269i	$-0.934843\pi$
$151$		−	61.3872i		−	0.406538i	0.979123	−	0.203269i	$-0.0651566\pi$
$152$		0			0
$153$			42.7259i			0.279254i
$154$		0			0
$155$	−62.7452			−0.404808
$156$		0			0
$157$	215.697			1.37386			0.686932	−	0.726722i	$-0.258957\pi$
$157$	215.697			1.37386			0.686932	+	0.726722i	$0.258957\pi$
$158$		0			0
$159$	270.218			1.69949
$160$		0			0
$161$		−	7.82283i		−	0.0485890i
$162$		0			0
$163$	−10.1956			−0.0625496			−0.0312748	−	0.999511i	$-0.509957\pi$
$163$	−10.1956			−0.0625496			−0.0312748	+	0.999511i	$0.509957\pi$
$164$		0			0
$165$	−1.83277	−	87.5804i	−0.0111077	−	0.530790i
$166$		0			0
$167$			192.569i			1.15311i	0.817059	+	0.576554i	$0.195603\pi$
$167$			192.569i			1.15311i	−0.817059	+	0.576554i	$0.804397\pi$
$168$		0			0
$169$	14.7148			0.0870696
$170$		0			0
$171$		−	89.0767i		−	0.520916i
$172$		0			0
$173$		−	181.024i		−	1.04638i	−0.852216	−	0.523189i	$-0.824742\pi$
$173$		−	181.024i		−	1.04638i	0.852216	−	0.523189i	$-0.175258\pi$
$174$		0			0
$175$		−	34.5027i		−	0.197158i
$176$		0			0
$177$	359.564			2.03144
$178$		0			0
$179$	−107.761			−0.602019			−0.301009	−	0.953621i	$-0.597324\pi$
$179$	−107.761			−0.602019			−0.301009	+	0.953621i	$0.597324\pi$
$180$		0			0
$181$	−324.494			−1.79279			−0.896394	−	0.443259i	$-0.853822\pi$
$181$	−324.494			−1.79279			−0.896394	+	0.443259i	$0.853822\pi$
$182$		0			0
$183$			314.868i			1.72059i
$184$		0			0
$185$	−106.097			−0.573499
$186$		0			0
$187$	143.556	−	3.00415i	0.767679	−	0.0160650i
$188$		0			0
$189$		−	34.9031i		−	0.184672i
$190$		0			0
$191$	−188.655			−0.987725			−0.493862	−	0.869540i	$-0.664415\pi$
$191$	−188.655			−0.987725			−0.493862	+	0.869540i	$0.664415\pi$
$192$		0			0
$193$			34.9664i			0.181173i	0.995889	+	0.0905866i	$0.0288742\pi$
$193$			34.9664i			0.181173i	−0.995889	+	0.0905866i	$0.971126\pi$
$194$		0			0
$195$			98.9171i			0.507267i
$196$		0			0
$197$		−	348.589i		−	1.76949i	−0.466078	−	0.884744i	$-0.654333\pi$
$197$		−	348.589i		−	1.76949i	0.466078	−	0.884744i	$-0.345667\pi$
$198$		0			0
$199$	−198.672			−0.998352			−0.499176	−	0.866501i	$-0.666364\pi$
$199$	−198.672			−0.998352			−0.499176	+	0.866501i	$0.666364\pi$
$200$		0			0
$201$	−92.1768			−0.458591
$202$		0			0
$203$	−44.3177			−0.218314
$204$		0			0
$205$		−	129.670i		−	0.632536i
$206$		0			0
$207$	−14.7184			−0.0711034
$208$		0			0
$209$	−299.291	+	6.26317i	−1.43201	+	0.0299673i
$210$		0			0
$211$			325.789i			1.54402i	0.635608	+	0.772012i	$0.280749\pi$
$211$			325.789i			1.54402i	−0.635608	+	0.772012i	$0.719251\pi$
$212$		0			0
$213$	19.7436			0.0926929
$214$		0			0
$215$			52.8730i			0.245921i
$216$		0			0
$217$		−	48.0198i		−	0.221289i
$218$		0			0
$219$		−	157.139i		−	0.717530i
$220$		0			0
$221$	−162.138			−0.733658
$222$		0			0
$223$	−204.550			−0.917263			−0.458631	−	0.888627i	$-0.651660\pi$
$223$	−204.550			−0.917263			−0.458631	+	0.888627i	$0.651660\pi$
$224$		0			0
$225$	−64.9158			−0.288515
$226$		0			0
$227$			169.646i			0.747341i	0.927561	+	0.373671i	$0.121901\pi$
$227$			169.646i			0.747341i	−0.927561	+	0.373671i	$0.878099\pi$
$228$		0			0
$229$	233.882			1.02132			0.510659	−	0.859783i	$-0.329401\pi$
$229$	233.882			1.02132			0.510659	+	0.859783i	$0.329401\pi$
$230$		0			0
$231$	67.0265	−	1.40264i	0.290158	−	0.00607205i
$232$		0			0
$233$			272.165i			1.16809i	0.811721	+	0.584046i	$0.198531\pi$
$233$			272.165i			1.16809i	−0.811721	+	0.584046i	$0.801469\pi$
$234$		0			0
$235$	85.8153			0.365172
$236$		0			0
$237$		−	491.144i		−	2.07234i
$238$		0			0
$239$		−	202.218i		−	0.846100i	−0.906106	−	0.423050i	$-0.860959\pi$
$239$		−	202.218i		−	0.846100i	0.906106	−	0.423050i	$-0.139041\pi$
$240$		0			0
$241$		−	44.4541i		−	0.184457i	−0.995738	−	0.0922284i	$-0.970601\pi$
$241$		−	44.4541i		−	0.184457i	0.995738	−	0.0922284i	$-0.0293990\pi$
$242$		0			0
$243$	−168.871			−0.694943
$244$		0			0
$245$	104.505			0.426552
$246$		0			0
$247$	338.032			1.36855
$248$		0			0
$249$			406.328i			1.63184i
$250$		0			0
$251$	−362.556			−1.44444			−0.722222	−	0.691661i	$-0.756879\pi$
$251$	−362.556			−1.44444			−0.722222	+	0.691661i	$0.756879\pi$
$252$		0			0
$253$	1.03488	+	49.4528i	0.00409045	+	0.195466i
$254$		0			0
$255$			103.952i			0.407655i
$256$		0			0
$257$	40.8146			0.158812			0.0794059	−	0.996842i	$-0.474698\pi$
$257$	40.8146			0.158812			0.0794059	+	0.996842i	$0.474698\pi$
$258$		0			0
$259$		−	81.1978i		−	0.313505i
$260$		0			0
$261$			83.3824i			0.319473i
$262$		0			0
$263$		−	245.411i		−	0.933121i	−0.884489	−	0.466560i	$-0.845493\pi$
$263$		−	245.411i		−	0.933121i	0.884489	−	0.466560i	$-0.154507\pi$
$264$		0			0
$265$	175.335			0.661640
$266$		0			0
$267$	375.625			1.40684
$268$		0			0
$269$	−246.762			−0.917329			−0.458665	−	0.888609i	$-0.651672\pi$
$269$	−246.762			−0.917329			−0.458665	+	0.888609i	$0.651672\pi$
$270$		0			0
$271$		−	387.820i		−	1.43107i	−0.698577	−	0.715535i	$-0.746183\pi$
$271$		−	387.820i		−	1.43107i	0.698577	−	0.715535i	$-0.253817\pi$
$272$		0			0
$273$	−75.7027			−0.277299
$274$		0			0
$275$	4.56437	+	218.112i	0.0165977	+	0.793135i
$276$		0			0
$277$			465.497i			1.68049i	0.542205	+	0.840246i	$0.317590\pi$
$277$			465.497i			1.68049i	−0.542205	+	0.840246i	$0.682410\pi$
$278$		0			0
$279$	−90.3478			−0.323827
$280$		0			0
$281$			31.0003i			0.110321i	0.998477	+	0.0551607i	$0.0175671\pi$
$281$			31.0003i			0.110321i	−0.998477	+	0.0551607i	$0.982433\pi$
$282$		0			0
$283$		−	44.7953i		−	0.158287i	−0.996863	−	0.0791436i	$-0.974781\pi$
$283$		−	44.7953i		−	0.158287i	0.996863	−	0.0791436i	$-0.0252186\pi$
$284$		0			0
$285$		−	216.723i		−	0.760432i
$286$		0			0
$287$	99.2382			0.345778
$288$		0			0
$289$	118.609			0.410411
$290$		0			0
$291$	−55.9306			−0.192201
$292$		0			0
$293$		−	113.519i		−	0.387436i	−0.981057	−	0.193718i	$-0.937945\pi$
$293$		−	113.519i		−	0.387436i	0.981057	−	0.193718i	$-0.0620546\pi$
$294$		0			0
$295$	233.308			0.790874
$296$		0			0
$297$	4.61733	+	220.643i	0.0155466	+	0.742906i
$298$		0			0
$299$		−	55.8541i		−	0.186803i
$300$		0			0
$301$	−40.4645			−0.134433
$302$		0			0
$303$		−	458.113i		−	1.51192i
$304$		0			0
$305$			204.306i			0.669856i
$306$		0			0
$307$			81.6657i			0.266012i	0.991115	+	0.133006i	$0.0424630\pi$
$307$			81.6657i			0.266012i	−0.991115	+	0.133006i	$0.957537\pi$
$308$		0			0
$309$	190.235			0.615646
$310$		0			0
$311$	−288.619			−0.928036			−0.464018	−	0.885826i	$-0.653593\pi$
$311$	−288.619			−0.928036			−0.464018	+	0.885826i	$0.653593\pi$
$312$		0			0
$313$	358.220			1.14447			0.572237	−	0.820089i	$-0.306076\pi$
$313$	358.220			1.14447			0.572237	+	0.820089i	$0.306076\pi$
$314$		0			0
$315$			12.9440i			0.0410921i
$316$		0			0
$317$	385.247			1.21529			0.607644	−	0.794209i	$-0.292115\pi$
$317$	385.247			1.21529			0.607644	+	0.794209i	$0.292115\pi$
$318$		0			0
$319$	280.159	−	5.86280i	0.878241	−	0.0183787i
$320$		0			0
$321$		−	2.25522i		−	0.00702562i
$322$		0			0
$323$	355.238			1.09981
$324$		0			0
$325$		−	246.345i		−	0.757986i
$326$		0			0
$327$			598.713i			1.83093i
$328$		0			0
$329$			65.6757i			0.199622i
$330$		0			0
$331$	224.841			0.679279			0.339639	−	0.940556i	$-0.389695\pi$
$331$	224.841			0.679279			0.339639	+	0.940556i	$0.389695\pi$
$332$		0			0
$333$	−152.771			−0.458773
$334$		0			0
$335$	−59.8101			−0.178538
$336$		0			0
$337$		−	434.610i		−	1.28964i	−0.764333	−	0.644822i	$-0.776932\pi$
$337$		−	434.610i		−	1.28964i	0.764333	−	0.644822i	$-0.223068\pi$
$338$		0			0
$339$	1.51349			0.00446458
$340$		0			0
$341$	6.35255	+	303.562i	0.0186292	+	0.890211i
$342$		0			0
$343$			165.224i			0.481703i
$344$		0			0
$345$	−35.8098			−0.103797
$346$		0			0
$347$		−	424.683i		−	1.22387i	−0.790908	−	0.611935i	$-0.790391\pi$
$347$		−	424.683i		−	1.22387i	0.790908	−	0.611935i	$-0.209609\pi$
$348$		0			0
$349$			267.504i			0.766486i	0.923647	+	0.383243i	$0.125193\pi$
$349$			267.504i			0.766486i	−0.923647	+	0.383243i	$0.874807\pi$
$350$		0			0
$351$		−	249.204i		−	0.709983i
$352$		0			0
$353$	−252.800			−0.716148			−0.358074	−	0.933693i	$-0.616566\pi$
$353$	−252.800			−0.716148			−0.358074	+	0.933693i	$0.616566\pi$
$354$		0			0
$355$	12.8109			0.0360870
$356$		0			0
$357$	−79.5559			−0.222846
$358$		0			0
$359$		−	190.991i		−	0.532007i	−0.963972	−	0.266004i	$-0.914297\pi$
$359$		−	190.991i		−	0.532007i	0.963972	−	0.266004i	$-0.0857033\pi$
$360$		0			0
$361$	−379.615			−1.05156
$362$		0			0
$363$	−423.529	+	17.7339i	−1.16675	+	0.0488537i
$364$		0			0
$365$		−	101.962i		−	0.279347i
$366$		0			0
$367$	−277.338			−0.755689			−0.377845	−	0.925869i	$-0.623335\pi$
$367$	−277.338			−0.755689			−0.377845	+	0.925869i	$0.623335\pi$
$368$		0			0
$369$		−	186.714i		−	0.505999i
$370$		0			0
$371$			134.186i			0.361687i
$372$		0			0
$373$			442.955i			1.18755i	0.804632	+	0.593774i	$0.202363\pi$
$373$			442.955i			1.18755i	−0.804632	+	0.593774i	$0.797637\pi$
$374$		0			0
$375$	−357.030			−0.952079
$376$		0			0
$377$	−316.424			−0.839320
$378$		0			0
$379$	−520.888			−1.37437			−0.687187	−	0.726481i	$-0.741155\pi$
$379$	−520.888			−1.37437			−0.687187	+	0.726481i	$0.741155\pi$
$380$		0			0
$381$			47.0924i			0.123602i
$382$		0			0
$383$	−117.258			−0.306158			−0.153079	−	0.988214i	$-0.548919\pi$
$383$	−117.258			−0.306158			−0.153079	+	0.988214i	$0.548919\pi$
$384$		0			0
$385$	43.4910	−	0.910122i	0.112964	−	0.00236395i
$386$		0			0
$387$			76.1327i			0.196725i
$388$		0			0
$389$	−329.077			−0.845956			−0.422978	−	0.906140i	$-0.639015\pi$
$389$	−329.077			−0.845956			−0.422978	+	0.906140i	$0.639015\pi$
$390$		0			0
$391$		−	58.6971i		−	0.150120i
$392$		0			0
$393$			212.884i			0.541689i
$394$		0			0
$395$		−	318.685i		−	0.806797i
$396$		0			0
$397$	−334.297			−0.842058			−0.421029	−	0.907047i	$-0.638331\pi$
$397$	−334.297			−0.842058			−0.421029	+	0.907047i	$0.638331\pi$
$398$		0			0
$399$	165.861			0.415693
$400$		0			0
$401$	399.291			0.995738			0.497869	−	0.867252i	$-0.334116\pi$
$401$	399.291			0.995738			0.497869	+	0.867252i	$0.334116\pi$
$402$		0			0
$403$		−	342.856i		−	0.850760i
$404$		0			0
$405$	−226.736			−0.559843
$406$		0			0
$407$	10.7417	+	513.300i	0.0263923	+	1.26118i
$408$		0			0
$409$		−	260.275i		−	0.636370i	−0.948029	−	0.318185i	$-0.896927\pi$
$409$		−	260.275i		−	0.636370i	0.948029	−	0.318185i	$-0.103073\pi$
$410$		0			0
$411$	243.334			0.592054
$412$		0			0
$413$			178.554i			0.432334i
$414$		0			0
$415$			263.651i			0.635304i
$416$		0			0
$417$		−	533.297i		−	1.27889i
$418$		0			0
$419$	128.534			0.306763			0.153382	−	0.988167i	$-0.450984\pi$
$419$	128.534			0.306763			0.153382	+	0.988167i	$0.450984\pi$
$420$		0			0
$421$	88.3502			0.209858			0.104929	−	0.994480i	$-0.466538\pi$
$421$	88.3502			0.209858			0.104929	+	0.994480i	$0.466538\pi$
$422$		0			0
$423$	123.567			0.292120
$424$		0			0
$425$		−	258.884i		−	0.609140i
$426$		0			0
$427$	−156.358			−0.366179
$428$		0			0
$429$	478.562	−	10.0147i	1.11553	−	0.0233443i
$430$		0			0
$431$		−	261.912i		−	0.607683i	−0.952723	−	0.303842i	$-0.901731\pi$
$431$		−	261.912i		−	0.607683i	0.952723	−	0.303842i	$-0.0982694\pi$
$432$		0			0
$433$	−161.189			−0.372261			−0.186130	−	0.982525i	$-0.559595\pi$
$433$	−161.189			−0.372261			−0.186130	+	0.982525i	$0.559595\pi$
$434$		0			0
$435$			202.869i			0.466366i
$436$		0			0
$437$			122.374i			0.280032i
$438$		0			0
$439$			796.202i			1.81367i	0.421484	+	0.906836i	$0.361510\pi$
$439$			796.202i			1.81367i	−0.421484	+	0.906836i	$0.638490\pi$
$440$		0			0
$441$	150.479			0.341222
$442$		0			0
$443$	842.927			1.90277			0.951385	−	0.308006i	$-0.0996615\pi$
$443$	842.927			1.90277			0.951385	+	0.308006i	$0.0996615\pi$
$444$		0			0
$445$	243.729			0.547706
$446$		0			0
$447$			386.601i			0.864880i
$448$		0			0
$449$	401.591			0.894411			0.447206	−	0.894431i	$-0.352419\pi$
$449$	401.591			0.894411			0.447206	+	0.894431i	$0.352419\pi$
$450$		0			0
$451$	−627.344	+	13.1282i	−1.39101	+	0.0291092i
$452$		0			0
$453$		−	215.058i		−	0.474742i
$454$		0			0
$455$	−49.1206			−0.107957
$456$		0			0
$457$		−	802.579i		−	1.75619i	−0.478486	−	0.878095i	$-0.658814\pi$
$457$		−	802.579i		−	1.75619i	0.478486	−	0.878095i	$-0.341186\pi$
$458$		0			0
$459$		−	261.889i		−	0.570563i
$460$		0			0
$461$		−	51.9184i		−	0.112621i	−0.998413	−	0.0563106i	$-0.982066\pi$
$461$		−	51.9184i		−	0.112621i	0.998413	−	0.0563106i	$-0.0179337\pi$
$462$		0			0
$463$	636.065			1.37379			0.686895	−	0.726756i	$-0.258973\pi$
$463$	636.065			1.37379			0.686895	+	0.726756i	$0.258973\pi$
$464$		0			0
$465$	−219.816			−0.472722
$466$		0			0
$467$	−527.855			−1.13031			−0.565155	−	0.824984i	$-0.691184\pi$
$467$	−527.855			−1.13031			−0.565155	+	0.824984i	$0.691184\pi$
$468$		0			0
$469$		−	45.7735i		−	0.0975981i
$470$		0			0
$471$	755.652			1.60436
$472$		0			0
$473$	255.800	−	5.35305i	0.540804	−	0.0113172i
$474$		0			0
$475$			539.733i			1.13628i
$476$		0			0
$477$	252.467			0.529281
$478$		0			0
$479$			468.191i			0.977434i	0.872442	+	0.488717i	$0.162535\pi$
$479$			468.191i			0.977434i	−0.872442	+	0.488717i	$0.837465\pi$
$480$		0			0
$481$		−	579.744i		−	1.20529i
$482$		0			0
$483$		−	27.4058i		−	0.0567407i
$484$		0			0
$485$	−36.2912			−0.0748273
$486$		0			0
$487$	−851.667			−1.74880			−0.874402	−	0.485203i	$-0.838746\pi$
$487$	−851.667			−1.74880			−0.874402	+	0.485203i	$0.838746\pi$
$488$		0			0
$489$	−35.7183			−0.0730435
$490$		0			0
$491$			358.667i			0.730483i	0.930913	+	0.365241i	$0.119014\pi$
$491$			358.667i			0.730483i	−0.930913	+	0.365241i	$0.880986\pi$
$492$		0			0
$493$	−332.529			−0.674502
$494$		0			0
$495$	−1.71237	−	81.8269i	−0.00345933	−	0.165307i
$496$		0			0
$497$			9.80435i			0.0197271i
$498$		0			0
$499$	833.169			1.66968			0.834838	−	0.550495i	$-0.185561\pi$
$499$	833.169			1.66968			0.834838	+	0.550495i	$0.185561\pi$
$500$		0			0
$501$			674.629i			1.34656i
$502$		0			0
$503$		−	317.397i		−	0.631008i	−0.948924	−	0.315504i	$-0.897826\pi$
$503$		−	317.397i		−	0.631008i	0.948924	−	0.315504i	$-0.102174\pi$
$504$		0			0
$505$		−	297.252i		−	0.588618i
$506$		0			0
$507$	51.5503			0.101677
$508$		0			0
$509$	55.9879			0.109996			0.0549980	−	0.998486i	$-0.482485\pi$
$509$	55.9879			0.109996			0.0549980	+	0.998486i	$0.482485\pi$
$510$		0			0
$511$	78.0326			0.152706
$512$		0			0
$513$			545.996i			1.06432i
$514$		0			0
$515$	123.436			0.239682
$516$		0			0
$517$	−8.68825	−	415.175i	−0.0168051	−	0.803047i
$518$		0			0
$519$		−	634.181i		−	1.22193i
$520$		0			0
$521$	427.253			0.820063			0.410031	−	0.912071i	$-0.365518\pi$
$521$	427.253			0.820063			0.410031	+	0.912071i	$0.365518\pi$
$522$		0			0
$523$			197.190i			0.377037i	0.982070	+	0.188519i	$0.0603686\pi$
$523$			197.190i			0.377037i	−0.982070	+	0.188519i	$0.939631\pi$
$524$		0			0
$525$		−	120.874i		−	0.230235i
$526$		0			0
$527$		−	360.307i		−	0.683695i
$528$		0			0
$529$	−508.780			−0.961776
$530$		0			0
$531$	335.943			0.632662
$532$		0			0
$533$	708.550			1.32936
$534$		0			0
$535$		−	1.46333i		−	0.00273520i
$536$		0			0
$537$	−377.521			−0.703019
$538$		0			0
$539$	−10.5805	−	505.598i	−0.0196299	−	0.938029i
$540$		0			0
$541$			654.393i			1.20960i	0.796378	+	0.604800i	$0.206747\pi$
$541$			654.393i			1.20960i	−0.796378	+	0.604800i	$0.793253\pi$
$542$		0			0
$543$	−1136.80			−2.09356
$544$		0			0
$545$			388.483i			0.712812i
$546$		0			0
$547$			37.5142i			0.0685817i	0.999412	+	0.0342908i	$0.0109173\pi$
$547$			37.5142i			0.0685817i	−0.999412	+	0.0342908i	$0.989083\pi$
$548$		0			0
$549$			294.184i			0.535854i
$550$		0			0
$551$	693.271			1.25820
$552$		0			0
$553$	243.894			0.441038
$554$		0			0
$555$	−371.692			−0.669715
$556$		0			0
$557$			715.749i			1.28501i	0.766283	+	0.642504i	$0.222104\pi$
$557$			715.749i			1.28501i	−0.766283	+	0.642504i	$0.777896\pi$
$558$		0			0
$559$	−288.912			−0.516837
$560$		0			0
$561$	502.920	−	10.5245i	0.896471	−	0.0187602i
$562$		0			0
$563$		−	691.989i		−	1.22911i	−0.788874	−	0.614555i	$-0.789335\pi$
$563$		−	691.989i		−	1.22911i	0.788874	−	0.614555i	$-0.210665\pi$
$564$		0			0
$565$	0.982049			0.00173814
$566$		0			0
$567$		−	173.525i		−	0.306040i
$568$		0			0
$569$		−	245.840i		−	0.432056i	−0.976387	−	0.216028i	$-0.930690\pi$
$569$		−	245.840i		−	0.432056i	0.976387	−	0.216028i	$-0.0693103\pi$
$570$		0			0
$571$			474.881i			0.831665i	0.909441	+	0.415832i	$0.136510\pi$
$571$			474.881i			0.831665i	−0.909441	+	0.415832i	$0.863490\pi$
$572$		0			0
$573$	−660.918			−1.15343
$574$		0			0
$575$	89.1817			0.155099
$576$		0			0
$577$	−541.321			−0.938165			−0.469082	−	0.883154i	$-0.655415\pi$
$577$	−541.321			−0.938165			−0.469082	+	0.883154i	$0.655415\pi$
$578$		0			0
$579$			122.498i			0.211568i
$580$		0			0
$581$	−201.776			−0.347291
$582$		0			0
$583$	−17.7515	−	848.270i	−0.0304485	−	1.45501i
$584$		0			0
$585$			92.4189i			0.157981i
$586$		0			0
$587$	−471.592			−0.803394			−0.401697	−	0.915773i	$-0.631580\pi$
$587$	−471.592			−0.803394			−0.401697	+	0.915773i	$0.631580\pi$
$588$		0			0
$589$			751.183i			1.27535i
$590$		0			0
$591$		−	1221.21i		−	2.06635i
$592$		0			0
$593$			460.953i			0.777323i	0.921381	+	0.388662i	$0.127062\pi$
$593$			460.953i			0.777323i	−0.921381	+	0.388662i	$0.872938\pi$
$594$		0			0
$595$	−51.6208			−0.0867577
$596$		0			0
$597$	−696.009			−1.16584
$598$		0			0
$599$	309.686			0.517006			0.258503	−	0.966011i	$-0.416771\pi$
$599$	309.686			0.517006			0.258503	+	0.966011i	$0.416771\pi$
$600$		0			0
$601$		−	766.230i		−	1.27493i	−0.770481	−	0.637463i	$-0.779984\pi$
$601$		−	766.230i		−	1.27493i	0.770481	−	0.637463i	$-0.220016\pi$
$602$		0			0
$603$	−86.1215			−0.142822
$604$		0			0
$605$	−274.812	+	11.5069i	−0.454235	+	0.0190196i
$606$		0			0
$607$			928.818i			1.53018i	0.643925	+	0.765089i	$0.277305\pi$
$607$			928.818i			1.53018i	−0.643925	+	0.765089i	$0.722695\pi$
$608$		0			0
$609$	−155.259			−0.254940
$610$		0			0
$611$			468.917i			0.767459i
$612$		0			0
$613$		−	224.968i		−	0.366995i	−0.983020	−	0.183497i	$-0.941258\pi$
$613$		−	224.968i		−	0.366995i	0.983020	−	0.183497i	$-0.0587419\pi$
$614$		0			0
$615$		−	454.273i		−	0.738656i
$616$		0			0
$617$	−809.197			−1.31150			−0.655751	−	0.754977i	$-0.727648\pi$
$617$	−809.197			−1.31150			−0.655751	+	0.754977i	$0.727648\pi$
$618$		0			0
$619$	168.012			0.271425			0.135713	−	0.990748i	$-0.456668\pi$
$619$	168.012			0.271425			0.135713	+	0.990748i	$0.456668\pi$
$620$		0			0
$621$	90.2165			0.145276
$622$		0			0
$623$			186.529i			0.299405i
$624$		0			0
$625$	264.155			0.422648
$626$		0			0
$627$	−1048.51	+	21.9418i	−1.67226	+	0.0349949i
$628$		0			0
$629$		−	609.252i		−	0.968605i
$630$		0			0
$631$	−779.311			−1.23504			−0.617521	−	0.786554i	$-0.711863\pi$
$631$	−779.311			−1.23504			−0.617521	+	0.786554i	$0.711863\pi$
$632$		0			0
$633$			1141.34i			1.80306i
$634$		0			0
$635$			30.5565i			0.0481204i
$636$		0			0
$637$			571.044i			0.896459i
$638$		0			0
$639$	18.4466			0.0288679
$640$		0			0
$641$	283.385			0.442098			0.221049	−	0.975263i	$-0.429052\pi$
$641$	283.385			0.442098			0.221049	+	0.975263i	$0.429052\pi$
$642$		0			0
$643$	−570.053			−0.886552			−0.443276	−	0.896385i	$-0.646184\pi$
$643$	−570.053			−0.886552			−0.443276	+	0.896385i	$0.646184\pi$
$644$		0			0
$645$			185.230i			0.287179i
$646$		0			0
$647$	557.941			0.862351			0.431175	−	0.902268i	$-0.358099\pi$
$647$	557.941			0.862351			0.431175	+	0.902268i	$0.358099\pi$
$648$		0			0
$649$	−23.6209	−	1128.75i	−0.0363958	−	1.73921i
$650$		0			0
$651$		−	168.228i		−	0.258415i
$652$		0			0
$653$	1111.31			1.70186			0.850928	−	0.525283i	$-0.176041\pi$
$653$	1111.31			1.70186			0.850928	+	0.525283i	$0.176041\pi$
$654$		0			0
$655$			138.132i			0.210889i
$656$		0			0
$657$		−	146.816i		−	0.223464i
$658$		0			0
$659$			10.0016i			0.0151769i	0.999971	+	0.00758847i	$0.00241551\pi$
$659$			10.0016i			0.0151769i	−0.999971	+	0.00758847i	$0.997584\pi$
$660$		0			0
$661$	330.126			0.499435			0.249717	−	0.968319i	$-0.419662\pi$
$661$	330.126			0.499435			0.249717	+	0.968319i	$0.419662\pi$
$662$		0			0
$663$	−568.020			−0.856743
$664$		0			0
$665$	107.621			0.161836
$666$		0			0
$667$		−	114.551i		−	0.171741i
$668$		0			0
$669$	−716.600			−1.07115
$670$		0			0
$671$	988.435	−	20.6847i	1.47308	−	0.0308267i
$672$		0			0
$673$		−	510.419i		−	0.758423i	−0.925310	−	0.379211i	$-0.876195\pi$
$673$		−	510.419i		−	0.758423i	0.925310	−	0.379211i	$-0.123805\pi$
$674$		0			0
$675$	397.901			0.589483
$676$		0			0
$677$		−	854.013i		−	1.26147i	−0.776000	−	0.630733i	$-0.782754\pi$
$677$		−	854.013i		−	1.26147i	0.776000	−	0.630733i	$-0.217246\pi$
$678$		0			0
$679$		−	27.7742i		−	0.0409046i
$680$		0			0
$681$			594.324i			0.872722i
$682$		0			0
$683$	1102.97			1.61489			0.807444	−	0.589944i	$-0.200850\pi$
$683$	1102.97			1.61489			0.807444	+	0.589944i	$0.200850\pi$
$684$		0			0
$685$	157.890			0.230497
$686$		0			0
$687$	819.361			1.19266
$688$		0			0
$689$			958.073i			1.39053i
$690$		0			0
$691$	1033.85			1.49617			0.748085	−	0.663603i	$-0.230974\pi$
$691$	1033.85			1.49617			0.748085	+	0.663603i	$0.230974\pi$
$692$		0			0
$693$	62.6233	−	1.31050i	0.0903655	−	0.00189105i
$694$		0			0
$695$		−	346.036i		−	0.497894i
$696$		0			0
$697$	744.615			1.06831
$698$		0			0
$699$			953.479i			1.36406i
$700$		0			0
$701$		−	188.420i		−	0.268788i	−0.990928	−	0.134394i	$-0.957091\pi$
$701$		−	188.420i		−	0.268788i	0.990928	−	0.134394i	$-0.0429088\pi$
$702$		0			0
$703$			1270.19i			1.80682i
$704$		0			0
$705$	300.638			0.426436
$706$		0			0
$707$	227.491			0.321770
$708$		0			0
$709$	585.088			0.825230			0.412615	−	0.910906i	$-0.364616\pi$
$709$	585.088			0.825230			0.412615	+	0.910906i	$0.364616\pi$
$710$		0			0
$711$		−	458.879i		−	0.645399i
$712$		0			0
$713$	124.120			0.174082
$714$		0			0
$715$	310.521	−	6.49817i	0.434295	−	0.00908835i
$716$		0			0
$717$		−	708.432i		−	0.988050i
$718$		0			0
$719$	−1237.24			−1.72078			−0.860392	−	0.509633i	$-0.829781\pi$
$719$	−1237.24			−1.72078			−0.860392	+	0.509633i	$0.829781\pi$
$720$		0			0
$721$			94.4674i			0.131023i
$722$		0			0
$723$		−	155.736i		−	0.215403i
$724$		0			0
$725$		−	505.230i		−	0.696869i
$726$		0			0
$727$	−257.146			−0.353709			−0.176854	−	0.984237i	$-0.556592\pi$
$727$	−257.146			−0.353709			−0.176854	+	0.984237i	$0.556592\pi$
$728$		0			0
$729$	306.096			0.419885
$730$		0			0
$731$	−303.617			−0.415345
$732$		0			0
$733$		−	716.541i		−	0.977545i	−0.872411	−	0.488773i	$-0.837445\pi$
$733$		−	716.541i		−	0.977545i	0.872411	−	0.488773i	$-0.162555\pi$
$734$		0			0
$735$	366.114			0.498115
$736$		0			0
$737$	6.05539	+	289.362i	0.00821626	+	0.392621i
$738$		0			0
$739$		−	938.114i		−	1.26944i	−0.772743	−	0.634719i	$-0.781116\pi$
$739$		−	938.114i		−	1.26944i	0.772743	−	0.634719i	$-0.218884\pi$
$740$		0			0
$741$	1184.23			1.59815
$742$		0			0
$743$			598.391i			0.805371i	0.915338	+	0.402686i	$0.131923\pi$
$743$			598.391i			0.805371i	−0.915338	+	0.402686i	$0.868077\pi$
$744$		0			0
$745$			250.851i			0.336713i
$746$		0			0
$747$			379.635i			0.508213i
$748$		0			0
$749$	1.11991			0.00149520
$750$		0			0
$751$	85.2179			0.113473			0.0567363	−	0.998389i	$-0.481931\pi$
$751$	85.2179			0.113473			0.0567363	+	0.998389i	$0.481931\pi$
$752$		0			0
$753$	−1270.14			−1.68678
$754$		0			0
$755$		−	139.543i		−	0.184825i
$756$		0			0
$757$	−1057.81			−1.39737			−0.698685	−	0.715430i	$-0.746231\pi$
$757$	−1057.81			−1.39737			−0.698685	+	0.715430i	$0.746231\pi$
$758$		0			0
$759$	3.62551	+	173.248i	0.00477670	+	0.228259i
$760$		0			0
$761$		−	584.173i		−	0.767639i	−0.923408	−	0.383820i	$-0.874608\pi$
$761$		−	584.173i		−	0.767639i	0.923408	−	0.383820i	$-0.125392\pi$
$762$		0			0
$763$	−297.311			−0.389661
$764$		0			0
$765$			97.1230i			0.126958i
$766$		0			0
$767$			1274.85i			1.66213i
$768$		0			0
$769$			1326.75i			1.72529i	0.505807	+	0.862647i	$0.331195\pi$
$769$			1326.75i			1.72529i	−0.505807	+	0.862647i	$0.668805\pi$
$770$		0			0
$771$	142.986			0.185456
$772$		0			0
$773$	−410.847			−0.531497			−0.265749	−	0.964042i	$-0.585619\pi$
$773$	−410.847			−0.531497			−0.265749	+	0.964042i	$0.585619\pi$
$774$		0			0
$775$	547.434			0.706367
$776$		0			0
$777$		−	284.461i		−	0.366102i
$778$		0			0
$779$	−1552.40			−1.99281
$780$		0			0
$781$	−1.29702	−	61.9791i	−0.00166072	−	0.0793587i
$782$		0			0
$783$		−	511.093i		−	0.652736i
$784$		0			0
$785$	490.314			0.624604
$786$		0			0
$787$		−	1208.02i		−	1.53497i	−0.641064	−	0.767487i	$-0.721507\pi$
$787$		−	1208.02i		−	1.53497i	0.641064	−	0.767487i	$-0.278493\pi$
$788$		0			0
$789$		−	859.749i		−	1.08967i
$790$		0			0
$791$			0.751576i			0.000950159i
$792$		0			0
$793$	−1116.38			−1.40780
$794$		0			0
$795$	614.251			0.772642
$796$		0			0
$797$	−222.786			−0.279531			−0.139765	−	0.990185i	$-0.544635\pi$
$797$	−222.786			−0.279531			−0.139765	+	0.990185i	$0.544635\pi$
$798$		0			0
$799$			492.785i			0.616752i
$800$		0			0
$801$	350.949			0.438139
$802$		0			0
$803$	−493.291	+	10.3230i	−0.614310	+	0.0128555i
$804$		0			0
$805$		−	17.7826i		−	0.0220902i
$806$		0			0
$807$	−864.482			−1.07123
$808$		0			0
$809$			629.428i			0.778033i	0.921231	+	0.389016i	$0.127185\pi$
$809$			629.428i			0.778033i	−0.921231	+	0.389016i	$0.872815\pi$
$810$		0			0
$811$			499.230i			0.615573i	0.951455	+	0.307787i	$0.0995883\pi$
$811$			499.230i			0.615573i	−0.951455	+	0.307787i	$0.900412\pi$
$812$		0			0
$813$		−	1358.65i		−	1.67116i
$814$		0			0
$815$	−23.1762			−0.0284371
$816$		0			0
$817$	632.994			0.774778
$818$		0			0
$819$	−70.7295			−0.0863608
$820$		0			0
$821$		−	666.128i		−	0.811361i	−0.914015	−	0.405681i	$-0.867035\pi$
$821$		−	666.128i		−	0.811361i	0.914015	−	0.405681i	$-0.132965\pi$
$822$		0			0
$823$	1424.55			1.73092			0.865462	−	0.500975i	$-0.167025\pi$
$823$	1424.55			1.73092			0.865462	+	0.500975i	$0.167025\pi$
$824$		0			0
$825$	15.9904	+	764.114i	0.0193823	+	0.926199i
$826$		0			0
$827$		−	596.146i		−	0.720853i	−0.932788	−	0.360427i	$-0.882631\pi$
$827$		−	596.146i		−	0.720853i	0.932788	−	0.360427i	$-0.117369\pi$
$828$		0			0
$829$	−861.706			−1.03945			−0.519726	−	0.854333i	$-0.673966\pi$
$829$	−861.706			−1.03945			−0.519726	+	0.854333i	$0.673966\pi$
$830$		0			0
$831$			1630.78i			1.96243i
$832$		0			0
$833$			600.110i			0.720420i
$834$		0			0
$835$			437.741i			0.524241i
$836$		0			0
$837$	553.787			0.661633
$838$		0			0
$839$	−1346.03			−1.60432			−0.802161	−	0.597108i	$-0.796316\pi$
$839$	−1346.03			−1.60432			−0.802161	+	0.597108i	$0.796316\pi$
$840$		0			0
$841$	192.047			0.228356
$842$		0			0
$843$			108.604i			0.128830i
$844$		0			0
$845$	33.4491			0.0395847
$846$		0			0
$847$	−8.80636	−	210.318i	−0.0103971	−	0.248309i
$848$		0			0
$849$		−	156.932i		−	0.184843i
$850$		0			0
$851$	209.878			0.246625
$852$		0			0
$853$		−	581.240i		−	0.681407i	−0.940171	−	0.340703i	$-0.889335\pi$
$853$		−	581.240i		−	0.681407i	0.940171	−	0.340703i	$-0.110665\pi$
$854$		0			0
$855$		−	202.486i		−	0.236826i
$856$		0			0
$857$		−	685.242i		−	0.799582i	−0.916606	−	0.399791i	$-0.869083\pi$
$857$		−	685.242i		−	0.799582i	0.916606	−	0.399791i	$-0.130917\pi$
$858$		0			0
$859$	23.1449			0.0269440			0.0134720	−	0.999909i	$-0.495712\pi$
$859$	23.1449			0.0269440			0.0134720	+	0.999909i	$0.495712\pi$
$860$		0			0
$861$	347.662			0.403789
$862$		0			0
$863$	−208.554			−0.241662			−0.120831	−	0.992673i	$-0.538556\pi$
$863$	−208.554			−0.241662			−0.120831	+	0.992673i	$0.538556\pi$
$864$		0			0
$865$		−	411.496i		−	0.475718i
$866$		0			0
$867$	415.524			0.479266
$868$		0			0
$869$	−1541.80	+	32.2648i	−1.77422	+	0.0371286i
$870$		0			0
$871$		−	326.818i		−	0.375221i
$872$		0			0
$873$	−52.2563			−0.0598583
$874$		0			0
$875$		−	177.295i		−	0.202623i
$876$		0			0
$877$			1249.16i			1.42436i	0.701997	+	0.712180i	$0.252292\pi$
$877$			1249.16i			1.42436i	−0.701997	+	0.712180i	$0.747708\pi$
$878$		0			0
$879$		−	397.691i		−	0.452435i
$880$		0			0
$881$	−1347.42			−1.52942			−0.764712	−	0.644372i	$-0.777119\pi$
$881$	−1347.42			−1.52942			−0.764712	+	0.644372i	$0.777119\pi$
$882$		0			0
$883$	−718.134			−0.813289			−0.406644	−	0.913587i	$-0.633301\pi$
$883$	−718.134			−0.813289			−0.406644	+	0.913587i	$0.633301\pi$
$884$		0			0
$885$	817.349			0.923558
$886$		0			0
$887$			453.536i			0.511315i	0.966767	+	0.255658i	$0.0822919\pi$
$887$			453.536i			0.511315i	−0.966767	+	0.255658i	$0.917708\pi$
$888$		0			0
$889$	−23.3853			−0.0263052
$890$		0			0
$891$	22.9556	+	1096.95i	0.0257639	+	1.23115i
$892$		0			0
$893$		−	1027.38i		−	1.15048i
$894$		0			0
$895$	−244.959			−0.273697
$896$		0			0
$897$		−	195.674i		−	0.218143i
$898$		0			0
$899$		−	703.163i		−	0.782162i
$900$		0			0
$901$			1006.84i			1.11747i
$902$		0			0
$903$	−141.759			−0.156987
$904$		0			0
$905$	−737.629			−0.815060
$906$		0			0
$907$	−503.384			−0.554999			−0.277500	−	0.960726i	$-0.589506\pi$
$907$	−503.384			−0.554999			−0.277500	+	0.960726i	$0.589506\pi$
$908$		0			0
$909$		−	428.018i		−	0.470867i
$910$		0			0
$911$	−771.136			−0.846472			−0.423236	−	0.906019i	$-0.639106\pi$
$911$	−771.136			−0.846472			−0.423236	+	0.906019i	$0.639106\pi$
$912$		0			0
$913$	1275.55	−	26.6930i	1.39709	−	0.0292366i
$914$		0			0
$915$			715.747i			0.782238i
$916$		0			0
$917$	−105.715			−0.115283
$918$		0			0
$919$		−	1422.23i		−	1.54758i	−0.633440	−	0.773792i	$-0.718358\pi$
$919$		−	1422.23i		−	1.54758i	0.633440	−	0.773792i	$-0.281642\pi$
$920$		0			0
$921$			286.100i			0.310641i
$922$		0			0
$923$			70.0019i			0.0758418i
$924$		0			0
$925$	925.670			1.00072
$926$		0			0
$927$	177.738			0.191734
$928$		0			0
$929$	−1097.58			−1.18146			−0.590732	−	0.806868i	$-0.701161\pi$
$929$	−1097.58			−1.18146			−0.590732	+	0.806868i	$0.701161\pi$
$930$		0			0
$931$		−	1251.13i		−	1.34386i
$932$		0			0
$933$	−1011.12			−1.08373
$934$		0			0
$935$	326.326	−	6.82893i	0.349012	−	0.00730367i
$936$		0			0
$937$			1501.84i			1.60282i	0.598117	+	0.801408i	$0.295916\pi$
$937$			1501.84i			1.60282i	−0.598117	+	0.801408i	$0.704084\pi$
$938$		0			0
$939$	1254.96			1.33648
$940$		0			0
$941$			280.903i			0.298515i	0.988798	+	0.149257i	$0.0476883\pi$
$941$			280.903i			0.298515i	−0.988798	+	0.149257i	$0.952312\pi$
$942$		0			0
$943$			256.508i			0.272013i
$944$		0			0
$945$		−	79.3404i		−	0.0839581i
$946$		0			0
$947$	−106.220			−0.112165			−0.0560824	−	0.998426i	$-0.517861\pi$
$947$	−106.220			−0.112165			−0.0560824	+	0.998426i	$0.517861\pi$
$948$		0			0
$949$	557.144			0.587086
$950$		0			0
$951$	1349.64			1.41918
$952$		0			0
$953$			1649.29i			1.73063i	0.501227	+	0.865316i	$0.332882\pi$
$953$			1649.29i			1.73063i	−0.501227	+	0.865316i	$0.667118\pi$
$954$		0			0
$955$	−428.845			−0.449052
$956$		0			0
$957$	981.482	−	20.5392i	1.02558	−	0.0214620i
$958$		0			0
$959$			120.836i			0.126002i
$960$		0			0
$961$	−199.098			−0.207177
$962$		0			0
$963$		−	2.10707i		−	0.00218803i
$964$		0			0
$965$			79.4844i			0.0823672i
$966$		0			0
$967$		−	1394.32i		−	1.44190i	−0.692986	−	0.720951i	$-0.743705\pi$
$967$		−	1394.32i		−	1.44190i	0.692986	−	0.720951i	$-0.256295\pi$
$968$		0			0
$969$	1244.51			1.28432
$970$		0			0
$971$	1171.15			1.20612			0.603062	−	0.797694i	$-0.293947\pi$
$971$	1171.15			1.20612			0.603062	+	0.797694i	$0.293947\pi$
$972$		0			0
$973$	264.826			0.272175
$974$		0			0
$975$		−	863.024i		−	0.885153i
$976$		0			0
$977$	1145.62			1.17259			0.586293	−	0.810099i	$-0.300587\pi$
$977$	1145.62			1.17259			0.586293	+	0.810099i	$0.300587\pi$
$978$		0			0
$979$	−24.6760	−	1179.16i	−0.0252053	−	1.20446i
$980$		0			0
$981$			559.382i			0.570216i
$982$		0			0
$983$	−116.438			−0.118452			−0.0592258	−	0.998245i	$-0.518863\pi$
$983$	−116.438			−0.118452			−0.0592258	+	0.998245i	$0.518863\pi$
$984$		0			0
$985$		−	792.400i		−	0.804467i
$986$		0			0
$987$			230.082i			0.233113i
$988$		0			0
$989$		−	104.591i		−	0.105755i
$990$		0			0
$991$	214.558			0.216506			0.108253	−	0.994123i	$-0.465474\pi$
$991$	214.558			0.216506			0.108253	+	0.994123i	$0.465474\pi$
$992$		0			0
$993$	787.688			0.793241
$994$		0			0
$995$	−451.614			−0.453884
$996$		0			0
$997$			1553.37i			1.55805i	0.626994	+	0.779024i	$0.284285\pi$
$997$			1553.37i			1.55805i	−0.626994	+	0.779024i	$0.715715\pi$
$998$		0			0
$999$	936.412			0.937349

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.3.h.a.65.6	yes	6
3.2	odd	2		792.3.j.a.505.4		6
4.3	odd	2		176.3.h.d.65.1		6
8.3	odd	2		704.3.h.g.65.5		6
8.5	even	2		704.3.h.h.65.2		6
11.10	odd	2	inner	88.3.h.a.65.5	✓	6
12.11	even	2		1584.3.j.k.1297.3		6
33.32	even	2		792.3.j.a.505.3		6
44.43	even	2		176.3.h.d.65.2		6
88.21	odd	2		704.3.h.h.65.1		6
88.43	even	2		704.3.h.g.65.6		6
132.131	odd	2		1584.3.j.k.1297.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.3.h.a.65.5	✓	6	11.10	odd	2	inner
88.3.h.a.65.6	yes	6	1.1	even	1	trivial
176.3.h.d.65.1		6	4.3	odd	2
176.3.h.d.65.2		6	44.43	even	2
704.3.h.g.65.5		6	8.3	odd	2
704.3.h.g.65.6		6	88.43	even	2
704.3.h.h.65.1		6	88.21	odd	2
704.3.h.h.65.2		6	8.5	even	2
792.3.j.a.505.3		6	33.32	even	2
792.3.j.a.505.4		6	3.2	odd	2
1584.3.j.k.1297.3		6	12.11	even	2
1584.3.j.k.1297.4		6	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	88.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.50331 q^{3} +2.27316 q^{5} +1.73969i q^{7} +3.27316 q^{9} +(-0.230143 - 10.9976i) q^{11} +12.4212i q^{13} +7.96359 q^{15} +13.0534i q^{17} -27.2142i q^{19} +6.09465i q^{21} -4.49669 q^{23} -19.8327 q^{25} -20.0629 q^{27} +25.4746i q^{29} -27.6026 q^{31} +(-0.806263 - 38.5279i) q^{33} +3.95459i q^{35} -46.6739 q^{37} +43.5151i q^{39} -57.0438i q^{41} +23.2596i q^{43} +7.44044 q^{45} +37.7515 q^{47} +45.9735 q^{49} +45.7301i q^{51} +77.1323 q^{53} +(-0.523153 - 24.9993i) q^{55} -95.3399i q^{57} +102.636 q^{59} +89.8774i q^{61} +5.69428i q^{63} +28.2353i q^{65} -26.3114 q^{67} -15.7533 q^{69} +5.63570 q^{71} -44.8545i q^{73} -69.4801 q^{75} +(19.1323 - 0.400377i) q^{77} -140.194i q^{79} -99.7449 q^{81} +115.984i q^{83} +29.6725i q^{85} +89.2452i q^{87} +107.220 q^{89} -21.6089 q^{91} -96.7003 q^{93} -61.8624i q^{95} -15.9651 q^{97} +(-0.753297 - 35.9969i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 6 q^{9} + 10 q^{11} - 52 q^{23} + 22 q^{25} + 32 q^{27} - 36 q^{31} - 64 q^{33} - 48 q^{37} + 172 q^{45} - 60 q^{47} - 170 q^{49} + 108 q^{53} + 172 q^{55} + 236 q^{59} - 292 q^{67} + 92 q^{69}+ \cdots + 182 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 6 * q^9 + 10 * q^11 - 52 * q^23 + 22 * q^25 + 32 * q^27 - 36 * q^31 - 64 * q^33 - 48 * q^37 + 172 * q^45 - 60 * q^47 - 170 * q^49 + 108 * q^53 + 172 * q^55 + 236 * q^59 - 292 * q^67 + 92 * q^69 + 300 * q^71 - 244 * q^75 - 240 * q^77 - 362 * q^81 + 384 * q^89 - 48 * q^91 - 148 * q^93 + 400 * q^97 + 182 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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