LMFDB - Embedded newform 1170.2.b.f.181.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1170,2,Mod(181,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.181"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1170, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,4,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.4
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	1170.181
Dual form			1170.2.b.f.181.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{10} -5.12311i q^{11} +(3.56155 - 0.561553i) q^{13} -3.12311 q^{14} +1.00000 q^{16} -2.00000 q^{17} -6.00000i q^{19} +1.00000i q^{20} +5.12311 q^{22} +5.12311 q^{23} -1.00000 q^{25} +(0.561553 + 3.56155i) q^{26} -3.12311i q^{28} -2.00000 q^{29} +3.12311i q^{31} +1.00000i q^{32} -2.00000i q^{34} +3.12311 q^{35} -5.12311i q^{37} +6.00000 q^{38} -1.00000 q^{40} -0.876894i q^{41} +6.24621 q^{43} +5.12311i q^{44} +5.12311i q^{46} +6.24621i q^{47} -2.75379 q^{49} -1.00000i q^{50} +(-3.56155 + 0.561553i) q^{52} +13.3693 q^{53} -5.12311 q^{55} +3.12311 q^{56} -2.00000i q^{58} +1.12311i q^{59} +10.0000 q^{61} -3.12311 q^{62} -1.00000 q^{64} +(-0.561553 - 3.56155i) q^{65} +4.87689i q^{67} +2.00000 q^{68} +3.12311i q^{70} -10.2462i q^{71} -13.1231i q^{73} +5.12311 q^{74} +6.00000i q^{76} +16.0000 q^{77} +8.00000 q^{79} -1.00000i q^{80} +0.876894 q^{82} +6.24621i q^{83} +2.00000i q^{85} +6.24621i q^{86} -5.12311 q^{88} -3.12311i q^{89} +(1.75379 + 11.1231i) q^{91} -5.12311 q^{92} -6.24621 q^{94} -6.00000 q^{95} -13.1231i q^{97} -2.75379i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{10} + 6 q^{13} + 4 q^{14} + 4 q^{16} - 8 q^{17} + 4 q^{22} + 4 q^{23} - 4 q^{25} - 6 q^{26} - 8 q^{29} - 4 q^{35} + 24 q^{38} - 4 q^{40} - 8 q^{43} - 44 q^{49} - 6 q^{52} + 4 q^{53}+ \cdots - 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^10 + 6 * q^13 + 4 * q^14 + 4 * q^16 - 8 * q^17 + 4 * q^22 + 4 * q^23 - 4 * q^25 - 6 * q^26 - 8 * q^29 - 4 * q^35 + 24 * q^38 - 4 * q^40 - 8 * q^43 - 44 * q^49 - 6 * q^52 + 4 * q^53 - 4 * q^55 - 4 * q^56 + 40 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 + 8 * q^68 + 4 * q^74 + 64 * q^77 + 32 * q^79 + 20 * q^82 - 4 * q^88 + 40 * q^91 - 4 * q^92 + 8 * q^94 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			3.12311i			1.18042i	0.807249	+	0.590211i	$0.200956\pi$
$7$			3.12311i			1.18042i	−0.807249	+	0.590211i	$0.799044\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	1.00000			0.316228
$11$		−	5.12311i		−	1.54467i	−0.635213	−	0.772337i	$-0.719088\pi$
$11$		−	5.12311i		−	1.54467i	0.635213	−	0.772337i	$-0.280912\pi$
$12$		0			0
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$13$	3.56155	−	0.561553i	0.987797	−	0.155747i
$14$	−3.12311			−0.834685
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$			1.00000i			0.223607i
$21$		0			0
$22$	5.12311			1.09225
$23$	5.12311			1.06824			0.534121	−	0.845408i	$-0.320643\pi$
$23$	5.12311			1.06824			0.534121	+	0.845408i	$0.320643\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	0.561553	+	3.56155i	0.110130	+	0.698478i
$27$		0			0
$28$		−	3.12311i		−	0.590211i
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$			3.12311i			0.560926i	0.959865	+	0.280463i	$0.0904881\pi$
$31$			3.12311i			0.560926i	−0.959865	+	0.280463i	$0.909512\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	2.00000i		−	0.342997i
$35$	3.12311			0.527901
$36$		0			0
$37$		−	5.12311i		−	0.842233i	−0.907006	−	0.421117i	$-0.861638\pi$
$37$		−	5.12311i		−	0.842233i	0.907006	−	0.421117i	$-0.138362\pi$
$38$	6.00000			0.973329
$39$		0			0
$40$	−1.00000			−0.158114
$41$		−	0.876894i		−	0.136948i	−0.997653	−	0.0684739i	$-0.978187\pi$
$41$		−	0.876894i		−	0.136948i	0.997653	−	0.0684739i	$-0.0218130\pi$
$42$		0			0
$43$	6.24621			0.952538			0.476269	−	0.879300i	$-0.341989\pi$
$43$	6.24621			0.952538			0.476269	+	0.879300i	$0.341989\pi$
$44$			5.12311i			0.772337i
$45$		0			0
$46$			5.12311i			0.755361i
$47$			6.24621i			0.911104i	0.890209	+	0.455552i	$0.150558\pi$
$47$			6.24621i			0.911104i	−0.890209	+	0.455552i	$0.849442\pi$
$48$		0			0
$49$	−2.75379			−0.393398
$50$		−	1.00000i		−	0.141421i
$51$		0			0
$52$	−3.56155	+	0.561553i	−0.493899	+	0.0778734i
$53$	13.3693			1.83642			0.918208	−	0.396098i	$-0.129636\pi$
$53$	13.3693			1.83642			0.918208	+	0.396098i	$0.129636\pi$
$54$		0			0
$55$	−5.12311			−0.690799
$56$	3.12311			0.417343
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$			1.12311i			0.146216i	0.997324	+	0.0731079i	$0.0232918\pi$
$59$			1.12311i			0.146216i	−0.997324	+	0.0731079i	$0.976708\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$	−3.12311			−0.396635
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−0.561553	−	3.56155i	−0.0696521	−	0.441756i
$66$		0			0
$67$			4.87689i			0.595807i	0.954596	+	0.297904i	$0.0962874\pi$
$67$			4.87689i			0.595807i	−0.954596	+	0.297904i	$0.903713\pi$
$68$	2.00000			0.242536
$69$		0			0
$70$			3.12311i			0.373283i
$71$		−	10.2462i		−	1.21600i	−0.793936	−	0.608001i	$-0.791972\pi$
$71$		−	10.2462i		−	1.21600i	0.793936	−	0.608001i	$-0.208028\pi$
$72$		0			0
$73$		−	13.1231i		−	1.53594i	−0.640484	−	0.767972i	$-0.721266\pi$
$73$		−	13.1231i		−	1.53594i	0.640484	−	0.767972i	$-0.278734\pi$
$74$	5.12311			0.595549
$75$		0			0
$76$			6.00000i			0.688247i
$77$	16.0000			1.82337
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		−	1.00000i		−	0.111803i
$81$		0			0
$82$	0.876894			0.0968368
$83$			6.24621i			0.685611i	0.939406	+	0.342805i	$0.111377\pi$
$83$			6.24621i			0.685611i	−0.939406	+	0.342805i	$0.888623\pi$
$84$		0			0
$85$			2.00000i			0.216930i
$86$			6.24621i			0.673546i
$87$		0			0
$88$	−5.12311			−0.546125
$89$		−	3.12311i		−	0.331049i	−0.986206	−	0.165524i	$-0.947068\pi$
$89$		−	3.12311i		−	0.331049i	0.986206	−	0.165524i	$-0.0529316\pi$
$90$		0			0
$91$	1.75379	+	11.1231i	0.183847	+	1.16602i
$92$	−5.12311			−0.534121
$93$		0			0
$94$	−6.24621			−0.644247
$95$	−6.00000			−0.615587
$96$		0			0
$97$		−	13.1231i		−	1.33245i	−0.745751	−	0.666225i	$-0.767909\pi$
$97$		−	13.1231i		−	1.33245i	0.745751	−	0.666225i	$-0.232091\pi$
$98$		−	2.75379i		−	0.278175i
$99$		0			0
$100$	1.00000			0.100000
$101$	12.2462			1.21854			0.609272	−	0.792961i	$-0.291462\pi$
$101$	12.2462			1.21854			0.609272	+	0.792961i	$0.291462\pi$
$102$		0			0
$103$	−13.1231			−1.29306			−0.646529	−	0.762889i	$-0.723780\pi$
$103$	−13.1231			−1.29306			−0.646529	+	0.762889i	$0.723780\pi$
$104$	−0.561553	−	3.56155i	−0.0550648	−	0.349239i
$105$		0			0
$106$			13.3693i			1.29854i
$107$	8.00000			0.773389			0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000			0.773389			0.386695	+	0.922208i	$0.373617\pi$
$108$		0			0
$109$			2.87689i			0.275557i	0.990463	+	0.137778i	$0.0439961\pi$
$109$			2.87689i			0.275557i	−0.990463	+	0.137778i	$0.956004\pi$
$110$		−	5.12311i		−	0.488469i
$111$		0			0
$112$			3.12311i			0.295106i
$113$	−12.2462			−1.15203			−0.576013	−	0.817440i	$-0.695392\pi$
$113$	−12.2462			−1.15203			−0.576013	+	0.817440i	$0.695392\pi$
$114$		0			0
$115$		−	5.12311i		−	0.477732i
$116$	2.00000			0.185695
$117$		0			0
$118$	−1.12311			−0.103390
$119$		−	6.24621i		−	0.572589i
$120$		0			0
$121$	−15.2462			−1.38602
$122$			10.0000i			0.905357i
$123$		0			0
$124$		−	3.12311i		−	0.280463i
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−13.1231			−1.16449			−0.582244	−	0.813014i	$-0.697825\pi$
$127$	−13.1231			−1.16449			−0.582244	+	0.813014i	$0.697825\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	3.56155	−	0.561553i	0.312369	−	0.0492514i
$131$	−4.00000			−0.349482			−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000			−0.349482			−0.174741	+	0.984614i	$0.555909\pi$
$132$		0			0
$133$	18.7386			1.62485
$134$	−4.87689			−0.421300
$135$		0			0
$136$			2.00000i			0.171499i
$137$			10.4924i			0.896428i	0.893926	+	0.448214i	$0.147940\pi$
$137$			10.4924i			0.896428i	−0.893926	+	0.448214i	$0.852060\pi$
$138$		0			0
$139$	−16.4924			−1.39887			−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924			−1.39887			−0.699435	+	0.714697i	$0.746565\pi$
$140$	−3.12311			−0.263951
$141$		0			0
$142$	10.2462			0.859843
$143$	−2.87689	−	18.2462i	−0.240578	−	1.52582i
$144$		0			0
$145$			2.00000i			0.166091i
$146$	13.1231			1.08608
$147$		0			0
$148$			5.12311i			0.421117i
$149$			14.0000i			1.14692i	0.819232	+	0.573462i	$0.194400\pi$
$149$			14.0000i			1.14692i	−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$			13.3693i			1.08798i	0.839092	+	0.543990i	$0.183087\pi$
$151$			13.3693i			1.08798i	−0.839092	+	0.543990i	$0.816913\pi$
$152$	−6.00000			−0.486664
$153$		0			0
$154$			16.0000i			1.28932i
$155$	3.12311			0.250854
$156$		0			0
$157$	21.3693			1.70546			0.852729	−	0.522354i	$-0.174946\pi$
$157$	21.3693			1.70546			0.852729	+	0.522354i	$0.174946\pi$
$158$			8.00000i			0.636446i
$159$		0			0
$160$	1.00000			0.0790569
$161$			16.0000i			1.26098i
$162$		0			0
$163$		−	7.12311i		−	0.557925i	−0.960302	−	0.278962i	$-0.910010\pi$
$163$		−	7.12311i		−	0.557925i	0.960302	−	0.278962i	$-0.0899905\pi$
$164$			0.876894i			0.0684739i
$165$		0			0
$166$	−6.24621			−0.484800
$167$			22.2462i			1.72146i	0.509059	+	0.860732i	$0.329994\pi$
$167$			22.2462i			1.72146i	−0.509059	+	0.860732i	$0.670006\pi$
$168$		0			0
$169$	12.3693	−	4.00000i	0.951486	−	0.307692i
$170$	−2.00000			−0.153393
$171$		0			0
$172$	−6.24621			−0.476269
$173$	−23.1231			−1.75802			−0.879009	−	0.476806i	$-0.841794\pi$
$173$	−23.1231			−1.75802			−0.879009	+	0.476806i	$0.841794\pi$
$174$		0			0
$175$		−	3.12311i		−	0.236085i
$176$		−	5.12311i		−	0.386169i
$177$		0			0
$178$	3.12311			0.234087
$179$	−16.4924			−1.23270			−0.616351	−	0.787472i	$-0.711390\pi$
$179$	−16.4924			−1.23270			−0.616351	+	0.787472i	$0.711390\pi$
$180$		0			0
$181$	−20.2462			−1.50489			−0.752445	−	0.658656i	$-0.771125\pi$
$181$	−20.2462			−1.50489			−0.752445	+	0.658656i	$0.771125\pi$
$182$	−11.1231	+	1.75379i	−0.824499	+	0.129999i
$183$		0			0
$184$		−	5.12311i		−	0.377680i
$185$	−5.12311			−0.376658
$186$		0			0
$187$			10.2462i			0.749277i
$188$		−	6.24621i		−	0.455552i
$189$		0			0
$190$		−	6.00000i		−	0.435286i
$191$	−16.4924			−1.19335			−0.596675	−	0.802483i	$-0.703512\pi$
$191$	−16.4924			−1.19335			−0.596675	+	0.802483i	$0.703512\pi$
$192$		0			0
$193$		−	25.1231i		−	1.80840i	−0.427109	−	0.904200i	$-0.640468\pi$
$193$		−	25.1231i		−	1.80840i	0.427109	−	0.904200i	$-0.359532\pi$
$194$	13.1231			0.942184
$195$		0			0
$196$	2.75379			0.196699
$197$		−	16.2462i		−	1.15749i	−0.815507	−	0.578747i	$-0.803542\pi$
$197$		−	16.2462i		−	1.15749i	0.815507	−	0.578747i	$-0.196458\pi$
$198$		0			0
$199$	18.2462			1.29344			0.646720	−	0.762728i	$-0.276140\pi$
$199$	18.2462			1.29344			0.646720	+	0.762728i	$0.276140\pi$
$200$			1.00000i			0.0707107i
$201$		0			0
$202$			12.2462i			0.861640i
$203$		−	6.24621i		−	0.438398i
$204$		0			0
$205$	−0.876894			−0.0612450
$206$		−	13.1231i		−	0.914330i
$207$		0			0
$208$	3.56155	−	0.561553i	0.246949	−	0.0389367i
$209$	−30.7386			−2.12624
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$	−13.3693			−0.918208
$213$		0			0
$214$			8.00000i			0.546869i
$215$		−	6.24621i		−	0.425988i
$216$		0			0
$217$	−9.75379			−0.662130
$218$	−2.87689			−0.194848
$219$		0			0
$220$	5.12311			0.345400
$221$	−7.12311	+	1.12311i	−0.479152	+	0.0755483i
$222$		0			0
$223$			9.36932i			0.627416i	0.949520	+	0.313708i	$0.101571\pi$
$223$			9.36932i			0.627416i	−0.949520	+	0.313708i	$0.898429\pi$
$224$	−3.12311			−0.208671
$225$		0			0
$226$		−	12.2462i		−	0.814606i
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$		−	5.12311i		−	0.338544i	−0.985569	−	0.169272i	$-0.945858\pi$
$229$		−	5.12311i		−	0.338544i	0.985569	−	0.169272i	$-0.0541417\pi$
$230$	5.12311			0.337808
$231$		0			0
$232$			2.00000i			0.131306i
$233$	7.75379			0.507968			0.253984	−	0.967208i	$-0.418259\pi$
$233$	7.75379			0.507968			0.253984	+	0.967208i	$0.418259\pi$
$234$		0			0
$235$	6.24621			0.407458
$236$		−	1.12311i		−	0.0731079i
$237$		0			0
$238$	6.24621			0.404882
$239$		−	4.49242i		−	0.290591i	−0.989388	−	0.145295i	$-0.953587\pi$
$239$		−	4.49242i		−	0.290591i	0.989388	−	0.145295i	$-0.0464132\pi$
$240$		0			0
$241$		−	14.2462i		−	0.917679i	−0.888519	−	0.458840i	$-0.848265\pi$
$241$		−	14.2462i		−	0.917679i	0.888519	−	0.458840i	$-0.151735\pi$
$242$		−	15.2462i		−	0.980064i
$243$		0			0
$244$	−10.0000			−0.640184
$245$			2.75379i			0.175933i
$246$		0			0
$247$	−3.36932	−	21.3693i	−0.214384	−	1.35970i
$248$	3.12311			0.198317
$249$		0			0
$250$	−1.00000			−0.0632456
$251$	−26.2462			−1.65665			−0.828323	−	0.560251i	$-0.810705\pi$
$251$	−26.2462			−1.65665			−0.828323	+	0.560251i	$0.810705\pi$
$252$		0			0
$253$		−	26.2462i		−	1.65009i
$254$		−	13.1231i		−	0.823417i
$255$		0			0
$256$	1.00000			0.0625000
$257$	12.2462			0.763898			0.381949	−	0.924183i	$-0.375253\pi$
$257$	12.2462			0.763898			0.381949	+	0.924183i	$0.375253\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$	0.561553	+	3.56155i	0.0348260	+	0.220878i
$261$		0			0
$262$		−	4.00000i		−	0.247121i
$263$	27.3693			1.68766			0.843832	−	0.536607i	$-0.180294\pi$
$263$	27.3693			1.68766			0.843832	+	0.536607i	$0.180294\pi$
$264$		0			0
$265$		−	13.3693i		−	0.821271i
$266$			18.7386i			1.14894i
$267$		0			0
$268$		−	4.87689i		−	0.297904i
$269$	16.2462			0.990549			0.495274	−	0.868737i	$-0.335067\pi$
$269$	16.2462			0.990549			0.495274	+	0.868737i	$0.335067\pi$
$270$		0			0
$271$			23.1231i			1.40463i	0.711867	+	0.702314i	$0.247850\pi$
$271$			23.1231i			1.40463i	−0.711867	+	0.702314i	$0.752150\pi$
$272$	−2.00000			−0.121268
$273$		0			0
$274$	−10.4924			−0.633870
$275$			5.12311i			0.308935i
$276$		0			0
$277$	−29.8617			−1.79422			−0.897109	−	0.441809i	$-0.854337\pi$
$277$	−29.8617			−1.79422			−0.897109	+	0.441809i	$0.854337\pi$
$278$		−	16.4924i		−	0.989150i
$279$		0			0
$280$		−	3.12311i		−	0.186641i
$281$			3.12311i			0.186309i	0.995652	+	0.0931544i	$0.0296950\pi$
$281$			3.12311i			0.186309i	−0.995652	+	0.0931544i	$0.970305\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$			10.2462i			0.608001i
$285$		0			0
$286$	18.2462	−	2.87689i	1.07892	−	0.170114i
$287$	2.73863			0.161656
$288$		0			0
$289$	−13.0000			−0.764706
$290$	−2.00000			−0.117444
$291$		0			0
$292$			13.1231i			0.767972i
$293$		−	28.7386i		−	1.67893i	−0.543415	−	0.839464i	$-0.682869\pi$
$293$		−	28.7386i		−	1.67893i	0.543415	−	0.839464i	$-0.317131\pi$
$294$		0			0
$295$	1.12311			0.0653897
$296$	−5.12311			−0.297774
$297$		0			0
$298$	−14.0000			−0.810998
$299$	18.2462	−	2.87689i	1.05521	−	0.166375i
$300$		0			0
$301$			19.5076i			1.12440i
$302$	−13.3693			−0.769318
$303$		0			0
$304$		−	6.00000i		−	0.344124i
$305$		−	10.0000i		−	0.572598i
$306$		0			0
$307$			31.1231i			1.77629i	0.459564	+	0.888145i	$0.348006\pi$
$307$			31.1231i			1.77629i	−0.459564	+	0.888145i	$0.651994\pi$
$308$	−16.0000			−0.911685
$309$		0			0
$310$			3.12311i			0.177380i
$311$	−8.49242			−0.481561			−0.240781	−	0.970580i	$-0.577403\pi$
$311$	−8.49242			−0.481561			−0.240781	+	0.970580i	$0.577403\pi$
$312$		0			0
$313$	16.2462			0.918290			0.459145	−	0.888361i	$-0.348156\pi$
$313$	16.2462			0.918290			0.459145	+	0.888361i	$0.348156\pi$
$314$			21.3693i			1.20594i
$315$		0			0
$316$	−8.00000			−0.450035
$317$		−	6.00000i		−	0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$		−	6.00000i		−	0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$		0			0
$319$			10.2462i			0.573678i
$320$			1.00000i			0.0559017i
$321$		0			0
$322$	−16.0000			−0.891645
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−3.56155	+	0.561553i	−0.197559	+	0.0311493i
$326$	7.12311			0.394512
$327$		0			0
$328$	−0.876894			−0.0484184
$329$	−19.5076			−1.07549
$330$		0			0
$331$		−	7.75379i		−	0.426187i	−0.977032	−	0.213093i	$-0.931646\pi$
$331$		−	7.75379i		−	0.426187i	0.977032	−	0.213093i	$-0.0683539\pi$
$332$		−	6.24621i		−	0.342805i
$333$		0			0
$334$	−22.2462			−1.21726
$335$	4.87689			0.266453
$336$		0			0
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$	4.00000	+	12.3693i	0.217571	+	0.672802i
$339$		0			0
$340$		−	2.00000i		−	0.108465i
$341$	16.0000			0.866449
$342$		0			0
$343$			13.2614i			0.716046i
$344$		−	6.24621i		−	0.336773i
$345$		0			0
$346$		−	23.1231i		−	1.24311i
$347$	14.2462			0.764777			0.382388	−	0.924002i	$-0.375102\pi$
$347$	14.2462			0.764777			0.382388	+	0.924002i	$0.375102\pi$
$348$		0			0
$349$			19.3693i			1.03682i	0.855133	+	0.518408i	$0.173475\pi$
$349$			19.3693i			1.03682i	−0.855133	+	0.518408i	$0.826525\pi$
$350$	3.12311			0.166937
$351$		0			0
$352$	5.12311			0.273062
$353$			12.2462i			0.651800i	0.945404	+	0.325900i	$0.105667\pi$
$353$			12.2462i			0.651800i	−0.945404	+	0.325900i	$0.894333\pi$
$354$		0			0
$355$	−10.2462			−0.543812
$356$			3.12311i			0.165524i
$357$		0			0
$358$		−	16.4924i		−	0.871652i
$359$		−	17.7538i		−	0.937009i	−0.883461	−	0.468505i	$-0.844793\pi$
$359$		−	17.7538i		−	0.937009i	0.883461	−	0.468505i	$-0.155207\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		−	20.2462i		−	1.06412i
$363$		0			0
$364$	−1.75379	−	11.1231i	−0.0919235	−	0.583009i
$365$	−13.1231			−0.686895
$366$		0			0
$367$	8.63068			0.450518			0.225259	−	0.974299i	$-0.427677\pi$
$367$	8.63068			0.450518			0.225259	+	0.974299i	$0.427677\pi$
$368$	5.12311			0.267060
$369$		0			0
$370$		−	5.12311i		−	0.266338i
$371$			41.7538i			2.16775i
$372$		0			0
$373$	7.12311			0.368820			0.184410	−	0.982849i	$-0.440963\pi$
$373$	7.12311			0.368820			0.184410	+	0.982849i	$0.440963\pi$
$374$	−10.2462			−0.529819
$375$		0			0
$376$	6.24621			0.322124
$377$	−7.12311	+	1.12311i	−0.366859	+	0.0578429i
$378$		0			0
$379$		−	6.00000i		−	0.308199i	−0.988055	−	0.154100i	$-0.950752\pi$
$379$		−	6.00000i		−	0.308199i	0.988055	−	0.154100i	$-0.0492477\pi$
$380$	6.00000			0.307794
$381$		0			0
$382$		−	16.4924i		−	0.843826i
$383$		−	1.75379i		−	0.0896144i	−0.998996	−	0.0448072i	$-0.985733\pi$
$383$		−	1.75379i		−	0.0896144i	0.998996	−	0.0448072i	$-0.0142674\pi$
$384$		0			0
$385$		−	16.0000i		−	0.815436i
$386$	25.1231			1.27873
$387$		0			0
$388$			13.1231i			0.666225i
$389$	0.246211			0.0124834			0.00624170	−	0.999981i	$-0.498013\pi$
$389$	0.246211			0.0124834			0.00624170	+	0.999981i	$0.498013\pi$
$390$		0			0
$391$	−10.2462			−0.518173
$392$			2.75379i			0.139087i
$393$		0			0
$394$	16.2462			0.818472
$395$		−	8.00000i		−	0.402524i
$396$		0			0
$397$		−	15.3693i		−	0.771364i	−0.922632	−	0.385682i	$-0.873966\pi$
$397$		−	15.3693i		−	0.771364i	0.922632	−	0.385682i	$-0.126034\pi$
$398$			18.2462i			0.914600i
$399$		0			0
$400$	−1.00000			−0.0500000
$401$			1.36932i			0.0683804i	0.999415	+	0.0341902i	$0.0108852\pi$
$401$			1.36932i			0.0683804i	−0.999415	+	0.0341902i	$0.989115\pi$
$402$		0			0
$403$	1.75379	+	11.1231i	0.0873624	+	0.554081i
$404$	−12.2462			−0.609272
$405$		0			0
$406$	6.24621			0.309994
$407$	−26.2462			−1.30098
$408$		0			0
$409$			8.49242i			0.419923i	0.977710	+	0.209962i	$0.0673339\pi$
$409$			8.49242i			0.419923i	−0.977710	+	0.209962i	$0.932666\pi$
$410$		−	0.876894i		−	0.0433067i
$411$		0			0
$412$	13.1231			0.646529
$413$	−3.50758			−0.172597
$414$		0			0
$415$	6.24621			0.306614
$416$	0.561553	+	3.56155i	0.0275324	+	0.174619i
$417$		0			0
$418$		−	30.7386i		−	1.50348i
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$		−	0.630683i		−	0.0307376i	−0.999882	−	0.0153688i	$-0.995108\pi$
$421$		−	0.630683i		−	0.0307376i	0.999882	−	0.0153688i	$-0.00489224\pi$
$422$			4.00000i			0.194717i
$423$		0			0
$424$		−	13.3693i		−	0.649271i
$425$	2.00000			0.0970143
$426$		0			0
$427$			31.2311i			1.51138i
$428$	−8.00000			−0.386695
$429$		0			0
$430$	6.24621			0.301219
$431$			32.4924i			1.56510i	0.622585	+	0.782552i	$0.286083\pi$
$431$			32.4924i			1.56510i	−0.622585	+	0.782552i	$0.713917\pi$
$432$		0			0
$433$	−18.0000			−0.865025			−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000			−0.865025			−0.432512	+	0.901628i	$0.642373\pi$
$434$		−	9.75379i		−	0.468197i
$435$		0			0
$436$		−	2.87689i		−	0.137778i
$437$		−	30.7386i		−	1.47043i
$438$		0			0
$439$	−36.4924			−1.74169			−0.870844	−	0.491559i	$-0.836427\pi$
$439$	−36.4924			−1.74169			−0.870844	+	0.491559i	$0.836427\pi$
$440$			5.12311i			0.244234i
$441$		0			0
$442$	−1.12311	−	7.12311i	−0.0534207	−	0.338812i
$443$	3.50758			0.166650			0.0833250	−	0.996522i	$-0.473446\pi$
$443$	3.50758			0.166650			0.0833250	+	0.996522i	$0.473446\pi$
$444$		0			0
$445$	−3.12311			−0.148049
$446$	−9.36932			−0.443650
$447$		0			0
$448$		−	3.12311i		−	0.147553i
$449$			28.8769i			1.36278i	0.731918	+	0.681392i	$0.238625\pi$
$449$			28.8769i			1.36278i	−0.731918	+	0.681392i	$0.761375\pi$
$450$		0			0
$451$	−4.49242			−0.211540
$452$	12.2462			0.576013
$453$		0			0
$454$	8.00000			0.375459
$455$	11.1231	−	1.75379i	0.521459	−	0.0822189i
$456$		0			0
$457$			31.3693i			1.46739i	0.679476	+	0.733697i	$0.262207\pi$
$457$			31.3693i			1.46739i	−0.679476	+	0.733697i	$0.737793\pi$
$458$	5.12311			0.239387
$459$		0			0
$460$			5.12311i			0.238866i
$461$		−	18.4924i		−	0.861278i	−0.902524	−	0.430639i	$-0.858288\pi$
$461$		−	18.4924i		−	0.861278i	0.902524	−	0.430639i	$-0.141712\pi$
$462$		0			0
$463$		−	21.8617i		−	1.01600i	−0.861357	−	0.508001i	$-0.830385\pi$
$463$		−	21.8617i		−	1.01600i	0.861357	−	0.508001i	$-0.169615\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$			7.75379i			0.359187i
$467$	22.2462			1.02943			0.514716	−	0.857361i	$-0.327897\pi$
$467$	22.2462			1.02943			0.514716	+	0.857361i	$0.327897\pi$
$468$		0			0
$469$	−15.2311			−0.703305
$470$			6.24621i			0.288116i
$471$		0			0
$472$	1.12311			0.0516951
$473$		−	32.0000i		−	1.47136i
$474$		0			0
$475$			6.00000i			0.275299i
$476$			6.24621i			0.286295i
$477$		0			0
$478$	4.49242			0.205479
$479$			12.4924i			0.570793i	0.958409	+	0.285397i	$0.0921253\pi$
$479$			12.4924i			0.570793i	−0.958409	+	0.285397i	$0.907875\pi$
$480$		0			0
$481$	−2.87689	−	18.2462i	−0.131175	−	0.831956i
$482$	14.2462			0.648897
$483$		0			0
$484$	15.2462			0.693010
$485$	−13.1231			−0.595890
$486$		0			0
$487$			17.3693i			0.787079i	0.919308	+	0.393539i	$0.128750\pi$
$487$			17.3693i			0.787079i	−0.919308	+	0.393539i	$0.871250\pi$
$488$		−	10.0000i		−	0.452679i
$489$		0			0
$490$	−2.75379			−0.124403
$491$	−38.7386			−1.74825			−0.874125	−	0.485701i	$-0.838564\pi$
$491$	−38.7386			−1.74825			−0.874125	+	0.485701i	$0.838564\pi$
$492$		0			0
$493$	4.00000			0.180151
$494$	21.3693	−	3.36932i	0.961451	−	0.151593i
$495$		0			0
$496$			3.12311i			0.140232i
$497$	32.0000			1.43540
$498$		0			0
$499$			34.4924i			1.54409i	0.635566	+	0.772046i	$0.280767\pi$
$499$			34.4924i			1.54409i	−0.635566	+	0.772046i	$0.719233\pi$
$500$		−	1.00000i		−	0.0447214i
$501$		0			0
$502$		−	26.2462i		−	1.17143i
$503$	−35.3693			−1.57704			−0.788520	−	0.615009i	$-0.789152\pi$
$503$	−35.3693			−1.57704			−0.788520	+	0.615009i	$0.789152\pi$
$504$		0			0
$505$		−	12.2462i		−	0.544949i
$506$	26.2462			1.16679
$507$		0			0
$508$	13.1231			0.582244
$509$		−	32.2462i		−	1.42929i	−0.699488	−	0.714644i	$-0.746589\pi$
$509$		−	32.2462i		−	1.42929i	0.699488	−	0.714644i	$-0.253411\pi$
$510$		0			0
$511$	40.9848			1.81306
$512$			1.00000i			0.0441942i
$513$		0			0
$514$			12.2462i			0.540157i
$515$			13.1231i			0.578273i
$516$		0			0
$517$	32.0000			1.40736
$518$			16.0000i			0.703000i
$519$		0			0
$520$	−3.56155	+	0.561553i	−0.156184	+	0.0246257i
$521$	−0.246211			−0.0107867			−0.00539336	−	0.999985i	$-0.501717\pi$
$521$	−0.246211			−0.0107867			−0.00539336	+	0.999985i	$0.501717\pi$
$522$		0			0
$523$	−26.7386			−1.16920			−0.584599	−	0.811322i	$-0.698748\pi$
$523$	−26.7386			−1.16920			−0.584599	+	0.811322i	$0.698748\pi$
$524$	4.00000			0.174741
$525$		0			0
$526$			27.3693i			1.19336i
$527$		−	6.24621i		−	0.272089i
$528$		0			0
$529$	3.24621			0.141140
$530$	13.3693			0.580726
$531$		0			0
$532$	−18.7386			−0.812423
$533$	−0.492423	−	3.12311i	−0.0213292	−	0.135277i
$534$		0			0
$535$		−	8.00000i		−	0.345870i
$536$	4.87689			0.210650
$537$		0			0
$538$			16.2462i			0.700424i
$539$			14.1080i			0.607672i
$540$		0			0
$541$			10.8769i			0.467634i	0.972281	+	0.233817i	$0.0751217\pi$
$541$			10.8769i			0.467634i	−0.972281	+	0.233817i	$0.924878\pi$
$542$	−23.1231			−0.993222
$543$		0			0
$544$		−	2.00000i		−	0.0857493i
$545$	2.87689			0.123233
$546$		0			0
$547$	20.9848			0.897247			0.448624	−	0.893721i	$-0.351914\pi$
$547$	20.9848			0.897247			0.448624	+	0.893721i	$0.351914\pi$
$548$		−	10.4924i		−	0.448214i
$549$		0			0
$550$	−5.12311			−0.218450
$551$			12.0000i			0.511217i
$552$		0			0
$553$			24.9848i			1.06246i
$554$		−	29.8617i		−	1.26870i
$555$		0			0
$556$	16.4924			0.699435
$557$		−	11.7538i		−	0.498024i	−0.968500	−	0.249012i	$-0.919894\pi$
$557$		−	11.7538i		−	0.498024i	0.968500	−	0.249012i	$-0.0801059\pi$
$558$		0			0
$559$	22.2462	−	3.50758i	0.940914	−	0.148355i
$560$	3.12311			0.131975
$561$		0			0
$562$	−3.12311			−0.131740
$563$	32.9848			1.39015			0.695073	−	0.718939i	$-0.255372\pi$
$563$	32.9848			1.39015			0.695073	+	0.718939i	$0.255372\pi$
$564$		0			0
$565$			12.2462i			0.515202i
$566$		−	4.00000i		−	0.168133i
$567$		0			0
$568$	−10.2462			−0.429921
$569$	12.7386			0.534031			0.267016	−	0.963692i	$-0.413962\pi$
$569$	12.7386			0.534031			0.267016	+	0.963692i	$0.413962\pi$
$570$		0			0
$571$	−8.49242			−0.355397			−0.177698	−	0.984085i	$-0.556865\pi$
$571$	−8.49242			−0.355397			−0.177698	+	0.984085i	$0.556865\pi$
$572$	2.87689	+	18.2462i	0.120289	+	0.762912i
$573$		0			0
$574$			2.73863i			0.114308i
$575$	−5.12311			−0.213648
$576$		0			0
$577$			27.3693i			1.13940i	0.821853	+	0.569700i	$0.192941\pi$
$577$			27.3693i			1.13940i	−0.821853	+	0.569700i	$0.807059\pi$
$578$		−	13.0000i		−	0.540729i
$579$		0			0
$580$		−	2.00000i		−	0.0830455i
$581$	−19.5076			−0.809311
$582$		0			0
$583$		−	68.4924i		−	2.83667i
$584$	−13.1231			−0.543038
$585$		0			0
$586$	28.7386			1.18718
$587$			16.4924i			0.680715i	0.940296	+	0.340358i	$0.110548\pi$
$587$			16.4924i			0.680715i	−0.940296	+	0.340358i	$0.889452\pi$
$588$		0			0
$589$	18.7386			0.772112
$590$			1.12311i			0.0462375i
$591$		0			0
$592$		−	5.12311i		−	0.210558i
$593$			5.50758i			0.226169i	0.993585	+	0.113085i	$0.0360731\pi$
$593$			5.50758i			0.226169i	−0.993585	+	0.113085i	$0.963927\pi$
$594$		0			0
$595$	−6.24621			−0.256070
$596$		−	14.0000i		−	0.573462i
$597$		0			0
$598$	2.87689	+	18.2462i	0.117645	+	0.746143i
$599$	36.4924			1.49104			0.745520	−	0.666483i	$-0.232201\pi$
$599$	36.4924			1.49104			0.745520	+	0.666483i	$0.232201\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$	−19.5076			−0.795070
$603$		0			0
$604$		−	13.3693i		−	0.543990i
$605$			15.2462i			0.619847i
$606$		0			0
$607$	15.3693			0.623821			0.311911	−	0.950111i	$-0.399031\pi$
$607$	15.3693			0.623821			0.311911	+	0.950111i	$0.399031\pi$
$608$	6.00000			0.243332
$609$		0			0
$610$	10.0000			0.404888
$611$	3.50758	+	22.2462i	0.141901	+	0.899985i
$612$		0			0
$613$			39.3693i			1.59011i	0.606536	+	0.795056i	$0.292559\pi$
$613$			39.3693i			1.59011i	−0.606536	+	0.795056i	$0.707441\pi$
$614$	−31.1231			−1.25603
$615$		0			0
$616$		−	16.0000i		−	0.644658i
$617$			40.7386i			1.64008i	0.572309	+	0.820038i	$0.306048\pi$
$617$			40.7386i			1.64008i	−0.572309	+	0.820038i	$0.693952\pi$
$618$		0			0
$619$			38.9848i			1.56693i	0.621434	+	0.783467i	$0.286550\pi$
$619$			38.9848i			1.56693i	−0.621434	+	0.783467i	$0.713450\pi$
$620$	−3.12311			−0.125427
$621$		0			0
$622$		−	8.49242i		−	0.340515i
$623$	9.75379			0.390777
$624$		0			0
$625$	1.00000			0.0400000
$626$			16.2462i			0.649329i
$627$		0			0
$628$	−21.3693			−0.852729
$629$			10.2462i			0.408543i
$630$		0			0
$631$		−	35.6155i		−	1.41783i	−0.705293	−	0.708916i	$-0.749185\pi$
$631$		−	35.6155i		−	1.41783i	0.705293	−	0.708916i	$-0.250815\pi$
$632$		−	8.00000i		−	0.318223i
$633$		0			0
$634$	6.00000			0.238290
$635$			13.1231i			0.520775i
$636$		0			0
$637$	−9.80776	+	1.54640i	−0.388598	+	0.0612705i
$638$	−10.2462			−0.405651
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$			17.3693i			0.684979i	0.939522	+	0.342489i	$0.111270\pi$
$643$			17.3693i			0.684979i	−0.939522	+	0.342489i	$0.888730\pi$
$644$		−	16.0000i		−	0.630488i
$645$		0			0
$646$	−12.0000			−0.472134
$647$	−29.6155			−1.16431			−0.582153	−	0.813079i	$-0.697790\pi$
$647$	−29.6155			−1.16431			−0.582153	+	0.813079i	$0.697790\pi$
$648$		0			0
$649$	5.75379			0.225856
$650$	−0.561553	−	3.56155i	−0.0220259	−	0.139696i
$651$		0			0
$652$			7.12311i			0.278962i
$653$	13.8617			0.542452			0.271226	−	0.962516i	$-0.412571\pi$
$653$	13.8617			0.542452			0.271226	+	0.962516i	$0.412571\pi$
$654$		0			0
$655$			4.00000i			0.156293i
$656$		−	0.876894i		−	0.0342370i
$657$		0			0
$658$		−	19.5076i		−	0.760485i
$659$	21.7538			0.847407			0.423704	−	0.905801i	$-0.360730\pi$
$659$	21.7538			0.847407			0.423704	+	0.905801i	$0.360730\pi$
$660$		0			0
$661$			9.12311i			0.354848i	0.984135	+	0.177424i	$0.0567763\pi$
$661$			9.12311i			0.354848i	−0.984135	+	0.177424i	$0.943224\pi$
$662$	7.75379			0.301360
$663$		0			0
$664$	6.24621			0.242400
$665$		−	18.7386i		−	0.726653i
$666$		0			0
$667$	−10.2462			−0.396735
$668$		−	22.2462i		−	0.860732i
$669$		0			0
$670$			4.87689i			0.188411i
$671$		−	51.2311i		−	1.97775i
$672$		0			0
$673$	26.9848			1.04019			0.520095	−	0.854109i	$-0.325897\pi$
$673$	26.9848			1.04019			0.520095	+	0.854109i	$0.325897\pi$
$674$			6.00000i			0.231111i
$675$		0			0
$676$	−12.3693	+	4.00000i	−0.475743	+	0.153846i
$677$	−12.8769			−0.494899			−0.247450	−	0.968901i	$-0.579592\pi$
$677$	−12.8769			−0.494899			−0.247450	+	0.968901i	$0.579592\pi$
$678$		0			0
$679$	40.9848			1.57285
$680$	2.00000			0.0766965
$681$		0			0
$682$			16.0000i			0.612672i
$683$			16.9848i			0.649907i	0.945730	+	0.324954i	$0.105349\pi$
$683$			16.9848i			0.649907i	−0.945730	+	0.324954i	$0.894651\pi$
$684$		0			0
$685$	10.4924			0.400895
$686$	−13.2614			−0.506321
$687$		0			0
$688$	6.24621			0.238135
$689$	47.6155	−	7.50758i	1.81401	−	0.286016i
$690$		0			0
$691$			28.7386i			1.09327i	0.837371	+	0.546635i	$0.184091\pi$
$691$			28.7386i			1.09327i	−0.837371	+	0.546635i	$0.815909\pi$
$692$	23.1231			0.879009
$693$		0			0
$694$			14.2462i			0.540779i
$695$			16.4924i			0.625593i
$696$		0			0
$697$			1.75379i			0.0664295i
$698$	−19.3693			−0.733139
$699$		0			0
$700$			3.12311i			0.118042i
$701$	14.0000			0.528773			0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000			0.528773			0.264386	+	0.964417i	$0.414831\pi$
$702$		0			0
$703$	−30.7386			−1.15933
$704$			5.12311i			0.193084i
$705$		0			0
$706$	−12.2462			−0.460892
$707$			38.2462i			1.43840i
$708$		0			0
$709$		−	1.61553i		−	0.0606724i	−0.999540	−	0.0303362i	$-0.990342\pi$
$709$		−	1.61553i		−	0.0606724i	0.999540	−	0.0303362i	$-0.00965780\pi$
$710$		−	10.2462i		−	0.384533i
$711$		0			0
$712$	−3.12311			−0.117043
$713$			16.0000i			0.599205i
$714$		0			0
$715$	−18.2462	+	2.87689i	−0.682370	+	0.107590i
$716$	16.4924			0.616351
$717$		0			0
$718$	17.7538			0.662566
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$		0			0
$721$		−	40.9848i		−	1.52636i
$722$		−	17.0000i		−	0.632674i
$723$		0			0
$724$	20.2462			0.752445
$725$	2.00000			0.0742781
$726$		0			0
$727$	19.8617			0.736631			0.368316	−	0.929701i	$-0.379935\pi$
$727$	19.8617			0.736631			0.368316	+	0.929701i	$0.379935\pi$
$728$	11.1231	−	1.75379i	0.412250	−	0.0649997i
$729$		0			0
$730$		−	13.1231i		−	0.485708i
$731$	−12.4924			−0.462049
$732$		0			0
$733$			35.3693i			1.30640i	0.757187	+	0.653198i	$0.226573\pi$
$733$			35.3693i			1.30640i	−0.757187	+	0.653198i	$0.773427\pi$
$734$			8.63068i			0.318564i
$735$		0			0
$736$			5.12311i			0.188840i
$737$	24.9848			0.920329
$738$		0			0
$739$		−	37.2311i		−	1.36957i	−0.728747	−	0.684783i	$-0.759897\pi$
$739$		−	37.2311i		−	1.36957i	0.728747	−	0.684783i	$-0.240103\pi$
$740$	5.12311			0.188329
$741$		0			0
$742$	−41.7538			−1.53283
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	14.0000			0.512920
$746$			7.12311i			0.260795i
$747$		0			0
$748$		−	10.2462i		−	0.374639i
$749$			24.9848i			0.912926i
$750$		0			0
$751$	−26.2462			−0.957738			−0.478869	−	0.877886i	$-0.658953\pi$
$751$	−26.2462			−0.957738			−0.478869	+	0.877886i	$0.658953\pi$
$752$			6.24621i			0.227776i
$753$		0			0
$754$	−1.12311	−	7.12311i	−0.0409011	−	0.259408i
$755$	13.3693			0.486559
$756$		0			0
$757$	−3.12311			−0.113511			−0.0567556	−	0.998388i	$-0.518076\pi$
$757$	−3.12311			−0.113511			−0.0567556	+	0.998388i	$0.518076\pi$
$758$	6.00000			0.217930
$759$		0			0
$760$			6.00000i			0.217643i
$761$			3.12311i			0.113212i	0.998397	+	0.0566062i	$0.0180280\pi$
$761$			3.12311i			0.113212i	−0.998397	+	0.0566062i	$0.981972\pi$
$762$		0			0
$763$	−8.98485			−0.325273
$764$	16.4924			0.596675
$765$		0			0
$766$	1.75379			0.0633670
$767$	0.630683	+	4.00000i	0.0227726	+	0.144432i
$768$		0			0
$769$		−	32.9848i		−	1.18946i	−0.803924	−	0.594732i	$-0.797258\pi$
$769$		−	32.9848i		−	1.18946i	0.803924	−	0.594732i	$-0.202742\pi$
$770$	16.0000			0.576600
$771$		0			0
$772$			25.1231i			0.904200i
$773$			16.2462i			0.584336i	0.956367	+	0.292168i	$0.0943766\pi$
$773$			16.2462i			0.584336i	−0.956367	+	0.292168i	$0.905623\pi$
$774$		0			0
$775$		−	3.12311i		−	0.112185i
$776$	−13.1231			−0.471092
$777$		0			0
$778$			0.246211i			0.00882710i
$779$	−5.26137			−0.188508
$780$		0			0
$781$	−52.4924			−1.87833
$782$		−	10.2462i		−	0.366404i
$783$		0			0
$784$	−2.75379			−0.0983496
$785$		−	21.3693i		−	0.762704i
$786$		0			0
$787$		−	12.3845i		−	0.441459i	−0.975335	−	0.220729i	$-0.929156\pi$
$787$		−	12.3845i		−	0.441459i	0.975335	−	0.220729i	$-0.0708438\pi$
$788$			16.2462i			0.578747i
$789$		0			0
$790$	8.00000			0.284627
$791$		−	38.2462i		−	1.35988i
$792$		0			0
$793$	35.6155	−	5.61553i	1.26474	−	0.199413i
$794$	15.3693			0.545437
$795$		0			0
$796$	−18.2462			−0.646720
$797$	−25.8617			−0.916070			−0.458035	−	0.888934i	$-0.651447\pi$
$797$	−25.8617			−0.916070			−0.458035	+	0.888934i	$0.651447\pi$
$798$		0			0
$799$		−	12.4924i		−	0.441950i
$800$		−	1.00000i		−	0.0353553i
$801$		0			0
$802$	−1.36932			−0.0483523
$803$	−67.2311			−2.37253
$804$		0			0
$805$	16.0000			0.563926
$806$	−11.1231	+	1.75379i	−0.391795	+	0.0617746i
$807$		0			0
$808$		−	12.2462i		−	0.430820i
$809$	13.5076			0.474901			0.237451	−	0.971400i	$-0.423688\pi$
$809$	13.5076			0.474901			0.237451	+	0.971400i	$0.423688\pi$
$810$		0			0
$811$		−	27.7538i		−	0.974567i	−0.873244	−	0.487284i	$-0.837988\pi$
$811$		−	27.7538i		−	0.974567i	0.873244	−	0.487284i	$-0.162012\pi$
$812$			6.24621i			0.219199i
$813$		0			0
$814$		−	26.2462i		−	0.919929i
$815$	−7.12311			−0.249512
$816$		0			0
$817$		−	37.4773i		−	1.31116i
$818$	−8.49242			−0.296931
$819$		0			0
$820$	0.876894			0.0306225
$821$		−	44.2462i		−	1.54420i	−0.635499	−	0.772102i	$-0.719206\pi$
$821$		−	44.2462i		−	1.54420i	0.635499	−	0.772102i	$-0.280794\pi$
$822$		0			0
$823$	−42.8769			−1.49459			−0.747297	−	0.664490i	$-0.768649\pi$
$823$	−42.8769			−1.49459			−0.747297	+	0.664490i	$0.768649\pi$
$824$			13.1231i			0.457165i
$825$		0			0
$826$		−	3.50758i		−	0.122044i
$827$		−	1.26137i		−	0.0438620i	−0.999759	−	0.0219310i	$-0.993019\pi$
$827$		−	1.26137i		−	0.0438620i	0.999759	−	0.0219310i	$-0.00698141\pi$
$828$		0			0
$829$	−24.2462			−0.842106			−0.421053	−	0.907036i	$-0.638339\pi$
$829$	−24.2462			−0.842106			−0.421053	+	0.907036i	$0.638339\pi$
$830$			6.24621i			0.216809i
$831$		0			0
$832$	−3.56155	+	0.561553i	−0.123475	+	0.0194683i
$833$	5.50758			0.190826
$834$		0			0
$835$	22.2462			0.769862
$836$	30.7386			1.06312
$837$		0			0
$838$			28.0000i			0.967244i
$839$			46.7386i			1.61360i	0.590827	+	0.806798i	$0.298802\pi$
$839$			46.7386i			1.61360i	−0.590827	+	0.806798i	$0.701198\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$	0.630683			0.0217348
$843$		0			0
$844$	−4.00000			−0.137686
$845$	−4.00000	−	12.3693i	−0.137604	−	0.425517i
$846$		0			0
$847$		−	47.6155i		−	1.63609i
$848$	13.3693			0.459104
$849$		0			0
$850$			2.00000i			0.0685994i
$851$		−	26.2462i		−	0.899709i
$852$		0			0
$853$			35.8617i			1.22788i	0.789352	+	0.613941i	$0.210417\pi$
$853$			35.8617i			1.22788i	−0.789352	+	0.613941i	$0.789583\pi$
$854$	−31.2311			−1.06870
$855$		0			0
$856$		−	8.00000i		−	0.273434i
$857$	30.4924			1.04160			0.520801	−	0.853678i	$-0.325633\pi$
$857$	30.4924			1.04160			0.520801	+	0.853678i	$0.325633\pi$
$858$		0			0
$859$	−12.0000			−0.409435			−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000			−0.409435			−0.204717	+	0.978821i	$0.565628\pi$
$860$			6.24621i			0.212994i
$861$		0			0
$862$	−32.4924			−1.10670
$863$			6.24621i			0.212624i	0.994333	+	0.106312i	$0.0339042\pi$
$863$			6.24621i			0.212624i	−0.994333	+	0.106312i	$0.966096\pi$
$864$		0			0
$865$			23.1231i			0.786209i
$866$		−	18.0000i		−	0.611665i
$867$		0			0
$868$	9.75379			0.331065
$869$		−	40.9848i		−	1.39032i
$870$		0			0
$871$	2.73863	+	17.3693i	0.0927951	+	0.588537i
$872$	2.87689			0.0974239
$873$		0			0
$874$	30.7386			1.03975
$875$	−3.12311			−0.105580
$876$		0			0
$877$		−	18.8769i		−	0.637427i	−0.947851	−	0.318714i	$-0.896749\pi$
$877$		−	18.8769i		−	0.637427i	0.947851	−	0.318714i	$-0.103251\pi$
$878$		−	36.4924i		−	1.23156i
$879$		0			0
$880$	−5.12311			−0.172700
$881$	−28.2462			−0.951639			−0.475820	−	0.879543i	$-0.657848\pi$
$881$	−28.2462			−0.951639			−0.475820	+	0.879543i	$0.657848\pi$
$882$		0			0
$883$	−9.75379			−0.328241			−0.164121	−	0.986440i	$-0.552479\pi$
$883$	−9.75379			−0.328241			−0.164121	+	0.986440i	$0.552479\pi$
$884$	7.12311	−	1.12311i	0.239576	−	0.0377741i
$885$		0			0
$886$			3.50758i			0.117839i
$887$	18.8769			0.633824			0.316912	−	0.948455i	$-0.397354\pi$
$887$	18.8769			0.633824			0.316912	+	0.948455i	$0.397354\pi$
$888$		0			0
$889$		−	40.9848i		−	1.37459i
$890$		−	3.12311i		−	0.104687i
$891$		0			0
$892$		−	9.36932i		−	0.313708i
$893$	37.4773			1.25413
$894$		0			0
$895$			16.4924i			0.551281i
$896$	3.12311			0.104336
$897$		0			0
$898$	−28.8769			−0.963634
$899$		−	6.24621i		−	0.208323i
$900$		0			0
$901$	−26.7386			−0.890793
$902$		−	4.49242i		−	0.149581i
$903$		0			0
$904$			12.2462i			0.407303i
$905$			20.2462i			0.673007i
$906$		0			0
$907$	25.7538			0.855141			0.427570	−	0.903982i	$-0.359370\pi$
$907$	25.7538			0.855141			0.427570	+	0.903982i	$0.359370\pi$
$908$			8.00000i			0.265489i
$909$		0			0
$910$	1.75379	+	11.1231i	0.0581375	+	0.368727i
$911$	−4.00000			−0.132526			−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000			−0.132526			−0.0662630	+	0.997802i	$0.521108\pi$
$912$		0			0
$913$	32.0000			1.05905
$914$	−31.3693			−1.03760
$915$		0			0
$916$			5.12311i			0.169272i
$917$		−	12.4924i		−	0.412536i
$918$		0			0
$919$	−21.7538			−0.717591			−0.358796	−	0.933416i	$-0.616813\pi$
$919$	−21.7538			−0.717591			−0.358796	+	0.933416i	$0.616813\pi$
$920$	−5.12311			−0.168904
$921$		0			0
$922$	18.4924			0.609016
$923$	−5.75379	−	36.4924i	−0.189388	−	1.20116i
$924$		0			0
$925$			5.12311i			0.168447i
$926$	21.8617			0.718421
$927$		0			0
$928$		−	2.00000i		−	0.0656532i
$929$		−	28.1080i		−	0.922192i	−0.887350	−	0.461096i	$-0.847456\pi$
$929$		−	28.1080i		−	0.922192i	0.887350	−	0.461096i	$-0.152544\pi$
$930$		0			0
$931$			16.5227i			0.541511i
$932$	−7.75379			−0.253984
$933$		0			0
$934$			22.2462i			0.727918i
$935$	10.2462			0.335087
$936$		0			0
$937$	−20.2462			−0.661415			−0.330707	−	0.943733i	$-0.607287\pi$
$937$	−20.2462			−0.661415			−0.330707	+	0.943733i	$0.607287\pi$
$938$		−	15.2311i		−	0.497312i
$939$		0			0
$940$	−6.24621			−0.203729
$941$			14.0000i			0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$			14.0000i			0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$		0			0
$943$		−	4.49242i		−	0.146293i
$944$			1.12311i			0.0365540i
$945$		0			0
$946$	32.0000			1.04041
$947$		−	8.49242i		−	0.275967i	−0.990435	−	0.137983i	$-0.955938\pi$
$947$		−	8.49242i		−	0.275967i	0.990435	−	0.137983i	$-0.0440620\pi$
$948$		0			0
$949$	−7.36932	−	46.7386i	−0.239218	−	1.51720i
$950$	−6.00000			−0.194666
$951$		0			0
$952$	−6.24621			−0.202441
$953$	−22.9848			−0.744552			−0.372276	−	0.928122i	$-0.621423\pi$
$953$	−22.9848			−0.744552			−0.372276	+	0.928122i	$0.621423\pi$
$954$		0			0
$955$			16.4924i			0.533682i
$956$			4.49242i			0.145295i
$957$		0			0
$958$	−12.4924			−0.403612
$959$	−32.7689			−1.05816
$960$		0			0
$961$	21.2462			0.685362
$962$	18.2462	−	2.87689i	0.588281	−	0.0927548i
$963$		0			0
$964$			14.2462i			0.458840i
$965$	−25.1231			−0.808741
$966$		0			0
$967$		−	47.6155i		−	1.53121i	−0.643310	−	0.765606i	$-0.722439\pi$
$967$		−	47.6155i		−	1.53121i	0.643310	−	0.765606i	$-0.277561\pi$
$968$			15.2462i			0.490032i
$969$		0			0
$970$		−	13.1231i		−	0.421358i
$971$	−37.7538			−1.21158			−0.605788	−	0.795626i	$-0.707142\pi$
$971$	−37.7538			−1.21158			−0.605788	+	0.795626i	$0.707142\pi$
$972$		0			0
$973$		−	51.5076i		−	1.65126i
$974$	−17.3693			−0.556549
$975$		0			0
$976$	10.0000			0.320092
$977$			4.73863i			0.151602i	0.997123	+	0.0758012i	$0.0241514\pi$
$977$			4.73863i			0.151602i	−0.997123	+	0.0758012i	$0.975849\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$		−	2.75379i		−	0.0879666i
$981$		0			0
$982$		−	38.7386i		−	1.23620i
$983$			60.4924i			1.92941i	0.263335	+	0.964704i	$0.415177\pi$
$983$			60.4924i			1.92941i	−0.263335	+	0.964704i	$0.584823\pi$
$984$		0			0
$985$	−16.2462			−0.517647
$986$			4.00000i			0.127386i
$987$		0			0
$988$	3.36932	+	21.3693i	0.107192	+	0.679849i
$989$	32.0000			1.01754
$990$		0			0
$991$	−30.7386			−0.976445			−0.488222	−	0.872719i	$-0.662355\pi$
$991$	−30.7386			−0.976445			−0.488222	+	0.872719i	$0.662355\pi$
$992$	−3.12311			−0.0991587
$993$		0			0
$994$			32.0000i			1.01498i
$995$		−	18.2462i		−	0.578444i
$996$		0			0
$997$	38.3542			1.21469			0.607344	−	0.794439i	$-0.292235\pi$
$997$	38.3542			1.21469			0.607344	+	0.794439i	$0.292235\pi$
$998$	−34.4924			−1.09184
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.b.f.181.4		4
3.2	odd	2		390.2.b.d.181.2	✓	4
12.11	even	2		3120.2.g.o.961.3		4
13.12	even	2	inner	1170.2.b.f.181.1		4
15.2	even	4		1950.2.f.o.649.1		4
15.8	even	4		1950.2.f.l.649.4		4
15.14	odd	2		1950.2.b.h.1351.3		4
39.5	even	4		5070.2.a.bd.1.1		2
39.8	even	4		5070.2.a.bh.1.2		2
39.38	odd	2		390.2.b.d.181.3	yes	4
156.155	even	2		3120.2.g.o.961.2		4
195.38	even	4		1950.2.f.o.649.3		4
195.77	even	4		1950.2.f.l.649.2		4
195.194	odd	2		1950.2.b.h.1351.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.b.d.181.2	✓	4	3.2	odd	2
390.2.b.d.181.3	yes	4	39.38	odd	2
1170.2.b.f.181.1		4	13.12	even	2	inner
1170.2.b.f.181.4		4	1.1	even	1	trivial
1950.2.b.h.1351.2		4	195.194	odd	2
1950.2.b.h.1351.3		4	15.14	odd	2
1950.2.f.l.649.2		4	195.77	even	4
1950.2.f.l.649.4		4	15.8	even	4
1950.2.f.o.649.1		4	15.2	even	4
1950.2.f.o.649.3		4	195.38	even	4
3120.2.g.o.961.2		4	156.155	even	2
3120.2.g.o.961.3		4	12.11	even	2
5070.2.a.bd.1.1		2	39.5	even	4
5070.2.a.bh.1.2		2	39.8	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} +3.12311i q^{7} -1.00000i q^{8} +1.00000 q^{10} -5.12311i q^{11} +(3.56155 - 0.561553i) q^{13} -3.12311 q^{14} +1.00000 q^{16} -2.00000 q^{17} -6.00000i q^{19} +1.00000i q^{20} +5.12311 q^{22} +5.12311 q^{23} -1.00000 q^{25} +(0.561553 + 3.56155i) q^{26} -3.12311i q^{28} -2.00000 q^{29} +3.12311i q^{31} +1.00000i q^{32} -2.00000i q^{34} +3.12311 q^{35} -5.12311i q^{37} +6.00000 q^{38} -1.00000 q^{40} -0.876894i q^{41} +6.24621 q^{43} +5.12311i q^{44} +5.12311i q^{46} +6.24621i q^{47} -2.75379 q^{49} -1.00000i q^{50} +(-3.56155 + 0.561553i) q^{52} +13.3693 q^{53} -5.12311 q^{55} +3.12311 q^{56} -2.00000i q^{58} +1.12311i q^{59} +10.0000 q^{61} -3.12311 q^{62} -1.00000 q^{64} +(-0.561553 - 3.56155i) q^{65} +4.87689i q^{67} +2.00000 q^{68} +3.12311i q^{70} -10.2462i q^{71} -13.1231i q^{73} +5.12311 q^{74} +6.00000i q^{76} +16.0000 q^{77} +8.00000 q^{79} -1.00000i q^{80} +0.876894 q^{82} +6.24621i q^{83} +2.00000i q^{85} +6.24621i q^{86} -5.12311 q^{88} -3.12311i q^{89} +(1.75379 + 11.1231i) q^{91} -5.12311 q^{92} -6.24621 q^{94} -6.00000 q^{95} -13.1231i q^{97} -2.75379i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{10} + 6 q^{13} + 4 q^{14} + 4 q^{16} - 8 q^{17} + 4 q^{22} + 4 q^{23} - 4 q^{25} - 6 q^{26} - 8 q^{29} - 4 q^{35} + 24 q^{38} - 4 q^{40} - 8 q^{43} - 44 q^{49} - 6 q^{52} + 4 q^{53}+ \cdots - 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^10 + 6 * q^13 + 4 * q^14 + 4 * q^16 - 8 * q^17 + 4 * q^22 + 4 * q^23 - 4 * q^25 - 6 * q^26 - 8 * q^29 - 4 * q^35 + 24 * q^38 - 4 * q^40 - 8 * q^43 - 44 * q^49 - 6 * q^52 + 4 * q^53 - 4 * q^55 - 4 * q^56 + 40 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 + 8 * q^68 + 4 * q^74 + 64 * q^77 + 32 * q^79 + 20 * q^82 - 4 * q^88 + 40 * q^91 - 4 * q^92 + 8 * q^94 - 24 * q^95

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