LMFDB - Embedded newform 1170.2.e.f.469.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1170,2,Mod(469,1170)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1170.469"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6,0,0,0,0,0,-4,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.3534400.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 $ x^6 - 2x^5 - 3x^4 + 16x^3 + x^2 - 12x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.6
Root			$0.627553 - 1.14620i$ of defining polynomial
Character	$\chi$	$=$	1170.469
Dual form			1170.2.e.f.469.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.14620 - 1.91995i) q^{5} +1.25511i q^{7} -1.00000i q^{8} +(1.91995 + 1.14620i) q^{10} -2.00000 q^{11} +1.00000i q^{13} -1.25511 q^{14} +1.00000 q^{16} +4.80261i q^{17} +5.09501 q^{19} +(-1.14620 + 1.91995i) q^{20} -2.00000i q^{22} +2.58480i q^{23} +(-2.37245 - 4.40131i) q^{25} -1.00000 q^{26} -1.25511i q^{28} +5.09501 q^{29} +8.58480 q^{31} +1.00000i q^{32} -4.80261 q^{34} +(2.40974 + 1.43860i) q^{35} +7.83991i q^{37} +5.09501i q^{38} +(-1.91995 - 1.14620i) q^{40} +9.67982 q^{41} -10.8772i q^{43} +2.00000 q^{44} -2.58480 q^{46} -2.74489i q^{47} +5.42471 q^{49} +(4.40131 - 2.37245i) q^{50} -1.00000i q^{52} +2.58480i q^{53} +(-2.29240 + 3.83991i) q^{55} +1.25511 q^{56} +5.09501i q^{58} -5.09501 q^{59} +13.6798 q^{61} +8.58480i q^{62} -1.00000 q^{64} +(1.91995 + 1.14620i) q^{65} +8.58480i q^{67} -4.80261i q^{68} +(-1.43860 + 2.40974i) q^{70} -5.38741 q^{71} -6.00000i q^{73} -7.83991 q^{74} -5.09501 q^{76} -2.51021i q^{77} -15.0950 q^{79} +(1.14620 - 1.91995i) q^{80} +9.67982i q^{82} +11.0950i q^{83} +(9.22079 + 5.50476i) q^{85} +10.8772 q^{86} +2.00000i q^{88} +5.09501 q^{89} -1.25511 q^{91} -2.58480i q^{92} +2.74489 q^{94} +(5.83991 - 9.78219i) q^{95} -6.26462i q^{97} +5.42471i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 4 q^{10} - 12 q^{11} - 4 q^{14} + 6 q^{16} - 4 q^{19} - 16 q^{25} - 6 q^{26} - 4 q^{29} + 24 q^{31} - 8 q^{34} + 6 q^{35} + 4 q^{40} - 4 q^{41} + 12 q^{44} + 12 q^{46} - 26 q^{49} + 16 q^{50}+ \cdots + 4 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^10 - 12 * q^11 - 4 * q^14 + 6 * q^16 - 4 * q^19 - 16 * q^25 - 6 * q^26 - 4 * q^29 + 24 * q^31 - 8 * q^34 + 6 * q^35 + 4 * q^40 - 4 * q^41 + 12 * q^44 + 12 * q^46 - 26 * q^49 + 16 * q^50 + 4 * q^56 + 4 * q^59 + 20 * q^61 - 6 * q^64 - 4 * q^65 + 12 * q^70 + 16 * q^71 - 16 * q^74 + 4 * q^76 - 56 * q^79 + 28 * q^85 + 24 * q^86 - 4 * q^89 - 4 * q^91 + 20 * q^94 + 4 * 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
$5$	1.14620	−	1.91995i	0.512597	−	0.858630i
$5$	1.14620	−	1.91995i	0.512597	−	0.858630i
$6$		0			0
$7$			1.25511i			0.474385i	0.971463	+	0.237193i	$0.0762272\pi$
$7$			1.25511i			0.474385i	−0.971463	+	0.237193i	$0.923773\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	1.91995	+	1.14620i	0.607143	+	0.362461i
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$	−1.25511			−0.335441
$15$		0			0
$16$	1.00000			0.250000
$17$			4.80261i			1.16480i	0.812901	+	0.582402i	$0.197887\pi$
$17$			4.80261i			1.16480i	−0.812901	+	0.582402i	$0.802113\pi$
$18$		0			0
$19$	5.09501			1.16888			0.584438	−	0.811438i	$-0.301315\pi$
$19$	5.09501			1.16888			0.584438	+	0.811438i	$0.301315\pi$
$20$	−1.14620	+	1.91995i	−0.256298	+	0.429315i
$21$		0			0
$22$		−	2.00000i		−	0.426401i
$23$			2.58480i			0.538969i	0.963005	+	0.269484i	$0.0868533\pi$
$23$			2.58480i			0.538969i	−0.963005	+	0.269484i	$0.913147\pi$
$24$		0			0
$25$	−2.37245	−	4.40131i	−0.474489	−	0.880261i
$26$	−1.00000			−0.196116
$27$		0			0
$28$		−	1.25511i		−	0.237193i
$29$	5.09501			0.946120			0.473060	−	0.881030i	$-0.343149\pi$
$29$	5.09501			0.946120			0.473060	+	0.881030i	$0.343149\pi$
$30$		0			0
$31$	8.58480			1.54188			0.770938	−	0.636910i	$-0.219788\pi$
$31$	8.58480			1.54188			0.770938	+	0.636910i	$0.219788\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−4.80261			−0.823641
$35$	2.40974	+	1.43860i	0.407321	+	0.243168i
$36$		0			0
$37$			7.83991i			1.28887i	0.764658	+	0.644436i	$0.222908\pi$
$37$			7.83991i			1.28887i	−0.764658	+	0.644436i	$0.777092\pi$
$38$			5.09501i			0.826520i
$39$		0			0
$40$	−1.91995	−	1.14620i	−0.303571	−	0.181230i
$41$	9.67982			1.51173			0.755867	−	0.654726i	$-0.227216\pi$
$41$	9.67982			1.51173			0.755867	+	0.654726i	$0.227216\pi$
$42$		0			0
$43$		−	10.8772i		−	1.65876i	−0.558686	−	0.829379i	$-0.688694\pi$
$43$		−	10.8772i		−	1.65876i	0.558686	−	0.829379i	$-0.311306\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$	−2.58480			−0.381108
$47$		−	2.74489i		−	0.400384i	−0.979757	−	0.200192i	$-0.935843\pi$
$47$		−	2.74489i		−	0.400384i	0.979757	−	0.200192i	$-0.0641566\pi$
$48$		0			0
$49$	5.42471			0.774959
$50$	4.40131	−	2.37245i	0.622439	−	0.335515i
$51$		0			0
$52$		−	1.00000i		−	0.138675i
$53$			2.58480i			0.355050i	0.984116	+	0.177525i	$0.0568091\pi$
$53$			2.58480i			0.355050i	−0.984116	+	0.177525i	$0.943191\pi$
$54$		0			0
$55$	−2.29240	+	3.83991i	−0.309107	+	0.517773i
$56$	1.25511			0.167720
$57$		0			0
$58$			5.09501i			0.669008i
$59$	−5.09501			−0.663314			−0.331657	−	0.943400i	$-0.607608\pi$
$59$	−5.09501			−0.663314			−0.331657	+	0.943400i	$0.607608\pi$
$60$		0			0
$61$	13.6798			1.75152			0.875761	−	0.482746i	$-0.160361\pi$
$61$	13.6798			1.75152			0.875761	+	0.482746i	$0.160361\pi$
$62$			8.58480i			1.09027i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.91995	+	1.14620i	0.238141	+	0.142169i
$66$		0			0
$67$			8.58480i			1.04880i	0.851472	+	0.524400i	$0.175710\pi$
$67$			8.58480i			1.04880i	−0.851472	+	0.524400i	$0.824290\pi$
$68$		−	4.80261i		−	0.582402i
$69$		0			0
$70$	−1.43860	+	2.40974i	−0.171946	+	0.288020i
$71$	−5.38741			−0.639369			−0.319684	−	0.947524i	$-0.603577\pi$
$71$	−5.38741			−0.639369			−0.319684	+	0.947524i	$0.603577\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−7.83991			−0.911371
$75$		0			0
$76$	−5.09501			−0.584438
$77$		−	2.51021i		−	0.286065i
$78$		0			0
$79$	−15.0950			−1.69832			−0.849161	−	0.528134i	$-0.822892\pi$
$79$	−15.0950			−1.69832			−0.849161	+	0.528134i	$0.822892\pi$
$80$	1.14620	−	1.91995i	0.128149	−	0.214657i
$81$		0			0
$82$			9.67982i			1.06896i
$83$			11.0950i			1.21784i	0.793233	+	0.608918i	$0.208396\pi$
$83$			11.0950i			1.21784i	−0.793233	+	0.608918i	$0.791604\pi$
$84$		0			0
$85$	9.22079	+	5.50476i	1.00014	+	0.597075i
$86$	10.8772			1.17292
$87$		0			0
$88$			2.00000i			0.213201i
$89$	5.09501			0.540070			0.270035	−	0.962850i	$-0.412965\pi$
$89$	5.09501			0.540070			0.270035	+	0.962850i	$0.412965\pi$
$90$		0			0
$91$	−1.25511			−0.131571
$92$		−	2.58480i		−	0.269484i
$93$		0			0
$94$	2.74489			0.283114
$95$	5.83991	−	9.78219i	0.599162	−	1.00363i
$96$		0			0
$97$		−	6.26462i		−	0.636076i	−0.948078	−	0.318038i	$-0.896976\pi$
$97$		−	6.26462i		−	0.636076i	0.948078	−	0.318038i	$-0.103024\pi$
$98$			5.42471i			0.547979i
$99$		0			0
$100$	2.37245	+	4.40131i	0.237245	+	0.440131i
$101$	12.6594			1.25966			0.629829	−	0.776734i	$-0.283125\pi$
$101$	12.6594			1.25966			0.629829	+	0.776734i	$0.283125\pi$
$102$		0			0
$103$			7.60522i			0.749365i	0.927153	+	0.374682i	$0.122248\pi$
$103$			7.60522i			0.749365i	−0.927153	+	0.374682i	$0.877752\pi$
$104$	1.00000			0.0980581
$105$		0			0
$106$	−2.58480			−0.251058
$107$			4.58480i			0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$107$			4.58480i			0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$108$		0			0
$109$	−14.8772			−1.42498			−0.712489	−	0.701683i	$-0.752432\pi$
$109$	−14.8772			−1.42498			−0.712489	+	0.701683i	$0.752432\pi$
$110$	−3.83991	−	2.29240i	−0.366121	−	0.218572i
$111$		0			0
$112$			1.25511i			0.118596i
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$	4.96270	+	2.96270i	0.462774	+	0.276274i
$116$	−5.09501			−0.473060
$117$		0			0
$118$		−	5.09501i		−	0.469034i
$119$	−6.02778			−0.552566
$120$		0			0
$121$	−7.00000			−0.636364
$122$			13.6798i			1.23851i
$123$		0			0
$124$	−8.58480			−0.770938
$125$	−11.1696	−	0.489790i	−0.999040	−	0.0438081i
$126$		0			0
$127$		−	1.41520i		−	0.125578i	−0.998027	−	0.0627892i	$-0.980000\pi$
$127$		−	1.41520i		−	0.125578i	0.998027	−	0.0627892i	$-0.0199996\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−1.14620	+	1.91995i	−0.100528	+	0.168391i
$131$	−11.0095			−0.961906			−0.480953	−	0.876746i	$-0.659709\pi$
$131$	−11.0095			−0.961906			−0.480953	+	0.876746i	$0.659709\pi$
$132$		0			0
$133$			6.39478i			0.554497i
$134$	−8.58480			−0.741614
$135$		0			0
$136$	4.80261			0.411821
$137$			14.5848i			1.24606i	0.782196	+	0.623032i	$0.214099\pi$
$137$			14.5848i			1.24606i	−0.782196	+	0.623032i	$0.785901\pi$
$138$		0			0
$139$	−7.32970			−0.621697			−0.310848	−	0.950459i	$-0.600613\pi$
$139$	−7.32970			−0.621697			−0.310848	+	0.950459i	$0.600613\pi$
$140$	−2.40974	−	1.43860i	−0.203661	−	0.121584i
$141$		0			0
$142$		−	5.38741i		−	0.452102i
$143$		−	2.00000i		−	0.167248i
$144$		0			0
$145$	5.83991	−	9.78219i	0.484978	−	0.812367i
$146$	6.00000			0.496564
$147$		0			0
$148$		−	7.83991i		−	0.644436i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−14.5570			−1.18463			−0.592317	−	0.805705i	$-0.701787\pi$
$151$	−14.5570			−1.18463			−0.592317	+	0.805705i	$0.701787\pi$
$152$		−	5.09501i		−	0.413260i
$153$		0			0
$154$	2.51021			0.202278
$155$	9.83991	−	16.4824i	0.790360	−	1.32390i
$156$		0			0
$157$		−	13.0950i		−	1.04510i	−0.852610	−	0.522548i	$-0.824982\pi$
$157$		−	13.0950i		−	1.04510i	0.852610	−	0.522548i	$-0.175018\pi$
$158$		−	15.0950i		−	1.20089i
$159$		0			0
$160$	1.91995	+	1.14620i	0.151786	+	0.0906151i
$161$	−3.24420			−0.255679
$162$		0			0
$163$			12.2646i			0.960639i	0.877094	+	0.480320i	$0.159479\pi$
$163$			12.2646i			0.960639i	−0.877094	+	0.480320i	$0.840521\pi$
$164$	−9.67982			−0.755867
$165$		0			0
$166$	−11.0950			−0.861140
$167$		−	18.3392i		−	1.41913i	−0.704640	−	0.709565i	$-0.748891\pi$
$167$		−	18.3392i		−	1.41913i	0.704640	−	0.709565i	$-0.251109\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$	−5.50476	+	9.22079i	−0.422196	+	0.707203i
$171$		0			0
$172$			10.8772i			0.829379i
$173$			9.41520i			0.715824i	0.933755	+	0.357912i	$0.116511\pi$
$173$			9.41520i			0.715824i	−0.933755	+	0.357912i	$0.883489\pi$
$174$		0			0
$175$	5.52410	−	2.97767i	0.417583	−	0.225091i
$176$	−2.00000			−0.150756
$177$		0			0
$178$			5.09501i			0.381887i
$179$	−7.32970			−0.547847			−0.273924	−	0.961751i	$-0.588322\pi$
$179$	−7.32970			−0.547847			−0.273924	+	0.961751i	$0.588322\pi$
$180$		0			0
$181$	11.7544			0.873698			0.436849	−	0.899535i	$-0.356094\pi$
$181$	11.7544			0.873698			0.436849	+	0.899535i	$0.356094\pi$
$182$		−	1.25511i		−	0.0930346i
$183$		0			0
$184$	2.58480			0.190554
$185$	15.0523	+	8.98611i	1.10666	+	0.660672i
$186$		0			0
$187$		−	9.60522i		−	0.702403i
$188$			2.74489i			0.200192i
$189$		0			0
$190$	9.78219	+	5.83991i	0.709675	+	0.423671i
$191$	1.60522			0.116150			0.0580749	−	0.998312i	$-0.481504\pi$
$191$	1.60522			0.116150			0.0580749	+	0.998312i	$0.481504\pi$
$192$		0			0
$193$		−	15.7544i		−	1.13403i	−0.823708	−	0.567014i	$-0.808099\pi$
$193$		−	15.7544i		−	1.13403i	0.823708	−	0.567014i	$-0.191901\pi$
$194$	6.26462			0.449773
$195$		0			0
$196$	−5.42471			−0.387479
$197$		−	1.00951i		−	0.0719249i	−0.999353	−	0.0359625i	$-0.988550\pi$
$197$		−	1.00951i		−	0.0719249i	0.999353	−	0.0359625i	$-0.0114497\pi$
$198$		0			0
$199$	−0.435617			−0.0308801			−0.0154400	−	0.999881i	$-0.504915\pi$
$199$	−0.435617			−0.0308801			−0.0154400	+	0.999881i	$0.504915\pi$
$200$	−4.40131	+	2.37245i	−0.311219	+	0.167757i
$201$		0			0
$202$			12.6594i			0.890712i
$203$			6.39478i			0.448825i
$204$		0			0
$205$	11.0950	−	18.5848i	0.774909	−	1.29802i
$206$	−7.60522			−0.529881
$207$		0			0
$208$			1.00000i			0.0693375i
$209$	−10.1900			−0.704859
$210$		0			0
$211$	16.3501			1.12559			0.562794	−	0.826597i	$-0.309726\pi$
$211$	16.3501			1.12559			0.562794	+	0.826597i	$0.309726\pi$
$212$		−	2.58480i		−	0.177525i
$213$		0			0
$214$	−4.58480			−0.313411
$215$	−20.8837	−	12.4675i	−1.42426	−	0.850274i
$216$		0			0
$217$			10.7748i			0.731443i
$218$		−	14.8772i		−	1.00761i
$219$		0			0
$220$	2.29240	−	3.83991i	0.154554	−	0.258887i
$221$	−4.80261			−0.323059
$222$		0			0
$223$		−	21.7843i		−	1.45879i	−0.684094	−	0.729394i	$-0.739802\pi$
$223$		−	21.7843i		−	1.45879i	0.684094	−	0.729394i	$-0.260198\pi$
$224$	−1.25511			−0.0838602
$225$		0			0
$226$	4.00000			0.266076
$227$		−	9.02042i		−	0.598706i	−0.954142	−	0.299353i	$-0.903229\pi$
$227$		−	9.02042i		−	0.598706i	0.954142	−	0.299353i	$-0.0967709\pi$
$228$		0			0
$229$	−4.68718			−0.309737			−0.154869	−	0.987935i	$-0.549495\pi$
$229$	−4.68718			−0.309737			−0.154869	+	0.987935i	$0.549495\pi$
$230$	−2.96270	+	4.96270i	−0.195355	+	0.327231i
$231$		0			0
$232$		−	5.09501i		−	0.334504i
$233$		−	13.5366i		−	0.886812i	−0.896321	−	0.443406i	$-0.853770\pi$
$233$		−	13.5366i		−	0.886812i	0.896321	−	0.443406i	$-0.146230\pi$
$234$		0			0
$235$	−5.27007	−	3.14620i	−0.343782	−	0.205236i
$236$	5.09501			0.331657
$237$		0			0
$238$		−	6.02778i		−	0.390723i
$239$	3.19739			0.206822			0.103411	−	0.994639i	$-0.467024\pi$
$239$	3.19739			0.206822			0.103411	+	0.994639i	$0.467024\pi$
$240$		0			0
$241$	14.1154			0.909255			0.454627	−	0.890682i	$-0.349772\pi$
$241$	14.1154			0.909255			0.454627	+	0.890682i	$0.349772\pi$
$242$		−	7.00000i		−	0.449977i
$243$		0			0
$244$	−13.6798			−0.875761
$245$	6.21781	−	10.4152i	0.397241	−	0.665403i
$246$		0			0
$247$			5.09501i			0.324188i
$248$		−	8.58480i		−	0.545136i
$249$		0			0
$250$	0.489790	−	11.1696i	0.0309770	−	0.706428i
$251$	21.1696			1.33621			0.668107	−	0.744065i	$-0.267105\pi$
$251$	21.1696			1.33621			0.668107	+	0.744065i	$0.267105\pi$
$252$		0			0
$253$		−	5.16961i		−	0.325010i
$254$	1.41520			0.0887973
$255$		0			0
$256$	1.00000			0.0625000
$257$			8.21781i			0.512613i	0.966596	+	0.256306i	$0.0825056\pi$
$257$			8.21781i			0.512613i	−0.966596	+	0.256306i	$0.917494\pi$
$258$		0			0
$259$	−9.83991			−0.611422
$260$	−1.91995	−	1.14620i	−0.119070	−	0.0710844i
$261$		0			0
$262$		−	11.0095i		−	0.680170i
$263$			4.51021i			0.278111i	0.990285	+	0.139056i	$0.0444067\pi$
$263$			4.51021i			0.278111i	−0.990285	+	0.139056i	$0.955593\pi$
$264$		0			0
$265$	4.96270	+	2.96270i	0.304856	+	0.181997i
$266$	−6.39478			−0.392089
$267$		0			0
$268$		−	8.58480i		−	0.524400i
$269$	−8.51021			−0.518877			−0.259438	−	0.965760i	$-0.583537\pi$
$269$	−8.51021			−0.518877			−0.259438	+	0.965760i	$0.583537\pi$
$270$		0			0
$271$	4.95180			0.300800			0.150400	−	0.988625i	$-0.451944\pi$
$271$	4.95180			0.300800			0.150400	+	0.988625i	$0.451944\pi$
$272$			4.80261i			0.291201i
$273$		0			0
$274$	−14.5848			−0.881100
$275$	4.74489	+	8.80261i	0.286128	+	0.530817i
$276$		0			0
$277$		−	19.8698i		−	1.19386i	−0.802292	−	0.596932i	$-0.796386\pi$
$277$		−	19.8698i		−	1.19386i	0.802292	−	0.596932i	$-0.203614\pi$
$278$		−	7.32970i		−	0.439606i
$279$		0			0
$280$	1.43860	−	2.40974i	0.0859729	−	0.144010i
$281$	−30.2646			−1.80544			−0.902718	−	0.430233i	$-0.858431\pi$
$281$	−30.2646			−1.80544			−0.902718	+	0.430233i	$0.858431\pi$
$282$		0			0
$283$			26.9240i			1.60047i	0.599689	+	0.800233i	$0.295291\pi$
$283$			26.9240i			1.60047i	−0.599689	+	0.800233i	$0.704709\pi$
$284$	5.38741			0.319684
$285$		0			0
$286$	2.00000			0.118262
$287$			12.1492i			0.717144i
$288$		0			0
$289$	−6.06508			−0.356769
$290$	9.78219	+	5.83991i	0.574430	+	0.342931i
$291$		0			0
$292$			6.00000i			0.351123i
$293$		−	5.18051i		−	0.302649i	−0.988484	−	0.151324i	$-0.951646\pi$
$293$		−	5.18051i		−	0.302649i	0.988484	−	0.151324i	$-0.0483538\pi$
$294$		0			0
$295$	−5.83991	+	9.78219i	−0.340013	+	0.569541i
$296$	7.83991			0.455685
$297$		0			0
$298$		0			0
$299$	−2.58480			−0.149483
$300$		0			0
$301$	13.6520			0.786890
$302$		−	14.5570i		−	0.837662i
$303$		0			0
$304$	5.09501			0.292219
$305$	15.6798	−	26.2646i	0.897824	−	1.50391i
$306$		0			0
$307$			0.320184i			0.0182738i	0.999958	+	0.00913692i	$0.00290841\pi$
$307$			0.320184i			0.0182738i	−0.999958	+	0.00913692i	$0.997092\pi$
$308$			2.51021i			0.143032i
$309$		0			0
$310$	16.4824	+	9.83991i	0.936139	+	0.558869i
$311$	9.86984			0.559667			0.279834	−	0.960048i	$-0.409721\pi$
$311$	9.86984			0.559667			0.279834	+	0.960048i	$0.409721\pi$
$312$		0			0
$313$			10.6126i			0.599859i	0.953961	+	0.299929i	$0.0969631\pi$
$313$			10.6126i			0.599859i	−0.953961	+	0.299929i	$0.903037\pi$
$314$	13.0950			0.738994
$315$		0			0
$316$	15.0950			0.849161
$317$		−	12.1900i		−	0.684660i	−0.939580	−	0.342330i	$-0.888784\pi$
$317$		−	12.1900i		−	0.684660i	0.939580	−	0.342330i	$-0.111216\pi$
$318$		0			0
$319$	−10.1900			−0.570532
$320$	−1.14620	+	1.91995i	−0.0640746	+	0.107329i
$321$		0			0
$322$		−	3.24420i		−	0.180792i
$323$			24.4694i			1.36151i
$324$		0			0
$325$	4.40131	−	2.37245i	0.244141	−	0.131600i
$326$	−12.2646			−0.679274
$327$		0			0
$328$		−	9.67982i		−	0.534478i
$329$	3.44513			0.189936
$330$		0			0
$331$	−17.9444			−0.986315			−0.493158	−	0.869940i	$-0.664157\pi$
$331$	−17.9444			−0.986315			−0.493158	+	0.869940i	$0.664157\pi$
$332$		−	11.0950i		−	0.608918i
$333$		0			0
$334$	18.3392			1.00348
$335$	16.4824	+	9.83991i	0.900531	+	0.537612i
$336$		0			0
$337$		−	20.9518i		−	1.14132i	−0.821187	−	0.570659i	$-0.806688\pi$
$337$		−	20.9518i		−	1.14132i	0.821187	−	0.570659i	$-0.193312\pi$
$338$		−	1.00000i		−	0.0543928i
$339$		0			0
$340$	−9.22079	−	5.50476i	−0.500068	−	0.298537i
$341$	−17.1696			−0.929786
$342$		0			0
$343$			15.5943i			0.842014i
$344$	−10.8772			−0.586460
$345$		0			0
$346$	−9.41520			−0.506164
$347$		−	3.51757i		−	0.188833i	−0.995533	−	0.0944166i	$-0.969901\pi$
$347$		−	3.51757i		−	0.188833i	0.995533	−	0.0944166i	$-0.0300986\pi$
$348$		0			0
$349$	−17.8568			−0.955852			−0.477926	−	0.878400i	$-0.658611\pi$
$349$	−17.8568			−0.955852			−0.477926	+	0.878400i	$0.658611\pi$
$350$	2.97767	+	5.52410i	0.159163	+	0.295276i
$351$		0			0
$352$		−	2.00000i		−	0.106600i
$353$		−	36.9240i		−	1.96527i	−0.185557	−	0.982634i	$-0.559409\pi$
$353$		−	36.9240i		−	1.96527i	0.185557	−	0.982634i	$-0.440591\pi$
$354$		0			0
$355$	−6.17506	+	10.3436i	−0.327738	+	0.548981i
$356$	−5.09501			−0.270035
$357$		0			0
$358$		−	7.32970i		−	0.387387i
$359$	−6.77483			−0.357562			−0.178781	−	0.983889i	$-0.557215\pi$
$359$	−6.77483			−0.357562			−0.178781	+	0.983889i	$0.557215\pi$
$360$		0			0
$361$	6.95916			0.366272
$362$			11.7544i			0.617798i
$363$		0			0
$364$	1.25511			0.0657854
$365$	−11.5197	−	6.87720i	−0.602970	−	0.359969i
$366$		0			0
$367$		−	7.80997i		−	0.407677i	−0.979004	−	0.203839i	$-0.934658\pi$
$367$		−	7.80997i		−	0.407677i	0.979004	−	0.203839i	$-0.0653418\pi$
$368$			2.58480i			0.134742i
$369$		0			0
$370$	−8.98611	+	15.0523i	−0.467166	+	0.782530i
$371$	−3.24420			−0.168430
$372$		0			0
$373$			15.8698i			0.821709i	0.911701	+	0.410855i	$0.134770\pi$
$373$			15.8698i			0.821709i	−0.911701	+	0.410855i	$0.865230\pi$
$374$	9.60522			0.496674
$375$		0			0
$376$	−2.74489			−0.141557
$377$			5.09501i			0.262407i
$378$		0			0
$379$	16.7748			0.861665			0.430833	−	0.902432i	$-0.358220\pi$
$379$	16.7748			0.861665			0.430833	+	0.902432i	$0.358220\pi$
$380$	−5.83991	+	9.78219i	−0.299581	+	0.501816i
$381$		0			0
$382$			1.60522i			0.0821304i
$383$		−	16.6147i		−	0.848973i	−0.905434	−	0.424487i	$-0.860455\pi$
$383$		−	16.6147i		−	0.848973i	0.905434	−	0.424487i	$-0.139545\pi$
$384$		0			0
$385$	−4.81949	−	2.87720i	−0.245624	−	0.146636i
$386$	15.7544			0.801878
$387$		0			0
$388$			6.26462i			0.318038i
$389$	36.7193			1.86174			0.930870	−	0.365350i	$-0.119051\pi$
$389$	36.7193			1.86174			0.930870	+	0.365350i	$0.119051\pi$
$390$		0			0
$391$	−12.4138			−0.627793
$392$		−	5.42471i		−	0.273989i
$393$		0			0
$394$	1.00951			0.0508586
$395$	−17.3019	+	28.9817i	−0.870554	+	1.45823i
$396$		0			0
$397$			29.2104i			1.46603i	0.680212	+	0.733015i	$0.261888\pi$
$397$			29.2104i			1.46603i	−0.680212	+	0.733015i	$0.738112\pi$
$398$		−	0.435617i		−	0.0218355i
$399$		0			0
$400$	−2.37245	−	4.40131i	−0.118622	−	0.220065i
$401$	−15.6052			−0.779288			−0.389644	−	0.920966i	$-0.627402\pi$
$401$	−15.6052			−0.779288			−0.389644	+	0.920966i	$0.627402\pi$
$402$		0			0
$403$			8.58480i			0.427640i
$404$	−12.6594			−0.629829
$405$		0			0
$406$	−6.39478			−0.317367
$407$		−	15.6798i		−	0.777220i
$408$		0			0
$409$	−13.7952			−0.682131			−0.341066	−	0.940039i	$-0.610788\pi$
$409$	−13.7952			−0.682131			−0.341066	+	0.940039i	$0.610788\pi$
$410$	18.5848	+	11.0950i	0.917838	+	0.547944i
$411$		0			0
$412$		−	7.60522i		−	0.374682i
$413$		−	6.39478i		−	0.314666i
$414$		0			0
$415$	21.3019	+	12.7171i	1.04567	+	0.624259i
$416$	−1.00000			−0.0490290
$417$		0			0
$418$		−	10.1900i		−	0.498410i
$419$	−30.6893			−1.49927			−0.749636	−	0.661850i	$-0.769771\pi$
$419$	−30.6893			−1.49927			−0.749636	+	0.661850i	$0.769771\pi$
$420$		0			0
$421$	−16.9180			−0.824535			−0.412268	−	0.911063i	$-0.635263\pi$
$421$	−16.9180			−0.824535			−0.412268	+	0.911063i	$0.635263\pi$
$422$			16.3501i			0.795911i
$423$		0			0
$424$	2.58480			0.125529
$425$	21.1378	−	11.3939i	1.02533	−	0.552687i
$426$		0			0
$427$			17.1696i			0.830895i
$428$		−	4.58480i		−	0.221615i
$429$		0			0
$430$	12.4675	−	20.8837i	0.601234	−	1.00710i
$431$	17.9722			0.865691			0.432846	−	0.901468i	$-0.357510\pi$
$431$	17.9722			0.865691			0.432846	+	0.901468i	$0.357510\pi$
$432$		0			0
$433$		−	2.61259i		−	0.125553i	−0.998028	−	0.0627764i	$-0.980004\pi$
$433$		−	2.61259i		−	0.125553i	0.998028	−	0.0627764i	$-0.0199955\pi$
$434$	−10.7748			−0.517208
$435$		0			0
$436$	14.8772			0.712489
$437$			13.1696i			0.629988i
$438$		0			0
$439$	−14.6594			−0.699655			−0.349827	−	0.936814i	$-0.613760\pi$
$439$	−14.6594			−0.699655			−0.349827	+	0.936814i	$0.613760\pi$
$440$	3.83991	+	2.29240i	0.183060	+	0.109286i
$441$		0			0
$442$		−	4.80261i		−	0.228437i
$443$			8.33324i			0.395924i	0.980210	+	0.197962i	$0.0634323\pi$
$443$			8.33324i			0.395924i	−0.980210	+	0.197962i	$0.936568\pi$
$444$		0			0
$445$	5.83991	−	9.78219i	0.276838	−	0.463720i
$446$	21.7843			1.03152
$447$		0			0
$448$		−	1.25511i		−	0.0592981i
$449$	−4.04084			−0.190699			−0.0953495	−	0.995444i	$-0.530397\pi$
$449$	−4.04084			−0.190699			−0.0953495	+	0.995444i	$0.530397\pi$
$450$		0			0
$451$	−19.3596			−0.911610
$452$			4.00000i			0.188144i
$453$		0			0
$454$	9.02042			0.423349
$455$	−1.43860	+	2.40974i	−0.0674427	+	0.112971i
$456$		0			0
$457$		−	0.979580i		−	0.0458228i	−0.999737	−	0.0229114i	$-0.992706\pi$
$457$		−	0.979580i		−	0.0458228i	0.999737	−	0.0229114i	$-0.00729357\pi$
$458$		−	4.68718i		−	0.219017i
$459$		0			0
$460$	−4.96270	−	2.96270i	−0.231387	−	0.138137i
$461$	−10.7280			−0.499654			−0.249827	−	0.968291i	$-0.580374\pi$
$461$	−10.7280			−0.499654			−0.249827	+	0.968291i	$0.580374\pi$
$462$		0			0
$463$		−	23.2104i		−	1.07868i	−0.842088	−	0.539340i	$-0.818674\pi$
$463$		−	23.2104i		−	1.07868i	0.842088	−	0.539340i	$-0.181326\pi$
$464$	5.09501			0.236530
$465$		0			0
$466$	13.5366			0.627071
$467$		−	28.5848i		−	1.32275i	−0.750057	−	0.661373i	$-0.769974\pi$
$467$		−	28.5848i		−	1.32275i	0.750057	−	0.661373i	$-0.230026\pi$
$468$		0			0
$469$	−10.7748			−0.497535
$470$	3.14620	−	5.27007i	0.145123	−	0.243090i
$471$		0			0
$472$			5.09501i			0.234517i
$473$			21.7544i			1.00027i
$474$		0			0
$475$	−12.0877	−	22.4247i	−0.554619	−	1.02892i
$476$	6.02778			0.276283
$477$		0			0
$478$			3.19739i			0.146245i
$479$	−30.4078			−1.38937			−0.694685	−	0.719314i	$-0.744456\pi$
$479$	−30.4078			−1.38937			−0.694685	+	0.719314i	$0.744456\pi$
$480$		0			0
$481$	−7.83991			−0.357469
$482$			14.1154i			0.642940i
$483$		0			0
$484$	7.00000			0.318182
$485$	−12.0278	−	7.18051i	−0.546153	−	0.326050i
$486$		0			0
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$		−	13.6798i		−	0.619256i
$489$		0			0
$490$	10.4152	+	6.21781i	0.470511	+	0.280892i
$491$	−22.6893			−1.02396			−0.511978	−	0.858999i	$-0.671087\pi$
$491$	−22.6893			−1.02396			−0.511978	+	0.858999i	$0.671087\pi$
$492$		0			0
$493$			24.4694i			1.10204i
$494$	−5.09501			−0.229235
$495$		0			0
$496$	8.58480			0.385469
$497$		−	6.76177i		−	0.303307i
$498$		0			0
$499$	−31.8698			−1.42669			−0.713345	−	0.700813i	$-0.752821\pi$
$499$	−31.8698			−1.42669			−0.713345	+	0.700813i	$0.752821\pi$
$500$	11.1696	+	0.489790i	0.499520	+	0.0219041i
$501$		0			0
$502$			21.1696i			0.944846i
$503$			14.2646i			0.636028i	0.948086	+	0.318014i	$0.103016\pi$
$503$			14.2646i			0.636028i	−0.948086	+	0.318014i	$0.896984\pi$
$504$		0			0
$505$	14.5102	−	24.3055i	0.645696	−	1.08158i
$506$	5.16961			0.229817
$507$		0			0
$508$			1.41520i			0.0627892i
$509$	23.3596			1.03540			0.517699	−	0.855563i	$-0.326789\pi$
$509$	23.3596			1.03540			0.517699	+	0.855563i	$0.326789\pi$
$510$		0			0
$511$	7.53063			0.333135
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−8.21781			−0.362472
$515$	14.6017	+	8.71711i	0.643427	+	0.384122i
$516$		0			0
$517$			5.48979i			0.241441i
$518$		−	9.83991i		−	0.432341i
$519$		0			0
$520$	1.14620	−	1.91995i	0.0502642	−	0.0841956i
$521$	40.4247			1.77104			0.885519	−	0.464602i	$-0.153803\pi$
$521$	40.4247			1.77104			0.885519	+	0.464602i	$0.153803\pi$
$522$		0			0
$523$			37.9853i			1.66098i	0.557033	+	0.830490i	$0.311940\pi$
$523$			37.9853i			1.66098i	−0.557033	+	0.830490i	$0.688060\pi$
$524$	11.0095			0.480953
$525$		0			0
$526$	−4.51021			−0.196655
$527$			41.2295i			1.79598i
$528$		0			0
$529$	16.3188			0.709513
$530$	−2.96270	+	4.96270i	−0.128692	+	0.215566i
$531$		0			0
$532$		−	6.39478i		−	0.277249i
$533$			9.67982i			0.419279i
$534$		0			0
$535$	8.80261	+	5.25511i	0.380570	+	0.227198i
$536$	8.58480			0.370807
$537$		0			0
$538$		−	8.51021i		−	0.366901i
$539$	−10.8494			−0.467318
$540$		0			0
$541$	26.8772			1.15554			0.577771	−	0.816199i	$-0.303923\pi$
$541$	26.8772			1.15554			0.577771	+	0.816199i	$0.303923\pi$
$542$			4.95180i			0.212698i
$543$		0			0
$544$	−4.80261			−0.205910
$545$	−17.0523	+	28.5636i	−0.730439	+	1.22353i
$546$		0			0
$547$		−	19.8976i		−	0.850761i	−0.905015	−	0.425381i	$-0.860140\pi$
$547$		−	19.8976i		−	0.850761i	0.905015	−	0.425381i	$-0.139860\pi$
$548$		−	14.5848i		−	0.623032i
$549$		0			0
$550$	−8.80261	+	4.74489i	−0.375345	+	0.202323i
$551$	25.9592			1.10590
$552$		0			0
$553$		−	18.9458i		−	0.805659i
$554$	19.8698			0.844189
$555$		0			0
$556$	7.32970			0.310848
$557$		−	1.50070i		−	0.0635865i	−0.999494	−	0.0317933i	$-0.989878\pi$
$557$		−	1.50070i		−	0.0635865i	0.999494	−	0.0317933i	$-0.0101218\pi$
$558$		0			0
$559$	10.8772			0.460057
$560$	2.40974	+	1.43860i	0.101830	+	0.0607920i
$561$		0			0
$562$		−	30.2646i		−	1.27664i
$563$			20.6872i			0.871861i	0.899980	+	0.435930i	$0.143581\pi$
$563$			20.6872i			0.871861i	−0.899980	+	0.435930i	$0.856419\pi$
$564$		0			0
$565$	−7.67982	−	4.58480i	−0.323092	−	0.192884i
$566$	−26.9240			−1.13170
$567$		0			0
$568$			5.38741i			0.226051i
$569$	−6.08550			−0.255117			−0.127559	−	0.991831i	$-0.540714\pi$
$569$	−6.08550			−0.255117			−0.127559	+	0.991831i	$0.540714\pi$
$570$		0			0
$571$	−9.98909			−0.418031			−0.209015	−	0.977912i	$-0.567026\pi$
$571$	−9.98909			−0.418031			−0.209015	+	0.977912i	$0.567026\pi$
$572$			2.00000i			0.0836242i
$573$		0			0
$574$	−12.1492			−0.507097
$575$	11.3765	−	6.13231i	0.474433	−	0.255735i
$576$		0			0
$577$			8.33921i			0.347166i	0.984819	+	0.173583i	$0.0555345\pi$
$577$			8.33921i			0.347166i	−0.984819	+	0.173583i	$0.944466\pi$
$578$		−	6.06508i		−	0.252274i
$579$		0			0
$580$	−5.83991	+	9.78219i	−0.242489	+	0.406183i
$581$	−13.9254			−0.577723
$582$		0			0
$583$		−	5.16961i		−	0.214103i
$584$	−6.00000			−0.248282
$585$		0			0
$586$	5.18051			0.214005
$587$		−	23.2442i		−	0.959391i	−0.877435	−	0.479695i	$-0.840747\pi$
$587$		−	23.2442i		−	0.959391i	0.877435	−	0.479695i	$-0.159253\pi$
$588$		0			0
$589$	43.7397			1.80226
$590$	−9.78219	−	5.83991i	−0.402726	−	0.240425i
$591$		0			0
$592$			7.83991i			0.322218i
$593$			34.2646i			1.40708i	0.710656	+	0.703540i	$0.248398\pi$
$593$			34.2646i			1.40708i	−0.710656	+	0.703540i	$0.751602\pi$
$594$		0			0
$595$	−6.90905	+	11.5731i	−0.283243	+	0.474449i
$596$		0			0
$597$		0			0
$598$		−	2.58480i		−	0.105700i
$599$	−31.1886			−1.27433			−0.637167	−	0.770726i	$-0.719894\pi$
$599$	−31.1886			−1.27433			−0.637167	+	0.770726i	$0.719894\pi$
$600$		0			0
$601$	−3.40429			−0.138864			−0.0694320	−	0.997587i	$-0.522119\pi$
$601$	−3.40429			−0.138864			−0.0694320	+	0.997587i	$0.522119\pi$
$602$			13.6520i			0.556415i
$603$		0			0
$604$	14.5570			0.592317
$605$	−8.02341	+	13.4397i	−0.326198	+	0.546401i
$606$		0			0
$607$			0.945827i			0.0383899i	0.999816	+	0.0191950i	$0.00611032\pi$
$607$			0.945827i			0.0383899i	−0.999816	+	0.0191950i	$0.993890\pi$
$608$			5.09501i			0.206630i
$609$		0			0
$610$	26.2646	+	15.6798i	1.06342	+	0.634857i
$611$	2.74489			0.111047
$612$		0			0
$613$			40.7193i			1.64464i	0.569028	+	0.822318i	$0.307319\pi$
$613$			40.7193i			1.64464i	−0.569028	+	0.822318i	$0.692681\pi$
$614$	−0.320184			−0.0129216
$615$		0			0
$616$	−2.51021			−0.101139
$617$			12.6594i			0.509648i	0.966987	+	0.254824i	$0.0820176\pi$
$617$			12.6594i			0.509648i	−0.966987	+	0.254824i	$0.917982\pi$
$618$		0			0
$619$	−5.03945			−0.202553			−0.101276	−	0.994858i	$-0.532293\pi$
$619$	−5.03945			−0.202553			−0.101276	+	0.994858i	$0.532293\pi$
$620$	−9.83991	+	16.4824i	−0.395180	+	0.661950i
$621$		0			0
$622$			9.86984i			0.395745i
$623$			6.39478i			0.256201i
$624$		0			0
$625$	−13.7430	+	20.8837i	−0.549719	+	0.835349i
$626$	−10.6126			−0.424164
$627$		0			0
$628$			13.0950i			0.522548i
$629$	−37.6520			−1.50128
$630$		0			0
$631$	−1.00736			−0.0401024			−0.0200512	−	0.999799i	$-0.506383\pi$
$631$	−1.00736			−0.0401024			−0.0200512	+	0.999799i	$0.506383\pi$
$632$			15.0950i			0.600447i
$633$		0			0
$634$	12.1900			0.484128
$635$	−2.71711	−	1.62210i	−0.107825	−	0.0643711i
$636$		0			0
$637$			5.42471i			0.214935i
$638$		−	10.1900i		−	0.403427i
$639$		0			0
$640$	−1.91995	−	1.14620i	−0.0758928	−	0.0453076i
$641$	41.3596			1.63361			0.816804	−	0.576916i	$-0.195744\pi$
$641$	41.3596			1.63361			0.816804	+	0.576916i	$0.195744\pi$
$642$		0			0
$643$		−	3.26601i		−	0.128799i	−0.997924	−	0.0643994i	$-0.979487\pi$
$643$		−	3.26601i		−	0.128799i	0.997924	−	0.0643994i	$-0.0205132\pi$
$644$	3.24420			0.127839
$645$		0			0
$646$	−24.4694			−0.962734
$647$			19.4342i			0.764038i	0.924154	+	0.382019i	$0.124771\pi$
$647$			19.4342i			0.764038i	−0.924154	+	0.382019i	$0.875229\pi$
$648$		0			0
$649$	10.1900			0.399994
$650$	2.37245	+	4.40131i	0.0930550	+	0.172633i
$651$		0			0
$652$		−	12.2646i		−	0.480320i
$653$			4.51021i			0.176498i	0.996098	+	0.0882491i	$0.0281271\pi$
$653$			4.51021i			0.176498i	−0.996098	+	0.0882491i	$0.971873\pi$
$654$		0			0
$655$	−12.6191	+	21.1378i	−0.493070	+	0.825921i
$656$	9.67982			0.377933
$657$		0			0
$658$			3.44513i			0.134305i
$659$	27.2104			1.05997			0.529984	−	0.848007i	$-0.322198\pi$
$659$	27.2104			1.05997			0.529984	+	0.848007i	$0.322198\pi$
$660$		0			0
$661$	−24.5292			−0.954077			−0.477038	−	0.878882i	$-0.658290\pi$
$661$	−24.5292			−0.954077			−0.477038	+	0.878882i	$0.658290\pi$
$662$		−	17.9444i		−	0.697430i
$663$		0			0
$664$	11.0950			0.430570
$665$	12.2777	+	7.32970i	0.476108	+	0.284234i
$666$		0			0
$667$			13.1696i			0.509929i
$668$			18.3392i			0.709565i
$669$		0			0
$670$	−9.83991	+	16.4824i	−0.380149	+	0.636772i
$671$	−27.3596			−1.05621
$672$		0			0
$673$		−	8.95180i		−	0.345066i	−0.985004	−	0.172533i	$-0.944805\pi$
$673$		−	8.95180i		−	0.345066i	0.985004	−	0.172533i	$-0.0551952\pi$
$674$	20.9518			0.807033
$675$		0			0
$676$	1.00000			0.0384615
$677$		−	32.6256i		−	1.25391i	−0.779057	−	0.626953i	$-0.784302\pi$
$677$		−	32.6256i		−	1.25391i	0.779057	−	0.626953i	$-0.215698\pi$
$678$		0			0
$679$	7.86276			0.301745
$680$	5.50476	−	9.22079i	0.211098	−	0.353601i
$681$		0			0
$682$		−	17.1696i		−	0.657458i
$683$			11.0950i			0.424539i	0.977211	+	0.212269i	$0.0680854\pi$
$683$			11.0950i			0.424539i	−0.977211	+	0.212269i	$0.931915\pi$
$684$		0			0
$685$	28.0022	+	16.7171i	1.06991	+	0.638728i
$686$	−15.5943			−0.595394
$687$		0			0
$688$		−	10.8772i		−	0.414690i
$689$	−2.58480			−0.0984732
$690$		0			0
$691$	−13.5306			−0.514729			−0.257365	−	0.966314i	$-0.582854\pi$
$691$	−13.5306			−0.514729			−0.257365	+	0.966314i	$0.582854\pi$
$692$		−	9.41520i		−	0.357912i
$693$		0			0
$694$	3.51757			0.133525
$695$	−8.40131	+	14.0727i	−0.318680	+	0.533807i
$696$		0			0
$697$			46.4884i			1.76087i
$698$		−	17.8568i		−	0.675889i
$699$		0			0
$700$	−5.52410	+	2.97767i	−0.208791	+	0.112545i
$701$	48.1345			1.81801			0.909007	−	0.416781i	$-0.136842\pi$
$701$	48.1345			1.81801			0.909007	+	0.416781i	$0.136842\pi$
$702$		0			0
$703$			39.9444i			1.50653i
$704$	2.00000			0.0753778
$705$		0			0
$706$	36.9240			1.38965
$707$			15.8889i			0.597563i
$708$		0			0
$709$	−16.5292			−0.620769			−0.310384	−	0.950611i	$-0.600458\pi$
$709$	−16.5292			−0.620769			−0.310384	+	0.950611i	$0.600458\pi$
$710$	−10.3436	−	6.17506i	−0.388188	−	0.231746i
$711$		0			0
$712$		−	5.09501i		−	0.190944i
$713$			22.1900i			0.831023i
$714$		0			0
$715$	−3.83991	−	2.29240i	−0.143604	−	0.0857310i
$716$	7.32970			0.273924
$717$		0			0
$718$		−	6.77483i		−	0.252834i
$719$	−6.39478			−0.238485			−0.119242	−	0.992865i	$-0.538047\pi$
$719$	−6.39478			−0.238485			−0.119242	+	0.992865i	$0.538047\pi$
$720$		0			0
$721$	−9.54535			−0.355488
$722$			6.95916i			0.258993i
$723$		0			0
$724$	−11.7544			−0.436849
$725$	−12.0877	−	22.4247i	−0.448924	−	0.832833i
$726$		0			0
$727$		−	35.0204i		−	1.29884i	−0.760432	−	0.649418i	$-0.775013\pi$
$727$		−	35.0204i		−	1.29884i	0.760432	−	0.649418i	$-0.224987\pi$
$728$			1.25511i			0.0465173i
$729$		0			0
$730$	6.87720	−	11.5197i	0.254537	−	0.426364i
$731$	52.2390			1.93213
$732$		0			0
$733$		−	22.6485i		−	0.836541i	−0.908322	−	0.418271i	$-0.862636\pi$
$733$		−	22.6485i		−	0.836541i	0.908322	−	0.418271i	$-0.137364\pi$
$734$	7.80997			0.288271
$735$		0			0
$736$	−2.58480			−0.0952771
$737$		−	17.1696i		−	0.632451i
$738$		0			0
$739$	−25.0394			−0.921091			−0.460546	−	0.887636i	$-0.652346\pi$
$739$	−25.0394			−0.921091			−0.460546	+	0.887636i	$0.652346\pi$
$740$	−15.0523	−	8.98611i	−0.553332	−	0.330336i
$741$		0			0
$742$		−	3.24420i		−	0.119098i
$743$			35.8252i			1.31430i	0.753760	+	0.657149i	$0.228238\pi$
$743$			35.8252i			1.31430i	−0.753760	+	0.657149i	$0.771762\pi$
$744$		0			0
$745$		0			0
$746$	−15.8698			−0.581036
$747$		0			0
$748$			9.60522i			0.351202i
$749$	−5.75441			−0.210262
$750$		0			0
$751$	41.6988			1.52161			0.760806	−	0.648979i	$-0.224804\pi$
$751$	41.6988			1.52161			0.760806	+	0.648979i	$0.224804\pi$
$752$		−	2.74489i		−	0.100096i
$753$		0			0
$754$	−5.09501			−0.185549
$755$	−16.6853	+	27.9488i	−0.607239	+	1.01716i
$756$		0			0
$757$		−	11.6052i		−	0.421799i	−0.977508	−	0.210900i	$-0.932361\pi$
$757$		−	11.6052i		−	0.421799i	0.977508	−	0.210900i	$-0.0676393\pi$
$758$			16.7748i			0.609289i
$759$		0			0
$760$	−9.78219	−	5.83991i	−0.354837	−	0.211836i
$761$	−30.2646			−1.09709			−0.548546	−	0.836121i	$-0.684818\pi$
$761$	−30.2646			−1.09709			−0.548546	+	0.836121i	$0.684818\pi$
$762$		0			0
$763$		−	18.6725i		−	0.675988i
$764$	−1.60522			−0.0580749
$765$		0			0
$766$	16.6147			0.600315
$767$		−	5.09501i		−	0.183970i
$768$		0			0
$769$	−24.6594			−0.889241			−0.444620	−	0.895719i	$-0.646661\pi$
$769$	−24.6594			−0.889241			−0.444620	+	0.895719i	$0.646661\pi$
$770$	2.87720	−	4.81949i	0.103687	−	0.173682i
$771$		0			0
$772$			15.7544i			0.567014i
$773$			30.3501i			1.09162i	0.837910	+	0.545809i	$0.183778\pi$
$773$			30.3501i			1.09162i	−0.837910	+	0.545809i	$0.816222\pi$
$774$		0			0
$775$	−20.3670	−	37.7843i	−0.731604	−	1.35725i
$776$	−6.26462			−0.224887
$777$		0			0
$778$			36.7193i			1.31645i
$779$	49.3188			1.76703
$780$		0			0
$781$	10.7748			0.385554
$782$		−	12.4138i		−	0.443917i
$783$		0			0
$784$	5.42471			0.193740
$785$	−25.1418	−	15.0095i	−0.897350	−	0.535713i
$786$		0			0
$787$			11.1288i			0.396698i	0.980131	+	0.198349i	$0.0635579\pi$
$787$			11.1288i			0.396698i	−0.980131	+	0.198349i	$0.936442\pi$
$788$			1.00951i			0.0359625i
$789$		0			0
$790$	−28.9817	−	17.3019i	−1.03112	−	0.615575i
$791$	5.02042			0.178506
$792$		0			0
$793$			13.6798i			0.485785i
$794$	−29.2104			−1.03664
$795$		0			0
$796$	0.435617			0.0154400
$797$		−	46.7939i		−	1.65752i	−0.559602	−	0.828762i	$-0.689046\pi$
$797$		−	46.7939i		−	1.65752i	0.559602	−	0.828762i	$-0.310954\pi$
$798$		0			0
$799$	13.1827			0.466369
$800$	4.40131	−	2.37245i	0.155610	−	0.0838787i
$801$		0			0
$802$		−	15.6052i		−	0.551040i
$803$			12.0000i			0.423471i
$804$		0			0
$805$	−3.71850	+	6.22871i	−0.131060	+	0.219533i
$806$	−8.58480			−0.302387
$807$		0			0
$808$		−	12.6594i		−	0.445356i
$809$	50.4437			1.77351			0.886754	−	0.462242i	$-0.152955\pi$
$809$	50.4437			1.77351			0.886754	+	0.462242i	$0.152955\pi$
$810$		0			0
$811$	−36.3991			−1.27814			−0.639072	−	0.769147i	$-0.720682\pi$
$811$	−36.3991			−1.27814			−0.639072	+	0.769147i	$0.720682\pi$
$812$		−	6.39478i		−	0.224413i
$813$		0			0
$814$	15.6798			0.549577
$815$	23.5475	+	14.0577i	0.824833	+	0.492420i
$816$		0			0
$817$		−	55.4195i		−	1.93888i
$818$		−	13.7952i		−	0.482340i
$819$		0			0
$820$	−11.0950	+	18.5848i	−0.387455	+	0.649009i
$821$	6.23684			0.217667			0.108834	−	0.994060i	$-0.465288\pi$
$821$	6.23684			0.217667			0.108834	+	0.994060i	$0.465288\pi$
$822$		0			0
$823$		−	33.7734i		−	1.17727i	−0.808400	−	0.588634i	$-0.799666\pi$
$823$		−	33.7734i		−	1.17727i	0.808400	−	0.588634i	$-0.200334\pi$
$824$	7.60522			0.264941
$825$		0			0
$826$	6.39478			0.222503
$827$		−	37.2295i		−	1.29460i	−0.762237	−	0.647298i	$-0.775899\pi$
$827$		−	37.2295i		−	1.29460i	0.762237	−	0.647298i	$-0.224101\pi$
$828$		0			0
$829$	38.2646			1.32899			0.664493	−	0.747295i	$-0.268648\pi$
$829$	38.2646			1.32899			0.664493	+	0.747295i	$0.268648\pi$
$830$	−12.7171	+	21.3019i	−0.441417	+	0.739400i
$831$		0			0
$832$		−	1.00000i		−	0.0346688i
$833$			26.0528i			0.902675i
$834$		0			0
$835$	−35.2104	−	21.0204i	−1.21851	−	0.727442i
$836$	10.1900			0.352429
$837$		0			0
$838$		−	30.6893i		−	1.06015i
$839$	50.1345			1.73083			0.865417	−	0.501052i	$-0.167054\pi$
$839$	50.1345			1.73083			0.865417	+	0.501052i	$0.167054\pi$
$840$		0			0
$841$	−3.04084			−0.104857
$842$		−	16.9180i		−	0.583034i
$843$		0			0
$844$	−16.3501			−0.562794
$845$	−1.14620	+	1.91995i	−0.0394305	+	0.0660484i
$846$		0			0
$847$		−	8.78574i		−	0.301881i
$848$			2.58480i			0.0887625i
$849$		0			0
$850$	11.3939	+	21.1378i	0.390809	+	0.725019i
$851$	−20.2646			−0.694662
$852$		0			0
$853$			12.3910i			0.424258i	0.977242	+	0.212129i	$0.0680398\pi$
$853$			12.3910i			0.424258i	−0.977242	+	0.212129i	$0.931960\pi$
$854$	−17.1696			−0.587532
$855$		0			0
$856$	4.58480			0.156705
$857$			25.4005i			0.867664i	0.900994	+	0.433832i	$0.142839\pi$
$857$			25.4005i			0.867664i	−0.900994	+	0.433832i	$0.857161\pi$
$858$		0			0
$859$	50.1900			1.71246			0.856231	−	0.516593i	$-0.172800\pi$
$859$	50.1900			1.71246			0.856231	+	0.516593i	$0.172800\pi$
$860$	20.8837	+	12.4675i	0.712129	+	0.425137i
$861$		0			0
$862$			17.9722i			0.612136i
$863$		−	43.4451i		−	1.47889i	−0.673217	−	0.739445i	$-0.735088\pi$
$863$		−	43.4451i		−	1.47889i	0.673217	−	0.739445i	$-0.264912\pi$
$864$		0			0
$865$	18.0767	+	10.7917i	0.614628	+	0.366929i
$866$	2.61259			0.0887793
$867$		0			0
$868$		−	10.7748i		−	0.365722i
$869$	30.1900			1.02413
$870$		0			0
$871$	−8.58480			−0.290885
$872$			14.8772i			0.503806i
$873$		0			0
$874$	−13.1696			−0.445469
$875$	0.614738	−	14.0190i	0.0207819	−	0.473930i
$876$		0			0
$877$		−	11.6689i		−	0.394031i	−0.980400	−	0.197016i	$-0.936875\pi$
$877$		−	11.6689i		−	0.394031i	0.980400	−	0.197016i	$-0.0631250\pi$
$878$		−	14.6594i		−	0.494731i
$879$		0			0
$880$	−2.29240	+	3.83991i	−0.0772768	+	0.129443i
$881$	0.0446585			0.00150458			0.000752291	−	1.00000i	$-0.499761\pi$
$881$	0.0446585			0.00150458			0.000752291		1.00000i	$0.499761\pi$
$882$		0			0
$883$		−	24.4269i		−	0.822029i	−0.911629	−	0.411015i	$-0.865174\pi$
$883$		−	24.4269i		−	0.822029i	0.911629	−	0.411015i	$-0.134826\pi$
$884$	4.80261			0.161529
$885$		0			0
$886$	−8.33324			−0.279961
$887$		−	47.2295i		−	1.58581i	−0.609345	−	0.792905i	$-0.708567\pi$
$887$		−	47.2295i		−	1.58581i	0.609345	−	0.792905i	$-0.291433\pi$
$888$		0			0
$889$	1.77622			0.0595725
$890$	9.78219	+	5.83991i	0.327900	+	0.195754i
$891$		0			0
$892$			21.7843i			0.729394i
$893$		−	13.9853i		−	0.467999i
$894$		0			0
$895$	−8.40131	+	14.0727i	−0.280825	+	0.470398i
$896$	1.25511			0.0419301
$897$		0			0
$898$		−	4.04084i		−	0.134845i
$899$	43.7397			1.45880
$900$		0			0
$901$	−12.4138			−0.413564
$902$		−	19.3596i		−	0.644605i
$903$		0			0
$904$	−4.00000			−0.133038
$905$	13.4729	−	22.5679i	0.447855	−	0.750183i
$906$		0			0
$907$			44.7133i			1.48468i	0.670023	+	0.742340i	$0.266284\pi$
$907$			44.7133i			1.48468i	−0.670023	+	0.742340i	$0.733716\pi$
$908$			9.02042i			0.299353i
$909$		0			0
$910$	−2.40974	−	1.43860i	−0.0798822	−	0.0476892i
$911$	−51.9444			−1.72100			−0.860498	−	0.509454i	$-0.829847\pi$
$911$	−51.9444			−1.72100			−0.860498	+	0.509454i	$0.829847\pi$
$912$		0			0
$913$		−	22.1900i		−	0.734383i
$914$	0.979580			0.0324016
$915$		0			0
$916$	4.68718			0.154869
$917$		−	13.8181i		−	0.456314i
$918$		0			0
$919$	13.5497			0.446962			0.223481	−	0.974708i	$-0.428258\pi$
$919$	13.5497			0.446962			0.223481	+	0.974708i	$0.428258\pi$
$920$	2.96270	−	4.96270i	0.0976774	−	0.163615i
$921$		0			0
$922$		−	10.7280i		−	0.353308i
$923$		−	5.38741i		−	0.177329i
$924$		0			0
$925$	34.5058	−	18.5998i	1.13454	−	0.611557i
$926$	23.2104			0.762743
$927$		0			0
$928$			5.09501i			0.167252i
$929$	−44.1682			−1.44911			−0.724556	−	0.689216i	$-0.757955\pi$
$929$	−44.1682			−1.44911			−0.724556	+	0.689216i	$0.757955\pi$
$930$		0			0
$931$	27.6390			0.905831
$932$			13.5366i			0.443406i
$933$		0			0
$934$	28.5848			0.935323
$935$	−18.4416	−	11.0095i	−0.603104	−	0.360050i
$936$		0			0
$937$			41.6988i			1.36224i	0.732171	+	0.681121i	$0.238507\pi$
$937$			41.6988i			1.36224i	−0.732171	+	0.681121i	$0.761493\pi$
$938$		−	10.7748i		−	0.351811i
$939$		0			0
$940$	5.27007	+	3.14620i	0.171891	+	0.102618i
$941$	−37.4473			−1.22075			−0.610373	−	0.792114i	$-0.708981\pi$
$941$	−37.4473			−1.22075			−0.610373	+	0.792114i	$0.708981\pi$
$942$		0			0
$943$			25.0204i			0.814777i
$944$	−5.09501			−0.165829
$945$		0			0
$946$	−21.7544			−0.707297
$947$		−	11.5644i		−	0.375792i	−0.982189	−	0.187896i	$-0.939833\pi$
$947$		−	11.5644i		−	0.375792i	0.982189	−	0.187896i	$-0.0601668\pi$
$948$		0			0
$949$	6.00000			0.194768
$950$	22.4247	−	12.0877i	0.727554	−	0.392175i
$951$		0			0
$952$			6.02778i			0.195362i
$953$		−	53.4810i		−	1.73242i	−0.499680	−	0.866210i	$-0.666549\pi$
$953$		−	53.4810i		−	1.73242i	0.499680	−	0.866210i	$-0.333451\pi$
$954$		0			0
$955$	1.83991	−	3.08196i	0.0595380	−	0.0997297i
$956$	−3.19739			−0.103411
$957$		0			0
$958$		−	30.4078i		−	0.982433i
$959$	−18.3055			−0.591114
$960$		0			0
$961$	42.6988			1.37738
$962$		−	7.83991i		−	0.252769i
$963$		0			0
$964$	−14.1154			−0.454627
$965$	−30.2477	−	18.0577i	−0.973709	−	0.581298i
$966$		0			0
$967$		−	34.8941i		−	1.12212i	−0.827776	−	0.561059i	$-0.810394\pi$
$967$		−	34.8941i		−	1.12212i	0.827776	−	0.561059i	$-0.189606\pi$
$968$			7.00000i			0.224989i
$969$		0			0
$970$	7.18051	−	12.0278i	0.230552	−	0.386189i
$971$	−31.8807			−1.02310			−0.511551	−	0.859253i	$-0.670929\pi$
$971$	−31.8807			−1.02310			−0.511551	+	0.859253i	$0.670929\pi$
$972$		0			0
$973$		−	9.19954i		−	0.294924i
$974$	8.00000			0.256337
$975$		0			0
$976$	13.6798			0.437880
$977$			2.03375i			0.0650655i	0.999471	+	0.0325328i	$0.0103573\pi$
$977$			2.03375i			0.0650655i	−0.999471	+	0.0325328i	$0.989643\pi$
$978$		0			0
$979$	−10.1900			−0.325675
$980$	−6.21781	+	10.4152i	−0.198621	+	0.332701i
$981$		0			0
$982$		−	22.6893i		−	0.724046i
$983$		−	29.4043i		−	0.937851i	−0.883238	−	0.468926i	$-0.844641\pi$
$983$		−	29.4043i		−	0.937851i	0.883238	−	0.468926i	$-0.155359\pi$
$984$		0			0
$985$	−1.93822	−	1.15711i	−0.0617569	−	0.0368685i
$986$	−24.4694			−0.779263
$987$		0			0
$988$		−	5.09501i		−	0.162094i
$989$	28.1154			0.894019
$990$		0			0
$991$	32.9986			1.04824			0.524118	−	0.851646i	$-0.324395\pi$
$991$	32.9986			1.04824			0.524118	+	0.851646i	$0.324395\pi$
$992$			8.58480i			0.272568i
$993$		0			0
$994$	6.76177			0.214470
$995$	−0.499304	+	0.836365i	−0.0158290	+	0.0265145i
$996$		0			0
$997$		−	31.1696i		−	0.987151i	−0.869703	−	0.493576i	$-0.835690\pi$
$997$		−	31.1696i		−	0.987151i	0.869703	−	0.493576i	$-0.164310\pi$
$998$		−	31.8698i		−	1.00882i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.e.f.469.6		6
3.2	odd	2		130.2.b.a.79.3	✓	6
5.2	odd	4		5850.2.a.cp.1.2		3
5.3	odd	4		5850.2.a.cs.1.2		3
5.4	even	2	inner	1170.2.e.f.469.3		6
12.11	even	2		1040.2.d.b.209.2		6
15.2	even	4		650.2.a.o.1.3		3
15.8	even	4		650.2.a.n.1.1		3
15.14	odd	2		130.2.b.a.79.4	yes	6
39.5	even	4		1690.2.c.a.1689.5		6
39.8	even	4		1690.2.c.d.1689.5		6
39.38	odd	2		1690.2.b.a.339.6		6
60.23	odd	4		5200.2.a.ce.1.3		3
60.47	odd	4		5200.2.a.cf.1.1		3
60.59	even	2		1040.2.d.b.209.5		6
195.38	even	4		8450.2.a.cc.1.1		3
195.44	even	4		1690.2.c.d.1689.2		6
195.77	even	4		8450.2.a.bs.1.3		3
195.164	even	4		1690.2.c.a.1689.2		6
195.194	odd	2		1690.2.b.a.339.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.b.a.79.3	✓	6	3.2	odd	2
130.2.b.a.79.4	yes	6	15.14	odd	2
650.2.a.n.1.1		3	15.8	even	4
650.2.a.o.1.3		3	15.2	even	4
1040.2.d.b.209.2		6	12.11	even	2
1040.2.d.b.209.5		6	60.59	even	2
1170.2.e.f.469.3		6	5.4	even	2	inner
1170.2.e.f.469.6		6	1.1	even	1	trivial
1690.2.b.a.339.1		6	195.194	odd	2
1690.2.b.a.339.6		6	39.38	odd	2
1690.2.c.a.1689.2		6	195.164	even	4
1690.2.c.a.1689.5		6	39.5	even	4
1690.2.c.d.1689.2		6	195.44	even	4
1690.2.c.d.1689.5		6	39.8	even	4
5200.2.a.ce.1.3		3	60.23	odd	4
5200.2.a.cf.1.1		3	60.47	odd	4
5850.2.a.cp.1.2		3	5.2	odd	4
5850.2.a.cs.1.2		3	5.3	odd	4
8450.2.a.bs.1.3		3	195.77	even	4
8450.2.a.cc.1.1		3	195.38	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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.14620 - 1.91995i) q^{5} +1.25511i q^{7} -1.00000i q^{8} +(1.91995 + 1.14620i) q^{10} -2.00000 q^{11} +1.00000i q^{13} -1.25511 q^{14} +1.00000 q^{16} +4.80261i q^{17} +5.09501 q^{19} +(-1.14620 + 1.91995i) q^{20} -2.00000i q^{22} +2.58480i q^{23} +(-2.37245 - 4.40131i) q^{25} -1.00000 q^{26} -1.25511i q^{28} +5.09501 q^{29} +8.58480 q^{31} +1.00000i q^{32} -4.80261 q^{34} +(2.40974 + 1.43860i) q^{35} +7.83991i q^{37} +5.09501i q^{38} +(-1.91995 - 1.14620i) q^{40} +9.67982 q^{41} -10.8772i q^{43} +2.00000 q^{44} -2.58480 q^{46} -2.74489i q^{47} +5.42471 q^{49} +(4.40131 - 2.37245i) q^{50} -1.00000i q^{52} +2.58480i q^{53} +(-2.29240 + 3.83991i) q^{55} +1.25511 q^{56} +5.09501i q^{58} -5.09501 q^{59} +13.6798 q^{61} +8.58480i q^{62} -1.00000 q^{64} +(1.91995 + 1.14620i) q^{65} +8.58480i q^{67} -4.80261i q^{68} +(-1.43860 + 2.40974i) q^{70} -5.38741 q^{71} -6.00000i q^{73} -7.83991 q^{74} -5.09501 q^{76} -2.51021i q^{77} -15.0950 q^{79} +(1.14620 - 1.91995i) q^{80} +9.67982i q^{82} +11.0950i q^{83} +(9.22079 + 5.50476i) q^{85} +10.8772 q^{86} +2.00000i q^{88} +5.09501 q^{89} -1.25511 q^{91} -2.58480i q^{92} +2.74489 q^{94} +(5.83991 - 9.78219i) q^{95} -6.26462i q^{97} +5.42471i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{10} - 12 q^{11} - 4 q^{14} + 6 q^{16} - 4 q^{19} - 16 q^{25} - 6 q^{26} - 4 q^{29} + 24 q^{31} - 8 q^{34} + 6 q^{35} + 4 q^{40} - 4 q^{41} + 12 q^{44} + 12 q^{46} - 26 q^{49} + 16 q^{50}+ \cdots + 4 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^10 - 12 * q^11 - 4 * q^14 + 6 * q^16 - 4 * q^19 - 16 * q^25 - 6 * q^26 - 4 * q^29 + 24 * q^31 - 8 * q^34 + 6 * q^35 + 4 * q^40 - 4 * q^41 + 12 * q^44 + 12 * q^46 - 26 * q^49 + 16 * q^50 + 4 * q^56 + 4 * q^59 + 20 * q^61 - 6 * q^64 - 4 * q^65 + 12 * q^70 + 16 * q^71 - 16 * q^74 + 4 * q^76 - 56 * q^79 + 28 * q^85 + 24 * q^86 - 4 * q^89 - 4 * q^91 + 20 * q^94 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more