LMFDB - Embedded newform 1134.3.c.d.811.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1134,3,Mod(811,1134)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1134.811"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1134.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,32,0,0,2,0,0,0,-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:	$30.8992619785$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 21 x^{14} + 48 x^{13} + 1111 x^{12} - 3102 x^{11} - 3 x^{10} - 28164 x^{9} + \cdots + 937890625 $ x^16 - 6x^15 - 21x^14 + 48x^13 + 1111x^12 - 3102x^11 - 3x^10 - 28164x^9 + 127089x^8 - 674100x^7 + 3789075x^6 - 9927750x^5 + 30244375x^4 - 95550000x^3 + 218203125x^2 - 482343750*x + 937890625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.5
Root			$-0.0370234 + 2.81559i$ of defining polynomial
Character	$\chi$	$=$	1134.811
Dual form			1134.3.c.d.811.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421 q^{2} +2.00000 q^{4} +1.38272i q^{5} +(4.59851 + 5.27766i) q^{7} -2.82843 q^{8} -1.95546i q^{10} +16.3713 q^{11} -18.5572i q^{13} +(-6.50328 - 7.46374i) q^{14} +4.00000 q^{16} -1.67115i q^{17} -28.9316i q^{19} +2.76543i q^{20} -23.1525 q^{22} -24.5167 q^{23} +23.0881 q^{25} +26.2439i q^{26} +(9.19702 + 10.5553i) q^{28} -41.4015 q^{29} -43.0220i q^{31} -5.65685 q^{32} +2.36336i q^{34} +(-7.29751 + 6.35844i) q^{35} -18.5784 q^{37} +40.9155i q^{38} -3.91091i q^{40} -11.3233i q^{41} +57.9277 q^{43} +32.7425 q^{44} +34.6719 q^{46} +12.9768i q^{47} +(-6.70739 + 48.5388i) q^{49} -32.6515 q^{50} -37.1145i q^{52} +45.2169 q^{53} +22.6368i q^{55} +(-13.0066 - 14.9275i) q^{56} +58.5506 q^{58} -24.9870i q^{59} +16.0979i q^{61} +60.8422i q^{62} +8.00000 q^{64} +25.6594 q^{65} +44.2782 q^{67} -3.34229i q^{68} +(10.3202 - 8.99219i) q^{70} -80.7195 q^{71} -79.4212i q^{73} +26.2738 q^{74} -57.8633i q^{76} +(75.2835 + 86.4020i) q^{77} -39.2311 q^{79} +5.53087i q^{80} +16.0136i q^{82} -115.894i q^{83} +2.31072 q^{85} -81.9221 q^{86} -46.3050 q^{88} -30.2467i q^{89} +(97.9389 - 85.3357i) q^{91} -49.0335 q^{92} -18.3519i q^{94} +40.0043 q^{95} -82.7997i q^{97} +(9.48569 - 68.6442i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} + 2 q^{7} - 12 q^{11} - 12 q^{14} + 64 q^{16} + 12 q^{23} - 80 q^{25} + 4 q^{28} - 48 q^{29} - 174 q^{35} - 44 q^{37} - 32 q^{43} - 24 q^{44} + 24 q^{46} - 50 q^{49} + 48 q^{50} + 432 q^{53}+ \cdots - 312 q^{98}+O(q^{100}) $ 16 * q + 32 * q^4 + 2 * q^7 - 12 * q^11 - 12 * q^14 + 64 * q^16 + 12 * q^23 - 80 * q^25 + 4 * q^28 - 48 * q^29 - 174 * q^35 - 44 * q^37 - 32 * q^43 - 24 * q^44 + 24 * q^46 - 50 * q^49 + 48 * q^50 + 432 * q^53 - 24 * q^56 - 48 * q^58 + 128 * q^64 + 120 * q^65 + 140 * q^67 + 108 * q^70 - 276 * q^71 + 144 * q^74 - 258 * q^77 + 176 * q^79 + 60 * q^85 + 96 * q^86 + 186 * q^91 + 24 * q^92 + 528 * q^95 - 312 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$.

$n$	$325$	$407$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			1.38272i			0.276543i	0.990394	+	0.138272i	$0.0441547\pi$
$5$			1.38272i			0.276543i	−0.990394	+	0.138272i	$0.955845\pi$
$6$		0			0
$7$	4.59851	+	5.27766i	0.656930	+	0.753951i
$7$	4.59851	+	5.27766i	0.656930	+	0.753951i
$8$	−2.82843			−0.353553
$9$		0			0
$10$		−	1.95546i		−	0.195546i
$11$	16.3713			1.48830			0.744149	−	0.668014i	$-0.232855\pi$
$11$	16.3713			1.48830			0.744149	+	0.668014i	$0.232855\pi$
$12$		0			0
$13$		−	18.5572i		−	1.42748i	−0.700410	−	0.713740i	$-0.747000\pi$
$13$		−	18.5572i		−	1.42748i	0.700410	−	0.713740i	$-0.253000\pi$
$14$	−6.50328	−	7.46374i	−0.464520	−	0.533124i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	1.67115i		−	0.0983028i	−0.998791	−	0.0491514i	$-0.984348\pi$
$17$		−	1.67115i		−	0.0983028i	0.998791	−	0.0491514i	$-0.0156517\pi$
$18$		0			0
$19$		−	28.9316i		−	1.52272i	−0.648330	−	0.761359i	$-0.724532\pi$
$19$		−	28.9316i		−	1.52272i	0.648330	−	0.761359i	$-0.275468\pi$
$20$			2.76543i			0.138272i
$21$		0			0
$22$	−23.1525			−1.05239
$23$	−24.5167			−1.06594			−0.532972	−	0.846133i	$-0.678925\pi$
$23$	−24.5167			−1.06594			−0.532972	+	0.846133i	$0.678925\pi$
$24$		0			0
$25$	23.0881			0.923524
$26$			26.2439i			1.00938i
$27$		0			0
$28$	9.19702	+	10.5553i	0.328465	+	0.376976i
$29$	−41.4015			−1.42764			−0.713820	−	0.700330i	$-0.753036\pi$
$29$	−41.4015			−1.42764			−0.713820	+	0.700330i	$0.753036\pi$
$30$		0			0
$31$		−	43.0220i		−	1.38781i	−0.720069	−	0.693903i	$-0.755890\pi$
$31$		−	43.0220i		−	1.38781i	0.720069	−	0.693903i	$-0.244110\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$			2.36336i			0.0695105i
$35$	−7.29751	+	6.35844i	−0.208500	+	0.181670i
$36$		0			0
$37$	−18.5784			−0.502118			−0.251059	−	0.967972i	$-0.580779\pi$
$37$	−18.5784			−0.502118			−0.251059	+	0.967972i	$0.580779\pi$
$38$			40.9155i			1.07672i
$39$		0			0
$40$		−	3.91091i		−	0.0977728i
$41$		−	11.3233i		−	0.276179i	−0.990420	−	0.138090i	$-0.955904\pi$
$41$		−	11.3233i		−	0.276179i	0.990420	−	0.138090i	$-0.0440962\pi$
$42$		0			0
$43$	57.9277			1.34716			0.673578	−	0.739116i	$-0.264757\pi$
$43$	57.9277			1.34716			0.673578	+	0.739116i	$0.264757\pi$
$44$	32.7425			0.744149
$45$		0			0
$46$	34.6719			0.753737
$47$			12.9768i			0.276101i	0.990425	+	0.138051i	$0.0440837\pi$
$47$			12.9768i			0.276101i	−0.990425	+	0.138051i	$0.955916\pi$
$48$		0			0
$49$	−6.70739	+	48.5388i	−0.136886	+	0.990587i
$50$	−32.6515			−0.653030
$51$		0			0
$52$		−	37.1145i		−	0.713740i
$53$	45.2169			0.853148			0.426574	−	0.904453i	$-0.359720\pi$
$53$	45.2169			0.853148			0.426574	+	0.904453i	$0.359720\pi$
$54$		0			0
$55$			22.6368i			0.411579i
$56$	−13.0066	−	14.9275i	−0.232260	−	0.266562i
$57$		0			0
$58$	58.5506			1.00949
$59$		−	24.9870i		−	0.423509i	−0.977323	−	0.211754i	$-0.932082\pi$
$59$		−	24.9870i		−	0.423509i	0.977323	−	0.211754i	$-0.0679177\pi$
$60$		0			0
$61$			16.0979i			0.263900i	0.991256	+	0.131950i	$0.0421239\pi$
$61$			16.0979i			0.263900i	−0.991256	+	0.131950i	$0.957876\pi$
$62$			60.8422i			0.981327i
$63$		0			0
$64$	8.00000			0.125000
$65$	25.6594			0.394760
$66$		0			0
$67$	44.2782			0.660869			0.330434	−	0.943829i	$-0.392805\pi$
$67$	44.2782			0.660869			0.330434	+	0.943829i	$0.392805\pi$
$68$		−	3.34229i		−	0.0491514i
$69$		0			0
$70$	10.3202	−	8.99219i	0.147432	−	0.128460i
$71$	−80.7195			−1.13690			−0.568448	−	0.822720i	$-0.692456\pi$
$71$	−80.7195			−1.13690			−0.568448	+	0.822720i	$0.692456\pi$
$72$		0			0
$73$		−	79.4212i		−	1.08796i	−0.839098	−	0.543981i	$-0.816917\pi$
$73$		−	79.4212i		−	1.08796i	0.839098	−	0.543981i	$-0.183083\pi$
$74$	26.2738			0.355051
$75$		0			0
$76$		−	57.8633i		−	0.761359i
$77$	75.2835	+	86.4020i	0.977708	+	1.12210i
$78$		0			0
$79$	−39.2311			−0.496596			−0.248298	−	0.968684i	$-0.579871\pi$
$79$	−39.2311			−0.496596			−0.248298	+	0.968684i	$0.579871\pi$
$80$			5.53087i			0.0691358i
$81$		0			0
$82$			16.0136i			0.195288i
$83$		−	115.894i		−	1.39631i	−0.715947	−	0.698155i	$-0.754005\pi$
$83$		−	115.894i		−	1.39631i	0.715947	−	0.698155i	$-0.245995\pi$
$84$		0			0
$85$	2.31072			0.0271850
$86$	−81.9221			−0.952583
$87$		0			0
$88$	−46.3050			−0.526193
$89$		−	30.2467i		−	0.339850i	−0.985457	−	0.169925i	$-0.945647\pi$
$89$		−	30.2467i		−	0.339850i	0.985457	−	0.169925i	$-0.0543526\pi$
$90$		0			0
$91$	97.9389	−	85.3357i	1.07625	−	0.937755i
$92$	−49.0335			−0.532972
$93$		0			0
$94$		−	18.3519i		−	0.195233i
$95$	40.0043			0.421098
$96$		0			0
$97$		−	82.7997i		−	0.853605i	−0.904345	−	0.426803i	$-0.859640\pi$
$97$		−	82.7997i		−	0.853605i	0.904345	−	0.426803i	$-0.140360\pi$
$98$	9.48569	−	68.6442i	0.0967927	−	0.700451i
$99$		0			0
$100$	46.1762			0.461762
$101$			131.198i			1.29899i	0.760367	+	0.649493i	$0.225019\pi$
$101$			131.198i			1.29899i	−0.760367	+	0.649493i	$0.774981\pi$
$102$		0			0
$103$		−	17.1016i		−	0.166035i	−0.996548	−	0.0830174i	$-0.973544\pi$
$103$		−	17.1016i		−	0.166035i	0.996548	−	0.0830174i	$-0.0264557\pi$
$104$			52.4878i			0.504691i
$105$		0			0
$106$	−63.9463			−0.603267
$107$	−19.9716			−0.186650			−0.0933252	−	0.995636i	$-0.529750\pi$
$107$	−19.9716			−0.186650			−0.0933252	+	0.995636i	$0.529750\pi$
$108$		0			0
$109$	−97.1390			−0.891184			−0.445592	−	0.895236i	$-0.647007\pi$
$109$	−97.1390			−0.891184			−0.445592	+	0.895236i	$0.647007\pi$
$110$		−	32.0133i		−	0.291030i
$111$		0			0
$112$	18.3940	+	21.1106i	0.164233	+	0.188488i
$113$	79.4528			0.703122			0.351561	−	0.936165i	$-0.385651\pi$
$113$	79.4528			0.703122			0.351561	+	0.936165i	$0.385651\pi$
$114$		0			0
$115$		−	33.8997i		−	0.294780i
$116$	−82.8031			−0.713820
$117$		0			0
$118$			35.3370i			0.299466i
$119$	8.81974	−	7.68479i	0.0741155	−	0.0645780i
$120$		0			0
$121$	147.019			1.21503
$122$		−	22.7659i		−	0.186606i
$123$		0			0
$124$		−	86.0439i		−	0.693903i
$125$			66.4922i			0.531938i
$126$		0			0
$127$	69.1372			0.544388			0.272194	−	0.962242i	$-0.412251\pi$
$127$	69.1372			0.544388			0.272194	+	0.962242i	$0.412251\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	−36.2879			−0.279138
$131$			117.754i			0.898885i	0.893309	+	0.449443i	$0.148377\pi$
$131$			117.754i			0.898885i	−0.893309	+	0.449443i	$0.851623\pi$
$132$		0			0
$133$	152.691	−	133.042i	1.14806	−	1.00032i
$134$	−62.6188			−0.467305
$135$		0			0
$136$			4.72672i			0.0347553i
$137$	−1.91179			−0.0139547			−0.00697733	−	0.999976i	$-0.502221\pi$
$137$	−1.91179			−0.0139547			−0.00697733	+	0.999976i	$0.502221\pi$
$138$		0			0
$139$		−	251.133i		−	1.80671i	−0.428892	−	0.903356i	$-0.641096\pi$
$139$		−	251.133i		−	1.80671i	0.428892	−	0.903356i	$-0.358904\pi$
$140$	−14.5950	+	12.7169i	−0.104250	+	0.0908348i
$141$		0			0
$142$	114.155			0.803906
$143$		−	303.806i		−	2.12452i
$144$		0			0
$145$		−	57.2466i		−	0.394804i
$146$			112.318i			0.769305i
$147$		0			0
$148$	−37.1567			−0.251059
$149$	156.226			1.04850			0.524248	−	0.851566i	$-0.324347\pi$
$149$	156.226			1.04850			0.524248	+	0.851566i	$0.324347\pi$
$150$		0			0
$151$	27.0753			0.179306			0.0896532	−	0.995973i	$-0.471424\pi$
$151$	27.0753			0.179306			0.0896532	+	0.995973i	$0.471424\pi$
$152$			81.8311i			0.538362i
$153$		0			0
$154$	−106.467	−	122.191i	−0.691344	−	0.793448i
$155$	59.4872			0.383788
$156$		0			0
$157$			149.775i			0.953983i	0.878908	+	0.476992i	$0.158273\pi$
$157$			149.775i			0.953983i	−0.878908	+	0.476992i	$0.841727\pi$
$158$	55.4812			0.351147
$159$		0			0
$160$		−	7.82183i		−	0.0488864i
$161$	−112.740	−	129.391i	−0.700251	−	0.803671i
$162$		0			0
$163$	105.699			0.648463			0.324231	−	0.945978i	$-0.394894\pi$
$163$	105.699			0.648463			0.324231	+	0.945978i	$0.394894\pi$
$164$		−	22.6467i		−	0.138090i
$165$		0			0
$166$			163.898i			0.987340i
$167$		−	259.204i		−	1.55212i	−0.630659	−	0.776060i	$-0.717216\pi$
$167$		−	259.204i		−	1.55212i	0.630659	−	0.776060i	$-0.282784\pi$
$168$		0			0
$169$	−175.371			−1.03770
$170$	−3.26786			−0.0192227
$171$		0			0
$172$	115.855			0.673578
$173$		−	235.867i		−	1.36339i	−0.731636	−	0.681696i	$-0.761243\pi$
$173$		−	235.867i		−	1.36339i	0.731636	−	0.681696i	$-0.238757\pi$
$174$		0			0
$175$	106.171	+	121.851i	0.606691	+	0.696292i
$176$	65.4851			0.372074
$177$		0			0
$178$			42.7753i			0.240311i
$179$	238.022			1.32973			0.664865	−	0.746963i	$-0.268489\pi$
$179$	238.022			1.32973			0.664865	+	0.746963i	$0.268489\pi$
$180$		0			0
$181$			163.801i			0.904980i	0.891770	+	0.452490i	$0.149464\pi$
$181$			163.801i			0.904980i	−0.891770	+	0.452490i	$0.850536\pi$
$182$	−138.506	+	120.683i	−0.761024	+	0.663093i
$183$		0			0
$184$	69.3438			0.376868
$185$		−	25.6886i		−	0.138857i
$186$		0			0
$187$		−	27.3588i		−	0.146304i
$188$			25.9535i			0.138051i
$189$		0			0
$190$	−56.5746			−0.297761
$191$	253.517			1.32731			0.663656	−	0.748038i	$-0.269004\pi$
$191$	253.517			1.32731			0.663656	+	0.748038i	$0.269004\pi$
$192$		0			0
$193$	31.1312			0.161301			0.0806507	−	0.996742i	$-0.474300\pi$
$193$	31.1312			0.161301			0.0806507	+	0.996742i	$0.474300\pi$
$194$			117.096i			0.603590i
$195$		0			0
$196$	−13.4148	+	97.0775i	−0.0684428	+	0.495293i
$197$	197.604			1.00307			0.501533	−	0.865138i	$-0.332770\pi$
$197$	197.604			1.00307			0.501533	+	0.865138i	$0.332770\pi$
$198$		0			0
$199$			161.524i			0.811680i	0.913944	+	0.405840i	$0.133021\pi$
$199$			161.524i			0.811680i	−0.913944	+	0.405840i	$0.866979\pi$
$200$	−65.3030			−0.326515
$201$		0			0
$202$		−	185.542i		−	0.918522i
$203$	−190.385	−	218.503i	−0.937859	−	1.07637i
$204$		0			0
$205$	15.6570			0.0763755
$206$			24.1853i			0.117404i
$207$		0			0
$208$		−	74.2290i		−	0.356870i
$209$		−	473.648i		−	2.26626i
$210$		0			0
$211$	14.4107			0.0682971			0.0341486	−	0.999417i	$-0.489128\pi$
$211$	14.4107			0.0682971			0.0341486	+	0.999417i	$0.489128\pi$
$212$	90.4337			0.426574
$213$		0			0
$214$	28.2441			0.131982
$215$			80.0976i			0.372547i
$216$		0			0
$217$	227.055	−	197.837i	1.04634	−	0.911691i
$218$	137.375			0.630162
$219$		0			0
$220$			45.2737i			0.205789i
$221$	−31.0119			−0.140325
$222$		0			0
$223$			187.618i			0.841337i	0.907214	+	0.420668i	$0.138204\pi$
$223$			187.618i			0.841337i	−0.907214	+	0.420668i	$0.861796\pi$
$224$	−26.0131	−	29.8550i	−0.116130	−	0.133281i
$225$		0			0
$226$	−112.363			−0.497182
$227$			276.147i			1.21650i	0.793744	+	0.608252i	$0.208129\pi$
$227$			276.147i			1.21650i	−0.793744	+	0.608252i	$0.791871\pi$
$228$		0			0
$229$		−	361.226i		−	1.57741i	−0.614775	−	0.788703i	$-0.710753\pi$
$229$		−	361.226i		−	1.57741i	0.614775	−	0.788703i	$-0.289247\pi$
$230$			47.9414i			0.208441i
$231$		0			0
$232$	117.101			0.504747
$233$	313.585			1.34586			0.672929	−	0.739707i	$-0.265036\pi$
$233$	313.585			1.34586			0.672929	+	0.739707i	$0.265036\pi$
$234$		0			0
$235$	−17.9432			−0.0763540
$236$		−	49.9740i		−	0.211754i
$237$		0			0
$238$	−12.4730	+	10.8679i	−0.0524076	+	0.0456636i
$239$	−401.379			−1.67941			−0.839705	−	0.543043i	$-0.817272\pi$
$239$	−401.379			−1.67941			−0.839705	+	0.543043i	$0.817272\pi$
$240$		0			0
$241$			99.4418i			0.412622i	0.978487	+	0.206311i	$0.0661458\pi$
$241$			99.4418i			0.412622i	−0.978487	+	0.206311i	$0.933854\pi$
$242$	−207.916			−0.859156
$243$		0			0
$244$			32.1958i			0.131950i
$245$	−67.1153	−	9.27443i	−0.273940	−	0.0378548i
$246$		0			0
$247$	−536.892			−2.17365
$248$			121.684i			0.490663i
$249$		0			0
$250$		−	94.0342i		−	0.376137i
$251$			450.528i			1.79493i	0.441082	+	0.897467i	$0.354595\pi$
$251$			450.528i			1.79493i	−0.441082	+	0.897467i	$0.645405\pi$
$252$		0			0
$253$	−401.370			−1.58644
$254$	−97.7748			−0.384940
$255$		0			0
$256$	16.0000			0.0625000
$257$		−	367.887i		−	1.43147i	−0.698373	−	0.715734i	$-0.746092\pi$
$257$		−	367.887i		−	1.43147i	0.698373	−	0.715734i	$-0.253908\pi$
$258$		0			0
$259$	−85.4328	−	98.0503i	−0.329857	−	0.378573i
$260$	51.3188			0.197380
$261$		0			0
$262$		−	166.529i		−	0.635608i
$263$	37.9448			0.144277			0.0721385	−	0.997395i	$-0.477018\pi$
$263$	37.9448			0.144277			0.0721385	+	0.997395i	$0.477018\pi$
$264$		0			0
$265$			62.5221i			0.235932i
$266$	−215.938	+	188.151i	−0.811798	+	0.707333i
$267$		0			0
$268$	88.5564			0.330434
$269$			25.3987i			0.0944190i	0.998885	+	0.0472095i	$0.0150328\pi$
$269$			25.3987i			0.0944190i	−0.998885	+	0.0472095i	$0.984967\pi$
$270$		0			0
$271$			171.660i			0.633432i	0.948520	+	0.316716i	$0.102580\pi$
$271$			171.660i			0.633432i	−0.948520	+	0.316716i	$0.897420\pi$
$272$		−	6.68459i		−	0.0245757i
$273$		0			0
$274$	2.70368			0.00986743
$275$	377.982			1.37448
$276$		0			0
$277$	−190.029			−0.686027			−0.343013	−	0.939331i	$-0.611448\pi$
$277$	−190.029			−0.686027			−0.343013	+	0.939331i	$0.611448\pi$
$278$			355.156i			1.27754i
$279$		0			0
$280$	20.6405	−	17.9844i	0.0737160	−	0.0642299i
$281$	−104.297			−0.371164			−0.185582	−	0.982629i	$-0.559417\pi$
$281$	−104.297			−0.371164			−0.185582	+	0.982629i	$0.559417\pi$
$282$		0			0
$283$			151.697i			0.536032i	0.963414	+	0.268016i	$0.0863681\pi$
$283$			151.697i			0.536032i	−0.963414	+	0.268016i	$0.913632\pi$
$284$	−161.439			−0.568448
$285$		0			0
$286$			429.646i			1.50226i
$287$	59.7608	−	52.0705i	0.208226	−	0.181430i
$288$		0			0
$289$	286.207			0.990337
$290$			80.9589i			0.279169i
$291$		0			0
$292$		−	158.842i		−	0.543981i
$293$		−	59.7746i		−	0.204009i	−0.994784	−	0.102004i	$-0.967474\pi$
$293$		−	59.7746i		−	0.204009i	0.994784	−	0.102004i	$-0.0325256\pi$
$294$		0			0
$295$	34.5500			0.117119
$296$	52.5476			0.177526
$297$		0			0
$298$	−220.937			−0.741398
$299$			454.963i			1.52162i
$300$		0			0
$301$	266.381	+	305.723i	0.884987	+	1.01569i
$302$	−38.2902			−0.126789
$303$		0			0
$304$		−	115.727i		−	0.380680i
$305$	−22.2589			−0.0729798
$306$		0			0
$307$			474.281i			1.54489i	0.635082	+	0.772445i	$0.280967\pi$
$307$			474.281i			1.54489i	−0.635082	+	0.772445i	$0.719033\pi$
$308$	150.567	+	172.804i	0.488854	+	0.561052i
$309$		0			0
$310$	−84.1276			−0.271379
$311$		−	104.558i		−	0.336200i	−0.985770	−	0.168100i	$-0.946237\pi$
$311$		−	104.558i		−	0.336200i	0.985770	−	0.168100i	$-0.0537632\pi$
$312$		0			0
$313$			288.084i			0.920397i	0.887816	+	0.460198i	$0.152222\pi$
$313$			288.084i			0.920397i	−0.887816	+	0.460198i	$0.847778\pi$
$314$		−	211.814i		−	0.674568i
$315$		0			0
$316$	−78.4622			−0.248298
$317$	123.488			0.389552			0.194776	−	0.980848i	$-0.437602\pi$
$317$	123.488			0.389552			0.194776	+	0.980848i	$0.437602\pi$
$318$		0			0
$319$	−677.796			−2.12475
$320$			11.0617i			0.0345679i
$321$		0			0
$322$	159.439	+	182.987i	0.495153	+	0.568281i
$323$	−48.3490			−0.149687
$324$		0			0
$325$		−	428.452i		−	1.31831i
$326$	−149.482			−0.458533
$327$		0			0
$328$			32.0272i			0.0976440i
$329$	−68.4870	+	59.6738i	−0.208167	+	0.181379i
$330$		0			0
$331$	400.180			1.20900			0.604502	−	0.796604i	$-0.293372\pi$
$331$	400.180			1.20900			0.604502	+	0.796604i	$0.293372\pi$
$332$		−	231.787i		−	0.698155i
$333$		0			0
$334$			366.570i			1.09751i
$335$			61.2242i			0.182759i
$336$		0			0
$337$	−344.372			−1.02187			−0.510937	−	0.859618i	$-0.670702\pi$
$337$	−344.372			−1.02187			−0.510937	+	0.859618i	$0.670702\pi$
$338$	248.013			0.733766
$339$		0			0
$340$	4.62144			0.0135925
$341$		−	704.324i		−	2.06547i
$342$		0			0
$343$	−287.015	+	187.807i	−0.836779	+	0.547541i
$344$	−163.844			−0.476292
$345$		0			0
$346$			333.566i			0.964064i
$347$	−90.3115			−0.260264			−0.130132	−	0.991497i	$-0.541540\pi$
$347$	−90.3115			−0.260264			−0.130132	+	0.991497i	$0.541540\pi$
$348$		0			0
$349$		−	147.918i		−	0.423834i	−0.977288	−	0.211917i	$-0.932029\pi$
$349$		−	147.918i		−	0.423834i	0.977288	−	0.211917i	$-0.0679707\pi$
$350$	−150.148	−	172.324i	−0.428995	−	0.492353i
$351$		0			0
$352$	−92.6099			−0.263096
$353$		−	352.158i		−	0.997615i	−0.866713	−	0.498808i	$-0.833771\pi$
$353$		−	352.158i		−	0.997615i	0.866713	−	0.498808i	$-0.166229\pi$
$354$		0			0
$355$		−	111.612i		−	0.314401i
$356$		−	60.4934i		−	0.169925i
$357$		0			0
$358$	−336.614			−0.940261
$359$	−122.724			−0.341850			−0.170925	−	0.985284i	$-0.554676\pi$
$359$	−122.724			−0.341850			−0.170925	+	0.985284i	$0.554676\pi$
$360$		0			0
$361$	−476.040			−1.31867
$362$		−	231.650i		−	0.639917i
$363$		0			0
$364$	195.878	−	170.671i	0.538126	−	0.468878i
$365$	109.817			0.300868
$366$		0			0
$367$			50.2068i			0.136803i	0.997658	+	0.0684017i	$0.0217899\pi$
$367$			50.2068i			0.136803i	−0.997658	+	0.0684017i	$0.978210\pi$
$368$	−98.0669			−0.266486
$369$		0			0
$370$			36.3292i			0.0981870i
$371$	207.930	+	238.639i	0.560459	+	0.643232i
$372$		0			0
$373$	528.187			1.41605			0.708025	−	0.706187i	$-0.249586\pi$
$373$	528.187			1.41605			0.708025	+	0.706187i	$0.249586\pi$
$374$			38.6912i			0.103452i
$375$		0			0
$376$		−	36.7038i		−	0.0976166i
$377$			768.299i			2.03793i
$378$		0			0
$379$	122.654			0.323624			0.161812	−	0.986822i	$-0.448266\pi$
$379$	122.654			0.323624			0.161812	+	0.986822i	$0.448266\pi$
$380$	80.0085			0.210549
$381$		0			0
$382$	−358.527			−0.938551
$383$			237.126i			0.619127i	0.950879	+	0.309563i	$0.100183\pi$
$383$			237.126i			0.619127i	−0.950879	+	0.309563i	$0.899817\pi$
$384$		0			0
$385$	−119.470	+	104.096i	−0.310310	+	0.270379i
$386$	−44.0261			−0.114057
$387$		0			0
$388$		−	165.599i		−	0.426803i
$389$	172.514			0.443480			0.221740	−	0.975106i	$-0.428826\pi$
$389$	172.514			0.443480			0.221740	+	0.975106i	$0.428826\pi$
$390$		0			0
$391$			40.9711i			0.104785i
$392$	18.9714	−	137.288i	0.0483964	−	0.350225i
$393$		0			0
$394$	−279.454			−0.709275
$395$		−	54.2455i		−	0.137330i
$396$		0			0
$397$			188.846i			0.475683i	0.971304	+	0.237842i	$0.0764400\pi$
$397$			188.846i			0.475683i	−0.971304	+	0.237842i	$0.923560\pi$
$398$		−	228.430i		−	0.573945i
$399$		0			0
$400$	92.3524			0.230881
$401$	−65.0129			−0.162127			−0.0810634	−	0.996709i	$-0.525832\pi$
$401$	−65.0129			−0.162127			−0.0810634	+	0.996709i	$0.525832\pi$
$402$		0			0
$403$	−798.369			−1.98107
$404$			262.395i			0.649493i
$405$		0			0
$406$	269.246	+	309.010i	0.663167	+	0.761109i
$407$	−304.152			−0.747301
$408$		0			0
$409$			439.887i			1.07552i	0.843098	+	0.537760i	$0.180729\pi$
$409$			439.887i			1.07552i	−0.843098	+	0.537760i	$0.819271\pi$
$410$	−22.1423			−0.0540056
$411$		0			0
$412$		−	34.2032i		−	0.0830174i
$413$	131.873	−	114.903i	0.319305	−	0.278216i
$414$		0			0
$415$	160.248			0.386140
$416$			104.976i			0.252345i
$417$		0			0
$418$			669.839i			1.60249i
$419$			613.028i			1.46307i	0.681802	+	0.731537i	$0.261197\pi$
$419$			613.028i			1.46307i	−0.681802	+	0.731537i	$0.738803\pi$
$420$		0			0
$421$	−314.404			−0.746803			−0.373402	−	0.927670i	$-0.621809\pi$
$421$	−314.404			−0.746803			−0.373402	+	0.927670i	$0.621809\pi$
$422$	−20.3798			−0.0482934
$423$		0			0
$424$	−127.893			−0.301633
$425$		−	38.5836i		−	0.0907849i
$426$		0			0
$427$	−84.9593	+	74.0264i	−0.198968	+	0.173364i
$428$	−39.9432			−0.0933252
$429$		0			0
$430$		−	113.275i		−	0.263431i
$431$	−545.839			−1.26645			−0.633224	−	0.773968i	$-0.718269\pi$
$431$	−545.839			−1.26645			−0.633224	+	0.773968i	$0.718269\pi$
$432$		0			0
$433$			243.306i			0.561908i	0.959721	+	0.280954i	$0.0906509\pi$
$433$			243.306i			0.561908i	−0.959721	+	0.280954i	$0.909349\pi$
$434$	−321.105	+	279.784i	−0.739873	+	0.644663i
$435$		0			0
$436$	−194.278			−0.445592
$437$			709.310i			1.62313i
$438$		0			0
$439$			786.048i			1.79054i	0.445522	+	0.895271i	$0.353018\pi$
$439$			786.048i			1.79054i	−0.445522	+	0.895271i	$0.646982\pi$
$440$		−	64.0266i		−	0.145515i
$441$		0			0
$442$	43.8574			0.0992250
$443$	−545.786			−1.23202			−0.616011	−	0.787737i	$-0.711252\pi$
$443$	−545.786			−1.23202			−0.616011	+	0.787737i	$0.711252\pi$
$444$		0			0
$445$	41.8226			0.0939834
$446$		−	265.332i		−	0.594915i
$447$		0			0
$448$	36.7881	+	42.2213i	0.0821163	+	0.0942439i
$449$	−789.261			−1.75782			−0.878910	−	0.476987i	$-0.841729\pi$
$449$	−789.261			−1.75782			−0.878910	+	0.476987i	$0.841729\pi$
$450$		0			0
$451$		−	185.378i		−	0.411037i
$452$	158.906			0.351561
$453$		0			0
$454$		−	390.530i		−	0.860199i
$455$	117.995	+	135.422i	0.259330	+	0.297630i
$456$		0			0
$457$	−88.2385			−0.193082			−0.0965411	−	0.995329i	$-0.530778\pi$
$457$	−88.2385			−0.193082			−0.0965411	+	0.995329i	$0.530778\pi$
$458$			510.851i			1.11539i
$459$		0			0
$460$		−	67.7994i		−	0.147390i
$461$			27.7852i			0.0602715i	0.999546	+	0.0301358i	$0.00959396\pi$
$461$			27.7852i			0.0602715i	−0.999546	+	0.0301358i	$0.990406\pi$
$462$		0			0
$463$	466.865			1.00835			0.504174	−	0.863602i	$-0.331797\pi$
$463$	466.865			1.00835			0.504174	+	0.863602i	$0.331797\pi$
$464$	−165.606			−0.356910
$465$		0			0
$466$	−443.476			−0.951666
$467$			163.433i			0.349963i	0.984572	+	0.174981i	$0.0559865\pi$
$467$			163.433i			0.349963i	−0.984572	+	0.174981i	$0.944014\pi$
$468$		0			0
$469$	203.614	+	233.685i	0.434145	+	0.498263i
$470$	25.3755			0.0539904
$471$		0			0
$472$			70.6740i			0.149733i
$473$	948.350			2.00497
$474$		0			0
$475$		−	667.977i		−	1.40627i
$476$	17.6395	−	15.3696i	0.0370578	−	0.0322890i
$477$		0			0
$478$	567.635			1.18752
$479$		−	224.810i		−	0.469332i	−0.972076	−	0.234666i	$-0.924600\pi$
$479$		−	224.810i		−	0.469332i	0.972076	−	0.234666i	$-0.0753997\pi$
$480$		0			0
$481$			344.763i			0.716764i
$482$		−	140.632i		−	0.291768i
$483$		0			0
$484$	294.037			0.607515
$485$	114.489			0.236059
$486$		0			0
$487$	−350.781			−0.720290			−0.360145	−	0.932896i	$-0.617273\pi$
$487$	−350.781			−0.720290			−0.360145	+	0.932896i	$0.617273\pi$
$488$		−	45.5318i		−	0.0933028i
$489$		0			0
$490$	94.9154	+	13.1160i	0.193705	+	0.0267674i
$491$	−734.288			−1.49550			−0.747748	−	0.663983i	$-0.768865\pi$
$491$	−734.288			−1.49550			−0.747748	+	0.663983i	$0.768865\pi$
$492$		0			0
$493$			69.1881i			0.140341i
$494$	759.280			1.53700
$495$		0			0
$496$		−	172.088i		−	0.346951i
$497$	−371.190	−	426.010i	−0.746861	−	0.857164i
$498$		0			0
$499$	935.150			1.87405			0.937024	−	0.349265i	$-0.113569\pi$
$499$	935.150			1.87405			0.937024	+	0.349265i	$0.113569\pi$
$500$			132.984i			0.265969i
$501$		0			0
$502$		−	637.143i		−	1.26921i
$503$		−	670.586i		−	1.33317i	−0.745427	−	0.666587i	$-0.767755\pi$
$503$		−	670.586i		−	1.33317i	0.745427	−	0.666587i	$-0.232245\pi$
$504$		0			0
$505$	−181.409			−0.359226
$506$	567.623			1.12178
$507$		0			0
$508$	138.274			0.272194
$509$		−	7.40802i		−	0.0145541i	−0.999974	−	0.00727703i	$-0.997684\pi$
$509$		−	7.40802i		−	0.0145541i	0.999974	−	0.00727703i	$-0.00231637\pi$
$510$		0			0
$511$	419.158	−	365.219i	0.820270	−	0.714714i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$			520.271i			1.01220i
$515$	23.6467			0.0459158
$516$		0			0
$517$			212.446i			0.410921i
$518$	120.820	+	138.664i	0.233244	+	0.267691i
$519$		0			0
$520$	−72.5758			−0.139569
$521$		−	855.964i		−	1.64292i	−0.570263	−	0.821462i	$-0.693159\pi$
$521$		−	855.964i		−	1.64292i	0.570263	−	0.821462i	$-0.306841\pi$
$522$		0			0
$523$		−	26.4383i		−	0.0505512i	−0.999681	−	0.0252756i	$-0.991954\pi$
$523$		−	26.4383i		−	0.0505512i	0.999681	−	0.0252756i	$-0.00804633\pi$
$524$			235.508i			0.449443i
$525$		0			0
$526$	−53.6621			−0.102019
$527$	−71.8960			−0.136425
$528$		0			0
$529$	72.0703			0.136239
$530$		−	88.4196i		−	0.166829i
$531$		0			0
$532$	305.383	−	266.085i	0.574028	−	0.500160i
$533$	−210.130			−0.394240
$534$		0			0
$535$		−	27.6151i		−	0.0516169i
$536$	−125.238			−0.233652
$537$		0			0
$538$		−	35.9192i		−	0.0667643i
$539$	−109.809	+	794.641i	−0.203727	+	1.47429i
$540$		0			0
$541$	−607.504			−1.12293			−0.561464	−	0.827501i	$-0.689762\pi$
$541$	−607.504			−1.12293			−0.561464	+	0.827501i	$0.689762\pi$
$542$		−	242.764i		−	0.447904i
$543$		0			0
$544$			9.45343i			0.0173776i
$545$		−	134.316i		−	0.246451i
$546$		0			0
$547$	−246.647			−0.450909			−0.225455	−	0.974254i	$-0.572387\pi$
$547$	−246.647			−0.450909			−0.225455	+	0.974254i	$0.572387\pi$
$548$	−3.82358			−0.00697733
$549$		0			0
$550$	−534.547			−0.971903
$551$			1197.81i			2.17389i
$552$		0			0
$553$	−180.405	−	207.048i	−0.326229	−	0.374409i
$554$	268.742			0.485094
$555$		0			0
$556$		−	502.266i		−	0.903356i
$557$	947.646			1.70134			0.850670	−	0.525700i	$-0.176197\pi$
$557$	947.646			1.70134			0.850670	+	0.525700i	$0.176197\pi$
$558$		0			0
$559$		−	1074.98i		−	1.92304i
$560$	−29.1900	+	25.4338i	−0.0521251	+	0.0454174i
$561$		0			0
$562$	147.499			0.262453
$563$		−	181.336i		−	0.322089i	−0.986947	−	0.161045i	$-0.948514\pi$
$563$		−	181.336i		−	0.322089i	0.986947	−	0.161045i	$-0.0514863\pi$
$564$		0			0
$565$			109.861i			0.194444i
$566$		−	214.532i		−	0.379032i
$567$		0			0
$568$	228.309			0.401953
$569$	578.055			1.01591			0.507957	−	0.861383i	$-0.330401\pi$
$569$	578.055			1.01591			0.507957	+	0.861383i	$0.330401\pi$
$570$		0			0
$571$	−753.049			−1.31883			−0.659413	−	0.751781i	$-0.729195\pi$
$571$	−753.049			−1.31883			−0.659413	+	0.751781i	$0.729195\pi$
$572$		−	607.612i		−	1.06226i
$573$		0			0
$574$	−84.5145	+	73.6388i	−0.147238	+	0.128291i
$575$	−566.045			−0.984426
$576$		0			0
$577$			11.7220i			0.0203154i	0.999948	+	0.0101577i	$0.00323335\pi$
$577$			11.7220i			0.0203154i	−0.999948	+	0.0101577i	$0.996767\pi$
$578$	−404.758			−0.700274
$579$		0			0
$580$		−	114.493i		−	0.197402i
$581$	611.648	−	532.938i	1.05275	−	0.917278i
$582$		0			0
$583$	740.258			1.26974
$584$			224.637i			0.384652i
$585$		0			0
$586$			84.5340i			0.144256i
$587$		−	619.395i		−	1.05519i	−0.849497	−	0.527594i	$-0.823094\pi$
$587$		−	619.395i		−	1.05519i	0.849497	−	0.527594i	$-0.176906\pi$
$588$		0			0
$589$	−1244.70			−2.11324
$590$	−48.8610			−0.0828153
$591$		0			0
$592$	−74.3135			−0.125530
$593$		−	242.416i		−	0.408797i	−0.978888	−	0.204398i	$-0.934476\pi$
$593$		−	242.416i		−	0.408797i	0.978888	−	0.204398i	$-0.0655238\pi$
$594$		0			0
$595$	10.6259	+	12.1952i	0.0178586	+	0.0204961i
$596$	312.452			0.524248
$597$		0			0
$598$		−	643.415i		−	1.07594i
$599$	98.2526			0.164028			0.0820139	−	0.996631i	$-0.473865\pi$
$599$	98.2526			0.164028			0.0820139	+	0.996631i	$0.473865\pi$
$600$		0			0
$601$			148.886i			0.247730i	0.992299	+	0.123865i	$0.0395290\pi$
$601$			148.886i			0.247730i	−0.992299	+	0.123865i	$0.960471\pi$
$602$	−376.720	−	432.357i	−0.625781	−	0.718201i
$603$		0			0
$604$	54.1505			0.0896532
$605$			203.285i			0.336008i
$606$		0			0
$607$		−	1194.81i		−	1.96839i	−0.177080	−	0.984196i	$-0.556665\pi$
$607$		−	1194.81i		−	1.96839i	0.177080	−	0.984196i	$-0.443335\pi$
$608$			163.662i			0.269181i
$609$		0			0
$610$	31.4788			0.0516045
$611$	240.813			0.394129
$612$		0			0
$613$	333.307			0.543731			0.271866	−	0.962335i	$-0.412359\pi$
$613$	333.307			0.543731			0.271866	+	0.962335i	$0.412359\pi$
$614$		−	670.735i		−	1.09240i
$615$		0			0
$616$	−212.934	−	244.382i	−0.345672	−	0.396724i
$617$	361.482			0.585871			0.292935	−	0.956132i	$-0.405368\pi$
$617$	361.482			0.585871			0.292935	+	0.956132i	$0.405368\pi$
$618$		0			0
$619$			186.138i			0.300707i	0.988632	+	0.150353i	$0.0480412\pi$
$619$			186.138i			0.300707i	−0.988632	+	0.150353i	$0.951959\pi$
$620$	118.974			0.191894
$621$		0			0
$622$			147.868i			0.237730i
$623$	159.632	−	139.090i	0.256231	−	0.223258i
$624$		0			0
$625$	485.262			0.776420
$626$		−	407.413i		−	0.650819i
$627$		0			0
$628$			299.551i			0.476992i
$629$			31.0472i			0.0493596i
$630$		0			0
$631$	72.1705			0.114375			0.0571874	−	0.998363i	$-0.481787\pi$
$631$	72.1705			0.114375			0.0571874	+	0.998363i	$0.481787\pi$
$632$	110.962			0.175573
$633$		0			0
$634$	−174.638			−0.275455
$635$			95.5972i			0.150547i
$636$		0			0
$637$	900.746	+	124.471i	1.41404	+	0.195402i
$638$	958.548			1.50243
$639$		0			0
$640$		−	15.6437i		−	0.0244432i
$641$	882.583			1.37688			0.688442	−	0.725291i	$-0.258295\pi$
$641$	882.583			1.37688			0.688442	+	0.725291i	$0.258295\pi$
$642$		0			0
$643$			1027.08i			1.59732i	0.601782	+	0.798660i	$0.294458\pi$
$643$			1027.08i			1.59732i	−0.601782	+	0.798660i	$0.705542\pi$
$644$	−225.481	−	258.782i	−0.350126	−	0.401835i
$645$		0			0
$646$	68.3759			0.105845
$647$			1016.79i			1.57154i	0.618516	+	0.785772i	$0.287734\pi$
$647$			1016.79i			1.57154i	−0.618516	+	0.785772i	$0.712266\pi$
$648$		0			0
$649$		−	409.069i		−	0.630307i
$650$			605.922i			0.932188i
$651$		0			0
$652$	211.399			0.324231
$653$	−765.713			−1.17261			−0.586304	−	0.810091i	$-0.699418\pi$
$653$	−765.713			−1.17261			−0.586304	+	0.810091i	$0.699418\pi$
$654$		0			0
$655$	−162.820			−0.248581
$656$		−	45.2934i		−	0.0690448i
$657$		0			0
$658$	96.8552	−	84.3915i	0.147196	−	0.128255i
$659$	−909.582			−1.38025			−0.690123	−	0.723692i	$-0.742444\pi$
$659$	−909.582			−1.38025			−0.690123	+	0.723692i	$0.742444\pi$
$660$		0			0
$661$		−	719.961i		−	1.08920i	−0.838696	−	0.544600i	$-0.816681\pi$
$661$		−	719.961i		−	1.08920i	0.838696	−	0.544600i	$-0.183319\pi$
$662$	−565.940			−0.854895
$663$		0			0
$664$			327.797i			0.493670i
$665$	183.960	+	211.129i	0.276632	+	0.317487i
$666$		0			0
$667$	1015.03			1.52179
$668$		−	518.408i		−	0.776060i
$669$		0			0
$670$		−	86.5841i		−	0.129230i
$671$			263.543i			0.392762i
$672$		0			0
$673$	908.284			1.34961			0.674803	−	0.737998i	$-0.264229\pi$
$673$	908.284			1.34961			0.674803	+	0.737998i	$0.264229\pi$
$674$	487.015			0.722574
$675$		0			0
$676$	−350.743			−0.518851
$677$		−	146.915i		−	0.217010i	−0.994096	−	0.108505i	$-0.965394\pi$
$677$		−	146.915i		−	0.217010i	0.994096	−	0.108505i	$-0.0346063\pi$
$678$		0			0
$679$	436.989	−	380.755i	0.643577	−	0.560759i
$680$	−6.53571			−0.00961134
$681$		0			0
$682$			996.065i			1.46051i
$683$	−532.770			−0.780044			−0.390022	−	0.920805i	$-0.627533\pi$
$683$	−532.770			−0.780044			−0.390022	+	0.920805i	$0.627533\pi$
$684$		0			0
$685$		−	2.64346i		−	0.00385907i
$686$	405.901	−	265.599i	0.591692	−	0.387170i
$687$		0			0
$688$	231.711			0.336789
$689$		−	839.101i		−	1.21785i
$690$		0			0
$691$			38.9893i			0.0564244i	0.999602	+	0.0282122i	$0.00898142\pi$
$691$			38.9893i			0.0564244i	−0.999602	+	0.0282122i	$0.991019\pi$
$692$		−	471.734i		−	0.681696i
$693$		0			0
$694$	127.720			0.184034
$695$	347.246			0.499634
$696$		0			0
$697$	−18.9230			−0.0271492
$698$			209.188i			0.299696i
$699$		0			0
$700$	212.342	+	243.702i	0.303345	+	0.348146i
$701$	−508.129			−0.724863			−0.362432	−	0.932010i	$-0.618053\pi$
$701$	−508.129			−0.724863			−0.362432	+	0.932010i	$0.618053\pi$
$702$		0			0
$703$			537.503i			0.764584i
$704$	130.970			0.186037
$705$		0			0
$706$			498.027i			0.705421i
$707$	−692.417	+	603.314i	−0.979373	+	0.853344i
$708$		0			0
$709$	607.768			0.857219			0.428609	−	0.903490i	$-0.359004\pi$
$709$	607.768			0.857219			0.428609	+	0.903490i	$0.359004\pi$
$710$			157.844i			0.222315i
$711$		0			0
$712$			85.5506i			0.120155i
$713$			1054.76i			1.47932i
$714$		0			0
$715$	420.077			0.587521
$716$	476.043			0.664865
$717$		0			0
$718$	173.558			0.241725
$719$		−	313.462i		−	0.435969i	−0.975952	−	0.217985i	$-0.930052\pi$
$719$		−	313.462i		−	0.435969i	0.975952	−	0.217985i	$-0.0699483\pi$
$720$		0			0
$721$	90.2564	−	78.6419i	0.125182	−	0.109073i
$722$	673.223			0.932441
$723$		0			0
$724$			327.603i			0.452490i
$725$	−955.883			−1.31846
$726$		0			0
$727$			729.179i			1.00300i	0.865158	+	0.501499i	$0.167218\pi$
$727$			729.179i			1.00300i	−0.865158	+	0.501499i	$0.832782\pi$
$728$	−277.013	+	241.366i	−0.380512	+	0.331546i
$729$		0			0
$730$	−155.305			−0.212746
$731$		−	96.8057i		−	0.132429i
$732$		0			0
$733$		−	345.407i		−	0.471224i	−0.971847	−	0.235612i	$-0.924291\pi$
$733$		−	345.407i		−	0.471224i	0.971847	−	0.235612i	$-0.0757094\pi$
$734$		−	71.0032i		−	0.0967346i
$735$		0			0
$736$	138.688			0.188434
$737$	724.891			0.983569
$738$		0			0
$739$	−1005.34			−1.36041			−0.680203	−	0.733024i	$-0.738108\pi$
$739$	−1005.34			−1.36041			−0.680203	+	0.733024i	$0.738108\pi$
$740$		−	51.3772i		−	0.0694287i
$741$		0			0
$742$	−294.058	−	337.487i	−0.396304	−	0.454834i
$743$	139.031			0.187122			0.0935609	−	0.995614i	$-0.470175\pi$
$743$	139.031			0.187122			0.0935609	+	0.995614i	$0.470175\pi$
$744$		0			0
$745$			216.016i			0.289954i
$746$	−746.969			−1.00130
$747$		0			0
$748$		−	54.7176i		−	0.0731519i
$749$	−91.8396	−	105.403i	−0.122616	−	0.140725i
$750$		0			0
$751$	35.6531			0.0474741			0.0237371	−	0.999718i	$-0.492444\pi$
$751$	35.6531			0.0474741			0.0237371	+	0.999718i	$0.492444\pi$
$752$			51.9071i			0.0690254i
$753$		0			0
$754$		−	1086.54i		−	1.44103i
$755$			37.4374i			0.0495860i
$756$		0			0
$757$	719.557			0.950538			0.475269	−	0.879841i	$-0.342351\pi$
$757$	719.557			0.950538			0.475269	+	0.879841i	$0.342351\pi$
$758$	−173.458			−0.228837
$759$		0			0
$760$	−113.149			−0.148880
$761$			186.180i			0.244651i	0.992490	+	0.122326i	$0.0390352\pi$
$761$			186.180i			0.244651i	−0.992490	+	0.122326i	$0.960965\pi$
$762$		0			0
$763$	−446.695	−	512.667i	−0.585445	−	0.671909i
$764$	507.033			0.663656
$765$		0			0
$766$		−	335.346i		−	0.437789i
$767$	−463.690			−0.604551
$768$		0			0
$769$		−	61.2943i		−	0.0797065i	−0.999206	−	0.0398533i	$-0.987311\pi$
$769$		−	61.2943i		−	0.0797065i	0.999206	−	0.0398533i	$-0.0126890\pi$
$770$	168.955	−	147.214i	0.219423	−	0.191186i
$771$		0			0
$772$	62.2623			0.0806507
$773$			484.245i			0.626449i	0.949679	+	0.313225i	$0.101409\pi$
$773$			484.245i			0.626449i	−0.949679	+	0.313225i	$0.898591\pi$
$774$		0			0
$775$		−	993.295i		−	1.28167i
$776$			234.193i			0.301795i
$777$		0			0
$778$	−243.971			−0.313588
$779$	−327.603			−0.420543
$780$		0			0
$781$	−1321.48			−1.69204
$782$		−	57.9418i		−	0.0740944i
$783$		0			0
$784$	−26.8296	+	194.155i	−0.0342214	+	0.247647i
$785$	−207.097			−0.263818
$786$		0			0
$787$		−	679.207i		−	0.863034i	−0.902105	−	0.431517i	$-0.857979\pi$
$787$		−	679.207i		−	0.863034i	0.902105	−	0.431517i	$-0.142021\pi$
$788$	395.208			0.501533
$789$		0			0
$790$			76.7147i			0.0971073i
$791$	365.364	+	419.325i	0.461902	+	0.530120i
$792$		0			0
$793$	298.733			0.376712
$794$		−	267.069i		−	0.336359i
$795$		0			0
$796$			323.049i			0.405840i
$797$		−	1481.08i		−	1.85832i	−0.369677	−	0.929160i	$-0.620532\pi$
$797$		−	1481.08i		−	1.85832i	0.369677	−	0.929160i	$-0.379468\pi$
$798$		0			0
$799$	21.6861			0.0271415
$800$	−130.606			−0.163257
$801$		0			0
$802$	91.9421			0.114641
$803$		−	1300.23i		−	1.61921i
$804$		0			0
$805$	178.911	−	155.888i	0.222250	−	0.193650i
$806$	1129.06			1.40082
$807$		0			0
$808$		−	371.083i		−	0.459261i
$809$	178.357			0.220466			0.110233	−	0.993906i	$-0.464840\pi$
$809$	178.357			0.220466			0.110233	+	0.993906i	$0.464840\pi$
$810$		0			0
$811$		−	1087.09i		−	1.34043i	−0.742166	−	0.670216i	$-0.766201\pi$
$811$		−	1087.09i		−	1.34043i	0.742166	−	0.670216i	$-0.233799\pi$
$812$	−380.771	−	437.007i	−0.468930	−	0.538185i
$813$		0			0
$814$	430.135			0.528422
$815$			146.152i			0.179328i
$816$		0			0
$817$		−	1675.94i		−	2.05134i
$818$		−	622.095i		−	0.760507i
$819$		0			0
$820$	31.3139			0.0381877
$821$	−726.271			−0.884618			−0.442309	−	0.896863i	$-0.645841\pi$
$821$	−726.271			−0.884618			−0.442309	+	0.896863i	$0.645841\pi$
$822$		0			0
$823$	−755.729			−0.918261			−0.459130	−	0.888369i	$-0.651839\pi$
$823$	−755.729			−0.918261			−0.459130	+	0.888369i	$0.651839\pi$
$824$			48.3706i			0.0587022i
$825$		0			0
$826$	−186.497	+	162.498i	−0.225783	+	0.196728i
$827$	890.828			1.07718			0.538590	−	0.842568i	$-0.318957\pi$
$827$	890.828			1.07718			0.538590	+	0.842568i	$0.318957\pi$
$828$		0			0
$829$			1129.80i			1.36285i	0.731888	+	0.681425i	$0.238640\pi$
$829$			1129.80i			1.36285i	−0.731888	+	0.681425i	$0.761360\pi$
$830$	−226.625			−0.273042
$831$		0			0
$832$		−	148.458i		−	0.178435i
$833$	81.1154	+	11.2090i	0.0973774	+	0.0134562i
$834$		0			0
$835$	358.406			0.429228
$836$		−	947.296i		−	1.13313i
$837$		0			0
$838$		−	866.952i		−	1.03455i
$839$			946.196i			1.12777i	0.825855	+	0.563883i	$0.190693\pi$
$839$			946.196i			1.12777i	−0.825855	+	0.563883i	$0.809307\pi$
$840$		0			0
$841$	873.088			1.03815
$842$	444.635			0.528070
$843$		0			0
$844$	28.8214			0.0341486
$845$		−	242.489i		−	0.286969i
$846$		0			0
$847$	676.067	+	775.914i	0.798190	+	0.916074i
$848$	180.867			0.213287
$849$		0			0
$850$			54.5654i			0.0641946i
$851$	455.481			0.535230
$852$		0			0
$853$		−	1657.51i		−	1.94316i	−0.236719	−	0.971578i	$-0.576072\pi$
$853$		−	1657.51i		−	1.94316i	0.236719	−	0.971578i	$-0.423928\pi$
$854$	120.151	−	104.689i	0.140692	−	0.122587i
$855$		0			0
$856$	56.4882			0.0659909
$857$			72.5068i			0.0846053i	0.999105	+	0.0423027i	$0.0134694\pi$
$857$			72.5068i			0.0846053i	−0.999105	+	0.0423027i	$0.986531\pi$
$858$		0			0
$859$			1229.82i			1.43169i	0.698259	+	0.715845i	$0.253958\pi$
$859$			1229.82i			1.43169i	−0.698259	+	0.715845i	$0.746042\pi$
$860$			160.195i			0.186274i
$861$		0			0
$862$	771.933			0.895514
$863$	−1190.72			−1.37974			−0.689871	−	0.723932i	$-0.742333\pi$
$863$	−1190.72			−1.37974			−0.689871	+	0.723932i	$0.742333\pi$
$864$		0			0
$865$	326.137			0.377037
$866$		−	344.087i		−	0.397329i
$867$		0			0
$868$	454.111	−	395.674i	0.523169	−	0.455846i
$869$	−642.263			−0.739083
$870$		0			0
$871$		−	821.682i		−	0.943377i
$872$	274.751			0.315081
$873$		0			0
$874$		−	1003.12i		−	1.14773i
$875$	−350.923	+	305.765i	−0.401055	+	0.349446i
$876$		0			0
$877$	630.661			0.719111			0.359556	−	0.933124i	$-0.382928\pi$
$877$	630.661			0.719111			0.359556	+	0.933124i	$0.382928\pi$
$878$		−	1111.64i		−	1.26610i
$879$		0			0
$880$			90.5473i			0.102895i
$881$		−	1025.23i		−	1.16372i	−0.813290	−	0.581858i	$-0.802326\pi$
$881$		−	1025.23i		−	1.16372i	0.813290	−	0.581858i	$-0.197674\pi$
$882$		0			0
$883$	−571.513			−0.647241			−0.323620	−	0.946187i	$-0.604900\pi$
$883$	−571.513			−0.647241			−0.323620	+	0.946187i	$0.604900\pi$
$884$	−62.0238			−0.0701626
$885$		0			0
$886$	771.858			0.871172
$887$			1033.52i			1.16519i	0.812763	+	0.582594i	$0.197962\pi$
$887$			1033.52i			1.16519i	−0.812763	+	0.582594i	$0.802038\pi$
$888$		0			0
$889$	317.928	+	364.883i	0.357625	+	0.410442i
$890$	−59.1461			−0.0664563
$891$		0			0
$892$			375.236i			0.420668i
$893$	375.439			0.420425
$894$		0			0
$895$			329.117i			0.367728i
$896$	−52.0262	−	59.7099i	−0.0580650	−	0.0666405i
$897$		0			0
$898$	1116.18			1.24297
$899$			1781.18i			1.98129i
$900$		0			0
$901$		−	75.5640i		−	0.0838668i
$902$			262.163i			0.290647i
$903$		0			0
$904$	−224.726			−0.248591
$905$	−226.491			−0.250266
$906$		0			0
$907$	−931.384			−1.02688			−0.513442	−	0.858124i	$-0.671630\pi$
$907$	−931.384			−1.02688			−0.513442	+	0.858124i	$0.671630\pi$
$908$			552.293i			0.608252i
$909$		0			0
$910$	−166.870	−	191.515i	−0.183374	−	0.210456i
$911$	1249.63			1.37171			0.685854	−	0.727739i	$-0.259429\pi$
$911$	1249.63			1.37171			0.685854	+	0.727739i	$0.259429\pi$
$912$		0			0
$913$		−	1897.33i		−	2.07812i
$914$	124.788			0.136530
$915$		0			0
$916$		−	722.452i		−	0.788703i
$917$	−621.466	+	541.493i	−0.677716	+	0.590505i
$918$		0			0
$919$	659.705			0.717851			0.358925	−	0.933366i	$-0.383143\pi$
$919$	659.705			0.717851			0.358925	+	0.933366i	$0.383143\pi$
$920$			95.8828i			0.104220i
$921$		0			0
$922$		−	39.2942i		−	0.0426184i
$923$			1497.93i			1.62290i
$924$		0			0
$925$	−428.939			−0.463718
$926$	−660.247			−0.713009
$927$		0			0
$928$	234.202			0.252373
$929$			1199.26i			1.29092i	0.763796	+	0.645458i	$0.223333\pi$
$929$			1199.26i			1.29092i	−0.763796	+	0.645458i	$0.776667\pi$
$930$		0			0
$931$	1404.31	+	194.056i	1.50838	+	0.208438i
$932$	627.170			0.672929
$933$		0			0
$934$		−	231.128i		−	0.247461i
$935$	37.8295			0.0404593
$936$		0			0
$937$			651.336i			0.695130i	0.937656	+	0.347565i	$0.112991\pi$
$937$			651.336i			0.695130i	−0.937656	+	0.347565i	$0.887009\pi$
$938$	−287.953	−	330.481i	−0.306987	−	0.352325i
$939$		0			0
$940$	−35.8864			−0.0381770
$941$			1458.28i			1.54972i	0.632135	+	0.774858i	$0.282179\pi$
$941$			1458.28i			1.54972i	−0.632135	+	0.774858i	$0.717821\pi$
$942$		0			0
$943$			277.611i			0.294392i
$944$		−	99.9481i		−	0.105877i
$945$		0			0
$946$	−1341.17			−1.41773
$947$	−1634.00			−1.72545			−0.862724	−	0.505675i	$-0.831244\pi$
$947$	−1634.00			−1.72545			−0.862724	+	0.505675i	$0.831244\pi$
$948$		0			0
$949$	−1473.84			−1.55304
$950$			944.662i			0.994381i
$951$		0			0
$952$	−24.9460	+	21.7359i	−0.0262038	+	0.0228318i
$953$	1002.99			1.05245			0.526226	−	0.850345i	$-0.323607\pi$
$953$	1002.99			1.05245			0.526226	+	0.850345i	$0.323607\pi$
$954$		0			0
$955$			350.542i			0.367059i
$956$	−802.758			−0.839705
$957$		0			0
$958$			317.930i			0.331868i
$959$	−8.79138	−	10.0898i	−0.00916724	−	0.0105211i
$960$		0			0
$961$	−889.890			−0.926004
$962$		−	487.569i		−	0.506829i
$963$		0			0
$964$			198.884i			0.206311i
$965$			43.0456i			0.0446068i
$966$		0			0
$967$	675.599			0.698654			0.349327	−	0.937001i	$-0.386410\pi$
$967$	675.599			0.698654			0.349327	+	0.937001i	$0.386410\pi$
$968$	−415.831			−0.429578
$969$		0			0
$970$	−161.911			−0.166919
$971$		−	103.021i		−	0.106098i	−0.998592	−	0.0530489i	$-0.983106\pi$
$971$		−	103.021i		−	0.106098i	0.998592	−	0.0530489i	$-0.0168939\pi$
$972$		0			0
$973$	1325.39	−	1154.84i	1.36217	−	1.18688i
$974$	496.079			0.509322
$975$		0			0
$976$			64.3916i			0.0659750i
$977$	290.183			0.297014			0.148507	−	0.988911i	$-0.452553\pi$
$977$	290.183			0.297014			0.148507	+	0.988911i	$0.452553\pi$
$978$		0			0
$979$		−	495.177i		−	0.505799i
$980$	−134.231	−	18.5489i	−0.136970	−	0.0189274i
$981$		0			0
$982$	1038.44			1.05747
$983$			714.870i			0.727233i	0.931549	+	0.363616i	$0.118458\pi$
$983$			714.870i			0.727233i	−0.931549	+	0.363616i	$0.881542\pi$
$984$		0			0
$985$			273.230i			0.277391i
$986$		−	97.8467i		−	0.0992360i
$987$		0			0
$988$	−1073.78			−1.08683
$989$	−1420.20			−1.43599
$990$		0			0
$991$	−1129.26			−1.13951			−0.569756	−	0.821814i	$-0.692962\pi$
$991$	−1129.26			−1.13951			−0.569756	+	0.821814i	$0.692962\pi$
$992$			243.369i			0.245332i
$993$		0			0
$994$	524.942	+	602.470i	0.528110	+	0.606106i
$995$	−223.342			−0.224465
$996$		0			0
$997$			407.106i			0.408331i	0.978936	+	0.204166i	$0.0654481\pi$
$997$			407.106i			0.408331i	−0.978936	+	0.204166i	$0.934552\pi$
$998$	−1322.50			−1.32515
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.3.c.d.811.5		16
3.2	odd	2		1134.3.c.e.811.12		16
7.6	odd	2	inner	1134.3.c.d.811.4		16
9.2	odd	6		126.3.o.a.13.6	yes	32
9.4	even	3		378.3.o.a.181.13		32
9.5	odd	6		126.3.o.a.97.3	yes	32
9.7	even	3		378.3.o.a.307.12		32
21.20	even	2		1134.3.c.e.811.13		16
63.13	odd	6		378.3.o.a.181.12		32
63.20	even	6		126.3.o.a.13.3	✓	32
63.34	odd	6		378.3.o.a.307.13		32
63.41	even	6		126.3.o.a.97.6	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.o.a.13.3	✓	32	63.20	even	6
126.3.o.a.13.6	yes	32	9.2	odd	6
126.3.o.a.97.3	yes	32	9.5	odd	6
126.3.o.a.97.6	yes	32	63.41	even	6
378.3.o.a.181.12		32	63.13	odd	6
378.3.o.a.181.13		32	9.4	even	3
378.3.o.a.307.12		32	9.7	even	3
378.3.o.a.307.13		32	63.34	odd	6
1134.3.c.d.811.4		16	7.6	odd	2	inner
1134.3.c.d.811.5		16	1.1	even	1	trivial
1134.3.c.e.811.12		16	3.2	odd	2
1134.3.c.e.811.13		16	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1134.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +1.38272i q^{5} +(4.59851 + 5.27766i) q^{7} -2.82843 q^{8} -1.95546i q^{10} +16.3713 q^{11} -18.5572i q^{13} +(-6.50328 - 7.46374i) q^{14} +4.00000 q^{16} -1.67115i q^{17} -28.9316i q^{19} +2.76543i q^{20} -23.1525 q^{22} -24.5167 q^{23} +23.0881 q^{25} +26.2439i q^{26} +(9.19702 + 10.5553i) q^{28} -41.4015 q^{29} -43.0220i q^{31} -5.65685 q^{32} +2.36336i q^{34} +(-7.29751 + 6.35844i) q^{35} -18.5784 q^{37} +40.9155i q^{38} -3.91091i q^{40} -11.3233i q^{41} +57.9277 q^{43} +32.7425 q^{44} +34.6719 q^{46} +12.9768i q^{47} +(-6.70739 + 48.5388i) q^{49} -32.6515 q^{50} -37.1145i q^{52} +45.2169 q^{53} +22.6368i q^{55} +(-13.0066 - 14.9275i) q^{56} +58.5506 q^{58} -24.9870i q^{59} +16.0979i q^{61} +60.8422i q^{62} +8.00000 q^{64} +25.6594 q^{65} +44.2782 q^{67} -3.34229i q^{68} +(10.3202 - 8.99219i) q^{70} -80.7195 q^{71} -79.4212i q^{73} +26.2738 q^{74} -57.8633i q^{76} +(75.2835 + 86.4020i) q^{77} -39.2311 q^{79} +5.53087i q^{80} +16.0136i q^{82} -115.894i q^{83} +2.31072 q^{85} -81.9221 q^{86} -46.3050 q^{88} -30.2467i q^{89} +(97.9389 - 85.3357i) q^{91} -49.0335 q^{92} -18.3519i q^{94} +40.0043 q^{95} -82.7997i q^{97} +(9.48569 - 68.6442i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} + 2 q^{7} - 12 q^{11} - 12 q^{14} + 64 q^{16} + 12 q^{23} - 80 q^{25} + 4 q^{28} - 48 q^{29} - 174 q^{35} - 44 q^{37} - 32 q^{43} - 24 q^{44} + 24 q^{46} - 50 q^{49} + 48 q^{50} + 432 q^{53}+ \cdots - 312 q^{98}+O(q^{100}) \) 16 * q + 32 * q^4 + 2 * q^7 - 12 * q^11 - 12 * q^14 + 64 * q^16 + 12 * q^23 - 80 * q^25 + 4 * q^28 - 48 * q^29 - 174 * q^35 - 44 * q^37 - 32 * q^43 - 24 * q^44 + 24 * q^46 - 50 * q^49 + 48 * q^50 + 432 * q^53 - 24 * q^56 - 48 * q^58 + 128 * q^64 + 120 * q^65 + 140 * q^67 + 108 * q^70 - 276 * q^71 + 144 * q^74 - 258 * q^77 + 176 * q^79 + 60 * q^85 + 96 * q^86 + 186 * q^91 + 24 * q^92 + 528 * q^95 - 312 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more