LMFDB - Embedded newform 116.6.c.a.57.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,6,Mod(57,116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("116.57");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 116 = 2^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	116.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.6045230983$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 1783 x^{10} + 1098222 x^{8} + 308936830 x^{6} + 41608294477 x^{4} + 2492454459339 x^{2} + 53357557623300 $ x^12 + 1783x^10 + 1098222x^8 + 308936830x^6 + 41608294477x^4 + 2492454459339*x^2 + 53357557623300
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			57.2
Root			$-17.8714i$ of defining polynomial
Character	$\chi$	$=$	116.57
Dual form			116.6.c.a.57.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.8714i q^{3} -82.5785 q^{5} -124.027 q^{7} -76.3873 q^{9} +O(q^{10})$	$q-17.8714i q^{3} -82.5785 q^{5} -124.027 q^{7} -76.3873 q^{9} +383.748i q^{11} +582.766 q^{13} +1475.79i q^{15} +476.845i q^{17} -1680.34i q^{19} +2216.54i q^{21} +2970.85 q^{23} +3694.21 q^{25} -2977.60i q^{27} +(-3454.56 + 2928.68i) q^{29} -2725.41i q^{31} +6858.13 q^{33} +10242.0 q^{35} +10061.9i q^{37} -10414.8i q^{39} +6210.68i q^{41} +19941.2i q^{43} +6307.95 q^{45} +25654.1i q^{47} -1424.33 q^{49} +8521.89 q^{51} +4946.76 q^{53} -31689.4i q^{55} -30030.1 q^{57} +12408.3 q^{59} +42285.7i q^{61} +9474.08 q^{63} -48123.9 q^{65} -33004.8 q^{67} -53093.2i q^{69} +50715.4 q^{71} -48544.2i q^{73} -66020.8i q^{75} -47595.1i q^{77} +25370.6i q^{79} -71776.1 q^{81} +33488.1 q^{83} -39377.1i q^{85} +(52339.7 + 61737.8i) q^{87} -4976.06i q^{89} -72278.6 q^{91} -48707.0 q^{93} +138760. i q^{95} -103525. i q^{97} -29313.5i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9}+O(q^{10}) $ 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9	$ 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9} - 194 q^{13} + 3996 q^{23} + 1782 q^{25} - 1900 q^{29} + 7266 q^{33} + 12348 q^{35} - 13988 q^{45} - 8396 q^{49} - 10788 q^{51} - 28626 q^{53} + 11140 q^{57} + 27356 q^{59} + 55392 q^{63} - 55126 q^{65} - 13184 q^{67} + 44352 q^{71} + 74264 q^{81} - 192628 q^{83} + 98908 q^{87} + 13580 q^{91} + 59710 q^{93}+O(q^{100}) $ 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9 - 194 * q^13 + 3996 * q^23 + 1782 * q^25 - 1900 * q^29 + 7266 * q^33 + 12348 * q^35 - 13988 * q^45 - 8396 * q^49 - 10788 * q^51 - 28626 * q^53 + 11140 * q^57 + 27356 * q^59 + 55392 * q^63 - 55126 * q^65 - 13184 * q^67 + 44352 * q^71 + 74264 * q^81 - 192628 * q^83 + 98908 * q^87 + 13580 * q^91 + 59710 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/116\mathbb{Z}\right)^\times$.

$n$	$59$	$89$
$\chi(n)$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	17.8714i		−	1.14645i	−0.819397	−	0.573226i	$-0.805692\pi$
$3$		−	17.8714i		−	1.14645i	0.819397	−	0.573226i	$-0.194308\pi$
$4$		0			0
$5$	−82.5785			−1.47721			−0.738605	−	0.674139i	$-0.764515\pi$
$5$	−82.5785			−1.47721			−0.738605	+	0.674139i	$0.764515\pi$
$6$		0			0
$7$	−124.027			−0.956689			−0.478345	−	0.878172i	$-0.658763\pi$
$7$	−124.027			−0.956689			−0.478345	+	0.878172i	$0.658763\pi$
$8$		0			0
$9$	−76.3873			−0.314351
$10$		0			0
$11$			383.748i			0.956236i	0.878296	+	0.478118i	$0.158681\pi$
$11$			383.748i			0.956236i	−0.878296	+	0.478118i	$0.841319\pi$
$12$		0			0
$13$	582.766			0.956391			0.478196	−	0.878253i	$-0.341291\pi$
$13$	582.766			0.956391			0.478196	+	0.878253i	$0.341291\pi$
$14$		0			0
$15$			1475.79i			1.69355i
$16$		0			0
$17$			476.845i			0.400179i	0.979778	+	0.200090i	$0.0641233\pi$
$17$			476.845i			0.400179i	−0.979778	+	0.200090i	$0.935877\pi$
$18$		0			0
$19$		−	1680.34i		−	1.06786i	−0.845529	−	0.533929i	$-0.820715\pi$
$19$		−	1680.34i		−	1.06786i	0.845529	−	0.533929i	$-0.179285\pi$
$20$		0			0
$21$			2216.54i			1.09680i
$22$		0			0
$23$	2970.85			1.17101			0.585505	−	0.810669i	$-0.300896\pi$
$23$	2970.85			1.17101			0.585505	+	0.810669i	$0.300896\pi$
$24$		0			0
$25$	3694.21			1.18215
$26$		0			0
$27$		−	2977.60i		−	0.786063i
$28$		0			0
$29$	−3454.56	+	2928.68i	−0.762777	+	0.646662i
$29$	−3454.56	+	2928.68i	−0.762777	+	0.646662i
$30$		0			0
$31$		−	2725.41i		−	0.509364i	−0.967025	−	0.254682i	$-0.918029\pi$
$31$		−	2725.41i		−	0.509364i	0.967025	−	0.254682i	$-0.0819707\pi$
$32$		0			0
$33$	6858.13			1.09628
$34$		0			0
$35$	10242.0			1.41323
$36$		0			0
$37$			10061.9i			1.20830i	0.796869	+	0.604152i	$0.206488\pi$
$37$			10061.9i			1.20830i	−0.796869	+	0.604152i	$0.793512\pi$
$38$		0			0
$39$		−	10414.8i		−	1.09646i
$40$		0			0
$41$			6210.68i			0.577005i	0.957479	+	0.288503i	$0.0931574\pi$
$41$			6210.68i			0.577005i	−0.957479	+	0.288503i	$0.906843\pi$
$42$		0			0
$43$			19941.2i			1.64467i	0.569000	+	0.822337i	$0.307330\pi$
$43$			19941.2i			1.64467i	−0.569000	+	0.822337i	$0.692670\pi$
$44$		0			0
$45$	6307.95			0.464363
$46$		0			0
$47$			25654.1i			1.69400i	0.531595	+	0.846998i	$0.321593\pi$
$47$			25654.1i			1.69400i	−0.531595	+	0.846998i	$0.678407\pi$
$48$		0			0
$49$	−1424.33			−0.0847461
$50$		0			0
$51$	8521.89			0.458786
$52$		0			0
$53$	4946.76			0.241897			0.120949	−	0.992659i	$-0.461406\pi$
$53$	4946.76			0.241897			0.120949	+	0.992659i	$0.461406\pi$
$54$		0			0
$55$		−	31689.4i		−	1.41256i
$56$		0			0
$57$	−30030.1			−1.22425
$58$		0			0
$59$	12408.3			0.464067			0.232034	−	0.972708i	$-0.425462\pi$
$59$	12408.3			0.464067			0.232034	+	0.972708i	$0.425462\pi$
$60$		0			0
$61$			42285.7i			1.45502i	0.686096	+	0.727511i	$0.259323\pi$
$61$			42285.7i			1.45502i	−0.686096	+	0.727511i	$0.740677\pi$
$62$		0			0
$63$	9474.08			0.300736
$64$		0			0
$65$	−48123.9			−1.41279
$66$		0			0
$67$	−33004.8			−0.898235			−0.449117	−	0.893473i	$-0.648261\pi$
$67$	−33004.8			−0.898235			−0.449117	+	0.893473i	$0.648261\pi$
$68$		0			0
$69$		−	53093.2i		−	1.34251i
$70$		0			0
$71$	50715.4			1.19397			0.596986	−	0.802252i	$-0.296365\pi$
$71$	50715.4			1.19397			0.596986	+	0.802252i	$0.296365\pi$
$72$		0			0
$73$		−	48544.2i		−	1.06618i	−0.846059	−	0.533089i	$-0.821031\pi$
$73$		−	48544.2i		−	1.06618i	0.846059	−	0.533089i	$-0.178969\pi$
$74$		0			0
$75$		−	66020.8i		−	1.35528i
$76$		0			0
$77$		−	47595.1i		−	0.914820i
$78$		0			0
$79$			25370.6i			0.457365i	0.973501	+	0.228683i	$0.0734418\pi$
$79$			25370.6i			0.457365i	−0.973501	+	0.228683i	$0.926558\pi$
$80$		0			0
$81$	−71776.1			−1.21553
$82$		0			0
$83$	33488.1			0.533574			0.266787	−	0.963755i	$-0.414038\pi$
$83$	33488.1			0.533574			0.266787	+	0.963755i	$0.414038\pi$
$84$		0			0
$85$		−	39377.1i		−	0.591149i
$86$		0			0
$87$	52339.7	+	61737.8i	0.741366	+	0.874487i
$88$		0			0
$89$		−	4976.06i		−	0.0665903i	−0.999446	−	0.0332951i	$-0.989400\pi$
$89$		−	4976.06i		−	0.0665903i	0.999446	−	0.0332951i	$-0.0106001\pi$
$90$		0			0
$91$	−72278.6			−0.914969
$92$		0			0
$93$	−48707.0			−0.583961
$94$		0			0
$95$			138760.i			1.57745i
$96$		0			0
$97$		−	103525.i		−	1.11716i	−0.829450	−	0.558581i	$-0.811346\pi$
$97$		−	103525.i		−	1.11716i	0.829450	−	0.558581i	$-0.188654\pi$
$98$		0			0
$99$		−	29313.5i		−	0.300594i
$100$		0			0
$101$			45890.2i			0.447628i	0.974632	+	0.223814i	$0.0718508\pi$
$101$			45890.2i			0.447628i	−0.974632	+	0.223814i	$0.928149\pi$
$102$		0			0
$103$	−73216.2			−0.680008			−0.340004	−	0.940424i	$-0.610428\pi$
$103$	−73216.2			−0.680008			−0.340004	+	0.940424i	$0.610428\pi$
$104$		0			0
$105$		−	183038.i		−	1.62020i
$106$		0			0
$107$	−100448.			−0.848171			−0.424085	−	0.905622i	$-0.639404\pi$
$107$	−100448.			−0.848171			−0.424085	+	0.905622i	$0.639404\pi$
$108$		0			0
$109$	−146442.			−1.18059			−0.590297	−	0.807186i	$-0.700989\pi$
$109$	−146442.			−1.18059			−0.590297	+	0.807186i	$0.700989\pi$
$110$		0			0
$111$	179821.			1.38526
$112$		0			0
$113$			154350.i			1.13713i	0.822639	+	0.568564i	$0.192501\pi$
$113$			154350.i			1.13713i	−0.822639	+	0.568564i	$0.807499\pi$
$114$		0			0
$115$	−245328.			−1.72983
$116$		0			0
$117$	−44515.9			−0.300643
$118$		0			0
$119$		−	59141.6i		−	0.382847i
$120$		0			0
$121$	13788.1			0.0856135
$122$		0			0
$123$	110994.			0.661508
$124$		0			0
$125$	−47004.9			−0.269072
$126$		0			0
$127$			91348.0i			0.502562i	0.967914	+	0.251281i	$0.0808518\pi$
$127$			91348.0i			0.502562i	−0.967914	+	0.251281i	$0.919148\pi$
$128$		0			0
$129$	356377.			1.88554
$130$		0			0
$131$		−	53789.1i		−	0.273852i	−0.990581	−	0.136926i	$-0.956278\pi$
$131$		−	53789.1i		−	0.273852i	0.990581	−	0.136926i	$-0.0437222\pi$
$132$		0			0
$133$			208407.i			1.02161i
$134$		0			0
$135$			245886.i			1.16118i
$136$		0			0
$137$		−	267293.i		−	1.21671i	−0.793666	−	0.608353i	$-0.791830\pi$
$137$		−	267293.i		−	1.21671i	0.793666	−	0.608353i	$-0.208170\pi$
$138$		0			0
$139$	179377.			0.787462			0.393731	−	0.919226i	$-0.371184\pi$
$139$	179377.			0.787462			0.393731	+	0.919226i	$0.371184\pi$
$140$		0			0
$141$	458475.			1.94209
$142$		0			0
$143$			223635.i			0.914535i
$144$		0			0
$145$	285272.	−	241846.i	1.12678	−	0.955255i
$146$		0			0
$147$			25454.7i			0.0971573i
$148$		0			0
$149$	−86872.6			−0.320566			−0.160283	−	0.987071i	$-0.551241\pi$
$149$	−86872.6			−0.320566			−0.160283	+	0.987071i	$0.551241\pi$
$150$		0			0
$151$	469855.			1.67696			0.838478	−	0.544935i	$-0.183446\pi$
$151$	469855.			1.67696			0.838478	+	0.544935i	$0.183446\pi$
$152$		0			0
$153$		−	36424.9i		−	0.125797i
$154$		0			0
$155$			225061.i			0.752437i
$156$		0			0
$157$			356077.i			1.15291i	0.817129	+	0.576455i	$0.195564\pi$
$157$			356077.i			1.15291i	−0.817129	+	0.576455i	$0.804436\pi$
$158$		0			0
$159$		−	88405.6i		−	0.277324i
$160$		0			0
$161$	−368465.			−1.12029
$162$		0			0
$163$		−	130451.i		−	0.384573i	−0.981339	−	0.192286i	$-0.938410\pi$
$163$		−	130451.i		−	0.384573i	0.981339	−	0.192286i	$-0.0615902\pi$
$164$		0			0
$165$	−566334.			−1.61943
$166$		0			0
$167$	558923.			1.55082			0.775409	−	0.631459i	$-0.217544\pi$
$167$	558923.			1.55082			0.775409	+	0.631459i	$0.217544\pi$
$168$		0			0
$169$	−31677.3			−0.0853161
$170$		0			0
$171$			128357.i			0.335682i
$172$		0			0
$173$	−588269.			−1.49438			−0.747189	−	0.664611i	$-0.768597\pi$
$173$	−588269.			−1.49438			−0.747189	+	0.664611i	$0.768597\pi$
$174$		0			0
$175$	−458182.			−1.13095
$176$		0			0
$177$		−	221753.i		−	0.532031i
$178$		0			0
$179$	508446.			1.18608			0.593038	−	0.805175i	$-0.297929\pi$
$179$	508446.			1.18608			0.593038	+	0.805175i	$0.297929\pi$
$180$		0			0
$181$	−437171.			−0.991870			−0.495935	−	0.868360i	$-0.665175\pi$
$181$	−437171.			−0.991870			−0.495935	+	0.868360i	$0.665175\pi$
$182$		0			0
$183$	755706.			1.66811
$184$		0			0
$185$		−	830898.i		−	1.78492i
$186$		0			0
$187$	−182988.			−0.382666
$188$		0			0
$189$			369303.i			0.752018i
$190$		0			0
$191$			924995.i			1.83466i	0.398127	+	0.917330i	$0.369660\pi$
$191$			924995.i			1.83466i	−0.398127	+	0.917330i	$0.630340\pi$
$192$		0			0
$193$			449717.i			0.869053i	0.900659	+	0.434526i	$0.143084\pi$
$193$			449717.i			0.869053i	−0.900659	+	0.434526i	$0.856916\pi$
$194$		0			0
$195$			860042.i			1.61970i
$196$		0			0
$197$	−665685.			−1.22209			−0.611045	−	0.791596i	$-0.709251\pi$
$197$	−665685.			−1.22209			−0.611045	+	0.791596i	$0.709251\pi$
$198$		0			0
$199$	458831.			0.821334			0.410667	−	0.911785i	$-0.365296\pi$
$199$	458831.			0.821334			0.410667	+	0.911785i	$0.365296\pi$
$200$		0			0
$201$			589842.i			1.02978i
$202$		0			0
$203$	428458.	−	363235.i	0.729740	−	0.618654i
$204$		0			0
$205$		−	512869.i		−	0.852358i
$206$		0			0
$207$	−226935.			−0.368109
$208$		0			0
$209$	644828.			1.02112
$210$		0			0
$211$		−	299044.i		−	0.462412i	−0.972905	−	0.231206i	$-0.925733\pi$
$211$		−	299044.i		−	0.462412i	0.972905	−	0.231206i	$-0.0742672\pi$
$212$		0			0
$213$		−	906356.i		−	1.36883i
$214$		0			0
$215$		−	1.64671e6i		−	2.42953i
$216$		0			0
$217$			338024.i			0.487303i
$218$		0			0
$219$	−867552.			−1.22232
$220$		0			0
$221$			277889.i			0.382728i
$222$		0			0
$223$	−1.32411e6			−1.78304			−0.891522	−	0.452977i	$-0.850362\pi$
$223$	−1.32411e6			−1.78304			−0.891522	+	0.452977i	$0.850362\pi$
$224$		0			0
$225$	−282191.			−0.371610
$226$		0			0
$227$	604546.			0.778690			0.389345	−	0.921092i	$-0.372701\pi$
$227$	604546.			0.778690			0.389345	+	0.921092i	$0.372701\pi$
$228$		0			0
$229$		−	1.08361e6i		−	1.36547i	−0.730665	−	0.682737i	$-0.760790\pi$
$229$		−	1.08361e6i		−	1.36547i	0.730665	−	0.682737i	$-0.239210\pi$
$230$		0			0
$231$	−850592.			−1.04880
$232$		0			0
$233$	−750226.			−0.905320			−0.452660	−	0.891683i	$-0.649525\pi$
$233$	−750226.			−0.905320			−0.452660	+	0.891683i	$0.649525\pi$
$234$		0			0
$235$		−	2.11848e6i		−	2.50239i
$236$		0			0
$237$	453409.			0.524347
$238$		0			0
$239$	432770.			0.490075			0.245037	−	0.969514i	$-0.421200\pi$
$239$	432770.			0.490075			0.245037	+	0.969514i	$0.421200\pi$
$240$		0			0
$241$	−541632.			−0.600706			−0.300353	−	0.953828i	$-0.597104\pi$
$241$	−541632.			−0.600706			−0.300353	+	0.953828i	$0.597104\pi$
$242$		0			0
$243$			559182.i			0.607488i
$244$		0			0
$245$	117619.			0.125188
$246$		0			0
$247$		−	979245.i		−	1.02129i
$248$		0			0
$249$		−	598479.i		−	0.611717i
$250$		0			0
$251$			481904.i			0.482809i	0.970425	+	0.241405i	$0.0776081\pi$
$251$			481904.i			0.482809i	−0.970425	+	0.241405i	$0.922392\pi$
$252$		0			0
$253$			1.14006e6i			1.11976i
$254$		0			0
$255$	−703725.			−0.677723
$256$		0			0
$257$	1.03496e6			0.977444			0.488722	−	0.872440i	$-0.337463\pi$
$257$	1.03496e6			0.977444			0.488722	+	0.872440i	$0.337463\pi$
$258$		0			0
$259$		−	1.24795e6i		−	1.15597i
$260$		0			0
$261$	263884.	−	223714.i	0.239780	−	0.203279i
$262$		0			0
$263$			1.76834e6i			1.57644i	0.615394	+	0.788220i	$0.288997\pi$
$263$			1.76834e6i			1.57644i	−0.615394	+	0.788220i	$0.711003\pi$
$264$		0			0
$265$	−408496.			−0.357333
$266$		0			0
$267$	−88929.3			−0.0763425
$268$		0			0
$269$			2.14834e6i			1.81019i	0.425213	+	0.905093i	$0.360199\pi$
$269$			2.14834e6i			1.81019i	−0.425213	+	0.905093i	$0.639801\pi$
$270$		0			0
$271$		−	980534.i		−	0.811034i	−0.914087	−	0.405517i	$-0.867091\pi$
$271$		−	980534.i		−	0.811034i	0.914087	−	0.405517i	$-0.132909\pi$
$272$		0			0
$273$			1.29172e6i			1.04897i
$274$		0			0
$275$			1.41765e6i			1.13041i
$276$		0			0
$277$	−1.80193e6			−1.41104			−0.705519	−	0.708691i	$-0.749286\pi$
$277$	−1.80193e6			−1.41104			−0.705519	+	0.708691i	$0.749286\pi$
$278$		0			0
$279$			208187.i			0.160119i
$280$		0			0
$281$	1.59676e6			1.20635			0.603174	−	0.797609i	$-0.293902\pi$
$281$	1.59676e6			1.20635			0.603174	+	0.797609i	$0.293902\pi$
$282$		0			0
$283$	−1.49532e6			−1.10986			−0.554930	−	0.831897i	$-0.687255\pi$
$283$	−1.49532e6			−1.10986			−0.554930	+	0.831897i	$0.687255\pi$
$284$		0			0
$285$	2.47984e6			1.80847
$286$		0			0
$287$		−	770292.i		−	0.552014i
$288$		0			0
$289$	1.19248e6			0.839856
$290$		0			0
$291$	−1.85014e6			−1.28077
$292$		0			0
$293$			1.74573e6i			1.18797i	0.804475	+	0.593987i	$0.202447\pi$
$293$			1.74573e6i			1.18797i	−0.804475	+	0.593987i	$0.797553\pi$
$294$		0			0
$295$	−1.02466e6			−0.685525
$296$		0			0
$297$	1.14265e6			0.751662
$298$		0			0
$299$	1.73131e6			1.11994
$300$		0			0
$301$		−	2.47324e6i		−	1.57344i
$302$		0			0
$303$	820123.			0.513183
$304$		0			0
$305$		−	3.49190e6i		−	2.14937i
$306$		0			0
$307$			2.52100e6i			1.52661i	0.646039	+	0.763304i	$0.276424\pi$
$307$			2.52100e6i			1.52661i	−0.646039	+	0.763304i	$0.723576\pi$
$308$		0			0
$309$			1.30848e6i			0.779596i
$310$		0			0
$311$			2.03733e6i			1.19443i	0.802082	+	0.597213i	$0.203725\pi$
$311$			2.03733e6i			1.19443i	−0.802082	+	0.597213i	$0.796275\pi$
$312$		0			0
$313$	−1.25982e6			−0.726853			−0.363427	−	0.931623i	$-0.618393\pi$
$313$	−1.25982e6			−0.726853			−0.363427	+	0.931623i	$0.618393\pi$
$314$		0			0
$315$	−782356.			−0.444251
$316$		0			0
$317$			724626.i			0.405010i	0.979281	+	0.202505i	$0.0649082\pi$
$317$			724626.i			0.405010i	−0.979281	+	0.202505i	$0.935092\pi$
$318$		0			0
$319$	−1.12388e6	−	1.32568e6i	−0.618361	−	0.729394i
$320$		0			0
$321$			1.79515e6i			0.972387i
$322$		0			0
$323$	801262.			0.427335
$324$		0			0
$325$	2.15286e6			1.13060
$326$		0			0
$327$			2.61713e6i			1.35349i
$328$		0			0
$329$		−	3.18180e6i		−	1.62063i
$330$		0			0
$331$		−	1.04953e6i		−	0.526533i	−0.964723	−	0.263267i	$-0.915200\pi$
$331$		−	1.04953e6i		−	0.526533i	0.964723	−	0.263267i	$-0.0847999\pi$
$332$		0			0
$333$		−	768603.i		−	0.379832i
$334$		0			0
$335$	2.72549e6			1.32688
$336$		0			0
$337$		−	1.88330e6i		−	0.903328i	−0.892188	−	0.451664i	$-0.850831\pi$
$337$		−	1.88330e6i		−	0.903328i	0.892188	−	0.451664i	$-0.149169\pi$
$338$		0			0
$339$	2.75844e6			1.30366
$340$		0			0
$341$	1.04587e6			0.487072
$342$		0			0
$343$	2.26118e6			1.03776
$344$		0			0
$345$			4.38436e6i			1.98316i
$346$		0			0
$347$	−517043.			−0.230517			−0.115258	−	0.993336i	$-0.536770\pi$
$347$	−517043.			−0.230517			−0.115258	+	0.993336i	$0.536770\pi$
$348$		0			0
$349$	2.43940e6			1.07206			0.536031	−	0.844198i	$-0.319923\pi$
$349$	2.43940e6			1.07206			0.536031	+	0.844198i	$0.319923\pi$
$350$		0			0
$351$		−	1.73524e6i		−	0.751784i
$352$		0			0
$353$	3.91774e6			1.67340			0.836699	−	0.547663i	$-0.184482\pi$
$353$	3.91774e6			1.67340			0.836699	+	0.547663i	$0.184482\pi$
$354$		0			0
$355$	−4.18801e6			−1.76375
$356$		0			0
$357$	−1.05694e6			−0.438916
$358$		0			0
$359$		−	1.58915e6i		−	0.650771i	−0.945582	−	0.325385i	$-0.894506\pi$
$359$		−	1.58915e6i		−	0.650771i	0.945582	−	0.325385i	$-0.105494\pi$
$360$		0			0
$361$	−347447.			−0.140320
$362$		0			0
$363$		−	246413.i		−	0.0981517i
$364$		0			0
$365$			4.00870e6i			1.57497i
$366$		0			0
$367$			1.17876e6i			0.456837i	0.973563	+	0.228419i	$0.0733555\pi$
$367$			1.17876e6i			0.456837i	−0.973563	+	0.228419i	$0.926645\pi$
$368$		0			0
$369$		−	474417.i		−	0.181382i
$370$		0			0
$371$	−613531.			−0.231421
$372$		0			0
$373$	−4.22519e6			−1.57244			−0.786220	−	0.617947i	$-0.787965\pi$
$373$	−4.22519e6			−1.57244			−0.786220	+	0.617947i	$0.787965\pi$
$374$		0			0
$375$			840043.i			0.308478i
$376$		0			0
$377$	−2.01320e6	+	1.70674e6i	−0.729513	+	0.618462i
$378$		0			0
$379$		−	2.61337e6i		−	0.934551i	−0.884112	−	0.467276i	$-0.845236\pi$
$379$		−	2.61337e6i		−	0.934551i	0.884112	−	0.467276i	$-0.154764\pi$
$380$		0			0
$381$	1.63252e6			0.576163
$382$		0			0
$383$	−1.44184e6			−0.502251			−0.251125	−	0.967955i	$-0.580801\pi$
$383$	−1.44184e6			−0.502251			−0.251125	+	0.967955i	$0.580801\pi$
$384$		0			0
$385$			3.93034e6i			1.35138i
$386$		0			0
$387$		−	1.52325e6i		−	0.517005i
$388$		0			0
$389$		−	491479.i		−	0.164676i	−0.996604	−	0.0823382i	$-0.973761\pi$
$389$		−	491479.i		−	0.164676i	0.996604	−	0.0823382i	$-0.0262388\pi$
$390$		0			0
$391$			1.41663e6i			0.468614i
$392$		0			0
$393$	−961287.			−0.313958
$394$		0			0
$395$		−	2.09507e6i		−	0.675624i
$396$		0			0
$397$	−2.13262e6			−0.679106			−0.339553	−	0.940587i	$-0.610276\pi$
$397$	−2.13262e6			−0.679106			−0.339553	+	0.940587i	$0.610276\pi$
$398$		0			0
$399$	3.72454e6			1.17122
$400$		0			0
$401$	−4.35320e6			−1.35191			−0.675954	−	0.736944i	$-0.736268\pi$
$401$	−4.35320e6			−1.35191			−0.675954	+	0.736944i	$0.736268\pi$
$402$		0			0
$403$		−	1.58828e6i		−	0.487151i
$404$		0			0
$405$	5.92716e6			1.79560
$406$		0			0
$407$	−3.86124e6			−1.15542
$408$		0			0
$409$		−	3.52885e6i		−	1.04310i	−0.853221	−	0.521549i	$-0.825354\pi$
$409$		−	3.52885e6i		−	1.04310i	0.853221	−	0.521549i	$-0.174646\pi$
$410$		0			0
$411$	−4.77690e6			−1.39489
$412$		0			0
$413$	−1.53896e6			−0.443968
$414$		0			0
$415$	−2.76540e6			−0.788201
$416$		0			0
$417$		−	3.20572e6i		−	0.902787i
$418$		0			0
$419$	−3.24417e6			−0.902753			−0.451377	−	0.892334i	$-0.649067\pi$
$419$	−3.24417e6			−0.902753			−0.451377	+	0.892334i	$0.649067\pi$
$420$		0			0
$421$			1.58898e6i			0.436932i	0.975845	+	0.218466i	$0.0701053\pi$
$421$			1.58898e6i			0.436932i	−0.975845	+	0.218466i	$0.929895\pi$
$422$		0			0
$423$		−	1.95965e6i		−	0.532510i
$424$		0			0
$425$			1.76157e6i			0.473071i
$426$		0			0
$427$		−	5.24457e6i		−	1.39200i
$428$		0			0
$429$	3.99668e6			1.04847
$430$		0			0
$431$	−2.88062e6			−0.746951			−0.373476	−	0.927640i	$-0.621834\pi$
$431$	−2.88062e6			−0.746951			−0.373476	+	0.927640i	$0.621834\pi$
$432$		0			0
$433$			7.01507e6i			1.79809i	0.437852	+	0.899047i	$0.355739\pi$
$433$			7.01507e6i			1.79809i	−0.437852	+	0.899047i	$0.644261\pi$
$434$		0			0
$435$	−4.32213e6	−	5.09822e6i	−1.09515	−	1.29180i
$436$		0			0
$437$		−	4.99204e6i		−	1.25047i
$438$		0			0
$439$	−1.23889e6			−0.306812			−0.153406	−	0.988163i	$-0.549024\pi$
$439$	−1.23889e6			−0.306812			−0.153406	+	0.988163i	$0.549024\pi$
$440$		0			0
$441$	108801.			0.0266400
$442$		0			0
$443$			234461.i			0.0567625i	0.999597	+	0.0283813i	$0.00903525\pi$
$443$			234461.i			0.0567625i	−0.999597	+	0.0283813i	$0.990965\pi$
$444$		0			0
$445$			410916.i			0.0983678i
$446$		0			0
$447$			1.55254e6i			0.367513i
$448$		0			0
$449$			7.66103e6i			1.79337i	0.442664	+	0.896687i	$0.354033\pi$
$449$			7.66103e6i			1.79337i	−0.442664	+	0.896687i	$0.645967\pi$
$450$		0			0
$451$	−2.38334e6			−0.551753
$452$		0			0
$453$		−	8.39698e6i		−	1.92255i
$454$		0			0
$455$	5.96866e6			1.35160
$456$		0			0
$457$	−3.08204e6			−0.690315			−0.345157	−	0.938545i	$-0.612174\pi$
$457$	−3.08204e6			−0.690315			−0.345157	+	0.938545i	$0.612174\pi$
$458$		0			0
$459$	1.41985e6			0.314566
$460$		0			0
$461$		−	6.89305e6i		−	1.51063i	−0.655360	−	0.755316i	$-0.727483\pi$
$461$		−	6.89305e6i		−	1.51063i	0.655360	−	0.755316i	$-0.272517\pi$
$462$		0			0
$463$	7.23192e6			1.56784			0.783919	−	0.620863i	$-0.213218\pi$
$463$	7.23192e6			1.56784			0.783919	+	0.620863i	$0.213218\pi$
$464$		0			0
$465$	4.02215e6			0.862632
$466$		0			0
$467$		−	5.58246e6i		−	1.18450i	−0.805756	−	0.592248i	$-0.798241\pi$
$467$		−	5.58246e6i		−	1.18450i	0.805756	−	0.592248i	$-0.201759\pi$
$468$		0			0
$469$	4.09348e6			0.859331
$470$		0			0
$471$	6.36360e6			1.32175
$472$		0			0
$473$	−7.65240e6			−1.57270
$474$		0			0
$475$		−	6.20754e6i		−	1.26237i
$476$		0			0
$477$	−377870.			−0.0760407
$478$		0			0
$479$			3.85893e6i			0.768473i	0.923235	+	0.384236i	$0.125535\pi$
$479$			3.85893e6i			0.768473i	−0.923235	+	0.384236i	$0.874465\pi$
$480$		0			0
$481$			5.86374e6i			1.15561i
$482$		0			0
$483$			6.58499e6i			1.28436i
$484$		0			0
$485$			8.54894e6i			1.65028i
$486$		0			0
$487$	7.45448e6			1.42428			0.712140	−	0.702038i	$-0.247726\pi$
$487$	7.45448e6			1.42428			0.712140	+	0.702038i	$0.247726\pi$
$488$		0			0
$489$	−2.33134e6			−0.440894
$490$		0			0
$491$		−	790498.i		−	0.147978i	−0.997259	−	0.0739890i	$-0.976427\pi$
$491$		−	790498.i		−	0.147978i	0.997259	−	0.0739890i	$-0.0235730\pi$
$492$		0			0
$493$	−1.39653e6	−	1.64729e6i	−0.258781	−	0.305248i
$494$		0			0
$495$			2.42067e6i			0.444040i
$496$		0			0
$497$	−6.29008e6			−1.14226
$498$		0			0
$499$	−9.37327e6			−1.68515			−0.842577	−	0.538576i	$-0.818963\pi$
$499$	−9.37327e6			−1.68515			−0.842577	+	0.538576i	$0.818963\pi$
$500$		0			0
$501$		−	9.98875e6i		−	1.77794i
$502$		0			0
$503$		−	3.67729e6i		−	0.648050i	−0.946049	−	0.324025i	$-0.894964\pi$
$503$		−	3.67729e6i		−	0.648050i	0.946049	−	0.324025i	$-0.105036\pi$
$504$		0			0
$505$		−	3.78955e6i		−	0.661240i
$506$		0			0
$507$			566117.i			0.0978107i
$508$		0			0
$509$	2.96089e6			0.506557			0.253278	−	0.967393i	$-0.418491\pi$
$509$	2.96089e6			0.506557			0.253278	+	0.967393i	$0.418491\pi$
$510$		0			0
$511$			6.02078e6i			1.02000i
$512$		0			0
$513$	−5.00339e6			−0.839404
$514$		0			0
$515$	6.04608e6			1.00451
$516$		0			0
$517$	−9.84473e6			−1.61986
$518$		0			0
$519$			1.05132e7i			1.71323i
$520$		0			0
$521$	7.10737e6			1.14714			0.573568	−	0.819158i	$-0.305559\pi$
$521$	7.10737e6			1.14714			0.573568	+	0.819158i	$0.305559\pi$
$522$		0			0
$523$	−857355.			−0.137059			−0.0685294	−	0.997649i	$-0.521831\pi$
$523$	−857355.			−0.137059			−0.0685294	+	0.997649i	$0.521831\pi$
$524$		0			0
$525$			8.18836e6i			1.29658i
$526$		0			0
$527$	1.29960e6			0.203837
$528$		0			0
$529$	2.38960e6			0.371266
$530$		0			0
$531$	−947834.			−0.145880
$532$		0			0
$533$			3.61937e6i			0.551843i
$534$		0			0
$535$	8.29488e6			1.25293
$536$		0			0
$537$		−	9.08665e6i		−	1.35978i
$538$		0			0
$539$		−	546584.i		−	0.0810372i
$540$		0			0
$541$			2.62756e6i			0.385975i	0.981201	+	0.192988i	$0.0618177\pi$
$541$			2.62756e6i			0.385975i	−0.981201	+	0.192988i	$0.938182\pi$
$542$		0			0
$543$			7.81286e6i			1.13713i
$544$		0			0
$545$	1.20930e7			1.74398
$546$		0			0
$547$	−1.03300e7			−1.47615			−0.738077	−	0.674716i	$-0.764266\pi$
$547$	−1.03300e7			−1.47615			−0.738077	+	0.674716i	$0.764266\pi$
$548$		0			0
$549$		−	3.23010e6i		−	0.457388i
$550$		0			0
$551$	4.92118e6	+	5.80484e6i	0.690543	+	0.814537i
$552$		0			0
$553$		−	3.14664e6i		−	0.437556i
$554$		0			0
$555$	−1.48493e7			−2.04632
$556$		0			0
$557$	−2.92518e6			−0.399498			−0.199749	−	0.979847i	$-0.564013\pi$
$557$	−2.92518e6			−0.399498			−0.199749	+	0.979847i	$0.564013\pi$
$558$		0			0
$559$			1.16210e7i			1.57295i
$560$		0			0
$561$			3.27026e6i			0.438708i
$562$		0			0
$563$			3.99693e6i			0.531441i	0.964050	+	0.265720i	$0.0856098\pi$
$563$			3.99693e6i			0.531441i	−0.964050	+	0.265720i	$0.914390\pi$
$564$		0			0
$565$		−	1.27460e7i		−	1.67978i
$566$		0			0
$567$	8.90217e6			1.16289
$568$		0			0
$569$			1.36215e7i			1.76377i	0.471462	+	0.881887i	$0.343727\pi$
$569$			1.36215e7i			1.76377i	−0.471462	+	0.881887i	$0.656273\pi$
$570$		0			0
$571$	5.89102e6			0.756136			0.378068	−	0.925778i	$-0.376589\pi$
$571$	5.89102e6			0.756136			0.378068	+	0.925778i	$0.376589\pi$
$572$		0			0
$573$	1.65310e7			2.10335
$574$		0			0
$575$	1.09749e7			1.38431
$576$		0			0
$577$		−	3.94840e6i		−	0.493721i	−0.969051	−	0.246860i	$-0.920601\pi$
$577$		−	3.94840e6i		−	0.493721i	0.969051	−	0.246860i	$-0.0793989\pi$
$578$		0			0
$579$	8.03708e6			0.996327
$580$		0			0
$581$	−4.15342e6			−0.510465
$582$		0			0
$583$			1.89831e6i			0.231311i
$584$		0			0
$585$	3.67606e6			0.444112
$586$		0			0
$587$	3.33096e6			0.399001			0.199501	−	0.979898i	$-0.436068\pi$
$587$	3.33096e6			0.399001			0.199501	+	0.979898i	$0.436068\pi$
$588$		0			0
$589$	−4.57962e6			−0.543928
$590$		0			0
$591$			1.18967e7i			1.40107i
$592$		0			0
$593$	3.86488e6			0.451335			0.225667	−	0.974204i	$-0.427544\pi$
$593$	3.86488e6			0.451335			0.225667	+	0.974204i	$0.427544\pi$
$594$		0			0
$595$			4.88382e6i			0.565546i
$596$		0			0
$597$		−	8.19995e6i		−	0.941619i
$598$		0			0
$599$		−	1.45145e7i		−	1.65285i	−0.563044	−	0.826427i	$-0.690370\pi$
$599$		−	1.45145e7i		−	1.65285i	0.563044	−	0.826427i	$-0.309630\pi$
$600$		0			0
$601$		−	3.15999e6i		−	0.356862i	−0.983952	−	0.178431i	$-0.942898\pi$
$601$		−	3.15999e6i		−	0.356862i	0.983952	−	0.178431i	$-0.0571021\pi$
$602$		0			0
$603$	2.52115e6			0.282361
$604$		0			0
$605$	−1.13860e6			−0.126469
$606$		0			0
$607$		−	6.94460e6i		−	0.765025i	−0.923950	−	0.382513i	$-0.875059\pi$
$607$		−	6.94460e6i		−	0.765025i	0.923950	−	0.382513i	$-0.124941\pi$
$608$		0			0
$609$	−6.49153e6	−	7.65715e6i	−0.709257	−	0.836612i
$610$		0			0
$611$			1.49503e7i			1.62012i
$612$		0			0
$613$	4.85595e6			0.521943			0.260972	−	0.965346i	$-0.415957\pi$
$613$	4.85595e6			0.521943			0.260972	+	0.965346i	$0.415957\pi$
$614$		0			0
$615$	−9.16569e6			−0.977187
$616$		0			0
$617$			6.10134e6i			0.645226i	0.946531	+	0.322613i	$0.104561\pi$
$617$			6.10134e6i			0.645226i	−0.946531	+	0.322613i	$0.895439\pi$
$618$		0			0
$619$		−	5.73640e6i		−	0.601745i	−0.953664	−	0.300873i	$-0.902722\pi$
$619$		−	5.73640e6i		−	0.601745i	0.953664	−	0.300873i	$-0.0972779\pi$
$620$		0			0
$621$		−	8.84601e6i		−	0.920488i
$622$		0			0
$623$			617166.i			0.0637062i
$624$		0			0
$625$	−7.66283e6			−0.784673
$626$		0			0
$627$		−	1.15240e7i		−	1.17067i
$628$		0			0
$629$	−4.79797e6			−0.483538
$630$		0			0
$631$	1.64205e7			1.64178			0.820888	−	0.571089i	$-0.193479\pi$
$631$	1.64205e7			1.64178			0.820888	+	0.571089i	$0.193479\pi$
$632$		0			0
$633$	−5.34434e6			−0.530133
$634$		0			0
$635$		−	7.54338e6i		−	0.742389i
$636$		0			0
$637$	−830049.			−0.0810504
$638$		0			0
$639$	−3.87402e6			−0.375326
$640$		0			0
$641$		−	1.38230e7i		−	1.32879i	−0.747381	−	0.664396i	$-0.768689\pi$
$641$		−	1.38230e7i		−	1.32879i	0.747381	−	0.664396i	$-0.231311\pi$
$642$		0			0
$643$	−4.19753e6			−0.400374			−0.200187	−	0.979758i	$-0.564155\pi$
$643$	−4.19753e6			−0.400374			−0.200187	+	0.979758i	$0.564155\pi$
$644$		0			0
$645$	−2.94291e7			−2.78534
$646$		0			0
$647$	−1.11302e7			−1.04530			−0.522651	−	0.852547i	$-0.675057\pi$
$647$	−1.11302e7			−1.04530			−0.522651	+	0.852547i	$0.675057\pi$
$648$		0			0
$649$			4.76165e6i			0.443758i
$650$		0			0
$651$	6.04097e6			0.558669
$652$		0			0
$653$		−	1.08532e7i		−	0.996040i	−0.867166	−	0.498020i	$-0.834061\pi$
$653$		−	1.08532e7i		−	0.996040i	0.867166	−	0.498020i	$-0.165939\pi$
$654$		0			0
$655$			4.44182e6i			0.404537i
$656$		0			0
$657$			3.70816e6i			0.335154i
$658$		0			0
$659$		−	9.45078e6i		−	0.847723i	−0.905727	−	0.423862i	$-0.860674\pi$
$659$		−	9.45078e6i		−	0.847723i	0.905727	−	0.423862i	$-0.139326\pi$
$660$		0			0
$661$	−39195.1			−0.00348922			−0.00174461	−	0.999998i	$-0.500555\pi$
$661$	−39195.1			−0.00348922			−0.00174461	+	0.999998i	$0.500555\pi$
$662$		0			0
$663$	4.96626e6			0.438779
$664$		0			0
$665$		−	1.72100e7i		−	1.50913i
$666$		0			0
$667$	−1.02630e7	+	8.70067e6i	−0.893220	+	0.757248i
$668$		0			0
$669$			2.36637e7i			2.04417i
$670$		0			0
$671$	−1.62271e7			−1.39134
$672$		0			0
$673$	1.73985e7			1.48072			0.740360	−	0.672211i	$-0.234655\pi$
$673$	1.73985e7			1.48072			0.740360	+	0.672211i	$0.234655\pi$
$674$		0			0
$675$		−	1.09999e7i		−	0.929243i
$676$		0			0
$677$			1.58614e7i			1.33005i	0.746819	+	0.665027i	$0.231580\pi$
$677$			1.58614e7i			1.33005i	−0.746819	+	0.665027i	$0.768420\pi$
$678$		0			0
$679$			1.28399e7i			1.06878i
$680$		0			0
$681$		−	1.08041e7i		−	0.892730i
$682$		0			0
$683$	−368754.			−0.0302472			−0.0151236	−	0.999886i	$-0.504814\pi$
$683$	−368754.			−0.0302472			−0.0151236	+	0.999886i	$0.504814\pi$
$684$		0			0
$685$			2.20726e7i			1.79733i
$686$		0			0
$687$	−1.93656e7			−1.56545
$688$		0			0
$689$	2.88280e6			0.231348
$690$		0			0
$691$	2.26412e7			1.80387			0.901933	−	0.431876i	$-0.142148\pi$
$691$	2.26412e7			1.80387			0.901933	+	0.431876i	$0.142148\pi$
$692$		0			0
$693$			3.63566e6i			0.287575i
$694$		0			0
$695$	−1.48127e7			−1.16325
$696$		0			0
$697$	−2.96153e6			−0.230906
$698$		0			0
$699$			1.34076e7i			1.03791i
$700$		0			0
$701$	5.63957e6			0.433462			0.216731	−	0.976231i	$-0.430461\pi$
$701$	5.63957e6			0.433462			0.216731	+	0.976231i	$0.430461\pi$
$702$		0			0
$703$	1.69075e7			1.29030
$704$		0			0
$705$	−3.78602e7			−2.86887
$706$		0			0
$707$		−	5.69162e6i		−	0.428240i
$708$		0			0
$709$	1.04032e7			0.777231			0.388615	−	0.921400i	$-0.372954\pi$
$709$	1.04032e7			0.777231			0.388615	+	0.921400i	$0.372954\pi$
$710$		0			0
$711$		−	1.93799e6i		−	0.143773i
$712$		0			0
$713$		−	8.09678e6i		−	0.596470i
$714$		0			0
$715$		−	1.84675e7i		−	1.35096i
$716$		0			0
$717$		−	7.73421e6i		−	0.561847i
$718$		0			0
$719$	1.17058e7			0.844457			0.422228	−	0.906490i	$-0.361248\pi$
$719$	1.17058e7			0.844457			0.422228	+	0.906490i	$0.361248\pi$
$720$		0			0
$721$	9.08077e6			0.650556
$722$		0			0
$723$			9.67973e6i			0.688680i
$724$		0			0
$725$	−1.27619e7	+	1.08192e7i	−0.901716	+	0.764450i
$726$		0			0
$727$		−	2.25143e7i		−	1.57988i	−0.613187	−	0.789938i	$-0.710113\pi$
$727$		−	2.25143e7i		−	1.57988i	0.613187	−	0.789938i	$-0.289887\pi$
$728$		0			0
$729$	−7.44821e6			−0.519079
$730$		0			0
$731$	−9.50885e6			−0.658165
$732$		0			0
$733$		−	2.84058e7i		−	1.95275i	−0.216074	−	0.976377i	$-0.569325\pi$
$733$		−	2.84058e7i		−	1.95275i	0.216074	−	0.976377i	$-0.430675\pi$
$734$		0			0
$735$		−	2.10202e6i		−	0.143522i
$736$		0			0
$737$		−	1.26655e7i		−	0.858924i
$738$		0			0
$739$			1.26863e7i			0.854523i	0.904128	+	0.427262i	$0.140522\pi$
$739$			1.26863e7i			0.854523i	−0.904128	+	0.427262i	$0.859478\pi$
$740$		0			0
$741$	−1.75005e7			−1.17086
$742$		0			0
$743$		−	1.52737e7i		−	1.01502i	−0.861647	−	0.507508i	$-0.830567\pi$
$743$		−	1.52737e7i		−	1.01502i	0.861647	−	0.507508i	$-0.169433\pi$
$744$		0			0
$745$	7.17382e6			0.473543
$746$		0			0
$747$	−2.55806e6			−0.167730
$748$		0			0
$749$	1.24583e7			0.811436
$750$		0			0
$751$		−	3.64779e6i		−	0.236010i	−0.993013	−	0.118005i	$-0.962350\pi$
$751$		−	3.64779e6i		−	0.236010i	0.993013	−	0.118005i	$-0.0376498\pi$
$752$		0			0
$753$	8.61230e6			0.553518
$754$		0			0
$755$	−3.88000e7			−2.47722
$756$		0			0
$757$			1.73535e7i			1.10065i	0.834952	+	0.550323i	$0.185496\pi$
$757$			1.73535e7i			1.10065i	−0.834952	+	0.550323i	$0.814504\pi$
$758$		0			0
$759$	2.03745e7			1.28375
$760$		0			0
$761$	−1.31411e7			−0.822564			−0.411282	−	0.911508i	$-0.634919\pi$
$761$	−1.31411e7			−0.822564			−0.411282	+	0.911508i	$0.634919\pi$
$762$		0			0
$763$	1.81628e7			1.12946
$764$		0			0
$765$			3.00791e6i			0.185828i
$766$		0			0
$767$	7.23111e6			0.443830
$768$		0			0
$769$			3.12336e7i			1.90461i	0.305146	+	0.952306i	$0.401295\pi$
$769$			3.12336e7i			1.90461i	−0.305146	+	0.952306i	$0.598705\pi$
$770$		0			0
$771$		−	1.84962e7i		−	1.12059i
$772$		0			0
$773$		−	2.56908e7i		−	1.54643i	−0.634146	−	0.773213i	$-0.718648\pi$
$773$		−	2.56908e7i		−	1.54643i	0.634146	−	0.773213i	$-0.281352\pi$
$774$		0			0
$775$		−	1.00683e7i		−	0.602143i
$776$		0			0
$777$	−2.23026e7			−1.32527
$778$		0			0
$779$	1.04361e7			0.616159
$780$		0			0
$781$			1.94620e7i			1.14172i
$782$		0			0
$783$	8.72045e6	+	1.02863e7i	0.508317	+	0.599591i
$784$		0			0
$785$		−	2.94043e7i		−	1.70309i
$786$		0			0
$787$	−2.04792e7			−1.17863			−0.589314	−	0.807904i	$-0.700602\pi$
$787$	−2.04792e7			−1.17863			−0.589314	+	0.807904i	$0.700602\pi$
$788$		0			0
$789$	3.16028e7			1.80731
$790$		0			0
$791$		−	1.91435e7i		−	1.08788i
$792$		0			0
$793$			2.46427e7i			1.39157i
$794$		0			0
$795$			7.30040e6i			0.409665i
$796$		0			0
$797$		−	6.67390e6i		−	0.372164i	−0.982534	−	0.186082i	$-0.940421\pi$
$797$		−	6.67390e6i		−	0.372164i	0.982534	−	0.186082i	$-0.0595790\pi$
$798$		0			0
$799$	−1.22330e7			−0.677903
$800$		0			0
$801$			380108.i			0.0209327i
$802$		0			0
$803$	1.86287e7			1.01952
$804$		0			0
$805$	3.04273e7			1.65491
$806$		0			0
$807$	3.83940e7			2.07529
$808$		0			0
$809$			1.95334e7i			1.04932i	0.851312	+	0.524659i	$0.175807\pi$
$809$			1.95334e7i			1.04932i	−0.851312	+	0.524659i	$0.824193\pi$
$810$		0			0
$811$	3.21877e7			1.71845			0.859227	−	0.511595i	$-0.170945\pi$
$811$	3.21877e7			1.71845			0.859227	+	0.511595i	$0.170945\pi$
$812$		0			0
$813$	−1.75235e7			−0.929812
$814$		0			0
$815$			1.07724e7i			0.568094i
$816$		0			0
$817$	3.35080e7			1.75628
$818$		0			0
$819$	5.52117e6			0.287621
$820$		0			0
$821$	−1.65356e7			−0.856173			−0.428087	−	0.903738i	$-0.640812\pi$
$821$	−1.65356e7			−0.856173			−0.428087	+	0.903738i	$0.640812\pi$
$822$		0			0
$823$			3.22979e7i			1.66217i	0.556146	+	0.831085i	$0.312280\pi$
$823$			3.22979e7i			1.66217i	−0.556146	+	0.831085i	$0.687720\pi$
$824$		0			0
$825$	2.53354e7			1.29596
$826$		0			0
$827$			2.34191e7i			1.19071i	0.803462	+	0.595356i	$0.202989\pi$
$827$			2.34191e7i			1.19071i	−0.803462	+	0.595356i	$0.797011\pi$
$828$		0			0
$829$			6.36511e6i			0.321677i	0.986981	+	0.160838i	$0.0514198\pi$
$829$			6.36511e6i			0.321677i	−0.986981	+	0.160838i	$0.948580\pi$
$830$		0			0
$831$			3.22030e7i			1.61769i
$832$		0			0
$833$		−	679183.i		−	0.0339136i
$834$		0			0
$835$	−4.61551e7			−2.29088
$836$		0			0
$837$	−8.11520e6			−0.400392
$838$		0			0
$839$			2.21934e7i			1.08847i	0.838931	+	0.544237i	$0.183181\pi$
$839$			2.21934e7i			1.08847i	−0.838931	+	0.544237i	$0.816819\pi$
$840$		0			0
$841$	3.35679e6	−	2.02346e7i	0.163657	−	0.986517i
$842$		0			0
$843$		−	2.85363e7i		−	1.38302i
$844$		0			0
$845$	2.61586e6			0.126030
$846$		0			0
$847$	−1.71010e6			−0.0819055
$848$		0			0
$849$			2.67235e7i			1.27240i
$850$		0			0
$851$			2.98924e7i			1.41494i
$852$		0			0
$853$			2.70083e7i			1.27094i	0.772125	+	0.635470i	$0.219194\pi$
$853$			2.70083e7i			1.27094i	−0.772125	+	0.635470i	$0.780806\pi$
$854$		0			0
$855$		−	1.05995e7i		−	0.495873i
$856$		0			0
$857$	−2.90868e7			−1.35283			−0.676417	−	0.736519i	$-0.736468\pi$
$857$	−2.90868e7			−1.35283			−0.676417	+	0.736519i	$0.736468\pi$
$858$		0			0
$859$			6.50109e6i			0.300610i	0.988640	+	0.150305i	$0.0480255\pi$
$859$			6.50109e6i			0.300610i	−0.988640	+	0.150305i	$0.951974\pi$
$860$		0			0
$861$	−1.37662e7			−0.632858
$862$		0			0
$863$	−1.76657e7			−0.807427			−0.403714	−	0.914885i	$-0.632281\pi$
$863$	−1.76657e7			−0.807427			−0.403714	+	0.914885i	$0.632281\pi$
$864$		0			0
$865$	4.85784e7			2.20751
$866$		0			0
$867$		−	2.13112e7i		−	0.962855i
$868$		0			0
$869$	−9.73593e6			−0.437349
$870$		0			0
$871$	−1.92340e7			−0.859064
$872$		0			0
$873$			7.90800e6i			0.351181i
$874$		0			0
$875$	5.82987e6			0.257418
$876$		0			0
$877$	−1.65170e7			−0.725157			−0.362579	−	0.931953i	$-0.618104\pi$
$877$	−1.65170e7			−0.725157			−0.362579	+	0.931953i	$0.618104\pi$
$878$		0			0
$879$	3.11986e7			1.36195
$880$		0			0
$881$		−	8.29360e6i		−	0.360001i	−0.983667	−	0.180000i	$-0.942390\pi$
$881$		−	8.29360e6i		−	0.360001i	0.983667	−	0.180000i	$-0.0576099\pi$
$882$		0			0
$883$	308478.			0.0133144			0.00665721	−	0.999978i	$-0.497881\pi$
$883$	308478.			0.0133144			0.00665721	+	0.999978i	$0.497881\pi$
$884$		0			0
$885$			1.83121e7i			0.785921i
$886$		0			0
$887$		−	3.88210e7i		−	1.65675i	−0.560171	−	0.828377i	$-0.689265\pi$
$887$		−	3.88210e7i		−	1.65675i	0.560171	−	0.828377i	$-0.310735\pi$
$888$		0			0
$889$		−	1.13296e7i		−	0.480796i
$890$		0			0
$891$		−	2.75440e7i		−	1.16234i
$892$		0			0
$893$	4.31077e7			1.80895
$894$		0			0
$895$	−4.19867e7			−1.75208
$896$		0			0
$897$		−	3.09409e7i		−	1.28396i
$898$		0			0
$899$	7.98186e6	+	9.41509e6i	0.329386	+	0.388531i
$900$		0			0
$901$			2.35884e6i			0.0968023i
$902$		0			0
$903$	−4.42004e7			−1.80388
$904$		0			0
$905$	3.61009e7			1.46520
$906$		0			0
$907$		−	1.46696e7i		−	0.592108i	−0.955171	−	0.296054i	$-0.904329\pi$
$907$		−	1.46696e7i		−	0.592108i	0.955171	−	0.296054i	$-0.0956708\pi$
$908$		0			0
$909$		−	3.50543e6i		−	0.140712i
$910$		0			0
$911$			7.19229e6i			0.287125i	0.989641	+	0.143563i	$0.0458558\pi$
$911$			7.19229e6i			0.287125i	−0.989641	+	0.143563i	$0.954144\pi$
$912$		0			0
$913$			1.28510e7i			0.510223i
$914$		0			0
$915$	−6.24051e7			−2.46415
$916$		0			0
$917$			6.67129e6i			0.261991i
$918$		0			0
$919$	−9.93221e6			−0.387934			−0.193967	−	0.981008i	$-0.562135\pi$
$919$	−9.93221e6			−0.387934			−0.193967	+	0.981008i	$0.562135\pi$
$920$		0			0
$921$	4.50539e7			1.75018
$922$		0			0
$923$	2.95552e7			1.14190
$924$		0			0
$925$			3.71709e7i			1.42840i
$926$		0			0
$927$	5.59279e6			0.213761
$928$		0			0
$929$	−4.49659e7			−1.70940			−0.854701	−	0.519120i	$-0.826260\pi$
$929$	−4.49659e7			−1.70940			−0.854701	+	0.519120i	$0.826260\pi$
$930$		0			0
$931$			2.39336e6i			0.0904968i
$932$		0			0
$933$	3.64099e7			1.36935
$934$		0			0
$935$	1.51109e7			0.565278
$936$		0			0
$937$	2.67471e7			0.995241			0.497621	−	0.867395i	$-0.334207\pi$
$937$	2.67471e7			0.995241			0.497621	+	0.867395i	$0.334207\pi$
$938$		0			0
$939$			2.25147e7i			0.833302i
$940$		0			0
$941$	−1.60955e7			−0.592559			−0.296279	−	0.955101i	$-0.595746\pi$
$941$	−1.60955e7			−0.592559			−0.296279	+	0.955101i	$0.595746\pi$
$942$		0			0
$943$			1.84510e7i			0.675679i
$944$		0			0
$945$		−	3.04965e7i		−	1.11089i
$946$		0			0
$947$			1.51931e7i			0.550519i	0.961370	+	0.275260i	$0.0887638\pi$
$947$			1.51931e7i			0.550519i	−0.961370	+	0.275260i	$0.911236\pi$
$948$		0			0
$949$		−	2.82899e7i		−	1.01968i
$950$		0			0
$951$	1.29501e7			0.464324
$952$		0			0
$953$	1.37193e7			0.489327			0.244663	−	0.969608i	$-0.421323\pi$
$953$	1.37193e7			0.489327			0.244663	+	0.969608i	$0.421323\pi$
$954$		0			0
$955$		−	7.63847e7i		−	2.71018i
$956$		0			0
$957$	−2.36918e7	+	2.00853e7i	−0.836215	+	0.708921i
$958$		0			0
$959$			3.31515e7i			1.16401i
$960$		0			0
$961$	2.12013e7			0.740549
$962$		0			0
$963$	7.67298e6			0.266623
$964$		0			0
$965$		−	3.71370e7i		−	1.28377i
$966$		0			0
$967$		−	2.03925e7i		−	0.701300i	−0.936507	−	0.350650i	$-0.885961\pi$
$967$		−	2.03925e7i		−	0.701300i	0.936507	−	0.350650i	$-0.114039\pi$
$968$		0			0
$969$		−	1.43197e7i		−	0.489918i
$970$		0			0
$971$		−	1.78600e7i		−	0.607903i	−0.952687	−	0.303952i	$-0.901694\pi$
$971$		−	1.78600e7i		−	0.607903i	0.952687	−	0.303952i	$-0.0983061\pi$
$972$		0			0
$973$	−2.22476e7			−0.753356
$974$		0			0
$975$		−	3.84747e7i		−	1.29617i
$976$		0			0
$977$	−4.01364e7			−1.34525			−0.672623	−	0.739986i	$-0.734832\pi$
$977$	−4.01364e7			−1.34525			−0.672623	+	0.739986i	$0.734832\pi$
$978$		0			0
$979$	1.90956e6			0.0636760
$980$		0			0
$981$	1.11863e7			0.371121
$982$		0			0
$983$		−	1.29137e7i		−	0.426251i	−0.977025	−	0.213126i	$-0.931636\pi$
$983$		−	1.29137e7i		−	0.426251i	0.977025	−	0.213126i	$-0.0683643\pi$
$984$		0			0
$985$	5.49713e7			1.80528
$986$		0			0
$987$	−5.68633e7			−1.85797
$988$		0			0
$989$			5.92423e7i			1.92593i
$990$		0			0
$991$	−1.20813e7			−0.390777			−0.195388	−	0.980726i	$-0.562597\pi$
$991$	−1.20813e7			−0.390777			−0.195388	+	0.980726i	$0.562597\pi$
$992$		0			0
$993$	−1.87566e7			−0.603645
$994$		0			0
$995$	−3.78896e7			−1.21328
$996$		0			0
$997$		−	1.49094e7i		−	0.475031i	−0.971384	−	0.237515i	$-0.923667\pi$
$997$		−	1.49094e7i		−	0.475031i	0.971384	−	0.237515i	$-0.0763330\pi$
$998$		0			0
$999$	2.99604e7			0.949804

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	116.6.c.a.57.2	✓	12
3.2	odd	2		1044.6.h.a.289.11		12
4.3	odd	2		464.6.e.a.289.11		12
29.28	even	2	inner	116.6.c.a.57.11	yes	12
87.86	odd	2		1044.6.h.a.289.12		12
116.115	odd	2		464.6.e.a.289.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.6.c.a.57.2	✓	12	1.1	even	1	trivial
116.6.c.a.57.11	yes	12	29.28	even	2	inner
464.6.e.a.289.2		12	116.115	odd	2
464.6.e.a.289.11		12	4.3	odd	2
1044.6.h.a.289.11		12	3.2	odd	2
1044.6.h.a.289.12		12	87.86	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	116.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-17.8714i q^{3} -82.5785 q^{5} -124.027 q^{7} -76.3873 q^{9} +O(q^{10})\)	\(q-17.8714i q^{3} -82.5785 q^{5} -124.027 q^{7} -76.3873 q^{9} +383.748i q^{11} +582.766 q^{13} +1475.79i q^{15} +476.845i q^{17} -1680.34i q^{19} +2216.54i q^{21} +2970.85 q^{23} +3694.21 q^{25} -2977.60i q^{27} +(-3454.56 + 2928.68i) q^{29} -2725.41i q^{31} +6858.13 q^{33} +10242.0 q^{35} +10061.9i q^{37} -10414.8i q^{39} +6210.68i q^{41} +19941.2i q^{43} +6307.95 q^{45} +25654.1i q^{47} -1424.33 q^{49} +8521.89 q^{51} +4946.76 q^{53} -31689.4i q^{55} -30030.1 q^{57} +12408.3 q^{59} +42285.7i q^{61} +9474.08 q^{63} -48123.9 q^{65} -33004.8 q^{67} -53093.2i q^{69} +50715.4 q^{71} -48544.2i q^{73} -66020.8i q^{75} -47595.1i q^{77} +25370.6i q^{79} -71776.1 q^{81} +33488.1 q^{83} -39377.1i q^{85} +(52339.7 + 61737.8i) q^{87} -4976.06i q^{89} -72278.6 q^{91} -48707.0 q^{93} +138760. i q^{95} -103525. i q^{97} -29313.5i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9}+O(q^{10}) \) 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9	\( 12 q - 10 q^{5} + 76 q^{7} - 650 q^{9} - 194 q^{13} + 3996 q^{23} + 1782 q^{25} - 1900 q^{29} + 7266 q^{33} + 12348 q^{35} - 13988 q^{45} - 8396 q^{49} - 10788 q^{51} - 28626 q^{53} + 11140 q^{57} + 27356 q^{59} + 55392 q^{63} - 55126 q^{65} - 13184 q^{67} + 44352 q^{71} + 74264 q^{81} - 192628 q^{83} + 98908 q^{87} + 13580 q^{91} + 59710 q^{93}+O(q^{100}) \) 12 * q - 10 * q^5 + 76 * q^7 - 650 * q^9 - 194 * q^13 + 3996 * q^23 + 1782 * q^25 - 1900 * q^29 + 7266 * q^33 + 12348 * q^35 - 13988 * q^45 - 8396 * q^49 - 10788 * q^51 - 28626 * q^53 + 11140 * q^57 + 27356 * q^59 + 55392 * q^63 - 55126 * q^65 - 13184 * q^67 + 44352 * q^71 + 74264 * q^81 - 192628 * q^83 + 98908 * q^87 + 13580 * q^91 + 59710 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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