LMFDB - Embedded newform 684.3.h.e.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.219615408.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 22x^{4} - 39x^{3} + 112x^{2} - 93x + 39 $ x^6 - 3x^5 + 22x^4 - 39x^3 + 112x^2 - 93*x + 39
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3 $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.3
Root			$0.500000 + 3.19918i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.3.h.e.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.556406 q^{5} -9.52587 q^{7} +O(q^{10})$	$q-0.556406 q^{5} -9.52587 q^{7} +13.5259 q^{11} -10.5348i q^{13} -2.63868 q^{17} +(0.112811 + 18.9997i) q^{19} +22.0212 q^{23} -24.6904 q^{25} +48.7824i q^{29} -0.248288i q^{31} +5.30025 q^{35} +59.3172i q^{37} +69.0705i q^{41} -27.5869 q^{43} -44.8855 q^{47} +41.7421 q^{49} -3.85489i q^{53} -7.52587 q^{55} +59.3539i q^{59} +96.4554 q^{61} +5.86162i q^{65} -10.7831i q^{67} -52.6373i q^{71} +13.7362 q^{73} -128.846 q^{77} -51.3224i q^{79} +63.7744 q^{83} +1.46818 q^{85} +44.9275i q^{89} +100.353i q^{91} +(-0.0627687 - 10.5715i) q^{95} +72.6403i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 6 * q + 2 * q^5 - 2 * q^7	$ 6 q + 2 q^{5} - 2 q^{7} + 26 q^{11} + 50 q^{17} - 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 210 q^{43} - 22 q^{47} - 36 q^{49} + 10 q^{55} + 214 q^{61} + 102 q^{73} - 266 q^{77} + 404 q^{83} + 370 q^{85} - 358 q^{95}+O(q^{100}) $ 6 * q + 2 * q^5 - 2 * q^7 + 26 * q^11 + 50 * q^17 - 10 * q^19 - 28 * q^23 + 28 * q^25 - 2 * q^35 - 210 * q^43 - 22 * q^47 - 36 * q^49 + 10 * q^55 + 214 * q^61 + 102 * q^73 - 266 * q^77 + 404 * q^83 + 370 * q^85 - 358 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.556406			−0.111281			−0.0556406	−	0.998451i	$-0.517720\pi$
$5$	−0.556406			−0.111281			−0.0556406	+	0.998451i	$0.517720\pi$
$6$		0			0
$7$	−9.52587			−1.36084			−0.680419	−	0.732823i	$-0.738202\pi$
$7$	−9.52587			−1.36084			−0.680419	+	0.732823i	$0.738202\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	13.5259			1.22962			0.614812	−	0.788674i	$-0.289232\pi$
$11$	13.5259			1.22962			0.614812	+	0.788674i	$0.289232\pi$
$12$		0			0
$13$		−	10.5348i		−	0.810370i	−0.914235	−	0.405185i	$-0.867207\pi$
$13$		−	10.5348i		−	0.810370i	0.914235	−	0.405185i	$-0.132793\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.63868			−0.155216			−0.0776082	−	0.996984i	$-0.524728\pi$
$17$	−2.63868			−0.155216			−0.0776082	+	0.996984i	$0.524728\pi$
$18$		0			0
$19$	0.112811	+	18.9997i	0.00593742	+	0.999982i
$19$	0.112811	+	18.9997i	0.00593742	+	0.999982i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	22.0212			0.957443			0.478722	−	0.877967i	$-0.341100\pi$
$23$	22.0212			0.957443			0.478722	+	0.877967i	$0.341100\pi$
$24$		0			0
$25$	−24.6904			−0.987617
$26$		0			0
$27$		0			0
$28$		0			0
$29$			48.7824i			1.68215i	0.540917	+	0.841076i	$0.318077\pi$
$29$			48.7824i			1.68215i	−0.540917	+	0.841076i	$0.681923\pi$
$30$		0			0
$31$		−	0.248288i		−	0.00800930i	−0.999992	−	0.00400465i	$-0.998725\pi$
$31$		−	0.248288i		−	0.00800930i	0.999992	−	0.00400465i	$-0.00127472\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.30025			0.151436
$36$		0			0
$37$			59.3172i			1.60317i	0.597882	+	0.801584i	$0.296009\pi$
$37$			59.3172i			1.60317i	−0.597882	+	0.801584i	$0.703991\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			69.0705i			1.68465i	0.538974	+	0.842323i	$0.318812\pi$
$41$			69.0705i			1.68465i	−0.538974	+	0.842323i	$0.681188\pi$
$42$		0			0
$43$	−27.5869			−0.641557			−0.320778	−	0.947154i	$-0.603944\pi$
$43$	−27.5869			−0.641557			−0.320778	+	0.947154i	$0.603944\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−44.8855			−0.955011			−0.477505	−	0.878629i	$-0.658459\pi$
$47$	−44.8855			−0.955011			−0.477505	+	0.878629i	$0.658459\pi$
$48$		0			0
$49$	41.7421			0.851881
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	3.85489i		−	0.0727338i	−0.999339	−	0.0363669i	$-0.988422\pi$
$53$		−	3.85489i		−	0.0727338i	0.999339	−	0.0363669i	$-0.0115785\pi$
$54$		0			0
$55$	−7.52587			−0.136834
$56$		0			0
$57$		0			0
$58$		0			0
$59$			59.3539i			1.00600i	0.864287	+	0.503000i	$0.167770\pi$
$59$			59.3539i			1.00600i	−0.864287	+	0.503000i	$0.832230\pi$
$60$		0			0
$61$	96.4554			1.58124			0.790618	−	0.612309i	$-0.209759\pi$
$61$	96.4554			1.58124			0.790618	+	0.612309i	$0.209759\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			5.86162i			0.0901788i
$66$		0			0
$67$		−	10.7831i		−	0.160942i	−0.996757	−	0.0804708i	$-0.974358\pi$
$67$		−	10.7831i		−	0.160942i	0.996757	−	0.0804708i	$-0.0256424\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	52.6373i		−	0.741371i	−0.928759	−	0.370685i	$-0.879123\pi$
$71$		−	52.6373i		−	0.741371i	0.928759	−	0.370685i	$-0.120877\pi$
$72$		0			0
$73$	13.7362			0.188167			0.0940835	−	0.995564i	$-0.470008\pi$
$73$	13.7362			0.188167			0.0940835	+	0.995564i	$0.470008\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−128.846			−1.67332
$78$		0			0
$79$		−	51.3224i		−	0.649651i	−0.945774	−	0.324826i	$-0.894694\pi$
$79$		−	51.3224i		−	0.649651i	0.945774	−	0.324826i	$-0.105306\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	63.7744			0.768366			0.384183	−	0.923257i	$-0.374483\pi$
$83$	63.7744			0.768366			0.384183	+	0.923257i	$0.374483\pi$
$84$		0			0
$85$	1.46818			0.0172726
$86$		0			0
$87$		0			0
$88$		0			0
$89$			44.9275i			0.504804i	0.967622	+	0.252402i	$0.0812205\pi$
$89$			44.9275i			0.504804i	−0.967622	+	0.252402i	$0.918780\pi$
$90$		0			0
$91$			100.353i			1.10278i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.0627687	−	10.5715i	−0.000660723	−	0.111279i
$96$		0			0
$97$			72.6403i			0.748870i	0.927253	+	0.374435i	$0.122163\pi$
$97$			72.6403i			0.748870i	−0.927253	+	0.374435i	$0.877837\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−116.783			−1.15627			−0.578133	−	0.815943i	$-0.696218\pi$
$101$	−116.783			−1.15627			−0.578133	+	0.815943i	$0.696218\pi$
$102$		0			0
$103$		−	72.3921i		−	0.702836i	−0.936219	−	0.351418i	$-0.885700\pi$
$103$		−	72.3921i		−	0.702836i	0.936219	−	0.351418i	$-0.114300\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			154.057i			1.43979i	0.694085	+	0.719893i	$0.255809\pi$
$107$			154.057i			1.43979i	−0.694085	+	0.719893i	$0.744191\pi$
$108$		0			0
$109$			159.670i			1.46487i	0.680839	+	0.732433i	$0.261615\pi$
$109$			159.670i			1.46487i	−0.680839	+	0.732433i	$0.738385\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			79.6420i			0.704796i	0.935850	+	0.352398i	$0.114634\pi$
$113$			79.6420i			0.704796i	−0.935850	+	0.352398i	$0.885366\pi$
$114$		0			0
$115$	−12.2527			−0.106545
$116$		0			0
$117$		0			0
$118$		0			0
$119$	25.1357			0.211224
$120$		0			0
$121$	61.9491			0.511976
$122$		0			0
$123$		0			0
$124$		0			0
$125$	27.6480			0.221184
$126$		0			0
$127$			30.2528i			0.238211i	0.992882	+	0.119106i	$0.0380027\pi$
$127$			30.2528i			0.238211i	−0.992882	+	0.119106i	$0.961997\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	142.842			1.09040			0.545199	−	0.838306i	$-0.316454\pi$
$131$	142.842			1.09040			0.545199	+	0.838306i	$0.316454\pi$
$132$		0			0
$133$	−1.07462	−	180.988i	−0.00807987	−	1.36081i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	165.488			1.20795			0.603973	−	0.797005i	$-0.293584\pi$
$137$	165.488			1.20795			0.603973	+	0.797005i	$0.293584\pi$
$138$		0			0
$139$	−169.324			−1.21816			−0.609079	−	0.793110i	$-0.708461\pi$
$139$	−169.324			−1.21816			−0.609079	+	0.793110i	$0.708461\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	142.492i		−	0.996450i
$144$		0			0
$145$		−	27.1428i		−	0.187192i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	139.406			0.935612			0.467806	−	0.883831i	$-0.345045\pi$
$149$	139.406			0.935612			0.467806	+	0.883831i	$0.345045\pi$
$150$		0			0
$151$		−	152.742i		−	1.01154i	−0.862669	−	0.505769i	$-0.831209\pi$
$151$		−	152.742i		−	1.01154i	0.862669	−	0.505769i	$-0.168791\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			0.138149i			0.000891284i
$156$		0			0
$157$	−295.932			−1.88492			−0.942459	−	0.334321i	$-0.891493\pi$
$157$	−295.932			−1.88492			−0.942459	+	0.334321i	$0.891493\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−209.771			−1.30293
$162$		0			0
$163$	−112.731			−0.691602			−0.345801	−	0.938308i	$-0.612393\pi$
$163$	−112.731			−0.691602			−0.345801	+	0.938308i	$0.612393\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	297.205i		−	1.77967i	−0.456284	−	0.889834i	$-0.650820\pi$
$167$		−	297.205i		−	1.77967i	0.456284	−	0.889834i	$-0.349180\pi$
$168$		0			0
$169$	58.0179			0.343301
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	262.849i		−	1.51936i	−0.650299	−	0.759678i	$-0.725357\pi$
$173$		−	262.849i		−	1.51936i	0.650299	−	0.759678i	$-0.274643\pi$
$174$		0			0
$175$	235.198			1.34399
$176$		0			0
$177$		0			0
$178$		0			0
$179$			73.9185i			0.412953i	0.978452	+	0.206476i	$0.0661996\pi$
$179$			73.9185i			0.412953i	−0.978452	+	0.206476i	$0.933800\pi$
$180$		0			0
$181$		−	34.8893i		−	0.192759i	−0.995345	−	0.0963793i	$-0.969274\pi$
$181$		−	34.8893i		−	0.192759i	0.995345	−	0.0963793i	$-0.0307262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	33.0044i		−	0.178402i
$186$		0			0
$187$	−35.6904			−0.190858
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−111.001			−0.581156			−0.290578	−	0.956851i	$-0.593848\pi$
$191$	−111.001			−0.581156			−0.290578	+	0.956851i	$0.593848\pi$
$192$		0			0
$193$		−	83.5616i		−	0.432962i	−0.976287	−	0.216481i	$-0.930542\pi$
$193$		−	83.5616i		−	0.432962i	0.976287	−	0.216481i	$-0.0694579\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	365.319			1.85441			0.927205	−	0.374554i	$-0.122204\pi$
$197$	365.319			1.85441			0.927205	+	0.374554i	$0.122204\pi$
$198$		0			0
$199$	−10.6303			−0.0534184			−0.0267092	−	0.999643i	$-0.508503\pi$
$199$	−10.6303			−0.0534184			−0.0267092	+	0.999643i	$0.508503\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	464.695i		−	2.28914i
$204$		0			0
$205$		−	38.4312i		−	0.187469i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.52587	+	256.987i	0.00730080	+	1.22960i
$210$		0			0
$211$			230.194i			1.09097i	0.838122	+	0.545483i	$0.183654\pi$
$211$			230.194i			1.09097i	−0.838122	+	0.545483i	$0.816346\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.3495			0.0713932
$216$		0			0
$217$			2.36516i			0.0108994i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			27.7980i			0.125783i
$222$		0			0
$223$		−	263.814i		−	1.18302i	−0.806297	−	0.591511i	$-0.798532\pi$
$223$		−	263.814i		−	1.18302i	0.806297	−	0.591511i	$-0.201468\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	213.928i		−	0.942414i	−0.882023	−	0.471207i	$-0.843818\pi$
$227$		−	213.928i		−	0.942414i	0.882023	−	0.471207i	$-0.156182\pi$
$228$		0			0
$229$	−63.4698			−0.277161			−0.138580	−	0.990351i	$-0.544254\pi$
$229$	−63.4698			−0.277161			−0.138580	+	0.990351i	$0.544254\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−288.449			−1.23798			−0.618989	−	0.785400i	$-0.712457\pi$
$233$	−288.449			−1.23798			−0.618989	+	0.785400i	$0.712457\pi$
$234$		0			0
$235$	24.9745			0.106275
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−279.128			−1.16790			−0.583950	−	0.811790i	$-0.698493\pi$
$239$	−279.128			−1.16790			−0.583950	+	0.811790i	$0.698493\pi$
$240$		0			0
$241$			242.687i			1.00700i	0.863995	+	0.503500i	$0.167955\pi$
$241$			242.687i			1.00700i	−0.863995	+	0.503500i	$0.832045\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−23.2256			−0.0947982
$246$		0			0
$247$	200.158	−	1.18844i	0.810355	−	0.00481151i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	162.678			0.648118			0.324059	−	0.946037i	$-0.394952\pi$
$251$	162.678			0.648118			0.324059	+	0.946037i	$0.394952\pi$
$252$		0			0
$253$	297.856			1.17730
$254$		0			0
$255$		0			0
$256$		0			0
$257$			256.629i			0.998555i	0.866442	+	0.499277i	$0.166401\pi$
$257$			256.629i			0.998555i	−0.866442	+	0.499277i	$0.833599\pi$
$258$		0			0
$259$		−	565.048i		−	2.18165i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−71.2027			−0.270733			−0.135366	−	0.990796i	$-0.543221\pi$
$263$	−71.2027			−0.270733			−0.135366	+	0.990796i	$0.543221\pi$
$264$		0			0
$265$			2.14488i			0.00809390i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			219.411i			0.815653i	0.913059	+	0.407827i	$0.133713\pi$
$269$			219.411i			0.815653i	−0.913059	+	0.407827i	$0.866287\pi$
$270$		0			0
$271$	323.913			1.19525			0.597626	−	0.801775i	$-0.296111\pi$
$271$	323.913			1.19525			0.597626	+	0.801775i	$0.296111\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−333.959			−1.21440
$276$		0			0
$277$	−37.4953			−0.135362			−0.0676810	−	0.997707i	$-0.521560\pi$
$277$	−37.4953			−0.135362			−0.0676810	+	0.997707i	$0.521560\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	177.483i		−	0.631613i	−0.948824	−	0.315806i	$-0.897725\pi$
$281$		−	177.483i		−	0.631613i	0.948824	−	0.315806i	$-0.102275\pi$
$282$		0			0
$283$	72.4063			0.255853			0.127926	−	0.991784i	$-0.459168\pi$
$283$	72.4063			0.255853			0.127926	+	0.991784i	$0.459168\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	657.956i		−	2.29253i
$288$		0			0
$289$	−282.037			−0.975908
$290$		0			0
$291$		0			0
$292$		0			0
$293$			57.9819i			0.197891i	0.995093	+	0.0989453i	$0.0315469\pi$
$293$			57.9819i			0.197891i	−0.995093	+	0.0989453i	$0.968453\pi$
$294$		0			0
$295$		−	33.0249i		−	0.111949i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	231.989i		−	0.775883i
$300$		0			0
$301$	262.790			0.873055
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−53.6683			−0.175962
$306$		0			0
$307$		−	484.604i		−	1.57851i	−0.614062	−	0.789257i	$-0.710466\pi$
$307$		−	484.604i		−	1.57851i	0.614062	−	0.789257i	$-0.289534\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−241.304			−0.775898			−0.387949	−	0.921681i	$-0.626816\pi$
$311$	−241.304			−0.775898			−0.387949	+	0.921681i	$0.626816\pi$
$312$		0			0
$313$	−313.749			−1.00239			−0.501196	−	0.865334i	$-0.667107\pi$
$313$	−313.749			−1.00239			−0.501196	+	0.865334i	$0.667107\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			184.282i			0.581331i	0.956825	+	0.290665i	$0.0938767\pi$
$317$			184.282i			0.581331i	−0.956825	+	0.290665i	$0.906123\pi$
$318$		0			0
$319$			659.825i			2.06842i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−0.297672	−	50.1340i	−0.000921585	−	0.155214i
$324$		0			0
$325$			260.109i			0.800334i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	427.573			1.29961
$330$		0			0
$331$		−	460.335i		−	1.39074i	−0.718652	−	0.695370i	$-0.755241\pi$
$331$		−	460.335i		−	1.39074i	0.718652	−	0.695370i	$-0.244759\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			5.99977i			0.0179098i
$336$		0			0
$337$			493.633i			1.46479i	0.680882	+	0.732393i	$0.261597\pi$
$337$			493.633i			1.46479i	−0.680882	+	0.732393i	$0.738403\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	3.35831i		−	0.00984843i
$342$		0			0
$343$	69.1373			0.201567
$344$		0			0
$345$		0			0
$346$		0			0
$347$	415.426			1.19719			0.598596	−	0.801051i	$-0.295725\pi$
$347$	415.426			1.19719			0.598596	+	0.801051i	$0.295725\pi$
$348$		0			0
$349$	34.1332			0.0978030			0.0489015	−	0.998804i	$-0.484428\pi$
$349$	34.1332			0.0978030			0.0489015	+	0.998804i	$0.484428\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	154.833			0.438620			0.219310	−	0.975655i	$-0.429619\pi$
$353$	154.833			0.438620			0.219310	+	0.975655i	$0.429619\pi$
$354$		0			0
$355$			29.2877i			0.0825005i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−565.544			−1.57533			−0.787665	−	0.616103i	$-0.788710\pi$
$359$	−565.544			−1.57533			−0.787665	+	0.616103i	$0.788710\pi$
$360$		0			0
$361$	−360.975	+	4.28674i	−0.999929	+	0.0118746i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7.64289			−0.0209394
$366$		0			0
$367$	197.695			0.538678			0.269339	−	0.963045i	$-0.413195\pi$
$367$	197.695			0.538678			0.269339	+	0.963045i	$0.413195\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			36.7212i			0.0989789i
$372$		0			0
$373$		−	198.960i		−	0.533404i	−0.963779	−	0.266702i	$-0.914066\pi$
$373$		−	198.960i		−	0.533404i	0.963779	−	0.266702i	$-0.0859339\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	513.913			1.36317
$378$		0			0
$379$			432.419i			1.14095i	0.821316	+	0.570474i	$0.193240\pi$
$379$			432.419i			1.14095i	−0.821316	+	0.570474i	$0.806760\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			543.758i			1.41973i	0.704336	+	0.709867i	$0.251245\pi$
$383$			543.758i			1.41973i	−0.704336	+	0.709867i	$0.748755\pi$
$384$		0			0
$385$	71.6904			0.186209
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−109.783			−0.282218			−0.141109	−	0.989994i	$-0.545067\pi$
$389$	−109.783			−0.282218			−0.141109	+	0.989994i	$0.545067\pi$
$390$		0			0
$391$	−58.1069			−0.148611
$392$		0			0
$393$		0			0
$394$		0			0
$395$			28.5561i			0.0722939i
$396$		0			0
$397$	−540.791			−1.36219			−0.681097	−	0.732193i	$-0.738497\pi$
$397$	−540.791			−1.36219			−0.681097	+	0.732193i	$0.738497\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			224.500i			0.559849i	0.960022	+	0.279925i	$0.0903095\pi$
$401$			224.500i			0.559849i	−0.960022	+	0.279925i	$0.909691\pi$
$402$		0			0
$403$	−2.61567			−0.00649049
$404$		0			0
$405$		0			0
$406$		0			0
$407$			802.317i			1.97129i
$408$		0			0
$409$		−	397.737i		−	0.972463i	−0.873830	−	0.486231i	$-0.838371\pi$
$409$		−	397.737i		−	0.972463i	0.873830	−	0.486231i	$-0.161629\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	565.398i		−	1.36900i
$414$		0			0
$415$	−35.4844			−0.0855046
$416$		0			0
$417$		0			0
$418$		0			0
$419$	769.815			1.83727			0.918634	−	0.395111i	$-0.129294\pi$
$419$	769.815			1.83727			0.918634	+	0.395111i	$0.129294\pi$
$420$		0			0
$421$		−	515.309i		−	1.22401i	−0.790853	−	0.612006i	$-0.790363\pi$
$421$		−	515.309i		−	1.22401i	0.790853	−	0.612006i	$-0.209637\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	65.1501			0.153294
$426$		0			0
$427$	−918.822			−2.15181
$428$		0			0
$429$		0			0
$430$		0			0
$431$			484.686i			1.12456i	0.826946	+	0.562281i	$0.190076\pi$
$431$			484.686i			1.12456i	−0.826946	+	0.562281i	$0.809924\pi$
$432$		0			0
$433$			734.040i			1.69524i	0.530602	+	0.847621i	$0.321966\pi$
$433$			734.040i			1.69524i	−0.530602	+	0.847621i	$0.678034\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.48423	+	418.395i	0.00568475	+	0.957426i
$438$		0			0
$439$		−	108.108i		−	0.246260i	−0.992391	−	0.123130i	$-0.960707\pi$
$439$		−	108.108i		−	0.246260i	0.992391	−	0.123130i	$-0.0392933\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	519.843			1.17346			0.586730	−	0.809782i	$-0.300415\pi$
$443$	519.843			1.17346			0.586730	+	0.809782i	$0.300415\pi$
$444$		0			0
$445$		−	24.9979i		−	0.0561751i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			810.462i			1.80504i	0.430651	+	0.902519i	$0.358284\pi$
$449$			810.462i			1.80504i	−0.430651	+	0.902519i	$0.641716\pi$
$450$		0			0
$451$			934.238i			2.07148i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	55.8371i		−	0.122719i
$456$		0			0
$457$	231.412			0.506373			0.253186	−	0.967418i	$-0.418521\pi$
$457$	231.412			0.506373			0.253186	+	0.967418i	$0.418521\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−766.366			−1.66240			−0.831200	−	0.555974i	$-0.812346\pi$
$461$	−766.366			−1.66240			−0.831200	+	0.555974i	$0.812346\pi$
$462$		0			0
$463$	152.091			0.328490			0.164245	−	0.986420i	$-0.447481\pi$
$463$	152.091			0.328490			0.164245	+	0.986420i	$0.447481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−123.992			−0.265508			−0.132754	−	0.991149i	$-0.542382\pi$
$467$	−123.992			−0.265508			−0.132754	+	0.991149i	$0.542382\pi$
$468$		0			0
$469$			102.718i			0.219016i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−373.137			−0.788874
$474$		0			0
$475$	−2.78535	−	469.110i	−0.00586390	−	0.987599i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	443.502			0.925892			0.462946	−	0.886387i	$-0.346792\pi$
$479$	443.502			0.925892			0.462946	+	0.886387i	$0.346792\pi$
$480$		0			0
$481$	624.895			1.29916
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	40.4175i		−	0.0833350i
$486$		0			0
$487$			628.314i			1.29017i	0.764110	+	0.645086i	$0.223179\pi$
$487$			628.314i			1.29017i	−0.764110	+	0.645086i	$0.776821\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−877.289			−1.78674			−0.893370	−	0.449322i	$-0.851666\pi$
$491$	−877.289			−1.78674			−0.893370	+	0.449322i	$0.851666\pi$
$492$		0			0
$493$		−	128.721i		−	0.261098i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			501.416i			1.00889i
$498$		0			0
$499$	206.303			0.413433			0.206716	−	0.978401i	$-0.433722\pi$
$499$	206.303			0.413433			0.206716	+	0.978401i	$0.433722\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	276.617			0.549934			0.274967	−	0.961454i	$-0.411333\pi$
$503$	276.617			0.549934			0.274967	+	0.961454i	$0.411333\pi$
$504$		0			0
$505$	64.9786			0.128671
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	728.875i		−	1.43197i	−0.698114	−	0.715987i	$-0.745977\pi$
$509$		−	728.875i		−	1.43197i	0.698114	−	0.715987i	$-0.254023\pi$
$510$		0			0
$511$	−130.849			−0.256065
$512$		0			0
$513$		0			0
$514$		0			0
$515$			40.2793i			0.0782123i
$516$		0			0
$517$	−607.115			−1.17430
$518$		0			0
$519$		0			0
$520$		0			0
$521$			14.2470i			0.0273455i	0.999907	+	0.0136727i	$0.00435230\pi$
$521$			14.2470i			0.0273455i	−0.999907	+	0.0136727i	$0.995648\pi$
$522$		0			0
$523$			535.561i			1.02402i	0.858981	+	0.512008i	$0.171098\pi$
$523$			535.561i			1.02402i	−0.858981	+	0.512008i	$0.828902\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			0.655153i			0.00124317i
$528$		0			0
$529$	−44.0669			−0.0833023
$530$		0			0
$531$		0			0
$532$		0			0
$533$	727.644			1.36519
$534$		0			0
$535$		−	85.7182i		−	0.160221i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	564.599			1.04749
$540$		0			0
$541$	1000.18			1.84875			0.924376	−	0.381482i	$-0.124586\pi$
$541$	1000.18			1.84875			0.924376	+	0.381482i	$0.124586\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	88.8415i		−	0.163012i
$546$		0			0
$547$		−	675.400i		−	1.23473i	−0.786675	−	0.617367i	$-0.788199\pi$
$547$		−	675.400i		−	1.23473i	0.786675	−	0.617367i	$-0.211801\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−926.850	+	5.50320i	−1.68212	+	0.00998765i
$552$		0			0
$553$			488.891i			0.884070i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	366.913			0.658730			0.329365	−	0.944203i	$-0.393165\pi$
$557$	366.913			0.658730			0.329365	+	0.944203i	$0.393165\pi$
$558$		0			0
$559$			290.623i			0.519898i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			50.3339i			0.0894029i	0.999000	+	0.0447015i	$0.0142337\pi$
$563$			50.3339i			0.0894029i	−0.999000	+	0.0447015i	$0.985766\pi$
$564$		0			0
$565$		−	44.3132i		−	0.0784305i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	61.6165i		−	0.108289i	−0.998533	−	0.0541446i	$-0.982757\pi$
$569$		−	61.6165i		−	0.108289i	0.998533	−	0.0541446i	$-0.0172432\pi$
$570$		0			0
$571$	256.504			0.449218			0.224609	−	0.974449i	$-0.427889\pi$
$571$	256.504			0.449218			0.224609	+	0.974449i	$0.427889\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−543.712			−0.945587
$576$		0			0
$577$	429.278			0.743983			0.371991	−	0.928236i	$-0.378675\pi$
$577$	429.278			0.743983			0.371991	+	0.928236i	$0.378675\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−607.506			−1.04562
$582$		0			0
$583$		−	52.1407i		−	0.0894352i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−491.725			−0.837691			−0.418845	−	0.908058i	$-0.637565\pi$
$587$	−491.725			−0.837691			−0.418845	+	0.908058i	$0.637565\pi$
$588$		0			0
$589$	4.71739	−	0.0280097i	0.00800916	−	4.75546e-5i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−954.134			−1.60899			−0.804497	−	0.593956i	$-0.797565\pi$
$593$	−954.134			−1.60899			−0.804497	+	0.593956i	$0.797565\pi$
$594$		0			0
$595$	−13.9856			−0.0235053
$596$		0			0
$597$		0			0
$598$		0			0
$599$			533.310i			0.890334i	0.895448	+	0.445167i	$0.146856\pi$
$599$			533.310i			0.890334i	−0.895448	+	0.445167i	$0.853144\pi$
$600$		0			0
$601$		−	830.453i		−	1.38179i	−0.722957	−	0.690893i	$-0.757218\pi$
$601$		−	830.453i		−	1.38179i	0.722957	−	0.690893i	$-0.242782\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−34.4688			−0.0569732
$606$		0			0
$607$		−	584.294i		−	0.962594i	−0.876558	−	0.481297i	$-0.840166\pi$
$607$		−	584.294i		−	0.962594i	0.876558	−	0.481297i	$-0.159834\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			472.860i			0.773912i
$612$		0			0
$613$	−152.216			−0.248314			−0.124157	−	0.992263i	$-0.539623\pi$
$613$	−152.216			−0.248314			−0.124157	+	0.992263i	$0.539623\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−647.334			−1.04916			−0.524582	−	0.851360i	$-0.675778\pi$
$617$	−647.334			−1.04916			−0.524582	+	0.851360i	$0.675778\pi$
$618$		0			0
$619$	−104.700			−0.169145			−0.0845723	−	0.996417i	$-0.526952\pi$
$619$	−104.700			−0.169145			−0.0845723	+	0.996417i	$0.526952\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	427.974i		−	0.686956i
$624$		0			0
$625$	601.877			0.963003
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	156.519i		−	0.248838i
$630$		0			0
$631$	628.425			0.995919			0.497959	−	0.867200i	$-0.334083\pi$
$631$	628.425			0.995919			0.497959	+	0.867200i	$0.334083\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	16.8328i		−	0.0265084i
$636$		0			0
$637$		−	439.745i		−	0.690338i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			462.053i			0.720832i	0.932792	+	0.360416i	$0.117365\pi$
$641$			462.053i			0.720832i	−0.932792	+	0.360416i	$0.882635\pi$
$642$		0			0
$643$	−898.660			−1.39761			−0.698803	−	0.715314i	$-0.746284\pi$
$643$	−898.660			−1.39761			−0.698803	+	0.715314i	$0.746284\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	773.813			1.19600			0.598001	−	0.801495i	$-0.295962\pi$
$647$	773.813			1.19600			0.598001	+	0.801495i	$0.295962\pi$
$648$		0			0
$649$			802.814i			1.23700i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	701.376			1.07408			0.537041	−	0.843556i	$-0.319542\pi$
$653$	701.376			1.07408			0.537041	+	0.843556i	$0.319542\pi$
$654$		0			0
$655$	−79.4782			−0.121341
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1255.87i		−	1.90572i	−0.303415	−	0.952858i	$-0.598127\pi$
$659$		−	1255.87i		−	1.90572i	0.303415	−	0.952858i	$-0.401873\pi$
$660$		0			0
$661$			399.716i			0.604714i	0.953195	+	0.302357i	$0.0977735\pi$
$661$			399.716i			0.604714i	−0.953195	+	0.302357i	$0.902227\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.597926	+	100.703i	0.000899137	+	0.151433i
$666$		0			0
$667$			1074.25i			1.61057i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1304.64			1.94433
$672$		0			0
$673$		−	116.954i		−	0.173779i	−0.996218	−	0.0868897i	$-0.972307\pi$
$673$		−	116.954i		−	0.173779i	0.996218	−	0.0868897i	$-0.0276928\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			497.520i			0.734890i	0.930045	+	0.367445i	$0.119767\pi$
$677$			497.520i			0.734890i	−0.930045	+	0.367445i	$0.880233\pi$
$678$		0			0
$679$		−	691.962i		−	1.01909i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			803.013i			1.17572i	0.808964	+	0.587858i	$0.200028\pi$
$683$			803.013i			1.17572i	−0.808964	+	0.587858i	$0.799972\pi$
$684$		0			0
$685$	−92.0787			−0.134421
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−40.6105			−0.0589413
$690$		0			0
$691$	891.667			1.29040			0.645200	−	0.764014i	$-0.276774\pi$
$691$	891.667			1.29040			0.645200	+	0.764014i	$0.276774\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	94.2128			0.135558
$696$		0			0
$697$		−	182.255i		−	0.261485i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1110.82			−1.58463			−0.792313	−	0.610114i	$-0.791124\pi$
$701$	−1110.82			−1.58463			−0.792313	+	0.610114i	$0.791124\pi$
$702$		0			0
$703$	−1127.01	+	6.69164i	−1.60314	+	0.00951869i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1112.46			1.57349
$708$		0			0
$709$	−9.63354			−0.0135875			−0.00679375	−	0.999977i	$-0.502163\pi$
$709$	−9.63354			−0.0135875			−0.00679375	+	0.999977i	$0.502163\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	5.46760i		−	0.00766845i
$714$		0			0
$715$			79.2835i			0.110886i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−97.4316			−0.135510			−0.0677549	−	0.997702i	$-0.521584\pi$
$719$	−97.4316			−0.135510			−0.0677549	+	0.997702i	$0.521584\pi$
$720$		0			0
$721$			689.597i			0.956445i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1204.46i		−	1.66132i
$726$		0			0
$727$	−466.358			−0.641483			−0.320742	−	0.947167i	$-0.603932\pi$
$727$	−466.358			−0.641483			−0.320742	+	0.947167i	$0.603932\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	72.7931			0.0995801
$732$		0			0
$733$	452.839			0.617789			0.308895	−	0.951096i	$-0.400041\pi$
$733$	452.839			0.617789			0.308895	+	0.951096i	$0.400041\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	145.851i		−	0.197898i
$738$		0			0
$739$	−1346.02			−1.82140			−0.910701	−	0.413066i	$-0.864458\pi$
$739$	−1346.02			−1.82140			−0.910701	+	0.413066i	$0.864458\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	787.553i		−	1.05996i	−0.848009	−	0.529982i	$-0.822199\pi$
$743$		−	787.553i		−	1.05996i	0.848009	−	0.529982i	$-0.177801\pi$
$744$		0			0
$745$	−77.5664			−0.104116
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1467.53i		−	1.95932i
$750$		0			0
$751$		−	120.158i		−	0.159997i	−0.996795	−	0.0799986i	$-0.974508\pi$
$751$		−	120.158i		−	0.159997i	0.996795	−	0.0799986i	$-0.0254916\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			84.9866i			0.112565i
$756$		0			0
$757$	263.516			0.348105			0.174053	−	0.984736i	$-0.444314\pi$
$757$	263.516			0.348105			0.174053	+	0.984736i	$0.444314\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	60.7691			0.0798543			0.0399271	−	0.999203i	$-0.487287\pi$
$761$	60.7691			0.0798543			0.0399271	+	0.999203i	$0.487287\pi$
$762$		0			0
$763$		−	1521.00i		−	1.99345i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	625.282			0.815231
$768$		0			0
$769$	−713.480			−0.927802			−0.463901	−	0.885887i	$-0.653551\pi$
$769$	−713.480			−0.927802			−0.463901	+	0.885887i	$0.653551\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	971.062i		−	1.25622i	−0.778123	−	0.628112i	$-0.783828\pi$
$773$		−	971.062i		−	1.25622i	0.778123	−	0.628112i	$-0.216172\pi$
$774$		0			0
$775$			6.13034i			0.00791012i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1312.32	+	7.79191i	−1.68462	+	0.0100025i
$780$		0			0
$781$		−	711.965i		−	0.911607i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	164.658			0.209756
$786$		0			0
$787$		−	903.790i		−	1.14840i	−0.818716	−	0.574199i	$-0.805313\pi$
$787$		−	903.790i		−	1.14840i	0.818716	−	0.574199i	$-0.194687\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	758.659i		−	0.959114i
$792$		0			0
$793$		−	1016.14i		−	1.28139i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	836.376i		−	1.04941i	−0.851286	−	0.524703i	$-0.824177\pi$
$797$		−	836.376i		−	1.04941i	0.851286	−	0.524703i	$-0.175823\pi$
$798$		0			0
$799$	118.438			0.148233
$800$		0			0
$801$		0			0
$802$		0			0
$803$	185.794			0.231375
$804$		0			0
$805$	116.718			0.144991
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1274.76			1.57572			0.787861	−	0.615853i	$-0.211189\pi$
$809$	1274.76			1.57572			0.787861	+	0.615853i	$0.211189\pi$
$810$		0			0
$811$			1452.92i			1.79152i	0.444539	+	0.895759i	$0.353367\pi$
$811$			1452.92i			1.79152i	−0.444539	+	0.895759i	$0.646633\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	62.7242			0.0769622
$816$		0			0
$817$	−3.11211	−	524.143i	−0.00380919	−	0.641546i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	579.401			0.705726			0.352863	−	0.935675i	$-0.385208\pi$
$821$	579.401			0.705726			0.352863	+	0.935675i	$0.385208\pi$
$822$		0			0
$823$	451.801			0.548968			0.274484	−	0.961592i	$-0.411493\pi$
$823$	451.801			0.548968			0.274484	+	0.961592i	$0.411493\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	740.460i		−	0.895356i	−0.894195	−	0.447678i	$-0.852251\pi$
$827$		−	740.460i		−	0.895356i	0.894195	−	0.447678i	$-0.147749\pi$
$828$		0			0
$829$		−	1456.61i		−	1.75707i	−0.477675	−	0.878536i	$-0.658520\pi$
$829$		−	1456.61i		−	1.75707i	0.477675	−	0.878536i	$-0.341480\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−110.144			−0.132226
$834$		0			0
$835$			165.366i			0.198043i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			87.8279i			0.104682i	0.998629	+	0.0523408i	$0.0166682\pi$
$839$			87.8279i			0.104682i	−0.998629	+	0.0523408i	$0.983332\pi$
$840$		0			0
$841$	−1538.72			−1.82964
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−32.2815			−0.0382029
$846$		0			0
$847$	−590.119			−0.696716
$848$		0			0
$849$		0			0
$850$		0			0
$851$			1306.24i			1.53494i
$852$		0			0
$853$	−870.193			−1.02016			−0.510078	−	0.860128i	$-0.670383\pi$
$853$	−870.193			−1.02016			−0.510078	+	0.860128i	$0.670383\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	143.885i		−	0.167894i	−0.996470	−	0.0839470i	$-0.973247\pi$
$857$		−	143.885i		−	0.167894i	0.996470	−	0.0839470i	$-0.0267527\pi$
$858$		0			0
$859$	1378.57			1.60485			0.802426	−	0.596752i	$-0.203542\pi$
$859$	1378.57			1.60485			0.802426	+	0.596752i	$0.203542\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	61.6014i		−	0.0713805i	−0.999363	−	0.0356903i	$-0.988637\pi$
$863$		−	61.6014i		−	0.0713805i	0.999363	−	0.0356903i	$-0.0113630\pi$
$864$		0			0
$865$			146.250i			0.169076i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	694.181i		−	0.798827i
$870$		0			0
$871$	−113.598			−0.130422
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−263.371			−0.300996
$876$		0			0
$877$			1405.80i			1.60297i	0.598016	+	0.801484i	$0.295956\pi$
$877$			1405.80i			1.60297i	−0.598016	+	0.801484i	$0.704044\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1449.85			1.64569			0.822846	−	0.568264i	$-0.192385\pi$
$881$	1449.85			1.64569			0.822846	+	0.568264i	$0.192385\pi$
$882$		0			0
$883$	1177.85			1.33392			0.666958	−	0.745095i	$-0.267596\pi$
$883$	1177.85			1.33392			0.666958	+	0.745095i	$0.267596\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1051.90i			1.18590i	0.805238	+	0.592952i	$0.202038\pi$
$887$			1051.90i			1.18590i	−0.805238	+	0.592952i	$0.797962\pi$
$888$		0			0
$889$		−	288.185i		−	0.324167i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.06358	−	852.809i	−0.00567030	−	0.954994i
$894$		0			0
$895$		−	41.1287i		−	0.0459538i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	12.1121			0.0134729
$900$		0			0
$901$			10.1718i			0.0112895i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			19.4126i			0.0214504i
$906$		0			0
$907$			222.998i			0.245863i	0.992415	+	0.122932i	$0.0392296\pi$
$907$			222.998i			0.245863i	−0.992415	+	0.122932i	$0.960770\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1400.58i			1.53741i	0.639603	+	0.768705i	$0.279099\pi$
$911$			1400.58i			1.53741i	−0.639603	+	0.768705i	$0.720901\pi$
$912$		0			0
$913$	862.604			0.944802
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1360.70			−1.48386
$918$		0			0
$919$	−241.714			−0.263018			−0.131509	−	0.991315i	$-0.541982\pi$
$919$	−241.714			−0.263018			−0.131509	+	0.991315i	$0.541982\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−554.524			−0.600784
$924$		0			0
$925$		−	1464.57i		−	1.58332i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−704.699			−0.758556			−0.379278	−	0.925283i	$-0.623828\pi$
$929$	−704.699			−0.758556			−0.379278	+	0.925283i	$0.623828\pi$
$930$		0			0
$931$	4.70898	+	793.087i	0.00505798	+	0.851866i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	19.8583			0.0212389
$936$		0			0
$937$	−212.822			−0.227131			−0.113566	−	0.993531i	$-0.536227\pi$
$937$	−212.822			−0.227131			−0.113566	+	0.993531i	$0.536227\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	863.837i		−	0.917999i	−0.888437	−	0.458999i	$-0.848208\pi$
$941$		−	863.837i		−	0.917999i	0.888437	−	0.458999i	$-0.151792\pi$
$942$		0			0
$943$			1521.01i			1.61295i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−595.986			−0.629341			−0.314671	−	0.949201i	$-0.601894\pi$
$947$	−595.986			−0.629341			−0.314671	+	0.949201i	$0.601894\pi$
$948$		0			0
$949$		−	144.708i		−	0.152485i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1057.19i		−	1.10933i	−0.832073	−	0.554666i	$-0.812846\pi$
$953$		−	1057.19i		−	1.10933i	0.832073	−	0.554666i	$-0.187154\pi$
$954$		0			0
$955$	61.7614			0.0646717
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1576.42			−1.64382
$960$		0			0
$961$	960.938			0.999936
$962$		0			0
$963$		0			0
$964$		0			0
$965$			46.4941i			0.0481804i
$966$		0			0
$967$	635.416			0.657100			0.328550	−	0.944487i	$-0.393440\pi$
$967$	635.416			0.657100			0.328550	+	0.944487i	$0.393440\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1498.44i			1.54320i	0.636111	+	0.771598i	$0.280542\pi$
$971$			1498.44i			1.54320i	−0.636111	+	0.771598i	$0.719458\pi$
$972$		0			0
$973$	1612.96			1.65772
$974$		0			0
$975$		0			0
$976$		0			0
$977$			892.040i			0.913039i	0.889713	+	0.456520i	$0.150904\pi$
$977$			892.040i			0.913039i	−0.889713	+	0.456520i	$0.849096\pi$
$978$		0			0
$979$			607.684i			0.620719i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	708.551i		−	0.720805i	−0.932797	−	0.360402i	$-0.882639\pi$
$983$		−	708.551i		−	0.720805i	0.932797	−	0.360402i	$-0.117361\pi$
$984$		0			0
$985$	−203.265			−0.206361
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−607.498			−0.614254
$990$		0			0
$991$			730.885i			0.737523i	0.929524	+	0.368762i	$0.120218\pi$
$991$			730.885i			0.737523i	−0.929524	+	0.368762i	$0.879782\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	5.91474			0.00594446
$996$		0			0
$997$	−841.726			−0.844259			−0.422129	−	0.906536i	$-0.638717\pi$
$997$	−841.726			−0.844259			−0.422129	+	0.906536i	$0.638717\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.h.e.37.3		6
3.2	odd	2		228.3.h.a.37.2	✓	6
4.3	odd	2		2736.3.o.m.721.3		6
12.11	even	2		912.3.o.c.721.5		6
19.18	odd	2	inner	684.3.h.e.37.4		6
57.56	even	2		228.3.h.a.37.5	yes	6
76.75	even	2		2736.3.o.m.721.4		6
228.227	odd	2		912.3.o.c.721.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.3.h.a.37.2	✓	6	3.2	odd	2
228.3.h.a.37.5	yes	6	57.56	even	2
684.3.h.e.37.3		6	1.1	even	1	trivial
684.3.h.e.37.4		6	19.18	odd	2	inner
912.3.o.c.721.2		6	228.227	odd	2
912.3.o.c.721.5		6	12.11	even	2
2736.3.o.m.721.3		6	4.3	odd	2
2736.3.o.m.721.4		6	76.75	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.556406 q^{5} -9.52587 q^{7} +O(q^{10})\)	\(q-0.556406 q^{5} -9.52587 q^{7} +13.5259 q^{11} -10.5348i q^{13} -2.63868 q^{17} +(0.112811 + 18.9997i) q^{19} +22.0212 q^{23} -24.6904 q^{25} +48.7824i q^{29} -0.248288i q^{31} +5.30025 q^{35} +59.3172i q^{37} +69.0705i q^{41} -27.5869 q^{43} -44.8855 q^{47} +41.7421 q^{49} -3.85489i q^{53} -7.52587 q^{55} +59.3539i q^{59} +96.4554 q^{61} +5.86162i q^{65} -10.7831i q^{67} -52.6373i q^{71} +13.7362 q^{73} -128.846 q^{77} -51.3224i q^{79} +63.7744 q^{83} +1.46818 q^{85} +44.9275i q^{89} +100.353i q^{91} +(-0.0627687 - 10.5715i) q^{95} +72.6403i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 6 * q + 2 * q^5 - 2 * q^7	\( 6 q + 2 q^{5} - 2 q^{7} + 26 q^{11} + 50 q^{17} - 10 q^{19} - 28 q^{23} + 28 q^{25} - 2 q^{35} - 210 q^{43} - 22 q^{47} - 36 q^{49} + 10 q^{55} + 214 q^{61} + 102 q^{73} - 266 q^{77} + 404 q^{83} + 370 q^{85} - 358 q^{95}+O(q^{100}) \) 6 * q + 2 * q^5 - 2 * q^7 + 26 * q^11 + 50 * q^17 - 10 * q^19 - 28 * q^23 + 28 * q^25 - 2 * q^35 - 210 * q^43 - 22 * q^47 - 36 * q^49 + 10 * q^55 + 214 * q^61 + 102 * q^73 - 266 * q^77 + 404 * q^83 + 370 * q^85 - 358 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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