LMFDB - Embedded newform 684.3.e.a.305.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(305,684)]

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, 1, 0]))

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

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

chi := DirichletCharacter("684.305");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.e (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:	$12$
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} + 156x^{10} + 8721x^{8} + 208784x^{6} + 2024760x^{4} + 7117056x^{2} + 6533136 $ x^12 + 156x^10 + 8721x^8 + 208784x^6 + 2024760x^4 + 7117056*x^2 + 6533136
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			305.2
Root			$-7.04185i$ of defining polynomial
Character	$\chi$	$=$	684.305
Dual form			684.3.e.a.305.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.04185i q^{5} -3.07744 q^{7} +O(q^{10})$	$q-7.04185i q^{5} -3.07744 q^{7} +7.06299i q^{11} -21.0065 q^{13} -2.77958i q^{17} -4.35890 q^{19} +18.0741i q^{23} -24.5876 q^{25} +19.2486i q^{29} +7.72584 q^{31} +21.6709i q^{35} +7.60723 q^{37} +36.9442i q^{41} +45.9411 q^{43} +58.6732i q^{47} -39.5294 q^{49} +23.4889i q^{53} +49.7365 q^{55} +31.2355i q^{59} -72.8380 q^{61} +147.925i q^{65} -86.3135 q^{67} -117.795i q^{71} -59.1047 q^{73} -21.7359i q^{77} -42.9053 q^{79} +30.4726i q^{83} -19.5734 q^{85} -41.2269i q^{89} +64.6464 q^{91} +30.6947i q^{95} -67.6398 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{7}+O(q^{10}) $ 12 * q + 16 * q^7	$ 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97}+O(q^{100}) $ 12 * q + 16 * q^7 - 16 * q^13 - 12 * q^25 - 40 * q^31 - 32 * q^37 + 92 * q^43 - 84 * q^55 - 48 * q^61 - 88 * q^67 + 148 * q^73 - 56 * q^79 + 228 * q^85 - 8 * q^91 + 72 * q^97

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
$5$	− 7.04185i	− 1.40837i	−0.710017	−	0.704185i	$-0.751313\pi$
$5$	− 7.04185i	− 1.40837i	0.710017	−	0.704185i	$-0.248687\pi$
$6$	0	0
$7$	−3.07744	−0.439634	−0.219817	−	0.975541i	$-0.570546\pi$
$7$	−3.07744	−0.439634	−0.219817	+	0.975541i	$0.570546\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7.06299i	0.642090i	0.947064	+	0.321045i	$0.104034\pi$
$11$	7.06299i	0.642090i	−0.947064	+	0.321045i	$0.895966\pi$
$12$	0	0
$13$	−21.0065	−1.61589	−0.807944	−	0.589259i	$-0.799420\pi$
$13$	−21.0065	−1.61589	−0.807944	+	0.589259i	$0.799420\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.77958i	− 0.163505i	−0.996653	−	0.0817525i	$-0.973948\pi$
$17$	− 2.77958i	− 0.163505i	0.996653	−	0.0817525i	$-0.0260517\pi$
$18$	0	0
$19$	−4.35890	−0.229416
$19$	−4.35890	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	18.0741i	0.785829i	0.919575	+	0.392914i	$0.128533\pi$
$23$	18.0741i	0.785829i	−0.919575	+	0.392914i	$0.871467\pi$
$24$	0	0
$25$	−24.5876	−0.983504
$26$	0	0
$27$	0	0
$28$	0	0
$29$	19.2486i	0.663744i	0.943325	+	0.331872i	$0.107680\pi$
$29$	19.2486i	0.663744i	−0.943325	+	0.331872i	$0.892320\pi$
$30$	0	0
$31$	7.72584	0.249220	0.124610	−	0.992206i	$-0.460232\pi$
$31$	7.72584	0.249220	0.124610	+	0.992206i	$0.460232\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	21.6709i	0.619168i
$36$	0	0
$37$	7.60723	0.205601	0.102800	−	0.994702i	$-0.467220\pi$
$37$	7.60723	0.205601	0.102800	+	0.994702i	$0.467220\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	36.9442i	0.901078i	0.892757	+	0.450539i	$0.148768\pi$
$41$	36.9442i	0.901078i	−0.892757	+	0.450539i	$0.851232\pi$
$42$	0	0
$43$	45.9411	1.06840	0.534199	−	0.845359i	$-0.320613\pi$
$43$	45.9411	1.06840	0.534199	+	0.845359i	$0.320613\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	58.6732i	1.24837i	0.781278	+	0.624183i	$0.214568\pi$
$47$	58.6732i	1.24837i	−0.781278	+	0.624183i	$0.785432\pi$
$48$	0	0
$49$	−39.5294	−0.806722
$50$	0	0
$51$	0	0
$52$	0	0
$53$	23.4889i	0.443186i	0.975139	+	0.221593i	$0.0711256\pi$
$53$	23.4889i	0.443186i	−0.975139	+	0.221593i	$0.928874\pi$
$54$	0	0
$55$	49.7365	0.904299
$56$	0	0
$57$	0	0
$58$	0	0
$59$	31.2355i	0.529416i	0.964329	+	0.264708i	$0.0852756\pi$
$59$	31.2355i	0.529416i	−0.964329	+	0.264708i	$0.914724\pi$
$60$	0	0
$61$	−72.8380	−1.19407	−0.597033	−	0.802217i	$-0.703654\pi$
$61$	−72.8380	−1.19407	−0.597033	+	0.802217i	$0.703654\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	147.925i	2.27577i
$66$	0	0
$67$	−86.3135	−1.28826	−0.644130	−	0.764916i	$-0.722781\pi$
$67$	−86.3135	−1.28826	−0.644130	+	0.764916i	$0.722781\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 117.795i	− 1.65909i	−0.558442	−	0.829543i	$-0.688601\pi$
$71$	− 117.795i	− 1.65909i	0.558442	−	0.829543i	$-0.311399\pi$
$72$	0	0
$73$	−59.1047	−0.809654	−0.404827	−	0.914393i	$-0.632668\pi$
$73$	−59.1047	−0.809654	−0.404827	+	0.914393i	$0.632668\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 21.7359i	− 0.282285i
$78$	0	0
$79$	−42.9053	−0.543105	−0.271553	−	0.962424i	$-0.587537\pi$
$79$	−42.9053	−0.543105	−0.271553	+	0.962424i	$0.587537\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	30.4726i	0.367140i	0.983007	+	0.183570i	$0.0587654\pi$
$83$	30.4726i	0.367140i	−0.983007	+	0.183570i	$0.941235\pi$
$84$	0	0
$85$	−19.5734	−0.230275
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 41.2269i	− 0.463223i	−0.972808	−	0.231612i	$-0.925600\pi$
$89$	− 41.2269i	− 0.463223i	0.972808	−	0.231612i	$-0.0743999\pi$
$90$	0	0
$91$	64.6464	0.710400
$92$	0	0
$93$	0	0
$94$	0	0
$95$	30.6947i	0.323102i
$96$	0	0
$97$	−67.6398	−0.697318	−0.348659	−	0.937250i	$-0.613363\pi$
$97$	−67.6398	−0.697318	−0.348659	+	0.937250i	$0.613363\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 105.103i	− 1.04062i	−0.853977	−	0.520311i	$-0.825816\pi$
$101$	− 105.103i	− 1.04062i	0.853977	−	0.520311i	$-0.174184\pi$
$102$	0	0
$103$	−66.1143	−0.641886	−0.320943	−	0.947098i	$-0.604000\pi$
$103$	−66.1143	−0.641886	−0.320943	+	0.947098i	$0.604000\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 45.2225i	− 0.422641i	−0.977417	−	0.211320i	$-0.932224\pi$
$107$	− 45.2225i	− 0.422641i	0.977417	−	0.211320i	$-0.0677763\pi$
$108$	0	0
$109$	−76.3847	−0.700777	−0.350388	−	0.936604i	$-0.613950\pi$
$109$	−76.3847	−0.700777	−0.350388	+	0.936604i	$0.613950\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	215.576i	1.90775i	0.300205	+	0.953875i	$0.402945\pi$
$113$	215.576i	1.90775i	−0.300205	+	0.953875i	$0.597055\pi$
$114$	0	0
$115$	127.275	1.10674
$116$	0	0
$117$	0	0
$118$	0	0
$119$	8.55400i	0.0718824i
$120$	0	0
$121$	71.1142	0.587721
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.90405i	− 0.0232324i
$126$	0	0
$127$	−171.843	−1.35309	−0.676547	−	0.736399i	$-0.736525\pi$
$127$	−171.843	−1.35309	−0.676547	+	0.736399i	$0.736525\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 0.269663i	− 0.00205850i	−0.999999	−	0.00102925i	$-0.999672\pi$
$131$	− 0.269663i	− 0.00205850i	0.999999	−	0.00102925i	$-0.000327620\pi$
$132$	0	0
$133$	13.4143	0.100859
$134$	0	0
$135$	0	0
$136$	0	0
$137$	251.827i	1.83815i	0.394080	+	0.919076i	$0.371063\pi$
$137$	251.827i	1.83815i	−0.394080	+	0.919076i	$0.628937\pi$
$138$	0	0
$139$	−146.403	−1.05326	−0.526629	−	0.850095i	$-0.676544\pi$
$139$	−146.403	−1.05326	−0.526629	+	0.850095i	$0.676544\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 148.369i	− 1.03755i
$144$	0	0
$145$	135.545	0.934796
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 143.774i	− 0.964929i	−0.875915	−	0.482465i	$-0.839742\pi$
$149$	− 143.774i	− 0.964929i	0.875915	−	0.482465i	$-0.160258\pi$
$150$	0	0
$151$	272.909	1.80735	0.903674	−	0.428222i	$-0.140860\pi$
$151$	272.909	1.80735	0.903674	+	0.428222i	$0.140860\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 54.4041i	− 0.350994i
$156$	0	0
$157$	111.257	0.708646	0.354323	−	0.935123i	$-0.384711\pi$
$157$	111.257	0.708646	0.354323	+	0.935123i	$0.384711\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 55.6219i	− 0.345477i
$162$	0	0
$163$	72.4508	0.444483	0.222242	−	0.974992i	$-0.428663\pi$
$163$	72.4508	0.444483	0.222242	+	0.974992i	$0.428663\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 34.8428i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$167$	− 34.8428i	− 0.208640i	0.994544	−	0.104320i	$-0.0332665\pi$
$168$	0	0
$169$	272.275	1.61110
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 60.1855i	− 0.347893i	−0.984755	−	0.173946i	$-0.944348\pi$
$173$	− 60.1855i	− 0.347893i	0.984755	−	0.173946i	$-0.0556520\pi$
$174$	0	0
$175$	75.6669	0.432382
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.98991i	0.0390498i	0.999809	+	0.0195249i	$0.00621536\pi$
$179$	6.98991i	0.0390498i	−0.999809	+	0.0195249i	$0.993785\pi$
$180$	0	0
$181$	−196.486	−1.08556	−0.542780	−	0.839875i	$-0.682628\pi$
$181$	−196.486	−1.08556	−0.542780	+	0.839875i	$0.682628\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 53.5689i	− 0.289562i
$186$	0	0
$187$	19.6322	0.104985
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 291.798i	− 1.52774i	−0.645371	−	0.763870i	$-0.723297\pi$
$191$	− 291.798i	− 1.52774i	0.645371	−	0.763870i	$-0.276703\pi$
$192$	0	0
$193$	70.2298	0.363885	0.181943	−	0.983309i	$-0.441761\pi$
$193$	70.2298	0.363885	0.181943	+	0.983309i	$0.441761\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27.0811i	0.137468i	0.997635	+	0.0687338i	$0.0218959\pi$
$197$	27.0811i	0.137468i	−0.997635	+	0.0687338i	$0.978104\pi$
$198$	0	0
$199$	360.292	1.81051	0.905257	−	0.424865i	$-0.139678\pi$
$199$	360.292	1.81051	0.905257	+	0.424865i	$0.139678\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 59.2363i	− 0.291805i
$204$	0	0
$205$	260.155	1.26905
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 30.7868i	− 0.147305i
$210$	0	0
$211$	−221.488	−1.04971	−0.524853	−	0.851193i	$-0.675880\pi$
$211$	−221.488	−1.04971	−0.524853	+	0.851193i	$0.675880\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 323.510i	− 1.50470i
$216$	0	0
$217$	−23.7758	−0.109566
$218$	0	0
$219$	0	0
$220$	0	0
$221$	58.3895i	0.264206i
$222$	0	0
$223$	−263.056	−1.17962	−0.589811	−	0.807541i	$-0.700798\pi$
$223$	−263.056	−1.17962	−0.589811	+	0.807541i	$0.700798\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 342.449i	− 1.50858i	−0.656539	−	0.754292i	$-0.727980\pi$
$227$	− 342.449i	− 1.50858i	0.656539	−	0.754292i	$-0.272020\pi$
$228$	0	0
$229$	−395.295	−1.72618	−0.863089	−	0.505051i	$-0.831474\pi$
$229$	−395.295	−1.72618	−0.863089	+	0.505051i	$0.831474\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	335.823i	1.44130i	0.693299	+	0.720650i	$0.256157\pi$
$233$	335.823i	1.44130i	−0.693299	+	0.720650i	$0.743843\pi$
$234$	0	0
$235$	413.168	1.75816
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 1.83821i	− 0.00769127i	−0.999993	−	0.00384564i	$-0.998776\pi$
$239$	− 1.83821i	− 0.00769127i	0.999993	−	0.00384564i	$-0.00122411\pi$
$240$	0	0
$241$	−248.216	−1.02994	−0.514972	−	0.857207i	$-0.672198\pi$
$241$	−248.216	−1.02994	−0.514972	+	0.857207i	$0.672198\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	278.360i	1.13616i
$246$	0	0
$247$	91.5654	0.370710
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 196.925i	− 0.784560i	−0.919846	−	0.392280i	$-0.871686\pi$
$251$	− 196.925i	− 0.784560i	0.919846	−	0.392280i	$-0.128314\pi$
$252$	0	0
$253$	−127.657	−0.504572
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 311.674i	− 1.21274i	−0.795183	−	0.606369i	$-0.792625\pi$
$257$	− 311.674i	− 1.21274i	0.795183	−	0.606369i	$-0.207375\pi$
$258$	0	0
$259$	−23.4108	−0.0903892
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 309.831i	− 1.17806i	−0.808110	−	0.589032i	$-0.799509\pi$
$263$	− 309.831i	− 1.17806i	0.808110	−	0.589032i	$-0.200491\pi$
$264$	0	0
$265$	165.405	0.624170
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 113.375i	− 0.421469i	−0.977543	−	0.210735i	$-0.932414\pi$
$269$	− 113.375i	− 0.421469i	0.977543	−	0.210735i	$-0.0675856\pi$
$270$	0	0
$271$	−5.96552	−0.0220130	−0.0110065	−	0.999939i	$-0.503504\pi$
$271$	−5.96552	−0.0220130	−0.0110065	+	0.999939i	$0.503504\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 173.662i	− 0.631498i
$276$	0	0
$277$	−396.971	−1.43311	−0.716553	−	0.697532i	$-0.754281\pi$
$277$	−396.971	−1.43311	−0.716553	+	0.697532i	$0.754281\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.29305i	0.0259539i	0.999916	+	0.0129770i	$0.00413081\pi$
$281$	7.29305i	0.0259539i	−0.999916	+	0.0129770i	$0.995869\pi$
$282$	0	0
$283$	176.922	0.625167	0.312583	−	0.949890i	$-0.398806\pi$
$283$	176.922	0.625167	0.312583	+	0.949890i	$0.398806\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 113.694i	− 0.396145i
$288$	0	0
$289$	281.274	0.973266
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 56.3193i	− 0.192216i	−0.995371	−	0.0961080i	$-0.969361\pi$
$293$	− 56.3193i	− 0.192216i	0.995371	−	0.0961080i	$-0.0306394\pi$
$294$	0	0
$295$	219.956	0.745613
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 379.674i	− 1.26981i
$300$	0	0
$301$	−141.381	−0.469705
$302$	0	0
$303$	0	0
$304$	0	0
$305$	512.914i	1.68169i
$306$	0	0
$307$	429.504	1.39904	0.699518	−	0.714615i	$-0.253398\pi$
$307$	429.504	1.39904	0.699518	+	0.714615i	$0.253398\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 20.9058i	− 0.0672212i	−0.999435	−	0.0336106i	$-0.989299\pi$
$311$	− 20.9058i	− 0.0672212i	0.999435	−	0.0336106i	$-0.0107006\pi$
$312$	0	0
$313$	−155.172	−0.495757	−0.247879	−	0.968791i	$-0.579733\pi$
$313$	−155.172	−0.495757	−0.247879	+	0.968791i	$0.579733\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 162.182i	− 0.511615i	−0.966728	−	0.255807i	$-0.917659\pi$
$317$	− 162.182i	− 0.511615i	0.966728	−	0.255807i	$-0.0823413\pi$
$318$	0	0
$319$	−135.952	−0.426183
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.1159i	0.0375106i
$324$	0	0
$325$	516.501	1.58923
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 180.563i	− 0.548825i
$330$	0	0
$331$	116.628	0.352350	0.176175	−	0.984359i	$-0.443628\pi$
$331$	116.628	0.352350	0.176175	+	0.984359i	$0.443628\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	607.806i	1.81435i
$336$	0	0
$337$	−52.1533	−0.154758	−0.0773788	−	0.997002i	$-0.524655\pi$
$337$	−52.1533	−0.154758	−0.0773788	+	0.997002i	$0.524655\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	54.5675i	0.160022i
$342$	0	0
$343$	272.444	0.794297
$344$	0	0
$345$	0	0
$346$	0	0
$347$	347.670i	1.00193i	0.865467	+	0.500966i	$0.167022\pi$
$347$	347.670i	1.00193i	−0.865467	+	0.500966i	$0.832978\pi$
$348$	0	0
$349$	288.791	0.827482	0.413741	−	0.910395i	$-0.364222\pi$
$349$	288.791	0.827482	0.413741	+	0.910395i	$0.364222\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	561.145i	1.58965i	0.606842	+	0.794823i	$0.292436\pi$
$353$	561.145i	1.58965i	−0.606842	+	0.794823i	$0.707564\pi$
$354$	0	0
$355$	−829.496	−2.33661
$356$	0	0
$357$	0	0
$358$	0	0
$359$	610.514i	1.70060i	0.526301	+	0.850299i	$0.323579\pi$
$359$	610.514i	1.70060i	−0.526301	+	0.850299i	$0.676421\pi$
$360$	0	0
$361$	19.0000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	416.206i	1.14029i
$366$	0	0
$367$	484.120	1.31913	0.659564	−	0.751649i	$-0.270741\pi$
$367$	484.120	1.31913	0.659564	+	0.751649i	$0.270741\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 72.2856i	− 0.194840i
$372$	0	0
$373$	−517.646	−1.38779	−0.693895	−	0.720076i	$-0.744107\pi$
$373$	−517.646	−1.38779	−0.693895	+	0.720076i	$0.744107\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 404.346i	− 1.07254i
$378$	0	0
$379$	82.1501	0.216755	0.108377	−	0.994110i	$-0.465435\pi$
$379$	82.1501	0.216755	0.108377	+	0.994110i	$0.465435\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	76.9697i	0.200965i	0.994939	+	0.100483i	$0.0320387\pi$
$383$	76.9697i	0.200965i	−0.994939	+	0.100483i	$0.967961\pi$
$384$	0	0
$385$	−153.061	−0.397561
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 476.823i	− 1.22577i	−0.790174	−	0.612883i	$-0.790010\pi$
$389$	− 476.823i	− 1.22577i	0.790174	−	0.612883i	$-0.209990\pi$
$390$	0	0
$391$	50.2384	0.128487
$392$	0	0
$393$	0	0
$394$	0	0
$395$	302.133i	0.764893i
$396$	0	0
$397$	−147.142	−0.370636	−0.185318	−	0.982679i	$-0.559331\pi$
$397$	−147.142	−0.370636	−0.185318	+	0.982679i	$0.559331\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 195.664i	− 0.487941i	−0.969783	−	0.243970i	$-0.921550\pi$
$401$	− 195.664i	− 0.487941i	0.969783	−	0.243970i	$-0.0784500\pi$
$402$	0	0
$403$	−162.293	−0.402712
$404$	0	0
$405$	0	0
$406$	0	0
$407$	53.7297i	0.132014i
$408$	0	0
$409$	−548.329	−1.34066	−0.670328	−	0.742064i	$-0.733847\pi$
$409$	−548.329	−1.34066	−0.670328	+	0.742064i	$0.733847\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 96.1256i	− 0.232750i
$414$	0	0
$415$	214.583	0.517068
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 198.847i	− 0.474576i	−0.971439	−	0.237288i	$-0.923741\pi$
$419$	− 198.847i	− 0.474576i	0.971439	−	0.237288i	$-0.0762586\pi$
$420$	0	0
$421$	111.151	0.264017	0.132009	−	0.991249i	$-0.457857\pi$
$421$	111.151	0.264017	0.132009	+	0.991249i	$0.457857\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	68.3433i	0.160808i
$426$	0	0
$427$	224.155	0.524953
$428$	0	0
$429$	0	0
$430$	0	0
$431$	649.455i	1.50686i	0.657530	+	0.753428i	$0.271601\pi$
$431$	649.455i	1.50686i	−0.657530	+	0.753428i	$0.728399\pi$
$432$	0	0
$433$	−510.850	−1.17979	−0.589896	−	0.807479i	$-0.700831\pi$
$433$	−510.850	−1.17979	−0.589896	+	0.807479i	$0.700831\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 78.7830i	− 0.180281i
$438$	0	0
$439$	−264.808	−0.603206	−0.301603	−	0.953434i	$-0.597522\pi$
$439$	−264.808	−0.603206	−0.301603	+	0.953434i	$0.597522\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	68.9203i	0.155576i	0.996970	+	0.0777882i	$0.0247858\pi$
$443$	68.9203i	0.155576i	−0.996970	+	0.0777882i	$0.975214\pi$
$444$	0	0
$445$	−290.313	−0.652389
$446$	0	0
$447$	0	0
$448$	0	0
$449$	385.762i	0.859159i	0.903029	+	0.429579i	$0.141338\pi$
$449$	385.762i	0.859159i	−0.903029	+	0.429579i	$0.858662\pi$
$450$	0	0
$451$	−260.936	−0.578573
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 455.230i	− 1.00051i
$456$	0	0
$457$	−597.187	−1.30675	−0.653377	−	0.757033i	$-0.726648\pi$
$457$	−597.187	−1.30675	−0.653377	+	0.757033i	$0.726648\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	50.9835i	0.110593i	0.998470	+	0.0552966i	$0.0176104\pi$
$461$	50.9835i	0.110593i	−0.998470	+	0.0552966i	$0.982390\pi$
$462$	0	0
$463$	−769.797	−1.66263	−0.831315	−	0.555802i	$-0.812411\pi$
$463$	−769.797	−1.66263	−0.831315	+	0.555802i	$0.812411\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	245.589i	0.525886i	0.964811	+	0.262943i	$0.0846931\pi$
$467$	245.589i	0.525886i	−0.964811	+	0.262943i	$0.915307\pi$
$468$	0	0
$469$	265.625	0.566364
$470$	0	0
$471$	0	0
$472$	0	0
$473$	324.482i	0.686008i
$474$	0	0
$475$	107.175	0.225631
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 214.267i	− 0.447321i	−0.974667	−	0.223661i	$-0.928199\pi$
$479$	− 214.267i	− 0.447321i	0.974667	−	0.223661i	$-0.0718007\pi$
$480$	0	0
$481$	−159.802	−0.332228
$482$	0	0
$483$	0	0
$484$	0	0
$485$	476.309i	0.982081i
$486$	0	0
$487$	249.000	0.511294	0.255647	−	0.966770i	$-0.417712\pi$
$487$	249.000	0.511294	0.255647	+	0.966770i	$0.417712\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	391.907i	0.798182i	0.916911	+	0.399091i	$0.130674\pi$
$491$	391.907i	0.798182i	−0.916911	+	0.399091i	$0.869326\pi$
$492$	0	0
$493$	53.5030	0.108525
$494$	0	0
$495$	0	0
$496$	0	0
$497$	362.508i	0.729392i
$498$	0	0
$499$	−305.503	−0.612230	−0.306115	−	0.951995i	$-0.599029\pi$
$499$	−305.503	−0.612230	−0.306115	+	0.951995i	$0.599029\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 169.437i	− 0.336852i	−0.985714	−	0.168426i	$-0.946132\pi$
$503$	− 169.437i	− 0.336852i	0.985714	−	0.168426i	$-0.0538684\pi$
$504$	0	0
$505$	−740.117	−1.46558
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 725.512i	− 1.42537i	−0.701485	−	0.712684i	$-0.747479\pi$
$509$	− 725.512i	− 1.42537i	0.701485	−	0.712684i	$-0.252521\pi$
$510$	0	0
$511$	181.891	0.355952
$512$	0	0
$513$	0	0
$514$	0	0
$515$	465.567i	0.904013i
$516$	0	0
$517$	−414.408	−0.801563
$518$	0	0
$519$	0	0
$520$	0	0
$521$	969.950i	1.86171i	0.365391	+	0.930854i	$0.380935\pi$
$521$	969.950i	1.86171i	−0.365391	+	0.930854i	$0.619065\pi$
$522$	0	0
$523$	−312.287	−0.597107	−0.298554	−	0.954393i	$-0.596504\pi$
$523$	−312.287	−0.597107	−0.298554	+	0.954393i	$0.596504\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 21.4746i	− 0.0407488i
$528$	0	0
$529$	202.328	0.382473
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 776.070i	− 1.45604i
$534$	0	0
$535$	−318.450	−0.595234
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 279.195i	− 0.517988i
$540$	0	0
$541$	884.389	1.63473	0.817365	−	0.576120i	$-0.195434\pi$
$541$	884.389	1.63473	0.817365	+	0.576120i	$0.195434\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	537.889i	0.986953i
$546$	0	0
$547$	427.915	0.782295	0.391147	−	0.920328i	$-0.372078\pi$
$547$	427.915	0.782295	0.391147	+	0.920328i	$0.372078\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 83.9025i	− 0.152273i
$552$	0	0
$553$	132.039	0.238768
$554$	0	0
$555$	0	0
$556$	0	0
$557$	290.044i	0.520726i	0.965511	+	0.260363i	$0.0838422\pi$
$557$	290.044i	0.520726i	−0.965511	+	0.260363i	$0.916158\pi$
$558$	0	0
$559$	−965.065	−1.72641
$560$	0	0
$561$	0	0
$562$	0	0
$563$	634.658i	1.12728i	0.826021	+	0.563640i	$0.190599\pi$
$563$	634.658i	1.12728i	−0.826021	+	0.563640i	$0.809401\pi$
$564$	0	0
$565$	1518.05	2.68682
$566$	0	0
$567$	0	0
$568$	0	0
$569$	673.573i	1.18378i	0.806017	+	0.591892i	$0.201619\pi$
$569$	673.573i	1.18378i	−0.806017	+	0.591892i	$0.798381\pi$
$570$	0	0
$571$	554.163	0.970513	0.485256	−	0.874372i	$-0.338726\pi$
$571$	554.163	0.970513	0.485256	+	0.874372i	$0.338726\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 444.398i	− 0.772866i
$576$	0	0
$577$	635.933	1.10214	0.551069	−	0.834460i	$-0.314220\pi$
$577$	635.933	1.10214	0.551069	+	0.834460i	$0.314220\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 93.7776i	− 0.161407i
$582$	0	0
$583$	−165.901	−0.284565
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1083.18i	− 1.84528i	−0.385657	−	0.922642i	$-0.626025\pi$
$587$	− 1083.18i	− 1.84528i	0.385657	−	0.922642i	$-0.373975\pi$
$588$	0	0
$589$	−33.6761	−0.0571751
$590$	0	0
$591$	0	0
$592$	0	0
$593$	735.810i	1.24083i	0.784275	+	0.620413i	$0.213035\pi$
$593$	735.810i	1.24083i	−0.784275	+	0.620413i	$0.786965\pi$
$594$	0	0
$595$	60.2360	0.101237
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1114.89i	1.86125i	0.365972	+	0.930626i	$0.380737\pi$
$599$	1114.89i	1.86125i	−0.365972	+	0.930626i	$0.619263\pi$
$600$	0	0
$601$	−630.224	−1.04862	−0.524312	−	0.851526i	$-0.675678\pi$
$601$	−630.224	−1.04862	−0.524312	+	0.851526i	$0.675678\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 500.776i	− 0.827728i
$606$	0	0
$607$	−981.329	−1.61669	−0.808344	−	0.588711i	$-0.799636\pi$
$607$	−981.329	−1.61669	−0.808344	+	0.588711i	$0.799636\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1232.52i	− 2.01722i
$612$	0	0
$613$	244.598	0.399017	0.199509	−	0.979896i	$-0.436065\pi$
$613$	244.598	0.399017	0.199509	+	0.979896i	$0.436065\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1049.10i	− 1.70033i	−0.526516	−	0.850165i	$-0.676502\pi$
$617$	− 1049.10i	− 1.70033i	0.526516	−	0.850165i	$-0.323498\pi$
$618$	0	0
$619$	937.314	1.51424	0.757120	−	0.653276i	$-0.226606\pi$
$619$	937.314	1.51424	0.757120	+	0.653276i	$0.226606\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	126.873i	0.203649i
$624$	0	0
$625$	−635.140	−1.01622
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 21.1449i	− 0.0336167i
$630$	0	0
$631$	65.8241	0.104317	0.0521586	−	0.998639i	$-0.483390\pi$
$631$	65.8241	0.104317	0.0521586	+	0.998639i	$0.483390\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1210.09i	1.90566i
$636$	0	0
$637$	830.375	1.30357
$638$	0	0
$639$	0	0
$640$	0	0
$641$	83.9190i	0.130919i	0.997855	+	0.0654595i	$0.0208513\pi$
$641$	83.9190i	0.130919i	−0.997855	+	0.0654595i	$0.979149\pi$
$642$	0	0
$643$	247.887	0.385517	0.192758	−	0.981246i	$-0.438257\pi$
$643$	247.887	0.385517	0.192758	+	0.981246i	$0.438257\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 85.4397i	− 0.132055i	−0.997818	−	0.0660276i	$-0.978967\pi$
$647$	− 85.4397i	− 0.132055i	0.997818	−	0.0660276i	$-0.0210325\pi$
$648$	0	0
$649$	−220.616	−0.339933
$650$	0	0
$651$	0	0
$652$	0	0
$653$	328.337i	0.502813i	0.967882	+	0.251407i	$0.0808931\pi$
$653$	328.337i	0.502813i	−0.967882	+	0.251407i	$0.919107\pi$
$654$	0	0
$655$	−1.89893	−0.00289913
$656$	0	0
$657$	0	0
$658$	0	0
$659$	796.546i	1.20872i	0.796712	+	0.604359i	$0.206571\pi$
$659$	796.546i	1.20872i	−0.796712	+	0.604359i	$0.793429\pi$
$660$	0	0
$661$	443.332	0.670699	0.335349	−	0.942094i	$-0.391146\pi$
$661$	443.332	0.670699	0.335349	+	0.942094i	$0.391146\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 94.4611i	− 0.142047i
$666$	0	0
$667$	−347.900	−0.521589
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 514.454i	− 0.766698i
$672$	0	0
$673$	−879.902	−1.30743	−0.653717	−	0.756740i	$-0.726791\pi$
$673$	−879.902	−1.30743	−0.653717	+	0.756740i	$0.726791\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 142.912i	− 0.211097i	−0.994414	−	0.105548i	$-0.966340\pi$
$677$	− 142.912i	− 0.211097i	0.994414	−	0.105548i	$-0.0336598\pi$
$678$	0	0
$679$	208.158	0.306565
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.7535i	0.0216010i	0.999942	+	0.0108005i	$0.00343797\pi$
$683$	14.7535i	0.0216010i	−0.999942	+	0.0108005i	$0.996562\pi$
$684$	0	0
$685$	1773.33	2.58880
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 493.420i	− 0.716139i
$690$	0	0
$691$	−959.483	−1.38854	−0.694271	−	0.719713i	$-0.744273\pi$
$691$	−959.483	−1.38854	−0.694271	+	0.719713i	$0.744273\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1030.95i	1.48338i
$696$	0	0
$697$	102.690	0.147331
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1061.81i	− 1.51471i	−0.653005	−	0.757354i	$-0.726492\pi$
$701$	− 1061.81i	− 1.51471i	0.653005	−	0.757354i	$-0.273508\pi$
$702$	0	0
$703$	−33.1591	−0.0471680
$704$	0	0
$705$	0	0
$706$	0	0
$707$	323.448i	0.457493i
$708$	0	0
$709$	1319.08	1.86048	0.930240	−	0.366951i	$-0.119598\pi$
$709$	1319.08	1.86048	0.930240	+	0.366951i	$0.119598\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	139.637i	0.195845i
$714$	0	0
$715$	−1044.79	−1.46125
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 486.575i	− 0.676739i	−0.941013	−	0.338369i	$-0.890125\pi$
$719$	− 486.575i	− 0.676739i	0.941013	−	0.338369i	$-0.109875\pi$
$720$	0	0
$721$	203.463	0.282195
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 473.276i	− 0.652795i
$726$	0	0
$727$	−34.3469	−0.0472448	−0.0236224	−	0.999721i	$-0.507520\pi$
$727$	−34.3469	−0.0472448	−0.0236224	+	0.999721i	$0.507520\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 127.697i	− 0.174688i
$732$	0	0
$733$	−734.420	−1.00194	−0.500969	−	0.865465i	$-0.667023\pi$
$733$	−734.420	−1.00194	−0.500969	+	0.865465i	$0.667023\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 609.631i	− 0.827179i
$738$	0	0
$739$	−715.718	−0.968495	−0.484247	−	0.874931i	$-0.660907\pi$
$739$	−715.718	−0.968495	−0.484247	+	0.874931i	$0.660907\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 775.760i	− 1.04409i	−0.852917	−	0.522046i	$-0.825169\pi$
$743$	− 775.760i	− 1.04409i	0.852917	−	0.522046i	$-0.174831\pi$
$744$	0	0
$745$	−1012.44	−1.35898
$746$	0	0
$747$	0	0
$748$	0	0
$749$	139.170i	0.185807i
$750$	0	0
$751$	159.327	0.212153	0.106076	−	0.994358i	$-0.466171\pi$
$751$	159.327	0.212153	0.106076	+	0.994358i	$0.466171\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1921.79i	− 2.54541i
$756$	0	0
$757$	893.090	1.17978	0.589888	−	0.807485i	$-0.299172\pi$
$757$	893.090	1.17978	0.589888	+	0.807485i	$0.299172\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1106.19i	− 1.45360i	−0.686849	−	0.726800i	$-0.741007\pi$
$761$	− 1106.19i	− 1.45360i	0.686849	−	0.726800i	$-0.258993\pi$
$762$	0	0
$763$	235.069	0.308086
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 656.151i	− 0.855477i
$768$	0	0
$769$	−63.7902	−0.0829522	−0.0414761	−	0.999139i	$-0.513206\pi$
$769$	−63.7902	−0.0829522	−0.0414761	+	0.999139i	$0.513206\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	882.727i	1.14195i	0.820968	+	0.570975i	$0.193435\pi$
$773$	882.727i	1.14195i	−0.820968	+	0.570975i	$0.806565\pi$
$774$	0	0
$775$	−189.960	−0.245109
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 161.036i	− 0.206722i
$780$	0	0
$781$	831.986	1.06528
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 783.457i	− 0.998035i
$786$	0	0
$787$	−216.807	−0.275486	−0.137743	−	0.990468i	$-0.543985\pi$
$787$	−216.807	−0.275486	−0.137743	+	0.990468i	$0.543985\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 663.421i	− 0.838712i
$792$	0	0
$793$	1530.08	1.92948
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 457.336i	− 0.573821i	−0.957957	−	0.286911i	$-0.907372\pi$
$797$	− 457.336i	− 0.573821i	0.957957	−	0.286911i	$-0.0926283\pi$
$798$	0	0
$799$	163.087	0.204114
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 417.456i	− 0.519870i
$804$	0	0
$805$	−391.681	−0.486560
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1055.85i	− 1.30513i	−0.757731	−	0.652567i	$-0.773692\pi$
$809$	− 1055.85i	− 1.30513i	0.757731	−	0.652567i	$-0.226308\pi$
$810$	0	0
$811$	1409.94	1.73852	0.869261	−	0.494353i	$-0.164595\pi$
$811$	1409.94	1.73852	0.869261	+	0.494353i	$0.164595\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 510.187i	− 0.625997i
$816$	0	0
$817$	−200.253	−0.245107
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 223.959i	− 0.272788i	−0.990655	−	0.136394i	$-0.956449\pi$
$821$	− 223.959i	− 0.272788i	0.990655	−	0.136394i	$-0.0435513\pi$
$822$	0	0
$823$	383.573	0.466067	0.233033	−	0.972469i	$-0.425135\pi$
$823$	383.573	0.466067	0.233033	+	0.972469i	$0.425135\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1199.31i	1.45019i	0.688647	+	0.725097i	$0.258205\pi$
$827$	1199.31i	1.45019i	−0.688647	+	0.725097i	$0.741795\pi$
$828$	0	0
$829$	−219.245	−0.264469	−0.132234	−	0.991218i	$-0.542215\pi$
$829$	−219.245	−0.264469	−0.132234	+	0.991218i	$0.542215\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	109.875i	0.131903i
$834$	0	0
$835$	−245.358	−0.293842
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1134.25i	1.35191i	0.736944	+	0.675954i	$0.236268\pi$
$839$	1134.25i	1.35191i	−0.736944	+	0.675954i	$0.763732\pi$
$840$	0	0
$841$	470.493	0.559444
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1917.32i	− 2.26902i
$846$	0	0
$847$	−218.850	−0.258382
$848$	0	0
$849$	0	0
$850$	0	0
$851$	137.493i	0.161567i
$852$	0	0
$853$	−1031.86	−1.20968	−0.604841	−	0.796346i	$-0.706763\pi$
$853$	−1031.86	−1.20968	−0.604841	+	0.796346i	$0.706763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	274.317i	0.320090i	0.987110	+	0.160045i	$0.0511639\pi$
$857$	274.317i	0.320090i	−0.987110	+	0.160045i	$0.948836\pi$
$858$	0	0
$859$	−1458.84	−1.69830	−0.849152	−	0.528149i	$-0.822886\pi$
$859$	−1458.84	−1.69830	−0.849152	+	0.528149i	$0.822886\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1504.73i	1.74361i	0.489855	+	0.871804i	$0.337050\pi$
$863$	1504.73i	1.74361i	−0.489855	+	0.871804i	$0.662950\pi$
$864$	0	0
$865$	−423.817	−0.489962
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 303.040i	− 0.348722i
$870$	0	0
$871$	1813.15	2.08169
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.93703i	0.0102138i
$876$	0	0
$877$	−224.480	−0.255964	−0.127982	−	0.991776i	$-0.540850\pi$
$877$	−224.480	−0.255964	−0.127982	+	0.991776i	$0.540850\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 214.475i	− 0.243444i	−0.992564	−	0.121722i	$-0.961158\pi$
$881$	− 214.475i	− 0.243444i	0.992564	−	0.121722i	$-0.0388417\pi$
$882$	0	0
$883$	1499.64	1.69835	0.849175	−	0.528111i	$-0.177100\pi$
$883$	1499.64	1.69835	0.849175	+	0.528111i	$0.177100\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1721.15i	1.94042i	0.242273	+	0.970208i	$0.422107\pi$
$887$	1721.15i	1.94042i	−0.242273	+	0.970208i	$0.577893\pi$
$888$	0	0
$889$	528.837	0.594867
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 255.751i	− 0.286395i
$894$	0	0
$895$	49.2219	0.0549965
$896$	0	0
$897$	0	0
$898$	0	0
$899$	148.711i	0.165419i
$900$	0	0
$901$	65.2892	0.0724631
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1383.63i	1.52887i
$906$	0	0
$907$	1479.21	1.63088	0.815439	−	0.578844i	$-0.196496\pi$
$907$	1479.21	1.63088	0.815439	+	0.578844i	$0.196496\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1776.54i	− 1.95010i	−0.221983	−	0.975050i	$-0.571253\pi$
$911$	− 1776.54i	− 1.95010i	0.221983	−	0.975050i	$-0.428747\pi$
$912$	0	0
$913$	−215.228	−0.235737
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.829873i	0 0.000904987i
$918$	0	0
$919$	1028.64	1.11931	0.559653	−	0.828727i	$-0.310935\pi$
$919$	1028.64	1.11931	0.559653	+	0.828727i	$0.310935\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2474.47i	2.68090i
$924$	0	0
$925$	−187.043	−0.202209
$926$	0	0
$927$	0	0
$928$	0	0
$929$	165.560i	0.178213i	0.996022	+	0.0891064i	$0.0284011\pi$
$929$	165.560i	0.178213i	−0.996022	+	0.0891064i	$0.971599\pi$
$930$	0	0
$931$	172.304	0.185075
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 138.247i	− 0.147857i
$936$	0	0
$937$	727.213	0.776108	0.388054	−	0.921637i	$-0.373147\pi$
$937$	727.213	0.776108	0.388054	+	0.921637i	$0.373147\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1199.11i	− 1.27429i	−0.770743	−	0.637146i	$-0.780115\pi$
$941$	− 1199.11i	− 1.27429i	0.770743	−	0.637146i	$-0.219885\pi$
$942$	0	0
$943$	−667.732	−0.708093
$944$	0	0
$945$	0	0
$946$	0	0
$947$	427.273i	0.451186i	0.974222	+	0.225593i	$0.0724319\pi$
$947$	427.273i	0.451186i	−0.974222	+	0.225593i	$0.927568\pi$
$948$	0	0
$949$	1241.59	1.30831
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1401.89i	− 1.47103i	−0.677511	−	0.735513i	$-0.736941\pi$
$953$	− 1401.89i	− 1.47103i	0.677511	−	0.735513i	$-0.263059\pi$
$954$	0	0
$955$	−2054.80	−2.15162
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 774.982i	− 0.808115i
$960$	0	0
$961$	−901.311	−0.937889
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 494.548i	− 0.512485i
$966$	0	0
$967$	65.4636	0.0676976	0.0338488	−	0.999427i	$-0.489224\pi$
$967$	65.4636	0.0676976	0.0338488	+	0.999427i	$0.489224\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 45.8136i	− 0.0471819i	−0.999722	−	0.0235909i	$-0.992490\pi$
$971$	− 45.8136i	− 0.0471819i	0.999722	−	0.0235909i	$-0.00750993\pi$
$972$	0	0
$973$	450.546	0.463049
$974$	0	0
$975$	0	0
$976$	0	0
$977$	958.764i	0.981335i	0.871347	+	0.490667i	$0.163247\pi$
$977$	958.764i	0.981335i	−0.871347	+	0.490667i	$0.836753\pi$
$978$	0	0
$979$	291.185	0.297431
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1366.12i	1.38975i	0.719131	+	0.694874i	$0.244540\pi$
$983$	1366.12i	1.38975i	−0.719131	+	0.694874i	$0.755460\pi$
$984$	0	0
$985$	190.701	0.193605
$986$	0	0
$987$	0	0
$988$	0	0
$989$	830.343i	0.839578i
$990$	0	0
$991$	725.869	0.732461	0.366231	−	0.930524i	$-0.380648\pi$
$991$	725.869	0.732461	0.366231	+	0.930524i	$0.380648\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2537.12i	− 2.54987i
$996$	0	0
$997$	−159.814	−0.160294	−0.0801472	−	0.996783i	$-0.525539\pi$
$997$	−159.814	−0.160294	−0.0801472	+	0.996783i	$0.525539\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.e.a.305.2	✓	12
3.2	odd	2	inner	684.3.e.a.305.11	yes	12
4.3	odd	2		2736.3.h.c.305.2		12
12.11	even	2		2736.3.h.c.305.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.e.a.305.2	✓	12	1.1	even	1	trivial
684.3.e.a.305.11	yes	12	3.2	odd	2	inner
2736.3.h.c.305.2		12	4.3	odd	2
2736.3.h.c.305.11		12	12.11	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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.04185i q^{5} -3.07744 q^{7} +O(q^{10})\)	\(q-7.04185i q^{5} -3.07744 q^{7} +7.06299i q^{11} -21.0065 q^{13} -2.77958i q^{17} -4.35890 q^{19} +18.0741i q^{23} -24.5876 q^{25} +19.2486i q^{29} +7.72584 q^{31} +21.6709i q^{35} +7.60723 q^{37} +36.9442i q^{41} +45.9411 q^{43} +58.6732i q^{47} -39.5294 q^{49} +23.4889i q^{53} +49.7365 q^{55} +31.2355i q^{59} -72.8380 q^{61} +147.925i q^{65} -86.3135 q^{67} -117.795i q^{71} -59.1047 q^{73} -21.7359i q^{77} -42.9053 q^{79} +30.4726i q^{83} -19.5734 q^{85} -41.2269i q^{89} +64.6464 q^{91} +30.6947i q^{95} -67.6398 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{7}+O(q^{10}) \) 12 * q + 16 * q^7	\( 12 q + 16 q^{7} - 16 q^{13} - 12 q^{25} - 40 q^{31} - 32 q^{37} + 92 q^{43} - 84 q^{55} - 48 q^{61} - 88 q^{67} + 148 q^{73} - 56 q^{79} + 228 q^{85} - 8 q^{91} + 72 q^{97}+O(q^{100}) \) 12 * q + 16 * q^7 - 16 * q^13 - 12 * q^25 - 40 * q^31 - 32 * q^37 + 92 * q^43 - 84 * q^55 - 48 * q^61 - 88 * q^67 + 148 * q^73 - 56 * q^79 + 228 * q^85 - 8 * q^91 + 72 * q^97

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