LMFDB - Embedded newform 684.3.e.a.305.4

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			305.4
Root			$-3.29597i$ of defining polynomial
Character	$\chi$	$=$	684.305
Dual form			684.3.e.a.305.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.29597i q^{5} -2.46307 q^{7} +O(q^{10})$	$q-3.29597i q^{5} -2.46307 q^{7} +10.4157i q^{11} +2.93243 q^{13} -27.8783i q^{17} -4.35890 q^{19} -24.3461i q^{23} +14.1366 q^{25} -7.80708i q^{29} -17.4419 q^{31} +8.11820i q^{35} -48.3596 q^{37} -51.2260i q^{41} -82.7891 q^{43} +22.2146i q^{47} -42.9333 q^{49} -16.6275i q^{53} +34.3300 q^{55} -49.5384i q^{59} +16.9475 q^{61} -9.66522i q^{65} -1.05167 q^{67} -79.0160i q^{71} -53.4664 q^{73} -25.6547i q^{77} +137.382 q^{79} +4.56292i q^{83} -91.8862 q^{85} -120.627i q^{89} -7.22278 q^{91} +14.3668i q^{95} +13.4947 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 3.29597i	− 0.659195i	−0.944122	−	0.329597i	$-0.893087\pi$
$5$	− 3.29597i	− 0.659195i	0.944122	−	0.329597i	$-0.106913\pi$
$6$	0	0
$7$	−2.46307	−0.351867	−0.175933	−	0.984402i	$-0.556294\pi$
$7$	−2.46307	−0.351867	−0.175933	+	0.984402i	$0.556294\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	10.4157i	0.946886i	0.880825	+	0.473443i	$0.156989\pi$
$11$	10.4157i	0.946886i	−0.880825	+	0.473443i	$0.843011\pi$
$12$	0	0
$13$	2.93243	0.225572	0.112786	−	0.993619i	$-0.464023\pi$
$13$	2.93243	0.225572	0.112786	+	0.993619i	$0.464023\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 27.8783i	− 1.63990i	−0.572433	−	0.819951i	$-0.694000\pi$
$17$	− 27.8783i	− 1.63990i	0.572433	−	0.819951i	$-0.306000\pi$
$18$	0	0
$19$	−4.35890	−0.229416
$19$	−4.35890	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 24.3461i	− 1.05853i	−0.848458	−	0.529263i	$-0.822468\pi$
$23$	− 24.3461i	− 1.05853i	0.848458	−	0.529263i	$-0.177532\pi$
$24$	0	0
$25$	14.1366	0.565463
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 7.80708i	− 0.269210i	−0.990899	−	0.134605i	$-0.957024\pi$
$29$	− 7.80708i	− 0.269210i	0.990899	−	0.134605i	$-0.0429765\pi$
$30$	0	0
$31$	−17.4419	−0.562642	−0.281321	−	0.959614i	$-0.590773\pi$
$31$	−17.4419	−0.562642	−0.281321	+	0.959614i	$0.590773\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.11820i	0.231949i
$36$	0	0
$37$	−48.3596	−1.30702	−0.653508	−	0.756919i	$-0.726704\pi$
$37$	−48.3596	−1.30702	−0.653508	+	0.756919i	$0.726704\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 51.2260i	− 1.24942i	−0.780859	−	0.624708i	$-0.785218\pi$
$41$	− 51.2260i	− 1.24942i	0.780859	−	0.624708i	$-0.214782\pi$
$42$	0	0
$43$	−82.7891	−1.92533	−0.962664	−	0.270700i	$-0.912745\pi$
$43$	−82.7891	−1.92533	−0.962664	+	0.270700i	$0.912745\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	22.2146i	0.472651i	0.971674	+	0.236325i	$0.0759432\pi$
$47$	22.2146i	0.472651i	−0.971674	+	0.236325i	$0.924057\pi$
$48$	0	0
$49$	−42.9333	−0.876190
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 16.6275i	− 0.313726i	−0.987620	−	0.156863i	$-0.949862\pi$
$53$	− 16.6275i	− 0.313726i	0.987620	−	0.156863i	$-0.0501382\pi$
$54$	0	0
$55$	34.3300	0.624182
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 49.5384i	− 0.839634i	−0.907609	−	0.419817i	$-0.862094\pi$
$59$	− 49.5384i	− 0.839634i	0.907609	−	0.419817i	$-0.137906\pi$
$60$	0	0
$61$	16.9475	0.277828	0.138914	−	0.990304i	$-0.455639\pi$
$61$	16.9475	0.277828	0.138914	+	0.990304i	$0.455639\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 9.66522i	− 0.148696i
$66$	0	0
$67$	−1.05167	−0.0156965	−0.00784826	−	0.999969i	$-0.502498\pi$
$67$	−1.05167	−0.0156965	−0.00784826	+	0.999969i	$0.502498\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 79.0160i	− 1.11290i	−0.830881	−	0.556451i	$-0.812163\pi$
$71$	− 79.0160i	− 1.11290i	0.830881	−	0.556451i	$-0.187837\pi$
$72$	0	0
$73$	−53.4664	−0.732417	−0.366208	−	0.930533i	$-0.619344\pi$
$73$	−53.4664	−0.732417	−0.366208	+	0.930533i	$0.619344\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 25.6547i	− 0.333178i
$78$	0	0
$79$	137.382	1.73902	0.869508	−	0.493919i	$-0.164436\pi$
$79$	137.382	1.73902	0.869508	+	0.493919i	$0.164436\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.56292i	0.0549750i	0.999622	+	0.0274875i	$0.00875064\pi$
$83$	4.56292i	0.0549750i	−0.999622	+	0.0274875i	$0.991249\pi$
$84$	0	0
$85$	−91.8862	−1.08101
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 120.627i	− 1.35536i	−0.735358	−	0.677679i	$-0.762986\pi$
$89$	− 120.627i	− 1.35536i	0.735358	−	0.677679i	$-0.237014\pi$
$90$	0	0
$91$	−7.22278	−0.0793712
$92$	0	0
$93$	0	0
$94$	0	0
$95$	14.3668i	0.151230i
$96$	0	0
$97$	13.4947	0.139121	0.0695604	−	0.997578i	$-0.477840\pi$
$97$	13.4947	0.139121	0.0695604	+	0.997578i	$0.477840\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 101.454i	− 1.00449i	−0.864724	−	0.502247i	$-0.832507\pi$
$101$	− 101.454i	− 1.00449i	0.864724	−	0.502247i	$-0.167493\pi$
$102$	0	0
$103$	−141.381	−1.37263	−0.686313	−	0.727306i	$-0.740772\pi$
$103$	−141.381	−1.37263	−0.686313	+	0.727306i	$0.740772\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	21.6171i	0.202029i	0.994885	+	0.101014i	$0.0322088\pi$
$107$	21.6171i	0.202029i	−0.994885	+	0.101014i	$0.967791\pi$
$108$	0	0
$109$	30.9232	0.283699	0.141850	−	0.989888i	$-0.454695\pi$
$109$	30.9232	0.283699	0.141850	+	0.989888i	$0.454695\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	154.363i	1.36605i	0.730397	+	0.683023i	$0.239335\pi$
$113$	154.363i	1.36605i	−0.730397	+	0.683023i	$0.760665\pi$
$114$	0	0
$115$	−80.2441	−0.697775
$116$	0	0
$117$	0	0
$118$	0	0
$119$	68.6662i	0.577027i
$120$	0	0
$121$	12.5123	0.103407
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 128.993i	− 1.03194i
$126$	0	0
$127$	71.0621	0.559544	0.279772	−	0.960066i	$-0.409741\pi$
$127$	71.0621	0.559544	0.279772	+	0.960066i	$0.409741\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	61.8603i	0.472216i	0.971727	+	0.236108i	$0.0758719\pi$
$131$	61.8603i	0.472216i	−0.971727	+	0.236108i	$0.924128\pi$
$132$	0	0
$133$	10.7363	0.0807238
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 86.2875i	− 0.629836i	−0.949119	−	0.314918i	$-0.898023\pi$
$137$	− 86.2875i	− 0.629836i	0.949119	−	0.314918i	$-0.101977\pi$
$138$	0	0
$139$	173.550	1.24856	0.624280	−	0.781201i	$-0.285393\pi$
$139$	173.550	1.24856	0.624280	+	0.781201i	$0.285393\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	30.5435i	0.213591i
$144$	0	0
$145$	−25.7319	−0.177461
$146$	0	0
$147$	0	0
$148$	0	0
$149$	46.9468i	0.315079i	0.987513	+	0.157540i	$0.0503562\pi$
$149$	46.9468i	0.315079i	−0.987513	+	0.157540i	$0.949644\pi$
$150$	0	0
$151$	−1.01794	−0.00674136	−0.00337068	−	0.999994i	$-0.501073\pi$
$151$	−1.01794	−0.00674136	−0.00337068	+	0.999994i	$0.501073\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	57.4880i	0.370890i
$156$	0	0
$157$	6.17959	0.0393604	0.0196802	−	0.999806i	$-0.493735\pi$
$157$	6.17959	0.0393604	0.0196802	+	0.999806i	$0.493735\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	59.9661i	0.372460i
$162$	0	0
$163$	37.5244	0.230211	0.115106	−	0.993353i	$-0.463279\pi$
$163$	37.5244	0.230211	0.115106	+	0.993353i	$0.463279\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	146.277i	0.875907i	0.898998	+	0.437954i	$0.144297\pi$
$167$	146.277i	0.875907i	−0.898998	+	0.437954i	$0.855703\pi$
$168$	0	0
$169$	−160.401	−0.949117
$170$	0	0
$171$	0	0
$172$	0	0
$173$	43.4528i	0.251172i	0.992083	+	0.125586i	$0.0400811\pi$
$173$	43.4528i	0.251172i	−0.992083	+	0.125586i	$0.959919\pi$
$174$	0	0
$175$	−34.8193	−0.198968
$176$	0	0
$177$	0	0
$178$	0	0
$179$	99.6202i	0.556537i	0.960503	+	0.278269i	$0.0897606\pi$
$179$	99.6202i	0.556537i	−0.960503	+	0.278269i	$0.910239\pi$
$180$	0	0
$181$	294.049	1.62458	0.812291	−	0.583252i	$-0.198220\pi$
$181$	294.049	1.62458	0.812291	+	0.583252i	$0.198220\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	159.392i	0.861578i
$186$	0	0
$187$	290.374	1.55280
$188$	0	0
$189$	0	0
$190$	0	0
$191$	149.910i	0.784869i	0.919780	+	0.392435i	$0.128367\pi$
$191$	149.910i	0.784869i	−0.919780	+	0.392435i	$0.871633\pi$
$192$	0	0
$193$	−376.836	−1.95252	−0.976259	−	0.216608i	$-0.930501\pi$
$193$	−376.836	−1.95252	−0.976259	+	0.216608i	$0.930501\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	100.556i	0.510439i	0.966883	+	0.255219i	$0.0821477\pi$
$197$	100.556i	0.510439i	−0.966883	+	0.255219i	$0.917852\pi$
$198$	0	0
$199$	150.210	0.754822	0.377411	−	0.926046i	$-0.376814\pi$
$199$	150.210	0.754822	0.377411	+	0.926046i	$0.376814\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	19.2294i	0.0947259i
$204$	0	0
$205$	−168.840	−0.823608
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 45.4012i	− 0.217231i
$210$	0	0
$211$	23.1583	0.109755	0.0548774	−	0.998493i	$-0.482523\pi$
$211$	23.1583	0.109755	0.0548774	+	0.998493i	$0.482523\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	272.871i	1.26917i
$216$	0	0
$217$	42.9606	0.197975
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 81.7514i	− 0.369916i
$222$	0	0
$223$	438.633	1.96696	0.983481	−	0.181010i	$-0.0579365\pi$
$223$	438.633	1.96696	0.983481	+	0.181010i	$0.0579365\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	123.454i	0.543850i	0.962318	+	0.271925i	$0.0876603\pi$
$227$	123.454i	0.543850i	−0.962318	+	0.271925i	$0.912340\pi$
$228$	0	0
$229$	159.655	0.697182	0.348591	−	0.937275i	$-0.386660\pi$
$229$	159.655	0.697182	0.348591	+	0.937275i	$0.386660\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 257.960i	− 1.10712i	−0.832808	−	0.553561i	$-0.813268\pi$
$233$	− 257.960i	− 1.10712i	0.832808	−	0.553561i	$-0.186732\pi$
$234$	0	0
$235$	73.2187	0.311569
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 15.7567i	− 0.0659276i	−0.999457	−	0.0329638i	$-0.989505\pi$
$239$	− 15.7567i	− 0.0659276i	0.999457	−	0.0329638i	$-0.0104946\pi$
$240$	0	0
$241$	−29.0324	−0.120466	−0.0602331	−	0.998184i	$-0.519184\pi$
$241$	−29.0324	−0.120466	−0.0602331	+	0.998184i	$0.519184\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	141.507i	0.577579i
$246$	0	0
$247$	−12.7822	−0.0517497
$248$	0	0
$249$	0	0
$250$	0	0
$251$	72.1205i	0.287333i	0.989626	+	0.143666i	$0.0458892\pi$
$251$	72.1205i	0.287333i	−0.989626	+	0.143666i	$0.954111\pi$
$252$	0	0
$253$	253.583	1.00230
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 252.576i	− 0.982787i	−0.870938	−	0.491393i	$-0.836488\pi$
$257$	− 252.576i	− 0.982787i	0.870938	−	0.491393i	$-0.163512\pi$
$258$	0	0
$259$	119.113	0.459896
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 47.7320i	− 0.181491i	−0.995874	−	0.0907453i	$-0.971075\pi$
$263$	− 47.7320i	− 0.181491i	0.995874	−	0.0907453i	$-0.0289249\pi$
$264$	0	0
$265$	−54.8038	−0.206807
$266$	0	0
$267$	0	0
$268$	0	0
$269$	443.061i	1.64707i	0.567266	+	0.823534i	$0.308001\pi$
$269$	443.061i	1.64707i	−0.567266	+	0.823534i	$0.691999\pi$
$270$	0	0
$271$	101.973	0.376286	0.188143	−	0.982142i	$-0.439753\pi$
$271$	101.973	0.376286	0.188143	+	0.982142i	$0.439753\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	147.243i	0.535429i
$276$	0	0
$277$	193.897	0.699989	0.349995	−	0.936752i	$-0.386183\pi$
$277$	193.897	0.699989	0.349995	+	0.936752i	$0.386183\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	287.031i	1.02146i	0.859741	+	0.510731i	$0.170625\pi$
$281$	287.031i	1.02146i	−0.859741	+	0.510731i	$0.829375\pi$
$282$	0	0
$283$	−253.702	−0.896474	−0.448237	−	0.893915i	$-0.647948\pi$
$283$	−253.702	−0.896474	−0.448237	+	0.893915i	$0.647948\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	126.173i	0.439628i
$288$	0	0
$289$	−488.202	−1.68928
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 563.387i	− 1.92282i	−0.275115	−	0.961411i	$-0.588716\pi$
$293$	− 563.387i	− 1.92282i	0.275115	−	0.961411i	$-0.411284\pi$
$294$	0	0
$295$	−163.277	−0.553482
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 71.3934i	− 0.238774i
$300$	0	0
$301$	203.915	0.677459
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 55.8586i	− 0.183143i
$306$	0	0
$307$	224.823	0.732321	0.366161	−	0.930552i	$-0.380672\pi$
$307$	224.823	0.732321	0.366161	+	0.930552i	$0.380672\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	495.020i	1.59171i	0.605490	+	0.795853i	$0.292977\pi$
$311$	495.020i	1.59171i	−0.605490	+	0.795853i	$0.707023\pi$
$312$	0	0
$313$	317.555	1.01455	0.507276	−	0.861784i	$-0.330653\pi$
$313$	317.555	1.01455	0.507276	+	0.861784i	$0.330653\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 290.705i	− 0.917051i	−0.888681	−	0.458525i	$-0.848378\pi$
$317$	− 290.705i	− 0.917051i	0.888681	−	0.458525i	$-0.151622\pi$
$318$	0	0
$319$	81.3165	0.254911
$320$	0	0
$321$	0	0
$322$	0	0
$323$	121.519i	0.376219i
$324$	0	0
$325$	41.4545	0.127552
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 54.7160i	− 0.166310i
$330$	0	0
$331$	3.55596	0.0107431	0.00537154	−	0.999986i	$-0.498290\pi$
$331$	3.55596	0.0107431	0.00537154	+	0.999986i	$0.498290\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.46626i	0.0103471i
$336$	0	0
$337$	−447.418	−1.32765	−0.663825	−	0.747888i	$-0.731068\pi$
$337$	−447.418	−1.32765	−0.663825	+	0.747888i	$0.731068\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 181.670i	− 0.532757i
$342$	0	0
$343$	226.438	0.660169
$344$	0	0
$345$	0	0
$346$	0	0
$347$	154.029i	0.443889i	0.975059	+	0.221944i	$0.0712403\pi$
$347$	154.029i	0.443889i	−0.975059	+	0.221944i	$0.928760\pi$
$348$	0	0
$349$	−356.174	−1.02056	−0.510278	−	0.860009i	$-0.670458\pi$
$349$	−356.174	−1.02056	−0.510278	+	0.860009i	$0.670458\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 251.734i	− 0.713129i	−0.934271	−	0.356564i	$-0.883948\pi$
$353$	− 251.734i	− 0.713129i	0.934271	−	0.356564i	$-0.116052\pi$
$354$	0	0
$355$	−260.435	−0.733618
$356$	0	0
$357$	0	0
$358$	0	0
$359$	488.041i	1.35944i	0.733470	+	0.679722i	$0.237900\pi$
$359$	488.041i	1.35944i	−0.733470	+	0.679722i	$0.762100\pi$
$360$	0	0
$361$	19.0000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	176.224i	0.482805i
$366$	0	0
$367$	−281.123	−0.766002	−0.383001	−	0.923748i	$-0.625109\pi$
$367$	−281.123	−0.766002	−0.383001	+	0.923748i	$0.625109\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	40.9547i	0.110390i
$372$	0	0
$373$	−343.161	−0.920001	−0.460001	−	0.887919i	$-0.652151\pi$
$373$	−343.161	−0.920001	−0.460001	+	0.887919i	$0.652151\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 22.8937i	− 0.0607261i
$378$	0	0
$379$	410.802	1.08391	0.541955	−	0.840408i	$-0.317684\pi$
$379$	410.802	1.08391	0.541955	+	0.840408i	$0.317684\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 370.444i	− 0.967217i	−0.875285	−	0.483608i	$-0.839326\pi$
$383$	− 370.444i	− 0.967217i	0.875285	−	0.483608i	$-0.160674\pi$
$384$	0	0
$385$	−84.5571	−0.219629
$386$	0	0
$387$	0	0
$388$	0	0
$389$	389.355i	1.00091i	0.865762	+	0.500456i	$0.166834\pi$
$389$	389.355i	1.00091i	−0.865762	+	0.500456i	$0.833166\pi$
$390$	0	0
$391$	−678.729	−1.73588
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 452.808i	− 1.14635i
$396$	0	0
$397$	−25.0875	−0.0631928	−0.0315964	−	0.999501i	$-0.510059\pi$
$397$	−25.0875	−0.0631928	−0.0315964	+	0.999501i	$0.510059\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 602.238i	− 1.50184i	−0.660393	−	0.750920i	$-0.729610\pi$
$401$	− 602.238i	− 1.50184i	0.660393	−	0.750920i	$-0.270390\pi$
$402$	0	0
$403$	−51.1472	−0.126916
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 503.701i	− 1.23760i
$408$	0	0
$409$	−47.6877	−0.116596	−0.0582979	−	0.998299i	$-0.518567\pi$
$409$	−47.6877	−0.116596	−0.0582979	+	0.998299i	$0.518567\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	122.016i	0.295439i
$414$	0	0
$415$	15.0393	0.0362392
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.3011i	0.0460647i	0.999735	+	0.0230323i	$0.00733207\pi$
$419$	19.3011i	0.0460647i	−0.999735	+	0.0230323i	$0.992668\pi$
$420$	0	0
$421$	81.7466	0.194172	0.0970862	−	0.995276i	$-0.469048\pi$
$421$	81.7466	0.194172	0.0970862	+	0.995276i	$0.469048\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 394.104i	− 0.927304i
$426$	0	0
$427$	−41.7429	−0.0977586
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 675.971i	− 1.56838i	−0.620522	−	0.784189i	$-0.713079\pi$
$431$	− 675.971i	− 1.56838i	0.620522	−	0.784189i	$-0.286921\pi$
$432$	0	0
$433$	391.222	0.903515	0.451758	−	0.892141i	$-0.350797\pi$
$433$	391.222	0.903515	0.451758	+	0.892141i	$0.350797\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	106.122i	0.242843i
$438$	0	0
$439$	120.944	0.275498	0.137749	−	0.990467i	$-0.456013\pi$
$439$	120.944	0.275498	0.137749	+	0.990467i	$0.456013\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 65.2480i	− 0.147287i	−0.997285	−	0.0736433i	$-0.976537\pi$
$443$	− 65.2480i	− 0.147287i	0.997285	−	0.0736433i	$-0.0234626\pi$
$444$	0	0
$445$	−397.583	−0.893444
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 486.760i	− 1.08410i	−0.840347	−	0.542049i	$-0.817649\pi$
$449$	− 486.760i	− 1.08410i	0.840347	−	0.542049i	$-0.182351\pi$
$450$	0	0
$451$	533.557	1.18305
$452$	0	0
$453$	0	0
$454$	0	0
$455$	23.8061i	0.0523211i
$456$	0	0
$457$	523.679	1.14591	0.572953	−	0.819588i	$-0.305798\pi$
$457$	523.679	1.14591	0.572953	+	0.819588i	$0.305798\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	142.495i	0.309101i	0.987985	+	0.154550i	$0.0493929\pi$
$461$	142.495i	0.309101i	−0.987985	+	0.154550i	$0.950607\pi$
$462$	0	0
$463$	−240.182	−0.518752	−0.259376	−	0.965776i	$-0.583517\pi$
$463$	−240.182	−0.518752	−0.259376	+	0.965776i	$0.583517\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 264.429i	− 0.566230i	−0.959086	−	0.283115i	$-0.908632\pi$
$467$	− 264.429i	− 0.566230i	0.959086	−	0.283115i	$-0.0913678\pi$
$468$	0	0
$469$	2.59033	0.00552308
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 862.310i	− 1.82307i
$474$	0	0
$475$	−61.6199	−0.129726
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 723.453i	− 1.51034i	−0.655529	−	0.755170i	$-0.727554\pi$
$479$	− 723.453i	− 1.51034i	0.655529	−	0.755170i	$-0.272446\pi$
$480$	0	0
$481$	−141.811	−0.294826
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 44.4782i	− 0.0917077i
$486$	0	0
$487$	−728.561	−1.49602	−0.748009	−	0.663688i	$-0.768990\pi$
$487$	−728.561	−1.49602	−0.748009	+	0.663688i	$0.768990\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	888.765i	1.81011i	0.425293	+	0.905056i	$0.360171\pi$
$491$	888.765i	1.81011i	−0.425293	+	0.905056i	$0.639829\pi$
$492$	0	0
$493$	−217.648	−0.441477
$494$	0	0
$495$	0	0
$496$	0	0
$497$	194.622i	0.391593i
$498$	0	0
$499$	−740.968	−1.48491	−0.742453	−	0.669899i	$-0.766338\pi$
$499$	−740.968	−1.48491	−0.742453	+	0.669899i	$0.766338\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	195.451i	0.388571i	0.980945	+	0.194285i	$0.0622388\pi$
$503$	195.451i	0.388571i	−0.980945	+	0.194285i	$0.937761\pi$
$504$	0	0
$505$	−334.389	−0.662156
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 232.031i	− 0.455857i	−0.973678	−	0.227929i	$-0.926805\pi$
$509$	− 232.031i	− 0.455857i	0.973678	−	0.227929i	$-0.0731953\pi$
$510$	0	0
$511$	131.691	0.257713
$512$	0	0
$513$	0	0
$514$	0	0
$515$	465.986i	0.904828i
$516$	0	0
$517$	−231.382	−0.447546
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 313.989i	− 0.602666i	−0.953519	−	0.301333i	$-0.902568\pi$
$521$	− 313.989i	− 0.602666i	0.953519	−	0.301333i	$-0.0974316\pi$
$522$	0	0
$523$	−414.025	−0.791635	−0.395817	−	0.918329i	$-0.629539\pi$
$523$	−414.025	−0.791635	−0.395817	+	0.918329i	$0.629539\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	486.251i	0.922678i
$528$	0	0
$529$	−63.7334	−0.120479
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 150.217i	− 0.281833i
$534$	0	0
$535$	71.2493	0.133176
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 447.182i	− 0.829652i
$540$	0	0
$541$	569.365	1.05243	0.526215	−	0.850351i	$-0.323611\pi$
$541$	569.365	1.05243	0.526215	+	0.850351i	$0.323611\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 101.922i	− 0.187013i
$546$	0	0
$547$	527.255	0.963902	0.481951	−	0.876198i	$-0.339928\pi$
$547$	527.255	0.963902	0.481951	+	0.876198i	$0.339928\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	34.0303i	0.0617609i
$552$	0	0
$553$	−338.382	−0.611902
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1092.85i	1.96203i	0.193930	+	0.981015i	$0.437877\pi$
$557$	1092.85i	1.96203i	−0.193930	+	0.981015i	$0.562123\pi$
$558$	0	0
$559$	−242.774	−0.434300
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 662.414i	− 1.17658i	−0.808650	−	0.588290i	$-0.799801\pi$
$563$	− 662.414i	− 1.17658i	0.808650	−	0.588290i	$-0.200199\pi$
$564$	0	0
$565$	508.777	0.900490
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.5128i	0.0413230i	0.999787	+	0.0206615i	$0.00657723\pi$
$569$	23.5128i	0.0413230i	−0.999787	+	0.0206615i	$0.993423\pi$
$570$	0	0
$571$	−531.954	−0.931618	−0.465809	−	0.884885i	$-0.654237\pi$
$571$	−531.954	−0.931618	−0.465809	+	0.884885i	$0.654237\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 344.170i	− 0.598557i
$576$	0	0
$577$	441.929	0.765909	0.382954	−	0.923767i	$-0.374907\pi$
$577$	441.929	0.765909	0.382954	+	0.923767i	$0.374907\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 11.2388i	− 0.0193439i
$582$	0	0
$583$	173.188	0.297063
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 716.074i	− 1.21989i	−0.792445	−	0.609944i	$-0.791192\pi$
$587$	− 716.074i	− 1.21989i	0.792445	−	0.609944i	$-0.208808\pi$
$588$	0	0
$589$	76.0275	0.129079
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 598.295i	− 1.00893i	−0.863432	−	0.504465i	$-0.831690\pi$
$593$	− 598.295i	− 1.00893i	0.863432	−	0.504465i	$-0.168310\pi$
$594$	0	0
$595$	226.322	0.380373
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 600.674i	− 1.00279i	−0.865217	−	0.501397i	$-0.832819\pi$
$599$	− 600.674i	− 1.00279i	0.865217	−	0.501397i	$-0.167181\pi$
$600$	0	0
$601$	−663.305	−1.10367	−0.551834	−	0.833954i	$-0.686072\pi$
$601$	−663.305	−1.10367	−0.551834	+	0.833954i	$0.686072\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 41.2401i	− 0.0681655i
$606$	0	0
$607$	89.6567	0.147705	0.0738523	−	0.997269i	$-0.476471\pi$
$607$	89.6567	0.147705	0.0738523	+	0.997269i	$0.476471\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	65.1428i	0.106617i
$612$	0	0
$613$	−129.918	−0.211938	−0.105969	−	0.994369i	$-0.533794\pi$
$613$	−129.918	−0.211938	−0.105969	+	0.994369i	$0.533794\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	262.827i	0.425975i	0.977055	+	0.212988i	$0.0683194\pi$
$617$	262.827i	0.425975i	−0.977055	+	0.212988i	$0.931681\pi$
$618$	0	0
$619$	−536.855	−0.867293	−0.433647	−	0.901083i	$-0.642773\pi$
$619$	−536.855	−0.867293	−0.433647	+	0.901083i	$0.642773\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	297.112i	0.476905i
$624$	0	0
$625$	−71.7434	−0.114789
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1348.19i	2.14338i
$630$	0	0
$631$	1013.65	1.60641	0.803207	−	0.595700i	$-0.203125\pi$
$631$	1013.65	1.60641	0.803207	+	0.595700i	$0.203125\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 234.219i	− 0.368848i
$636$	0	0
$637$	−125.899	−0.197644
$638$	0	0
$639$	0	0
$640$	0	0
$641$	801.720i	1.25073i	0.780331	+	0.625367i	$0.215051\pi$
$641$	801.720i	1.25073i	−0.780331	+	0.625367i	$0.784949\pi$
$642$	0	0
$643$	−309.838	−0.481863	−0.240931	−	0.970542i	$-0.577453\pi$
$643$	−309.838	−0.481863	−0.240931	+	0.970542i	$0.577453\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	829.721i	1.28241i	0.767369	+	0.641206i	$0.221566\pi$
$647$	829.721i	1.28241i	−0.767369	+	0.641206i	$0.778434\pi$
$648$	0	0
$649$	515.979	0.795037
$650$	0	0
$651$	0	0
$652$	0	0
$653$	785.167i	1.20240i	0.799099	+	0.601200i	$0.205311\pi$
$653$	785.167i	1.20240i	−0.799099	+	0.601200i	$0.794689\pi$
$654$	0	0
$655$	203.890	0.311282
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1098.52i	− 1.66695i	−0.552558	−	0.833474i	$-0.686348\pi$
$659$	− 1098.52i	− 1.66695i	0.552558	−	0.833474i	$-0.313652\pi$
$660$	0	0
$661$	−294.514	−0.445558	−0.222779	−	0.974869i	$-0.571513\pi$
$661$	−294.514	−0.445558	−0.222779	+	0.974869i	$0.571513\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 35.3864i	− 0.0532127i
$666$	0	0
$667$	−190.072	−0.284966
$668$	0	0
$669$	0	0
$670$	0	0
$671$	176.521i	0.263072i
$672$	0	0
$673$	−924.048	−1.37303	−0.686514	−	0.727117i	$-0.740860\pi$
$673$	−924.048	−1.37303	−0.686514	+	0.727117i	$0.740860\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 615.975i	− 0.909860i	−0.890527	−	0.454930i	$-0.849664\pi$
$677$	− 615.975i	− 0.909860i	0.890527	−	0.454930i	$-0.150336\pi$
$678$	0	0
$679$	−33.2384	−0.0489520
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 49.5710i	− 0.0725783i	−0.999341	−	0.0362891i	$-0.988446\pi$
$683$	− 49.5710i	− 0.0725783i	0.999341	−	0.0362891i	$-0.0115537\pi$
$684$	0	0
$685$	−284.401	−0.415184
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 48.7590i	− 0.0707678i
$690$	0	0
$691$	1306.23	1.89035	0.945175	−	0.326564i	$-0.105891\pi$
$691$	1306.23	1.89035	0.945175	+	0.326564i	$0.105891\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 572.015i	− 0.823044i
$696$	0	0
$697$	−1428.10	−2.04892
$698$	0	0
$699$	0	0
$700$	0	0
$701$	814.490i	1.16190i	0.813940	+	0.580949i	$0.197318\pi$
$701$	814.490i	1.16190i	−0.813940	+	0.580949i	$0.802682\pi$
$702$	0	0
$703$	210.795	0.299850
$704$	0	0
$705$	0	0
$706$	0	0
$707$	249.888i	0.353448i
$708$	0	0
$709$	873.275	1.23170	0.615849	−	0.787864i	$-0.288813\pi$
$709$	873.275	1.23170	0.615849	+	0.787864i	$0.288813\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	424.642i	0.595571i
$714$	0	0
$715$	100.670	0.140798
$716$	0	0
$717$	0	0
$718$	0	0
$719$	448.892i	0.624328i	0.950028	+	0.312164i	$0.101054\pi$
$719$	448.892i	0.624328i	−0.950028	+	0.312164i	$0.898946\pi$
$720$	0	0
$721$	348.230	0.482982
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 110.365i	− 0.152228i
$726$	0	0
$727$	772.702	1.06286	0.531432	−	0.847101i	$-0.321654\pi$
$727$	772.702	1.06286	0.531432	+	0.847101i	$0.321654\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2308.02i	3.15735i
$732$	0	0
$733$	544.850	0.743316	0.371658	−	0.928370i	$-0.378789\pi$
$733$	544.850	0.743316	0.371658	+	0.928370i	$0.378789\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 10.9539i	− 0.0148628i
$738$	0	0
$739$	840.323	1.13711	0.568554	−	0.822646i	$-0.307503\pi$
$739$	840.323	1.13711	0.568554	+	0.822646i	$0.307503\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1361.90i	− 1.83297i	−0.400070	−	0.916485i	$-0.631014\pi$
$743$	− 1361.90i	− 1.83297i	0.400070	−	0.916485i	$-0.368986\pi$
$744$	0	0
$745$	154.735	0.207699
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 53.2443i	− 0.0710872i
$750$	0	0
$751$	−850.123	−1.13199	−0.565994	−	0.824409i	$-0.691507\pi$
$751$	−850.123	−1.13199	−0.565994	+	0.824409i	$0.691507\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.35512i	0.00444387i
$756$	0	0
$757$	1375.84	1.81750	0.908748	−	0.417345i	$-0.137039\pi$
$757$	1375.84	1.81750	0.908748	+	0.417345i	$0.137039\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	562.980i	0.739790i	0.929074	+	0.369895i	$0.120606\pi$
$761$	562.980i	0.739790i	−0.929074	+	0.369895i	$0.879394\pi$
$762$	0	0
$763$	−76.1659	−0.0998243
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 145.268i	− 0.189398i
$768$	0	0
$769$	1034.71	1.34552	0.672760	−	0.739861i	$-0.265109\pi$
$769$	1034.71	1.34552	0.672760	+	0.739861i	$0.265109\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	144.532i	0.186975i	0.995620	+	0.0934875i	$0.0298015\pi$
$773$	144.532i	0.186975i	−0.995620	+	0.0934875i	$0.970198\pi$
$774$	0	0
$775$	−246.568	−0.318153
$776$	0	0
$777$	0	0
$778$	0	0
$779$	223.289i	0.286636i
$780$	0	0
$781$	823.010	1.05379
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 20.3677i	− 0.0259462i
$786$	0	0
$787$	−589.006	−0.748419	−0.374210	−	0.927344i	$-0.622086\pi$
$787$	−589.006	−0.748419	−0.374210	+	0.927344i	$0.622086\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 380.207i	− 0.480666i
$792$	0	0
$793$	49.6975	0.0626703
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1241.83i	1.55813i	0.626944	+	0.779064i	$0.284305\pi$
$797$	1241.83i	1.55813i	−0.626944	+	0.779064i	$0.715695\pi$
$798$	0	0
$799$	619.306	0.775101
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 556.893i	− 0.693515i
$804$	0	0
$805$	197.647	0.245524
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1350.09i	− 1.66883i	−0.551135	−	0.834416i	$-0.685805\pi$
$809$	− 1350.09i	− 1.66883i	0.551135	−	0.834416i	$-0.314195\pi$
$810$	0	0
$811$	−1180.46	−1.45555	−0.727777	−	0.685813i	$-0.759447\pi$
$811$	−1180.46	−1.45555	−0.727777	+	0.685813i	$0.759447\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 123.680i	− 0.151754i
$816$	0	0
$817$	360.869	0.441700
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.71685i	− 0.00209117i	−0.999999	−	0.00104558i	$-0.999667\pi$
$821$	− 1.71685i	− 0.00209117i	0.999999	−	0.00104558i	$-0.000332820\pi$
$822$	0	0
$823$	−243.690	−0.296099	−0.148050	−	0.988980i	$-0.547300\pi$
$823$	−243.690	−0.296099	−0.148050	+	0.988980i	$0.547300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 586.109i	− 0.708717i	−0.935110	−	0.354359i	$-0.884699\pi$
$827$	− 586.109i	− 0.708717i	0.935110	−	0.354359i	$-0.115301\pi$
$828$	0	0
$829$	1097.09	1.32339	0.661697	−	0.749772i	$-0.269837\pi$
$829$	1097.09	1.32339	0.661697	+	0.749772i	$0.269837\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1196.91i	1.43687i
$834$	0	0
$835$	482.123	0.577393
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1628.39i	1.94087i	0.241357	+	0.970436i	$0.422407\pi$
$839$	1628.39i	1.94087i	−0.241357	+	0.970436i	$0.577593\pi$
$840$	0	0
$841$	780.050	0.927526
$842$	0	0
$843$	0	0
$844$	0	0
$845$	528.677i	0.625653i
$846$	0	0
$847$	−30.8186	−0.0363856
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1177.37i	1.38351i
$852$	0	0
$853$	−379.377	−0.444757	−0.222378	−	0.974960i	$-0.571382\pi$
$853$	−379.377	−0.444757	−0.222378	+	0.974960i	$0.571382\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1094.67i	1.27733i	0.769485	+	0.638665i	$0.220513\pi$
$857$	1094.67i	1.27733i	−0.769485	+	0.638665i	$0.779487\pi$
$858$	0	0
$859$	−1482.81	−1.72621	−0.863103	−	0.505028i	$-0.831482\pi$
$859$	−1482.81	−1.72621	−0.863103	+	0.505028i	$0.831482\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1083.41i	− 1.25540i	−0.778457	−	0.627698i	$-0.783997\pi$
$863$	− 1083.41i	− 1.25540i	0.778457	−	0.627698i	$-0.216003\pi$
$864$	0	0
$865$	143.219	0.165571
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1430.94i	1.64665i
$870$	0	0
$871$	−3.08394	−0.00354069
$872$	0	0
$873$	0	0
$874$	0	0
$875$	317.719i	0.363107i
$876$	0	0
$877$	1265.49	1.44297	0.721487	−	0.692428i	$-0.243459\pi$
$877$	1265.49	1.44297	0.721487	+	0.692428i	$0.243459\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	78.8518i	0.0895027i	0.998998	+	0.0447513i	$0.0142495\pi$
$881$	78.8518i	0.0895027i	−0.998998	+	0.0447513i	$0.985750\pi$
$882$	0	0
$883$	86.6337	0.0981129	0.0490565	−	0.998796i	$-0.484379\pi$
$883$	86.6337	0.0981129	0.0490565	+	0.998796i	$0.484379\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	449.343i	0.506587i	0.967389	+	0.253293i	$0.0815138\pi$
$887$	449.343i	0.506587i	−0.967389	+	0.253293i	$0.918486\pi$
$888$	0	0
$889$	−175.031	−0.196885
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 96.8312i	− 0.108434i
$894$	0	0
$895$	328.345	0.366866
$896$	0	0
$897$	0	0
$898$	0	0
$899$	136.170i	0.151469i
$900$	0	0
$901$	−463.547	−0.514481
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 969.179i	− 1.07092i
$906$	0	0
$907$	−1271.12	−1.40145	−0.700727	−	0.713429i	$-0.747141\pi$
$907$	−1271.12	−1.40145	−0.700727	+	0.713429i	$0.747141\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	428.669i	0.470548i	0.971929	+	0.235274i	$0.0755987\pi$
$911$	428.669i	0.470548i	−0.971929	+	0.235274i	$0.924401\pi$
$912$	0	0
$913$	−47.5263	−0.0520550
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 152.366i	− 0.166157i
$918$	0	0
$919$	−302.820	−0.329511	−0.164755	−	0.986334i	$-0.552683\pi$
$919$	−302.820	−0.329511	−0.164755	+	0.986334i	$0.552683\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 231.709i	− 0.251039i
$924$	0	0
$925$	−683.639	−0.739069
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1191.45i	− 1.28250i	−0.767331	−	0.641252i	$-0.778415\pi$
$929$	− 1191.45i	− 1.28250i	0.767331	−	0.641252i	$-0.221585\pi$
$930$	0	0
$931$	187.142	0.201012
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 957.064i	− 1.02360i
$936$	0	0
$937$	438.222	0.467686	0.233843	−	0.972274i	$-0.424870\pi$
$937$	438.222	0.467686	0.233843	+	0.972274i	$0.424870\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	796.939i	0.846907i	0.905918	+	0.423453i	$0.139182\pi$
$941$	796.939i	0.846907i	−0.905918	+	0.423453i	$0.860818\pi$
$942$	0	0
$943$	−1247.15	−1.32254
$944$	0	0
$945$	0	0
$946$	0	0
$947$	744.367i	0.786026i	0.919533	+	0.393013i	$0.128567\pi$
$947$	744.367i	0.786026i	−0.919533	+	0.393013i	$0.871433\pi$
$948$	0	0
$949$	−156.787	−0.165213
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 370.138i	− 0.388392i	−0.980963	−	0.194196i	$-0.937790\pi$
$953$	− 370.138i	− 0.388392i	0.980963	−	0.194196i	$-0.0622098\pi$
$954$	0	0
$955$	494.099	0.517382
$956$	0	0
$957$	0	0
$958$	0	0
$959$	212.532i	0.221618i
$960$	0	0
$961$	−656.780	−0.683434
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1242.04i	1.28709i
$966$	0	0
$967$	509.275	0.526654	0.263327	−	0.964707i	$-0.415180\pi$
$967$	509.275	0.526654	0.263327	+	0.964707i	$0.415180\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 987.400i	− 1.01689i	−0.861095	−	0.508445i	$-0.830221\pi$
$971$	− 987.400i	− 1.01689i	0.861095	−	0.508445i	$-0.169779\pi$
$972$	0	0
$973$	−427.465	−0.439327
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1679.85i	− 1.71939i	−0.510804	−	0.859697i	$-0.670652\pi$
$977$	− 1679.85i	− 1.71939i	0.510804	−	0.859697i	$-0.329348\pi$
$978$	0	0
$979$	1256.42	1.28337
$980$	0	0
$981$	0	0
$982$	0	0
$983$	575.062i	0.585007i	0.956264	+	0.292503i	$0.0944883\pi$
$983$	575.062i	0.585007i	−0.956264	+	0.292503i	$0.905512\pi$
$984$	0	0
$985$	331.431	0.336478
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2015.59i	2.03801i
$990$	0	0
$991$	−544.246	−0.549188	−0.274594	−	0.961560i	$-0.588544\pi$
$991$	−544.246	−0.549188	−0.274594	+	0.961560i	$0.588544\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 495.087i	− 0.497575i
$996$	0	0
$997$	1569.52	1.57424	0.787120	−	0.616800i	$-0.211571\pi$
$997$	1569.52	1.57424	0.787120	+	0.616800i	$0.211571\pi$
$998$	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.305.4	✓	12
3.2	odd	2	inner	684.3.e.a.305.9	yes	12
4.3	odd	2		2736.3.h.c.305.4		12
12.11	even	2		2736.3.h.c.305.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.e.a.305.4	✓	12	1.1	even	1	trivial
684.3.e.a.305.9	yes	12	3.2	odd	2	inner
2736.3.h.c.305.4		12	4.3	odd	2
2736.3.h.c.305.9		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)

\(f(q)\)	\(=\)	\(q-3.29597i q^{5} -2.46307 q^{7} +O(q^{10})\)	\(q-3.29597i q^{5} -2.46307 q^{7} +10.4157i q^{11} +2.93243 q^{13} -27.8783i q^{17} -4.35890 q^{19} -24.3461i q^{23} +14.1366 q^{25} -7.80708i q^{29} -17.4419 q^{31} +8.11820i q^{35} -48.3596 q^{37} -51.2260i q^{41} -82.7891 q^{43} +22.2146i q^{47} -42.9333 q^{49} -16.6275i q^{53} +34.3300 q^{55} -49.5384i q^{59} +16.9475 q^{61} -9.66522i q^{65} -1.05167 q^{67} -79.0160i q^{71} -53.4664 q^{73} -25.6547i q^{77} +137.382 q^{79} +4.56292i q^{83} -91.8862 q^{85} -120.627i q^{89} -7.22278 q^{91} +14.3668i q^{95} +13.4947 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{7}+O(q^{10}) \) 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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