LMFDB - Embedded newform 2736.3.h.c.305.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(305,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2736.305");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.h (of order $2$, degree $1$, not 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:	$74.5506003290$
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:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			305.4
Root			$-3.29597i$ of defining polynomial
Character	$\chi$	$=$	2736.305
Dual form			2736.3.h.c.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 $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$-1$	$1$	$1$

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	$-0.443706\pi$
$7$	2.46307	0.351867	0.175933	+	0.984402i	$0.443706\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 10.4157i	− 0.946886i	−0.880825	−	0.473443i	$-0.843011\pi$
$11$	− 10.4157i	− 0.946886i	0.880825	−	0.473443i	$-0.156989\pi$
$12$	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	$0.177532\pi$
$23$	24.3461i	1.05853i	−0.848458	+	0.529263i	$0.822468\pi$
$24$	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	$-0.409227\pi$
$31$	17.4419	0.562642	0.281321	+	0.959614i	$0.409227\pi$
$32$	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	$-0.0872550\pi$
$43$	82.7891	1.92533	0.962664	+	0.270700i	$0.0872550\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 22.2146i	− 0.472651i	−0.971674	−	0.236325i	$-0.924057\pi$
$47$	− 22.2146i	− 0.472651i	0.971674	−	0.236325i	$-0.0759432\pi$
$48$	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	$0.137906\pi$
$59$	49.5384i	0.839634i	−0.907609	+	0.419817i	$0.862094\pi$
$60$	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	$-0.497502\pi$
$67$	1.05167	0.0156965	0.00784826	+	0.999969i	$0.497502\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	79.0160i	1.11290i	0.830881	+	0.556451i	$0.187837\pi$
$71$	79.0160i	1.11290i	−0.830881	+	0.556451i	$0.812163\pi$
$72$	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	$-0.835564\pi$
$79$	−137.382	−1.73902	−0.869508	+	0.493919i	$0.835564\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.56292i	− 0.0549750i	−0.999622	−	0.0274875i	$-0.991249\pi$
$83$	− 4.56292i	− 0.0549750i	0.999622	−	0.0274875i	$-0.00875064\pi$
$84$	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	$-0.259228\pi$
$103$	141.381	1.37263	0.686313	+	0.727306i	$0.259228\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 21.6171i	− 0.202029i	−0.994885	−	0.101014i	$-0.967791\pi$
$107$	− 21.6171i	− 0.202029i	0.994885	−	0.101014i	$-0.0322088\pi$
$108$	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	$-0.590259\pi$
$127$	−71.0621	−0.559544	−0.279772	+	0.960066i	$0.590259\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 61.8603i	− 0.472216i	−0.971727	−	0.236108i	$-0.924128\pi$
$131$	− 61.8603i	− 0.472216i	0.971727	−	0.236108i	$-0.0758719\pi$
$132$	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	$-0.714607\pi$
$139$	−173.550	−1.24856	−0.624280	+	0.781201i	$0.714607\pi$
$140$	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	$-0.498927\pi$
$151$	1.01794	0.00674136	0.00337068	+	0.999994i	$0.498927\pi$
$152$	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	$-0.536721\pi$
$163$	−37.5244	−0.230211	−0.115106	+	0.993353i	$0.536721\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 146.277i	− 0.875907i	−0.898998	−	0.437954i	$-0.855703\pi$
$167$	− 146.277i	− 0.875907i	0.898998	−	0.437954i	$-0.144297\pi$
$168$	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	$-0.910239\pi$
$179$	− 99.6202i	− 0.556537i	0.960503	−	0.278269i	$-0.0897606\pi$
$180$	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	$-0.871633\pi$
$191$	− 149.910i	− 0.784869i	0.919780	−	0.392435i	$-0.128367\pi$
$192$	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	$-0.623186\pi$
$199$	−150.210	−0.754822	−0.377411	+	0.926046i	$0.623186\pi$
$200$	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	$-0.517477\pi$
$211$	−23.1583	−0.109755	−0.0548774	+	0.998493i	$0.517477\pi$
$212$	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	$-0.942063\pi$
$223$	−438.633	−1.96696	−0.983481	+	0.181010i	$0.942063\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 123.454i	− 0.543850i	−0.962318	−	0.271925i	$-0.912340\pi$
$227$	− 123.454i	− 0.543850i	0.962318	−	0.271925i	$-0.0876603\pi$
$228$	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	$0.0104946\pi$
$239$	15.7567i	0.0659276i	−0.999457	+	0.0329638i	$0.989505\pi$
$240$	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	$-0.954111\pi$
$251$	− 72.1205i	− 0.287333i	0.989626	−	0.143666i	$-0.0458892\pi$
$252$	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	$0.0289249\pi$
$263$	47.7320i	0.181491i	−0.995874	+	0.0907453i	$0.971075\pi$
$264$	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	$-0.560247\pi$
$271$	−101.973	−0.376286	−0.188143	+	0.982142i	$0.560247\pi$
$272$	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	$-0.352052\pi$
$283$	253.702	0.896474	0.448237	+	0.893915i	$0.352052\pi$
$284$	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	$-0.619328\pi$
$307$	−224.823	−0.732321	−0.366161	+	0.930552i	$0.619328\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 495.020i	− 1.59171i	−0.605490	−	0.795853i	$-0.707023\pi$
$311$	− 495.020i	− 1.59171i	0.605490	−	0.795853i	$-0.292977\pi$
$312$	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	$-0.501710\pi$
$331$	−3.55596	−0.0107431	−0.00537154	+	0.999986i	$0.501710\pi$
$332$	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	$-0.928760\pi$
$347$	− 154.029i	− 0.443889i	0.975059	−	0.221944i	$-0.0712403\pi$
$348$	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	$-0.762100\pi$
$359$	− 488.041i	− 1.35944i	0.733470	−	0.679722i	$-0.237900\pi$
$360$	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	$-0.374891\pi$
$367$	281.123	0.766002	0.383001	+	0.923748i	$0.374891\pi$
$368$	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	$-0.682316\pi$
$379$	−410.802	−1.08391	−0.541955	+	0.840408i	$0.682316\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	370.444i	0.967217i	0.875285	+	0.483608i	$0.160674\pi$
$383$	370.444i	0.967217i	−0.875285	+	0.483608i	$0.839326\pi$
$384$	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	$-0.992668\pi$
$419$	− 19.3011i	− 0.0460647i	0.999735	−	0.0230323i	$-0.00733207\pi$
$420$	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	$0.286921\pi$
$431$	675.971i	1.56838i	−0.620522	+	0.784189i	$0.713079\pi$
$432$	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	$-0.543987\pi$
$439$	−120.944	−0.275498	−0.137749	+	0.990467i	$0.543987\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	65.2480i	0.147287i	0.997285	+	0.0736433i	$0.0234626\pi$
$443$	65.2480i	0.147287i	−0.997285	+	0.0736433i	$0.976537\pi$
$444$	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	$-0.416483\pi$
$463$	240.182	0.518752	0.259376	+	0.965776i	$0.416483\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	264.429i	0.566230i	0.959086	+	0.283115i	$0.0913678\pi$
$467$	264.429i	0.566230i	−0.959086	+	0.283115i	$0.908632\pi$
$468$	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	$0.272446\pi$
$479$	723.453i	1.51034i	−0.655529	+	0.755170i	$0.727554\pi$
$480$	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	$-0.231010\pi$
$487$	728.561	1.49602	0.748009	+	0.663688i	$0.231010\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 888.765i	− 1.81011i	−0.425293	−	0.905056i	$-0.639829\pi$
$491$	− 888.765i	− 1.81011i	0.425293	−	0.905056i	$-0.360171\pi$
$492$	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	$-0.233662\pi$
$499$	740.968	1.48491	0.742453	+	0.669899i	$0.233662\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 195.451i	− 0.388571i	−0.980945	−	0.194285i	$-0.937761\pi$
$503$	− 195.451i	− 0.388571i	0.980945	−	0.194285i	$-0.0622388\pi$
$504$	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	$-0.370461\pi$
$523$	414.025	0.791635	0.395817	+	0.918329i	$0.370461\pi$
$524$	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	$-0.660072\pi$
$547$	−527.255	−0.963902	−0.481951	+	0.876198i	$0.660072\pi$
$548$	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	$0.200199\pi$
$563$	662.414i	1.17658i	−0.808650	+	0.588290i	$0.799801\pi$
$564$	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	$-0.345763\pi$
$571$	531.954	0.931618	0.465809	+	0.884885i	$0.345763\pi$
$572$	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	$0.208808\pi$
$587$	716.074i	1.21989i	−0.792445	+	0.609944i	$0.791192\pi$
$588$	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	$0.167181\pi$
$599$	600.674i	1.00279i	−0.865217	+	0.501397i	$0.832819\pi$
$600$	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	$-0.523529\pi$
$607$	−89.6567	−0.147705	−0.0738523	+	0.997269i	$0.523529\pi$
$608$	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	$-0.357227\pi$
$619$	536.855	0.867293	0.433647	+	0.901083i	$0.357227\pi$
$620$	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	$-0.796875\pi$
$631$	−1013.65	−1.60641	−0.803207	+	0.595700i	$0.796875\pi$
$632$	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	$-0.422547\pi$
$643$	309.838	0.481863	0.240931	+	0.970542i	$0.422547\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 829.721i	− 1.28241i	−0.767369	−	0.641206i	$-0.778434\pi$
$647$	− 829.721i	− 1.28241i	0.767369	−	0.641206i	$-0.221566\pi$
$648$	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	$0.313652\pi$
$659$	1098.52i	1.66695i	−0.552558	+	0.833474i	$0.686348\pi$
$660$	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	$0.0115537\pi$
$683$	49.5710i	0.0725783i	−0.999341	+	0.0362891i	$0.988446\pi$
$684$	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	$-0.894109\pi$
$691$	−1306.23	−1.89035	−0.945175	+	0.326564i	$0.894109\pi$
$692$	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	$-0.898946\pi$
$719$	− 448.892i	− 0.624328i	0.950028	−	0.312164i	$-0.101054\pi$
$720$	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	$-0.678346\pi$
$727$	−772.702	−1.06286	−0.531432	+	0.847101i	$0.678346\pi$
$728$	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	$-0.692497\pi$
$739$	−840.323	−1.13711	−0.568554	+	0.822646i	$0.692497\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1361.90i	1.83297i	0.400070	+	0.916485i	$0.368986\pi$
$743$	1361.90i	1.83297i	−0.400070	+	0.916485i	$0.631014\pi$
$744$	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	$-0.308493\pi$
$751$	850.123	1.13199	0.565994	+	0.824409i	$0.308493\pi$
$752$	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	$-0.377914\pi$
$787$	589.006	0.748419	0.374210	+	0.927344i	$0.377914\pi$
$788$	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	$-0.240553\pi$
$811$	1180.46	1.45555	0.727777	+	0.685813i	$0.240553\pi$
$812$	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	$-0.452700\pi$
$823$	243.690	0.296099	0.148050	+	0.988980i	$0.452700\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	586.109i	0.708717i	0.935110	+	0.354359i	$0.115301\pi$
$827$	586.109i	0.708717i	−0.935110	+	0.354359i	$0.884699\pi$
$828$	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	$-0.577593\pi$
$839$	− 1628.39i	− 1.94087i	0.241357	−	0.970436i	$-0.422407\pi$
$840$	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	$-0.168518\pi$
$859$	1482.81	1.72621	0.863103	+	0.505028i	$0.168518\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1083.41i	1.25540i	0.778457	+	0.627698i	$0.216003\pi$
$863$	1083.41i	1.25540i	−0.778457	+	0.627698i	$0.783997\pi$
$864$	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	$-0.515621\pi$
$883$	−86.6337	−0.0981129	−0.0490565	+	0.998796i	$0.515621\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 449.343i	− 0.506587i	−0.967389	−	0.253293i	$-0.918486\pi$
$887$	− 449.343i	− 0.506587i	0.967389	−	0.253293i	$-0.0815138\pi$
$888$	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	$-0.252859\pi$
$907$	1271.12	1.40145	0.700727	+	0.713429i	$0.252859\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 428.669i	− 0.470548i	−0.971929	−	0.235274i	$-0.924401\pi$
$911$	− 428.669i	− 0.470548i	0.971929	−	0.235274i	$-0.0755987\pi$
$912$	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	$-0.447317\pi$
$919$	302.820	0.329511	0.164755	+	0.986334i	$0.447317\pi$
$920$	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	$-0.871433\pi$
$947$	− 744.367i	− 0.786026i	0.919533	−	0.393013i	$-0.128567\pi$
$948$	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	$-0.584820\pi$
$967$	−509.275	−0.526654	−0.263327	+	0.964707i	$0.584820\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	987.400i	1.01689i	0.861095	+	0.508445i	$0.169779\pi$
$971$	987.400i	1.01689i	−0.861095	+	0.508445i	$0.830221\pi$
$972$	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	$-0.905512\pi$
$983$	− 575.062i	− 0.585007i	0.956264	−	0.292503i	$-0.0944883\pi$
$984$	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	$-0.411456\pi$
$991$	544.246	0.549188	0.274594	+	0.961560i	$0.411456\pi$
$992$	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	2736.3.h.c.305.4		12
3.2	odd	2	inner	2736.3.h.c.305.9		12
4.3	odd	2		684.3.e.a.305.4	✓	12
12.11	even	2		684.3.e.a.305.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.e.a.305.4	✓	12	4.3	odd	2
684.3.e.a.305.9	yes	12	12.11	even	2
2736.3.h.c.305.4		12	1.1	even	1	trivial
2736.3.h.c.305.9		12	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.h (of order \(2\), degree \(1\), not 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more