LMFDB - Embedded newform 2704.2.f.m.337.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(337,2704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2704.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,0,0,0,0,16,0,0,0,0,0,0,0,-10,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 338)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Root			$-0.445042i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.m.337.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04892 q^{3} +3.60388i q^{5} -1.10992i q^{7} +1.19806 q^{9} +2.35690i q^{11} +7.38404i q^{15} -5.96077 q^{17} +0.911854i q^{19} -2.27413i q^{21} +3.38404 q^{23} -7.98792 q^{25} -3.69202 q^{27} -3.78017 q^{29} +8.49396i q^{31} +4.82908i q^{33} +4.00000 q^{35} +4.89008i q^{37} +7.18598i q^{41} -0.515729 q^{43} +4.31767i q^{45} -6.98792i q^{47} +5.76809 q^{49} -12.2131 q^{51} -3.38404 q^{53} -8.49396 q^{55} +1.86831i q^{57} -10.1468i q^{59} -0.439665 q^{61} -1.32975i q^{63} +2.14675i q^{67} +6.93362 q^{69} +0.615957i q^{71} +6.32304i q^{73} -16.3666 q^{75} +2.61596 q^{77} +15.4819 q^{79} -11.1588 q^{81} -0.911854i q^{83} -21.4819i q^{85} -7.74525 q^{87} +3.75063i q^{89} +17.4034i q^{93} -3.28621 q^{95} +14.6746i q^{97} +2.82371i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 16 q^{9} - 10 q^{17} - 10 q^{25} - 12 q^{27} - 20 q^{29} + 24 q^{35} + 22 q^{43} - 6 q^{49} - 32 q^{51} - 32 q^{55} - 8 q^{61} + 28 q^{69} - 46 q^{75} + 36 q^{77} + 36 q^{79} - 50 q^{81}+ \cdots - 36 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 16 * q^9 - 10 * q^17 - 10 * q^25 - 12 * q^27 - 20 * q^29 + 24 * q^35 + 22 * q^43 - 6 * q^49 - 32 * q^51 - 32 * q^55 - 8 * q^61 + 28 * q^69 - 46 * q^75 + 36 * q^77 + 36 * q^79 - 50 * q^81 + 20 * q^87 - 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	2.04892	1.18294	0.591471	−	0.806326i	$-0.298547\pi$
$3$	2.04892	1.18294	0.591471	+	0.806326i	$0.298547\pi$
$4$	0	0
$5$	3.60388i	1.61170i	0.592118	+	0.805851i	$0.298292\pi$
$5$	3.60388i	1.61170i	−0.592118	+	0.805851i	$0.701708\pi$
$6$	0	0
$7$	− 1.10992i	− 0.419509i	−0.977754	−	0.209754i	$-0.932734\pi$
$7$	− 1.10992i	− 0.419509i	0.977754	−	0.209754i	$-0.0672665\pi$
$8$	0	0
$9$	1.19806	0.399354
$10$	0	0
$11$	2.35690i	0.710631i	0.934746	+	0.355315i	$0.115627\pi$
$11$	2.35690i	0.710631i	−0.934746	+	0.355315i	$0.884373\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	7.38404i	1.90655i
$16$	0	0
$17$	−5.96077	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$17$	−5.96077	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$18$	0	0
$19$	0.911854i	0.209194i	0.994515	+	0.104597i	$0.0333552\pi$
$19$	0.911854i	0.209194i	−0.994515	+	0.104597i	$0.966645\pi$
$20$	0	0
$21$	− 2.27413i	− 0.496255i
$22$	0	0
$23$	3.38404	0.705622	0.352811	−	0.935695i	$-0.385226\pi$
$23$	3.38404	0.705622	0.352811	+	0.935695i	$0.385226\pi$
$24$	0	0
$25$	−7.98792	−1.59758
$26$	0	0
$27$	−3.69202	−0.710530
$28$	0	0
$29$	−3.78017	−0.701959	−0.350980	−	0.936383i	$-0.614151\pi$
$29$	−3.78017	−0.701959	−0.350980	+	0.936383i	$0.614151\pi$
$30$	0	0
$31$	8.49396i	1.52556i	0.646658	+	0.762780i	$0.276166\pi$
$31$	8.49396i	1.52556i	−0.646658	+	0.762780i	$0.723834\pi$
$32$	0	0
$33$	4.82908i	0.840636i
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	4.89008i	0.803925i	0.915656	+	0.401962i	$0.131672\pi$
$37$	4.89008i	0.803925i	−0.915656	+	0.401962i	$0.868328\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.18598i	1.12226i	0.827727	+	0.561131i	$0.189634\pi$
$41$	7.18598i	1.12226i	−0.827727	+	0.561131i	$0.810366\pi$
$42$	0	0
$43$	−0.515729	−0.0786480	−0.0393240	−	0.999227i	$-0.512520\pi$
$43$	−0.515729	−0.0786480	−0.0393240	+	0.999227i	$0.512520\pi$
$44$	0	0
$45$	4.31767i	0.643640i
$46$	0	0
$47$	− 6.98792i	− 1.01929i	−0.860384	−	0.509646i	$-0.829776\pi$
$47$	− 6.98792i	− 1.01929i	0.860384	−	0.509646i	$-0.170224\pi$
$48$	0	0
$49$	5.76809	0.824012
$50$	0	0
$51$	−12.2131	−1.71018
$52$	0	0
$53$	−3.38404	−0.464834	−0.232417	−	0.972616i	$-0.574663\pi$
$53$	−3.38404	−0.464834	−0.232417	+	0.972616i	$0.574663\pi$
$54$	0	0
$55$	−8.49396	−1.14533
$56$	0	0
$57$	1.86831i	0.247464i
$58$	0	0
$59$	− 10.1468i	− 1.32099i	−0.750828	−	0.660497i	$-0.770345\pi$
$59$	− 10.1468i	− 1.32099i	0.750828	−	0.660497i	$-0.229655\pi$
$60$	0	0
$61$	−0.439665	−0.0562933	−0.0281467	−	0.999604i	$-0.508961\pi$
$61$	−0.439665	−0.0562933	−0.0281467	+	0.999604i	$0.508961\pi$
$62$	0	0
$63$	− 1.32975i	− 0.167533i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.14675i	0.262268i	0.991365	+	0.131134i	$0.0418617\pi$
$67$	2.14675i	0.262268i	−0.991365	+	0.131134i	$0.958138\pi$
$68$	0	0
$69$	6.93362	0.834710
$70$	0	0
$71$	0.615957i	0.0731007i	0.999332	+	0.0365503i	$0.0116369\pi$
$71$	0.615957i	0.0731007i	−0.999332	+	0.0365503i	$0.988363\pi$
$72$	0	0
$73$	6.32304i	0.740056i	0.929021	+	0.370028i	$0.120652\pi$
$73$	6.32304i	0.740056i	−0.929021	+	0.370028i	$0.879348\pi$
$74$	0	0
$75$	−16.3666	−1.88985
$76$	0	0
$77$	2.61596	0.298116
$78$	0	0
$79$	15.4819	1.74185	0.870924	−	0.491418i	$-0.163521\pi$
$79$	15.4819	1.74185	0.870924	+	0.491418i	$0.163521\pi$
$80$	0	0
$81$	−11.1588	−1.23987
$82$	0	0
$83$	− 0.911854i	− 0.100089i	−0.998747	−	0.0500445i	$-0.984064\pi$
$83$	− 0.911854i	− 0.100089i	0.998747	−	0.0500445i	$-0.0159363\pi$
$84$	0	0
$85$	− 21.4819i	− 2.33004i
$86$	0	0
$87$	−7.74525	−0.830378
$88$	0	0
$89$	3.75063i	0.397566i	0.980044	+	0.198783i	$0.0636988\pi$
$89$	3.75063i	0.397566i	−0.980044	+	0.198783i	$0.936301\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	17.4034i	1.80465i
$94$	0	0
$95$	−3.28621	−0.337158
$96$	0	0
$97$	14.6746i	1.48998i	0.667078	+	0.744988i	$0.267545\pi$
$97$	14.6746i	1.48998i	−0.667078	+	0.744988i	$0.732455\pi$
$98$	0	0
$99$	2.82371i	0.283793i
$100$	0	0
$101$	−8.76809	−0.872457	−0.436229	−	0.899836i	$-0.643686\pi$
$101$	−8.76809	−0.872457	−0.436229	+	0.899836i	$0.643686\pi$
$102$	0	0
$103$	18.8116	1.85356	0.926782	−	0.375599i	$-0.122563\pi$
$103$	18.8116	1.85356	0.926782	+	0.375599i	$0.122563\pi$
$104$	0	0
$105$	8.19567	0.799815
$106$	0	0
$107$	−18.0519	−1.74514	−0.872572	−	0.488486i	$-0.837549\pi$
$107$	−18.0519	−1.74514	−0.872572	+	0.488486i	$0.837549\pi$
$108$	0	0
$109$	− 6.09783i	− 0.584067i	−0.956408	−	0.292033i	$-0.905668\pi$
$109$	− 6.09783i	− 0.584067i	0.956408	−	0.292033i	$-0.0943318\pi$
$110$	0	0
$111$	10.0194i	0.950997i
$112$	0	0
$113$	−12.2010	−1.14778	−0.573889	−	0.818933i	$-0.694566\pi$
$113$	−12.2010	−1.14778	−0.573889	+	0.818933i	$0.694566\pi$
$114$	0	0
$115$	12.1957i	1.13725i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.61596i	0.606484i
$120$	0	0
$121$	5.44504	0.495004
$122$	0	0
$123$	14.7235i	1.32757i
$124$	0	0
$125$	− 10.7681i	− 0.963127i
$126$	0	0
$127$	11.4276	1.01403	0.507017	−	0.861936i	$-0.330748\pi$
$127$	11.4276	1.01403	0.507017	+	0.861936i	$0.330748\pi$
$128$	0	0
$129$	−1.05669	−0.0930361
$130$	0	0
$131$	−2.29590	−0.200593	−0.100297	−	0.994958i	$-0.531979\pi$
$131$	−2.29590	−0.200593	−0.100297	+	0.994958i	$0.531979\pi$
$132$	0	0
$133$	1.01208	0.0877586
$134$	0	0
$135$	− 13.3056i	− 1.14516i
$136$	0	0
$137$	9.08038i	0.775789i	0.921704	+	0.387894i	$0.126797\pi$
$137$	9.08038i	0.775789i	−0.921704	+	0.387894i	$0.873203\pi$
$138$	0	0
$139$	−18.9051	−1.60351	−0.801757	−	0.597650i	$-0.796101\pi$
$139$	−18.9051	−1.60351	−0.801757	+	0.597650i	$0.796101\pi$
$140$	0	0
$141$	− 14.3177i	− 1.20577i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 13.6233i	− 1.13135i
$146$	0	0
$147$	11.8183	0.974760
$148$	0	0
$149$	− 18.6896i	− 1.53111i	−0.643368	−	0.765557i	$-0.722463\pi$
$149$	− 18.6896i	− 1.53111i	0.643368	−	0.765557i	$-0.277537\pi$
$150$	0	0
$151$	0.317667i	0.0258514i	0.999916	+	0.0129257i	$0.00411449\pi$
$151$	0.317667i	0.0258514i	−0.999916	+	0.0129257i	$0.995886\pi$
$152$	0	0
$153$	−7.14138	−0.577346
$154$	0	0
$155$	−30.6112	−2.45875
$156$	0	0
$157$	18.8901	1.50759	0.753796	−	0.657108i	$-0.228220\pi$
$157$	18.8901	1.50759	0.753796	+	0.657108i	$0.228220\pi$
$158$	0	0
$159$	−6.93362	−0.549872
$160$	0	0
$161$	− 3.75600i	− 0.296015i
$162$	0	0
$163$	4.33273i	0.339366i	0.985499	+	0.169683i	$0.0542744\pi$
$163$	4.33273i	0.339366i	−0.985499	+	0.169683i	$0.945726\pi$
$164$	0	0
$165$	−17.4034	−1.35485
$166$	0	0
$167$	− 14.0000i	− 1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$	− 14.0000i	− 1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.09246i	0.0835423i
$172$	0	0
$173$	10.9879	0.835396	0.417698	−	0.908586i	$-0.362837\pi$
$173$	10.9879	0.835396	0.417698	+	0.908586i	$0.362837\pi$
$174$	0	0
$175$	8.86592i	0.670201i
$176$	0	0
$177$	− 20.7899i	− 1.56266i
$178$	0	0
$179$	4.65519	0.347945	0.173972	−	0.984751i	$-0.444340\pi$
$179$	4.65519	0.347945	0.173972	+	0.984751i	$0.444340\pi$
$180$	0	0
$181$	1.06638	0.0792631	0.0396315	−	0.999214i	$-0.487382\pi$
$181$	1.06638	0.0792631	0.0396315	+	0.999214i	$0.487382\pi$
$182$	0	0
$183$	−0.900837	−0.0665918
$184$	0	0
$185$	−17.6233	−1.29569
$186$	0	0
$187$	− 14.0489i	− 1.02736i
$188$	0	0
$189$	4.09783i	0.298074i
$190$	0	0
$191$	−0.890084	−0.0644042	−0.0322021	−	0.999481i	$-0.510252\pi$
$191$	−0.890084	−0.0644042	−0.0322021	+	0.999481i	$0.510252\pi$
$192$	0	0
$193$	16.2174i	1.16736i	0.811985	+	0.583678i	$0.198387\pi$
$193$	16.2174i	1.16736i	−0.811985	+	0.583678i	$0.801613\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.4711i	0.817284i	0.912695	+	0.408642i	$0.133997\pi$
$197$	11.4711i	0.817284i	−0.912695	+	0.408642i	$0.866003\pi$
$198$	0	0
$199$	−3.79954	−0.269343	−0.134671	−	0.990890i	$-0.542998\pi$
$199$	−3.79954	−0.269343	−0.134671	+	0.990890i	$0.542998\pi$
$200$	0	0
$201$	4.39852i	0.310247i
$202$	0	0
$203$	4.19567i	0.294478i
$204$	0	0
$205$	−25.8974	−1.80875
$206$	0	0
$207$	4.05429	0.281793
$208$	0	0
$209$	−2.14914	−0.148659
$210$	0	0
$211$	25.0465	1.72427	0.862137	−	0.506675i	$-0.169126\pi$
$211$	25.0465	1.72427	0.862137	+	0.506675i	$0.169126\pi$
$212$	0	0
$213$	1.26205i	0.0864739i
$214$	0	0
$215$	− 1.85862i	− 0.126757i
$216$	0	0
$217$	9.42758	0.639986
$218$	0	0
$219$	12.9554i	0.875444i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	12.9879i	0.869735i	0.900494	+	0.434868i	$0.143205\pi$
$223$	12.9879i	0.869735i	−0.900494	+	0.434868i	$0.856795\pi$
$224$	0	0
$225$	−9.57002	−0.638002
$226$	0	0
$227$	− 13.8049i	− 0.916265i	−0.888884	−	0.458132i	$-0.848519\pi$
$227$	− 13.8049i	− 0.916265i	0.888884	−	0.458132i	$-0.151481\pi$
$228$	0	0
$229$	− 11.5603i	− 0.763928i	−0.924177	−	0.381964i	$-0.875248\pi$
$229$	− 11.5603i	− 0.763928i	0.924177	−	0.381964i	$-0.124752\pi$
$230$	0	0
$231$	5.35988	0.352654
$232$	0	0
$233$	−9.77479	−0.640368	−0.320184	−	0.947355i	$-0.603745\pi$
$233$	−9.77479	−0.640368	−0.320184	+	0.947355i	$0.603745\pi$
$234$	0	0
$235$	25.1836	1.64280
$236$	0	0
$237$	31.7211	2.06051
$238$	0	0
$239$	− 0.944378i	− 0.0610867i	−0.999533	−	0.0305434i	$-0.990276\pi$
$239$	− 0.944378i	− 0.0610867i	0.999533	−	0.0305434i	$-0.00972377\pi$
$240$	0	0
$241$	− 0.219833i	− 0.0141607i	−0.999975	−	0.00708033i	$-0.997746\pi$
$241$	− 0.219833i	− 0.0141607i	0.999975	−	0.00708033i	$-0.00225376\pi$
$242$	0	0
$243$	−11.7875	−0.756166
$244$	0	0
$245$	20.7875i	1.32806i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 1.86831i	− 0.118400i
$250$	0	0
$251$	16.2543	1.02596	0.512980	−	0.858400i	$-0.328541\pi$
$251$	16.2543	1.02596	0.512980	+	0.858400i	$0.328541\pi$
$252$	0	0
$253$	7.97584i	0.501437i
$254$	0	0
$255$	− 44.0146i	− 2.75630i
$256$	0	0
$257$	22.4373	1.39960	0.699799	−	0.714340i	$-0.253273\pi$
$257$	22.4373	1.39960	0.699799	+	0.714340i	$0.253273\pi$
$258$	0	0
$259$	5.42758	0.337254
$260$	0	0
$261$	−4.52888	−0.280330
$262$	0	0
$263$	10.4940	0.647085	0.323543	−	0.946214i	$-0.395126\pi$
$263$	10.4940	0.647085	0.323543	+	0.946214i	$0.395126\pi$
$264$	0	0
$265$	− 12.1957i	− 0.749174i
$266$	0	0
$267$	7.68473i	0.470298i
$268$	0	0
$269$	26.4155	1.61058	0.805291	−	0.592880i	$-0.202009\pi$
$269$	26.4155	1.61058	0.805291	+	0.592880i	$0.202009\pi$
$270$	0	0
$271$	22.0301i	1.33824i	0.743157	+	0.669118i	$0.233328\pi$
$271$	22.0301i	1.33824i	−0.743157	+	0.669118i	$0.766672\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 18.8267i	− 1.13529i
$276$	0	0
$277$	−2.17629	−0.130761	−0.0653804	−	0.997860i	$-0.520826\pi$
$277$	−2.17629	−0.130761	−0.0653804	+	0.997860i	$0.520826\pi$
$278$	0	0
$279$	10.1763i	0.609239i
$280$	0	0
$281$	25.0030i	1.49155i	0.666196	+	0.745776i	$0.267921\pi$
$281$	25.0030i	1.49155i	−0.666196	+	0.745776i	$0.732079\pi$
$282$	0	0
$283$	−16.3153	−0.969842	−0.484921	−	0.874558i	$-0.661152\pi$
$283$	−16.3153	−0.969842	−0.484921	+	0.874558i	$0.661152\pi$
$284$	0	0
$285$	−6.73317	−0.398839
$286$	0	0
$287$	7.97584	0.470799
$288$	0	0
$289$	18.5308	1.09005
$290$	0	0
$291$	30.0670i	1.76256i
$292$	0	0
$293$	− 1.87800i	− 0.109714i	−0.998494	−	0.0548570i	$-0.982530\pi$
$293$	− 1.87800i	− 0.109714i	0.998494	−	0.0548570i	$-0.0174703\pi$
$294$	0	0
$295$	36.5676	2.12905
$296$	0	0
$297$	− 8.70171i	− 0.504924i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0.572417i	0.0329935i
$302$	0	0
$303$	−17.9651	−1.03207
$304$	0	0
$305$	− 1.58450i	− 0.0907281i
$306$	0	0
$307$	23.9801i	1.36862i	0.729192	+	0.684310i	$0.239896\pi$
$307$	23.9801i	1.36862i	−0.729192	+	0.684310i	$0.760104\pi$
$308$	0	0
$309$	38.5435	2.19266
$310$	0	0
$311$	−5.38404	−0.305301	−0.152651	−	0.988280i	$-0.548781\pi$
$311$	−5.38404	−0.305301	−0.152651	+	0.988280i	$0.548781\pi$
$312$	0	0
$313$	18.9487	1.07104	0.535522	−	0.844522i	$-0.320115\pi$
$313$	18.9487	1.07104	0.535522	+	0.844522i	$0.320115\pi$
$314$	0	0
$315$	4.79225	0.270013
$316$	0	0
$317$	11.5013i	0.645975i	0.946403	+	0.322987i	$0.104687\pi$
$317$	11.5013i	0.645975i	−0.946403	+	0.322987i	$0.895313\pi$
$318$	0	0
$319$	− 8.90946i	− 0.498834i
$320$	0	0
$321$	−36.9869	−2.06440
$322$	0	0
$323$	− 5.43535i	− 0.302431i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 12.4940i	− 0.690918i
$328$	0	0
$329$	−7.75600	−0.427602
$330$	0	0
$331$	− 34.6112i	− 1.90240i	−0.308572	−	0.951201i	$-0.599851\pi$
$331$	− 34.6112i	− 1.90240i	0.308572	−	0.951201i	$-0.400149\pi$
$332$	0	0
$333$	5.85862i	0.321051i
$334$	0	0
$335$	−7.73663	−0.422697
$336$	0	0
$337$	−1.95407	−0.106445	−0.0532224	−	0.998583i	$-0.516949\pi$
$337$	−1.95407	−0.106445	−0.0532224	+	0.998583i	$0.516949\pi$
$338$	0	0
$339$	−24.9989	−1.35776
$340$	0	0
$341$	−20.0194	−1.08411
$342$	0	0
$343$	− 14.1715i	− 0.765189i
$344$	0	0
$345$	24.9879i	1.34530i
$346$	0	0
$347$	6.41550	0.344402	0.172201	−	0.985062i	$-0.444912\pi$
$347$	6.41550	0.344402	0.172201	+	0.985062i	$0.444912\pi$
$348$	0	0
$349$	− 1.08575i	− 0.0581190i	−0.999578	−	0.0290595i	$-0.990749\pi$
$349$	− 1.08575i	− 0.0581190i	0.999578	−	0.0290595i	$-0.00925123\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4.28919i	0.228291i	0.993464	+	0.114145i	$0.0364130\pi$
$353$	4.28919i	0.228291i	−0.993464	+	0.114145i	$0.963587\pi$
$354$	0	0
$355$	−2.21983	−0.117816
$356$	0	0
$357$	13.5555i	0.717436i
$358$	0	0
$359$	− 15.5060i	− 0.818378i	−0.912450	−	0.409189i	$-0.865812\pi$
$359$	− 15.5060i	− 0.818378i	0.912450	−	0.409189i	$-0.134188\pi$
$360$	0	0
$361$	18.1685	0.956238
$362$	0	0
$363$	11.1564	0.585561
$364$	0	0
$365$	−22.7875	−1.19275
$366$	0	0
$367$	−17.4276	−0.909712	−0.454856	−	0.890565i	$-0.650309\pi$
$367$	−17.4276	−0.909712	−0.454856	+	0.890565i	$0.650309\pi$
$368$	0	0
$369$	8.60925i	0.448180i
$370$	0	0
$371$	3.75600i	0.195002i
$372$	0	0
$373$	8.19567	0.424356	0.212178	−	0.977231i	$-0.431944\pi$
$373$	8.19567	0.424356	0.212178	+	0.977231i	$0.431944\pi$
$374$	0	0
$375$	− 22.0629i	− 1.13932i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	15.0476i	0.772943i	0.922301	+	0.386471i	$0.126306\pi$
$379$	15.0476i	0.772943i	−0.922301	+	0.386471i	$0.873694\pi$
$380$	0	0
$381$	23.4142	1.19954
$382$	0	0
$383$	11.1207i	0.568240i	0.958789	+	0.284120i	$0.0917014\pi$
$383$	11.1207i	0.568240i	−0.958789	+	0.284120i	$0.908299\pi$
$384$	0	0
$385$	9.42758i	0.480474i
$386$	0	0
$387$	−0.617876	−0.0314084
$388$	0	0
$389$	8.04354	0.407824	0.203912	−	0.978989i	$-0.434634\pi$
$389$	8.04354	0.407824	0.203912	+	0.978989i	$0.434634\pi$
$390$	0	0
$391$	−20.1715	−1.02012
$392$	0	0
$393$	−4.70410	−0.237291
$394$	0	0
$395$	55.7948i	2.80734i
$396$	0	0
$397$	21.9081i	1.09954i	0.835317	+	0.549769i	$0.185284\pi$
$397$	21.9081i	1.09954i	−0.835317	+	0.549769i	$0.814716\pi$
$398$	0	0
$399$	2.07367	0.103813
$400$	0	0
$401$	− 17.4426i	− 0.871044i	−0.900178	−	0.435522i	$-0.856564\pi$
$401$	− 17.4426i	− 0.871044i	0.900178	−	0.435522i	$-0.143436\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 40.2150i	− 1.99830i
$406$	0	0
$407$	−11.5254	−0.571294
$408$	0	0
$409$	− 17.4330i	− 0.862004i	−0.902351	−	0.431002i	$-0.858160\pi$
$409$	− 17.4330i	− 0.862004i	0.902351	−	0.431002i	$-0.141840\pi$
$410$	0	0
$411$	18.6049i	0.917714i
$412$	0	0
$413$	−11.2620	−0.554169
$414$	0	0
$415$	3.28621	0.161314
$416$	0	0
$417$	−38.7351	−1.89687
$418$	0	0
$419$	−9.97584	−0.487352	−0.243676	−	0.969857i	$-0.578353\pi$
$419$	−9.97584	−0.487352	−0.243676	+	0.969857i	$0.578353\pi$
$420$	0	0
$421$	0.615957i	0.0300199i	0.999887	+	0.0150100i	$0.00477800\pi$
$421$	0.615957i	0.0300199i	−0.999887	+	0.0150100i	$0.995222\pi$
$422$	0	0
$423$	− 8.37196i	− 0.407059i
$424$	0	0
$425$	47.6142	2.30963
$426$	0	0
$427$	0.487991i	0.0236156i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.7922i	0.712518i	0.934387	+	0.356259i	$0.115948\pi$
$431$	14.7922i	0.712518i	−0.934387	+	0.356259i	$0.884052\pi$
$432$	0	0
$433$	−16.5321	−0.794483	−0.397242	−	0.917714i	$-0.630032\pi$
$433$	−16.5321	−0.794483	−0.397242	+	0.917714i	$0.630032\pi$
$434$	0	0
$435$	− 27.9129i	− 1.33832i
$436$	0	0
$437$	3.08575i	0.147612i
$438$	0	0
$439$	3.50125	0.167106	0.0835529	−	0.996503i	$-0.473373\pi$
$439$	3.50125	0.167106	0.0835529	+	0.996503i	$0.473373\pi$
$440$	0	0
$441$	6.91053	0.329073
$442$	0	0
$443$	−17.4077	−0.827066	−0.413533	−	0.910489i	$-0.635705\pi$
$443$	−17.4077	−0.827066	−0.413533	+	0.910489i	$0.635705\pi$
$444$	0	0
$445$	−13.5168	−0.640758
$446$	0	0
$447$	− 38.2935i	− 1.81122i
$448$	0	0
$449$	− 34.1497i	− 1.61163i	−0.592170	−	0.805813i	$-0.701729\pi$
$449$	− 34.1497i	− 1.61163i	0.592170	−	0.805813i	$-0.298271\pi$
$450$	0	0
$451$	−16.9366	−0.797514
$452$	0	0
$453$	0.650874i	0.0305807i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.40342i	0.439873i	0.975514	+	0.219937i	$0.0705851\pi$
$457$	9.40342i	0.439873i	−0.975514	+	0.219937i	$0.929415\pi$
$458$	0	0
$459$	22.0073	1.02721
$460$	0	0
$461$	− 0.733169i	− 0.0341471i	−0.999854	−	0.0170735i	$-0.994565\pi$
$461$	− 0.733169i	− 0.0341471i	0.999854	−	0.0170735i	$-0.00543494\pi$
$462$	0	0
$463$	7.24267i	0.336595i	0.985736	+	0.168298i	$0.0538270\pi$
$463$	7.24267i	0.336595i	−0.985736	+	0.168298i	$0.946173\pi$
$464$	0	0
$465$	−62.7198	−2.90856
$466$	0	0
$467$	30.2446	1.39955	0.699776	−	0.714362i	$-0.253283\pi$
$467$	30.2446	1.39955	0.699776	+	0.714362i	$0.253283\pi$
$468$	0	0
$469$	2.38271	0.110024
$470$	0	0
$471$	38.7042	1.78340
$472$	0	0
$473$	− 1.21552i	− 0.0558897i
$474$	0	0
$475$	− 7.28382i	− 0.334204i
$476$	0	0
$477$	−4.05429	−0.185633
$478$	0	0
$479$	− 36.7198i	− 1.67777i	−0.544310	−	0.838884i	$-0.683208\pi$
$479$	− 36.7198i	− 1.67777i	0.544310	−	0.838884i	$-0.316792\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 7.69574i	− 0.350168i
$484$	0	0
$485$	−52.8853	−2.40140
$486$	0	0
$487$	28.6547i	1.29847i	0.760588	+	0.649234i	$0.224911\pi$
$487$	28.6547i	1.29847i	−0.760588	+	0.649234i	$0.775089\pi$
$488$	0	0
$489$	8.87741i	0.401450i
$490$	0	0
$491$	30.4295	1.37326	0.686632	−	0.727005i	$-0.259088\pi$
$491$	30.4295	1.37326	0.686632	+	0.727005i	$0.259088\pi$
$492$	0	0
$493$	22.5327	1.01482
$494$	0	0
$495$	−10.1763	−0.457390
$496$	0	0
$497$	0.683661	0.0306664
$498$	0	0
$499$	− 15.9715i	− 0.714984i	−0.933916	−	0.357492i	$-0.883632\pi$
$499$	− 15.9715i	− 0.714984i	0.933916	−	0.357492i	$-0.116368\pi$
$500$	0	0
$501$	− 28.6848i	− 1.28154i
$502$	0	0
$503$	41.9711	1.87140	0.935698	−	0.352801i	$-0.114771\pi$
$503$	41.9711	1.87140	0.935698	+	0.352801i	$0.114771\pi$
$504$	0	0
$505$	− 31.5991i	− 1.40614i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.914247i	0.0405233i	0.999795	+	0.0202616i	$0.00644992\pi$
$509$	0.914247i	0.0405233i	−0.999795	+	0.0202616i	$0.993550\pi$
$510$	0	0
$511$	7.01805	0.310460
$512$	0	0
$513$	− 3.36658i	− 0.148638i
$514$	0	0
$515$	67.7948i	2.98739i
$516$	0	0
$517$	16.4698	0.724341
$518$	0	0
$519$	22.5133	0.988226
$520$	0	0
$521$	−3.31096	−0.145056	−0.0725279	−	0.997366i	$-0.523107\pi$
$521$	−3.31096	−0.145056	−0.0725279	+	0.997366i	$0.523107\pi$
$522$	0	0
$523$	0.850855	0.0372053	0.0186026	−	0.999827i	$-0.494078\pi$
$523$	0.850855	0.0372053	0.0186026	+	0.999827i	$0.494078\pi$
$524$	0	0
$525$	18.1655i	0.792809i
$526$	0	0
$527$	− 50.6305i	− 2.20550i
$528$	0	0
$529$	−11.5483	−0.502098
$530$	0	0
$531$	− 12.1564i	− 0.527545i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 65.0568i	− 2.81265i
$536$	0	0
$537$	9.53809	0.411599
$538$	0	0
$539$	13.5948i	0.585569i
$540$	0	0
$541$	40.8853i	1.75780i	0.477010	+	0.878898i	$0.341721\pi$
$541$	40.8853i	1.75780i	−0.477010	+	0.878898i	$0.658279\pi$
$542$	0	0
$543$	2.18492	0.0937637
$544$	0	0
$545$	21.9758	0.941341
$546$	0	0
$547$	2.39075	0.102221	0.0511105	−	0.998693i	$-0.483724\pi$
$547$	2.39075	0.102221	0.0511105	+	0.998693i	$0.483724\pi$
$548$	0	0
$549$	−0.526746	−0.0224810
$550$	0	0
$551$	− 3.44696i	− 0.146845i
$552$	0	0
$553$	− 17.1836i	− 0.730720i
$554$	0	0
$555$	−36.1086	−1.53272
$556$	0	0
$557$	− 27.1508i	− 1.15042i	−0.818007	−	0.575208i	$-0.804921\pi$
$557$	− 27.1508i	− 1.15042i	0.818007	−	0.575208i	$-0.195079\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 28.7851i	− 1.21531i
$562$	0	0
$563$	−6.52409	−0.274958	−0.137479	−	0.990505i	$-0.543900\pi$
$563$	−6.52409	−0.274958	−0.137479	+	0.990505i	$0.543900\pi$
$564$	0	0
$565$	− 43.9711i	− 1.84988i
$566$	0	0
$567$	12.3854i	0.520137i
$568$	0	0
$569$	−7.30021	−0.306041	−0.153020	−	0.988223i	$-0.548900\pi$
$569$	−7.30021	−0.306041	−0.153020	+	0.988223i	$0.548900\pi$
$570$	0	0
$571$	−43.6722	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$571$	−43.6722	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$572$	0	0
$573$	−1.82371	−0.0761865
$574$	0	0
$575$	−27.0315	−1.12729
$576$	0	0
$577$	16.8528i	0.701590i	0.936452	+	0.350795i	$0.114089\pi$
$577$	16.8528i	0.701590i	−0.936452	+	0.350795i	$0.885911\pi$
$578$	0	0
$579$	33.2282i	1.38092i
$580$	0	0
$581$	−1.01208	−0.0419882
$582$	0	0
$583$	− 7.97584i	− 0.330325i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.1825i	0.915571i	0.889063	+	0.457785i	$0.151357\pi$
$587$	22.1825i	0.915571i	−0.889063	+	0.457785i	$0.848643\pi$
$588$	0	0
$589$	−7.74525	−0.319137
$590$	0	0
$591$	23.5034i	0.966800i
$592$	0	0
$593$	− 3.98493i	− 0.163642i	−0.996647	−	0.0818208i	$-0.973926\pi$
$593$	− 3.98493i	− 0.163642i	0.996647	−	0.0818208i	$-0.0260735\pi$
$594$	0	0
$595$	−23.8431	−0.977471
$596$	0	0
$597$	−7.78495	−0.318617
$598$	0	0
$599$	−33.2379	−1.35806	−0.679032	−	0.734109i	$-0.737600\pi$
$599$	−33.2379	−1.35806	−0.679032	+	0.734109i	$0.737600\pi$
$600$	0	0
$601$	9.79715	0.399634	0.199817	−	0.979833i	$-0.435965\pi$
$601$	9.79715	0.399634	0.199817	+	0.979833i	$0.435965\pi$
$602$	0	0
$603$	2.57194i	0.104738i
$604$	0	0
$605$	19.6233i	0.797799i
$606$	0	0
$607$	24.2258	0.983295	0.491647	−	0.870794i	$-0.336395\pi$
$607$	24.2258	0.983295	0.491647	+	0.870794i	$0.336395\pi$
$608$	0	0
$609$	8.59658i	0.348351i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 15.0556i	− 0.608091i	−0.952658	−	0.304045i	$-0.901663\pi$
$613$	− 15.0556i	− 0.608091i	0.952658	−	0.304045i	$-0.0983375\pi$
$614$	0	0
$615$	−53.0616	−2.13965
$616$	0	0
$617$	− 2.01879i	− 0.0812733i	−0.999174	−	0.0406366i	$-0.987061\pi$
$617$	− 2.01879i	− 0.0812733i	0.999174	−	0.0406366i	$-0.0129386\pi$
$618$	0	0
$619$	− 7.84309i	− 0.315240i	−0.987500	−	0.157620i	$-0.949618\pi$
$619$	− 7.84309i	− 0.315240i	0.987500	−	0.157620i	$-0.0503821\pi$
$620$	0	0
$621$	−12.4940	−0.501365
$622$	0	0
$623$	4.16288	0.166782
$624$	0	0
$625$	−1.13275	−0.0453101
$626$	0	0
$627$	−4.40342	−0.175856
$628$	0	0
$629$	− 29.1487i	− 1.16223i
$630$	0	0
$631$	24.5327i	0.976632i	0.872667	+	0.488316i	$0.162389\pi$
$631$	24.5327i	0.976632i	−0.872667	+	0.488316i	$0.837611\pi$
$632$	0	0
$633$	51.3183	2.03972
$634$	0	0
$635$	41.1836i	1.63432i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.737955i	0.0291930i
$640$	0	0
$641$	41.6015	1.64316	0.821580	−	0.570093i	$-0.193093\pi$
$641$	41.6015	1.64316	0.821580	+	0.570093i	$0.193093\pi$
$642$	0	0
$643$	− 45.4118i	− 1.79087i	−0.445196	−	0.895433i	$-0.646866\pi$
$643$	− 45.4118i	− 1.79087i	0.445196	−	0.895433i	$-0.353134\pi$
$644$	0	0
$645$	− 3.80817i	− 0.149946i
$646$	0	0
$647$	35.8345	1.40880	0.704399	−	0.709804i	$-0.251217\pi$
$647$	35.8345	1.40880	0.704399	+	0.709804i	$0.251217\pi$
$648$	0	0
$649$	23.9148	0.938740
$650$	0	0
$651$	19.3163	0.757067
$652$	0	0
$653$	18.5590	0.726270	0.363135	−	0.931737i	$-0.381706\pi$
$653$	18.5590	0.726270	0.363135	+	0.931737i	$0.381706\pi$
$654$	0	0
$655$	− 8.27413i	− 0.323297i
$656$	0	0
$657$	7.57540i	0.295545i
$658$	0	0
$659$	3.97525	0.154854	0.0774268	−	0.996998i	$-0.475330\pi$
$659$	3.97525	0.154854	0.0774268	+	0.996998i	$0.475330\pi$
$660$	0	0
$661$	− 1.23191i	− 0.0479159i	−0.999713	−	0.0239580i	$-0.992373\pi$
$661$	− 1.23191i	− 0.0479159i	0.999713	−	0.0239580i	$-0.00762678\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.64742i	0.141441i
$666$	0	0
$667$	−12.7922	−0.495318
$668$	0	0
$669$	26.6112i	1.02885i
$670$	0	0
$671$	− 1.03624i	− 0.0400038i
$672$	0	0
$673$	36.8256	1.41952	0.709762	−	0.704442i	$-0.248803\pi$
$673$	36.8256	1.41952	0.709762	+	0.704442i	$0.248803\pi$
$674$	0	0
$675$	29.4916	1.13513
$676$	0	0
$677$	−25.9215	−0.996246	−0.498123	−	0.867106i	$-0.665977\pi$
$677$	−25.9215	−0.996246	−0.498123	+	0.867106i	$0.665977\pi$
$678$	0	0
$679$	16.2875	0.625058
$680$	0	0
$681$	− 28.2851i	− 1.08389i
$682$	0	0
$683$	37.5472i	1.43670i	0.695680	+	0.718352i	$0.255103\pi$
$683$	37.5472i	1.43670i	−0.695680	+	0.718352i	$0.744897\pi$
$684$	0	0
$685$	−32.7245	−1.25034
$686$	0	0
$687$	− 23.6862i	− 0.903684i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 45.2549i	− 1.72158i	−0.508963	−	0.860788i	$-0.669971\pi$
$691$	− 45.2549i	− 1.72158i	0.508963	−	0.860788i	$-0.330029\pi$
$692$	0	0
$693$	3.13408	0.119054
$694$	0	0
$695$	− 68.1318i	− 2.58439i
$696$	0	0
$697$	− 42.8340i	− 1.62245i
$698$	0	0
$699$	−20.0277	−0.757519
$700$	0	0
$701$	−36.0823	−1.36281	−0.681405	−	0.731907i	$-0.738631\pi$
$701$	−36.0823	−1.36281	−0.681405	+	0.731907i	$0.738631\pi$
$702$	0	0
$703$	−4.45904	−0.168176
$704$	0	0
$705$	51.5991	1.94333
$706$	0	0
$707$	9.73184i	0.366004i
$708$	0	0
$709$	19.0664i	0.716053i	0.933711	+	0.358026i	$0.116550\pi$
$709$	19.0664i	0.716053i	−0.933711	+	0.358026i	$0.883450\pi$
$710$	0	0
$711$	18.5483	0.695614
$712$	0	0
$713$	28.7439i	1.07647i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.93495i	− 0.0722621i
$718$	0	0
$719$	−15.3056	−0.570802	−0.285401	−	0.958408i	$-0.592127\pi$
$719$	−15.3056	−0.570802	−0.285401	+	0.958408i	$0.592127\pi$
$720$	0	0
$721$	− 20.8793i	− 0.777587i
$722$	0	0
$723$	− 0.450419i	− 0.0167513i
$724$	0	0
$725$	30.1957	1.12144
$726$	0	0
$727$	3.46250	0.128417	0.0642085	−	0.997937i	$-0.479548\pi$
$727$	3.46250	0.128417	0.0642085	+	0.997937i	$0.479548\pi$
$728$	0	0
$729$	9.32496	0.345369
$730$	0	0
$731$	3.07415	0.113701
$732$	0	0
$733$	− 26.0930i	− 0.963769i	−0.876235	−	0.481884i	$-0.839953\pi$
$733$	− 26.0930i	− 0.963769i	0.876235	−	0.481884i	$-0.160047\pi$
$734$	0	0
$735$	42.5918i	1.57102i
$736$	0	0
$737$	−5.05967	−0.186375
$738$	0	0
$739$	− 26.4993i	− 0.974794i	−0.873181	−	0.487397i	$-0.837946\pi$
$739$	− 26.4993i	− 0.974794i	0.873181	−	0.487397i	$-0.162054\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.415502i	0.0152433i	0.999971	+	0.00762164i	$0.00242607\pi$
$743$	0.415502i	0.0152433i	−0.999971	+	0.00762164i	$0.997574\pi$
$744$	0	0
$745$	67.3551	2.46770
$746$	0	0
$747$	− 1.09246i	− 0.0399709i
$748$	0	0
$749$	20.0361i	0.732103i
$750$	0	0
$751$	2.90946	0.106168	0.0530839	−	0.998590i	$-0.483095\pi$
$751$	2.90946	0.106168	0.0530839	+	0.998590i	$0.483095\pi$
$752$	0	0
$753$	33.3037	1.21365
$754$	0	0
$755$	−1.14483	−0.0416647
$756$	0	0
$757$	12.3720	0.449667	0.224833	−	0.974397i	$-0.427816\pi$
$757$	12.3720	0.449667	0.224833	+	0.974397i	$0.427816\pi$
$758$	0	0
$759$	16.3418i	0.593171i
$760$	0	0
$761$	− 42.4306i	− 1.53811i	−0.639184	−	0.769053i	$-0.720728\pi$
$761$	− 42.4306i	− 1.53811i	0.639184	−	0.769053i	$-0.279272\pi$
$762$	0	0
$763$	−6.76809	−0.245021
$764$	0	0
$765$	− 25.7366i	− 0.930510i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	13.3341i	0.480839i	0.970669	+	0.240419i	$0.0772849\pi$
$769$	13.3341i	0.480839i	−0.970669	+	0.240419i	$0.922715\pi$
$770$	0	0
$771$	45.9721	1.65565
$772$	0	0
$773$	− 5.85384i	− 0.210548i	−0.994443	−	0.105274i	$-0.966428\pi$
$773$	− 5.85384i	− 0.210548i	0.994443	−	0.105274i	$-0.0335720\pi$
$774$	0	0
$775$	− 67.8491i	− 2.43721i
$776$	0	0
$777$	11.1207	0.398952
$778$	0	0
$779$	−6.55257	−0.234770
$780$	0	0
$781$	−1.45175	−0.0519476
$782$	0	0
$783$	13.9565	0.498763
$784$	0	0
$785$	68.0775i	2.42979i
$786$	0	0
$787$	− 23.2965i	− 0.830430i	−0.909723	−	0.415215i	$-0.863706\pi$
$787$	− 23.2965i	− 0.830430i	0.909723	−	0.415215i	$-0.136294\pi$
$788$	0	0
$789$	21.5013	0.765465
$790$	0	0
$791$	13.5421i	0.481503i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 24.9879i	− 0.886230i
$796$	0	0
$797$	−35.8103	−1.26847	−0.634233	−	0.773142i	$-0.718684\pi$
$797$	−35.8103	−1.26847	−0.634233	+	0.773142i	$0.718684\pi$
$798$	0	0
$799$	41.6534i	1.47359i
$800$	0	0
$801$	4.49349i	0.158769i
$802$	0	0
$803$	−14.9028	−0.525907
$804$	0	0
$805$	13.5362	0.477087
$806$	0	0
$807$	54.1232	1.90523
$808$	0	0
$809$	28.3744	0.997589	0.498795	−	0.866720i	$-0.333776\pi$
$809$	28.3744	0.997589	0.498795	+	0.866720i	$0.333776\pi$
$810$	0	0
$811$	− 5.20344i	− 0.182717i	−0.995818	−	0.0913587i	$-0.970879\pi$
$811$	− 5.20344i	− 0.182717i	0.995818	−	0.0913587i	$-0.0291210\pi$
$812$	0	0
$813$	45.1379i	1.58306i
$814$	0	0
$815$	−15.6146	−0.546957
$816$	0	0
$817$	− 0.470270i	− 0.0164527i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 5.65338i	− 0.197304i	−0.995122	−	0.0986522i	$-0.968547\pi$
$821$	− 5.65338i	− 0.197304i	0.995122	−	0.0986522i	$-0.0314531\pi$
$822$	0	0
$823$	39.0616	1.36160	0.680801	−	0.732469i	$-0.261632\pi$
$823$	39.0616	1.36160	0.680801	+	0.732469i	$0.261632\pi$
$824$	0	0
$825$	− 38.5743i	− 1.34299i
$826$	0	0
$827$	− 5.40283i	− 0.187875i	−0.995578	−	0.0939374i	$-0.970055\pi$
$827$	− 5.40283i	− 0.187875i	0.995578	−	0.0939374i	$-0.0299454\pi$
$828$	0	0
$829$	8.38537	0.291236	0.145618	−	0.989341i	$-0.453483\pi$
$829$	8.38537	0.291236	0.145618	+	0.989341i	$0.453483\pi$
$830$	0	0
$831$	−4.45904	−0.154682
$832$	0	0
$833$	−34.3822	−1.19127
$834$	0	0
$835$	50.4543	1.74604
$836$	0	0
$837$	− 31.3599i	− 1.08396i
$838$	0	0
$839$	− 3.98062i	− 0.137426i	−0.997636	−	0.0687132i	$-0.978111\pi$
$839$	− 3.98062i	− 0.137426i	0.997636	−	0.0687132i	$-0.0218893\pi$
$840$	0	0
$841$	−14.7103	−0.507253
$842$	0	0
$843$	51.2290i	1.76442i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.04354i	− 0.207659i
$848$	0	0
$849$	−33.4286	−1.14727
$850$	0	0
$851$	16.5483i	0.567267i
$852$	0	0
$853$	− 6.29350i	− 0.215485i	−0.994179	−	0.107743i	$-0.965638\pi$
$853$	− 6.29350i	− 0.215485i	0.994179	−	0.107743i	$-0.0343623\pi$
$854$	0	0
$855$	−3.93708	−0.134645
$856$	0	0
$857$	−4.37627	−0.149491	−0.0747453	−	0.997203i	$-0.523814\pi$
$857$	−4.37627	−0.149491	−0.0747453	+	0.997203i	$0.523814\pi$
$858$	0	0
$859$	−15.0261	−0.512683	−0.256342	−	0.966586i	$-0.582517\pi$
$859$	−15.0261	−0.512683	−0.256342	+	0.966586i	$0.582517\pi$
$860$	0	0
$861$	16.3418	0.556928
$862$	0	0
$863$	− 6.21121i	− 0.211432i	−0.994396	−	0.105716i	$-0.966287\pi$
$863$	− 6.21121i	− 0.211432i	0.994396	−	0.105716i	$-0.0337134\pi$
$864$	0	0
$865$	39.5991i	1.34641i
$866$	0	0
$867$	37.9681	1.28946
$868$	0	0
$869$	36.4892i	1.23781i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	17.5810i	0.595028i
$874$	0	0
$875$	−11.9517	−0.404040
$876$	0	0
$877$	38.2198i	1.29059i	0.763933	+	0.645296i	$0.223266\pi$
$877$	38.2198i	1.29059i	−0.763933	+	0.645296i	$0.776734\pi$
$878$	0	0
$879$	− 3.84787i	− 0.129785i
$880$	0	0
$881$	−26.7832	−0.902347	−0.451174	−	0.892436i	$-0.648994\pi$
$881$	−26.7832	−0.902347	−0.451174	+	0.892436i	$0.648994\pi$
$882$	0	0
$883$	−34.4956	−1.16087	−0.580435	−	0.814307i	$-0.697117\pi$
$883$	−34.4956	−1.16087	−0.580435	+	0.814307i	$0.697117\pi$
$884$	0	0
$885$	74.9241	2.51854
$886$	0	0
$887$	23.9866	0.805391	0.402695	−	0.915334i	$-0.368073\pi$
$887$	23.9866	0.805391	0.402695	+	0.915334i	$0.368073\pi$
$888$	0	0
$889$	− 12.6837i	− 0.425396i
$890$	0	0
$891$	− 26.3002i	− 0.881090i
$892$	0	0
$893$	6.37196	0.213230
$894$	0	0
$895$	16.7767i	0.560784i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 32.1086i	− 1.07088i
$900$	0	0
$901$	20.1715	0.672010
$902$	0	0
$903$	1.17283i	0.0390295i
$904$	0	0
$905$	3.84309i	0.127748i
$906$	0	0
$907$	23.9269	0.794480	0.397240	−	0.917715i	$-0.369968\pi$
$907$	23.9269	0.794480	0.397240	+	0.917715i	$0.369968\pi$
$908$	0	0
$909$	−10.5047	−0.348419
$910$	0	0
$911$	35.8866	1.18898	0.594488	−	0.804104i	$-0.297355\pi$
$911$	35.8866	1.18898	0.594488	+	0.804104i	$0.297355\pi$
$912$	0	0
$913$	2.14914	0.0711263
$914$	0	0
$915$	− 3.24651i	− 0.107326i
$916$	0	0
$917$	2.54825i	0.0841507i
$918$	0	0
$919$	−33.2465	−1.09670	−0.548351	−	0.836249i	$-0.684744\pi$
$919$	−33.2465	−1.09670	−0.548351	+	0.836249i	$0.684744\pi$
$920$	0	0
$921$	49.1333i	1.61900i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 39.0616i	− 1.28434i
$926$	0	0
$927$	22.5375	0.740229
$928$	0	0
$929$	54.2583i	1.78016i	0.455806	+	0.890079i	$0.349351\pi$
$929$	54.2583i	1.78016i	−0.455806	+	0.890079i	$0.650649\pi$
$930$	0	0
$931$	5.25965i	0.172378i
$932$	0	0
$933$	−11.0315	−0.361154
$934$	0	0
$935$	50.6305	1.65580
$936$	0	0
$937$	16.5265	0.539897	0.269948	−	0.962875i	$-0.412993\pi$
$937$	16.5265	0.539897	0.269948	+	0.962875i	$0.412993\pi$
$938$	0	0
$939$	38.8243	1.26698
$940$	0	0
$941$	41.7017i	1.35944i	0.733473	+	0.679718i	$0.237898\pi$
$941$	41.7017i	1.35944i	−0.733473	+	0.679718i	$0.762102\pi$
$942$	0	0
$943$	24.3177i	0.791892i
$944$	0	0
$945$	−14.7681	−0.480406
$946$	0	0
$947$	− 3.00106i	− 0.0975215i	−0.998810	−	0.0487608i	$-0.984473\pi$
$947$	− 3.00106i	− 0.0975215i	0.998810	−	0.0487608i	$-0.0155272\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	23.5651i	0.764151i
$952$	0	0
$953$	−38.1450	−1.23564	−0.617818	−	0.786321i	$-0.711983\pi$
$953$	−38.1450	−1.23564	−0.617818	+	0.786321i	$0.711983\pi$
$954$	0	0
$955$	− 3.20775i	− 0.103800i
$956$	0	0
$957$	− 18.2547i	− 0.590092i
$958$	0	0
$959$	10.0785	0.325450
$960$	0	0
$961$	−41.1473	−1.32733
$962$	0	0
$963$	−21.6273	−0.696930
$964$	0	0
$965$	−58.4456	−1.88143
$966$	0	0
$967$	− 26.8793i	− 0.864381i	−0.901782	−	0.432190i	$-0.857741\pi$
$967$	− 26.8793i	− 0.864381i	0.901782	−	0.432190i	$-0.142259\pi$
$968$	0	0
$969$	− 11.1366i	− 0.357759i
$970$	0	0
$971$	−3.13647	−0.100654	−0.0503271	−	0.998733i	$-0.516026\pi$
$971$	−3.13647	−0.100654	−0.0503271	+	0.998733i	$0.516026\pi$
$972$	0	0
$973$	20.9831i	0.672688i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.8864i	− 1.14811i	−0.818818	−	0.574053i	$-0.805370\pi$
$977$	− 35.8864i	− 1.14811i	0.818818	−	0.574053i	$-0.194630\pi$
$978$	0	0
$979$	−8.83984	−0.282522
$980$	0	0
$981$	− 7.30559i	− 0.233249i
$982$	0	0
$983$	30.4370i	0.970790i	0.874295	+	0.485395i	$0.161324\pi$
$983$	30.4370i	0.970790i	−0.874295	+	0.485395i	$0.838676\pi$
$984$	0	0
$985$	−41.3405	−1.31722
$986$	0	0
$987$	−15.8914	−0.505829
$988$	0	0
$989$	−1.74525	−0.0554957
$990$	0	0
$991$	−31.4470	−0.998946	−0.499473	−	0.866330i	$-0.666473\pi$
$991$	−31.4470	−0.998946	−0.499473	+	0.866330i	$0.666473\pi$
$992$	0	0
$993$	− 70.9154i	− 2.25043i
$994$	0	0
$995$	− 13.6931i	− 0.434100i
$996$	0	0
$997$	19.1099	0.605217	0.302609	−	0.953115i	$-0.402143\pi$
$997$	19.1099	0.605217	0.302609	+	0.953115i	$0.402143\pi$
$998$	0	0
$999$	− 18.0543i	− 0.571213i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.m.337.6		6
4.3	odd	2		338.2.b.d.337.4		6
12.11	even	2		3042.2.b.n.1351.1		6
13.5	odd	4		2704.2.a.w.1.3		3
13.8	odd	4		2704.2.a.v.1.3		3
13.12	even	2	inner	2704.2.f.m.337.5		6
52.3	odd	6		338.2.e.e.147.3		12
52.7	even	12		338.2.c.i.315.3		6
52.11	even	12		338.2.c.i.191.3		6
52.15	even	12		338.2.c.h.191.3		6
52.19	even	12		338.2.c.h.315.3		6
52.23	odd	6		338.2.e.e.147.6		12
52.31	even	4		338.2.a.h.1.1	yes	3
52.35	odd	6		338.2.e.e.23.6		12
52.43	odd	6		338.2.e.e.23.3		12
52.47	even	4		338.2.a.g.1.1	✓	3
52.51	odd	2		338.2.b.d.337.1		6
156.47	odd	4		3042.2.a.bi.1.3		3
156.83	odd	4		3042.2.a.z.1.1		3
156.155	even	2		3042.2.b.n.1351.6		6
260.99	even	4		8450.2.a.bx.1.3		3
260.239	even	4		8450.2.a.bn.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.1	✓	3	52.47	even	4
338.2.a.h.1.1	yes	3	52.31	even	4
338.2.b.d.337.1		6	52.51	odd	2
338.2.b.d.337.4		6	4.3	odd	2
338.2.c.h.191.3		6	52.15	even	12
338.2.c.h.315.3		6	52.19	even	12
338.2.c.i.191.3		6	52.11	even	12
338.2.c.i.315.3		6	52.7	even	12
338.2.e.e.23.3		12	52.43	odd	6
338.2.e.e.23.6		12	52.35	odd	6
338.2.e.e.147.3		12	52.3	odd	6
338.2.e.e.147.6		12	52.23	odd	6
2704.2.a.v.1.3		3	13.8	odd	4
2704.2.a.w.1.3		3	13.5	odd	4
2704.2.f.m.337.5		6	13.12	even	2	inner
2704.2.f.m.337.6		6	1.1	even	1	trivial
3042.2.a.z.1.1		3	156.83	odd	4
3042.2.a.bi.1.3		3	156.47	odd	4
3042.2.b.n.1351.1		6	12.11	even	2
3042.2.b.n.1351.6		6	156.155	even	2
8450.2.a.bn.1.3		3	260.239	even	4
8450.2.a.bx.1.3		3	260.99	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.04892 q^{3} +3.60388i q^{5} -1.10992i q^{7} +1.19806 q^{9} +2.35690i q^{11} +7.38404i q^{15} -5.96077 q^{17} +0.911854i q^{19} -2.27413i q^{21} +3.38404 q^{23} -7.98792 q^{25} -3.69202 q^{27} -3.78017 q^{29} +8.49396i q^{31} +4.82908i q^{33} +4.00000 q^{35} +4.89008i q^{37} +7.18598i q^{41} -0.515729 q^{43} +4.31767i q^{45} -6.98792i q^{47} +5.76809 q^{49} -12.2131 q^{51} -3.38404 q^{53} -8.49396 q^{55} +1.86831i q^{57} -10.1468i q^{59} -0.439665 q^{61} -1.32975i q^{63} +2.14675i q^{67} +6.93362 q^{69} +0.615957i q^{71} +6.32304i q^{73} -16.3666 q^{75} +2.61596 q^{77} +15.4819 q^{79} -11.1588 q^{81} -0.911854i q^{83} -21.4819i q^{85} -7.74525 q^{87} +3.75063i q^{89} +17.4034i q^{93} -3.28621 q^{95} +14.6746i q^{97} +2.82371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 16 q^{9} - 10 q^{17} - 10 q^{25} - 12 q^{27} - 20 q^{29} + 24 q^{35} + 22 q^{43} - 6 q^{49} - 32 q^{51} - 32 q^{55} - 8 q^{61} + 28 q^{69} - 46 q^{75} + 36 q^{77} + 36 q^{79} - 50 q^{81}+ \cdots - 36 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 16 * q^9 - 10 * q^17 - 10 * q^25 - 12 * q^27 - 20 * q^29 + 24 * q^35 + 22 * q^43 - 6 * q^49 - 32 * q^51 - 32 * q^55 - 8 * q^61 + 28 * q^69 - 46 * q^75 + 36 * q^77 + 36 * q^79 - 50 * q^81 + 20 * q^87 - 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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