LMFDB - Embedded newform 3042.2.b.n.1351.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1351,3042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("3042.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6,0,0,0,0,0,-4,0,0,0,8,0,6,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$24.2904922949$
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			1351.1
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.n.1351.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -3.60388i q^{5} +1.10992i q^{7} +1.00000i q^{8} -3.60388 q^{10} +2.35690i q^{11} +1.10992 q^{14} +1.00000 q^{16} +5.96077 q^{17} -0.911854i q^{19} +3.60388i q^{20} +2.35690 q^{22} +3.38404 q^{23} -7.98792 q^{25} -1.10992i q^{28} +3.78017 q^{29} -8.49396i q^{31} -1.00000i q^{32} -5.96077i q^{34} +4.00000 q^{35} +4.89008i q^{37} -0.911854 q^{38} +3.60388 q^{40} -7.18598i q^{41} +0.515729 q^{43} -2.35690i q^{44} -3.38404i q^{46} -6.98792i q^{47} +5.76809 q^{49} +7.98792i q^{50} +3.38404 q^{53} +8.49396 q^{55} -1.10992 q^{56} -3.78017i q^{58} -10.1468i q^{59} -0.439665 q^{61} -8.49396 q^{62} -1.00000 q^{64} -2.14675i q^{67} -5.96077 q^{68} -4.00000i q^{70} +0.615957i q^{71} +6.32304i q^{73} +4.89008 q^{74} +0.911854i q^{76} -2.61596 q^{77} -15.4819 q^{79} -3.60388i q^{80} -7.18598 q^{82} -0.911854i q^{83} -21.4819i q^{85} -0.515729i q^{86} -2.35690 q^{88} -3.75063i q^{89} -3.38404 q^{92} -6.98792 q^{94} -3.28621 q^{95} +14.6746i q^{97} -5.76809i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 4 q^{10} + 8 q^{14} + 6 q^{16} + 10 q^{17} + 6 q^{22} - 10 q^{25} + 20 q^{29} + 24 q^{35} + 2 q^{38} + 4 q^{40} - 22 q^{43} - 6 q^{49} + 32 q^{55} - 8 q^{56} - 8 q^{61} - 32 q^{62} - 6 q^{64}+ \cdots - 36 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^10 + 8 * q^14 + 6 * q^16 + 10 * q^17 + 6 * q^22 - 10 * q^25 + 20 * q^29 + 24 * q^35 + 2 * q^38 + 4 * q^40 - 22 * q^43 - 6 * q^49 + 32 * q^55 - 8 * q^56 - 8 * q^61 - 32 * q^62 - 6 * q^64 - 10 * q^68 + 28 * q^74 - 36 * q^77 - 36 * q^79 - 14 * q^82 - 6 * q^88 - 4 * q^94 - 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	− 3.60388i	− 1.61170i	−0.592118	−	0.805851i	$-0.701708\pi$
$5$	− 3.60388i	− 1.61170i	0.592118	−	0.805851i	$-0.298292\pi$
$6$	0	0
$7$	1.10992i	0.419509i	0.977754	+	0.209754i	$0.0672665\pi$
$7$	1.10992i	0.419509i	−0.977754	+	0.209754i	$0.932734\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	−3.60388	−1.13965
$11$	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
$14$	1.10992	0.296638
$15$	0	0
$16$	1.00000	0.250000
$17$	5.96077	1.44570	0.722850	−	0.691005i	$-0.242832\pi$
$17$	5.96077	1.44570	0.722850	+	0.691005i	$0.242832\pi$
$18$	0	0
$19$	− 0.911854i	− 0.209194i	−0.994515	−	0.104597i	$-0.966645\pi$
$19$	− 0.911854i	− 0.209194i	0.994515	−	0.104597i	$-0.0333552\pi$
$20$	3.60388i	0.805851i
$21$	0	0
$22$	2.35690	0.502492
$23$	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
$27$	0	0
$28$	− 1.10992i	− 0.209754i
$29$	3.78017	0.701959	0.350980	−	0.936383i	$-0.385849\pi$
$29$	3.78017	0.701959	0.350980	+	0.936383i	$0.385849\pi$
$30$	0	0
$31$	− 8.49396i	− 1.52556i	−0.646658	−	0.762780i	$-0.723834\pi$
$31$	− 8.49396i	− 1.52556i	0.646658	−	0.762780i	$-0.276166\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 5.96077i	− 1.02226i
$35$	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	$0.868328\pi$
$38$	−0.911854	−0.147922
$39$	0	0
$40$	3.60388	0.569823
$41$	− 7.18598i	− 1.12226i	−0.827727	−	0.561131i	$-0.810366\pi$
$41$	− 7.18598i	− 1.12226i	0.827727	−	0.561131i	$-0.189634\pi$
$42$	0	0
$43$	0.515729	0.0786480	0.0393240	−	0.999227i	$-0.487480\pi$
$43$	0.515729	0.0786480	0.0393240	+	0.999227i	$0.487480\pi$
$44$	− 2.35690i	− 0.355315i
$45$	0	0
$46$	− 3.38404i	− 0.498950i
$47$	− 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
$50$	7.98792i	1.12966i
$51$	0	0
$52$	0	0
$53$	3.38404	0.464834	0.232417	−	0.972616i	$-0.425337\pi$
$53$	3.38404	0.464834	0.232417	+	0.972616i	$0.425337\pi$
$54$	0	0
$55$	8.49396	1.14533
$56$	−1.10992	−0.148319
$57$	0	0
$58$	− 3.78017i	− 0.496360i
$59$	− 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	$0.508961\pi$
$62$	−8.49396	−1.07873
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 2.14675i	− 0.262268i	−0.991365	−	0.131134i	$-0.958138\pi$
$67$	− 2.14675i	− 0.262268i	0.991365	−	0.131134i	$-0.0418617\pi$
$68$	−5.96077	−0.722850
$69$	0	0
$70$	− 4.00000i	− 0.478091i
$71$	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	$0.879348\pi$
$74$	4.89008	0.568461
$75$	0	0
$76$	0.911854i	0.104597i
$77$	−2.61596	−0.298116
$78$	0	0
$79$	−15.4819	−1.74185	−0.870924	−	0.491418i	$-0.836479\pi$
$79$	−15.4819	−1.74185	−0.870924	+	0.491418i	$0.836479\pi$
$80$	− 3.60388i	− 0.402926i
$81$	0	0
$82$	−7.18598	−0.793559
$83$	− 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
$86$	− 0.515729i	− 0.0556125i
$87$	0	0
$88$	−2.35690	−0.251246
$89$	− 3.75063i	− 0.397566i	−0.980044	−	0.198783i	$-0.936301\pi$
$89$	− 3.75063i	− 0.397566i	0.980044	−	0.198783i	$-0.0636988\pi$
$90$	0	0
$91$	0	0
$92$	−3.38404	−0.352811
$93$	0	0
$94$	−6.98792	−0.720749
$95$	−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	$0.732455\pi$
$98$	− 5.76809i	− 0.582665i
$99$	0	0
$100$	7.98792	0.798792
$101$	8.76809	0.872457	0.436229	−	0.899836i	$-0.356314\pi$
$101$	8.76809	0.872457	0.436229	+	0.899836i	$0.356314\pi$
$102$	0	0
$103$	−18.8116	−1.85356	−0.926782	−	0.375599i	$-0.877437\pi$
$103$	−18.8116	−1.85356	−0.926782	+	0.375599i	$0.877437\pi$
$104$	0	0
$105$	0	0
$106$	− 3.38404i	− 0.328687i
$107$	−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	$-0.0943318\pi$
$110$	− 8.49396i	− 0.809867i
$111$	0	0
$112$	1.10992i	0.104877i
$113$	12.2010	1.14778	0.573889	−	0.818933i	$-0.305434\pi$
$113$	12.2010	1.14778	0.573889	+	0.818933i	$0.305434\pi$
$114$	0	0
$115$	− 12.1957i	− 1.13725i
$116$	−3.78017	−0.350980
$117$	0	0
$118$	−10.1468	−0.934084
$119$	6.61596i	0.606484i
$120$	0	0
$121$	5.44504	0.495004
$122$	0.439665i	0.0398054i
$123$	0	0
$124$	8.49396i	0.762780i
$125$	10.7681i	0.963127i
$126$	0	0
$127$	−11.4276	−1.01403	−0.507017	−	0.861936i	$-0.669252\pi$
$127$	−11.4276	−1.01403	−0.507017	+	0.861936i	$0.669252\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	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
$134$	−2.14675	−0.185451
$135$	0	0
$136$	5.96077i	0.511132i
$137$	− 9.08038i	− 0.775789i	−0.921704	−	0.387894i	$-0.873203\pi$
$137$	− 9.08038i	− 0.775789i	0.921704	−	0.387894i	$-0.126797\pi$
$138$	0	0
$139$	18.9051	1.60351	0.801757	−	0.597650i	$-0.203899\pi$
$139$	18.9051	1.60351	0.801757	+	0.597650i	$0.203899\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	0.615957	0.0516900
$143$	0	0
$144$	0	0
$145$	− 13.6233i	− 1.13135i
$146$	6.32304	0.523299
$147$	0	0
$148$	− 4.89008i	− 0.401962i
$149$	18.6896i	1.53111i	0.643368	+	0.765557i	$0.277537\pi$
$149$	18.6896i	1.53111i	−0.643368	+	0.765557i	$0.722463\pi$
$150$	0	0
$151$	− 0.317667i	− 0.0258514i	−0.999916	−	0.0129257i	$-0.995886\pi$
$151$	− 0.317667i	− 0.0258514i	0.999916	−	0.0129257i	$-0.00411449\pi$
$152$	0.911854	0.0739611
$153$	0	0
$154$	2.61596i	0.210800i
$155$	−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	$0.228220\pi$
$158$	15.4819i	1.23167i
$159$	0	0
$160$	−3.60388	−0.284911
$161$	3.75600i	0.296015i
$162$	0	0
$163$	− 4.33273i	− 0.339366i	−0.985499	−	0.169683i	$-0.945726\pi$
$163$	− 4.33273i	− 0.339366i	0.985499	−	0.169683i	$-0.0542744\pi$
$164$	7.18598i	0.561131i
$165$	0	0
$166$	−0.911854	−0.0707736
$167$	− 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
$170$	−21.4819	−1.64758
$171$	0	0
$172$	−0.515729	−0.0393240
$173$	−10.9879	−0.835396	−0.417698	−	0.908586i	$-0.637163\pi$
$173$	−10.9879	−0.835396	−0.417698	+	0.908586i	$0.637163\pi$
$174$	0	0
$175$	− 8.86592i	− 0.670201i
$176$	2.35690i	0.177658i
$177$	0	0
$178$	−3.75063	−0.281121
$179$	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	0
$184$	3.38404i	0.249475i
$185$	17.6233	1.29569
$186$	0	0
$187$	14.0489i	1.02736i
$188$	6.98792i	0.509646i
$189$	0	0
$190$	3.28621i	0.238407i
$191$	−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	$0.801613\pi$
$194$	14.6746	1.05357
$195$	0	0
$196$	−5.76809	−0.412006
$197$	− 11.4711i	− 0.817284i	−0.912695	−	0.408642i	$-0.866003\pi$
$197$	− 11.4711i	− 0.817284i	0.912695	−	0.408642i	$-0.133997\pi$
$198$	0	0
$199$	3.79954	0.269343	0.134671	−	0.990890i	$-0.457002\pi$
$199$	3.79954	0.269343	0.134671	+	0.990890i	$0.457002\pi$
$200$	− 7.98792i	− 0.564831i
$201$	0	0
$202$	− 8.76809i	− 0.616920i
$203$	4.19567i	0.294478i
$204$	0	0
$205$	−25.8974	−1.80875
$206$	18.8116i	1.31067i
$207$	0	0
$208$	0	0
$209$	2.14914	0.148659
$210$	0	0
$211$	−25.0465	−1.72427	−0.862137	−	0.506675i	$-0.830874\pi$
$211$	−25.0465	−1.72427	−0.862137	+	0.506675i	$0.830874\pi$
$212$	−3.38404	−0.232417
$213$	0	0
$214$	18.0519i	1.23400i
$215$	− 1.85862i	− 0.126757i
$216$	0	0
$217$	9.42758	0.639986
$218$	−6.09783	−0.412997
$219$	0	0
$220$	−8.49396	−0.572663
$221$	0	0
$222$	0	0
$223$	− 12.9879i	− 0.869735i	−0.900494	−	0.434868i	$-0.856795\pi$
$223$	− 12.9879i	− 0.869735i	0.900494	−	0.434868i	$-0.143205\pi$
$224$	1.10992	0.0741594
$225$	0	0
$226$	− 12.2010i	− 0.811602i
$227$	− 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	$-0.124752\pi$
$230$	−12.1957	−0.804159
$231$	0	0
$232$	3.78017i	0.248180i
$233$	9.77479	0.640368	0.320184	−	0.947355i	$-0.396255\pi$
$233$	9.77479	0.640368	0.320184	+	0.947355i	$0.396255\pi$
$234$	0	0
$235$	−25.1836	−1.64280
$236$	10.1468i	0.660497i
$237$	0	0
$238$	6.61596	0.428849
$239$	− 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	$-0.00225376\pi$
$242$	− 5.44504i	− 0.350021i
$243$	0	0
$244$	0.439665	0.0281467
$245$	− 20.7875i	− 1.32806i
$246$	0	0
$247$	0	0
$248$	8.49396	0.539367
$249$	0	0
$250$	10.7681	0.681034
$251$	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
$254$	11.4276i	0.717030i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.4373	−1.39960	−0.699799	−	0.714340i	$-0.746727\pi$
$257$	−22.4373	−1.39960	−0.699799	+	0.714340i	$0.746727\pi$
$258$	0	0
$259$	−5.42758	−0.337254
$260$	0	0
$261$	0	0
$262$	2.29590i	0.141841i
$263$	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
$266$	− 1.01208i	− 0.0620547i
$267$	0	0
$268$	2.14675i	0.131134i
$269$	−26.4155	−1.61058	−0.805291	−	0.592880i	$-0.797991\pi$
$269$	−26.4155	−1.61058	−0.805291	+	0.592880i	$0.797991\pi$
$270$	0	0
$271$	− 22.0301i	− 1.33824i	−0.743157	−	0.669118i	$-0.766672\pi$
$271$	− 22.0301i	− 1.33824i	0.743157	−	0.669118i	$-0.233328\pi$
$272$	5.96077	0.361425
$273$	0	0
$274$	−9.08038	−0.548566
$275$	− 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	$0.520826\pi$
$278$	− 18.9051i	− 1.13386i
$279$	0	0
$280$	4.00000i	0.239046i
$281$	− 25.0030i	− 1.49155i	−0.666196	−	0.745776i	$-0.732079\pi$
$281$	− 25.0030i	− 1.49155i	0.666196	−	0.745776i	$-0.267921\pi$
$282$	0	0
$283$	16.3153	0.969842	0.484921	−	0.874558i	$-0.338848\pi$
$283$	16.3153	0.969842	0.484921	+	0.874558i	$0.338848\pi$
$284$	− 0.615957i	− 0.0365503i
$285$	0	0
$286$	0	0
$287$	7.97584	0.470799
$288$	0	0
$289$	18.5308	1.09005
$290$	−13.6233	−0.799985
$291$	0	0
$292$	− 6.32304i	− 0.370028i
$293$	1.87800i	0.109714i	0.998494	+	0.0548570i	$0.0174703\pi$
$293$	1.87800i	0.109714i	−0.998494	+	0.0548570i	$0.982530\pi$
$294$	0	0
$295$	−36.5676	−2.12905
$296$	−4.89008	−0.284230
$297$	0	0
$298$	18.6896	1.08266
$299$	0	0
$300$	0	0
$301$	0.572417i	0.0329935i
$302$	−0.317667	−0.0182797
$303$	0	0
$304$	− 0.911854i	− 0.0522984i
$305$	1.58450i	0.0907281i
$306$	0	0
$307$	− 23.9801i	− 1.36862i	−0.729192	−	0.684310i	$-0.760104\pi$
$307$	− 23.9801i	− 1.36862i	0.729192	−	0.684310i	$-0.239896\pi$
$308$	2.61596	0.149058
$309$	0	0
$310$	30.6112i	1.73860i
$311$	−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	$0.320115\pi$
$314$	− 18.8901i	− 1.06603i
$315$	0	0
$316$	15.4819	0.870924
$317$	− 11.5013i	− 0.645975i	−0.946403	−	0.322987i	$-0.895313\pi$
$317$	− 11.5013i	− 0.645975i	0.946403	−	0.322987i	$-0.104687\pi$
$318$	0	0
$319$	8.90946i	0.498834i
$320$	3.60388i	0.201463i
$321$	0	0
$322$	3.75600	0.209314
$323$	− 5.43535i	− 0.302431i
$324$	0	0
$325$	0	0
$326$	−4.33273	−0.239968
$327$	0	0
$328$	7.18598	0.396779
$329$	7.75600	0.427602
$330$	0	0
$331$	34.6112i	1.90240i	0.308572	+	0.951201i	$0.400149\pi$
$331$	34.6112i	1.90240i	−0.308572	+	0.951201i	$0.599851\pi$
$332$	0.911854i	0.0500445i
$333$	0	0
$334$	−14.0000	−0.766046
$335$	−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
$339$	0	0
$340$	21.4819i	1.16502i
$341$	20.0194	1.08411
$342$	0	0
$343$	14.1715i	0.765189i
$344$	0.515729i	0.0278063i
$345$	0	0
$346$	10.9879i	0.590714i
$347$	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	$-0.00925123\pi$
$350$	−8.86592	−0.473903
$351$	0	0
$352$	2.35690	0.125623
$353$	− 4.28919i	− 0.228291i	−0.993464	−	0.114145i	$-0.963587\pi$
$353$	− 4.28919i	− 0.228291i	0.993464	−	0.114145i	$-0.0364130\pi$
$354$	0	0
$355$	2.21983	0.117816
$356$	3.75063i	0.198783i
$357$	0	0
$358$	− 4.65519i	− 0.246034i
$359$	− 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
$362$	− 1.06638i	− 0.0560475i
$363$	0	0
$364$	0	0
$365$	22.7875	1.19275
$366$	0	0
$367$	17.4276	0.909712	0.454856	−	0.890565i	$-0.349691\pi$
$367$	17.4276	0.909712	0.454856	+	0.890565i	$0.349691\pi$
$368$	3.38404	0.176405
$369$	0	0
$370$	− 17.6233i	− 0.916189i
$371$	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	$0.431944\pi$
$374$	14.0489	0.726452
$375$	0	0
$376$	6.98792	0.360374
$377$	0	0
$378$	0	0
$379$	− 15.0476i	− 0.772943i	−0.922301	−	0.386471i	$-0.873694\pi$
$379$	− 15.0476i	− 0.772943i	0.922301	−	0.386471i	$-0.126306\pi$
$380$	3.28621	0.168579
$381$	0	0
$382$	0.890084i	0.0455406i
$383$	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
$386$	16.2174	0.825446
$387$	0	0
$388$	− 14.6746i	− 0.744988i
$389$	−8.04354	−0.407824	−0.203912	−	0.978989i	$-0.565366\pi$
$389$	−8.04354	−0.407824	−0.203912	+	0.978989i	$0.565366\pi$
$390$	0	0
$391$	20.1715	1.02012
$392$	5.76809i	0.291332i
$393$	0	0
$394$	−11.4711	−0.577907
$395$	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	$0.814716\pi$
$398$	− 3.79954i	− 0.190454i
$399$	0	0
$400$	−7.98792	−0.399396
$401$	17.4426i	0.871044i	0.900178	+	0.435522i	$0.143436\pi$
$401$	17.4426i	0.871044i	−0.900178	+	0.435522i	$0.856564\pi$
$402$	0	0
$403$	0	0
$404$	−8.76809	−0.436229
$405$	0	0
$406$	4.19567	0.208228
$407$	−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	$-0.141840\pi$
$410$	25.8974i	1.27898i
$411$	0	0
$412$	18.8116	0.926782
$413$	11.2620	0.554169
$414$	0	0
$415$	−3.28621	−0.161314
$416$	0	0
$417$	0	0
$418$	− 2.14914i	− 0.105118i
$419$	−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	$0.995222\pi$
$422$	25.0465i	1.21925i
$423$	0	0
$424$	3.38404i	0.164344i
$425$	−47.6142	−2.30963
$426$	0	0
$427$	− 0.487991i	− 0.0236156i
$428$	18.0519	0.872572
$429$	0	0
$430$	−1.85862	−0.0896308
$431$	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	$0.630032\pi$
$434$	− 9.42758i	− 0.452538i
$435$	0	0
$436$	6.09783i	0.292033i
$437$	− 3.08575i	− 0.147612i
$438$	0	0
$439$	−3.50125	−0.167106	−0.0835529	−	0.996503i	$-0.526627\pi$
$439$	−3.50125	−0.167106	−0.0835529	+	0.996503i	$0.526627\pi$
$440$	8.49396i	0.404934i
$441$	0	0
$442$	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
$446$	−12.9879	−0.614996
$447$	0	0
$448$	− 1.10992i	− 0.0524386i
$449$	34.1497i	1.61163i	0.592170	+	0.805813i	$0.298271\pi$
$449$	34.1497i	1.61163i	−0.592170	+	0.805813i	$0.701729\pi$
$450$	0	0
$451$	16.9366	0.797514
$452$	−12.2010	−0.573889
$453$	0	0
$454$	−13.8049	−0.647897
$455$	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	$0.929415\pi$
$458$	−11.5603	−0.540179
$459$	0	0
$460$	12.1957i	0.568626i
$461$	0.733169i	0.0341471i	0.999854	+	0.0170735i	$0.00543494\pi$
$461$	0.733169i	0.0341471i	−0.999854	+	0.0170735i	$0.994565\pi$
$462$	0	0
$463$	− 7.24267i	− 0.336595i	−0.985736	−	0.168298i	$-0.946173\pi$
$463$	− 7.24267i	− 0.336595i	0.985736	−	0.168298i	$-0.0538270\pi$
$464$	3.78017	0.175490
$465$	0	0
$466$	− 9.77479i	− 0.452808i
$467$	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
$470$	25.1836i	1.16163i
$471$	0	0
$472$	10.1468	0.467042
$473$	1.21552i	0.0558897i
$474$	0	0
$475$	7.28382i	0.334204i
$476$	− 6.61596i	− 0.303242i
$477$	0	0
$478$	−0.944378	−0.0431948
$479$	− 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
$482$	−0.219833	−0.0100131
$483$	0	0
$484$	−5.44504	−0.247502
$485$	52.8853	2.40140
$486$	0	0
$487$	− 28.6547i	− 1.29847i	−0.760588	−	0.649234i	$-0.775089\pi$
$487$	− 28.6547i	− 1.29847i	0.760588	−	0.649234i	$-0.224911\pi$
$488$	− 0.439665i	− 0.0199027i
$489$	0	0
$490$	−20.7875	−0.939082
$491$	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
$495$	0	0
$496$	− 8.49396i	− 0.381390i
$497$	−0.683661	−0.0306664
$498$	0	0
$499$	15.9715i	0.714984i	0.933916	+	0.357492i	$0.116368\pi$
$499$	15.9715i	0.714984i	−0.933916	+	0.357492i	$0.883632\pi$
$500$	− 10.7681i	− 0.481563i
$501$	0	0
$502$	− 16.2543i	− 0.725464i
$503$	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
$506$	7.97584	0.354569
$507$	0	0
$508$	11.4276	0.507017
$509$	− 0.914247i	− 0.0405233i	−0.999795	−	0.0202616i	$-0.993550\pi$
$509$	− 0.914247i	− 0.0405233i	0.999795	−	0.0202616i	$-0.00644992\pi$
$510$	0	0
$511$	−7.01805	−0.310460
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	22.4373i	0.989666i
$515$	67.7948i	2.98739i
$516$	0	0
$517$	16.4698	0.724341
$518$	5.42758i	0.238474i
$519$	0	0
$520$	0	0
$521$	3.31096	0.145056	0.0725279	−	0.997366i	$-0.476893\pi$
$521$	3.31096	0.145056	0.0725279	+	0.997366i	$0.476893\pi$
$522$	0	0
$523$	−0.850855	−0.0372053	−0.0186026	−	0.999827i	$-0.505922\pi$
$523$	−0.850855	−0.0372053	−0.0186026	+	0.999827i	$0.505922\pi$
$524$	2.29590	0.100297
$525$	0	0
$526$	− 10.4940i	− 0.457558i
$527$	− 50.6305i	− 2.20550i
$528$	0	0
$529$	−11.5483	−0.502098
$530$	−12.1957	−0.529746
$531$	0	0
$532$	−1.01208	−0.0438793
$533$	0	0
$534$	0	0
$535$	65.0568i	2.81265i
$536$	2.14675	0.0927256
$537$	0	0
$538$	26.4155i	1.13885i
$539$	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	$0.658279\pi$
$542$	−22.0301	−0.946275
$543$	0	0
$544$	− 5.96077i	− 0.255566i
$545$	−21.9758	−0.941341
$546$	0	0
$547$	−2.39075	−0.102221	−0.0511105	−	0.998693i	$-0.516276\pi$
$547$	−2.39075	−0.102221	−0.0511105	+	0.998693i	$0.516276\pi$
$548$	9.08038i	0.387894i
$549$	0	0
$550$	−18.8267	−0.802773
$551$	− 3.44696i	− 0.146845i
$552$	0	0
$553$	− 17.1836i	− 0.730720i
$554$	2.17629i	0.0924618i
$555$	0	0
$556$	−18.9051	−0.801757
$557$	27.1508i	1.15042i	0.818007	+	0.575208i	$0.195079\pi$
$557$	27.1508i	1.15042i	−0.818007	+	0.575208i	$0.804921\pi$
$558$	0	0
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	−25.0030	−1.05469
$563$	−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
$566$	− 16.3153i	− 0.685782i
$567$	0	0
$568$	−0.615957	−0.0258450
$569$	7.30021	0.306041	0.153020	−	0.988223i	$-0.451100\pi$
$569$	7.30021	0.306041	0.153020	+	0.988223i	$0.451100\pi$
$570$	0	0
$571$	43.6722	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$571$	43.6722	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$572$	0	0
$573$	0	0
$574$	− 7.97584i	− 0.332905i
$575$	−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	$0.885911\pi$
$578$	− 18.5308i	− 0.770779i
$579$	0	0
$580$	13.6233i	0.565675i
$581$	1.01208	0.0419882
$582$	0	0
$583$	7.97584i	0.330325i
$584$	−6.32304	−0.261649
$585$	0	0
$586$	1.87800	0.0775796
$587$	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
$590$	36.5676i	1.50547i
$591$	0	0
$592$	4.89008i	0.200981i
$593$	3.98493i	0.163642i	0.996647	+	0.0818208i	$0.0260735\pi$
$593$	3.98493i	0.163642i	−0.996647	+	0.0818208i	$0.973926\pi$
$594$	0	0
$595$	23.8431	0.977471
$596$	− 18.6896i	− 0.765557i
$597$	0	0
$598$	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	$0.435965\pi$
$602$	0.572417	0.0233300
$603$	0	0
$604$	0.317667i	0.0129257i
$605$	− 19.6233i	− 0.797799i
$606$	0	0
$607$	−24.2258	−0.983295	−0.491647	−	0.870794i	$-0.663605\pi$
$607$	−24.2258	−0.983295	−0.491647	+	0.870794i	$0.663605\pi$
$608$	−0.911854	−0.0369806
$609$	0	0
$610$	1.58450	0.0641545
$611$	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	$-0.0983375\pi$
$614$	−23.9801	−0.967760
$615$	0	0
$616$	− 2.61596i	− 0.105400i
$617$	2.01879i	0.0812733i	0.999174	+	0.0406366i	$0.0129386\pi$
$617$	2.01879i	0.0812733i	−0.999174	+	0.0406366i	$0.987061\pi$
$618$	0	0
$619$	7.84309i	0.315240i	0.987500	+	0.157620i	$0.0503821\pi$
$619$	7.84309i	0.315240i	−0.987500	+	0.157620i	$0.949618\pi$
$620$	30.6112	1.22937
$621$	0	0
$622$	5.38404i	0.215880i
$623$	4.16288	0.166782
$624$	0	0
$625$	−1.13275	−0.0453101
$626$	− 18.9487i	− 0.757342i
$627$	0	0
$628$	−18.8901	−0.753796
$629$	29.1487i	1.16223i
$630$	0	0
$631$	− 24.5327i	− 0.976632i	−0.872667	−	0.488316i	$-0.837611\pi$
$631$	− 24.5327i	− 0.976632i	0.872667	−	0.488316i	$-0.162389\pi$
$632$	− 15.4819i	− 0.615836i
$633$	0	0
$634$	−11.5013	−0.456773
$635$	41.1836i	1.63432i
$636$	0	0
$637$	0	0
$638$	8.90946	0.352729
$639$	0	0
$640$	3.60388	0.142456
$641$	−41.6015	−1.64316	−0.821580	−	0.570093i	$-0.806907\pi$
$641$	−41.6015	−1.64316	−0.821580	+	0.570093i	$0.806907\pi$
$642$	0	0
$643$	45.4118i	1.79087i	0.445196	+	0.895433i	$0.353134\pi$
$643$	45.4118i	1.79087i	−0.445196	+	0.895433i	$0.646866\pi$
$644$	− 3.75600i	− 0.148007i
$645$	0	0
$646$	−5.43535	−0.213851
$647$	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
$651$	0	0
$652$	4.33273i	0.169683i
$653$	−18.5590	−0.726270	−0.363135	−	0.931737i	$-0.618294\pi$
$653$	−18.5590	−0.726270	−0.363135	+	0.931737i	$0.618294\pi$
$654$	0	0
$655$	8.27413i	0.323297i
$656$	− 7.18598i	− 0.280565i
$657$	0	0
$658$	− 7.75600i	− 0.302361i
$659$	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	$-0.00762678\pi$
$662$	34.6112	1.34520
$663$	0	0
$664$	0.911854	0.0353868
$665$	− 3.64742i	− 0.141441i
$666$	0	0
$667$	12.7922	0.495318
$668$	14.0000i	0.541676i
$669$	0	0
$670$	7.73663i	0.298892i
$671$	− 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	$0.248803\pi$
$674$	1.95407i	0.0752678i
$675$	0	0
$676$	0	0
$677$	25.9215	0.996246	0.498123	−	0.867106i	$-0.334023\pi$
$677$	25.9215	0.996246	0.498123	+	0.867106i	$0.334023\pi$
$678$	0	0
$679$	−16.2875	−0.625058
$680$	21.4819	0.823792
$681$	0	0
$682$	− 20.0194i	− 0.766582i
$683$	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
$686$	14.1715	0.541071
$687$	0	0
$688$	0.515729	0.0196620
$689$	0	0
$690$	0	0
$691$	45.2549i	1.72158i	0.508963	+	0.860788i	$0.330029\pi$
$691$	45.2549i	1.72158i	−0.508963	+	0.860788i	$0.669971\pi$
$692$	10.9879	0.417698
$693$	0	0
$694$	− 6.41550i	− 0.243529i
$695$	− 68.1318i	− 2.58439i
$696$	0	0
$697$	− 42.8340i	− 1.62245i
$698$	−1.08575	−0.0410964
$699$	0	0
$700$	8.86592i	0.335100i
$701$	36.0823	1.36281	0.681405	−	0.731907i	$-0.261369\pi$
$701$	36.0823	1.36281	0.681405	+	0.731907i	$0.261369\pi$
$702$	0	0
$703$	4.45904	0.168176
$704$	− 2.35690i	− 0.0888289i
$705$	0	0
$706$	−4.28919	−0.161426
$707$	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	$0.883450\pi$
$710$	− 2.21983i	− 0.0833088i
$711$	0	0
$712$	3.75063	0.140561
$713$	− 28.7439i	− 1.07647i
$714$	0	0
$715$	0	0
$716$	−4.65519	−0.173972
$717$	0	0
$718$	−15.5060	−0.578680
$719$	−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
$722$	− 18.1685i	− 0.676162i
$723$	0	0
$724$	−1.06638	−0.0396315
$725$	−30.1957	−1.12144
$726$	0	0
$727$	−3.46250	−0.128417	−0.0642085	−	0.997937i	$-0.520452\pi$
$727$	−3.46250	−0.128417	−0.0642085	+	0.997937i	$0.520452\pi$
$728$	0	0
$729$	0	0
$730$	− 22.7875i	− 0.843402i
$731$	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	$-0.160047\pi$
$734$	− 17.4276i	− 0.643264i
$735$	0	0
$736$	− 3.38404i	− 0.124737i
$737$	5.05967	0.186375
$738$	0	0
$739$	26.4993i	0.974794i	0.873181	+	0.487397i	$0.162054\pi$
$739$	26.4993i	0.974794i	−0.873181	+	0.487397i	$0.837946\pi$
$740$	−17.6233	−0.647844
$741$	0	0
$742$	3.75600	0.137887
$743$	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
$746$	− 8.19567i	− 0.300065i
$747$	0	0
$748$	− 14.0489i	− 0.513679i
$749$	− 20.0361i	− 0.732103i
$750$	0	0
$751$	−2.90946	−0.106168	−0.0530839	−	0.998590i	$-0.516905\pi$
$751$	−2.90946	−0.106168	−0.0530839	+	0.998590i	$0.516905\pi$
$752$	− 6.98792i	− 0.254823i
$753$	0	0
$754$	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	$0.427816\pi$
$758$	−15.0476	−0.546553
$759$	0	0
$760$	− 3.28621i	− 0.119203i
$761$	42.4306i	1.53811i	0.639184	+	0.769053i	$0.279272\pi$
$761$	42.4306i	1.53811i	−0.639184	+	0.769053i	$0.720728\pi$
$762$	0	0
$763$	6.76809	0.245021
$764$	0.890084	0.0322021
$765$	0	0
$766$	11.1207	0.401806
$767$	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	$0.922715\pi$
$770$	9.42758	0.339747
$771$	0	0
$772$	− 16.2174i	− 0.583678i
$773$	5.85384i	0.210548i	0.994443	+	0.105274i	$0.0335720\pi$
$773$	5.85384i	0.210548i	−0.994443	+	0.105274i	$0.966428\pi$
$774$	0	0
$775$	67.8491i	2.43721i
$776$	−14.6746	−0.526786
$777$	0	0
$778$	8.04354i	0.288375i
$779$	−6.55257	−0.234770
$780$	0	0
$781$	−1.45175	−0.0519476
$782$	− 20.1715i	− 0.721332i
$783$	0	0
$784$	5.76809	0.206003
$785$	− 68.0775i	− 2.42979i
$786$	0	0
$787$	23.2965i	0.830430i	0.909723	+	0.415215i	$0.136294\pi$
$787$	23.2965i	0.830430i	−0.909723	+	0.415215i	$0.863706\pi$
$788$	11.4711i	0.408642i
$789$	0	0
$790$	55.7948	1.98509
$791$	13.5421i	0.481503i
$792$	0	0
$793$	0	0
$794$	21.9081	0.777491
$795$	0	0
$796$	−3.79954	−0.134671
$797$	35.8103	1.26847	0.634233	−	0.773142i	$-0.281316\pi$
$797$	35.8103	1.26847	0.634233	+	0.773142i	$0.281316\pi$
$798$	0	0
$799$	− 41.6534i	− 1.47359i
$800$	7.98792i	0.282416i
$801$	0	0
$802$	17.4426	0.615921
$803$	−14.9028	−0.525907
$804$	0	0
$805$	13.5362	0.477087
$806$	0	0
$807$	0	0
$808$	8.76809i	0.308460i
$809$	−28.3744	−0.997589	−0.498795	−	0.866720i	$-0.666224\pi$
$809$	−28.3744	−0.997589	−0.498795	+	0.866720i	$0.666224\pi$
$810$	0	0
$811$	5.20344i	0.182717i	0.995818	+	0.0913587i	$0.0291210\pi$
$811$	5.20344i	0.182717i	−0.995818	+	0.0913587i	$0.970879\pi$
$812$	− 4.19567i	− 0.147239i
$813$	0	0
$814$	11.5254i	0.403966i
$815$	−15.6146	−0.546957
$816$	0	0
$817$	− 0.470270i	− 0.0164527i
$818$	−17.4330	−0.609529
$819$	0	0
$820$	25.8974	0.904376
$821$	5.65338i	0.197304i	0.995122	+	0.0986522i	$0.0314531\pi$
$821$	5.65338i	0.197304i	−0.995122	+	0.0986522i	$0.968547\pi$
$822$	0	0
$823$	−39.0616	−1.36160	−0.680801	−	0.732469i	$-0.738368\pi$
$823$	−39.0616	−1.36160	−0.680801	+	0.732469i	$0.738368\pi$
$824$	− 18.8116i	− 0.655334i
$825$	0	0
$826$	− 11.2620i	− 0.391857i
$827$	− 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	$0.453483\pi$
$830$	3.28621i	0.114066i
$831$	0	0
$832$	0	0
$833$	34.3822	1.19127
$834$	0	0
$835$	−50.4543	−1.74604
$836$	−2.14914	−0.0743297
$837$	0	0
$838$	9.97584i	0.344610i
$839$	− 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
$842$	0.615957	0.0212273
$843$	0	0
$844$	25.0465	0.862137
$845$	0	0
$846$	0	0
$847$	6.04354i	0.207659i
$848$	3.38404	0.116209
$849$	0	0
$850$	47.6142i	1.63315i
$851$	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	$-0.0343623\pi$
$854$	−0.487991	−0.0166987
$855$	0	0
$856$	− 18.0519i	− 0.617001i
$857$	4.37627	0.149491	0.0747453	−	0.997203i	$-0.476186\pi$
$857$	4.37627	0.149491	0.0747453	+	0.997203i	$0.476186\pi$
$858$	0	0
$859$	15.0261	0.512683	0.256342	−	0.966586i	$-0.417483\pi$
$859$	15.0261	0.512683	0.256342	+	0.966586i	$0.417483\pi$
$860$	1.85862i	0.0633786i
$861$	0	0
$862$	14.7922	0.503826
$863$	− 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
$866$	16.5321i	0.561784i
$867$	0	0
$868$	−9.42758	−0.319993
$869$	− 36.4892i	− 1.23781i
$870$	0	0
$871$	0	0
$872$	6.09783	0.206499
$873$	0	0
$874$	−3.08575	−0.104377
$875$	−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	$0.776734\pi$
$878$	3.50125i	0.118162i
$879$	0	0
$880$	8.49396	0.286331
$881$	26.7832	0.902347	0.451174	−	0.892436i	$-0.351006\pi$
$881$	26.7832	0.902347	0.451174	+	0.892436i	$0.351006\pi$
$882$	0	0
$883$	34.4956	1.16087	0.580435	−	0.814307i	$-0.302883\pi$
$883$	34.4956	1.16087	0.580435	+	0.814307i	$0.302883\pi$
$884$	0	0
$885$	0	0
$886$	17.4077i	0.584824i
$887$	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
$890$	13.5168i	0.453084i
$891$	0	0
$892$	12.9879i	0.434868i
$893$	−6.37196	−0.213230
$894$	0	0
$895$	− 16.7767i	− 0.560784i
$896$	−1.10992	−0.0370797
$897$	0	0
$898$	34.1497	1.13959
$899$	− 32.1086i	− 1.07088i
$900$	0	0
$901$	20.1715	0.672010
$902$	− 16.9366i	− 0.563927i
$903$	0	0
$904$	12.2010i	0.405801i
$905$	− 3.84309i	− 0.127748i
$906$	0	0
$907$	−23.9269	−0.794480	−0.397240	−	0.917715i	$-0.630032\pi$
$907$	−23.9269	−0.794480	−0.397240	+	0.917715i	$0.630032\pi$
$908$	13.8049i	0.458132i
$909$	0	0
$910$	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
$914$	9.40342	0.311037
$915$	0	0
$916$	11.5603i	0.381964i
$917$	− 2.54825i	− 0.0841507i
$918$	0	0
$919$	33.2465	1.09670	0.548351	−	0.836249i	$-0.315256\pi$
$919$	33.2465	1.09670	0.548351	+	0.836249i	$0.315256\pi$
$920$	12.1957	0.402079
$921$	0	0
$922$	0.733169	0.0241456
$923$	0	0
$924$	0	0
$925$	− 39.0616i	− 1.28434i
$926$	−7.24267	−0.238009
$927$	0	0
$928$	− 3.78017i	− 0.124090i
$929$	− 54.2583i	− 1.78016i	−0.455806	−	0.890079i	$-0.650649\pi$
$929$	− 54.2583i	− 1.78016i	0.455806	−	0.890079i	$-0.349351\pi$
$930$	0	0
$931$	− 5.25965i	− 0.172378i
$932$	−9.77479	−0.320184
$933$	0	0
$934$	− 30.2446i	− 0.989633i
$935$	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	$0.412993\pi$
$938$	− 2.38271i	− 0.0777984i
$939$	0	0
$940$	25.1836	0.821398
$941$	− 41.7017i	− 1.35944i	−0.733473	−	0.679718i	$-0.762102\pi$
$941$	− 41.7017i	− 1.35944i	0.733473	−	0.679718i	$-0.237898\pi$
$942$	0	0
$943$	− 24.3177i	− 0.791892i
$944$	− 10.1468i	− 0.330249i
$945$	0	0
$946$	1.21552	0.0395200
$947$	− 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
$950$	7.28382	0.236318
$951$	0	0
$952$	−6.61596	−0.214424
$953$	38.1450	1.23564	0.617818	−	0.786321i	$-0.288017\pi$
$953$	38.1450	1.23564	0.617818	+	0.786321i	$0.288017\pi$
$954$	0	0
$955$	3.20775i	0.103800i
$956$	0.944378i	0.0305434i
$957$	0	0
$958$	−36.7198	−1.18636
$959$	10.0785	0.325450
$960$	0	0
$961$	−41.1473	−1.32733
$962$	0	0
$963$	0	0
$964$	0.219833i	0.00708033i
$965$	58.4456	1.88143
$966$	0	0
$967$	26.8793i	0.864381i	0.901782	+	0.432190i	$0.142259\pi$
$967$	26.8793i	0.864381i	−0.901782	+	0.432190i	$0.857741\pi$
$968$	5.44504i	0.175010i
$969$	0	0
$970$	− 52.8853i	− 1.69804i
$971$	−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
$974$	−28.6547	−0.918156
$975$	0	0
$976$	−0.439665	−0.0140733
$977$	35.8864i	1.14811i	0.818818	+	0.574053i	$0.194630\pi$
$977$	35.8864i	1.14811i	−0.818818	+	0.574053i	$0.805370\pi$
$978$	0	0
$979$	8.83984	0.282522
$980$	20.7875i	0.664031i
$981$	0	0
$982$	− 30.4295i	− 0.971044i
$983$	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
$986$	− 22.5327i	− 0.717588i
$987$	0	0
$988$	0	0
$989$	1.74525	0.0554957
$990$	0	0
$991$	31.4470	0.998946	0.499473	−	0.866330i	$-0.333527\pi$
$991$	31.4470	0.998946	0.499473	+	0.866330i	$0.333527\pi$
$992$	−8.49396	−0.269683
$993$	0	0
$994$	0.683661i	0.0216844i
$995$	− 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	$0.402143\pi$
$998$	15.9715	0.505570
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.n.1351.1		6
3.2	odd	2		338.2.b.d.337.4		6
12.11	even	2		2704.2.f.m.337.6		6
13.5	odd	4		3042.2.a.z.1.1		3
13.8	odd	4		3042.2.a.bi.1.3		3
13.12	even	2	inner	3042.2.b.n.1351.6		6
39.2	even	12		338.2.c.h.191.3		6
39.5	even	4		338.2.a.h.1.1	yes	3
39.8	even	4		338.2.a.g.1.1	✓	3
39.11	even	12		338.2.c.i.191.3		6
39.17	odd	6		338.2.e.e.23.3		12
39.20	even	12		338.2.c.i.315.3		6
39.23	odd	6		338.2.e.e.147.6		12
39.29	odd	6		338.2.e.e.147.3		12
39.32	even	12		338.2.c.h.315.3		6
39.35	odd	6		338.2.e.e.23.6		12
39.38	odd	2		338.2.b.d.337.1		6
156.47	odd	4		2704.2.a.v.1.3		3
156.83	odd	4		2704.2.a.w.1.3		3
156.155	even	2		2704.2.f.m.337.5		6
195.44	even	4		8450.2.a.bn.1.3		3
195.164	even	4		8450.2.a.bx.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.2.a.g.1.1	✓	3	39.8	even	4
338.2.a.h.1.1	yes	3	39.5	even	4
338.2.b.d.337.1		6	39.38	odd	2
338.2.b.d.337.4		6	3.2	odd	2
338.2.c.h.191.3		6	39.2	even	12
338.2.c.h.315.3		6	39.32	even	12
338.2.c.i.191.3		6	39.11	even	12
338.2.c.i.315.3		6	39.20	even	12
338.2.e.e.23.3		12	39.17	odd	6
338.2.e.e.23.6		12	39.35	odd	6
338.2.e.e.147.3		12	39.29	odd	6
338.2.e.e.147.6		12	39.23	odd	6
2704.2.a.v.1.3		3	156.47	odd	4
2704.2.a.w.1.3		3	156.83	odd	4
2704.2.f.m.337.5		6	156.155	even	2
2704.2.f.m.337.6		6	12.11	even	2
3042.2.a.z.1.1		3	13.5	odd	4
3042.2.a.bi.1.3		3	13.8	odd	4
3042.2.b.n.1351.1		6	1.1	even	1	trivial
3042.2.b.n.1351.6		6	13.12	even	2	inner
8450.2.a.bn.1.3		3	195.44	even	4
8450.2.a.bx.1.3		3	195.164	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.60388i q^{5} +1.10992i q^{7} +1.00000i q^{8} -3.60388 q^{10} +2.35690i q^{11} +1.10992 q^{14} +1.00000 q^{16} +5.96077 q^{17} -0.911854i q^{19} +3.60388i q^{20} +2.35690 q^{22} +3.38404 q^{23} -7.98792 q^{25} -1.10992i q^{28} +3.78017 q^{29} -8.49396i q^{31} -1.00000i q^{32} -5.96077i q^{34} +4.00000 q^{35} +4.89008i q^{37} -0.911854 q^{38} +3.60388 q^{40} -7.18598i q^{41} +0.515729 q^{43} -2.35690i q^{44} -3.38404i q^{46} -6.98792i q^{47} +5.76809 q^{49} +7.98792i q^{50} +3.38404 q^{53} +8.49396 q^{55} -1.10992 q^{56} -3.78017i q^{58} -10.1468i q^{59} -0.439665 q^{61} -8.49396 q^{62} -1.00000 q^{64} -2.14675i q^{67} -5.96077 q^{68} -4.00000i q^{70} +0.615957i q^{71} +6.32304i q^{73} +4.89008 q^{74} +0.911854i q^{76} -2.61596 q^{77} -15.4819 q^{79} -3.60388i q^{80} -7.18598 q^{82} -0.911854i q^{83} -21.4819i q^{85} -0.515729i q^{86} -2.35690 q^{88} -3.75063i q^{89} -3.38404 q^{92} -6.98792 q^{94} -3.28621 q^{95} +14.6746i q^{97} -5.76809i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{10} + 8 q^{14} + 6 q^{16} + 10 q^{17} + 6 q^{22} - 10 q^{25} + 20 q^{29} + 24 q^{35} + 2 q^{38} + 4 q^{40} - 22 q^{43} - 6 q^{49} + 32 q^{55} - 8 q^{56} - 8 q^{61} - 32 q^{62} - 6 q^{64}+ \cdots - 36 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^10 + 8 * q^14 + 6 * q^16 + 10 * q^17 + 6 * q^22 - 10 * q^25 + 20 * q^29 + 24 * q^35 + 2 * q^38 + 4 * q^40 - 22 * q^43 - 6 * q^49 + 32 * q^55 - 8 * q^56 - 8 * q^61 - 32 * q^62 - 6 * q^64 - 10 * q^68 + 28 * q^74 - 36 * q^77 - 36 * q^79 - 14 * q^82 - 6 * q^88 - 4 * q^94 - 36 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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