LMFDB - Embedded newform 338.2.b.d.337.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,2,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	338.b (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:	$2.69894358832$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.2.b.d.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -2.04892 q^{3} -1.00000 q^{4} +3.60388i q^{5} -2.04892i q^{6} +1.10992i q^{7} -1.00000i q^{8} +1.19806 q^{9} -3.60388 q^{10} -2.35690i q^{11} +2.04892 q^{12} -1.10992 q^{14} -7.38404i q^{15} +1.00000 q^{16} -5.96077 q^{17} +1.19806i q^{18} -0.911854i q^{19} -3.60388i q^{20} -2.27413i q^{21} +2.35690 q^{22} -3.38404 q^{23} +2.04892i q^{24} -7.98792 q^{25} +3.69202 q^{27} -1.10992i q^{28} -3.78017 q^{29} +7.38404 q^{30} -8.49396i q^{31} +1.00000i q^{32} +4.82908i q^{33} -5.96077i q^{34} -4.00000 q^{35} -1.19806 q^{36} +4.89008i q^{37} +0.911854 q^{38} +3.60388 q^{40} +7.18598i q^{41} +2.27413 q^{42} +0.515729 q^{43} +2.35690i q^{44} +4.31767i q^{45} -3.38404i q^{46} +6.98792i q^{47} -2.04892 q^{48} +5.76809 q^{49} -7.98792i q^{50} +12.2131 q^{51} -3.38404 q^{53} +3.69202i q^{54} +8.49396 q^{55} +1.10992 q^{56} +1.86831i q^{57} -3.78017i q^{58} +10.1468i q^{59} +7.38404i q^{60} -0.439665 q^{61} +8.49396 q^{62} +1.32975i q^{63} -1.00000 q^{64} -4.82908 q^{66} -2.14675i q^{67} +5.96077 q^{68} +6.93362 q^{69} -4.00000i q^{70} -0.615957i q^{71} -1.19806i q^{72} +6.32304i q^{73} -4.89008 q^{74} +16.3666 q^{75} +0.911854i q^{76} +2.61596 q^{77} -15.4819 q^{79} +3.60388i q^{80} -11.1588 q^{81} -7.18598 q^{82} +0.911854i q^{83} +2.27413i q^{84} -21.4819i q^{85} +0.515729i q^{86} +7.74525 q^{87} -2.35690 q^{88} +3.75063i q^{89} -4.31767 q^{90} +3.38404 q^{92} +17.4034i q^{93} -6.98792 q^{94} +3.28621 q^{95} -2.04892i q^{96} +14.6746i q^{97} +5.76809i q^{98} -2.82371i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 6 q^{4} + 16 q^{9} - 4 q^{10} - 6 q^{12} - 8 q^{14} + 6 q^{16} - 10 q^{17} + 6 q^{22} - 10 q^{25} + 12 q^{27} - 20 q^{29} + 24 q^{30} - 24 q^{35} - 16 q^{36} - 2 q^{38} + 4 q^{40} - 8 q^{42}+ \cdots + 36 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 - 6 * q^4 + 16 * q^9 - 4 * q^10 - 6 * q^12 - 8 * q^14 + 6 * q^16 - 10 * q^17 + 6 * q^22 - 10 * q^25 + 12 * q^27 - 20 * q^29 + 24 * q^30 - 24 * q^35 - 16 * q^36 - 2 * q^38 + 4 * q^40 - 8 * q^42 - 22 * q^43 + 6 * q^48 - 6 * q^49 + 32 * q^51 + 32 * q^55 + 8 * q^56 - 8 * q^61 + 32 * q^62 - 6 * q^64 - 8 * q^66 + 10 * q^68 + 28 * q^69 - 28 * q^74 + 46 * q^75 + 36 * q^77 - 36 * q^79 - 50 * q^81 - 14 * q^82 - 20 * q^87 - 6 * q^88 + 8 * q^90 - 4 * q^94 + 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$.

$n$	$171$
$\chi(n)$	$-1$

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
$3$	−2.04892	−1.18294	−0.591471	−	0.806326i	$-0.701453\pi$
$3$	−2.04892	−1.18294	−0.591471	+	0.806326i	$0.701453\pi$
$4$	−1.00000	−0.500000
$5$	3.60388i	1.61170i	0.592118	+	0.805851i	$0.298292\pi$
$5$	3.60388i	1.61170i	−0.592118	+	0.805851i	$0.701708\pi$
$6$	− 2.04892i	− 0.836467i
$7$	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
$9$	1.19806	0.399354
$10$	−3.60388	−1.13965
$11$	− 2.35690i	− 0.710631i	−0.934746	−	0.355315i	$-0.884373\pi$
$11$	− 2.35690i	− 0.710631i	0.934746	−	0.355315i	$-0.115627\pi$
$12$	2.04892	0.591471
$13$	0	0
$13$	0	0
$14$	−1.10992	−0.296638
$15$	− 7.38404i	− 1.90655i
$16$	1.00000	0.250000
$17$	−5.96077	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$17$	−5.96077	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$18$	1.19806i	0.282386i
$19$	− 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
$21$	− 2.27413i	− 0.496255i
$22$	2.35690	0.502492
$23$	−3.38404	−0.705622	−0.352811	−	0.935695i	$-0.614774\pi$
$23$	−3.38404	−0.705622	−0.352811	+	0.935695i	$0.614774\pi$
$24$	2.04892i	0.418234i
$25$	−7.98792	−1.59758
$26$	0	0
$27$	3.69202	0.710530
$28$	− 1.10992i	− 0.209754i
$29$	−3.78017	−0.701959	−0.350980	−	0.936383i	$-0.614151\pi$
$29$	−3.78017	−0.701959	−0.350980	+	0.936383i	$0.614151\pi$
$30$	7.38404	1.34814
$31$	− 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
$33$	4.82908i	0.840636i
$34$	− 5.96077i	− 1.02226i
$35$	−4.00000	−0.676123
$36$	−1.19806	−0.199677
$37$	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	$0.189634\pi$
$41$	7.18598i	1.12226i	−0.827727	+	0.561131i	$0.810366\pi$
$42$	2.27413	0.350905
$43$	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
$45$	4.31767i	0.643640i
$46$	− 3.38404i	− 0.498950i
$47$	6.98792i	1.01929i	0.860384	+	0.509646i	$0.170224\pi$
$47$	6.98792i	1.01929i	−0.860384	+	0.509646i	$0.829776\pi$
$48$	−2.04892	−0.295736
$49$	5.76809	0.824012
$50$	− 7.98792i	− 1.12966i
$51$	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	$0.574663\pi$
$54$	3.69202i	0.502420i
$55$	8.49396	1.14533
$56$	1.10992	0.148319
$57$	1.86831i	0.247464i
$58$	− 3.78017i	− 0.496360i
$59$	10.1468i	1.32099i	0.750828	+	0.660497i	$0.229655\pi$
$59$	10.1468i	1.32099i	−0.750828	+	0.660497i	$0.770345\pi$
$60$	7.38404i	0.953276i
$61$	−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
$63$	1.32975i	0.167533i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.82908	−0.594419
$67$	− 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
$69$	6.93362	0.834710
$70$	− 4.00000i	− 0.478091i
$71$	− 0.615957i	− 0.0731007i	−0.999332	−	0.0365503i	$-0.988363\pi$
$71$	− 0.615957i	− 0.0731007i	0.999332	−	0.0365503i	$-0.0116369\pi$
$72$	− 1.19806i	− 0.141193i
$73$	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
$75$	16.3666	1.88985
$76$	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
$81$	−11.1588	−1.23987
$82$	−7.18598	−0.793559
$83$	0.911854i	0.100089i	0.998747	+	0.0500445i	$0.0159363\pi$
$83$	0.911854i	0.100089i	−0.998747	+	0.0500445i	$0.984064\pi$
$84$	2.27413i	0.248128i
$85$	− 21.4819i	− 2.33004i
$86$	0.515729i	0.0556125i
$87$	7.74525	0.830378
$88$	−2.35690	−0.251246
$89$	3.75063i	0.397566i	0.980044	+	0.198783i	$0.0636988\pi$
$89$	3.75063i	0.397566i	−0.980044	+	0.198783i	$0.936301\pi$
$90$	−4.31767	−0.455122
$91$	0	0
$92$	3.38404	0.352811
$93$	17.4034i	1.80465i
$94$	−6.98792	−0.720749
$95$	3.28621	0.337158
$96$	− 2.04892i	− 0.209117i
$97$	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
$99$	− 2.82371i	− 0.283793i
$100$	7.98792	0.798792
$101$	−8.76809	−0.872457	−0.436229	−	0.899836i	$-0.643686\pi$
$101$	−8.76809	−0.872457	−0.436229	+	0.899836i	$0.643686\pi$
$102$	12.2131i	1.20928i
$103$	−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
$105$	8.19567	0.799815
$106$	− 3.38404i	− 0.328687i
$107$	18.0519	1.74514	0.872572	−	0.488486i	$-0.162451\pi$
$107$	18.0519	1.74514	0.872572	+	0.488486i	$0.162451\pi$
$108$	−3.69202	−0.355265
$109$	− 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
$111$	− 10.0194i	− 0.950997i
$112$	1.10992i	0.104877i
$113$	−12.2010	−1.14778	−0.573889	−	0.818933i	$-0.694566\pi$
$113$	−12.2010	−1.14778	−0.573889	+	0.818933i	$0.694566\pi$
$114$	−1.86831	−0.174984
$115$	− 12.1957i	− 1.13725i
$116$	3.78017	0.350980
$117$	0	0
$118$	−10.1468	−0.934084
$119$	− 6.61596i	− 0.606484i
$120$	−7.38404	−0.674068
$121$	5.44504	0.495004
$122$	− 0.439665i	− 0.0398054i
$123$	− 14.7235i	− 1.32757i
$124$	8.49396i	0.762780i
$125$	− 10.7681i	− 0.963127i
$126$	−1.32975	−0.118463
$127$	−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
$129$	−1.05669	−0.0930361
$130$	0	0
$131$	2.29590	0.200593	0.100297	−	0.994958i	$-0.468021\pi$
$131$	2.29590	0.200593	0.100297	+	0.994958i	$0.468021\pi$
$132$	− 4.82908i	− 0.420318i
$133$	1.01208	0.0877586
$134$	2.14675	0.185451
$135$	13.3056i	1.14516i
$136$	5.96077i	0.511132i
$137$	9.08038i	0.775789i	0.921704	+	0.387894i	$0.126797\pi$
$137$	9.08038i	0.775789i	−0.921704	+	0.387894i	$0.873203\pi$
$138$	6.93362i	0.590229i
$139$	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
$141$	− 14.3177i	− 1.20577i
$142$	0.615957	0.0516900
$143$	0	0
$144$	1.19806	0.0998385
$145$	− 13.6233i	− 1.13135i
$146$	−6.32304	−0.523299
$147$	−11.8183	−0.974760
$148$	− 4.89008i	− 0.401962i
$149$	− 18.6896i	− 1.53111i	−0.643368	−	0.765557i	$-0.722463\pi$
$149$	− 18.6896i	− 1.53111i	0.643368	−	0.765557i	$-0.277537\pi$
$150$	16.3666i	1.33633i
$151$	− 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
$153$	−7.14138	−0.577346
$154$	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
$159$	6.93362	0.549872
$160$	−3.60388	−0.284911
$161$	− 3.75600i	− 0.296015i
$162$	− 11.1588i	− 0.876721i
$163$	− 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
$165$	−17.4034	−1.35485
$166$	−0.911854	−0.0707736
$167$	14.0000i	1.08335i	0.840587	+	0.541676i	$0.182210\pi$
$167$	14.0000i	1.08335i	−0.840587	+	0.541676i	$0.817790\pi$
$168$	−2.27413	−0.175453
$169$	0	0
$170$	21.4819	1.64758
$171$	− 1.09246i	− 0.0835423i
$172$	−0.515729	−0.0393240
$173$	10.9879	0.835396	0.417698	−	0.908586i	$-0.362837\pi$
$173$	10.9879	0.835396	0.417698	+	0.908586i	$0.362837\pi$
$174$	7.74525i	0.587166i
$175$	− 8.86592i	− 0.670201i
$176$	− 2.35690i	− 0.177658i
$177$	− 20.7899i	− 1.56266i
$178$	−3.75063	−0.281121
$179$	−4.65519	−0.347945	−0.173972	−	0.984751i	$-0.555660\pi$
$179$	−4.65519	−0.347945	−0.173972	+	0.984751i	$0.555660\pi$
$180$	− 4.31767i	− 0.321820i
$181$	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
$184$	3.38404i	0.249475i
$185$	−17.6233	−1.29569
$186$	−17.4034	−1.27608
$187$	14.0489i	1.02736i
$188$	− 6.98792i	− 0.509646i
$189$	4.09783i	0.298074i
$190$	3.28621i	0.238407i
$191$	0.890084	0.0644042	0.0322021	−	0.999481i	$-0.489748\pi$
$191$	0.890084	0.0644042	0.0322021	+	0.999481i	$0.489748\pi$
$192$	2.04892	0.147868
$193$	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	$0.133997\pi$
$197$	11.4711i	0.817284i	−0.912695	+	0.408642i	$0.866003\pi$
$198$	2.82371	0.200672
$199$	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
$201$	4.39852i	0.310247i
$202$	− 8.76809i	− 0.616920i
$203$	− 4.19567i	− 0.294478i
$204$	−12.2131	−0.855090
$205$	−25.8974	−1.80875
$206$	− 18.8116i	− 1.31067i
$207$	−4.05429	−0.281793
$208$	0	0
$209$	−2.14914	−0.148659
$210$	8.19567i	0.565555i
$211$	−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
$213$	1.26205i	0.0864739i
$214$	18.0519i	1.23400i
$215$	1.85862i	0.126757i
$216$	− 3.69202i	− 0.251210i
$217$	9.42758	0.639986
$218$	6.09783	0.412997
$219$	− 12.9554i	− 0.875444i
$220$	−8.49396	−0.572663
$221$	0	0
$222$	10.0194	0.672457
$223$	− 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
$225$	−9.57002	−0.638002
$226$	− 12.2010i	− 0.811602i
$227$	13.8049i	0.916265i	0.888884	+	0.458132i	$0.151481\pi$
$227$	13.8049i	0.916265i	−0.888884	+	0.458132i	$0.848519\pi$
$228$	− 1.86831i	− 0.123732i
$229$	− 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
$231$	−5.35988	−0.352654
$232$	3.78017i	0.248180i
$233$	−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
$236$	− 10.1468i	− 0.660497i
$237$	31.7211	2.06051
$238$	6.61596	0.428849
$239$	0.944378i	0.0610867i	0.999533	+	0.0305434i	$0.00972377\pi$
$239$	0.944378i	0.0610867i	−0.999533	+	0.0305434i	$0.990276\pi$
$240$	− 7.38404i	− 0.476638i
$241$	− 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
$243$	11.7875	0.756166
$244$	0.439665	0.0281467
$245$	20.7875i	1.32806i
$246$	14.7235	0.938735
$247$	0	0
$248$	−8.49396	−0.539367
$249$	− 1.86831i	− 0.118400i
$250$	10.7681	0.681034
$251$	−16.2543	−1.02596	−0.512980	−	0.858400i	$-0.671459\pi$
$251$	−16.2543	−1.02596	−0.512980	+	0.858400i	$0.671459\pi$
$252$	− 1.32975i	− 0.0837663i
$253$	7.97584i	0.501437i
$254$	− 11.4276i	− 0.717030i
$255$	44.0146i	2.75630i
$256$	1.00000	0.0625000
$257$	22.4373	1.39960	0.699799	−	0.714340i	$-0.253273\pi$
$257$	22.4373	1.39960	0.699799	+	0.714340i	$0.253273\pi$
$258$	− 1.05669i	− 0.0657865i
$259$	−5.42758	−0.337254
$260$	0	0
$261$	−4.52888	−0.280330
$262$	2.29590i	0.141841i
$263$	−10.4940	−0.647085	−0.323543	−	0.946214i	$-0.604874\pi$
$263$	−10.4940	−0.647085	−0.323543	+	0.946214i	$0.604874\pi$
$264$	4.82908	0.297210
$265$	− 12.1957i	− 0.749174i
$266$	1.01208i	0.0620547i
$267$	− 7.68473i	− 0.470298i
$268$	2.14675i	0.131134i
$269$	26.4155	1.61058	0.805291	−	0.592880i	$-0.202009\pi$
$269$	26.4155	1.61058	0.805291	+	0.592880i	$0.202009\pi$
$270$	−13.3056	−0.809752
$271$	− 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
$276$	−6.93362	−0.417355
$277$	−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
$279$	− 10.1763i	− 0.609239i
$280$	4.00000i	0.239046i
$281$	25.0030i	1.49155i	0.666196	+	0.745776i	$0.267921\pi$
$281$	25.0030i	1.49155i	−0.666196	+	0.745776i	$0.732079\pi$
$282$	14.3177	0.852605
$283$	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
$285$	−6.73317	−0.398839
$286$	0	0
$287$	−7.97584	−0.470799
$288$	1.19806i	0.0705965i
$289$	18.5308	1.09005
$290$	13.6233	0.799985
$291$	− 30.0670i	− 1.76256i
$292$	− 6.32304i	− 0.370028i
$293$	− 1.87800i	− 0.109714i	−0.998494	−	0.0548570i	$-0.982530\pi$
$293$	− 1.87800i	− 0.109714i	0.998494	−	0.0548570i	$-0.0174703\pi$
$294$	− 11.8183i	− 0.689259i
$295$	−36.5676	−2.12905
$296$	4.89008	0.284230
$297$	− 8.70171i	− 0.504924i
$298$	18.6896	1.08266
$299$	0	0
$300$	−16.3666	−0.944925
$301$	0.572417i	0.0329935i
$302$	0.317667	0.0182797
$303$	17.9651	1.03207
$304$	− 0.911854i	− 0.0522984i
$305$	− 1.58450i	− 0.0907281i
$306$	− 7.14138i	− 0.408245i
$307$	− 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
$309$	38.5435	2.19266
$310$	30.6112i	1.73860i
$311$	5.38404	0.305301	0.152651	−	0.988280i	$-0.451219\pi$
$311$	5.38404	0.305301	0.152651	+	0.988280i	$0.451219\pi$
$312$	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
$315$	−4.79225	−0.270013
$316$	15.4819	0.870924
$317$	11.5013i	0.645975i	0.946403	+	0.322987i	$0.104687\pi$
$317$	11.5013i	0.645975i	−0.946403	+	0.322987i	$0.895313\pi$
$318$	6.93362i	0.388818i
$319$	8.90946i	0.498834i
$320$	− 3.60388i	− 0.201463i
$321$	−36.9869	−2.06440
$322$	3.75600	0.209314
$323$	5.43535i	0.302431i
$324$	11.1588	0.619935
$325$	0	0
$326$	4.33273	0.239968
$327$	12.4940i	0.690918i
$328$	7.18598	0.396779
$329$	−7.75600	−0.427602
$330$	− 17.4034i	− 0.958027i
$331$	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
$333$	5.85862i	0.321051i
$334$	−14.0000	−0.766046
$335$	7.73663	0.422697
$336$	− 2.27413i	− 0.124064i
$337$	−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
$340$	21.4819i	1.16502i
$341$	−20.0194	−1.08411
$342$	1.09246	0.0590734
$343$	14.1715i	0.765189i
$344$	− 0.515729i	− 0.0278063i
$345$	24.9879i	1.34530i
$346$	10.9879i	0.590714i
$347$	−6.41550	−0.344402	−0.172201	−	0.985062i	$-0.555088\pi$
$347$	−6.41550	−0.344402	−0.172201	+	0.985062i	$0.555088\pi$
$348$	−7.74525	−0.415189
$349$	− 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	$0.0364130\pi$
$353$	4.28919i	0.228291i	−0.993464	+	0.114145i	$0.963587\pi$
$354$	20.7899	1.10497
$355$	2.21983	0.117816
$356$	− 3.75063i	− 0.198783i
$357$	13.5555i	0.717436i
$358$	− 4.65519i	− 0.246034i
$359$	15.5060i	0.818378i	0.912450	+	0.409189i	$0.134188\pi$
$359$	15.5060i	0.818378i	−0.912450	+	0.409189i	$0.865812\pi$
$360$	4.31767	0.227561
$361$	18.1685	0.956238
$362$	1.06638i	0.0560475i
$363$	−11.1564	−0.585561
$364$	0	0
$365$	−22.7875	−1.19275
$366$	0.900837i	0.0470875i
$367$	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
$369$	8.60925i	0.448180i
$370$	− 17.6233i	− 0.916189i
$371$	− 3.75600i	− 0.195002i
$372$	− 17.4034i	− 0.902325i
$373$	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
$375$	22.0629i	1.13932i
$376$	6.98792	0.360374
$377$	0	0
$378$	−4.09783	−0.210770
$379$	− 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
$381$	23.4142	1.19954
$382$	0.890084i	0.0455406i
$383$	− 11.1207i	− 0.568240i	−0.958789	−	0.284120i	$-0.908299\pi$
$383$	− 11.1207i	− 0.568240i	0.958789	−	0.284120i	$-0.0917014\pi$
$384$	2.04892i	0.104558i
$385$	9.42758i	0.480474i
$386$	−16.2174	−0.825446
$387$	0.617876	0.0314084
$388$	− 14.6746i	− 0.744988i
$389$	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
$392$	− 5.76809i	− 0.291332i
$393$	−4.70410	−0.237291
$394$	−11.4711	−0.577907
$395$	− 55.7948i	− 2.80734i
$396$	2.82371i	0.141897i
$397$	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
$399$	−2.07367	−0.103813
$400$	−7.98792	−0.399396
$401$	− 17.4426i	− 0.871044i	−0.900178	−	0.435522i	$-0.856564\pi$
$401$	− 17.4426i	− 0.871044i	0.900178	−	0.435522i	$-0.143436\pi$
$402$	−4.39852	−0.219378
$403$	0	0
$404$	8.76809	0.436229
$405$	− 40.2150i	− 1.99830i
$406$	4.19567	0.208228
$407$	11.5254	0.571294
$408$	− 12.2131i	− 0.604640i
$409$	− 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
$411$	− 18.6049i	− 0.917714i
$412$	18.8116	0.926782
$413$	−11.2620	−0.554169
$414$	− 4.05429i	− 0.199258i
$415$	−3.28621	−0.161314
$416$	0	0
$417$	−38.7351	−1.89687
$418$	− 2.14914i	− 0.105118i
$419$	9.97584	0.487352	0.243676	−	0.969857i	$-0.421647\pi$
$419$	9.97584	0.487352	0.243676	+	0.969857i	$0.421647\pi$
$420$	−8.19567	−0.399908
$421$	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
$423$	8.37196i	0.407059i
$424$	3.38404i	0.164344i
$425$	47.6142	2.30963
$426$	−1.26205	−0.0611463
$427$	− 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	$-0.884052\pi$
$431$	− 14.7922i	− 0.712518i	0.934387	−	0.356259i	$-0.115948\pi$
$432$	3.69202	0.177632
$433$	−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
$435$	27.9129i	1.33832i
$436$	6.09783i	0.292033i
$437$	3.08575i	0.147612i
$438$	12.9554	0.619033
$439$	−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
$441$	6.91053	0.329073
$442$	0	0
$443$	17.4077	0.827066	0.413533	−	0.910489i	$-0.364295\pi$
$443$	17.4077	0.827066	0.413533	+	0.910489i	$0.364295\pi$
$444$	10.0194i	0.475499i
$445$	−13.5168	−0.640758
$446$	12.9879	0.614996
$447$	38.2935i	1.81122i
$448$	− 1.10992i	− 0.0524386i
$449$	− 34.1497i	− 1.61163i	−0.592170	−	0.805813i	$-0.701729\pi$
$449$	− 34.1497i	− 1.61163i	0.592170	−	0.805813i	$-0.298271\pi$
$450$	− 9.57002i	− 0.451135i
$451$	16.9366	0.797514
$452$	12.2010	0.573889
$453$	0.650874i	0.0305807i
$454$	−13.8049	−0.647897
$455$	0	0
$456$	1.86831	0.0874918
$457$	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
$459$	−22.0073	−1.02721
$460$	12.1957i	0.568626i
$461$	− 0.733169i	− 0.0341471i	−0.999854	−	0.0170735i	$-0.994565\pi$
$461$	− 0.733169i	− 0.0341471i	0.999854	−	0.0170735i	$-0.00543494\pi$
$462$	− 5.35988i	− 0.249364i
$463$	− 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
$465$	−62.7198	−2.90856
$466$	− 9.77479i	− 0.452808i
$467$	−30.2446	−1.39955	−0.699776	−	0.714362i	$-0.746717\pi$
$467$	−30.2446	−1.39955	−0.699776	+	0.714362i	$0.746717\pi$
$468$	0	0
$469$	2.38271	0.110024
$470$	− 25.1836i	− 1.16163i
$471$	−38.7042	−1.78340
$472$	10.1468	0.467042
$473$	− 1.21552i	− 0.0558897i
$474$	31.7211i	1.45700i
$475$	7.28382i	0.334204i
$476$	6.61596i	0.303242i
$477$	−4.05429	−0.185633
$478$	−0.944378	−0.0431948
$479$	36.7198i	1.67777i	0.544310	+	0.838884i	$0.316792\pi$
$479$	36.7198i	1.67777i	−0.544310	+	0.838884i	$0.683208\pi$
$480$	7.38404	0.337034
$481$	0	0
$482$	0.219833	0.0100131
$483$	7.69574i	0.350168i
$484$	−5.44504	−0.247502
$485$	−52.8853	−2.40140
$486$	11.7875i	0.534690i
$487$	− 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
$489$	8.87741i	0.401450i
$490$	−20.7875	−0.939082
$491$	−30.4295	−1.37326	−0.686632	−	0.727005i	$-0.740912\pi$
$491$	−30.4295	−1.37326	−0.686632	+	0.727005i	$0.740912\pi$
$492$	14.7235i	0.663786i
$493$	22.5327	1.01482
$494$	0	0
$495$	10.1763	0.457390
$496$	− 8.49396i	− 0.381390i
$497$	0.683661	0.0306664
$498$	1.86831	0.0837211
$499$	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
$501$	− 28.6848i	− 1.28154i
$502$	− 16.2543i	− 0.725464i
$503$	−41.9711	−1.87140	−0.935698	−	0.352801i	$-0.885229\pi$
$503$	−41.9711	−1.87140	−0.935698	+	0.352801i	$0.885229\pi$
$504$	1.32975	0.0592317
$505$	− 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	$0.00644992\pi$
$509$	0.914247i	0.0405233i	−0.999795	+	0.0202616i	$0.993550\pi$
$510$	−44.0146	−1.94900
$511$	−7.01805	−0.310460
$512$	1.00000i	0.0441942i
$513$	− 3.36658i	− 0.148638i
$514$	22.4373i	0.989666i
$515$	− 67.7948i	− 2.98739i
$516$	1.05669	0.0465181
$517$	16.4698	0.724341
$518$	− 5.42758i	− 0.238474i
$519$	−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	$0.523107\pi$
$522$	− 4.52888i	− 0.198224i
$523$	−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
$525$	18.1655i	0.792809i
$526$	− 10.4940i	− 0.457558i
$527$	50.6305i	2.20550i
$528$	4.82908i	0.210159i
$529$	−11.5483	−0.502098
$530$	12.1957	0.529746
$531$	12.1564i	0.527545i
$532$	−1.01208	−0.0438793
$533$	0	0
$534$	7.68473	0.332551
$535$	65.0568i	2.81265i
$536$	−2.14675	−0.0927256
$537$	9.53809	0.411599
$538$	26.4155i	1.13885i
$539$	− 13.5948i	− 0.585569i
$540$	− 13.3056i	− 0.572581i
$541$	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
$543$	−2.18492	−0.0937637
$544$	− 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
$549$	−0.526746	−0.0224810
$550$	−18.8267	−0.802773
$551$	3.44696i	0.146845i
$552$	− 6.93362i	− 0.295115i
$553$	− 17.1836i	− 0.730720i
$554$	− 2.17629i	− 0.0924618i
$555$	36.1086	1.53272
$556$	−18.9051	−0.801757
$557$	− 27.1508i	− 1.15042i	−0.818007	−	0.575208i	$-0.804921\pi$
$557$	− 27.1508i	− 1.15042i	0.818007	−	0.575208i	$-0.195079\pi$
$558$	10.1763	0.430797
$559$	0	0
$560$	−4.00000	−0.169031
$561$	− 28.7851i	− 1.21531i
$562$	−25.0030	−1.05469
$563$	6.52409	0.274958	0.137479	−	0.990505i	$-0.456100\pi$
$563$	6.52409	0.274958	0.137479	+	0.990505i	$0.456100\pi$
$564$	14.3177i	0.602883i
$565$	− 43.9711i	− 1.84988i
$566$	16.3153i	0.685782i
$567$	− 12.3854i	− 0.520137i
$568$	−0.615957	−0.0258450
$569$	−7.30021	−0.306041	−0.153020	−	0.988223i	$-0.548900\pi$
$569$	−7.30021	−0.306041	−0.153020	+	0.988223i	$0.548900\pi$
$570$	− 6.73317i	− 0.282021i
$571$	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
$573$	−1.82371	−0.0761865
$574$	− 7.97584i	− 0.332905i
$575$	27.0315	1.12729
$576$	−1.19806	−0.0499193
$577$	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
$579$	− 33.2282i	− 1.38092i
$580$	13.6233i	0.565675i
$581$	−1.01208	−0.0419882
$582$	30.0670	1.24632
$583$	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	$-0.848643\pi$
$587$	− 22.1825i	− 0.915571i	0.889063	−	0.457785i	$-0.151357\pi$
$588$	11.8183	0.487380
$589$	−7.74525	−0.319137
$590$	− 36.5676i	− 1.50547i
$591$	− 23.5034i	− 0.966800i
$592$	4.89008i	0.200981i
$593$	− 3.98493i	− 0.163642i	−0.996647	−	0.0818208i	$-0.973926\pi$
$593$	− 3.98493i	− 0.163642i	0.996647	−	0.0818208i	$-0.0260735\pi$
$594$	8.70171	0.357035
$595$	23.8431	0.977471
$596$	18.6896i	0.765557i
$597$	−7.78495	−0.318617
$598$	0	0
$599$	33.2379	1.35806	0.679032	−	0.734109i	$-0.262400\pi$
$599$	33.2379	1.35806	0.679032	+	0.734109i	$0.262400\pi$
$600$	− 16.3666i	− 0.668163i
$601$	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
$603$	− 2.57194i	− 0.104738i
$604$	0.317667i	0.0129257i
$605$	19.6233i	0.797799i
$606$	17.9651i	0.729782i
$607$	−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
$609$	8.59658i	0.348351i
$610$	1.58450	0.0641545
$611$	0	0
$612$	7.14138	0.288673
$613$	− 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
$615$	53.0616	2.13965
$616$	− 2.61596i	− 0.105400i
$617$	− 2.01879i	− 0.0812733i	−0.999174	−	0.0406366i	$-0.987061\pi$
$617$	− 2.01879i	− 0.0812733i	0.999174	−	0.0406366i	$-0.0129386\pi$
$618$	38.5435i	1.55045i
$619$	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
$621$	−12.4940	−0.501365
$622$	5.38404i	0.215880i
$623$	−4.16288	−0.166782
$624$	0	0
$625$	−1.13275	−0.0453101
$626$	18.9487i	0.757342i
$627$	4.40342	0.175856
$628$	−18.8901	−0.753796
$629$	− 29.1487i	− 1.16223i
$630$	− 4.79225i	− 0.190928i
$631$	− 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
$633$	51.3183	2.03972
$634$	−11.5013	−0.456773
$635$	− 41.1836i	− 1.63432i
$636$	−6.93362	−0.274936
$637$	0	0
$638$	−8.90946	−0.352729
$639$	− 0.737955i	− 0.0291930i
$640$	3.60388	0.142456
$641$	41.6015	1.64316	0.821580	−	0.570093i	$-0.193093\pi$
$641$	41.6015	1.64316	0.821580	+	0.570093i	$0.193093\pi$
$642$	− 36.9869i	− 1.45975i
$643$	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
$645$	− 3.80817i	− 0.149946i
$646$	−5.43535	−0.213851
$647$	−35.8345	−1.40880	−0.704399	−	0.709804i	$-0.748783\pi$
$647$	−35.8345	−1.40880	−0.704399	+	0.709804i	$0.748783\pi$
$648$	11.1588i	0.438360i
$649$	23.9148	0.938740
$650$	0	0
$651$	−19.3163	−0.757067
$652$	4.33273i	0.169683i
$653$	18.5590	0.726270	0.363135	−	0.931737i	$-0.381706\pi$
$653$	18.5590	0.726270	0.363135	+	0.931737i	$0.381706\pi$
$654$	−12.4940	−0.488552
$655$	8.27413i	0.323297i
$656$	7.18598i	0.280565i
$657$	7.57540i	0.295545i
$658$	− 7.75600i	− 0.302361i
$659$	−3.97525	−0.154854	−0.0774268	−	0.996998i	$-0.524670\pi$
$659$	−3.97525	−0.154854	−0.0774268	+	0.996998i	$0.524670\pi$
$660$	17.4034	0.677427
$661$	− 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
$666$	−5.85862	−0.227017
$667$	12.7922	0.495318
$668$	− 14.0000i	− 0.541676i
$669$	26.6112i	1.02885i
$670$	7.73663i	0.298892i
$671$	1.03624i	0.0400038i
$672$	2.27413	0.0877263
$673$	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
$675$	−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	$0.665977\pi$
$678$	24.9989i	0.960078i
$679$	−16.2875	−0.625058
$680$	−21.4819	−0.823792
$681$	− 28.2851i	− 1.08389i
$682$	− 20.0194i	− 0.766582i
$683$	− 37.5472i	− 1.43670i	−0.695680	−	0.718352i	$-0.744897\pi$
$683$	− 37.5472i	− 1.43670i	0.695680	−	0.718352i	$-0.255103\pi$
$684$	1.09246i	0.0417712i
$685$	−32.7245	−1.25034
$686$	−14.1715	−0.541071
$687$	23.6862i	0.903684i
$688$	0.515729	0.0196620
$689$	0	0
$690$	−24.9879	−0.951274
$691$	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
$693$	3.13408	0.119054
$694$	− 6.41550i	− 0.243529i
$695$	68.1318i	2.58439i
$696$	− 7.74525i	− 0.293583i
$697$	− 42.8340i	− 1.62245i
$698$	1.08575	0.0410964
$699$	20.0277	0.757519
$700$	8.86592i	0.335100i
$701$	−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
$704$	2.35690i	0.0888289i
$705$	51.5991	1.94333
$706$	−4.28919	−0.161426
$707$	− 9.73184i	− 0.366004i
$708$	20.7899i	0.781331i
$709$	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
$711$	−18.5483	−0.695614
$712$	3.75063	0.140561
$713$	28.7439i	1.07647i
$714$	−13.5555	−0.507304
$715$	0	0
$716$	4.65519	0.173972
$717$	− 1.93495i	− 0.0722621i
$718$	−15.5060	−0.578680
$719$	15.3056	0.570802	0.285401	−	0.958408i	$-0.407873\pi$
$719$	15.3056	0.570802	0.285401	+	0.958408i	$0.407873\pi$
$720$	4.31767i	0.160910i
$721$	− 20.8793i	− 0.777587i
$722$	18.1685i	0.676162i
$723$	0.450419i	0.0167513i
$724$	−1.06638	−0.0396315
$725$	30.1957	1.12144
$726$	− 11.1564i	− 0.414054i
$727$	−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
$729$	9.32496	0.345369
$730$	− 22.7875i	− 0.843402i
$731$	−3.07415	−0.113701
$732$	−0.900837	−0.0332959
$733$	− 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
$735$	− 42.5918i	− 1.57102i
$736$	− 3.38404i	− 0.124737i
$737$	−5.05967	−0.186375
$738$	−8.60925	−0.316911
$739$	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	$-0.997574\pi$
$743$	− 0.415502i	− 0.0152433i	0.999971	−	0.00762164i	$-0.00242607\pi$
$744$	17.4034	0.638040
$745$	67.3551	2.46770
$746$	8.19567i	0.300065i
$747$	1.09246i	0.0399709i
$748$	− 14.0489i	− 0.513679i
$749$	20.0361i	0.732103i
$750$	−22.0629	−0.805624
$751$	−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
$753$	33.3037	1.21365
$754$	0	0
$755$	1.14483	0.0416647
$756$	− 4.09783i	− 0.149037i
$757$	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
$759$	− 16.3418i	− 0.593171i
$760$	− 3.28621i	− 0.119203i
$761$	− 42.4306i	− 1.53811i	−0.639184	−	0.769053i	$-0.720728\pi$
$761$	− 42.4306i	− 1.53811i	0.639184	−	0.769053i	$-0.279272\pi$
$762$	23.4142i	0.848206i
$763$	6.76809	0.245021
$764$	−0.890084	−0.0322021
$765$	− 25.7366i	− 0.930510i
$766$	11.1207	0.401806
$767$	0	0
$768$	−2.04892	−0.0739339
$769$	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
$771$	−45.9721	−1.65565
$772$	− 16.2174i	− 0.583678i
$773$	− 5.85384i	− 0.210548i	−0.994443	−	0.105274i	$-0.966428\pi$
$773$	− 5.85384i	− 0.210548i	0.994443	−	0.105274i	$-0.0335720\pi$
$774$	0.617876i	0.0222091i
$775$	67.8491i	2.43721i
$776$	14.6746	0.526786
$777$	11.1207	0.398952
$778$	8.04354i	0.288375i
$779$	6.55257	0.234770
$780$	0	0
$781$	−1.45175	−0.0519476
$782$	20.1715i	0.721332i
$783$	−13.9565	−0.498763
$784$	5.76809	0.206003
$785$	68.0775i	2.42979i
$786$	− 4.70410i	− 0.167790i
$787$	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
$789$	21.5013	0.765465
$790$	55.7948	1.98509
$791$	− 13.5421i	− 0.481503i
$792$	−2.82371	−0.100336
$793$	0	0
$794$	−21.9081	−0.777491
$795$	24.9879i	0.886230i
$796$	−3.79954	−0.134671
$797$	−35.8103	−1.26847	−0.634233	−	0.773142i	$-0.718684\pi$
$797$	−35.8103	−1.26847	−0.634233	+	0.773142i	$0.718684\pi$
$798$	− 2.07367i	− 0.0734072i
$799$	− 41.6534i	− 1.47359i
$800$	− 7.98792i	− 0.282416i
$801$	4.49349i	0.158769i
$802$	17.4426	0.615921
$803$	14.9028	0.525907
$804$	− 4.39852i	− 0.155124i
$805$	13.5362	0.477087
$806$	0	0
$807$	−54.1232	−1.90523
$808$	8.76809i	0.308460i
$809$	28.3744	0.997589	0.498795	−	0.866720i	$-0.333776\pi$
$809$	28.3744	0.997589	0.498795	+	0.866720i	$0.333776\pi$
$810$	40.2150	1.41301
$811$	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
$813$	45.1379i	1.58306i
$814$	11.5254i	0.403966i
$815$	15.6146	0.546957
$816$	12.2131	0.427545
$817$	− 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	$-0.968547\pi$
$821$	− 5.65338i	− 0.197304i	0.995122	−	0.0986522i	$-0.0314531\pi$
$822$	18.6049	0.648922
$823$	−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
$825$	− 38.5743i	− 1.34299i
$826$	− 11.2620i	− 0.391857i
$827$	5.40283i	0.187875i	0.995578	+	0.0939374i	$0.0299454\pi$
$827$	5.40283i	0.187875i	−0.995578	+	0.0939374i	$0.970055\pi$
$828$	4.05429	0.140896
$829$	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
$831$	4.45904	0.154682
$832$	0	0
$833$	−34.3822	−1.19127
$834$	− 38.7351i	− 1.34129i
$835$	−50.4543	−1.74604
$836$	2.14914	0.0743297
$837$	− 31.3599i	− 1.08396i
$838$	9.97584i	0.344610i
$839$	3.98062i	0.137426i	0.997636	+	0.0687132i	$0.0218893\pi$
$839$	3.98062i	0.137426i	−0.997636	+	0.0687132i	$0.978111\pi$
$840$	− 8.19567i	− 0.282777i
$841$	−14.7103	−0.507253
$842$	−0.615957	−0.0212273
$843$	− 51.2290i	− 1.76442i
$844$	25.0465	0.862137
$845$	0	0
$846$	−8.37196	−0.287834
$847$	6.04354i	0.207659i
$848$	−3.38404	−0.116209
$849$	−33.4286	−1.14727
$850$	47.6142i	1.63315i
$851$	− 16.5483i	− 0.567267i
$852$	− 1.26205i	− 0.0432370i
$853$	− 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
$855$	3.93708	0.134645
$856$	− 18.0519i	− 0.617001i
$857$	−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	$-0.417483\pi$
$859$	15.0261	0.512683	0.256342	+	0.966586i	$0.417483\pi$
$860$	− 1.85862i	− 0.0633786i
$861$	16.3418	0.556928
$862$	14.7922	0.503826
$863$	6.21121i	0.211432i	0.994396	+	0.105716i	$0.0337134\pi$
$863$	6.21121i	0.211432i	−0.994396	+	0.105716i	$0.966287\pi$
$864$	3.69202i	0.125605i
$865$	39.5991i	1.34641i
$866$	− 16.5321i	− 0.561784i
$867$	−37.9681	−1.28946
$868$	−9.42758	−0.319993
$869$	36.4892i	1.23781i
$870$	−27.9129	−0.946337
$871$	0	0
$872$	−6.09783	−0.206499
$873$	17.5810i	0.595028i
$874$	−3.08575	−0.104377
$875$	11.9517	0.404040
$876$	12.9554i	0.437722i
$877$	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
$879$	3.84787i	0.129785i
$880$	8.49396	0.286331
$881$	−26.7832	−0.902347	−0.451174	−	0.892436i	$-0.648994\pi$
$881$	−26.7832	−0.902347	−0.451174	+	0.892436i	$0.648994\pi$
$882$	6.91053i	0.232690i
$883$	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
$885$	74.9241	2.51854
$886$	17.4077i	0.584824i
$887$	−23.9866	−0.805391	−0.402695	−	0.915334i	$-0.631927\pi$
$887$	−23.9866	−0.805391	−0.402695	+	0.915334i	$0.631927\pi$
$888$	−10.0194	−0.336228
$889$	− 12.6837i	− 0.425396i
$890$	− 13.5168i	− 0.453084i
$891$	26.3002i	0.881090i
$892$	12.9879i	0.434868i
$893$	6.37196	0.213230
$894$	−38.2935	−1.28073
$895$	− 16.7767i	− 0.560784i
$896$	1.10992	0.0370797
$897$	0	0
$898$	34.1497	1.13959
$899$	32.1086i	1.07088i
$900$	9.57002	0.319001
$901$	20.1715	0.672010
$902$	16.9366i	0.563927i
$903$	− 1.17283i	− 0.0390295i
$904$	12.2010i	0.405801i
$905$	3.84309i	0.127748i
$906$	−0.650874	−0.0216238
$907$	−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
$909$	−10.5047	−0.348419
$910$	0	0
$911$	−35.8866	−1.18898	−0.594488	−	0.804104i	$-0.702645\pi$
$911$	−35.8866	−1.18898	−0.594488	+	0.804104i	$0.702645\pi$
$912$	1.86831i	0.0618660i
$913$	2.14914	0.0711263
$914$	−9.40342	−0.311037
$915$	3.24651i	0.107326i
$916$	11.5603i	0.381964i
$917$	2.54825i	0.0841507i
$918$	− 22.0073i	− 0.726349i
$919$	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
$921$	49.1333i	1.61900i
$922$	0.733169	0.0241456
$923$	0	0
$924$	5.35988	0.176327
$925$	− 39.0616i	− 1.28434i
$926$	7.24267	0.238009
$927$	−22.5375	−0.740229
$928$	− 3.78017i	− 0.124090i
$929$	54.2583i	1.78016i	0.455806	+	0.890079i	$0.349351\pi$
$929$	54.2583i	1.78016i	−0.455806	+	0.890079i	$0.650649\pi$
$930$	− 62.7198i	− 2.05666i
$931$	− 5.25965i	− 0.172378i
$932$	9.77479	0.320184
$933$	−11.0315	−0.361154
$934$	− 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
$939$	−38.8243	−1.26698
$940$	25.1836	0.821398
$941$	41.7017i	1.35944i	0.733473	+	0.679718i	$0.237898\pi$
$941$	41.7017i	1.35944i	−0.733473	+	0.679718i	$0.762102\pi$
$942$	− 38.7042i	− 1.26105i
$943$	− 24.3177i	− 0.791892i
$944$	10.1468i	0.330249i
$945$	−14.7681	−0.480406
$946$	1.21552	0.0395200
$947$	3.00106i	0.0975215i	0.998810	+	0.0487608i	$0.0155272\pi$
$947$	3.00106i	0.0975215i	−0.998810	+	0.0487608i	$0.984473\pi$
$948$	−31.7211	−1.03025
$949$	0	0
$950$	−7.28382	−0.236318
$951$	− 23.5651i	− 0.764151i
$952$	−6.61596	−0.214424
$953$	−38.1450	−1.23564	−0.617818	−	0.786321i	$-0.711983\pi$
$953$	−38.1450	−1.23564	−0.617818	+	0.786321i	$0.711983\pi$
$954$	− 4.05429i	− 0.131263i
$955$	3.20775i	0.103800i
$956$	− 0.944378i	− 0.0305434i
$957$	− 18.2547i	− 0.590092i
$958$	−36.7198	−1.18636
$959$	−10.0785	−0.325450
$960$	7.38404i	0.238319i
$961$	−41.1473	−1.32733
$962$	0	0
$963$	21.6273	0.696930
$964$	0.219833i	0.00708033i
$965$	−58.4456	−1.88143
$966$	−7.69574	−0.247606
$967$	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
$969$	− 11.1366i	− 0.357759i
$970$	− 52.8853i	− 1.69804i
$971$	3.13647	0.100654	0.0503271	−	0.998733i	$-0.483974\pi$
$971$	3.13647	0.100654	0.0503271	+	0.998733i	$0.483974\pi$
$972$	−11.7875	−0.378083
$973$	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	$-0.805370\pi$
$977$	− 35.8864i	− 1.14811i	0.818818	−	0.574053i	$-0.194630\pi$
$978$	−8.87741	−0.283868
$979$	8.83984	0.282522
$980$	− 20.7875i	− 0.664031i
$981$	− 7.30559i	− 0.233249i
$982$	− 30.4295i	− 0.971044i
$983$	− 30.4370i	− 0.970790i	−0.874295	−	0.485395i	$-0.838676\pi$
$983$	− 30.4370i	− 0.970790i	0.874295	−	0.485395i	$-0.161324\pi$
$984$	−14.7235	−0.469367
$985$	−41.3405	−1.31722
$986$	22.5327i	0.717588i
$987$	15.8914	0.505829
$988$	0	0
$989$	−1.74525	−0.0554957
$990$	10.1763i	0.323424i
$991$	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
$993$	− 70.9154i	− 2.25043i
$994$	0.683661i	0.0216844i
$995$	13.6931i	0.434100i
$996$	1.86831i	0.0591998i
$997$	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
$999$	18.0543i	0.571213i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.2.b.d.337.4		6
3.2	odd	2		3042.2.b.n.1351.1		6
4.3	odd	2		2704.2.f.m.337.6		6
13.2	odd	12		338.2.c.h.191.3		6
13.3	even	3		338.2.e.e.147.3		12
13.4	even	6		338.2.e.e.23.3		12
13.5	odd	4		338.2.a.h.1.1	yes	3
13.6	odd	12		338.2.c.h.315.3		6
13.7	odd	12		338.2.c.i.315.3		6
13.8	odd	4		338.2.a.g.1.1	✓	3
13.9	even	3		338.2.e.e.23.6		12
13.10	even	6		338.2.e.e.147.6		12
13.11	odd	12		338.2.c.i.191.3		6
13.12	even	2	inner	338.2.b.d.337.1		6
39.5	even	4		3042.2.a.z.1.1		3
39.8	even	4		3042.2.a.bi.1.3		3
39.38	odd	2		3042.2.b.n.1351.6		6
52.31	even	4		2704.2.a.w.1.3		3
52.47	even	4		2704.2.a.v.1.3		3
52.51	odd	2		2704.2.f.m.337.5		6
65.34	odd	4		8450.2.a.bx.1.3		3
65.44	odd	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	13.8	odd	4
338.2.a.h.1.1	yes	3	13.5	odd	4
338.2.b.d.337.1		6	13.12	even	2	inner
338.2.b.d.337.4		6	1.1	even	1	trivial
338.2.c.h.191.3		6	13.2	odd	12
338.2.c.h.315.3		6	13.6	odd	12
338.2.c.i.191.3		6	13.11	odd	12
338.2.c.i.315.3		6	13.7	odd	12
338.2.e.e.23.3		12	13.4	even	6
338.2.e.e.23.6		12	13.9	even	3
338.2.e.e.147.3		12	13.3	even	3
338.2.e.e.147.6		12	13.10	even	6
2704.2.a.v.1.3		3	52.47	even	4
2704.2.a.w.1.3		3	52.31	even	4
2704.2.f.m.337.5		6	52.51	odd	2
2704.2.f.m.337.6		6	4.3	odd	2
3042.2.a.z.1.1		3	39.5	even	4
3042.2.a.bi.1.3		3	39.8	even	4
3042.2.b.n.1351.1		6	3.2	odd	2
3042.2.b.n.1351.6		6	39.38	odd	2
8450.2.a.bn.1.3		3	65.44	odd	4
8450.2.a.bx.1.3		3	65.34	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	338.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.04892 q^{3} -1.00000 q^{4} +3.60388i q^{5} -2.04892i q^{6} +1.10992i q^{7} -1.00000i q^{8} +1.19806 q^{9} -3.60388 q^{10} -2.35690i q^{11} +2.04892 q^{12} -1.10992 q^{14} -7.38404i q^{15} +1.00000 q^{16} -5.96077 q^{17} +1.19806i q^{18} -0.911854i q^{19} -3.60388i q^{20} -2.27413i q^{21} +2.35690 q^{22} -3.38404 q^{23} +2.04892i q^{24} -7.98792 q^{25} +3.69202 q^{27} -1.10992i q^{28} -3.78017 q^{29} +7.38404 q^{30} -8.49396i q^{31} +1.00000i q^{32} +4.82908i q^{33} -5.96077i q^{34} -4.00000 q^{35} -1.19806 q^{36} +4.89008i q^{37} +0.911854 q^{38} +3.60388 q^{40} +7.18598i q^{41} +2.27413 q^{42} +0.515729 q^{43} +2.35690i q^{44} +4.31767i q^{45} -3.38404i q^{46} +6.98792i q^{47} -2.04892 q^{48} +5.76809 q^{49} -7.98792i q^{50} +12.2131 q^{51} -3.38404 q^{53} +3.69202i q^{54} +8.49396 q^{55} +1.10992 q^{56} +1.86831i q^{57} -3.78017i q^{58} +10.1468i q^{59} +7.38404i q^{60} -0.439665 q^{61} +8.49396 q^{62} +1.32975i q^{63} -1.00000 q^{64} -4.82908 q^{66} -2.14675i q^{67} +5.96077 q^{68} +6.93362 q^{69} -4.00000i q^{70} -0.615957i q^{71} -1.19806i q^{72} +6.32304i q^{73} -4.89008 q^{74} +16.3666 q^{75} +0.911854i q^{76} +2.61596 q^{77} -15.4819 q^{79} +3.60388i q^{80} -11.1588 q^{81} -7.18598 q^{82} +0.911854i q^{83} +2.27413i q^{84} -21.4819i q^{85} +0.515729i q^{86} +7.74525 q^{87} -2.35690 q^{88} +3.75063i q^{89} -4.31767 q^{90} +3.38404 q^{92} +17.4034i q^{93} -6.98792 q^{94} +3.28621 q^{95} -2.04892i q^{96} +14.6746i q^{97} +5.76809i q^{98} -2.82371i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 6 q^{4} + 16 q^{9} - 4 q^{10} - 6 q^{12} - 8 q^{14} + 6 q^{16} - 10 q^{17} + 6 q^{22} - 10 q^{25} + 12 q^{27} - 20 q^{29} + 24 q^{30} - 24 q^{35} - 16 q^{36} - 2 q^{38} + 4 q^{40} - 8 q^{42}+ \cdots + 36 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 - 6 * q^4 + 16 * q^9 - 4 * q^10 - 6 * q^12 - 8 * q^14 + 6 * q^16 - 10 * q^17 + 6 * q^22 - 10 * q^25 + 12 * q^27 - 20 * q^29 + 24 * q^30 - 24 * q^35 - 16 * q^36 - 2 * q^38 + 4 * q^40 - 8 * q^42 - 22 * q^43 + 6 * q^48 - 6 * q^49 + 32 * q^51 + 32 * q^55 + 8 * q^56 - 8 * q^61 + 32 * q^62 - 6 * q^64 - 8 * q^66 + 10 * q^68 + 28 * q^69 - 28 * q^74 + 46 * q^75 + 36 * q^77 - 36 * q^79 - 50 * q^81 - 14 * q^82 - 20 * q^87 - 6 * q^88 + 8 * q^90 - 4 * q^94 + 36 * q^95

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