LMFDB - Embedded newform 3381.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.a (trivial)

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:	yes
Analytic conductor:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.329727$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.329727 q^{2} +1.00000 q^{3} -1.89128 q^{4} +2.73589 q^{5} -0.329727 q^{6} +1.28306 q^{8} +1.00000 q^{9} -0.902098 q^{10} -2.50407 q^{11} -1.89128 q^{12} -1.48511 q^{13} +2.73589 q^{15} +3.35950 q^{16} -0.902098 q^{17} -0.329727 q^{18} -2.50407 q^{19} -5.17434 q^{20} +0.825659 q^{22} +1.00000 q^{23} +1.28306 q^{24} +2.48511 q^{25} +0.489682 q^{26} +1.00000 q^{27} +6.68466 q^{29} -0.902098 q^{30} -1.09790 q^{31} -3.67384 q^{32} -2.50407 q^{33} +0.297446 q^{34} -1.89128 q^{36} +9.91023 q^{37} +0.825659 q^{38} -1.48511 q^{39} +3.51032 q^{40} +2.30826 q^{41} +5.83380 q^{43} +4.73589 q^{44} +2.73589 q^{45} -0.329727 q^{46} +6.74671 q^{47} +3.35950 q^{48} -0.819409 q^{50} -0.902098 q^{51} +2.80877 q^{52} -4.91648 q^{53} -0.329727 q^{54} -6.85086 q^{55} -2.50407 q^{57} -2.20411 q^{58} +7.32973 q^{59} -5.17434 q^{60} +1.00625 q^{61} +0.362008 q^{62} -5.50764 q^{64} -4.06311 q^{65} +0.825659 q^{66} +11.2929 q^{67} +1.70612 q^{68} +1.00000 q^{69} -0.362008 q^{71} +1.28306 q^{72} +5.34411 q^{73} -3.26767 q^{74} +2.48511 q^{75} +4.73589 q^{76} +0.489682 q^{78} +10.4672 q^{79} +9.19123 q^{80} +1.00000 q^{81} -0.761098 q^{82} -7.59946 q^{83} -2.46805 q^{85} -1.92356 q^{86} +6.68466 q^{87} -3.21287 q^{88} -10.6245 q^{89} -0.902098 q^{90} -1.89128 q^{92} -1.09790 q^{93} -2.22457 q^{94} -6.85086 q^{95} -3.67384 q^{96} -13.9102 q^{97} -2.50407 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 9 q^{8} + 4 q^{9} - 2 q^{10} + q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 2 q^{17} + 2 q^{18} + q^{19} - 13 q^{20} + 11 q^{22} + 4 q^{23}+ \cdots + q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 9 * q^8 + 4 * q^9 - 2 * q^10 + q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 2 * q^17 + 2 * q^18 + q^19 - 13 * q^20 + 11 * q^22 + 4 * q^23 + 9 * q^24 + 11 * q^25 + 19 * q^26 + 4 * q^27 + 2 * q^29 - 2 * q^30 - 6 * q^31 + 20 * q^32 + q^33 - 23 * q^34 + 4 * q^36 + 16 * q^37 + 11 * q^38 - 7 * q^39 - 3 * q^40 - 5 * q^41 + 9 * q^43 + 3 * q^44 - 5 * q^45 + 2 * q^46 + 21 * q^47 + 8 * q^48 - 17 * q^50 - 2 * q^51 + 24 * q^52 + 10 * q^53 + 2 * q^54 - 17 * q^55 + q^57 + q^58 + 26 * q^59 - 13 * q^60 - 2 * q^61 + 19 * q^62 + 27 * q^64 + 26 * q^65 + 11 * q^66 + 5 * q^67 - 7 * q^68 + 4 * q^69 - 19 * q^71 + 9 * q^72 - 10 * q^73 + 9 * q^74 + 11 * q^75 + 3 * q^76 + 19 * q^78 - 6 * q^79 + 24 * q^80 + 4 * q^81 + 31 * q^82 + 2 * q^83 - 7 * q^85 - 17 * q^86 + 2 * q^87 - 20 * q^88 + 17 * q^89 - 2 * q^90 + 4 * q^92 - 6 * q^93 + 44 * q^94 - 17 * q^95 + 20 * q^96 - 32 * q^97 + q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.329727	−0.233152	−0.116576	−	0.993182i	$-0.537192\pi$
$2$	−0.329727	−0.233152	−0.116576	+	0.993182i	$0.537192\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.89128	−0.945640
$5$	2.73589	1.22353	0.611764	−	0.791040i	$-0.290460\pi$
$5$	2.73589	1.22353	0.611764	+	0.791040i	$0.290460\pi$
$6$	−0.329727	−0.134611
$7$	0	0
$7$	0	0
$8$	1.28306	0.453630
$9$	1.00000	0.333333
$10$	−0.902098	−0.285269
$11$	−2.50407	−0.755005	−0.377502	−	0.926009i	$-0.623217\pi$
$11$	−2.50407	−0.755005	−0.377502	+	0.926009i	$0.623217\pi$
$12$	−1.89128	−0.545966
$13$	−1.48511	−0.411896	−0.205948	−	0.978563i	$-0.566028\pi$
$13$	−1.48511	−0.411896	−0.205948	+	0.978563i	$0.566028\pi$
$14$	0	0
$15$	2.73589	0.706405
$16$	3.35950	0.839875
$17$	−0.902098	−0.218791	−0.109396	−	0.993998i	$-0.534892\pi$
$17$	−0.902098	−0.218791	−0.109396	+	0.993998i	$0.534892\pi$
$18$	−0.329727	−0.0777174
$19$	−2.50407	−0.574473	−0.287236	−	0.957860i	$-0.592736\pi$
$19$	−2.50407	−0.574473	−0.287236	+	0.957860i	$0.592736\pi$
$20$	−5.17434	−1.15702
$21$	0	0
$22$	0.825659	0.176031
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.28306	0.261904
$25$	2.48511	0.497023
$26$	0.489682	0.0960346
$27$	1.00000	0.192450
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	−0.902098	−0.164700
$31$	−1.09790	−0.197189	−0.0985945	−	0.995128i	$-0.531435\pi$
$31$	−1.09790	−0.197189	−0.0985945	+	0.995128i	$0.531435\pi$
$32$	−3.67384	−0.649449
$33$	−2.50407	−0.435902
$34$	0.297446	0.0510116
$35$	0	0
$36$	−1.89128	−0.315213
$37$	9.91023	1.62923	0.814616	−	0.580000i	$-0.196948\pi$
$37$	9.91023	1.62923	0.814616	+	0.580000i	$0.196948\pi$
$38$	0.825659	0.133940
$39$	−1.48511	−0.237808
$40$	3.51032	0.555030
$41$	2.30826	0.360490	0.180245	−	0.983622i	$-0.442311\pi$
$41$	2.30826	0.360490	0.180245	+	0.983622i	$0.442311\pi$
$42$	0	0
$43$	5.83380	0.889645	0.444823	−	0.895619i	$-0.353267\pi$
$43$	5.83380	0.889645	0.444823	+	0.895619i	$0.353267\pi$
$44$	4.73589	0.713963
$45$	2.73589	0.407843
$46$	−0.329727	−0.0486156
$47$	6.74671	0.984109	0.492055	−	0.870564i	$-0.336246\pi$
$47$	6.74671	0.984109	0.492055	+	0.870564i	$0.336246\pi$
$48$	3.35950	0.484902
$49$	0	0
$50$	−0.819409	−0.115882
$51$	−0.902098	−0.126319
$52$	2.80877	0.389506
$53$	−4.91648	−0.675331	−0.337666	−	0.941266i	$-0.609637\pi$
$53$	−4.91648	−0.675331	−0.337666	+	0.941266i	$0.609637\pi$
$54$	−0.329727	−0.0448702
$55$	−6.85086	−0.923770
$56$	0	0
$57$	−2.50407	−0.331672
$58$	−2.20411	−0.289414
$59$	7.32973	0.954249	0.477125	−	0.878836i	$-0.341679\pi$
$59$	7.32973	0.954249	0.477125	+	0.878836i	$0.341679\pi$
$60$	−5.17434	−0.668005
$61$	1.00625	0.128837	0.0644185	−	0.997923i	$-0.479481\pi$
$61$	1.00625	0.128837	0.0644185	+	0.997923i	$0.479481\pi$
$62$	0.362008	0.0459751
$63$	0	0
$64$	−5.50764	−0.688454
$65$	−4.06311	−0.503967
$66$	0.825659	0.101632
$67$	11.2929	1.37964	0.689822	−	0.723979i	$-0.257689\pi$
$67$	11.2929	1.37964	0.689822	+	0.723979i	$0.257689\pi$
$68$	1.70612	0.206898
$69$	1.00000	0.120386
$70$	0	0
$71$	−0.362008	−0.0429624	−0.0214812	−	0.999769i	$-0.506838\pi$
$71$	−0.362008	−0.0429624	−0.0214812	+	0.999769i	$0.506838\pi$
$72$	1.28306	0.151210
$73$	5.34411	0.625481	0.312741	−	0.949839i	$-0.398753\pi$
$73$	5.34411	0.625481	0.312741	+	0.949839i	$0.398753\pi$
$74$	−3.26767	−0.379859
$75$	2.48511	0.286956
$76$	4.73589	0.543244
$77$	0	0
$78$	0.489682	0.0554456
$79$	10.4672	1.17765	0.588827	−	0.808259i	$-0.299590\pi$
$79$	10.4672	1.17765	0.588827	+	0.808259i	$0.299590\pi$
$80$	9.19123	1.02761
$81$	1.00000	0.111111
$82$	−0.761098	−0.0840492
$83$	−7.59946	−0.834149	−0.417075	−	0.908872i	$-0.636945\pi$
$83$	−7.59946	−0.834149	−0.417075	+	0.908872i	$0.636945\pi$
$84$	0	0
$85$	−2.46805	−0.267697
$86$	−1.92356	−0.207423
$87$	6.68466	0.716671
$88$	−3.21287	−0.342493
$89$	−10.6245	−1.12619	−0.563097	−	0.826391i	$-0.690390\pi$
$89$	−10.6245	−1.12619	−0.563097	+	0.826391i	$0.690390\pi$
$90$	−0.902098	−0.0950895
$91$	0	0
$92$	−1.89128	−0.197180
$93$	−1.09790	−0.113847
$94$	−2.22457	−0.229447
$95$	−6.85086	−0.702884
$96$	−3.67384	−0.374960
$97$	−13.9102	−1.41237	−0.706185	−	0.708027i	$-0.749585\pi$
$97$	−13.9102	−1.41237	−0.706185	+	0.708027i	$0.749585\pi$
$98$	0	0
$99$	−2.50407	−0.251668
$100$	−4.70005	−0.470005
$101$	−1.81502	−0.180601	−0.0903004	−	0.995915i	$-0.528783\pi$
$101$	−1.81502	−0.180601	−0.0903004	+	0.995915i	$0.528783\pi$
$102$	0.297446	0.0294516
$103$	16.6605	1.64161	0.820805	−	0.571209i	$-0.193525\pi$
$103$	16.6605	1.64161	0.820805	+	0.571209i	$0.193525\pi$
$104$	−1.90549	−0.186849
$105$	0	0
$106$	1.62110	0.157455
$107$	1.68092	0.162500	0.0812502	−	0.996694i	$-0.474109\pi$
$107$	1.68092	0.162500	0.0812502	+	0.996694i	$0.474109\pi$
$108$	−1.89128	−0.181989
$109$	11.4333	1.09511	0.547554	−	0.836771i	$-0.315559\pi$
$109$	11.4333	1.09511	0.547554	+	0.836771i	$0.315559\pi$
$110$	2.25892	0.215379
$111$	9.91023	0.940638
$112$	0	0
$113$	3.14914	0.296246	0.148123	−	0.988969i	$-0.452677\pi$
$113$	3.14914	0.296246	0.148123	+	0.988969i	$0.452677\pi$
$114$	0.825659	0.0773301
$115$	2.73589	0.255123
$116$	−12.6426	−1.17383
$117$	−1.48511	−0.137299
$118$	−2.41681	−0.222485
$119$	0	0
$120$	3.51032	0.320447
$121$	−4.72964	−0.429968
$122$	−0.331788	−0.0300387
$123$	2.30826	0.208129
$124$	2.07644	0.186470
$125$	−6.88046	−0.615407
$126$	0	0
$127$	0.449266	0.0398659	0.0199329	−	0.999801i	$-0.493655\pi$
$127$	0.449266	0.0398659	0.0199329	+	0.999801i	$0.493655\pi$
$128$	9.16370	0.809964
$129$	5.83380	0.513637
$130$	1.33972	0.117501
$131$	19.9759	1.74530	0.872649	−	0.488347i	$-0.162400\pi$
$131$	19.9759	1.74530	0.872649	+	0.488347i	$0.162400\pi$
$132$	4.73589	0.412207
$133$	0	0
$134$	−3.72357	−0.321667
$135$	2.73589	0.235468
$136$	−1.15745	−0.0992503
$137$	0.270356	0.0230981	0.0115490	−	0.999933i	$-0.496324\pi$
$137$	0.270356	0.0230981	0.0115490	+	0.999933i	$0.496324\pi$
$138$	−0.329727	−0.0280682
$139$	−4.42512	−0.375334	−0.187667	−	0.982233i	$-0.560093\pi$
$139$	−4.42512	−0.375334	−0.187667	+	0.982233i	$0.560093\pi$
$140$	0	0
$141$	6.74671	0.568176
$142$	0.119364	0.0100168
$143$	3.71883	0.310984
$144$	3.35950	0.279958
$145$	18.2885	1.51878
$146$	−1.76210	−0.145832
$147$	0	0
$148$	−18.7430	−1.54067
$149$	−3.69530	−0.302731	−0.151365	−	0.988478i	$-0.548367\pi$
$149$	−3.69530	−0.302731	−0.151365	+	0.988478i	$0.548367\pi$
$150$	−0.819409	−0.0669045
$151$	8.63174	0.702441	0.351221	−	0.936293i	$-0.385767\pi$
$151$	8.63174	0.702441	0.351221	+	0.936293i	$0.385767\pi$
$152$	−3.21287	−0.260598
$153$	−0.902098	−0.0729303
$154$	0	0
$155$	−3.00374	−0.241266
$156$	2.80877	0.224881
$157$	−9.93438	−0.792850	−0.396425	−	0.918067i	$-0.629749\pi$
$157$	−9.93438	−0.792850	−0.396425	+	0.918067i	$0.629749\pi$
$158$	−3.45133	−0.274573
$159$	−4.91648	−0.389903
$160$	−10.0512	−0.794620
$161$	0	0
$162$	−0.329727	−0.0259058
$163$	14.7082	1.15203	0.576017	−	0.817438i	$-0.304606\pi$
$163$	14.7082	1.15203	0.576017	+	0.817438i	$0.304606\pi$
$164$	−4.36558	−0.340894
$165$	−6.85086	−0.533339
$166$	2.50575	0.194484
$167$	5.47430	0.423614	0.211807	−	0.977312i	$-0.432065\pi$
$167$	5.47430	0.423614	0.211807	+	0.977312i	$0.432065\pi$
$168$	0	0
$169$	−10.7944	−0.830341
$170$	0.813782	0.0624142
$171$	−2.50407	−0.191491
$172$	−11.0333	−0.841284
$173$	−18.1727	−1.38165	−0.690823	−	0.723024i	$-0.742752\pi$
$173$	−18.1727	−1.38165	−0.690823	+	0.723024i	$0.742752\pi$
$174$	−2.20411	−0.167093
$175$	0	0
$176$	−8.41242	−0.634110
$177$	7.32973	0.550936
$178$	3.50318	0.262575
$179$	10.1860	0.761341	0.380670	−	0.924711i	$-0.375693\pi$
$179$	10.1860	0.761341	0.380670	+	0.924711i	$0.375693\pi$
$180$	−5.17434	−0.385673
$181$	−13.5186	−1.00483	−0.502416	−	0.864626i	$-0.667555\pi$
$181$	−13.5186	−1.00483	−0.502416	+	0.864626i	$0.667555\pi$
$182$	0	0
$183$	1.00625	0.0743841
$184$	1.28306	0.0945885
$185$	27.1133	1.99341
$186$	0.362008	0.0265437
$187$	2.25892	0.165188
$188$	−12.7599	−0.930613
$189$	0	0
$190$	2.25892	0.163879
$191$	−0.631742	−0.0457113	−0.0228556	−	0.999739i	$-0.507276\pi$
$191$	−0.631742	−0.0457113	−0.0228556	+	0.999739i	$0.507276\pi$
$192$	−5.50764	−0.397479
$193$	1.36826	0.0984893	0.0492447	−	0.998787i	$-0.484319\pi$
$193$	1.36826	0.0984893	0.0492447	+	0.998787i	$0.484319\pi$
$194$	4.58658	0.329297
$195$	−4.06311	−0.290966
$196$	0	0
$197$	24.7611	1.76416	0.882078	−	0.471104i	$-0.156144\pi$
$197$	24.7611	1.76416	0.882078	+	0.471104i	$0.156144\pi$
$198$	0.825659	0.0586770
$199$	−1.44470	−0.102412	−0.0512059	−	0.998688i	$-0.516306\pi$
$199$	−1.44470	−0.102412	−0.0512059	+	0.998688i	$0.516306\pi$
$200$	3.18855	0.225465
$201$	11.2929	0.796538
$202$	0.598460	0.0421075
$203$	0	0
$204$	1.70612	0.119452
$205$	6.31517	0.441070
$206$	−5.49342	−0.382745
$207$	1.00000	0.0695048
$208$	−4.98924	−0.345941
$209$	6.27036	0.433730
$210$	0	0
$211$	−23.0990	−1.59020	−0.795099	−	0.606480i	$-0.792581\pi$
$211$	−23.0990	−1.59020	−0.795099	+	0.606480i	$0.792581\pi$
$212$	9.29845	0.638620
$213$	−0.362008	−0.0248044
$214$	−0.554244	−0.0378873
$215$	15.9606	1.08851
$216$	1.28306	0.0873012
$217$	0	0
$218$	−3.76986	−0.255327
$219$	5.34411	0.361122
$220$	12.9569	0.873554
$221$	1.33972	0.0901192
$222$	−3.26767	−0.219312
$223$	18.6497	1.24888	0.624438	−	0.781074i	$-0.285328\pi$
$223$	18.6497	1.24888	0.624438	+	0.781074i	$0.285328\pi$
$224$	0	0
$225$	2.48511	0.165674
$226$	−1.03836	−0.0690704
$227$	−20.6334	−1.36949	−0.684744	−	0.728783i	$-0.740086\pi$
$227$	−20.6334	−1.36949	−0.684744	+	0.728783i	$0.740086\pi$
$228$	4.73589	0.313642
$229$	−14.9544	−0.988214	−0.494107	−	0.869401i	$-0.664505\pi$
$229$	−14.9544	−0.988214	−0.494107	+	0.869401i	$0.664505\pi$
$230$	−0.902098	−0.0594826
$231$	0	0
$232$	8.57682	0.563096
$233$	28.1052	1.84123	0.920617	−	0.390467i	$-0.127687\pi$
$233$	28.1052	1.84123	0.920617	+	0.390467i	$0.127687\pi$
$234$	0.489682	0.0320115
$235$	18.4583	1.20409
$236$	−13.8626	−0.902376
$237$	10.4672	0.679919
$238$	0	0
$239$	25.8536	1.67233	0.836166	−	0.548476i	$-0.184792\pi$
$239$	25.8536	1.67233	0.836166	+	0.548476i	$0.184792\pi$
$240$	9.19123	0.593292
$241$	14.8311	0.955356	0.477678	−	0.878535i	$-0.341479\pi$
$241$	14.8311	0.955356	0.477678	+	0.878535i	$0.341479\pi$
$242$	1.55949	0.100248
$243$	1.00000	0.0641500
$244$	−1.90310	−0.121833
$245$	0	0
$246$	−0.761098	−0.0485258
$247$	3.71883	0.236623
$248$	−1.40867	−0.0894509
$249$	−7.59946	−0.481596
$250$	2.26868	0.143484
$251$	16.2841	1.02784	0.513922	−	0.857837i	$-0.328192\pi$
$251$	16.2841	1.02784	0.513922	+	0.857837i	$0.328192\pi$
$252$	0	0
$253$	−2.50407	−0.157429
$254$	−0.148135	−0.00929482
$255$	−2.46805	−0.154555
$256$	7.99375	0.499609
$257$	−22.9446	−1.43125	−0.715623	−	0.698486i	$-0.753857\pi$
$257$	−22.9446	−1.43125	−0.715623	+	0.698486i	$0.753857\pi$
$258$	−1.92356	−0.119756
$259$	0	0
$260$	7.68448	0.476571
$261$	6.68466	0.413770
$262$	−6.58658	−0.406920
$263$	−0.776932	−0.0479077	−0.0239538	−	0.999713i	$-0.507625\pi$
$263$	−0.776932	−0.0479077	−0.0239538	+	0.999713i	$0.507625\pi$
$264$	−3.21287	−0.197739
$265$	−13.4510	−0.826287
$266$	0	0
$267$	−10.6245	−0.650208
$268$	−21.3580	−1.30465
$269$	−3.05937	−0.186533	−0.0932666	−	0.995641i	$-0.529731\pi$
$269$	−3.05937	−0.186533	−0.0932666	+	0.995641i	$0.529731\pi$
$270$	−0.902098	−0.0549000
$271$	16.0631	0.975765	0.487882	−	0.872909i	$-0.337770\pi$
$271$	16.0631	0.975765	0.487882	+	0.872909i	$0.337770\pi$
$272$	−3.03060	−0.183757
$273$	0	0
$274$	−0.0891438	−0.00538537
$275$	−6.22289	−0.375255
$276$	−1.89128	−0.113842
$277$	−11.6103	−0.697594	−0.348797	−	0.937198i	$-0.613410\pi$
$277$	−11.6103	−0.697594	−0.348797	+	0.937198i	$0.613410\pi$
$278$	1.45908	0.0875100
$279$	−1.09790	−0.0657296
$280$	0	0
$281$	9.89753	0.590437	0.295219	−	0.955430i	$-0.404608\pi$
$281$	9.89753	0.590437	0.295219	+	0.955430i	$0.404608\pi$
$282$	−2.22457	−0.132471
$283$	−13.9994	−0.832177	−0.416088	−	0.909324i	$-0.636599\pi$
$283$	−13.9994	−0.832177	−0.416088	+	0.909324i	$0.636599\pi$
$284$	0.684658	0.0406270
$285$	−6.85086	−0.405810
$286$	−1.22620	−0.0725066
$287$	0	0
$288$	−3.67384	−0.216483
$289$	−16.1862	−0.952130
$290$	−6.03022	−0.354107
$291$	−13.9102	−0.815432
$292$	−10.1072	−0.591480
$293$	8.14038	0.475566	0.237783	−	0.971318i	$-0.423579\pi$
$293$	8.14038	0.475566	0.237783	+	0.971318i	$0.423579\pi$
$294$	0	0
$295$	20.0534	1.16755
$296$	12.7154	0.739070
$297$	−2.50407	−0.145301
$298$	1.21844	0.0705824
$299$	−1.48511	−0.0858863
$300$	−4.70005	−0.271357
$301$	0	0
$302$	−2.84612	−0.163776
$303$	−1.81502	−0.104270
$304$	−8.41242	−0.482485
$305$	2.75299	0.157636
$306$	0.297446	0.0170039
$307$	−13.5095	−0.771027	−0.385514	−	0.922702i	$-0.625976\pi$
$307$	−13.5095	−0.771027	−0.385514	+	0.922702i	$0.625976\pi$
$308$	0	0
$309$	16.6605	0.947783
$310$	0.990415	0.0562518
$311$	24.3035	1.37813	0.689063	−	0.724701i	$-0.258022\pi$
$311$	24.3035	1.37813	0.689063	+	0.724701i	$0.258022\pi$
$312$	−1.90549	−0.107877
$313$	33.7836	1.90956	0.954782	−	0.297308i	$-0.0960889\pi$
$313$	33.7836	1.90956	0.954782	+	0.297308i	$0.0960889\pi$
$314$	3.27563	0.184855
$315$	0	0
$316$	−19.7964	−1.11364
$317$	32.1594	1.80625	0.903126	−	0.429376i	$-0.141267\pi$
$317$	32.1594	1.80625	0.903126	+	0.429376i	$0.141267\pi$
$318$	1.62110	0.0909067
$319$	−16.7388	−0.937195
$320$	−15.0683	−0.842344
$321$	1.68092	0.0938196
$322$	0	0
$323$	2.25892	0.125689
$324$	−1.89128	−0.105071
$325$	−3.69068	−0.204722
$326$	−4.84969	−0.268599
$327$	11.4333	0.632261
$328$	2.96164	0.163529
$329$	0	0
$330$	2.25892	0.124349
$331$	−4.84004	−0.266033	−0.133016	−	0.991114i	$-0.542466\pi$
$331$	−4.84004	−0.266033	−0.133016	+	0.991114i	$0.542466\pi$
$332$	14.3727	0.788805
$333$	9.91023	0.543077
$334$	−1.80502	−0.0987665
$335$	30.8961	1.68803
$336$	0	0
$337$	−2.59195	−0.141192	−0.0705962	−	0.997505i	$-0.522490\pi$
$337$	−2.59195	−0.141192	−0.0705962	+	0.997505i	$0.522490\pi$
$338$	3.55922	0.193596
$339$	3.14914	0.171038
$340$	4.66776	0.253145
$341$	2.74922	0.148879
$342$	0.825659	0.0446465
$343$	0	0
$344$	7.48511	0.403570
$345$	2.73589	0.147296
$346$	5.99204	0.322134
$347$	−15.9390	−0.855651	−0.427825	−	0.903861i	$-0.640720\pi$
$347$	−15.9390	−0.855651	−0.427825	+	0.903861i	$0.640720\pi$
$348$	−12.6426	−0.677712
$349$	8.02603	0.429624	0.214812	−	0.976655i	$-0.431086\pi$
$349$	8.02603	0.429624	0.214812	+	0.976655i	$0.431086\pi$
$350$	0	0
$351$	−1.48511	−0.0792695
$352$	9.19954	0.490337
$353$	−10.7871	−0.574141	−0.287070	−	0.957909i	$-0.592681\pi$
$353$	−10.7871	−0.574141	−0.287070	+	0.957909i	$0.592681\pi$
$354$	−2.41681	−0.128452
$355$	−0.990415	−0.0525658
$356$	20.0939	1.06497
$357$	0	0
$358$	−3.35862	−0.177508
$359$	−2.17891	−0.114998	−0.0574992	−	0.998346i	$-0.518313\pi$
$359$	−2.17891	−0.114998	−0.0574992	+	0.998346i	$0.518313\pi$
$360$	3.51032	0.185010
$361$	−12.7296	−0.669981
$362$	4.45746	0.234279
$363$	−4.72964	−0.248242
$364$	0	0
$365$	14.6209	0.765294
$366$	−0.331788	−0.0173428
$367$	−27.0388	−1.41141	−0.705707	−	0.708504i	$-0.749370\pi$
$367$	−27.0388	−1.41141	−0.705707	+	0.708504i	$0.749370\pi$
$368$	3.35950	0.175126
$369$	2.30826	0.120163
$370$	−8.94001	−0.464769
$371$	0	0
$372$	2.07644	0.107658
$373$	16.4189	0.850140	0.425070	−	0.905161i	$-0.360249\pi$
$373$	16.4189	0.850140	0.425070	+	0.905161i	$0.360249\pi$
$374$	−0.744826	−0.0385140
$375$	−6.88046	−0.355306
$376$	8.65644	0.446422
$377$	−9.92748	−0.511291
$378$	0	0
$379$	−34.6399	−1.77933	−0.889667	−	0.456610i	$-0.849064\pi$
$379$	−34.6399	−1.77933	−0.889667	+	0.456610i	$0.849064\pi$
$380$	12.9569	0.664675
$381$	0.449266	0.0230166
$382$	0.208303	0.0106577
$383$	10.6722	0.545322	0.272661	−	0.962110i	$-0.412096\pi$
$383$	10.6722	0.545322	0.272661	+	0.962110i	$0.412096\pi$
$384$	9.16370	0.467633
$385$	0	0
$386$	−0.451152	−0.0229630
$387$	5.83380	0.296548
$388$	26.3081	1.33559
$389$	0.0302201	0.00153222	0.000766110	−	1.00000i	$-0.499756\pi$
$389$	0.0302201	0.00153222	0.000766110		1.00000i	$0.499756\pi$
$390$	1.33972	0.0678393
$391$	−0.902098	−0.0456211
$392$	0	0
$393$	19.9759	1.00765
$394$	−8.16441	−0.411317
$395$	28.6372	1.44089
$396$	4.73589	0.237988
$397$	−20.4749	−1.02761	−0.513803	−	0.857908i	$-0.671764\pi$
$397$	−20.4749	−1.02761	−0.513803	+	0.857908i	$0.671764\pi$
$398$	0.476356	0.0238776
$399$	0	0
$400$	8.34874	0.417437
$401$	−36.4304	−1.81925	−0.909623	−	0.415435i	$-0.863629\pi$
$401$	−36.4304	−1.81925	−0.909623	+	0.415435i	$0.863629\pi$
$402$	−3.72357	−0.185715
$403$	1.63051	0.0812214
$404$	3.43270	0.170783
$405$	2.73589	0.135948
$406$	0	0
$407$	−24.8159	−1.23008
$408$	−1.15745	−0.0573022
$409$	−37.4054	−1.84958	−0.924790	−	0.380479i	$-0.875759\pi$
$409$	−37.4054	−1.84958	−0.924790	+	0.380479i	$0.875759\pi$
$410$	−2.08228	−0.102837
$411$	0.270356	0.0133357
$412$	−31.5097	−1.55237
$413$	0	0
$414$	−0.329727	−0.0162052
$415$	−20.7913	−1.02061
$416$	5.45607	0.267506
$417$	−4.42512	−0.216699
$418$	−2.06751	−0.101125
$419$	37.2017	1.81742	0.908710	−	0.417428i	$-0.137068\pi$
$419$	37.2017	1.81742	0.908710	+	0.417428i	$0.137068\pi$
$420$	0	0
$421$	27.6326	1.34673	0.673366	−	0.739309i	$-0.264848\pi$
$421$	27.6326	1.34673	0.673366	+	0.739309i	$0.264848\pi$
$422$	7.61636	0.370758
$423$	6.74671	0.328036
$424$	−6.30815	−0.306351
$425$	−2.24182	−0.108744
$426$	0.119364	0.00578320
$427$	0	0
$428$	−3.17908	−0.153667
$429$	3.71883	0.179547
$430$	−5.26266	−0.253788
$431$	35.0676	1.68915	0.844573	−	0.535441i	$-0.179855\pi$
$431$	35.0676	1.68915	0.844573	+	0.535441i	$0.179855\pi$
$432$	3.35950	0.161634
$433$	1.00708	0.0483970	0.0241985	−	0.999707i	$-0.492297\pi$
$433$	1.00708	0.0483970	0.0241985	+	0.999707i	$0.492297\pi$
$434$	0	0
$435$	18.2885	0.876867
$436$	−21.6235	−1.03558
$437$	−2.50407	−0.119786
$438$	−1.76210	−0.0841964
$439$	4.04649	0.193129	0.0965643	−	0.995327i	$-0.469215\pi$
$439$	4.04649	0.193129	0.0965643	+	0.995327i	$0.469215\pi$
$440$	−8.79007	−0.419050
$441$	0	0
$442$	−0.441742	−0.0210115
$443$	−29.3947	−1.39659	−0.698293	−	0.715812i	$-0.746057\pi$
$443$	−29.3947	−1.39659	−0.698293	+	0.715812i	$0.746057\pi$
$444$	−18.7430	−0.889505
$445$	−29.0675	−1.37793
$446$	−6.14931	−0.291178
$447$	−3.69530	−0.174782
$448$	0	0
$449$	−4.87627	−0.230126	−0.115063	−	0.993358i	$-0.536707\pi$
$449$	−4.87627	−0.230126	−0.115063	+	0.993358i	$0.536707\pi$
$450$	−0.819409	−0.0386273
$451$	−5.78005	−0.272172
$452$	−5.95590	−0.280142
$453$	8.63174	0.405555
$454$	6.80340	0.319299
$455$	0	0
$456$	−3.21287	−0.150456
$457$	−35.5212	−1.66161	−0.830806	−	0.556562i	$-0.812120\pi$
$457$	−35.5212	−1.66161	−0.830806	+	0.556562i	$0.812120\pi$
$458$	4.93087	0.230404
$459$	−0.902098	−0.0421064
$460$	−5.17434	−0.241255
$461$	18.9442	0.882319	0.441160	−	0.897429i	$-0.354567\pi$
$461$	18.9442	0.882319	0.441160	+	0.897429i	$0.354567\pi$
$462$	0	0
$463$	14.1933	0.659618	0.329809	−	0.944048i	$-0.393016\pi$
$463$	14.1933	0.659618	0.329809	+	0.944048i	$0.393016\pi$
$464$	22.4571	1.04255
$465$	−3.00374	−0.139295
$466$	−9.26705	−0.429288
$467$	31.0355	1.43615	0.718075	−	0.695966i	$-0.245024\pi$
$467$	31.0355	1.43615	0.718075	+	0.695966i	$0.245024\pi$
$468$	2.80877	0.129835
$469$	0	0
$470$	−6.08620	−0.280735
$471$	−9.93438	−0.457752
$472$	9.40449	0.432877
$473$	−14.6082	−0.671687
$474$	−3.45133	−0.158525
$475$	−6.22289	−0.285526
$476$	0	0
$477$	−4.91648	−0.225110
$478$	−8.52465	−0.389908
$479$	−10.5214	−0.480735	−0.240367	−	0.970682i	$-0.577268\pi$
$479$	−10.5214	−0.480735	−0.240367	+	0.970682i	$0.577268\pi$
$480$	−10.0512	−0.458774
$481$	−14.7178	−0.671075
$482$	−4.89022	−0.222743
$483$	0	0
$484$	8.94508	0.406595
$485$	−38.0569	−1.72808
$486$	−0.329727	−0.0149567
$487$	−31.7378	−1.43818	−0.719089	−	0.694918i	$-0.755441\pi$
$487$	−31.7378	−1.43818	−0.719089	+	0.694918i	$0.755441\pi$
$488$	1.29108	0.0584444
$489$	14.7082	0.665127
$490$	0	0
$491$	−31.8788	−1.43867	−0.719336	−	0.694662i	$-0.755554\pi$
$491$	−31.8788	−1.43867	−0.719336	+	0.694662i	$0.755554\pi$
$492$	−4.36558	−0.196815
$493$	−6.03022	−0.271587
$494$	−1.22620	−0.0551692
$495$	−6.85086	−0.307923
$496$	−3.68840	−0.165614
$497$	0	0
$498$	2.50575	0.112285
$499$	−39.4279	−1.76503	−0.882517	−	0.470280i	$-0.844153\pi$
$499$	−39.4279	−1.76503	−0.882517	+	0.470280i	$0.844153\pi$
$500$	13.0129	0.581954
$501$	5.47430	0.244573
$502$	−5.36932	−0.239644
$503$	−24.4589	−1.09057	−0.545284	−	0.838251i	$-0.683578\pi$
$503$	−24.4589	−1.09057	−0.545284	+	0.838251i	$0.683578\pi$
$504$	0	0
$505$	−4.96569	−0.220970
$506$	0.825659	0.0367050
$507$	−10.7944	−0.479398
$508$	−0.849687	−0.0376988
$509$	−20.3927	−0.903889	−0.451944	−	0.892046i	$-0.649269\pi$
$509$	−20.3927	−0.903889	−0.451944	+	0.892046i	$0.649269\pi$
$510$	0.813782	0.0360349
$511$	0	0
$512$	−20.9632	−0.926449
$513$	−2.50407	−0.110557
$514$	7.56547	0.333699
$515$	45.5814	2.00856
$516$	−11.0333	−0.485716
$517$	−16.8942	−0.743007
$518$	0	0
$519$	−18.1727	−0.797694
$520$	−5.21322	−0.228615
$521$	−26.3058	−1.15248	−0.576238	−	0.817282i	$-0.695480\pi$
$521$	−26.3058	−1.15248	−0.576238	+	0.817282i	$0.695480\pi$
$522$	−2.20411	−0.0964714
$523$	40.4256	1.76769	0.883843	−	0.467783i	$-0.154947\pi$
$523$	40.4256	1.76769	0.883843	+	0.467783i	$0.154947\pi$
$524$	−37.7799	−1.65042
$525$	0	0
$526$	0.256176	0.0111698
$527$	0.990415	0.0431432
$528$	−8.41242	−0.366103
$529$	1.00000	0.0434783
$530$	4.43515	0.192651
$531$	7.32973	0.318083
$532$	0	0
$533$	−3.42804	−0.148485
$534$	3.50318	0.151598
$535$	4.59881	0.198824
$536$	14.4894	0.625849
$537$	10.1860	0.439560
$538$	1.00876	0.0434906
$539$	0	0
$540$	−5.17434	−0.222668
$541$	0.360122	0.0154828	0.00774142	−	0.999970i	$-0.497536\pi$
$541$	0.360122	0.0154828	0.00774142	+	0.999970i	$0.497536\pi$
$542$	−5.29644	−0.227502
$543$	−13.5186	−0.580140
$544$	3.31417	0.142094
$545$	31.2802	1.33990
$546$	0	0
$547$	20.2023	0.863789	0.431894	−	0.901924i	$-0.357845\pi$
$547$	20.2023	0.863789	0.431894	+	0.901924i	$0.357845\pi$
$548$	−0.511319	−0.0218425
$549$	1.00625	0.0429457
$550$	2.05186	0.0874915
$551$	−16.7388	−0.713099
$552$	1.28306	0.0546107
$553$	0	0
$554$	3.82822	0.162646
$555$	27.1133	1.15090
$556$	8.36914	0.354931
$557$	9.71086	0.411463	0.205731	−	0.978609i	$-0.434043\pi$
$557$	9.71086	0.411463	0.205731	+	0.978609i	$0.434043\pi$
$558$	0.362008	0.0153250
$559$	−8.66385	−0.366442
$560$	0	0
$561$	2.25892	0.0953715
$562$	−3.26348	−0.137662
$563$	−27.2789	−1.14967	−0.574835	−	0.818269i	$-0.694934\pi$
$563$	−27.2789	−1.14967	−0.574835	+	0.818269i	$0.694934\pi$
$564$	−12.7599	−0.537290
$565$	8.61570	0.362465
$566$	4.61598	0.194024
$567$	0	0
$568$	−0.464478	−0.0194891
$569$	19.6132	0.822229	0.411115	−	0.911584i	$-0.365140\pi$
$569$	19.6132	0.822229	0.411115	+	0.911584i	$0.365140\pi$
$570$	2.25892	0.0946156
$571$	21.1859	0.886601	0.443301	−	0.896373i	$-0.353807\pi$
$571$	21.1859	0.886601	0.443301	+	0.896373i	$0.353807\pi$
$572$	−7.03334	−0.294079
$573$	−0.631742	−0.0263914
$574$	0	0
$575$	2.48511	0.103636
$576$	−5.50764	−0.229485
$577$	12.4079	0.516547	0.258273	−	0.966072i	$-0.416846\pi$
$577$	12.4079	0.516547	0.258273	+	0.966072i	$0.416846\pi$
$578$	5.33704	0.221991
$579$	1.36826	0.0568629
$580$	−34.5887	−1.43622
$581$	0	0
$582$	4.58658	0.190120
$583$	12.3112	0.509878
$584$	6.85682	0.283737
$585$	−4.06311	−0.167989
$586$	−2.68410	−0.110879
$587$	9.35493	0.386119	0.193060	−	0.981187i	$-0.438159\pi$
$587$	9.35493	0.386119	0.193060	+	0.981187i	$0.438159\pi$
$588$	0	0
$589$	2.74922	0.113280
$590$	−6.61214	−0.272217
$591$	24.7611	1.01854
$592$	33.2934	1.36835
$593$	16.7836	0.689218	0.344609	−	0.938746i	$-0.388011\pi$
$593$	16.7836	0.689218	0.344609	+	0.938746i	$0.388011\pi$
$594$	0.825659	0.0338772
$595$	0	0
$596$	6.98885	0.286275
$597$	−1.44470	−0.0591275
$598$	0.489682	0.0200246
$599$	−16.0512	−0.655836	−0.327918	−	0.944706i	$-0.606347\pi$
$599$	−16.0512	−0.655836	−0.327918	+	0.944706i	$0.606347\pi$
$600$	3.18855	0.130172
$601$	24.4364	0.996781	0.498390	−	0.866953i	$-0.333925\pi$
$601$	24.4364	0.996781	0.498390	+	0.866953i	$0.333925\pi$
$602$	0	0
$603$	11.2929	0.459882
$604$	−16.3250	−0.664257
$605$	−12.9398	−0.526078
$606$	0.598460	0.0243108
$607$	−28.2329	−1.14594	−0.572969	−	0.819577i	$-0.694208\pi$
$607$	−28.2329	−1.14594	−0.572969	+	0.819577i	$0.694208\pi$
$608$	9.19954	0.373091
$609$	0	0
$610$	−0.907736	−0.0367532
$611$	−10.0196	−0.405351
$612$	1.70612	0.0689658
$613$	17.9873	0.726500	0.363250	−	0.931692i	$-0.381667\pi$
$613$	17.9873	0.726500	0.363250	+	0.931692i	$0.381667\pi$
$614$	4.45445	0.179767
$615$	6.31517	0.254652
$616$	0	0
$617$	12.9998	0.523353	0.261677	−	0.965156i	$-0.415725\pi$
$617$	12.9998	0.523353	0.261677	+	0.965156i	$0.415725\pi$
$618$	−5.49342	−0.220978
$619$	1.26266	0.0507505	0.0253752	−	0.999678i	$-0.491922\pi$
$619$	1.26266	0.0507505	0.0253752	+	0.999678i	$0.491922\pi$
$620$	5.68092	0.228151
$621$	1.00000	0.0401286
$622$	−8.01353	−0.321313
$623$	0	0
$624$	−4.98924	−0.199729
$625$	−31.2498	−1.24999
$626$	−11.1394	−0.445219
$627$	6.27036	0.250414
$628$	18.7887	0.749750
$629$	−8.94001	−0.356461
$630$	0	0
$631$	−24.4568	−0.973611	−0.486806	−	0.873510i	$-0.661838\pi$
$631$	−24.4568	−0.973611	−0.486806	+	0.873510i	$0.661838\pi$
$632$	13.4301	0.534220
$633$	−23.0990	−0.918101
$634$	−10.6038	−0.421132
$635$	1.22914	0.0487771
$636$	9.29845	0.368707
$637$	0	0
$638$	5.51925	0.218509
$639$	−0.362008	−0.0143208
$640$	25.0709	0.991014
$641$	−1.29037	−0.0509665	−0.0254833	−	0.999675i	$-0.508112\pi$
$641$	−1.29037	−0.0509665	−0.0254833	+	0.999675i	$0.508112\pi$
$642$	−0.554244	−0.0218743
$643$	−34.3110	−1.35309	−0.676547	−	0.736399i	$-0.736524\pi$
$643$	−34.3110	−1.35309	−0.676547	+	0.736399i	$0.736524\pi$
$644$	0	0
$645$	15.9606	0.628450
$646$	−0.744826	−0.0293048
$647$	2.62092	0.103039	0.0515196	−	0.998672i	$-0.483594\pi$
$647$	2.62092	0.103039	0.0515196	+	0.998672i	$0.483594\pi$
$648$	1.28306	0.0504034
$649$	−18.3541	−0.720463
$650$	1.21692	0.0477314
$651$	0	0
$652$	−27.8173	−1.08941
$653$	7.53697	0.294944	0.147472	−	0.989066i	$-0.452886\pi$
$653$	7.53697	0.294944	0.147472	+	0.989066i	$0.452886\pi$
$654$	−3.76986	−0.147413
$655$	54.6518	2.13542
$656$	7.75462	0.302767
$657$	5.34411	0.208494
$658$	0	0
$659$	22.8213	0.888990	0.444495	−	0.895781i	$-0.353383\pi$
$659$	22.8213	0.888990	0.444495	+	0.895781i	$0.353383\pi$
$660$	12.9569	0.504347
$661$	−27.3668	−1.06445	−0.532223	−	0.846604i	$-0.678643\pi$
$661$	−27.3668	−1.06445	−0.532223	+	0.846604i	$0.678643\pi$
$662$	1.59589	0.0620262
$663$	1.33972	0.0520304
$664$	−9.75057	−0.378396
$665$	0	0
$666$	−3.26767	−0.126620
$667$	6.68466	0.258831
$668$	−10.3534	−0.400586
$669$	18.6497	0.721039
$670$	−10.1873	−0.393569
$671$	−2.51972	−0.0972726
$672$	0	0
$673$	22.9790	0.885774	0.442887	−	0.896577i	$-0.353954\pi$
$673$	22.9790	0.885774	0.442887	+	0.896577i	$0.353954\pi$
$674$	0.854636	0.0329193
$675$	2.48511	0.0956521
$676$	20.4153	0.785204
$677$	8.80814	0.338524	0.169262	−	0.985571i	$-0.445862\pi$
$677$	8.80814	0.338524	0.169262	+	0.985571i	$0.445862\pi$
$678$	−1.03836	−0.0398778
$679$	0	0
$680$	−3.16665	−0.121436
$681$	−20.6334	−0.790674
$682$	−0.906493	−0.0347114
$683$	34.4889	1.31968	0.659840	−	0.751406i	$-0.270624\pi$
$683$	34.4889	1.31968	0.659840	+	0.751406i	$0.270624\pi$
$684$	4.73589	0.181081
$685$	0.739666	0.0282612
$686$	0	0
$687$	−14.9544	−0.570546
$688$	19.5986	0.747191
$689$	7.30154	0.278166
$690$	−0.902098	−0.0343423
$691$	−37.6110	−1.43079	−0.715395	−	0.698721i	$-0.753753\pi$
$691$	−37.6110	−1.43079	−0.715395	+	0.698721i	$0.753753\pi$
$692$	34.3697	1.30654
$693$	0	0
$694$	5.25552	0.199497
$695$	−12.1067	−0.459232
$696$	8.57682	0.325104
$697$	−2.08228	−0.0788721
$698$	−2.64640	−0.100168
$699$	28.1052	1.06304
$700$	0	0
$701$	−47.5348	−1.79536	−0.897682	−	0.440644i	$-0.854750\pi$
$701$	−47.5348	−1.79536	−0.897682	+	0.440644i	$0.854750\pi$
$702$	0.489682	0.0184819
$703$	−24.8159	−0.935949
$704$	13.7915	0.519786
$705$	18.4583	0.695179
$706$	3.55681	0.133862
$707$	0	0
$708$	−13.8626	−0.520987
$709$	−41.1332	−1.54479	−0.772395	−	0.635143i	$-0.780941\pi$
$709$	−41.1332	−1.54479	−0.772395	+	0.635143i	$0.780941\pi$
$710$	0.326567	0.0122558
$711$	10.4672	0.392551
$712$	−13.6319	−0.510876
$713$	−1.09790	−0.0411167
$714$	0	0
$715$	10.1743	0.380498
$716$	−19.2647	−0.719954
$717$	25.8536	0.965522
$718$	0.718446	0.0268122
$719$	−8.25857	−0.307993	−0.153996	−	0.988071i	$-0.549214\pi$
$719$	−8.25857	−0.307993	−0.153996	+	0.988071i	$0.549214\pi$
$720$	9.19123	0.342537
$721$	0	0
$722$	4.19731	0.156208
$723$	14.8311	0.551575
$724$	25.5675	0.950209
$725$	16.6121	0.616959
$726$	1.55949	0.0578782
$727$	−0.985235	−0.0365403	−0.0182702	−	0.999833i	$-0.505816\pi$
$727$	−0.985235	−0.0365403	−0.0182702	+	0.999833i	$0.505816\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−4.82092	−0.178430
$731$	−5.26266	−0.194646
$732$	−1.90310	−0.0703406
$733$	−27.7847	−1.02625	−0.513125	−	0.858314i	$-0.671512\pi$
$733$	−27.7847	−1.02625	−0.513125	+	0.858314i	$0.671512\pi$
$734$	8.91542	0.329074
$735$	0	0
$736$	−3.67384	−0.135420
$737$	−28.2781	−1.04164
$738$	−0.761098	−0.0280164
$739$	30.0540	1.10555	0.552777	−	0.833329i	$-0.313568\pi$
$739$	30.0540	1.10555	0.552777	+	0.833329i	$0.313568\pi$
$740$	−51.2789	−1.88505
$741$	3.71883	0.136614
$742$	0	0
$743$	40.1617	1.47339	0.736695	−	0.676225i	$-0.236385\pi$
$743$	40.1617	1.47339	0.736695	+	0.676225i	$0.236385\pi$
$744$	−1.40867	−0.0516445
$745$	−10.1100	−0.370400
$746$	−5.41377	−0.198212
$747$	−7.59946	−0.278050
$748$	−4.27224	−0.156209
$749$	0	0
$750$	2.26868	0.0828403
$751$	−2.15350	−0.0785823	−0.0392912	−	0.999228i	$-0.512510\pi$
$751$	−2.15350	−0.0785823	−0.0392912	+	0.999228i	$0.512510\pi$
$752$	22.6656	0.826529
$753$	16.2841	0.593426
$754$	3.27336	0.119209
$755$	23.6155	0.859457
$756$	0	0
$757$	−32.0577	−1.16516	−0.582579	−	0.812774i	$-0.697956\pi$
$757$	−32.0577	−1.16516	−0.582579	+	0.812774i	$0.697956\pi$
$758$	11.4217	0.414856
$759$	−2.50407	−0.0908919
$760$	−8.79007	−0.318849
$761$	−0.669651	−0.0242748	−0.0121374	−	0.999926i	$-0.503864\pi$
$761$	−0.669651	−0.0242748	−0.0121374	+	0.999926i	$0.503864\pi$
$762$	−0.148135	−0.00536637
$763$	0	0
$764$	1.19480	0.0432264
$765$	−2.46805	−0.0892324
$766$	−3.51890	−0.127143
$767$	−10.8855	−0.393052
$768$	7.99375	0.288450
$769$	−43.5115	−1.56906	−0.784532	−	0.620089i	$-0.787097\pi$
$769$	−43.5115	−1.56906	−0.784532	+	0.620089i	$0.787097\pi$
$770$	0	0
$771$	−22.9446	−0.826331
$772$	−2.58776	−0.0931355
$773$	−21.0046	−0.755482	−0.377741	−	0.925911i	$-0.623299\pi$
$773$	−21.0046	−0.755482	−0.377741	+	0.925911i	$0.623299\pi$
$774$	−1.92356	−0.0691410
$775$	−2.72841	−0.0980074
$776$	−17.8477	−0.640694
$777$	0	0
$778$	−0.00996438	−0.000357240 0
$779$	−5.78005	−0.207092
$780$	7.68448	0.275149
$781$	0.906493	0.0324369
$782$	0.297446	0.0106367
$783$	6.68466	0.238890
$784$	0	0
$785$	−27.1794	−0.970075
$786$	−6.58658	−0.234936
$787$	−22.1975	−0.791255	−0.395627	−	0.918411i	$-0.629473\pi$
$787$	−22.1975	−0.791255	−0.395627	+	0.918411i	$0.629473\pi$
$788$	−46.8302	−1.66826
$789$	−0.776932	−0.0276595
$790$	−9.44246	−0.335948
$791$	0	0
$792$	−3.21287	−0.114164
$793$	−1.49440	−0.0530675
$794$	6.75113	0.239589
$795$	−13.4510	−0.477057
$796$	2.73233	0.0968447
$797$	−22.8502	−0.809397	−0.404699	−	0.914450i	$-0.632624\pi$
$797$	−22.8502	−0.809397	−0.404699	+	0.914450i	$0.632624\pi$
$798$	0	0
$799$	−6.08620	−0.215314
$800$	−9.12991	−0.322791
$801$	−10.6245	−0.375398
$802$	12.0121	0.424161
$803$	−13.3820	−0.472241
$804$	−21.3580	−0.753238
$805$	0	0
$806$	−0.537623	−0.0189370
$807$	−3.05937	−0.107695
$808$	−2.32878	−0.0819260
$809$	−38.0621	−1.33819	−0.669097	−	0.743175i	$-0.733319\pi$
$809$	−38.0621	−1.33819	−0.669097	+	0.743175i	$0.733319\pi$
$810$	−0.902098	−0.0316965
$811$	−21.0208	−0.738142	−0.369071	−	0.929401i	$-0.620324\pi$
$811$	−21.0208	−0.738142	−0.369071	+	0.929401i	$0.620324\pi$
$812$	0	0
$813$	16.0631	0.563358
$814$	8.18248	0.286796
$815$	40.2400	1.40955
$816$	−3.03060	−0.106092
$817$	−14.6082	−0.511077
$818$	12.3336	0.431234
$819$	0	0
$820$	−11.9437	−0.417094
$821$	−10.4132	−0.363425	−0.181712	−	0.983352i	$-0.558164\pi$
$821$	−10.4132	−0.363425	−0.181712	+	0.983352i	$0.558164\pi$
$822$	−0.0891438	−0.00310925
$823$	30.0396	1.04711	0.523557	−	0.851991i	$-0.324605\pi$
$823$	30.0396	1.04711	0.523557	+	0.851991i	$0.324605\pi$
$824$	21.3765	0.744684
$825$	−6.22289	−0.216653
$826$	0	0
$827$	28.6978	0.997920	0.498960	−	0.866625i	$-0.333715\pi$
$827$	28.6978	0.997920	0.498960	+	0.866625i	$0.333715\pi$
$828$	−1.89128	−0.0657265
$829$	−16.4384	−0.570931	−0.285465	−	0.958389i	$-0.592148\pi$
$829$	−16.4384	−0.570931	−0.285465	+	0.958389i	$0.592148\pi$
$830$	6.85546	0.237957
$831$	−11.6103	−0.402756
$832$	8.17946	0.283572
$833$	0	0
$834$	1.45908	0.0505239
$835$	14.9771	0.518304
$836$	−11.8590	−0.410152
$837$	−1.09790	−0.0379490
$838$	−12.2664	−0.423736
$839$	12.5437	0.433055	0.216528	−	0.976277i	$-0.430527\pi$
$839$	12.5437	0.433055	0.216528	+	0.976277i	$0.430527\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	−9.11123	−0.313994
$843$	9.89753	0.340889
$844$	43.6866	1.50375
$845$	−29.5324	−1.01595
$846$	−2.22457	−0.0764824
$847$	0	0
$848$	−16.5169	−0.567194
$849$	−13.9994	−0.480457
$850$	0.739188	0.0253539
$851$	9.91023	0.339718
$852$	0.684658	0.0234560
$853$	22.0613	0.755365	0.377683	−	0.925935i	$-0.376721\pi$
$853$	22.0613	0.755365	0.377683	+	0.925935i	$0.376721\pi$
$854$	0	0
$855$	−6.85086	−0.234295
$856$	2.15672	0.0737151
$857$	−6.15359	−0.210203	−0.105101	−	0.994462i	$-0.533517\pi$
$857$	−6.15359	−0.210203	−0.105101	+	0.994462i	$0.533517\pi$
$858$	−1.22620	−0.0418617
$859$	−23.0698	−0.787131	−0.393566	−	0.919296i	$-0.628759\pi$
$859$	−23.0698	−0.787131	−0.393566	+	0.919296i	$0.628759\pi$
$860$	−30.1860	−1.02934
$861$	0	0
$862$	−11.5627	−0.393828
$863$	−5.22145	−0.177740	−0.0888702	−	0.996043i	$-0.528326\pi$
$863$	−5.22145	−0.177740	−0.0888702	+	0.996043i	$0.528326\pi$
$864$	−3.67384	−0.124987
$865$	−49.7186	−1.69048
$866$	−0.332061	−0.0112839
$867$	−16.1862	−0.549713
$868$	0	0
$869$	−26.2106	−0.889135
$870$	−6.03022	−0.204444
$871$	−16.7712	−0.568271
$872$	14.6696	0.496774
$873$	−13.9102	−0.470790
$874$	0.825659	0.0279283
$875$	0	0
$876$	−10.1072	−0.341491
$877$	49.8716	1.68404	0.842022	−	0.539443i	$-0.181365\pi$
$877$	49.8716	1.68404	0.842022	+	0.539443i	$0.181365\pi$
$878$	−1.33424	−0.0450284
$879$	8.14038	0.274568
$880$	−23.0155	−0.775852
$881$	20.3054	0.684107	0.342053	−	0.939681i	$-0.388878\pi$
$881$	20.3054	0.684107	0.342053	+	0.939681i	$0.388878\pi$
$882$	0	0
$883$	−50.3500	−1.69441	−0.847206	−	0.531265i	$-0.821717\pi$
$883$	−50.3500	−1.69441	−0.847206	+	0.531265i	$0.821717\pi$
$884$	−2.53378	−0.0852203
$885$	20.0534	0.674086
$886$	9.69224	0.325617
$887$	−47.2694	−1.58715	−0.793576	−	0.608471i	$-0.791783\pi$
$887$	−47.2694	−1.58715	−0.793576	+	0.608471i	$0.791783\pi$
$888$	12.7154	0.426702
$889$	0	0
$890$	9.58434	0.321268
$891$	−2.50407	−0.0838894
$892$	−35.2718	−1.18099
$893$	−16.8942	−0.565344
$894$	1.21844	0.0407508
$895$	27.8679	0.931522
$896$	0	0
$897$	−1.48511	−0.0495865
$898$	1.60784	0.0536543
$899$	−7.33910	−0.244773
$900$	−4.70005	−0.156668
$901$	4.43515	0.147756
$902$	1.90584	0.0634575
$903$	0	0
$904$	4.04053	0.134386
$905$	−36.9855	−1.22944
$906$	−2.84612	−0.0945560
$907$	−5.65035	−0.187617	−0.0938084	−	0.995590i	$-0.529904\pi$
$907$	−5.65035	−0.187617	−0.0938084	+	0.995590i	$0.529904\pi$
$908$	39.0236	1.29504
$909$	−1.81502	−0.0602003
$910$	0	0
$911$	18.2196	0.603641	0.301820	−	0.953365i	$-0.402406\pi$
$911$	18.2196	0.603641	0.301820	+	0.953365i	$0.402406\pi$
$912$	−8.41242	−0.278563
$913$	19.0296	0.629787
$914$	11.7123	0.387409
$915$	2.75299	0.0910111
$916$	28.2829	0.934495
$917$	0	0
$918$	0.297446	0.00981719
$919$	7.79561	0.257154	0.128577	−	0.991700i	$-0.458959\pi$
$919$	7.79561	0.257154	0.128577	+	0.991700i	$0.458959\pi$
$920$	3.51032	0.115732
$921$	−13.5095	−0.445153
$922$	−6.24642	−0.205715
$923$	0.537623	0.0176961
$924$	0	0
$925$	24.6281	0.809766
$926$	−4.67992	−0.153792
$927$	16.6605	0.547203
$928$	−24.5584	−0.806168
$929$	−11.5912	−0.380293	−0.190147	−	0.981756i	$-0.560896\pi$
$929$	−11.5912	−0.380293	−0.190147	+	0.981756i	$0.560896\pi$
$930$	0.990415	0.0324770
$931$	0	0
$932$	−53.1548	−1.74114
$933$	24.3035	0.795662
$934$	−10.2332	−0.334842
$935$	6.18015	0.202113
$936$	−1.90549	−0.0622829
$937$	32.0050	1.04556	0.522778	−	0.852469i	$-0.324895\pi$
$937$	32.0050	1.04556	0.522778	+	0.852469i	$0.324895\pi$
$938$	0	0
$939$	33.7836	1.10249
$940$	−34.9098	−1.13863
$941$	−11.1090	−0.362141	−0.181071	−	0.983470i	$-0.557956\pi$
$941$	−11.1090	−0.362141	−0.181071	+	0.983470i	$0.557956\pi$
$942$	3.27563	0.106726
$943$	2.30826	0.0751674
$944$	24.6242	0.801450
$945$	0	0
$946$	4.81673	0.156605
$947$	−12.1262	−0.394049	−0.197025	−	0.980399i	$-0.563128\pi$
$947$	−12.1262	−0.394049	−0.197025	+	0.980399i	$0.563128\pi$
$948$	−19.7964	−0.642959
$949$	−7.93661	−0.257633
$950$	2.05186	0.0665710
$951$	32.1594	1.04284
$952$	0	0
$953$	27.6645	0.896140	0.448070	−	0.893999i	$-0.352112\pi$
$953$	27.6645	0.896140	0.448070	+	0.893999i	$0.352112\pi$
$954$	1.62110	0.0524850
$955$	−1.72838	−0.0559291
$956$	−48.8965	−1.58142
$957$	−16.7388	−0.541090
$958$	3.46919	0.112084
$959$	0	0
$960$	−15.0683	−0.486327
$961$	−29.7946	−0.961117
$962$	4.85287	0.156463
$963$	1.68092	0.0541668
$964$	−28.0498	−0.903423
$965$	3.74341	0.120505
$966$	0	0
$967$	−56.1838	−1.80675	−0.903374	−	0.428853i	$-0.858918\pi$
$967$	−56.1838	−1.80675	−0.903374	+	0.428853i	$0.858918\pi$
$968$	−6.06842	−0.195046
$969$	2.25892	0.0725668
$970$	12.5484	0.402905
$971$	43.9884	1.41166	0.705828	−	0.708383i	$-0.250575\pi$
$971$	43.9884	1.41166	0.705828	+	0.708383i	$0.250575\pi$
$972$	−1.89128	−0.0606628
$973$	0	0
$974$	10.4648	0.335315
$975$	−3.69068	−0.118196
$976$	3.38050	0.108207
$977$	−6.47804	−0.207251	−0.103625	−	0.994616i	$-0.533044\pi$
$977$	−6.47804	−0.207251	−0.103625	+	0.994616i	$0.533044\pi$
$978$	−4.84969	−0.155076
$979$	26.6044	0.850282
$980$	0	0
$981$	11.4333	0.365036
$982$	10.5113	0.335430
$983$	−37.8961	−1.20870	−0.604349	−	0.796720i	$-0.706567\pi$
$983$	−37.8961	−1.20870	−0.604349	+	0.796720i	$0.706567\pi$
$984$	2.96164	0.0944138
$985$	67.7437	2.15849
$986$	1.98833	0.0633212
$987$	0	0
$988$	−7.03334	−0.223760
$989$	5.83380	0.185504
$990$	2.25892	0.0717931
$991$	53.3499	1.69472	0.847358	−	0.531022i	$-0.178192\pi$
$991$	53.3499	1.69472	0.847358	+	0.531022i	$0.178192\pi$
$992$	4.03351	0.128064
$993$	−4.84004	−0.153594
$994$	0	0
$995$	−3.95254	−0.125304
$996$	14.3727	0.455417
$997$	−24.8754	−0.787813	−0.393907	−	0.919150i	$-0.628877\pi$
$997$	−24.8754	−0.787813	−0.393907	+	0.919150i	$0.628877\pi$
$998$	13.0004	0.411522
$999$	9.91023	0.313546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.x.1.2		4
7.6	odd	2		483.2.a.j.1.2	✓	4
21.20	even	2		1449.2.a.o.1.3		4
28.27	even	2		7728.2.a.ce.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.2	✓	4	7.6	odd	2
1449.2.a.o.1.3		4	21.20	even	2
3381.2.a.x.1.2		4	1.1	even	1	trivial
7728.2.a.ce.1.1		4	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-0.329727 q^{2} +1.00000 q^{3} -1.89128 q^{4} +2.73589 q^{5} -0.329727 q^{6} +1.28306 q^{8} +1.00000 q^{9} -0.902098 q^{10} -2.50407 q^{11} -1.89128 q^{12} -1.48511 q^{13} +2.73589 q^{15} +3.35950 q^{16} -0.902098 q^{17} -0.329727 q^{18} -2.50407 q^{19} -5.17434 q^{20} +0.825659 q^{22} +1.00000 q^{23} +1.28306 q^{24} +2.48511 q^{25} +0.489682 q^{26} +1.00000 q^{27} +6.68466 q^{29} -0.902098 q^{30} -1.09790 q^{31} -3.67384 q^{32} -2.50407 q^{33} +0.297446 q^{34} -1.89128 q^{36} +9.91023 q^{37} +0.825659 q^{38} -1.48511 q^{39} +3.51032 q^{40} +2.30826 q^{41} +5.83380 q^{43} +4.73589 q^{44} +2.73589 q^{45} -0.329727 q^{46} +6.74671 q^{47} +3.35950 q^{48} -0.819409 q^{50} -0.902098 q^{51} +2.80877 q^{52} -4.91648 q^{53} -0.329727 q^{54} -6.85086 q^{55} -2.50407 q^{57} -2.20411 q^{58} +7.32973 q^{59} -5.17434 q^{60} +1.00625 q^{61} +0.362008 q^{62} -5.50764 q^{64} -4.06311 q^{65} +0.825659 q^{66} +11.2929 q^{67} +1.70612 q^{68} +1.00000 q^{69} -0.362008 q^{71} +1.28306 q^{72} +5.34411 q^{73} -3.26767 q^{74} +2.48511 q^{75} +4.73589 q^{76} +0.489682 q^{78} +10.4672 q^{79} +9.19123 q^{80} +1.00000 q^{81} -0.761098 q^{82} -7.59946 q^{83} -2.46805 q^{85} -1.92356 q^{86} +6.68466 q^{87} -3.21287 q^{88} -10.6245 q^{89} -0.902098 q^{90} -1.89128 q^{92} -1.09790 q^{93} -2.22457 q^{94} -6.85086 q^{95} -3.67384 q^{96} -13.9102 q^{97} -2.50407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{4} - 5 q^{5} + 2 q^{6} + 9 q^{8} + 4 q^{9} - 2 q^{10} + q^{11} + 4 q^{12} - 7 q^{13} - 5 q^{15} + 8 q^{16} - 2 q^{17} + 2 q^{18} + q^{19} - 13 q^{20} + 11 q^{22} + 4 q^{23}+ \cdots + q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^4 - 5 * q^5 + 2 * q^6 + 9 * q^8 + 4 * q^9 - 2 * q^10 + q^11 + 4 * q^12 - 7 * q^13 - 5 * q^15 + 8 * q^16 - 2 * q^17 + 2 * q^18 + q^19 - 13 * q^20 + 11 * q^22 + 4 * q^23 + 9 * q^24 + 11 * q^25 + 19 * q^26 + 4 * q^27 + 2 * q^29 - 2 * q^30 - 6 * q^31 + 20 * q^32 + q^33 - 23 * q^34 + 4 * q^36 + 16 * q^37 + 11 * q^38 - 7 * q^39 - 3 * q^40 - 5 * q^41 + 9 * q^43 + 3 * q^44 - 5 * q^45 + 2 * q^46 + 21 * q^47 + 8 * q^48 - 17 * q^50 - 2 * q^51 + 24 * q^52 + 10 * q^53 + 2 * q^54 - 17 * q^55 + q^57 + q^58 + 26 * q^59 - 13 * q^60 - 2 * q^61 + 19 * q^62 + 27 * q^64 + 26 * q^65 + 11 * q^66 + 5 * q^67 - 7 * q^68 + 4 * q^69 - 19 * q^71 + 9 * q^72 - 10 * q^73 + 9 * q^74 + 11 * q^75 + 3 * q^76 + 19 * q^78 - 6 * q^79 + 24 * q^80 + 4 * q^81 + 31 * q^82 + 2 * q^83 - 7 * q^85 - 17 * q^86 + 2 * q^87 - 20 * q^88 + 17 * q^89 - 2 * q^90 + 4 * q^92 - 6 * q^93 + 44 * q^94 - 17 * q^95 + 20 * q^96 - 32 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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