LMFDB - Embedded newform 2366.2.a.bg.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2366,2,Mod(1,2366)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2366.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,6,1,6,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.8926051182$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6052921.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 7x^{3} + 33x^{2} - 9x - 27 $ x^6 - x^5 - 11x^4 + 7x^3 + 33x^2 - 9x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.85334$ of defining polynomial
Character	$\chi$	$=$	2366.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.85334 q^{3} +1.00000 q^{4} +0.532083 q^{5} +2.85334 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.14154 q^{9} +0.532083 q^{10} +2.71490 q^{11} +2.85334 q^{12} +1.00000 q^{14} +1.51821 q^{15} +1.00000 q^{16} -2.94148 q^{17} +5.14154 q^{18} +5.83604 q^{19} +0.532083 q^{20} +2.85334 q^{21} +2.71490 q^{22} -5.52440 q^{23} +2.85334 q^{24} -4.71689 q^{25} +6.11053 q^{27} +1.00000 q^{28} -8.19741 q^{29} +1.51821 q^{30} +3.42217 q^{31} +1.00000 q^{32} +7.74652 q^{33} -2.94148 q^{34} +0.532083 q^{35} +5.14154 q^{36} +6.00763 q^{37} +5.83604 q^{38} +0.532083 q^{40} -7.76238 q^{41} +2.85334 q^{42} -9.13544 q^{43} +2.71490 q^{44} +2.73572 q^{45} -5.52440 q^{46} -9.98779 q^{47} +2.85334 q^{48} +1.00000 q^{49} -4.71689 q^{50} -8.39305 q^{51} +12.8839 q^{53} +6.11053 q^{54} +1.44455 q^{55} +1.00000 q^{56} +16.6522 q^{57} -8.19741 q^{58} -1.24624 q^{59} +1.51821 q^{60} +1.36564 q^{61} +3.42217 q^{62} +5.14154 q^{63} +1.00000 q^{64} +7.74652 q^{66} -9.49764 q^{67} -2.94148 q^{68} -15.7630 q^{69} +0.532083 q^{70} +1.34074 q^{71} +5.14154 q^{72} +11.2652 q^{73} +6.00763 q^{74} -13.4589 q^{75} +5.83604 q^{76} +2.71490 q^{77} +7.62525 q^{79} +0.532083 q^{80} +2.01079 q^{81} -7.76238 q^{82} -3.49046 q^{83} +2.85334 q^{84} -1.56511 q^{85} -9.13544 q^{86} -23.3900 q^{87} +2.71490 q^{88} -14.5067 q^{89} +2.73572 q^{90} -5.52440 q^{92} +9.76460 q^{93} -9.98779 q^{94} +3.10526 q^{95} +2.85334 q^{96} +7.63555 q^{97} +1.00000 q^{98} +13.9587 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 4 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 5 q^{9} + 4 q^{10} + 6 q^{11} + q^{12} + 6 q^{14} - 5 q^{15} + 6 q^{16} - 9 q^{17} + 5 q^{18} + 10 q^{19} + 4 q^{20} + q^{21}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + q^3 + 6 * q^4 + 4 * q^5 + q^6 + 6 * q^7 + 6 * q^8 + 5 * q^9 + 4 * q^10 + 6 * q^11 + q^12 + 6 * q^14 - 5 * q^15 + 6 * q^16 - 9 * q^17 + 5 * q^18 + 10 * q^19 + 4 * q^20 + q^21 + 6 * q^22 + 21 * q^23 + q^24 + 8 * q^25 + 7 * q^27 + 6 * q^28 - q^29 - 5 * q^30 + 20 * q^31 + 6 * q^32 + 9 * q^33 - 9 * q^34 + 4 * q^35 + 5 * q^36 + 16 * q^37 + 10 * q^38 + 4 * q^40 + 2 * q^41 + q^42 + 2 * q^43 + 6 * q^44 + 17 * q^45 + 21 * q^46 - 5 * q^47 + q^48 + 6 * q^49 + 8 * q^50 - 15 * q^51 + 28 * q^53 + 7 * q^54 - 29 * q^55 + 6 * q^56 + 22 * q^57 - q^58 - 12 * q^59 - 5 * q^60 - 27 * q^61 + 20 * q^62 + 5 * q^63 + 6 * q^64 + 9 * q^66 + 16 * q^67 - 9 * q^68 - 15 * q^69 + 4 * q^70 + 5 * q^72 + 38 * q^73 + 16 * q^74 - 7 * q^75 + 10 * q^76 + 6 * q^77 + 6 * q^79 + 4 * q^80 - 26 * q^81 + 2 * q^82 - 6 * q^83 + q^84 + 9 * q^85 + 2 * q^86 - 39 * q^87 + 6 * q^88 - q^89 + 17 * q^90 + 21 * q^92 - 7 * q^93 - 5 * q^94 + 3 * q^95 + q^96 + 16 * q^97 + 6 * q^98 - 4 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.85334	1.64738	0.823688	−	0.567044i	$-0.191913\pi$
$3$	2.85334	1.64738	0.823688	+	0.567044i	$0.191913\pi$
$4$	1.00000	0.500000
$5$	0.532083	0.237955	0.118977	−	0.992897i	$-0.462038\pi$
$5$	0.532083	0.237955	0.118977	+	0.992897i	$0.462038\pi$
$6$	2.85334	1.16487
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	5.14154	1.71385
$10$	0.532083	0.168259
$11$	2.71490	0.818572	0.409286	−	0.912406i	$-0.365778\pi$
$11$	2.71490	0.818572	0.409286	+	0.912406i	$0.365778\pi$
$12$	2.85334	0.823688
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	1.51821	0.392001
$16$	1.00000	0.250000
$17$	−2.94148	−0.713415	−0.356707	−	0.934216i	$-0.616101\pi$
$17$	−2.94148	−0.713415	−0.356707	+	0.934216i	$0.616101\pi$
$18$	5.14154	1.21187
$19$	5.83604	1.33888	0.669440	−	0.742866i	$-0.266534\pi$
$19$	5.83604	1.33888	0.669440	+	0.742866i	$0.266534\pi$
$20$	0.532083	0.118977
$21$	2.85334	0.622649
$22$	2.71490	0.578818
$23$	−5.52440	−1.15192	−0.575959	−	0.817479i	$-0.695371\pi$
$23$	−5.52440	−1.15192	−0.575959	+	0.817479i	$0.695371\pi$
$24$	2.85334	0.582435
$25$	−4.71689	−0.943378
$26$	0	0
$27$	6.11053	1.17597
$28$	1.00000	0.188982
$29$	−8.19741	−1.52222	−0.761111	−	0.648622i	$-0.775346\pi$
$29$	−8.19741	−1.52222	−0.761111	+	0.648622i	$0.775346\pi$
$30$	1.51821	0.277186
$31$	3.42217	0.614639	0.307320	−	0.951606i	$-0.400568\pi$
$31$	3.42217	0.614639	0.307320	+	0.951606i	$0.400568\pi$
$32$	1.00000	0.176777
$33$	7.74652	1.34850
$34$	−2.94148	−0.504460
$35$	0.532083	0.0899384
$36$	5.14154	0.856923
$37$	6.00763	0.987648	0.493824	−	0.869562i	$-0.335599\pi$
$37$	6.00763	0.987648	0.493824	+	0.869562i	$0.335599\pi$
$38$	5.83604	0.946731
$39$	0	0
$40$	0.532083	0.0841297
$41$	−7.76238	−1.21228	−0.606140	−	0.795358i	$-0.707283\pi$
$41$	−7.76238	−1.21228	−0.606140	+	0.795358i	$0.707283\pi$
$42$	2.85334	0.440280
$43$	−9.13544	−1.39314	−0.696570	−	0.717488i	$-0.745292\pi$
$43$	−9.13544	−1.39314	−0.696570	+	0.717488i	$0.745292\pi$
$44$	2.71490	0.409286
$45$	2.73572	0.407818
$46$	−5.52440	−0.814529
$47$	−9.98779	−1.45687	−0.728435	−	0.685115i	$-0.759752\pi$
$47$	−9.98779	−1.45687	−0.728435	+	0.685115i	$0.759752\pi$
$48$	2.85334	0.411844
$49$	1.00000	0.142857
$50$	−4.71689	−0.667069
$51$	−8.39305	−1.17526
$52$	0	0
$53$	12.8839	1.76973	0.884867	−	0.465844i	$-0.154249\pi$
$53$	12.8839	1.76973	0.884867	+	0.465844i	$0.154249\pi$
$54$	6.11053	0.831538
$55$	1.44455	0.194783
$56$	1.00000	0.133631
$57$	16.6522	2.20564
$58$	−8.19741	−1.07637
$59$	−1.24624	−0.162246	−0.0811232	−	0.996704i	$-0.525851\pi$
$59$	−1.24624	−0.162246	−0.0811232	+	0.996704i	$0.525851\pi$
$60$	1.51821	0.196000
$61$	1.36564	0.174852	0.0874262	−	0.996171i	$-0.472136\pi$
$61$	1.36564	0.174852	0.0874262	+	0.996171i	$0.472136\pi$
$62$	3.42217	0.434616
$63$	5.14154	0.647773
$64$	1.00000	0.125000
$65$	0	0
$66$	7.74652	0.953530
$67$	−9.49764	−1.16032	−0.580161	−	0.814502i	$-0.697010\pi$
$67$	−9.49764	−1.16032	−0.580161	+	0.814502i	$0.697010\pi$
$68$	−2.94148	−0.356707
$69$	−15.7630	−1.89764
$70$	0.532083	0.0635961
$71$	1.34074	0.159116	0.0795580	−	0.996830i	$-0.474649\pi$
$71$	1.34074	0.159116	0.0795580	+	0.996830i	$0.474649\pi$
$72$	5.14154	0.605936
$73$	11.2652	1.31849	0.659244	−	0.751929i	$-0.270876\pi$
$73$	11.2652	1.31849	0.659244	+	0.751929i	$0.270876\pi$
$74$	6.00763	0.698372
$75$	−13.4589	−1.55410
$76$	5.83604	0.669440
$77$	2.71490	0.309391
$78$	0	0
$79$	7.62525	0.857907	0.428954	−	0.903327i	$-0.358882\pi$
$79$	7.62525	0.857907	0.428954	+	0.903327i	$0.358882\pi$
$80$	0.532083	0.0594887
$81$	2.01079	0.223422
$82$	−7.76238	−0.857211
$83$	−3.49046	−0.383128	−0.191564	−	0.981480i	$-0.561356\pi$
$83$	−3.49046	−0.383128	−0.191564	+	0.981480i	$0.561356\pi$
$84$	2.85334	0.311325
$85$	−1.56511	−0.169760
$86$	−9.13544	−0.985099
$87$	−23.3900	−2.50767
$88$	2.71490	0.289409
$89$	−14.5067	−1.53771	−0.768854	−	0.639424i	$-0.779173\pi$
$89$	−14.5067	−1.53771	−0.768854	+	0.639424i	$0.779173\pi$
$90$	2.73572	0.288371
$91$	0	0
$92$	−5.52440	−0.575959
$93$	9.76460	1.01254
$94$	−9.98779	−1.03016
$95$	3.10526	0.318593
$96$	2.85334	0.291218
$97$	7.63555	0.775273	0.387636	−	0.921812i	$-0.373292\pi$
$97$	7.63555	0.775273	0.387636	+	0.921812i	$0.373292\pi$
$98$	1.00000	0.101015
$99$	13.9587	1.40291
$100$	−4.71689	−0.471689
$101$	9.28861	0.924252	0.462126	−	0.886814i	$-0.347087\pi$
$101$	9.28861	0.924252	0.462126	+	0.886814i	$0.347087\pi$
$102$	−8.39305	−0.831036
$103$	11.8102	1.16369	0.581846	−	0.813299i	$-0.302331\pi$
$103$	11.8102	1.16369	0.581846	+	0.813299i	$0.302331\pi$
$104$	0	0
$105$	1.51821	0.148162
$106$	12.8839	1.25139
$107$	14.9803	1.44820	0.724102	−	0.689693i	$-0.242254\pi$
$107$	14.9803	1.44820	0.724102	+	0.689693i	$0.242254\pi$
$108$	6.11053	0.587986
$109$	12.9227	1.23777	0.618884	−	0.785482i	$-0.287585\pi$
$109$	12.9227	1.23777	0.618884	+	0.785482i	$0.287585\pi$
$110$	1.44455	0.137732
$111$	17.1418	1.62703
$112$	1.00000	0.0944911
$113$	−13.9606	−1.31330	−0.656649	−	0.754196i	$-0.728027\pi$
$113$	−13.9606	−1.31330	−0.656649	+	0.754196i	$0.728027\pi$
$114$	16.6522	1.55962
$115$	−2.93944	−0.274104
$116$	−8.19741	−0.761111
$117$	0	0
$118$	−1.24624	−0.114726
$119$	−2.94148	−0.269645
$120$	1.51821	0.138593
$121$	−3.62934	−0.329940
$122$	1.36564	0.123639
$123$	−22.1487	−1.99708
$124$	3.42217	0.307320
$125$	−5.17019	−0.462436
$126$	5.14154	0.458045
$127$	5.30082	0.470372	0.235186	−	0.971950i	$-0.424430\pi$
$127$	5.30082	0.470372	0.235186	+	0.971950i	$0.424430\pi$
$128$	1.00000	0.0883883
$129$	−26.0665	−2.29503
$130$	0	0
$131$	−10.7726	−0.941209	−0.470605	−	0.882344i	$-0.655964\pi$
$131$	−10.7726	−0.941209	−0.470605	+	0.882344i	$0.655964\pi$
$132$	7.74652	0.674248
$133$	5.83604	0.506049
$134$	−9.49764	−0.820471
$135$	3.25131	0.279828
$136$	−2.94148	−0.252230
$137$	19.5853	1.67329	0.836644	−	0.547748i	$-0.184515\pi$
$137$	19.5853	1.67329	0.836644	+	0.547748i	$0.184515\pi$
$138$	−15.7630	−1.34184
$139$	−12.1467	−1.03027	−0.515135	−	0.857109i	$-0.672258\pi$
$139$	−12.1467	−1.03027	−0.515135	+	0.857109i	$0.672258\pi$
$140$	0.532083	0.0449692
$141$	−28.4985	−2.40001
$142$	1.34074	0.112512
$143$	0	0
$144$	5.14154	0.428461
$145$	−4.36170	−0.362220
$146$	11.2652	0.932312
$147$	2.85334	0.235339
$148$	6.00763	0.493824
$149$	7.13948	0.584889	0.292444	−	0.956283i	$-0.405531\pi$
$149$	7.13948	0.584889	0.292444	+	0.956283i	$0.405531\pi$
$150$	−13.4589	−1.09891
$151$	−9.61643	−0.782574	−0.391287	−	0.920269i	$-0.627970\pi$
$151$	−9.61643	−0.782574	−0.391287	+	0.920269i	$0.627970\pi$
$152$	5.83604	0.473365
$153$	−15.1237	−1.22268
$154$	2.71490	0.218773
$155$	1.82088	0.146256
$156$	0	0
$157$	−19.5652	−1.56147	−0.780736	−	0.624861i	$-0.785156\pi$
$157$	−19.5652	−1.56147	−0.780736	+	0.624861i	$0.785156\pi$
$158$	7.62525	0.606632
$159$	36.7620	2.91542
$160$	0.532083	0.0420648
$161$	−5.52440	−0.435384
$162$	2.01079	0.157983
$163$	−12.9875	−1.01726	−0.508628	−	0.860986i	$-0.669847\pi$
$163$	−12.9875	−1.01726	−0.508628	+	0.860986i	$0.669847\pi$
$164$	−7.76238	−0.606140
$165$	4.12179	0.320881
$166$	−3.49046	−0.270912
$167$	17.7973	1.37719	0.688597	−	0.725144i	$-0.258227\pi$
$167$	17.7973	1.37719	0.688597	+	0.725144i	$0.258227\pi$
$168$	2.85334	0.220140
$169$	0	0
$170$	−1.56511	−0.120039
$171$	30.0062	2.29463
$172$	−9.13544	−0.696570
$173$	4.08496	0.310574	0.155287	−	0.987869i	$-0.450370\pi$
$173$	4.08496	0.310574	0.155287	+	0.987869i	$0.450370\pi$
$174$	−23.3900	−1.77319
$175$	−4.71689	−0.356563
$176$	2.71490	0.204643
$177$	−3.55594	−0.267281
$178$	−14.5067	−1.08732
$179$	4.10649	0.306934	0.153467	−	0.988154i	$-0.450956\pi$
$179$	4.10649	0.306934	0.153467	+	0.988154i	$0.450956\pi$
$180$	2.73572	0.203909
$181$	20.3643	1.51367	0.756834	−	0.653607i	$-0.226745\pi$
$181$	20.3643	1.51367	0.756834	+	0.653607i	$0.226745\pi$
$182$	0	0
$183$	3.89664	0.288048
$184$	−5.52440	−0.407264
$185$	3.19656	0.235015
$186$	9.76460	0.715975
$187$	−7.98582	−0.583981
$188$	−9.98779	−0.728435
$189$	6.11053	0.444476
$190$	3.10526	0.225279
$191$	3.10166	0.224429	0.112214	−	0.993684i	$-0.464206\pi$
$191$	3.10166	0.224429	0.112214	+	0.993684i	$0.464206\pi$
$192$	2.85334	0.205922
$193$	−5.26716	−0.379138	−0.189569	−	0.981867i	$-0.560709\pi$
$193$	−5.26716	−0.379138	−0.189569	+	0.981867i	$0.560709\pi$
$194$	7.63555	0.548201
$195$	0	0
$196$	1.00000	0.0714286
$197$	−15.0025	−1.06889	−0.534443	−	0.845204i	$-0.679479\pi$
$197$	−15.0025	−1.06889	−0.534443	+	0.845204i	$0.679479\pi$
$198$	13.9587	0.992005
$199$	9.16998	0.650043	0.325021	−	0.945707i	$-0.394628\pi$
$199$	9.16998	0.650043	0.325021	+	0.945707i	$0.394628\pi$
$200$	−4.71689	−0.333534
$201$	−27.1000	−1.91148
$202$	9.28861	0.653545
$203$	−8.19741	−0.575345
$204$	−8.39305	−0.587631
$205$	−4.13023	−0.288468
$206$	11.8102	0.822854
$207$	−28.4039	−1.97421
$208$	0	0
$209$	15.8443	1.09597
$210$	1.51821	0.104767
$211$	−24.5871	−1.69264	−0.846321	−	0.532673i	$-0.821188\pi$
$211$	−24.5871	−1.69264	−0.846321	+	0.532673i	$0.821188\pi$
$212$	12.8839	0.884867
$213$	3.82557	0.262124
$214$	14.9803	1.02404
$215$	−4.86081	−0.331504
$216$	6.11053	0.415769
$217$	3.42217	0.232312
$218$	12.9227	0.875234
$219$	32.1433	2.17204
$220$	1.44455	0.0973915
$221$	0	0
$222$	17.1418	1.15048
$223$	2.77472	0.185809	0.0929045	−	0.995675i	$-0.470385\pi$
$223$	2.77472	0.185809	0.0929045	+	0.995675i	$0.470385\pi$
$224$	1.00000	0.0668153
$225$	−24.2521	−1.61680
$226$	−13.9606	−0.928642
$227$	−19.6685	−1.30544	−0.652721	−	0.757598i	$-0.726373\pi$
$227$	−19.6685	−1.30544	−0.652721	+	0.757598i	$0.726373\pi$
$228$	16.6522	1.10282
$229$	−24.2762	−1.60422	−0.802109	−	0.597178i	$-0.796289\pi$
$229$	−24.2762	−1.60422	−0.802109	+	0.597178i	$0.796289\pi$
$230$	−2.93944	−0.193821
$231$	7.74652	0.509683
$232$	−8.19741	−0.538186
$233$	16.5986	1.08741	0.543706	−	0.839276i	$-0.317021\pi$
$233$	16.5986	1.08741	0.543706	+	0.839276i	$0.317021\pi$
$234$	0	0
$235$	−5.31433	−0.346669
$236$	−1.24624	−0.0811232
$237$	21.7574	1.41330
$238$	−2.94148	−0.190668
$239$	−3.69206	−0.238820	−0.119410	−	0.992845i	$-0.538100\pi$
$239$	−3.69206	−0.238820	−0.119410	+	0.992845i	$0.538100\pi$
$240$	1.51821	0.0980002
$241$	6.84568	0.440969	0.220485	−	0.975390i	$-0.429236\pi$
$241$	6.84568	0.440969	0.220485	+	0.975390i	$0.429236\pi$
$242$	−3.62934	−0.233303
$243$	−12.5941	−0.807913
$244$	1.36564	0.0874262
$245$	0.532083	0.0339935
$246$	−22.1487	−1.41215
$247$	0	0
$248$	3.42217	0.217308
$249$	−9.95947	−0.631156
$250$	−5.17019	−0.326992
$251$	2.48250	0.156694	0.0783470	−	0.996926i	$-0.475036\pi$
$251$	2.48250	0.156694	0.0783470	+	0.996926i	$0.475036\pi$
$252$	5.14154	0.323886
$253$	−14.9982	−0.942928
$254$	5.30082	0.332603
$255$	−4.46580	−0.279659
$256$	1.00000	0.0625000
$257$	−23.6908	−1.47779	−0.738895	−	0.673820i	$-0.764652\pi$
$257$	−23.6908	−1.47779	−0.738895	+	0.673820i	$0.764652\pi$
$258$	−26.0665	−1.62283
$259$	6.00763	0.373296
$260$	0	0
$261$	−42.1473	−2.60885
$262$	−10.7726	−0.665536
$263$	−12.4047	−0.764908	−0.382454	−	0.923975i	$-0.624921\pi$
$263$	−12.4047	−0.764908	−0.382454	+	0.923975i	$0.624921\pi$
$264$	7.74652	0.476765
$265$	6.85528	0.421116
$266$	5.83604	0.357831
$267$	−41.3926	−2.53318
$268$	−9.49764	−0.580161
$269$	−19.4334	−1.18487	−0.592437	−	0.805617i	$-0.701834\pi$
$269$	−19.4334	−1.18487	−0.592437	+	0.805617i	$0.701834\pi$
$270$	3.25131	0.197868
$271$	27.4623	1.66822	0.834108	−	0.551602i	$-0.185983\pi$
$271$	27.4623	1.66822	0.834108	+	0.551602i	$0.185983\pi$
$272$	−2.94148	−0.178354
$273$	0	0
$274$	19.5853	1.18319
$275$	−12.8059	−0.772223
$276$	−15.7630	−0.948821
$277$	−11.6541	−0.700227	−0.350114	−	0.936707i	$-0.613857\pi$
$277$	−11.6541	−0.700227	−0.350114	+	0.936707i	$0.613857\pi$
$278$	−12.1467	−0.728511
$279$	17.5952	1.05340
$280$	0.532083	0.0317980
$281$	22.4177	1.33733	0.668663	−	0.743566i	$-0.266867\pi$
$281$	22.4177	1.33733	0.668663	+	0.743566i	$0.266867\pi$
$282$	−28.4985	−1.69706
$283$	−22.7926	−1.35488	−0.677439	−	0.735579i	$-0.736910\pi$
$283$	−22.7926	−1.35488	−0.677439	+	0.735579i	$0.736910\pi$
$284$	1.34074	0.0795580
$285$	8.86035	0.524842
$286$	0	0
$287$	−7.76238	−0.458199
$288$	5.14154	0.302968
$289$	−8.34767	−0.491040
$290$	−4.36170	−0.256128
$291$	21.7868	1.27717
$292$	11.2652	0.659244
$293$	6.47704	0.378393	0.189196	−	0.981939i	$-0.439412\pi$
$293$	6.47704	0.378393	0.189196	+	0.981939i	$0.439412\pi$
$294$	2.85334	0.166410
$295$	−0.663102	−0.0386073
$296$	6.00763	0.349186
$297$	16.5895	0.962618
$298$	7.13948	0.413579
$299$	0	0
$300$	−13.4589	−0.777048
$301$	−9.13544	−0.526558
$302$	−9.61643	−0.553363
$303$	26.5036	1.52259
$304$	5.83604	0.334720
$305$	0.726635	0.0416070
$306$	−15.1237	−0.864567
$307$	12.8345	0.732505	0.366253	−	0.930515i	$-0.380641\pi$
$307$	12.8345	0.732505	0.366253	+	0.930515i	$0.380641\pi$
$308$	2.71490	0.154696
$309$	33.6984	1.91704
$310$	1.82088	0.103419
$311$	−7.16543	−0.406314	−0.203157	−	0.979146i	$-0.565120\pi$
$311$	−7.16543	−0.406314	−0.203157	+	0.979146i	$0.565120\pi$
$312$	0	0
$313$	20.0799	1.13498	0.567491	−	0.823380i	$-0.307914\pi$
$313$	20.0799	1.13498	0.567491	+	0.823380i	$0.307914\pi$
$314$	−19.5652	−1.10413
$315$	2.73572	0.154141
$316$	7.62525	0.428954
$317$	18.5590	1.04238	0.521190	−	0.853441i	$-0.325488\pi$
$317$	18.5590	1.04238	0.521190	+	0.853441i	$0.325488\pi$
$318$	36.7620	2.06151
$319$	−22.2551	−1.24605
$320$	0.532083	0.0297443
$321$	42.7440	2.38574
$322$	−5.52440	−0.307863
$323$	−17.1666	−0.955176
$324$	2.01079	0.111711
$325$	0	0
$326$	−12.9875	−0.719308
$327$	36.8728	2.03907
$328$	−7.76238	−0.428606
$329$	−9.98779	−0.550645
$330$	4.12179	0.226897
$331$	27.2308	1.49674	0.748371	−	0.663280i	$-0.230836\pi$
$331$	27.2308	1.49674	0.748371	+	0.663280i	$0.230836\pi$
$332$	−3.49046	−0.191564
$333$	30.8884	1.69268
$334$	17.7973	0.973823
$335$	−5.05353	−0.276104
$336$	2.85334	0.155662
$337$	−0.559670	−0.0304872	−0.0152436	−	0.999884i	$-0.504852\pi$
$337$	−0.559670	−0.0304872	−0.0152436	+	0.999884i	$0.504852\pi$
$338$	0	0
$339$	−39.8342	−2.16350
$340$	−1.56511	−0.0848802
$341$	9.29083	0.503127
$342$	30.0062	1.62255
$343$	1.00000	0.0539949
$344$	−9.13544	−0.492550
$345$	−8.38722	−0.451553
$346$	4.08496	0.219609
$347$	−10.2001	−0.547573	−0.273786	−	0.961791i	$-0.588276\pi$
$347$	−10.2001	−0.547573	−0.273786	+	0.961791i	$0.588276\pi$
$348$	−23.3900	−1.25383
$349$	−2.44922	−0.131104	−0.0655519	−	0.997849i	$-0.520881\pi$
$349$	−2.44922	−0.131104	−0.0655519	+	0.997849i	$0.520881\pi$
$350$	−4.71689	−0.252128
$351$	0	0
$352$	2.71490	0.144704
$353$	0.674466	0.0358982	0.0179491	−	0.999839i	$-0.494286\pi$
$353$	0.674466	0.0358982	0.0179491	+	0.999839i	$0.494286\pi$
$354$	−3.55594	−0.188996
$355$	0.713383	0.0378624
$356$	−14.5067	−0.768854
$357$	−8.39305	−0.444207
$358$	4.10649	0.217035
$359$	−14.9727	−0.790230	−0.395115	−	0.918632i	$-0.629295\pi$
$359$	−14.9727	−0.790230	−0.395115	+	0.918632i	$0.629295\pi$
$360$	2.73572	0.144185
$361$	15.0594	0.792599
$362$	20.3643	1.07033
$363$	−10.3557	−0.543534
$364$	0	0
$365$	5.99400	0.313740
$366$	3.89664	0.203680
$367$	−26.8987	−1.40410	−0.702050	−	0.712128i	$-0.747732\pi$
$367$	−26.8987	−1.40410	−0.702050	+	0.712128i	$0.747732\pi$
$368$	−5.52440	−0.287979
$369$	−39.9106	−2.07766
$370$	3.19656	0.166181
$371$	12.8839	0.668896
$372$	9.76460	0.506271
$373$	26.4013	1.36701	0.683504	−	0.729947i	$-0.260455\pi$
$373$	26.4013	1.36701	0.683504	+	0.729947i	$0.260455\pi$
$374$	−7.98582	−0.412937
$375$	−14.7523	−0.761805
$376$	−9.98779	−0.515081
$377$	0	0
$378$	6.11053	0.314292
$379$	1.23207	0.0632872	0.0316436	−	0.999499i	$-0.489926\pi$
$379$	1.23207	0.0632872	0.0316436	+	0.999499i	$0.489926\pi$
$380$	3.10526	0.159296
$381$	15.1250	0.774879
$382$	3.10166	0.158695
$383$	−10.7559	−0.549602	−0.274801	−	0.961501i	$-0.588612\pi$
$383$	−10.7559	−0.549602	−0.274801	+	0.961501i	$0.588612\pi$
$384$	2.85334	0.145609
$385$	1.44455	0.0736211
$386$	−5.26716	−0.268091
$387$	−46.9702	−2.38763
$388$	7.63555	0.387636
$389$	24.5394	1.24420	0.622099	−	0.782938i	$-0.286280\pi$
$389$	24.5394	1.24420	0.622099	+	0.782938i	$0.286280\pi$
$390$	0	0
$391$	16.2499	0.821795
$392$	1.00000	0.0505076
$393$	−30.7380	−1.55053
$394$	−15.0025	−0.755817
$395$	4.05726	0.204143
$396$	13.9587	0.701453
$397$	−29.1779	−1.46440	−0.732199	−	0.681091i	$-0.761506\pi$
$397$	−29.1779	−1.46440	−0.732199	+	0.681091i	$0.761506\pi$
$398$	9.16998	0.459650
$399$	16.6522	0.833653
$400$	−4.71689	−0.235844
$401$	11.9570	0.597106	0.298553	−	0.954393i	$-0.403496\pi$
$401$	11.9570	0.597106	0.298553	+	0.954393i	$0.403496\pi$
$402$	−27.1000	−1.35162
$403$	0	0
$404$	9.28861	0.462126
$405$	1.06991	0.0531642
$406$	−8.19741	−0.406831
$407$	16.3101	0.808461
$408$	−8.39305	−0.415518
$409$	32.0698	1.58575	0.792874	−	0.609386i	$-0.208584\pi$
$409$	32.0698	1.58575	0.792874	+	0.609386i	$0.208584\pi$
$410$	−4.13023	−0.203977
$411$	55.8835	2.75653
$412$	11.8102	0.581846
$413$	−1.24624	−0.0613234
$414$	−28.4039	−1.39598
$415$	−1.85722	−0.0911671
$416$	0	0
$417$	−34.6587	−1.69724
$418$	15.8443	0.774968
$419$	30.4134	1.48579	0.742895	−	0.669408i	$-0.233452\pi$
$419$	30.4134	1.48579	0.742895	+	0.669408i	$0.233452\pi$
$420$	1.51821	0.0740812
$421$	31.4025	1.53046	0.765232	−	0.643755i	$-0.222624\pi$
$421$	31.4025	1.53046	0.765232	+	0.643755i	$0.222624\pi$
$422$	−24.5871	−1.19688
$423$	−51.3526	−2.49685
$424$	12.8839	0.625695
$425$	13.8746	0.673019
$426$	3.82557	0.185350
$427$	1.36564	0.0660880
$428$	14.9803	0.724102
$429$	0	0
$430$	−4.86081	−0.234409
$431$	−7.96279	−0.383554	−0.191777	−	0.981439i	$-0.561425\pi$
$431$	−7.96279	−0.383554	−0.191777	+	0.981439i	$0.561425\pi$
$432$	6.11053	0.293993
$433$	−12.7044	−0.610533	−0.305266	−	0.952267i	$-0.598746\pi$
$433$	−12.7044	−0.610533	−0.305266	+	0.952267i	$0.598746\pi$
$434$	3.42217	0.164269
$435$	−12.4454	−0.596712
$436$	12.9227	0.618884
$437$	−32.2407	−1.54228
$438$	32.1433	1.53587
$439$	22.4990	1.07382	0.536910	−	0.843640i	$-0.319592\pi$
$439$	22.4990	1.07382	0.536910	+	0.843640i	$0.319592\pi$
$440$	1.44455	0.0688662
$441$	5.14154	0.244835
$442$	0	0
$443$	9.31651	0.442641	0.221320	−	0.975201i	$-0.428963\pi$
$443$	9.31651	0.442641	0.221320	+	0.975201i	$0.428963\pi$
$444$	17.1418	0.813513
$445$	−7.71877	−0.365905
$446$	2.77472	0.131387
$447$	20.3713	0.963531
$448$	1.00000	0.0472456
$449$	6.54442	0.308850	0.154425	−	0.988004i	$-0.450647\pi$
$449$	6.54442	0.308850	0.154425	+	0.988004i	$0.450647\pi$
$450$	−24.2521	−1.14325
$451$	−21.0741	−0.992339
$452$	−13.9606	−0.656649
$453$	−27.4389	−1.28919
$454$	−19.6685	−0.923087
$455$	0	0
$456$	16.6522	0.779811
$457$	−14.8031	−0.692459	−0.346230	−	0.938150i	$-0.612538\pi$
$457$	−14.8031	−0.692459	−0.346230	+	0.938150i	$0.612538\pi$
$458$	−24.2762	−1.13435
$459$	−17.9740	−0.838956
$460$	−2.93944	−0.137052
$461$	1.50317	0.0700097	0.0350048	−	0.999387i	$-0.488855\pi$
$461$	1.50317	0.0700097	0.0350048	+	0.999387i	$0.488855\pi$
$462$	7.74652	0.360401
$463$	32.7578	1.52238	0.761192	−	0.648526i	$-0.224614\pi$
$463$	32.7578	1.52238	0.761192	+	0.648526i	$0.224614\pi$
$464$	−8.19741	−0.380555
$465$	5.19558	0.240939
$466$	16.5986	0.768916
$467$	16.7557	0.775361	0.387681	−	0.921794i	$-0.373276\pi$
$467$	16.7557	0.775361	0.387681	+	0.921794i	$0.373276\pi$
$468$	0	0
$469$	−9.49764	−0.438560
$470$	−5.31433	−0.245132
$471$	−55.8261	−2.57233
$472$	−1.24624	−0.0573628
$473$	−24.8018	−1.14039
$474$	21.7574	0.999351
$475$	−27.5280	−1.26307
$476$	−2.94148	−0.134823
$477$	66.2428	3.03305
$478$	−3.69206	−0.168871
$479$	−5.95833	−0.272243	−0.136122	−	0.990692i	$-0.543464\pi$
$479$	−5.95833	−0.272243	−0.136122	+	0.990692i	$0.543464\pi$
$480$	1.51821	0.0692966
$481$	0	0
$482$	6.84568	0.311812
$483$	−15.7630	−0.717241
$484$	−3.62934	−0.164970
$485$	4.06275	0.184480
$486$	−12.5941	−0.571281
$487$	16.3486	0.740827	0.370413	−	0.928867i	$-0.379216\pi$
$487$	16.3486	0.740827	0.370413	+	0.928867i	$0.379216\pi$
$488$	1.36564	0.0618197
$489$	−37.0576	−1.67580
$490$	0.532083	0.0240371
$491$	−21.9658	−0.991304	−0.495652	−	0.868521i	$-0.665071\pi$
$491$	−21.9658	−0.991304	−0.495652	+	0.868521i	$0.665071\pi$
$492$	−22.1487	−0.998540
$493$	24.1126	1.08597
$494$	0	0
$495$	7.42721	0.333828
$496$	3.42217	0.153660
$497$	1.34074	0.0601402
$498$	−9.95947	−0.446294
$499$	10.0994	0.452112	0.226056	−	0.974114i	$-0.427417\pi$
$499$	10.0994	0.452112	0.226056	+	0.974114i	$0.427417\pi$
$500$	−5.17019	−0.231218
$501$	50.7816	2.26876
$502$	2.48250	0.110799
$503$	13.4575	0.600038	0.300019	−	0.953933i	$-0.403007\pi$
$503$	13.4575	0.600038	0.300019	+	0.953933i	$0.403007\pi$
$504$	5.14154	0.229022
$505$	4.94231	0.219930
$506$	−14.9982	−0.666751
$507$	0	0
$508$	5.30082	0.235186
$509$	−24.2795	−1.07617	−0.538086	−	0.842890i	$-0.680852\pi$
$509$	−24.2795	−1.07617	−0.538086	+	0.842890i	$0.680852\pi$
$510$	−4.46580	−0.197749
$511$	11.2652	0.498342
$512$	1.00000	0.0441942
$513$	35.6613	1.57449
$514$	−23.6908	−1.04496
$515$	6.28399	0.276906
$516$	−26.0665	−1.14751
$517$	−27.1158	−1.19255
$518$	6.00763	0.263960
$519$	11.6558	0.511632
$520$	0	0
$521$	−27.6257	−1.21030	−0.605151	−	0.796110i	$-0.706887\pi$
$521$	−27.6257	−1.21030	−0.605151	+	0.796110i	$0.706887\pi$
$522$	−42.1473	−1.84474
$523$	−5.32290	−0.232754	−0.116377	−	0.993205i	$-0.537128\pi$
$523$	−5.32290	−0.232754	−0.116377	+	0.993205i	$0.537128\pi$
$524$	−10.7726	−0.470605
$525$	−13.4589	−0.587393
$526$	−12.4047	−0.540872
$527$	−10.0662	−0.438493
$528$	7.74652	0.337124
$529$	7.51904	0.326915
$530$	6.85528	0.297774
$531$	−6.40758	−0.278065
$532$	5.83604	0.253024
$533$	0	0
$534$	−41.3926	−1.79123
$535$	7.97078	0.344607
$536$	−9.49764	−0.410235
$537$	11.7172	0.505635
$538$	−19.4334	−0.837832
$539$	2.71490	0.116939
$540$	3.25131	0.139914
$541$	27.5719	1.18541	0.592704	−	0.805420i	$-0.298060\pi$
$541$	27.5719	1.18541	0.592704	+	0.805420i	$0.298060\pi$
$542$	27.4623	1.17961
$543$	58.1063	2.49358
$544$	−2.94148	−0.126115
$545$	6.87594	0.294533
$546$	0	0
$547$	28.4744	1.21748	0.608738	−	0.793371i	$-0.291676\pi$
$547$	28.4744	1.21748	0.608738	+	0.793371i	$0.291676\pi$
$548$	19.5853	0.836644
$549$	7.02150	0.299670
$550$	−12.8059	−0.546044
$551$	−47.8404	−2.03807
$552$	−15.7630	−0.670918
$553$	7.62525	0.324258
$554$	−11.6541	−0.495135
$555$	9.12085	0.387159
$556$	−12.1467	−0.515135
$557$	2.56189	0.108551	0.0542753	−	0.998526i	$-0.482715\pi$
$557$	2.56189	0.108551	0.0542753	+	0.998526i	$0.482715\pi$
$558$	17.5952	0.744864
$559$	0	0
$560$	0.532083	0.0224846
$561$	−22.7863	−0.962037
$562$	22.4177	0.945632
$563$	23.5957	0.994442	0.497221	−	0.867624i	$-0.334354\pi$
$563$	23.5957	0.994442	0.497221	+	0.867624i	$0.334354\pi$
$564$	−28.4985	−1.20001
$565$	−7.42817	−0.312506
$566$	−22.7926	−0.958044
$567$	2.01079	0.0844454
$568$	1.34074	0.0562560
$569$	7.62433	0.319629	0.159814	−	0.987147i	$-0.448910\pi$
$569$	7.62433	0.319629	0.159814	+	0.987147i	$0.448910\pi$
$570$	8.86035	0.371119
$571$	15.4711	0.647444	0.323722	−	0.946152i	$-0.395066\pi$
$571$	15.4711	0.647444	0.323722	+	0.946152i	$0.395066\pi$
$572$	0	0
$573$	8.85010	0.369718
$574$	−7.76238	−0.323995
$575$	26.0580	1.08669
$576$	5.14154	0.214231
$577$	−2.37245	−0.0987665	−0.0493832	−	0.998780i	$-0.515726\pi$
$577$	−2.37245	−0.0987665	−0.0493832	+	0.998780i	$0.515726\pi$
$578$	−8.34767	−0.347217
$579$	−15.0290	−0.624583
$580$	−4.36170	−0.181110
$581$	−3.49046	−0.144809
$582$	21.7868	0.903092
$583$	34.9783	1.44865
$584$	11.2652	0.466156
$585$	0	0
$586$	6.47704	0.267564
$587$	−22.2865	−0.919864	−0.459932	−	0.887954i	$-0.652126\pi$
$587$	−22.2865	−0.919864	−0.459932	+	0.887954i	$0.652126\pi$
$588$	2.85334	0.117670
$589$	19.9719	0.822928
$590$	−0.663102	−0.0272995
$591$	−42.8073	−1.76086
$592$	6.00763	0.246912
$593$	−12.2613	−0.503513	−0.251757	−	0.967791i	$-0.581008\pi$
$593$	−12.2613	−0.503513	−0.251757	+	0.967791i	$0.581008\pi$
$594$	16.5895	0.680674
$595$	−1.56511	−0.0641634
$596$	7.13948	0.292444
$597$	26.1651	1.07086
$598$	0	0
$599$	22.5567	0.921641	0.460820	−	0.887493i	$-0.347555\pi$
$599$	22.5567	0.921641	0.460820	+	0.887493i	$0.347555\pi$
$600$	−13.4589	−0.549456
$601$	−23.8571	−0.973153	−0.486576	−	0.873638i	$-0.661754\pi$
$601$	−23.8571	−0.973153	−0.486576	+	0.873638i	$0.661754\pi$
$602$	−9.13544	−0.372333
$603$	−48.8325	−1.98861
$604$	−9.61643	−0.391287
$605$	−1.93111	−0.0785107
$606$	26.5036	1.07663
$607$	−14.0867	−0.571760	−0.285880	−	0.958265i	$-0.592286\pi$
$607$	−14.0867	−0.571760	−0.285880	+	0.958265i	$0.592286\pi$
$608$	5.83604	0.236683
$609$	−23.3900	−0.947810
$610$	0.726635	0.0294206
$611$	0	0
$612$	−15.1237	−0.611341
$613$	−10.2670	−0.414681	−0.207341	−	0.978269i	$-0.566481\pi$
$613$	−10.2670	−0.414681	−0.207341	+	0.978269i	$0.566481\pi$
$614$	12.8345	0.517959
$615$	−11.7849	−0.475215
$616$	2.71490	0.109386
$617$	−8.82812	−0.355407	−0.177703	−	0.984084i	$-0.556867\pi$
$617$	−8.82812	−0.355407	−0.177703	+	0.984084i	$0.556867\pi$
$618$	33.6984	1.35555
$619$	−16.5151	−0.663799	−0.331899	−	0.943315i	$-0.607689\pi$
$619$	−16.5151	−0.663799	−0.331899	+	0.943315i	$0.607689\pi$
$620$	1.82088	0.0731282
$621$	−33.7570	−1.35462
$622$	−7.16543	−0.287308
$623$	−14.5067	−0.581199
$624$	0	0
$625$	20.8335	0.833339
$626$	20.0799	0.802553
$627$	45.2090	1.80547
$628$	−19.5652	−0.780736
$629$	−17.6713	−0.704602
$630$	2.73572	0.108994
$631$	16.0493	0.638913	0.319456	−	0.947601i	$-0.396500\pi$
$631$	16.0493	0.638913	0.319456	+	0.947601i	$0.396500\pi$
$632$	7.62525	0.303316
$633$	−70.1552	−2.78842
$634$	18.5590	0.737074
$635$	2.82048	0.111927
$636$	36.7620	1.45771
$637$	0	0
$638$	−22.2551	−0.881089
$639$	6.89344	0.272700
$640$	0.532083	0.0210324
$641$	−15.2785	−0.603464	−0.301732	−	0.953393i	$-0.597565\pi$
$641$	−15.2785	−0.603464	−0.301732	+	0.953393i	$0.597565\pi$
$642$	42.7440	1.68697
$643$	−31.7804	−1.25330	−0.626649	−	0.779302i	$-0.715574\pi$
$643$	−31.7804	−1.25330	−0.626649	+	0.779302i	$0.715574\pi$
$644$	−5.52440	−0.217692
$645$	−13.8695	−0.546112
$646$	−17.1666	−0.675412
$647$	−24.2231	−0.952309	−0.476155	−	0.879362i	$-0.657970\pi$
$647$	−24.2231	−0.952309	−0.476155	+	0.879362i	$0.657970\pi$
$648$	2.01079	0.0789914
$649$	−3.38341	−0.132810
$650$	0	0
$651$	9.76460	0.382705
$652$	−12.9875	−0.508628
$653$	−10.6301	−0.415989	−0.207994	−	0.978130i	$-0.566694\pi$
$653$	−10.6301	−0.415989	−0.207994	+	0.978130i	$0.566694\pi$
$654$	36.8728	1.44184
$655$	−5.73193	−0.223965
$656$	−7.76238	−0.303070
$657$	57.9203	2.25968
$658$	−9.98779	−0.389365
$659$	23.3239	0.908571	0.454285	−	0.890856i	$-0.349895\pi$
$659$	23.3239	0.908571	0.454285	+	0.890856i	$0.349895\pi$
$660$	4.12179	0.160440
$661$	2.32110	0.0902803	0.0451401	−	0.998981i	$-0.485627\pi$
$661$	2.32110	0.0902803	0.0451401	+	0.998981i	$0.485627\pi$
$662$	27.2308	1.05836
$663$	0	0
$664$	−3.49046	−0.135456
$665$	3.10526	0.120417
$666$	30.8884	1.19690
$667$	45.2858	1.75347
$668$	17.7973	0.688597
$669$	7.91721	0.306097
$670$	−5.05353	−0.195235
$671$	3.70758	0.143129
$672$	2.85334	0.110070
$673$	−22.1349	−0.853237	−0.426619	−	0.904432i	$-0.640295\pi$
$673$	−22.1349	−0.853237	−0.426619	+	0.904432i	$0.640295\pi$
$674$	−0.559670	−0.0215577
$675$	−28.8227	−1.10939
$676$	0	0
$677$	−27.8052	−1.06864	−0.534320	−	0.845282i	$-0.679432\pi$
$677$	−27.8052	−1.06864	−0.534320	+	0.845282i	$0.679432\pi$
$678$	−39.8342	−1.52982
$679$	7.63555	0.293026
$680$	−1.56511	−0.0600194
$681$	−56.1208	−2.15055
$682$	9.29083	0.355764
$683$	36.3786	1.39199	0.695994	−	0.718048i	$-0.254964\pi$
$683$	36.3786	1.39199	0.695994	+	0.718048i	$0.254964\pi$
$684$	30.0062	1.14732
$685$	10.4210	0.398167
$686$	1.00000	0.0381802
$687$	−69.2683	−2.64275
$688$	−9.13544	−0.348285
$689$	0	0
$690$	−8.38722	−0.319296
$691$	−19.4303	−0.739163	−0.369581	−	0.929198i	$-0.620499\pi$
$691$	−19.4303	−0.739163	−0.369581	+	0.929198i	$0.620499\pi$
$692$	4.08496	0.155287
$693$	13.9587	0.530249
$694$	−10.2001	−0.387192
$695$	−6.46306	−0.245158
$696$	−23.3900	−0.886595
$697$	22.8329	0.864858
$698$	−2.44922	−0.0927044
$699$	47.3615	1.79138
$700$	−4.71689	−0.178282
$701$	−36.0908	−1.36313	−0.681566	−	0.731757i	$-0.738701\pi$
$701$	−36.0908	−1.36313	−0.681566	+	0.731757i	$0.738701\pi$
$702$	0	0
$703$	35.0608	1.32234
$704$	2.71490	0.102322
$705$	−15.1636	−0.571094
$706$	0.674466	0.0253839
$707$	9.28861	0.349334
$708$	−3.55594	−0.133640
$709$	−42.3319	−1.58981	−0.794905	−	0.606734i	$-0.792479\pi$
$709$	−42.3319	−1.58981	−0.794905	+	0.606734i	$0.792479\pi$
$710$	0.713383	0.0267728
$711$	39.2055	1.47032
$712$	−14.5067	−0.543662
$713$	−18.9054	−0.708014
$714$	−8.39305	−0.314102
$715$	0	0
$716$	4.10649	0.153467
$717$	−10.5347	−0.393425
$718$	−14.9727	−0.558777
$719$	−3.73654	−0.139350	−0.0696748	−	0.997570i	$-0.522196\pi$
$719$	−3.73654	−0.139350	−0.0696748	+	0.997570i	$0.522196\pi$
$720$	2.73572	0.101954
$721$	11.8102	0.439834
$722$	15.0594	0.560452
$723$	19.5330	0.726442
$724$	20.3643	0.756834
$725$	38.6663	1.43603
$726$	−10.3557	−0.384337
$727$	−41.0128	−1.52108	−0.760540	−	0.649291i	$-0.775066\pi$
$727$	−41.0128	−1.52108	−0.760540	+	0.649291i	$0.775066\pi$
$728$	0	0
$729$	−41.9676	−1.55436
$730$	5.99400	0.221848
$731$	26.8717	0.993887
$732$	3.89664	0.144024
$733$	6.15408	0.227306	0.113653	−	0.993521i	$-0.463745\pi$
$733$	6.15408	0.227306	0.113653	+	0.993521i	$0.463745\pi$
$734$	−26.8987	−0.992849
$735$	1.51821	0.0560001
$736$	−5.52440	−0.203632
$737$	−25.7851	−0.949806
$738$	−39.9106	−1.46913
$739$	−19.5498	−0.719149	−0.359575	−	0.933116i	$-0.617078\pi$
$739$	−19.5498	−0.719149	−0.359575	+	0.933116i	$0.617078\pi$
$740$	3.19656	0.117508
$741$	0	0
$742$	12.8839	0.472981
$743$	16.6884	0.612239	0.306119	−	0.951993i	$-0.400969\pi$
$743$	16.6884	0.612239	0.306119	+	0.951993i	$0.400969\pi$
$744$	9.76460	0.357988
$745$	3.79879	0.139177
$746$	26.4013	0.966620
$747$	−17.9463	−0.656622
$748$	−7.98582	−0.291991
$749$	14.9803	0.547370
$750$	−14.7523	−0.538678
$751$	7.63652	0.278661	0.139330	−	0.990246i	$-0.455505\pi$
$751$	7.63652	0.278661	0.139330	+	0.990246i	$0.455505\pi$
$752$	−9.98779	−0.364217
$753$	7.08341	0.258134
$754$	0	0
$755$	−5.11674	−0.186217
$756$	6.11053	0.222238
$757$	−8.74215	−0.317739	−0.158870	−	0.987300i	$-0.550785\pi$
$757$	−8.74215	−0.317739	−0.158870	+	0.987300i	$0.550785\pi$
$758$	1.23207	0.0447508
$759$	−42.7949	−1.55336
$760$	3.10526	0.112640
$761$	1.38527	0.0502160	0.0251080	−	0.999685i	$-0.492007\pi$
$761$	1.38527	0.0502160	0.0251080	+	0.999685i	$0.492007\pi$
$762$	15.1250	0.547922
$763$	12.9227	0.467832
$764$	3.10166	0.112214
$765$	−8.04709	−0.290943
$766$	−10.7559	−0.388627
$767$	0	0
$768$	2.85334	0.102961
$769$	29.5853	1.06687	0.533436	−	0.845840i	$-0.320900\pi$
$769$	29.5853	1.06687	0.533436	+	0.845840i	$0.320900\pi$
$770$	1.44455	0.0520580
$771$	−67.5978	−2.43448
$772$	−5.26716	−0.189569
$773$	−4.51998	−0.162572	−0.0812862	−	0.996691i	$-0.525903\pi$
$773$	−4.51998	−0.162572	−0.0812862	+	0.996691i	$0.525903\pi$
$774$	−46.9702	−1.68831
$775$	−16.1420	−0.579837
$776$	7.63555	0.274100
$777$	17.1418	0.614958
$778$	24.5394	0.879781
$779$	−45.3016	−1.62310
$780$	0	0
$781$	3.63996	0.130248
$782$	16.2499	0.581097
$783$	−50.0905	−1.79009
$784$	1.00000	0.0357143
$785$	−10.4103	−0.371560
$786$	−30.7380	−1.09639
$787$	3.37079	0.120156	0.0600779	−	0.998194i	$-0.480865\pi$
$787$	3.37079	0.120156	0.0600779	+	0.998194i	$0.480865\pi$
$788$	−15.0025	−0.534443
$789$	−35.3949	−1.26009
$790$	4.05726	0.144351
$791$	−13.9606	−0.496380
$792$	13.9587	0.496002
$793$	0	0
$794$	−29.1779	−1.03549
$795$	19.5604	0.693737
$796$	9.16998	0.325021
$797$	12.2603	0.434281	0.217140	−	0.976140i	$-0.430327\pi$
$797$	12.2603	0.434281	0.217140	+	0.976140i	$0.430327\pi$
$798$	16.6522	0.589481
$799$	29.3789	1.03935
$800$	−4.71689	−0.166767
$801$	−74.5868	−2.63540
$802$	11.9570	0.422217
$803$	30.5838	1.07928
$804$	−27.1000	−0.955742
$805$	−2.93944	−0.103602
$806$	0	0
$807$	−55.4500	−1.95193
$808$	9.28861	0.326772
$809$	23.6424	0.831224	0.415612	−	0.909542i	$-0.363567\pi$
$809$	23.6424	0.831224	0.415612	+	0.909542i	$0.363567\pi$
$810$	1.06991	0.0375928
$811$	56.5089	1.98430	0.992148	−	0.125071i	$-0.0399159\pi$
$811$	56.5089	1.98430	0.992148	+	0.125071i	$0.0399159\pi$
$812$	−8.19741	−0.287673
$813$	78.3592	2.74818
$814$	16.3101	0.571668
$815$	−6.91040	−0.242061
$816$	−8.39305	−0.293815
$817$	−53.3148	−1.86525
$818$	32.0698	1.12129
$819$	0	0
$820$	−4.13023	−0.144234
$821$	21.8291	0.761840	0.380920	−	0.924608i	$-0.375607\pi$
$821$	21.8291	0.761840	0.380920	+	0.924608i	$0.375607\pi$
$822$	55.8835	1.94916
$823$	41.6120	1.45050	0.725251	−	0.688484i	$-0.241724\pi$
$823$	41.6120	1.45050	0.725251	+	0.688484i	$0.241724\pi$
$824$	11.8102	0.411427
$825$	−36.5395	−1.27214
$826$	−1.24624	−0.0433622
$827$	23.0557	0.801726	0.400863	−	0.916138i	$-0.368710\pi$
$827$	23.0557	0.801726	0.400863	+	0.916138i	$0.368710\pi$
$828$	−28.4039	−0.987105
$829$	30.1019	1.04548	0.522741	−	0.852492i	$-0.324910\pi$
$829$	30.1019	1.04548	0.522741	+	0.852492i	$0.324910\pi$
$830$	−1.85722	−0.0644649
$831$	−33.2531	−1.15354
$832$	0	0
$833$	−2.94148	−0.101916
$834$	−34.6587	−1.20013
$835$	9.46962	0.327710
$836$	15.8443	0.547985
$837$	20.9113	0.722799
$838$	30.4134	1.05061
$839$	13.0262	0.449714	0.224857	−	0.974392i	$-0.427808\pi$
$839$	13.0262	0.449714	0.224857	+	0.974392i	$0.427808\pi$
$840$	1.51821	0.0523833
$841$	38.1976	1.31716
$842$	31.4025	1.08220
$843$	63.9651	2.20308
$844$	−24.5871	−0.846321
$845$	0	0
$846$	−51.3526	−1.76554
$847$	−3.62934	−0.124705
$848$	12.8839	0.442433
$849$	−65.0350	−2.23199
$850$	13.8746	0.475897
$851$	−33.1886	−1.13769
$852$	3.82557	0.131062
$853$	35.1170	1.20238	0.601191	−	0.799105i	$-0.294693\pi$
$853$	35.1170	1.20238	0.601191	+	0.799105i	$0.294693\pi$
$854$	1.36564	0.0467313
$855$	15.9658	0.546019
$856$	14.9803	0.512018
$857$	−37.6715	−1.28683	−0.643416	−	0.765516i	$-0.722484\pi$
$857$	−37.6715	−1.28683	−0.643416	+	0.765516i	$0.722484\pi$
$858$	0	0
$859$	−42.9068	−1.46396	−0.731981	−	0.681325i	$-0.761404\pi$
$859$	−42.9068	−1.46396	−0.731981	+	0.681325i	$0.761404\pi$
$860$	−4.86081	−0.165752
$861$	−22.1487	−0.754825
$862$	−7.96279	−0.271214
$863$	−39.5392	−1.34593	−0.672965	−	0.739674i	$-0.734980\pi$
$863$	−39.5392	−1.34593	−0.672965	+	0.739674i	$0.734980\pi$
$864$	6.11053	0.207884
$865$	2.17354	0.0739025
$866$	−12.7044	−0.431712
$867$	−23.8187	−0.808926
$868$	3.42217	0.116156
$869$	20.7018	0.702259
$870$	−12.4454	−0.421939
$871$	0	0
$872$	12.9227	0.437617
$873$	39.2585	1.32870
$874$	−32.2407	−1.09056
$875$	−5.17019	−0.174784
$876$	32.1433	1.08602
$877$	20.1984	0.682051	0.341026	−	0.940054i	$-0.389226\pi$
$877$	20.1984	0.682051	0.341026	+	0.940054i	$0.389226\pi$
$878$	22.4990	0.759305
$879$	18.4812	0.623355
$880$	1.44455	0.0486958
$881$	17.7169	0.596899	0.298449	−	0.954425i	$-0.403531\pi$
$881$	17.7169	0.596899	0.298449	+	0.954425i	$0.403531\pi$
$882$	5.14154	0.173125
$883$	31.8330	1.07127	0.535633	−	0.844451i	$-0.320073\pi$
$883$	31.8330	1.07127	0.535633	+	0.844451i	$0.320073\pi$
$884$	0	0
$885$	−1.89206	−0.0636007
$886$	9.31651	0.312994
$887$	−30.0786	−1.00994	−0.504970	−	0.863137i	$-0.668496\pi$
$887$	−30.0786	−1.00994	−0.504970	+	0.863137i	$0.668496\pi$
$888$	17.1418	0.575241
$889$	5.30082	0.177784
$890$	−7.71877	−0.258734
$891$	5.45910	0.182887
$892$	2.77472	0.0929045
$893$	−58.2892	−1.95057
$894$	20.3713	0.681319
$895$	2.18499	0.0730363
$896$	1.00000	0.0334077
$897$	0	0
$898$	6.54442	0.218390
$899$	−28.0529	−0.935617
$900$	−24.2521	−0.808402
$901$	−37.8976	−1.26255
$902$	−21.0741	−0.701689
$903$	−26.0665	−0.867438
$904$	−13.9606	−0.464321
$905$	10.8355	0.360185
$906$	−27.4389	−0.911597
$907$	−47.4949	−1.57704	−0.788521	−	0.615008i	$-0.789153\pi$
$907$	−47.4949	−1.57704	−0.788521	+	0.615008i	$0.789153\pi$
$908$	−19.6685	−0.652721
$909$	47.7578	1.58402
$910$	0	0
$911$	42.6543	1.41320	0.706600	−	0.707613i	$-0.250228\pi$
$911$	42.6543	1.41320	0.706600	+	0.707613i	$0.250228\pi$
$912$	16.6522	0.551409
$913$	−9.47625	−0.313618
$914$	−14.8031	−0.489642
$915$	2.07333	0.0685423
$916$	−24.2762	−0.802109
$917$	−10.7726	−0.355744
$918$	−17.9740	−0.593231
$919$	0.153611	0.00506716	0.00253358	−	0.999997i	$-0.499194\pi$
$919$	0.153611	0.00506716	0.00253358	+	0.999997i	$0.499194\pi$
$920$	−2.93944	−0.0969105
$921$	36.6212	1.20671
$922$	1.50317	0.0495043
$923$	0	0
$924$	7.74652	0.254842
$925$	−28.3373	−0.931725
$926$	32.7578	1.07649
$927$	60.7225	1.99439
$928$	−8.19741	−0.269093
$929$	54.1539	1.77673	0.888366	−	0.459136i	$-0.151841\pi$
$929$	54.1539	1.77673	0.888366	+	0.459136i	$0.151841\pi$
$930$	5.19558	0.170370
$931$	5.83604	0.191269
$932$	16.5986	0.543706
$933$	−20.4454	−0.669352
$934$	16.7557	0.548263
$935$	−4.24912	−0.138961
$936$	0	0
$937$	46.1549	1.50781	0.753907	−	0.656981i	$-0.228167\pi$
$937$	46.1549	1.50781	0.753907	+	0.656981i	$0.228167\pi$
$938$	−9.49764	−0.310109
$939$	57.2947	1.86974
$940$	−5.31433	−0.173334
$941$	−10.2273	−0.333402	−0.166701	−	0.986008i	$-0.553311\pi$
$941$	−10.2273	−0.333402	−0.166701	+	0.986008i	$0.553311\pi$
$942$	−55.8261	−1.81891
$943$	42.8825	1.39645
$944$	−1.24624	−0.0405616
$945$	3.25131	0.105765
$946$	−24.8018	−0.806375
$947$	8.29170	0.269444	0.134722	−	0.990883i	$-0.456986\pi$
$947$	8.29170	0.269444	0.134722	+	0.990883i	$0.456986\pi$
$948$	21.7574	0.706648
$949$	0	0
$950$	−27.5280	−0.893125
$951$	52.9552	1.71719
$952$	−2.94148	−0.0953340
$953$	52.5932	1.70366	0.851830	−	0.523818i	$-0.175493\pi$
$953$	52.5932	1.70366	0.851830	+	0.523818i	$0.175493\pi$
$954$	66.2428	2.14469
$955$	1.65034	0.0534038
$956$	−3.69206	−0.119410
$957$	−63.5014	−2.05271
$958$	−5.95833	−0.192505
$959$	19.5853	0.632443
$960$	1.51821	0.0490001
$961$	−19.2888	−0.622219
$962$	0	0
$963$	77.0220	2.48200
$964$	6.84568	0.220485
$965$	−2.80256	−0.0902177
$966$	−15.7630	−0.507166
$967$	−44.2570	−1.42321	−0.711605	−	0.702580i	$-0.752031\pi$
$967$	−44.2570	−1.42321	−0.711605	+	0.702580i	$0.752031\pi$
$968$	−3.62934	−0.116651
$969$	−48.9822	−1.57353
$970$	4.06275	0.130447
$971$	−27.7388	−0.890182	−0.445091	−	0.895485i	$-0.646829\pi$
$971$	−27.7388	−0.890182	−0.445091	+	0.895485i	$0.646829\pi$
$972$	−12.5941	−0.403956
$973$	−12.1467	−0.389406
$974$	16.3486	0.523844
$975$	0	0
$976$	1.36564	0.0437131
$977$	−10.4341	−0.333817	−0.166908	−	0.985972i	$-0.553378\pi$
$977$	−10.4341	−0.333817	−0.166908	+	0.985972i	$0.553378\pi$
$978$	−37.0576	−1.18497
$979$	−39.3842	−1.25873
$980$	0.532083	0.0169968
$981$	66.4424	2.12134
$982$	−21.9658	−0.700958
$983$	−26.7410	−0.852906	−0.426453	−	0.904510i	$-0.640237\pi$
$983$	−26.7410	−0.852906	−0.426453	+	0.904510i	$0.640237\pi$
$984$	−22.1487	−0.706074
$985$	−7.98259	−0.254347
$986$	24.1126	0.767900
$987$	−28.4985	−0.907119
$988$	0	0
$989$	50.4678	1.60478
$990$	7.42721	0.236052
$991$	7.10312	0.225638	0.112819	−	0.993616i	$-0.464012\pi$
$991$	7.10312	0.225638	0.112819	+	0.993616i	$0.464012\pi$
$992$	3.42217	0.108654
$993$	77.6988	2.46570
$994$	1.34074	0.0425256
$995$	4.87919	0.154681
$996$	−9.95947	−0.315578
$997$	−13.6792	−0.433223	−0.216612	−	0.976258i	$-0.569501\pi$
$997$	−13.6792	−0.433223	−0.216612	+	0.976258i	$0.569501\pi$
$998$	10.0994	0.319691
$999$	36.7098	1.16145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.a.bg.1.6	yes	6
13.5	odd	4		2366.2.d.q.337.6		12
13.8	odd	4		2366.2.d.q.337.12		12
13.12	even	2		2366.2.a.be.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2366.2.a.be.1.6	✓	6	13.12	even	2
2366.2.a.bg.1.6	yes	6	1.1	even	1	trivial
2366.2.d.q.337.6		12	13.5	odd	4
2366.2.d.q.337.12		12	13.8	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.85334 q^{3} +1.00000 q^{4} +0.532083 q^{5} +2.85334 q^{6} +1.00000 q^{7} +1.00000 q^{8} +5.14154 q^{9} +0.532083 q^{10} +2.71490 q^{11} +2.85334 q^{12} +1.00000 q^{14} +1.51821 q^{15} +1.00000 q^{16} -2.94148 q^{17} +5.14154 q^{18} +5.83604 q^{19} +0.532083 q^{20} +2.85334 q^{21} +2.71490 q^{22} -5.52440 q^{23} +2.85334 q^{24} -4.71689 q^{25} +6.11053 q^{27} +1.00000 q^{28} -8.19741 q^{29} +1.51821 q^{30} +3.42217 q^{31} +1.00000 q^{32} +7.74652 q^{33} -2.94148 q^{34} +0.532083 q^{35} +5.14154 q^{36} +6.00763 q^{37} +5.83604 q^{38} +0.532083 q^{40} -7.76238 q^{41} +2.85334 q^{42} -9.13544 q^{43} +2.71490 q^{44} +2.73572 q^{45} -5.52440 q^{46} -9.98779 q^{47} +2.85334 q^{48} +1.00000 q^{49} -4.71689 q^{50} -8.39305 q^{51} +12.8839 q^{53} +6.11053 q^{54} +1.44455 q^{55} +1.00000 q^{56} +16.6522 q^{57} -8.19741 q^{58} -1.24624 q^{59} +1.51821 q^{60} +1.36564 q^{61} +3.42217 q^{62} +5.14154 q^{63} +1.00000 q^{64} +7.74652 q^{66} -9.49764 q^{67} -2.94148 q^{68} -15.7630 q^{69} +0.532083 q^{70} +1.34074 q^{71} +5.14154 q^{72} +11.2652 q^{73} +6.00763 q^{74} -13.4589 q^{75} +5.83604 q^{76} +2.71490 q^{77} +7.62525 q^{79} +0.532083 q^{80} +2.01079 q^{81} -7.76238 q^{82} -3.49046 q^{83} +2.85334 q^{84} -1.56511 q^{85} -9.13544 q^{86} -23.3900 q^{87} +2.71490 q^{88} -14.5067 q^{89} +2.73572 q^{90} -5.52440 q^{92} +9.76460 q^{93} -9.98779 q^{94} +3.10526 q^{95} +2.85334 q^{96} +7.63555 q^{97} +1.00000 q^{98} +13.9587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 4 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 5 q^{9} + 4 q^{10} + 6 q^{11} + q^{12} + 6 q^{14} - 5 q^{15} + 6 q^{16} - 9 q^{17} + 5 q^{18} + 10 q^{19} + 4 q^{20} + q^{21}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + q^3 + 6 * q^4 + 4 * q^5 + q^6 + 6 * q^7 + 6 * q^8 + 5 * q^9 + 4 * q^10 + 6 * q^11 + q^12 + 6 * q^14 - 5 * q^15 + 6 * q^16 - 9 * q^17 + 5 * q^18 + 10 * q^19 + 4 * q^20 + q^21 + 6 * q^22 + 21 * q^23 + q^24 + 8 * q^25 + 7 * q^27 + 6 * q^28 - q^29 - 5 * q^30 + 20 * q^31 + 6 * q^32 + 9 * q^33 - 9 * q^34 + 4 * q^35 + 5 * q^36 + 16 * q^37 + 10 * q^38 + 4 * q^40 + 2 * q^41 + q^42 + 2 * q^43 + 6 * q^44 + 17 * q^45 + 21 * q^46 - 5 * q^47 + q^48 + 6 * q^49 + 8 * q^50 - 15 * q^51 + 28 * q^53 + 7 * q^54 - 29 * q^55 + 6 * q^56 + 22 * q^57 - q^58 - 12 * q^59 - 5 * q^60 - 27 * q^61 + 20 * q^62 + 5 * q^63 + 6 * q^64 + 9 * q^66 + 16 * q^67 - 9 * q^68 - 15 * q^69 + 4 * q^70 + 5 * q^72 + 38 * q^73 + 16 * q^74 - 7 * q^75 + 10 * q^76 + 6 * q^77 + 6 * q^79 + 4 * q^80 - 26 * q^81 + 2 * q^82 - 6 * q^83 + q^84 + 9 * q^85 + 2 * q^86 - 39 * q^87 + 6 * q^88 - q^89 + 17 * q^90 + 21 * q^92 - 7 * q^93 - 5 * q^94 + 3 * q^95 + q^96 + 16 * q^97 + 6 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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