LMFDB - Embedded newform 8967.2.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8967,2,Mod(1,8967)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8967, 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("8967.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8967 = 3 \cdot 7^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8967.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:	$71.6018554925$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 23 x^{17} + 104 x^{16} + 198 x^{15} - 1098 x^{14} - 729 x^{13} + 6066 x^{12} + \cdots + 350 $ x^19 - 4x^18 - 23x^17 + 104x^16 + 198x^15 - 1098x^14 - 729x^13 + 6066x^12 + 529x^11 - 18926x^10 + 4020x^9 + 33838x^8 - 12274x^7 - 33918x^6 + 14107x^5 + 18090x^4 - 7188x^3 - 4490x^2 + 1343x + 350
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			1.4
Root			$-1.70252$ of defining polynomial
Character	$\chi$	$=$	8967.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.70252 q^{2} -1.00000 q^{3} +0.898581 q^{4} +0.715975 q^{5} +1.70252 q^{6} +1.87519 q^{8} +1.00000 q^{9} -1.21896 q^{10} +5.02281 q^{11} -0.898581 q^{12} +0.797072 q^{13} -0.715975 q^{15} -4.98971 q^{16} -7.91925 q^{17} -1.70252 q^{18} -1.11057 q^{19} +0.643361 q^{20} -8.55145 q^{22} +5.76080 q^{23} -1.87519 q^{24} -4.48738 q^{25} -1.35703 q^{26} -1.00000 q^{27} -6.92066 q^{29} +1.21896 q^{30} +0.0330926 q^{31} +4.74472 q^{32} -5.02281 q^{33} +13.4827 q^{34} +0.898581 q^{36} +9.85357 q^{37} +1.89077 q^{38} -0.797072 q^{39} +1.34259 q^{40} -8.44884 q^{41} -4.57304 q^{43} +4.51340 q^{44} +0.715975 q^{45} -9.80790 q^{46} -5.76215 q^{47} +4.98971 q^{48} +7.63986 q^{50} +7.91925 q^{51} +0.716234 q^{52} -6.14590 q^{53} +1.70252 q^{54} +3.59621 q^{55} +1.11057 q^{57} +11.7826 q^{58} +0.612378 q^{59} -0.643361 q^{60} -1.00000 q^{61} -0.0563409 q^{62} +1.90144 q^{64} +0.570683 q^{65} +8.55145 q^{66} +7.19225 q^{67} -7.11609 q^{68} -5.76080 q^{69} +12.6813 q^{71} +1.87519 q^{72} -14.3106 q^{73} -16.7759 q^{74} +4.48738 q^{75} -0.997939 q^{76} +1.35703 q^{78} +5.69237 q^{79} -3.57251 q^{80} +1.00000 q^{81} +14.3843 q^{82} +8.11052 q^{83} -5.66999 q^{85} +7.78570 q^{86} +6.92066 q^{87} +9.41873 q^{88} -4.96719 q^{89} -1.21896 q^{90} +5.17655 q^{92} -0.0330926 q^{93} +9.81018 q^{94} -0.795141 q^{95} -4.74472 q^{96} +8.89123 q^{97} +5.02281 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{2} - 19 q^{3} + 24 q^{4} + 2 q^{5} - 4 q^{6} + 12 q^{8} + 19 q^{9} - 10 q^{10} + 4 q^{11} - 24 q^{12} - 2 q^{15} + 34 q^{16} - 5 q^{17} + 4 q^{18} - 18 q^{19} + 24 q^{20} + 6 q^{22} + 18 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) $ 19 * q + 4 * q^2 - 19 * q^3 + 24 * q^4 + 2 * q^5 - 4 * q^6 + 12 * q^8 + 19 * q^9 - 10 * q^10 + 4 * q^11 - 24 * q^12 - 2 * q^15 + 34 * q^16 - 5 * q^17 + 4 * q^18 - 18 * q^19 + 24 * q^20 + 6 * q^22 + 18 * q^23 - 12 * q^24 + 49 * q^25 + 8 * q^26 - 19 * q^27 + 20 * q^29 + 10 * q^30 - 29 * q^31 + 38 * q^32 - 4 * q^33 + 16 * q^34 + 24 * q^36 + 22 * q^37 - 30 * q^38 - 22 * q^40 + 20 * q^41 + 24 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 34 * q^48 + 26 * q^50 + 5 * q^51 - 4 * q^52 + 16 * q^53 - 4 * q^54 - 10 * q^55 + 18 * q^57 + 16 * q^58 - 7 * q^59 - 24 * q^60 - 19 * q^61 + 44 * q^62 + 60 * q^64 + 42 * q^65 - 6 * q^66 - 14 * q^68 - 18 * q^69 + 8 * q^71 + 12 * q^72 + 2 * q^73 - 46 * q^74 - 49 * q^75 - 46 * q^76 - 8 * q^78 + 34 * q^79 - 4 * q^80 + 19 * q^81 - 34 * q^82 + 6 * q^83 + 28 * q^85 + 60 * q^86 - 20 * q^87 - 18 * q^88 - 22 * q^89 - 10 * q^90 + 46 * q^92 + 29 * q^93 - 14 * q^95 - 38 * q^96 + 12 * q^97 + 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 $

$2$	−1.70252	−1.20386	−0.601932	−	0.798547i	$-0.705602\pi$
$2$	−1.70252	−1.20386	−0.601932	+	0.798547i	$0.705602\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.898581	0.449290
$5$	0.715975	0.320194	0.160097	−	0.987101i	$-0.448819\pi$
$5$	0.715975	0.320194	0.160097	+	0.987101i	$0.448819\pi$
$6$	1.70252	0.695052
$7$	0	0
$7$	0	0
$8$	1.87519	0.662980
$9$	1.00000	0.333333
$10$	−1.21896	−0.385470
$11$	5.02281	1.51443	0.757217	−	0.653163i	$-0.226558\pi$
$11$	5.02281	1.51443	0.757217	+	0.653163i	$0.226558\pi$
$12$	−0.898581	−0.259398
$13$	0.797072	0.221068	0.110534	−	0.993872i	$-0.464744\pi$
$13$	0.797072	0.221068	0.110534	+	0.993872i	$0.464744\pi$
$14$	0	0
$15$	−0.715975	−0.184864
$16$	−4.98971	−1.24743
$17$	−7.91925	−1.92070	−0.960351	−	0.278795i	$-0.910065\pi$
$17$	−7.91925	−1.92070	−0.960351	+	0.278795i	$0.910065\pi$
$18$	−1.70252	−0.401288
$19$	−1.11057	−0.254783	−0.127391	−	0.991853i	$-0.540660\pi$
$19$	−1.11057	−0.254783	−0.127391	+	0.991853i	$0.540660\pi$
$20$	0.643361	0.143860
$21$	0	0
$22$	−8.55145	−1.82317
$23$	5.76080	1.20121	0.600605	−	0.799546i	$-0.294926\pi$
$23$	5.76080	1.20121	0.600605	+	0.799546i	$0.294926\pi$
$24$	−1.87519	−0.382772
$25$	−4.48738	−0.897476
$26$	−1.35703	−0.266136
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.92066	−1.28513	−0.642567	−	0.766230i	$-0.722130\pi$
$29$	−6.92066	−1.28513	−0.642567	+	0.766230i	$0.722130\pi$
$30$	1.21896	0.222551
$31$	0.0330926	0.00594361	0.00297181	−	0.999996i	$-0.499054\pi$
$31$	0.0330926	0.00594361	0.00297181	+	0.999996i	$0.499054\pi$
$32$	4.74472	0.838755
$33$	−5.02281	−0.874359
$34$	13.4827	2.31226
$35$	0	0
$36$	0.898581	0.149763
$37$	9.85357	1.61992	0.809959	−	0.586487i	$-0.199489\pi$
$37$	9.85357	1.61992	0.809959	+	0.586487i	$0.199489\pi$
$38$	1.89077	0.306724
$39$	−0.797072	−0.127634
$40$	1.34259	0.212282
$41$	−8.44884	−1.31949	−0.659744	−	0.751491i	$-0.729335\pi$
$41$	−8.44884	−1.31949	−0.659744	+	0.751491i	$0.729335\pi$
$42$	0	0
$43$	−4.57304	−0.697382	−0.348691	−	0.937238i	$-0.613374\pi$
$43$	−4.57304	−0.697382	−0.348691	+	0.937238i	$0.613374\pi$
$44$	4.51340	0.680421
$45$	0.715975	0.106731
$46$	−9.80790	−1.44610
$47$	−5.76215	−0.840495	−0.420248	−	0.907409i	$-0.638057\pi$
$47$	−5.76215	−0.840495	−0.420248	+	0.907409i	$0.638057\pi$
$48$	4.98971	0.720203
$49$	0	0
$50$	7.63986	1.08044
$51$	7.91925	1.10892
$52$	0.716234	0.0993237
$53$	−6.14590	−0.844205	−0.422102	−	0.906548i	$-0.638708\pi$
$53$	−6.14590	−0.844205	−0.422102	+	0.906548i	$0.638708\pi$
$54$	1.70252	0.231684
$55$	3.59621	0.484912
$56$	0	0
$57$	1.11057	0.147099
$58$	11.7826	1.54713
$59$	0.612378	0.0797248	0.0398624	−	0.999205i	$-0.487308\pi$
$59$	0.612378	0.0797248	0.0398624	+	0.999205i	$0.487308\pi$
$60$	−0.643361	−0.0830576
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.0563409	−0.00715531
$63$	0	0
$64$	1.90144	0.237680
$65$	0.570683	0.0707846
$66$	8.55145	1.05261
$67$	7.19225	0.878673	0.439337	−	0.898323i	$-0.355214\pi$
$67$	7.19225	0.878673	0.439337	+	0.898323i	$0.355214\pi$
$68$	−7.11609	−0.862953
$69$	−5.76080	−0.693519
$70$	0	0
$71$	12.6813	1.50499	0.752496	−	0.658597i	$-0.228850\pi$
$71$	12.6813	1.50499	0.752496	+	0.658597i	$0.228850\pi$
$72$	1.87519	0.220993
$73$	−14.3106	−1.67493	−0.837465	−	0.546491i	$-0.815963\pi$
$73$	−14.3106	−1.67493	−0.837465	+	0.546491i	$0.815963\pi$
$74$	−16.7759	−1.95016
$75$	4.48738	0.518158
$76$	−0.997939	−0.114471
$77$	0	0
$78$	1.35703	0.153654
$79$	5.69237	0.640441	0.320221	−	0.947343i	$-0.396243\pi$
$79$	5.69237	0.640441	0.320221	+	0.947343i	$0.396243\pi$
$80$	−3.57251	−0.399419
$81$	1.00000	0.111111
$82$	14.3843	1.58848
$83$	8.11052	0.890245	0.445123	−	0.895470i	$-0.353160\pi$
$83$	8.11052	0.890245	0.445123	+	0.895470i	$0.353160\pi$
$84$	0	0
$85$	−5.66999	−0.614996
$86$	7.78570	0.839554
$87$	6.92066	0.741972
$88$	9.41873	1.00404
$89$	−4.96719	−0.526521	−0.263261	−	0.964725i	$-0.584798\pi$
$89$	−4.96719	−0.526521	−0.263261	+	0.964725i	$0.584798\pi$
$90$	−1.21896	−0.128490
$91$	0	0
$92$	5.17655	0.539692
$93$	−0.0330926	−0.00343155
$94$	9.81018	1.01184
$95$	−0.795141	−0.0815798
$96$	−4.74472	−0.484256
$97$	8.89123	0.902767	0.451384	−	0.892330i	$-0.350931\pi$
$97$	8.89123	0.902767	0.451384	+	0.892330i	$0.350931\pi$
$98$	0	0
$99$	5.02281	0.504812
$100$	−4.03227	−0.403227
$101$	2.34654	0.233490	0.116745	−	0.993162i	$-0.462754\pi$
$101$	2.34654	0.233490	0.116745	+	0.993162i	$0.462754\pi$
$102$	−13.4827	−1.33499
$103$	9.64415	0.950266	0.475133	−	0.879914i	$-0.342400\pi$
$103$	9.64415	0.950266	0.475133	+	0.879914i	$0.342400\pi$
$104$	1.49466	0.146564
$105$	0	0
$106$	10.4635	1.01631
$107$	11.3426	1.09653	0.548267	−	0.836303i	$-0.315288\pi$
$107$	11.3426	1.09653	0.548267	+	0.836303i	$0.315288\pi$
$108$	−0.898581	−0.0864660
$109$	−3.61005	−0.345780	−0.172890	−	0.984941i	$-0.555310\pi$
$109$	−3.61005	−0.345780	−0.172890	+	0.984941i	$0.555310\pi$
$110$	−6.12262	−0.583769
$111$	−9.85357	−0.935260
$112$	0	0
$113$	−2.12570	−0.199969	−0.0999843	−	0.994989i	$-0.531879\pi$
$113$	−2.12570	−0.199969	−0.0999843	+	0.994989i	$0.531879\pi$
$114$	−1.89077	−0.177087
$115$	4.12459	0.384620
$116$	−6.21877	−0.577398
$117$	0.797072	0.0736893
$118$	−1.04259	−0.0959778
$119$	0	0
$120$	−1.34259	−0.122561
$121$	14.2286	1.29351
$122$	1.70252	0.154139
$123$	8.44884	0.761806
$124$	0.0297364	0.00267041
$125$	−6.79272	−0.607560
$126$	0	0
$127$	−0.110580	−0.00981240	−0.00490620	−	0.999988i	$-0.501562\pi$
$127$	−0.110580	−0.00981240	−0.00490620	+	0.999988i	$0.501562\pi$
$128$	−12.7267	−1.12489
$129$	4.57304	0.402634
$130$	−0.971601	−0.0852150
$131$	16.3331	1.42703	0.713514	−	0.700641i	$-0.247103\pi$
$131$	16.3331	1.42703	0.713514	+	0.700641i	$0.247103\pi$
$132$	−4.51340	−0.392841
$133$	0	0
$134$	−12.2450	−1.05780
$135$	−0.715975	−0.0616213
$136$	−14.8501	−1.27339
$137$	2.13706	0.182582	0.0912909	−	0.995824i	$-0.470901\pi$
$137$	2.13706	0.182582	0.0912909	+	0.995824i	$0.470901\pi$
$138$	9.80790	0.834903
$139$	−3.49893	−0.296775	−0.148388	−	0.988929i	$-0.547408\pi$
$139$	−3.49893	−0.296775	−0.148388	+	0.988929i	$0.547408\pi$
$140$	0	0
$141$	5.76215	0.485260
$142$	−21.5902	−1.81181
$143$	4.00354	0.334793
$144$	−4.98971	−0.415810
$145$	−4.95501	−0.411492
$146$	24.3641	2.01639
$147$	0	0
$148$	8.85423	0.727813
$149$	11.3975	0.933718	0.466859	−	0.884332i	$-0.345386\pi$
$149$	11.3975	0.933718	0.466859	+	0.884332i	$0.345386\pi$
$150$	−7.63986	−0.623792
$151$	5.93538	0.483014	0.241507	−	0.970399i	$-0.422358\pi$
$151$	5.93538	0.483014	0.241507	+	0.970399i	$0.422358\pi$
$152$	−2.08253	−0.168916
$153$	−7.91925	−0.640234
$154$	0	0
$155$	0.0236935	0.00190311
$156$	−0.716234	−0.0573446
$157$	8.17089	0.652108	0.326054	−	0.945351i	$-0.394281\pi$
$157$	8.17089	0.652108	0.326054	+	0.945351i	$0.394281\pi$
$158$	−9.69138	−0.771005
$159$	6.14590	0.487402
$160$	3.39710	0.268564
$161$	0	0
$162$	−1.70252	−0.133763
$163$	−0.316974	−0.0248273	−0.0124136	−	0.999923i	$-0.503951\pi$
$163$	−0.316974	−0.0248273	−0.0124136	+	0.999923i	$0.503951\pi$
$164$	−7.59197	−0.592833
$165$	−3.59621	−0.279964
$166$	−13.8083	−1.07173
$167$	17.8139	1.37848	0.689241	−	0.724532i	$-0.257944\pi$
$167$	17.8139	1.37848	0.689241	+	0.724532i	$0.257944\pi$
$168$	0	0
$169$	−12.3647	−0.951129
$170$	9.65328	0.740372
$171$	−1.11057	−0.0849275
$172$	−4.10925	−0.313327
$173$	22.5077	1.71123	0.855613	−	0.517616i	$-0.173180\pi$
$173$	22.5077	1.71123	0.855613	+	0.517616i	$0.173180\pi$
$174$	−11.7826	−0.893234
$175$	0	0
$176$	−25.0624	−1.88915
$177$	−0.612378	−0.0460291
$178$	8.45675	0.633860
$179$	−3.34758	−0.250210	−0.125105	−	0.992144i	$-0.539927\pi$
$179$	−3.34758	−0.250210	−0.125105	+	0.992144i	$0.539927\pi$
$180$	0.643361	0.0479533
$181$	12.2063	0.907286	0.453643	−	0.891183i	$-0.350124\pi$
$181$	12.2063	0.907286	0.453643	+	0.891183i	$0.350124\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	10.8026	0.796378
$185$	7.05491	0.518687
$186$	0.0563409	0.00413112
$187$	−39.7769	−2.90878
$188$	−5.17775	−0.377627
$189$	0	0
$190$	1.35375	0.0982110
$191$	−2.91525	−0.210940	−0.105470	−	0.994422i	$-0.533635\pi$
$191$	−2.91525	−0.210940	−0.105470	+	0.994422i	$0.533635\pi$
$192$	−1.90144	−0.137225
$193$	−16.4879	−1.18682	−0.593411	−	0.804900i	$-0.702219\pi$
$193$	−16.4879	−1.18682	−0.593411	+	0.804900i	$0.702219\pi$
$194$	−15.1375	−1.08681
$195$	−0.570683	−0.0408675
$196$	0	0
$197$	−3.08933	−0.220106	−0.110053	−	0.993926i	$-0.535102\pi$
$197$	−3.08933	−0.220106	−0.110053	+	0.993926i	$0.535102\pi$
$198$	−8.55145	−0.607725
$199$	−3.32564	−0.235748	−0.117874	−	0.993029i	$-0.537608\pi$
$199$	−3.32564	−0.235748	−0.117874	+	0.993029i	$0.537608\pi$
$200$	−8.41469	−0.595009
$201$	−7.19225	−0.507302
$202$	−3.99504	−0.281090
$203$	0	0
$204$	7.11609	0.498226
$205$	−6.04916	−0.422491
$206$	−16.4194	−1.14399
$207$	5.76080	0.400404
$208$	−3.97716	−0.275767
$209$	−5.57819	−0.385852
$210$	0	0
$211$	5.28003	0.363492	0.181746	−	0.983345i	$-0.441825\pi$
$211$	5.28003	0.363492	0.181746	+	0.983345i	$0.441825\pi$
$212$	−5.52259	−0.379293
$213$	−12.6813	−0.868907
$214$	−19.3111	−1.32008
$215$	−3.27418	−0.223297
$216$	−1.87519	−0.127591
$217$	0	0
$218$	6.14618	0.416272
$219$	14.3106	0.967021
$220$	3.23148	0.217866
$221$	−6.31222	−0.424606
$222$	16.7759	1.12593
$223$	−5.64363	−0.377925	−0.188963	−	0.981984i	$-0.560512\pi$
$223$	−5.64363	−0.377925	−0.188963	+	0.981984i	$0.560512\pi$
$224$	0	0
$225$	−4.48738	−0.299159
$226$	3.61904	0.240735
$227$	24.6696	1.63738	0.818688	−	0.574238i	$-0.194702\pi$
$227$	24.6696	1.63738	0.818688	+	0.574238i	$0.194702\pi$
$228$	0.997939	0.0660901
$229$	−17.4454	−1.15283	−0.576413	−	0.817158i	$-0.695548\pi$
$229$	−17.4454	−1.15283	−0.576413	+	0.817158i	$0.695548\pi$
$230$	−7.02220	−0.463030
$231$	0	0
$232$	−12.9775	−0.852018
$233$	−12.0618	−0.790196	−0.395098	−	0.918639i	$-0.629289\pi$
$233$	−12.0618	−0.790196	−0.395098	+	0.918639i	$0.629289\pi$
$234$	−1.35703	−0.0887120
$235$	−4.12555	−0.269121
$236$	0.550271	0.0358196
$237$	−5.69237	−0.369759
$238$	0	0
$239$	2.43011	0.157191	0.0785953	−	0.996907i	$-0.474957\pi$
$239$	2.43011	0.157191	0.0785953	+	0.996907i	$0.474957\pi$
$240$	3.57251	0.230604
$241$	−17.1573	−1.10520	−0.552598	−	0.833448i	$-0.686364\pi$
$241$	−17.1573	−1.10520	−0.552598	+	0.833448i	$0.686364\pi$
$242$	−24.2246	−1.55721
$243$	−1.00000	−0.0641500
$244$	−0.898581	−0.0575257
$245$	0	0
$246$	−14.3843	−0.917112
$247$	−0.885206	−0.0563243
$248$	0.0620550	0.00394049
$249$	−8.11052	−0.513983
$250$	11.5648	0.731420
$251$	−14.5612	−0.919097	−0.459548	−	0.888153i	$-0.651989\pi$
$251$	−14.5612	−0.919097	−0.459548	+	0.888153i	$0.651989\pi$
$252$	0	0
$253$	28.9354	1.81916
$254$	0.188265	0.0118128
$255$	5.66999	0.355068
$256$	17.8646	1.11654
$257$	11.2133	0.699467	0.349734	−	0.936849i	$-0.386272\pi$
$257$	11.2133	0.699467	0.349734	+	0.936849i	$0.386272\pi$
$258$	−7.78570	−0.484716
$259$	0	0
$260$	0.512805	0.0318028
$261$	−6.92066	−0.428378
$262$	−27.8074	−1.71795
$263$	−24.6102	−1.51753	−0.758765	−	0.651365i	$-0.774197\pi$
$263$	−24.6102	−1.51753	−0.758765	+	0.651365i	$0.774197\pi$
$264$	−9.41873	−0.579683
$265$	−4.40031	−0.270309
$266$	0	0
$267$	4.96719	0.303987
$268$	6.46282	0.394779
$269$	18.5064	1.12835	0.564177	−	0.825654i	$-0.309194\pi$
$269$	18.5064	1.12835	0.564177	+	0.825654i	$0.309194\pi$
$270$	1.21896	0.0741837
$271$	−22.8306	−1.38686	−0.693431	−	0.720523i	$-0.743902\pi$
$271$	−22.8306	−1.38686	−0.693431	+	0.720523i	$0.743902\pi$
$272$	39.5148	2.39594
$273$	0	0
$274$	−3.63840	−0.219804
$275$	−22.5393	−1.35917
$276$	−5.17655	−0.311592
$277$	−14.3030	−0.859387	−0.429693	−	0.902975i	$-0.641378\pi$
$277$	−14.3030	−0.859387	−0.429693	+	0.902975i	$0.641378\pi$
$278$	5.95701	0.357277
$279$	0.0330926	0.00198120
$280$	0	0
$281$	2.93959	0.175361	0.0876807	−	0.996149i	$-0.472054\pi$
$281$	2.93959	0.175361	0.0876807	+	0.996149i	$0.472054\pi$
$282$	−9.81018	−0.584188
$283$	−7.11484	−0.422934	−0.211467	−	0.977385i	$-0.567824\pi$
$283$	−7.11484	−0.422934	−0.211467	+	0.977385i	$0.567824\pi$
$284$	11.3952	0.676178
$285$	0.795141	0.0471001
$286$	−6.81612	−0.403046
$287$	0	0
$288$	4.74472	0.279585
$289$	45.7146	2.68909
$290$	8.43602	0.495380
$291$	−8.89123	−0.521213
$292$	−12.8592	−0.752530
$293$	−4.10621	−0.239887	−0.119944	−	0.992781i	$-0.538271\pi$
$293$	−4.10621	−0.239887	−0.119944	+	0.992781i	$0.538271\pi$
$294$	0	0
$295$	0.438447	0.0255274
$296$	18.4773	1.07397
$297$	−5.02281	−0.291453
$298$	−19.4045	−1.12407
$299$	4.59178	0.265549
$300$	4.03227	0.232803
$301$	0	0
$302$	−10.1051	−0.581484
$303$	−2.34654	−0.134805
$304$	5.54144	0.317823
$305$	−0.715975	−0.0409966
$306$	13.4827	0.770755
$307$	23.2377	1.32624	0.663122	−	0.748512i	$-0.269231\pi$
$307$	23.2377	1.32624	0.663122	+	0.748512i	$0.269231\pi$
$308$	0	0
$309$	−9.64415	−0.548636
$310$	−0.0403387	−0.00229108
$311$	27.2896	1.54745	0.773726	−	0.633521i	$-0.218391\pi$
$311$	27.2896	1.54745	0.773726	+	0.633521i	$0.218391\pi$
$312$	−1.49466	−0.0846186
$313$	−10.4593	−0.591195	−0.295597	−	0.955313i	$-0.595519\pi$
$313$	−10.4593	−0.591195	−0.295597	+	0.955313i	$0.595519\pi$
$314$	−13.9111	−0.785050
$315$	0	0
$316$	5.11505	0.287744
$317$	10.5615	0.593194	0.296597	−	0.955003i	$-0.404148\pi$
$317$	10.5615	0.593194	0.296597	+	0.955003i	$0.404148\pi$
$318$	−10.4635	−0.586766
$319$	−34.7612	−1.94625
$320$	1.36138	0.0761037
$321$	−11.3426	−0.633085
$322$	0	0
$323$	8.79490	0.489361
$324$	0.898581	0.0499212
$325$	−3.57677	−0.198403
$326$	0.539654	0.0298887
$327$	3.61005	0.199636
$328$	−15.8432	−0.874794
$329$	0	0
$330$	6.12262	0.337039
$331$	15.3791	0.845313	0.422657	−	0.906290i	$-0.361098\pi$
$331$	15.3791	0.845313	0.422657	+	0.906290i	$0.361098\pi$
$332$	7.28796	0.399979
$333$	9.85357	0.539973
$334$	−30.3286	−1.65951
$335$	5.14947	0.281346
$336$	0	0
$337$	−3.11038	−0.169433	−0.0847167	−	0.996405i	$-0.526999\pi$
$337$	−3.11038	−0.169433	−0.0847167	+	0.996405i	$0.526999\pi$
$338$	21.0511	1.14503
$339$	2.12570	0.115452
$340$	−5.09494	−0.276312
$341$	0.166218	0.00900121
$342$	1.89077	0.102241
$343$	0	0
$344$	−8.57532	−0.462350
$345$	−4.12459	−0.222060
$346$	−38.3198	−2.06008
$347$	29.8835	1.60423	0.802116	−	0.597168i	$-0.203707\pi$
$347$	29.8835	1.60423	0.802116	+	0.597168i	$0.203707\pi$
$348$	6.21877	0.333361
$349$	26.3186	1.40880	0.704400	−	0.709803i	$-0.251216\pi$
$349$	26.3186	1.40880	0.704400	+	0.709803i	$0.251216\pi$
$350$	0	0
$351$	−0.797072	−0.0425446
$352$	23.8318	1.27024
$353$	−30.9112	−1.64524	−0.822619	−	0.568594i	$-0.807488\pi$
$353$	−30.9112	−1.64524	−0.822619	+	0.568594i	$0.807488\pi$
$354$	1.04259	0.0554128
$355$	9.07948	0.481889
$356$	−4.46342	−0.236561
$357$	0	0
$358$	5.69932	0.301219
$359$	9.08987	0.479745	0.239872	−	0.970804i	$-0.422894\pi$
$359$	9.08987	0.479745	0.239872	+	0.970804i	$0.422894\pi$
$360$	1.34259	0.0707606
$361$	−17.7666	−0.935086
$362$	−20.7815	−1.09225
$363$	−14.2286	−0.746810
$364$	0	0
$365$	−10.2460	−0.536302
$366$	−1.70252	−0.0889922
$367$	−21.3134	−1.11255	−0.556275	−	0.830998i	$-0.687770\pi$
$367$	−21.3134	−1.11255	−0.556275	+	0.830998i	$0.687770\pi$
$368$	−28.7448	−1.49842
$369$	−8.44884	−0.439829
$370$	−12.0111	−0.624429
$371$	0	0
$372$	−0.0297364	−0.00154176
$373$	11.5896	0.600088	0.300044	−	0.953925i	$-0.402999\pi$
$373$	11.5896	0.600088	0.300044	+	0.953925i	$0.402999\pi$
$374$	67.7211	3.50177
$375$	6.79272	0.350775
$376$	−10.8051	−0.557231
$377$	−5.51626	−0.284102
$378$	0	0
$379$	−12.0906	−0.621052	−0.310526	−	0.950565i	$-0.600505\pi$
$379$	−12.0906	−0.621052	−0.310526	+	0.950565i	$0.600505\pi$
$380$	−0.714499	−0.0366530
$381$	0.110580	0.00566519
$382$	4.96328	0.253943
$383$	11.2910	0.576945	0.288472	−	0.957488i	$-0.406853\pi$
$383$	11.2910	0.576945	0.288472	+	0.957488i	$0.406853\pi$
$384$	12.7267	0.649456
$385$	0	0
$386$	28.0709	1.42877
$387$	−4.57304	−0.232461
$388$	7.98949	0.405605
$389$	27.7403	1.40649	0.703244	−	0.710949i	$-0.251734\pi$
$389$	27.7403	1.40649	0.703244	+	0.710949i	$0.251734\pi$
$390$	0.971601	0.0491989
$391$	−45.6213	−2.30717
$392$	0	0
$393$	−16.3331	−0.823895
$394$	5.25965	0.264977
$395$	4.07559	0.205065
$396$	4.51340	0.226807
$397$	34.8643	1.74979	0.874895	−	0.484312i	$-0.160930\pi$
$397$	34.8643	1.74979	0.874895	+	0.484312i	$0.160930\pi$
$398$	5.66197	0.283809
$399$	0	0
$400$	22.3907	1.11954
$401$	28.0508	1.40079	0.700394	−	0.713757i	$-0.253008\pi$
$401$	28.0508	1.40079	0.700394	+	0.713757i	$0.253008\pi$
$402$	12.2450	0.610723
$403$	0.0263772	0.00131394
$404$	2.10856	0.104905
$405$	0.715975	0.0355771
$406$	0	0
$407$	49.4926	2.45326
$408$	14.8501	0.735190
$409$	−14.5900	−0.721430	−0.360715	−	0.932676i	$-0.617467\pi$
$409$	−14.5900	−0.721430	−0.360715	+	0.932676i	$0.617467\pi$
$410$	10.2988	0.508623
$411$	−2.13706	−0.105414
$412$	8.66605	0.426945
$413$	0	0
$414$	−9.80790	−0.482032
$415$	5.80693	0.285051
$416$	3.78188	0.185422
$417$	3.49893	0.171343
$418$	9.49700	0.464513
$419$	−22.6052	−1.10434	−0.552168	−	0.833733i	$-0.686199\pi$
$419$	−22.6052	−1.10434	−0.552168	+	0.833733i	$0.686199\pi$
$420$	0	0
$421$	−22.6932	−1.10600	−0.553000	−	0.833181i	$-0.686517\pi$
$421$	−22.6932	−1.10600	−0.553000	+	0.833181i	$0.686517\pi$
$422$	−8.98936	−0.437595
$423$	−5.76215	−0.280165
$424$	−11.5247	−0.559691
$425$	35.5367	1.72378
$426$	21.5902	1.04605
$427$	0	0
$428$	10.1923	0.492663
$429$	−4.00354	−0.193293
$430$	5.57436	0.268820
$431$	15.4307	0.743272	0.371636	−	0.928379i	$-0.378797\pi$
$431$	15.4307	0.743272	0.371636	+	0.928379i	$0.378797\pi$
$432$	4.98971	0.240068
$433$	27.6596	1.32924	0.664618	−	0.747183i	$-0.268594\pi$
$433$	27.6596	1.32924	0.664618	+	0.747183i	$0.268594\pi$
$434$	0	0
$435$	4.95501	0.237575
$436$	−3.24392	−0.155356
$437$	−6.39779	−0.306048
$438$	−24.3641	−1.16416
$439$	0.302563	0.0144405	0.00722026	−	0.999974i	$-0.497702\pi$
$439$	0.302563	0.0144405	0.00722026	+	0.999974i	$0.497702\pi$
$440$	6.74357	0.321487
$441$	0	0
$442$	10.7467	0.511168
$443$	15.5661	0.739568	0.369784	−	0.929118i	$-0.379432\pi$
$443$	15.5661	0.739568	0.369784	+	0.929118i	$0.379432\pi$
$444$	−8.85423	−0.420203
$445$	−3.55638	−0.168589
$446$	9.60840	0.454971
$447$	−11.3975	−0.539082
$448$	0	0
$449$	11.3581	0.536023	0.268011	−	0.963416i	$-0.413634\pi$
$449$	11.3581	0.536023	0.268011	+	0.963416i	$0.413634\pi$
$450$	7.63986	0.360147
$451$	−42.4369	−1.99828
$452$	−1.91011	−0.0898440
$453$	−5.93538	−0.278868
$454$	−42.0005	−1.97118
$455$	0	0
$456$	2.08253	0.0975236
$457$	40.3117	1.88570	0.942850	−	0.333218i	$-0.108134\pi$
$457$	40.3117	1.88570	0.942850	+	0.333218i	$0.108134\pi$
$458$	29.7012	1.38785
$459$	7.91925	0.369639
$460$	3.70628	0.172806
$461$	−3.03620	−0.141410	−0.0707050	−	0.997497i	$-0.522525\pi$
$461$	−3.03620	−0.141410	−0.0707050	+	0.997497i	$0.522525\pi$
$462$	0	0
$463$	18.5030	0.859906	0.429953	−	0.902851i	$-0.358530\pi$
$463$	18.5030	0.859906	0.429953	+	0.902851i	$0.358530\pi$
$464$	34.5321	1.60311
$465$	−0.0236935	−0.00109876
$466$	20.5355	0.951289
$467$	8.89364	0.411549	0.205774	−	0.978599i	$-0.434029\pi$
$467$	8.89364	0.411549	0.205774	+	0.978599i	$0.434029\pi$
$468$	0.716234	0.0331079
$469$	0	0
$470$	7.02384	0.323986
$471$	−8.17089	−0.376495
$472$	1.14832	0.0528559
$473$	−22.9695	−1.05614
$474$	9.69138	0.445140
$475$	4.98356	0.228661
$476$	0	0
$477$	−6.14590	−0.281402
$478$	−4.13731	−0.189236
$479$	−36.4225	−1.66419	−0.832093	−	0.554635i	$-0.812858\pi$
$479$	−36.4225	−1.66419	−0.832093	+	0.554635i	$0.812858\pi$
$480$	−3.39710	−0.155056
$481$	7.85401	0.358112
$482$	29.2106	1.33051
$483$	0	0
$484$	12.7856	0.581163
$485$	6.36589	0.289060
$486$	1.70252	0.0772280
$487$	34.3756	1.55771	0.778853	−	0.627207i	$-0.215802\pi$
$487$	34.3756	1.55771	0.778853	+	0.627207i	$0.215802\pi$
$488$	−1.87519	−0.0848859
$489$	0.316974	0.0143340
$490$	0	0
$491$	−27.2807	−1.23116	−0.615581	−	0.788074i	$-0.711079\pi$
$491$	−27.2807	−1.23116	−0.615581	+	0.788074i	$0.711079\pi$
$492$	7.59197	0.342272
$493$	54.8064	2.46836
$494$	1.50708	0.0678068
$495$	3.59621	0.161637
$496$	−0.165123	−0.00741423
$497$	0	0
$498$	13.8083	0.618766
$499$	−13.9728	−0.625510	−0.312755	−	0.949834i	$-0.601252\pi$
$499$	−13.9728	−0.625510	−0.312755	+	0.949834i	$0.601252\pi$
$500$	−6.10381	−0.272971
$501$	−17.8139	−0.795867
$502$	24.7908	1.10647
$503$	18.0122	0.803126	0.401563	−	0.915831i	$-0.368467\pi$
$503$	18.0122	0.803126	0.401563	+	0.915831i	$0.368467\pi$
$504$	0	0
$505$	1.68007	0.0747620
$506$	−49.2632	−2.19002
$507$	12.3647	0.549135
$508$	−0.0993652	−0.00440862
$509$	25.8573	1.14610	0.573052	−	0.819519i	$-0.305759\pi$
$509$	25.8573	1.14610	0.573052	+	0.819519i	$0.305759\pi$
$510$	−9.65328	−0.427454
$511$	0	0
$512$	−4.96146	−0.219268
$513$	1.11057	0.0490329
$514$	−19.0909	−0.842064
$515$	6.90496	0.304269
$516$	4.10925	0.180899
$517$	−28.9422	−1.27288
$518$	0	0
$519$	−22.5077	−0.987977
$520$	1.07014	0.0469287
$521$	29.7281	1.30241	0.651205	−	0.758902i	$-0.274264\pi$
$521$	29.7281	1.30241	0.651205	+	0.758902i	$0.274264\pi$
$522$	11.7826	0.515709
$523$	−43.7715	−1.91400	−0.956998	−	0.290095i	$-0.906313\pi$
$523$	−43.7715	−1.91400	−0.956998	+	0.290095i	$0.906313\pi$
$524$	14.6766	0.641150
$525$	0	0
$526$	41.8994	1.82690
$527$	−0.262069	−0.0114159
$528$	25.0624	1.09070
$529$	10.1869	0.442907
$530$	7.49163	0.325415
$531$	0.612378	0.0265749
$532$	0	0
$533$	−6.73434	−0.291696
$534$	−8.45675	−0.365960
$535$	8.12105	0.351103
$536$	13.4868	0.582543
$537$	3.34758	0.144459
$538$	−31.5075	−1.35838
$539$	0	0
$540$	−0.643361	−0.0276859
$541$	−40.8984	−1.75836	−0.879179	−	0.476491i	$-0.841908\pi$
$541$	−40.8984	−1.75836	−0.879179	+	0.476491i	$0.841908\pi$
$542$	38.8696	1.66959
$543$	−12.2063	−0.523822
$544$	−37.5746	−1.61100
$545$	−2.58470	−0.110716
$546$	0	0
$547$	7.91121	0.338259	0.169129	−	0.985594i	$-0.445904\pi$
$547$	7.91121	0.338259	0.169129	+	0.985594i	$0.445904\pi$
$548$	1.92033	0.0820322
$549$	−1.00000	−0.0426790
$550$	38.3736	1.63626
$551$	7.68589	0.327430
$552$	−10.8026	−0.459789
$553$	0	0
$554$	24.3512	1.03459
$555$	−7.05491	−0.299464
$556$	−3.14407	−0.133338
$557$	40.6542	1.72257	0.861286	−	0.508121i	$-0.169660\pi$
$557$	40.6542	1.72257	0.861286	+	0.508121i	$0.169660\pi$
$558$	−0.0563409	−0.00238510
$559$	−3.64504	−0.154169
$560$	0	0
$561$	39.7769	1.67938
$562$	−5.00472	−0.211111
$563$	−19.3344	−0.814848	−0.407424	−	0.913239i	$-0.633573\pi$
$563$	−19.3344	−0.814848	−0.407424	+	0.913239i	$0.633573\pi$
$564$	5.17775	0.218023
$565$	−1.52194	−0.0640287
$566$	12.1132	0.509155
$567$	0	0
$568$	23.7798	0.997779
$569$	−31.1132	−1.30433	−0.652167	−	0.758075i	$-0.726140\pi$
$569$	−31.1132	−1.30433	−0.652167	+	0.758075i	$0.726140\pi$
$570$	−1.35375	−0.0567022
$571$	7.23244	0.302668	0.151334	−	0.988483i	$-0.451643\pi$
$571$	7.23244	0.302668	0.151334	+	0.988483i	$0.451643\pi$
$572$	3.59751	0.150419
$573$	2.91525	0.121786
$574$	0	0
$575$	−25.8509	−1.07806
$576$	1.90144	0.0792268
$577$	−9.34896	−0.389202	−0.194601	−	0.980882i	$-0.562341\pi$
$577$	−9.34896	−0.389202	−0.194601	+	0.980882i	$0.562341\pi$
$578$	−77.8301	−3.23731
$579$	16.4879	0.685212
$580$	−4.45248	−0.184879
$581$	0	0
$582$	15.1375	0.627470
$583$	−30.8697	−1.27849
$584$	−26.8351	−1.11044
$585$	0.570683	0.0235949
$586$	6.99092	0.288792
$587$	45.3176	1.87046	0.935228	−	0.354046i	$-0.115194\pi$
$587$	45.3176	1.87046	0.935228	+	0.354046i	$0.115194\pi$
$588$	0	0
$589$	−0.0367517	−0.00151433
$590$	−0.746465	−0.0307315
$591$	3.08933	0.127078
$592$	−49.1665	−2.02073
$593$	−29.4132	−1.20786	−0.603928	−	0.797039i	$-0.706399\pi$
$593$	−29.4132	−1.20786	−0.603928	+	0.797039i	$0.706399\pi$
$594$	8.55145	0.350870
$595$	0	0
$596$	10.2416	0.419510
$597$	3.32564	0.136109
$598$	−7.81760	−0.319685
$599$	19.3224	0.789493	0.394746	−	0.918790i	$-0.370832\pi$
$599$	19.3224	0.789493	0.394746	+	0.918790i	$0.370832\pi$
$600$	8.41469	0.343528
$601$	14.9795	0.611025	0.305513	−	0.952188i	$-0.401172\pi$
$601$	14.9795	0.611025	0.305513	+	0.952188i	$0.401172\pi$
$602$	0	0
$603$	7.19225	0.292891
$604$	5.33342	0.217014
$605$	10.1873	0.414174
$606$	3.99504	0.162288
$607$	−38.3167	−1.55523	−0.777613	−	0.628743i	$-0.783570\pi$
$607$	−38.3167	−1.55523	−0.777613	+	0.628743i	$0.783570\pi$
$608$	−5.26935	−0.213700
$609$	0	0
$610$	1.21896	0.0493544
$611$	−4.59285	−0.185807
$612$	−7.11609	−0.287651
$613$	0.941209	0.0380151	0.0190075	−	0.999819i	$-0.493949\pi$
$613$	0.941209	0.0380151	0.0190075	+	0.999819i	$0.493949\pi$
$614$	−39.5626	−1.59662
$615$	6.04916	0.243926
$616$	0	0
$617$	37.7881	1.52129	0.760646	−	0.649167i	$-0.224883\pi$
$617$	37.7881	1.52129	0.760646	+	0.649167i	$0.224883\pi$
$618$	16.4194	0.660484
$619$	8.85132	0.355764	0.177882	−	0.984052i	$-0.443075\pi$
$619$	8.85132	0.355764	0.177882	+	0.984052i	$0.443075\pi$
$620$	0.0212905	0.000855048 0
$621$	−5.76080	−0.231173
$622$	−46.4611	−1.86292
$623$	0	0
$624$	3.97716	0.159214
$625$	17.5735	0.702939
$626$	17.8072	0.711719
$627$	5.57819	0.222772
$628$	7.34221	0.292986
$629$	−78.0330	−3.11138
$630$	0	0
$631$	34.8029	1.38548	0.692740	−	0.721188i	$-0.256403\pi$
$631$	34.8029	1.38548	0.692740	+	0.721188i	$0.256403\pi$
$632$	10.6743	0.424600
$633$	−5.28003	−0.209862
$634$	−17.9812	−0.714126
$635$	−0.0791726	−0.00314187
$636$	5.52259	0.218985
$637$	0	0
$638$	59.1816	2.34302
$639$	12.6813	0.501664
$640$	−9.11198	−0.360183
$641$	−34.9944	−1.38220	−0.691098	−	0.722761i	$-0.742873\pi$
$641$	−34.9944	−1.38220	−0.691098	+	0.722761i	$0.742873\pi$
$642$	19.3111	0.762148
$643$	−2.10935	−0.0831846	−0.0415923	−	0.999135i	$-0.513243\pi$
$643$	−2.10935	−0.0831846	−0.0415923	+	0.999135i	$0.513243\pi$
$644$	0	0
$645$	3.27418	0.128921
$646$	−14.9735	−0.589125
$647$	24.2631	0.953881	0.476941	−	0.878936i	$-0.341746\pi$
$647$	24.2631	0.953881	0.476941	+	0.878936i	$0.341746\pi$
$648$	1.87519	0.0736644
$649$	3.07586	0.120738
$650$	6.08952	0.238851
$651$	0	0
$652$	−0.284826	−0.0111547
$653$	−16.7175	−0.654208	−0.327104	−	0.944988i	$-0.606073\pi$
$653$	−16.7175	−0.654208	−0.327104	+	0.944988i	$0.606073\pi$
$654$	−6.14618	−0.240335
$655$	11.6941	0.456925
$656$	42.1573	1.64597
$657$	−14.3106	−0.558310
$658$	0	0
$659$	−11.6639	−0.454361	−0.227181	−	0.973853i	$-0.572951\pi$
$659$	−11.6639	−0.454361	−0.227181	+	0.973853i	$0.572951\pi$
$660$	−3.23148	−0.125785
$661$	−22.8255	−0.887811	−0.443905	−	0.896074i	$-0.646407\pi$
$661$	−22.8255	−0.887811	−0.443905	+	0.896074i	$0.646407\pi$
$662$	−26.1833	−1.01764
$663$	6.31222	0.245146
$664$	15.2088	0.590215
$665$	0	0
$666$	−16.7759	−0.650054
$667$	−39.8685	−1.54372
$668$	16.0073	0.619339
$669$	5.64363	0.218195
$670$	−8.76708	−0.338702
$671$	−5.02281	−0.193903
$672$	0	0
$673$	−5.73065	−0.220900	−0.110450	−	0.993882i	$-0.535229\pi$
$673$	−5.73065	−0.220900	−0.110450	+	0.993882i	$0.535229\pi$
$674$	5.29549	0.203975
$675$	4.48738	0.172719
$676$	−11.1107	−0.427333
$677$	7.16870	0.275515	0.137758	−	0.990466i	$-0.456010\pi$
$677$	7.16870	0.275515	0.137758	+	0.990466i	$0.456010\pi$
$678$	−3.61904	−0.138989
$679$	0	0
$680$	−10.6323	−0.407730
$681$	−24.6696	−0.945340
$682$	−0.282990	−0.0108362
$683$	−7.97371	−0.305105	−0.152553	−	0.988295i	$-0.548749\pi$
$683$	−7.97371	−0.305105	−0.152553	+	0.988295i	$0.548749\pi$
$684$	−0.997939	−0.0381571
$685$	1.53008	0.0584615
$686$	0	0
$687$	17.4454	0.665585
$688$	22.8182	0.869934
$689$	−4.89873	−0.186627
$690$	7.02220	0.267331
$691$	25.2094	0.959009	0.479505	−	0.877539i	$-0.340816\pi$
$691$	25.2094	0.959009	0.479505	+	0.877539i	$0.340816\pi$
$692$	20.2250	0.768838
$693$	0	0
$694$	−50.8774	−1.93128
$695$	−2.50515	−0.0950256
$696$	12.9775	0.491913
$697$	66.9085	2.53434
$698$	−44.8079	−1.69600
$699$	12.0618	0.456220
$700$	0	0
$701$	−29.8776	−1.12846	−0.564230	−	0.825617i	$-0.690827\pi$
$701$	−29.8776	−1.12846	−0.564230	+	0.825617i	$0.690827\pi$
$702$	1.35703	0.0512179
$703$	−10.9431	−0.412727
$704$	9.55059	0.359951
$705$	4.12555	0.155377
$706$	52.6270	1.98064
$707$	0	0
$708$	−0.550271	−0.0206804
$709$	5.82246	0.218667	0.109334	−	0.994005i	$-0.465128\pi$
$709$	5.82246	0.218667	0.109334	+	0.994005i	$0.465128\pi$
$710$	−15.4580	−0.580129
$711$	5.69237	0.213480
$712$	−9.31443	−0.349073
$713$	0.190640	0.00713953
$714$	0	0
$715$	2.86644	0.107199
$716$	−3.00807	−0.112417
$717$	−2.43011	−0.0907540
$718$	−15.4757	−0.577548
$719$	47.7077	1.77920	0.889599	−	0.456742i	$-0.150984\pi$
$719$	47.7077	1.77920	0.889599	+	0.456742i	$0.150984\pi$
$720$	−3.57251	−0.133140
$721$	0	0
$722$	30.2481	1.12572
$723$	17.1573	0.638085
$724$	10.9683	0.407635
$725$	31.0556	1.15338
$726$	24.2246	0.899058
$727$	2.01118	0.0745907	0.0372953	−	0.999304i	$-0.488126\pi$
$727$	2.01118	0.0745907	0.0372953	+	0.999304i	$0.488126\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	17.4441	0.645635
$731$	36.2151	1.33946
$732$	0.898581	0.0332125
$733$	6.73511	0.248767	0.124383	−	0.992234i	$-0.460305\pi$
$733$	6.73511	0.248767	0.124383	+	0.992234i	$0.460305\pi$
$734$	36.2865	1.33936
$735$	0	0
$736$	27.3334	1.00752
$737$	36.1253	1.33069
$738$	14.3843	0.529495
$739$	17.8702	0.657367	0.328684	−	0.944440i	$-0.393395\pi$
$739$	17.8702	0.657367	0.328684	+	0.944440i	$0.393395\pi$
$740$	6.33941	0.233041
$741$	0.885206	0.0325188
$742$	0	0
$743$	36.9318	1.35490	0.677448	−	0.735570i	$-0.263086\pi$
$743$	36.9318	1.35490	0.677448	+	0.735570i	$0.263086\pi$
$744$	−0.0620550	−0.00227505
$745$	8.16030	0.298970
$746$	−19.7316	−0.722424
$747$	8.11052	0.296748
$748$	−35.7428	−1.30689
$749$	0	0
$750$	−11.5648	−0.422285
$751$	−15.0806	−0.550298	−0.275149	−	0.961402i	$-0.588727\pi$
$751$	−15.0806	−0.550298	−0.275149	+	0.961402i	$0.588727\pi$
$752$	28.7515	1.04846
$753$	14.5612	0.530641
$754$	9.39156	0.342020
$755$	4.24958	0.154658
$756$	0	0
$757$	−3.32546	−0.120866	−0.0604329	−	0.998172i	$-0.519248\pi$
$757$	−3.32546	−0.120866	−0.0604329	+	0.998172i	$0.519248\pi$
$758$	20.5845	0.747663
$759$	−28.9354	−1.05029
$760$	−1.49104	−0.0540857
$761$	−47.3280	−1.71564	−0.857820	−	0.513950i	$-0.828182\pi$
$761$	−47.3280	−1.71564	−0.857820	+	0.513950i	$0.828182\pi$
$762$	−0.188265	−0.00682013
$763$	0	0
$764$	−2.61959	−0.0947734
$765$	−5.66999	−0.204999
$766$	−19.2232	−0.694563
$767$	0.488109	0.0176246
$768$	−17.8646	−0.644632
$769$	−21.8179	−0.786772	−0.393386	−	0.919373i	$-0.628696\pi$
$769$	−21.8179	−0.786772	−0.393386	+	0.919373i	$0.628696\pi$
$770$	0	0
$771$	−11.2133	−0.403838
$772$	−14.8157	−0.533228
$773$	8.69294	0.312663	0.156332	−	0.987705i	$-0.450033\pi$
$773$	8.69294	0.312663	0.156332	+	0.987705i	$0.450033\pi$
$774$	7.78570	0.279851
$775$	−0.148499	−0.00533425
$776$	16.6727	0.598516
$777$	0	0
$778$	−47.2284	−1.69322
$779$	9.38305	0.336182
$780$	−0.512805	−0.0183614
$781$	63.6957	2.27921
$782$	77.6712	2.77752
$783$	6.92066	0.247324
$784$	0	0
$785$	5.85015	0.208801
$786$	27.8074	0.991858
$787$	−5.38176	−0.191839	−0.0959195	−	0.995389i	$-0.530579\pi$
$787$	−5.38176	−0.191839	−0.0959195	+	0.995389i	$0.530579\pi$
$788$	−2.77601	−0.0988913
$789$	24.6102	0.876146
$790$	−6.93878	−0.246871
$791$	0	0
$792$	9.41873	0.334680
$793$	−0.797072	−0.0283049
$794$	−59.3573	−2.10651
$795$	4.40031	0.156063
$796$	−2.98836	−0.105919
$797$	50.2321	1.77931	0.889657	−	0.456630i	$-0.150943\pi$
$797$	50.2321	1.77931	0.889657	+	0.456630i	$0.150943\pi$
$798$	0	0
$799$	45.6319	1.61434
$800$	−21.2914	−0.752763
$801$	−4.96719	−0.175507
$802$	−47.7570	−1.68636
$803$	−71.8795	−2.53657
$804$	−6.46282	−0.227926
$805$	0	0
$806$	−0.0449078	−0.00158181
$807$	−18.5064	−0.651455
$808$	4.40022	0.154799
$809$	−27.1109	−0.953169	−0.476584	−	0.879129i	$-0.658125\pi$
$809$	−27.1109	−0.953169	−0.476584	+	0.879129i	$0.658125\pi$
$810$	−1.21896	−0.0428300
$811$	39.7898	1.39721	0.698605	−	0.715508i	$-0.253805\pi$
$811$	39.7898	1.39721	0.698605	+	0.715508i	$0.253805\pi$
$812$	0	0
$813$	22.8306	0.800705
$814$	−84.2623	−2.95339
$815$	−0.226945	−0.00794954
$816$	−39.5148	−1.38330
$817$	5.07869	0.177681
$818$	24.8398	0.868504
$819$	0	0
$820$	−5.43566	−0.189821
$821$	23.3428	0.814670	0.407335	−	0.913279i	$-0.366458\pi$
$821$	23.3428	0.814670	0.407335	+	0.913279i	$0.366458\pi$
$822$	3.63840	0.126904
$823$	−10.5596	−0.368083	−0.184042	−	0.982918i	$-0.558918\pi$
$823$	−10.5596	−0.368083	−0.184042	+	0.982918i	$0.558918\pi$
$824$	18.0846	0.630007
$825$	22.5393	0.784717
$826$	0	0
$827$	1.17250	0.0407717	0.0203859	−	0.999792i	$-0.493511\pi$
$827$	1.17250	0.0407717	0.0203859	+	0.999792i	$0.493511\pi$
$828$	5.17655	0.179897
$829$	37.1523	1.29035	0.645177	−	0.764034i	$-0.276784\pi$
$829$	37.1523	1.29035	0.645177	+	0.764034i	$0.276784\pi$
$830$	−9.88642	−0.343163
$831$	14.3030	0.496167
$832$	1.51559	0.0525435
$833$	0	0
$834$	−5.95701	−0.206274
$835$	12.7543	0.441381
$836$	−5.01246	−0.173359
$837$	−0.0330926	−0.00114385
$838$	38.4859	1.32947
$839$	−4.08386	−0.140990	−0.0704952	−	0.997512i	$-0.522458\pi$
$839$	−4.08386	−0.140990	−0.0704952	+	0.997512i	$0.522458\pi$
$840$	0	0
$841$	18.8955	0.651568
$842$	38.6357	1.33147
$843$	−2.93959	−0.101245
$844$	4.74453	0.163314
$845$	−8.85280	−0.304545
$846$	9.81018	0.337281
$847$	0	0
$848$	30.6663	1.05308
$849$	7.11484	0.244181
$850$	−60.5020	−2.07520
$851$	56.7645	1.94586
$852$	−11.3952	−0.390392
$853$	49.8635	1.70729	0.853647	−	0.520853i	$-0.174386\pi$
$853$	49.8635	1.70729	0.853647	+	0.520853i	$0.174386\pi$
$854$	0	0
$855$	−0.795141	−0.0271933
$856$	21.2696	0.726980
$857$	−40.5246	−1.38429	−0.692147	−	0.721757i	$-0.743335\pi$
$857$	−40.5246	−1.38429	−0.692147	+	0.721757i	$0.743335\pi$
$858$	6.81612	0.232698
$859$	0.0749403	0.00255693	0.00127846	−	0.999999i	$-0.499593\pi$
$859$	0.0749403	0.00255693	0.00127846	+	0.999999i	$0.499593\pi$
$860$	−2.94212	−0.100325
$861$	0	0
$862$	−26.2711	−0.894798
$863$	−34.0515	−1.15913	−0.579563	−	0.814927i	$-0.696777\pi$
$863$	−34.0515	−1.15913	−0.579563	+	0.814927i	$0.696777\pi$
$864$	−4.74472	−0.161419
$865$	16.1149	0.547924
$866$	−47.0911	−1.60022
$867$	−45.7146	−1.55255
$868$	0	0
$869$	28.5917	0.969907
$870$	−8.43602	−0.286008
$871$	5.73274	0.194247
$872$	−6.76953	−0.229245
$873$	8.89123	0.300922
$874$	10.8924	0.368440
$875$	0	0
$876$	12.8592	0.434473
$877$	27.3300	0.922868	0.461434	−	0.887175i	$-0.347335\pi$
$877$	27.3300	0.922868	0.461434	+	0.887175i	$0.347335\pi$
$878$	−0.515119	−0.0173844
$879$	4.10621	0.138499
$880$	−17.9440	−0.604893
$881$	−46.2965	−1.55977	−0.779884	−	0.625924i	$-0.784722\pi$
$881$	−46.2965	−1.55977	−0.779884	+	0.625924i	$0.784722\pi$
$882$	0	0
$883$	−6.70623	−0.225683	−0.112841	−	0.993613i	$-0.535995\pi$
$883$	−6.70623	−0.225683	−0.112841	+	0.993613i	$0.535995\pi$
$884$	−5.67204	−0.190771
$885$	−0.438447	−0.0147382
$886$	−26.5016	−0.890340
$887$	−5.84579	−0.196283	−0.0981413	−	0.995172i	$-0.531290\pi$
$887$	−5.84579	−0.196283	−0.0981413	+	0.995172i	$0.531290\pi$
$888$	−18.4773	−0.620058
$889$	0	0
$890$	6.05482	0.202958
$891$	5.02281	0.168271
$892$	−5.07125	−0.169798
$893$	6.39928	0.214144
$894$	19.4045	0.648982
$895$	−2.39678	−0.0801155
$896$	0	0
$897$	−4.59178	−0.153315
$898$	−19.3374	−0.645299
$899$	−0.229023	−0.00763834
$900$	−4.03227	−0.134409
$901$	48.6710	1.62147
$902$	72.2498	2.40566
$903$	0	0
$904$	−3.98609	−0.132575
$905$	8.73939	0.290507
$906$	10.1051	0.335720
$907$	−16.2984	−0.541180	−0.270590	−	0.962695i	$-0.587219\pi$
$907$	−16.2984	−0.541180	−0.270590	+	0.962695i	$0.587219\pi$
$908$	22.1676	0.735658
$909$	2.34654	0.0778300
$910$	0	0
$911$	−5.51928	−0.182862	−0.0914309	−	0.995811i	$-0.529144\pi$
$911$	−5.51928	−0.182862	−0.0914309	+	0.995811i	$0.529144\pi$
$912$	−5.54144	−0.183495
$913$	40.7376	1.34822
$914$	−68.6315	−2.27013
$915$	0.715975	0.0236694
$916$	−15.6761	−0.517954
$917$	0	0
$918$	−13.4827	−0.444996
$919$	−37.6925	−1.24336	−0.621681	−	0.783270i	$-0.713550\pi$
$919$	−37.6925	−1.24336	−0.621681	+	0.783270i	$0.713550\pi$
$920$	7.73439	0.254995
$921$	−23.2377	−0.765707
$922$	5.16920	0.170239
$923$	10.1079	0.332705
$924$	0	0
$925$	−44.2167	−1.45384
$926$	−31.5017	−1.03521
$927$	9.64415	0.316755
$928$	−32.8366	−1.07791
$929$	−59.7371	−1.95991	−0.979956	−	0.199214i	$-0.936161\pi$
$929$	−59.7371	−1.95991	−0.979956	+	0.199214i	$0.936161\pi$
$930$	0.0403387	0.00132276
$931$	0	0
$932$	−10.8385	−0.355027
$933$	−27.2896	−0.893421
$934$	−15.1416	−0.495449
$935$	−28.4793	−0.931372
$936$	1.49466	0.0488545
$937$	10.0508	0.328345	0.164173	−	0.986432i	$-0.447505\pi$
$937$	10.0508	0.328345	0.164173	+	0.986432i	$0.447505\pi$
$938$	0	0
$939$	10.4593	0.341327
$940$	−3.70714	−0.120914
$941$	50.8909	1.65900	0.829498	−	0.558509i	$-0.188626\pi$
$941$	50.8909	1.65900	0.829498	+	0.558509i	$0.188626\pi$
$942$	13.9111	0.453249
$943$	−48.6721	−1.58498
$944$	−3.05559	−0.0994510
$945$	0	0
$946$	39.1061	1.27145
$947$	60.2902	1.95917	0.979585	−	0.201032i	$-0.0644295\pi$
$947$	60.2902	1.95917	0.979585	+	0.201032i	$0.0644295\pi$
$948$	−5.11505	−0.166129
$949$	−11.4066	−0.370273
$950$	−8.48462	−0.275277
$951$	−10.5615	−0.342481
$952$	0	0
$953$	−17.3439	−0.561824	−0.280912	−	0.959733i	$-0.590637\pi$
$953$	−17.3439	−0.561824	−0.280912	+	0.959733i	$0.590637\pi$
$954$	10.4635	0.338769
$955$	−2.08725	−0.0675417
$956$	2.18365	0.0706242
$957$	34.7612	1.12367
$958$	62.0101	2.00346
$959$	0	0
$960$	−1.36138	−0.0439385
$961$	−30.9989	−0.999965
$962$	−13.3716	−0.431118
$963$	11.3426	0.365512
$964$	−15.4172	−0.496554
$965$	−11.8049	−0.380013
$966$	0	0
$967$	21.5858	0.694154	0.347077	−	0.937837i	$-0.387174\pi$
$967$	21.5858	0.694154	0.347077	+	0.937837i	$0.387174\pi$
$968$	26.6814	0.857573
$969$	−8.79490	−0.282533
$970$	−10.8381	−0.347990
$971$	−12.1881	−0.391135	−0.195568	−	0.980690i	$-0.562655\pi$
$971$	−12.1881	−0.391135	−0.195568	+	0.980690i	$0.562655\pi$
$972$	−0.898581	−0.0288220
$973$	0	0
$974$	−58.5252	−1.87527
$975$	3.57677	0.114548
$976$	4.98971	0.159717
$977$	45.5427	1.45704	0.728520	−	0.685025i	$-0.240209\pi$
$977$	45.5427	1.45704	0.728520	+	0.685025i	$0.240209\pi$
$978$	−0.539654	−0.0172562
$979$	−24.9493	−0.797382
$980$	0	0
$981$	−3.61005	−0.115260
$982$	46.4460	1.48215
$983$	34.4219	1.09789	0.548943	−	0.835860i	$-0.315030\pi$
$983$	34.4219	1.09789	0.548943	+	0.835860i	$0.315030\pi$
$984$	15.8432	0.505062
$985$	−2.21188	−0.0704764
$986$	−93.3092	−2.97157
$987$	0	0
$988$	−0.795429	−0.0253060
$989$	−26.3444	−0.837703
$990$	−6.12262	−0.194590
$991$	54.7416	1.73893	0.869463	−	0.493998i	$-0.164465\pi$
$991$	54.7416	1.73893	0.869463	+	0.493998i	$0.164465\pi$
$992$	0.157015	0.00498524
$993$	−15.3791	−0.488042
$994$	0	0
$995$	−2.38107	−0.0754851
$996$	−7.28796	−0.230928
$997$	−35.7253	−1.13143	−0.565716	−	0.824600i	$-0.691400\pi$
$997$	−35.7253	−1.13143	−0.565716	+	0.824600i	$0.691400\pi$
$998$	23.7891	0.753029
$999$	−9.85357	−0.311753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8967.2.a.bi.1.4	✓	19
7.6	odd	2		8967.2.a.bj.1.4	yes	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8967.2.a.bi.1.4	✓	19	1.1	even	1	trivial
8967.2.a.bj.1.4	yes	19	7.6	odd	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:	\( N \)	\(=\)	\( 8967 = 3 \cdot 7^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8967.a (trivial)

\(f(q)\)	\(=\)	\(q-1.70252 q^{2} -1.00000 q^{3} +0.898581 q^{4} +0.715975 q^{5} +1.70252 q^{6} +1.87519 q^{8} +1.00000 q^{9} -1.21896 q^{10} +5.02281 q^{11} -0.898581 q^{12} +0.797072 q^{13} -0.715975 q^{15} -4.98971 q^{16} -7.91925 q^{17} -1.70252 q^{18} -1.11057 q^{19} +0.643361 q^{20} -8.55145 q^{22} +5.76080 q^{23} -1.87519 q^{24} -4.48738 q^{25} -1.35703 q^{26} -1.00000 q^{27} -6.92066 q^{29} +1.21896 q^{30} +0.0330926 q^{31} +4.74472 q^{32} -5.02281 q^{33} +13.4827 q^{34} +0.898581 q^{36} +9.85357 q^{37} +1.89077 q^{38} -0.797072 q^{39} +1.34259 q^{40} -8.44884 q^{41} -4.57304 q^{43} +4.51340 q^{44} +0.715975 q^{45} -9.80790 q^{46} -5.76215 q^{47} +4.98971 q^{48} +7.63986 q^{50} +7.91925 q^{51} +0.716234 q^{52} -6.14590 q^{53} +1.70252 q^{54} +3.59621 q^{55} +1.11057 q^{57} +11.7826 q^{58} +0.612378 q^{59} -0.643361 q^{60} -1.00000 q^{61} -0.0563409 q^{62} +1.90144 q^{64} +0.570683 q^{65} +8.55145 q^{66} +7.19225 q^{67} -7.11609 q^{68} -5.76080 q^{69} +12.6813 q^{71} +1.87519 q^{72} -14.3106 q^{73} -16.7759 q^{74} +4.48738 q^{75} -0.997939 q^{76} +1.35703 q^{78} +5.69237 q^{79} -3.57251 q^{80} +1.00000 q^{81} +14.3843 q^{82} +8.11052 q^{83} -5.66999 q^{85} +7.78570 q^{86} +6.92066 q^{87} +9.41873 q^{88} -4.96719 q^{89} -1.21896 q^{90} +5.17655 q^{92} -0.0330926 q^{93} +9.81018 q^{94} -0.795141 q^{95} -4.74472 q^{96} +8.89123 q^{97} +5.02281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{2} - 19 q^{3} + 24 q^{4} + 2 q^{5} - 4 q^{6} + 12 q^{8} + 19 q^{9} - 10 q^{10} + 4 q^{11} - 24 q^{12} - 2 q^{15} + 34 q^{16} - 5 q^{17} + 4 q^{18} - 18 q^{19} + 24 q^{20} + 6 q^{22} + 18 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 19 * q + 4 * q^2 - 19 * q^3 + 24 * q^4 + 2 * q^5 - 4 * q^6 + 12 * q^8 + 19 * q^9 - 10 * q^10 + 4 * q^11 - 24 * q^12 - 2 * q^15 + 34 * q^16 - 5 * q^17 + 4 * q^18 - 18 * q^19 + 24 * q^20 + 6 * q^22 + 18 * q^23 - 12 * q^24 + 49 * q^25 + 8 * q^26 - 19 * q^27 + 20 * q^29 + 10 * q^30 - 29 * q^31 + 38 * q^32 - 4 * q^33 + 16 * q^34 + 24 * q^36 + 22 * q^37 - 30 * q^38 - 22 * q^40 + 20 * q^41 + 24 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 34 * q^48 + 26 * q^50 + 5 * q^51 - 4 * q^52 + 16 * q^53 - 4 * q^54 - 10 * q^55 + 18 * q^57 + 16 * q^58 - 7 * q^59 - 24 * q^60 - 19 * q^61 + 44 * q^62 + 60 * q^64 + 42 * q^65 - 6 * q^66 - 14 * q^68 - 18 * q^69 + 8 * q^71 + 12 * q^72 + 2 * q^73 - 46 * q^74 - 49 * q^75 - 46 * q^76 - 8 * q^78 + 34 * q^79 - 4 * q^80 + 19 * q^81 - 34 * q^82 + 6 * q^83 + 28 * q^85 + 60 * q^86 - 20 * q^87 - 18 * q^88 - 22 * q^89 - 10 * q^90 + 46 * q^92 + 29 * q^93 - 14 * q^95 - 38 * q^96 + 12 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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