LMFDB - Embedded newform 6897.2.a.ba.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6897,2,Mod(1,6897)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6897.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6897, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6897 = 3 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6897.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,2,-7,6,-2,-2,6,12,7,-6,0,-6,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.0728222741$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 8x^{5} + 12x^{4} + 21x^{3} - 14x^{2} - 19x - 4 $ x^7 - 2x^6 - 8x^5 + 12x^4 + 21x^3 - 14x^2 - 19x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.53474$ of defining polynomial
Character	$\chi$	$=$	6897.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.53474 q^{2} -1.00000 q^{3} +4.42489 q^{4} -1.92344 q^{5} -2.53474 q^{6} -3.31812 q^{7} +6.14646 q^{8} +1.00000 q^{9} -4.87540 q^{10} -4.42489 q^{12} +2.37848 q^{13} -8.41057 q^{14} +1.92344 q^{15} +6.72988 q^{16} +0.204137 q^{17} +2.53474 q^{18} +1.00000 q^{19} -8.51100 q^{20} +3.31812 q^{21} -1.65771 q^{23} -6.14646 q^{24} -1.30039 q^{25} +6.02882 q^{26} -1.00000 q^{27} -14.6823 q^{28} +1.12757 q^{29} +4.87540 q^{30} +8.50443 q^{31} +4.76556 q^{32} +0.517434 q^{34} +6.38220 q^{35} +4.42489 q^{36} +5.13064 q^{37} +2.53474 q^{38} -2.37848 q^{39} -11.8223 q^{40} +5.48072 q^{41} +8.41057 q^{42} -4.47173 q^{43} -1.92344 q^{45} -4.20186 q^{46} -8.54180 q^{47} -6.72988 q^{48} +4.00993 q^{49} -3.29616 q^{50} -0.204137 q^{51} +10.5245 q^{52} +8.38239 q^{53} -2.53474 q^{54} -20.3947 q^{56} -1.00000 q^{57} +2.85810 q^{58} +5.15021 q^{59} +8.51100 q^{60} -1.45119 q^{61} +21.5565 q^{62} -3.31812 q^{63} -1.38033 q^{64} -4.57485 q^{65} +5.76760 q^{67} +0.903285 q^{68} +1.65771 q^{69} +16.1772 q^{70} +8.08303 q^{71} +6.14646 q^{72} +12.6433 q^{73} +13.0048 q^{74} +1.30039 q^{75} +4.42489 q^{76} -6.02882 q^{78} +4.05754 q^{79} -12.9445 q^{80} +1.00000 q^{81} +13.8922 q^{82} +17.4366 q^{83} +14.6823 q^{84} -0.392645 q^{85} -11.3347 q^{86} -1.12757 q^{87} -3.36049 q^{89} -4.87540 q^{90} -7.89209 q^{91} -7.33520 q^{92} -8.50443 q^{93} -21.6512 q^{94} -1.92344 q^{95} -4.76556 q^{96} -17.0429 q^{97} +10.1641 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} - 7 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} + 6 q^{7} + 12 q^{8} + 7 q^{9} - 6 q^{10} - 6 q^{12} + q^{13} - 4 q^{14} + 2 q^{15} + 5 q^{17} + 2 q^{18} + 7 q^{19} + 10 q^{20} - 6 q^{21} + 2 q^{23}+ \cdots + 32 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 + 6 * q^7 + 12 * q^8 + 7 * q^9 - 6 * q^10 - 6 * q^12 + q^13 - 4 * q^14 + 2 * q^15 + 5 * q^17 + 2 * q^18 + 7 * q^19 + 10 * q^20 - 6 * q^21 + 2 * q^23 - 12 * q^24 + 3 * q^25 + 4 * q^26 - 7 * q^27 + 6 * q^30 + 10 * q^31 - 4 * q^32 - 6 * q^34 - 2 * q^35 + 6 * q^36 + 14 * q^37 + 2 * q^38 - q^39 - 10 * q^40 + 28 * q^41 + 4 * q^42 + 14 * q^43 - 2 * q^45 - 24 * q^46 + 6 * q^47 + 13 * q^49 - 12 * q^50 - 5 * q^51 + 38 * q^52 + 7 * q^53 - 2 * q^54 - 4 * q^56 - 7 * q^57 - 2 * q^58 + 17 * q^59 - 10 * q^60 - 14 * q^61 + 2 * q^62 + 6 * q^63 - 14 * q^64 + 30 * q^65 + 18 * q^67 - 10 * q^68 - 2 * q^69 + 36 * q^70 + 7 * q^71 + 12 * q^72 + 4 * q^73 + 56 * q^74 - 3 * q^75 + 6 * q^76 - 4 * q^78 - 21 * q^79 - 24 * q^80 + 7 * q^81 + 9 * q^83 - 14 * q^85 + 4 * q^86 + q^89 - 6 * q^90 + 14 * q^91 - 22 * q^92 - 10 * q^93 + 14 * q^94 - 2 * q^95 + 4 * q^96 - 24 * q^97 + 32 * q^98

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 $

$2$	2.53474	1.79233	0.896165	−	0.443721i	$-0.146342\pi$
$2$	2.53474	1.79233	0.896165	+	0.443721i	$0.146342\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.42489	2.21245
$5$	−1.92344	−0.860187	−0.430093	−	0.902784i	$-0.641519\pi$
$5$	−1.92344	−0.860187	−0.430093	+	0.902784i	$0.641519\pi$
$6$	−2.53474	−1.03480
$7$	−3.31812	−1.25413	−0.627066	−	0.778966i	$-0.715744\pi$
$7$	−3.31812	−1.25413	−0.627066	+	0.778966i	$0.715744\pi$
$8$	6.14646	2.17310
$9$	1.00000	0.333333
$10$	−4.87540	−1.54174
$11$	0	0
$11$	0	0
$12$	−4.42489	−1.27736
$13$	2.37848	0.659672	0.329836	−	0.944038i	$-0.393007\pi$
$13$	2.37848	0.659672	0.329836	+	0.944038i	$0.393007\pi$
$14$	−8.41057	−2.24782
$15$	1.92344	0.496629
$16$	6.72988	1.68247
$17$	0.204137	0.0495105	0.0247553	−	0.999694i	$-0.492119\pi$
$17$	0.204137	0.0495105	0.0247553	+	0.999694i	$0.492119\pi$
$18$	2.53474	0.597443
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−8.51100	−1.90312
$21$	3.31812	0.724074
$22$	0	0
$23$	−1.65771	−0.345657	−0.172828	−	0.984952i	$-0.555291\pi$
$23$	−1.65771	−0.345657	−0.172828	+	0.984952i	$0.555291\pi$
$24$	−6.14646	−1.25464
$25$	−1.30039	−0.260079
$26$	6.02882	1.18235
$27$	−1.00000	−0.192450
$28$	−14.6823	−2.77470
$29$	1.12757	0.209385	0.104693	−	0.994505i	$-0.466614\pi$
$29$	1.12757	0.209385	0.104693	+	0.994505i	$0.466614\pi$
$30$	4.87540	0.890123
$31$	8.50443	1.52744	0.763721	−	0.645547i	$-0.223370\pi$
$31$	8.50443	1.52744	0.763721	+	0.645547i	$0.223370\pi$
$32$	4.76556	0.842440
$33$	0	0
$34$	0.517434	0.0887392
$35$	6.38220	1.07879
$36$	4.42489	0.737482
$37$	5.13064	0.843471	0.421736	−	0.906719i	$-0.361421\pi$
$37$	5.13064	0.843471	0.421736	+	0.906719i	$0.361421\pi$
$38$	2.53474	0.411189
$39$	−2.37848	−0.380862
$40$	−11.8223	−1.86927
$41$	5.48072	0.855944	0.427972	−	0.903792i	$-0.359228\pi$
$41$	5.48072	0.855944	0.427972	+	0.903792i	$0.359228\pi$
$42$	8.41057	1.29778
$43$	−4.47173	−0.681932	−0.340966	−	0.940076i	$-0.610754\pi$
$43$	−4.47173	−0.681932	−0.340966	+	0.940076i	$0.610754\pi$
$44$	0	0
$45$	−1.92344	−0.286729
$46$	−4.20186	−0.619531
$47$	−8.54180	−1.24595	−0.622975	−	0.782242i	$-0.714076\pi$
$47$	−8.54180	−1.24595	−0.622975	+	0.782242i	$0.714076\pi$
$48$	−6.72988	−0.971375
$49$	4.00993	0.572848
$50$	−3.29616	−0.466147
$51$	−0.204137	−0.0285849
$52$	10.5245	1.45949
$53$	8.38239	1.15141	0.575705	−	0.817657i	$-0.304728\pi$
$53$	8.38239	1.15141	0.575705	+	0.817657i	$0.304728\pi$
$54$	−2.53474	−0.344934
$55$	0	0
$56$	−20.3947	−2.72536
$57$	−1.00000	−0.132453
$58$	2.85810	0.375287
$59$	5.15021	0.670500	0.335250	−	0.942129i	$-0.391179\pi$
$59$	5.15021	0.670500	0.335250	+	0.942129i	$0.391179\pi$
$60$	8.51100	1.09876
$61$	−1.45119	−0.185805	−0.0929026	−	0.995675i	$-0.529615\pi$
$61$	−1.45119	−0.185805	−0.0929026	+	0.995675i	$0.529615\pi$
$62$	21.5565	2.73768
$63$	−3.31812	−0.418044
$64$	−1.38033	−0.172541
$65$	−4.57485	−0.567441
$66$	0	0
$67$	5.76760	0.704625	0.352313	−	0.935882i	$-0.385395\pi$
$67$	5.76760	0.704625	0.352313	+	0.935882i	$0.385395\pi$
$68$	0.903285	0.109539
$69$	1.65771	0.199565
$70$	16.1772	1.93354
$71$	8.08303	0.959279	0.479640	−	0.877466i	$-0.340767\pi$
$71$	8.08303	0.959279	0.479640	+	0.877466i	$0.340767\pi$
$72$	6.14646	0.724368
$73$	12.6433	1.47979	0.739893	−	0.672725i	$-0.234876\pi$
$73$	12.6433	1.47979	0.739893	+	0.672725i	$0.234876\pi$
$74$	13.0048	1.51178
$75$	1.30039	0.150157
$76$	4.42489	0.507570
$77$	0	0
$78$	−6.02882	−0.682630
$79$	4.05754	0.456509	0.228255	−	0.973601i	$-0.426698\pi$
$79$	4.05754	0.456509	0.228255	+	0.973601i	$0.426698\pi$
$80$	−12.9445	−1.44724
$81$	1.00000	0.111111
$82$	13.8922	1.53413
$83$	17.4366	1.91392	0.956958	−	0.290226i	$-0.0937305\pi$
$83$	17.4366	1.91392	0.956958	+	0.290226i	$0.0937305\pi$
$84$	14.6823	1.60197
$85$	−0.392645	−0.0425883
$86$	−11.3347	−1.22225
$87$	−1.12757	−0.120889
$88$	0	0
$89$	−3.36049	−0.356211	−0.178105	−	0.984011i	$-0.556997\pi$
$89$	−3.36049	−0.356211	−0.178105	+	0.984011i	$0.556997\pi$
$90$	−4.87540	−0.513913
$91$	−7.89209	−0.827316
$92$	−7.33520	−0.764747
$93$	−8.50443	−0.881869
$94$	−21.6512	−2.23315
$95$	−1.92344	−0.197340
$96$	−4.76556	−0.486383
$97$	−17.0429	−1.73044	−0.865220	−	0.501393i	$-0.832821\pi$
$97$	−17.0429	−1.73044	−0.865220	+	0.501393i	$0.832821\pi$
$98$	10.1641	1.02673
$99$	0	0
$100$	−5.75410	−0.575410
$101$	−4.72280	−0.469936	−0.234968	−	0.972003i	$-0.575499\pi$
$101$	−4.72280	−0.469936	−0.234968	+	0.972003i	$0.575499\pi$
$102$	−0.517434	−0.0512336
$103$	15.6721	1.54422	0.772110	−	0.635488i	$-0.219201\pi$
$103$	15.6721	1.54422	0.772110	+	0.635488i	$0.219201\pi$
$104$	14.6192	1.43353
$105$	−6.38220	−0.622838
$106$	21.2472	2.06371
$107$	13.6773	1.32223	0.661115	−	0.750285i	$-0.270084\pi$
$107$	13.6773	1.32223	0.661115	+	0.750285i	$0.270084\pi$
$108$	−4.42489	−0.425785
$109$	−1.52679	−0.146240	−0.0731200	−	0.997323i	$-0.523296\pi$
$109$	−1.52679	−0.146240	−0.0731200	+	0.997323i	$0.523296\pi$
$110$	0	0
$111$	−5.13064	−0.486978
$112$	−22.3306	−2.11004
$113$	7.96153	0.748958	0.374479	−	0.927235i	$-0.377822\pi$
$113$	7.96153	0.748958	0.374479	+	0.927235i	$0.377822\pi$
$114$	−2.53474	−0.237400
$115$	3.18850	0.297329
$116$	4.98939	0.463253
$117$	2.37848	0.219891
$118$	13.0544	1.20176
$119$	−0.677352	−0.0620928
$120$	11.8223	1.07923
$121$	0	0
$122$	−3.67837	−0.333024
$123$	−5.48072	−0.494180
$124$	37.6312	3.37938
$125$	12.1184	1.08390
$126$	−8.41057	−0.749273
$127$	−6.44545	−0.571942	−0.285971	−	0.958238i	$-0.592316\pi$
$127$	−6.44545	−0.571942	−0.285971	+	0.958238i	$0.592316\pi$
$128$	−13.0299	−1.15169
$129$	4.47173	0.393714
$130$	−11.5961	−1.01704
$131$	13.1752	1.15112	0.575561	−	0.817759i	$-0.304784\pi$
$131$	13.1752	1.15112	0.575561	+	0.817759i	$0.304784\pi$
$132$	0	0
$133$	−3.31812	−0.287718
$134$	14.6194	1.26292
$135$	1.92344	0.165543
$136$	1.25472	0.107591
$137$	−3.81690	−0.326100	−0.163050	−	0.986618i	$-0.552133\pi$
$137$	−3.81690	−0.326100	−0.163050	+	0.986618i	$0.552133\pi$
$138$	4.20186	0.357686
$139$	13.2851	1.12683	0.563414	−	0.826175i	$-0.309488\pi$
$139$	13.2851	1.12683	0.563414	+	0.826175i	$0.309488\pi$
$140$	28.2405	2.38676
$141$	8.54180	0.719349
$142$	20.4884	1.71934
$143$	0	0
$144$	6.72988	0.560824
$145$	−2.16881	−0.180110
$146$	32.0474	2.65226
$147$	−4.00993	−0.330734
$148$	22.7025	1.86613
$149$	−1.62183	−0.132866	−0.0664328	−	0.997791i	$-0.521162\pi$
$149$	−1.62183	−0.132866	−0.0664328	+	0.997791i	$0.521162\pi$
$150$	3.29616	0.269130
$151$	−20.4540	−1.66453	−0.832263	−	0.554382i	$-0.812955\pi$
$151$	−20.4540	−1.66453	−0.832263	+	0.554382i	$0.812955\pi$
$152$	6.14646	0.498544
$153$	0.204137	0.0165035
$154$	0	0
$155$	−16.3577	−1.31388
$156$	−10.5245	−0.842636
$157$	4.12547	0.329248	0.164624	−	0.986356i	$-0.447359\pi$
$157$	4.12547	0.329248	0.164624	+	0.986356i	$0.447359\pi$
$158$	10.2848	0.818215
$159$	−8.38239	−0.664767
$160$	−9.16625	−0.724656
$161$	5.50049	0.433499
$162$	2.53474	0.199148
$163$	11.9021	0.932244	0.466122	−	0.884721i	$-0.345651\pi$
$163$	11.9021	0.932244	0.466122	+	0.884721i	$0.345651\pi$
$164$	24.2516	1.89373
$165$	0	0
$166$	44.1972	3.43037
$167$	20.2408	1.56628	0.783142	−	0.621843i	$-0.213616\pi$
$167$	20.2408	1.56628	0.783142	+	0.621843i	$0.213616\pi$
$168$	20.3947	1.57349
$169$	−7.34283	−0.564833
$170$	−0.995251	−0.0763323
$171$	1.00000	0.0764719
$172$	−19.7869	−1.50874
$173$	−2.33910	−0.177839	−0.0889193	−	0.996039i	$-0.528341\pi$
$173$	−2.33910	−0.177839	−0.0889193	+	0.996039i	$0.528341\pi$
$174$	−2.85810	−0.216672
$175$	4.31487	0.326173
$176$	0	0
$177$	−5.15021	−0.387114
$178$	−8.51795	−0.638447
$179$	5.91659	0.442227	0.221113	−	0.975248i	$-0.429031\pi$
$179$	5.91659	0.442227	0.221113	+	0.975248i	$0.429031\pi$
$180$	−8.51100	−0.634372
$181$	−24.6664	−1.83344	−0.916720	−	0.399529i	$-0.869174\pi$
$181$	−24.6664	−1.83344	−0.916720	+	0.399529i	$0.869174\pi$
$182$	−20.0044	−1.48282
$183$	1.45119	0.107275
$184$	−10.1891	−0.751148
$185$	−9.86845	−0.725543
$186$	−21.5565	−1.58060
$187$	0	0
$188$	−37.7965	−2.75660
$189$	3.31812	0.241358
$190$	−4.87540	−0.353699
$191$	9.22647	0.667604	0.333802	−	0.942643i	$-0.391668\pi$
$191$	9.22647	0.667604	0.333802	+	0.942643i	$0.391668\pi$
$192$	1.38033	0.0996166
$193$	21.7223	1.56360	0.781801	−	0.623528i	$-0.214301\pi$
$193$	21.7223	1.56360	0.781801	+	0.623528i	$0.214301\pi$
$194$	−43.1992	−3.10152
$195$	4.57485	0.327612
$196$	17.7435	1.26739
$197$	0.898238	0.0639968	0.0319984	−	0.999488i	$-0.489813\pi$
$197$	0.898238	0.0639968	0.0319984	+	0.999488i	$0.489813\pi$
$198$	0	0
$199$	4.88142	0.346035	0.173018	−	0.984919i	$-0.444648\pi$
$199$	4.88142	0.346035	0.173018	+	0.984919i	$0.444648\pi$
$200$	−7.99282	−0.565178
$201$	−5.76760	−0.406815
$202$	−11.9711	−0.842280
$203$	−3.74143	−0.262597
$204$	−0.903285	−0.0632426
$205$	−10.5418	−0.736272
$206$	39.7247	2.76775
$207$	−1.65771	−0.115219
$208$	16.0069	1.10988
$209$	0	0
$210$	−16.1772	−1.11633
$211$	−25.6524	−1.76598	−0.882992	−	0.469389i	$-0.844474\pi$
$211$	−25.6524	−1.76598	−0.882992	+	0.469389i	$0.844474\pi$
$212$	37.0912	2.54743
$213$	−8.08303	−0.553840
$214$	34.6682	2.36987
$215$	8.60108	0.586589
$216$	−6.14646	−0.418214
$217$	−28.2187	−1.91561
$218$	−3.87001	−0.262110
$219$	−12.6433	−0.854354
$220$	0	0
$221$	0.485536	0.0326607
$222$	−13.0048	−0.872826
$223$	21.5751	1.44478	0.722389	−	0.691486i	$-0.243044\pi$
$223$	21.5751	1.44478	0.722389	+	0.691486i	$0.243044\pi$
$224$	−15.8127	−1.05653
$225$	−1.30039	−0.0866929
$226$	20.1804	1.34238
$227$	−28.5013	−1.89170	−0.945849	−	0.324608i	$-0.894768\pi$
$227$	−28.5013	−1.89170	−0.945849	+	0.324608i	$0.894768\pi$
$228$	−4.42489	−0.293046
$229$	13.6556	0.902390	0.451195	−	0.892425i	$-0.350998\pi$
$229$	13.6556	0.902390	0.451195	+	0.892425i	$0.350998\pi$
$230$	8.08202	0.532912
$231$	0	0
$232$	6.93059	0.455015
$233$	11.2396	0.736329	0.368165	−	0.929761i	$-0.379986\pi$
$233$	11.2396	0.736329	0.368165	+	0.929761i	$0.379986\pi$
$234$	6.02882	0.394116
$235$	16.4296	1.07175
$236$	22.7891	1.48345
$237$	−4.05754	−0.263566
$238$	−1.71691	−0.111291
$239$	−11.5540	−0.747369	−0.373685	−	0.927556i	$-0.621906\pi$
$239$	−11.5540	−0.747369	−0.373685	+	0.927556i	$0.621906\pi$
$240$	12.9445	0.835564
$241$	14.5771	0.938993	0.469496	−	0.882934i	$-0.344435\pi$
$241$	14.5771	0.938993	0.469496	+	0.882934i	$0.344435\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−6.42134	−0.411084
$245$	−7.71285	−0.492756
$246$	−13.8922	−0.885733
$247$	2.37848	0.151339
$248$	52.2722	3.31929
$249$	−17.4366	−1.10500
$250$	30.7170	1.94271
$251$	−7.05787	−0.445489	−0.222744	−	0.974877i	$-0.571501\pi$
$251$	−7.05787	−0.445489	−0.222744	+	0.974877i	$0.571501\pi$
$252$	−14.6823	−0.924900
$253$	0	0
$254$	−16.3375	−1.02511
$255$	0.392645	0.0245884
$256$	−30.2667	−1.89167
$257$	−21.7061	−1.35399	−0.676995	−	0.735987i	$-0.736718\pi$
$257$	−21.7061	−1.35399	−0.676995	+	0.735987i	$0.736718\pi$
$258$	11.3347	0.705665
$259$	−17.0241	−1.05782
$260$	−20.2432	−1.25543
$261$	1.12757	0.0697950
$262$	33.3956	2.06319
$263$	−7.47167	−0.460723	−0.230362	−	0.973105i	$-0.573991\pi$
$263$	−7.47167	−0.460723	−0.230362	+	0.973105i	$0.573991\pi$
$264$	0	0
$265$	−16.1230	−0.990428
$266$	−8.41057	−0.515685
$267$	3.36049	0.205658
$268$	25.5210	1.55894
$269$	26.0700	1.58952	0.794759	−	0.606925i	$-0.207597\pi$
$269$	26.0700	1.58952	0.794759	+	0.606925i	$0.207597\pi$
$270$	4.87540	0.296708
$271$	22.4290	1.36247	0.681234	−	0.732066i	$-0.261444\pi$
$271$	22.4290	1.36247	0.681234	+	0.732066i	$0.261444\pi$
$272$	1.37382	0.0833000
$273$	7.89209	0.477651
$274$	−9.67484	−0.584479
$275$	0	0
$276$	7.33520	0.441527
$277$	6.53588	0.392703	0.196351	−	0.980534i	$-0.437091\pi$
$277$	6.53588	0.392703	0.196351	+	0.980534i	$0.437091\pi$
$278$	33.6742	2.01965
$279$	8.50443	0.509147
$280$	39.2279	2.34432
$281$	−12.5012	−0.745758	−0.372879	−	0.927880i	$-0.621629\pi$
$281$	−12.5012	−0.745758	−0.372879	+	0.927880i	$0.621629\pi$
$282$	21.6512	1.28931
$283$	8.56414	0.509085	0.254543	−	0.967062i	$-0.418075\pi$
$283$	8.56414	0.509085	0.254543	+	0.967062i	$0.418075\pi$
$284$	35.7665	2.12235
$285$	1.92344	0.113935
$286$	0	0
$287$	−18.1857	−1.07347
$288$	4.76556	0.280813
$289$	−16.9583	−0.997549
$290$	−5.49738	−0.322817
$291$	17.0429	0.999070
$292$	55.9452	3.27394
$293$	−12.2494	−0.715620	−0.357810	−	0.933794i	$-0.616476\pi$
$293$	−12.2494	−0.715620	−0.357810	+	0.933794i	$0.616476\pi$
$294$	−10.1641	−0.592784
$295$	−9.90610	−0.576756
$296$	31.5353	1.83295
$297$	0	0
$298$	−4.11092	−0.238139
$299$	−3.94284	−0.228020
$300$	5.75410	0.332213
$301$	14.8377	0.855233
$302$	−51.8456	−2.98338
$303$	4.72280	0.271318
$304$	6.72988	0.385985
$305$	2.79126	0.159827
$306$	0.517434	0.0295797
$307$	3.88709	0.221848	0.110924	−	0.993829i	$-0.464619\pi$
$307$	3.88709	0.221848	0.110924	+	0.993829i	$0.464619\pi$
$308$	0	0
$309$	−15.6721	−0.891556
$310$	−41.4626	−2.35491
$311$	−1.32432	−0.0750953	−0.0375476	−	0.999295i	$-0.511955\pi$
$311$	−1.32432	−0.0750953	−0.0375476	+	0.999295i	$0.511955\pi$
$312$	−14.6192	−0.827652
$313$	−27.2169	−1.53839	−0.769195	−	0.639014i	$-0.779342\pi$
$313$	−27.2169	−1.53839	−0.769195	+	0.639014i	$0.779342\pi$
$314$	10.4570	0.590122
$315$	6.38220	0.359596
$316$	17.9542	1.01000
$317$	−34.5074	−1.93813	−0.969064	−	0.246808i	$-0.920618\pi$
$317$	−34.5074	−1.93813	−0.969064	+	0.246808i	$0.920618\pi$
$318$	−21.2472	−1.19148
$319$	0	0
$320$	2.65497	0.148417
$321$	−13.6773	−0.763390
$322$	13.9423	0.776974
$323$	0.204137	0.0113585
$324$	4.42489	0.245827
$325$	−3.09296	−0.171567
$326$	30.1687	1.67089
$327$	1.52679	0.0844317
$328$	33.6870	1.86006
$329$	28.3427	1.56259
$330$	0	0
$331$	22.3412	1.22798	0.613992	−	0.789312i	$-0.289563\pi$
$331$	22.3412	1.22798	0.613992	+	0.789312i	$0.289563\pi$
$332$	77.1551	4.23444
$333$	5.13064	0.281157
$334$	51.3052	2.80730
$335$	−11.0936	−0.606109
$336$	22.3306	1.21823
$337$	7.06395	0.384798	0.192399	−	0.981317i	$-0.438373\pi$
$337$	7.06395	0.384798	0.192399	+	0.981317i	$0.438373\pi$
$338$	−18.6121	−1.01237
$339$	−7.96153	−0.432411
$340$	−1.73741	−0.0942243
$341$	0	0
$342$	2.53474	0.137063
$343$	9.92141	0.535706
$344$	−27.4853	−1.48191
$345$	−3.18850	−0.171663
$346$	−5.92900	−0.318745
$347$	16.3256	0.876403	0.438202	−	0.898877i	$-0.355616\pi$
$347$	16.3256	0.876403	0.438202	+	0.898877i	$0.355616\pi$
$348$	−4.98939	−0.267459
$349$	−26.9640	−1.44335	−0.721674	−	0.692233i	$-0.756627\pi$
$349$	−26.9640	−1.44335	−0.721674	+	0.692233i	$0.756627\pi$
$350$	10.9370	0.584610
$351$	−2.37848	−0.126954
$352$	0	0
$353$	−14.1278	−0.751947	−0.375974	−	0.926630i	$-0.622692\pi$
$353$	−14.1278	−0.751947	−0.375974	+	0.926630i	$0.622692\pi$
$354$	−13.0544	−0.693835
$355$	−15.5472	−0.825159
$356$	−14.8698	−0.788097
$357$	0.677352	0.0358493
$358$	14.9970	0.792616
$359$	−19.0050	−1.00304	−0.501522	−	0.865145i	$-0.667226\pi$
$359$	−19.0050	−1.00304	−0.501522	+	0.865145i	$0.667226\pi$
$360$	−11.8223	−0.623091
$361$	1.00000	0.0526316
$362$	−62.5229	−3.28613
$363$	0	0
$364$	−34.9216	−1.83039
$365$	−24.3186	−1.27289
$366$	3.67837	0.192272
$367$	14.7495	0.769920	0.384960	−	0.922933i	$-0.374215\pi$
$367$	14.7495	0.769920	0.384960	+	0.922933i	$0.374215\pi$
$368$	−11.1562	−0.581558
$369$	5.48072	0.285315
$370$	−25.0139	−1.30041
$371$	−27.8138	−1.44402
$372$	−37.6312	−1.95109
$373$	−0.505843	−0.0261916	−0.0130958	−	0.999914i	$-0.504169\pi$
$373$	−0.505843	−0.0261916	−0.0130958	+	0.999914i	$0.504169\pi$
$374$	0	0
$375$	−12.1184	−0.625792
$376$	−52.5019	−2.70758
$377$	2.68191	0.138125
$378$	8.41057	0.432593
$379$	−29.1936	−1.49957	−0.749786	−	0.661680i	$-0.769844\pi$
$379$	−29.1936	−1.49957	−0.749786	+	0.661680i	$0.769844\pi$
$380$	−8.51100	−0.436605
$381$	6.44545	0.330211
$382$	23.3867	1.19657
$383$	6.99648	0.357503	0.178752	−	0.983894i	$-0.442794\pi$
$383$	6.99648	0.357503	0.178752	+	0.983894i	$0.442794\pi$
$384$	13.0299	0.664929
$385$	0	0
$386$	55.0602	2.80249
$387$	−4.47173	−0.227311
$388$	−75.4128	−3.82850
$389$	31.5975	1.60206	0.801028	−	0.598627i	$-0.204287\pi$
$389$	31.5975	1.60206	0.801028	+	0.598627i	$0.204287\pi$
$390$	11.5961	0.587189
$391$	−0.338401	−0.0171137
$392$	24.6469	1.24486
$393$	−13.1752	−0.664601
$394$	2.27680	0.114703
$395$	−7.80442	−0.392683
$396$	0	0
$397$	−0.451857	−0.0226781	−0.0113390	−	0.999936i	$-0.503609\pi$
$397$	−0.451857	−0.0226781	−0.0113390	+	0.999936i	$0.503609\pi$
$398$	12.3731	0.620209
$399$	3.31812	0.166114
$400$	−8.75150	−0.437575
$401$	−8.42479	−0.420714	−0.210357	−	0.977625i	$-0.567463\pi$
$401$	−8.42479	−0.420714	−0.210357	+	0.977625i	$0.567463\pi$
$402$	−14.6194	−0.729147
$403$	20.2276	1.00761
$404$	−20.8979	−1.03971
$405$	−1.92344	−0.0955763
$406$	−9.48353	−0.470660
$407$	0	0
$408$	−1.25472	−0.0621180
$409$	−32.5480	−1.60939	−0.804697	−	0.593685i	$-0.797673\pi$
$409$	−32.5480	−1.60939	−0.804697	+	0.593685i	$0.797673\pi$
$410$	−26.7207	−1.31964
$411$	3.81690	0.188274
$412$	69.3475	3.41651
$413$	−17.0890	−0.840896
$414$	−4.20186	−0.206510
$415$	−33.5382	−1.64633
$416$	11.3348	0.555734
$417$	−13.2851	−0.650574
$418$	0	0
$419$	16.6499	0.813398	0.406699	−	0.913562i	$-0.366680\pi$
$419$	16.6499	0.813398	0.406699	+	0.913562i	$0.366680\pi$
$420$	−28.2405	−1.37800
$421$	−8.88468	−0.433013	−0.216506	−	0.976281i	$-0.569466\pi$
$421$	−8.88468	−0.433013	−0.216506	+	0.976281i	$0.569466\pi$
$422$	−65.0221	−3.16522
$423$	−8.54180	−0.415317
$424$	51.5221	2.50213
$425$	−0.265459	−0.0128766
$426$	−20.4884	−0.992664
$427$	4.81521	0.233024
$428$	60.5204	2.92536
$429$	0	0
$430$	21.8015	1.05136
$431$	−12.3273	−0.593783	−0.296892	−	0.954911i	$-0.595950\pi$
$431$	−12.3273	−0.593783	−0.296892	+	0.954911i	$0.595950\pi$
$432$	−6.72988	−0.323792
$433$	1.56520	0.0752187	0.0376093	−	0.999293i	$-0.488026\pi$
$433$	1.56520	0.0752187	0.0376093	+	0.999293i	$0.488026\pi$
$434$	−71.5271	−3.43341
$435$	2.16881	0.103987
$436$	−6.75588	−0.323548
$437$	−1.65771	−0.0792991
$438$	−32.0474	−1.53128
$439$	8.81832	0.420875	0.210438	−	0.977607i	$-0.432511\pi$
$439$	8.81832	0.420875	0.210438	+	0.977607i	$0.432511\pi$
$440$	0	0
$441$	4.00993	0.190949
$442$	1.23071	0.0585387
$443$	5.79732	0.275439	0.137720	−	0.990471i	$-0.456023\pi$
$443$	5.79732	0.275439	0.137720	+	0.990471i	$0.456023\pi$
$444$	−22.7025	−1.07741
$445$	6.46368	0.306408
$446$	54.6873	2.58952
$447$	1.62183	0.0767100
$448$	4.58010	0.216389
$449$	−0.352416	−0.0166316	−0.00831578	−	0.999965i	$-0.502647\pi$
$449$	−0.352416	−0.0166316	−0.00831578	+	0.999965i	$0.502647\pi$
$450$	−3.29616	−0.155382
$451$	0	0
$452$	35.2289	1.65703
$453$	20.4540	0.961014
$454$	−72.2433	−3.39055
$455$	15.1799	0.711646
$456$	−6.14646	−0.287834
$457$	0.290406	0.0135846	0.00679231	−	0.999977i	$-0.497838\pi$
$457$	0.290406	0.0135846	0.00679231	+	0.999977i	$0.497838\pi$
$458$	34.6134	1.61738
$459$	−0.204137	−0.00952831
$460$	14.1088	0.657825
$461$	29.0862	1.35468	0.677339	−	0.735671i	$-0.263133\pi$
$461$	29.0862	1.35468	0.677339	+	0.735671i	$0.263133\pi$
$462$	0	0
$463$	0.641382	0.0298075	0.0149038	−	0.999889i	$-0.495256\pi$
$463$	0.641382	0.0298075	0.0149038	+	0.999889i	$0.495256\pi$
$464$	7.58844	0.352284
$465$	16.3577	0.758572
$466$	28.4894	1.31974
$467$	18.2106	0.842686	0.421343	−	0.906901i	$-0.361559\pi$
$467$	18.2106	0.842686	0.421343	+	0.906901i	$0.361559\pi$
$468$	10.5245	0.486496
$469$	−19.1376	−0.883693
$470$	41.6447	1.92093
$471$	−4.12547	−0.190092
$472$	31.6556	1.45707
$473$	0	0
$474$	−10.2848	−0.472397
$475$	−1.30039	−0.0596662
$476$	−2.99721	−0.137377
$477$	8.38239	0.383803
$478$	−29.2865	−1.33953
$479$	5.96494	0.272545	0.136273	−	0.990671i	$-0.456488\pi$
$479$	5.96494	0.272545	0.136273	+	0.990671i	$0.456488\pi$
$480$	9.16625	0.418380
$481$	12.2031	0.556414
$482$	36.9491	1.68298
$483$	−5.50049	−0.250281
$484$	0	0
$485$	32.7808	1.48850
$486$	−2.53474	−0.114978
$487$	−29.5886	−1.34079	−0.670394	−	0.742005i	$-0.733875\pi$
$487$	−29.5886	−1.34079	−0.670394	+	0.742005i	$0.733875\pi$
$488$	−8.91966	−0.403774
$489$	−11.9021	−0.538231
$490$	−19.5500	−0.883181
$491$	−8.93065	−0.403035	−0.201517	−	0.979485i	$-0.564587\pi$
$491$	−8.93065	−0.403035	−0.201517	+	0.979485i	$0.564587\pi$
$492$	−24.2516	−1.09335
$493$	0.230180	0.0103668
$494$	6.02882	0.271250
$495$	0	0
$496$	57.2338	2.56988
$497$	−26.8205	−1.20306
$498$	−44.1972	−1.98052
$499$	−27.3756	−1.22550	−0.612750	−	0.790277i	$-0.709937\pi$
$499$	−27.3756	−1.22550	−0.612750	+	0.790277i	$0.709937\pi$
$500$	53.6226	2.39808
$501$	−20.2408	−0.904294
$502$	−17.8898	−0.798462
$503$	−13.2472	−0.590664	−0.295332	−	0.955395i	$-0.595430\pi$
$503$	−13.2472	−0.590664	−0.295332	+	0.955395i	$0.595430\pi$
$504$	−20.3947	−0.908453
$505$	9.08400	0.404233
$506$	0	0
$507$	7.34283	0.326107
$508$	−28.5204	−1.26539
$509$	27.7018	1.22786	0.613930	−	0.789360i	$-0.289588\pi$
$509$	27.7018	1.22786	0.613930	+	0.789360i	$0.289588\pi$
$510$	0.995251	0.0440705
$511$	−41.9520	−1.85585
$512$	−50.6583	−2.23880
$513$	−1.00000	−0.0441511
$514$	−55.0193	−2.42680
$515$	−30.1443	−1.32832
$516$	19.7869	0.871070
$517$	0	0
$518$	−43.1516	−1.89597
$519$	2.33910	0.102675
$520$	−28.1192	−1.23311
$521$	−36.1146	−1.58221	−0.791105	−	0.611681i	$-0.790494\pi$
$521$	−36.1146	−1.58221	−0.791105	+	0.611681i	$0.790494\pi$
$522$	2.85810	0.125096
$523$	8.16148	0.356877	0.178438	−	0.983951i	$-0.442896\pi$
$523$	8.16148	0.356877	0.178438	+	0.983951i	$0.442896\pi$
$524$	58.2988	2.54680
$525$	−4.31487	−0.188316
$526$	−18.9387	−0.825768
$527$	1.73607	0.0756244
$528$	0	0
$529$	−20.2520	−0.880521
$530$	−40.8676	−1.77517
$531$	5.15021	0.223500
$532$	−14.6823	−0.636560
$533$	13.0358	0.564642
$534$	8.51795	0.368608
$535$	−26.3073	−1.13736
$536$	35.4504	1.53122
$537$	−5.91659	−0.255320
$538$	66.0807	2.84894
$539$	0	0
$540$	8.51100	0.366255
$541$	30.1687	1.29706	0.648528	−	0.761191i	$-0.275385\pi$
$541$	30.1687	1.29706	0.648528	+	0.761191i	$0.275385\pi$
$542$	56.8517	2.44199
$543$	24.6664	1.05854
$544$	0.972828	0.0417096
$545$	2.93668	0.125794
$546$	20.0044	0.856108
$547$	−12.8476	−0.549323	−0.274661	−	0.961541i	$-0.588566\pi$
$547$	−12.8476	−0.549323	−0.274661	+	0.961541i	$0.588566\pi$
$548$	−16.8894	−0.721479
$549$	−1.45119	−0.0619351
$550$	0	0
$551$	1.12757	0.0480362
$552$	10.1891	0.433676
$553$	−13.4634	−0.572523
$554$	16.5667	0.703853
$555$	9.86845	0.418892
$556$	58.7851	2.49304
$557$	−1.81968	−0.0771022	−0.0385511	−	0.999257i	$-0.512274\pi$
$557$	−1.81968	−0.0771022	−0.0385511	+	0.999257i	$0.512274\pi$
$558$	21.5565	0.912559
$559$	−10.6359	−0.449851
$560$	42.9514	1.81503
$561$	0	0
$562$	−31.6872	−1.33664
$563$	43.8640	1.84865	0.924324	−	0.381608i	$-0.124630\pi$
$563$	43.8640	1.84865	0.924324	+	0.381608i	$0.124630\pi$
$564$	37.7965	1.59152
$565$	−15.3135	−0.644243
$566$	21.7078	0.912448
$567$	−3.31812	−0.139348
$568$	49.6820	2.08461
$569$	−44.3888	−1.86088	−0.930438	−	0.366450i	$-0.880573\pi$
$569$	−44.3888	−1.86088	−0.930438	+	0.366450i	$0.880573\pi$
$570$	4.87540	0.204208
$571$	−11.4927	−0.480956	−0.240478	−	0.970655i	$-0.577304\pi$
$571$	−11.4927	−0.480956	−0.240478	+	0.970655i	$0.577304\pi$
$572$	0	0
$573$	−9.22647	−0.385441
$574$	−46.0959	−1.92401
$575$	2.15568	0.0898980
$576$	−1.38033	−0.0575137
$577$	−11.4614	−0.477143	−0.238572	−	0.971125i	$-0.576679\pi$
$577$	−11.4614	−0.477143	−0.238572	+	0.971125i	$0.576679\pi$
$578$	−42.9849	−1.78794
$579$	−21.7223	−0.902746
$580$	−9.59677	−0.398484
$581$	−57.8568	−2.40030
$582$	43.1992	1.79066
$583$	0	0
$584$	77.7115	3.21573
$585$	−4.57485	−0.189147
$586$	−31.0491	−1.28263
$587$	47.0909	1.94365	0.971824	−	0.235709i	$-0.0757411\pi$
$587$	47.0909	1.94365	0.971824	+	0.235709i	$0.0757411\pi$
$588$	−17.7435	−0.731730
$589$	8.50443	0.350419
$590$	−25.1094	−1.03374
$591$	−0.898238	−0.0369486
$592$	34.5286	1.41912
$593$	33.6175	1.38051	0.690253	−	0.723568i	$-0.257499\pi$
$593$	33.6175	1.38051	0.690253	+	0.723568i	$0.257499\pi$
$594$	0	0
$595$	1.30284	0.0534114
$596$	−7.17643	−0.293958
$597$	−4.88142	−0.199783
$598$	−9.99405	−0.408687
$599$	−31.2271	−1.27590	−0.637952	−	0.770076i	$-0.720218\pi$
$599$	−31.2271	−1.27590	−0.637952	+	0.770076i	$0.720218\pi$
$600$	7.99282	0.326306
$601$	34.8825	1.42289	0.711444	−	0.702743i	$-0.248041\pi$
$601$	34.8825	1.42289	0.711444	+	0.702743i	$0.248041\pi$
$602$	37.6098	1.53286
$603$	5.76760	0.234875
$604$	−90.5069	−3.68267
$605$	0	0
$606$	11.9711	0.486291
$607$	42.4971	1.72491	0.862453	−	0.506138i	$-0.168927\pi$
$607$	42.4971	1.72491	0.862453	+	0.506138i	$0.168927\pi$
$608$	4.76556	0.193269
$609$	3.74143	0.151610
$610$	7.07512	0.286463
$611$	−20.3165	−0.821918
$612$	0.903285	0.0365131
$613$	−19.8708	−0.802573	−0.401287	−	0.915953i	$-0.631437\pi$
$613$	−19.8708	−0.802573	−0.401287	+	0.915953i	$0.631437\pi$
$614$	9.85275	0.397625
$615$	10.5418	0.425087
$616$	0	0
$617$	35.8003	1.44126	0.720632	−	0.693317i	$-0.243852\pi$
$617$	35.8003	1.44126	0.720632	+	0.693317i	$0.243852\pi$
$618$	−39.7247	−1.59796
$619$	−38.3015	−1.53947	−0.769733	−	0.638366i	$-0.779611\pi$
$619$	−38.3015	−1.53947	−0.769733	+	0.638366i	$0.779611\pi$
$620$	−72.3812	−2.90690
$621$	1.65771	0.0665217
$622$	−3.35680	−0.134595
$623$	11.1505	0.446735
$624$	−16.0069	−0.640789
$625$	−16.8070	−0.672280
$626$	−68.9877	−2.75730
$627$	0	0
$628$	18.2548	0.728444
$629$	1.04735	0.0417607
$630$	16.1772	0.644515
$631$	−13.1847	−0.524874	−0.262437	−	0.964949i	$-0.584526\pi$
$631$	−13.1847	−0.524874	−0.262437	+	0.964949i	$0.584526\pi$
$632$	24.9395	0.992042
$633$	25.6524	1.01959
$634$	−87.4672	−3.47377
$635$	12.3974	0.491977
$636$	−37.0912	−1.47076
$637$	9.53755	0.377891
$638$	0	0
$639$	8.08303	0.319760
$640$	25.0622	0.990669
$641$	−4.22177	−0.166750	−0.0833749	−	0.996518i	$-0.526570\pi$
$641$	−4.22177	−0.166750	−0.0833749	+	0.996518i	$0.526570\pi$
$642$	−34.6682	−1.36825
$643$	−28.5440	−1.12566	−0.562832	−	0.826571i	$-0.690288\pi$
$643$	−28.5440	−1.12566	−0.562832	+	0.826571i	$0.690288\pi$
$644$	24.3391	0.959094
$645$	−8.60108	−0.338667
$646$	0.517434	0.0203582
$647$	−7.99104	−0.314160	−0.157080	−	0.987586i	$-0.550208\pi$
$647$	−7.99104	−0.314160	−0.157080	+	0.987586i	$0.550208\pi$
$648$	6.14646	0.241456
$649$	0	0
$650$	−7.83984	−0.307504
$651$	28.2187	1.10598
$652$	52.6655	2.06254
$653$	−0.607074	−0.0237567	−0.0118783	−	0.999929i	$-0.503781\pi$
$653$	−0.607074	−0.0237567	−0.0118783	+	0.999929i	$0.503781\pi$
$654$	3.87001	0.151329
$655$	−25.3416	−0.990180
$656$	36.8846	1.44010
$657$	12.6433	0.493262
$658$	71.8414	2.80067
$659$	−27.4217	−1.06820	−0.534099	−	0.845422i	$-0.679349\pi$
$659$	−27.4217	−1.06820	−0.534099	+	0.845422i	$0.679349\pi$
$660$	0	0
$661$	−1.01032	−0.0392970	−0.0196485	−	0.999807i	$-0.506255\pi$
$661$	−1.01032	−0.0392970	−0.0196485	+	0.999807i	$0.506255\pi$
$662$	56.6291	2.20095
$663$	−0.485536	−0.0188567
$664$	107.173	4.15914
$665$	6.38220	0.247491
$666$	13.0048	0.503926
$667$	−1.86919	−0.0723754
$668$	89.5635	3.46532
$669$	−21.5751	−0.834143
$670$	−28.1194	−1.08635
$671$	0	0
$672$	15.8127	0.609988
$673$	11.3802	0.438674	0.219337	−	0.975649i	$-0.429611\pi$
$673$	11.3802	0.438674	0.219337	+	0.975649i	$0.429611\pi$
$674$	17.9053	0.689685
$675$	1.30039	0.0500522
$676$	−32.4912	−1.24966
$677$	41.9674	1.61294	0.806469	−	0.591276i	$-0.201376\pi$
$677$	41.9674	1.61294	0.806469	+	0.591276i	$0.201376\pi$
$678$	−20.1804	−0.775023
$679$	56.5503	2.17020
$680$	−2.41338	−0.0925488
$681$	28.5013	1.09217
$682$	0	0
$683$	−51.6706	−1.97712	−0.988560	−	0.150831i	$-0.951805\pi$
$683$	−51.6706	−1.97712	−0.988560	+	0.150831i	$0.951805\pi$
$684$	4.42489	0.169190
$685$	7.34157	0.280507
$686$	25.1482	0.960161
$687$	−13.6556	−0.520995
$688$	−30.0942	−1.14733
$689$	19.9374	0.759553
$690$	−8.08202	−0.307677
$691$	−23.3632	−0.888779	−0.444390	−	0.895834i	$-0.646579\pi$
$691$	−23.3632	−0.888779	−0.444390	+	0.895834i	$0.646579\pi$
$692$	−10.3503	−0.393458
$693$	0	0
$694$	41.3811	1.57080
$695$	−25.5530	−0.969282
$696$	−6.93059	−0.262703
$697$	1.11882	0.0423783
$698$	−68.3466	−2.58696
$699$	−11.2396	−0.425120
$700$	19.0928	0.721641
$701$	24.7481	0.934723	0.467361	−	0.884066i	$-0.345205\pi$
$701$	24.7481	0.934723	0.467361	+	0.884066i	$0.345205\pi$
$702$	−6.02882	−0.227543
$703$	5.13064	0.193506
$704$	0	0
$705$	−16.4296	−0.618775
$706$	−35.8103	−1.34774
$707$	15.6708	0.589362
$708$	−22.7891	−0.856468
$709$	12.0601	0.452927	0.226464	−	0.974020i	$-0.427284\pi$
$709$	12.0601	0.452927	0.226464	+	0.974020i	$0.427284\pi$
$710$	−39.4080	−1.47896
$711$	4.05754	0.152170
$712$	−20.6551	−0.774083
$713$	−14.0979	−0.527971
$714$	1.71691	0.0642537
$715$	0	0
$716$	26.1803	0.978402
$717$	11.5540	0.431494
$718$	−48.1726	−1.79778
$719$	47.2054	1.76047	0.880233	−	0.474542i	$-0.157386\pi$
$719$	47.2054	1.76047	0.880233	+	0.474542i	$0.157386\pi$
$720$	−12.9445	−0.482413
$721$	−52.0020	−1.93666
$722$	2.53474	0.0943331
$723$	−14.5771	−0.542128
$724$	−109.146	−4.05639
$725$	−1.46629	−0.0544566
$726$	0	0
$727$	11.3041	0.419247	0.209623	−	0.977782i	$-0.432776\pi$
$727$	11.3041	0.419247	0.209623	+	0.977782i	$0.432776\pi$
$728$	−48.5084	−1.79784
$729$	1.00000	0.0370370
$730$	−61.6411	−2.28144
$731$	−0.912846	−0.0337628
$732$	6.42134	0.237339
$733$	12.1202	0.447671	0.223836	−	0.974627i	$-0.428142\pi$
$733$	12.1202	0.447671	0.223836	+	0.974627i	$0.428142\pi$
$734$	37.3862	1.37995
$735$	7.71285	0.284493
$736$	−7.89993	−0.291195
$737$	0	0
$738$	13.8922	0.511378
$739$	−2.99221	−0.110070	−0.0550351	−	0.998484i	$-0.517527\pi$
$739$	−2.99221	−0.110070	−0.0550351	+	0.998484i	$0.517527\pi$
$740$	−43.6668	−1.60522
$741$	−2.37848	−0.0873757
$742$	−70.5007	−2.58816
$743$	25.7045	0.943007	0.471503	−	0.881864i	$-0.343711\pi$
$743$	25.7045	0.943007	0.471503	+	0.881864i	$0.343711\pi$
$744$	−52.2722	−1.91639
$745$	3.11949	0.114289
$746$	−1.28218	−0.0469439
$747$	17.4366	0.637972
$748$	0	0
$749$	−45.3828	−1.65825
$750$	−30.7170	−1.12163
$751$	−5.09324	−0.185855	−0.0929275	−	0.995673i	$-0.529622\pi$
$751$	−5.09324	−0.185855	−0.0929275	+	0.995673i	$0.529622\pi$
$752$	−57.4853	−2.09627
$753$	7.05787	0.257203
$754$	6.79794	0.247566
$755$	39.3420	1.43180
$756$	14.6823	0.533991
$757$	−52.9352	−1.92396	−0.961981	−	0.273118i	$-0.911945\pi$
$757$	−52.9352	−1.92396	−0.961981	+	0.273118i	$0.911945\pi$
$758$	−73.9980	−2.68773
$759$	0	0
$760$	−11.8223	−0.428841
$761$	−30.9096	−1.12047	−0.560236	−	0.828333i	$-0.689290\pi$
$761$	−30.9096	−1.12047	−0.560236	+	0.828333i	$0.689290\pi$
$762$	16.3375	0.591846
$763$	5.06608	0.183404
$764$	40.8261	1.47704
$765$	−0.392645	−0.0141961
$766$	17.7342	0.640764
$767$	12.2497	0.442310
$768$	30.2667	1.09215
$769$	−9.46378	−0.341273	−0.170636	−	0.985334i	$-0.554582\pi$
$769$	−9.46378	−0.341273	−0.170636	+	0.985334i	$0.554582\pi$
$770$	0	0
$771$	21.7061	0.781727
$772$	96.1186	3.45939
$773$	−0.267329	−0.00961517	−0.00480758	−	0.999988i	$-0.501530\pi$
$773$	−0.267329	−0.00961517	−0.00480758	+	0.999988i	$0.501530\pi$
$774$	−11.3347	−0.407416
$775$	−11.0591	−0.397255
$776$	−104.753	−3.76042
$777$	17.0241	0.610735
$778$	80.0913	2.87141
$779$	5.48072	0.196367
$780$	20.2432	0.724824
$781$	0	0
$782$	−0.857757	−0.0306733
$783$	−1.12757	−0.0402962
$784$	26.9864	0.963799
$785$	−7.93508	−0.283215
$786$	−33.3956	−1.19118
$787$	−21.5186	−0.767054	−0.383527	−	0.923530i	$-0.625291\pi$
$787$	−21.5186	−0.767054	−0.383527	+	0.923530i	$0.625291\pi$
$788$	3.97461	0.141589
$789$	7.47167	0.265999
$790$	−19.7822	−0.703818
$791$	−26.4173	−0.939292
$792$	0	0
$793$	−3.45162	−0.122570
$794$	−1.14534	−0.0406466
$795$	16.1230	0.571824
$796$	21.5998	0.765584
$797$	−9.55923	−0.338605	−0.169303	−	0.985564i	$-0.554152\pi$
$797$	−9.55923	−0.338605	−0.169303	+	0.985564i	$0.554152\pi$
$798$	8.41057	0.297731
$799$	−1.74370	−0.0616876
$800$	−6.19710	−0.219101
$801$	−3.36049	−0.118737
$802$	−21.3546	−0.754059
$803$	0	0
$804$	−25.5210	−0.900057
$805$	−10.5798	−0.372890
$806$	51.2717	1.80597
$807$	−26.0700	−0.917709
$808$	−29.0285	−1.02122
$809$	14.5330	0.510953	0.255476	−	0.966815i	$-0.417768\pi$
$809$	14.5330	0.510953	0.255476	+	0.966815i	$0.417768\pi$
$810$	−4.87540	−0.171304
$811$	27.4722	0.964679	0.482339	−	0.875984i	$-0.339787\pi$
$811$	27.4722	0.964679	0.482339	+	0.875984i	$0.339787\pi$
$812$	−16.5554	−0.580981
$813$	−22.4290	−0.786621
$814$	0	0
$815$	−22.8929	−0.801904
$816$	−1.37382	−0.0480933
$817$	−4.47173	−0.156446
$818$	−82.5006	−2.88457
$819$	−7.89209	−0.275772
$820$	−46.6464	−1.62896
$821$	18.1080	0.631974	0.315987	−	0.948763i	$-0.397664\pi$
$821$	18.1080	0.631974	0.315987	+	0.948763i	$0.397664\pi$
$822$	9.67484	0.337449
$823$	17.8968	0.623842	0.311921	−	0.950108i	$-0.399028\pi$
$823$	17.8968	0.623842	0.311921	+	0.950108i	$0.399028\pi$
$824$	96.3282	3.35575
$825$	0	0
$826$	−43.3162	−1.50716
$827$	39.5793	1.37631	0.688153	−	0.725565i	$-0.258422\pi$
$827$	39.5793	1.37631	0.688153	+	0.725565i	$0.258422\pi$
$828$	−7.33520	−0.254916
$829$	−37.7530	−1.31122	−0.655609	−	0.755101i	$-0.727588\pi$
$829$	−37.7530	−1.31122	−0.655609	+	0.755101i	$0.727588\pi$
$830$	−85.0105	−2.95076
$831$	−6.53588	−0.226727
$832$	−3.28308	−0.113820
$833$	0.818576	0.0283620
$834$	−33.6742	−1.16604
$835$	−38.9320	−1.34730
$836$	0	0
$837$	−8.50443	−0.293956
$838$	42.2030	1.45788
$839$	−39.2603	−1.35542	−0.677709	−	0.735331i	$-0.737027\pi$
$839$	−39.2603	−1.35542	−0.677709	+	0.735331i	$0.737027\pi$
$840$	−39.2279	−1.35349
$841$	−27.7286	−0.956158
$842$	−22.5203	−0.776102
$843$	12.5012	0.430564
$844$	−113.509	−3.90714
$845$	14.1235	0.485862
$846$	−21.6512	−0.744384
$847$	0	0
$848$	56.4125	1.93721
$849$	−8.56414	−0.293920
$850$	−0.672868	−0.0230792
$851$	−8.50512	−0.291552
$852$	−35.7665	−1.22534
$853$	−9.19860	−0.314954	−0.157477	−	0.987523i	$-0.550336\pi$
$853$	−9.19860	−0.314954	−0.157477	+	0.987523i	$0.550336\pi$
$854$	12.2053	0.417656
$855$	−1.92344	−0.0657801
$856$	84.0667	2.87334
$857$	32.5470	1.11178	0.555892	−	0.831255i	$-0.312377\pi$
$857$	32.5470	1.11178	0.555892	+	0.831255i	$0.312377\pi$
$858$	0	0
$859$	18.4471	0.629405	0.314703	−	0.949190i	$-0.398095\pi$
$859$	18.4471	0.629405	0.314703	+	0.949190i	$0.398095\pi$
$860$	38.0589	1.29780
$861$	18.1857	0.619767
$862$	−31.2464	−1.06426
$863$	4.27006	0.145354	0.0726772	−	0.997356i	$-0.476846\pi$
$863$	4.27006	0.145354	0.0726772	+	0.997356i	$0.476846\pi$
$864$	−4.76556	−0.162128
$865$	4.49911	0.152974
$866$	3.96737	0.134817
$867$	16.9583	0.575935
$868$	−124.865	−4.23819
$869$	0	0
$870$	5.49738	0.186378
$871$	13.7181	0.464821
$872$	−9.38436	−0.317795
$873$	−17.0429	−0.576813
$874$	−4.20186	−0.142130
$875$	−40.2103	−1.35936
$876$	−55.9452	−1.89021
$877$	8.12092	0.274224	0.137112	−	0.990556i	$-0.456218\pi$
$877$	8.12092	0.274224	0.137112	+	0.990556i	$0.456218\pi$
$878$	22.3521	0.754347
$879$	12.2494	0.413163
$880$	0	0
$881$	−32.3928	−1.09134	−0.545670	−	0.838000i	$-0.683725\pi$
$881$	−32.3928	−1.09134	−0.545670	+	0.838000i	$0.683725\pi$
$882$	10.1641	0.342244
$883$	−30.0730	−1.01204	−0.506018	−	0.862523i	$-0.668883\pi$
$883$	−30.0730	−1.01204	−0.506018	+	0.862523i	$0.668883\pi$
$884$	2.14845	0.0722600
$885$	9.90610	0.332990
$886$	14.6947	0.493678
$887$	−1.67977	−0.0564011	−0.0282006	−	0.999602i	$-0.508978\pi$
$887$	−1.67977	−0.0564011	−0.0282006	+	0.999602i	$0.508978\pi$
$888$	−31.5353	−1.05825
$889$	21.3868	0.717290
$890$	16.3837	0.549184
$891$	0	0
$892$	95.4677	3.19649
$893$	−8.54180	−0.285840
$894$	4.11092	0.137490
$895$	−11.3802	−0.380397
$896$	43.2348	1.44437
$897$	3.94284	0.131647
$898$	−0.893283	−0.0298092
$899$	9.58937	0.319823
$900$	−5.75410	−0.191803
$901$	1.71116	0.0570069
$902$	0	0
$903$	−14.8377	−0.493769
$904$	48.9352	1.62756
$905$	47.4443	1.57710
$906$	51.8456	1.72245
$907$	22.5438	0.748554	0.374277	−	0.927317i	$-0.377891\pi$
$907$	22.5438	0.748554	0.374277	+	0.927317i	$0.377891\pi$
$908$	−126.115	−4.18528
$909$	−4.72280	−0.156645
$910$	38.4771	1.27550
$911$	−55.1249	−1.82637	−0.913184	−	0.407547i	$-0.866384\pi$
$911$	−55.1249	−1.82637	−0.913184	+	0.407547i	$0.866384\pi$
$912$	−6.72988	−0.222849
$913$	0	0
$914$	0.736103	0.0243481
$915$	−2.79126	−0.0922763
$916$	60.4247	1.99649
$917$	−43.7169	−1.44366
$918$	−0.517434	−0.0170779
$919$	2.64135	0.0871302	0.0435651	−	0.999051i	$-0.486128\pi$
$919$	2.64135	0.0871302	0.0435651	+	0.999051i	$0.486128\pi$
$920$	19.5980	0.646128
$921$	−3.88709	−0.128084
$922$	73.7258	2.42803
$923$	19.2253	0.632809
$924$	0	0
$925$	−6.67185	−0.219369
$926$	1.62573	0.0534249
$927$	15.6721	0.514740
$928$	5.37352	0.176394
$929$	8.09079	0.265450	0.132725	−	0.991153i	$-0.457627\pi$
$929$	8.09079	0.265450	0.132725	+	0.991153i	$0.457627\pi$
$930$	41.4626	1.35961
$931$	4.00993	0.131420
$932$	49.7339	1.62909
$933$	1.32432	0.0433563
$934$	46.1591	1.51037
$935$	0	0
$936$	14.6192	0.477845
$937$	−11.2602	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$937$	−11.2602	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$938$	−48.5088	−1.58387
$939$	27.2169	0.888190
$940$	72.6992	2.37119
$941$	−25.9678	−0.846527	−0.423264	−	0.906007i	$-0.639116\pi$
$941$	−25.9678	−0.846527	−0.423264	+	0.906007i	$0.639116\pi$
$942$	−10.4570	−0.340707
$943$	−9.08545	−0.295863
$944$	34.6603	1.12810
$945$	−6.38220	−0.207613
$946$	0	0
$947$	15.2346	0.495057	0.247528	−	0.968881i	$-0.420382\pi$
$947$	15.2346	0.495057	0.247528	+	0.968881i	$0.420382\pi$
$948$	−17.9542	−0.583125
$949$	30.0718	0.976172
$950$	−3.29616	−0.106941
$951$	34.5074	1.11898
$952$	−4.16332	−0.134934
$953$	35.5911	1.15291	0.576454	−	0.817130i	$-0.304436\pi$
$953$	35.5911	1.15291	0.576454	+	0.817130i	$0.304436\pi$
$954$	21.2472	0.687902
$955$	−17.7465	−0.574264
$956$	−51.1254	−1.65351
$957$	0	0
$958$	15.1196	0.488491
$959$	12.6649	0.408973
$960$	−2.65497	−0.0856889
$961$	41.3254	1.33308
$962$	30.9317	0.997278
$963$	13.6773	0.440743
$964$	64.5020	2.07747
$965$	−41.7814	−1.34499
$966$	−13.9423	−0.448586
$967$	26.6658	0.857513	0.428756	−	0.903420i	$-0.358952\pi$
$967$	26.6658	0.857513	0.428756	+	0.903420i	$0.358952\pi$
$968$	0	0
$969$	−0.204137	−0.00655783
$970$	83.0908	2.66789
$971$	−0.383507	−0.0123073	−0.00615366	−	0.999981i	$-0.501959\pi$
$971$	−0.383507	−0.0123073	−0.00615366	+	0.999981i	$0.501959\pi$
$972$	−4.42489	−0.141928
$973$	−44.0816	−1.41319
$974$	−74.9994	−2.40314
$975$	3.09296	0.0990540
$976$	−9.76631	−0.312612
$977$	−9.15129	−0.292776	−0.146388	−	0.989227i	$-0.546765\pi$
$977$	−9.15129	−0.292776	−0.146388	+	0.989227i	$0.546765\pi$
$978$	−30.1687	−0.964688
$979$	0	0
$980$	−34.1285	−1.09020
$981$	−1.52679	−0.0487467
$982$	−22.6369	−0.722371
$983$	−37.4856	−1.19560	−0.597802	−	0.801644i	$-0.703959\pi$
$983$	−37.4856	−1.19560	−0.597802	+	0.801644i	$0.703959\pi$
$984$	−33.6870	−1.07390
$985$	−1.72770	−0.0550492
$986$	0.583445	0.0185807
$987$	−28.3427	−0.902159
$988$	10.5245	0.334830
$989$	7.41284	0.235715
$990$	0	0
$991$	49.4486	1.57079	0.785393	−	0.618998i	$-0.212461\pi$
$991$	49.4486	1.57079	0.785393	+	0.618998i	$0.212461\pi$
$992$	40.5284	1.28678
$993$	−22.3412	−0.708977
$994$	−67.9829	−2.15629
$995$	−9.38911	−0.297655
$996$	−77.1551	−2.44475
$997$	47.1446	1.49308	0.746542	−	0.665339i	$-0.231713\pi$
$997$	47.1446	1.49308	0.746542	+	0.665339i	$0.231713\pi$
$998$	−69.3899	−2.19650
$999$	−5.13064	−0.162326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.ba.1.7	yes	7
11.10	odd	2		6897.2.a.z.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6897.2.a.z.1.1	✓	7	11.10	odd	2
6897.2.a.ba.1.7	yes	7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6897.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53474 q^{2} -1.00000 q^{3} +4.42489 q^{4} -1.92344 q^{5} -2.53474 q^{6} -3.31812 q^{7} +6.14646 q^{8} +1.00000 q^{9} -4.87540 q^{10} -4.42489 q^{12} +2.37848 q^{13} -8.41057 q^{14} +1.92344 q^{15} +6.72988 q^{16} +0.204137 q^{17} +2.53474 q^{18} +1.00000 q^{19} -8.51100 q^{20} +3.31812 q^{21} -1.65771 q^{23} -6.14646 q^{24} -1.30039 q^{25} +6.02882 q^{26} -1.00000 q^{27} -14.6823 q^{28} +1.12757 q^{29} +4.87540 q^{30} +8.50443 q^{31} +4.76556 q^{32} +0.517434 q^{34} +6.38220 q^{35} +4.42489 q^{36} +5.13064 q^{37} +2.53474 q^{38} -2.37848 q^{39} -11.8223 q^{40} +5.48072 q^{41} +8.41057 q^{42} -4.47173 q^{43} -1.92344 q^{45} -4.20186 q^{46} -8.54180 q^{47} -6.72988 q^{48} +4.00993 q^{49} -3.29616 q^{50} -0.204137 q^{51} +10.5245 q^{52} +8.38239 q^{53} -2.53474 q^{54} -20.3947 q^{56} -1.00000 q^{57} +2.85810 q^{58} +5.15021 q^{59} +8.51100 q^{60} -1.45119 q^{61} +21.5565 q^{62} -3.31812 q^{63} -1.38033 q^{64} -4.57485 q^{65} +5.76760 q^{67} +0.903285 q^{68} +1.65771 q^{69} +16.1772 q^{70} +8.08303 q^{71} +6.14646 q^{72} +12.6433 q^{73} +13.0048 q^{74} +1.30039 q^{75} +4.42489 q^{76} -6.02882 q^{78} +4.05754 q^{79} -12.9445 q^{80} +1.00000 q^{81} +13.8922 q^{82} +17.4366 q^{83} +14.6823 q^{84} -0.392645 q^{85} -11.3347 q^{86} -1.12757 q^{87} -3.36049 q^{89} -4.87540 q^{90} -7.89209 q^{91} -7.33520 q^{92} -8.50443 q^{93} -21.6512 q^{94} -1.92344 q^{95} -4.76556 q^{96} -17.0429 q^{97} +10.1641 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} - 7 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} + 6 q^{7} + 12 q^{8} + 7 q^{9} - 6 q^{10} - 6 q^{12} + q^{13} - 4 q^{14} + 2 q^{15} + 5 q^{17} + 2 q^{18} + 7 q^{19} + 10 q^{20} - 6 q^{21} + 2 q^{23}+ \cdots + 32 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 - 7 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 + 6 * q^7 + 12 * q^8 + 7 * q^9 - 6 * q^10 - 6 * q^12 + q^13 - 4 * q^14 + 2 * q^15 + 5 * q^17 + 2 * q^18 + 7 * q^19 + 10 * q^20 - 6 * q^21 + 2 * q^23 - 12 * q^24 + 3 * q^25 + 4 * q^26 - 7 * q^27 + 6 * q^30 + 10 * q^31 - 4 * q^32 - 6 * q^34 - 2 * q^35 + 6 * q^36 + 14 * q^37 + 2 * q^38 - q^39 - 10 * q^40 + 28 * q^41 + 4 * q^42 + 14 * q^43 - 2 * q^45 - 24 * q^46 + 6 * q^47 + 13 * q^49 - 12 * q^50 - 5 * q^51 + 38 * q^52 + 7 * q^53 - 2 * q^54 - 4 * q^56 - 7 * q^57 - 2 * q^58 + 17 * q^59 - 10 * q^60 - 14 * q^61 + 2 * q^62 + 6 * q^63 - 14 * q^64 + 30 * q^65 + 18 * q^67 - 10 * q^68 - 2 * q^69 + 36 * q^70 + 7 * q^71 + 12 * q^72 + 4 * q^73 + 56 * q^74 - 3 * q^75 + 6 * q^76 - 4 * q^78 - 21 * q^79 - 24 * q^80 + 7 * q^81 + 9 * q^83 - 14 * q^85 + 4 * q^86 + q^89 - 6 * q^90 + 14 * q^91 - 22 * q^92 - 10 * q^93 + 14 * q^94 - 2 * q^95 + 4 * q^96 - 24 * q^97 + 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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