LMFDB - Embedded newform 3969.2.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.114612039936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 34x^{4} - 36x^{2} + 9 $ x^8 - 11x^6 + 34x^4 - 36*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.27020$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27020 q^{2} -0.386601 q^{4} -1.55350 q^{5} +3.03145 q^{8} +O(q^{10})$	$q-1.27020 q^{2} -0.386601 q^{4} -1.55350 q^{5} +3.03145 q^{8} +1.97325 q^{10} -3.21001 q^{11} -4.78669 q^{13} -3.07734 q^{16} +2.11836 q^{17} +4.86403 q^{19} +0.600584 q^{20} +4.07734 q^{22} +3.70758 q^{23} -2.58665 q^{25} +6.08004 q^{26} +7.37944 q^{29} -5.50418 q^{31} -2.15408 q^{32} -2.69074 q^{34} -0.186556 q^{37} -6.17827 q^{38} -4.70935 q^{40} +10.7972 q^{41} +4.86916 q^{43} +1.24099 q^{44} -4.70935 q^{46} -1.77187 q^{47} +3.28555 q^{50} +1.85054 q^{52} +1.66886 q^{53} +4.98674 q^{55} -9.37334 q^{58} -5.82594 q^{59} -6.87729 q^{61} +6.99139 q^{62} +8.89078 q^{64} +7.43611 q^{65} +12.2374 q^{67} -0.818961 q^{68} -13.8101 q^{71} -11.8640 q^{73} +0.236963 q^{74} -1.88044 q^{76} -1.30926 q^{79} +4.78064 q^{80} -13.7146 q^{82} +0.346488 q^{83} -3.29087 q^{85} -6.18478 q^{86} -9.73098 q^{88} +17.4064 q^{89} -1.43335 q^{92} +2.25063 q^{94} -7.55625 q^{95} +10.5683 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4}+O(q^{10}) $ 8 * q + 6 * q^4	$ 8 q + 6 q^{4} - 14 q^{10} - 6 q^{13} + 6 q^{16} - 24 q^{19} + 2 q^{22} - 20 q^{31} - 4 q^{37} - 36 q^{40} + 10 q^{43} - 36 q^{46} - 34 q^{52} - 4 q^{55} - 22 q^{58} - 36 q^{61} + 38 q^{64} - 18 q^{67} - 32 q^{73} - 58 q^{76} - 32 q^{79} + 2 q^{82} + 30 q^{85} - 72 q^{88} - 54 q^{94} - 14 q^{97}+O(q^{100}) $ 8 * q + 6 * q^4 - 14 * q^10 - 6 * q^13 + 6 * q^16 - 24 * q^19 + 2 * q^22 - 20 * q^31 - 4 * q^37 - 36 * q^40 + 10 * q^43 - 36 * q^46 - 34 * q^52 - 4 * q^55 - 22 * q^58 - 36 * q^61 + 38 * q^64 - 18 * q^67 - 32 * q^73 - 58 * q^76 - 32 * q^79 + 2 * q^82 + 30 * q^85 - 72 * q^88 - 54 * q^94 - 14 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−1.27020	−0.898165	−0.449082	−	0.893490i	$-0.648249\pi$
$2$	−1.27020	−0.898165	−0.449082	+	0.893490i	$0.648249\pi$
$3$	0	0
$3$	0	0
$4$	−0.386601	−0.193301
$5$	−1.55350	−0.694745	−0.347373	−	0.937727i	$-0.612926\pi$
$5$	−1.55350	−0.694745	−0.347373	+	0.937727i	$0.612926\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.03145	1.07178
$9$	0	0
$10$	1.97325	0.623995
$11$	−3.21001	−0.967853	−0.483927	−	0.875109i	$-0.660790\pi$
$11$	−3.21001	−0.967853	−0.483927	+	0.875109i	$0.660790\pi$
$12$	0	0
$13$	−4.78669	−1.32759	−0.663795	−	0.747915i	$-0.731055\pi$
$13$	−4.78669	−1.32759	−0.663795	+	0.747915i	$0.731055\pi$
$14$	0	0
$15$	0	0
$16$	−3.07734	−0.769334
$17$	2.11836	0.513778	0.256889	−	0.966441i	$-0.417302\pi$
$17$	2.11836	0.513778	0.256889	+	0.966441i	$0.417302\pi$
$18$	0	0
$19$	4.86403	1.11588	0.557942	−	0.829880i	$-0.311591\pi$
$19$	4.86403	1.11588	0.557942	+	0.829880i	$0.311591\pi$
$20$	0.600584	0.134295
$21$	0	0
$22$	4.07734	0.869291
$23$	3.70758	0.773084	0.386542	−	0.922272i	$-0.373670\pi$
$23$	3.70758	0.773084	0.386542	+	0.922272i	$0.373670\pi$
$24$	0	0
$25$	−2.58665	−0.517329
$26$	6.08004	1.19239
$27$	0	0
$28$	0	0
$29$	7.37944	1.37033	0.685164	−	0.728389i	$-0.259731\pi$
$29$	7.37944	1.37033	0.685164	+	0.728389i	$0.259731\pi$
$30$	0	0
$31$	−5.50418	−0.988580	−0.494290	−	0.869297i	$-0.664572\pi$
$31$	−5.50418	−0.988580	−0.494290	+	0.869297i	$0.664572\pi$
$32$	−2.15408	−0.380791
$33$	0	0
$34$	−2.69074	−0.461457
$35$	0	0
$36$	0	0
$37$	−0.186556	−0.0306697	−0.0153348	−	0.999882i	$-0.504881\pi$
$37$	−0.186556	−0.0306697	−0.0153348	+	0.999882i	$0.504881\pi$
$38$	−6.17827	−1.00225
$39$	0	0
$40$	−4.70935	−0.744614
$41$	10.7972	1.68624	0.843120	−	0.537726i	$-0.180716\pi$
$41$	10.7972	1.68624	0.843120	+	0.537726i	$0.180716\pi$
$42$	0	0
$43$	4.86916	0.742539	0.371270	−	0.928525i	$-0.378923\pi$
$43$	4.86916	0.742539	0.371270	+	0.928525i	$0.378923\pi$
$44$	1.24099	0.187087
$45$	0	0
$46$	−4.70935	−0.694356
$47$	−1.77187	−0.258455	−0.129227	−	0.991615i	$-0.541250\pi$
$47$	−1.77187	−0.258455	−0.129227	+	0.991615i	$0.541250\pi$
$48$	0	0
$49$	0	0
$50$	3.28555	0.464647
$51$	0	0
$52$	1.85054	0.256624
$53$	1.66886	0.229236	0.114618	−	0.993410i	$-0.463436\pi$
$53$	1.66886	0.229236	0.114618	+	0.993410i	$0.463436\pi$
$54$	0	0
$55$	4.98674	0.672411
$56$	0	0
$57$	0	0
$58$	−9.37334	−1.23078
$59$	−5.82594	−0.758473	−0.379236	−	0.925300i	$-0.623813\pi$
$59$	−5.82594	−0.758473	−0.379236	+	0.925300i	$0.623813\pi$
$60$	0	0
$61$	−6.87729	−0.880547	−0.440274	−	0.897864i	$-0.645119\pi$
$61$	−6.87729	−0.880547	−0.440274	+	0.897864i	$0.645119\pi$
$62$	6.99139	0.887907
$63$	0	0
$64$	8.89078	1.11135
$65$	7.43611	0.922336
$66$	0	0
$67$	12.2374	1.49503	0.747516	−	0.664244i	$-0.231246\pi$
$67$	12.2374	1.49503	0.747516	+	0.664244i	$0.231246\pi$
$68$	−0.818961	−0.0993136
$69$	0	0
$70$	0	0
$71$	−13.8101	−1.63895	−0.819477	−	0.573112i	$-0.805736\pi$
$71$	−13.8101	−1.63895	−0.819477	+	0.573112i	$0.805736\pi$
$72$	0	0
$73$	−11.8640	−1.38858	−0.694290	−	0.719696i	$-0.744281\pi$
$73$	−11.8640	−1.38858	−0.694290	+	0.719696i	$0.744281\pi$
$74$	0.236963	0.0275464
$75$	0	0
$76$	−1.88044	−0.215701
$77$	0	0
$78$	0	0
$79$	−1.30926	−0.147304	−0.0736518	−	0.997284i	$-0.523465\pi$
$79$	−1.30926	−0.147304	−0.0736518	+	0.997284i	$0.523465\pi$
$80$	4.78064	0.534491
$81$	0	0
$82$	−13.7146	−1.51452
$83$	0.346488	0.0380320	0.0190160	−	0.999819i	$-0.493947\pi$
$83$	0.346488	0.0380320	0.0190160	+	0.999819i	$0.493947\pi$
$84$	0	0
$85$	−3.29087	−0.356945
$86$	−6.18478	−0.666922
$87$	0	0
$88$	−9.73098	−1.03733
$89$	17.4064	1.84507	0.922537	−	0.385910i	$-0.126112\pi$
$89$	17.4064	1.84507	0.922537	+	0.385910i	$0.126112\pi$
$90$	0	0
$91$	0	0
$92$	−1.43335	−0.149437
$93$	0	0
$94$	2.25063	0.232135
$95$	−7.55625	−0.775255
$96$	0	0
$97$	10.5683	1.07304	0.536522	−	0.843886i	$-0.319738\pi$
$97$	10.5683	1.07304	0.536522	+	0.843886i	$0.319738\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	2.03394	0.202384	0.101192	−	0.994867i	$-0.467734\pi$
$101$	2.03394	0.202384	0.101192	+	0.994867i	$0.467734\pi$
$102$	0	0
$103$	−19.9816	−1.96885	−0.984423	−	0.175816i	$-0.943744\pi$
$103$	−19.9816	−1.96885	−0.984423	+	0.175816i	$0.943744\pi$
$104$	−14.5106	−1.42288
$105$	0	0
$106$	−2.11979	−0.205892
$107$	16.9064	1.63441	0.817204	−	0.576349i	$-0.195523\pi$
$107$	16.9064	1.63441	0.817204	+	0.576349i	$0.195523\pi$
$108$	0	0
$109$	−10.6991	−1.02479	−0.512394	−	0.858751i	$-0.671241\pi$
$109$	−10.6991	−1.02479	−0.512394	+	0.858751i	$0.671241\pi$
$110$	−6.33413	−0.603936
$111$	0	0
$112$	0	0
$113$	8.58471	0.807582	0.403791	−	0.914851i	$-0.367692\pi$
$113$	8.58471	0.807582	0.403791	+	0.914851i	$0.367692\pi$
$114$	0	0
$115$	−5.75971	−0.537096
$116$	−2.85290	−0.264885
$117$	0	0
$118$	7.40009	0.681233
$119$	0	0
$120$	0	0
$121$	−0.695865	−0.0632604
$122$	8.73551	0.790876
$123$	0	0
$124$	2.12792	0.191093
$125$	11.7858	1.05416
$126$	0	0
$127$	1.95162	0.173178	0.0865892	−	0.996244i	$-0.472403\pi$
$127$	1.95162	0.173178	0.0865892	+	0.996244i	$0.472403\pi$
$128$	−6.98488	−0.617382
$129$	0	0
$130$	−9.44532	−0.828410
$131$	11.7443	1.02610	0.513051	−	0.858358i	$-0.328515\pi$
$131$	11.7443	1.02610	0.513051	+	0.858358i	$0.328515\pi$
$132$	0	0
$133$	0	0
$134$	−15.5439	−1.34278
$135$	0	0
$136$	6.42171	0.550657
$137$	−16.1321	−1.37826	−0.689128	−	0.724639i	$-0.742006\pi$
$137$	−16.1321	−1.37826	−0.689128	+	0.724639i	$0.742006\pi$
$138$	0	0
$139$	−13.4425	−1.14018	−0.570091	−	0.821582i	$-0.693092\pi$
$139$	−13.4425	−1.14018	−0.570091	+	0.821582i	$0.693092\pi$
$140$	0	0
$141$	0	0
$142$	17.5415	1.47205
$143$	15.3653	1.28491
$144$	0	0
$145$	−11.4639	−0.952028
$146$	15.0696	1.24717
$147$	0	0
$148$	0.0721229	0.00592846
$149$	−3.21174	−0.263116	−0.131558	−	0.991308i	$-0.541998\pi$
$149$	−3.21174	−0.263116	−0.131558	+	0.991308i	$0.541998\pi$
$150$	0	0
$151$	17.0322	1.38606	0.693030	−	0.720909i	$-0.256275\pi$
$151$	17.0322	1.38606	0.693030	+	0.720909i	$0.256275\pi$
$152$	14.7451	1.19598
$153$	0	0
$154$	0	0
$155$	8.55073	0.686811
$156$	0	0
$157$	−9.43418	−0.752929	−0.376465	−	0.926431i	$-0.622860\pi$
$157$	−9.43418	−0.752929	−0.376465	+	0.926431i	$0.622860\pi$
$158$	1.66302	0.132303
$159$	0	0
$160$	3.34636	0.264553
$161$	0	0
$162$	0	0
$163$	3.67747	0.288042	0.144021	−	0.989575i	$-0.453997\pi$
$163$	3.67747	0.288042	0.144021	+	0.989575i	$0.453997\pi$
$164$	−4.17421	−0.325951
$165$	0	0
$166$	−0.440107	−0.0341590
$167$	−1.61017	−0.124599	−0.0622994	−	0.998058i	$-0.519843\pi$
$167$	−1.61017	−0.124599	−0.0622994	+	0.998058i	$0.519843\pi$
$168$	0	0
$169$	9.91241	0.762493
$170$	4.18005	0.320595
$171$	0	0
$172$	−1.88242	−0.143533
$173$	−16.1843	−1.23047	−0.615233	−	0.788345i	$-0.710938\pi$
$173$	−16.1843	−1.23047	−0.615233	+	0.788345i	$0.710938\pi$
$174$	0	0
$175$	0	0
$176$	9.87827	0.744603
$177$	0	0
$178$	−22.1095	−1.65718
$179$	17.4064	1.30101	0.650507	−	0.759500i	$-0.274556\pi$
$179$	17.4064	1.30101	0.650507	+	0.759500i	$0.274556\pi$
$180$	0	0
$181$	−8.89591	−0.661228	−0.330614	−	0.943766i	$-0.607256\pi$
$181$	−8.89591	−0.661228	−0.330614	+	0.943766i	$0.607256\pi$
$182$	0	0
$183$	0	0
$184$	11.2393	0.828576
$185$	0.289815	0.0213076
$186$	0	0
$187$	−6.79996	−0.497262
$188$	0.685009	0.0499594
$189$	0	0
$190$	9.59793	0.696307
$191$	−14.7922	−1.07033	−0.535163	−	0.844749i	$-0.679750\pi$
$191$	−14.7922	−1.07033	−0.535163	+	0.844749i	$0.679750\pi$
$192$	0	0
$193$	1.82158	0.131120	0.0655601	−	0.997849i	$-0.479117\pi$
$193$	1.82158	0.131120	0.0655601	+	0.997849i	$0.479117\pi$
$194$	−13.4238	−0.963770
$195$	0	0
$196$	0	0
$197$	−7.71970	−0.550006	−0.275003	−	0.961443i	$-0.588679\pi$
$197$	−7.71970	−0.550006	−0.275003	+	0.961443i	$0.588679\pi$
$198$	0	0
$199$	−2.21331	−0.156897	−0.0784487	−	0.996918i	$-0.524997\pi$
$199$	−2.21331	−0.156897	−0.0784487	+	0.996918i	$0.524997\pi$
$200$	−7.84129	−0.554463
$201$	0	0
$202$	−2.58350	−0.181774
$203$	0	0
$204$	0	0
$205$	−16.7734	−1.17151
$206$	25.3806	1.76835
$207$	0	0
$208$	14.7303	1.02136
$209$	−15.6136	−1.08001
$210$	0	0
$211$	−1.81344	−0.124843	−0.0624213	−	0.998050i	$-0.519882\pi$
$211$	−1.81344	−0.124843	−0.0624213	+	0.998050i	$0.519882\pi$
$212$	−0.645185	−0.0443115
$213$	0	0
$214$	−21.4745	−1.46797
$215$	−7.56422	−0.515876
$216$	0	0
$217$	0	0
$218$	13.5900	0.920428
$219$	0	0
$220$	−1.92788	−0.129977
$221$	−10.1399	−0.682087
$222$	0	0
$223$	−9.50616	−0.636580	−0.318290	−	0.947993i	$-0.603109\pi$
$223$	−9.50616	−0.636580	−0.318290	+	0.947993i	$0.603109\pi$
$224$	0	0
$225$	0	0
$226$	−10.9043	−0.725341
$227$	−10.8329	−0.719006	−0.359503	−	0.933144i	$-0.617054\pi$
$227$	−10.8329	−0.719006	−0.359503	+	0.933144i	$0.617054\pi$
$228$	0	0
$229$	−12.5197	−0.827322	−0.413661	−	0.910431i	$-0.635750\pi$
$229$	−12.5197	−0.827322	−0.413661	+	0.910431i	$0.635750\pi$
$230$	7.31597	0.482401
$231$	0	0
$232$	22.3704	1.46869
$233$	−15.2809	−1.00108	−0.500542	−	0.865712i	$-0.666866\pi$
$233$	−15.2809	−1.00108	−0.500542	+	0.865712i	$0.666866\pi$
$234$	0	0
$235$	2.75260	0.179560
$236$	2.25231	0.146613
$237$	0	0
$238$	0	0
$239$	−10.7158	−0.693149	−0.346574	−	0.938023i	$-0.612655\pi$
$239$	−10.7158	−0.693149	−0.346574	+	0.938023i	$0.612655\pi$
$240$	0	0
$241$	15.0002	0.966249	0.483125	−	0.875552i	$-0.339502\pi$
$241$	15.0002	0.966249	0.483125	+	0.875552i	$0.339502\pi$
$242$	0.883885	0.0568183
$243$	0	0
$244$	2.65877	0.170210
$245$	0	0
$246$	0	0
$247$	−23.2826	−1.48144
$248$	−16.6857	−1.05954
$249$	0	0
$250$	−14.9703	−0.946806
$251$	−16.5665	−1.04567	−0.522833	−	0.852435i	$-0.675125\pi$
$251$	−16.5665	−1.04567	−0.522833	+	0.852435i	$0.675125\pi$
$252$	0	0
$253$	−11.9013	−0.748231
$254$	−2.47894	−0.155543
$255$	0	0
$256$	−8.90940	−0.556837
$257$	−22.5772	−1.40833	−0.704163	−	0.710038i	$-0.748678\pi$
$257$	−22.5772	−1.40833	−0.704163	+	0.710038i	$0.748678\pi$
$258$	0	0
$259$	0	0
$260$	−2.87481	−0.178288
$261$	0	0
$262$	−14.9175	−0.921608
$263$	22.3461	1.37792	0.688959	−	0.724800i	$-0.258068\pi$
$263$	22.3461	1.37792	0.688959	+	0.724800i	$0.258068\pi$
$264$	0	0
$265$	−2.59258	−0.159261
$266$	0	0
$267$	0	0
$268$	−4.73098	−0.288990
$269$	−27.8949	−1.70078	−0.850392	−	0.526149i	$-0.823635\pi$
$269$	−27.8949	−1.70078	−0.850392	+	0.526149i	$0.823635\pi$
$270$	0	0
$271$	−9.87428	−0.599820	−0.299910	−	0.953967i	$-0.596957\pi$
$271$	−9.87428	−0.599820	−0.299910	+	0.953967i	$0.596957\pi$
$272$	−6.51892	−0.395267
$273$	0	0
$274$	20.4909	1.23790
$275$	8.30315	0.500699
$276$	0	0
$277$	−8.71457	−0.523608	−0.261804	−	0.965121i	$-0.584317\pi$
$277$	−8.71457	−0.523608	−0.261804	+	0.965121i	$0.584317\pi$
$278$	17.0747	1.02407
$279$	0	0
$280$	0	0
$281$	−6.20641	−0.370243	−0.185122	−	0.982716i	$-0.559268\pi$
$281$	−6.20641	−0.370243	−0.185122	+	0.982716i	$0.559268\pi$
$282$	0	0
$283$	−19.6589	−1.16860	−0.584299	−	0.811539i	$-0.698630\pi$
$283$	−19.6589	−1.16860	−0.584299	+	0.811539i	$0.698630\pi$
$284$	5.33899	0.316811
$285$	0	0
$286$	−19.5170	−1.15406
$287$	0	0
$288$	0	0
$289$	−12.5125	−0.736032
$290$	14.5615	0.855078
$291$	0	0
$292$	4.58665	0.268413
$293$	14.4968	0.846913	0.423456	−	0.905916i	$-0.360817\pi$
$293$	14.4968	0.846913	0.423456	+	0.905916i	$0.360817\pi$
$294$	0	0
$295$	9.05058	0.526945
$296$	−0.565537	−0.0328711
$297$	0	0
$298$	4.07954	0.236322
$299$	−17.7470	−1.02634
$300$	0	0
$301$	0	0
$302$	−21.6342	−1.24491
$303$	0	0
$304$	−14.9683	−0.858488
$305$	10.6839	0.611756
$306$	0	0
$307$	−22.2776	−1.27145	−0.635725	−	0.771916i	$-0.719299\pi$
$307$	−22.2776	−1.27145	−0.635725	+	0.771916i	$0.719299\pi$
$308$	0	0
$309$	0	0
$310$	−10.8611	−0.616869
$311$	−17.4005	−0.986694	−0.493347	−	0.869833i	$-0.664227\pi$
$311$	−17.4005	−0.986694	−0.493347	+	0.869833i	$0.664227\pi$
$312$	0	0
$313$	0.200045	0.0113072	0.00565360	−	0.999984i	$-0.498200\pi$
$313$	0.200045	0.0113072	0.00565360	+	0.999984i	$0.498200\pi$
$314$	11.9833	0.676254
$315$	0	0
$316$	0.506163	0.0284739
$317$	8.23412	0.462474	0.231237	−	0.972897i	$-0.425723\pi$
$317$	8.23412	0.462474	0.231237	+	0.972897i	$0.425723\pi$
$318$	0	0
$319$	−23.6880	−1.32628
$320$	−13.8118	−0.772103
$321$	0	0
$322$	0	0
$323$	10.3038	0.573317
$324$	0	0
$325$	12.3815	0.686801
$326$	−4.67111	−0.258709
$327$	0	0
$328$	32.7312	1.80728
$329$	0	0
$330$	0	0
$331$	−16.6323	−0.914195	−0.457098	−	0.889417i	$-0.651111\pi$
$331$	−16.6323	−0.914195	−0.457098	+	0.889417i	$0.651111\pi$
$332$	−0.133952	−0.00735160
$333$	0	0
$334$	2.04523	0.111910
$335$	−19.0107	−1.03867
$336$	0	0
$337$	15.9095	0.866645	0.433322	−	0.901239i	$-0.357341\pi$
$337$	15.9095	0.866645	0.433322	+	0.901239i	$0.357341\pi$
$338$	−12.5907	−0.684844
$339$	0	0
$340$	1.27225	0.0689977
$341$	17.6684	0.956800
$342$	0	0
$343$	0	0
$344$	14.7606	0.795839
$345$	0	0
$346$	20.5572	1.10516
$347$	6.28925	0.337625	0.168812	−	0.985648i	$-0.446007\pi$
$347$	6.28925	0.337625	0.168812	+	0.985648i	$0.446007\pi$
$348$	0	0
$349$	−1.61955	−0.0866927	−0.0433463	−	0.999060i	$-0.513802\pi$
$349$	−1.61955	−0.0866927	−0.0433463	+	0.999060i	$0.513802\pi$
$350$	0	0
$351$	0	0
$352$	6.91461	0.368550
$353$	−32.5118	−1.73043	−0.865213	−	0.501405i	$-0.832817\pi$
$353$	−32.5118	−1.73043	−0.865213	+	0.501405i	$0.832817\pi$
$354$	0	0
$355$	21.4539	1.13866
$356$	−6.72933	−0.356654
$357$	0	0
$358$	−22.1095	−1.16852
$359$	11.1918	0.590679	0.295339	−	0.955392i	$-0.404567\pi$
$359$	11.1918	0.590679	0.295339	+	0.955392i	$0.404567\pi$
$360$	0	0
$361$	4.65877	0.245198
$362$	11.2996	0.593891
$363$	0	0
$364$	0	0
$365$	18.4307	0.964709
$366$	0	0
$367$	5.18678	0.270748	0.135374	−	0.990795i	$-0.456776\pi$
$367$	5.18678	0.270748	0.135374	+	0.990795i	$0.456776\pi$
$368$	−11.4095	−0.594760
$369$	0	0
$370$	−0.368122	−0.0191377
$371$	0	0
$372$	0	0
$373$	32.8643	1.70165	0.850825	−	0.525448i	$-0.176102\pi$
$373$	32.8643	1.70165	0.850825	+	0.525448i	$0.176102\pi$
$374$	8.63728	0.446623
$375$	0	0
$376$	−5.37135	−0.277006
$377$	−35.3231	−1.81923
$378$	0	0
$379$	13.9362	0.715856	0.357928	−	0.933749i	$-0.383483\pi$
$379$	13.9362	0.715856	0.357928	+	0.933749i	$0.383483\pi$
$380$	2.92126	0.149857
$381$	0	0
$382$	18.7890	0.961328
$383$	1.71520	0.0876427	0.0438214	−	0.999039i	$-0.486047\pi$
$383$	1.71520	0.0876427	0.0438214	+	0.999039i	$0.486047\pi$
$384$	0	0
$385$	0	0
$386$	−2.31376	−0.117768
$387$	0	0
$388$	−4.08570	−0.207420
$389$	−3.14856	−0.159638	−0.0798191	−	0.996809i	$-0.525434\pi$
$389$	−3.14856	−0.159638	−0.0798191	+	0.996809i	$0.525434\pi$
$390$	0	0
$391$	7.85400	0.397194
$392$	0	0
$393$	0	0
$394$	9.80553	0.493996
$395$	2.03394	0.102339
$396$	0	0
$397$	−27.4345	−1.37690	−0.688449	−	0.725285i	$-0.741708\pi$
$397$	−27.4345	−1.37690	−0.688449	+	0.725285i	$0.741708\pi$
$398$	2.81134	0.140920
$399$	0	0
$400$	7.95998	0.397999
$401$	29.0370	1.45004	0.725020	−	0.688728i	$-0.241830\pi$
$401$	29.0370	1.45004	0.725020	+	0.688728i	$0.241830\pi$
$402$	0	0
$403$	26.3468	1.31243
$404$	−0.786322	−0.0391210
$405$	0	0
$406$	0	0
$407$	0.598847	0.0296837
$408$	0	0
$409$	27.7300	1.37116	0.685581	−	0.727996i	$-0.259548\pi$
$409$	27.7300	1.37116	0.685581	+	0.727996i	$0.259548\pi$
$410$	21.3055	1.05221
$411$	0	0
$412$	7.72491	0.380579
$413$	0	0
$414$	0	0
$415$	−0.538268	−0.0264225
$416$	10.3109	0.505534
$417$	0	0
$418$	19.8323	0.970029
$419$	−30.3598	−1.48317	−0.741586	−	0.670857i	$-0.765926\pi$
$419$	−30.3598	−1.48317	−0.741586	+	0.670857i	$0.765926\pi$
$420$	0	0
$421$	−27.7735	−1.35360	−0.676799	−	0.736168i	$-0.736633\pi$
$421$	−27.7735	−1.35360	−0.676799	+	0.736168i	$0.736633\pi$
$422$	2.30343	0.112129
$423$	0	0
$424$	5.05908	0.245691
$425$	−5.47945	−0.265793
$426$	0	0
$427$	0	0
$428$	−6.53605	−0.315932
$429$	0	0
$430$	9.60805	0.463341
$431$	−7.25929	−0.349668	−0.174834	−	0.984598i	$-0.555939\pi$
$431$	−7.25929	−0.349668	−0.174834	+	0.984598i	$0.555939\pi$
$432$	0	0
$433$	−15.0375	−0.722658	−0.361329	−	0.932438i	$-0.617677\pi$
$433$	−15.0375	−0.722658	−0.361329	+	0.932438i	$0.617677\pi$
$434$	0	0
$435$	0	0
$436$	4.13628	0.198092
$437$	18.0338	0.862672
$438$	0	0
$439$	−1.54119	−0.0735570	−0.0367785	−	0.999323i	$-0.511710\pi$
$439$	−1.54119	−0.0735570	−0.0367785	+	0.999323i	$0.511710\pi$
$440$	15.1170	0.720677
$441$	0	0
$442$	12.8797	0.612626
$443$	−20.3379	−0.966281	−0.483141	−	0.875543i	$-0.660504\pi$
$443$	−20.3379	−0.966281	−0.483141	+	0.875543i	$0.660504\pi$
$444$	0	0
$445$	−27.0408	−1.28186
$446$	12.0747	0.571753
$447$	0	0
$448$	0	0
$449$	−26.4527	−1.24838	−0.624190	−	0.781272i	$-0.714571\pi$
$449$	−26.4527	−1.24838	−0.624190	+	0.781272i	$0.714571\pi$
$450$	0	0
$451$	−34.6591	−1.63203
$452$	−3.31886	−0.156106
$453$	0	0
$454$	13.7599	0.645786
$455$	0	0
$456$	0	0
$457$	−6.49361	−0.303758	−0.151879	−	0.988399i	$-0.548532\pi$
$457$	−6.49361	−0.303758	−0.151879	+	0.988399i	$0.548532\pi$
$458$	15.9024	0.743071
$459$	0	0
$460$	2.22671	0.103821
$461$	15.5916	0.726175	0.363088	−	0.931755i	$-0.381723\pi$
$461$	15.5916	0.726175	0.363088	+	0.931755i	$0.381723\pi$
$462$	0	0
$463$	7.65585	0.355797	0.177899	−	0.984049i	$-0.443070\pi$
$463$	7.65585	0.355797	0.177899	+	0.984049i	$0.443070\pi$
$464$	−22.7090	−1.05424
$465$	0	0
$466$	19.4097	0.899138
$467$	−41.6273	−1.92628	−0.963142	−	0.268994i	$-0.913309\pi$
$467$	−41.6273	−1.92628	−0.963142	+	0.268994i	$0.913309\pi$
$468$	0	0
$469$	0	0
$470$	−3.49635	−0.161274
$471$	0	0
$472$	−17.6611	−0.812916
$473$	−15.6300	−0.718669
$474$	0	0
$475$	−12.5815	−0.577280
$476$	0	0
$477$	0	0
$478$	13.6112	0.622561
$479$	2.31579	0.105811	0.0529055	−	0.998600i	$-0.483152\pi$
$479$	2.31579	0.105811	0.0529055	+	0.998600i	$0.483152\pi$
$480$	0	0
$481$	0.892987	0.0407167
$482$	−19.0532	−0.867851
$483$	0	0
$484$	0.269022	0.0122283
$485$	−16.4178	−0.745492
$486$	0	0
$487$	18.1279	0.821455	0.410727	−	0.911758i	$-0.365275\pi$
$487$	18.1279	0.821455	0.410727	+	0.911758i	$0.365275\pi$
$488$	−20.8482	−0.943753
$489$	0	0
$490$	0	0
$491$	21.5944	0.974542	0.487271	−	0.873251i	$-0.337992\pi$
$491$	21.5944	0.974542	0.487271	+	0.873251i	$0.337992\pi$
$492$	0	0
$493$	15.6323	0.704045
$494$	29.5735	1.33057
$495$	0	0
$496$	16.9382	0.760549
$497$	0	0
$498$	0	0
$499$	27.1654	1.21609	0.608045	−	0.793903i	$-0.291954\pi$
$499$	27.1654	1.21609	0.608045	+	0.793903i	$0.291954\pi$
$500$	−4.55642	−0.203769
$501$	0	0
$502$	21.0427	0.939180
$503$	−11.8850	−0.529927	−0.264964	−	0.964258i	$-0.585360\pi$
$503$	−11.8850	−0.529927	−0.264964	+	0.964258i	$0.585360\pi$
$504$	0	0
$505$	−3.15972	−0.140606
$506$	15.1170	0.672035
$507$	0	0
$508$	−0.754499	−0.0334755
$509$	41.4595	1.83766	0.918829	−	0.394656i	$-0.129136\pi$
$509$	41.4595	1.83766	0.918829	+	0.394656i	$0.129136\pi$
$510$	0	0
$511$	0	0
$512$	25.2864	1.11751
$513$	0	0
$514$	28.6775	1.26491
$515$	31.0414	1.36785
$516$	0	0
$517$	5.68773	0.250146
$518$	0	0
$519$	0	0
$520$	22.5422	0.988542
$521$	−3.30442	−0.144769	−0.0723846	−	0.997377i	$-0.523061\pi$
$521$	−3.30442	−0.144769	−0.0723846	+	0.997377i	$0.523061\pi$
$522$	0	0
$523$	2.48967	0.108865	0.0544327	−	0.998517i	$-0.482665\pi$
$523$	2.48967	0.108865	0.0544327	+	0.998517i	$0.482665\pi$
$524$	−4.54035	−0.198346
$525$	0	0
$526$	−28.3839	−1.23760
$527$	−11.6598	−0.507911
$528$	0	0
$529$	−9.25386	−0.402342
$530$	3.29308	0.143042
$531$	0	0
$532$	0	0
$533$	−51.6829	−2.23863
$534$	0	0
$535$	−26.2641	−1.13550
$536$	37.0970	1.60235
$537$	0	0
$538$	35.4321	1.52758
$539$	0	0
$540$	0	0
$541$	−22.4794	−0.966465	−0.483233	−	0.875492i	$-0.660537\pi$
$541$	−22.4794	−0.966465	−0.483233	+	0.875492i	$0.660537\pi$
$542$	12.5423	0.538737
$543$	0	0
$544$	−4.56312	−0.195642
$545$	16.6210	0.711966
$546$	0	0
$547$	−14.5545	−0.622307	−0.311154	−	0.950360i	$-0.600715\pi$
$547$	−14.5545	−0.622307	−0.311154	+	0.950360i	$0.600715\pi$
$548$	6.23668	0.266418
$549$	0	0
$550$	−10.5466	−0.449710
$551$	35.8938	1.52913
$552$	0	0
$553$	0	0
$554$	11.0692	0.470286
$555$	0	0
$556$	5.19690	0.220398
$557$	15.5466	0.658731	0.329366	−	0.944202i	$-0.393165\pi$
$557$	15.5466	0.658731	0.329366	+	0.944202i	$0.393165\pi$
$558$	0	0
$559$	−23.3071	−0.985787
$560$	0	0
$561$	0	0
$562$	7.88336	0.332539
$563$	14.7383	0.621144	0.310572	−	0.950550i	$-0.399479\pi$
$563$	14.7383	0.621144	0.310572	+	0.950550i	$0.399479\pi$
$564$	0	0
$565$	−13.3363	−0.561063
$566$	24.9706	1.04959
$567$	0	0
$568$	−41.8646	−1.75660
$569$	12.9708	0.543764	0.271882	−	0.962331i	$-0.412354\pi$
$569$	12.9708	0.543764	0.271882	+	0.962331i	$0.412354\pi$
$570$	0	0
$571$	12.8586	0.538115	0.269058	−	0.963124i	$-0.413288\pi$
$571$	12.8586	0.538115	0.269058	+	0.963124i	$0.413288\pi$
$572$	−5.94024	−0.248374
$573$	0	0
$574$	0	0
$575$	−9.59019	−0.399939
$576$	0	0
$577$	10.5256	0.438186	0.219093	−	0.975704i	$-0.429690\pi$
$577$	10.5256	0.438186	0.219093	+	0.975704i	$0.429690\pi$
$578$	15.8934	0.661078
$579$	0	0
$580$	4.43197	0.184028
$581$	0	0
$582$	0	0
$583$	−5.35706	−0.221867
$584$	−35.9652	−1.48825
$585$	0	0
$586$	−18.4138	−0.760667
$587$	6.20018	0.255909	0.127954	−	0.991780i	$-0.459159\pi$
$587$	6.20018	0.255909	0.127954	+	0.991780i	$0.459159\pi$
$588$	0	0
$589$	−26.7725	−1.10314
$590$	−11.4960	−0.473284
$591$	0	0
$592$	0.574097	0.0235952
$593$	−37.5259	−1.54100	−0.770502	−	0.637438i	$-0.779994\pi$
$593$	−37.5259	−1.54100	−0.770502	+	0.637438i	$0.779994\pi$
$594$	0	0
$595$	0	0
$596$	1.24166	0.0508605
$597$	0	0
$598$	22.5422	0.921820
$599$	−35.0920	−1.43382	−0.716909	−	0.697166i	$-0.754444\pi$
$599$	−35.0920	−1.43382	−0.716909	+	0.697166i	$0.754444\pi$
$600$	0	0
$601$	−26.2342	−1.07012	−0.535058	−	0.844815i	$-0.679710\pi$
$601$	−26.2342	−1.07012	−0.535058	+	0.844815i	$0.679710\pi$
$602$	0	0
$603$	0	0
$604$	−6.58466	−0.267926
$605$	1.08102	0.0439499
$606$	0	0
$607$	2.21866	0.0900527	0.0450263	−	0.998986i	$-0.485663\pi$
$607$	2.21866	0.0900527	0.0450263	+	0.998986i	$0.485663\pi$
$608$	−10.4775	−0.424919
$609$	0	0
$610$	−13.5706	−0.549457
$611$	8.48142	0.343121
$612$	0	0
$613$	14.8662	0.600442	0.300221	−	0.953870i	$-0.402940\pi$
$613$	14.8662	0.600442	0.300221	+	0.953870i	$0.402940\pi$
$614$	28.2969	1.14197
$615$	0	0
$616$	0	0
$617$	−23.2046	−0.934181	−0.467091	−	0.884210i	$-0.654698\pi$
$617$	−23.2046	−0.934181	−0.467091	+	0.884210i	$0.654698\pi$
$618$	0	0
$619$	8.31125	0.334057	0.167029	−	0.985952i	$-0.446583\pi$
$619$	8.31125	0.334057	0.167029	+	0.985952i	$0.446583\pi$
$620$	−3.30572	−0.132761
$621$	0	0
$622$	22.1021	0.886214
$623$	0	0
$624$	0	0
$625$	−5.37603	−0.215041
$626$	−0.254096	−0.0101557
$627$	0	0
$628$	3.64726	0.145542
$629$	−0.395194	−0.0157574
$630$	0	0
$631$	−24.5415	−0.976982	−0.488491	−	0.872569i	$-0.662452\pi$
$631$	−24.5415	−0.976982	−0.488491	+	0.872569i	$0.662452\pi$
$632$	−3.96897	−0.157877
$633$	0	0
$634$	−10.4589	−0.415378
$635$	−3.03184	−0.120315
$636$	0	0
$637$	0	0
$638$	30.0885	1.19121
$639$	0	0
$640$	10.8510	0.428923
$641$	−7.97525	−0.315003	−0.157502	−	0.987519i	$-0.550344\pi$
$641$	−7.97525	−0.315003	−0.157502	+	0.987519i	$0.550344\pi$
$642$	0	0
$643$	3.12279	0.123151	0.0615755	−	0.998102i	$-0.480388\pi$
$643$	3.12279	0.123151	0.0615755	+	0.998102i	$0.480388\pi$
$644$	0	0
$645$	0	0
$646$	−13.0878	−0.514933
$647$	−12.2655	−0.482205	−0.241102	−	0.970500i	$-0.577509\pi$
$647$	−12.2655	−0.482205	−0.241102	+	0.970500i	$0.577509\pi$
$648$	0	0
$649$	18.7013	0.734090
$650$	−15.7269	−0.616860
$651$	0	0
$652$	−1.42171	−0.0556786
$653$	−16.5095	−0.646067	−0.323034	−	0.946387i	$-0.604703\pi$
$653$	−16.5095	−0.646067	−0.323034	+	0.946387i	$0.604703\pi$
$654$	0	0
$655$	−18.2447	−0.712879
$656$	−33.2266	−1.29728
$657$	0	0
$658$	0	0
$659$	−10.7388	−0.418324	−0.209162	−	0.977881i	$-0.567074\pi$
$659$	−10.7388	−0.418324	−0.209162	+	0.977881i	$0.567074\pi$
$660$	0	0
$661$	13.3763	0.520280	0.260140	−	0.965571i	$-0.416231\pi$
$661$	13.3763	0.520280	0.260140	+	0.965571i	$0.416231\pi$
$662$	21.1263	0.821098
$663$	0	0
$664$	1.05036	0.0407619
$665$	0	0
$666$	0	0
$667$	27.3598	1.05938
$668$	0.622494	0.0240850
$669$	0	0
$670$	24.1473	0.932893
$671$	22.0761	0.852240
$672$	0	0
$673$	40.8986	1.57653	0.788264	−	0.615338i	$-0.210980\pi$
$673$	40.8986	1.57653	0.788264	+	0.615338i	$0.210980\pi$
$674$	−20.2082	−0.778390
$675$	0	0
$676$	−3.83215	−0.147390
$677$	−21.2500	−0.816705	−0.408353	−	0.912824i	$-0.633897\pi$
$677$	−21.2500	−0.816705	−0.408353	+	0.912824i	$0.633897\pi$
$678$	0	0
$679$	0	0
$680$	−9.97612	−0.382567
$681$	0	0
$682$	−22.4424	−0.859364
$683$	−41.3417	−1.58190	−0.790948	−	0.611884i	$-0.790412\pi$
$683$	−41.3417	−1.58190	−0.790948	+	0.611884i	$0.790412\pi$
$684$	0	0
$685$	25.0612	0.957537
$686$	0	0
$687$	0	0
$688$	−14.9840	−0.571261
$689$	−7.98834	−0.304331
$690$	0	0
$691$	1.52037	0.0578375	0.0289187	−	0.999582i	$-0.490794\pi$
$691$	1.52037	0.0578375	0.0289187	+	0.999582i	$0.490794\pi$
$692$	6.25685	0.237850
$693$	0	0
$694$	−7.98858	−0.303242
$695$	20.8829	0.792135
$696$	0	0
$697$	22.8724	0.866353
$698$	2.05715	0.0778643
$699$	0	0
$700$	0	0
$701$	32.5344	1.22881	0.614404	−	0.788991i	$-0.289396\pi$
$701$	32.5344	1.22881	0.614404	+	0.788991i	$0.289396\pi$
$702$	0	0
$703$	−0.907415	−0.0342238
$704$	−28.5395	−1.07562
$705$	0	0
$706$	41.2963	1.55421
$707$	0	0
$708$	0	0
$709$	22.5621	0.847337	0.423669	−	0.905817i	$-0.360742\pi$
$709$	22.5621	0.847337	0.423669	+	0.905817i	$0.360742\pi$
$710$	−27.2507	−1.02270
$711$	0	0
$712$	52.7666	1.97751
$713$	−20.4072	−0.764255
$714$	0	0
$715$	−23.8700	−0.892686
$716$	−6.72933	−0.251487
$717$	0	0
$718$	−14.2157	−0.530527
$719$	−6.50168	−0.242472	−0.121236	−	0.992624i	$-0.538686\pi$
$719$	−6.50168	−0.242472	−0.121236	+	0.992624i	$0.538686\pi$
$720$	0	0
$721$	0	0
$722$	−5.91755	−0.220228
$723$	0	0
$724$	3.43917	0.127816
$725$	−19.0880	−0.708910
$726$	0	0
$727$	7.83215	0.290478	0.145239	−	0.989397i	$-0.453605\pi$
$727$	7.83215	0.290478	0.145239	+	0.989397i	$0.453605\pi$
$728$	0	0
$729$	0	0
$730$	−23.4107	−0.866467
$731$	10.3146	0.381501
$732$	0	0
$733$	7.49560	0.276856	0.138428	−	0.990372i	$-0.455795\pi$
$733$	7.49560	0.276856	0.138428	+	0.990372i	$0.455795\pi$
$734$	−6.58823	−0.243176
$735$	0	0
$736$	−7.98643	−0.294384
$737$	−39.2820	−1.44697
$738$	0	0
$739$	24.0960	0.886387	0.443194	−	0.896426i	$-0.353845\pi$
$739$	24.0960	0.886387	0.443194	+	0.896426i	$0.353845\pi$
$740$	−0.112043	−0.00411877
$741$	0	0
$742$	0	0
$743$	−19.2882	−0.707616	−0.353808	−	0.935318i	$-0.615113\pi$
$743$	−19.2882	−0.707616	−0.353808	+	0.935318i	$0.615113\pi$
$744$	0	0
$745$	4.98943	0.182799
$746$	−41.7442	−1.52836
$747$	0	0
$748$	2.62887	0.0961210
$749$	0	0
$750$	0	0
$751$	−33.2089	−1.21181	−0.605906	−	0.795536i	$-0.707189\pi$
$751$	−33.2089	−1.21181	−0.605906	+	0.795536i	$0.707189\pi$
$752$	5.45266	0.198838
$753$	0	0
$754$	44.8673	1.63397
$755$	−26.4595	−0.962959
$756$	0	0
$757$	9.70935	0.352892	0.176446	−	0.984310i	$-0.443540\pi$
$757$	9.70935	0.352892	0.176446	+	0.984310i	$0.443540\pi$
$758$	−17.7018	−0.642957
$759$	0	0
$760$	−22.9064	−0.830903
$761$	2.72609	0.0988207	0.0494104	−	0.998779i	$-0.484266\pi$
$761$	2.72609	0.0988207	0.0494104	+	0.998779i	$0.484266\pi$
$762$	0	0
$763$	0	0
$764$	5.71868	0.206894
$765$	0	0
$766$	−2.17864	−0.0787176
$767$	27.8870	1.00694
$768$	0	0
$769$	−50.0460	−1.80470	−0.902352	−	0.430999i	$-0.858161\pi$
$769$	−50.0460	−1.80470	−0.902352	+	0.430999i	$0.858161\pi$
$770$	0	0
$771$	0	0
$772$	−0.704225	−0.0253456
$773$	19.0406	0.684842	0.342421	−	0.939547i	$-0.388753\pi$
$773$	19.0406	0.684842	0.342421	+	0.939547i	$0.388753\pi$
$774$	0	0
$775$	14.2374	0.511421
$776$	32.0372	1.15007
$777$	0	0
$778$	3.99928	0.143381
$779$	52.5179	1.88165
$780$	0	0
$781$	44.3304	1.58627
$782$	−9.97612	−0.356745
$783$	0	0
$784$	0	0
$785$	14.6560	0.523094
$786$	0	0
$787$	33.2023	1.18353	0.591766	−	0.806110i	$-0.298431\pi$
$787$	33.2023	1.18353	0.591766	+	0.806110i	$0.298431\pi$
$788$	2.98444	0.106316
$789$	0	0
$790$	−2.58350	−0.0919168
$791$	0	0
$792$	0	0
$793$	32.9195	1.16900
$794$	34.8472	1.23668
$795$	0	0
$796$	0.855668	0.0303283
$797$	2.08944	0.0740118	0.0370059	−	0.999315i	$-0.488218\pi$
$797$	2.08944	0.0740118	0.0370059	+	0.999315i	$0.488218\pi$
$798$	0	0
$799$	−3.75347	−0.132788
$800$	5.57184	0.196994
$801$	0	0
$802$	−36.8827	−1.30237
$803$	38.0836	1.34394
$804$	0	0
$805$	0	0
$806$	−33.4656	−1.17878
$807$	0	0
$808$	6.16578	0.216912
$809$	−0.482809	−0.0169746	−0.00848732	−	0.999964i	$-0.502702\pi$
$809$	−0.482809	−0.0169746	−0.00848732	+	0.999964i	$0.502702\pi$
$810$	0	0
$811$	−17.1671	−0.602820	−0.301410	−	0.953495i	$-0.597457\pi$
$811$	−17.1671	−0.602820	−0.301410	+	0.953495i	$0.597457\pi$
$812$	0	0
$813$	0	0
$814$	−0.760653	−0.0266609
$815$	−5.71294	−0.200116
$816$	0	0
$817$	23.6837	0.828588
$818$	−35.2226	−1.23153
$819$	0	0
$820$	6.48462	0.226453
$821$	12.8217	0.447481	0.223741	−	0.974649i	$-0.428173\pi$
$821$	12.8217	0.447481	0.223741	+	0.974649i	$0.428173\pi$
$822$	0	0
$823$	24.5945	0.857311	0.428655	−	0.903468i	$-0.358987\pi$
$823$	24.5945	0.857311	0.428655	+	0.903468i	$0.358987\pi$
$824$	−60.5733	−2.11017
$825$	0	0
$826$	0	0
$827$	−0.527165	−0.0183313	−0.00916567	−	0.999958i	$-0.502918\pi$
$827$	−0.527165	−0.0183313	−0.00916567	+	0.999958i	$0.502918\pi$
$828$	0	0
$829$	46.2030	1.60470	0.802348	−	0.596856i	$-0.203584\pi$
$829$	46.2030	1.60470	0.802348	+	0.596856i	$0.203584\pi$
$830$	0.683706	0.0237318
$831$	0	0
$832$	−42.5574	−1.47541
$833$	0	0
$834$	0	0
$835$	2.50140	0.0865644
$836$	6.03622	0.208767
$837$	0	0
$838$	38.5629	1.33213
$839$	−10.8258	−0.373747	−0.186874	−	0.982384i	$-0.559835\pi$
$839$	−10.8258	−0.373747	−0.186874	+	0.982384i	$0.559835\pi$
$840$	0	0
$841$	25.4561	0.877797
$842$	35.2778	1.21575
$843$	0	0
$844$	0.701079	0.0241321
$845$	−15.3989	−0.529738
$846$	0	0
$847$	0	0
$848$	−5.13566	−0.176359
$849$	0	0
$850$	6.95998	0.238725
$851$	−0.691672	−0.0237102
$852$	0	0
$853$	43.1922	1.47887	0.739437	−	0.673226i	$-0.235092\pi$
$853$	43.1922	1.47887	0.739437	+	0.673226i	$0.235092\pi$
$854$	0	0
$855$	0	0
$856$	51.2511	1.75173
$857$	1.57445	0.0537822	0.0268911	−	0.999638i	$-0.491439\pi$
$857$	1.57445	0.0537822	0.0268911	+	0.999638i	$0.491439\pi$
$858$	0	0
$859$	−6.82180	−0.232757	−0.116378	−	0.993205i	$-0.537129\pi$
$859$	−6.82180	−0.232757	−0.116378	+	0.993205i	$0.537129\pi$
$860$	2.92434	0.0997190
$861$	0	0
$862$	9.22073	0.314059
$863$	−9.02774	−0.307308	−0.153654	−	0.988125i	$-0.549104\pi$
$863$	−9.02774	−0.307308	−0.153654	+	0.988125i	$0.549104\pi$
$864$	0	0
$865$	25.1422	0.854861
$866$	19.1006	0.649066
$867$	0	0
$868$	0	0
$869$	4.20274	0.142568
$870$	0	0
$871$	−58.5765	−1.98479
$872$	−32.4338	−1.09835
$873$	0	0
$874$	−22.9064	−0.774821
$875$	0	0
$876$	0	0
$877$	9.66920	0.326506	0.163253	−	0.986584i	$-0.447801\pi$
$877$	9.66920	0.326506	0.163253	+	0.986584i	$0.447801\pi$
$878$	1.95761	0.0660663
$879$	0	0
$880$	−15.3459	−0.517309
$881$	−21.4721	−0.723415	−0.361707	−	0.932292i	$-0.617806\pi$
$881$	−21.4721	−0.723415	−0.361707	+	0.932292i	$0.617806\pi$
$882$	0	0
$883$	−26.5380	−0.893076	−0.446538	−	0.894765i	$-0.647343\pi$
$883$	−26.5380	−0.893076	−0.446538	+	0.894765i	$0.647343\pi$
$884$	3.92011	0.131848
$885$	0	0
$886$	25.8331	0.867880
$887$	41.5223	1.39418	0.697091	−	0.716982i	$-0.254477\pi$
$887$	41.5223	1.39418	0.697091	+	0.716982i	$0.254477\pi$
$888$	0	0
$889$	0	0
$890$	34.3471	1.15132
$891$	0	0
$892$	3.67509	0.123051
$893$	−8.61845	−0.288405
$894$	0	0
$895$	−27.0408	−0.903873
$896$	0	0
$897$	0	0
$898$	33.6001	1.12125
$899$	−40.6178	−1.35468
$900$	0	0
$901$	3.53526	0.117777
$902$	44.0238	1.46583
$903$	0	0
$904$	26.0241	0.865550
$905$	13.8198	0.459385
$906$	0	0
$907$	−4.64361	−0.154188	−0.0770942	−	0.997024i	$-0.524564\pi$
$907$	−4.64361	−0.154188	−0.0770942	+	0.997024i	$0.524564\pi$
$908$	4.18802	0.138984
$909$	0	0
$910$	0	0
$911$	1.34811	0.0446648	0.0223324	−	0.999751i	$-0.492891\pi$
$911$	1.34811	0.0446648	0.0223324	+	0.999751i	$0.492891\pi$
$912$	0	0
$913$	−1.11223	−0.0368093
$914$	8.24816	0.272825
$915$	0	0
$916$	4.84011	0.159922
$917$	0	0
$918$	0	0
$919$	−40.4944	−1.33579	−0.667893	−	0.744257i	$-0.732804\pi$
$919$	−40.4944	−1.33579	−0.667893	+	0.744257i	$0.732804\pi$
$920$	−17.4603	−0.575649
$921$	0	0
$922$	−19.8045	−0.652225
$923$	66.1045	2.17586
$924$	0	0
$925$	0.482555	0.0158663
$926$	−9.72443	−0.319565
$927$	0	0
$928$	−15.8959	−0.521809
$929$	22.3138	0.732093	0.366046	−	0.930597i	$-0.380711\pi$
$929$	22.3138	0.732093	0.366046	+	0.930597i	$0.380711\pi$
$930$	0	0
$931$	0	0
$932$	5.90760	0.193510
$933$	0	0
$934$	52.8749	1.73012
$935$	10.5637	0.345470
$936$	0	0
$937$	1.13943	0.0372235	0.0186117	−	0.999827i	$-0.494075\pi$
$937$	1.13943	0.0372235	0.0186117	+	0.999827i	$0.494075\pi$
$938$	0	0
$939$	0	0
$940$	−1.06416	−0.0347090
$941$	47.1875	1.53827	0.769134	−	0.639087i	$-0.220688\pi$
$941$	47.1875	1.53827	0.769134	+	0.639087i	$0.220688\pi$
$942$	0	0
$943$	40.0315	1.30360
$944$	17.9284	0.583519
$945$	0	0
$946$	19.8532	0.645483
$947$	−24.6460	−0.800888	−0.400444	−	0.916321i	$-0.631144\pi$
$947$	−24.6460	−0.800888	−0.400444	+	0.916321i	$0.631144\pi$
$948$	0	0
$949$	56.7894	1.84346
$950$	15.9810	0.518492
$951$	0	0
$952$	0	0
$953$	56.2821	1.82316	0.911579	−	0.411125i	$-0.134864\pi$
$953$	56.2821	1.82316	0.911579	+	0.411125i	$0.134864\pi$
$954$	0	0
$955$	22.9796	0.743603
$956$	4.14275	0.133986
$957$	0	0
$958$	−2.94150	−0.0950356
$959$	0	0
$960$	0	0
$961$	−0.704001	−0.0227097
$962$	−1.13427	−0.0365703
$963$	0	0
$964$	−5.79910	−0.186777
$965$	−2.82982	−0.0910951
$966$	0	0
$967$	−58.5977	−1.88438	−0.942188	−	0.335084i	$-0.891235\pi$
$967$	−58.5977	−1.88438	−0.942188	+	0.335084i	$0.891235\pi$
$968$	−2.10948	−0.0678013
$969$	0	0
$970$	20.8538	0.669574
$971$	−6.09982	−0.195753	−0.0978763	−	0.995199i	$-0.531205\pi$
$971$	−6.09982	−0.195753	−0.0978763	+	0.995199i	$0.531205\pi$
$972$	0	0
$973$	0	0
$974$	−23.0260	−0.737801
$975$	0	0
$976$	21.1638	0.677435
$977$	5.15976	0.165075	0.0825377	−	0.996588i	$-0.473698\pi$
$977$	5.15976	0.165075	0.0825377	+	0.996588i	$0.473698\pi$
$978$	0	0
$979$	−55.8746	−1.78576
$980$	0	0
$981$	0	0
$982$	−27.4291	−0.875299
$983$	62.1071	1.98091	0.990455	−	0.137837i	$-0.0440150\pi$
$983$	62.1071	1.98091	0.990455	+	0.137837i	$0.0440150\pi$
$984$	0	0
$985$	11.9925	0.382114
$986$	−19.8561	−0.632348
$987$	0	0
$988$	9.00108	0.286362
$989$	18.0528	0.574045
$990$	0	0
$991$	−21.6325	−0.687181	−0.343590	−	0.939120i	$-0.611643\pi$
$991$	−21.6325	−0.687181	−0.343590	+	0.939120i	$0.611643\pi$
$992$	11.8565	0.376443
$993$	0	0
$994$	0	0
$995$	3.43837	0.109004
$996$	0	0
$997$	40.7362	1.29013	0.645064	−	0.764128i	$-0.276831\pi$
$997$	40.7362	1.29013	0.645064	+	0.764128i	$0.276831\pi$
$998$	−34.5054	−1.09225
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bf.1.3		8
3.2	odd	2	inner	3969.2.a.bf.1.6		8
7.3	odd	6		567.2.e.g.163.6	yes	16
7.5	odd	6		567.2.e.g.487.6	yes	16
7.6	odd	2		3969.2.a.bg.1.3		8
21.5	even	6		567.2.e.g.487.3	yes	16
21.17	even	6		567.2.e.g.163.3	✓	16
21.20	even	2		3969.2.a.bg.1.6		8
63.5	even	6		567.2.h.l.298.6		16
63.31	odd	6		567.2.g.l.541.6		16
63.38	even	6		567.2.h.l.352.6		16
63.40	odd	6		567.2.h.l.298.3		16
63.47	even	6		567.2.g.l.109.3		16
63.52	odd	6		567.2.h.l.352.3		16
63.59	even	6		567.2.g.l.541.3		16
63.61	odd	6		567.2.g.l.109.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.g.163.3	✓	16	21.17	even	6
567.2.e.g.163.6	yes	16	7.3	odd	6
567.2.e.g.487.3	yes	16	21.5	even	6
567.2.e.g.487.6	yes	16	7.5	odd	6
567.2.g.l.109.3		16	63.47	even	6
567.2.g.l.109.6		16	63.61	odd	6
567.2.g.l.541.3		16	63.59	even	6
567.2.g.l.541.6		16	63.31	odd	6
567.2.h.l.298.3		16	63.40	odd	6
567.2.h.l.298.6		16	63.5	even	6
567.2.h.l.352.3		16	63.52	odd	6
567.2.h.l.352.6		16	63.38	even	6
3969.2.a.bf.1.3		8	1.1	even	1	trivial
3969.2.a.bf.1.6		8	3.2	odd	2	inner
3969.2.a.bg.1.3		8	7.6	odd	2
3969.2.a.bg.1.6		8	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-1.27020 q^{2} -0.386601 q^{4} -1.55350 q^{5} +3.03145 q^{8} +O(q^{10})\)	\(q-1.27020 q^{2} -0.386601 q^{4} -1.55350 q^{5} +3.03145 q^{8} +1.97325 q^{10} -3.21001 q^{11} -4.78669 q^{13} -3.07734 q^{16} +2.11836 q^{17} +4.86403 q^{19} +0.600584 q^{20} +4.07734 q^{22} +3.70758 q^{23} -2.58665 q^{25} +6.08004 q^{26} +7.37944 q^{29} -5.50418 q^{31} -2.15408 q^{32} -2.69074 q^{34} -0.186556 q^{37} -6.17827 q^{38} -4.70935 q^{40} +10.7972 q^{41} +4.86916 q^{43} +1.24099 q^{44} -4.70935 q^{46} -1.77187 q^{47} +3.28555 q^{50} +1.85054 q^{52} +1.66886 q^{53} +4.98674 q^{55} -9.37334 q^{58} -5.82594 q^{59} -6.87729 q^{61} +6.99139 q^{62} +8.89078 q^{64} +7.43611 q^{65} +12.2374 q^{67} -0.818961 q^{68} -13.8101 q^{71} -11.8640 q^{73} +0.236963 q^{74} -1.88044 q^{76} -1.30926 q^{79} +4.78064 q^{80} -13.7146 q^{82} +0.346488 q^{83} -3.29087 q^{85} -6.18478 q^{86} -9.73098 q^{88} +17.4064 q^{89} -1.43335 q^{92} +2.25063 q^{94} -7.55625 q^{95} +10.5683 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4}+O(q^{10}) \) 8 * q + 6 * q^4	\( 8 q + 6 q^{4} - 14 q^{10} - 6 q^{13} + 6 q^{16} - 24 q^{19} + 2 q^{22} - 20 q^{31} - 4 q^{37} - 36 q^{40} + 10 q^{43} - 36 q^{46} - 34 q^{52} - 4 q^{55} - 22 q^{58} - 36 q^{61} + 38 q^{64} - 18 q^{67} - 32 q^{73} - 58 q^{76} - 32 q^{79} + 2 q^{82} + 30 q^{85} - 72 q^{88} - 54 q^{94} - 14 q^{97}+O(q^{100}) \) 8 * q + 6 * q^4 - 14 * q^10 - 6 * q^13 + 6 * q^16 - 24 * q^19 + 2 * q^22 - 20 * q^31 - 4 * q^37 - 36 * q^40 + 10 * q^43 - 36 * q^46 - 34 * q^52 - 4 * q^55 - 22 * q^58 - 36 * q^61 + 38 * q^64 - 18 * q^67 - 32 * q^73 - 58 * q^76 - 32 * q^79 + 2 * q^82 + 30 * q^85 - 72 * q^88 - 54 * q^94 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more