LMFDB - Embedded newform 7728.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.93543 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.93543 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.18953 q^{11} +0.745898 q^{13} -2.93543 q^{15} +0.189534 q^{17} -4.69774 q^{19} +1.00000 q^{21} -1.00000 q^{23} +3.61676 q^{25} +1.00000 q^{27} +3.55220 q^{29} -3.68133 q^{31} +2.18953 q^{33} -2.93543 q^{35} -0.664924 q^{37} +0.745898 q^{39} -7.17313 q^{41} +1.25410 q^{43} -2.93543 q^{45} -5.49180 q^{47} +1.00000 q^{49} +0.189534 q^{51} -1.06457 q^{53} -6.42723 q^{55} -4.69774 q^{57} -7.91903 q^{59} +11.8503 q^{61} +1.00000 q^{63} -2.18953 q^{65} -9.31450 q^{67} -1.00000 q^{69} +16.8339 q^{71} -1.20594 q^{73} +3.61676 q^{75} +2.18953 q^{77} +13.5522 q^{79} +1.00000 q^{81} -2.66492 q^{83} -0.556364 q^{85} +3.55220 q^{87} -16.8667 q^{89} +0.745898 q^{91} -3.68133 q^{93} +13.7899 q^{95} -9.20594 q^{97} +2.18953 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{13} - q^{15} - 8 q^{17} - 4 q^{19} + 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} - 16 q^{41} + 3 q^{43} - q^{45} - 18 q^{47} + 3 q^{49} - 8 q^{51} - 11 q^{53} - 13 q^{55} - 4 q^{57} - 19 q^{59} - 9 q^{61} + 3 q^{63} + 2 q^{65} - 3 q^{67} - 3 q^{69} + 9 q^{71} + 8 q^{73} - 4 q^{75} - 2 q^{77} + 18 q^{79} + 3 q^{81} - 4 q^{83} - 11 q^{85} - 12 q^{87} - 3 q^{89} + 3 q^{91} - 4 q^{93} + 21 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^13 - q^15 - 8 * q^17 - 4 * q^19 + 3 * q^21 - 3 * q^23 - 4 * q^25 + 3 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^33 - q^35 + 2 * q^37 + 3 * q^39 - 16 * q^41 + 3 * q^43 - q^45 - 18 * q^47 + 3 * q^49 - 8 * q^51 - 11 * q^53 - 13 * q^55 - 4 * q^57 - 19 * q^59 - 9 * q^61 + 3 * q^63 + 2 * q^65 - 3 * q^67 - 3 * q^69 + 9 * q^71 + 8 * q^73 - 4 * q^75 - 2 * q^77 + 18 * q^79 + 3 * q^81 - 4 * q^83 - 11 * q^85 - 12 * q^87 - 3 * q^89 + 3 * q^91 - 4 * q^93 + 21 * q^95 - 16 * q^97 - 2 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.93543	−1.31277	−0.656383	−	0.754428i	$-0.727914\pi$
$5$	−2.93543	−1.31277	−0.656383	+	0.754428i	$0.727914\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.18953	0.660169	0.330085	−	0.943951i	$-0.392923\pi$
$11$	2.18953	0.660169	0.330085	+	0.943951i	$0.392923\pi$
$12$	0	0
$13$	0.745898	0.206875	0.103437	−	0.994636i	$-0.467016\pi$
$13$	0.745898	0.206875	0.103437	+	0.994636i	$0.467016\pi$
$14$	0	0
$15$	−2.93543	−0.757925
$16$	0	0
$17$	0.189534	0.0459688	0.0229844	−	0.999736i	$-0.492683\pi$
$17$	0.189534	0.0459688	0.0229844	+	0.999736i	$0.492683\pi$
$18$	0	0
$19$	−4.69774	−1.07773	−0.538867	−	0.842391i	$-0.681148\pi$
$19$	−4.69774	−1.07773	−0.538867	+	0.842391i	$0.681148\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.61676	0.723353
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.55220	0.659626	0.329813	−	0.944046i	$-0.393014\pi$
$29$	3.55220	0.659626	0.329813	+	0.944046i	$0.393014\pi$
$30$	0	0
$31$	−3.68133	−0.661187	−0.330593	−	0.943773i	$-0.607249\pi$
$31$	−3.68133	−0.661187	−0.330593	+	0.943773i	$0.607249\pi$
$32$	0	0
$33$	2.18953	0.381149
$34$	0	0
$35$	−2.93543	−0.496179
$36$	0	0
$37$	−0.664924	−0.109313	−0.0546564	−	0.998505i	$-0.517406\pi$
$37$	−0.664924	−0.109313	−0.0546564	+	0.998505i	$0.517406\pi$
$38$	0	0
$39$	0.745898	0.119439
$40$	0	0
$41$	−7.17313	−1.12025	−0.560127	−	0.828407i	$-0.689248\pi$
$41$	−7.17313	−1.12025	−0.560127	+	0.828407i	$0.689248\pi$
$42$	0	0
$43$	1.25410	0.191249	0.0956244	−	0.995417i	$-0.469515\pi$
$43$	1.25410	0.191249	0.0956244	+	0.995417i	$0.469515\pi$
$44$	0	0
$45$	−2.93543	−0.437588
$46$	0	0
$47$	−5.49180	−0.801061	−0.400530	−	0.916283i	$-0.631174\pi$
$47$	−5.49180	−0.801061	−0.400530	+	0.916283i	$0.631174\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.189534	0.0265401
$52$	0	0
$53$	−1.06457	−0.146230	−0.0731148	−	0.997324i	$-0.523294\pi$
$53$	−1.06457	−0.146230	−0.0731148	+	0.997324i	$0.523294\pi$
$54$	0	0
$55$	−6.42723	−0.866647
$56$	0	0
$57$	−4.69774	−0.622231
$58$	0	0
$59$	−7.91903	−1.03097	−0.515485	−	0.856899i	$-0.672388\pi$
$59$	−7.91903	−1.03097	−0.515485	+	0.856899i	$0.672388\pi$
$60$	0	0
$61$	11.8503	1.51727	0.758637	−	0.651514i	$-0.225866\pi$
$61$	11.8503	1.51727	0.758637	+	0.651514i	$0.225866\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−2.18953	−0.271578
$66$	0	0
$67$	−9.31450	−1.13795	−0.568974	−	0.822356i	$-0.692659\pi$
$67$	−9.31450	−1.13795	−0.568974	+	0.822356i	$0.692659\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	16.8339	1.99781	0.998907	−	0.0467391i	$-0.0148829\pi$
$71$	16.8339	1.99781	0.998907	+	0.0467391i	$0.0148829\pi$
$72$	0	0
$73$	−1.20594	−0.141145	−0.0705723	−	0.997507i	$-0.522483\pi$
$73$	−1.20594	−0.141145	−0.0705723	+	0.997507i	$0.522483\pi$
$74$	0	0
$75$	3.61676	0.417628
$76$	0	0
$77$	2.18953	0.249521
$78$	0	0
$79$	13.5522	1.52474	0.762371	−	0.647141i	$-0.224035\pi$
$79$	13.5522	1.52474	0.762371	+	0.647141i	$0.224035\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.66492	−0.292513	−0.146257	−	0.989247i	$-0.546723\pi$
$83$	−2.66492	−0.292513	−0.146257	+	0.989247i	$0.546723\pi$
$84$	0	0
$85$	−0.556364	−0.0603462
$86$	0	0
$87$	3.55220	0.380835
$88$	0	0
$89$	−16.8667	−1.78787	−0.893933	−	0.448200i	$-0.852065\pi$
$89$	−16.8667	−1.78787	−0.893933	+	0.448200i	$0.852065\pi$
$90$	0	0
$91$	0.745898	0.0781914
$92$	0	0
$93$	−3.68133	−0.381736
$94$	0	0
$95$	13.7899	1.41481
$96$	0	0
$97$	−9.20594	−0.934722	−0.467361	−	0.884067i	$-0.654795\pi$
$97$	−9.20594	−0.934722	−0.467361	+	0.884067i	$0.654795\pi$
$98$	0	0
$99$	2.18953	0.220056
$100$	0	0
$101$	3.78989	0.377108	0.188554	−	0.982063i	$-0.439620\pi$
$101$	3.78989	0.377108	0.188554	+	0.982063i	$0.439620\pi$
$102$	0	0
$103$	−2.03281	−0.200299	−0.100150	−	0.994972i	$-0.531932\pi$
$103$	−2.03281	−0.200299	−0.100150	+	0.994972i	$0.531932\pi$
$104$	0	0
$105$	−2.93543	−0.286469
$106$	0	0
$107$	6.10856	0.590537	0.295268	−	0.955414i	$-0.404591\pi$
$107$	6.10856	0.590537	0.295268	+	0.955414i	$0.404591\pi$
$108$	0	0
$109$	3.31450	0.317472	0.158736	−	0.987321i	$-0.449258\pi$
$109$	3.31450	0.317472	0.158736	+	0.987321i	$0.449258\pi$
$110$	0	0
$111$	−0.664924	−0.0631118
$112$	0	0
$113$	−17.2817	−1.62572	−0.812862	−	0.582456i	$-0.802092\pi$
$113$	−17.2817	−1.62572	−0.812862	+	0.582456i	$0.802092\pi$
$114$	0	0
$115$	2.93543	0.273730
$116$	0	0
$117$	0.745898	0.0689583
$118$	0	0
$119$	0.189534	0.0173746
$120$	0	0
$121$	−6.20594	−0.564176
$122$	0	0
$123$	−7.17313	−0.646779
$124$	0	0
$125$	4.06040	0.363173
$126$	0	0
$127$	−9.47122	−0.840435	−0.420217	−	0.907423i	$-0.638046\pi$
$127$	−9.47122	−0.840435	−0.420217	+	0.907423i	$0.638046\pi$
$128$	0	0
$129$	1.25410	0.110417
$130$	0	0
$131$	−13.2939	−1.16150	−0.580748	−	0.814084i	$-0.697240\pi$
$131$	−13.2939	−1.16150	−0.580748	+	0.814084i	$0.697240\pi$
$132$	0	0
$133$	−4.69774	−0.407345
$134$	0	0
$135$	−2.93543	−0.252642
$136$	0	0
$137$	6.91486	0.590776	0.295388	−	0.955377i	$-0.404551\pi$
$137$	6.91486	0.590776	0.295388	+	0.955377i	$0.404551\pi$
$138$	0	0
$139$	14.6443	1.24212	0.621059	−	0.783764i	$-0.286703\pi$
$139$	14.6443	1.24212	0.621059	+	0.783764i	$0.286703\pi$
$140$	0	0
$141$	−5.49180	−0.462493
$142$	0	0
$143$	1.63317	0.136573
$144$	0	0
$145$	−10.4272	−0.865934
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−16.8873	−1.38346	−0.691730	−	0.722157i	$-0.743151\pi$
$149$	−16.8873	−1.38346	−0.691730	+	0.722157i	$0.743151\pi$
$150$	0	0
$151$	−0.475390	−0.0386867	−0.0193433	−	0.999813i	$-0.506158\pi$
$151$	−0.475390	−0.0386867	−0.0193433	+	0.999813i	$0.506158\pi$
$152$	0	0
$153$	0.189534	0.0153229
$154$	0	0
$155$	10.8063	0.867983
$156$	0	0
$157$	13.5246	1.07938	0.539691	−	0.841863i	$-0.318541\pi$
$157$	13.5246	1.07938	0.539691	+	0.841863i	$0.318541\pi$
$158$	0	0
$159$	−1.06457	−0.0844257
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−0.394415	−0.0308930	−0.0154465	−	0.999881i	$-0.504917\pi$
$163$	−0.394415	−0.0308930	−0.0154465	+	0.999881i	$0.504917\pi$
$164$	0	0
$165$	−6.42723	−0.500359
$166$	0	0
$167$	−16.6566	−1.28893	−0.644463	−	0.764636i	$-0.722919\pi$
$167$	−16.6566	−1.28893	−0.644463	+	0.764636i	$0.722919\pi$
$168$	0	0
$169$	−12.4436	−0.957203
$170$	0	0
$171$	−4.69774	−0.359245
$172$	0	0
$173$	−26.0192	−1.97821	−0.989103	−	0.147223i	$-0.952967\pi$
$173$	−26.0192	−1.97821	−0.989103	+	0.147223i	$0.952967\pi$
$174$	0	0
$175$	3.61676	0.273402
$176$	0	0
$177$	−7.91903	−0.595230
$178$	0	0
$179$	−1.78989	−0.133783	−0.0668913	−	0.997760i	$-0.521308\pi$
$179$	−1.78989	−0.133783	−0.0668913	+	0.997760i	$0.521308\pi$
$180$	0	0
$181$	−0.189534	−0.0140880	−0.00704398	−	0.999975i	$-0.502242\pi$
$181$	−0.189534	−0.0140880	−0.00704398	+	0.999975i	$0.502242\pi$
$182$	0	0
$183$	11.8503	0.875999
$184$	0	0
$185$	1.95184	0.143502
$186$	0	0
$187$	0.414991	0.0303472
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	10.5082	0.760347	0.380173	−	0.924915i	$-0.375864\pi$
$191$	10.5082	0.760347	0.380173	+	0.924915i	$0.375864\pi$
$192$	0	0
$193$	6.54102	0.470833	0.235416	−	0.971895i	$-0.424355\pi$
$193$	6.54102	0.470833	0.235416	+	0.971895i	$0.424355\pi$
$194$	0	0
$195$	−2.18953	−0.156796
$196$	0	0
$197$	−21.3421	−1.52056	−0.760280	−	0.649595i	$-0.774938\pi$
$197$	−21.3421	−1.52056	−0.760280	+	0.649595i	$0.774938\pi$
$198$	0	0
$199$	17.5316	1.24278	0.621392	−	0.783500i	$-0.286568\pi$
$199$	17.5316	1.24278	0.621392	+	0.783500i	$0.286568\pi$
$200$	0	0
$201$	−9.31450	−0.656994
$202$	0	0
$203$	3.55220	0.249315
$204$	0	0
$205$	21.0562	1.47063
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−10.2859	−0.711488
$210$	0	0
$211$	12.0604	0.830272	0.415136	−	0.909759i	$-0.363734\pi$
$211$	12.0604	0.830272	0.415136	+	0.909759i	$0.363734\pi$
$212$	0	0
$213$	16.8339	1.15344
$214$	0	0
$215$	−3.68133	−0.251065
$216$	0	0
$217$	−3.68133	−0.249905
$218$	0	0
$219$	−1.20594	−0.0814899
$220$	0	0
$221$	0.141373	0.00950978
$222$	0	0
$223$	−2.80630	−0.187924	−0.0939618	−	0.995576i	$-0.529953\pi$
$223$	−2.80630	−0.187924	−0.0939618	+	0.995576i	$0.529953\pi$
$224$	0	0
$225$	3.61676	0.241118
$226$	0	0
$227$	17.6279	1.17001	0.585004	−	0.811031i	$-0.301093\pi$
$227$	17.6279	1.17001	0.585004	+	0.811031i	$0.301093\pi$
$228$	0	0
$229$	−12.1690	−0.804147	−0.402074	−	0.915607i	$-0.631710\pi$
$229$	−12.1690	−0.804147	−0.402074	+	0.915607i	$0.631710\pi$
$230$	0	0
$231$	2.18953	0.144061
$232$	0	0
$233$	−5.22652	−0.342400	−0.171200	−	0.985236i	$-0.554765\pi$
$233$	−5.22652	−0.342400	−0.171200	+	0.985236i	$0.554765\pi$
$234$	0	0
$235$	16.1208	1.05160
$236$	0	0
$237$	13.5522	0.880310
$238$	0	0
$239$	−23.7816	−1.53830	−0.769150	−	0.639068i	$-0.779320\pi$
$239$	−23.7816	−1.53830	−0.769150	+	0.639068i	$0.779320\pi$
$240$	0	0
$241$	−2.79406	−0.179981	−0.0899906	−	0.995943i	$-0.528684\pi$
$241$	−2.79406	−0.179981	−0.0899906	+	0.995943i	$0.528684\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.93543	−0.187538
$246$	0	0
$247$	−3.50403	−0.222956
$248$	0	0
$249$	−2.66492	−0.168883
$250$	0	0
$251$	3.83805	0.242256	0.121128	−	0.992637i	$-0.461349\pi$
$251$	3.83805	0.242256	0.121128	+	0.992637i	$0.461349\pi$
$252$	0	0
$253$	−2.18953	−0.137655
$254$	0	0
$255$	−0.556364	−0.0348409
$256$	0	0
$257$	−14.9201	−0.930689	−0.465345	−	0.885130i	$-0.654070\pi$
$257$	−14.9201	−0.930689	−0.465345	+	0.885130i	$0.654070\pi$
$258$	0	0
$259$	−0.664924	−0.0413164
$260$	0	0
$261$	3.55220	0.219875
$262$	0	0
$263$	−2.28586	−0.140952	−0.0704760	−	0.997513i	$-0.522452\pi$
$263$	−2.28586	−0.140952	−0.0704760	+	0.997513i	$0.522452\pi$
$264$	0	0
$265$	3.12497	0.191965
$266$	0	0
$267$	−16.8667	−1.03223
$268$	0	0
$269$	0.108560	0.00661900	0.00330950	−	0.999995i	$-0.498947\pi$
$269$	0.108560	0.00661900	0.00330950	+	0.999995i	$0.498947\pi$
$270$	0	0
$271$	−13.8656	−0.842277	−0.421139	−	0.906996i	$-0.638369\pi$
$271$	−13.8656	−0.842277	−0.421139	+	0.906996i	$0.638369\pi$
$272$	0	0
$273$	0.745898	0.0451438
$274$	0	0
$275$	7.91903	0.477535
$276$	0	0
$277$	25.3421	1.52266	0.761329	−	0.648366i	$-0.224547\pi$
$277$	25.3421	1.52266	0.761329	+	0.648366i	$0.224547\pi$
$278$	0	0
$279$	−3.68133	−0.220396
$280$	0	0
$281$	−13.5246	−0.806811	−0.403405	−	0.915021i	$-0.632174\pi$
$281$	−13.5246	−0.806811	−0.403405	+	0.915021i	$0.632174\pi$
$282$	0	0
$283$	−13.2489	−0.787564	−0.393782	−	0.919204i	$-0.628833\pi$
$283$	−13.2489	−0.787564	−0.393782	+	0.919204i	$0.628833\pi$
$284$	0	0
$285$	13.7899	0.816843
$286$	0	0
$287$	−7.17313	−0.423416
$288$	0	0
$289$	−16.9641	−0.997887
$290$	0	0
$291$	−9.20594	−0.539662
$292$	0	0
$293$	−16.4119	−0.958792	−0.479396	−	0.877599i	$-0.659144\pi$
$293$	−16.4119	−0.958792	−0.479396	+	0.877599i	$0.659144\pi$
$294$	0	0
$295$	23.2458	1.35342
$296$	0	0
$297$	2.18953	0.127050
$298$	0	0
$299$	−0.745898	−0.0431364
$300$	0	0
$301$	1.25410	0.0722852
$302$	0	0
$303$	3.78989	0.217724
$304$	0	0
$305$	−34.7857	−1.99182
$306$	0	0
$307$	−10.3515	−0.590790	−0.295395	−	0.955375i	$-0.595451\pi$
$307$	−10.3515	−0.590790	−0.295395	+	0.955375i	$0.595451\pi$
$308$	0	0
$309$	−2.03281	−0.115643
$310$	0	0
$311$	−11.6660	−0.661517	−0.330759	−	0.943715i	$-0.607305\pi$
$311$	−11.6660	−0.661517	−0.330759	+	0.943715i	$0.607305\pi$
$312$	0	0
$313$	−2.37907	−0.134473	−0.0672364	−	0.997737i	$-0.521418\pi$
$313$	−2.37907	−0.134473	−0.0672364	+	0.997737i	$0.521418\pi$
$314$	0	0
$315$	−2.93543	−0.165393
$316$	0	0
$317$	−15.7899	−0.886849	−0.443424	−	0.896312i	$-0.646236\pi$
$317$	−15.7899	−0.886849	−0.443424	+	0.896312i	$0.646236\pi$
$318$	0	0
$319$	7.77765	0.435465
$320$	0	0
$321$	6.10856	0.340947
$322$	0	0
$323$	−0.890381	−0.0495421
$324$	0	0
$325$	2.69774	0.149644
$326$	0	0
$327$	3.31450	0.183292
$328$	0	0
$329$	−5.49180	−0.302773
$330$	0	0
$331$	12.1208	0.666219	0.333110	−	0.942888i	$-0.391902\pi$
$331$	12.1208	0.666219	0.333110	+	0.942888i	$0.391902\pi$
$332$	0	0
$333$	−0.664924	−0.0364376
$334$	0	0
$335$	27.3421	1.49386
$336$	0	0
$337$	−16.7459	−0.912207	−0.456104	−	0.889927i	$-0.650755\pi$
$337$	−16.7459	−0.912207	−0.456104	+	0.889927i	$0.650755\pi$
$338$	0	0
$339$	−17.2817	−0.938612
$340$	0	0
$341$	−8.06040	−0.436495
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.93543	0.158038
$346$	0	0
$347$	−5.76931	−0.309713	−0.154857	−	0.987937i	$-0.549492\pi$
$347$	−5.76931	−0.309713	−0.154857	+	0.987937i	$0.549492\pi$
$348$	0	0
$349$	−3.57277	−0.191246	−0.0956230	−	0.995418i	$-0.530484\pi$
$349$	−3.57277	−0.191246	−0.0956230	+	0.995418i	$0.530484\pi$
$350$	0	0
$351$	0.745898	0.0398131
$352$	0	0
$353$	−4.18953	−0.222986	−0.111493	−	0.993765i	$-0.535563\pi$
$353$	−4.18953	−0.222986	−0.111493	+	0.993765i	$0.535563\pi$
$354$	0	0
$355$	−49.4147	−2.62266
$356$	0	0
$357$	0.189534	0.0100312
$358$	0	0
$359$	0.814635	0.0429948	0.0214974	−	0.999769i	$-0.493157\pi$
$359$	0.814635	0.0429948	0.0214974	+	0.999769i	$0.493157\pi$
$360$	0	0
$361$	3.06874	0.161512
$362$	0	0
$363$	−6.20594	−0.325727
$364$	0	0
$365$	3.53996	0.185290
$366$	0	0
$367$	6.01224	0.313836	0.156918	−	0.987612i	$-0.449844\pi$
$367$	6.01224	0.313836	0.156918	+	0.987612i	$0.449844\pi$
$368$	0	0
$369$	−7.17313	−0.373418
$370$	0	0
$371$	−1.06457	−0.0552696
$372$	0	0
$373$	0.609754	0.0315718	0.0157859	−	0.999875i	$-0.494975\pi$
$373$	0.609754	0.0315718	0.0157859	+	0.999875i	$0.494975\pi$
$374$	0	0
$375$	4.06040	0.209678
$376$	0	0
$377$	2.64958	0.136460
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−9.47122	−0.485225
$382$	0	0
$383$	−27.8021	−1.42062	−0.710311	−	0.703888i	$-0.751446\pi$
$383$	−27.8021	−1.42062	−0.710311	+	0.703888i	$0.751446\pi$
$384$	0	0
$385$	−6.42723	−0.327562
$386$	0	0
$387$	1.25410	0.0637496
$388$	0	0
$389$	−2.38430	−0.120889	−0.0604443	−	0.998172i	$-0.519252\pi$
$389$	−2.38430	−0.120889	−0.0604443	+	0.998172i	$0.519252\pi$
$390$	0	0
$391$	−0.189534	−0.00958515
$392$	0	0
$393$	−13.2939	−0.670590
$394$	0	0
$395$	−39.7816	−2.00163
$396$	0	0
$397$	12.1291	0.608744	0.304372	−	0.952553i	$-0.401553\pi$
$397$	12.1291	0.608744	0.304372	+	0.952553i	$0.401553\pi$
$398$	0	0
$399$	−4.69774	−0.235181
$400$	0	0
$401$	−12.6649	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$401$	−12.6649	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$402$	0	0
$403$	−2.74590	−0.136783
$404$	0	0
$405$	−2.93543	−0.145863
$406$	0	0
$407$	−1.45587	−0.0721650
$408$	0	0
$409$	−14.0604	−0.695242	−0.347621	−	0.937635i	$-0.613010\pi$
$409$	−14.0604	−0.695242	−0.347621	+	0.937635i	$0.613010\pi$
$410$	0	0
$411$	6.91486	0.341085
$412$	0	0
$413$	−7.91903	−0.389670
$414$	0	0
$415$	7.82270	0.384001
$416$	0	0
$417$	14.6443	0.717137
$418$	0	0
$419$	−32.1606	−1.57115	−0.785575	−	0.618767i	$-0.787632\pi$
$419$	−32.1606	−1.57115	−0.785575	+	0.618767i	$0.787632\pi$
$420$	0	0
$421$	−14.2929	−0.696592	−0.348296	−	0.937385i	$-0.613240\pi$
$421$	−14.2929	−0.696592	−0.348296	+	0.937385i	$0.613240\pi$
$422$	0	0
$423$	−5.49180	−0.267020
$424$	0	0
$425$	0.685500	0.0332516
$426$	0	0
$427$	11.8503	0.573476
$428$	0	0
$429$	1.63317	0.0788502
$430$	0	0
$431$	11.8586	0.571210	0.285605	−	0.958347i	$-0.407805\pi$
$431$	11.8586	0.571210	0.285605	+	0.958347i	$0.407805\pi$
$432$	0	0
$433$	25.0081	1.20181	0.600906	−	0.799320i	$-0.294807\pi$
$433$	25.0081	1.20181	0.600906	+	0.799320i	$0.294807\pi$
$434$	0	0
$435$	−10.4272	−0.499947
$436$	0	0
$437$	4.69774	0.224723
$438$	0	0
$439$	6.78572	0.323865	0.161932	−	0.986802i	$-0.448227\pi$
$439$	6.78572	0.323865	0.161932	+	0.986802i	$0.448227\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	5.70892	0.271239	0.135619	−	0.990761i	$-0.456698\pi$
$443$	5.70892	0.271239	0.135619	+	0.990761i	$0.456698\pi$
$444$	0	0
$445$	49.5110	2.34705
$446$	0	0
$447$	−16.8873	−0.798741
$448$	0	0
$449$	−5.06457	−0.239012	−0.119506	−	0.992833i	$-0.538131\pi$
$449$	−5.06457	−0.239012	−0.119506	+	0.992833i	$0.538131\pi$
$450$	0	0
$451$	−15.7058	−0.739558
$452$	0	0
$453$	−0.475390	−0.0223358
$454$	0	0
$455$	−2.18953	−0.102647
$456$	0	0
$457$	−31.0562	−1.45275	−0.726375	−	0.687299i	$-0.758796\pi$
$457$	−31.0562	−1.45275	−0.726375	+	0.687299i	$0.758796\pi$
$458$	0	0
$459$	0.189534	0.00884669
$460$	0	0
$461$	4.54803	0.211823	0.105911	−	0.994376i	$-0.466224\pi$
$461$	4.54803	0.211823	0.105911	+	0.994376i	$0.466224\pi$
$462$	0	0
$463$	−38.6238	−1.79500	−0.897499	−	0.441016i	$-0.854618\pi$
$463$	−38.6238	−1.79500	−0.897499	+	0.441016i	$0.854618\pi$
$464$	0	0
$465$	10.8063	0.501130
$466$	0	0
$467$	38.8409	1.79734	0.898671	−	0.438623i	$-0.144534\pi$
$467$	38.8409	1.79734	0.898671	+	0.438623i	$0.144534\pi$
$468$	0	0
$469$	−9.31450	−0.430104
$470$	0	0
$471$	13.5246	0.623181
$472$	0	0
$473$	2.74590	0.126257
$474$	0	0
$475$	−16.9906	−0.779582
$476$	0	0
$477$	−1.06457	−0.0487432
$478$	0	0
$479$	−33.2939	−1.52124	−0.760619	−	0.649198i	$-0.775104\pi$
$479$	−33.2939	−1.52124	−0.760619	+	0.649198i	$0.775104\pi$
$480$	0	0
$481$	−0.495966	−0.0226141
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	27.0234	1.22707
$486$	0	0
$487$	−12.6014	−0.571025	−0.285512	−	0.958375i	$-0.592164\pi$
$487$	−12.6014	−0.571025	−0.285512	+	0.958375i	$0.592164\pi$
$488$	0	0
$489$	−0.394415	−0.0178361
$490$	0	0
$491$	−13.5124	−0.609805	−0.304902	−	0.952384i	$-0.598624\pi$
$491$	−13.5124	−0.609805	−0.304902	+	0.952384i	$0.598624\pi$
$492$	0	0
$493$	0.673262	0.0303222
$494$	0	0
$495$	−6.42723	−0.288882
$496$	0	0
$497$	16.8339	0.755103
$498$	0	0
$499$	6.64435	0.297442	0.148721	−	0.988879i	$-0.452484\pi$
$499$	6.64435	0.297442	0.148721	+	0.988879i	$0.452484\pi$
$500$	0	0
$501$	−16.6566	−0.744161
$502$	0	0
$503$	26.9682	1.20245	0.601227	−	0.799078i	$-0.294679\pi$
$503$	26.9682	1.20245	0.601227	+	0.799078i	$0.294679\pi$
$504$	0	0
$505$	−11.1250	−0.495055
$506$	0	0
$507$	−12.4436	−0.552641
$508$	0	0
$509$	6.59619	0.292371	0.146185	−	0.989257i	$-0.453300\pi$
$509$	6.59619	0.292371	0.146185	+	0.989257i	$0.453300\pi$
$510$	0	0
$511$	−1.20594	−0.0533477
$512$	0	0
$513$	−4.69774	−0.207410
$514$	0	0
$515$	5.96719	0.262946
$516$	0	0
$517$	−12.0245	−0.528836
$518$	0	0
$519$	−26.0192	−1.14212
$520$	0	0
$521$	−14.9508	−0.655006	−0.327503	−	0.944850i	$-0.606207\pi$
$521$	−14.9508	−0.655006	−0.327503	+	0.944850i	$0.606207\pi$
$522$	0	0
$523$	−3.36266	−0.147039	−0.0735195	−	0.997294i	$-0.523423\pi$
$523$	−3.36266	−0.147039	−0.0735195	+	0.997294i	$0.523423\pi$
$524$	0	0
$525$	3.61676	0.157848
$526$	0	0
$527$	−0.697737	−0.0303939
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−7.91903	−0.343656
$532$	0	0
$533$	−5.35042	−0.231753
$534$	0	0
$535$	−17.9313	−0.775236
$536$	0	0
$537$	−1.78989	−0.0772395
$538$	0	0
$539$	2.18953	0.0943099
$540$	0	0
$541$	−6.42022	−0.276027	−0.138013	−	0.990430i	$-0.544072\pi$
$541$	−6.42022	−0.276027	−0.138013	+	0.990430i	$0.544072\pi$
$542$	0	0
$543$	−0.189534	−0.00813368
$544$	0	0
$545$	−9.72949	−0.416766
$546$	0	0
$547$	35.1442	1.50266	0.751329	−	0.659928i	$-0.229413\pi$
$547$	35.1442	1.50266	0.751329	+	0.659928i	$0.229413\pi$
$548$	0	0
$549$	11.8503	0.505758
$550$	0	0
$551$	−16.6873	−0.710902
$552$	0	0
$553$	13.5522	0.576298
$554$	0	0
$555$	1.95184	0.0828510
$556$	0	0
$557$	−19.6678	−0.833350	−0.416675	−	0.909056i	$-0.636805\pi$
$557$	−19.6678	−0.833350	−0.416675	+	0.909056i	$0.636805\pi$
$558$	0	0
$559$	0.935432	0.0395646
$560$	0	0
$561$	0.414991	0.0175209
$562$	0	0
$563$	14.2294	0.599696	0.299848	−	0.953987i	$-0.403064\pi$
$563$	14.2294	0.599696	0.299848	+	0.953987i	$0.403064\pi$
$564$	0	0
$565$	50.7292	2.13419
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−32.3463	−1.35603	−0.678013	−	0.735050i	$-0.737159\pi$
$569$	−32.3463	−1.35603	−0.678013	+	0.735050i	$0.737159\pi$
$570$	0	0
$571$	−14.2447	−0.596122	−0.298061	−	0.954547i	$-0.596340\pi$
$571$	−14.2447	−0.596122	−0.298061	+	0.954547i	$0.596340\pi$
$572$	0	0
$573$	10.5082	0.438986
$574$	0	0
$575$	−3.61676	−0.150829
$576$	0	0
$577$	30.4999	1.26973	0.634863	−	0.772625i	$-0.281057\pi$
$577$	30.4999	1.26973	0.634863	+	0.772625i	$0.281057\pi$
$578$	0	0
$579$	6.54102	0.271835
$580$	0	0
$581$	−2.66492	−0.110560
$582$	0	0
$583$	−2.33091	−0.0965363
$584$	0	0
$585$	−2.18953	−0.0905261
$586$	0	0
$587$	−16.9302	−0.698784	−0.349392	−	0.936977i	$-0.613612\pi$
$587$	−16.9302	−0.698784	−0.349392	+	0.936977i	$0.613612\pi$
$588$	0	0
$589$	17.2939	0.712584
$590$	0	0
$591$	−21.3421	−0.877896
$592$	0	0
$593$	1.52461	0.0626082	0.0313041	−	0.999510i	$-0.490034\pi$
$593$	1.52461	0.0626082	0.0313041	+	0.999510i	$0.490034\pi$
$594$	0	0
$595$	−0.556364	−0.0228087
$596$	0	0
$597$	17.5316	0.717521
$598$	0	0
$599$	41.7128	1.70434	0.852170	−	0.523265i	$-0.175286\pi$
$599$	41.7128	1.70434	0.852170	+	0.523265i	$0.175286\pi$
$600$	0	0
$601$	−11.7899	−0.480920	−0.240460	−	0.970659i	$-0.577298\pi$
$601$	−11.7899	−0.480920	−0.240460	+	0.970659i	$0.577298\pi$
$602$	0	0
$603$	−9.31450	−0.379316
$604$	0	0
$605$	18.2171	0.740631
$606$	0	0
$607$	11.9383	0.484560	0.242280	−	0.970206i	$-0.422105\pi$
$607$	11.9383	0.484560	0.242280	+	0.970206i	$0.422105\pi$
$608$	0	0
$609$	3.55220	0.143942
$610$	0	0
$611$	−4.09632	−0.165719
$612$	0	0
$613$	18.1567	0.733343	0.366672	−	0.930350i	$-0.380497\pi$
$613$	18.1567	0.733343	0.366672	+	0.930350i	$0.380497\pi$
$614$	0	0
$615$	21.0562	0.849069
$616$	0	0
$617$	34.4381	1.38643	0.693214	−	0.720732i	$-0.256194\pi$
$617$	34.4381	1.38643	0.693214	+	0.720732i	$0.256194\pi$
$618$	0	0
$619$	33.1718	1.33329	0.666644	−	0.745377i	$-0.267730\pi$
$619$	33.1718	1.33329	0.666644	+	0.745377i	$0.267730\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−16.8667	−0.675750
$624$	0	0
$625$	−30.0028	−1.20011
$626$	0	0
$627$	−10.2859	−0.410778
$628$	0	0
$629$	−0.126026	−0.00502497
$630$	0	0
$631$	8.37384	0.333357	0.166679	−	0.986011i	$-0.446696\pi$
$631$	8.37384	0.333357	0.166679	+	0.986011i	$0.446696\pi$
$632$	0	0
$633$	12.0604	0.479358
$634$	0	0
$635$	27.8021	1.10329
$636$	0	0
$637$	0.745898	0.0295536
$638$	0	0
$639$	16.8339	0.665938
$640$	0	0
$641$	−2.10022	−0.0829538	−0.0414769	−	0.999139i	$-0.513206\pi$
$641$	−2.10022	−0.0829538	−0.0414769	+	0.999139i	$0.513206\pi$
$642$	0	0
$643$	23.9794	0.945656	0.472828	−	0.881155i	$-0.343233\pi$
$643$	23.9794	0.945656	0.472828	+	0.881155i	$0.343233\pi$
$644$	0	0
$645$	−3.68133	−0.144952
$646$	0	0
$647$	2.92709	0.115076	0.0575380	−	0.998343i	$-0.481675\pi$
$647$	2.92709	0.115076	0.0575380	+	0.998343i	$0.481675\pi$
$648$	0	0
$649$	−17.3390	−0.680614
$650$	0	0
$651$	−3.68133	−0.144283
$652$	0	0
$653$	32.8063	1.28381	0.641905	−	0.766784i	$-0.278144\pi$
$653$	32.8063	1.28381	0.641905	+	0.766784i	$0.278144\pi$
$654$	0	0
$655$	39.0234	1.52477
$656$	0	0
$657$	−1.20594	−0.0470482
$658$	0	0
$659$	7.64018	0.297619	0.148810	−	0.988866i	$-0.452456\pi$
$659$	7.64018	0.297619	0.148810	+	0.988866i	$0.452456\pi$
$660$	0	0
$661$	7.05233	0.274304	0.137152	−	0.990550i	$-0.456205\pi$
$661$	7.05233	0.274304	0.137152	+	0.990550i	$0.456205\pi$
$662$	0	0
$663$	0.141373	0.00549048
$664$	0	0
$665$	13.7899	0.534749
$666$	0	0
$667$	−3.55220	−0.137542
$668$	0	0
$669$	−2.80630	−0.108498
$670$	0	0
$671$	25.9466	1.00166
$672$	0	0
$673$	−12.5030	−0.481954	−0.240977	−	0.970531i	$-0.577468\pi$
$673$	−12.5030	−0.481954	−0.240977	+	0.970531i	$0.577468\pi$
$674$	0	0
$675$	3.61676	0.139209
$676$	0	0
$677$	−8.44648	−0.324624	−0.162312	−	0.986739i	$-0.551895\pi$
$677$	−8.44648	−0.324624	−0.162312	+	0.986739i	$0.551895\pi$
$678$	0	0
$679$	−9.20594	−0.353292
$680$	0	0
$681$	17.6279	0.675504
$682$	0	0
$683$	37.2528	1.42544	0.712719	−	0.701450i	$-0.247464\pi$
$683$	37.2528	1.42544	0.712719	+	0.701450i	$0.247464\pi$
$684$	0	0
$685$	−20.2981	−0.775550
$686$	0	0
$687$	−12.1690	−0.464275
$688$	0	0
$689$	−0.794059	−0.0302512
$690$	0	0
$691$	−13.4712	−0.512469	−0.256235	−	0.966615i	$-0.582482\pi$
$691$	−13.4712	−0.512469	−0.256235	+	0.966615i	$0.582482\pi$
$692$	0	0
$693$	2.18953	0.0831735
$694$	0	0
$695$	−42.9875	−1.63061
$696$	0	0
$697$	−1.35955	−0.0514967
$698$	0	0
$699$	−5.22652	−0.197685
$700$	0	0
$701$	27.2733	1.03010	0.515050	−	0.857160i	$-0.327773\pi$
$701$	27.2733	1.03010	0.515050	+	0.857160i	$0.327773\pi$
$702$	0	0
$703$	3.12364	0.117810
$704$	0	0
$705$	16.1208	0.607144
$706$	0	0
$707$	3.78989	0.142533
$708$	0	0
$709$	41.4629	1.55717	0.778586	−	0.627538i	$-0.215937\pi$
$709$	41.4629	1.55717	0.778586	+	0.627538i	$0.215937\pi$
$710$	0	0
$711$	13.5522	0.508247
$712$	0	0
$713$	3.68133	0.137867
$714$	0	0
$715$	−4.79406	−0.179288
$716$	0	0
$717$	−23.7816	−0.888138
$718$	0	0
$719$	−48.9393	−1.82513	−0.912565	−	0.408933i	$-0.865901\pi$
$719$	−48.9393	−1.82513	−0.912565	+	0.408933i	$0.865901\pi$
$720$	0	0
$721$	−2.03281	−0.0757059
$722$	0	0
$723$	−2.79406	−0.103912
$724$	0	0
$725$	12.8474	0.477142
$726$	0	0
$727$	12.4395	0.461354	0.230677	−	0.973030i	$-0.425906\pi$
$727$	12.4395	0.461354	0.230677	+	0.973030i	$0.425906\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.237695	0.00879146
$732$	0	0
$733$	−8.68417	−0.320757	−0.160379	−	0.987056i	$-0.551272\pi$
$733$	−8.68417	−0.320757	−0.160379	+	0.987056i	$0.551272\pi$
$734$	0	0
$735$	−2.93543	−0.108275
$736$	0	0
$737$	−20.3944	−0.751238
$738$	0	0
$739$	−28.6154	−1.05264	−0.526318	−	0.850288i	$-0.676428\pi$
$739$	−28.6154	−1.05264	−0.526318	+	0.850288i	$0.676428\pi$
$740$	0	0
$741$	−3.50403	−0.128724
$742$	0	0
$743$	−12.6360	−0.463570	−0.231785	−	0.972767i	$-0.574457\pi$
$743$	−12.6360	−0.463570	−0.231785	+	0.972767i	$0.574457\pi$
$744$	0	0
$745$	49.5714	1.81616
$746$	0	0
$747$	−2.66492	−0.0975045
$748$	0	0
$749$	6.10856	0.223202
$750$	0	0
$751$	9.25933	0.337878	0.168939	−	0.985627i	$-0.445966\pi$
$751$	9.25933	0.337878	0.168939	+	0.985627i	$0.445966\pi$
$752$	0	0
$753$	3.83805	0.139866
$754$	0	0
$755$	1.39547	0.0507865
$756$	0	0
$757$	10.1979	0.370648	0.185324	−	0.982677i	$-0.440667\pi$
$757$	10.1979	0.370648	0.185324	+	0.982677i	$0.440667\pi$
$758$	0	0
$759$	−2.18953	−0.0794750
$760$	0	0
$761$	12.6702	0.459293	0.229646	−	0.973274i	$-0.426243\pi$
$761$	12.6702	0.459293	0.229646	+	0.973274i	$0.426243\pi$
$762$	0	0
$763$	3.31450	0.119993
$764$	0	0
$765$	−0.556364	−0.0201154
$766$	0	0
$767$	−5.90679	−0.213282
$768$	0	0
$769$	17.2007	0.620274	0.310137	−	0.950692i	$-0.399625\pi$
$769$	17.2007	0.620274	0.310137	+	0.950692i	$0.399625\pi$
$770$	0	0
$771$	−14.9201	−0.537334
$772$	0	0
$773$	3.90679	0.140517	0.0702587	−	0.997529i	$-0.477618\pi$
$773$	3.90679	0.140517	0.0702587	+	0.997529i	$0.477618\pi$
$774$	0	0
$775$	−13.3145	−0.478271
$776$	0	0
$777$	−0.664924	−0.0238540
$778$	0	0
$779$	33.6975	1.20734
$780$	0	0
$781$	36.8584	1.31890
$782$	0	0
$783$	3.55220	0.126945
$784$	0	0
$785$	−39.7006	−1.41697
$786$	0	0
$787$	−42.0622	−1.49935	−0.749677	−	0.661803i	$-0.769791\pi$
$787$	−42.0622	−1.49935	−0.749677	+	0.661803i	$0.769791\pi$
$788$	0	0
$789$	−2.28586	−0.0813786
$790$	0	0
$791$	−17.2817	−0.614466
$792$	0	0
$793$	8.83911	0.313886
$794$	0	0
$795$	3.12497	0.110831
$796$	0	0
$797$	−49.6454	−1.75853	−0.879265	−	0.476332i	$-0.841966\pi$
$797$	−49.6454	−1.75853	−0.879265	+	0.476332i	$0.841966\pi$
$798$	0	0
$799$	−1.04088	−0.0368238
$800$	0	0
$801$	−16.8667	−0.595955
$802$	0	0
$803$	−2.64045	−0.0931794
$804$	0	0
$805$	2.93543	0.103460
$806$	0	0
$807$	0.108560	0.00382148
$808$	0	0
$809$	38.2210	1.34378	0.671890	−	0.740651i	$-0.265483\pi$
$809$	38.2210	1.34378	0.671890	+	0.740651i	$0.265483\pi$
$810$	0	0
$811$	26.7061	0.937777	0.468889	−	0.883257i	$-0.344655\pi$
$811$	26.7061	0.937777	0.468889	+	0.883257i	$0.344655\pi$
$812$	0	0
$813$	−13.8656	−0.486289
$814$	0	0
$815$	1.15778	0.0405553
$816$	0	0
$817$	−5.89144	−0.206115
$818$	0	0
$819$	0.745898	0.0260638
$820$	0	0
$821$	−28.3051	−0.987855	−0.493927	−	0.869503i	$-0.664439\pi$
$821$	−28.3051	−0.987855	−0.493927	+	0.869503i	$0.664439\pi$
$822$	0	0
$823$	27.0479	0.942830	0.471415	−	0.881911i	$-0.343743\pi$
$823$	27.0479	0.942830	0.471415	+	0.881911i	$0.343743\pi$
$824$	0	0
$825$	7.91903	0.275705
$826$	0	0
$827$	37.1770	1.29277	0.646386	−	0.763011i	$-0.276280\pi$
$827$	37.1770	1.29277	0.646386	+	0.763011i	$0.276280\pi$
$828$	0	0
$829$	−27.3736	−0.950723	−0.475362	−	0.879790i	$-0.657683\pi$
$829$	−27.3736	−0.950723	−0.475362	+	0.879790i	$0.657683\pi$
$830$	0	0
$831$	25.3421	0.879107
$832$	0	0
$833$	0.189534	0.00656696
$834$	0	0
$835$	48.8943	1.69206
$836$	0	0
$837$	−3.68133	−0.127245
$838$	0	0
$839$	43.8175	1.51275	0.756374	−	0.654140i	$-0.226969\pi$
$839$	43.8175	1.51275	0.756374	+	0.654140i	$0.226969\pi$
$840$	0	0
$841$	−16.3819	−0.564893
$842$	0	0
$843$	−13.5246	−0.465812
$844$	0	0
$845$	36.5275	1.25658
$846$	0	0
$847$	−6.20594	−0.213239
$848$	0	0
$849$	−13.2489	−0.454700
$850$	0	0
$851$	0.664924	0.0227933
$852$	0	0
$853$	−5.13509	−0.175822	−0.0879110	−	0.996128i	$-0.528019\pi$
$853$	−5.13509	−0.175822	−0.0879110	+	0.996128i	$0.528019\pi$
$854$	0	0
$855$	13.7899	0.471604
$856$	0	0
$857$	−0.832101	−0.0284240	−0.0142120	−	0.999899i	$-0.504524\pi$
$857$	−0.832101	−0.0284240	−0.0142120	+	0.999899i	$0.504524\pi$
$858$	0	0
$859$	−33.3298	−1.13720	−0.568600	−	0.822614i	$-0.692515\pi$
$859$	−33.3298	−1.13720	−0.568600	+	0.822614i	$0.692515\pi$
$860$	0	0
$861$	−7.17313	−0.244460
$862$	0	0
$863$	−13.8052	−0.469936	−0.234968	−	0.972003i	$-0.575498\pi$
$863$	−13.8052	−0.469936	−0.234968	+	0.972003i	$0.575498\pi$
$864$	0	0
$865$	76.3777	2.59692
$866$	0	0
$867$	−16.9641	−0.576130
$868$	0	0
$869$	29.6730	1.00659
$870$	0	0
$871$	−6.94767	−0.235413
$872$	0	0
$873$	−9.20594	−0.311574
$874$	0	0
$875$	4.06040	0.137267
$876$	0	0
$877$	26.6207	0.898916	0.449458	−	0.893301i	$-0.351617\pi$
$877$	26.6207	0.898916	0.449458	+	0.893301i	$0.351617\pi$
$878$	0	0
$879$	−16.4119	−0.553559
$880$	0	0
$881$	−30.2827	−1.02025	−0.510126	−	0.860100i	$-0.670401\pi$
$881$	−30.2827	−1.02025	−0.510126	+	0.860100i	$0.670401\pi$
$882$	0	0
$883$	−33.1823	−1.11667	−0.558336	−	0.829615i	$-0.688560\pi$
$883$	−33.1823	−1.11667	−0.558336	+	0.829615i	$0.688560\pi$
$884$	0	0
$885$	23.2458	0.781398
$886$	0	0
$887$	−25.4405	−0.854209	−0.427105	−	0.904202i	$-0.640466\pi$
$887$	−25.4405	−0.854209	−0.427105	+	0.904202i	$0.640466\pi$
$888$	0	0
$889$	−9.47122	−0.317655
$890$	0	0
$891$	2.18953	0.0733521
$892$	0	0
$893$	25.7990	0.863331
$894$	0	0
$895$	5.25410	0.175625
$896$	0	0
$897$	−0.745898	−0.0249048
$898$	0	0
$899$	−13.0768	−0.436136
$900$	0	0
$901$	−0.201772	−0.00672199
$902$	0	0
$903$	1.25410	0.0417339
$904$	0	0
$905$	0.556364	0.0184942
$906$	0	0
$907$	2.90262	0.0963799	0.0481899	−	0.998838i	$-0.484655\pi$
$907$	2.90262	0.0963799	0.0481899	+	0.998838i	$0.484655\pi$
$908$	0	0
$909$	3.78989	0.125703
$910$	0	0
$911$	35.5079	1.17643	0.588215	−	0.808704i	$-0.299831\pi$
$911$	35.5079	1.17643	0.588215	+	0.808704i	$0.299831\pi$
$912$	0	0
$913$	−5.83494	−0.193108
$914$	0	0
$915$	−34.7857	−1.14998
$916$	0	0
$917$	−13.2939	−0.439004
$918$	0	0
$919$	3.68133	0.121436	0.0607179	−	0.998155i	$-0.480661\pi$
$919$	3.68133	0.121436	0.0607179	+	0.998155i	$0.480661\pi$
$920$	0	0
$921$	−10.3515	−0.341093
$922$	0	0
$923$	12.5564	0.413298
$924$	0	0
$925$	−2.40487	−0.0790717
$926$	0	0
$927$	−2.03281	−0.0667664
$928$	0	0
$929$	53.0890	1.74180	0.870898	−	0.491465i	$-0.163538\pi$
$929$	53.0890	1.74180	0.870898	+	0.491465i	$0.163538\pi$
$930$	0	0
$931$	−4.69774	−0.153962
$932$	0	0
$933$	−11.6660	−0.381927
$934$	0	0
$935$	−1.21818	−0.0398387
$936$	0	0
$937$	−12.4618	−0.407110	−0.203555	−	0.979064i	$-0.565250\pi$
$937$	−12.4618	−0.407110	−0.203555	+	0.979064i	$0.565250\pi$
$938$	0	0
$939$	−2.37907	−0.0776380
$940$	0	0
$941$	−12.1536	−0.396196	−0.198098	−	0.980182i	$-0.563476\pi$
$941$	−12.1536	−0.396196	−0.198098	+	0.980182i	$0.563476\pi$
$942$	0	0
$943$	7.17313	0.233589
$944$	0	0
$945$	−2.93543	−0.0954896
$946$	0	0
$947$	−26.8461	−0.872382	−0.436191	−	0.899854i	$-0.643673\pi$
$947$	−26.8461	−0.872382	−0.436191	+	0.899854i	$0.643673\pi$
$948$	0	0
$949$	−0.899509	−0.0291993
$950$	0	0
$951$	−15.7899	−0.512022
$952$	0	0
$953$	24.4876	0.793232	0.396616	−	0.917985i	$-0.370184\pi$
$953$	24.4876	0.793232	0.396616	+	0.917985i	$0.370184\pi$
$954$	0	0
$955$	−30.8461	−0.998157
$956$	0	0
$957$	7.77765	0.251416
$958$	0	0
$959$	6.91486	0.223292
$960$	0	0
$961$	−17.4478	−0.562832
$962$	0	0
$963$	6.10856	0.196846
$964$	0	0
$965$	−19.2007	−0.618093
$966$	0	0
$967$	−2.34625	−0.0754505	−0.0377252	−	0.999288i	$-0.512011\pi$
$967$	−2.34625	−0.0754505	−0.0377252	+	0.999288i	$0.512011\pi$
$968$	0	0
$969$	−0.890381	−0.0286032
$970$	0	0
$971$	15.2213	0.488474	0.244237	−	0.969716i	$-0.421462\pi$
$971$	15.2213	0.488474	0.244237	+	0.969716i	$0.421462\pi$
$972$	0	0
$973$	14.6443	0.469476
$974$	0	0
$975$	2.69774	0.0863967
$976$	0	0
$977$	21.0593	0.673748	0.336874	−	0.941550i	$-0.390630\pi$
$977$	21.0593	0.673748	0.336874	+	0.941550i	$0.390630\pi$
$978$	0	0
$979$	−36.9302	−1.18029
$980$	0	0
$981$	3.31450	0.105824
$982$	0	0
$983$	12.4395	0.396757	0.198379	−	0.980125i	$-0.436432\pi$
$983$	12.4395	0.396757	0.198379	+	0.980125i	$0.436432\pi$
$984$	0	0
$985$	62.6482	1.99614
$986$	0	0
$987$	−5.49180	−0.174806
$988$	0	0
$989$	−1.25410	−0.0398781
$990$	0	0
$991$	45.2130	1.43624	0.718118	−	0.695921i	$-0.245004\pi$
$991$	45.2130	1.43624	0.718118	+	0.695921i	$0.245004\pi$
$992$	0	0
$993$	12.1208	0.384642
$994$	0	0
$995$	−51.4629	−1.63148
$996$	0	0
$997$	51.0768	1.61762	0.808809	−	0.588071i	$-0.200112\pi$
$997$	51.0768	1.61762	0.808809	+	0.588071i	$0.200112\pi$
$998$	0	0
$999$	−0.664924	−0.0210373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bw.1.1		3
4.3	odd	2		3864.2.a.l.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.l.1.1	✓	3	4.3	odd	2
7728.2.a.bw.1.1		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.93543 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.93543 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.18953 q^{11} +0.745898 q^{13} -2.93543 q^{15} +0.189534 q^{17} -4.69774 q^{19} +1.00000 q^{21} -1.00000 q^{23} +3.61676 q^{25} +1.00000 q^{27} +3.55220 q^{29} -3.68133 q^{31} +2.18953 q^{33} -2.93543 q^{35} -0.664924 q^{37} +0.745898 q^{39} -7.17313 q^{41} +1.25410 q^{43} -2.93543 q^{45} -5.49180 q^{47} +1.00000 q^{49} +0.189534 q^{51} -1.06457 q^{53} -6.42723 q^{55} -4.69774 q^{57} -7.91903 q^{59} +11.8503 q^{61} +1.00000 q^{63} -2.18953 q^{65} -9.31450 q^{67} -1.00000 q^{69} +16.8339 q^{71} -1.20594 q^{73} +3.61676 q^{75} +2.18953 q^{77} +13.5522 q^{79} +1.00000 q^{81} -2.66492 q^{83} -0.556364 q^{85} +3.55220 q^{87} -16.8667 q^{89} +0.745898 q^{91} -3.68133 q^{93} +13.7899 q^{95} -9.20594 q^{97} +2.18953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{13} - q^{15} - 8 q^{17} - 4 q^{19} + 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} - 16 q^{41} + 3 q^{43} - q^{45} - 18 q^{47} + 3 q^{49} - 8 q^{51} - 11 q^{53} - 13 q^{55} - 4 q^{57} - 19 q^{59} - 9 q^{61} + 3 q^{63} + 2 q^{65} - 3 q^{67} - 3 q^{69} + 9 q^{71} + 8 q^{73} - 4 q^{75} - 2 q^{77} + 18 q^{79} + 3 q^{81} - 4 q^{83} - 11 q^{85} - 12 q^{87} - 3 q^{89} + 3 q^{91} - 4 q^{93} + 21 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9 - 2 * q^11 + 3 * q^13 - q^15 - 8 * q^17 - 4 * q^19 + 3 * q^21 - 3 * q^23 - 4 * q^25 + 3 * q^27 - 12 * q^29 - 4 * q^31 - 2 * q^33 - q^35 + 2 * q^37 + 3 * q^39 - 16 * q^41 + 3 * q^43 - q^45 - 18 * q^47 + 3 * q^49 - 8 * q^51 - 11 * q^53 - 13 * q^55 - 4 * q^57 - 19 * q^59 - 9 * q^61 + 3 * q^63 + 2 * q^65 - 3 * q^67 - 3 * q^69 + 9 * q^71 + 8 * q^73 - 4 * q^75 - 2 * q^77 + 18 * q^79 + 3 * q^81 - 4 * q^83 - 11 * q^85 - 12 * q^87 - 3 * q^89 + 3 * q^91 - 4 * q^93 + 21 * q^95 - 16 * q^97 - 2 * q^99

Introduction

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