LMFDB - Embedded newform 4928.2.a.cj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4928,2,Mod(1,4928)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4928, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4928.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4928 = 2^{6} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4928.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:	$39.3502781161$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2464)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.31743$ of defining polynomial
Character	$\chi$	$=$	4928.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.26438 q^{3} -0.401352 q^{5} -1.00000 q^{7} -1.40135 q^{9} +O(q^{10})$	$q+1.26438 q^{3} -0.401352 q^{5} -1.00000 q^{7} -1.40135 q^{9} -1.00000 q^{11} +6.30059 q^{13} -0.507460 q^{15} +6.93545 q^{17} -7.96632 q^{19} -1.26438 q^{21} +4.76226 q^{23} -4.83892 q^{25} -5.56497 q^{27} -0.528753 q^{29} -3.37048 q^{31} -1.26438 q^{33} +0.401352 q^{35} +9.45886 q^{37} +7.96632 q^{39} -3.13698 q^{41} +9.23604 q^{43} +0.562435 q^{45} +6.19448 q^{47} +1.00000 q^{49} +8.76902 q^{51} -4.49507 q^{53} +0.401352 q^{55} -10.0724 q^{57} +11.3368 q^{59} +12.8293 q^{61} +1.40135 q^{63} -2.52875 q^{65} -5.56497 q^{67} +6.02129 q^{69} +0.507460 q^{71} -7.73815 q^{73} -6.11821 q^{75} +1.00000 q^{77} -1.63909 q^{79} -2.83216 q^{81} +7.96632 q^{83} -2.78356 q^{85} -0.668543 q^{87} -4.02129 q^{89} -6.30059 q^{91} -4.26156 q^{93} +3.19730 q^{95} +7.35275 q^{97} +1.40135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 5 q^{5} - 4 q^{7} + q^{9}+O(q^{10}) $ 4 * q + q^3 + 5 * q^5 - 4 * q^7 + q^9	$ 4 q + q^{3} + 5 q^{5} - 4 q^{7} + q^{9} - 4 q^{11} + 6 q^{13} + 5 q^{15} - 2 q^{19} - q^{21} + 9 q^{23} + 9 q^{25} + q^{27} + 6 q^{29} - 9 q^{31} - q^{33} - 5 q^{35} + 15 q^{37} + 2 q^{39} - 10 q^{41} - 10 q^{43} + 24 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 20 q^{53} - 5 q^{55} - 10 q^{57} + 11 q^{59} + 24 q^{61} - q^{63} - 2 q^{65} + q^{67} + 23 q^{69} - 5 q^{71} + 10 q^{73} + 22 q^{75} + 4 q^{77} - 10 q^{79} - 16 q^{81} + 2 q^{83} - 24 q^{87} - 15 q^{89} - 6 q^{91} + 19 q^{93} + 26 q^{95} + 7 q^{97} - q^{99}+O(q^{100}) $ 4 * q + q^3 + 5 * q^5 - 4 * q^7 + q^9 - 4 * q^11 + 6 * q^13 + 5 * q^15 - 2 * q^19 - q^21 + 9 * q^23 + 9 * q^25 + q^27 + 6 * q^29 - 9 * q^31 - q^33 - 5 * q^35 + 15 * q^37 + 2 * q^39 - 10 * q^41 - 10 * q^43 + 24 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 20 * q^53 - 5 * q^55 - 10 * q^57 + 11 * q^59 + 24 * q^61 - q^63 - 2 * q^65 + q^67 + 23 * q^69 - 5 * q^71 + 10 * q^73 + 22 * q^75 + 4 * q^77 - 10 * q^79 - 16 * q^81 + 2 * q^83 - 24 * q^87 - 15 * q^89 - 6 * q^91 + 19 * q^93 + 26 * q^95 + 7 * q^97 - 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
$3$	1.26438	0.729988	0.364994	−	0.931010i	$-0.381071\pi$
$3$	1.26438	0.729988	0.364994	+	0.931010i	$0.381071\pi$
$4$	0	0
$5$	−0.401352	−0.179490	−0.0897450	−	0.995965i	$-0.528605\pi$
$5$	−0.401352	−0.179490	−0.0897450	+	0.995965i	$0.528605\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.40135	−0.467117
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.30059	1.74747	0.873735	−	0.486403i	$-0.161691\pi$
$13$	6.30059	1.74747	0.873735	+	0.486403i	$0.161691\pi$
$14$	0	0
$15$	−0.507460	−0.131026
$16$	0	0
$17$	6.93545	1.68209	0.841047	−	0.540962i	$-0.181940\pi$
$17$	6.93545	1.68209	0.841047	+	0.540962i	$0.181940\pi$
$18$	0	0
$19$	−7.96632	−1.82760	−0.913799	−	0.406166i	$-0.866866\pi$
$19$	−7.96632	−1.82760	−0.913799	+	0.406166i	$0.866866\pi$
$20$	0	0
$21$	−1.26438	−0.275910
$22$	0	0
$23$	4.76226	0.993000	0.496500	−	0.868037i	$-0.334618\pi$
$23$	4.76226	0.993000	0.496500	+	0.868037i	$0.334618\pi$
$24$	0	0
$25$	−4.83892	−0.967783
$26$	0	0
$27$	−5.56497	−1.07098
$28$	0	0
$29$	−0.528753	−0.0981870	−0.0490935	−	0.998794i	$-0.515633\pi$
$29$	−0.528753	−0.0981870	−0.0490935	+	0.998794i	$0.515633\pi$
$30$	0	0
$31$	−3.37048	−0.605357	−0.302678	−	0.953093i	$-0.597881\pi$
$31$	−3.37048	−0.605357	−0.302678	+	0.953093i	$0.597881\pi$
$32$	0	0
$33$	−1.26438	−0.220100
$34$	0	0
$35$	0.401352	0.0678408
$36$	0	0
$37$	9.45886	1.55503	0.777513	−	0.628866i	$-0.216481\pi$
$37$	9.45886	1.55503	0.777513	+	0.628866i	$0.216481\pi$
$38$	0	0
$39$	7.96632	1.27563
$40$	0	0
$41$	−3.13698	−0.489913	−0.244957	−	0.969534i	$-0.578774\pi$
$41$	−3.13698	−0.489913	−0.244957	+	0.969534i	$0.578774\pi$
$42$	0	0
$43$	9.23604	1.40848	0.704241	−	0.709961i	$-0.251287\pi$
$43$	9.23604	1.40848	0.704241	+	0.709961i	$0.251287\pi$
$44$	0	0
$45$	0.562435	0.0838429
$46$	0	0
$47$	6.19448	0.903558	0.451779	−	0.892130i	$-0.350790\pi$
$47$	6.19448	0.903558	0.451779	+	0.892130i	$0.350790\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.76902	1.22791
$52$	0	0
$53$	−4.49507	−0.617446	−0.308723	−	0.951152i	$-0.599902\pi$
$53$	−4.49507	−0.617446	−0.308723	+	0.951152i	$0.599902\pi$
$54$	0	0
$55$	0.401352	0.0541183
$56$	0	0
$57$	−10.0724	−1.33413
$58$	0	0
$59$	11.3368	1.47593	0.737963	−	0.674841i	$-0.235788\pi$
$59$	11.3368	1.47593	0.737963	+	0.674841i	$0.235788\pi$
$60$	0	0
$61$	12.8293	1.64263	0.821315	−	0.570476i	$-0.193241\pi$
$61$	12.8293	1.64263	0.821315	+	0.570476i	$0.193241\pi$
$62$	0	0
$63$	1.40135	0.176554
$64$	0	0
$65$	−2.52875	−0.313653
$66$	0	0
$67$	−5.56497	−0.679869	−0.339934	−	0.940449i	$-0.610405\pi$
$67$	−5.56497	−0.679869	−0.339934	+	0.940449i	$0.610405\pi$
$68$	0	0
$69$	6.02129	0.724879
$70$	0	0
$71$	0.507460	0.0602244	0.0301122	−	0.999547i	$-0.490414\pi$
$71$	0.507460	0.0602244	0.0301122	+	0.999547i	$0.490414\pi$
$72$	0	0
$73$	−7.73815	−0.905682	−0.452841	−	0.891591i	$-0.649590\pi$
$73$	−7.73815	−0.905682	−0.452841	+	0.891591i	$0.649590\pi$
$74$	0	0
$75$	−6.11821	−0.706470
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−1.63909	−0.184412	−0.0922060	−	0.995740i	$-0.529392\pi$
$79$	−1.63909	−0.184412	−0.0922060	+	0.995740i	$0.529392\pi$
$80$	0	0
$81$	−2.83216	−0.314684
$82$	0	0
$83$	7.96632	0.874417	0.437209	−	0.899360i	$-0.355967\pi$
$83$	7.96632	0.874417	0.437209	+	0.899360i	$0.355967\pi$
$84$	0	0
$85$	−2.78356	−0.301919
$86$	0	0
$87$	−0.668543	−0.0716753
$88$	0	0
$89$	−4.02129	−0.426256	−0.213128	−	0.977024i	$-0.568365\pi$
$89$	−4.02129	−0.426256	−0.213128	+	0.977024i	$0.568365\pi$
$90$	0	0
$91$	−6.30059	−0.660481
$92$	0	0
$93$	−4.26156	−0.441903
$94$	0	0
$95$	3.19730	0.328036
$96$	0	0
$97$	7.35275	0.746559	0.373279	−	0.927719i	$-0.378233\pi$
$97$	7.35275	0.746559	0.373279	+	0.927719i	$0.378233\pi$
$98$	0	0
$99$	1.40135	0.140841
$100$	0	0
$101$	9.49789	0.945075	0.472537	−	0.881311i	$-0.343338\pi$
$101$	9.49789	0.945075	0.472537	+	0.881311i	$0.343338\pi$
$102$	0	0
$103$	−18.7957	−1.85199	−0.925996	−	0.377534i	$-0.876772\pi$
$103$	−18.7957	−1.85199	−0.925996	+	0.377534i	$0.876772\pi$
$104$	0	0
$105$	0.507460	0.0495230
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−2.60118	−0.249148	−0.124574	−	0.992210i	$-0.539756\pi$
$109$	−2.60118	−0.249148	−0.124574	+	0.992210i	$0.539756\pi$
$110$	0	0
$111$	11.9596	1.13515
$112$	0	0
$113$	10.8964	1.02505	0.512525	−	0.858673i	$-0.328710\pi$
$113$	10.8964	1.02505	0.512525	+	0.858673i	$0.328710\pi$
$114$	0	0
$115$	−1.91134	−0.178234
$116$	0	0
$117$	−8.82934	−0.816273
$118$	0	0
$119$	−6.93545	−0.635772
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.96632	−0.357631
$124$	0	0
$125$	3.94887	0.353197
$126$	0	0
$127$	15.6778	1.39118	0.695591	−	0.718438i	$-0.255142\pi$
$127$	15.6778	1.39118	0.695591	+	0.718438i	$0.255142\pi$
$128$	0	0
$129$	11.6778	1.02818
$130$	0	0
$131$	−5.77325	−0.504411	−0.252206	−	0.967674i	$-0.581156\pi$
$131$	−5.77325	−0.504411	−0.252206	+	0.967674i	$0.581156\pi$
$132$	0	0
$133$	7.96632	0.690767
$134$	0	0
$135$	2.23351	0.192230
$136$	0	0
$137$	1.67107	0.142770	0.0713848	−	0.997449i	$-0.477258\pi$
$137$	1.67107	0.142770	0.0713848	+	0.997449i	$0.477258\pi$
$138$	0	0
$139$	20.7017	1.75589	0.877946	−	0.478760i	$-0.158914\pi$
$139$	20.7017	1.75589	0.877946	+	0.478760i	$0.158914\pi$
$140$	0	0
$141$	7.83216	0.659587
$142$	0	0
$143$	−6.30059	−0.526882
$144$	0	0
$145$	0.212216	0.0176236
$146$	0	0
$147$	1.26438	0.104284
$148$	0	0
$149$	22.1874	1.81767	0.908833	−	0.417160i	$-0.136975\pi$
$149$	22.1874	1.81767	0.908833	+	0.417160i	$0.136975\pi$
$150$	0	0
$151$	0.0336819	0.00274100	0.00137050	−	0.999999i	$-0.499564\pi$
$151$	0.0336819	0.00274100	0.00137050	+	0.999999i	$0.499564\pi$
$152$	0	0
$153$	−9.71901	−0.785735
$154$	0	0
$155$	1.35275	0.108656
$156$	0	0
$157$	13.4163	1.07074	0.535368	−	0.844619i	$-0.320173\pi$
$157$	13.4163	1.07074	0.535368	+	0.844619i	$0.320173\pi$
$158$	0	0
$159$	−5.68346	−0.450728
$160$	0	0
$161$	−4.76226	−0.375319
$162$	0	0
$163$	3.54367	0.277562	0.138781	−	0.990323i	$-0.455682\pi$
$163$	3.54367	0.277562	0.138781	+	0.990323i	$0.455682\pi$
$164$	0	0
$165$	0.507460	0.0395057
$166$	0	0
$167$	−9.52453	−0.737030	−0.368515	−	0.929622i	$-0.620134\pi$
$167$	−9.52453	−0.737030	−0.368515	+	0.929622i	$0.620134\pi$
$168$	0	0
$169$	26.6974	2.05365
$170$	0	0
$171$	11.1636	0.853703
$172$	0	0
$173$	5.24308	0.398624	0.199312	−	0.979936i	$-0.436129\pi$
$173$	5.24308	0.398624	0.199312	+	0.979936i	$0.436129\pi$
$174$	0	0
$175$	4.83892	0.365788
$176$	0	0
$177$	14.3340	1.07741
$178$	0	0
$179$	18.4209	1.37685	0.688423	−	0.725309i	$-0.258303\pi$
$179$	18.4209	1.37685	0.688423	+	0.725309i	$0.258303\pi$
$180$	0	0
$181$	19.6654	1.46172	0.730861	−	0.682527i	$-0.239119\pi$
$181$	19.6654	1.46172	0.730861	+	0.682527i	$0.239119\pi$
$182$	0	0
$183$	16.2211	1.19910
$184$	0	0
$185$	−3.79633	−0.279112
$186$	0	0
$187$	−6.93545	−0.507170
$188$	0	0
$189$	5.56497	0.404792
$190$	0	0
$191$	−16.4294	−1.18879	−0.594395	−	0.804173i	$-0.702608\pi$
$191$	−16.4294	−1.18879	−0.594395	+	0.804173i	$0.702608\pi$
$192$	0	0
$193$	−6.39965	−0.460657	−0.230329	−	0.973113i	$-0.573980\pi$
$193$	−6.39965	−0.460657	−0.230329	+	0.973113i	$0.573980\pi$
$194$	0	0
$195$	−3.19730	−0.228963
$196$	0	0
$197$	12.8751	0.917315	0.458658	−	0.888613i	$-0.348330\pi$
$197$	12.8751	0.917315	0.458658	+	0.888613i	$0.348330\pi$
$198$	0	0
$199$	−8.25622	−0.585268	−0.292634	−	0.956225i	$-0.594532\pi$
$199$	−8.25622	−0.585268	−0.292634	+	0.956225i	$0.594532\pi$
$200$	0	0
$201$	−7.03621	−0.496296
$202$	0	0
$203$	0.528753	0.0371112
$204$	0	0
$205$	1.25903	0.0879345
$206$	0	0
$207$	−6.67361	−0.463848
$208$	0	0
$209$	7.96632	0.551042
$210$	0	0
$211$	−3.66432	−0.252262	−0.126131	−	0.992014i	$-0.540256\pi$
$211$	−3.66432	−0.252262	−0.126131	+	0.992014i	$0.540256\pi$
$212$	0	0
$213$	0.641620	0.0439631
$214$	0	0
$215$	−3.70690	−0.252809
$216$	0	0
$217$	3.37048	0.228803
$218$	0	0
$219$	−9.78394	−0.661137
$220$	0	0
$221$	43.6974	2.93941
$222$	0	0
$223$	10.9035	0.730150	0.365075	−	0.930978i	$-0.381043\pi$
$223$	10.9035	0.730150	0.365075	+	0.930978i	$0.381043\pi$
$224$	0	0
$225$	6.78102	0.452068
$226$	0	0
$227$	−21.8798	−1.45221	−0.726107	−	0.687582i	$-0.758672\pi$
$227$	−21.8798	−1.45221	−0.726107	+	0.687582i	$0.758672\pi$
$228$	0	0
$229$	−2.59442	−0.171444	−0.0857221	−	0.996319i	$-0.527320\pi$
$229$	−2.59442	−0.171444	−0.0857221	+	0.996319i	$0.527320\pi$
$230$	0	0
$231$	1.26438	0.0831899
$232$	0	0
$233$	5.79848	0.379871	0.189935	−	0.981797i	$-0.439172\pi$
$233$	5.79848	0.379871	0.189935	+	0.981797i	$0.439172\pi$
$234$	0	0
$235$	−2.48617	−0.162180
$236$	0	0
$237$	−2.07243	−0.134619
$238$	0	0
$239$	30.0051	1.94087	0.970433	−	0.241370i	$-0.0775969\pi$
$239$	30.0051	1.94087	0.970433	+	0.241370i	$0.0775969\pi$
$240$	0	0
$241$	−11.3244	−0.729470	−0.364735	−	0.931111i	$-0.618840\pi$
$241$	−11.3244	−0.729470	−0.364735	+	0.931111i	$0.618840\pi$
$242$	0	0
$243$	13.1140	0.841263
$244$	0	0
$245$	−0.401352	−0.0256414
$246$	0	0
$247$	−50.1925	−3.19367
$248$	0	0
$249$	10.0724	0.638314
$250$	0	0
$251$	−12.8613	−0.811800	−0.405900	−	0.913918i	$-0.633042\pi$
$251$	−12.8613	−0.811800	−0.405900	+	0.913918i	$0.633042\pi$
$252$	0	0
$253$	−4.76226	−0.299401
$254$	0	0
$255$	−3.51946	−0.220397
$256$	0	0
$257$	−31.1350	−1.94215	−0.971074	−	0.238780i	$-0.923253\pi$
$257$	−31.1350	−1.94215	−0.971074	+	0.238780i	$0.923253\pi$
$258$	0	0
$259$	−9.45886	−0.587745
$260$	0	0
$261$	0.740969	0.0458648
$262$	0	0
$263$	21.4146	1.32048	0.660240	−	0.751055i	$-0.270455\pi$
$263$	21.4146	1.32048	0.660240	+	0.751055i	$0.270455\pi$
$264$	0	0
$265$	1.80411	0.110825
$266$	0	0
$267$	−5.08443	−0.311162
$268$	0	0
$269$	−21.6059	−1.31733	−0.658666	−	0.752435i	$-0.728879\pi$
$269$	−21.6059	−1.31733	−0.658666	+	0.752435i	$0.728879\pi$
$270$	0	0
$271$	−7.92194	−0.481224	−0.240612	−	0.970621i	$-0.577348\pi$
$271$	−7.92194	−0.481224	−0.240612	+	0.970621i	$0.577348\pi$
$272$	0	0
$273$	−7.96632	−0.482144
$274$	0	0
$275$	4.83892	0.291798
$276$	0	0
$277$	−26.8667	−1.61426	−0.807131	−	0.590372i	$-0.798981\pi$
$277$	−26.8667	−1.61426	−0.807131	+	0.590372i	$0.798981\pi$
$278$	0	0
$279$	4.72323	0.282773
$280$	0	0
$281$	2.60118	0.155173	0.0775867	−	0.996986i	$-0.475279\pi$
$281$	2.60118	0.155173	0.0775867	+	0.996986i	$0.475279\pi$
$282$	0	0
$283$	13.9472	0.829073	0.414537	−	0.910033i	$-0.363944\pi$
$283$	13.9472	0.829073	0.414537	+	0.910033i	$0.363944\pi$
$284$	0	0
$285$	4.04259	0.239462
$286$	0	0
$287$	3.13698	0.185170
$288$	0	0
$289$	31.1005	1.82944
$290$	0	0
$291$	9.29665	0.544979
$292$	0	0
$293$	−22.5689	−1.31849	−0.659245	−	0.751928i	$-0.729124\pi$
$293$	−22.5689	−1.31849	−0.659245	+	0.751928i	$0.729124\pi$
$294$	0	0
$295$	−4.55005	−0.264914
$296$	0	0
$297$	5.56497	0.322912
$298$	0	0
$299$	30.0051	1.73524
$300$	0	0
$301$	−9.23604	−0.532357
$302$	0	0
$303$	12.0089	0.689894
$304$	0	0
$305$	−5.14908	−0.294835
$306$	0	0
$307$	−20.3553	−1.16174	−0.580869	−	0.813997i	$-0.697287\pi$
$307$	−20.3553	−1.16174	−0.580869	+	0.813997i	$0.697287\pi$
$308$	0	0
$309$	−23.7648	−1.35193
$310$	0	0
$311$	−7.92053	−0.449132	−0.224566	−	0.974459i	$-0.572096\pi$
$311$	−7.92053	−0.449132	−0.224566	+	0.974459i	$0.572096\pi$
$312$	0	0
$313$	13.5841	0.767820	0.383910	−	0.923371i	$-0.374577\pi$
$313$	13.5841	0.767820	0.383910	+	0.923371i	$0.374577\pi$
$314$	0	0
$315$	−0.562435	−0.0316896
$316$	0	0
$317$	12.8240	0.720267	0.360134	−	0.932901i	$-0.382731\pi$
$317$	12.8240	0.720267	0.360134	+	0.932901i	$0.382731\pi$
$318$	0	0
$319$	0.528753	0.0296045
$320$	0	0
$321$	−5.05751	−0.282282
$322$	0	0
$323$	−55.2500	−3.07419
$324$	0	0
$325$	−30.4880	−1.69117
$326$	0	0
$327$	−3.28887	−0.181875
$328$	0	0
$329$	−6.19448	−0.341513
$330$	0	0
$331$	15.5032	0.852135	0.426067	−	0.904691i	$-0.359899\pi$
$331$	15.5032	0.852135	0.426067	+	0.904691i	$0.359899\pi$
$332$	0	0
$333$	−13.2552	−0.726380
$334$	0	0
$335$	2.23351	0.122030
$336$	0	0
$337$	3.13556	0.170805	0.0854025	−	0.996347i	$-0.472782\pi$
$337$	3.13556	0.170805	0.0854025	+	0.996347i	$0.472782\pi$
$338$	0	0
$339$	13.7772	0.748274
$340$	0	0
$341$	3.37048	0.182522
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.41666	−0.130108
$346$	0	0
$347$	−1.21466	−0.0652062	−0.0326031	−	0.999468i	$-0.510380\pi$
$347$	−1.21466	−0.0652062	−0.0326031	+	0.999468i	$0.510380\pi$
$348$	0	0
$349$	14.0991	0.754706	0.377353	−	0.926070i	$-0.376834\pi$
$349$	14.0991	0.754706	0.377353	+	0.926070i	$0.376834\pi$
$350$	0	0
$351$	−35.0626	−1.87150
$352$	0	0
$353$	−14.5607	−0.774990	−0.387495	−	0.921872i	$-0.626660\pi$
$353$	−14.5607	−0.774990	−0.387495	+	0.921872i	$0.626660\pi$
$354$	0	0
$355$	−0.203670	−0.0108097
$356$	0	0
$357$	−8.76902	−0.464106
$358$	0	0
$359$	−14.2935	−0.754385	−0.377192	−	0.926135i	$-0.623110\pi$
$359$	−14.2935	−0.754385	−0.377192	+	0.926135i	$0.623110\pi$
$360$	0	0
$361$	44.4622	2.34012
$362$	0	0
$363$	1.26438	0.0663626
$364$	0	0
$365$	3.10572	0.162561
$366$	0	0
$367$	6.43305	0.335803	0.167901	−	0.985804i	$-0.446301\pi$
$367$	6.43305	0.335803	0.167901	+	0.985804i	$0.446301\pi$
$368$	0	0
$369$	4.39601	0.228847
$370$	0	0
$371$	4.49507	0.233372
$372$	0	0
$373$	24.8504	1.28670	0.643351	−	0.765571i	$-0.277544\pi$
$373$	24.8504	1.28670	0.643351	+	0.765571i	$0.277544\pi$
$374$	0	0
$375$	4.99286	0.257830
$376$	0	0
$377$	−3.33146	−0.171579
$378$	0	0
$379$	9.98940	0.513121	0.256560	−	0.966528i	$-0.417411\pi$
$379$	9.98940	0.513121	0.256560	+	0.966528i	$0.417411\pi$
$380$	0	0
$381$	19.8227	1.01555
$382$	0	0
$383$	−0.618824	−0.0316204	−0.0158102	−	0.999875i	$-0.505033\pi$
$383$	−0.618824	−0.0316204	−0.0158102	+	0.999875i	$0.505033\pi$
$384$	0	0
$385$	−0.401352	−0.0204548
$386$	0	0
$387$	−12.9429	−0.657927
$388$	0	0
$389$	27.1726	1.37771	0.688853	−	0.724901i	$-0.258115\pi$
$389$	27.1726	1.37771	0.688853	+	0.724901i	$0.258115\pi$
$390$	0	0
$391$	33.0284	1.67032
$392$	0	0
$393$	−7.29956	−0.368214
$394$	0	0
$395$	0.657851	0.0331001
$396$	0	0
$397$	20.3042	1.01904	0.509520	−	0.860459i	$-0.329823\pi$
$397$	20.3042	1.01904	0.509520	+	0.860459i	$0.329823\pi$
$398$	0	0
$399$	10.0724	0.504252
$400$	0	0
$401$	−23.2024	−1.15867	−0.579335	−	0.815089i	$-0.696688\pi$
$401$	−23.2024	−1.15867	−0.579335	+	0.815089i	$0.696688\pi$
$402$	0	0
$403$	−21.2360	−1.05784
$404$	0	0
$405$	1.13669	0.0564827
$406$	0	0
$407$	−9.45886	−0.468858
$408$	0	0
$409$	22.3478	1.10503	0.552514	−	0.833504i	$-0.313669\pi$
$409$	22.3478	1.10503	0.552514	+	0.833504i	$0.313669\pi$
$410$	0	0
$411$	2.11287	0.104220
$412$	0	0
$413$	−11.3368	−0.557848
$414$	0	0
$415$	−3.19730	−0.156949
$416$	0	0
$417$	26.1747	1.28178
$418$	0	0
$419$	−8.56608	−0.418481	−0.209240	−	0.977864i	$-0.567099\pi$
$419$	−8.56608	−0.418481	−0.209240	+	0.977864i	$0.567099\pi$
$420$	0	0
$421$	−3.22535	−0.157194	−0.0785969	−	0.996906i	$-0.525044\pi$
$421$	−3.22535	−0.157194	−0.0785969	+	0.996906i	$0.525044\pi$
$422$	0	0
$423$	−8.68065	−0.422068
$424$	0	0
$425$	−33.5601	−1.62790
$426$	0	0
$427$	−12.8293	−0.620855
$428$	0	0
$429$	−7.96632	−0.384617
$430$	0	0
$431$	−28.0303	−1.35017	−0.675086	−	0.737739i	$-0.735893\pi$
$431$	−28.0303	−1.35017	−0.675086	+	0.737739i	$0.735893\pi$
$432$	0	0
$433$	30.4401	1.46286	0.731429	−	0.681918i	$-0.238854\pi$
$433$	30.4401	1.46286	0.731429	+	0.681918i	$0.238854\pi$
$434$	0	0
$435$	0.268321	0.0128650
$436$	0	0
$437$	−37.9377	−1.81481
$438$	0	0
$439$	2.44787	0.116831	0.0584153	−	0.998292i	$-0.481395\pi$
$439$	2.44787	0.116831	0.0584153	+	0.998292i	$0.481395\pi$
$440$	0	0
$441$	−1.40135	−0.0667310
$442$	0	0
$443$	−38.3110	−1.82021	−0.910105	−	0.414377i	$-0.863999\pi$
$443$	−38.3110	−1.82021	−0.910105	+	0.414377i	$0.863999\pi$
$444$	0	0
$445$	1.61395	0.0765087
$446$	0	0
$447$	28.0533	1.32687
$448$	0	0
$449$	29.9820	1.41494	0.707469	−	0.706745i	$-0.249837\pi$
$449$	29.9820	1.41494	0.707469	+	0.706745i	$0.249837\pi$
$450$	0	0
$451$	3.13698	0.147714
$452$	0	0
$453$	0.0425867	0.00200090
$454$	0	0
$455$	2.52875	0.118550
$456$	0	0
$457$	−33.7844	−1.58037	−0.790184	−	0.612870i	$-0.790015\pi$
$457$	−33.7844	−1.58037	−0.790184	+	0.612870i	$0.790015\pi$
$458$	0	0
$459$	−38.5955	−1.80149
$460$	0	0
$461$	6.88825	0.320818	0.160409	−	0.987051i	$-0.448719\pi$
$461$	6.88825	0.320818	0.160409	+	0.987051i	$0.448719\pi$
$462$	0	0
$463$	−25.0520	−1.16426	−0.582132	−	0.813094i	$-0.697781\pi$
$463$	−25.0520	−1.16426	−0.582132	+	0.813094i	$0.697781\pi$
$464$	0	0
$465$	1.71039	0.0793172
$466$	0	0
$467$	27.7931	1.28611	0.643056	−	0.765819i	$-0.277666\pi$
$467$	27.7931	1.28611	0.643056	+	0.765819i	$0.277666\pi$
$468$	0	0
$469$	5.56497	0.256966
$470$	0	0
$471$	16.9632	0.781624
$472$	0	0
$473$	−9.23604	−0.424674
$474$	0	0
$475$	38.5483	1.76872
$476$	0	0
$477$	6.29918	0.288419
$478$	0	0
$479$	−13.8283	−0.631832	−0.315916	−	0.948787i	$-0.602312\pi$
$479$	−13.8283	−0.631832	−0.315916	+	0.948787i	$0.602312\pi$
$480$	0	0
$481$	59.5964	2.71736
$482$	0	0
$483$	−6.02129	−0.273978
$484$	0	0
$485$	−2.95104	−0.134000
$486$	0	0
$487$	21.5224	0.975272	0.487636	−	0.873047i	$-0.337859\pi$
$487$	21.5224	0.975272	0.487636	+	0.873047i	$0.337859\pi$
$488$	0	0
$489$	4.48054	0.202617
$490$	0	0
$491$	−37.5291	−1.69367	−0.846833	−	0.531859i	$-0.821494\pi$
$491$	−37.5291	−1.69367	−0.846833	+	0.531859i	$0.821494\pi$
$492$	0	0
$493$	−3.66714	−0.165160
$494$	0	0
$495$	−0.562435	−0.0252796
$496$	0	0
$497$	−0.507460	−0.0227627
$498$	0	0
$499$	−9.14908	−0.409569	−0.204785	−	0.978807i	$-0.565649\pi$
$499$	−9.14908	−0.409569	−0.204785	+	0.978807i	$0.565649\pi$
$500$	0	0
$501$	−12.0426	−0.538023
$502$	0	0
$503$	−23.1158	−1.03068	−0.515342	−	0.856984i	$-0.672335\pi$
$503$	−23.1158	−1.03068	−0.515342	+	0.856984i	$0.672335\pi$
$504$	0	0
$505$	−3.81199	−0.169632
$506$	0	0
$507$	33.7556	1.49914
$508$	0	0
$509$	−11.3723	−0.504070	−0.252035	−	0.967718i	$-0.581100\pi$
$509$	−11.3723	−0.504070	−0.252035	+	0.967718i	$0.581100\pi$
$510$	0	0
$511$	7.73815	0.342316
$512$	0	0
$513$	44.3323	1.95732
$514$	0	0
$515$	7.54367	0.332414
$516$	0	0
$517$	−6.19448	−0.272433
$518$	0	0
$519$	6.62923	0.290991
$520$	0	0
$521$	14.0370	0.614974	0.307487	−	0.951552i	$-0.400512\pi$
$521$	14.0370	0.614974	0.307487	+	0.951552i	$0.400512\pi$
$522$	0	0
$523$	−32.5889	−1.42501	−0.712506	−	0.701666i	$-0.752440\pi$
$523$	−32.5889	−1.42501	−0.712506	+	0.701666i	$0.752440\pi$
$524$	0	0
$525$	6.11821	0.267021
$526$	0	0
$527$	−23.3758	−1.01827
$528$	0	0
$529$	−0.320855	−0.0139502
$530$	0	0
$531$	−15.8868	−0.689431
$532$	0	0
$533$	−19.7648	−0.856108
$534$	0	0
$535$	1.60541	0.0694078
$536$	0	0
$537$	23.2910	1.00508
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−1.70261	−0.0732009	−0.0366005	−	0.999330i	$-0.511653\pi$
$541$	−1.70261	−0.0732009	−0.0366005	+	0.999330i	$0.511653\pi$
$542$	0	0
$543$	24.8645	1.06704
$544$	0	0
$545$	1.04399	0.0447195
$546$	0	0
$547$	−17.4606	−0.746559	−0.373280	−	0.927719i	$-0.621767\pi$
$547$	−17.4606	−0.746559	−0.373280	+	0.927719i	$0.621767\pi$
$548$	0	0
$549$	−17.9784	−0.767300
$550$	0	0
$551$	4.21222	0.179446
$552$	0	0
$553$	1.63909	0.0697012
$554$	0	0
$555$	−4.79999	−0.203748
$556$	0	0
$557$	−16.7269	−0.708741	−0.354370	−	0.935105i	$-0.615305\pi$
$557$	−16.7269	−0.708741	−0.354370	+	0.935105i	$0.615305\pi$
$558$	0	0
$559$	58.1925	2.46128
$560$	0	0
$561$	−8.76902	−0.370228
$562$	0	0
$563$	39.6442	1.67080	0.835401	−	0.549641i	$-0.185236\pi$
$563$	39.6442	1.67080	0.835401	+	0.549641i	$0.185236\pi$
$564$	0	0
$565$	−4.37330	−0.183986
$566$	0	0
$567$	2.83216	0.118939
$568$	0	0
$569$	−7.80693	−0.327284	−0.163642	−	0.986520i	$-0.552324\pi$
$569$	−7.80693	−0.327284	−0.163642	+	0.986520i	$0.552324\pi$
$570$	0	0
$571$	−2.46196	−0.103030	−0.0515148	−	0.998672i	$-0.516405\pi$
$571$	−2.46196	−0.103030	−0.0515148	+	0.998672i	$0.516405\pi$
$572$	0	0
$573$	−20.7730	−0.867803
$574$	0	0
$575$	−23.0442	−0.961009
$576$	0	0
$577$	−5.21296	−0.217018	−0.108509	−	0.994095i	$-0.534608\pi$
$577$	−5.21296	−0.217018	−0.108509	+	0.994095i	$0.534608\pi$
$578$	0	0
$579$	−8.09157	−0.336274
$580$	0	0
$581$	−7.96632	−0.330499
$582$	0	0
$583$	4.49507	0.186167
$584$	0	0
$585$	3.54367	0.146513
$586$	0	0
$587$	16.3922	0.676577	0.338288	−	0.941042i	$-0.390152\pi$
$587$	16.3922	0.676577	0.338288	+	0.941042i	$0.390152\pi$
$588$	0	0
$589$	26.8504	1.10635
$590$	0	0
$591$	16.2790	0.669629
$592$	0	0
$593$	36.4707	1.49767	0.748835	−	0.662756i	$-0.230613\pi$
$593$	36.4707	1.49767	0.748835	+	0.662756i	$0.230613\pi$
$594$	0	0
$595$	2.78356	0.114115
$596$	0	0
$597$	−10.4390	−0.427238
$598$	0	0
$599$	21.5352	0.879905	0.439953	−	0.898021i	$-0.354995\pi$
$599$	21.5352	0.879905	0.439953	+	0.898021i	$0.354995\pi$
$600$	0	0
$601$	−18.9887	−0.774566	−0.387283	−	0.921961i	$-0.626586\pi$
$601$	−18.9887	−0.774566	−0.387283	+	0.921961i	$0.626586\pi$
$602$	0	0
$603$	7.79848	0.317578
$604$	0	0
$605$	−0.401352	−0.0163173
$606$	0	0
$607$	−0.148246	−0.00601712	−0.00300856	−	0.999995i	$-0.500958\pi$
$607$	−0.148246	−0.00601712	−0.00300856	+	0.999995i	$0.500958\pi$
$608$	0	0
$609$	0.668543	0.0270907
$610$	0	0
$611$	39.0289	1.57894
$612$	0	0
$613$	−43.3365	−1.75035	−0.875173	−	0.483811i	$-0.839252\pi$
$613$	−43.3365	−1.75035	−0.875173	+	0.483811i	$0.839252\pi$
$614$	0	0
$615$	1.59189	0.0641912
$616$	0	0
$617$	−21.6413	−0.871247	−0.435623	−	0.900129i	$-0.643472\pi$
$617$	−21.6413	−0.871247	−0.435623	+	0.900129i	$0.643472\pi$
$618$	0	0
$619$	32.0189	1.28695	0.643473	−	0.765468i	$-0.277493\pi$
$619$	32.0189	1.28695	0.643473	+	0.765468i	$0.277493\pi$
$620$	0	0
$621$	−26.5018	−1.06348
$622$	0	0
$623$	4.02129	0.161110
$624$	0	0
$625$	22.6097	0.904388
$626$	0	0
$627$	10.0724	0.402254
$628$	0	0
$629$	65.6014	2.61570
$630$	0	0
$631$	−26.9553	−1.07307	−0.536537	−	0.843877i	$-0.680268\pi$
$631$	−26.9553	−1.07307	−0.536537	+	0.843877i	$0.680268\pi$
$632$	0	0
$633$	−4.63307	−0.184148
$634$	0	0
$635$	−6.29233	−0.249703
$636$	0	0
$637$	6.30059	0.249638
$638$	0	0
$639$	−0.711130	−0.0281319
$640$	0	0
$641$	−15.2765	−0.603385	−0.301692	−	0.953405i	$-0.597552\pi$
$641$	−15.2765	−0.603385	−0.301692	+	0.953405i	$0.597552\pi$
$642$	0	0
$643$	48.9955	1.93219	0.966097	−	0.258180i	$-0.0831229\pi$
$643$	48.9955	1.93219	0.966097	+	0.258180i	$0.0831229\pi$
$644$	0	0
$645$	−4.68692	−0.184547
$646$	0	0
$647$	−23.1718	−0.910977	−0.455488	−	0.890242i	$-0.650535\pi$
$647$	−23.1718	−0.910977	−0.455488	+	0.890242i	$0.650535\pi$
$648$	0	0
$649$	−11.3368	−0.445008
$650$	0	0
$651$	4.26156	0.167024
$652$	0	0
$653$	−7.97871	−0.312231	−0.156115	−	0.987739i	$-0.549897\pi$
$653$	−7.97871	−0.312231	−0.156115	+	0.987739i	$0.549897\pi$
$654$	0	0
$655$	2.31710	0.0905368
$656$	0	0
$657$	10.8439	0.423060
$658$	0	0
$659$	−24.7264	−0.963205	−0.481603	−	0.876390i	$-0.659945\pi$
$659$	−24.7264	−0.963205	−0.481603	+	0.876390i	$0.659945\pi$
$660$	0	0
$661$	−21.8159	−0.848541	−0.424271	−	0.905535i	$-0.639469\pi$
$661$	−21.8159	−0.848541	−0.424271	+	0.905535i	$0.639469\pi$
$662$	0	0
$663$	55.2500	2.14573
$664$	0	0
$665$	−3.19730	−0.123986
$666$	0	0
$667$	−2.51806	−0.0974997
$668$	0	0
$669$	13.7861	0.533001
$670$	0	0
$671$	−12.8293	−0.495271
$672$	0	0
$673$	−4.08312	−0.157393	−0.0786963	−	0.996899i	$-0.525076\pi$
$673$	−4.08312	−0.157393	−0.0786963	+	0.996899i	$0.525076\pi$
$674$	0	0
$675$	26.9284	1.03647
$676$	0	0
$677$	−26.5796	−1.02154	−0.510769	−	0.859718i	$-0.670639\pi$
$677$	−26.5796	−1.02154	−0.510769	+	0.859718i	$0.670639\pi$
$678$	0	0
$679$	−7.35275	−0.282173
$680$	0	0
$681$	−27.6643	−1.06010
$682$	0	0
$683$	−25.4707	−0.974608	−0.487304	−	0.873232i	$-0.662020\pi$
$683$	−25.4707	−0.974608	−0.487304	+	0.873232i	$0.662020\pi$
$684$	0	0
$685$	−0.670689	−0.0256257
$686$	0	0
$687$	−3.28032	−0.125152
$688$	0	0
$689$	−28.3216	−1.07897
$690$	0	0
$691$	17.6257	0.670515	0.335257	−	0.942127i	$-0.391177\pi$
$691$	17.6257	0.670515	0.335257	+	0.942127i	$0.391177\pi$
$692$	0	0
$693$	−1.40135	−0.0532330
$694$	0	0
$695$	−8.30865	−0.315165
$696$	0	0
$697$	−21.7563	−0.824080
$698$	0	0
$699$	7.33146	0.277301
$700$	0	0
$701$	−10.9447	−0.413377	−0.206688	−	0.978407i	$-0.566269\pi$
$701$	−10.9447	−0.413377	−0.206688	+	0.978407i	$0.566269\pi$
$702$	0	0
$703$	−75.3523	−2.84196
$704$	0	0
$705$	−3.14345	−0.118389
$706$	0	0
$707$	−9.49789	−0.357205
$708$	0	0
$709$	7.34992	0.276032	0.138016	−	0.990430i	$-0.455927\pi$
$709$	7.34992	0.276032	0.138016	+	0.990430i	$0.455927\pi$
$710$	0	0
$711$	2.29694	0.0861420
$712$	0	0
$713$	−16.0511	−0.601120
$714$	0	0
$715$	2.52875	0.0945700
$716$	0	0
$717$	37.9377	1.41681
$718$	0	0
$719$	−36.7851	−1.37185	−0.685926	−	0.727672i	$-0.740602\pi$
$719$	−36.7851	−1.37185	−0.685926	+	0.727672i	$0.740602\pi$
$720$	0	0
$721$	18.7957	0.699987
$722$	0	0
$723$	−14.3183	−0.532504
$724$	0	0
$725$	2.55859	0.0950237
$726$	0	0
$727$	0.909095	0.0337165	0.0168582	−	0.999858i	$-0.494634\pi$
$727$	0.909095	0.0337165	0.0168582	+	0.999858i	$0.494634\pi$
$728$	0	0
$729$	25.0775	0.928796
$730$	0	0
$731$	64.0561	2.36920
$732$	0	0
$733$	−7.40068	−0.273350	−0.136675	−	0.990616i	$-0.543642\pi$
$733$	−7.40068	−0.273350	−0.136675	+	0.990616i	$0.543642\pi$
$734$	0	0
$735$	−0.507460	−0.0187179
$736$	0	0
$737$	5.56497	0.204988
$738$	0	0
$739$	−15.3039	−0.562961	−0.281481	−	0.959567i	$-0.590826\pi$
$739$	−15.3039	−0.562961	−0.281481	+	0.959567i	$0.590826\pi$
$740$	0	0
$741$	−63.4622	−2.33134
$742$	0	0
$743$	13.8176	0.506919	0.253460	−	0.967346i	$-0.418431\pi$
$743$	13.8176	0.506919	0.253460	+	0.967346i	$0.418431\pi$
$744$	0	0
$745$	−8.90497	−0.326253
$746$	0	0
$747$	−11.1636	−0.408455
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−26.6651	−0.973022	−0.486511	−	0.873674i	$-0.661731\pi$
$751$	−26.6651	−0.973022	−0.486511	+	0.873674i	$0.661731\pi$
$752$	0	0
$753$	−16.2616	−0.592604
$754$	0	0
$755$	−0.0135183	−0.000491982 0
$756$	0	0
$757$	−20.9986	−0.763207	−0.381604	−	0.924326i	$-0.624628\pi$
$757$	−20.9986	−0.763207	−0.381604	+	0.924326i	$0.624628\pi$
$758$	0	0
$759$	−6.02129	−0.218559
$760$	0	0
$761$	−35.0163	−1.26934	−0.634671	−	0.772783i	$-0.718864\pi$
$761$	−35.0163	−1.26934	−0.634671	+	0.772783i	$0.718864\pi$
$762$	0	0
$763$	2.60118	0.0941690
$764$	0	0
$765$	3.90074	0.141032
$766$	0	0
$767$	71.4285	2.57914
$768$	0	0
$769$	−5.24916	−0.189290	−0.0946448	−	0.995511i	$-0.530172\pi$
$769$	−5.24916	−0.189290	−0.0946448	+	0.995511i	$0.530172\pi$
$770$	0	0
$771$	−39.3664	−1.41774
$772$	0	0
$773$	25.9629	0.933821	0.466911	−	0.884304i	$-0.345367\pi$
$773$	25.9629	0.933821	0.466911	+	0.884304i	$0.345367\pi$
$774$	0	0
$775$	16.3095	0.585854
$776$	0	0
$777$	−11.9596	−0.429047
$778$	0	0
$779$	24.9901	0.895365
$780$	0	0
$781$	−0.507460	−0.0181583
$782$	0	0
$783$	2.94249	0.105156
$784$	0	0
$785$	−5.38465	−0.192186
$786$	0	0
$787$	−52.6557	−1.87697	−0.938486	−	0.345317i	$-0.887771\pi$
$787$	−52.6557	−1.87697	−0.938486	+	0.345317i	$0.887771\pi$
$788$	0	0
$789$	27.0761	0.963934
$790$	0	0
$791$	−10.8964	−0.387432
$792$	0	0
$793$	80.8324	2.87044
$794$	0	0
$795$	2.28107	0.0809012
$796$	0	0
$797$	−12.2957	−0.435536	−0.217768	−	0.976001i	$-0.569878\pi$
$797$	−12.2957	−0.435536	−0.217768	+	0.976001i	$0.569878\pi$
$798$	0	0
$799$	42.9615	1.51987
$800$	0	0
$801$	5.63525	0.199112
$802$	0	0
$803$	7.73815	0.273074
$804$	0	0
$805$	1.91134	0.0673660
$806$	0	0
$807$	−27.3179	−0.961637
$808$	0	0
$809$	−49.1052	−1.72645	−0.863223	−	0.504823i	$-0.831558\pi$
$809$	−49.1052	−1.72645	−0.863223	+	0.504823i	$0.831558\pi$
$810$	0	0
$811$	−23.1239	−0.811988	−0.405994	−	0.913876i	$-0.633075\pi$
$811$	−23.1239	−0.811988	−0.405994	+	0.913876i	$0.633075\pi$
$812$	0	0
$813$	−10.0163	−0.351288
$814$	0	0
$815$	−1.42226	−0.0498196
$816$	0	0
$817$	−73.5772	−2.57414
$818$	0	0
$819$	8.82934	0.308522
$820$	0	0
$821$	−50.1336	−1.74967	−0.874837	−	0.484417i	$-0.839032\pi$
$821$	−50.1336	−1.74967	−0.874837	+	0.484417i	$0.839032\pi$
$822$	0	0
$823$	−1.08726	−0.0378994	−0.0189497	−	0.999820i	$-0.506032\pi$
$823$	−1.08726	−0.0378994	−0.0189497	+	0.999820i	$0.506032\pi$
$824$	0	0
$825$	6.11821	0.213009
$826$	0	0
$827$	−40.1673	−1.39675	−0.698377	−	0.715731i	$-0.746094\pi$
$827$	−40.1673	−1.39675	−0.698377	+	0.715731i	$0.746094\pi$
$828$	0	0
$829$	0.108254	0.00375980	0.00187990	−	0.999998i	$-0.499402\pi$
$829$	0.108254	0.00375980	0.00187990	+	0.999998i	$0.499402\pi$
$830$	0	0
$831$	−33.9696	−1.17839
$832$	0	0
$833$	6.93545	0.240299
$834$	0	0
$835$	3.82269	0.132289
$836$	0	0
$837$	18.7566	0.648324
$838$	0	0
$839$	28.6856	0.990338	0.495169	−	0.868797i	$-0.335106\pi$
$839$	28.6856	0.990338	0.495169	+	0.868797i	$0.335106\pi$
$840$	0	0
$841$	−28.7204	−0.990359
$842$	0	0
$843$	3.28887	0.113275
$844$	0	0
$845$	−10.7151	−0.368609
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	17.6345	0.605214
$850$	0	0
$851$	45.0456	1.54414
$852$	0	0
$853$	12.2708	0.420142	0.210071	−	0.977686i	$-0.432630\pi$
$853$	12.2708	0.420142	0.210071	+	0.977686i	$0.432630\pi$
$854$	0	0
$855$	−4.48054	−0.153231
$856$	0	0
$857$	15.0448	0.513922	0.256961	−	0.966422i	$-0.417279\pi$
$857$	15.0448	0.513922	0.256961	+	0.966422i	$0.417279\pi$
$858$	0	0
$859$	−14.0637	−0.479847	−0.239923	−	0.970792i	$-0.577122\pi$
$859$	−14.0637	−0.479847	−0.239923	+	0.970792i	$0.577122\pi$
$860$	0	0
$861$	3.96632	0.135172
$862$	0	0
$863$	30.9143	1.05234	0.526168	−	0.850381i	$-0.323628\pi$
$863$	30.9143	1.05234	0.526168	+	0.850381i	$0.323628\pi$
$864$	0	0
$865$	−2.10432	−0.0715491
$866$	0	0
$867$	39.3227	1.33547
$868$	0	0
$869$	1.63909	0.0556023
$870$	0	0
$871$	−35.0626	−1.18805
$872$	0	0
$873$	−10.3038	−0.348730
$874$	0	0
$875$	−3.94887	−0.133496
$876$	0	0
$877$	29.4380	0.994051	0.497026	−	0.867736i	$-0.334426\pi$
$877$	29.4380	0.994051	0.497026	+	0.867736i	$0.334426\pi$
$878$	0	0
$879$	−28.5356	−0.962482
$880$	0	0
$881$	18.3344	0.617703	0.308851	−	0.951110i	$-0.400055\pi$
$881$	18.3344	0.617703	0.308851	+	0.951110i	$0.400055\pi$
$882$	0	0
$883$	15.1725	0.510596	0.255298	−	0.966862i	$-0.417826\pi$
$883$	15.1725	0.510596	0.255298	+	0.966862i	$0.417826\pi$
$884$	0	0
$885$	−5.75297	−0.193384
$886$	0	0
$887$	−36.1734	−1.21458	−0.607291	−	0.794479i	$-0.707744\pi$
$887$	−36.1734	−1.21458	−0.607291	+	0.794479i	$0.707744\pi$
$888$	0	0
$889$	−15.6778	−0.525818
$890$	0	0
$891$	2.83216	0.0948808
$892$	0	0
$893$	−49.3472	−1.65134
$894$	0	0
$895$	−7.39328	−0.247130
$896$	0	0
$897$	37.9377	1.26670
$898$	0	0
$899$	1.78215	0.0594382
$900$	0	0
$901$	−31.1753	−1.03860
$902$	0	0
$903$	−11.6778	−0.388614
$904$	0	0
$905$	−7.89276	−0.262364
$906$	0	0
$907$	24.1477	0.801811	0.400905	−	0.916120i	$-0.368696\pi$
$907$	24.1477	0.801811	0.400905	+	0.916120i	$0.368696\pi$
$908$	0	0
$909$	−13.3099	−0.441461
$910$	0	0
$911$	14.7275	0.487942	0.243971	−	0.969783i	$-0.421550\pi$
$911$	14.7275	0.487942	0.243971	+	0.969783i	$0.421550\pi$
$912$	0	0
$913$	−7.96632	−0.263647
$914$	0	0
$915$	−6.51038	−0.215226
$916$	0	0
$917$	5.77325	0.190650
$918$	0	0
$919$	53.1776	1.75417	0.877083	−	0.480338i	$-0.159486\pi$
$919$	53.1776	1.75417	0.877083	+	0.480338i	$0.159486\pi$
$920$	0	0
$921$	−25.7367	−0.848054
$922$	0	0
$923$	3.19730	0.105240
$924$	0	0
$925$	−45.7706	−1.50493
$926$	0	0
$927$	26.3393	0.865097
$928$	0	0
$929$	25.5998	0.839901	0.419951	−	0.907547i	$-0.362047\pi$
$929$	25.5998	0.839901	0.419951	+	0.907547i	$0.362047\pi$
$930$	0	0
$931$	−7.96632	−0.261086
$932$	0	0
$933$	−10.0145	−0.327861
$934$	0	0
$935$	2.78356	0.0910320
$936$	0	0
$937$	42.8966	1.40137	0.700685	−	0.713471i	$-0.252878\pi$
$937$	42.8966	1.40137	0.700685	+	0.713471i	$0.252878\pi$
$938$	0	0
$939$	17.1754	0.560499
$940$	0	0
$941$	7.02524	0.229016	0.114508	−	0.993422i	$-0.463471\pi$
$941$	7.02524	0.229016	0.114508	+	0.993422i	$0.463471\pi$
$942$	0	0
$943$	−14.9391	−0.486484
$944$	0	0
$945$	−2.23351	−0.0726561
$946$	0	0
$947$	−15.4145	−0.500903	−0.250452	−	0.968129i	$-0.580579\pi$
$947$	−15.4145	−0.500903	−0.250452	+	0.968129i	$0.580579\pi$
$948$	0	0
$949$	−48.7549	−1.58265
$950$	0	0
$951$	16.2144	0.525787
$952$	0	0
$953$	8.67078	0.280874	0.140437	−	0.990090i	$-0.455149\pi$
$953$	8.67078	0.280874	0.140437	+	0.990090i	$0.455149\pi$
$954$	0	0
$955$	6.59397	0.213376
$956$	0	0
$957$	0.668543	0.0216109
$958$	0	0
$959$	−1.67107	−0.0539618
$960$	0	0
$961$	−19.6398	−0.633543
$962$	0	0
$963$	5.60541	0.180632
$964$	0	0
$965$	2.56851	0.0826834
$966$	0	0
$967$	1.49084	0.0479423	0.0239711	−	0.999713i	$-0.492369\pi$
$967$	1.49084	0.0479423	0.0239711	+	0.999713i	$0.492369\pi$
$968$	0	0
$969$	−69.8568	−2.24412
$970$	0	0
$971$	12.6733	0.406706	0.203353	−	0.979105i	$-0.434816\pi$
$971$	12.6733	0.406706	0.203353	+	0.979105i	$0.434816\pi$
$972$	0	0
$973$	−20.7017	−0.663665
$974$	0	0
$975$	−38.5483	−1.23454
$976$	0	0
$977$	−8.31439	−0.266001	−0.133000	−	0.991116i	$-0.542461\pi$
$977$	−8.31439	−0.266001	−0.133000	+	0.991116i	$0.542461\pi$
$978$	0	0
$979$	4.02129	0.128521
$980$	0	0
$981$	3.64517	0.116381
$982$	0	0
$983$	8.61543	0.274790	0.137395	−	0.990516i	$-0.456127\pi$
$983$	8.61543	0.274790	0.137395	+	0.990516i	$0.456127\pi$
$984$	0	0
$985$	−5.16746	−0.164649
$986$	0	0
$987$	−7.83216	−0.249300
$988$	0	0
$989$	43.9844	1.39862
$990$	0	0
$991$	54.2244	1.72249	0.861247	−	0.508186i	$-0.169684\pi$
$991$	54.2244	1.72249	0.861247	+	0.508186i	$0.169684\pi$
$992$	0	0
$993$	19.6019	0.622048
$994$	0	0
$995$	3.31365	0.105050
$996$	0	0
$997$	17.5362	0.555376	0.277688	−	0.960671i	$-0.410432\pi$
$997$	17.5362	0.555376	0.277688	+	0.960671i	$0.410432\pi$
$998$	0	0
$999$	−52.6382	−1.66540

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4928.2.a.cj.1.3		4
4.3	odd	2		4928.2.a.cf.1.2		4
8.3	odd	2		2464.2.a.v.1.3	yes	4
8.5	even	2		2464.2.a.s.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2464.2.a.s.1.2	✓	4	8.5	even	2
2464.2.a.v.1.3	yes	4	8.3	odd	2
4928.2.a.cf.1.2		4	4.3	odd	2
4928.2.a.cj.1.3		4	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:	\( N \)	\(=\)	\( 4928 = 2^{6} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4928.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26438 q^{3} -0.401352 q^{5} -1.00000 q^{7} -1.40135 q^{9} +O(q^{10})\)	\(q+1.26438 q^{3} -0.401352 q^{5} -1.00000 q^{7} -1.40135 q^{9} -1.00000 q^{11} +6.30059 q^{13} -0.507460 q^{15} +6.93545 q^{17} -7.96632 q^{19} -1.26438 q^{21} +4.76226 q^{23} -4.83892 q^{25} -5.56497 q^{27} -0.528753 q^{29} -3.37048 q^{31} -1.26438 q^{33} +0.401352 q^{35} +9.45886 q^{37} +7.96632 q^{39} -3.13698 q^{41} +9.23604 q^{43} +0.562435 q^{45} +6.19448 q^{47} +1.00000 q^{49} +8.76902 q^{51} -4.49507 q^{53} +0.401352 q^{55} -10.0724 q^{57} +11.3368 q^{59} +12.8293 q^{61} +1.40135 q^{63} -2.52875 q^{65} -5.56497 q^{67} +6.02129 q^{69} +0.507460 q^{71} -7.73815 q^{73} -6.11821 q^{75} +1.00000 q^{77} -1.63909 q^{79} -2.83216 q^{81} +7.96632 q^{83} -2.78356 q^{85} -0.668543 q^{87} -4.02129 q^{89} -6.30059 q^{91} -4.26156 q^{93} +3.19730 q^{95} +7.35275 q^{97} +1.40135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 5 q^{5} - 4 q^{7} + q^{9}+O(q^{10}) \) 4 * q + q^3 + 5 * q^5 - 4 * q^7 + q^9	\( 4 q + q^{3} + 5 q^{5} - 4 q^{7} + q^{9} - 4 q^{11} + 6 q^{13} + 5 q^{15} - 2 q^{19} - q^{21} + 9 q^{23} + 9 q^{25} + q^{27} + 6 q^{29} - 9 q^{31} - q^{33} - 5 q^{35} + 15 q^{37} + 2 q^{39} - 10 q^{41} - 10 q^{43} + 24 q^{45} + 6 q^{47} + 4 q^{49} - 8 q^{51} + 20 q^{53} - 5 q^{55} - 10 q^{57} + 11 q^{59} + 24 q^{61} - q^{63} - 2 q^{65} + q^{67} + 23 q^{69} - 5 q^{71} + 10 q^{73} + 22 q^{75} + 4 q^{77} - 10 q^{79} - 16 q^{81} + 2 q^{83} - 24 q^{87} - 15 q^{89} - 6 q^{91} + 19 q^{93} + 26 q^{95} + 7 q^{97} - q^{99}+O(q^{100}) \) 4 * q + q^3 + 5 * q^5 - 4 * q^7 + q^9 - 4 * q^11 + 6 * q^13 + 5 * q^15 - 2 * q^19 - q^21 + 9 * q^23 + 9 * q^25 + q^27 + 6 * q^29 - 9 * q^31 - q^33 - 5 * q^35 + 15 * q^37 + 2 * q^39 - 10 * q^41 - 10 * q^43 + 24 * q^45 + 6 * q^47 + 4 * q^49 - 8 * q^51 + 20 * q^53 - 5 * q^55 - 10 * q^57 + 11 * q^59 + 24 * q^61 - q^63 - 2 * q^65 + q^67 + 23 * q^69 - 5 * q^71 + 10 * q^73 + 22 * q^75 + 4 * q^77 - 10 * q^79 - 16 * q^81 + 2 * q^83 - 24 * q^87 - 15 * q^89 - 6 * q^91 + 19 * q^93 + 26 * q^95 + 7 * q^97 - q^99

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