LMFDB - Embedded newform 7728.2.a.cc.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.75645.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 13x^{2} + 22x - 8 $ x^4 - x^3 - 13x^2 + 22x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.2
Root			$0.537835$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.537835 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.537835 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.264331 q^{11} +3.16373 q^{13} +0.537835 q^{15} -1.70156 q^{17} +1.73567 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.71073 q^{25} +1.00000 q^{27} -2.62589 q^{29} -2.62589 q^{31} +0.264331 q^{33} +0.537835 q^{35} +3.70156 q^{37} +3.16373 q^{39} +0.659999 q^{41} +7.71073 q^{43} +0.537835 q^{45} +2.89022 q^{47} +1.00000 q^{49} -1.70156 q^{51} +11.2269 q^{53} +0.142167 q^{55} +1.73567 q^{57} +7.97507 q^{59} -6.60096 q^{61} +1.00000 q^{63} +1.70156 q^{65} -7.26096 q^{67} +1.00000 q^{69} +10.6351 q^{71} +7.17290 q^{73} -4.71073 q^{75} +0.264331 q^{77} -7.50035 q^{79} +1.00000 q^{81} +14.0290 q^{83} -0.915159 q^{85} -2.62589 q^{87} +6.08806 q^{89} +3.16373 q^{91} -2.62589 q^{93} +0.933503 q^{95} -11.5004 q^{97} +0.264331 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + q^{5} + 4 q^{7} + 4 q^{9} + 7 q^{11} + q^{13} + q^{15} + 6 q^{17} + q^{19} + 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 7 q^{33} + q^{35} + 2 q^{37} + q^{39} - q^{41} + 5 q^{43} + q^{45} + 7 q^{47} + 4 q^{49} + 6 q^{51} + 4 q^{53} + 9 q^{55} + q^{57} + 12 q^{59} + 4 q^{61} + 4 q^{63} - 6 q^{65} + 5 q^{67} + 4 q^{69} + 19 q^{71} + 4 q^{73} + 7 q^{75} + 7 q^{77} + 18 q^{79} + 4 q^{81} + 20 q^{83} - 19 q^{85} + 15 q^{89} + q^{91} - 7 q^{95} + 2 q^{97} + 7 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + q^5 + 4 * q^7 + 4 * q^9 + 7 * q^11 + q^13 + q^15 + 6 * q^17 + q^19 + 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 7 * q^33 + q^35 + 2 * q^37 + q^39 - q^41 + 5 * q^43 + q^45 + 7 * q^47 + 4 * q^49 + 6 * q^51 + 4 * q^53 + 9 * q^55 + q^57 + 12 * q^59 + 4 * q^61 + 4 * q^63 - 6 * q^65 + 5 * q^67 + 4 * q^69 + 19 * q^71 + 4 * q^73 + 7 * q^75 + 7 * q^77 + 18 * q^79 + 4 * q^81 + 20 * q^83 - 19 * q^85 + 15 * q^89 + q^91 - 7 * q^95 + 2 * q^97 + 7 * 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
$5$	0.537835	0.240527	0.120263	−	0.992742i	$-0.461626\pi$
$5$	0.537835	0.240527	0.120263	+	0.992742i	$0.461626\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.264331	0.0796989	0.0398495	−	0.999206i	$-0.487312\pi$
$11$	0.264331	0.0796989	0.0398495	+	0.999206i	$0.487312\pi$
$12$	0	0
$13$	3.16373	0.877460	0.438730	−	0.898619i	$-0.355428\pi$
$13$	3.16373	0.877460	0.438730	+	0.898619i	$0.355428\pi$
$14$	0	0
$15$	0.537835	0.138868
$16$	0	0
$17$	−1.70156	−0.412689	−0.206345	−	0.978479i	$-0.566157\pi$
$17$	−1.70156	−0.412689	−0.206345	+	0.978479i	$0.566157\pi$
$18$	0	0
$19$	1.73567	0.398190	0.199095	−	0.979980i	$-0.436200\pi$
$19$	1.73567	0.398190	0.199095	+	0.979980i	$0.436200\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.71073	−0.942147
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.62589	−0.487616	−0.243808	−	0.969824i	$-0.578397\pi$
$29$	−2.62589	−0.487616	−0.243808	+	0.969824i	$0.578397\pi$
$30$	0	0
$31$	−2.62589	−0.471624	−0.235812	−	0.971799i	$-0.575775\pi$
$31$	−2.62589	−0.471624	−0.235812	+	0.971799i	$0.575775\pi$
$32$	0	0
$33$	0.264331	0.0460142
$34$	0	0
$35$	0.537835	0.0909106
$36$	0	0
$37$	3.70156	0.608533	0.304267	−	0.952587i	$-0.401589\pi$
$37$	3.70156	0.608533	0.304267	+	0.952587i	$0.401589\pi$
$38$	0	0
$39$	3.16373	0.506602
$40$	0	0
$41$	0.659999	0.103075	0.0515373	−	0.998671i	$-0.483588\pi$
$41$	0.659999	0.103075	0.0515373	+	0.998671i	$0.483588\pi$
$42$	0	0
$43$	7.71073	1.17588	0.587938	−	0.808906i	$-0.299940\pi$
$43$	7.71073	1.17588	0.587938	+	0.808906i	$0.299940\pi$
$44$	0	0
$45$	0.537835	0.0801756
$46$	0	0
$47$	2.89022	0.421583	0.210791	−	0.977531i	$-0.432396\pi$
$47$	2.89022	0.421583	0.210791	+	0.977531i	$0.432396\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.70156	−0.238266
$52$	0	0
$53$	11.2269	1.54213	0.771063	−	0.636758i	$-0.219725\pi$
$53$	11.2269	1.54213	0.771063	+	0.636758i	$0.219725\pi$
$54$	0	0
$55$	0.142167	0.0191697
$56$	0	0
$57$	1.73567	0.229895
$58$	0	0
$59$	7.97507	1.03827	0.519133	−	0.854694i	$-0.326255\pi$
$59$	7.97507	1.03827	0.519133	+	0.854694i	$0.326255\pi$
$60$	0	0
$61$	−6.60096	−0.845166	−0.422583	−	0.906324i	$-0.638877\pi$
$61$	−6.60096	−0.845166	−0.422583	+	0.906324i	$0.638877\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.70156	0.211053
$66$	0	0
$67$	−7.26096	−0.887067	−0.443534	−	0.896258i	$-0.646275\pi$
$67$	−7.26096	−0.887067	−0.443534	+	0.896258i	$0.646275\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	10.6351	1.26215	0.631075	−	0.775722i	$-0.282614\pi$
$71$	10.6351	1.26215	0.631075	+	0.775722i	$0.282614\pi$
$72$	0	0
$73$	7.17290	0.839524	0.419762	−	0.907634i	$-0.362113\pi$
$73$	7.17290	0.839524	0.419762	+	0.907634i	$0.362113\pi$
$74$	0	0
$75$	−4.71073	−0.543949
$76$	0	0
$77$	0.264331	0.0301234
$78$	0	0
$79$	−7.50035	−0.843856	−0.421928	−	0.906629i	$-0.638646\pi$
$79$	−7.50035	−0.843856	−0.421928	+	0.906629i	$0.638646\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.0290	1.53988	0.769942	−	0.638113i	$-0.220285\pi$
$83$	14.0290	1.53988	0.769942	+	0.638113i	$0.220285\pi$
$84$	0	0
$85$	−0.915159	−0.0992629
$86$	0	0
$87$	−2.62589	−0.281525
$88$	0	0
$89$	6.08806	0.645333	0.322666	−	0.946513i	$-0.395421\pi$
$89$	6.08806	0.645333	0.322666	+	0.946513i	$0.395421\pi$
$90$	0	0
$91$	3.16373	0.331649
$92$	0	0
$93$	−2.62589	−0.272292
$94$	0	0
$95$	0.933503	0.0957753
$96$	0	0
$97$	−11.5004	−1.16768	−0.583842	−	0.811867i	$-0.698451\pi$
$97$	−11.5004	−1.16768	−0.583842	+	0.811867i	$0.698451\pi$
$98$	0	0
$99$	0.264331	0.0265663
$100$	0	0
$101$	7.37819	0.734157	0.367079	−	0.930190i	$-0.380358\pi$
$101$	7.37819	0.734157	0.367079	+	0.930190i	$0.380358\pi$
$102$	0	0
$103$	8.84035	0.871066	0.435533	−	0.900173i	$-0.356560\pi$
$103$	8.84035	0.871066	0.435533	+	0.900173i	$0.356560\pi$
$104$	0	0
$105$	0.537835	0.0524873
$106$	0	0
$107$	9.55940	0.924142	0.462071	−	0.886843i	$-0.347107\pi$
$107$	9.55940	0.924142	0.462071	+	0.886843i	$0.347107\pi$
$108$	0	0
$109$	12.1354	1.16236	0.581181	−	0.813774i	$-0.302591\pi$
$109$	12.1354	1.16236	0.581181	+	0.813774i	$0.302591\pi$
$110$	0	0
$111$	3.70156	0.351337
$112$	0	0
$113$	−1.06650	−0.100328	−0.0501638	−	0.998741i	$-0.515974\pi$
$113$	−1.06650	−0.100328	−0.0501638	+	0.998741i	$0.515974\pi$
$114$	0	0
$115$	0.537835	0.0501533
$116$	0	0
$117$	3.16373	0.292487
$118$	0	0
$119$	−1.70156	−0.155982
$120$	0	0
$121$	−10.9301	−0.993648
$122$	0	0
$123$	0.659999	0.0595101
$124$	0	0
$125$	−5.22277	−0.467139
$126$	0	0
$127$	−2.20529	−0.195688	−0.0978439	−	0.995202i	$-0.531195\pi$
$127$	−2.20529	−0.195688	−0.0978439	+	0.995202i	$0.531195\pi$
$128$	0	0
$129$	7.71073	0.678892
$130$	0	0
$131$	−16.3407	−1.42769	−0.713847	−	0.700301i	$-0.753049\pi$
$131$	−16.3407	−1.42769	−0.713847	+	0.700301i	$0.753049\pi$
$132$	0	0
$133$	1.73567	0.150502
$134$	0	0
$135$	0.537835	0.0462894
$136$	0	0
$137$	−14.3906	−1.22947	−0.614735	−	0.788734i	$-0.710737\pi$
$137$	−14.3906	−1.22947	−0.614735	+	0.788734i	$0.710737\pi$
$138$	0	0
$139$	9.39395	0.796785	0.398392	−	0.917215i	$-0.369568\pi$
$139$	9.39395	0.796785	0.398392	+	0.917215i	$0.369568\pi$
$140$	0	0
$141$	2.89022	0.243401
$142$	0	0
$143$	0.836272	0.0699326
$144$	0	0
$145$	−1.41230	−0.117285
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−16.6391	−1.36313	−0.681566	−	0.731757i	$-0.738701\pi$
$149$	−16.6391	−1.36313	−0.681566	+	0.731757i	$0.738701\pi$
$150$	0	0
$151$	20.3300	1.65443	0.827217	−	0.561882i	$-0.189922\pi$
$151$	20.3300	1.65443	0.827217	+	0.561882i	$0.189922\pi$
$152$	0	0
$153$	−1.70156	−0.137563
$154$	0	0
$155$	−1.41230	−0.113438
$156$	0	0
$157$	−21.8909	−1.74709	−0.873543	−	0.486746i	$-0.838184\pi$
$157$	−21.8909	−1.74709	−0.873543	+	0.486746i	$0.838184\pi$
$158$	0	0
$159$	11.2269	0.890347
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−5.77386	−0.452243	−0.226122	−	0.974099i	$-0.572605\pi$
$163$	−5.77386	−0.452243	−0.226122	+	0.974099i	$0.572605\pi$
$164$	0	0
$165$	0.142167	0.0110677
$166$	0	0
$167$	−7.46625	−0.577756	−0.288878	−	0.957366i	$-0.593282\pi$
$167$	−7.46625	−0.577756	−0.288878	+	0.957366i	$0.593282\pi$
$168$	0	0
$169$	−2.99083	−0.230064
$170$	0	0
$171$	1.73567	0.132730
$172$	0	0
$173$	−5.15456	−0.391894	−0.195947	−	0.980615i	$-0.562778\pi$
$173$	−5.15456	−0.391894	−0.195947	+	0.980615i	$0.562778\pi$
$174$	0	0
$175$	−4.71073	−0.356098
$176$	0	0
$177$	7.97507	0.599443
$178$	0	0
$179$	18.6641	1.39502	0.697509	−	0.716576i	$-0.254292\pi$
$179$	18.6641	1.39502	0.697509	+	0.716576i	$0.254292\pi$
$180$	0	0
$181$	11.2808	0.838495	0.419248	−	0.907872i	$-0.362294\pi$
$181$	11.2808	0.838495	0.419248	+	0.907872i	$0.362294\pi$
$182$	0	0
$183$	−6.60096	−0.487957
$184$	0	0
$185$	1.99083	0.146369
$186$	0	0
$187$	−0.449776	−0.0328909
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	11.3042	0.817946	0.408973	−	0.912547i	$-0.365887\pi$
$191$	11.3042	0.817946	0.408973	+	0.912547i	$0.365887\pi$
$192$	0	0
$193$	2.36156	0.169989	0.0849945	−	0.996381i	$-0.472913\pi$
$193$	2.36156	0.169989	0.0849945	+	0.996381i	$0.472913\pi$
$194$	0	0
$195$	1.70156	0.121851
$196$	0	0
$197$	−14.9625	−1.06604	−0.533018	−	0.846104i	$-0.678942\pi$
$197$	−14.9625	−1.06604	−0.533018	+	0.846104i	$0.678942\pi$
$198$	0	0
$199$	−13.7739	−0.976403	−0.488201	−	0.872731i	$-0.662347\pi$
$199$	−13.7739	−0.976403	−0.488201	+	0.872731i	$0.662347\pi$
$200$	0	0
$201$	−7.26096	−0.512148
$202$	0	0
$203$	−2.62589	−0.184302
$204$	0	0
$205$	0.354971	0.0247922
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	0.458792	0.0317353
$210$	0	0
$211$	18.7862	1.29330	0.646649	−	0.762787i	$-0.276170\pi$
$211$	18.7862	1.29330	0.646649	+	0.762787i	$0.276170\pi$
$212$	0	0
$213$	10.6351	0.728703
$214$	0	0
$215$	4.14710	0.282830
$216$	0	0
$217$	−2.62589	−0.178257
$218$	0	0
$219$	7.17290	0.484700
$220$	0	0
$221$	−5.38328	−0.362119
$222$	0	0
$223$	5.08484	0.340506	0.170253	−	0.985400i	$-0.445541\pi$
$223$	5.08484	0.340506	0.170253	+	0.985400i	$0.445541\pi$
$224$	0	0
$225$	−4.71073	−0.314049
$226$	0	0
$227$	9.08484	0.602982	0.301491	−	0.953469i	$-0.402516\pi$
$227$	9.08484	0.602982	0.301491	+	0.953469i	$0.402516\pi$
$228$	0	0
$229$	0.334046	0.0220744	0.0110372	−	0.999939i	$-0.496487\pi$
$229$	0.334046	0.0220744	0.0110372	+	0.999939i	$0.496487\pi$
$230$	0	0
$231$	0.264331	0.0173917
$232$	0	0
$233$	13.8080	0.904590	0.452295	−	0.891868i	$-0.350605\pi$
$233$	13.8080	0.904590	0.452295	+	0.891868i	$0.350605\pi$
$234$	0	0
$235$	1.55446	0.101402
$236$	0	0
$237$	−7.50035	−0.487200
$238$	0	0
$239$	23.8975	1.54580	0.772901	−	0.634526i	$-0.218805\pi$
$239$	23.8975	1.54580	0.772901	+	0.634526i	$0.218805\pi$
$240$	0	0
$241$	22.2576	1.43374	0.716869	−	0.697208i	$-0.245575\pi$
$241$	22.2576	1.43374	0.716869	+	0.697208i	$0.245575\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.537835	0.0343610
$246$	0	0
$247$	5.49118	0.349396
$248$	0	0
$249$	14.0290	0.889053
$250$	0	0
$251$	7.67254	0.484287	0.242143	−	0.970240i	$-0.422150\pi$
$251$	7.67254	0.484287	0.242143	+	0.970240i	$0.422150\pi$
$252$	0	0
$253$	0.264331	0.0166184
$254$	0	0
$255$	−0.915159	−0.0573095
$256$	0	0
$257$	2.95844	0.184542	0.0922711	−	0.995734i	$-0.470587\pi$
$257$	2.95844	0.184542	0.0922711	+	0.995734i	$0.470587\pi$
$258$	0	0
$259$	3.70156	0.230004
$260$	0	0
$261$	−2.62589	−0.162539
$262$	0	0
$263$	8.87955	0.547537	0.273768	−	0.961796i	$-0.411730\pi$
$263$	8.87955	0.547537	0.273768	+	0.961796i	$0.411730\pi$
$264$	0	0
$265$	6.03819	0.370923
$266$	0	0
$267$	6.08806	0.372583
$268$	0	0
$269$	13.4912	0.822572	0.411286	−	0.911506i	$-0.365080\pi$
$269$	13.4912	0.822572	0.411286	+	0.911506i	$0.365080\pi$
$270$	0	0
$271$	−21.9791	−1.33514	−0.667569	−	0.744548i	$-0.732665\pi$
$271$	−21.9791	−1.33514	−0.667569	+	0.744548i	$0.732665\pi$
$272$	0	0
$273$	3.16373	0.191478
$274$	0	0
$275$	−1.24519	−0.0750881
$276$	0	0
$277$	−19.6084	−1.17816	−0.589078	−	0.808076i	$-0.700509\pi$
$277$	−19.6084	−1.17816	−0.589078	+	0.808076i	$0.700509\pi$
$278$	0	0
$279$	−2.62589	−0.157208
$280$	0	0
$281$	10.0764	0.601106	0.300553	−	0.953765i	$-0.402829\pi$
$281$	10.0764	0.601106	0.300553	+	0.953765i	$0.402829\pi$
$282$	0	0
$283$	−28.6142	−1.70094	−0.850469	−	0.526025i	$-0.823682\pi$
$283$	−28.6142	−1.70094	−0.850469	+	0.526025i	$0.823682\pi$
$284$	0	0
$285$	0.933503	0.0552959
$286$	0	0
$287$	0.659999	0.0389585
$288$	0	0
$289$	−14.1047	−0.829687
$290$	0	0
$291$	−11.5004	−0.674163
$292$	0	0
$293$	−25.3716	−1.48222	−0.741112	−	0.671381i	$-0.765701\pi$
$293$	−25.3716	−1.48222	−0.741112	+	0.671381i	$0.765701\pi$
$294$	0	0
$295$	4.28927	0.249731
$296$	0	0
$297$	0.264331	0.0153381
$298$	0	0
$299$	3.16373	0.182963
$300$	0	0
$301$	7.71073	0.444439
$302$	0	0
$303$	7.37819	0.423866
$304$	0	0
$305$	−3.55022	−0.203285
$306$	0	0
$307$	12.8204	0.731696	0.365848	−	0.930675i	$-0.380779\pi$
$307$	12.8204	0.731696	0.365848	+	0.930675i	$0.380779\pi$
$308$	0	0
$309$	8.84035	0.502910
$310$	0	0
$311$	28.2633	1.60267	0.801333	−	0.598218i	$-0.204125\pi$
$311$	28.2633	1.60267	0.801333	+	0.598218i	$0.204125\pi$
$312$	0	0
$313$	9.96589	0.563306	0.281653	−	0.959516i	$-0.409117\pi$
$313$	9.96589	0.563306	0.281653	+	0.959516i	$0.409117\pi$
$314$	0	0
$315$	0.537835	0.0303035
$316$	0	0
$317$	12.6641	0.711286	0.355643	−	0.934622i	$-0.384262\pi$
$317$	12.6641	0.711286	0.355643	+	0.934622i	$0.384262\pi$
$318$	0	0
$319$	−0.694106	−0.0388625
$320$	0	0
$321$	9.55940	0.533553
$322$	0	0
$323$	−2.95335	−0.164329
$324$	0	0
$325$	−14.9035	−0.826696
$326$	0	0
$327$	12.1354	0.671090
$328$	0	0
$329$	2.89022	0.159343
$330$	0	0
$331$	17.3690	0.954688	0.477344	−	0.878717i	$-0.341600\pi$
$331$	17.3690	0.954688	0.477344	+	0.878717i	$0.341600\pi$
$332$	0	0
$333$	3.70156	0.202844
$334$	0	0
$335$	−3.90519	−0.213364
$336$	0	0
$337$	20.2143	1.10114	0.550572	−	0.834788i	$-0.314410\pi$
$337$	20.2143	1.10114	0.550572	+	0.834788i	$0.314410\pi$
$338$	0	0
$339$	−1.06650	−0.0579242
$340$	0	0
$341$	−0.694106	−0.0375879
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.537835	0.0289560
$346$	0	0
$347$	1.57531	0.0845673	0.0422837	−	0.999106i	$-0.486537\pi$
$347$	1.57531	0.0845673	0.0422837	+	0.999106i	$0.486537\pi$
$348$	0	0
$349$	−10.2535	−0.548857	−0.274429	−	0.961607i	$-0.588489\pi$
$349$	−10.2535	−0.548857	−0.274429	+	0.961607i	$0.588489\pi$
$350$	0	0
$351$	3.16373	0.168867
$352$	0	0
$353$	−2.02902	−0.107994	−0.0539968	−	0.998541i	$-0.517196\pi$
$353$	−2.02902	−0.107994	−0.0539968	+	0.998541i	$0.517196\pi$
$354$	0	0
$355$	5.71991	0.303581
$356$	0	0
$357$	−1.70156	−0.0900562
$358$	0	0
$359$	−5.89109	−0.310920	−0.155460	−	0.987842i	$-0.549686\pi$
$359$	−5.89109	−0.310920	−0.155460	+	0.987842i	$0.549686\pi$
$360$	0	0
$361$	−15.9875	−0.841445
$362$	0	0
$363$	−10.9301	−0.573683
$364$	0	0
$365$	3.85783	0.201928
$366$	0	0
$367$	−2.14726	−0.112086	−0.0560429	−	0.998428i	$-0.517848\pi$
$367$	−2.14726	−0.112086	−0.0560429	+	0.998428i	$0.517848\pi$
$368$	0	0
$369$	0.659999	0.0343582
$370$	0	0
$371$	11.2269	0.582869
$372$	0	0
$373$	−9.08977	−0.470651	−0.235325	−	0.971917i	$-0.575616\pi$
$373$	−9.08977	−0.470651	−0.235325	+	0.971917i	$0.575616\pi$
$374$	0	0
$375$	−5.22277	−0.269703
$376$	0	0
$377$	−8.30761	−0.427864
$378$	0	0
$379$	4.06821	0.208970	0.104485	−	0.994526i	$-0.466681\pi$
$379$	4.06821	0.208970	0.104485	+	0.994526i	$0.466681\pi$
$380$	0	0
$381$	−2.20529	−0.112980
$382$	0	0
$383$	−4.99678	−0.255324	−0.127662	−	0.991818i	$-0.540747\pi$
$383$	−4.99678	−0.255324	−0.127662	+	0.991818i	$0.540747\pi$
$384$	0	0
$385$	0.142167	0.00724548
$386$	0	0
$387$	7.71073	0.391959
$388$	0	0
$389$	−14.3317	−0.726646	−0.363323	−	0.931663i	$-0.618358\pi$
$389$	−14.3317	−0.726646	−0.363323	+	0.931663i	$0.618358\pi$
$390$	0	0
$391$	−1.70156	−0.0860517
$392$	0	0
$393$	−16.3407	−0.824280
$394$	0	0
$395$	−4.03395	−0.202970
$396$	0	0
$397$	8.54701	0.428962	0.214481	−	0.976728i	$-0.431194\pi$
$397$	8.54701	0.428962	0.214481	+	0.976728i	$0.431194\pi$
$398$	0	0
$399$	1.73567	0.0868921
$400$	0	0
$401$	−24.4920	−1.22307	−0.611537	−	0.791216i	$-0.709449\pi$
$401$	−24.4920	−1.22307	−0.611537	+	0.791216i	$0.709449\pi$
$402$	0	0
$403$	−8.30761	−0.413832
$404$	0	0
$405$	0.537835	0.0267252
$406$	0	0
$407$	0.978439	0.0484994
$408$	0	0
$409$	−0.374816	−0.0185335	−0.00926673	−	0.999957i	$-0.502950\pi$
$409$	−0.374816	−0.0185335	−0.00926673	+	0.999957i	$0.502950\pi$
$410$	0	0
$411$	−14.3906	−0.709835
$412$	0	0
$413$	7.97507	0.392427
$414$	0	0
$415$	7.54529	0.370384
$416$	0	0
$417$	9.39395	0.460024
$418$	0	0
$419$	31.9094	1.55888	0.779439	−	0.626478i	$-0.215504\pi$
$419$	31.9094	1.55888	0.779439	+	0.626478i	$0.215504\pi$
$420$	0	0
$421$	21.2218	1.03429	0.517143	−	0.855899i	$-0.326996\pi$
$421$	21.2218	1.03429	0.517143	+	0.855899i	$0.326996\pi$
$422$	0	0
$423$	2.89022	0.140528
$424$	0	0
$425$	8.01561	0.388814
$426$	0	0
$427$	−6.60096	−0.319443
$428$	0	0
$429$	0.836272	0.0403756
$430$	0	0
$431$	−9.74484	−0.469392	−0.234696	−	0.972069i	$-0.575410\pi$
$431$	−9.74484	−0.469392	−0.234696	+	0.972069i	$0.575410\pi$
$432$	0	0
$433$	31.3623	1.50717	0.753587	−	0.657348i	$-0.228322\pi$
$433$	31.3623	1.50717	0.753587	+	0.657348i	$0.228322\pi$
$434$	0	0
$435$	−1.41230	−0.0677144
$436$	0	0
$437$	1.73567	0.0830283
$438$	0	0
$439$	−8.44303	−0.402964	−0.201482	−	0.979492i	$-0.564576\pi$
$439$	−8.44303	−0.402964	−0.201482	+	0.979492i	$0.564576\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	11.8636	0.563655	0.281828	−	0.959465i	$-0.409059\pi$
$443$	11.8636	0.563655	0.281828	+	0.959465i	$0.409059\pi$
$444$	0	0
$445$	3.27437	0.155220
$446$	0	0
$447$	−16.6391	−0.787005
$448$	0	0
$449$	−25.3373	−1.19574	−0.597871	−	0.801592i	$-0.703987\pi$
$449$	−25.3373	−1.19574	−0.597871	+	0.801592i	$0.703987\pi$
$450$	0	0
$451$	0.174459	0.00821493
$452$	0	0
$453$	20.3300	0.955188
$454$	0	0
$455$	1.70156	0.0797705
$456$	0	0
$457$	21.8975	1.02432	0.512161	−	0.858889i	$-0.328845\pi$
$457$	21.8975	1.02432	0.512161	+	0.858889i	$0.328845\pi$
$458$	0	0
$459$	−1.70156	−0.0794221
$460$	0	0
$461$	−29.3624	−1.36754	−0.683772	−	0.729695i	$-0.739662\pi$
$461$	−29.3624	−1.36754	−0.683772	+	0.729695i	$0.739662\pi$
$462$	0	0
$463$	−23.4415	−1.08942	−0.544709	−	0.838625i	$-0.683360\pi$
$463$	−23.4415	−1.08942	−0.544709	+	0.838625i	$0.683360\pi$
$464$	0	0
$465$	−1.41230	−0.0654937
$466$	0	0
$467$	−2.46812	−0.114211	−0.0571055	−	0.998368i	$-0.518187\pi$
$467$	−2.46812	−0.114211	−0.0571055	+	0.998368i	$0.518187\pi$
$468$	0	0
$469$	−7.26096	−0.335280
$470$	0	0
$471$	−21.8909	−1.00868
$472$	0	0
$473$	2.03819	0.0937160
$474$	0	0
$475$	−8.17627	−0.375153
$476$	0	0
$477$	11.2269	0.514042
$478$	0	0
$479$	1.39889	0.0639167	0.0319584	−	0.999489i	$-0.489826\pi$
$479$	1.39889	0.0639167	0.0319584	+	0.999489i	$0.489826\pi$
$480$	0	0
$481$	11.7107	0.533964
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−6.18529	−0.280859
$486$	0	0
$487$	−41.9541	−1.90112	−0.950560	−	0.310541i	$-0.899490\pi$
$487$	−41.9541	−1.90112	−0.950560	+	0.310541i	$0.899490\pi$
$488$	0	0
$489$	−5.77386	−0.261103
$490$	0	0
$491$	31.8553	1.43761	0.718805	−	0.695211i	$-0.244689\pi$
$491$	31.8553	1.43761	0.718805	+	0.695211i	$0.244689\pi$
$492$	0	0
$493$	4.46812	0.201234
$494$	0	0
$495$	0.142167	0.00638991
$496$	0	0
$497$	10.6351	0.477048
$498$	0	0
$499$	29.0099	1.29866	0.649330	−	0.760507i	$-0.275049\pi$
$499$	29.0099	1.29866	0.649330	+	0.760507i	$0.275049\pi$
$500$	0	0
$501$	−7.46625	−0.333567
$502$	0	0
$503$	42.8769	1.91179	0.955893	−	0.293715i	$-0.0948918\pi$
$503$	42.8769	1.91179	0.955893	+	0.293715i	$0.0948918\pi$
$504$	0	0
$505$	3.96825	0.176585
$506$	0	0
$507$	−2.99083	−0.132827
$508$	0	0
$509$	1.48710	0.0659146	0.0329573	−	0.999457i	$-0.489507\pi$
$509$	1.48710	0.0659146	0.0329573	+	0.999457i	$0.489507\pi$
$510$	0	0
$511$	7.17290	0.317310
$512$	0	0
$513$	1.73567	0.0766316
$514$	0	0
$515$	4.75465	0.209515
$516$	0	0
$517$	0.763977	0.0335997
$518$	0	0
$519$	−5.15456	−0.226260
$520$	0	0
$521$	−7.06892	−0.309695	−0.154848	−	0.987938i	$-0.549489\pi$
$521$	−7.06892	−0.309695	−0.154848	+	0.987938i	$0.549489\pi$
$522$	0	0
$523$	−23.9659	−1.04796	−0.523978	−	0.851732i	$-0.675552\pi$
$523$	−23.9659	−1.04796	−0.523978	+	0.851732i	$0.675552\pi$
$524$	0	0
$525$	−4.71073	−0.205593
$526$	0	0
$527$	4.46812	0.194634
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.97507	0.346088
$532$	0	0
$533$	2.08806	0.0904438
$534$	0	0
$535$	5.14137	0.222281
$536$	0	0
$537$	18.6641	0.805415
$538$	0	0
$539$	0.264331	0.0113856
$540$	0	0
$541$	−17.7280	−0.762186	−0.381093	−	0.924537i	$-0.624452\pi$
$541$	−17.7280	−0.762186	−0.381093	+	0.924537i	$0.624452\pi$
$542$	0	0
$543$	11.2808	0.484106
$544$	0	0
$545$	6.52685	0.279579
$546$	0	0
$547$	9.41230	0.402441	0.201220	−	0.979546i	$-0.435509\pi$
$547$	9.41230	0.402441	0.201220	+	0.979546i	$0.435509\pi$
$548$	0	0
$549$	−6.60096	−0.281722
$550$	0	0
$551$	−4.55768	−0.194164
$552$	0	0
$553$	−7.50035	−0.318948
$554$	0	0
$555$	1.99083	0.0845059
$556$	0	0
$557$	−15.8884	−0.673211	−0.336606	−	0.941646i	$-0.609279\pi$
$557$	−15.8884	−0.673211	−0.336606	+	0.941646i	$0.609279\pi$
$558$	0	0
$559$	24.3947	1.03178
$560$	0	0
$561$	−0.449776	−0.0189896
$562$	0	0
$563$	16.7246	0.704859	0.352429	−	0.935838i	$-0.385356\pi$
$563$	16.7246	0.704859	0.352429	+	0.935838i	$0.385356\pi$
$564$	0	0
$565$	−0.573599	−0.0241315
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	25.8411	1.08331	0.541657	−	0.840600i	$-0.317797\pi$
$569$	25.8411	1.08331	0.541657	+	0.840600i	$0.317797\pi$
$570$	0	0
$571$	−19.1081	−0.799650	−0.399825	−	0.916592i	$-0.630929\pi$
$571$	−19.1081	−0.799650	−0.399825	+	0.916592i	$0.630929\pi$
$572$	0	0
$573$	11.3042	0.472241
$574$	0	0
$575$	−4.71073	−0.196451
$576$	0	0
$577$	41.4612	1.72605	0.863025	−	0.505161i	$-0.168567\pi$
$577$	41.4612	1.72605	0.863025	+	0.505161i	$0.168567\pi$
$578$	0	0
$579$	2.36156	0.0981431
$580$	0	0
$581$	14.0290	0.582022
$582$	0	0
$583$	2.96761	0.122906
$584$	0	0
$585$	1.70156	0.0703509
$586$	0	0
$587$	−12.3563	−0.510000	−0.255000	−	0.966941i	$-0.582075\pi$
$587$	−12.3563	−0.510000	−0.255000	+	0.966941i	$0.582075\pi$
$588$	0	0
$589$	−4.55768	−0.187796
$590$	0	0
$591$	−14.9625	−0.615476
$592$	0	0
$593$	18.4788	0.758833	0.379417	−	0.925226i	$-0.376125\pi$
$593$	18.4788	0.758833	0.379417	+	0.925226i	$0.376125\pi$
$594$	0	0
$595$	−0.915159	−0.0375179
$596$	0	0
$597$	−13.7739	−0.563726
$598$	0	0
$599$	−35.2401	−1.43987	−0.719936	−	0.694041i	$-0.755829\pi$
$599$	−35.2401	−1.43987	−0.719936	+	0.694041i	$0.755829\pi$
$600$	0	0
$601$	−8.09873	−0.330354	−0.165177	−	0.986264i	$-0.552820\pi$
$601$	−8.09873	−0.330354	−0.165177	+	0.986264i	$0.552820\pi$
$602$	0	0
$603$	−7.26096	−0.295689
$604$	0	0
$605$	−5.87860	−0.238999
$606$	0	0
$607$	13.8553	0.562370	0.281185	−	0.959654i	$-0.409272\pi$
$607$	13.8553	0.562370	0.281185	+	0.959654i	$0.409272\pi$
$608$	0	0
$609$	−2.62589	−0.106407
$610$	0	0
$611$	9.14388	0.369922
$612$	0	0
$613$	−27.4753	−1.10972	−0.554858	−	0.831945i	$-0.687227\pi$
$613$	−27.4753	−1.10972	−0.554858	+	0.831945i	$0.687227\pi$
$614$	0	0
$615$	0.354971	0.0143138
$616$	0	0
$617$	−6.48373	−0.261025	−0.130512	−	0.991447i	$-0.541662\pi$
$617$	−6.48373	−0.261025	−0.130512	+	0.991447i	$0.541662\pi$
$618$	0	0
$619$	−13.1821	−0.529832	−0.264916	−	0.964271i	$-0.585344\pi$
$619$	−13.1821	−0.529832	−0.264916	+	0.964271i	$0.585344\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	6.08806	0.243913
$624$	0	0
$625$	20.7447	0.829787
$626$	0	0
$627$	0.458792	0.0183224
$628$	0	0
$629$	−6.29844	−0.251135
$630$	0	0
$631$	1.30933	0.0521234	0.0260617	−	0.999660i	$-0.491703\pi$
$631$	1.30933	0.0521234	0.0260617	+	0.999660i	$0.491703\pi$
$632$	0	0
$633$	18.7862	0.746686
$634$	0	0
$635$	−1.18608	−0.0470682
$636$	0	0
$637$	3.16373	0.125351
$638$	0	0
$639$	10.6351	0.420717
$640$	0	0
$641$	1.89618	0.0748946	0.0374473	−	0.999299i	$-0.488077\pi$
$641$	1.89618	0.0748946	0.0374473	+	0.999299i	$0.488077\pi$
$642$	0	0
$643$	−40.1056	−1.58161	−0.790804	−	0.612069i	$-0.790337\pi$
$643$	−40.1056	−1.58161	−0.790804	+	0.612069i	$0.790337\pi$
$644$	0	0
$645$	4.14710	0.163292
$646$	0	0
$647$	−49.2691	−1.93697	−0.968485	−	0.249074i	$-0.919874\pi$
$647$	−49.2691	−1.93697	−0.968485	+	0.249074i	$0.919874\pi$
$648$	0	0
$649$	2.10806	0.0827486
$650$	0	0
$651$	−2.62589	−0.102917
$652$	0	0
$653$	−20.4296	−0.799473	−0.399736	−	0.916630i	$-0.630898\pi$
$653$	−20.4296	−0.799473	−0.399736	+	0.916630i	$0.630898\pi$
$654$	0	0
$655$	−8.78860	−0.343399
$656$	0	0
$657$	7.17290	0.279841
$658$	0	0
$659$	−20.9965	−0.817906	−0.408953	−	0.912555i	$-0.634106\pi$
$659$	−20.9965	−0.817906	−0.408953	+	0.912555i	$0.634106\pi$
$660$	0	0
$661$	32.2576	1.25467	0.627337	−	0.778748i	$-0.284145\pi$
$661$	32.2576	1.25467	0.627337	+	0.778748i	$0.284145\pi$
$662$	0	0
$663$	−5.38328	−0.209069
$664$	0	0
$665$	0.933503	0.0361997
$666$	0	0
$667$	−2.62589	−0.101675
$668$	0	0
$669$	5.08484	0.196591
$670$	0	0
$671$	−1.74484	−0.0673588
$672$	0	0
$673$	23.5928	0.909436	0.454718	−	0.890635i	$-0.349740\pi$
$673$	23.5928	0.909436	0.454718	+	0.890635i	$0.349740\pi$
$674$	0	0
$675$	−4.71073	−0.181316
$676$	0	0
$677$	6.45220	0.247978	0.123989	−	0.992284i	$-0.460431\pi$
$677$	6.45220	0.247978	0.123989	+	0.992284i	$0.460431\pi$
$678$	0	0
$679$	−11.5004	−0.441343
$680$	0	0
$681$	9.08484	0.348132
$682$	0	0
$683$	−3.35576	−0.128405	−0.0642024	−	0.997937i	$-0.520450\pi$
$683$	−3.35576	−0.128405	−0.0642024	+	0.997937i	$0.520450\pi$
$684$	0	0
$685$	−7.73975	−0.295721
$686$	0	0
$687$	0.334046	0.0127447
$688$	0	0
$689$	35.5187	1.35315
$690$	0	0
$691$	−23.5927	−0.897507	−0.448753	−	0.893656i	$-0.648132\pi$
$691$	−23.5927	−0.897507	−0.448753	+	0.893656i	$0.648132\pi$
$692$	0	0
$693$	0.264331	0.0100411
$694$	0	0
$695$	5.05239	0.191648
$696$	0	0
$697$	−1.12303	−0.0425378
$698$	0	0
$699$	13.8080	0.522265
$700$	0	0
$701$	30.4905	1.15161	0.575806	−	0.817586i	$-0.304688\pi$
$701$	30.4905	1.15161	0.575806	+	0.817586i	$0.304688\pi$
$702$	0	0
$703$	6.42469	0.242312
$704$	0	0
$705$	1.55446	0.0585444
$706$	0	0
$707$	7.37819	0.277485
$708$	0	0
$709$	−10.4257	−0.391545	−0.195773	−	0.980649i	$-0.562721\pi$
$709$	−10.4257	−0.391545	−0.195773	+	0.980649i	$0.562721\pi$
$710$	0	0
$711$	−7.50035	−0.281285
$712$	0	0
$713$	−2.62589	−0.0983405
$714$	0	0
$715$	0.449776	0.0168207
$716$	0	0
$717$	23.8975	0.892469
$718$	0	0
$719$	6.92826	0.258380	0.129190	−	0.991620i	$-0.458762\pi$
$719$	6.92826	0.258380	0.129190	+	0.991620i	$0.458762\pi$
$720$	0	0
$721$	8.84035	0.329232
$722$	0	0
$723$	22.2576	0.827768
$724$	0	0
$725$	12.3699	0.459406
$726$	0	0
$727$	−24.8195	−0.920504	−0.460252	−	0.887788i	$-0.652241\pi$
$727$	−24.8195	−0.920504	−0.460252	+	0.887788i	$0.652241\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.1203	−0.485272
$732$	0	0
$733$	−16.2277	−0.599384	−0.299692	−	0.954036i	$-0.596884\pi$
$733$	−16.2277	−0.599384	−0.299692	+	0.954036i	$0.596884\pi$
$734$	0	0
$735$	0.537835	0.0198383
$736$	0	0
$737$	−1.91930	−0.0706983
$738$	0	0
$739$	−43.8712	−1.61383	−0.806915	−	0.590668i	$-0.798864\pi$
$739$	−43.8712	−1.61383	−0.806915	+	0.590668i	$0.798864\pi$
$740$	0	0
$741$	5.49118	0.201724
$742$	0	0
$743$	−19.3198	−0.708776	−0.354388	−	0.935098i	$-0.615311\pi$
$743$	−19.3198	−0.708776	−0.354388	+	0.935098i	$0.615311\pi$
$744$	0	0
$745$	−8.94911	−0.327870
$746$	0	0
$747$	14.0290	0.513295
$748$	0	0
$749$	9.55940	0.349293
$750$	0	0
$751$	1.26739	0.0462478	0.0231239	−	0.999733i	$-0.492639\pi$
$751$	1.26739	0.0462478	0.0231239	+	0.999733i	$0.492639\pi$
$752$	0	0
$753$	7.67254	0.279603
$754$	0	0
$755$	10.9342	0.397936
$756$	0	0
$757$	−43.1312	−1.56763	−0.783815	−	0.620994i	$-0.786729\pi$
$757$	−43.1312	−1.56763	−0.783815	+	0.620994i	$0.786729\pi$
$758$	0	0
$759$	0.264331	0.00959462
$760$	0	0
$761$	−13.1495	−0.476668	−0.238334	−	0.971183i	$-0.576601\pi$
$761$	−13.1495	−0.476668	−0.238334	+	0.971183i	$0.576601\pi$
$762$	0	0
$763$	12.1354	0.439332
$764$	0	0
$765$	−0.915159	−0.0330876
$766$	0	0
$767$	25.2309	0.911036
$768$	0	0
$769$	−15.2450	−0.549750	−0.274875	−	0.961480i	$-0.588636\pi$
$769$	−15.2450	−0.549750	−0.274875	+	0.961480i	$0.588636\pi$
$770$	0	0
$771$	2.95844	0.106546
$772$	0	0
$773$	37.9975	1.36667	0.683337	−	0.730103i	$-0.260528\pi$
$773$	37.9975	1.36667	0.683337	+	0.730103i	$0.260528\pi$
$774$	0	0
$775$	12.3699	0.444339
$776$	0	0
$777$	3.70156	0.132793
$778$	0	0
$779$	1.14554	0.0410432
$780$	0	0
$781$	2.81118	0.100592
$782$	0	0
$783$	−2.62589	−0.0938418
$784$	0	0
$785$	−11.7737	−0.420221
$786$	0	0
$787$	−29.1621	−1.03952	−0.519758	−	0.854314i	$-0.673978\pi$
$787$	−29.1621	−1.03952	−0.519758	+	0.854314i	$0.673978\pi$
$788$	0	0
$789$	8.87955	0.316120
$790$	0	0
$791$	−1.06650	−0.0379203
$792$	0	0
$793$	−20.8836	−0.741600
$794$	0	0
$795$	6.03819	0.214152
$796$	0	0
$797$	39.2284	1.38954	0.694771	−	0.719231i	$-0.255506\pi$
$797$	39.2284	1.38954	0.694771	+	0.719231i	$0.255506\pi$
$798$	0	0
$799$	−4.91790	−0.173983
$800$	0	0
$801$	6.08806	0.215111
$802$	0	0
$803$	1.89602	0.0669092
$804$	0	0
$805$	0.537835	0.0189562
$806$	0	0
$807$	13.4912	0.474912
$808$	0	0
$809$	0.264488	0.00929890	0.00464945	−	0.999989i	$-0.498520\pi$
$809$	0.264488	0.00929890	0.00464945	+	0.999989i	$0.498520\pi$
$810$	0	0
$811$	−32.2818	−1.13357	−0.566784	−	0.823866i	$-0.691813\pi$
$811$	−32.2818	−1.13357	−0.566784	+	0.823866i	$0.691813\pi$
$812$	0	0
$813$	−21.9791	−0.770842
$814$	0	0
$815$	−3.10538	−0.108777
$816$	0	0
$817$	13.3833	0.468222
$818$	0	0
$819$	3.16373	0.110550
$820$	0	0
$821$	−11.0075	−0.384163	−0.192081	−	0.981379i	$-0.561524\pi$
$821$	−11.0075	−0.384163	−0.192081	+	0.981379i	$0.561524\pi$
$822$	0	0
$823$	−3.61435	−0.125988	−0.0629942	−	0.998014i	$-0.520065\pi$
$823$	−3.61435	−0.125988	−0.0629942	+	0.998014i	$0.520065\pi$
$824$	0	0
$825$	−1.24519	−0.0433521
$826$	0	0
$827$	43.0125	1.49569	0.747845	−	0.663873i	$-0.231089\pi$
$827$	43.0125	1.49569	0.747845	+	0.663873i	$0.231089\pi$
$828$	0	0
$829$	40.2133	1.39667	0.698333	−	0.715773i	$-0.253926\pi$
$829$	40.2133	1.39667	0.698333	+	0.715773i	$0.253926\pi$
$830$	0	0
$831$	−19.6084	−0.680208
$832$	0	0
$833$	−1.70156	−0.0589556
$834$	0	0
$835$	−4.01561	−0.138966
$836$	0	0
$837$	−2.62589	−0.0907641
$838$	0	0
$839$	−5.62761	−0.194287	−0.0971433	−	0.995270i	$-0.530971\pi$
$839$	−5.62761	−0.194287	−0.0971433	+	0.995270i	$0.530971\pi$
$840$	0	0
$841$	−22.1047	−0.762231
$842$	0	0
$843$	10.0764	0.347049
$844$	0	0
$845$	−1.60857	−0.0553365
$846$	0	0
$847$	−10.9301	−0.375564
$848$	0	0
$849$	−28.6142	−0.982037
$850$	0	0
$851$	3.70156	0.126888
$852$	0	0
$853$	−43.8317	−1.50077	−0.750385	−	0.661001i	$-0.770132\pi$
$853$	−43.8317	−1.50077	−0.750385	+	0.661001i	$0.770132\pi$
$854$	0	0
$855$	0.933503	0.0319251
$856$	0	0
$857$	25.9096	0.885055	0.442527	−	0.896755i	$-0.354082\pi$
$857$	25.9096	0.885055	0.442527	+	0.896755i	$0.354082\pi$
$858$	0	0
$859$	−34.5983	−1.18048	−0.590239	−	0.807228i	$-0.700967\pi$
$859$	−34.5983	−1.18048	−0.590239	+	0.807228i	$0.700967\pi$
$860$	0	0
$861$	0.659999	0.0224927
$862$	0	0
$863$	−37.3964	−1.27299	−0.636494	−	0.771282i	$-0.719616\pi$
$863$	−37.3964	−1.27299	−0.636494	+	0.771282i	$0.719616\pi$
$864$	0	0
$865$	−2.77230	−0.0942610
$866$	0	0
$867$	−14.1047	−0.479020
$868$	0	0
$869$	−1.98258	−0.0672544
$870$	0	0
$871$	−22.9717	−0.778366
$872$	0	0
$873$	−11.5004	−0.389228
$874$	0	0
$875$	−5.22277	−0.176562
$876$	0	0
$877$	14.3117	0.483272	0.241636	−	0.970367i	$-0.422316\pi$
$877$	14.3117	0.483272	0.241636	+	0.970367i	$0.422316\pi$
$878$	0	0
$879$	−25.3716	−0.855763
$880$	0	0
$881$	1.18057	0.0397744	0.0198872	−	0.999802i	$-0.493669\pi$
$881$	1.18057	0.0397744	0.0198872	+	0.999802i	$0.493669\pi$
$882$	0	0
$883$	−33.6725	−1.13317	−0.566584	−	0.824004i	$-0.691735\pi$
$883$	−33.6725	−1.13317	−0.566584	+	0.824004i	$0.691735\pi$
$884$	0	0
$885$	4.28927	0.144182
$886$	0	0
$887$	−29.2004	−0.980454	−0.490227	−	0.871595i	$-0.663086\pi$
$887$	−29.2004	−0.980454	−0.490227	+	0.871595i	$0.663086\pi$
$888$	0	0
$889$	−2.20529	−0.0739631
$890$	0	0
$891$	0.264331	0.00885543
$892$	0	0
$893$	5.01647	0.167870
$894$	0	0
$895$	10.0382	0.335540
$896$	0	0
$897$	3.16373	0.105634
$898$	0	0
$899$	6.89531	0.229972
$900$	0	0
$901$	−19.1032	−0.636419
$902$	0	0
$903$	7.71073	0.256597
$904$	0	0
$905$	6.06721	0.201681
$906$	0	0
$907$	11.4554	0.380371	0.190185	−	0.981748i	$-0.439091\pi$
$907$	11.4554	0.380371	0.190185	+	0.981748i	$0.439091\pi$
$908$	0	0
$909$	7.37819	0.244719
$910$	0	0
$911$	−11.6875	−0.387223	−0.193611	−	0.981078i	$-0.562020\pi$
$911$	−11.6875	−0.387223	−0.193611	+	0.981078i	$0.562020\pi$
$912$	0	0
$913$	3.70831	0.122727
$914$	0	0
$915$	−3.55022	−0.117367
$916$	0	0
$917$	−16.3407	−0.539618
$918$	0	0
$919$	21.6914	0.715533	0.357766	−	0.933811i	$-0.383538\pi$
$919$	21.6914	0.715533	0.357766	+	0.933811i	$0.383538\pi$
$920$	0	0
$921$	12.8204	0.422445
$922$	0	0
$923$	33.6464	1.10749
$924$	0	0
$925$	−17.4371	−0.573327
$926$	0	0
$927$	8.84035	0.290355
$928$	0	0
$929$	38.5524	1.26486	0.632432	−	0.774616i	$-0.282057\pi$
$929$	38.5524	1.26486	0.632432	+	0.774616i	$0.282057\pi$
$930$	0	0
$931$	1.73567	0.0568842
$932$	0	0
$933$	28.2633	0.925300
$934$	0	0
$935$	−0.241905	−0.00791115
$936$	0	0
$937$	−1.37969	−0.0450725	−0.0225363	−	0.999746i	$-0.507174\pi$
$937$	−1.37969	−0.0450725	−0.0225363	+	0.999746i	$0.507174\pi$
$938$	0	0
$939$	9.96589	0.325225
$940$	0	0
$941$	−19.6594	−0.640877	−0.320438	−	0.947269i	$-0.603830\pi$
$941$	−19.6594	−0.640877	−0.320438	+	0.947269i	$0.603830\pi$
$942$	0	0
$943$	0.659999	0.0214925
$944$	0	0
$945$	0.537835	0.0174958
$946$	0	0
$947$	41.8070	1.35854	0.679272	−	0.733887i	$-0.262296\pi$
$947$	41.8070	1.35854	0.679272	+	0.733887i	$0.262296\pi$
$948$	0	0
$949$	22.6931	0.736649
$950$	0	0
$951$	12.6641	0.410661
$952$	0	0
$953$	56.0869	1.81683	0.908416	−	0.418067i	$-0.137292\pi$
$953$	56.0869	1.81683	0.908416	+	0.418067i	$0.137292\pi$
$954$	0	0
$955$	6.07981	0.196738
$956$	0	0
$957$	−0.694106	−0.0224373
$958$	0	0
$959$	−14.3906	−0.464696
$960$	0	0
$961$	−24.1047	−0.777571
$962$	0	0
$963$	9.55940	0.308047
$964$	0	0
$965$	1.27013	0.0408869
$966$	0	0
$967$	21.7703	0.700085	0.350042	−	0.936734i	$-0.386167\pi$
$967$	21.7703	0.700085	0.350042	+	0.936734i	$0.386167\pi$
$968$	0	0
$969$	−2.95335	−0.0948752
$970$	0	0
$971$	−7.69882	−0.247067	−0.123534	−	0.992340i	$-0.539423\pi$
$971$	−7.69882	−0.247067	−0.123534	+	0.992340i	$0.539423\pi$
$972$	0	0
$973$	9.39395	0.301156
$974$	0	0
$975$	−14.9035	−0.477293
$976$	0	0
$977$	6.98503	0.223471	0.111735	−	0.993738i	$-0.464359\pi$
$977$	6.98503	0.223471	0.111735	+	0.993738i	$0.464359\pi$
$978$	0	0
$979$	1.60926	0.0514323
$980$	0	0
$981$	12.1354	0.387454
$982$	0	0
$983$	13.9757	0.445756	0.222878	−	0.974846i	$-0.428455\pi$
$983$	13.9757	0.445756	0.222878	+	0.974846i	$0.428455\pi$
$984$	0	0
$985$	−8.04736	−0.256410
$986$	0	0
$987$	2.89022	0.0919969
$988$	0	0
$989$	7.71073	0.245187
$990$	0	0
$991$	4.15885	0.132110	0.0660551	−	0.997816i	$-0.478959\pi$
$991$	4.15885	0.132110	0.0660551	+	0.997816i	$0.478959\pi$
$992$	0	0
$993$	17.3690	0.551189
$994$	0	0
$995$	−7.40806	−0.234851
$996$	0	0
$997$	−57.0148	−1.80568	−0.902839	−	0.429980i	$-0.858521\pi$
$997$	−57.0148	−1.80568	−0.902839	+	0.429980i	$0.858521\pi$
$998$	0	0
$999$	3.70156	0.117112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cc.1.2		4
4.3	odd	2		3864.2.a.s.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.s.1.2	✓	4	4.3	odd	2
7728.2.a.cc.1.2		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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.537835 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.537835 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.264331 q^{11} +3.16373 q^{13} +0.537835 q^{15} -1.70156 q^{17} +1.73567 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.71073 q^{25} +1.00000 q^{27} -2.62589 q^{29} -2.62589 q^{31} +0.264331 q^{33} +0.537835 q^{35} +3.70156 q^{37} +3.16373 q^{39} +0.659999 q^{41} +7.71073 q^{43} +0.537835 q^{45} +2.89022 q^{47} +1.00000 q^{49} -1.70156 q^{51} +11.2269 q^{53} +0.142167 q^{55} +1.73567 q^{57} +7.97507 q^{59} -6.60096 q^{61} +1.00000 q^{63} +1.70156 q^{65} -7.26096 q^{67} +1.00000 q^{69} +10.6351 q^{71} +7.17290 q^{73} -4.71073 q^{75} +0.264331 q^{77} -7.50035 q^{79} +1.00000 q^{81} +14.0290 q^{83} -0.915159 q^{85} -2.62589 q^{87} +6.08806 q^{89} +3.16373 q^{91} -2.62589 q^{93} +0.933503 q^{95} -11.5004 q^{97} +0.264331 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + q^{5} + 4 q^{7} + 4 q^{9} + 7 q^{11} + q^{13} + q^{15} + 6 q^{17} + q^{19} + 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 7 q^{33} + q^{35} + 2 q^{37} + q^{39} - q^{41} + 5 q^{43} + q^{45} + 7 q^{47} + 4 q^{49} + 6 q^{51} + 4 q^{53} + 9 q^{55} + q^{57} + 12 q^{59} + 4 q^{61} + 4 q^{63} - 6 q^{65} + 5 q^{67} + 4 q^{69} + 19 q^{71} + 4 q^{73} + 7 q^{75} + 7 q^{77} + 18 q^{79} + 4 q^{81} + 20 q^{83} - 19 q^{85} + 15 q^{89} + q^{91} - 7 q^{95} + 2 q^{97} + 7 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + q^5 + 4 * q^7 + 4 * q^9 + 7 * q^11 + q^13 + q^15 + 6 * q^17 + q^19 + 4 * q^21 + 4 * q^23 + 7 * q^25 + 4 * q^27 + 7 * q^33 + q^35 + 2 * q^37 + q^39 - q^41 + 5 * q^43 + q^45 + 7 * q^47 + 4 * q^49 + 6 * q^51 + 4 * q^53 + 9 * q^55 + q^57 + 12 * q^59 + 4 * q^61 + 4 * q^63 - 6 * q^65 + 5 * q^67 + 4 * q^69 + 19 * q^71 + 4 * q^73 + 7 * q^75 + 7 * q^77 + 18 * q^79 + 4 * q^81 + 20 * q^83 - 19 * q^85 + 15 * q^89 + q^91 - 7 * q^95 + 2 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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