LMFDB - Embedded newform 7728.2.a.bc.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.70156$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -0.701562 q^{13} +2.70156 q^{15} +4.00000 q^{17} -7.40312 q^{19} +1.00000 q^{21} -1.00000 q^{23} +2.29844 q^{25} -1.00000 q^{27} +6.70156 q^{29} +2.00000 q^{31} -4.00000 q^{33} +2.70156 q^{35} +10.7016 q^{37} +0.701562 q^{39} -6.70156 q^{41} -4.70156 q^{43} -2.70156 q^{45} -8.10469 q^{47} +1.00000 q^{49} -4.00000 q^{51} +3.40312 q^{53} -10.8062 q^{55} +7.40312 q^{57} -5.40312 q^{59} -2.00000 q^{61} -1.00000 q^{63} +1.89531 q^{65} +10.8062 q^{67} +1.00000 q^{69} -5.40312 q^{71} -11.4031 q^{73} -2.29844 q^{75} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -0.596876 q^{83} -10.8062 q^{85} -6.70156 q^{87} +1.40312 q^{89} +0.701562 q^{91} -2.00000 q^{93} +20.0000 q^{95} +12.7016 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 8 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 11 q^{25} - 2 q^{27} + 7 q^{29} + 4 q^{31} - 8 q^{33} - q^{35} + 15 q^{37} - 5 q^{39} - 7 q^{41} - 3 q^{43} + q^{45} + 3 q^{47} + 2 q^{49} - 8 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 2 q^{59} - 4 q^{61} - 2 q^{63} + 23 q^{65} - 4 q^{67} + 2 q^{69} + 2 q^{71} - 10 q^{73} - 11 q^{75} - 8 q^{77} - 16 q^{79} + 2 q^{81} - 14 q^{83} + 4 q^{85} - 7 q^{87} - 10 q^{89} - 5 q^{91} - 4 q^{93} + 40 q^{95} + 19 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 8 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 11 * q^25 - 2 * q^27 + 7 * q^29 + 4 * q^31 - 8 * q^33 - q^35 + 15 * q^37 - 5 * q^39 - 7 * q^41 - 3 * q^43 + q^45 + 3 * q^47 + 2 * q^49 - 8 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 2 * q^59 - 4 * q^61 - 2 * q^63 + 23 * q^65 - 4 * q^67 + 2 * q^69 + 2 * q^71 - 10 * q^73 - 11 * q^75 - 8 * q^77 - 16 * q^79 + 2 * q^81 - 14 * q^83 + 4 * q^85 - 7 * q^87 - 10 * q^89 - 5 * q^91 - 4 * q^93 + 40 * q^95 + 19 * q^97 + 8 * 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
$5$	−2.70156	−1.20818	−0.604088	−	0.796918i	$-0.706462\pi$
$5$	−2.70156	−1.20818	−0.604088	+	0.796918i	$0.706462\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−0.701562	−0.194578	−0.0972892	−	0.995256i	$-0.531017\pi$
$13$	−0.701562	−0.194578	−0.0972892	+	0.995256i	$0.531017\pi$
$14$	0	0
$15$	2.70156	0.697540
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−7.40312	−1.69839	−0.849197	−	0.528077i	$-0.822913\pi$
$19$	−7.40312	−1.69839	−0.849197	+	0.528077i	$0.822913\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.29844	0.459688
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.70156	1.24445	0.622224	−	0.782839i	$-0.286229\pi$
$29$	6.70156	1.24445	0.622224	+	0.782839i	$0.286229\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	2.70156	0.456647
$36$	0	0
$37$	10.7016	1.75933	0.879663	−	0.475598i	$-0.157768\pi$
$37$	10.7016	1.75933	0.879663	+	0.475598i	$0.157768\pi$
$38$	0	0
$39$	0.701562	0.112340
$40$	0	0
$41$	−6.70156	−1.04661	−0.523304	−	0.852146i	$-0.675301\pi$
$41$	−6.70156	−1.04661	−0.523304	+	0.852146i	$0.675301\pi$
$42$	0	0
$43$	−4.70156	−0.716982	−0.358491	−	0.933533i	$-0.616709\pi$
$43$	−4.70156	−0.716982	−0.358491	+	0.933533i	$0.616709\pi$
$44$	0	0
$45$	−2.70156	−0.402725
$46$	0	0
$47$	−8.10469	−1.18219	−0.591095	−	0.806602i	$-0.701304\pi$
$47$	−8.10469	−1.18219	−0.591095	+	0.806602i	$0.701304\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	3.40312	0.467455	0.233728	−	0.972302i	$-0.424908\pi$
$53$	3.40312	0.467455	0.233728	+	0.972302i	$0.424908\pi$
$54$	0	0
$55$	−10.8062	−1.45711
$56$	0	0
$57$	7.40312	0.980568
$58$	0	0
$59$	−5.40312	−0.703427	−0.351713	−	0.936108i	$-0.614401\pi$
$59$	−5.40312	−0.703427	−0.351713	+	0.936108i	$0.614401\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.89531	0.235085
$66$	0	0
$67$	10.8062	1.32019	0.660097	−	0.751181i	$-0.270515\pi$
$67$	10.8062	1.32019	0.660097	+	0.751181i	$0.270515\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−5.40312	−0.641233	−0.320616	−	0.947209i	$-0.603890\pi$
$71$	−5.40312	−0.641233	−0.320616	+	0.947209i	$0.603890\pi$
$72$	0	0
$73$	−11.4031	−1.33463	−0.667317	−	0.744773i	$-0.732558\pi$
$73$	−11.4031	−1.33463	−0.667317	+	0.744773i	$0.732558\pi$
$74$	0	0
$75$	−2.29844	−0.265401
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.596876	−0.0655156	−0.0327578	−	0.999463i	$-0.510429\pi$
$83$	−0.596876	−0.0655156	−0.0327578	+	0.999463i	$0.510429\pi$
$84$	0	0
$85$	−10.8062	−1.17210
$86$	0	0
$87$	−6.70156	−0.718483
$88$	0	0
$89$	1.40312	0.148731	0.0743654	−	0.997231i	$-0.476307\pi$
$89$	1.40312	0.148731	0.0743654	+	0.997231i	$0.476307\pi$
$90$	0	0
$91$	0.701562	0.0735437
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	20.0000	2.05196
$96$	0	0
$97$	12.7016	1.28965	0.644824	−	0.764331i	$-0.276931\pi$
$97$	12.7016	1.28965	0.644824	+	0.764331i	$0.276931\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	9.40312	0.935646	0.467823	−	0.883822i	$-0.345039\pi$
$101$	9.40312	0.935646	0.467823	+	0.883822i	$0.345039\pi$
$102$	0	0
$103$	−8.70156	−0.857390	−0.428695	−	0.903449i	$-0.641027\pi$
$103$	−8.70156	−0.857390	−0.428695	+	0.903449i	$0.641027\pi$
$104$	0	0
$105$	−2.70156	−0.263645
$106$	0	0
$107$	6.80625	0.657985	0.328992	−	0.944333i	$-0.393291\pi$
$107$	6.80625	0.657985	0.328992	+	0.944333i	$0.393291\pi$
$108$	0	0
$109$	−1.29844	−0.124368	−0.0621839	−	0.998065i	$-0.519807\pi$
$109$	−1.29844	−0.124368	−0.0621839	+	0.998065i	$0.519807\pi$
$110$	0	0
$111$	−10.7016	−1.01575
$112$	0	0
$113$	−12.1047	−1.13871	−0.569357	−	0.822091i	$-0.692808\pi$
$113$	−12.1047	−1.13871	−0.569357	+	0.822091i	$0.692808\pi$
$114$	0	0
$115$	2.70156	0.251922
$116$	0	0
$117$	−0.701562	−0.0648594
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	6.70156	0.604260
$124$	0	0
$125$	7.29844	0.652792
$126$	0	0
$127$	−0.701562	−0.0622536	−0.0311268	−	0.999515i	$-0.509910\pi$
$127$	−0.701562	−0.0622536	−0.0311268	+	0.999515i	$0.509910\pi$
$128$	0	0
$129$	4.70156	0.413949
$130$	0	0
$131$	−5.40312	−0.472073	−0.236037	−	0.971744i	$-0.575849\pi$
$131$	−5.40312	−0.472073	−0.236037	+	0.971744i	$0.575849\pi$
$132$	0	0
$133$	7.40312	0.641932
$134$	0	0
$135$	2.70156	0.232513
$136$	0	0
$137$	−13.2984	−1.13616	−0.568081	−	0.822973i	$-0.692314\pi$
$137$	−13.2984	−1.13616	−0.568081	+	0.822973i	$0.692314\pi$
$138$	0	0
$139$	−11.2984	−0.958321	−0.479160	−	0.877727i	$-0.659059\pi$
$139$	−11.2984	−0.958321	−0.479160	+	0.877727i	$0.659059\pi$
$140$	0	0
$141$	8.10469	0.682538
$142$	0	0
$143$	−2.80625	−0.234670
$144$	0	0
$145$	−18.1047	−1.50351
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	18.2094	1.49177	0.745885	−	0.666075i	$-0.232027\pi$
$149$	18.2094	1.49177	0.745885	+	0.666075i	$0.232027\pi$
$150$	0	0
$151$	20.9109	1.70171	0.850854	−	0.525402i	$-0.176085\pi$
$151$	20.9109	1.70171	0.850854	+	0.525402i	$0.176085\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−5.40312	−0.433989
$156$	0	0
$157$	11.4031	0.910068	0.455034	−	0.890474i	$-0.349627\pi$
$157$	11.4031	0.910068	0.455034	+	0.890474i	$0.349627\pi$
$158$	0	0
$159$	−3.40312	−0.269885
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−14.8062	−1.15971	−0.579857	−	0.814718i	$-0.696892\pi$
$163$	−14.8062	−1.15971	−0.579857	+	0.814718i	$0.696892\pi$
$164$	0	0
$165$	10.8062	0.841265
$166$	0	0
$167$	0.596876	0.0461876	0.0230938	−	0.999733i	$-0.492648\pi$
$167$	0.596876	0.0461876	0.0230938	+	0.999733i	$0.492648\pi$
$168$	0	0
$169$	−12.5078	−0.962139
$170$	0	0
$171$	−7.40312	−0.566131
$172$	0	0
$173$	6.59688	0.501551	0.250776	−	0.968045i	$-0.419314\pi$
$173$	6.59688	0.501551	0.250776	+	0.968045i	$0.419314\pi$
$174$	0	0
$175$	−2.29844	−0.173746
$176$	0	0
$177$	5.40312	0.406124
$178$	0	0
$179$	5.89531	0.440636	0.220318	−	0.975428i	$-0.429290\pi$
$179$	5.89531	0.440636	0.220318	+	0.975428i	$0.429290\pi$
$180$	0	0
$181$	8.80625	0.654563	0.327282	−	0.944927i	$-0.393867\pi$
$181$	8.80625	0.654563	0.327282	+	0.944927i	$0.393867\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−28.9109	−2.12557
$186$	0	0
$187$	16.0000	1.17004
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	2.80625	0.203053	0.101527	−	0.994833i	$-0.467627\pi$
$191$	2.80625	0.203053	0.101527	+	0.994833i	$0.467627\pi$
$192$	0	0
$193$	−3.89531	−0.280391	−0.140195	−	0.990124i	$-0.544773\pi$
$193$	−3.89531	−0.280391	−0.140195	+	0.990124i	$0.544773\pi$
$194$	0	0
$195$	−1.89531	−0.135726
$196$	0	0
$197$	−9.50781	−0.677403	−0.338702	−	0.940894i	$-0.609988\pi$
$197$	−9.50781	−0.677403	−0.338702	+	0.940894i	$0.609988\pi$
$198$	0	0
$199$	8.70156	0.616837	0.308419	−	0.951251i	$-0.400200\pi$
$199$	8.70156	0.616837	0.308419	+	0.951251i	$0.400200\pi$
$200$	0	0
$201$	−10.8062	−0.762214
$202$	0	0
$203$	−6.70156	−0.470357
$204$	0	0
$205$	18.1047	1.26449
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−29.6125	−2.04834
$210$	0	0
$211$	14.8062	1.01930	0.509652	−	0.860381i	$-0.329774\pi$
$211$	14.8062	1.01930	0.509652	+	0.860381i	$0.329774\pi$
$212$	0	0
$213$	5.40312	0.370216
$214$	0	0
$215$	12.7016	0.866239
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	11.4031	0.770552
$220$	0	0
$221$	−2.80625	−0.188769
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	0	0
$225$	2.29844	0.153229
$226$	0	0
$227$	3.89531	0.258541	0.129271	−	0.991609i	$-0.458736\pi$
$227$	3.89531	0.258541	0.129271	+	0.991609i	$0.458736\pi$
$228$	0	0
$229$	−27.4031	−1.81085	−0.905425	−	0.424507i	$-0.860447\pi$
$229$	−27.4031	−1.81085	−0.905425	+	0.424507i	$0.860447\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	21.8953	1.42829
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	14.8062	0.957737	0.478868	−	0.877887i	$-0.341047\pi$
$239$	14.8062	0.957737	0.478868	+	0.877887i	$0.341047\pi$
$240$	0	0
$241$	28.9109	1.86232	0.931159	−	0.364615i	$-0.118799\pi$
$241$	28.9109	1.86232	0.931159	+	0.364615i	$0.118799\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.70156	−0.172596
$246$	0	0
$247$	5.19375	0.330470
$248$	0	0
$249$	0.596876	0.0378255
$250$	0	0
$251$	−13.2984	−0.839390	−0.419695	−	0.907665i	$-0.637863\pi$
$251$	−13.2984	−0.839390	−0.419695	+	0.907665i	$0.637863\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	10.8062	0.676714
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−10.7016	−0.664963
$260$	0	0
$261$	6.70156	0.414816
$262$	0	0
$263$	−3.29844	−0.203390	−0.101695	−	0.994816i	$-0.532427\pi$
$263$	−3.29844	−0.203390	−0.101695	+	0.994816i	$0.532427\pi$
$264$	0	0
$265$	−9.19375	−0.564768
$266$	0	0
$267$	−1.40312	−0.0858698
$268$	0	0
$269$	28.2094	1.71996	0.859978	−	0.510331i	$-0.170477\pi$
$269$	28.2094	1.71996	0.859978	+	0.510331i	$0.170477\pi$
$270$	0	0
$271$	11.4031	0.692690	0.346345	−	0.938107i	$-0.387423\pi$
$271$	11.4031	0.692690	0.346345	+	0.938107i	$0.387423\pi$
$272$	0	0
$273$	−0.701562	−0.0424605
$274$	0	0
$275$	9.19375	0.554404
$276$	0	0
$277$	−8.59688	−0.516536	−0.258268	−	0.966073i	$-0.583152\pi$
$277$	−8.59688	−0.516536	−0.258268	+	0.966073i	$0.583152\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−17.5078	−1.04443	−0.522214	−	0.852814i	$-0.674894\pi$
$281$	−17.5078	−1.04443	−0.522214	+	0.852814i	$0.674894\pi$
$282$	0	0
$283$	20.8062	1.23680	0.618402	−	0.785862i	$-0.287781\pi$
$283$	20.8062	1.23680	0.618402	+	0.785862i	$0.287781\pi$
$284$	0	0
$285$	−20.0000	−1.18470
$286$	0	0
$287$	6.70156	0.395581
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−12.7016	−0.744579
$292$	0	0
$293$	4.80625	0.280784	0.140392	−	0.990096i	$-0.455164\pi$
$293$	4.80625	0.280784	0.140392	+	0.990096i	$0.455164\pi$
$294$	0	0
$295$	14.5969	0.849863
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	0.701562	0.0405724
$300$	0	0
$301$	4.70156	0.270994
$302$	0	0
$303$	−9.40312	−0.540195
$304$	0	0
$305$	5.40312	0.309382
$306$	0	0
$307$	32.9109	1.87833	0.939163	−	0.343471i	$-0.111603\pi$
$307$	32.9109	1.87833	0.939163	+	0.343471i	$0.111603\pi$
$308$	0	0
$309$	8.70156	0.495015
$310$	0	0
$311$	−0.596876	−0.0338457	−0.0169229	−	0.999857i	$-0.505387\pi$
$311$	−0.596876	−0.0338457	−0.0169229	+	0.999857i	$0.505387\pi$
$312$	0	0
$313$	32.2094	1.82058	0.910291	−	0.413970i	$-0.135858\pi$
$313$	32.2094	1.82058	0.910291	+	0.413970i	$0.135858\pi$
$314$	0	0
$315$	2.70156	0.152216
$316$	0	0
$317$	−14.7016	−0.825722	−0.412861	−	0.910794i	$-0.635470\pi$
$317$	−14.7016	−0.825722	−0.412861	+	0.910794i	$0.635470\pi$
$318$	0	0
$319$	26.8062	1.50086
$320$	0	0
$321$	−6.80625	−0.379888
$322$	0	0
$323$	−29.6125	−1.64768
$324$	0	0
$325$	−1.61250	−0.0894452
$326$	0	0
$327$	1.29844	0.0718038
$328$	0	0
$329$	8.10469	0.446826
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	10.7016	0.586442
$334$	0	0
$335$	−29.1938	−1.59503
$336$	0	0
$337$	34.2094	1.86350	0.931752	−	0.363096i	$-0.118280\pi$
$337$	34.2094	1.86350	0.931752	+	0.363096i	$0.118280\pi$
$338$	0	0
$339$	12.1047	0.657436
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.70156	−0.145447
$346$	0	0
$347$	16.7016	0.896587	0.448293	−	0.893886i	$-0.352032\pi$
$347$	16.7016	0.896587	0.448293	+	0.893886i	$0.352032\pi$
$348$	0	0
$349$	25.4031	1.35980	0.679899	−	0.733306i	$-0.262024\pi$
$349$	25.4031	1.35980	0.679899	+	0.733306i	$0.262024\pi$
$350$	0	0
$351$	0.701562	0.0374466
$352$	0	0
$353$	10.7016	0.569587	0.284793	−	0.958589i	$-0.408075\pi$
$353$	10.7016	0.569587	0.284793	+	0.958589i	$0.408075\pi$
$354$	0	0
$355$	14.5969	0.774722
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	22.1047	1.16664	0.583320	−	0.812242i	$-0.301753\pi$
$359$	22.1047	1.16664	0.583320	+	0.812242i	$0.301753\pi$
$360$	0	0
$361$	35.8062	1.88454
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	30.8062	1.61247
$366$	0	0
$367$	−2.10469	−0.109864	−0.0549319	−	0.998490i	$-0.517494\pi$
$367$	−2.10469	−0.109864	−0.0549319	+	0.998490i	$0.517494\pi$
$368$	0	0
$369$	−6.70156	−0.348869
$370$	0	0
$371$	−3.40312	−0.176681
$372$	0	0
$373$	24.8062	1.28442	0.642209	−	0.766529i	$-0.278018\pi$
$373$	24.8062	1.28442	0.642209	+	0.766529i	$0.278018\pi$
$374$	0	0
$375$	−7.29844	−0.376890
$376$	0	0
$377$	−4.70156	−0.242143
$378$	0	0
$379$	−7.29844	−0.374896	−0.187448	−	0.982275i	$-0.560022\pi$
$379$	−7.29844	−0.374896	−0.187448	+	0.982275i	$0.560022\pi$
$380$	0	0
$381$	0.701562	0.0359421
$382$	0	0
$383$	21.6125	1.10435	0.552174	−	0.833729i	$-0.313799\pi$
$383$	21.6125	1.10435	0.552174	+	0.833729i	$0.313799\pi$
$384$	0	0
$385$	10.8062	0.550737
$386$	0	0
$387$	−4.70156	−0.238994
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	0	0
$393$	5.40312	0.272552
$394$	0	0
$395$	21.6125	1.08744
$396$	0	0
$397$	21.4031	1.07419	0.537096	−	0.843521i	$-0.319521\pi$
$397$	21.4031	1.07419	0.537096	+	0.843521i	$0.319521\pi$
$398$	0	0
$399$	−7.40312	−0.370620
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	−1.40312	−0.0698946
$404$	0	0
$405$	−2.70156	−0.134242
$406$	0	0
$407$	42.8062	2.12183
$408$	0	0
$409$	−7.40312	−0.366061	−0.183030	−	0.983107i	$-0.558591\pi$
$409$	−7.40312	−0.366061	−0.183030	+	0.983107i	$0.558591\pi$
$410$	0	0
$411$	13.2984	0.655964
$412$	0	0
$413$	5.40312	0.265870
$414$	0	0
$415$	1.61250	0.0791544
$416$	0	0
$417$	11.2984	0.553287
$418$	0	0
$419$	−38.2094	−1.86665	−0.933325	−	0.359033i	$-0.883107\pi$
$419$	−38.2094	−1.86665	−0.933325	+	0.359033i	$0.883107\pi$
$420$	0	0
$421$	−9.29844	−0.453178	−0.226589	−	0.973990i	$-0.572757\pi$
$421$	−9.29844	−0.453178	−0.226589	+	0.973990i	$0.572757\pi$
$422$	0	0
$423$	−8.10469	−0.394063
$424$	0	0
$425$	9.19375	0.445962
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	2.80625	0.135487
$430$	0	0
$431$	−40.9109	−1.97061	−0.985305	−	0.170803i	$-0.945364\pi$
$431$	−40.9109	−1.97061	−0.985305	+	0.170803i	$0.945364\pi$
$432$	0	0
$433$	−26.3141	−1.26457	−0.632286	−	0.774735i	$-0.717883\pi$
$433$	−26.3141	−1.26457	−0.632286	+	0.774735i	$0.717883\pi$
$434$	0	0
$435$	18.1047	0.868053
$436$	0	0
$437$	7.40312	0.354139
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	7.29844	0.346759	0.173380	−	0.984855i	$-0.444531\pi$
$443$	7.29844	0.346759	0.173380	+	0.984855i	$0.444531\pi$
$444$	0	0
$445$	−3.79063	−0.179693
$446$	0	0
$447$	−18.2094	−0.861274
$448$	0	0
$449$	38.4187	1.81309	0.906546	−	0.422106i	$-0.138709\pi$
$449$	38.4187	1.81309	0.906546	+	0.422106i	$0.138709\pi$
$450$	0	0
$451$	−26.8062	−1.26226
$452$	0	0
$453$	−20.9109	−0.982481
$454$	0	0
$455$	−1.89531	−0.0888537
$456$	0	0
$457$	−10.2094	−0.477574	−0.238787	−	0.971072i	$-0.576750\pi$
$457$	−10.2094	−0.477574	−0.238787	+	0.971072i	$0.576750\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	36.2094	1.68644	0.843219	−	0.537570i	$-0.180658\pi$
$461$	36.2094	1.68644	0.843219	+	0.537570i	$0.180658\pi$
$462$	0	0
$463$	28.7016	1.33387	0.666937	−	0.745114i	$-0.267605\pi$
$463$	28.7016	1.33387	0.666937	+	0.745114i	$0.267605\pi$
$464$	0	0
$465$	5.40312	0.250564
$466$	0	0
$467$	−2.70156	−0.125013	−0.0625067	−	0.998045i	$-0.519909\pi$
$467$	−2.70156	−0.125013	−0.0625067	+	0.998045i	$0.519909\pi$
$468$	0	0
$469$	−10.8062	−0.498986
$470$	0	0
$471$	−11.4031	−0.525428
$472$	0	0
$473$	−18.8062	−0.864712
$474$	0	0
$475$	−17.0156	−0.780730
$476$	0	0
$477$	3.40312	0.155818
$478$	0	0
$479$	−29.4031	−1.34346	−0.671732	−	0.740795i	$-0.734449\pi$
$479$	−29.4031	−1.34346	−0.671732	+	0.740795i	$0.734449\pi$
$480$	0	0
$481$	−7.50781	−0.342327
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−34.3141	−1.55812
$486$	0	0
$487$	12.7016	0.575563	0.287781	−	0.957696i	$-0.407082\pi$
$487$	12.7016	0.575563	0.287781	+	0.957696i	$0.407082\pi$
$488$	0	0
$489$	14.8062	0.669562
$490$	0	0
$491$	−41.6125	−1.87795	−0.938973	−	0.343991i	$-0.888221\pi$
$491$	−41.6125	−1.87795	−0.938973	+	0.343991i	$0.888221\pi$
$492$	0	0
$493$	26.8062	1.20729
$494$	0	0
$495$	−10.8062	−0.485705
$496$	0	0
$497$	5.40312	0.242363
$498$	0	0
$499$	10.5969	0.474381	0.237191	−	0.971463i	$-0.423773\pi$
$499$	10.5969	0.474381	0.237191	+	0.971463i	$0.423773\pi$
$500$	0	0
$501$	−0.596876	−0.0266664
$502$	0	0
$503$	−17.4031	−0.775967	−0.387983	−	0.921666i	$-0.626828\pi$
$503$	−17.4031	−0.775967	−0.387983	+	0.921666i	$0.626828\pi$
$504$	0	0
$505$	−25.4031	−1.13042
$506$	0	0
$507$	12.5078	0.555491
$508$	0	0
$509$	−13.6125	−0.603363	−0.301682	−	0.953409i	$-0.597548\pi$
$509$	−13.6125	−0.603363	−0.301682	+	0.953409i	$0.597548\pi$
$510$	0	0
$511$	11.4031	0.504445
$512$	0	0
$513$	7.40312	0.326856
$514$	0	0
$515$	23.5078	1.03588
$516$	0	0
$517$	−32.4187	−1.42577
$518$	0	0
$519$	−6.59688	−0.289571
$520$	0	0
$521$	−33.4031	−1.46342	−0.731709	−	0.681617i	$-0.761277\pi$
$521$	−33.4031	−1.46342	−0.731709	+	0.681617i	$0.761277\pi$
$522$	0	0
$523$	4.80625	0.210163	0.105081	−	0.994464i	$-0.466490\pi$
$523$	4.80625	0.210163	0.105081	+	0.994464i	$0.466490\pi$
$524$	0	0
$525$	2.29844	0.100312
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.40312	−0.234476
$532$	0	0
$533$	4.70156	0.203647
$534$	0	0
$535$	−18.3875	−0.794961
$536$	0	0
$537$	−5.89531	−0.254402
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	20.5969	0.885529	0.442764	−	0.896638i	$-0.353998\pi$
$541$	20.5969	0.885529	0.442764	+	0.896638i	$0.353998\pi$
$542$	0	0
$543$	−8.80625	−0.377912
$544$	0	0
$545$	3.50781	0.150258
$546$	0	0
$547$	35.0156	1.49716	0.748580	−	0.663045i	$-0.230736\pi$
$547$	35.0156	1.49716	0.748580	+	0.663045i	$0.230736\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−49.6125	−2.11356
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	28.9109	1.22720
$556$	0	0
$557$	−24.5969	−1.04220	−0.521102	−	0.853495i	$-0.674479\pi$
$557$	−24.5969	−1.04220	−0.521102	+	0.853495i	$0.674479\pi$
$558$	0	0
$559$	3.29844	0.139509
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	−16.1047	−0.678732	−0.339366	−	0.940654i	$-0.610212\pi$
$563$	−16.1047	−0.678732	−0.339366	+	0.940654i	$0.610212\pi$
$564$	0	0
$565$	32.7016	1.37577
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	25.2984	1.06057	0.530283	−	0.847821i	$-0.322086\pi$
$569$	25.2984	1.06057	0.530283	+	0.847821i	$0.322086\pi$
$570$	0	0
$571$	−10.8062	−0.452227	−0.226114	−	0.974101i	$-0.572602\pi$
$571$	−10.8062	−0.452227	−0.226114	+	0.974101i	$0.572602\pi$
$572$	0	0
$573$	−2.80625	−0.117233
$574$	0	0
$575$	−2.29844	−0.0958515
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	3.89531	0.161884
$580$	0	0
$581$	0.596876	0.0247626
$582$	0	0
$583$	13.6125	0.563772
$584$	0	0
$585$	1.89531	0.0783616
$586$	0	0
$587$	10.8062	0.446022	0.223011	−	0.974816i	$-0.428411\pi$
$587$	10.8062	0.446022	0.223011	+	0.974816i	$0.428411\pi$
$588$	0	0
$589$	−14.8062	−0.610081
$590$	0	0
$591$	9.50781	0.391099
$592$	0	0
$593$	37.2984	1.53166	0.765832	−	0.643041i	$-0.222328\pi$
$593$	37.2984	1.53166	0.765832	+	0.643041i	$0.222328\pi$
$594$	0	0
$595$	10.8062	0.443013
$596$	0	0
$597$	−8.70156	−0.356131
$598$	0	0
$599$	37.6125	1.53680	0.768402	−	0.639967i	$-0.221052\pi$
$599$	37.6125	1.53680	0.768402	+	0.639967i	$0.221052\pi$
$600$	0	0
$601$	−0.596876	−0.0243471	−0.0121735	−	0.999926i	$-0.503875\pi$
$601$	−0.596876	−0.0243471	−0.0121735	+	0.999926i	$0.503875\pi$
$602$	0	0
$603$	10.8062	0.440064
$604$	0	0
$605$	−13.5078	−0.549171
$606$	0	0
$607$	−11.6125	−0.471337	−0.235668	−	0.971834i	$-0.575728\pi$
$607$	−11.6125	−0.471337	−0.235668	+	0.971834i	$0.575728\pi$
$608$	0	0
$609$	6.70156	0.271561
$610$	0	0
$611$	5.68594	0.230029
$612$	0	0
$613$	−2.70156	−0.109115	−0.0545575	−	0.998511i	$-0.517375\pi$
$613$	−2.70156	−0.109115	−0.0545575	+	0.998511i	$0.517375\pi$
$614$	0	0
$615$	−18.1047	−0.730051
$616$	0	0
$617$	−44.8062	−1.80383	−0.901916	−	0.431912i	$-0.857839\pi$
$617$	−44.8062	−1.80383	−0.901916	+	0.431912i	$0.857839\pi$
$618$	0	0
$619$	32.8062	1.31859	0.659297	−	0.751882i	$-0.270854\pi$
$619$	32.8062	1.31859	0.659297	+	0.751882i	$0.270854\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−1.40312	−0.0562150
$624$	0	0
$625$	−31.2094	−1.24837
$626$	0	0
$627$	29.6125	1.18261
$628$	0	0
$629$	42.8062	1.70680
$630$	0	0
$631$	−21.6125	−0.860380	−0.430190	−	0.902738i	$-0.641553\pi$
$631$	−21.6125	−0.860380	−0.430190	+	0.902738i	$0.641553\pi$
$632$	0	0
$633$	−14.8062	−0.588496
$634$	0	0
$635$	1.89531	0.0752132
$636$	0	0
$637$	−0.701562	−0.0277969
$638$	0	0
$639$	−5.40312	−0.213744
$640$	0	0
$641$	49.7172	1.96371	0.981855	−	0.189631i	$-0.0607293\pi$
$641$	49.7172	1.96371	0.981855	+	0.189631i	$0.0607293\pi$
$642$	0	0
$643$	24.8062	0.978263	0.489131	−	0.872210i	$-0.337314\pi$
$643$	24.8062	0.978263	0.489131	+	0.872210i	$0.337314\pi$
$644$	0	0
$645$	−12.7016	−0.500124
$646$	0	0
$647$	19.4031	0.762816	0.381408	−	0.924407i	$-0.375439\pi$
$647$	19.4031	0.762816	0.381408	+	0.924407i	$0.375439\pi$
$648$	0	0
$649$	−21.6125	−0.848365
$650$	0	0
$651$	2.00000	0.0783862
$652$	0	0
$653$	−12.1047	−0.473693	−0.236846	−	0.971547i	$-0.576114\pi$
$653$	−12.1047	−0.473693	−0.236846	+	0.971547i	$0.576114\pi$
$654$	0	0
$655$	14.5969	0.570347
$656$	0	0
$657$	−11.4031	−0.444878
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−34.4187	−1.33873	−0.669367	−	0.742932i	$-0.733435\pi$
$661$	−34.4187	−1.33873	−0.669367	+	0.742932i	$0.733435\pi$
$662$	0	0
$663$	2.80625	0.108986
$664$	0	0
$665$	−20.0000	−0.775567
$666$	0	0
$667$	−6.70156	−0.259486
$668$	0	0
$669$	−22.0000	−0.850569
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	1.29844	0.0500511	0.0250256	−	0.999687i	$-0.492033\pi$
$673$	1.29844	0.0500511	0.0250256	+	0.999687i	$0.492033\pi$
$674$	0	0
$675$	−2.29844	−0.0884669
$676$	0	0
$677$	26.4187	1.01535	0.507677	−	0.861547i	$-0.330504\pi$
$677$	26.4187	1.01535	0.507677	+	0.861547i	$0.330504\pi$
$678$	0	0
$679$	−12.7016	−0.487441
$680$	0	0
$681$	−3.89531	−0.149269
$682$	0	0
$683$	17.6125	0.673923	0.336962	−	0.941518i	$-0.390601\pi$
$683$	17.6125	0.673923	0.336962	+	0.941518i	$0.390601\pi$
$684$	0	0
$685$	35.9266	1.37268
$686$	0	0
$687$	27.4031	1.04549
$688$	0	0
$689$	−2.38750	−0.0909566
$690$	0	0
$691$	−31.5078	−1.19861	−0.599307	−	0.800519i	$-0.704557\pi$
$691$	−31.5078	−1.19861	−0.599307	+	0.800519i	$0.704557\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	30.5234	1.15782
$696$	0	0
$697$	−26.8062	−1.01536
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	−17.0156	−0.642671	−0.321336	−	0.946965i	$-0.604132\pi$
$701$	−17.0156	−0.642671	−0.321336	+	0.946965i	$0.604132\pi$
$702$	0	0
$703$	−79.2250	−2.98803
$704$	0	0
$705$	−21.8953	−0.824625
$706$	0	0
$707$	−9.40312	−0.353641
$708$	0	0
$709$	48.8062	1.83296	0.916479	−	0.400084i	$-0.131019\pi$
$709$	48.8062	1.83296	0.916479	+	0.400084i	$0.131019\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	7.58125	0.283523
$716$	0	0
$717$	−14.8062	−0.552949
$718$	0	0
$719$	−38.9109	−1.45113	−0.725567	−	0.688152i	$-0.758422\pi$
$719$	−38.9109	−1.45113	−0.725567	+	0.688152i	$0.758422\pi$
$720$	0	0
$721$	8.70156	0.324063
$722$	0	0
$723$	−28.9109	−1.07521
$724$	0	0
$725$	15.4031	0.572058
$726$	0	0
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.8062	−0.695574
$732$	0	0
$733$	13.7906	0.509368	0.254684	−	0.967024i	$-0.418028\pi$
$733$	13.7906	0.509368	0.254684	+	0.967024i	$0.418028\pi$
$734$	0	0
$735$	2.70156	0.0996486
$736$	0	0
$737$	43.2250	1.59221
$738$	0	0
$739$	−35.0156	−1.28807	−0.644035	−	0.764996i	$-0.722741\pi$
$739$	−35.0156	−1.28807	−0.644035	+	0.764996i	$0.722741\pi$
$740$	0	0
$741$	−5.19375	−0.190797
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	−49.1938	−1.80232
$746$	0	0
$747$	−0.596876	−0.0218385
$748$	0	0
$749$	−6.80625	−0.248695
$750$	0	0
$751$	21.6125	0.788651	0.394326	−	0.918971i	$-0.370978\pi$
$751$	21.6125	0.788651	0.394326	+	0.918971i	$0.370978\pi$
$752$	0	0
$753$	13.2984	0.484622
$754$	0	0
$755$	−56.4922	−2.05596
$756$	0	0
$757$	50.4187	1.83250	0.916250	−	0.400606i	$-0.131201\pi$
$757$	50.4187	1.83250	0.916250	+	0.400606i	$0.131201\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	15.6125	0.565953	0.282976	−	0.959127i	$-0.408678\pi$
$761$	15.6125	0.565953	0.282976	+	0.959127i	$0.408678\pi$
$762$	0	0
$763$	1.29844	0.0470066
$764$	0	0
$765$	−10.8062	−0.390701
$766$	0	0
$767$	3.79063	0.136872
$768$	0	0
$769$	−28.9109	−1.04255	−0.521277	−	0.853387i	$-0.674544\pi$
$769$	−28.9109	−1.04255	−0.521277	+	0.853387i	$0.674544\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	0	0
$773$	12.1047	0.435375	0.217688	−	0.976018i	$-0.430149\pi$
$773$	12.1047	0.435375	0.217688	+	0.976018i	$0.430149\pi$
$774$	0	0
$775$	4.59688	0.165125
$776$	0	0
$777$	10.7016	0.383916
$778$	0	0
$779$	49.6125	1.77755
$780$	0	0
$781$	−21.6125	−0.773356
$782$	0	0
$783$	−6.70156	−0.239494
$784$	0	0
$785$	−30.8062	−1.09952
$786$	0	0
$787$	31.4031	1.11940	0.559700	−	0.828695i	$-0.310916\pi$
$787$	31.4031	1.11940	0.559700	+	0.828695i	$0.310916\pi$
$788$	0	0
$789$	3.29844	0.117427
$790$	0	0
$791$	12.1047	0.430393
$792$	0	0
$793$	1.40312	0.0498264
$794$	0	0
$795$	9.19375	0.326069
$796$	0	0
$797$	9.29844	0.329368	0.164684	−	0.986346i	$-0.447340\pi$
$797$	9.29844	0.329368	0.164684	+	0.986346i	$0.447340\pi$
$798$	0	0
$799$	−32.4187	−1.14689
$800$	0	0
$801$	1.40312	0.0495770
$802$	0	0
$803$	−45.6125	−1.60963
$804$	0	0
$805$	−2.70156	−0.0952176
$806$	0	0
$807$	−28.2094	−0.993017
$808$	0	0
$809$	−33.0156	−1.16077	−0.580384	−	0.814343i	$-0.697097\pi$
$809$	−33.0156	−1.16077	−0.580384	+	0.814343i	$0.697097\pi$
$810$	0	0
$811$	−20.4922	−0.719578	−0.359789	−	0.933034i	$-0.617151\pi$
$811$	−20.4922	−0.719578	−0.359789	+	0.933034i	$0.617151\pi$
$812$	0	0
$813$	−11.4031	−0.399925
$814$	0	0
$815$	40.0000	1.40114
$816$	0	0
$817$	34.8062	1.21772
$818$	0	0
$819$	0.701562	0.0245146
$820$	0	0
$821$	−23.6125	−0.824082	−0.412041	−	0.911165i	$-0.635184\pi$
$821$	−23.6125	−0.824082	−0.412041	+	0.911165i	$0.635184\pi$
$822$	0	0
$823$	7.29844	0.254408	0.127204	−	0.991877i	$-0.459400\pi$
$823$	7.29844	0.254408	0.127204	+	0.991877i	$0.459400\pi$
$824$	0	0
$825$	−9.19375	−0.320085
$826$	0	0
$827$	34.5969	1.20305	0.601526	−	0.798854i	$-0.294560\pi$
$827$	34.5969	1.20305	0.601526	+	0.798854i	$0.294560\pi$
$828$	0	0
$829$	42.5969	1.47945	0.739725	−	0.672909i	$-0.234955\pi$
$829$	42.5969	1.47945	0.739725	+	0.672909i	$0.234955\pi$
$830$	0	0
$831$	8.59688	0.298222
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	−1.61250	−0.0558028
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−3.79063	−0.130867	−0.0654335	−	0.997857i	$-0.520843\pi$
$839$	−3.79063	−0.130867	−0.0654335	+	0.997857i	$0.520843\pi$
$840$	0	0
$841$	15.9109	0.548653
$842$	0	0
$843$	17.5078	0.603001
$844$	0	0
$845$	33.7906	1.16243
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−20.8062	−0.714069
$850$	0	0
$851$	−10.7016	−0.366845
$852$	0	0
$853$	−10.1047	−0.345978	−0.172989	−	0.984924i	$-0.555342\pi$
$853$	−10.1047	−0.345978	−0.172989	+	0.984924i	$0.555342\pi$
$854$	0	0
$855$	20.0000	0.683986
$856$	0	0
$857$	−6.70156	−0.228921	−0.114461	−	0.993428i	$-0.536514\pi$
$857$	−6.70156	−0.228921	−0.114461	+	0.993428i	$0.536514\pi$
$858$	0	0
$859$	47.5078	1.62095	0.810473	−	0.585776i	$-0.199210\pi$
$859$	47.5078	1.62095	0.810473	+	0.585776i	$0.199210\pi$
$860$	0	0
$861$	−6.70156	−0.228389
$862$	0	0
$863$	57.6125	1.96115	0.980576	−	0.196139i	$-0.0628404\pi$
$863$	57.6125	1.96115	0.980576	+	0.196139i	$0.0628404\pi$
$864$	0	0
$865$	−17.8219	−0.605962
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−7.58125	−0.256881
$872$	0	0
$873$	12.7016	0.429883
$874$	0	0
$875$	−7.29844	−0.246732
$876$	0	0
$877$	−4.59688	−0.155225	−0.0776127	−	0.996984i	$-0.524730\pi$
$877$	−4.59688	−0.155225	−0.0776127	+	0.996984i	$0.524730\pi$
$878$	0	0
$879$	−4.80625	−0.162111
$880$	0	0
$881$	−32.2094	−1.08516	−0.542581	−	0.840004i	$-0.682553\pi$
$881$	−32.2094	−1.08516	−0.542581	+	0.840004i	$0.682553\pi$
$882$	0	0
$883$	37.4031	1.25872	0.629358	−	0.777116i	$-0.283318\pi$
$883$	37.4031	1.25872	0.629358	+	0.777116i	$0.283318\pi$
$884$	0	0
$885$	−14.5969	−0.490669
$886$	0	0
$887$	−10.2094	−0.342797	−0.171399	−	0.985202i	$-0.554829\pi$
$887$	−10.2094	−0.342797	−0.171399	+	0.985202i	$0.554829\pi$
$888$	0	0
$889$	0.701562	0.0235296
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	60.0000	2.00782
$894$	0	0
$895$	−15.9266	−0.532366
$896$	0	0
$897$	−0.701562	−0.0234245
$898$	0	0
$899$	13.4031	0.447019
$900$	0	0
$901$	13.6125	0.453498
$902$	0	0
$903$	−4.70156	−0.156458
$904$	0	0
$905$	−23.7906	−0.790827
$906$	0	0
$907$	11.5078	0.382111	0.191055	−	0.981579i	$-0.438809\pi$
$907$	11.5078	0.382111	0.191055	+	0.981579i	$0.438809\pi$
$908$	0	0
$909$	9.40312	0.311882
$910$	0	0
$911$	−33.8953	−1.12300	−0.561501	−	0.827476i	$-0.689776\pi$
$911$	−33.8953	−1.12300	−0.561501	+	0.827476i	$0.689776\pi$
$912$	0	0
$913$	−2.38750	−0.0790148
$914$	0	0
$915$	−5.40312	−0.178622
$916$	0	0
$917$	5.40312	0.178427
$918$	0	0
$919$	−24.4187	−0.805500	−0.402750	−	0.915310i	$-0.631946\pi$
$919$	−24.4187	−0.805500	−0.402750	+	0.915310i	$0.631946\pi$
$920$	0	0
$921$	−32.9109	−1.08445
$922$	0	0
$923$	3.79063	0.124770
$924$	0	0
$925$	24.5969	0.808740
$926$	0	0
$927$	−8.70156	−0.285797
$928$	0	0
$929$	−1.08907	−0.0357311	−0.0178655	−	0.999840i	$-0.505687\pi$
$929$	−1.08907	−0.0357311	−0.0178655	+	0.999840i	$0.505687\pi$
$930$	0	0
$931$	−7.40312	−0.242628
$932$	0	0
$933$	0.596876	0.0195408
$934$	0	0
$935$	−43.2250	−1.41361
$936$	0	0
$937$	5.89531	0.192592	0.0962958	−	0.995353i	$-0.469301\pi$
$937$	5.89531	0.192592	0.0962958	+	0.995353i	$0.469301\pi$
$938$	0	0
$939$	−32.2094	−1.05111
$940$	0	0
$941$	−32.3141	−1.05341	−0.526704	−	0.850049i	$-0.676572\pi$
$941$	−32.3141	−1.05341	−0.526704	+	0.850049i	$0.676572\pi$
$942$	0	0
$943$	6.70156	0.218233
$944$	0	0
$945$	−2.70156	−0.0878818
$946$	0	0
$947$	28.9109	0.939479	0.469740	−	0.882805i	$-0.344348\pi$
$947$	28.9109	0.939479	0.469740	+	0.882805i	$0.344348\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	14.7016	0.476731
$952$	0	0
$953$	−41.2250	−1.33541	−0.667704	−	0.744427i	$-0.732723\pi$
$953$	−41.2250	−1.33541	−0.667704	+	0.744427i	$0.732723\pi$
$954$	0	0
$955$	−7.58125	−0.245324
$956$	0	0
$957$	−26.8062	−0.866523
$958$	0	0
$959$	13.2984	0.429429
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	6.80625	0.219328
$964$	0	0
$965$	10.5234	0.338761
$966$	0	0
$967$	−10.8062	−0.347506	−0.173753	−	0.984789i	$-0.555589\pi$
$967$	−10.8062	−0.347506	−0.173753	+	0.984789i	$0.555589\pi$
$968$	0	0
$969$	29.6125	0.951290
$970$	0	0
$971$	39.8219	1.27794	0.638972	−	0.769230i	$-0.279360\pi$
$971$	39.8219	1.27794	0.638972	+	0.769230i	$0.279360\pi$
$972$	0	0
$973$	11.2984	0.362211
$974$	0	0
$975$	1.61250	0.0516412
$976$	0	0
$977$	−53.7172	−1.71856	−0.859282	−	0.511501i	$-0.829090\pi$
$977$	−53.7172	−1.71856	−0.859282	+	0.511501i	$0.829090\pi$
$978$	0	0
$979$	5.61250	0.179376
$980$	0	0
$981$	−1.29844	−0.0414559
$982$	0	0
$983$	−3.79063	−0.120902	−0.0604511	−	0.998171i	$-0.519254\pi$
$983$	−3.79063	−0.120902	−0.0604511	+	0.998171i	$0.519254\pi$
$984$	0	0
$985$	25.6859	0.818422
$986$	0	0
$987$	−8.10469	−0.257975
$988$	0	0
$989$	4.70156	0.149501
$990$	0	0
$991$	−28.0000	−0.889449	−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000	−0.889449	−0.444725	+	0.895667i	$0.646698\pi$
$992$	0	0
$993$	−20.0000	−0.634681
$994$	0	0
$995$	−23.5078	−0.745248
$996$	0	0
$997$	−10.5969	−0.335606	−0.167803	−	0.985821i	$-0.553667\pi$
$997$	−10.5969	−0.335606	−0.167803	+	0.985821i	$0.553667\pi$
$998$	0	0
$999$	−10.7016	−0.338582

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bc.1.1		2
4.3	odd	2		966.2.a.n.1.1	✓	2
12.11	even	2		2898.2.a.bb.1.2		2
28.27	even	2		6762.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.n.1.1	✓	2	4.3	odd	2
2898.2.a.bb.1.2		2	12.11	even	2
6762.2.a.bo.1.2		2	28.27	even	2
7728.2.a.bc.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.70156 q^{5} -1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -0.701562 q^{13} +2.70156 q^{15} +4.00000 q^{17} -7.40312 q^{19} +1.00000 q^{21} -1.00000 q^{23} +2.29844 q^{25} -1.00000 q^{27} +6.70156 q^{29} +2.00000 q^{31} -4.00000 q^{33} +2.70156 q^{35} +10.7016 q^{37} +0.701562 q^{39} -6.70156 q^{41} -4.70156 q^{43} -2.70156 q^{45} -8.10469 q^{47} +1.00000 q^{49} -4.00000 q^{51} +3.40312 q^{53} -10.8062 q^{55} +7.40312 q^{57} -5.40312 q^{59} -2.00000 q^{61} -1.00000 q^{63} +1.89531 q^{65} +10.8062 q^{67} +1.00000 q^{69} -5.40312 q^{71} -11.4031 q^{73} -2.29844 q^{75} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -0.596876 q^{83} -10.8062 q^{85} -6.70156 q^{87} +1.40312 q^{89} +0.701562 q^{91} -2.00000 q^{93} +20.0000 q^{95} +12.7016 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 8 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 11 q^{25} - 2 q^{27} + 7 q^{29} + 4 q^{31} - 8 q^{33} - q^{35} + 15 q^{37} - 5 q^{39} - 7 q^{41} - 3 q^{43} + q^{45} + 3 q^{47} + 2 q^{49} - 8 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 2 q^{59} - 4 q^{61} - 2 q^{63} + 23 q^{65} - 4 q^{67} + 2 q^{69} + 2 q^{71} - 10 q^{73} - 11 q^{75} - 8 q^{77} - 16 q^{79} + 2 q^{81} - 14 q^{83} + 4 q^{85} - 7 q^{87} - 10 q^{89} - 5 q^{91} - 4 q^{93} + 40 q^{95} + 19 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 8 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 11 * q^25 - 2 * q^27 + 7 * q^29 + 4 * q^31 - 8 * q^33 - q^35 + 15 * q^37 - 5 * q^39 - 7 * q^41 - 3 * q^43 + q^45 + 3 * q^47 + 2 * q^49 - 8 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 2 * q^59 - 4 * q^61 - 2 * q^63 + 23 * q^65 - 4 * q^67 + 2 * q^69 + 2 * q^71 - 10 * q^73 - 11 * q^75 - 8 * q^77 - 16 * q^79 + 2 * q^81 - 14 * q^83 + 4 * q^85 - 7 * q^87 - 10 * q^89 - 5 * q^91 - 4 * q^93 + 40 * q^95 + 19 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more