LMFDB - Embedded newform 7728.2.a.x.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +2.30278 q^{13} +0.697224 q^{15} -5.60555 q^{17} +1.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.51388 q^{25} -1.00000 q^{27} +6.21110 q^{29} -3.00000 q^{31} -5.00000 q^{33} -0.697224 q^{35} -9.00000 q^{37} -2.30278 q^{39} -12.2111 q^{41} -5.51388 q^{43} -0.697224 q^{45} +8.60555 q^{47} +1.00000 q^{49} +5.60555 q^{51} -12.5139 q^{53} -3.48612 q^{55} -1.60555 q^{57} -3.90833 q^{59} -1.09167 q^{61} +1.00000 q^{63} -1.60555 q^{65} +11.9083 q^{67} +1.00000 q^{69} -0.908327 q^{71} -2.21110 q^{73} +4.51388 q^{75} +5.00000 q^{77} +1.00000 q^{79} +1.00000 q^{81} -5.60555 q^{83} +3.90833 q^{85} -6.21110 q^{87} +10.9083 q^{89} +2.30278 q^{91} +3.00000 q^{93} -1.11943 q^{95} -17.6056 q^{97} +5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} + 10 q^{11} + q^{13} + 5 q^{15} - 4 q^{17} - 4 q^{19} - 2 q^{21} - 2 q^{23} + 9 q^{25} - 2 q^{27} - 2 q^{29} - 6 q^{31} - 10 q^{33} - 5 q^{35} - 18 q^{37} - q^{39} - 10 q^{41} + 7 q^{43} - 5 q^{45} + 10 q^{47} + 2 q^{49} + 4 q^{51} - 7 q^{53} - 25 q^{55} + 4 q^{57} + 3 q^{59} - 13 q^{61} + 2 q^{63} + 4 q^{65} + 13 q^{67} + 2 q^{69} + 9 q^{71} + 10 q^{73} - 9 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} - 4 q^{83} - 3 q^{85} + 2 q^{87} + 11 q^{89} + q^{91} + 6 q^{93} + 23 q^{95} - 28 q^{97} + 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 + 10 * q^11 + q^13 + 5 * q^15 - 4 * q^17 - 4 * q^19 - 2 * q^21 - 2 * q^23 + 9 * q^25 - 2 * q^27 - 2 * q^29 - 6 * q^31 - 10 * q^33 - 5 * q^35 - 18 * q^37 - q^39 - 10 * q^41 + 7 * q^43 - 5 * q^45 + 10 * q^47 + 2 * q^49 + 4 * q^51 - 7 * q^53 - 25 * q^55 + 4 * q^57 + 3 * q^59 - 13 * q^61 + 2 * q^63 + 4 * q^65 + 13 * q^67 + 2 * q^69 + 9 * q^71 + 10 * q^73 - 9 * q^75 + 10 * q^77 + 2 * q^79 + 2 * q^81 - 4 * q^83 - 3 * q^85 + 2 * q^87 + 11 * q^89 + q^91 + 6 * q^93 + 23 * q^95 - 28 * q^97 + 10 * 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
$5$	−0.697224	−0.311808	−0.155904	−	0.987772i	$-0.549829\pi$
$5$	−0.697224	−0.311808	−0.155904	+	0.987772i	$0.549829\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	2.30278	0.638675	0.319338	−	0.947641i	$-0.396540\pi$
$13$	2.30278	0.638675	0.319338	+	0.947641i	$0.396540\pi$
$14$	0	0
$15$	0.697224	0.180023
$16$	0	0
$17$	−5.60555	−1.35955	−0.679773	−	0.733423i	$-0.737922\pi$
$17$	−5.60555	−1.35955	−0.679773	+	0.733423i	$0.737922\pi$
$18$	0	0
$19$	1.60555	0.368339	0.184169	−	0.982895i	$-0.441041\pi$
$19$	1.60555	0.368339	0.184169	+	0.982895i	$0.441041\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.51388	−0.902776
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.21110	1.15337	0.576686	−	0.816966i	$-0.304345\pi$
$29$	6.21110	1.15337	0.576686	+	0.816966i	$0.304345\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−5.00000	−0.870388
$34$	0	0
$35$	−0.697224	−0.117852
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	0	0
$39$	−2.30278	−0.368739
$40$	0	0
$41$	−12.2111	−1.90705	−0.953527	−	0.301308i	$-0.902577\pi$
$41$	−12.2111	−1.90705	−0.953527	+	0.301308i	$0.902577\pi$
$42$	0	0
$43$	−5.51388	−0.840859	−0.420429	−	0.907325i	$-0.638121\pi$
$43$	−5.51388	−0.840859	−0.420429	+	0.907325i	$0.638121\pi$
$44$	0	0
$45$	−0.697224	−0.103936
$46$	0	0
$47$	8.60555	1.25525	0.627624	−	0.778516i	$-0.284027\pi$
$47$	8.60555	1.25525	0.627624	+	0.778516i	$0.284027\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.60555	0.784934
$52$	0	0
$53$	−12.5139	−1.71891	−0.859457	−	0.511209i	$-0.829198\pi$
$53$	−12.5139	−1.71891	−0.859457	+	0.511209i	$0.829198\pi$
$54$	0	0
$55$	−3.48612	−0.470069
$56$	0	0
$57$	−1.60555	−0.212660
$58$	0	0
$59$	−3.90833	−0.508821	−0.254410	−	0.967096i	$-0.581881\pi$
$59$	−3.90833	−0.508821	−0.254410	+	0.967096i	$0.581881\pi$
$60$	0	0
$61$	−1.09167	−0.139774	−0.0698872	−	0.997555i	$-0.522264\pi$
$61$	−1.09167	−0.139774	−0.0698872	+	0.997555i	$0.522264\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.60555	−0.199144
$66$	0	0
$67$	11.9083	1.45483	0.727417	−	0.686196i	$-0.240721\pi$
$67$	11.9083	1.45483	0.727417	+	0.686196i	$0.240721\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−0.908327	−0.107799	−0.0538993	−	0.998546i	$-0.517165\pi$
$71$	−0.908327	−0.107799	−0.0538993	+	0.998546i	$0.517165\pi$
$72$	0	0
$73$	−2.21110	−0.258790	−0.129395	−	0.991593i	$-0.541304\pi$
$73$	−2.21110	−0.258790	−0.129395	+	0.991593i	$0.541304\pi$
$74$	0	0
$75$	4.51388	0.521218
$76$	0	0
$77$	5.00000	0.569803
$78$	0	0
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.60555	−0.615289	−0.307645	−	0.951501i	$-0.599541\pi$
$83$	−5.60555	−0.615289	−0.307645	+	0.951501i	$0.599541\pi$
$84$	0	0
$85$	3.90833	0.423918
$86$	0	0
$87$	−6.21110	−0.665900
$88$	0	0
$89$	10.9083	1.15628	0.578140	−	0.815937i	$-0.303779\pi$
$89$	10.9083	1.15628	0.578140	+	0.815937i	$0.303779\pi$
$90$	0	0
$91$	2.30278	0.241396
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	−1.11943	−0.114851
$96$	0	0
$97$	−17.6056	−1.78757	−0.893786	−	0.448493i	$-0.851961\pi$
$97$	−17.6056	−1.78757	−0.893786	+	0.448493i	$0.851961\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	9.30278	0.925661	0.462830	−	0.886447i	$-0.346834\pi$
$101$	9.30278	0.925661	0.462830	+	0.886447i	$0.346834\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0.697224	0.0680421
$106$	0	0
$107$	−7.51388	−0.726394	−0.363197	−	0.931712i	$-0.618315\pi$
$107$	−7.51388	−0.726394	−0.363197	+	0.931712i	$0.618315\pi$
$108$	0	0
$109$	−3.90833	−0.374350	−0.187175	−	0.982327i	$-0.559933\pi$
$109$	−3.90833	−0.374350	−0.187175	+	0.982327i	$0.559933\pi$
$110$	0	0
$111$	9.00000	0.854242
$112$	0	0
$113$	5.30278	0.498843	0.249422	−	0.968395i	$-0.419760\pi$
$113$	5.30278	0.498843	0.249422	+	0.968395i	$0.419760\pi$
$114$	0	0
$115$	0.697224	0.0650165
$116$	0	0
$117$	2.30278	0.212892
$118$	0	0
$119$	−5.60555	−0.513860
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	12.2111	1.10104
$124$	0	0
$125$	6.63331	0.593301
$126$	0	0
$127$	−1.69722	−0.150604	−0.0753022	−	0.997161i	$-0.523992\pi$
$127$	−1.69722	−0.150604	−0.0753022	+	0.997161i	$0.523992\pi$
$128$	0	0
$129$	5.51388	0.485470
$130$	0	0
$131$	−10.3944	−0.908167	−0.454084	−	0.890959i	$-0.650033\pi$
$131$	−10.3944	−0.908167	−0.454084	+	0.890959i	$0.650033\pi$
$132$	0	0
$133$	1.60555	0.139219
$134$	0	0
$135$	0.697224	0.0600075
$136$	0	0
$137$	16.8167	1.43674	0.718372	−	0.695659i	$-0.244888\pi$
$137$	16.8167	1.43674	0.718372	+	0.695659i	$0.244888\pi$
$138$	0	0
$139$	−15.9083	−1.34933	−0.674663	−	0.738126i	$-0.735711\pi$
$139$	−15.9083	−1.34933	−0.674663	+	0.738126i	$0.735711\pi$
$140$	0	0
$141$	−8.60555	−0.724718
$142$	0	0
$143$	11.5139	0.962839
$144$	0	0
$145$	−4.33053	−0.359631
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	8.60555	0.704994	0.352497	−	0.935813i	$-0.385333\pi$
$149$	8.60555	0.704994	0.352497	+	0.935813i	$0.385333\pi$
$150$	0	0
$151$	−16.6056	−1.35134	−0.675670	−	0.737204i	$-0.736146\pi$
$151$	−16.6056	−1.35134	−0.675670	+	0.737204i	$0.736146\pi$
$152$	0	0
$153$	−5.60555	−0.453182
$154$	0	0
$155$	2.09167	0.168007
$156$	0	0
$157$	−3.81665	−0.304602	−0.152301	−	0.988334i	$-0.548668\pi$
$157$	−3.81665	−0.304602	−0.152301	+	0.988334i	$0.548668\pi$
$158$	0	0
$159$	12.5139	0.992415
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	13.7250	1.07502	0.537512	−	0.843256i	$-0.319364\pi$
$163$	13.7250	1.07502	0.537512	+	0.843256i	$0.319364\pi$
$164$	0	0
$165$	3.48612	0.271394
$166$	0	0
$167$	−2.81665	−0.217959	−0.108980	−	0.994044i	$-0.534758\pi$
$167$	−2.81665	−0.217959	−0.108980	+	0.994044i	$0.534758\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	0	0
$171$	1.60555	0.122780
$172$	0	0
$173$	−18.2111	−1.38456	−0.692282	−	0.721627i	$-0.743395\pi$
$173$	−18.2111	−1.38456	−0.692282	+	0.721627i	$0.743395\pi$
$174$	0	0
$175$	−4.51388	−0.341217
$176$	0	0
$177$	3.90833	0.293768
$178$	0	0
$179$	2.69722	0.201600	0.100800	−	0.994907i	$-0.467860\pi$
$179$	2.69722	0.201600	0.100800	+	0.994907i	$0.467860\pi$
$180$	0	0
$181$	−6.81665	−0.506678	−0.253339	−	0.967378i	$-0.581529\pi$
$181$	−6.81665	−0.506678	−0.253339	+	0.967378i	$0.581529\pi$
$182$	0	0
$183$	1.09167	0.0806988
$184$	0	0
$185$	6.27502	0.461349
$186$	0	0
$187$	−28.0278	−2.04959
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	4.60555	0.333246	0.166623	−	0.986021i	$-0.446714\pi$
$191$	4.60555	0.333246	0.166623	+	0.986021i	$0.446714\pi$
$192$	0	0
$193$	9.02776	0.649832	0.324916	−	0.945743i	$-0.394664\pi$
$193$	9.02776	0.649832	0.324916	+	0.945743i	$0.394664\pi$
$194$	0	0
$195$	1.60555	0.114976
$196$	0	0
$197$	−7.09167	−0.505261	−0.252630	−	0.967563i	$-0.581296\pi$
$197$	−7.09167	−0.505261	−0.252630	+	0.967563i	$0.581296\pi$
$198$	0	0
$199$	−12.5139	−0.887085	−0.443543	−	0.896253i	$-0.646279\pi$
$199$	−12.5139	−0.887085	−0.443543	+	0.896253i	$0.646279\pi$
$200$	0	0
$201$	−11.9083	−0.839949
$202$	0	0
$203$	6.21110	0.435934
$204$	0	0
$205$	8.51388	0.594635
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	8.02776	0.555292
$210$	0	0
$211$	27.4222	1.88782	0.943911	−	0.330199i	$-0.107116\pi$
$211$	27.4222	1.88782	0.943911	+	0.330199i	$0.107116\pi$
$212$	0	0
$213$	0.908327	0.0622375
$214$	0	0
$215$	3.84441	0.262187
$216$	0	0
$217$	−3.00000	−0.203653
$218$	0	0
$219$	2.21110	0.149412
$220$	0	0
$221$	−12.9083	−0.868308
$222$	0	0
$223$	−19.9083	−1.33316	−0.666580	−	0.745433i	$-0.732243\pi$
$223$	−19.9083	−1.33316	−0.666580	+	0.745433i	$0.732243\pi$
$224$	0	0
$225$	−4.51388	−0.300925
$226$	0	0
$227$	20.3305	1.34938	0.674692	−	0.738099i	$-0.264276\pi$
$227$	20.3305	1.34938	0.674692	+	0.738099i	$0.264276\pi$
$228$	0	0
$229$	2.51388	0.166122	0.0830609	−	0.996544i	$-0.473530\pi$
$229$	2.51388	0.166122	0.0830609	+	0.996544i	$0.473530\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	−14.3305	−0.938824	−0.469412	−	0.882979i	$-0.655534\pi$
$233$	−14.3305	−0.938824	−0.469412	+	0.882979i	$0.655534\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	0	0
$237$	−1.00000	−0.0649570
$238$	0	0
$239$	14.0917	0.911515	0.455757	−	0.890104i	$-0.349369\pi$
$239$	14.0917	0.911515	0.455757	+	0.890104i	$0.349369\pi$
$240$	0	0
$241$	12.0278	0.774776	0.387388	−	0.921917i	$-0.373377\pi$
$241$	12.0278	0.774776	0.387388	+	0.921917i	$0.373377\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.697224	−0.0445440
$246$	0	0
$247$	3.69722	0.235249
$248$	0	0
$249$	5.60555	0.355237
$250$	0	0
$251$	−23.8167	−1.50329	−0.751647	−	0.659566i	$-0.770740\pi$
$251$	−23.8167	−1.50329	−0.751647	+	0.659566i	$0.770740\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	0	0
$255$	−3.90833	−0.244749
$256$	0	0
$257$	27.0278	1.68595	0.842973	−	0.537957i	$-0.180804\pi$
$257$	27.0278	1.68595	0.842973	+	0.537957i	$0.180804\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	6.21110	0.384458
$262$	0	0
$263$	−10.3944	−0.640949	−0.320475	−	0.947257i	$-0.603842\pi$
$263$	−10.3944	−0.640949	−0.320475	+	0.947257i	$0.603842\pi$
$264$	0	0
$265$	8.72498	0.535971
$266$	0	0
$267$	−10.9083	−0.667579
$268$	0	0
$269$	0.908327	0.0553817	0.0276908	−	0.999617i	$-0.491185\pi$
$269$	0.908327	0.0553817	0.0276908	+	0.999617i	$0.491185\pi$
$270$	0	0
$271$	−3.60555	−0.219022	−0.109511	−	0.993986i	$-0.534928\pi$
$271$	−3.60555	−0.219022	−0.109511	+	0.993986i	$0.534928\pi$
$272$	0	0
$273$	−2.30278	−0.139370
$274$	0	0
$275$	−22.5694	−1.36099
$276$	0	0
$277$	−12.3028	−0.739202	−0.369601	−	0.929191i	$-0.620506\pi$
$277$	−12.3028	−0.739202	−0.369601	+	0.929191i	$0.620506\pi$
$278$	0	0
$279$	−3.00000	−0.179605
$280$	0	0
$281$	27.8167	1.65940	0.829701	−	0.558208i	$-0.188511\pi$
$281$	27.8167	1.65940	0.829701	+	0.558208i	$0.188511\pi$
$282$	0	0
$283$	−17.3028	−1.02854	−0.514272	−	0.857627i	$-0.671938\pi$
$283$	−17.3028	−1.02854	−0.514272	+	0.857627i	$0.671938\pi$
$284$	0	0
$285$	1.11943	0.0663093
$286$	0	0
$287$	−12.2111	−0.720799
$288$	0	0
$289$	14.4222	0.848365
$290$	0	0
$291$	17.6056	1.03206
$292$	0	0
$293$	−24.8444	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$293$	−24.8444	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$294$	0	0
$295$	2.72498	0.158655
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−2.30278	−0.133173
$300$	0	0
$301$	−5.51388	−0.317815
$302$	0	0
$303$	−9.30278	−0.534430
$304$	0	0
$305$	0.761141	0.0435828
$306$	0	0
$307$	−18.2111	−1.03936	−0.519681	−	0.854360i	$-0.673949\pi$
$307$	−18.2111	−1.03936	−0.519681	+	0.854360i	$0.673949\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	9.51388	0.539483	0.269741	−	0.962933i	$-0.413062\pi$
$311$	9.51388	0.539483	0.269741	+	0.962933i	$0.413062\pi$
$312$	0	0
$313$	1.21110	0.0684556	0.0342278	−	0.999414i	$-0.489103\pi$
$313$	1.21110	0.0684556	0.0342278	+	0.999414i	$0.489103\pi$
$314$	0	0
$315$	−0.697224	−0.0392841
$316$	0	0
$317$	−30.5139	−1.71383	−0.856915	−	0.515458i	$-0.827622\pi$
$317$	−30.5139	−1.71383	−0.856915	+	0.515458i	$0.827622\pi$
$318$	0	0
$319$	31.0555	1.73877
$320$	0	0
$321$	7.51388	0.419384
$322$	0	0
$323$	−9.00000	−0.500773
$324$	0	0
$325$	−10.3944	−0.576580
$326$	0	0
$327$	3.90833	0.216131
$328$	0	0
$329$	8.60555	0.474439
$330$	0	0
$331$	−17.6333	−0.969214	−0.484607	−	0.874732i	$-0.661037\pi$
$331$	−17.6333	−0.969214	−0.484607	+	0.874732i	$0.661037\pi$
$332$	0	0
$333$	−9.00000	−0.493197
$334$	0	0
$335$	−8.30278	−0.453629
$336$	0	0
$337$	−15.6972	−0.855082	−0.427541	−	0.903996i	$-0.640620\pi$
$337$	−15.6972	−0.855082	−0.427541	+	0.903996i	$0.640620\pi$
$338$	0	0
$339$	−5.30278	−0.288007
$340$	0	0
$341$	−15.0000	−0.812296
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.697224	−0.0375373
$346$	0	0
$347$	−14.6333	−0.785557	−0.392779	−	0.919633i	$-0.628486\pi$
$347$	−14.6333	−0.785557	−0.392779	+	0.919633i	$0.628486\pi$
$348$	0	0
$349$	−10.0917	−0.540195	−0.270097	−	0.962833i	$-0.587056\pi$
$349$	−10.0917	−0.540195	−0.270097	+	0.962833i	$0.587056\pi$
$350$	0	0
$351$	−2.30278	−0.122913
$352$	0	0
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	0	0
$355$	0.633308	0.0336125
$356$	0	0
$357$	5.60555	0.296677
$358$	0	0
$359$	35.5416	1.87582	0.937908	−	0.346884i	$-0.112760\pi$
$359$	35.5416	1.87582	0.937908	+	0.346884i	$0.112760\pi$
$360$	0	0
$361$	−16.4222	−0.864327
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	1.54163	0.0806928
$366$	0	0
$367$	−21.5139	−1.12302	−0.561508	−	0.827472i	$-0.689778\pi$
$367$	−21.5139	−1.12302	−0.561508	+	0.827472i	$0.689778\pi$
$368$	0	0
$369$	−12.2111	−0.635685
$370$	0	0
$371$	−12.5139	−0.649688
$372$	0	0
$373$	16.2111	0.839379	0.419690	−	0.907668i	$-0.362139\pi$
$373$	16.2111	0.839379	0.419690	+	0.907668i	$0.362139\pi$
$374$	0	0
$375$	−6.63331	−0.342543
$376$	0	0
$377$	14.3028	0.736630
$378$	0	0
$379$	22.4222	1.15175	0.575876	−	0.817537i	$-0.304661\pi$
$379$	22.4222	1.15175	0.575876	+	0.817537i	$0.304661\pi$
$380$	0	0
$381$	1.69722	0.0869514
$382$	0	0
$383$	−4.81665	−0.246120	−0.123060	−	0.992399i	$-0.539271\pi$
$383$	−4.81665	−0.246120	−0.123060	+	0.992399i	$0.539271\pi$
$384$	0	0
$385$	−3.48612	−0.177669
$386$	0	0
$387$	−5.51388	−0.280286
$388$	0	0
$389$	−36.6333	−1.85738	−0.928691	−	0.370854i	$-0.879065\pi$
$389$	−36.6333	−1.85738	−0.928691	+	0.370854i	$0.879065\pi$
$390$	0	0
$391$	5.60555	0.283485
$392$	0	0
$393$	10.3944	0.524331
$394$	0	0
$395$	−0.697224	−0.0350812
$396$	0	0
$397$	−27.3944	−1.37489	−0.687444	−	0.726237i	$-0.741267\pi$
$397$	−27.3944	−1.37489	−0.687444	+	0.726237i	$0.741267\pi$
$398$	0	0
$399$	−1.60555	−0.0803781
$400$	0	0
$401$	−13.4222	−0.670273	−0.335136	−	0.942170i	$-0.608782\pi$
$401$	−13.4222	−0.670273	−0.335136	+	0.942170i	$0.608782\pi$
$402$	0	0
$403$	−6.90833	−0.344128
$404$	0	0
$405$	−0.697224	−0.0346454
$406$	0	0
$407$	−45.0000	−2.23057
$408$	0	0
$409$	24.0278	1.18810	0.594048	−	0.804430i	$-0.297529\pi$
$409$	24.0278	1.18810	0.594048	+	0.804430i	$0.297529\pi$
$410$	0	0
$411$	−16.8167	−0.829504
$412$	0	0
$413$	−3.90833	−0.192316
$414$	0	0
$415$	3.90833	0.191852
$416$	0	0
$417$	15.9083	0.779034
$418$	0	0
$419$	−15.1194	−0.738632	−0.369316	−	0.929304i	$-0.620408\pi$
$419$	−15.1194	−0.738632	−0.369316	+	0.929304i	$0.620408\pi$
$420$	0	0
$421$	9.69722	0.472614	0.236307	−	0.971678i	$-0.424063\pi$
$421$	9.69722	0.472614	0.236307	+	0.971678i	$0.424063\pi$
$422$	0	0
$423$	8.60555	0.418416
$424$	0	0
$425$	25.3028	1.22736
$426$	0	0
$427$	−1.09167	−0.0528298
$428$	0	0
$429$	−11.5139	−0.555895
$430$	0	0
$431$	8.72498	0.420268	0.210134	−	0.977673i	$-0.432610\pi$
$431$	8.72498	0.420268	0.210134	+	0.977673i	$0.432610\pi$
$432$	0	0
$433$	−21.0278	−1.01053	−0.505265	−	0.862964i	$-0.668605\pi$
$433$	−21.0278	−1.01053	−0.505265	+	0.862964i	$0.668605\pi$
$434$	0	0
$435$	4.33053	0.207633
$436$	0	0
$437$	−1.60555	−0.0768039
$438$	0	0
$439$	−20.2111	−0.964623	−0.482312	−	0.876000i	$-0.660203\pi$
$439$	−20.2111	−0.964623	−0.482312	+	0.876000i	$0.660203\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	32.8444	1.56049	0.780243	−	0.625477i	$-0.215096\pi$
$443$	32.8444	1.56049	0.780243	+	0.625477i	$0.215096\pi$
$444$	0	0
$445$	−7.60555	−0.360538
$446$	0	0
$447$	−8.60555	−0.407029
$448$	0	0
$449$	−35.7250	−1.68597	−0.842983	−	0.537940i	$-0.819203\pi$
$449$	−35.7250	−1.68597	−0.842983	+	0.537940i	$0.819203\pi$
$450$	0	0
$451$	−61.0555	−2.87499
$452$	0	0
$453$	16.6056	0.780197
$454$	0	0
$455$	−1.60555	−0.0752694
$456$	0	0
$457$	−10.8806	−0.508972	−0.254486	−	0.967077i	$-0.581906\pi$
$457$	−10.8806	−0.508972	−0.254486	+	0.967077i	$0.581906\pi$
$458$	0	0
$459$	5.60555	0.261645
$460$	0	0
$461$	7.72498	0.359788	0.179894	−	0.983686i	$-0.442424\pi$
$461$	7.72498	0.359788	0.179894	+	0.983686i	$0.442424\pi$
$462$	0	0
$463$	18.8167	0.874484	0.437242	−	0.899344i	$-0.355955\pi$
$463$	18.8167	0.874484	0.437242	+	0.899344i	$0.355955\pi$
$464$	0	0
$465$	−2.09167	−0.0969990
$466$	0	0
$467$	−23.6056	−1.09233	−0.546167	−	0.837676i	$-0.683914\pi$
$467$	−23.6056	−1.09233	−0.546167	+	0.837676i	$0.683914\pi$
$468$	0	0
$469$	11.9083	0.549875
$470$	0	0
$471$	3.81665	0.175862
$472$	0	0
$473$	−27.5694	−1.26764
$474$	0	0
$475$	−7.24726	−0.332527
$476$	0	0
$477$	−12.5139	−0.572971
$478$	0	0
$479$	18.2111	0.832087	0.416043	−	0.909345i	$-0.363416\pi$
$479$	18.2111	0.832087	0.416043	+	0.909345i	$0.363416\pi$
$480$	0	0
$481$	−20.7250	−0.944978
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	12.2750	0.557380
$486$	0	0
$487$	0.816654	0.0370061	0.0185031	−	0.999829i	$-0.494110\pi$
$487$	0.816654	0.0370061	0.0185031	+	0.999829i	$0.494110\pi$
$488$	0	0
$489$	−13.7250	−0.620665
$490$	0	0
$491$	17.9361	0.809444	0.404722	−	0.914440i	$-0.367368\pi$
$491$	17.9361	0.809444	0.404722	+	0.914440i	$0.367368\pi$
$492$	0	0
$493$	−34.8167	−1.56806
$494$	0	0
$495$	−3.48612	−0.156690
$496$	0	0
$497$	−0.908327	−0.0407440
$498$	0	0
$499$	−31.9083	−1.42841	−0.714206	−	0.699935i	$-0.753212\pi$
$499$	−31.9083	−1.42841	−0.714206	+	0.699935i	$0.753212\pi$
$500$	0	0
$501$	2.81665	0.125839
$502$	0	0
$503$	−17.7250	−0.790318	−0.395159	−	0.918613i	$-0.629310\pi$
$503$	−17.7250	−0.790318	−0.395159	+	0.918613i	$0.629310\pi$
$504$	0	0
$505$	−6.48612	−0.288629
$506$	0	0
$507$	7.69722	0.341846
$508$	0	0
$509$	−33.4500	−1.48264	−0.741322	−	0.671150i	$-0.765801\pi$
$509$	−33.4500	−1.48264	−0.741322	+	0.671150i	$0.765801\pi$
$510$	0	0
$511$	−2.21110	−0.0978134
$512$	0	0
$513$	−1.60555	−0.0708868
$514$	0	0
$515$	−2.78890	−0.122894
$516$	0	0
$517$	43.0278	1.89236
$518$	0	0
$519$	18.2111	0.799379
$520$	0	0
$521$	21.6333	0.947772	0.473886	−	0.880586i	$-0.342851\pi$
$521$	21.6333	0.947772	0.473886	+	0.880586i	$0.342851\pi$
$522$	0	0
$523$	0.422205	0.0184617	0.00923087	−	0.999957i	$-0.497062\pi$
$523$	0.422205	0.0184617	0.00923087	+	0.999957i	$0.497062\pi$
$524$	0	0
$525$	4.51388	0.197002
$526$	0	0
$527$	16.8167	0.732545
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.90833	−0.169607
$532$	0	0
$533$	−28.1194	−1.21799
$534$	0	0
$535$	5.23886	0.226496
$536$	0	0
$537$	−2.69722	−0.116394
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	−5.81665	−0.250077	−0.125039	−	0.992152i	$-0.539906\pi$
$541$	−5.81665	−0.250077	−0.125039	+	0.992152i	$0.539906\pi$
$542$	0	0
$543$	6.81665	0.292531
$544$	0	0
$545$	2.72498	0.116725
$546$	0	0
$547$	7.72498	0.330296	0.165148	−	0.986269i	$-0.447190\pi$
$547$	7.72498	0.330296	0.165148	+	0.986269i	$0.447190\pi$
$548$	0	0
$549$	−1.09167	−0.0465915
$550$	0	0
$551$	9.97224	0.424832
$552$	0	0
$553$	1.00000	0.0425243
$554$	0	0
$555$	−6.27502	−0.266360
$556$	0	0
$557$	5.02776	0.213033	0.106516	−	0.994311i	$-0.466030\pi$
$557$	5.02776	0.213033	0.106516	+	0.994311i	$0.466030\pi$
$558$	0	0
$559$	−12.6972	−0.537035
$560$	0	0
$561$	28.0278	1.18333
$562$	0	0
$563$	43.3305	1.82616	0.913082	−	0.407776i	$-0.133695\pi$
$563$	43.3305	1.82616	0.913082	+	0.407776i	$0.133695\pi$
$564$	0	0
$565$	−3.69722	−0.155543
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	10.7889	0.452294	0.226147	−	0.974093i	$-0.427387\pi$
$569$	10.7889	0.452294	0.226147	+	0.974093i	$0.427387\pi$
$570$	0	0
$571$	−10.2111	−0.427321	−0.213661	−	0.976908i	$-0.568539\pi$
$571$	−10.2111	−0.427321	−0.213661	+	0.976908i	$0.568539\pi$
$572$	0	0
$573$	−4.60555	−0.192400
$574$	0	0
$575$	4.51388	0.188242
$576$	0	0
$577$	−36.4222	−1.51628	−0.758138	−	0.652094i	$-0.773891\pi$
$577$	−36.4222	−1.51628	−0.758138	+	0.652094i	$0.773891\pi$
$578$	0	0
$579$	−9.02776	−0.375181
$580$	0	0
$581$	−5.60555	−0.232557
$582$	0	0
$583$	−62.5694	−2.59136
$584$	0	0
$585$	−1.60555	−0.0663814
$586$	0	0
$587$	39.1472	1.61578	0.807889	−	0.589335i	$-0.200610\pi$
$587$	39.1472	1.61578	0.807889	+	0.589335i	$0.200610\pi$
$588$	0	0
$589$	−4.81665	−0.198467
$590$	0	0
$591$	7.09167	0.291712
$592$	0	0
$593$	37.0278	1.52055	0.760274	−	0.649603i	$-0.225065\pi$
$593$	37.0278	1.52055	0.760274	+	0.649603i	$0.225065\pi$
$594$	0	0
$595$	3.90833	0.160226
$596$	0	0
$597$	12.5139	0.512159
$598$	0	0
$599$	39.5139	1.61449	0.807247	−	0.590214i	$-0.200957\pi$
$599$	39.5139	1.61449	0.807247	+	0.590214i	$0.200957\pi$
$600$	0	0
$601$	40.9361	1.66982	0.834909	−	0.550388i	$-0.185520\pi$
$601$	40.9361	1.66982	0.834909	+	0.550388i	$0.185520\pi$
$602$	0	0
$603$	11.9083	0.484945
$604$	0	0
$605$	−9.76114	−0.396847
$606$	0	0
$607$	24.4861	0.993861	0.496931	−	0.867790i	$-0.334460\pi$
$607$	24.4861	0.993861	0.496931	+	0.867790i	$0.334460\pi$
$608$	0	0
$609$	−6.21110	−0.251687
$610$	0	0
$611$	19.8167	0.801696
$612$	0	0
$613$	2.81665	0.113764	0.0568818	−	0.998381i	$-0.481884\pi$
$613$	2.81665	0.113764	0.0568818	+	0.998381i	$0.481884\pi$
$614$	0	0
$615$	−8.51388	−0.343313
$616$	0	0
$617$	14.7250	0.592805	0.296403	−	0.955063i	$-0.404213\pi$
$617$	14.7250	0.592805	0.296403	+	0.955063i	$0.404213\pi$
$618$	0	0
$619$	−9.88057	−0.397134	−0.198567	−	0.980087i	$-0.563629\pi$
$619$	−9.88057	−0.397134	−0.198567	+	0.980087i	$0.563629\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	10.9083	0.437033
$624$	0	0
$625$	17.9445	0.717779
$626$	0	0
$627$	−8.02776	−0.320598
$628$	0	0
$629$	50.4500	2.01157
$630$	0	0
$631$	11.9722	0.476607	0.238304	−	0.971191i	$-0.423409\pi$
$631$	11.9722	0.476607	0.238304	+	0.971191i	$0.423409\pi$
$632$	0	0
$633$	−27.4222	−1.08993
$634$	0	0
$635$	1.18335	0.0469597
$636$	0	0
$637$	2.30278	0.0912393
$638$	0	0
$639$	−0.908327	−0.0359329
$640$	0	0
$641$	−10.5416	−0.416370	−0.208185	−	0.978090i	$-0.566756\pi$
$641$	−10.5416	−0.416370	−0.208185	+	0.978090i	$0.566756\pi$
$642$	0	0
$643$	−43.5694	−1.71821	−0.859105	−	0.511800i	$-0.828979\pi$
$643$	−43.5694	−1.71821	−0.859105	+	0.511800i	$0.828979\pi$
$644$	0	0
$645$	−3.84441	−0.151374
$646$	0	0
$647$	−16.3305	−0.642019	−0.321010	−	0.947076i	$-0.604022\pi$
$647$	−16.3305	−0.642019	−0.321010	+	0.947076i	$0.604022\pi$
$648$	0	0
$649$	−19.5416	−0.767076
$650$	0	0
$651$	3.00000	0.117579
$652$	0	0
$653$	30.5139	1.19410	0.597050	−	0.802204i	$-0.296339\pi$
$653$	30.5139	1.19410	0.597050	+	0.802204i	$0.296339\pi$
$654$	0	0
$655$	7.24726	0.283174
$656$	0	0
$657$	−2.21110	−0.0862633
$658$	0	0
$659$	−42.6333	−1.66076	−0.830379	−	0.557199i	$-0.811876\pi$
$659$	−42.6333	−1.66076	−0.830379	+	0.557199i	$0.811876\pi$
$660$	0	0
$661$	20.8167	0.809674	0.404837	−	0.914389i	$-0.367328\pi$
$661$	20.8167	0.809674	0.404837	+	0.914389i	$0.367328\pi$
$662$	0	0
$663$	12.9083	0.501318
$664$	0	0
$665$	−1.11943	−0.0434096
$666$	0	0
$667$	−6.21110	−0.240495
$668$	0	0
$669$	19.9083	0.769700
$670$	0	0
$671$	−5.45837	−0.210718
$672$	0	0
$673$	−26.6333	−1.02664	−0.513319	−	0.858198i	$-0.671584\pi$
$673$	−26.6333	−1.02664	−0.513319	+	0.858198i	$0.671584\pi$
$674$	0	0
$675$	4.51388	0.173739
$676$	0	0
$677$	37.1472	1.42768	0.713841	−	0.700308i	$-0.246954\pi$
$677$	37.1472	1.42768	0.713841	+	0.700308i	$0.246954\pi$
$678$	0	0
$679$	−17.6056	−0.675639
$680$	0	0
$681$	−20.3305	−0.779068
$682$	0	0
$683$	1.42221	0.0544192	0.0272096	−	0.999630i	$-0.491338\pi$
$683$	1.42221	0.0544192	0.0272096	+	0.999630i	$0.491338\pi$
$684$	0	0
$685$	−11.7250	−0.447988
$686$	0	0
$687$	−2.51388	−0.0959104
$688$	0	0
$689$	−28.8167	−1.09783
$690$	0	0
$691$	31.5139	1.19884	0.599422	−	0.800433i	$-0.295397\pi$
$691$	31.5139	1.19884	0.599422	+	0.800433i	$0.295397\pi$
$692$	0	0
$693$	5.00000	0.189934
$694$	0	0
$695$	11.0917	0.420731
$696$	0	0
$697$	68.4500	2.59273
$698$	0	0
$699$	14.3305	0.542031
$700$	0	0
$701$	−25.5416	−0.964694	−0.482347	−	0.875980i	$-0.660216\pi$
$701$	−25.5416	−0.964694	−0.482347	+	0.875980i	$0.660216\pi$
$702$	0	0
$703$	−14.4500	−0.544991
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	9.30278	0.349867
$708$	0	0
$709$	−47.3305	−1.77754	−0.888768	−	0.458358i	$-0.848438\pi$
$709$	−47.3305	−1.77754	−0.888768	+	0.458358i	$0.848438\pi$
$710$	0	0
$711$	1.00000	0.0375029
$712$	0	0
$713$	3.00000	0.112351
$714$	0	0
$715$	−8.02776	−0.300221
$716$	0	0
$717$	−14.0917	−0.526263
$718$	0	0
$719$	−14.2111	−0.529985	−0.264992	−	0.964251i	$-0.585369\pi$
$719$	−14.2111	−0.529985	−0.264992	+	0.964251i	$0.585369\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	−12.0278	−0.447317
$724$	0	0
$725$	−28.0362	−1.04124
$726$	0	0
$727$	46.8722	1.73839	0.869196	−	0.494467i	$-0.164637\pi$
$727$	46.8722	1.73839	0.869196	+	0.494467i	$0.164637\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	30.9083	1.14319
$732$	0	0
$733$	−26.6056	−0.982698	−0.491349	−	0.870963i	$-0.663496\pi$
$733$	−26.6056	−0.982698	−0.491349	+	0.870963i	$0.663496\pi$
$734$	0	0
$735$	0.697224	0.0257175
$736$	0	0
$737$	59.5416	2.19324
$738$	0	0
$739$	43.4222	1.59731	0.798656	−	0.601788i	$-0.205545\pi$
$739$	43.4222	1.59731	0.798656	+	0.601788i	$0.205545\pi$
$740$	0	0
$741$	−3.69722	−0.135821
$742$	0	0
$743$	−30.5139	−1.11945	−0.559723	−	0.828680i	$-0.689092\pi$
$743$	−30.5139	−1.11945	−0.559723	+	0.828680i	$0.689092\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	−5.60555	−0.205096
$748$	0	0
$749$	−7.51388	−0.274551
$750$	0	0
$751$	−8.69722	−0.317366	−0.158683	−	0.987330i	$-0.550725\pi$
$751$	−8.69722	−0.317366	−0.158683	+	0.987330i	$0.550725\pi$
$752$	0	0
$753$	23.8167	0.867927
$754$	0	0
$755$	11.5778	0.421359
$756$	0	0
$757$	−3.36669	−0.122365	−0.0611823	−	0.998127i	$-0.519487\pi$
$757$	−3.36669	−0.122365	−0.0611823	+	0.998127i	$0.519487\pi$
$758$	0	0
$759$	5.00000	0.181489
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	−3.90833	−0.141491
$764$	0	0
$765$	3.90833	0.141306
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	−27.4500	−0.989871	−0.494935	−	0.868930i	$-0.664808\pi$
$769$	−27.4500	−0.989871	−0.494935	+	0.868930i	$0.664808\pi$
$770$	0	0
$771$	−27.0278	−0.973381
$772$	0	0
$773$	31.8444	1.14536	0.572682	−	0.819778i	$-0.305903\pi$
$773$	31.8444	1.14536	0.572682	+	0.819778i	$0.305903\pi$
$774$	0	0
$775$	13.5416	0.486430
$776$	0	0
$777$	9.00000	0.322873
$778$	0	0
$779$	−19.6056	−0.702442
$780$	0	0
$781$	−4.54163	−0.162512
$782$	0	0
$783$	−6.21110	−0.221967
$784$	0	0
$785$	2.66106	0.0949774
$786$	0	0
$787$	−32.1472	−1.14592	−0.572962	−	0.819582i	$-0.694206\pi$
$787$	−32.1472	−1.14592	−0.572962	+	0.819582i	$0.694206\pi$
$788$	0	0
$789$	10.3944	0.370052
$790$	0	0
$791$	5.30278	0.188545
$792$	0	0
$793$	−2.51388	−0.0892704
$794$	0	0
$795$	−8.72498	−0.309443
$796$	0	0
$797$	−39.0278	−1.38243	−0.691217	−	0.722647i	$-0.742925\pi$
$797$	−39.0278	−1.38243	−0.691217	+	0.722647i	$0.742925\pi$
$798$	0	0
$799$	−48.2389	−1.70657
$800$	0	0
$801$	10.9083	0.385427
$802$	0	0
$803$	−11.0555	−0.390141
$804$	0	0
$805$	0.697224	0.0245739
$806$	0	0
$807$	−0.908327	−0.0319746
$808$	0	0
$809$	25.9361	0.911864	0.455932	−	0.890015i	$-0.349306\pi$
$809$	25.9361	0.911864	0.455932	+	0.890015i	$0.349306\pi$
$810$	0	0
$811$	10.3944	0.364998	0.182499	−	0.983206i	$-0.441581\pi$
$811$	10.3944	0.364998	0.182499	+	0.983206i	$0.441581\pi$
$812$	0	0
$813$	3.60555	0.126452
$814$	0	0
$815$	−9.56939	−0.335201
$816$	0	0
$817$	−8.85281	−0.309721
$818$	0	0
$819$	2.30278	0.0804655
$820$	0	0
$821$	−19.3944	−0.676871	−0.338435	−	0.940990i	$-0.609898\pi$
$821$	−19.3944	−0.676871	−0.338435	+	0.940990i	$0.609898\pi$
$822$	0	0
$823$	−6.51388	−0.227060	−0.113530	−	0.993535i	$-0.536216\pi$
$823$	−6.51388	−0.227060	−0.113530	+	0.993535i	$0.536216\pi$
$824$	0	0
$825$	22.5694	0.785765
$826$	0	0
$827$	9.90833	0.344546	0.172273	−	0.985049i	$-0.444889\pi$
$827$	9.90833	0.344546	0.172273	+	0.985049i	$0.444889\pi$
$828$	0	0
$829$	−30.4500	−1.05757	−0.528785	−	0.848756i	$-0.677352\pi$
$829$	−30.4500	−1.05757	−0.528785	+	0.848756i	$0.677352\pi$
$830$	0	0
$831$	12.3028	0.426779
$832$	0	0
$833$	−5.60555	−0.194221
$834$	0	0
$835$	1.96384	0.0679615
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	−38.3028	−1.32236	−0.661179	−	0.750228i	$-0.729944\pi$
$839$	−38.3028	−1.32236	−0.661179	+	0.750228i	$0.729944\pi$
$840$	0	0
$841$	9.57779	0.330269
$842$	0	0
$843$	−27.8167	−0.958056
$844$	0	0
$845$	5.36669	0.184620
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	17.3028	0.593830
$850$	0	0
$851$	9.00000	0.308516
$852$	0	0
$853$	−58.2389	−1.99406	−0.997030	−	0.0770106i	$-0.975462\pi$
$853$	−58.2389	−1.99406	−0.997030	+	0.0770106i	$0.975462\pi$
$854$	0	0
$855$	−1.11943	−0.0382837
$856$	0	0
$857$	−50.2389	−1.71613	−0.858063	−	0.513544i	$-0.828332\pi$
$857$	−50.2389	−1.71613	−0.858063	+	0.513544i	$0.828332\pi$
$858$	0	0
$859$	−43.2111	−1.47434	−0.737172	−	0.675705i	$-0.763839\pi$
$859$	−43.2111	−1.47434	−0.737172	+	0.675705i	$0.763839\pi$
$860$	0	0
$861$	12.2111	0.416153
$862$	0	0
$863$	5.76114	0.196112	0.0980558	−	0.995181i	$-0.468738\pi$
$863$	5.76114	0.196112	0.0980558	+	0.995181i	$0.468738\pi$
$864$	0	0
$865$	12.6972	0.431719
$866$	0	0
$867$	−14.4222	−0.489804
$868$	0	0
$869$	5.00000	0.169613
$870$	0	0
$871$	27.4222	0.929166
$872$	0	0
$873$	−17.6056	−0.595858
$874$	0	0
$875$	6.63331	0.224247
$876$	0	0
$877$	−14.7889	−0.499386	−0.249693	−	0.968325i	$-0.580330\pi$
$877$	−14.7889	−0.499386	−0.249693	+	0.968325i	$0.580330\pi$
$878$	0	0
$879$	24.8444	0.837981
$880$	0	0
$881$	−43.8167	−1.47622	−0.738110	−	0.674680i	$-0.764282\pi$
$881$	−43.8167	−1.47622	−0.738110	+	0.674680i	$0.764282\pi$
$882$	0	0
$883$	18.4861	0.622108	0.311054	−	0.950392i	$-0.399318\pi$
$883$	18.4861	0.622108	0.311054	+	0.950392i	$0.399318\pi$
$884$	0	0
$885$	−2.72498	−0.0915992
$886$	0	0
$887$	11.9361	0.400774	0.200387	−	0.979717i	$-0.435780\pi$
$887$	11.9361	0.400774	0.200387	+	0.979717i	$0.435780\pi$
$888$	0	0
$889$	−1.69722	−0.0569231
$890$	0	0
$891$	5.00000	0.167506
$892$	0	0
$893$	13.8167	0.462357
$894$	0	0
$895$	−1.88057	−0.0628605
$896$	0	0
$897$	2.30278	0.0768874
$898$	0	0
$899$	−18.6333	−0.621456
$900$	0	0
$901$	70.1472	2.33694
$902$	0	0
$903$	5.51388	0.183490
$904$	0	0
$905$	4.75274	0.157986
$906$	0	0
$907$	8.14719	0.270523	0.135261	−	0.990810i	$-0.456813\pi$
$907$	8.14719	0.270523	0.135261	+	0.990810i	$0.456813\pi$
$908$	0	0
$909$	9.30278	0.308554
$910$	0	0
$911$	24.2389	0.803069	0.401535	−	0.915844i	$-0.368477\pi$
$911$	24.2389	0.803069	0.401535	+	0.915844i	$0.368477\pi$
$912$	0	0
$913$	−28.0278	−0.927583
$914$	0	0
$915$	−0.761141	−0.0251625
$916$	0	0
$917$	−10.3944	−0.343255
$918$	0	0
$919$	10.3944	0.342881	0.171441	−	0.985194i	$-0.445158\pi$
$919$	10.3944	0.342881	0.171441	+	0.985194i	$0.445158\pi$
$920$	0	0
$921$	18.2111	0.600076
$922$	0	0
$923$	−2.09167	−0.0688483
$924$	0	0
$925$	40.6249	1.33574
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	40.3305	1.32320	0.661601	−	0.749856i	$-0.269877\pi$
$929$	40.3305	1.32320	0.661601	+	0.749856i	$0.269877\pi$
$930$	0	0
$931$	1.60555	0.0526198
$932$	0	0
$933$	−9.51388	−0.311470
$934$	0	0
$935$	19.5416	0.639080
$936$	0	0
$937$	−4.57779	−0.149550	−0.0747750	−	0.997200i	$-0.523824\pi$
$937$	−4.57779	−0.149550	−0.0747750	+	0.997200i	$0.523824\pi$
$938$	0	0
$939$	−1.21110	−0.0395228
$940$	0	0
$941$	−0.844410	−0.0275270	−0.0137635	−	0.999905i	$-0.504381\pi$
$941$	−0.844410	−0.0275270	−0.0137635	+	0.999905i	$0.504381\pi$
$942$	0	0
$943$	12.2111	0.397648
$944$	0	0
$945$	0.697224	0.0226807
$946$	0	0
$947$	13.6333	0.443023	0.221511	−	0.975158i	$-0.428901\pi$
$947$	13.6333	0.443023	0.221511	+	0.975158i	$0.428901\pi$
$948$	0	0
$949$	−5.09167	−0.165283
$950$	0	0
$951$	30.5139	0.989480
$952$	0	0
$953$	−51.3305	−1.66276	−0.831380	−	0.555705i	$-0.812448\pi$
$953$	−51.3305	−1.66276	−0.831380	+	0.555705i	$0.812448\pi$
$954$	0	0
$955$	−3.21110	−0.103909
$956$	0	0
$957$	−31.0555	−1.00388
$958$	0	0
$959$	16.8167	0.543038
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−7.51388	−0.242131
$964$	0	0
$965$	−6.29437	−0.202623
$966$	0	0
$967$	−41.2666	−1.32704	−0.663522	−	0.748156i	$-0.730939\pi$
$967$	−41.2666	−1.32704	−0.663522	+	0.748156i	$0.730939\pi$
$968$	0	0
$969$	9.00000	0.289122
$970$	0	0
$971$	35.0917	1.12615	0.563073	−	0.826407i	$-0.309619\pi$
$971$	35.0917	1.12615	0.563073	+	0.826407i	$0.309619\pi$
$972$	0	0
$973$	−15.9083	−0.509998
$974$	0	0
$975$	10.3944	0.332889
$976$	0	0
$977$	31.1472	0.996487	0.498243	−	0.867037i	$-0.333979\pi$
$977$	31.1472	0.996487	0.498243	+	0.867037i	$0.333979\pi$
$978$	0	0
$979$	54.5416	1.74316
$980$	0	0
$981$	−3.90833	−0.124783
$982$	0	0
$983$	22.8167	0.727738	0.363869	−	0.931450i	$-0.381456\pi$
$983$	22.8167	0.727738	0.363869	+	0.931450i	$0.381456\pi$
$984$	0	0
$985$	4.94449	0.157544
$986$	0	0
$987$	−8.60555	−0.273918
$988$	0	0
$989$	5.51388	0.175331
$990$	0	0
$991$	−35.5139	−1.12814	−0.564068	−	0.825729i	$-0.690764\pi$
$991$	−35.5139	−1.12814	−0.564068	+	0.825729i	$0.690764\pi$
$992$	0	0
$993$	17.6333	0.559576
$994$	0	0
$995$	8.72498	0.276600
$996$	0	0
$997$	−0.0277564	−0.000879053 0	−0.000439527	−	1.00000i	$-0.500140\pi$
$997$	−0.0277564	−0.000879053 0	−0.000439527		1.00000i	$0.500140\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.x.1.2		2
4.3	odd	2		483.2.a.f.1.1	✓	2
12.11	even	2		1449.2.a.j.1.2		2
28.27	even	2		3381.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.f.1.1	✓	2	4.3	odd	2
1449.2.a.j.1.2		2	12.11	even	2
3381.2.a.p.1.1		2	28.27	even	2
7728.2.a.x.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +2.30278 q^{13} +0.697224 q^{15} -5.60555 q^{17} +1.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.51388 q^{25} -1.00000 q^{27} +6.21110 q^{29} -3.00000 q^{31} -5.00000 q^{33} -0.697224 q^{35} -9.00000 q^{37} -2.30278 q^{39} -12.2111 q^{41} -5.51388 q^{43} -0.697224 q^{45} +8.60555 q^{47} +1.00000 q^{49} +5.60555 q^{51} -12.5139 q^{53} -3.48612 q^{55} -1.60555 q^{57} -3.90833 q^{59} -1.09167 q^{61} +1.00000 q^{63} -1.60555 q^{65} +11.9083 q^{67} +1.00000 q^{69} -0.908327 q^{71} -2.21110 q^{73} +4.51388 q^{75} +5.00000 q^{77} +1.00000 q^{79} +1.00000 q^{81} -5.60555 q^{83} +3.90833 q^{85} -6.21110 q^{87} +10.9083 q^{89} +2.30278 q^{91} +3.00000 q^{93} -1.11943 q^{95} -17.6056 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 5 q^{5} + 2 q^{7} + 2 q^{9} + 10 q^{11} + q^{13} + 5 q^{15} - 4 q^{17} - 4 q^{19} - 2 q^{21} - 2 q^{23} + 9 q^{25} - 2 q^{27} - 2 q^{29} - 6 q^{31} - 10 q^{33} - 5 q^{35} - 18 q^{37} - q^{39} - 10 q^{41} + 7 q^{43} - 5 q^{45} + 10 q^{47} + 2 q^{49} + 4 q^{51} - 7 q^{53} - 25 q^{55} + 4 q^{57} + 3 q^{59} - 13 q^{61} + 2 q^{63} + 4 q^{65} + 13 q^{67} + 2 q^{69} + 9 q^{71} + 10 q^{73} - 9 q^{75} + 10 q^{77} + 2 q^{79} + 2 q^{81} - 4 q^{83} - 3 q^{85} + 2 q^{87} + 11 q^{89} + q^{91} + 6 q^{93} + 23 q^{95} - 28 q^{97} + 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^5 + 2 * q^7 + 2 * q^9 + 10 * q^11 + q^13 + 5 * q^15 - 4 * q^17 - 4 * q^19 - 2 * q^21 - 2 * q^23 + 9 * q^25 - 2 * q^27 - 2 * q^29 - 6 * q^31 - 10 * q^33 - 5 * q^35 - 18 * q^37 - q^39 - 10 * q^41 + 7 * q^43 - 5 * q^45 + 10 * q^47 + 2 * q^49 + 4 * q^51 - 7 * q^53 - 25 * q^55 + 4 * q^57 + 3 * q^59 - 13 * q^61 + 2 * q^63 + 4 * q^65 + 13 * q^67 + 2 * q^69 + 9 * q^71 + 10 * q^73 - 9 * q^75 + 10 * q^77 + 2 * q^79 + 2 * q^81 - 4 * q^83 - 3 * q^85 + 2 * q^87 + 11 * q^89 + q^91 + 6 * q^93 + 23 * q^95 - 28 * q^97 + 10 * q^99

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