LMFDB - Embedded newform 7616.2.a.bo.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.58874$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.25886 q^{3} -0.557299 q^{5} -1.00000 q^{7} +7.62018 q^{9} -5.17748 q^{11} -2.00000 q^{13} -1.81616 q^{15} +1.00000 q^{17} -3.25886 q^{21} -6.51772 q^{23} -4.68942 q^{25} +15.0565 q^{27} -3.54515 q^{29} +4.84937 q^{31} -16.8727 q^{33} +0.557299 q^{35} +3.17748 q^{37} -6.51772 q^{39} -5.32174 q^{41} -4.59911 q^{43} -4.24672 q^{45} +0.746918 q^{47} +1.00000 q^{49} +3.25886 q^{51} -11.5056 q^{53} +2.88540 q^{55} +10.3550 q^{59} +1.25886 q^{61} -7.62018 q^{63} +1.11460 q^{65} -13.1379 q^{67} -21.2404 q^{69} +16.1258 q^{71} -14.0687 q^{73} -15.2822 q^{75} +5.17748 q^{77} -0.746918 q^{79} +26.2066 q^{81} -13.7824 q^{83} -0.557299 q^{85} -11.5532 q^{87} +2.20491 q^{89} +2.00000 q^{91} +15.8034 q^{93} -10.0233 q^{97} -39.4533 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * q^99

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
$3$	3.25886	1.88150	0.940752	−	0.339095i	$-0.110121\pi$
$3$	3.25886	1.88150	0.940752	+	0.339095i	$0.110121\pi$
$4$	0	0
$5$	−0.557299	−0.249232	−0.124616	−	0.992205i	$-0.539770\pi$
$5$	−0.557299	−0.249232	−0.124616	+	0.992205i	$0.539770\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.62018	2.54006
$10$	0	0
$11$	−5.17748	−1.56107	−0.780534	−	0.625114i	$-0.785053\pi$
$11$	−5.17748	−1.56107	−0.780534	+	0.625114i	$0.785053\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.81616	−0.468931
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−3.25886	−0.711142
$22$	0	0
$23$	−6.51772	−1.35904	−0.679520	−	0.733657i	$-0.737812\pi$
$23$	−6.51772	−1.35904	−0.679520	+	0.733657i	$0.737812\pi$
$24$	0	0
$25$	−4.68942	−0.937884
$26$	0	0
$27$	15.0565	2.89763
$28$	0	0
$29$	−3.54515	−0.658319	−0.329159	−	0.944274i	$-0.606765\pi$
$29$	−3.54515	−0.658319	−0.329159	+	0.944274i	$0.606765\pi$
$30$	0	0
$31$	4.84937	0.870973	0.435486	−	0.900195i	$-0.356576\pi$
$31$	4.84937	0.870973	0.435486	+	0.900195i	$0.356576\pi$
$32$	0	0
$33$	−16.8727	−2.93716
$34$	0	0
$35$	0.557299	0.0942007
$36$	0	0
$37$	3.17748	0.522374	0.261187	−	0.965288i	$-0.415886\pi$
$37$	3.17748	0.522374	0.261187	+	0.965288i	$0.415886\pi$
$38$	0	0
$39$	−6.51772	−1.04367
$40$	0	0
$41$	−5.32174	−0.831116	−0.415558	−	0.909567i	$-0.636414\pi$
$41$	−5.32174	−0.831116	−0.415558	+	0.909567i	$0.636414\pi$
$42$	0	0
$43$	−4.59911	−0.701357	−0.350679	−	0.936496i	$-0.614049\pi$
$43$	−4.59911	−0.701357	−0.350679	+	0.936496i	$0.614049\pi$
$44$	0	0
$45$	−4.24672	−0.633063
$46$	0	0
$47$	0.746918	0.108949	0.0544746	−	0.998515i	$-0.482652\pi$
$47$	0.746918	0.108949	0.0544746	+	0.998515i	$0.482652\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.25886	0.456332
$52$	0	0
$53$	−11.5056	−1.58041	−0.790206	−	0.612841i	$-0.790027\pi$
$53$	−11.5056	−1.58041	−0.790206	+	0.612841i	$0.790027\pi$
$54$	0	0
$55$	2.88540	0.389068
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3550	1.34810	0.674050	−	0.738686i	$-0.264553\pi$
$59$	10.3550	1.34810	0.674050	+	0.738686i	$0.264553\pi$
$60$	0	0
$61$	1.25886	0.161181	0.0805903	−	0.996747i	$-0.474319\pi$
$61$	1.25886	0.161181	0.0805903	+	0.996747i	$0.474319\pi$
$62$	0	0
$63$	−7.62018	−0.960052
$64$	0	0
$65$	1.11460	0.138249
$66$	0	0
$67$	−13.1379	−1.60505	−0.802525	−	0.596619i	$-0.796511\pi$
$67$	−13.1379	−1.60505	−0.802525	+	0.596619i	$0.796511\pi$
$68$	0	0
$69$	−21.2404	−2.55704
$70$	0	0
$71$	16.1258	1.91377	0.956887	−	0.290459i	$-0.0938080\pi$
$71$	16.1258	1.91377	0.956887	+	0.290459i	$0.0938080\pi$
$72$	0	0
$73$	−14.0687	−1.64661	−0.823306	−	0.567598i	$-0.807873\pi$
$73$	−14.0687	−1.64661	−0.823306	+	0.567598i	$0.807873\pi$
$74$	0	0
$75$	−15.2822	−1.76463
$76$	0	0
$77$	5.17748	0.590028
$78$	0	0
$79$	−0.746918	−0.0840349	−0.0420174	−	0.999117i	$-0.513379\pi$
$79$	−0.746918	−0.0840349	−0.0420174	+	0.999117i	$0.513379\pi$
$80$	0	0
$81$	26.2066	2.91184
$82$	0	0
$83$	−13.7824	−1.51281	−0.756405	−	0.654103i	$-0.773046\pi$
$83$	−13.7824	−1.51281	−0.756405	+	0.654103i	$0.773046\pi$
$84$	0	0
$85$	−0.557299	−0.0604476
$86$	0	0
$87$	−11.5532	−1.23863
$88$	0	0
$89$	2.20491	0.233720	0.116860	−	0.993148i	$-0.462717\pi$
$89$	2.20491	0.233720	0.116860	+	0.993148i	$0.462717\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	15.8034	1.63874
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−10.0233	−1.01771	−0.508856	−	0.860852i	$-0.669931\pi$
$97$	−10.0233	−1.01771	−0.508856	+	0.860852i	$0.669931\pi$
$98$	0	0
$99$	−39.4533	−3.96520
$100$	0	0
$101$	−5.63232	−0.560437	−0.280218	−	0.959936i	$-0.590407\pi$
$101$	−5.63232	−0.560437	−0.280218	+	0.959936i	$0.590407\pi$
$102$	0	0
$103$	−6.06642	−0.597743	−0.298871	−	0.954293i	$-0.596610\pi$
$103$	−6.06642	−0.597743	−0.298871	+	0.954293i	$0.596610\pi$
$104$	0	0
$105$	1.81616	0.177239
$106$	0	0
$107$	10.9483	1.05841	0.529205	−	0.848494i	$-0.322490\pi$
$107$	10.9483	1.05841	0.529205	+	0.848494i	$0.322490\pi$
$108$	0	0
$109$	−10.8098	−1.03539	−0.517695	−	0.855565i	$-0.673210\pi$
$109$	−10.8098	−1.03539	−0.517695	+	0.855565i	$0.673210\pi$
$110$	0	0
$111$	10.3550	0.982848
$112$	0	0
$113$	14.5050	1.36451	0.682257	−	0.731112i	$-0.260998\pi$
$113$	14.5050	1.36451	0.682257	+	0.731112i	$0.260998\pi$
$114$	0	0
$115$	3.63232	0.338716
$116$	0	0
$117$	−15.2404	−1.40897
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	15.8062	1.43693
$122$	0	0
$123$	−17.3428	−1.56375
$124$	0	0
$125$	5.39990	0.482982
$126$	0	0
$127$	9.66835	0.857927	0.428964	−	0.903322i	$-0.358879\pi$
$127$	9.66835	0.857927	0.428964	+	0.903322i	$0.358879\pi$
$128$	0	0
$129$	−14.9879	−1.31961
$130$	0	0
$131$	−10.5806	−0.924431	−0.462216	−	0.886768i	$-0.652945\pi$
$131$	−10.5806	−0.924431	−0.462216	+	0.886768i	$0.652945\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−8.39098	−0.722181
$136$	0	0
$137$	1.85219	0.158243	0.0791216	−	0.996865i	$-0.474788\pi$
$137$	1.85219	0.158243	0.0791216	+	0.996865i	$0.474788\pi$
$138$	0	0
$139$	−17.6592	−1.49783	−0.748916	−	0.662665i	$-0.769425\pi$
$139$	−17.6592	−1.49783	−0.748916	+	0.662665i	$0.769425\pi$
$140$	0	0
$141$	2.43410	0.204989
$142$	0	0
$143$	10.3550	0.865924
$144$	0	0
$145$	1.97571	0.164074
$146$	0	0
$147$	3.25886	0.268786
$148$	0	0
$149$	−5.85219	−0.479430	−0.239715	−	0.970843i	$-0.577054\pi$
$149$	−5.85219	−0.479430	−0.239715	+	0.970843i	$0.577054\pi$
$150$	0	0
$151$	−17.8816	−1.45518	−0.727592	−	0.686010i	$-0.759360\pi$
$151$	−17.8816	−1.45518	−0.727592	+	0.686010i	$0.759360\pi$
$152$	0	0
$153$	7.62018	0.616055
$154$	0	0
$155$	−2.70255	−0.217074
$156$	0	0
$157$	−0.722630	−0.0576721	−0.0288361	−	0.999584i	$-0.509180\pi$
$157$	−0.722630	−0.0576721	−0.0288361	+	0.999584i	$0.509180\pi$
$158$	0	0
$159$	−37.4951	−2.97355
$160$	0	0
$161$	6.51772	0.513668
$162$	0	0
$163$	3.03899	0.238032	0.119016	−	0.992892i	$-0.462026\pi$
$163$	3.03899	0.238032	0.119016	+	0.992892i	$0.462026\pi$
$164$	0	0
$165$	9.40312	0.732032
$166$	0	0
$167$	−11.1590	−0.863507	−0.431753	−	0.901992i	$-0.642105\pi$
$167$	−11.1590	−0.863507	−0.431753	+	0.901992i	$0.642105\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.3846	1.32173	0.660864	−	0.750506i	$-0.270190\pi$
$173$	17.3846	1.32173	0.660864	+	0.750506i	$0.270190\pi$
$174$	0	0
$175$	4.68942	0.354487
$176$	0	0
$177$	33.7453	2.53646
$178$	0	0
$179$	1.91861	0.143404	0.0717020	−	0.997426i	$-0.477157\pi$
$179$	1.91861	0.143404	0.0717020	+	0.997426i	$0.477157\pi$
$180$	0	0
$181$	−15.2611	−1.13435	−0.567174	−	0.823598i	$-0.691963\pi$
$181$	−15.2611	−1.13435	−0.567174	+	0.823598i	$0.691963\pi$
$182$	0	0
$183$	4.10245	0.303262
$184$	0	0
$185$	−1.77080	−0.130192
$186$	0	0
$187$	−5.17748	−0.378614
$188$	0	0
$189$	−15.0565	−1.09520
$190$	0	0
$191$	14.4574	1.04610	0.523051	−	0.852302i	$-0.324794\pi$
$191$	14.4574	1.04610	0.523051	+	0.852302i	$0.324794\pi$
$192$	0	0
$193$	23.1612	1.66718	0.833590	−	0.552384i	$-0.186282\pi$
$193$	23.1612	1.66718	0.833590	+	0.552384i	$0.186282\pi$
$194$	0	0
$195$	3.63232	0.260116
$196$	0	0
$197$	−1.85797	−0.132375	−0.0661874	−	0.997807i	$-0.521084\pi$
$197$	−1.85797	−0.132375	−0.0661874	+	0.997807i	$0.521084\pi$
$198$	0	0
$199$	−5.83946	−0.413948	−0.206974	−	0.978346i	$-0.566362\pi$
$199$	−5.83946	−0.413948	−0.206974	+	0.978346i	$0.566362\pi$
$200$	0	0
$201$	−42.8146	−3.01991
$202$	0	0
$203$	3.54515	0.248821
$204$	0	0
$205$	2.96580	0.207140
$206$	0	0
$207$	−49.6662	−3.45204
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−12.9726	−0.893068	−0.446534	−	0.894767i	$-0.647342\pi$
$211$	−12.9726	−0.893068	−0.446534	+	0.894767i	$0.647342\pi$
$212$	0	0
$213$	52.5516	3.60078
$214$	0	0
$215$	2.56308	0.174801
$216$	0	0
$217$	−4.84937	−0.329197
$218$	0	0
$219$	−45.8478	−3.09811
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	20.5841	1.37842	0.689208	−	0.724563i	$-0.257959\pi$
$223$	20.5841	1.37842	0.689208	+	0.724563i	$0.257959\pi$
$224$	0	0
$225$	−35.7342	−2.38228
$226$	0	0
$227$	12.8701	0.854219	0.427110	−	0.904200i	$-0.359532\pi$
$227$	12.8701	0.854219	0.427110	+	0.904200i	$0.359532\pi$
$228$	0	0
$229$	−25.3904	−1.67784	−0.838922	−	0.544251i	$-0.816814\pi$
$229$	−25.3904	−1.67784	−0.838922	+	0.544251i	$0.816814\pi$
$230$	0	0
$231$	16.8727	1.11014
$232$	0	0
$233$	−21.8417	−1.43090	−0.715448	−	0.698666i	$-0.753778\pi$
$233$	−21.8417	−1.43090	−0.715448	+	0.698666i	$0.753778\pi$
$234$	0	0
$235$	−0.416257	−0.0271536
$236$	0	0
$237$	−2.43410	−0.158112
$238$	0	0
$239$	23.9895	1.55175	0.775876	−	0.630885i	$-0.217308\pi$
$239$	23.9895	1.55175	0.775876	+	0.630885i	$0.217308\pi$
$240$	0	0
$241$	13.4928	0.869151	0.434575	−	0.900635i	$-0.356898\pi$
$241$	13.4928	0.869151	0.434575	+	0.900635i	$0.356898\pi$
$242$	0	0
$243$	40.2340	2.58101
$244$	0	0
$245$	−0.557299	−0.0356045
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−44.9148	−2.84636
$250$	0	0
$251$	19.8488	1.25284	0.626422	−	0.779484i	$-0.284519\pi$
$251$	19.8488	1.25284	0.626422	+	0.779484i	$0.284519\pi$
$252$	0	0
$253$	33.7453	2.12155
$254$	0	0
$255$	−1.81616	−0.113732
$256$	0	0
$257$	−0.343391	−0.0214201	−0.0107101	−	0.999943i	$-0.503409\pi$
$257$	−0.343391	−0.0214201	−0.0107101	+	0.999943i	$0.503409\pi$
$258$	0	0
$259$	−3.17748	−0.197439
$260$	0	0
$261$	−27.0147	−1.67217
$262$	0	0
$263$	28.4258	1.75281	0.876406	−	0.481573i	$-0.159934\pi$
$263$	28.4258	1.75281	0.876406	+	0.481573i	$0.159934\pi$
$264$	0	0
$265$	6.41205	0.393889
$266$	0	0
$267$	7.18549	0.439745
$268$	0	0
$269$	−12.9904	−0.792040	−0.396020	−	0.918242i	$-0.629609\pi$
$269$	−12.9904	−0.792040	−0.396020	+	0.918242i	$0.629609\pi$
$270$	0	0
$271$	24.9984	1.51855	0.759273	−	0.650772i	$-0.225555\pi$
$271$	24.9984	1.51855	0.759273	+	0.650772i	$0.225555\pi$
$272$	0	0
$273$	6.51772	0.394470
$274$	0	0
$275$	24.2793	1.46410
$276$	0	0
$277$	17.6952	1.06320	0.531601	−	0.846995i	$-0.321591\pi$
$277$	17.6952	1.06320	0.531601	+	0.846995i	$0.321591\pi$
$278$	0	0
$279$	36.9531	2.21232
$280$	0	0
$281$	12.2946	0.733436	0.366718	−	0.930332i	$-0.380481\pi$
$281$	12.2946	0.733436	0.366718	+	0.930332i	$0.380481\pi$
$282$	0	0
$283$	−18.3154	−1.08874	−0.544368	−	0.838847i	$-0.683230\pi$
$283$	−18.3154	−1.08874	−0.544368	+	0.838847i	$0.683230\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.32174	0.314132
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−32.6645	−1.91483
$292$	0	0
$293$	−21.9630	−1.28309	−0.641546	−	0.767085i	$-0.721706\pi$
$293$	−21.9630	−1.28309	−0.641546	+	0.767085i	$0.721706\pi$
$294$	0	0
$295$	−5.77080	−0.335989
$296$	0	0
$297$	−77.9547	−4.52339
$298$	0	0
$299$	13.0354	0.753859
$300$	0	0
$301$	4.59911	0.265088
$302$	0	0
$303$	−18.3550	−1.05446
$304$	0	0
$305$	−0.701562	−0.0401713
$306$	0	0
$307$	−12.5050	−0.713698	−0.356849	−	0.934162i	$-0.616149\pi$
$307$	−12.5050	−0.713698	−0.356849	+	0.934162i	$0.616149\pi$
$308$	0	0
$309$	−19.7696	−1.12466
$310$	0	0
$311$	−26.6202	−1.50949	−0.754746	−	0.656017i	$-0.772240\pi$
$311$	−26.6202	−1.50949	−0.754746	+	0.656017i	$0.772240\pi$
$312$	0	0
$313$	8.45740	0.478041	0.239020	−	0.971015i	$-0.423174\pi$
$313$	8.45740	0.478041	0.239020	+	0.971015i	$0.423174\pi$
$314$	0	0
$315$	4.24672	0.239275
$316$	0	0
$317$	5.20176	0.292160	0.146080	−	0.989273i	$-0.453334\pi$
$317$	5.20176	0.292160	0.146080	+	0.989273i	$0.453334\pi$
$318$	0	0
$319$	18.3550	1.02768
$320$	0	0
$321$	35.6789	1.99140
$322$	0	0
$323$	0	0
$324$	0	0
$325$	9.37884	0.520244
$326$	0	0
$327$	−35.2276	−1.94809
$328$	0	0
$329$	−0.746918	−0.0411789
$330$	0	0
$331$	5.13790	0.282404	0.141202	−	0.989981i	$-0.454903\pi$
$331$	5.13790	0.282404	0.141202	+	0.989981i	$0.454903\pi$
$332$	0	0
$333$	24.2129	1.32686
$334$	0	0
$335$	7.32174	0.400029
$336$	0	0
$337$	−23.4031	−1.27485	−0.637425	−	0.770513i	$-0.720000\pi$
$337$	−23.4031	−1.27485	−0.637425	+	0.770513i	$0.720000\pi$
$338$	0	0
$339$	47.2698	2.56734
$340$	0	0
$341$	−25.1075	−1.35965
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	11.8372	0.637295
$346$	0	0
$347$	13.1775	0.707404	0.353702	−	0.935358i	$-0.384923\pi$
$347$	13.1775	0.707404	0.353702	+	0.935358i	$0.384923\pi$
$348$	0	0
$349$	14.1373	0.756753	0.378377	−	0.925652i	$-0.376482\pi$
$349$	14.1373	0.756753	0.378377	+	0.925652i	$0.376482\pi$
$350$	0	0
$351$	−30.1130	−1.60731
$352$	0	0
$353$	18.5771	0.988757	0.494378	−	0.869247i	$-0.335396\pi$
$353$	18.5771	0.988757	0.494378	+	0.869247i	$0.335396\pi$
$354$	0	0
$355$	−8.98687	−0.476973
$356$	0	0
$357$	−3.25886	−0.172477
$358$	0	0
$359$	−18.7575	−0.989982	−0.494991	−	0.868898i	$-0.664829\pi$
$359$	−18.7575	−0.989982	−0.494991	+	0.868898i	$0.664829\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	51.5104	2.70359
$364$	0	0
$365$	7.84045	0.410388
$366$	0	0
$367$	20.6412	1.07746	0.538732	−	0.842477i	$-0.318904\pi$
$367$	20.6412	1.07746	0.538732	+	0.842477i	$0.318904\pi$
$368$	0	0
$369$	−40.5526	−2.11108
$370$	0	0
$371$	11.5056	0.597340
$372$	0	0
$373$	−17.2381	−0.892556	−0.446278	−	0.894894i	$-0.647251\pi$
$373$	−17.2381	−0.892556	−0.446278	+	0.894894i	$0.647251\pi$
$374$	0	0
$375$	17.5975	0.908733
$376$	0	0
$377$	7.09031	0.365170
$378$	0	0
$379$	25.6033	1.31515	0.657577	−	0.753387i	$-0.271581\pi$
$379$	25.6033	1.31515	0.657577	+	0.753387i	$0.271581\pi$
$380$	0	0
$381$	31.5078	1.61419
$382$	0	0
$383$	−27.6323	−1.41195	−0.705973	−	0.708239i	$-0.749490\pi$
$383$	−27.6323	−1.41195	−0.705973	+	0.708239i	$0.749490\pi$
$384$	0	0
$385$	−2.88540	−0.147054
$386$	0	0
$387$	−35.0460	−1.78149
$388$	0	0
$389$	36.8717	1.86947	0.934734	−	0.355347i	$-0.115637\pi$
$389$	36.8717	1.86947	0.934734	+	0.355347i	$0.115637\pi$
$390$	0	0
$391$	−6.51772	−0.329615
$392$	0	0
$393$	−34.4807	−1.73932
$394$	0	0
$395$	0.416257	0.0209442
$396$	0	0
$397$	−18.5024	−0.928611	−0.464305	−	0.885675i	$-0.653696\pi$
$397$	−18.5024	−0.928611	−0.464305	+	0.885675i	$0.653696\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0294	−1.19997	−0.599986	−	0.800011i	$-0.704827\pi$
$401$	−24.0294	−1.19997	−0.599986	+	0.800011i	$0.704827\pi$
$402$	0	0
$403$	−9.69874	−0.483129
$404$	0	0
$405$	−14.6049	−0.725723
$406$	0	0
$407$	−16.4513	−0.815461
$408$	0	0
$409$	−8.43410	−0.417040	−0.208520	−	0.978018i	$-0.566865\pi$
$409$	−8.43410	−0.417040	−0.208520	+	0.978018i	$0.566865\pi$
$410$	0	0
$411$	6.03603	0.297735
$412$	0	0
$413$	−10.3550	−0.509534
$414$	0	0
$415$	7.68090	0.377040
$416$	0	0
$417$	−57.5488	−2.81818
$418$	0	0
$419$	5.81038	0.283856	0.141928	−	0.989877i	$-0.454670\pi$
$419$	5.81038	0.283856	0.141928	+	0.989877i	$0.454670\pi$
$420$	0	0
$421$	−32.6700	−1.59224	−0.796119	−	0.605140i	$-0.793117\pi$
$421$	−32.6700	−1.59224	−0.796119	+	0.605140i	$0.793117\pi$
$422$	0	0
$423$	5.69165	0.276738
$424$	0	0
$425$	−4.68942	−0.227470
$426$	0	0
$427$	−1.25886	−0.0609206
$428$	0	0
$429$	33.7453	1.62924
$430$	0	0
$431$	−15.3438	−0.739085	−0.369542	−	0.929214i	$-0.620486\pi$
$431$	−15.3438	−0.739085	−0.369542	+	0.929214i	$0.620486\pi$
$432$	0	0
$433$	28.9436	1.39094	0.695469	−	0.718556i	$-0.255196\pi$
$433$	28.9436	1.39094	0.695469	+	0.718556i	$0.255196\pi$
$434$	0	0
$435$	6.43857	0.308706
$436$	0	0
$437$	0	0
$438$	0	0
$439$	11.4430	0.546146	0.273073	−	0.961993i	$-0.411960\pi$
$439$	11.4430	0.546146	0.273073	+	0.961993i	$0.411960\pi$
$440$	0	0
$441$	7.62018	0.362866
$442$	0	0
$443$	−4.12575	−0.196020	−0.0980102	−	0.995185i	$-0.531248\pi$
$443$	−4.12575	−0.196020	−0.0980102	+	0.995185i	$0.531248\pi$
$444$	0	0
$445$	−1.22879	−0.0582504
$446$	0	0
$447$	−19.0715	−0.902050
$448$	0	0
$449$	−20.3971	−0.962598	−0.481299	−	0.876556i	$-0.659835\pi$
$449$	−20.3971	−0.962598	−0.481299	+	0.876556i	$0.659835\pi$
$450$	0	0
$451$	27.5532	1.29743
$452$	0	0
$453$	−58.2736	−2.73793
$454$	0	0
$455$	−1.11460	−0.0522532
$456$	0	0
$457$	16.8971	0.790415	0.395208	−	0.918592i	$-0.370673\pi$
$457$	16.8971	0.790415	0.395208	+	0.918592i	$0.370673\pi$
$458$	0	0
$459$	15.0565	0.702778
$460$	0	0
$461$	−19.1982	−0.894150	−0.447075	−	0.894496i	$-0.647534\pi$
$461$	−19.1982	−0.894150	−0.447075	+	0.894496i	$0.647534\pi$
$462$	0	0
$463$	−20.8462	−0.968803	−0.484401	−	0.874846i	$-0.660963\pi$
$463$	−20.8462	−0.968803	−0.484401	+	0.874846i	$0.660963\pi$
$464$	0	0
$465$	−8.80724	−0.408426
$466$	0	0
$467$	26.6013	1.23096	0.615482	−	0.788151i	$-0.288962\pi$
$467$	26.6013	1.23096	0.615482	+	0.788151i	$0.288962\pi$
$468$	0	0
$469$	13.1379	0.606652
$470$	0	0
$471$	−2.35495	−0.108510
$472$	0	0
$473$	23.8118	1.09487
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−87.6745	−4.01434
$478$	0	0
$479$	−37.2183	−1.70055	−0.850274	−	0.526341i	$-0.823564\pi$
$479$	−37.2183	−1.70055	−0.850274	+	0.526341i	$0.823564\pi$
$480$	0	0
$481$	−6.35495	−0.289761
$482$	0	0
$483$	21.2404	0.966469
$484$	0	0
$485$	5.58598	0.253646
$486$	0	0
$487$	27.4625	1.24444	0.622221	−	0.782841i	$-0.286230\pi$
$487$	27.4625	1.24444	0.622221	+	0.782841i	$0.286230\pi$
$488$	0	0
$489$	9.90365	0.447859
$490$	0	0
$491$	26.5040	1.19611	0.598054	−	0.801455i	$-0.295941\pi$
$491$	26.5040	1.19611	0.598054	+	0.801455i	$0.295941\pi$
$492$	0	0
$493$	−3.54515	−0.159666
$494$	0	0
$495$	21.9873	0.988254
$496$	0	0
$497$	−16.1258	−0.723339
$498$	0	0
$499$	0.479134	0.0214490	0.0107245	−	0.999942i	$-0.496586\pi$
$499$	0.479134	0.0214490	0.0107245	+	0.999942i	$0.496586\pi$
$500$	0	0
$501$	−36.3655	−1.62469
$502$	0	0
$503$	16.8423	0.750960	0.375480	−	0.926830i	$-0.377478\pi$
$503$	16.8423	0.750960	0.375480	+	0.926830i	$0.377478\pi$
$504$	0	0
$505$	3.13889	0.139679
$506$	0	0
$507$	−29.3298	−1.30258
$508$	0	0
$509$	−31.2519	−1.38522	−0.692608	−	0.721314i	$-0.743539\pi$
$509$	−31.2519	−1.38522	−0.692608	+	0.721314i	$0.743539\pi$
$510$	0	0
$511$	14.0687	0.622361
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.38081	0.148976
$516$	0	0
$517$	−3.86715	−0.170077
$518$	0	0
$519$	56.6540	2.48684
$520$	0	0
$521$	42.6573	1.86885	0.934424	−	0.356161i	$-0.115915\pi$
$521$	42.6573	1.86885	0.934424	+	0.356161i	$0.115915\pi$
$522$	0	0
$523$	13.0239	0.569495	0.284747	−	0.958603i	$-0.408090\pi$
$523$	13.0239	0.569495	0.284747	+	0.958603i	$0.408090\pi$
$524$	0	0
$525$	15.2822	0.666968
$526$	0	0
$527$	4.84937	0.211242
$528$	0	0
$529$	19.4807	0.846987
$530$	0	0
$531$	78.9065	3.42425
$532$	0	0
$533$	10.6435	0.461020
$534$	0	0
$535$	−6.10147	−0.263789
$536$	0	0
$537$	6.25250	0.269815
$538$	0	0
$539$	−5.17748	−0.223010
$540$	0	0
$541$	−13.1181	−0.563993	−0.281997	−	0.959415i	$-0.590997\pi$
$541$	−13.1181	−0.563993	−0.281997	+	0.959415i	$0.590997\pi$
$542$	0	0
$543$	−49.7338	−2.13428
$544$	0	0
$545$	6.02429	0.258052
$546$	0	0
$547$	6.90614	0.295285	0.147643	−	0.989041i	$-0.452831\pi$
$547$	6.90614	0.295285	0.147643	+	0.989041i	$0.452831\pi$
$548$	0	0
$549$	9.59274	0.409408
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0.746918	0.0317622
$554$	0	0
$555$	−5.77080	−0.244957
$556$	0	0
$557$	−10.5841	−0.448465	−0.224232	−	0.974536i	$-0.571987\pi$
$557$	−10.5841	−0.448465	−0.224232	+	0.974536i	$0.571987\pi$
$558$	0	0
$559$	9.19822	0.389043
$560$	0	0
$561$	−16.8727	−0.712365
$562$	0	0
$563$	−3.34826	−0.141112	−0.0705562	−	0.997508i	$-0.522477\pi$
$563$	−3.34826	−0.141112	−0.0705562	+	0.997508i	$0.522477\pi$
$564$	0	0
$565$	−8.08362	−0.340080
$566$	0	0
$567$	−26.2066	−1.10057
$568$	0	0
$569$	−14.8832	−0.623935	−0.311967	−	0.950093i	$-0.600988\pi$
$569$	−14.8832	−0.623935	−0.311967	+	0.950093i	$0.600988\pi$
$570$	0	0
$571$	3.61604	0.151327	0.0756634	−	0.997133i	$-0.475893\pi$
$571$	3.61604	0.151327	0.0756634	+	0.997133i	$0.475893\pi$
$572$	0	0
$573$	47.1147	1.96824
$574$	0	0
$575$	30.5643	1.27462
$576$	0	0
$577$	2.99331	0.124613	0.0623065	−	0.998057i	$-0.480154\pi$
$577$	2.99331	0.124613	0.0623065	+	0.998057i	$0.480154\pi$
$578$	0	0
$579$	75.4791	3.13681
$580$	0	0
$581$	13.7824	0.571789
$582$	0	0
$583$	59.5698	2.46713
$584$	0	0
$585$	8.49343	0.351160
$586$	0	0
$587$	9.28893	0.383395	0.191698	−	0.981454i	$-0.438601\pi$
$587$	9.28893	0.383395	0.191698	+	0.981454i	$0.438601\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−6.05486	−0.249064
$592$	0	0
$593$	22.0421	0.905162	0.452581	−	0.891723i	$-0.350503\pi$
$593$	22.0421	0.905162	0.452581	+	0.891723i	$0.350503\pi$
$594$	0	0
$595$	0.557299	0.0228470
$596$	0	0
$597$	−19.0300	−0.778846
$598$	0	0
$599$	−3.97348	−0.162352	−0.0811760	−	0.996700i	$-0.525868\pi$
$599$	−3.97348	−0.162352	−0.0811760	+	0.996700i	$0.525868\pi$
$600$	0	0
$601$	18.4513	0.752644	0.376322	−	0.926489i	$-0.377189\pi$
$601$	18.4513	0.752644	0.376322	+	0.926489i	$0.377189\pi$
$602$	0	0
$603$	−100.113	−4.07692
$604$	0	0
$605$	−8.80881	−0.358129
$606$	0	0
$607$	6.11518	0.248208	0.124104	−	0.992269i	$-0.460394\pi$
$607$	6.11518	0.248208	0.124104	+	0.992269i	$0.460394\pi$
$608$	0	0
$609$	11.5532	0.468158
$610$	0	0
$611$	−1.49384	−0.0604342
$612$	0	0
$613$	1.65562	0.0668699	0.0334349	−	0.999441i	$-0.489355\pi$
$613$	1.65562	0.0668699	0.0334349	+	0.999441i	$0.489355\pi$
$614$	0	0
$615$	9.66513	0.389736
$616$	0	0
$617$	9.09740	0.366248	0.183124	−	0.983090i	$-0.441379\pi$
$617$	9.09740	0.366248	0.183124	+	0.983090i	$0.441379\pi$
$618$	0	0
$619$	12.2002	0.490367	0.245184	−	0.969477i	$-0.421152\pi$
$619$	12.2002	0.490367	0.245184	+	0.969477i	$0.421152\pi$
$620$	0	0
$621$	−98.1342	−3.93799
$622$	0	0
$623$	−2.20491	−0.0883378
$624$	0	0
$625$	20.4377	0.817509
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.17748	0.126694
$630$	0	0
$631$	39.7698	1.58321	0.791606	−	0.611032i	$-0.209245\pi$
$631$	39.7698	1.58321	0.791606	+	0.611032i	$0.209245\pi$
$632$	0	0
$633$	−42.2758	−1.68031
$634$	0	0
$635$	−5.38816	−0.213823
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	122.881	4.86110
$640$	0	0
$641$	21.5532	0.851299	0.425649	−	0.904888i	$-0.360046\pi$
$641$	21.5532	0.851299	0.425649	+	0.904888i	$0.360046\pi$
$642$	0	0
$643$	−28.6970	−1.13170	−0.565850	−	0.824508i	$-0.691452\pi$
$643$	−28.6970	−1.13170	−0.565850	+	0.824508i	$0.691452\pi$
$644$	0	0
$645$	8.35272	0.328888
$646$	0	0
$647$	40.0582	1.57485	0.787424	−	0.616411i	$-0.211414\pi$
$647$	40.0582	1.57485	0.787424	+	0.616411i	$0.211414\pi$
$648$	0	0
$649$	−53.6125	−2.10447
$650$	0	0
$651$	−15.8034	−0.619385
$652$	0	0
$653$	11.5522	0.452074	0.226037	−	0.974119i	$-0.427423\pi$
$653$	11.5522	0.452074	0.226037	+	0.974119i	$0.427423\pi$
$654$	0	0
$655$	5.89656	0.230398
$656$	0	0
$657$	−107.206	−4.18249
$658$	0	0
$659$	−7.65116	−0.298047	−0.149023	−	0.988834i	$-0.547613\pi$
$659$	−7.65116	−0.298047	−0.149023	+	0.988834i	$0.547613\pi$
$660$	0	0
$661$	−44.0709	−1.71416	−0.857079	−	0.515184i	$-0.827723\pi$
$661$	−44.0709	−1.71416	−0.857079	+	0.515184i	$0.827723\pi$
$662$	0	0
$663$	−6.51772	−0.253127
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.1063	0.894681
$668$	0	0
$669$	67.0809	2.59350
$670$	0	0
$671$	−6.51772	−0.251614
$672$	0	0
$673$	14.6983	0.566579	0.283290	−	0.959034i	$-0.408574\pi$
$673$	14.6983	0.566579	0.283290	+	0.959034i	$0.408574\pi$
$674$	0	0
$675$	−70.6063	−2.71764
$676$	0	0
$677$	29.1233	1.11930	0.559649	−	0.828730i	$-0.310936\pi$
$677$	29.1233	1.11930	0.559649	+	0.828730i	$0.310936\pi$
$678$	0	0
$679$	10.0233	0.384659
$680$	0	0
$681$	41.9419	1.60722
$682$	0	0
$683$	−30.0967	−1.15162	−0.575810	−	0.817584i	$-0.695313\pi$
$683$	−30.0967	−1.15162	−0.575810	+	0.817584i	$0.695313\pi$
$684$	0	0
$685$	−1.03222	−0.0394392
$686$	0	0
$687$	−82.7438	−3.15687
$688$	0	0
$689$	23.0112	0.876655
$690$	0	0
$691$	39.6432	1.50810	0.754050	−	0.656818i	$-0.228098\pi$
$691$	39.6432	1.50810	0.754050	+	0.656818i	$0.228098\pi$
$692$	0	0
$693$	39.4533	1.49871
$694$	0	0
$695$	9.84144	0.373307
$696$	0	0
$697$	−5.32174	−0.201575
$698$	0	0
$699$	−71.1790	−2.69224
$700$	0	0
$701$	13.2575	0.500731	0.250365	−	0.968151i	$-0.419449\pi$
$701$	13.2575	0.500731	0.250365	+	0.968151i	$0.419449\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−1.35652	−0.0510896
$706$	0	0
$707$	5.63232	0.211825
$708$	0	0
$709$	−24.9904	−0.938535	−0.469267	−	0.883056i	$-0.655482\pi$
$709$	−24.9904	−0.938535	−0.469267	+	0.883056i	$0.655482\pi$
$710$	0	0
$711$	−5.69165	−0.213454
$712$	0	0
$713$	−31.6069	−1.18369
$714$	0	0
$715$	−5.77080	−0.215816
$716$	0	0
$717$	78.1785	2.91963
$718$	0	0
$719$	−41.9723	−1.56530	−0.782651	−	0.622460i	$-0.786133\pi$
$719$	−41.9723	−1.56530	−0.782651	+	0.622460i	$0.786133\pi$
$720$	0	0
$721$	6.06642	0.225925
$722$	0	0
$723$	43.9713	1.63531
$724$	0	0
$725$	16.6247	0.617426
$726$	0	0
$727$	−13.1989	−0.489519	−0.244759	−	0.969584i	$-0.578709\pi$
$727$	−13.1989	−0.489519	−0.244759	+	0.969584i	$0.578709\pi$
$728$	0	0
$729$	52.4973	1.94434
$730$	0	0
$731$	−4.59911	−0.170104
$732$	0	0
$733$	−8.39197	−0.309964	−0.154982	−	0.987917i	$-0.549532\pi$
$733$	−8.39197	−0.309964	−0.154982	+	0.987917i	$0.549532\pi$
$734$	0	0
$735$	−1.81616	−0.0669901
$736$	0	0
$737$	68.0211	2.50559
$738$	0	0
$739$	−16.3106	−0.599994	−0.299997	−	0.953940i	$-0.596986\pi$
$739$	−16.3106	−0.599994	−0.299997	+	0.953940i	$0.596986\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−54.1661	−1.98716	−0.993580	−	0.113130i	$-0.963912\pi$
$743$	−54.1661	−1.98716	−0.993580	+	0.113130i	$0.963912\pi$
$744$	0	0
$745$	3.26142	0.119489
$746$	0	0
$747$	−105.024	−3.84263
$748$	0	0
$749$	−10.9483	−0.400041
$750$	0	0
$751$	−10.7648	−0.392812	−0.196406	−	0.980523i	$-0.562927\pi$
$751$	−10.7648	−0.392812	−0.196406	+	0.980523i	$0.562927\pi$
$752$	0	0
$753$	64.6844	2.35723
$754$	0	0
$755$	9.96540	0.362678
$756$	0	0
$757$	−12.6540	−0.459916	−0.229958	−	0.973200i	$-0.573859\pi$
$757$	−12.6540	−0.459916	−0.229958	+	0.973200i	$0.573859\pi$
$758$	0	0
$759$	109.971	3.99171
$760$	0	0
$761$	−23.4497	−0.850052	−0.425026	−	0.905181i	$-0.639735\pi$
$761$	−23.4497	−0.850052	−0.425026	+	0.905181i	$0.639735\pi$
$762$	0	0
$763$	10.8098	0.391341
$764$	0	0
$765$	−4.24672	−0.153540
$766$	0	0
$767$	−20.7099	−0.747791
$768$	0	0
$769$	−54.3422	−1.95963	−0.979815	−	0.199905i	$-0.935937\pi$
$769$	−54.3422	−1.95963	−0.979815	+	0.199905i	$0.935937\pi$
$770$	0	0
$771$	−1.11906	−0.0403021
$772$	0	0
$773$	34.7891	1.25128	0.625638	−	0.780114i	$-0.284839\pi$
$773$	34.7891	1.25128	0.625638	+	0.780114i	$0.284839\pi$
$774$	0	0
$775$	−22.7407	−0.816871
$776$	0	0
$777$	−10.3550	−0.371482
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−83.4907	−2.98753
$782$	0	0
$783$	−53.3777	−1.90756
$784$	0	0
$785$	0.402721	0.0143737
$786$	0	0
$787$	19.9487	0.711094	0.355547	−	0.934658i	$-0.384295\pi$
$787$	19.9487	0.711094	0.355547	+	0.934658i	$0.384295\pi$
$788$	0	0
$789$	92.6359	3.29792
$790$	0	0
$791$	−14.5050	−0.515738
$792$	0	0
$793$	−2.51772	−0.0894069
$794$	0	0
$795$	20.8960	0.741104
$796$	0	0
$797$	−14.5050	−0.513793	−0.256897	−	0.966439i	$-0.582700\pi$
$797$	−14.5050	−0.513793	−0.256897	+	0.966439i	$0.582700\pi$
$798$	0	0
$799$	0.746918	0.0264241
$800$	0	0
$801$	16.8018	0.593662
$802$	0	0
$803$	72.8401	2.57047
$804$	0	0
$805$	−3.63232	−0.128022
$806$	0	0
$807$	−42.3340	−1.49023
$808$	0	0
$809$	29.4325	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$809$	29.4325	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$810$	0	0
$811$	7.67314	0.269440	0.134720	−	0.990884i	$-0.456986\pi$
$811$	7.67314	0.269440	0.134720	+	0.990884i	$0.456986\pi$
$812$	0	0
$813$	81.4664	2.85715
$814$	0	0
$815$	−1.69363	−0.0593252
$816$	0	0
$817$	0	0
$818$	0	0
$819$	15.2404	0.532541
$820$	0	0
$821$	9.86506	0.344293	0.172147	−	0.985071i	$-0.444930\pi$
$821$	9.86506	0.344293	0.172147	+	0.985071i	$0.444930\pi$
$822$	0	0
$823$	21.7759	0.759061	0.379530	−	0.925179i	$-0.376086\pi$
$823$	21.7759	0.759061	0.379530	+	0.925179i	$0.376086\pi$
$824$	0	0
$825$	79.1230	2.75471
$826$	0	0
$827$	−14.5340	−0.505397	−0.252698	−	0.967545i	$-0.581318\pi$
$827$	−14.5340	−0.505397	−0.252698	+	0.967545i	$0.581318\pi$
$828$	0	0
$829$	−0.222102	−0.00771392	−0.00385696	−	0.999993i	$-0.501228\pi$
$829$	−0.222102	−0.00771392	−0.00385696	+	0.999993i	$0.501228\pi$
$830$	0	0
$831$	57.6662	2.00042
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	6.21888	0.215213
$836$	0	0
$837$	73.0146	2.52375
$838$	0	0
$839$	42.4807	1.46660	0.733298	−	0.679907i	$-0.237980\pi$
$839$	42.4807	1.46660	0.733298	+	0.679907i	$0.237980\pi$
$840$	0	0
$841$	−16.4319	−0.566616
$842$	0	0
$843$	40.0665	1.37996
$844$	0	0
$845$	5.01569	0.172545
$846$	0	0
$847$	−15.8062	−0.543109
$848$	0	0
$849$	−59.6873	−2.04846
$850$	0	0
$851$	−20.7099	−0.709926
$852$	0	0
$853$	8.27342	0.283276	0.141638	−	0.989918i	$-0.454763\pi$
$853$	8.27342	0.283276	0.141638	+	0.989918i	$0.454763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.3103	1.75272	0.876362	−	0.481652i	$-0.159963\pi$
$857$	51.3103	1.75272	0.876362	+	0.481652i	$0.159963\pi$
$858$	0	0
$859$	19.1434	0.653163	0.326581	−	0.945169i	$-0.394103\pi$
$859$	19.1434	0.653163	0.326581	+	0.945169i	$0.394103\pi$
$860$	0	0
$861$	17.3428	0.591041
$862$	0	0
$863$	7.43915	0.253232	0.126616	−	0.991952i	$-0.459588\pi$
$863$	7.43915	0.253232	0.126616	+	0.991952i	$0.459588\pi$
$864$	0	0
$865$	−9.68843	−0.329416
$866$	0	0
$867$	3.25886	0.110677
$868$	0	0
$869$	3.86715	0.131184
$870$	0	0
$871$	26.2758	0.890321
$872$	0	0
$873$	−76.3793	−2.58505
$874$	0	0
$875$	−5.39990	−0.182550
$876$	0	0
$877$	20.2058	0.682302	0.341151	−	0.940008i	$-0.389183\pi$
$877$	20.2058	0.682302	0.341151	+	0.940008i	$0.389183\pi$
$878$	0	0
$879$	−71.5743	−2.41414
$880$	0	0
$881$	−22.8124	−0.768568	−0.384284	−	0.923215i	$-0.625552\pi$
$881$	−22.8124	−0.768568	−0.384284	+	0.923215i	$0.625552\pi$
$882$	0	0
$883$	1.51202	0.0508835	0.0254418	−	0.999676i	$-0.491901\pi$
$883$	1.51202	0.0508835	0.0254418	+	0.999676i	$0.491901\pi$
$884$	0	0
$885$	−18.8062	−0.632165
$886$	0	0
$887$	47.4558	1.59341	0.796705	−	0.604368i	$-0.206574\pi$
$887$	47.4558	1.59341	0.796705	+	0.604368i	$0.206574\pi$
$888$	0	0
$889$	−9.66835	−0.324266
$890$	0	0
$891$	−135.684	−4.54558
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−1.06924	−0.0357408
$896$	0	0
$897$	42.4807	1.41839
$898$	0	0
$899$	−17.1918	−0.573378
$900$	0	0
$901$	−11.5056	−0.383306
$902$	0	0
$903$	14.9879	0.498765
$904$	0	0
$905$	8.50499	0.282716
$906$	0	0
$907$	−8.90051	−0.295537	−0.147768	−	0.989022i	$-0.547209\pi$
$907$	−8.90051	−0.295537	−0.147768	+	0.989022i	$0.547209\pi$
$908$	0	0
$909$	−42.9193	−1.42354
$910$	0	0
$911$	14.8484	0.491949	0.245974	−	0.969276i	$-0.420892\pi$
$911$	14.8484	0.491949	0.245974	+	0.969276i	$0.420892\pi$
$912$	0	0
$913$	71.3578	2.36160
$914$	0	0
$915$	−2.28629	−0.0755825
$916$	0	0
$917$	10.5806	0.349402
$918$	0	0
$919$	−34.4702	−1.13707	−0.568534	−	0.822660i	$-0.692489\pi$
$919$	−34.4702	−1.13707	−0.568534	+	0.822660i	$0.692489\pi$
$920$	0	0
$921$	−40.7520	−1.34283
$922$	0	0
$923$	−32.2515	−1.06157
$924$	0	0
$925$	−14.9005	−0.489926
$926$	0	0
$927$	−46.2272	−1.51830
$928$	0	0
$929$	−1.56688	−0.0514078	−0.0257039	−	0.999670i	$-0.508183\pi$
$929$	−1.56688	−0.0514078	−0.0257039	+	0.999670i	$0.508183\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−86.7515	−2.84012
$934$	0	0
$935$	2.88540	0.0943627
$936$	0	0
$937$	−14.6678	−0.479175	−0.239587	−	0.970875i	$-0.577012\pi$
$937$	−14.6678	−0.479175	−0.239587	+	0.970875i	$0.577012\pi$
$938$	0	0
$939$	27.5615	0.899436
$940$	0	0
$941$	−10.6926	−0.348568	−0.174284	−	0.984695i	$-0.555761\pi$
$941$	−10.6926	−0.348568	−0.174284	+	0.984695i	$0.555761\pi$
$942$	0	0
$943$	34.6856	1.12952
$944$	0	0
$945$	8.39098	0.272959
$946$	0	0
$947$	41.9105	1.36191	0.680954	−	0.732326i	$-0.261565\pi$
$947$	41.9105	1.36191	0.680954	+	0.732326i	$0.261565\pi$
$948$	0	0
$949$	28.1373	0.913376
$950$	0	0
$951$	16.9518	0.549701
$952$	0	0
$953$	−30.6656	−0.993356	−0.496678	−	0.867935i	$-0.665447\pi$
$953$	−30.6656	−0.993356	−0.496678	+	0.867935i	$0.665447\pi$
$954$	0	0
$955$	−8.05710	−0.260722
$956$	0	0
$957$	59.8162	1.93358
$958$	0	0
$959$	−1.85219	−0.0598103
$960$	0	0
$961$	−7.48359	−0.241406
$962$	0	0
$963$	83.4278	2.68842
$964$	0	0
$965$	−12.9077	−0.415514
$966$	0	0
$967$	−17.3460	−0.557811	−0.278905	−	0.960319i	$-0.589972\pi$
$967$	−17.3460	−0.557811	−0.278905	+	0.960319i	$0.589972\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.8909	0.509964	0.254982	−	0.966946i	$-0.417930\pi$
$971$	15.8909	0.509964	0.254982	+	0.966946i	$0.417930\pi$
$972$	0	0
$973$	17.6592	0.566127
$974$	0	0
$975$	30.5643	0.978842
$976$	0	0
$977$	−23.3544	−0.747172	−0.373586	−	0.927595i	$-0.621872\pi$
$977$	−23.3544	−0.747172	−0.373586	+	0.927595i	$0.621872\pi$
$978$	0	0
$979$	−11.4159	−0.364852
$980$	0	0
$981$	−82.3725	−2.62995
$982$	0	0
$983$	−53.3473	−1.70151	−0.850757	−	0.525559i	$-0.823856\pi$
$983$	−53.3473	−1.70151	−0.850757	+	0.525559i	$0.823856\pi$
$984$	0	0
$985$	1.03544	0.0329920
$986$	0	0
$987$	−2.43410	−0.0774784
$988$	0	0
$989$	29.9757	0.953172
$990$	0	0
$991$	46.9741	1.49218	0.746091	−	0.665844i	$-0.231928\pi$
$991$	46.9741	1.49218	0.746091	+	0.665844i	$0.231928\pi$
$992$	0	0
$993$	16.7437	0.531345
$994$	0	0
$995$	3.25433	0.103169
$996$	0	0
$997$	−6.74114	−0.213494	−0.106747	−	0.994286i	$-0.534043\pi$
$997$	−6.74114	−0.213494	−0.106747	+	0.994286i	$0.534043\pi$
$998$	0	0
$999$	47.8417	1.51364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bo.1.4		4
4.3	odd	2		7616.2.a.bi.1.1		4
8.3	odd	2		952.2.a.h.1.4	✓	4
8.5	even	2		1904.2.a.r.1.1		4
24.11	even	2		8568.2.a.bg.1.3		4
56.27	even	2		6664.2.a.n.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.h.1.4	✓	4	8.3	odd	2
1904.2.a.r.1.1		4	8.5	even	2
6664.2.a.n.1.1		4	56.27	even	2
7616.2.a.bi.1.1		4	4.3	odd	2
7616.2.a.bo.1.4		4	1.1	even	1	trivial
8568.2.a.bg.1.3		4	24.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+3.25886 q^{3} -0.557299 q^{5} -1.00000 q^{7} +7.62018 q^{9} -5.17748 q^{11} -2.00000 q^{13} -1.81616 q^{15} +1.00000 q^{17} -3.25886 q^{21} -6.51772 q^{23} -4.68942 q^{25} +15.0565 q^{27} -3.54515 q^{29} +4.84937 q^{31} -16.8727 q^{33} +0.557299 q^{35} +3.17748 q^{37} -6.51772 q^{39} -5.32174 q^{41} -4.59911 q^{43} -4.24672 q^{45} +0.746918 q^{47} +1.00000 q^{49} +3.25886 q^{51} -11.5056 q^{53} +2.88540 q^{55} +10.3550 q^{59} +1.25886 q^{61} -7.62018 q^{63} +1.11460 q^{65} -13.1379 q^{67} -21.2404 q^{69} +16.1258 q^{71} -14.0687 q^{73} -15.2822 q^{75} +5.17748 q^{77} -0.746918 q^{79} +26.2066 q^{81} -13.7824 q^{83} -0.557299 q^{85} -11.5532 q^{87} +2.20491 q^{89} +2.00000 q^{91} +15.8034 q^{93} -10.0233 q^{97} -39.4533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} - 8 q^{13} + 4 q^{17} - 3 q^{21} - 6 q^{23} + 3 q^{25} + 6 q^{27} - 8 q^{29} + 7 q^{31} - 6 q^{33} + 5 q^{35} - 8 q^{37} - 6 q^{39} + 15 q^{41} - 9 q^{43} + 2 q^{45}+ \cdots - 50 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 7 * q^9 - 8 * q^13 + 4 * q^17 - 3 * q^21 - 6 * q^23 + 3 * q^25 + 6 * q^27 - 8 * q^29 + 7 * q^31 - 6 * q^33 + 5 * q^35 - 8 * q^37 - 6 * q^39 + 15 * q^41 - 9 * q^43 + 2 * q^45 - 6 * q^47 + 4 * q^49 + 3 * q^51 - 17 * q^53 + 6 * q^55 - 5 * q^61 - 7 * q^63 + 10 * q^65 - 9 * q^67 - 38 * q^69 + 12 * q^71 - 11 * q^73 - 2 * q^75 + 6 * q^79 + 16 * q^81 - 6 * q^83 - 5 * q^85 + 14 * q^87 + 2 * q^89 + 8 * q^91 + 9 * q^97 - 50 * q^99

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