LMFDB - Embedded newform 2312.2.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,2,Mod(1,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2312.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-6,0,-3,0,0,0,6,0,-6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.4614129473$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3418281.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - 4x^{3} + 18x^{2} + 12x - 1 $ x^6 - 9x^4 - 4x^3 + 18x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.0750494$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0750494 q^{3} +2.06942 q^{5} +2.86317 q^{7} -2.99437 q^{9} -1.65231 q^{11} -3.96428 q^{13} -0.155308 q^{15} -5.57295 q^{19} -0.214879 q^{21} -5.28027 q^{23} -0.717513 q^{25} +0.449874 q^{27} -6.99818 q^{29} -6.39002 q^{31} +0.124005 q^{33} +5.92509 q^{35} +5.70355 q^{37} +0.297517 q^{39} -9.87650 q^{41} -6.67809 q^{43} -6.19660 q^{45} +4.43037 q^{47} +1.19773 q^{49} +8.27111 q^{53} -3.41932 q^{55} +0.418247 q^{57} -2.42912 q^{59} -1.81849 q^{61} -8.57338 q^{63} -8.20375 q^{65} +15.8315 q^{67} +0.396281 q^{69} +8.54703 q^{71} -2.30202 q^{73} +0.0538489 q^{75} -4.73084 q^{77} +14.0262 q^{79} +8.94934 q^{81} -1.00763 q^{83} +0.525209 q^{87} -16.2569 q^{89} -11.3504 q^{91} +0.479567 q^{93} -11.5328 q^{95} -13.8424 q^{97} +4.94762 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} - 3 q^{7} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} - 9 q^{23} + 6 q^{25} - 12 q^{27} - 27 q^{29} - 9 q^{31} + 3 q^{33} + 21 q^{35} - 15 q^{37} + 12 q^{39} - 21 q^{41} - 24 q^{45}+ \cdots + 42 q^{99}+O(q^{100}) $ 6 * q - 6 * q^5 - 3 * q^7 + 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 - 9 * q^23 + 6 * q^25 - 12 * q^27 - 27 * q^29 - 9 * q^31 + 3 * q^33 + 21 * q^35 - 15 * q^37 + 12 * q^39 - 21 * q^41 - 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 - 30 * q^57 - 18 * q^61 - 24 * q^63 + 9 * q^65 - 9 * q^67 - 3 * q^69 - 6 * q^71 - 27 * q^73 + 24 * q^75 - 45 * q^77 - 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 - 30 * q^91 - 30 * q^93 - 3 * q^95 - 51 * q^97 + 42 * 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
$3$	−0.0750494	−0.0433298	−0.0216649	−	0.999765i	$-0.506897\pi$
$3$	−0.0750494	−0.0433298	−0.0216649	+	0.999765i	$0.506897\pi$
$4$	0	0
$5$	2.06942	0.925471	0.462736	−	0.886496i	$-0.346868\pi$
$5$	2.06942	0.925471	0.462736	+	0.886496i	$0.346868\pi$
$6$	0	0
$7$	2.86317	1.08218	0.541088	−	0.840966i	$-0.318013\pi$
$7$	2.86317	1.08218	0.541088	+	0.840966i	$0.318013\pi$
$8$	0	0
$9$	−2.99437	−0.998123
$10$	0	0
$11$	−1.65231	−0.498190	−0.249095	−	0.968479i	$-0.580133\pi$
$11$	−1.65231	−0.498190	−0.249095	+	0.968479i	$0.580133\pi$
$12$	0	0
$13$	−3.96428	−1.09949	−0.549747	−	0.835331i	$-0.685276\pi$
$13$	−3.96428	−1.09949	−0.549747	+	0.835331i	$0.685276\pi$
$14$	0	0
$15$	−0.155308	−0.0401005
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−5.57295	−1.27852	−0.639262	−	0.768989i	$-0.720760\pi$
$19$	−5.57295	−1.27852	−0.639262	+	0.768989i	$0.720760\pi$
$20$	0	0
$21$	−0.214879	−0.0468904
$22$	0	0
$23$	−5.28027	−1.10101	−0.550507	−	0.834831i	$-0.685565\pi$
$23$	−5.28027	−1.10101	−0.550507	+	0.834831i	$0.685565\pi$
$24$	0	0
$25$	−0.717513	−0.143503
$26$	0	0
$27$	0.449874	0.0865782
$28$	0	0
$29$	−6.99818	−1.29953	−0.649765	−	0.760135i	$-0.725133\pi$
$29$	−6.99818	−1.29953	−0.649765	+	0.760135i	$0.725133\pi$
$30$	0	0
$31$	−6.39002	−1.14768	−0.573840	−	0.818967i	$-0.694547\pi$
$31$	−6.39002	−1.14768	−0.573840	+	0.818967i	$0.694547\pi$
$32$	0	0
$33$	0.124005	0.0215865
$34$	0	0
$35$	5.92509	1.00152
$36$	0	0
$37$	5.70355	0.937658	0.468829	−	0.883289i	$-0.344676\pi$
$37$	5.70355	0.937658	0.468829	+	0.883289i	$0.344676\pi$
$38$	0	0
$39$	0.297517	0.0476408
$40$	0	0
$41$	−9.87650	−1.54245	−0.771225	−	0.636562i	$-0.780356\pi$
$41$	−9.87650	−1.54245	−0.771225	+	0.636562i	$0.780356\pi$
$42$	0	0
$43$	−6.67809	−1.01840	−0.509199	−	0.860649i	$-0.670058\pi$
$43$	−6.67809	−1.01840	−0.509199	+	0.860649i	$0.670058\pi$
$44$	0	0
$45$	−6.19660	−0.923734
$46$	0	0
$47$	4.43037	0.646236	0.323118	−	0.946359i	$-0.395269\pi$
$47$	4.43037	0.646236	0.323118	+	0.946359i	$0.395269\pi$
$48$	0	0
$49$	1.19773	0.171104
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.27111	1.13612	0.568062	−	0.822985i	$-0.307693\pi$
$53$	8.27111	1.13612	0.568062	+	0.822985i	$0.307693\pi$
$54$	0	0
$55$	−3.41932	−0.461061
$56$	0	0
$57$	0.418247	0.0553981
$58$	0	0
$59$	−2.42912	−0.316244	−0.158122	−	0.987420i	$-0.550544\pi$
$59$	−2.42912	−0.316244	−0.158122	+	0.987420i	$0.550544\pi$
$60$	0	0
$61$	−1.81849	−0.232834	−0.116417	−	0.993200i	$-0.537141\pi$
$61$	−1.81849	−0.232834	−0.116417	+	0.993200i	$0.537141\pi$
$62$	0	0
$63$	−8.57338	−1.08014
$64$	0	0
$65$	−8.20375	−1.01755
$66$	0	0
$67$	15.8315	1.93413	0.967064	−	0.254535i	$-0.0819224\pi$
$67$	15.8315	1.93413	0.967064	+	0.254535i	$0.0819224\pi$
$68$	0	0
$69$	0.396281	0.0477067
$70$	0	0
$71$	8.54703	1.01435	0.507173	−	0.861844i	$-0.330691\pi$
$71$	8.54703	1.01435	0.507173	+	0.861844i	$0.330691\pi$
$72$	0	0
$73$	−2.30202	−0.269431	−0.134715	−	0.990884i	$-0.543012\pi$
$73$	−2.30202	−0.269431	−0.134715	+	0.990884i	$0.543012\pi$
$74$	0	0
$75$	0.0538489	0.00621794
$76$	0	0
$77$	−4.73084	−0.539129
$78$	0	0
$79$	14.0262	1.57807	0.789033	−	0.614351i	$-0.210582\pi$
$79$	14.0262	1.57807	0.789033	+	0.614351i	$0.210582\pi$
$80$	0	0
$81$	8.94934	0.994371
$82$	0	0
$83$	−1.00763	−0.110601	−0.0553006	−	0.998470i	$-0.517612\pi$
$83$	−1.00763	−0.110601	−0.0553006	+	0.998470i	$0.517612\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.525209	0.0563083
$88$	0	0
$89$	−16.2569	−1.72323	−0.861616	−	0.507561i	$-0.830547\pi$
$89$	−16.2569	−1.72323	−0.861616	+	0.507561i	$0.830547\pi$
$90$	0	0
$91$	−11.3504	−1.18985
$92$	0	0
$93$	0.479567	0.0497288
$94$	0	0
$95$	−11.5328	−1.18324
$96$	0	0
$97$	−13.8424	−1.40549	−0.702744	−	0.711443i	$-0.748042\pi$
$97$	−13.8424	−1.40549	−0.702744	+	0.711443i	$0.748042\pi$
$98$	0	0
$99$	4.94762	0.497255
$100$	0	0
$101$	14.9508	1.48766	0.743832	−	0.668366i	$-0.233006\pi$
$101$	14.9508	1.48766	0.743832	+	0.668366i	$0.233006\pi$
$102$	0	0
$103$	16.1276	1.58910	0.794552	−	0.607196i	$-0.207706\pi$
$103$	16.1276	1.58910	0.794552	+	0.607196i	$0.207706\pi$
$104$	0	0
$105$	−0.444674	−0.0433958
$106$	0	0
$107$	14.7416	1.42513	0.712563	−	0.701608i	$-0.247534\pi$
$107$	14.7416	1.42513	0.712563	+	0.701608i	$0.247534\pi$
$108$	0	0
$109$	−11.4726	−1.09887	−0.549437	−	0.835535i	$-0.685158\pi$
$109$	−11.4726	−1.09887	−0.549437	+	0.835535i	$0.685158\pi$
$110$	0	0
$111$	−0.428048	−0.0406285
$112$	0	0
$113$	−3.33361	−0.313599	−0.156800	−	0.987630i	$-0.550118\pi$
$113$	−3.33361	−0.313599	−0.156800	+	0.987630i	$0.550118\pi$
$114$	0	0
$115$	−10.9271	−1.01896
$116$	0	0
$117$	11.8705	1.09743
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.26987	−0.751806
$122$	0	0
$123$	0.741226	0.0668341
$124$	0	0
$125$	−11.8319	−1.05828
$126$	0	0
$127$	19.8464	1.76109	0.880544	−	0.473965i	$-0.157178\pi$
$127$	19.8464	1.76109	0.880544	+	0.473965i	$0.157178\pi$
$128$	0	0
$129$	0.501187	0.0441270
$130$	0	0
$131$	14.8864	1.30063	0.650316	−	0.759664i	$-0.274636\pi$
$131$	14.8864	1.30063	0.650316	+	0.759664i	$0.274636\pi$
$132$	0	0
$133$	−15.9563	−1.38359
$134$	0	0
$135$	0.930976	0.0801257
$136$	0	0
$137$	−2.48499	−0.212307	−0.106154	−	0.994350i	$-0.533853\pi$
$137$	−2.48499	−0.212307	−0.106154	+	0.994350i	$0.533853\pi$
$138$	0	0
$139$	3.28018	0.278221	0.139111	−	0.990277i	$-0.455576\pi$
$139$	3.28018	0.278221	0.139111	+	0.990277i	$0.455576\pi$
$140$	0	0
$141$	−0.332497	−0.0280013
$142$	0	0
$143$	6.55022	0.547757
$144$	0	0
$145$	−14.4822	−1.20268
$146$	0	0
$147$	−0.0898887	−0.00741390
$148$	0	0
$149$	8.18480	0.670525	0.335262	−	0.942125i	$-0.391175\pi$
$149$	8.18480	0.670525	0.335262	+	0.942125i	$0.391175\pi$
$150$	0	0
$151$	−15.8084	−1.28647	−0.643234	−	0.765670i	$-0.722408\pi$
$151$	−15.8084	−1.28647	−0.643234	+	0.765670i	$0.722408\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13.2236	−1.06215
$156$	0	0
$157$	3.84395	0.306781	0.153390	−	0.988166i	$-0.450981\pi$
$157$	3.84395	0.306781	0.153390	+	0.988166i	$0.450981\pi$
$158$	0	0
$159$	−0.620742	−0.0492281
$160$	0	0
$161$	−15.1183	−1.19149
$162$	0	0
$163$	−9.30898	−0.729136	−0.364568	−	0.931177i	$-0.618783\pi$
$163$	−9.30898	−0.729136	−0.364568	+	0.931177i	$0.618783\pi$
$164$	0	0
$165$	0.256618	0.0199777
$166$	0	0
$167$	−4.36977	−0.338143	−0.169071	−	0.985604i	$-0.554077\pi$
$167$	−4.36977	−0.338143	−0.169071	+	0.985604i	$0.554077\pi$
$168$	0	0
$169$	2.71552	0.208887
$170$	0	0
$171$	16.6875	1.27612
$172$	0	0
$173$	1.58579	0.120565	0.0602826	−	0.998181i	$-0.480800\pi$
$173$	1.58579	0.120565	0.0602826	+	0.998181i	$0.480800\pi$
$174$	0	0
$175$	−2.05436	−0.155295
$176$	0	0
$177$	0.182304	0.0137028
$178$	0	0
$179$	−22.0093	−1.64505	−0.822525	−	0.568729i	$-0.807435\pi$
$179$	−22.0093	−1.64505	−0.822525	+	0.568729i	$0.807435\pi$
$180$	0	0
$181$	−14.1859	−1.05443	−0.527216	−	0.849731i	$-0.676764\pi$
$181$	−14.1859	−1.05443	−0.527216	+	0.849731i	$0.676764\pi$
$182$	0	0
$183$	0.136477	0.0100886
$184$	0	0
$185$	11.8030	0.867776
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.28806	0.0936928
$190$	0	0
$191$	−9.40915	−0.680823	−0.340411	−	0.940277i	$-0.610566\pi$
$191$	−9.40915	−0.680823	−0.340411	+	0.940277i	$0.610566\pi$
$192$	0	0
$193$	7.63065	0.549266	0.274633	−	0.961549i	$-0.411444\pi$
$193$	7.63065	0.549266	0.274633	+	0.961549i	$0.411444\pi$
$194$	0	0
$195$	0.615687	0.0440902
$196$	0	0
$197$	9.04547	0.644463	0.322232	−	0.946661i	$-0.395567\pi$
$197$	9.04547	0.644463	0.322232	+	0.946661i	$0.395567\pi$
$198$	0	0
$199$	−11.4553	−0.812047	−0.406023	−	0.913863i	$-0.633085\pi$
$199$	−11.4553	−0.812047	−0.406023	+	0.913863i	$0.633085\pi$
$200$	0	0
$201$	−1.18815	−0.0838053
$202$	0	0
$203$	−20.0370	−1.40632
$204$	0	0
$205$	−20.4386	−1.42749
$206$	0	0
$207$	15.8111	1.09895
$208$	0	0
$209$	9.20825	0.636948
$210$	0	0
$211$	−9.18283	−0.632172	−0.316086	−	0.948731i	$-0.602369\pi$
$211$	−9.18283	−0.632172	−0.316086	+	0.948731i	$0.602369\pi$
$212$	0	0
$213$	−0.641449	−0.0439514
$214$	0	0
$215$	−13.8198	−0.942499
$216$	0	0
$217$	−18.2957	−1.24199
$218$	0	0
$219$	0.172765	0.0116744
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−7.34273	−0.491705	−0.245853	−	0.969307i	$-0.579068\pi$
$223$	−7.34273	−0.491705	−0.245853	+	0.969307i	$0.579068\pi$
$224$	0	0
$225$	2.14850	0.143233
$226$	0	0
$227$	−11.7173	−0.777703	−0.388852	−	0.921300i	$-0.627128\pi$
$227$	−11.7173	−0.777703	−0.388852	+	0.921300i	$0.627128\pi$
$228$	0	0
$229$	−8.45705	−0.558857	−0.279429	−	0.960166i	$-0.590145\pi$
$229$	−8.45705	−0.558857	−0.279429	+	0.960166i	$0.590145\pi$
$230$	0	0
$231$	0.355047	0.0233604
$232$	0	0
$233$	−18.7135	−1.22596	−0.612980	−	0.790099i	$-0.710029\pi$
$233$	−18.7135	−1.22596	−0.612980	+	0.790099i	$0.710029\pi$
$234$	0	0
$235$	9.16829	0.598073
$236$	0	0
$237$	−1.05265	−0.0683772
$238$	0	0
$239$	7.40650	0.479086	0.239543	−	0.970886i	$-0.423002\pi$
$239$	7.40650	0.479086	0.239543	+	0.970886i	$0.423002\pi$
$240$	0	0
$241$	−22.6670	−1.46011	−0.730054	−	0.683390i	$-0.760505\pi$
$241$	−22.6670	−1.46011	−0.730054	+	0.683390i	$0.760505\pi$
$242$	0	0
$243$	−2.02126	−0.129664
$244$	0	0
$245$	2.47860	0.158352
$246$	0	0
$247$	22.0928	1.40573
$248$	0	0
$249$	0.0756217	0.00479233
$250$	0	0
$251$	−5.67032	−0.357907	−0.178954	−	0.983857i	$-0.557271\pi$
$251$	−5.67032	−0.357907	−0.178954	+	0.983857i	$0.557271\pi$
$252$	0	0
$253$	8.72465	0.548514
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.2181	1.76020	0.880100	−	0.474789i	$-0.157476\pi$
$257$	28.2181	1.76020	0.880100	+	0.474789i	$0.157476\pi$
$258$	0	0
$259$	16.3302	1.01471
$260$	0	0
$261$	20.9551	1.29709
$262$	0	0
$263$	−4.06551	−0.250690	−0.125345	−	0.992113i	$-0.540004\pi$
$263$	−4.06551	−0.250690	−0.125345	+	0.992113i	$0.540004\pi$
$264$	0	0
$265$	17.1164	1.05145
$266$	0	0
$267$	1.22007	0.0746673
$268$	0	0
$269$	−1.95914	−0.119451	−0.0597255	−	0.998215i	$-0.519023\pi$
$269$	−1.95914	−0.119451	−0.0597255	+	0.998215i	$0.519023\pi$
$270$	0	0
$271$	6.00392	0.364712	0.182356	−	0.983233i	$-0.441628\pi$
$271$	6.00392	0.364712	0.182356	+	0.983233i	$0.441628\pi$
$272$	0	0
$273$	0.851841	0.0515557
$274$	0	0
$275$	1.18555	0.0714916
$276$	0	0
$277$	−5.74968	−0.345465	−0.172732	−	0.984969i	$-0.555260\pi$
$277$	−5.74968	−0.345465	−0.172732	+	0.984969i	$0.555260\pi$
$278$	0	0
$279$	19.1341	1.14553
$280$	0	0
$281$	20.4528	1.22011	0.610056	−	0.792359i	$-0.291147\pi$
$281$	20.4528	1.22011	0.610056	+	0.792359i	$0.291147\pi$
$282$	0	0
$283$	26.9965	1.60478	0.802389	−	0.596801i	$-0.203562\pi$
$283$	26.9965	1.60478	0.802389	+	0.596801i	$0.203562\pi$
$284$	0	0
$285$	0.865527	0.0512694
$286$	0	0
$287$	−28.2781	−1.66920
$288$	0	0
$289$	0	0
$290$	0	0
$291$	1.03887	0.0608995
$292$	0	0
$293$	−5.55130	−0.324310	−0.162155	−	0.986765i	$-0.551845\pi$
$293$	−5.55130	−0.324310	−0.162155	+	0.986765i	$0.551845\pi$
$294$	0	0
$295$	−5.02686	−0.292675
$296$	0	0
$297$	−0.743331	−0.0431324
$298$	0	0
$299$	20.9325	1.21056
$300$	0	0
$301$	−19.1205	−1.10209
$302$	0	0
$303$	−1.12205	−0.0644602
$304$	0	0
$305$	−3.76322	−0.215481
$306$	0	0
$307$	−16.1215	−0.920104	−0.460052	−	0.887892i	$-0.652169\pi$
$307$	−16.1215	−0.920104	−0.460052	+	0.887892i	$0.652169\pi$
$308$	0	0
$309$	−1.21037	−0.0688555
$310$	0	0
$311$	−18.2242	−1.03340	−0.516699	−	0.856167i	$-0.672839\pi$
$311$	−18.2242	−1.03340	−0.516699	+	0.856167i	$0.672839\pi$
$312$	0	0
$313$	−9.98052	−0.564132	−0.282066	−	0.959395i	$-0.591020\pi$
$313$	−9.98052	−0.564132	−0.282066	+	0.959395i	$0.591020\pi$
$314$	0	0
$315$	−17.7419	−0.999642
$316$	0	0
$317$	4.47278	0.251216	0.125608	−	0.992080i	$-0.459912\pi$
$317$	4.47278	0.251216	0.125608	+	0.992080i	$0.459912\pi$
$318$	0	0
$319$	11.5632	0.647413
$320$	0	0
$321$	−1.10635	−0.0617504
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.84442	0.157780
$326$	0	0
$327$	0.861010	0.0476140
$328$	0	0
$329$	12.6849	0.699341
$330$	0	0
$331$	0.514782	0.0282950	0.0141475	−	0.999900i	$-0.495497\pi$
$331$	0.514782	0.0282950	0.0141475	+	0.999900i	$0.495497\pi$
$332$	0	0
$333$	−17.0785	−0.935898
$334$	0	0
$335$	32.7620	1.78998
$336$	0	0
$337$	3.35929	0.182992	0.0914960	−	0.995805i	$-0.470835\pi$
$337$	3.35929	0.182992	0.0914960	+	0.995805i	$0.470835\pi$
$338$	0	0
$339$	0.250185	0.0135882
$340$	0	0
$341$	10.5583	0.571764
$342$	0	0
$343$	−16.6129	−0.897011
$344$	0	0
$345$	0.820071	0.0441512
$346$	0	0
$347$	23.5337	1.26336	0.631678	−	0.775231i	$-0.282366\pi$
$347$	23.5337	1.26336	0.631678	+	0.775231i	$0.282366\pi$
$348$	0	0
$349$	25.3356	1.35619	0.678093	−	0.734976i	$-0.262807\pi$
$349$	25.3356	1.35619	0.678093	+	0.734976i	$0.262807\pi$
$350$	0	0
$351$	−1.78343	−0.0951922
$352$	0	0
$353$	17.0119	0.905454	0.452727	−	0.891649i	$-0.350451\pi$
$353$	17.0119	0.905454	0.452727	+	0.891649i	$0.350451\pi$
$354$	0	0
$355$	17.6874	0.938748
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.00460	−0.316911	−0.158455	−	0.987366i	$-0.550651\pi$
$359$	−6.00460	−0.316911	−0.158455	+	0.987366i	$0.550651\pi$
$360$	0	0
$361$	12.0578	0.634621
$362$	0	0
$363$	0.620649	0.0325756
$364$	0	0
$365$	−4.76383	−0.249350
$366$	0	0
$367$	−7.43083	−0.387886	−0.193943	−	0.981013i	$-0.562128\pi$
$367$	−7.43083	−0.387886	−0.193943	+	0.981013i	$0.562128\pi$
$368$	0	0
$369$	29.5739	1.53955
$370$	0	0
$371$	23.6816	1.22949
$372$	0	0
$373$	−20.8566	−1.07991	−0.539957	−	0.841692i	$-0.681560\pi$
$373$	−20.8566	−1.07991	−0.539957	+	0.841692i	$0.681560\pi$
$374$	0	0
$375$	0.887978	0.0458550
$376$	0	0
$377$	27.7428	1.42882
$378$	0	0
$379$	−2.34051	−0.120224	−0.0601119	−	0.998192i	$-0.519146\pi$
$379$	−2.34051	−0.120224	−0.0601119	+	0.998192i	$0.519146\pi$
$380$	0	0
$381$	−1.48946	−0.0763075
$382$	0	0
$383$	31.7087	1.62024	0.810119	−	0.586265i	$-0.199402\pi$
$383$	31.7087	1.62024	0.810119	+	0.586265i	$0.199402\pi$
$384$	0	0
$385$	−9.79008	−0.498949
$386$	0	0
$387$	19.9967	1.01649
$388$	0	0
$389$	−10.8275	−0.548975	−0.274487	−	0.961591i	$-0.588508\pi$
$389$	−10.8275	−0.548975	−0.274487	+	0.961591i	$0.588508\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1.11722	−0.0563561
$394$	0	0
$395$	29.0260	1.46045
$396$	0	0
$397$	−5.82927	−0.292562	−0.146281	−	0.989243i	$-0.546730\pi$
$397$	−5.82927	−0.292562	−0.146281	+	0.989243i	$0.546730\pi$
$398$	0	0
$399$	1.19751	0.0599505
$400$	0	0
$401$	−25.5361	−1.27521	−0.637605	−	0.770363i	$-0.720075\pi$
$401$	−25.5361	−1.27521	−0.637605	+	0.770363i	$0.720075\pi$
$402$	0	0
$403$	25.3318	1.26187
$404$	0	0
$405$	18.5199	0.920262
$406$	0	0
$407$	−9.42404	−0.467132
$408$	0	0
$409$	−19.4417	−0.961329	−0.480664	−	0.876905i	$-0.659604\pi$
$409$	−19.4417	−0.961329	−0.480664	+	0.876905i	$0.659604\pi$
$410$	0	0
$411$	0.186497	0.00919922
$412$	0	0
$413$	−6.95497	−0.342232
$414$	0	0
$415$	−2.08520	−0.102358
$416$	0	0
$417$	−0.246176	−0.0120553
$418$	0	0
$419$	37.6604	1.83983	0.919916	−	0.392116i	$-0.128257\pi$
$419$	37.6604	1.83983	0.919916	+	0.392116i	$0.128257\pi$
$420$	0	0
$421$	−13.2597	−0.646238	−0.323119	−	0.946358i	$-0.604731\pi$
$421$	−13.2597	−0.646238	−0.323119	+	0.946358i	$0.604731\pi$
$422$	0	0
$423$	−13.2662	−0.645023
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.20664	−0.251967
$428$	0	0
$429$	−0.491590	−0.0237342
$430$	0	0
$431$	31.8730	1.53527	0.767635	−	0.640887i	$-0.221433\pi$
$431$	31.8730	1.53527	0.767635	+	0.640887i	$0.221433\pi$
$432$	0	0
$433$	37.8807	1.82043	0.910215	−	0.414137i	$-0.135917\pi$
$433$	37.8807	1.82043	0.910215	+	0.414137i	$0.135917\pi$
$434$	0	0
$435$	1.08688	0.0521118
$436$	0	0
$437$	29.4267	1.40767
$438$	0	0
$439$	4.95029	0.236264	0.118132	−	0.992998i	$-0.462309\pi$
$439$	4.95029	0.236264	0.118132	+	0.992998i	$0.462309\pi$
$440$	0	0
$441$	−3.58644	−0.170783
$442$	0	0
$443$	−25.7002	−1.22106	−0.610528	−	0.791995i	$-0.709043\pi$
$443$	−25.7002	−1.22106	−0.610528	+	0.791995i	$0.709043\pi$
$444$	0	0
$445$	−33.6424	−1.59480
$446$	0	0
$447$	−0.614264	−0.0290537
$448$	0	0
$449$	−13.9403	−0.657882	−0.328941	−	0.944350i	$-0.606692\pi$
$449$	−13.9403	−0.657882	−0.328941	+	0.944350i	$0.606692\pi$
$450$	0	0
$451$	16.3190	0.768434
$452$	0	0
$453$	1.18641	0.0557424
$454$	0	0
$455$	−23.4887	−1.10117
$456$	0	0
$457$	7.21894	0.337688	0.168844	−	0.985643i	$-0.445997\pi$
$457$	7.21894	0.337688	0.168844	+	0.985643i	$0.445997\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.0248950	−0.00115948	−0.000579739	−	1.00000i	$-0.500185\pi$
$461$	−0.0248950	−0.00115948	−0.000579739		1.00000i	$0.500185\pi$
$462$	0	0
$463$	−41.3218	−1.92039	−0.960194	−	0.279334i	$-0.909886\pi$
$463$	−41.3218	−1.92039	−0.960194	+	0.279334i	$0.909886\pi$
$464$	0	0
$465$	0.992424	0.0460226
$466$	0	0
$467$	−2.01187	−0.0930985	−0.0465492	−	0.998916i	$-0.514822\pi$
$467$	−2.01187	−0.0930985	−0.0465492	+	0.998916i	$0.514822\pi$
$468$	0	0
$469$	45.3283	2.09307
$470$	0	0
$471$	−0.288486	−0.0132927
$472$	0	0
$473$	11.0343	0.507357
$474$	0	0
$475$	3.99867	0.183471
$476$	0	0
$477$	−24.7668	−1.13399
$478$	0	0
$479$	−40.6285	−1.85637	−0.928183	−	0.372124i	$-0.878629\pi$
$479$	−40.6285	−1.85637	−0.928183	+	0.372124i	$0.878629\pi$
$480$	0	0
$481$	−22.6105	−1.03095
$482$	0	0
$483$	1.13462	0.0516270
$484$	0	0
$485$	−28.6458	−1.30074
$486$	0	0
$487$	−30.6041	−1.38680	−0.693401	−	0.720552i	$-0.743889\pi$
$487$	−30.6041	−1.38680	−0.693401	+	0.720552i	$0.743889\pi$
$488$	0	0
$489$	0.698633	0.0315933
$490$	0	0
$491$	15.3815	0.694157	0.347078	−	0.937836i	$-0.387174\pi$
$491$	15.3815	0.694157	0.347078	+	0.937836i	$0.387174\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	10.2387	0.460195
$496$	0	0
$497$	24.4716	1.09770
$498$	0	0
$499$	−12.2998	−0.550617	−0.275308	−	0.961356i	$-0.588780\pi$
$499$	−12.2998	−0.550617	−0.275308	+	0.961356i	$0.588780\pi$
$500$	0	0
$501$	0.327949	0.0146517
$502$	0	0
$503$	−23.2615	−1.03718	−0.518589	−	0.855024i	$-0.673543\pi$
$503$	−23.2615	−1.03718	−0.518589	+	0.855024i	$0.673543\pi$
$504$	0	0
$505$	30.9395	1.37679
$506$	0	0
$507$	−0.203799	−0.00905101
$508$	0	0
$509$	−17.9307	−0.794762	−0.397381	−	0.917654i	$-0.630081\pi$
$509$	−17.9307	−0.794762	−0.397381	+	0.917654i	$0.630081\pi$
$510$	0	0
$511$	−6.59106	−0.291571
$512$	0	0
$513$	−2.50712	−0.110692
$514$	0	0
$515$	33.3748	1.47067
$516$	0	0
$517$	−7.32035	−0.321949
$518$	0	0
$519$	−0.119012	−0.00522407
$520$	0	0
$521$	−12.5178	−0.548413	−0.274206	−	0.961671i	$-0.588415\pi$
$521$	−12.5178	−0.548413	−0.274206	+	0.961671i	$0.588415\pi$
$522$	0	0
$523$	−10.5810	−0.462677	−0.231338	−	0.972873i	$-0.574310\pi$
$523$	−10.5810	−0.462677	−0.231338	+	0.972873i	$0.574310\pi$
$524$	0	0
$525$	0.154179	0.00672890
$526$	0	0
$527$	0	0
$528$	0	0
$529$	4.88129	0.212230
$530$	0	0
$531$	7.27367	0.315651
$532$	0	0
$533$	39.1532	1.69591
$534$	0	0
$535$	30.5066	1.31891
$536$	0	0
$537$	1.65178	0.0712797
$538$	0	0
$539$	−1.97902	−0.0852423
$540$	0	0
$541$	31.4314	1.35134	0.675671	−	0.737203i	$-0.263854\pi$
$541$	31.4314	1.35134	0.675671	+	0.737203i	$0.263854\pi$
$542$	0	0
$543$	1.06465	0.0456883
$544$	0	0
$545$	−23.7415	−1.01698
$546$	0	0
$547$	−4.84354	−0.207095	−0.103547	−	0.994625i	$-0.533019\pi$
$547$	−4.84354	−0.207095	−0.103547	+	0.994625i	$0.533019\pi$
$548$	0	0
$549$	5.44523	0.232397
$550$	0	0
$551$	39.0005	1.66148
$552$	0	0
$553$	40.1592	1.70774
$554$	0	0
$555$	−0.885810	−0.0376005
$556$	0	0
$557$	13.3327	0.564927	0.282463	−	0.959278i	$-0.408848\pi$
$557$	13.3327	0.564927	0.282463	+	0.959278i	$0.408848\pi$
$558$	0	0
$559$	26.4738	1.11972
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.2597	1.27529	0.637647	−	0.770329i	$-0.279908\pi$
$563$	30.2597	1.27529	0.637647	+	0.770329i	$0.279908\pi$
$564$	0	0
$565$	−6.89862	−0.290227
$566$	0	0
$567$	25.6235	1.07608
$568$	0	0
$569$	−30.4438	−1.27627	−0.638136	−	0.769924i	$-0.720294\pi$
$569$	−30.4438	−1.27627	−0.638136	+	0.769924i	$0.720294\pi$
$570$	0	0
$571$	37.5161	1.57000	0.785000	−	0.619496i	$-0.212663\pi$
$571$	37.5161	1.57000	0.785000	+	0.619496i	$0.212663\pi$
$572$	0	0
$573$	0.706151	0.0294999
$574$	0	0
$575$	3.78867	0.157998
$576$	0	0
$577$	−24.2745	−1.01056	−0.505280	−	0.862955i	$-0.668611\pi$
$577$	−24.2745	−1.01056	−0.505280	+	0.862955i	$0.668611\pi$
$578$	0	0
$579$	−0.572676	−0.0237996
$580$	0	0
$581$	−2.88500	−0.119690
$582$	0	0
$583$	−13.6664	−0.566007
$584$	0	0
$585$	24.5650	1.01564
$586$	0	0
$587$	18.4536	0.761661	0.380831	−	0.924645i	$-0.375638\pi$
$587$	18.4536	0.761661	0.380831	+	0.924645i	$0.375638\pi$
$588$	0	0
$589$	35.6113	1.46734
$590$	0	0
$591$	−0.678857	−0.0279245
$592$	0	0
$593$	−12.0405	−0.494444	−0.247222	−	0.968959i	$-0.579518\pi$
$593$	−12.0405	−0.494444	−0.247222	+	0.968959i	$0.579518\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.859715	0.0351858
$598$	0	0
$599$	21.0725	0.860999	0.430499	−	0.902591i	$-0.358337\pi$
$599$	21.0725	0.860999	0.430499	+	0.902591i	$0.358337\pi$
$600$	0	0
$601$	−19.5781	−0.798609	−0.399305	−	0.916818i	$-0.630748\pi$
$601$	−19.5781	−0.798609	−0.399305	+	0.916818i	$0.630748\pi$
$602$	0	0
$603$	−47.4054	−1.93050
$604$	0	0
$605$	−17.1138	−0.695775
$606$	0	0
$607$	−7.14287	−0.289920	−0.144960	−	0.989438i	$-0.546305\pi$
$607$	−7.14287	−0.289920	−0.144960	+	0.989438i	$0.546305\pi$
$608$	0	0
$609$	1.50376	0.0609355
$610$	0	0
$611$	−17.5632	−0.710533
$612$	0	0
$613$	9.73218	0.393079	0.196539	−	0.980496i	$-0.437030\pi$
$613$	9.73218	0.393079	0.196539	+	0.980496i	$0.437030\pi$
$614$	0	0
$615$	1.53390	0.0618530
$616$	0	0
$617$	−15.7203	−0.632875	−0.316437	−	0.948613i	$-0.602487\pi$
$617$	−15.7203	−0.632875	−0.316437	+	0.948613i	$0.602487\pi$
$618$	0	0
$619$	−46.8385	−1.88260	−0.941299	−	0.337573i	$-0.890394\pi$
$619$	−46.8385	−1.88260	−0.941299	+	0.337573i	$0.890394\pi$
$620$	0	0
$621$	−2.37546	−0.0953238
$622$	0	0
$623$	−46.5463	−1.86484
$624$	0	0
$625$	−20.8976	−0.835904
$626$	0	0
$627$	−0.691073	−0.0275988
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−0.994345	−0.0395843	−0.0197921	−	0.999804i	$-0.506300\pi$
$631$	−0.994345	−0.0395843	−0.0197921	+	0.999804i	$0.506300\pi$
$632$	0	0
$633$	0.689166	0.0273919
$634$	0	0
$635$	41.0706	1.62984
$636$	0	0
$637$	−4.74813	−0.188128
$638$	0	0
$639$	−25.5929	−1.01244
$640$	0	0
$641$	−12.9583	−0.511821	−0.255911	−	0.966700i	$-0.582375\pi$
$641$	−12.9583	−0.511821	−0.255911	+	0.966700i	$0.582375\pi$
$642$	0	0
$643$	5.44186	0.214606	0.107303	−	0.994226i	$-0.465778\pi$
$643$	5.44186	0.214606	0.107303	+	0.994226i	$0.465778\pi$
$644$	0	0
$645$	1.03716	0.0408383
$646$	0	0
$647$	−13.4339	−0.528143	−0.264072	−	0.964503i	$-0.585065\pi$
$647$	−13.4339	−0.528143	−0.264072	+	0.964503i	$0.585065\pi$
$648$	0	0
$649$	4.01366	0.157550
$650$	0	0
$651$	1.37308	0.0538153
$652$	0	0
$653$	29.8838	1.16944	0.584722	−	0.811234i	$-0.301203\pi$
$653$	29.8838	1.16944	0.584722	+	0.811234i	$0.301203\pi$
$654$	0	0
$655$	30.8062	1.20370
$656$	0	0
$657$	6.89309	0.268925
$658$	0	0
$659$	26.0153	1.01341	0.506707	−	0.862119i	$-0.330863\pi$
$659$	26.0153	1.01341	0.506707	+	0.862119i	$0.330863\pi$
$660$	0	0
$661$	−29.6768	−1.15430	−0.577148	−	0.816640i	$-0.695834\pi$
$661$	−29.6768	−1.15430	−0.577148	+	0.816640i	$0.695834\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−33.0202	−1.28047
$666$	0	0
$667$	36.9523	1.43080
$668$	0	0
$669$	0.551067	0.0213055
$670$	0	0
$671$	3.00471	0.115996
$672$	0	0
$673$	27.1439	1.04632	0.523159	−	0.852235i	$-0.324753\pi$
$673$	27.1439	1.04632	0.523159	+	0.852235i	$0.324753\pi$
$674$	0	0
$675$	−0.322790	−0.0124242
$676$	0	0
$677$	−22.8605	−0.878602	−0.439301	−	0.898340i	$-0.644774\pi$
$677$	−22.8605	−0.878602	−0.439301	+	0.898340i	$0.644774\pi$
$678$	0	0
$679$	−39.6332	−1.52098
$680$	0	0
$681$	0.879375	0.0336977
$682$	0	0
$683$	9.49776	0.363422	0.181711	−	0.983352i	$-0.441837\pi$
$683$	9.49776	0.363422	0.181711	+	0.983352i	$0.441837\pi$
$684$	0	0
$685$	−5.14248	−0.196484
$686$	0	0
$687$	0.634696	0.0242152
$688$	0	0
$689$	−32.7890	−1.24916
$690$	0	0
$691$	34.6226	1.31711	0.658554	−	0.752534i	$-0.271169\pi$
$691$	34.6226	1.31711	0.658554	+	0.752534i	$0.271169\pi$
$692$	0	0
$693$	14.1659	0.538117
$694$	0	0
$695$	6.78806	0.257486
$696$	0	0
$697$	0	0
$698$	0	0
$699$	1.40443	0.0531206
$700$	0	0
$701$	−5.15517	−0.194708	−0.0973541	−	0.995250i	$-0.531038\pi$
$701$	−5.15517	−0.194708	−0.0973541	+	0.995250i	$0.531038\pi$
$702$	0	0
$703$	−31.7856	−1.19882
$704$	0	0
$705$	−0.688075	−0.0259144
$706$	0	0
$707$	42.8068	1.60991
$708$	0	0
$709$	2.45568	0.0922250	0.0461125	−	0.998936i	$-0.485317\pi$
$709$	2.45568	0.0922250	0.0461125	+	0.998936i	$0.485317\pi$
$710$	0	0
$711$	−41.9995	−1.57510
$712$	0	0
$713$	33.7410	1.26361
$714$	0	0
$715$	13.5551	0.506934
$716$	0	0
$717$	−0.555853	−0.0207587
$718$	0	0
$719$	−35.6674	−1.33017	−0.665086	−	0.746767i	$-0.731605\pi$
$719$	−35.6674	−1.33017	−0.665086	+	0.746767i	$0.731605\pi$
$720$	0	0
$721$	46.1761	1.71969
$722$	0	0
$723$	1.70114	0.0632661
$724$	0	0
$725$	5.02129	0.186486
$726$	0	0
$727$	−14.4438	−0.535692	−0.267846	−	0.963462i	$-0.586312\pi$
$727$	−14.4438	−0.535692	−0.267846	+	0.963462i	$0.586312\pi$
$728$	0	0
$729$	−26.6963	−0.988753
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−22.8514	−0.844036	−0.422018	−	0.906587i	$-0.638678\pi$
$733$	−22.8514	−0.844036	−0.422018	+	0.906587i	$0.638678\pi$
$734$	0	0
$735$	−0.186017	−0.00686135
$736$	0	0
$737$	−26.1586	−0.963563
$738$	0	0
$739$	−13.1134	−0.482383	−0.241192	−	0.970478i	$-0.577538\pi$
$739$	−13.1134	−0.482383	−0.241192	+	0.970478i	$0.577538\pi$
$740$	0	0
$741$	−1.65805	−0.0609099
$742$	0	0
$743$	−5.86298	−0.215092	−0.107546	−	0.994200i	$-0.534299\pi$
$743$	−5.86298	−0.215092	−0.107546	+	0.994200i	$0.534299\pi$
$744$	0	0
$745$	16.9378	0.620551
$746$	0	0
$747$	3.01720	0.110394
$748$	0	0
$749$	42.2077	1.54224
$750$	0	0
$751$	−10.5603	−0.385351	−0.192676	−	0.981262i	$-0.561717\pi$
$751$	−10.5603	−0.385351	−0.192676	+	0.981262i	$0.561717\pi$
$752$	0	0
$753$	0.425554	0.0155081
$754$	0	0
$755$	−32.7141	−1.19059
$756$	0	0
$757$	52.3557	1.90290	0.951451	−	0.307801i	$-0.0995930\pi$
$757$	52.3557	1.90290	0.951451	+	0.307801i	$0.0995930\pi$
$758$	0	0
$759$	−0.654780	−0.0237670
$760$	0	0
$761$	−1.60283	−0.0581027	−0.0290513	−	0.999578i	$-0.509249\pi$
$761$	−1.60283	−0.0581027	−0.0290513	+	0.999578i	$0.509249\pi$
$762$	0	0
$763$	−32.8479	−1.18917
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.62971	0.347709
$768$	0	0
$769$	19.0060	0.685373	0.342686	−	0.939450i	$-0.388663\pi$
$769$	19.0060	0.685373	0.342686	+	0.939450i	$0.388663\pi$
$770$	0	0
$771$	−2.11775	−0.0762691
$772$	0	0
$773$	42.9719	1.54559	0.772795	−	0.634655i	$-0.218858\pi$
$773$	42.9719	1.54559	0.772795	+	0.634655i	$0.218858\pi$
$774$	0	0
$775$	4.58492	0.164695
$776$	0	0
$777$	−1.22557	−0.0439672
$778$	0	0
$779$	55.0413	1.97206
$780$	0	0
$781$	−14.1223	−0.505337
$782$	0	0
$783$	−3.14830	−0.112511
$784$	0	0
$785$	7.95474	0.283917
$786$	0	0
$787$	−15.6985	−0.559592	−0.279796	−	0.960059i	$-0.590267\pi$
$787$	−15.6985	−0.559592	−0.279796	+	0.960059i	$0.590267\pi$
$788$	0	0
$789$	0.305114	0.0108623
$790$	0	0
$791$	−9.54468	−0.339370
$792$	0	0
$793$	7.20901	0.255999
$794$	0	0
$795$	−1.28457	−0.0455592
$796$	0	0
$797$	10.2757	0.363984	0.181992	−	0.983300i	$-0.441745\pi$
$797$	10.2757	0.363984	0.181992	+	0.983300i	$0.441745\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	48.6792	1.72000
$802$	0	0
$803$	3.80365	0.134228
$804$	0	0
$805$	−31.2861	−1.10269
$806$	0	0
$807$	0.147032	0.00517578
$808$	0	0
$809$	−7.07102	−0.248604	−0.124302	−	0.992244i	$-0.539669\pi$
$809$	−7.07102	−0.248604	−0.124302	+	0.992244i	$0.539669\pi$
$810$	0	0
$811$	−25.2175	−0.885506	−0.442753	−	0.896644i	$-0.645998\pi$
$811$	−25.2175	−0.885506	−0.442753	+	0.896644i	$0.645998\pi$
$812$	0	0
$813$	−0.450591	−0.0158029
$814$	0	0
$815$	−19.2642	−0.674794
$816$	0	0
$817$	37.2167	1.30205
$818$	0	0
$819$	33.9873	1.18761
$820$	0	0
$821$	17.1538	0.598672	0.299336	−	0.954148i	$-0.403235\pi$
$821$	17.1538	0.598672	0.299336	+	0.954148i	$0.403235\pi$
$822$	0	0
$823$	−19.1448	−0.667345	−0.333673	−	0.942689i	$-0.608288\pi$
$823$	−19.1448	−0.667345	−0.333673	+	0.942689i	$0.608288\pi$
$824$	0	0
$825$	−0.0889752	−0.00309772
$826$	0	0
$827$	12.2442	0.425772	0.212886	−	0.977077i	$-0.431714\pi$
$827$	12.2442	0.425772	0.212886	+	0.977077i	$0.431714\pi$
$828$	0	0
$829$	−22.8247	−0.792734	−0.396367	−	0.918092i	$-0.629729\pi$
$829$	−22.8247	−0.792734	−0.396367	+	0.918092i	$0.629729\pi$
$830$	0	0
$831$	0.431510	0.0149689
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−9.04287	−0.312942
$836$	0	0
$837$	−2.87470	−0.0993642
$838$	0	0
$839$	42.9945	1.48434	0.742168	−	0.670214i	$-0.233798\pi$
$839$	42.9945	1.48434	0.742168	+	0.670214i	$0.233798\pi$
$840$	0	0
$841$	19.9745	0.688777
$842$	0	0
$843$	−1.53497	−0.0528672
$844$	0	0
$845$	5.61955	0.193319
$846$	0	0
$847$	−23.6780	−0.813586
$848$	0	0
$849$	−2.02607	−0.0695347
$850$	0	0
$851$	−30.1163	−1.03237
$852$	0	0
$853$	−6.75339	−0.231232	−0.115616	−	0.993294i	$-0.536884\pi$
$853$	−6.75339	−0.231232	−0.115616	+	0.993294i	$0.536884\pi$
$854$	0	0
$855$	34.5333	1.18102
$856$	0	0
$857$	22.3142	0.762237	0.381119	−	0.924526i	$-0.375539\pi$
$857$	22.3142	0.762237	0.381119	+	0.924526i	$0.375539\pi$
$858$	0	0
$859$	−49.3533	−1.68391	−0.841956	−	0.539546i	$-0.818596\pi$
$859$	−49.3533	−1.68391	−0.841956	+	0.539546i	$0.818596\pi$
$860$	0	0
$861$	2.12225	0.0723262
$862$	0	0
$863$	9.00113	0.306402	0.153201	−	0.988195i	$-0.451042\pi$
$863$	9.00113	0.306402	0.153201	+	0.988195i	$0.451042\pi$
$864$	0	0
$865$	3.28166	0.111580
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.1756	−0.786177
$870$	0	0
$871$	−62.7606	−2.12656
$872$	0	0
$873$	41.4494	1.40285
$874$	0	0
$875$	−33.8768	−1.14524
$876$	0	0
$877$	−55.1621	−1.86269	−0.931346	−	0.364134i	$-0.881365\pi$
$877$	−55.1621	−1.86269	−0.931346	+	0.364134i	$0.881365\pi$
$878$	0	0
$879$	0.416622	0.0140523
$880$	0	0
$881$	−8.00493	−0.269693	−0.134846	−	0.990867i	$-0.543054\pi$
$881$	−8.00493	−0.269693	−0.134846	+	0.990867i	$0.543054\pi$
$882$	0	0
$883$	49.4099	1.66277	0.831387	−	0.555693i	$-0.187547\pi$
$883$	49.4099	1.66277	0.831387	+	0.555693i	$0.187547\pi$
$884$	0	0
$885$	0.377263	0.0126815
$886$	0	0
$887$	3.38687	0.113720	0.0568600	−	0.998382i	$-0.481891\pi$
$887$	3.38687	0.113720	0.0568600	+	0.998382i	$0.481891\pi$
$888$	0	0
$889$	56.8237	1.90581
$890$	0	0
$891$	−14.7871	−0.495386
$892$	0	0
$893$	−24.6903	−0.826228
$894$	0	0
$895$	−45.5464	−1.52245
$896$	0	0
$897$	−1.57097	−0.0524532
$898$	0	0
$899$	44.7185	1.49145
$900$	0	0
$901$	0	0
$902$	0	0
$903$	1.43498	0.0477532
$904$	0	0
$905$	−29.3566	−0.975846
$906$	0	0
$907$	−35.6952	−1.18524	−0.592619	−	0.805483i	$-0.701906\pi$
$907$	−35.6952	−1.18524	−0.592619	+	0.805483i	$0.701906\pi$
$908$	0	0
$909$	−44.7683	−1.48487
$910$	0	0
$911$	−30.1013	−0.997301	−0.498650	−	0.866803i	$-0.666171\pi$
$911$	−30.1013	−0.997301	−0.498650	+	0.866803i	$0.666171\pi$
$912$	0	0
$913$	1.66491	0.0551005
$914$	0	0
$915$	0.282427	0.00933675
$916$	0	0
$917$	42.6223	1.40751
$918$	0	0
$919$	−35.6600	−1.17632	−0.588158	−	0.808746i	$-0.700147\pi$
$919$	−35.6600	−1.17632	−0.588158	+	0.808746i	$0.700147\pi$
$920$	0	0
$921$	1.20991	0.0398679
$922$	0	0
$923$	−33.8828	−1.11527
$924$	0	0
$925$	−4.09237	−0.134556
$926$	0	0
$927$	−48.2921	−1.58612
$928$	0	0
$929$	27.7694	0.911086	0.455543	−	0.890214i	$-0.349445\pi$
$929$	27.7694	0.911086	0.455543	+	0.890214i	$0.349445\pi$
$930$	0	0
$931$	−6.67488	−0.218760
$932$	0	0
$933$	1.36771	0.0447769
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−34.9960	−1.14327	−0.571635	−	0.820508i	$-0.693691\pi$
$937$	−34.9960	−1.14327	−0.571635	+	0.820508i	$0.693691\pi$
$938$	0	0
$939$	0.749032	0.0244437
$940$	0	0
$941$	−20.3155	−0.662267	−0.331134	−	0.943584i	$-0.607431\pi$
$941$	−20.3155	−0.662267	−0.331134	+	0.943584i	$0.607431\pi$
$942$	0	0
$943$	52.1506	1.69826
$944$	0	0
$945$	2.66554	0.0867100
$946$	0	0
$947$	−18.7350	−0.608807	−0.304403	−	0.952543i	$-0.598457\pi$
$947$	−18.7350	−0.608807	−0.304403	+	0.952543i	$0.598457\pi$
$948$	0	0
$949$	9.12584	0.296237
$950$	0	0
$951$	−0.335679	−0.0108852
$952$	0	0
$953$	23.4719	0.760330	0.380165	−	0.924919i	$-0.375867\pi$
$953$	23.4719	0.760330	0.380165	+	0.924919i	$0.375867\pi$
$954$	0	0
$955$	−19.4715	−0.630082
$956$	0	0
$957$	−0.867809	−0.0280523
$958$	0	0
$959$	−7.11494	−0.229753
$960$	0	0
$961$	9.83232	0.317172
$962$	0	0
$963$	−44.1418	−1.42245
$964$	0	0
$965$	15.7910	0.508330
$966$	0	0
$967$	7.08838	0.227947	0.113974	−	0.993484i	$-0.463642\pi$
$967$	7.08838	0.227947	0.113974	+	0.993484i	$0.463642\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.1859	1.48218	0.741088	−	0.671408i	$-0.234310\pi$
$971$	46.1859	1.48218	0.741088	+	0.671408i	$0.234310\pi$
$972$	0	0
$973$	9.39171	0.301084
$974$	0	0
$975$	−0.213472	−0.00683659
$976$	0	0
$977$	21.3334	0.682516	0.341258	−	0.939970i	$-0.389147\pi$
$977$	21.3334	0.682516	0.341258	+	0.939970i	$0.389147\pi$
$978$	0	0
$979$	26.8615	0.858497
$980$	0	0
$981$	34.3531	1.09681
$982$	0	0
$983$	−46.3727	−1.47906	−0.739530	−	0.673124i	$-0.764952\pi$
$983$	−46.3727	−1.47906	−0.739530	+	0.673124i	$0.764952\pi$
$984$	0	0
$985$	18.7189	0.596432
$986$	0	0
$987$	−0.951994	−0.0303023
$988$	0	0
$989$	35.2621	1.12127
$990$	0	0
$991$	21.5269	0.683824	0.341912	−	0.939732i	$-0.388926\pi$
$991$	21.5269	0.683824	0.341912	+	0.939732i	$0.388926\pi$
$992$	0	0
$993$	−0.0386341	−0.00122602
$994$	0	0
$995$	−23.7058	−0.751526
$996$	0	0
$997$	5.98544	0.189561	0.0947804	−	0.995498i	$-0.469785\pi$
$997$	5.98544	0.189561	0.0947804	+	0.995498i	$0.469785\pi$
$998$	0	0
$999$	2.56588	0.0811808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.u.1.3	✓	6
4.3	odd	2		4624.2.a.br.1.4		6
17.4	even	4		2312.2.b.o.577.7		12
17.13	even	4		2312.2.b.o.577.6		12
17.16	even	2		2312.2.a.v.1.4	yes	6
68.67	odd	2		4624.2.a.bs.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.u.1.3	✓	6	1.1	even	1	trivial
2312.2.a.v.1.4	yes	6	17.16	even	2
2312.2.b.o.577.6		12	17.13	even	4
2312.2.b.o.577.7		12	17.4	even	4
4624.2.a.br.1.4		6	4.3	odd	2
4624.2.a.bs.1.3		6	68.67	odd	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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0750494 q^{3} +2.06942 q^{5} +2.86317 q^{7} -2.99437 q^{9} -1.65231 q^{11} -3.96428 q^{13} -0.155308 q^{15} -5.57295 q^{19} -0.214879 q^{21} -5.28027 q^{23} -0.717513 q^{25} +0.449874 q^{27} -6.99818 q^{29} -6.39002 q^{31} +0.124005 q^{33} +5.92509 q^{35} +5.70355 q^{37} +0.297517 q^{39} -9.87650 q^{41} -6.67809 q^{43} -6.19660 q^{45} +4.43037 q^{47} +1.19773 q^{49} +8.27111 q^{53} -3.41932 q^{55} +0.418247 q^{57} -2.42912 q^{59} -1.81849 q^{61} -8.57338 q^{63} -8.20375 q^{65} +15.8315 q^{67} +0.396281 q^{69} +8.54703 q^{71} -2.30202 q^{73} +0.0538489 q^{75} -4.73084 q^{77} +14.0262 q^{79} +8.94934 q^{81} -1.00763 q^{83} +0.525209 q^{87} -16.2569 q^{89} -11.3504 q^{91} +0.479567 q^{93} -11.5328 q^{95} -13.8424 q^{97} +4.94762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} - 3 q^{7} + 6 q^{11} - 6 q^{13} - 6 q^{15} + 6 q^{19} + 6 q^{21} - 9 q^{23} + 6 q^{25} - 12 q^{27} - 27 q^{29} - 9 q^{31} + 3 q^{33} + 21 q^{35} - 15 q^{37} + 12 q^{39} - 21 q^{41} - 24 q^{45}+ \cdots + 42 q^{99}+O(q^{100}) \) 6 * q - 6 * q^5 - 3 * q^7 + 6 * q^11 - 6 * q^13 - 6 * q^15 + 6 * q^19 + 6 * q^21 - 9 * q^23 + 6 * q^25 - 12 * q^27 - 27 * q^29 - 9 * q^31 + 3 * q^33 + 21 * q^35 - 15 * q^37 + 12 * q^39 - 21 * q^41 - 24 * q^45 + 3 * q^47 + 9 * q^49 + 33 * q^53 - 51 * q^55 - 30 * q^57 - 18 * q^61 - 24 * q^63 + 9 * q^65 - 9 * q^67 - 3 * q^69 - 6 * q^71 - 27 * q^73 + 24 * q^75 - 45 * q^77 - 33 * q^79 - 18 * q^81 - 12 * q^83 + 15 * q^87 + 3 * q^89 - 30 * q^91 - 30 * q^93 - 3 * q^95 - 51 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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