LMFDB - Embedded newform 2312.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,2,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$18.4614129473$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q+3.30278 q^{3} -2.30278 q^{5} -2.30278 q^{7} +7.90833 q^{9} +5.60555 q^{11} +0.697224 q^{13} -7.60555 q^{15} -1.30278 q^{19} -7.60555 q^{21} +4.30278 q^{23} +0.302776 q^{25} +16.2111 q^{27} -0.697224 q^{29} -4.21110 q^{31} +18.5139 q^{33} +5.30278 q^{35} +8.60555 q^{37} +2.30278 q^{39} +6.00000 q^{41} -4.21110 q^{43} -18.2111 q^{45} +11.6056 q^{47} -1.69722 q^{49} +3.30278 q^{53} -12.9083 q^{55} -4.30278 q^{57} +3.21110 q^{59} -1.60555 q^{61} -18.2111 q^{63} -1.60555 q^{65} +0.605551 q^{67} +14.2111 q^{69} -12.4222 q^{71} +2.39445 q^{73} +1.00000 q^{75} -12.9083 q^{77} +10.0000 q^{79} +29.8167 q^{81} -8.69722 q^{83} -2.30278 q^{87} +7.21110 q^{89} -1.60555 q^{91} -13.9083 q^{93} +3.00000 q^{95} +16.1194 q^{97} +44.3305 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - q^{5} - q^{7} + 5 q^{9} + 4 q^{11} + 5 q^{13} - 8 q^{15} + q^{19} - 8 q^{21} + 5 q^{23} - 3 q^{25} + 18 q^{27} - 5 q^{29} + 6 q^{31} + 19 q^{33} + 7 q^{35} + 10 q^{37} + q^{39} + 12 q^{41}+ \cdots + 49 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - q^5 - q^7 + 5 * q^9 + 4 * q^11 + 5 * q^13 - 8 * q^15 + q^19 - 8 * q^21 + 5 * q^23 - 3 * q^25 + 18 * q^27 - 5 * q^29 + 6 * q^31 + 19 * q^33 + 7 * q^35 + 10 * q^37 + q^39 + 12 * q^41 + 6 * q^43 - 22 * q^45 + 16 * q^47 - 7 * q^49 + 3 * q^53 - 15 * q^55 - 5 * q^57 - 8 * q^59 + 4 * q^61 - 22 * q^63 + 4 * q^65 - 6 * q^67 + 14 * q^69 + 4 * q^71 + 12 * q^73 + 2 * q^75 - 15 * q^77 + 20 * q^79 + 38 * q^81 - 21 * q^83 - q^87 + 4 * q^91 - 17 * q^93 + 6 * q^95 + 7 * q^97 + 49 * 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
$3$	3.30278	1.90686	0.953429	−	0.301617i	$-0.0975264\pi$
$3$	3.30278	1.90686	0.953429	+	0.301617i	$0.0975264\pi$
$4$	0	0
$5$	−2.30278	−1.02983	−0.514916	−	0.857240i	$-0.672177\pi$
$5$	−2.30278	−1.02983	−0.514916	+	0.857240i	$0.672177\pi$
$6$	0	0
$7$	−2.30278	−0.870367	−0.435184	−	0.900342i	$-0.643317\pi$
$7$	−2.30278	−0.870367	−0.435184	+	0.900342i	$0.643317\pi$
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	5.60555	1.69014	0.845069	−	0.534658i	$-0.179559\pi$
$11$	5.60555	1.69014	0.845069	+	0.534658i	$0.179559\pi$
$12$	0	0
$13$	0.697224	0.193375	0.0966876	−	0.995315i	$-0.469175\pi$
$13$	0.697224	0.193375	0.0966876	+	0.995315i	$0.469175\pi$
$14$	0	0
$15$	−7.60555	−1.96374
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.30278	−0.298877	−0.149439	−	0.988771i	$-0.547747\pi$
$19$	−1.30278	−0.298877	−0.149439	+	0.988771i	$0.547747\pi$
$20$	0	0
$21$	−7.60555	−1.65967
$22$	0	0
$23$	4.30278	0.897191	0.448595	−	0.893735i	$-0.351924\pi$
$23$	4.30278	0.897191	0.448595	+	0.893735i	$0.351924\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	16.2111	3.11983
$28$	0	0
$29$	−0.697224	−0.129471	−0.0647357	−	0.997902i	$-0.520620\pi$
$29$	−0.697224	−0.129471	−0.0647357	+	0.997902i	$0.520620\pi$
$30$	0	0
$31$	−4.21110	−0.756336	−0.378168	−	0.925737i	$-0.623446\pi$
$31$	−4.21110	−0.756336	−0.378168	+	0.925737i	$0.623446\pi$
$32$	0	0
$33$	18.5139	3.22285
$34$	0	0
$35$	5.30278	0.896333
$36$	0	0
$37$	8.60555	1.41474	0.707372	−	0.706842i	$-0.249881\pi$
$37$	8.60555	1.41474	0.707372	+	0.706842i	$0.249881\pi$
$38$	0	0
$39$	2.30278	0.368739
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−4.21110	−0.642187	−0.321094	−	0.947047i	$-0.604050\pi$
$43$	−4.21110	−0.642187	−0.321094	+	0.947047i	$0.604050\pi$
$44$	0	0
$45$	−18.2111	−2.71475
$46$	0	0
$47$	11.6056	1.69284	0.846422	−	0.532513i	$-0.178752\pi$
$47$	11.6056	1.69284	0.846422	+	0.532513i	$0.178752\pi$
$48$	0	0
$49$	−1.69722	−0.242461
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.30278	0.453671	0.226836	−	0.973933i	$-0.427162\pi$
$53$	3.30278	0.453671	0.226836	+	0.973933i	$0.427162\pi$
$54$	0	0
$55$	−12.9083	−1.74056
$56$	0	0
$57$	−4.30278	−0.569917
$58$	0	0
$59$	3.21110	0.418050	0.209025	−	0.977910i	$-0.432971\pi$
$59$	3.21110	0.418050	0.209025	+	0.977910i	$0.432971\pi$
$60$	0	0
$61$	−1.60555	−0.205570	−0.102785	−	0.994704i	$-0.532775\pi$
$61$	−1.60555	−0.205570	−0.102785	+	0.994704i	$0.532775\pi$
$62$	0	0
$63$	−18.2111	−2.29438
$64$	0	0
$65$	−1.60555	−0.199144
$66$	0	0
$67$	0.605551	0.0739799	0.0369899	−	0.999316i	$-0.488223\pi$
$67$	0.605551	0.0739799	0.0369899	+	0.999316i	$0.488223\pi$
$68$	0	0
$69$	14.2111	1.71082
$70$	0	0
$71$	−12.4222	−1.47424	−0.737122	−	0.675759i	$-0.763816\pi$
$71$	−12.4222	−1.47424	−0.737122	+	0.675759i	$0.763816\pi$
$72$	0	0
$73$	2.39445	0.280249	0.140125	−	0.990134i	$-0.455250\pi$
$73$	2.39445	0.280249	0.140125	+	0.990134i	$0.455250\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−12.9083	−1.47104
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	−8.69722	−0.954644	−0.477322	−	0.878728i	$-0.658393\pi$
$83$	−8.69722	−0.954644	−0.477322	+	0.878728i	$0.658393\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.30278	−0.246883
$88$	0	0
$89$	7.21110	0.764375	0.382188	−	0.924085i	$-0.375171\pi$
$89$	7.21110	0.764375	0.382188	+	0.924085i	$0.375171\pi$
$90$	0	0
$91$	−1.60555	−0.168308
$92$	0	0
$93$	−13.9083	−1.44223
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	16.1194	1.63668	0.818340	−	0.574734i	$-0.194895\pi$
$97$	16.1194	1.63668	0.818340	+	0.574734i	$0.194895\pi$
$98$	0	0
$99$	44.3305	4.45539
$100$	0	0
$101$	−6.81665	−0.678282	−0.339141	−	0.940735i	$-0.610136\pi$
$101$	−6.81665	−0.678282	−0.339141	+	0.940735i	$0.610136\pi$
$102$	0	0
$103$	−3.21110	−0.316399	−0.158200	−	0.987407i	$-0.550569\pi$
$103$	−3.21110	−0.316399	−0.158200	+	0.987407i	$0.550569\pi$
$104$	0	0
$105$	17.5139	1.70918
$106$	0	0
$107$	−10.8167	−1.04569	−0.522843	−	0.852429i	$-0.675128\pi$
$107$	−10.8167	−1.04569	−0.522843	+	0.852429i	$0.675128\pi$
$108$	0	0
$109$	15.0000	1.43674	0.718370	−	0.695662i	$-0.244889\pi$
$109$	15.0000	1.43674	0.718370	+	0.695662i	$0.244889\pi$
$110$	0	0
$111$	28.4222	2.69772
$112$	0	0
$113$	−0.486122	−0.0457305	−0.0228652	−	0.999739i	$-0.507279\pi$
$113$	−0.486122	−0.0457305	−0.0228652	+	0.999739i	$0.507279\pi$
$114$	0	0
$115$	−9.90833	−0.923956
$116$	0	0
$117$	5.51388	0.509758
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.4222	1.85656
$122$	0	0
$123$	19.8167	1.78681
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	−22.2111	−1.97092	−0.985458	−	0.169917i	$-0.945650\pi$
$127$	−22.2111	−1.97092	−0.985458	+	0.169917i	$0.945650\pi$
$128$	0	0
$129$	−13.9083	−1.22456
$130$	0	0
$131$	0.605551	0.0529073	0.0264536	−	0.999650i	$-0.491579\pi$
$131$	0.605551	0.0529073	0.0264536	+	0.999650i	$0.491579\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	0	0
$135$	−37.3305	−3.21290
$136$	0	0
$137$	−8.60555	−0.735222	−0.367611	−	0.929980i	$-0.619824\pi$
$137$	−8.60555	−0.735222	−0.367611	+	0.929980i	$0.619824\pi$
$138$	0	0
$139$	8.30278	0.704232	0.352116	−	0.935956i	$-0.385462\pi$
$139$	8.30278	0.704232	0.352116	+	0.935956i	$0.385462\pi$
$140$	0	0
$141$	38.3305	3.22801
$142$	0	0
$143$	3.90833	0.326831
$144$	0	0
$145$	1.60555	0.133334
$146$	0	0
$147$	−5.60555	−0.462338
$148$	0	0
$149$	−17.0278	−1.39497	−0.697484	−	0.716600i	$-0.745697\pi$
$149$	−17.0278	−1.39497	−0.697484	+	0.716600i	$0.745697\pi$
$150$	0	0
$151$	0.394449	0.0320998	0.0160499	−	0.999871i	$-0.494891\pi$
$151$	0.394449	0.0320998	0.0160499	+	0.999871i	$0.494891\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.69722	0.778900
$156$	0	0
$157$	−1.09167	−0.0871250	−0.0435625	−	0.999051i	$-0.513871\pi$
$157$	−1.09167	−0.0871250	−0.0435625	+	0.999051i	$0.513871\pi$
$158$	0	0
$159$	10.9083	0.865087
$160$	0	0
$161$	−9.90833	−0.780886
$162$	0	0
$163$	7.78890	0.610074	0.305037	−	0.952341i	$-0.401331\pi$
$163$	7.78890	0.610074	0.305037	+	0.952341i	$0.401331\pi$
$164$	0	0
$165$	−42.6333	−3.31900
$166$	0	0
$167$	19.3028	1.49369	0.746847	−	0.664996i	$-0.231567\pi$
$167$	19.3028	1.49369	0.746847	+	0.664996i	$0.231567\pi$
$168$	0	0
$169$	−12.5139	−0.962606
$170$	0	0
$171$	−10.3028	−0.787873
$172$	0	0
$173$	−22.8167	−1.73472	−0.867359	−	0.497683i	$-0.834184\pi$
$173$	−22.8167	−1.73472	−0.867359	+	0.497683i	$0.834184\pi$
$174$	0	0
$175$	−0.697224	−0.0527052
$176$	0	0
$177$	10.6056	0.797162
$178$	0	0
$179$	−10.1194	−0.756362	−0.378181	−	0.925732i	$-0.623450\pi$
$179$	−10.1194	−0.756362	−0.378181	+	0.925732i	$0.623450\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	−5.30278	−0.391992
$184$	0	0
$185$	−19.8167	−1.45695
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−37.3305	−2.71540
$190$	0	0
$191$	−1.30278	−0.0942655	−0.0471328	−	0.998889i	$-0.515008\pi$
$191$	−1.30278	−0.0942655	−0.0471328	+	0.998889i	$0.515008\pi$
$192$	0	0
$193$	−26.2111	−1.88672	−0.943358	−	0.331776i	$-0.892352\pi$
$193$	−26.2111	−1.88672	−0.943358	+	0.331776i	$0.892352\pi$
$194$	0	0
$195$	−5.30278	−0.379740
$196$	0	0
$197$	−1.09167	−0.0777785	−0.0388892	−	0.999244i	$-0.512382\pi$
$197$	−1.09167	−0.0777785	−0.0388892	+	0.999244i	$0.512382\pi$
$198$	0	0
$199$	10.2111	0.723846	0.361923	−	0.932208i	$-0.382120\pi$
$199$	10.2111	0.723846	0.361923	+	0.932208i	$0.382120\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	1.60555	0.112688
$204$	0	0
$205$	−13.8167	−0.964997
$206$	0	0
$207$	34.0278	2.36509
$208$	0	0
$209$	−7.30278	−0.505144
$210$	0	0
$211$	7.69722	0.529899	0.264949	−	0.964262i	$-0.414645\pi$
$211$	7.69722	0.529899	0.264949	+	0.964262i	$0.414645\pi$
$212$	0	0
$213$	−41.0278	−2.81118
$214$	0	0
$215$	9.69722	0.661345
$216$	0	0
$217$	9.69722	0.658290
$218$	0	0
$219$	7.90833	0.534395
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.9083	−0.864406	−0.432203	−	0.901776i	$-0.642264\pi$
$223$	−12.9083	−0.864406	−0.432203	+	0.901776i	$0.642264\pi$
$224$	0	0
$225$	2.39445	0.159630
$226$	0	0
$227$	−14.6333	−0.971247	−0.485623	−	0.874168i	$-0.661407\pi$
$227$	−14.6333	−0.971247	−0.485623	+	0.874168i	$0.661407\pi$
$228$	0	0
$229$	−18.4222	−1.21737	−0.608687	−	0.793411i	$-0.708303\pi$
$229$	−18.4222	−1.21737	−0.608687	+	0.793411i	$0.708303\pi$
$230$	0	0
$231$	−42.6333	−2.80507
$232$	0	0
$233$	−11.8167	−0.774135	−0.387067	−	0.922051i	$-0.626512\pi$
$233$	−11.8167	−0.774135	−0.387067	+	0.922051i	$0.626512\pi$
$234$	0	0
$235$	−26.7250	−1.74335
$236$	0	0
$237$	33.0278	2.14538
$238$	0	0
$239$	10.3944	0.672361	0.336180	−	0.941798i	$-0.390865\pi$
$239$	10.3944	0.672361	0.336180	+	0.941798i	$0.390865\pi$
$240$	0	0
$241$	−26.1194	−1.68250	−0.841250	−	0.540646i	$-0.818180\pi$
$241$	−26.1194	−1.68250	−0.841250	+	0.540646i	$0.818180\pi$
$242$	0	0
$243$	49.8444	3.19752
$244$	0	0
$245$	3.90833	0.249694
$246$	0	0
$247$	−0.908327	−0.0577955
$248$	0	0
$249$	−28.7250	−1.82037
$250$	0	0
$251$	−3.48612	−0.220042	−0.110021	−	0.993929i	$-0.535092\pi$
$251$	−3.48612	−0.220042	−0.110021	+	0.993929i	$0.535092\pi$
$252$	0	0
$253$	24.1194	1.51638
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.1194	−1.13026	−0.565129	−	0.825002i	$-0.691174\pi$
$257$	−18.1194	−1.13026	−0.565129	+	0.825002i	$0.691174\pi$
$258$	0	0
$259$	−19.8167	−1.23135
$260$	0	0
$261$	−5.51388	−0.341300
$262$	0	0
$263$	−10.6972	−0.659619	−0.329810	−	0.944047i	$-0.606985\pi$
$263$	−10.6972	−0.659619	−0.329810	+	0.944047i	$0.606985\pi$
$264$	0	0
$265$	−7.60555	−0.467205
$266$	0	0
$267$	23.8167	1.45756
$268$	0	0
$269$	−1.48612	−0.0906104	−0.0453052	−	0.998973i	$-0.514426\pi$
$269$	−1.48612	−0.0906104	−0.0453052	+	0.998973i	$0.514426\pi$
$270$	0	0
$271$	15.3305	0.931263	0.465632	−	0.884979i	$-0.345827\pi$
$271$	15.3305	0.931263	0.465632	+	0.884979i	$0.345827\pi$
$272$	0	0
$273$	−5.30278	−0.320939
$274$	0	0
$275$	1.69722	0.102346
$276$	0	0
$277$	−14.4222	−0.866546	−0.433273	−	0.901263i	$-0.642641\pi$
$277$	−14.4222	−0.866546	−0.433273	+	0.901263i	$0.642641\pi$
$278$	0	0
$279$	−33.3028	−1.99379
$280$	0	0
$281$	10.1194	0.603675	0.301837	−	0.953359i	$-0.402400\pi$
$281$	10.1194	0.603675	0.301837	+	0.953359i	$0.402400\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	9.90833	0.586919
$286$	0	0
$287$	−13.8167	−0.815571
$288$	0	0
$289$	0	0
$290$	0	0
$291$	53.2389	3.12092
$292$	0	0
$293$	−19.0278	−1.11161	−0.555807	−	0.831312i	$-0.687591\pi$
$293$	−19.0278	−1.11161	−0.555807	+	0.831312i	$0.687591\pi$
$294$	0	0
$295$	−7.39445	−0.430521
$296$	0	0
$297$	90.8722	5.27294
$298$	0	0
$299$	3.00000	0.173494
$300$	0	0
$301$	9.69722	0.558939
$302$	0	0
$303$	−22.5139	−1.29339
$304$	0	0
$305$	3.69722	0.211702
$306$	0	0
$307$	14.7250	0.840399	0.420200	−	0.907432i	$-0.361960\pi$
$307$	14.7250	0.840399	0.420200	+	0.907432i	$0.361960\pi$
$308$	0	0
$309$	−10.6056	−0.603329
$310$	0	0
$311$	−27.9083	−1.58254	−0.791268	−	0.611469i	$-0.790579\pi$
$311$	−27.9083	−1.58254	−0.791268	+	0.611469i	$0.790579\pi$
$312$	0	0
$313$	15.3944	0.870146	0.435073	−	0.900395i	$-0.356723\pi$
$313$	15.3944	0.870146	0.435073	+	0.900395i	$0.356723\pi$
$314$	0	0
$315$	41.9361	2.36283
$316$	0	0
$317$	24.0278	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$317$	24.0278	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$318$	0	0
$319$	−3.90833	−0.218824
$320$	0	0
$321$	−35.7250	−1.99397
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.211103	0.0117099
$326$	0	0
$327$	49.5416	2.73966
$328$	0	0
$329$	−26.7250	−1.47340
$330$	0	0
$331$	13.0917	0.719583	0.359792	−	0.933033i	$-0.382848\pi$
$331$	13.0917	0.719583	0.359792	+	0.933033i	$0.382848\pi$
$332$	0	0
$333$	68.0555	3.72942
$334$	0	0
$335$	−1.39445	−0.0761869
$336$	0	0
$337$	23.5416	1.28239	0.641197	−	0.767376i	$-0.278438\pi$
$337$	23.5416	1.28239	0.641197	+	0.767376i	$0.278438\pi$
$338$	0	0
$339$	−1.60555	−0.0872016
$340$	0	0
$341$	−23.6056	−1.27831
$342$	0	0
$343$	20.0278	1.08140
$344$	0	0
$345$	−32.7250	−1.76185
$346$	0	0
$347$	−9.90833	−0.531907	−0.265953	−	0.963986i	$-0.585687\pi$
$347$	−9.90833	−0.531907	−0.265953	+	0.963986i	$0.585687\pi$
$348$	0	0
$349$	−9.60555	−0.514173	−0.257087	−	0.966388i	$-0.582763\pi$
$349$	−9.60555	−0.514173	−0.257087	+	0.966388i	$0.582763\pi$
$350$	0	0
$351$	11.3028	0.603298
$352$	0	0
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	28.6056	1.51823
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.72498	0.460487	0.230243	−	0.973133i	$-0.426048\pi$
$359$	8.72498	0.460487	0.230243	+	0.973133i	$0.426048\pi$
$360$	0	0
$361$	−17.3028	−0.910672
$362$	0	0
$363$	67.4500	3.54020
$364$	0	0
$365$	−5.51388	−0.288610
$366$	0	0
$367$	24.2389	1.26526	0.632629	−	0.774455i	$-0.281976\pi$
$367$	24.2389	1.26526	0.632629	+	0.774455i	$0.281976\pi$
$368$	0	0
$369$	47.4500	2.47015
$370$	0	0
$371$	−7.60555	−0.394861
$372$	0	0
$373$	−23.4222	−1.21276	−0.606378	−	0.795177i	$-0.707378\pi$
$373$	−23.4222	−1.21276	−0.606378	+	0.795177i	$0.707378\pi$
$374$	0	0
$375$	35.7250	1.84483
$376$	0	0
$377$	−0.486122	−0.0250365
$378$	0	0
$379$	−25.4222	−1.30585	−0.652925	−	0.757422i	$-0.726459\pi$
$379$	−25.4222	−1.30585	−0.652925	+	0.757422i	$0.726459\pi$
$380$	0	0
$381$	−73.3583	−3.75826
$382$	0	0
$383$	0.697224	0.0356265	0.0178133	−	0.999841i	$-0.494330\pi$
$383$	0.697224	0.0356265	0.0178133	+	0.999841i	$0.494330\pi$
$384$	0	0
$385$	29.7250	1.51493
$386$	0	0
$387$	−33.3028	−1.69288
$388$	0	0
$389$	−9.42221	−0.477725	−0.238862	−	0.971053i	$-0.576774\pi$
$389$	−9.42221	−0.477725	−0.238862	+	0.971053i	$0.576774\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	2.00000	0.100887
$394$	0	0
$395$	−23.0278	−1.15865
$396$	0	0
$397$	32.4500	1.62862	0.814308	−	0.580432i	$-0.197116\pi$
$397$	32.4500	1.62862	0.814308	+	0.580432i	$0.197116\pi$
$398$	0	0
$399$	9.90833	0.496037
$400$	0	0
$401$	−25.5139	−1.27410	−0.637051	−	0.770822i	$-0.719846\pi$
$401$	−25.5139	−1.27410	−0.637051	+	0.770822i	$0.719846\pi$
$402$	0	0
$403$	−2.93608	−0.146257
$404$	0	0
$405$	−68.6611	−3.41180
$406$	0	0
$407$	48.2389	2.39111
$408$	0	0
$409$	22.0278	1.08920	0.544601	−	0.838695i	$-0.316681\pi$
$409$	22.0278	1.08920	0.544601	+	0.838695i	$0.316681\pi$
$410$	0	0
$411$	−28.4222	−1.40196
$412$	0	0
$413$	−7.39445	−0.363857
$414$	0	0
$415$	20.0278	0.983124
$416$	0	0
$417$	27.4222	1.34287
$418$	0	0
$419$	−0.394449	−0.0192701	−0.00963504	−	0.999954i	$-0.503067\pi$
$419$	−0.394449	−0.0192701	−0.00963504	+	0.999954i	$0.503067\pi$
$420$	0	0
$421$	23.3305	1.13706	0.568530	−	0.822662i	$-0.307512\pi$
$421$	23.3305	1.13706	0.568530	+	0.822662i	$0.307512\pi$
$422$	0	0
$423$	91.7805	4.46252
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.69722	0.178921
$428$	0	0
$429$	12.9083	0.623220
$430$	0	0
$431$	7.51388	0.361931	0.180965	−	0.983489i	$-0.442078\pi$
$431$	7.51388	0.361931	0.180965	+	0.983489i	$0.442078\pi$
$432$	0	0
$433$	−14.4222	−0.693087	−0.346543	−	0.938034i	$-0.612645\pi$
$433$	−14.4222	−0.693087	−0.346543	+	0.938034i	$0.612645\pi$
$434$	0	0
$435$	5.30278	0.254249
$436$	0	0
$437$	−5.60555	−0.268150
$438$	0	0
$439$	−12.1833	−0.581479	−0.290740	−	0.956802i	$-0.593901\pi$
$439$	−12.1833	−0.581479	−0.290740	+	0.956802i	$0.593901\pi$
$440$	0	0
$441$	−13.4222	−0.639153
$442$	0	0
$443$	−9.78890	−0.465085	−0.232542	−	0.972586i	$-0.574704\pi$
$443$	−9.78890	−0.465085	−0.232542	+	0.972586i	$0.574704\pi$
$444$	0	0
$445$	−16.6056	−0.787179
$446$	0	0
$447$	−56.2389	−2.66001
$448$	0	0
$449$	−23.4861	−1.10838	−0.554189	−	0.832391i	$-0.686972\pi$
$449$	−23.4861	−1.10838	−0.554189	+	0.832391i	$0.686972\pi$
$450$	0	0
$451$	33.6333	1.58373
$452$	0	0
$453$	1.30278	0.0612097
$454$	0	0
$455$	3.69722	0.173329
$456$	0	0
$457$	−2.72498	−0.127469	−0.0637346	−	0.997967i	$-0.520301\pi$
$457$	−2.72498	−0.127469	−0.0637346	+	0.997967i	$0.520301\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−27.8167	−1.29275	−0.646375	−	0.763020i	$-0.723716\pi$
$463$	−27.8167	−1.29275	−0.646375	+	0.763020i	$0.723716\pi$
$464$	0	0
$465$	32.0278	1.48525
$466$	0	0
$467$	−4.11943	−0.190624	−0.0953122	−	0.995447i	$-0.530385\pi$
$467$	−4.11943	−0.190624	−0.0953122	+	0.995447i	$0.530385\pi$
$468$	0	0
$469$	−1.39445	−0.0643897
$470$	0	0
$471$	−3.60555	−0.166135
$472$	0	0
$473$	−23.6056	−1.08538
$474$	0	0
$475$	−0.394449	−0.0180985
$476$	0	0
$477$	26.1194	1.19593
$478$	0	0
$479$	24.8167	1.13390	0.566951	−	0.823752i	$-0.308123\pi$
$479$	24.8167	1.13390	0.566951	+	0.823752i	$0.308123\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−32.7250	−1.48904
$484$	0	0
$485$	−37.1194	−1.68551
$486$	0	0
$487$	4.88057	0.221160	0.110580	−	0.993867i	$-0.464729\pi$
$487$	4.88057	0.221160	0.110580	+	0.993867i	$0.464729\pi$
$488$	0	0
$489$	25.7250	1.16332
$490$	0	0
$491$	−26.3028	−1.18703	−0.593514	−	0.804824i	$-0.702260\pi$
$491$	−26.3028	−1.18703	−0.593514	+	0.804824i	$0.702260\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−102.083	−4.58830
$496$	0	0
$497$	28.6056	1.28313
$498$	0	0
$499$	−16.9083	−0.756921	−0.378460	−	0.925618i	$-0.623546\pi$
$499$	−16.9083	−0.756921	−0.378460	+	0.925618i	$0.623546\pi$
$500$	0	0
$501$	63.7527	2.84826
$502$	0	0
$503$	23.2389	1.03617	0.518085	−	0.855329i	$-0.326645\pi$
$503$	23.2389	1.03617	0.518085	+	0.855329i	$0.326645\pi$
$504$	0	0
$505$	15.6972	0.698517
$506$	0	0
$507$	−41.3305	−1.83555
$508$	0	0
$509$	6.81665	0.302143	0.151071	−	0.988523i	$-0.451728\pi$
$509$	6.81665	0.302143	0.151071	+	0.988523i	$0.451728\pi$
$510$	0	0
$511$	−5.51388	−0.243920
$512$	0	0
$513$	−21.1194	−0.932446
$514$	0	0
$515$	7.39445	0.325838
$516$	0	0
$517$	65.0555	2.86114
$518$	0	0
$519$	−75.3583	−3.30786
$520$	0	0
$521$	−2.42221	−0.106119	−0.0530594	−	0.998591i	$-0.516897\pi$
$521$	−2.42221	−0.106119	−0.0530594	+	0.998591i	$0.516897\pi$
$522$	0	0
$523$	−26.3944	−1.15415	−0.577074	−	0.816692i	$-0.695806\pi$
$523$	−26.3944	−1.15415	−0.577074	+	0.816692i	$0.695806\pi$
$524$	0	0
$525$	−2.30278	−0.100501
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−4.48612	−0.195049
$530$	0	0
$531$	25.3944	1.10203
$532$	0	0
$533$	4.18335	0.181201
$534$	0	0
$535$	24.9083	1.07688
$536$	0	0
$537$	−33.4222	−1.44227
$538$	0	0
$539$	−9.51388	−0.409792
$540$	0	0
$541$	−19.9083	−0.855926	−0.427963	−	0.903796i	$-0.640769\pi$
$541$	−19.9083	−0.855926	−0.427963	+	0.903796i	$0.640769\pi$
$542$	0	0
$543$	−23.1194	−0.992150
$544$	0	0
$545$	−34.5416	−1.47960
$546$	0	0
$547$	−43.4222	−1.85660	−0.928300	−	0.371833i	$-0.878729\pi$
$547$	−43.4222	−1.85660	−0.928300	+	0.371833i	$0.878729\pi$
$548$	0	0
$549$	−12.6972	−0.541904
$550$	0	0
$551$	0.908327	0.0386960
$552$	0	0
$553$	−23.0278	−0.979240
$554$	0	0
$555$	−65.4500	−2.77820
$556$	0	0
$557$	−22.7889	−0.965597	−0.482798	−	0.875732i	$-0.660380\pi$
$557$	−22.7889	−0.965597	−0.482798	+	0.875732i	$0.660380\pi$
$558$	0	0
$559$	−2.93608	−0.124183
$560$	0	0
$561$	0	0
$562$	0	0
$563$	44.6056	1.87990	0.939950	−	0.341312i	$-0.110871\pi$
$563$	44.6056	1.87990	0.939950	+	0.341312i	$0.110871\pi$
$564$	0	0
$565$	1.11943	0.0470948
$566$	0	0
$567$	−68.6611	−2.88349
$568$	0	0
$569$	−2.81665	−0.118080	−0.0590401	−	0.998256i	$-0.518804\pi$
$569$	−2.81665	−0.118080	−0.0590401	+	0.998256i	$0.518804\pi$
$570$	0	0
$571$	−4.93608	−0.206569	−0.103284	−	0.994652i	$-0.532935\pi$
$571$	−4.93608	−0.206569	−0.103284	+	0.994652i	$0.532935\pi$
$572$	0	0
$573$	−4.30278	−0.179751
$574$	0	0
$575$	1.30278	0.0543295
$576$	0	0
$577$	4.02776	0.167678	0.0838388	−	0.996479i	$-0.473282\pi$
$577$	4.02776	0.167678	0.0838388	+	0.996479i	$0.473282\pi$
$578$	0	0
$579$	−86.5694	−3.59770
$580$	0	0
$581$	20.0278	0.830891
$582$	0	0
$583$	18.5139	0.766766
$584$	0	0
$585$	−12.6972	−0.524966
$586$	0	0
$587$	3.90833	0.161314	0.0806570	−	0.996742i	$-0.474298\pi$
$587$	3.90833	0.161314	0.0806570	+	0.996742i	$0.474298\pi$
$588$	0	0
$589$	5.48612	0.226052
$590$	0	0
$591$	−3.60555	−0.148313
$592$	0	0
$593$	16.2111	0.665710	0.332855	−	0.942978i	$-0.391988\pi$
$593$	16.2111	0.665710	0.332855	+	0.942978i	$0.391988\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	33.7250	1.38027
$598$	0	0
$599$	−9.60555	−0.392472	−0.196236	−	0.980557i	$-0.562872\pi$
$599$	−9.60555	−0.392472	−0.196236	+	0.980557i	$0.562872\pi$
$600$	0	0
$601$	−41.5416	−1.69452	−0.847259	−	0.531180i	$-0.821749\pi$
$601$	−41.5416	−1.69452	−0.847259	+	0.531180i	$0.821749\pi$
$602$	0	0
$603$	4.78890	0.195019
$604$	0	0
$605$	−47.0278	−1.91195
$606$	0	0
$607$	44.0278	1.78703	0.893516	−	0.449032i	$-0.148231\pi$
$607$	44.0278	1.78703	0.893516	+	0.449032i	$0.148231\pi$
$608$	0	0
$609$	5.30278	0.214879
$610$	0	0
$611$	8.09167	0.327354
$612$	0	0
$613$	19.0278	0.768524	0.384262	−	0.923224i	$-0.374456\pi$
$613$	19.0278	0.768524	0.384262	+	0.923224i	$0.374456\pi$
$614$	0	0
$615$	−45.6333	−1.84011
$616$	0	0
$617$	29.8444	1.20149	0.600745	−	0.799440i	$-0.294871\pi$
$617$	29.8444	1.20149	0.600745	+	0.799440i	$0.294871\pi$
$618$	0	0
$619$	38.8167	1.56017	0.780087	−	0.625672i	$-0.215175\pi$
$619$	38.8167	1.56017	0.780087	+	0.625672i	$0.215175\pi$
$620$	0	0
$621$	69.7527	2.79908
$622$	0	0
$623$	−16.6056	−0.665287
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	−24.1194	−0.963237
$628$	0	0
$629$	0	0
$630$	0	0
$631$	24.1472	0.961284	0.480642	−	0.876917i	$-0.340404\pi$
$631$	24.1472	0.961284	0.480642	+	0.876917i	$0.340404\pi$
$632$	0	0
$633$	25.4222	1.01044
$634$	0	0
$635$	51.1472	2.02971
$636$	0	0
$637$	−1.18335	−0.0468859
$638$	0	0
$639$	−98.2389	−3.88627
$640$	0	0
$641$	45.7250	1.80603	0.903014	−	0.429611i	$-0.141349\pi$
$641$	45.7250	1.80603	0.903014	+	0.429611i	$0.141349\pi$
$642$	0	0
$643$	−6.18335	−0.243847	−0.121924	−	0.992539i	$-0.538906\pi$
$643$	−6.18335	−0.243847	−0.121924	+	0.992539i	$0.538906\pi$
$644$	0	0
$645$	32.0278	1.26109
$646$	0	0
$647$	20.0278	0.787372	0.393686	−	0.919245i	$-0.371200\pi$
$647$	20.0278	0.787372	0.393686	+	0.919245i	$0.371200\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	32.0278	1.25527
$652$	0	0
$653$	−6.39445	−0.250234	−0.125117	−	0.992142i	$-0.539931\pi$
$653$	−6.39445	−0.250234	−0.125117	+	0.992142i	$0.539931\pi$
$654$	0	0
$655$	−1.39445	−0.0544856
$656$	0	0
$657$	18.9361	0.738767
$658$	0	0
$659$	13.0000	0.506408	0.253204	−	0.967413i	$-0.418516\pi$
$659$	13.0000	0.506408	0.253204	+	0.967413i	$0.418516\pi$
$660$	0	0
$661$	34.9083	1.35778	0.678888	−	0.734242i	$-0.262462\pi$
$661$	34.9083	1.35778	0.678888	+	0.734242i	$0.262462\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−6.90833	−0.267893
$666$	0	0
$667$	−3.00000	−0.116160
$668$	0	0
$669$	−42.6333	−1.64830
$670$	0	0
$671$	−9.00000	−0.347441
$672$	0	0
$673$	30.7889	1.18682	0.593412	−	0.804899i	$-0.297780\pi$
$673$	30.7889	1.18682	0.593412	+	0.804899i	$0.297780\pi$
$674$	0	0
$675$	4.90833	0.188922
$676$	0	0
$677$	38.9361	1.49644	0.748218	−	0.663453i	$-0.230910\pi$
$677$	38.9361	1.49644	0.748218	+	0.663453i	$0.230910\pi$
$678$	0	0
$679$	−37.1194	−1.42451
$680$	0	0
$681$	−48.3305	−1.85203
$682$	0	0
$683$	2.33053	0.0891753	0.0445877	−	0.999005i	$-0.485803\pi$
$683$	2.33053	0.0891753	0.0445877	+	0.999005i	$0.485803\pi$
$684$	0	0
$685$	19.8167	0.757155
$686$	0	0
$687$	−60.8444	−2.32136
$688$	0	0
$689$	2.30278	0.0877288
$690$	0	0
$691$	51.2666	1.95027	0.975137	−	0.221603i	$-0.0711289\pi$
$691$	51.2666	1.95027	0.975137	+	0.221603i	$0.0711289\pi$
$692$	0	0
$693$	−102.083	−3.87782
$694$	0	0
$695$	−19.1194	−0.725241
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−39.0278	−1.47617
$700$	0	0
$701$	33.0000	1.24639	0.623196	−	0.782065i	$-0.285834\pi$
$701$	33.0000	1.24639	0.623196	+	0.782065i	$0.285834\pi$
$702$	0	0
$703$	−11.2111	−0.422835
$704$	0	0
$705$	−88.2666	−3.32431
$706$	0	0
$707$	15.6972	0.590355
$708$	0	0
$709$	−32.3305	−1.21420	−0.607099	−	0.794626i	$-0.707667\pi$
$709$	−32.3305	−1.21420	−0.607099	+	0.794626i	$0.707667\pi$
$710$	0	0
$711$	79.0833	2.96585
$712$	0	0
$713$	−18.1194	−0.678578
$714$	0	0
$715$	−9.00000	−0.336581
$716$	0	0
$717$	34.3305	1.28210
$718$	0	0
$719$	9.84441	0.367135	0.183567	−	0.983007i	$-0.441235\pi$
$719$	9.84441	0.367135	0.183567	+	0.983007i	$0.441235\pi$
$720$	0	0
$721$	7.39445	0.275384
$722$	0	0
$723$	−86.2666	−3.20829
$724$	0	0
$725$	−0.211103	−0.00784015
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−38.1472	−1.40900	−0.704499	−	0.709705i	$-0.748828\pi$
$733$	−38.1472	−1.40900	−0.704499	+	0.709705i	$0.748828\pi$
$734$	0	0
$735$	12.9083	0.476131
$736$	0	0
$737$	3.39445	0.125036
$738$	0	0
$739$	43.2389	1.59057	0.795284	−	0.606238i	$-0.207322\pi$
$739$	43.2389	1.59057	0.795284	+	0.606238i	$0.207322\pi$
$740$	0	0
$741$	−3.00000	−0.110208
$742$	0	0
$743$	−43.0000	−1.57752	−0.788759	−	0.614703i	$-0.789276\pi$
$743$	−43.0000	−1.57752	−0.788759	+	0.614703i	$0.789276\pi$
$744$	0	0
$745$	39.2111	1.43658
$746$	0	0
$747$	−68.7805	−2.51655
$748$	0	0
$749$	24.9083	0.910130
$750$	0	0
$751$	0.394449	0.0143936	0.00719682	−	0.999974i	$-0.497709\pi$
$751$	0.394449	0.0143936	0.00719682	+	0.999974i	$0.497709\pi$
$752$	0	0
$753$	−11.5139	−0.419589
$754$	0	0
$755$	−0.908327	−0.0330574
$756$	0	0
$757$	29.0000	1.05402	0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000	1.05402	0.527011	+	0.849858i	$0.323312\pi$
$758$	0	0
$759$	79.6611	2.89151
$760$	0	0
$761$	−32.2111	−1.16765	−0.583826	−	0.811879i	$-0.698445\pi$
$761$	−32.2111	−1.16765	−0.583826	+	0.811879i	$0.698445\pi$
$762$	0	0
$763$	−34.5416	−1.25049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.23886	0.0808405
$768$	0	0
$769$	1.90833	0.0688160	0.0344080	−	0.999408i	$-0.489045\pi$
$769$	1.90833	0.0688160	0.0344080	+	0.999408i	$0.489045\pi$
$770$	0	0
$771$	−59.8444	−2.15524
$772$	0	0
$773$	−25.0555	−0.901184	−0.450592	−	0.892730i	$-0.648787\pi$
$773$	−25.0555	−0.901184	−0.450592	+	0.892730i	$0.648787\pi$
$774$	0	0
$775$	−1.27502	−0.0458000
$776$	0	0
$777$	−65.4500	−2.34800
$778$	0	0
$779$	−7.81665	−0.280061
$780$	0	0
$781$	−69.6333	−2.49168
$782$	0	0
$783$	−11.3028	−0.403928
$784$	0	0
$785$	2.51388	0.0897242
$786$	0	0
$787$	39.8167	1.41931	0.709655	−	0.704549i	$-0.248851\pi$
$787$	39.8167	1.41931	0.709655	+	0.704549i	$0.248851\pi$
$788$	0	0
$789$	−35.3305	−1.25780
$790$	0	0
$791$	1.11943	0.0398023
$792$	0	0
$793$	−1.11943	−0.0397521
$794$	0	0
$795$	−25.1194	−0.890894
$796$	0	0
$797$	35.0555	1.24173	0.620865	−	0.783918i	$-0.286782\pi$
$797$	35.0555	1.24173	0.620865	+	0.783918i	$0.286782\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	57.0278	2.01498
$802$	0	0
$803$	13.4222	0.473659
$804$	0	0
$805$	22.8167	0.804181
$806$	0	0
$807$	−4.90833	−0.172781
$808$	0	0
$809$	−38.6056	−1.35730	−0.678649	−	0.734462i	$-0.737434\pi$
$809$	−38.6056	−1.35730	−0.678649	+	0.734462i	$0.737434\pi$
$810$	0	0
$811$	36.6056	1.28539	0.642697	−	0.766120i	$-0.277815\pi$
$811$	36.6056	1.28539	0.642697	+	0.766120i	$0.277815\pi$
$812$	0	0
$813$	50.6333	1.77579
$814$	0	0
$815$	−17.9361	−0.628274
$816$	0	0
$817$	5.48612	0.191935
$818$	0	0
$819$	−12.6972	−0.443677
$820$	0	0
$821$	−49.6611	−1.73318	−0.866592	−	0.499018i	$-0.833694\pi$
$821$	−49.6611	−1.73318	−0.866592	+	0.499018i	$0.833694\pi$
$822$	0	0
$823$	−25.8444	−0.900880	−0.450440	−	0.892807i	$-0.648733\pi$
$823$	−25.8444	−0.900880	−0.450440	+	0.892807i	$0.648733\pi$
$824$	0	0
$825$	5.60555	0.195160
$826$	0	0
$827$	−31.7889	−1.10541	−0.552704	−	0.833378i	$-0.686404\pi$
$827$	−31.7889	−1.10541	−0.552704	+	0.833378i	$0.686404\pi$
$828$	0	0
$829$	13.3944	0.465208	0.232604	−	0.972571i	$-0.425275\pi$
$829$	13.3944	0.465208	0.232604	+	0.972571i	$0.425275\pi$
$830$	0	0
$831$	−47.6333	−1.65238
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−44.4500	−1.53825
$836$	0	0
$837$	−68.2666	−2.35964
$838$	0	0
$839$	−4.18335	−0.144425	−0.0722126	−	0.997389i	$-0.523006\pi$
$839$	−4.18335	−0.144425	−0.0722126	+	0.997389i	$0.523006\pi$
$840$	0	0
$841$	−28.5139	−0.983237
$842$	0	0
$843$	33.4222	1.15112
$844$	0	0
$845$	28.8167	0.991323
$846$	0	0
$847$	−47.0278	−1.61589
$848$	0	0
$849$	13.2111	0.453404
$850$	0	0
$851$	37.0278	1.26930
$852$	0	0
$853$	7.36669	0.252231	0.126115	−	0.992016i	$-0.459749\pi$
$853$	7.36669	0.252231	0.126115	+	0.992016i	$0.459749\pi$
$854$	0	0
$855$	23.7250	0.811377
$856$	0	0
$857$	−26.8167	−0.916039	−0.458020	−	0.888942i	$-0.651441\pi$
$857$	−26.8167	−0.916039	−0.458020	+	0.888942i	$0.651441\pi$
$858$	0	0
$859$	23.0278	0.785697	0.392848	−	0.919603i	$-0.371490\pi$
$859$	23.0278	0.785697	0.392848	+	0.919603i	$0.371490\pi$
$860$	0	0
$861$	−45.6333	−1.55518
$862$	0	0
$863$	9.27502	0.315725	0.157863	−	0.987461i	$-0.449540\pi$
$863$	9.27502	0.315725	0.157863	+	0.987461i	$0.449540\pi$
$864$	0	0
$865$	52.5416	1.78647
$866$	0	0
$867$	0	0
$868$	0	0
$869$	56.0555	1.90155
$870$	0	0
$871$	0.422205	0.0143059
$872$	0	0
$873$	127.478	4.31447
$874$	0	0
$875$	−24.9083	−0.842055
$876$	0	0
$877$	−14.5416	−0.491036	−0.245518	−	0.969392i	$-0.578958\pi$
$877$	−14.5416	−0.491036	−0.245518	+	0.969392i	$0.578958\pi$
$878$	0	0
$879$	−62.8444	−2.11969
$880$	0	0
$881$	−41.7250	−1.40575	−0.702875	−	0.711313i	$-0.748101\pi$
$881$	−41.7250	−1.40575	−0.702875	+	0.711313i	$0.748101\pi$
$882$	0	0
$883$	44.1472	1.48567	0.742836	−	0.669474i	$-0.233480\pi$
$883$	44.1472	1.48567	0.742836	+	0.669474i	$0.233480\pi$
$884$	0	0
$885$	−24.4222	−0.820943
$886$	0	0
$887$	−16.5416	−0.555414	−0.277707	−	0.960666i	$-0.589574\pi$
$887$	−16.5416	−0.555414	−0.277707	+	0.960666i	$0.589574\pi$
$888$	0	0
$889$	51.1472	1.71542
$890$	0	0
$891$	167.139	5.59936
$892$	0	0
$893$	−15.1194	−0.505952
$894$	0	0
$895$	23.3028	0.778926
$896$	0	0
$897$	9.90833	0.330829
$898$	0	0
$899$	2.93608	0.0979239
$900$	0	0
$901$	0	0
$902$	0	0
$903$	32.0278	1.06582
$904$	0	0
$905$	16.1194	0.535828
$906$	0	0
$907$	−23.3028	−0.773756	−0.386878	−	0.922131i	$-0.626447\pi$
$907$	−23.3028	−0.773756	−0.386878	+	0.922131i	$0.626447\pi$
$908$	0	0
$909$	−53.9083	−1.78803
$910$	0	0
$911$	−21.3305	−0.706712	−0.353356	−	0.935489i	$-0.614960\pi$
$911$	−21.3305	−0.706712	−0.353356	+	0.935489i	$0.614960\pi$
$912$	0	0
$913$	−48.7527	−1.61348
$914$	0	0
$915$	12.2111	0.403687
$916$	0	0
$917$	−1.39445	−0.0460488
$918$	0	0
$919$	26.0917	0.860685	0.430342	−	0.902666i	$-0.358393\pi$
$919$	26.0917	0.860685	0.430342	+	0.902666i	$0.358393\pi$
$920$	0	0
$921$	48.6333	1.60252
$922$	0	0
$923$	−8.66106	−0.285082
$924$	0	0
$925$	2.60555	0.0856700
$926$	0	0
$927$	−25.3944	−0.834063
$928$	0	0
$929$	−44.5694	−1.46227	−0.731137	−	0.682231i	$-0.761010\pi$
$929$	−44.5694	−1.46227	−0.731137	+	0.682231i	$0.761010\pi$
$930$	0	0
$931$	2.21110	0.0724660
$932$	0	0
$933$	−92.1749	−3.01767
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.3305	−0.566164	−0.283082	−	0.959096i	$-0.591357\pi$
$937$	−17.3305	−0.566164	−0.283082	+	0.959096i	$0.591357\pi$
$938$	0	0
$939$	50.8444	1.65924
$940$	0	0
$941$	−4.90833	−0.160007	−0.0800034	−	0.996795i	$-0.525493\pi$
$941$	−4.90833	−0.160007	−0.0800034	+	0.996795i	$0.525493\pi$
$942$	0	0
$943$	25.8167	0.840706
$944$	0	0
$945$	85.9638	2.79640
$946$	0	0
$947$	42.4500	1.37944	0.689719	−	0.724077i	$-0.257734\pi$
$947$	42.4500	1.37944	0.689719	+	0.724077i	$0.257734\pi$
$948$	0	0
$949$	1.66947	0.0541932
$950$	0	0
$951$	79.3583	2.57337
$952$	0	0
$953$	31.1472	1.00896	0.504478	−	0.863424i	$-0.331685\pi$
$953$	31.1472	1.00896	0.504478	+	0.863424i	$0.331685\pi$
$954$	0	0
$955$	3.00000	0.0970777
$956$	0	0
$957$	−12.9083	−0.417267
$958$	0	0
$959$	19.8167	0.639913
$960$	0	0
$961$	−13.2666	−0.427955
$962$	0	0
$963$	−85.5416	−2.75654
$964$	0	0
$965$	60.3583	1.94300
$966$	0	0
$967$	16.9083	0.543735	0.271868	−	0.962335i	$-0.412359\pi$
$967$	16.9083	0.543735	0.271868	+	0.962335i	$0.412359\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−52.1749	−1.67437	−0.837187	−	0.546917i	$-0.815801\pi$
$971$	−52.1749	−1.67437	−0.837187	+	0.546917i	$0.815801\pi$
$972$	0	0
$973$	−19.1194	−0.612941
$974$	0	0
$975$	0.697224	0.0223290
$976$	0	0
$977$	28.1472	0.900508	0.450254	−	0.892900i	$-0.351333\pi$
$977$	28.1472	0.900508	0.450254	+	0.892900i	$0.351333\pi$
$978$	0	0
$979$	40.4222	1.29190
$980$	0	0
$981$	118.625	3.78740
$982$	0	0
$983$	23.5778	0.752015	0.376007	−	0.926617i	$-0.377297\pi$
$983$	23.5778	0.752015	0.376007	+	0.926617i	$0.377297\pi$
$984$	0	0
$985$	2.51388	0.0800988
$986$	0	0
$987$	−88.2666	−2.80956
$988$	0	0
$989$	−18.1194	−0.576164
$990$	0	0
$991$	−35.5416	−1.12902	−0.564509	−	0.825427i	$-0.690934\pi$
$991$	−35.5416	−1.12902	−0.564509	+	0.825427i	$0.690934\pi$
$992$	0	0
$993$	43.2389	1.37214
$994$	0	0
$995$	−23.5139	−0.745440
$996$	0	0
$997$	−0.936083	−0.0296461	−0.0148230	−	0.999890i	$-0.504718\pi$
$997$	−0.936083	−0.0296461	−0.0148230	+	0.999890i	$0.504718\pi$
$998$	0	0
$999$	139.505	4.41376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.n.1.2	yes	2
4.3	odd	2		4624.2.a.g.1.1		2
17.4	even	4		2312.2.b.h.577.1		4
17.13	even	4		2312.2.b.h.577.4		4
17.16	even	2		2312.2.a.e.1.1	✓	2
68.67	odd	2		4624.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.e.1.1	✓	2	17.16	even	2
2312.2.a.n.1.2	yes	2	1.1	even	1	trivial
2312.2.b.h.577.1		4	17.4	even	4
2312.2.b.h.577.4		4	17.13	even	4
4624.2.a.g.1.1		2	4.3	odd	2
4624.2.a.y.1.2		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.30278 q^{3} -2.30278 q^{5} -2.30278 q^{7} +7.90833 q^{9} +5.60555 q^{11} +0.697224 q^{13} -7.60555 q^{15} -1.30278 q^{19} -7.60555 q^{21} +4.30278 q^{23} +0.302776 q^{25} +16.2111 q^{27} -0.697224 q^{29} -4.21110 q^{31} +18.5139 q^{33} +5.30278 q^{35} +8.60555 q^{37} +2.30278 q^{39} +6.00000 q^{41} -4.21110 q^{43} -18.2111 q^{45} +11.6056 q^{47} -1.69722 q^{49} +3.30278 q^{53} -12.9083 q^{55} -4.30278 q^{57} +3.21110 q^{59} -1.60555 q^{61} -18.2111 q^{63} -1.60555 q^{65} +0.605551 q^{67} +14.2111 q^{69} -12.4222 q^{71} +2.39445 q^{73} +1.00000 q^{75} -12.9083 q^{77} +10.0000 q^{79} +29.8167 q^{81} -8.69722 q^{83} -2.30278 q^{87} +7.21110 q^{89} -1.60555 q^{91} -13.9083 q^{93} +3.00000 q^{95} +16.1194 q^{97} +44.3305 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - q^{5} - q^{7} + 5 q^{9} + 4 q^{11} + 5 q^{13} - 8 q^{15} + q^{19} - 8 q^{21} + 5 q^{23} - 3 q^{25} + 18 q^{27} - 5 q^{29} + 6 q^{31} + 19 q^{33} + 7 q^{35} + 10 q^{37} + q^{39} + 12 q^{41}+ \cdots + 49 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - q^5 - q^7 + 5 * q^9 + 4 * q^11 + 5 * q^13 - 8 * q^15 + q^19 - 8 * q^21 + 5 * q^23 - 3 * q^25 + 18 * q^27 - 5 * q^29 + 6 * q^31 + 19 * q^33 + 7 * q^35 + 10 * q^37 + q^39 + 12 * q^41 + 6 * q^43 - 22 * q^45 + 16 * q^47 - 7 * q^49 + 3 * q^53 - 15 * q^55 - 5 * q^57 - 8 * q^59 + 4 * q^61 - 22 * q^63 + 4 * q^65 - 6 * q^67 + 14 * q^69 + 4 * q^71 + 12 * q^73 + 2 * q^75 - 15 * q^77 + 20 * q^79 + 38 * q^81 - 21 * q^83 - q^87 + 4 * q^91 - 17 * q^93 + 6 * q^95 + 7 * q^97 + 49 * q^99

Introduction

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Database

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