LMFDB - Embedded newform 2312.2.a.r.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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q+2.18479 q^{3} +1.65270 q^{5} +3.87939 q^{7} +1.77332 q^{9} -1.06418 q^{11} +6.10607 q^{13} +3.61081 q^{15} +1.41147 q^{19} +8.47565 q^{21} +6.92127 q^{23} -2.26857 q^{25} -2.68004 q^{27} -7.90167 q^{29} -9.00774 q^{31} -2.32501 q^{33} +6.41147 q^{35} +3.47565 q^{37} +13.3405 q^{39} -10.3473 q^{41} -8.86484 q^{43} +2.93077 q^{45} +3.22668 q^{47} +8.04963 q^{49} -1.06418 q^{53} -1.75877 q^{55} +3.08378 q^{57} -9.46110 q^{59} +11.3327 q^{61} +6.87939 q^{63} +10.0915 q^{65} +2.04963 q^{67} +15.1215 q^{69} +0.448311 q^{71} -1.83750 q^{73} -4.95636 q^{75} -4.12836 q^{77} +2.75877 q^{79} -11.1753 q^{81} +0.389185 q^{83} -17.2635 q^{87} +13.9290 q^{89} +23.6878 q^{91} -19.6800 q^{93} +2.33275 q^{95} +13.7588 q^{97} -1.88713 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{5} + 6 q^{7} + 12 q^{9} + 6 q^{11} + 6 q^{13} + 15 q^{15} - 6 q^{19} + 6 q^{21} + 12 q^{23} + 3 q^{25} + 12 q^{27} - 12 q^{29} - 3 q^{31} - 12 q^{33} + 9 q^{35} - 9 q^{37} - 3 q^{39}+ \cdots + 24 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^5 + 6 * q^7 + 12 * q^9 + 6 * q^11 + 6 * q^13 + 15 * q^15 - 6 * q^19 + 6 * q^21 + 12 * q^23 + 3 * q^25 + 12 * q^27 - 12 * q^29 - 3 * q^31 - 12 * q^33 + 9 * q^35 - 9 * q^37 - 3 * q^39 - 30 * q^41 - 3 * q^43 + 33 * q^45 + 3 * q^47 - 3 * q^49 + 6 * q^53 + 6 * q^55 + 3 * q^57 + 9 * q^59 + 15 * q^61 + 15 * q^63 - 21 * q^67 - 15 * q^69 + 3 * q^71 - 3 * q^73 + 39 * q^75 + 6 * q^77 - 3 * q^79 + 3 * q^81 - 3 * q^83 - 57 * q^87 + 9 * q^89 + 27 * q^91 - 39 * q^93 - 12 * q^95 + 30 * q^97 + 24 * 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
$3$	2.18479	1.26139	0.630695	−	0.776031i	$-0.282770\pi$
$3$	2.18479	1.26139	0.630695	+	0.776031i	$0.282770\pi$
$4$	0	0
$5$	1.65270	0.739112	0.369556	−	0.929209i	$-0.379510\pi$
$5$	1.65270	0.739112	0.369556	+	0.929209i	$0.379510\pi$
$6$	0	0
$7$	3.87939	1.46627	0.733135	−	0.680083i	$-0.238056\pi$
$7$	3.87939	1.46627	0.733135	+	0.680083i	$0.238056\pi$
$8$	0	0
$9$	1.77332	0.591106
$10$	0	0
$11$	−1.06418	−0.320862	−0.160431	−	0.987047i	$-0.551288\pi$
$11$	−1.06418	−0.320862	−0.160431	+	0.987047i	$0.551288\pi$
$12$	0	0
$13$	6.10607	1.69352	0.846759	−	0.531976i	$-0.178551\pi$
$13$	6.10607	1.69352	0.846759	+	0.531976i	$0.178551\pi$
$14$	0	0
$15$	3.61081	0.932308
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	1.41147	0.323814	0.161907	−	0.986806i	$-0.448235\pi$
$19$	1.41147	0.323814	0.161907	+	0.986806i	$0.448235\pi$
$20$	0	0
$21$	8.47565	1.84954
$22$	0	0
$23$	6.92127	1.44319	0.721593	−	0.692318i	$-0.243410\pi$
$23$	6.92127	1.44319	0.721593	+	0.692318i	$0.243410\pi$
$24$	0	0
$25$	−2.26857	−0.453714
$26$	0	0
$27$	−2.68004	−0.515775
$28$	0	0
$29$	−7.90167	−1.46730	−0.733652	−	0.679525i	$-0.762186\pi$
$29$	−7.90167	−1.46730	−0.733652	+	0.679525i	$0.762186\pi$
$30$	0	0
$31$	−9.00774	−1.61784	−0.808919	−	0.587920i	$-0.799947\pi$
$31$	−9.00774	−1.61784	−0.808919	+	0.587920i	$0.799947\pi$
$32$	0	0
$33$	−2.32501	−0.404732
$34$	0	0
$35$	6.41147	1.08374
$36$	0	0
$37$	3.47565	0.571394	0.285697	−	0.958320i	$-0.407775\pi$
$37$	3.47565	0.571394	0.285697	+	0.958320i	$0.407775\pi$
$38$	0	0
$39$	13.3405	2.13619
$40$	0	0
$41$	−10.3473	−1.61598	−0.807988	−	0.589199i	$-0.799443\pi$
$41$	−10.3473	−1.61598	−0.807988	+	0.589199i	$0.799443\pi$
$42$	0	0
$43$	−8.86484	−1.35188	−0.675938	−	0.736959i	$-0.736261\pi$
$43$	−8.86484	−1.35188	−0.675938	+	0.736959i	$0.736261\pi$
$44$	0	0
$45$	2.93077	0.436893
$46$	0	0
$47$	3.22668	0.470660	0.235330	−	0.971916i	$-0.424383\pi$
$47$	3.22668	0.470660	0.235330	+	0.971916i	$0.424383\pi$
$48$	0	0
$49$	8.04963	1.14995
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.06418	−0.146176	−0.0730880	−	0.997325i	$-0.523285\pi$
$53$	−1.06418	−0.146176	−0.0730880	+	0.997325i	$0.523285\pi$
$54$	0	0
$55$	−1.75877	−0.237153
$56$	0	0
$57$	3.08378	0.408456
$58$	0	0
$59$	−9.46110	−1.23173	−0.615865	−	0.787851i	$-0.711193\pi$
$59$	−9.46110	−1.23173	−0.615865	+	0.787851i	$0.711193\pi$
$60$	0	0
$61$	11.3327	1.45101	0.725505	−	0.688217i	$-0.241606\pi$
$61$	11.3327	1.45101	0.725505	+	0.688217i	$0.241606\pi$
$62$	0	0
$63$	6.87939	0.866721
$64$	0	0
$65$	10.0915	1.25170
$66$	0	0
$67$	2.04963	0.250402	0.125201	−	0.992131i	$-0.460042\pi$
$67$	2.04963	0.250402	0.125201	+	0.992131i	$0.460042\pi$
$68$	0	0
$69$	15.1215	1.82042
$70$	0	0
$71$	0.448311	0.0532047	0.0266023	−	0.999646i	$-0.491531\pi$
$71$	0.448311	0.0532047	0.0266023	+	0.999646i	$0.491531\pi$
$72$	0	0
$73$	−1.83750	−0.215063	−0.107531	−	0.994202i	$-0.534295\pi$
$73$	−1.83750	−0.215063	−0.107531	+	0.994202i	$0.534295\pi$
$74$	0	0
$75$	−4.95636	−0.572311
$76$	0	0
$77$	−4.12836	−0.470470
$78$	0	0
$79$	2.75877	0.310386	0.155193	−	0.987884i	$-0.450400\pi$
$79$	2.75877	0.310386	0.155193	+	0.987884i	$0.450400\pi$
$80$	0	0
$81$	−11.1753	−1.24170
$82$	0	0
$83$	0.389185	0.0427186	0.0213593	−	0.999772i	$-0.493201\pi$
$83$	0.389185	0.0427186	0.0213593	+	0.999772i	$0.493201\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−17.2635	−1.85084
$88$	0	0
$89$	13.9290	1.47647	0.738236	−	0.674542i	$-0.235659\pi$
$89$	13.9290	1.47647	0.738236	+	0.674542i	$0.235659\pi$
$90$	0	0
$91$	23.6878	2.48315
$92$	0	0
$93$	−19.6800	−2.04073
$94$	0	0
$95$	2.33275	0.239335
$96$	0	0
$97$	13.7588	1.39699	0.698496	−	0.715614i	$-0.253853\pi$
$97$	13.7588	1.39699	0.698496	+	0.715614i	$0.253853\pi$
$98$	0	0
$99$	−1.88713	−0.189663
$100$	0	0
$101$	−14.5672	−1.44949	−0.724744	−	0.689018i	$-0.758042\pi$
$101$	−14.5672	−1.44949	−0.724744	+	0.689018i	$0.758042\pi$
$102$	0	0
$103$	−14.1506	−1.39430	−0.697152	−	0.716923i	$-0.745550\pi$
$103$	−14.1506	−1.39430	−0.697152	+	0.716923i	$0.745550\pi$
$104$	0	0
$105$	14.0077	1.36702
$106$	0	0
$107$	1.38413	0.133809	0.0669046	−	0.997759i	$-0.478688\pi$
$107$	1.38413	0.133809	0.0669046	+	0.997759i	$0.478688\pi$
$108$	0	0
$109$	−8.25671	−0.790849	−0.395425	−	0.918498i	$-0.629403\pi$
$109$	−8.25671	−0.790849	−0.395425	+	0.918498i	$0.629403\pi$
$110$	0	0
$111$	7.59358	0.720751
$112$	0	0
$113$	−9.98545	−0.939352	−0.469676	−	0.882839i	$-0.655629\pi$
$113$	−9.98545	−0.939352	−0.469676	+	0.882839i	$0.655629\pi$
$114$	0	0
$115$	11.4388	1.06668
$116$	0	0
$117$	10.8280	1.00105
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.86753	−0.897048
$122$	0	0
$123$	−22.6067	−2.03838
$124$	0	0
$125$	−12.0128	−1.07446
$126$	0	0
$127$	9.04189	0.802338	0.401169	−	0.916004i	$-0.368604\pi$
$127$	9.04189	0.802338	0.401169	+	0.916004i	$0.368604\pi$
$128$	0	0
$129$	−19.3678	−1.70524
$130$	0	0
$131$	15.6159	1.36436	0.682182	−	0.731182i	$-0.261031\pi$
$131$	15.6159	1.36436	0.682182	+	0.731182i	$0.261031\pi$
$132$	0	0
$133$	5.47565	0.474799
$134$	0	0
$135$	−4.42932	−0.381215
$136$	0	0
$137$	−17.5226	−1.49706	−0.748528	−	0.663103i	$-0.769239\pi$
$137$	−17.5226	−1.49706	−0.748528	+	0.663103i	$0.769239\pi$
$138$	0	0
$139$	4.40373	0.373520	0.186760	−	0.982406i	$-0.440201\pi$
$139$	4.40373	0.373520	0.186760	+	0.982406i	$0.440201\pi$
$140$	0	0
$141$	7.04963	0.593686
$142$	0	0
$143$	−6.49794	−0.543385
$144$	0	0
$145$	−13.0591	−1.08450
$146$	0	0
$147$	17.5868	1.45053
$148$	0	0
$149$	0.467911	0.0383328	0.0191664	−	0.999816i	$-0.493899\pi$
$149$	0.467911	0.0383328	0.0191664	+	0.999816i	$0.493899\pi$
$150$	0	0
$151$	12.9881	1.05696	0.528480	−	0.848946i	$-0.322762\pi$
$151$	12.9881	1.05696	0.528480	+	0.848946i	$0.322762\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−14.8871	−1.19576
$156$	0	0
$157$	9.96316	0.795147	0.397574	−	0.917570i	$-0.369852\pi$
$157$	9.96316	0.795147	0.397574	+	0.917570i	$0.369852\pi$
$158$	0	0
$159$	−2.32501	−0.184385
$160$	0	0
$161$	26.8503	2.11610
$162$	0	0
$163$	8.41147	0.658838	0.329419	−	0.944184i	$-0.393147\pi$
$163$	8.41147	0.658838	0.329419	+	0.944184i	$0.393147\pi$
$164$	0	0
$165$	−3.84255	−0.299142
$166$	0	0
$167$	1.80066	0.139339	0.0696696	−	0.997570i	$-0.477805\pi$
$167$	1.80066	0.139339	0.0696696	+	0.997570i	$0.477805\pi$
$168$	0	0
$169$	24.2841	1.86800
$170$	0	0
$171$	2.50299	0.191409
$172$	0	0
$173$	14.3824	1.09347	0.546736	−	0.837305i	$-0.315870\pi$
$173$	14.3824	1.09347	0.546736	+	0.837305i	$0.315870\pi$
$174$	0	0
$175$	−8.80066	−0.665267
$176$	0	0
$177$	−20.6705	−1.55369
$178$	0	0
$179$	2.83750	0.212084	0.106042	−	0.994362i	$-0.466182\pi$
$179$	2.83750	0.212084	0.106042	+	0.994362i	$0.466182\pi$
$180$	0	0
$181$	9.01548	0.670115	0.335058	−	0.942198i	$-0.391244\pi$
$181$	9.01548	0.670115	0.335058	+	0.942198i	$0.391244\pi$
$182$	0	0
$183$	24.7597	1.83029
$184$	0	0
$185$	5.74422	0.422324
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−10.3969	−0.756265
$190$	0	0
$191$	8.25402	0.597240	0.298620	−	0.954372i	$-0.403474\pi$
$191$	8.25402	0.597240	0.298620	+	0.954372i	$0.403474\pi$
$192$	0	0
$193$	4.46791	0.321607	0.160804	−	0.986986i	$-0.448591\pi$
$193$	4.46791	0.321607	0.160804	+	0.986986i	$0.448591\pi$
$194$	0	0
$195$	22.0479	1.57888
$196$	0	0
$197$	−19.3628	−1.37954	−0.689770	−	0.724028i	$-0.742289\pi$
$197$	−19.3628	−1.37954	−0.689770	+	0.724028i	$0.742289\pi$
$198$	0	0
$199$	0.376392	0.0266817	0.0133409	−	0.999911i	$-0.495753\pi$
$199$	0.376392	0.0266817	0.0133409	+	0.999911i	$0.495753\pi$
$200$	0	0
$201$	4.47802	0.315855
$202$	0	0
$203$	−30.6536	−2.15146
$204$	0	0
$205$	−17.1010	−1.19439
$206$	0	0
$207$	12.2736	0.853076
$208$	0	0
$209$	−1.50206	−0.103900
$210$	0	0
$211$	19.6732	1.35436	0.677181	−	0.735817i	$-0.263202\pi$
$211$	19.6732	1.35436	0.677181	+	0.735817i	$0.263202\pi$
$212$	0	0
$213$	0.979466	0.0671119
$214$	0	0
$215$	−14.6509	−0.999186
$216$	0	0
$217$	−34.9445	−2.37219
$218$	0	0
$219$	−4.01455	−0.271278
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3.11287	0.208453	0.104227	−	0.994554i	$-0.466763\pi$
$223$	3.11287	0.208453	0.104227	+	0.994554i	$0.466763\pi$
$224$	0	0
$225$	−4.02290	−0.268193
$226$	0	0
$227$	16.8033	1.11528	0.557639	−	0.830084i	$-0.311707\pi$
$227$	16.8033	1.11528	0.557639	+	0.830084i	$0.311707\pi$
$228$	0	0
$229$	−22.9145	−1.51423	−0.757115	−	0.653281i	$-0.773392\pi$
$229$	−22.9145	−1.51423	−0.757115	+	0.653281i	$0.773392\pi$
$230$	0	0
$231$	−9.01960	−0.593446
$232$	0	0
$233$	−8.00505	−0.524428	−0.262214	−	0.965010i	$-0.584453\pi$
$233$	−8.00505	−0.524428	−0.262214	+	0.965010i	$0.584453\pi$
$234$	0	0
$235$	5.33275	0.347870
$236$	0	0
$237$	6.02734	0.391518
$238$	0	0
$239$	−3.48070	−0.225148	−0.112574	−	0.993643i	$-0.535910\pi$
$239$	−3.48070	−0.225148	−0.112574	+	0.993643i	$0.535910\pi$
$240$	0	0
$241$	−4.15570	−0.267692	−0.133846	−	0.991002i	$-0.542733\pi$
$241$	−4.15570	−0.267692	−0.133846	+	0.991002i	$0.542733\pi$
$242$	0	0
$243$	−16.3756	−1.05049
$244$	0	0
$245$	13.3037	0.849939
$246$	0	0
$247$	8.61856	0.548386
$248$	0	0
$249$	0.850289	0.0538849
$250$	0	0
$251$	−15.7493	−0.994085	−0.497043	−	0.867726i	$-0.665581\pi$
$251$	−15.7493	−0.994085	−0.497043	+	0.867726i	$0.665581\pi$
$252$	0	0
$253$	−7.36547	−0.463063
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.54488	−0.283502	−0.141751	−	0.989902i	$-0.545273\pi$
$257$	−4.54488	−0.283502	−0.141751	+	0.989902i	$0.545273\pi$
$258$	0	0
$259$	13.4834	0.837817
$260$	0	0
$261$	−14.0122	−0.867332
$262$	0	0
$263$	−13.4611	−0.830047	−0.415024	−	0.909811i	$-0.636227\pi$
$263$	−13.4611	−0.830047	−0.415024	+	0.909811i	$0.636227\pi$
$264$	0	0
$265$	−1.75877	−0.108040
$266$	0	0
$267$	30.4320	1.86241
$268$	0	0
$269$	−5.35410	−0.326445	−0.163223	−	0.986589i	$-0.552189\pi$
$269$	−5.35410	−0.326445	−0.163223	+	0.986589i	$0.552189\pi$
$270$	0	0
$271$	23.0993	1.40318	0.701590	−	0.712581i	$-0.252474\pi$
$271$	23.0993	1.40318	0.701590	+	0.712581i	$0.252474\pi$
$272$	0	0
$273$	51.7529	3.13223
$274$	0	0
$275$	2.41416	0.145579
$276$	0	0
$277$	−17.7469	−1.06631	−0.533154	−	0.846018i	$-0.678993\pi$
$277$	−17.7469	−1.06631	−0.533154	+	0.846018i	$0.678993\pi$
$278$	0	0
$279$	−15.9736	−0.956314
$280$	0	0
$281$	−2.28993	−0.136606	−0.0683028	−	0.997665i	$-0.521758\pi$
$281$	−2.28993	−0.136606	−0.0683028	+	0.997665i	$0.521758\pi$
$282$	0	0
$283$	8.13516	0.483585	0.241793	−	0.970328i	$-0.422265\pi$
$283$	8.13516	0.483585	0.241793	+	0.970328i	$0.422265\pi$
$284$	0	0
$285$	5.09657	0.301895
$286$	0	0
$287$	−40.1411	−2.36946
$288$	0	0
$289$	0	0
$290$	0	0
$291$	30.0601	1.76215
$292$	0	0
$293$	3.10338	0.181301	0.0906507	−	0.995883i	$-0.471105\pi$
$293$	3.10338	0.181301	0.0906507	+	0.995883i	$0.471105\pi$
$294$	0	0
$295$	−15.6364	−0.910386
$296$	0	0
$297$	2.85204	0.165492
$298$	0	0
$299$	42.2618	2.44406
$300$	0	0
$301$	−34.3901	−1.98221
$302$	0	0
$303$	−31.8262	−1.82837
$304$	0	0
$305$	18.7297	1.07246
$306$	0	0
$307$	0.580785	0.0331472	0.0165736	−	0.999863i	$-0.494724\pi$
$307$	0.580785	0.0331472	0.0165736	+	0.999863i	$0.494724\pi$
$308$	0	0
$309$	−30.9162	−1.75876
$310$	0	0
$311$	21.0128	1.19153	0.595763	−	0.803160i	$-0.296850\pi$
$311$	21.0128	1.19153	0.595763	+	0.803160i	$0.296850\pi$
$312$	0	0
$313$	9.12836	0.515965	0.257983	−	0.966150i	$-0.416942\pi$
$313$	9.12836	0.515965	0.257983	+	0.966150i	$0.416942\pi$
$314$	0	0
$315$	11.3696	0.640604
$316$	0	0
$317$	8.04189	0.451677	0.225839	−	0.974165i	$-0.427488\pi$
$317$	8.04189	0.451677	0.225839	+	0.974165i	$0.427488\pi$
$318$	0	0
$319$	8.40879	0.470802
$320$	0	0
$321$	3.02404	0.168786
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−13.8520	−0.768373
$326$	0	0
$327$	−18.0392	−0.997570
$328$	0	0
$329$	12.5175	0.690114
$330$	0	0
$331$	−4.03920	−0.222015	−0.111007	−	0.993820i	$-0.535408\pi$
$331$	−4.03920	−0.222015	−0.111007	+	0.993820i	$0.535408\pi$
$332$	0	0
$333$	6.16344	0.337754
$334$	0	0
$335$	3.38743	0.185075
$336$	0	0
$337$	14.9385	0.813753	0.406876	−	0.913483i	$-0.366618\pi$
$337$	14.9385	0.813753	0.406876	+	0.913483i	$0.366618\pi$
$338$	0	0
$339$	−21.8161	−1.18489
$340$	0	0
$341$	9.58584	0.519102
$342$	0	0
$343$	4.07192	0.219863
$344$	0	0
$345$	24.9914	1.34549
$346$	0	0
$347$	−2.17705	−0.116870	−0.0584351	−	0.998291i	$-0.518611\pi$
$347$	−2.17705	−0.116870	−0.0584351	+	0.998291i	$0.518611\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−16.3645	−0.873474
$352$	0	0
$353$	12.8726	0.685138	0.342569	−	0.939493i	$-0.388703\pi$
$353$	12.8726	0.685138	0.342569	+	0.939493i	$0.388703\pi$
$354$	0	0
$355$	0.740925	0.0393242
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.7529	1.93974	0.969872	−	0.243616i	$-0.0783335\pi$
$359$	36.7529	1.93974	0.969872	+	0.243616i	$0.0783335\pi$
$360$	0	0
$361$	−17.0077	−0.895144
$362$	0	0
$363$	−21.5585	−1.13153
$364$	0	0
$365$	−3.03684	−0.158955
$366$	0	0
$367$	19.9463	1.04119	0.520593	−	0.853805i	$-0.325711\pi$
$367$	19.9463	1.04119	0.520593	+	0.853805i	$0.325711\pi$
$368$	0	0
$369$	−18.3491	−0.955213
$370$	0	0
$371$	−4.12836	−0.214334
$372$	0	0
$373$	−10.4929	−0.543301	−0.271651	−	0.962396i	$-0.587569\pi$
$373$	−10.4929	−0.543301	−0.271651	+	0.962396i	$0.587569\pi$
$374$	0	0
$375$	−26.2455	−1.35531
$376$	0	0
$377$	−48.2481	−2.48491
$378$	0	0
$379$	10.6254	0.545788	0.272894	−	0.962044i	$-0.412019\pi$
$379$	10.6254	0.545788	0.272894	+	0.962044i	$0.412019\pi$
$380$	0	0
$381$	19.7547	1.01206
$382$	0	0
$383$	−12.4415	−0.635731	−0.317866	−	0.948136i	$-0.602966\pi$
$383$	−12.4415	−0.635731	−0.317866	+	0.948136i	$0.602966\pi$
$384$	0	0
$385$	−6.82295	−0.347730
$386$	0	0
$387$	−15.7202	−0.799102
$388$	0	0
$389$	−18.7297	−0.949632	−0.474816	−	0.880085i	$-0.657485\pi$
$389$	−18.7297	−0.949632	−0.474816	+	0.880085i	$0.657485\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	34.1174	1.72100
$394$	0	0
$395$	4.55943	0.229410
$396$	0	0
$397$	0.425088	0.0213346	0.0106673	−	0.999943i	$-0.496604\pi$
$397$	0.425088	0.0213346	0.0106673	+	0.999943i	$0.496604\pi$
$398$	0	0
$399$	11.9632	0.598907
$400$	0	0
$401$	−11.4979	−0.574180	−0.287090	−	0.957904i	$-0.592688\pi$
$401$	−11.4979	−0.574180	−0.287090	+	0.957904i	$0.592688\pi$
$402$	0	0
$403$	−55.0019	−2.73984
$404$	0	0
$405$	−18.4695	−0.917755
$406$	0	0
$407$	−3.69871	−0.183338
$408$	0	0
$409$	−22.9786	−1.13622	−0.568110	−	0.822952i	$-0.692325\pi$
$409$	−22.9786	−1.13622	−0.568110	+	0.822952i	$0.692325\pi$
$410$	0	0
$411$	−38.2832	−1.88837
$412$	0	0
$413$	−36.7033	−1.80605
$414$	0	0
$415$	0.643208	0.0315738
$416$	0	0
$417$	9.62124	0.471154
$418$	0	0
$419$	−27.4962	−1.34328	−0.671638	−	0.740879i	$-0.734409\pi$
$419$	−27.4962	−1.34328	−0.671638	+	0.740879i	$0.734409\pi$
$420$	0	0
$421$	−7.27631	−0.354626	−0.177313	−	0.984155i	$-0.556740\pi$
$421$	−7.27631	−0.354626	−0.177313	+	0.984155i	$0.556740\pi$
$422$	0	0
$423$	5.72193	0.278210
$424$	0	0
$425$	0	0
$426$	0	0
$427$	43.9641	2.12757
$428$	0	0
$429$	−14.1967	−0.685421
$430$	0	0
$431$	8.31820	0.400674	0.200337	−	0.979727i	$-0.435796\pi$
$431$	8.31820	0.400674	0.200337	+	0.979727i	$0.435796\pi$
$432$	0	0
$433$	−29.4047	−1.41310	−0.706549	−	0.707664i	$-0.749749\pi$
$433$	−29.4047	−1.41310	−0.706549	+	0.707664i	$0.749749\pi$
$434$	0	0
$435$	−28.5315	−1.36798
$436$	0	0
$437$	9.76920	0.467324
$438$	0	0
$439$	−3.65507	−0.174447	−0.0872234	−	0.996189i	$-0.527799\pi$
$439$	−3.65507	−0.174447	−0.0872234	+	0.996189i	$0.527799\pi$
$440$	0	0
$441$	14.2746	0.679741
$442$	0	0
$443$	34.9760	1.66176	0.830879	−	0.556453i	$-0.187838\pi$
$443$	34.9760	1.66176	0.830879	+	0.556453i	$0.187838\pi$
$444$	0	0
$445$	23.0205	1.09128
$446$	0	0
$447$	1.02229	0.0483526
$448$	0	0
$449$	35.8384	1.69132	0.845660	−	0.533722i	$-0.179207\pi$
$449$	35.8384	1.69132	0.845660	+	0.533722i	$0.179207\pi$
$450$	0	0
$451$	11.0114	0.518505
$452$	0	0
$453$	28.3764	1.33324
$454$	0	0
$455$	39.1489	1.83533
$456$	0	0
$457$	10.7820	0.504360	0.252180	−	0.967680i	$-0.418852\pi$
$457$	10.7820	0.504360	0.252180	+	0.967680i	$0.418852\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.297667	0.0138637	0.00693186	−	0.999976i	$-0.497794\pi$
$461$	0.297667	0.0138637	0.00693186	+	0.999976i	$0.497794\pi$
$462$	0	0
$463$	−1.30272	−0.0605425	−0.0302712	−	0.999542i	$-0.509637\pi$
$463$	−1.30272	−0.0605425	−0.0302712	+	0.999542i	$0.509637\pi$
$464$	0	0
$465$	−32.5253	−1.50832
$466$	0	0
$467$	25.1584	1.16419	0.582096	−	0.813120i	$-0.302233\pi$
$467$	25.1584	1.16419	0.582096	+	0.813120i	$0.302233\pi$
$468$	0	0
$469$	7.95130	0.367157
$470$	0	0
$471$	21.7674	1.00299
$472$	0	0
$473$	9.43376	0.433765
$474$	0	0
$475$	−3.20203	−0.146919
$476$	0	0
$477$	−1.88713	−0.0864056
$478$	0	0
$479$	−33.1634	−1.51528	−0.757638	−	0.652675i	$-0.773647\pi$
$479$	−33.1634	−1.51528	−0.757638	+	0.652675i	$0.773647\pi$
$480$	0	0
$481$	21.2226	0.967666
$482$	0	0
$483$	58.6623	2.66923
$484$	0	0
$485$	22.7392	1.03253
$486$	0	0
$487$	−25.1848	−1.14123	−0.570616	−	0.821217i	$-0.693295\pi$
$487$	−25.1848	−1.14123	−0.570616	+	0.821217i	$0.693295\pi$
$488$	0	0
$489$	18.3773	0.831051
$490$	0	0
$491$	37.9445	1.71241	0.856206	−	0.516635i	$-0.172816\pi$
$491$	37.9445	1.71241	0.856206	+	0.516635i	$0.172816\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−3.11886	−0.140182
$496$	0	0
$497$	1.73917	0.0780124
$498$	0	0
$499$	0.330060	0.0147755	0.00738776	−	0.999973i	$-0.497648\pi$
$499$	0.330060	0.0147755	0.00738776	+	0.999973i	$0.497648\pi$
$500$	0	0
$501$	3.93407	0.175761
$502$	0	0
$503$	−6.39187	−0.285000	−0.142500	−	0.989795i	$-0.545514\pi$
$503$	−6.39187	−0.285000	−0.142500	+	0.989795i	$0.545514\pi$
$504$	0	0
$505$	−24.0752	−1.07133
$506$	0	0
$507$	53.0556	2.35628
$508$	0	0
$509$	12.2172	0.541517	0.270759	−	0.962647i	$-0.412725\pi$
$509$	12.2172	0.541517	0.270759	+	0.962647i	$0.412725\pi$
$510$	0	0
$511$	−7.12836	−0.315340
$512$	0	0
$513$	−3.78281	−0.167015
$514$	0	0
$515$	−23.3868	−1.03055
$516$	0	0
$517$	−3.43376	−0.151017
$518$	0	0
$519$	31.4225	1.37930
$520$	0	0
$521$	−43.2763	−1.89597	−0.947985	−	0.318316i	$-0.896883\pi$
$521$	−43.2763	−1.89597	−0.947985	+	0.318316i	$0.896883\pi$
$522$	0	0
$523$	−28.0036	−1.22451	−0.612256	−	0.790659i	$-0.709738\pi$
$523$	−28.0036	−1.22451	−0.612256	+	0.790659i	$0.709738\pi$
$524$	0	0
$525$	−19.2276	−0.839162
$526$	0	0
$527$	0	0
$528$	0	0
$529$	24.9040	1.08278
$530$	0	0
$531$	−16.7775	−0.728084
$532$	0	0
$533$	−63.1813	−2.73669
$534$	0	0
$535$	2.28756	0.0988999
$536$	0	0
$537$	6.19934	0.267521
$538$	0	0
$539$	−8.56624	−0.368974
$540$	0	0
$541$	23.9982	1.03177	0.515883	−	0.856659i	$-0.327464\pi$
$541$	23.9982	1.03177	0.515883	+	0.856659i	$0.327464\pi$
$542$	0	0
$543$	19.6970	0.845277
$544$	0	0
$545$	−13.6459	−0.584526
$546$	0	0
$547$	34.8435	1.48980	0.744900	−	0.667176i	$-0.232497\pi$
$547$	34.8435	1.48980	0.744900	+	0.667176i	$0.232497\pi$
$548$	0	0
$549$	20.0966	0.857701
$550$	0	0
$551$	−11.1530	−0.475134
$552$	0	0
$553$	10.7023	0.455110
$554$	0	0
$555$	12.5499	0.532715
$556$	0	0
$557$	−10.3200	−0.437271	−0.218635	−	0.975807i	$-0.570160\pi$
$557$	−10.3200	−0.437271	−0.218635	+	0.975807i	$0.570160\pi$
$558$	0	0
$559$	−54.1293	−2.28942
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−29.9290	−1.26136	−0.630679	−	0.776044i	$-0.717223\pi$
$563$	−29.9290	−1.26136	−0.630679	+	0.776044i	$0.717223\pi$
$564$	0	0
$565$	−16.5030	−0.694286
$566$	0	0
$567$	−43.3533	−1.82067
$568$	0	0
$569$	4.71419	0.197629	0.0988146	−	0.995106i	$-0.468495\pi$
$569$	4.71419	0.197629	0.0988146	+	0.995106i	$0.468495\pi$
$570$	0	0
$571$	1.14290	0.0478290	0.0239145	−	0.999714i	$-0.492387\pi$
$571$	1.14290	0.0478290	0.0239145	+	0.999714i	$0.492387\pi$
$572$	0	0
$573$	18.0333	0.753353
$574$	0	0
$575$	−15.7014	−0.654794
$576$	0	0
$577$	1.68180	0.0700142	0.0350071	−	0.999387i	$-0.488855\pi$
$577$	1.68180	0.0700142	0.0350071	+	0.999387i	$0.488855\pi$
$578$	0	0
$579$	9.76146	0.405672
$580$	0	0
$581$	1.50980	0.0626371
$582$	0	0
$583$	1.13247	0.0469023
$584$	0	0
$585$	17.8955	0.739887
$586$	0	0
$587$	−34.5330	−1.42533	−0.712665	−	0.701504i	$-0.752512\pi$
$587$	−34.5330	−1.42533	−0.712665	+	0.701504i	$0.752512\pi$
$588$	0	0
$589$	−12.7142	−0.523879
$590$	0	0
$591$	−42.3037	−1.74014
$592$	0	0
$593$	−37.7847	−1.55163	−0.775815	−	0.630960i	$-0.782661\pi$
$593$	−37.7847	−1.55163	−0.775815	+	0.630960i	$0.782661\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.822339	0.0336561
$598$	0	0
$599$	23.2841	0.951361	0.475680	−	0.879618i	$-0.342202\pi$
$599$	23.2841	0.951361	0.475680	+	0.879618i	$0.342202\pi$
$600$	0	0
$601$	31.4620	1.28336	0.641682	−	0.766971i	$-0.278237\pi$
$601$	31.4620	1.28336	0.641682	+	0.766971i	$0.278237\pi$
$602$	0	0
$603$	3.63465	0.148014
$604$	0	0
$605$	−16.3081	−0.663018
$606$	0	0
$607$	−28.6527	−1.16298	−0.581489	−	0.813554i	$-0.697530\pi$
$607$	−28.6527	−1.16298	−0.581489	+	0.813554i	$0.697530\pi$
$608$	0	0
$609$	−66.9718	−2.71384
$610$	0	0
$611$	19.7023	0.797071
$612$	0	0
$613$	−48.3996	−1.95484	−0.977421	−	0.211301i	$-0.932230\pi$
$613$	−48.3996	−1.95484	−0.977421	+	0.211301i	$0.932230\pi$
$614$	0	0
$615$	−37.3622	−1.50659
$616$	0	0
$617$	−33.4989	−1.34861	−0.674307	−	0.738451i	$-0.735558\pi$
$617$	−33.4989	−1.34861	−0.674307	+	0.738451i	$0.735558\pi$
$618$	0	0
$619$	−2.07367	−0.0833480	−0.0416740	−	0.999131i	$-0.513269\pi$
$619$	−2.07367	−0.0833480	−0.0416740	+	0.999131i	$0.513269\pi$
$620$	0	0
$621$	−18.5493	−0.744359
$622$	0	0
$623$	54.0360	2.16491
$624$	0	0
$625$	−8.51073	−0.340429
$626$	0	0
$627$	−3.28169	−0.131058
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.52704	0.0607904	0.0303952	−	0.999538i	$-0.490323\pi$
$631$	1.52704	0.0607904	0.0303952	+	0.999538i	$0.490323\pi$
$632$	0	0
$633$	42.9819	1.70838
$634$	0	0
$635$	14.9436	0.593017
$636$	0	0
$637$	49.1516	1.94746
$638$	0	0
$639$	0.794998	0.0314496
$640$	0	0
$641$	14.5098	0.573103	0.286551	−	0.958065i	$-0.407491\pi$
$641$	14.5098	0.573103	0.286551	+	0.958065i	$0.407491\pi$
$642$	0	0
$643$	20.0155	0.789334	0.394667	−	0.918824i	$-0.370860\pi$
$643$	20.0155	0.789334	0.394667	+	0.918824i	$0.370860\pi$
$644$	0	0
$645$	−32.0093	−1.26036
$646$	0	0
$647$	−24.9486	−0.980831	−0.490416	−	0.871489i	$-0.663155\pi$
$647$	−24.9486	−0.980831	−0.490416	+	0.871489i	$0.663155\pi$
$648$	0	0
$649$	10.0683	0.395215
$650$	0	0
$651$	−76.3465	−2.99225
$652$	0	0
$653$	−4.97359	−0.194632	−0.0973159	−	0.995254i	$-0.531026\pi$
$653$	−4.97359	−0.194632	−0.0973159	+	0.995254i	$0.531026\pi$
$654$	0	0
$655$	25.8084	1.00842
$656$	0	0
$657$	−3.25847	−0.127125
$658$	0	0
$659$	26.9009	1.04791	0.523954	−	0.851746i	$-0.324456\pi$
$659$	26.9009	1.04791	0.523954	+	0.851746i	$0.324456\pi$
$660$	0	0
$661$	−4.86390	−0.189184	−0.0945920	−	0.995516i	$-0.530155\pi$
$661$	−4.86390	−0.189184	−0.0945920	+	0.995516i	$0.530155\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.04963	0.350930
$666$	0	0
$667$	−54.6897	−2.11759
$668$	0	0
$669$	6.80098	0.262941
$670$	0	0
$671$	−12.0601	−0.465573
$672$	0	0
$673$	30.4252	1.17281	0.586403	−	0.810020i	$-0.300544\pi$
$673$	30.4252	1.17281	0.586403	+	0.810020i	$0.300544\pi$
$674$	0	0
$675$	6.07987	0.234014
$676$	0	0
$677$	−9.97266	−0.383280	−0.191640	−	0.981465i	$-0.561381\pi$
$677$	−9.97266	−0.383280	−0.191640	+	0.981465i	$0.561381\pi$
$678$	0	0
$679$	53.3756	2.04837
$680$	0	0
$681$	36.7118	1.40680
$682$	0	0
$683$	−12.7784	−0.488951	−0.244475	−	0.969655i	$-0.578616\pi$
$683$	−12.7784	−0.488951	−0.244475	+	0.969655i	$0.578616\pi$
$684$	0	0
$685$	−28.9597	−1.10649
$686$	0	0
$687$	−50.0634	−1.91004
$688$	0	0
$689$	−6.49794	−0.247552
$690$	0	0
$691$	25.9564	0.987426	0.493713	−	0.869625i	$-0.335639\pi$
$691$	25.9564	0.987426	0.493713	+	0.869625i	$0.335639\pi$
$692$	0	0
$693$	−7.32089	−0.278098
$694$	0	0
$695$	7.27807	0.276073
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−17.4894	−0.661509
$700$	0	0
$701$	−37.7279	−1.42496	−0.712482	−	0.701690i	$-0.752429\pi$
$701$	−37.7279	−1.42496	−0.712482	+	0.701690i	$0.752429\pi$
$702$	0	0
$703$	4.90579	0.185025
$704$	0	0
$705$	11.6509	0.438800
$706$	0	0
$707$	−56.5117	−2.12534
$708$	0	0
$709$	32.5936	1.22408	0.612039	−	0.790828i	$-0.290350\pi$
$709$	32.5936	1.22408	0.612039	+	0.790828i	$0.290350\pi$
$710$	0	0
$711$	4.89218	0.183471
$712$	0	0
$713$	−62.3450	−2.33484
$714$	0	0
$715$	−10.7392	−0.401622
$716$	0	0
$717$	−7.60462	−0.284000
$718$	0	0
$719$	33.5708	1.25198	0.625990	−	0.779831i	$-0.284695\pi$
$719$	33.5708	1.25198	0.625990	+	0.779831i	$0.284695\pi$
$720$	0	0
$721$	−54.8958	−2.04443
$722$	0	0
$723$	−9.07934	−0.337664
$724$	0	0
$725$	17.9255	0.665737
$726$	0	0
$727$	−33.6313	−1.24732	−0.623659	−	0.781697i	$-0.714355\pi$
$727$	−33.6313	−1.24732	−0.623659	+	0.781697i	$0.714355\pi$
$728$	0	0
$729$	−2.25133	−0.0833828
$730$	0	0
$731$	0	0
$732$	0	0
$733$	6.53209	0.241268	0.120634	−	0.992697i	$-0.461507\pi$
$733$	6.53209	0.241268	0.120634	+	0.992697i	$0.461507\pi$
$734$	0	0
$735$	29.0657	1.07211
$736$	0	0
$737$	−2.18117	−0.0803444
$738$	0	0
$739$	9.57903	0.352370	0.176185	−	0.984357i	$-0.443624\pi$
$739$	9.57903	0.352370	0.176185	+	0.984357i	$0.443624\pi$
$740$	0	0
$741$	18.8298	0.691728
$742$	0	0
$743$	−9.98639	−0.366365	−0.183182	−	0.983079i	$-0.558640\pi$
$743$	−9.98639	−0.366365	−0.183182	+	0.983079i	$0.558640\pi$
$744$	0	0
$745$	0.773318	0.0283322
$746$	0	0
$747$	0.690150	0.0252513
$748$	0	0
$749$	5.36959	0.196200
$750$	0	0
$751$	−4.90261	−0.178899	−0.0894493	−	0.995991i	$-0.528511\pi$
$751$	−4.90261	−0.178899	−0.0894493	+	0.995991i	$0.528511\pi$
$752$	0	0
$753$	−34.4089	−1.25393
$754$	0	0
$755$	21.4655	0.781211
$756$	0	0
$757$	26.5340	0.964393	0.482197	−	0.876063i	$-0.339839\pi$
$757$	26.5340	0.964393	0.482197	+	0.876063i	$0.339839\pi$
$758$	0	0
$759$	−16.0920	−0.584103
$760$	0	0
$761$	−43.3628	−1.57190	−0.785950	−	0.618290i	$-0.787826\pi$
$761$	−43.3628	−1.57190	−0.785950	+	0.618290i	$0.787826\pi$
$762$	0	0
$763$	−32.0310	−1.15960
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−57.7701	−2.08596
$768$	0	0
$769$	−47.7333	−1.72131	−0.860653	−	0.509191i	$-0.829945\pi$
$769$	−47.7333	−1.72131	−0.860653	+	0.509191i	$0.829945\pi$
$770$	0	0
$771$	−9.92962	−0.357607
$772$	0	0
$773$	46.9205	1.68761	0.843806	−	0.536649i	$-0.180310\pi$
$773$	46.9205	1.68761	0.843806	+	0.536649i	$0.180310\pi$
$774$	0	0
$775$	20.4347	0.734036
$776$	0	0
$777$	29.4584	1.05681
$778$	0	0
$779$	−14.6049	−0.523276
$780$	0	0
$781$	−0.477082	−0.0170713
$782$	0	0
$783$	21.1768	0.756799
$784$	0	0
$785$	16.4662	0.587702
$786$	0	0
$787$	8.41828	0.300079	0.150040	−	0.988680i	$-0.452060\pi$
$787$	8.41828	0.300079	0.150040	+	0.988680i	$0.452060\pi$
$788$	0	0
$789$	−29.4097	−1.04701
$790$	0	0
$791$	−38.7374	−1.37734
$792$	0	0
$793$	69.1985	2.45731
$794$	0	0
$795$	−3.84255	−0.136281
$796$	0	0
$797$	−8.56355	−0.303336	−0.151668	−	0.988431i	$-0.548465\pi$
$797$	−8.56355	−0.303336	−0.151668	+	0.988431i	$0.548465\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	24.7006	0.872752
$802$	0	0
$803$	1.95542	0.0690054
$804$	0	0
$805$	44.3756	1.56403
$806$	0	0
$807$	−11.6976	−0.411775
$808$	0	0
$809$	13.1935	0.463858	0.231929	−	0.972733i	$-0.425496\pi$
$809$	13.1935	0.463858	0.231929	+	0.972733i	$0.425496\pi$
$810$	0	0
$811$	6.96854	0.244699	0.122349	−	0.992487i	$-0.460957\pi$
$811$	6.96854	0.244699	0.122349	+	0.992487i	$0.460957\pi$
$812$	0	0
$813$	50.4671	1.76996
$814$	0	0
$815$	13.9017	0.486954
$816$	0	0
$817$	−12.5125	−0.437757
$818$	0	0
$819$	42.0060	1.46781
$820$	0	0
$821$	10.3226	0.360263	0.180131	−	0.983643i	$-0.442348\pi$
$821$	10.3226	0.360263	0.180131	+	0.983643i	$0.442348\pi$
$822$	0	0
$823$	36.8800	1.28556	0.642778	−	0.766053i	$-0.277782\pi$
$823$	36.8800	1.28556	0.642778	+	0.766053i	$0.277782\pi$
$824$	0	0
$825$	5.27444	0.183633
$826$	0	0
$827$	−18.6391	−0.648145	−0.324072	−	0.946032i	$-0.605052\pi$
$827$	−18.6391	−0.648145	−0.324072	+	0.946032i	$0.605052\pi$
$828$	0	0
$829$	18.2594	0.634175	0.317088	−	0.948396i	$-0.397295\pi$
$829$	18.2594	0.634175	0.317088	+	0.948396i	$0.397295\pi$
$830$	0	0
$831$	−38.7733	−1.34503
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.97596	0.102987
$836$	0	0
$837$	24.1411	0.834440
$838$	0	0
$839$	−46.9421	−1.62062	−0.810311	−	0.586000i	$-0.800702\pi$
$839$	−46.9421	−1.62062	−0.810311	+	0.586000i	$0.800702\pi$
$840$	0	0
$841$	33.4365	1.15298
$842$	0	0
$843$	−5.00301	−0.172313
$844$	0	0
$845$	40.1343	1.38066
$846$	0	0
$847$	−38.2799	−1.31531
$848$	0	0
$849$	17.7736	0.609990
$850$	0	0
$851$	24.0559	0.824627
$852$	0	0
$853$	30.5790	1.04701	0.523503	−	0.852024i	$-0.324625\pi$
$853$	30.5790	1.04701	0.523503	+	0.852024i	$0.324625\pi$
$854$	0	0
$855$	4.13671	0.141472
$856$	0	0
$857$	43.1739	1.47479	0.737396	−	0.675461i	$-0.236055\pi$
$857$	43.1739	1.47479	0.737396	+	0.675461i	$0.236055\pi$
$858$	0	0
$859$	3.50030	0.119429	0.0597144	−	0.998216i	$-0.480981\pi$
$859$	3.50030	0.119429	0.0597144	+	0.998216i	$0.480981\pi$
$860$	0	0
$861$	−87.7001	−2.98881
$862$	0	0
$863$	−36.2894	−1.23531	−0.617653	−	0.786451i	$-0.711916\pi$
$863$	−36.2894	−1.23531	−0.617653	+	0.786451i	$0.711916\pi$
$864$	0	0
$865$	23.7698	0.808198
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.93582	−0.0995909
$870$	0	0
$871$	12.5152	0.424061
$872$	0	0
$873$	24.3987	0.825770
$874$	0	0
$875$	−46.6023	−1.57544
$876$	0	0
$877$	24.2517	0.818920	0.409460	−	0.912328i	$-0.365717\pi$
$877$	24.2517	0.818920	0.409460	+	0.912328i	$0.365717\pi$
$878$	0	0
$879$	6.78024	0.228692
$880$	0	0
$881$	26.9932	0.909424	0.454712	−	0.890639i	$-0.349742\pi$
$881$	26.9932	0.909424	0.454712	+	0.890639i	$0.349742\pi$
$882$	0	0
$883$	−45.9992	−1.54800	−0.773998	−	0.633188i	$-0.781746\pi$
$883$	−45.9992	−1.54800	−0.773998	+	0.633188i	$0.781746\pi$
$884$	0	0
$885$	−34.1623	−1.14835
$886$	0	0
$887$	1.78880	0.0600620	0.0300310	−	0.999549i	$-0.490439\pi$
$887$	1.78880	0.0600620	0.0300310	+	0.999549i	$0.490439\pi$
$888$	0	0
$889$	35.0770	1.17644
$890$	0	0
$891$	11.8925	0.398414
$892$	0	0
$893$	4.55438	0.152406
$894$	0	0
$895$	4.68954	0.156754
$896$	0	0
$897$	92.3332	3.08292
$898$	0	0
$899$	71.1762	2.37386
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−75.1353	−2.50035
$904$	0	0
$905$	14.8999	0.495290
$906$	0	0
$907$	15.3874	0.510931	0.255466	−	0.966818i	$-0.417771\pi$
$907$	15.3874	0.510931	0.255466	+	0.966818i	$0.417771\pi$
$908$	0	0
$909$	−25.8322	−0.856801
$910$	0	0
$911$	−49.7998	−1.64994	−0.824971	−	0.565175i	$-0.808809\pi$
$911$	−49.7998	−1.64994	−0.824971	+	0.565175i	$0.808809\pi$
$912$	0	0
$913$	−0.414162	−0.0137068
$914$	0	0
$915$	40.9205	1.35279
$916$	0	0
$917$	60.5800	2.00053
$918$	0	0
$919$	−53.7847	−1.77419	−0.887096	−	0.461584i	$-0.847281\pi$
$919$	−53.7847	−1.77419	−0.887096	+	0.461584i	$0.847281\pi$
$920$	0	0
$921$	1.26889	0.0418115
$922$	0	0
$923$	2.73742	0.0901031
$924$	0	0
$925$	−7.88476	−0.259249
$926$	0	0
$927$	−25.0936	−0.824182
$928$	0	0
$929$	−13.6486	−0.447796	−0.223898	−	0.974613i	$-0.571878\pi$
$929$	−13.6486	−0.447796	−0.223898	+	0.974613i	$0.571878\pi$
$930$	0	0
$931$	11.3618	0.372369
$932$	0	0
$933$	45.9086	1.50298
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−14.0847	−0.460127	−0.230064	−	0.973176i	$-0.573893\pi$
$937$	−14.0847	−0.460127	−0.230064	+	0.973176i	$0.573893\pi$
$938$	0	0
$939$	19.9436	0.650834
$940$	0	0
$941$	37.7606	1.23096	0.615481	−	0.788152i	$-0.288962\pi$
$941$	37.7606	1.23096	0.615481	+	0.788152i	$0.288962\pi$
$942$	0	0
$943$	−71.6165	−2.33215
$944$	0	0
$945$	−17.1830	−0.558964
$946$	0	0
$947$	−12.9145	−0.419664	−0.209832	−	0.977737i	$-0.567292\pi$
$947$	−12.9145	−0.419664	−0.209832	+	0.977737i	$0.567292\pi$
$948$	0	0
$949$	−11.2199	−0.364213
$950$	0	0
$951$	17.5699	0.569742
$952$	0	0
$953$	−16.8375	−0.545420	−0.272710	−	0.962096i	$-0.587920\pi$
$953$	−16.8375	−0.545420	−0.272710	+	0.962096i	$0.587920\pi$
$954$	0	0
$955$	13.6415	0.441427
$956$	0	0
$957$	18.3715	0.593865
$958$	0	0
$959$	−67.9769	−2.19509
$960$	0	0
$961$	50.1394	1.61740
$962$	0	0
$963$	2.45451	0.0790954
$964$	0	0
$965$	7.38413	0.237704
$966$	0	0
$967$	58.3746	1.87720	0.938601	−	0.345005i	$-0.112123\pi$
$967$	58.3746	1.87720	0.938601	+	0.345005i	$0.112123\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20.0419	0.643175	0.321587	−	0.946880i	$-0.395784\pi$
$971$	20.0419	0.643175	0.321587	+	0.946880i	$0.395784\pi$
$972$	0	0
$973$	17.0838	0.547681
$974$	0	0
$975$	−30.2638	−0.969219
$976$	0	0
$977$	−29.8229	−0.954121	−0.477060	−	0.878871i	$-0.658298\pi$
$977$	−29.8229	−0.954121	−0.477060	+	0.878871i	$0.658298\pi$
$978$	0	0
$979$	−14.8229	−0.473743
$980$	0	0
$981$	−14.6418	−0.467476
$982$	0	0
$983$	−4.80840	−0.153364	−0.0766821	−	0.997056i	$-0.524433\pi$
$983$	−4.80840	−0.153364	−0.0766821	+	0.997056i	$0.524433\pi$
$984$	0	0
$985$	−32.0009	−1.01963
$986$	0	0
$987$	27.3482	0.870504
$988$	0	0
$989$	−61.3560	−1.95101
$990$	0	0
$991$	2.14889	0.0682617	0.0341309	−	0.999417i	$-0.489134\pi$
$991$	2.14889	0.0682617	0.0341309	+	0.999417i	$0.489134\pi$
$992$	0	0
$993$	−8.82482	−0.280047
$994$	0	0
$995$	0.622065	0.0197208
$996$	0	0
$997$	5.84018	0.184960	0.0924802	−	0.995715i	$-0.470520\pi$
$997$	5.84018	0.184960	0.0924802	+	0.995715i	$0.470520\pi$
$998$	0	0
$999$	−9.31490	−0.294710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.r.1.2	yes	3
4.3	odd	2		4624.2.a.bb.1.2		3
17.4	even	4		2312.2.b.l.577.3		6
17.13	even	4		2312.2.b.l.577.4		6
17.16	even	2		2312.2.a.o.1.2	✓	3
68.67	odd	2		4624.2.a.bi.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.o.1.2	✓	3	17.16	even	2
2312.2.a.r.1.2	yes	3	1.1	even	1	trivial
2312.2.b.l.577.3		6	17.4	even	4
2312.2.b.l.577.4		6	17.13	even	4
4624.2.a.bb.1.2		3	4.3	odd	2
4624.2.a.bi.1.2		3	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)

\(f(q)\)	\(=\)	\(q+2.18479 q^{3} +1.65270 q^{5} +3.87939 q^{7} +1.77332 q^{9} -1.06418 q^{11} +6.10607 q^{13} +3.61081 q^{15} +1.41147 q^{19} +8.47565 q^{21} +6.92127 q^{23} -2.26857 q^{25} -2.68004 q^{27} -7.90167 q^{29} -9.00774 q^{31} -2.32501 q^{33} +6.41147 q^{35} +3.47565 q^{37} +13.3405 q^{39} -10.3473 q^{41} -8.86484 q^{43} +2.93077 q^{45} +3.22668 q^{47} +8.04963 q^{49} -1.06418 q^{53} -1.75877 q^{55} +3.08378 q^{57} -9.46110 q^{59} +11.3327 q^{61} +6.87939 q^{63} +10.0915 q^{65} +2.04963 q^{67} +15.1215 q^{69} +0.448311 q^{71} -1.83750 q^{73} -4.95636 q^{75} -4.12836 q^{77} +2.75877 q^{79} -11.1753 q^{81} +0.389185 q^{83} -17.2635 q^{87} +13.9290 q^{89} +23.6878 q^{91} -19.6800 q^{93} +2.33275 q^{95} +13.7588 q^{97} -1.88713 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{5} + 6 q^{7} + 12 q^{9} + 6 q^{11} + 6 q^{13} + 15 q^{15} - 6 q^{19} + 6 q^{21} + 12 q^{23} + 3 q^{25} + 12 q^{27} - 12 q^{29} - 3 q^{31} - 12 q^{33} + 9 q^{35} - 9 q^{37} - 3 q^{39}+ \cdots + 24 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^5 + 6 * q^7 + 12 * q^9 + 6 * q^11 + 6 * q^13 + 15 * q^15 - 6 * q^19 + 6 * q^21 + 12 * q^23 + 3 * q^25 + 12 * q^27 - 12 * q^29 - 3 * q^31 - 12 * q^33 + 9 * q^35 - 9 * q^37 - 3 * q^39 - 30 * q^41 - 3 * q^43 + 33 * q^45 + 3 * q^47 - 3 * q^49 + 6 * q^53 + 6 * q^55 + 3 * q^57 + 9 * q^59 + 15 * q^61 + 15 * q^63 - 21 * q^67 - 15 * q^69 + 3 * q^71 - 3 * q^73 + 39 * q^75 + 6 * q^77 - 3 * q^79 + 3 * q^81 - 3 * q^83 - 57 * q^87 + 9 * q^89 + 27 * q^91 - 39 * q^93 - 12 * q^95 + 30 * q^97 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more