LMFDB - Embedded newform 2312.2.a.f.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.61803 q^{3} -3.85410 q^{5} -1.38197 q^{7} -0.381966 q^{9} -0.236068 q^{11} +4.09017 q^{13} +6.23607 q^{15} +3.61803 q^{19} +2.23607 q^{21} +8.32624 q^{23} +9.85410 q^{25} +5.47214 q^{27} +8.85410 q^{29} -9.47214 q^{31} +0.381966 q^{33} +5.32624 q^{35} -9.23607 q^{37} -6.61803 q^{39} -2.00000 q^{41} -1.47214 q^{43} +1.47214 q^{45} -10.2361 q^{47} -5.09017 q^{49} -3.14590 q^{53} +0.909830 q^{55} -5.85410 q^{57} -0.472136 q^{59} -7.76393 q^{61} +0.527864 q^{63} -15.7639 q^{65} +3.70820 q^{67} -13.4721 q^{69} +10.9443 q^{71} -3.76393 q^{73} -15.9443 q^{75} +0.326238 q^{77} +10.0000 q^{79} -7.70820 q^{81} +15.3262 q^{83} -14.3262 q^{87} -4.47214 q^{89} -5.65248 q^{91} +15.3262 q^{93} -13.9443 q^{95} -3.85410 q^{97} +0.0901699 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} - 5 q^{7} - 3 q^{9} + 4 q^{11} - 3 q^{13} + 8 q^{15} + 5 q^{19} + q^{23} + 13 q^{25} + 2 q^{27} + 11 q^{29} - 10 q^{31} + 3 q^{33} - 5 q^{35} - 14 q^{37} - 11 q^{39} - 4 q^{41} + 6 q^{43}+ \cdots - 11 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 - 5 * q^7 - 3 * q^9 + 4 * q^11 - 3 * q^13 + 8 * q^15 + 5 * q^19 + q^23 + 13 * q^25 + 2 * q^27 + 11 * q^29 - 10 * q^31 + 3 * q^33 - 5 * q^35 - 14 * q^37 - 11 * q^39 - 4 * q^41 + 6 * q^43 - 6 * q^45 - 16 * q^47 + q^49 - 13 * q^53 + 13 * q^55 - 5 * q^57 + 8 * q^59 - 20 * q^61 + 10 * q^63 - 36 * q^65 - 6 * q^67 - 18 * q^69 + 4 * q^71 - 12 * q^73 - 14 * q^75 - 15 * q^77 + 20 * q^79 - 2 * q^81 + 15 * q^83 - 13 * q^87 + 20 * q^91 + 15 * q^93 - 10 * q^95 - q^97 - 11 * 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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	0	0
$7$	−1.38197	−0.522334	−0.261167	−	0.965294i	$-0.584107\pi$
$7$	−1.38197	−0.522334	−0.261167	+	0.965294i	$0.584107\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	4.09017	1.13441	0.567205	−	0.823577i	$-0.308025\pi$
$13$	4.09017	1.13441	0.567205	+	0.823577i	$0.308025\pi$
$14$	0	0
$15$	6.23607	1.61015
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.61803	0.830034	0.415017	−	0.909814i	$-0.363776\pi$
$19$	3.61803	0.830034	0.415017	+	0.909814i	$0.363776\pi$
$20$	0	0
$21$	2.23607	0.487950
$22$	0	0
$23$	8.32624	1.73614	0.868070	−	0.496441i	$-0.165360\pi$
$23$	8.32624	1.73614	0.868070	+	0.496441i	$0.165360\pi$
$24$	0	0
$25$	9.85410	1.97082
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	8.85410	1.64417	0.822083	−	0.569368i	$-0.192812\pi$
$29$	8.85410	1.64417	0.822083	+	0.569368i	$0.192812\pi$
$30$	0	0
$31$	−9.47214	−1.70125	−0.850623	−	0.525776i	$-0.823775\pi$
$31$	−9.47214	−1.70125	−0.850623	+	0.525776i	$0.823775\pi$
$32$	0	0
$33$	0.381966	0.0664917
$34$	0	0
$35$	5.32624	0.900299
$36$	0	0
$37$	−9.23607	−1.51840	−0.759200	−	0.650857i	$-0.774410\pi$
$37$	−9.23607	−1.51840	−0.759200	+	0.650857i	$0.774410\pi$
$38$	0	0
$39$	−6.61803	−1.05973
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−1.47214	−0.224499	−0.112249	−	0.993680i	$-0.535806\pi$
$43$	−1.47214	−0.224499	−0.112249	+	0.993680i	$0.535806\pi$
$44$	0	0
$45$	1.47214	0.219453
$46$	0	0
$47$	−10.2361	−1.49308	−0.746542	−	0.665338i	$-0.768287\pi$
$47$	−10.2361	−1.49308	−0.746542	+	0.665338i	$0.768287\pi$
$48$	0	0
$49$	−5.09017	−0.727167
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.14590	−0.432122	−0.216061	−	0.976380i	$-0.569321\pi$
$53$	−3.14590	−0.432122	−0.216061	+	0.976380i	$0.569321\pi$
$54$	0	0
$55$	0.909830	0.122681
$56$	0	0
$57$	−5.85410	−0.775395
$58$	0	0
$59$	−0.472136	−0.0614669	−0.0307334	−	0.999528i	$-0.509784\pi$
$59$	−0.472136	−0.0614669	−0.0307334	+	0.999528i	$0.509784\pi$
$60$	0	0
$61$	−7.76393	−0.994070	−0.497035	−	0.867731i	$-0.665578\pi$
$61$	−7.76393	−0.994070	−0.497035	+	0.867731i	$0.665578\pi$
$62$	0	0
$63$	0.527864	0.0665046
$64$	0	0
$65$	−15.7639	−1.95528
$66$	0	0
$67$	3.70820	0.453029	0.226515	−	0.974008i	$-0.427267\pi$
$67$	3.70820	0.453029	0.226515	+	0.974008i	$0.427267\pi$
$68$	0	0
$69$	−13.4721	−1.62185
$70$	0	0
$71$	10.9443	1.29885	0.649423	−	0.760427i	$-0.275010\pi$
$71$	10.9443	1.29885	0.649423	+	0.760427i	$0.275010\pi$
$72$	0	0
$73$	−3.76393	−0.440535	−0.220267	−	0.975440i	$-0.570693\pi$
$73$	−3.76393	−0.440535	−0.220267	+	0.975440i	$0.570693\pi$
$74$	0	0
$75$	−15.9443	−1.84109
$76$	0	0
$77$	0.326238	0.0371783
$78$	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
$81$	−7.70820	−0.856467
$82$	0	0
$83$	15.3262	1.68227	0.841137	−	0.540823i	$-0.181887\pi$
$83$	15.3262	1.68227	0.841137	+	0.540823i	$0.181887\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−14.3262	−1.53593
$88$	0	0
$89$	−4.47214	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$90$	0	0
$91$	−5.65248	−0.592540
$92$	0	0
$93$	15.3262	1.58926
$94$	0	0
$95$	−13.9443	−1.43065
$96$	0	0
$97$	−3.85410	−0.391325	−0.195662	−	0.980671i	$-0.562686\pi$
$97$	−3.85410	−0.391325	−0.195662	+	0.980671i	$0.562686\pi$
$98$	0	0
$99$	0.0901699	0.00906242
$100$	0	0
$101$	6.70820	0.667491	0.333746	−	0.942663i	$-0.391687\pi$
$101$	6.70820	0.667491	0.333746	+	0.942663i	$0.391687\pi$
$102$	0	0
$103$	−9.41641	−0.927826	−0.463913	−	0.885881i	$-0.653555\pi$
$103$	−9.41641	−0.927826	−0.463913	+	0.885881i	$0.653555\pi$
$104$	0	0
$105$	−8.61803	−0.841034
$106$	0	0
$107$	6.70820	0.648507	0.324253	−	0.945970i	$-0.394887\pi$
$107$	6.70820	0.648507	0.324253	+	0.945970i	$0.394887\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	14.9443	1.41845
$112$	0	0
$113$	−2.61803	−0.246284	−0.123142	−	0.992389i	$-0.539297\pi$
$113$	−2.61803	−0.246284	−0.123142	+	0.992389i	$0.539297\pi$
$114$	0	0
$115$	−32.0902	−2.99242
$116$	0	0
$117$	−1.56231	−0.144435
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.9443	−0.994934
$122$	0	0
$123$	3.23607	0.291786
$124$	0	0
$125$	−18.7082	−1.67331
$126$	0	0
$127$	−3.47214	−0.308102	−0.154051	−	0.988063i	$-0.549232\pi$
$127$	−3.47214	−0.308102	−0.154051	+	0.988063i	$0.549232\pi$
$128$	0	0
$129$	2.38197	0.209720
$130$	0	0
$131$	19.7082	1.72191	0.860957	−	0.508678i	$-0.169866\pi$
$131$	19.7082	1.72191	0.860957	+	0.508678i	$0.169866\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	0	0
$135$	−21.0902	−1.81515
$136$	0	0
$137$	−6.76393	−0.577882	−0.288941	−	0.957347i	$-0.593303\pi$
$137$	−6.76393	−0.577882	−0.288941	+	0.957347i	$0.593303\pi$
$138$	0	0
$139$	−8.61803	−0.730972	−0.365486	−	0.930817i	$-0.619097\pi$
$139$	−8.61803	−0.730972	−0.365486	+	0.930817i	$0.619097\pi$
$140$	0	0
$141$	16.5623	1.39480
$142$	0	0
$143$	−0.965558	−0.0807440
$144$	0	0
$145$	−34.1246	−2.83389
$146$	0	0
$147$	8.23607	0.679299
$148$	0	0
$149$	−9.70820	−0.795327	−0.397664	−	0.917531i	$-0.630179\pi$
$149$	−9.70820	−0.795327	−0.397664	+	0.917531i	$0.630179\pi$
$150$	0	0
$151$	−11.6525	−0.948265	−0.474133	−	0.880453i	$-0.657238\pi$
$151$	−11.6525	−0.948265	−0.474133	+	0.880453i	$0.657238\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	36.5066	2.93228
$156$	0	0
$157$	14.6180	1.16665	0.583323	−	0.812240i	$-0.301752\pi$
$157$	14.6180	1.16665	0.583323	+	0.812240i	$0.301752\pi$
$158$	0	0
$159$	5.09017	0.403677
$160$	0	0
$161$	−11.5066	−0.906845
$162$	0	0
$163$	−5.47214	−0.428611	−0.214305	−	0.976767i	$-0.568749\pi$
$163$	−5.47214	−0.428611	−0.214305	+	0.976767i	$0.568749\pi$
$164$	0	0
$165$	−1.47214	−0.114606
$166$	0	0
$167$	1.43769	0.111252	0.0556261	−	0.998452i	$-0.482285\pi$
$167$	1.43769	0.111252	0.0556261	+	0.998452i	$0.482285\pi$
$168$	0	0
$169$	3.72949	0.286884
$170$	0	0
$171$	−1.38197	−0.105682
$172$	0	0
$173$	−1.29180	−0.0982134	−0.0491067	−	0.998794i	$-0.515637\pi$
$173$	−1.29180	−0.0982134	−0.0491067	+	0.998794i	$0.515637\pi$
$174$	0	0
$175$	−13.6180	−1.02943
$176$	0	0
$177$	0.763932	0.0574206
$178$	0	0
$179$	7.38197	0.551754	0.275877	−	0.961193i	$-0.411032\pi$
$179$	7.38197	0.551754	0.275877	+	0.961193i	$0.411032\pi$
$180$	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
$183$	12.5623	0.928632
$184$	0	0
$185$	35.5967	2.61712
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−7.56231	−0.550077
$190$	0	0
$191$	13.5066	0.977302	0.488651	−	0.872479i	$-0.337489\pi$
$191$	13.5066	0.977302	0.488651	+	0.872479i	$0.337489\pi$
$192$	0	0
$193$	19.3607	1.39361	0.696806	−	0.717260i	$-0.254604\pi$
$193$	19.3607	1.39361	0.696806	+	0.717260i	$0.254604\pi$
$194$	0	0
$195$	25.5066	1.82656
$196$	0	0
$197$	−22.3262	−1.59068	−0.795339	−	0.606165i	$-0.792707\pi$
$197$	−22.3262	−1.59068	−0.795339	+	0.606165i	$0.792707\pi$
$198$	0	0
$199$	−0.527864	−0.0374193	−0.0187096	−	0.999825i	$-0.505956\pi$
$199$	−0.527864	−0.0374193	−0.0187096	+	0.999825i	$0.505956\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	−12.2361	−0.858804
$204$	0	0
$205$	7.70820	0.538364
$206$	0	0
$207$	−3.18034	−0.221049
$208$	0	0
$209$	−0.854102	−0.0590795
$210$	0	0
$211$	−7.38197	−0.508195	−0.254098	−	0.967179i	$-0.581778\pi$
$211$	−7.38197	−0.508195	−0.254098	+	0.967179i	$0.581778\pi$
$212$	0	0
$213$	−17.7082	−1.21335
$214$	0	0
$215$	5.67376	0.386947
$216$	0	0
$217$	13.0902	0.888619
$218$	0	0
$219$	6.09017	0.411536
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−15.0902	−1.01051	−0.505256	−	0.862969i	$-0.668602\pi$
$223$	−15.0902	−1.01051	−0.505256	+	0.862969i	$0.668602\pi$
$224$	0	0
$225$	−3.76393	−0.250929
$226$	0	0
$227$	3.47214	0.230454	0.115227	−	0.993339i	$-0.463240\pi$
$227$	3.47214	0.230454	0.115227	+	0.993339i	$0.463240\pi$
$228$	0	0
$229$	−4.94427	−0.326727	−0.163363	−	0.986566i	$-0.552234\pi$
$229$	−4.94427	−0.326727	−0.163363	+	0.986566i	$0.552234\pi$
$230$	0	0
$231$	−0.527864	−0.0347309
$232$	0	0
$233$	−8.18034	−0.535912	−0.267956	−	0.963431i	$-0.586348\pi$
$233$	−8.18034	−0.535912	−0.267956	+	0.963431i	$0.586348\pi$
$234$	0	0
$235$	39.4508	2.57349
$236$	0	0
$237$	−16.1803	−1.05103
$238$	0	0
$239$	8.23607	0.532747	0.266373	−	0.963870i	$-0.414175\pi$
$239$	8.23607	0.532747	0.266373	+	0.963870i	$0.414175\pi$
$240$	0	0
$241$	−24.0344	−1.54819	−0.774097	−	0.633067i	$-0.781796\pi$
$241$	−24.0344	−1.54819	−0.774097	+	0.633067i	$0.781796\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	19.6180	1.25335
$246$	0	0
$247$	14.7984	0.941598
$248$	0	0
$249$	−24.7984	−1.57153
$250$	0	0
$251$	13.7984	0.870946	0.435473	−	0.900202i	$-0.356581\pi$
$251$	13.7984	0.870946	0.435473	+	0.900202i	$0.356581\pi$
$252$	0	0
$253$	−1.96556	−0.123574
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.20163	−0.0749554	−0.0374777	−	0.999297i	$-0.511932\pi$
$257$	−1.20163	−0.0749554	−0.0374777	+	0.999297i	$0.511932\pi$
$258$	0	0
$259$	12.7639	0.793112
$260$	0	0
$261$	−3.38197	−0.209338
$262$	0	0
$263$	−28.5623	−1.76123	−0.880614	−	0.473835i	$-0.842869\pi$
$263$	−28.5623	−1.76123	−0.880614	+	0.473835i	$0.842869\pi$
$264$	0	0
$265$	12.1246	0.744809
$266$	0	0
$267$	7.23607	0.442840
$268$	0	0
$269$	−19.6180	−1.19613	−0.598066	−	0.801447i	$-0.704064\pi$
$269$	−19.6180	−1.19613	−0.598066	+	0.801447i	$0.704064\pi$
$270$	0	0
$271$	−10.7984	−0.655954	−0.327977	−	0.944686i	$-0.606367\pi$
$271$	−10.7984	−0.655954	−0.327977	+	0.944686i	$0.606367\pi$
$272$	0	0
$273$	9.14590	0.553535
$274$	0	0
$275$	−2.32624	−0.140277
$276$	0	0
$277$	−26.8328	−1.61223	−0.806114	−	0.591761i	$-0.798433\pi$
$277$	−26.8328	−1.61223	−0.806114	+	0.591761i	$0.798433\pi$
$278$	0	0
$279$	3.61803	0.216606
$280$	0	0
$281$	3.09017	0.184344	0.0921720	−	0.995743i	$-0.470619\pi$
$281$	3.09017	0.184344	0.0921720	+	0.995743i	$0.470619\pi$
$282$	0	0
$283$	13.8885	0.825588	0.412794	−	0.910824i	$-0.364553\pi$
$283$	13.8885	0.825588	0.412794	+	0.910824i	$0.364553\pi$
$284$	0	0
$285$	22.5623	1.33648
$286$	0	0
$287$	2.76393	0.163150
$288$	0	0
$289$	0	0
$290$	0	0
$291$	6.23607	0.365565
$292$	0	0
$293$	−19.7082	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$293$	−19.7082	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$294$	0	0
$295$	1.81966	0.105945
$296$	0	0
$297$	−1.29180	−0.0749576
$298$	0	0
$299$	34.0557	1.96949
$300$	0	0
$301$	2.03444	0.117263
$302$	0	0
$303$	−10.8541	−0.623552
$304$	0	0
$305$	29.9230	1.71339
$306$	0	0
$307$	−20.6180	−1.17673	−0.588367	−	0.808594i	$-0.700229\pi$
$307$	−20.6180	−1.17673	−0.588367	+	0.808594i	$0.700229\pi$
$308$	0	0
$309$	15.2361	0.866750
$310$	0	0
$311$	−23.0344	−1.30616	−0.653082	−	0.757287i	$-0.726524\pi$
$311$	−23.0344	−1.30616	−0.653082	+	0.757287i	$0.726524\pi$
$312$	0	0
$313$	1.23607	0.0698667	0.0349333	−	0.999390i	$-0.488878\pi$
$313$	1.23607	0.0698667	0.0349333	+	0.999390i	$0.488878\pi$
$314$	0	0
$315$	−2.03444	−0.114628
$316$	0	0
$317$	−25.1803	−1.41427	−0.707134	−	0.707079i	$-0.750012\pi$
$317$	−25.1803	−1.41427	−0.707134	+	0.707079i	$0.750012\pi$
$318$	0	0
$319$	−2.09017	−0.117027
$320$	0	0
$321$	−10.8541	−0.605817
$322$	0	0
$323$	0	0
$324$	0	0
$325$	40.3050	2.23572
$326$	0	0
$327$	1.61803	0.0894775
$328$	0	0
$329$	14.1459	0.779889
$330$	0	0
$331$	4.79837	0.263742	0.131871	−	0.991267i	$-0.457901\pi$
$331$	4.79837	0.263742	0.131871	+	0.991267i	$0.457901\pi$
$332$	0	0
$333$	3.52786	0.193326
$334$	0	0
$335$	−14.2918	−0.780844
$336$	0	0
$337$	4.20163	0.228877	0.114439	−	0.993430i	$-0.463493\pi$
$337$	4.20163	0.228877	0.114439	+	0.993430i	$0.463493\pi$
$338$	0	0
$339$	4.23607	0.230072
$340$	0	0
$341$	2.23607	0.121090
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	51.9230	2.79544
$346$	0	0
$347$	23.9098	1.28355	0.641773	−	0.766894i	$-0.278199\pi$
$347$	23.9098	1.28355	0.641773	+	0.766894i	$0.278199\pi$
$348$	0	0
$349$	26.1246	1.39842	0.699209	−	0.714917i	$-0.253536\pi$
$349$	26.1246	1.39842	0.699209	+	0.714917i	$0.253536\pi$
$350$	0	0
$351$	22.3820	1.19466
$352$	0	0
$353$	−14.8885	−0.792437	−0.396219	−	0.918156i	$-0.629678\pi$
$353$	−14.8885	−0.792437	−0.396219	+	0.918156i	$0.629678\pi$
$354$	0	0
$355$	−42.1803	−2.23870
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.5066	−1.08230	−0.541148	−	0.840927i	$-0.682010\pi$
$359$	−20.5066	−1.08230	−0.541148	+	0.840927i	$0.682010\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	0	0
$363$	17.7082	0.929440
$364$	0	0
$365$	14.5066	0.759309
$366$	0	0
$367$	−5.81966	−0.303784	−0.151892	−	0.988397i	$-0.548537\pi$
$367$	−5.81966	−0.303784	−0.151892	+	0.988397i	$0.548537\pi$
$368$	0	0
$369$	0.763932	0.0397687
$370$	0	0
$371$	4.34752	0.225712
$372$	0	0
$373$	31.9443	1.65401	0.827006	−	0.562193i	$-0.190042\pi$
$373$	31.9443	1.65401	0.827006	+	0.562193i	$0.190042\pi$
$374$	0	0
$375$	30.2705	1.56316
$376$	0	0
$377$	36.2148	1.86516
$378$	0	0
$379$	22.8885	1.17571	0.587853	−	0.808968i	$-0.299973\pi$
$379$	22.8885	1.17571	0.587853	+	0.808968i	$0.299973\pi$
$380$	0	0
$381$	5.61803	0.287821
$382$	0	0
$383$	−4.27051	−0.218213	−0.109106	−	0.994030i	$-0.534799\pi$
$383$	−4.27051	−0.218213	−0.109106	+	0.994030i	$0.534799\pi$
$384$	0	0
$385$	−1.25735	−0.0640807
$386$	0	0
$387$	0.562306	0.0285836
$388$	0	0
$389$	−29.8328	−1.51258	−0.756292	−	0.654234i	$-0.772991\pi$
$389$	−29.8328	−1.51258	−0.756292	+	0.654234i	$0.772991\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−31.8885	−1.60856
$394$	0	0
$395$	−38.5410	−1.93921
$396$	0	0
$397$	−6.23607	−0.312979	−0.156490	−	0.987680i	$-0.550018\pi$
$397$	−6.23607	−0.312979	−0.156490	+	0.987680i	$0.550018\pi$
$398$	0	0
$399$	8.09017	0.405015
$400$	0	0
$401$	−4.32624	−0.216042	−0.108021	−	0.994149i	$-0.534451\pi$
$401$	−4.32624	−0.216042	−0.108021	+	0.994149i	$0.534451\pi$
$402$	0	0
$403$	−38.7426	−1.92991
$404$	0	0
$405$	29.7082	1.47621
$406$	0	0
$407$	2.18034	0.108075
$408$	0	0
$409$	6.70820	0.331699	0.165850	−	0.986151i	$-0.446963\pi$
$409$	6.70820	0.331699	0.165850	+	0.986151i	$0.446963\pi$
$410$	0	0
$411$	10.9443	0.539841
$412$	0	0
$413$	0.652476	0.0321062
$414$	0	0
$415$	−59.0689	−2.89958
$416$	0	0
$417$	13.9443	0.682854
$418$	0	0
$419$	−6.23607	−0.304652	−0.152326	−	0.988330i	$-0.548676\pi$
$419$	−6.23607	−0.304652	−0.152326	+	0.988330i	$0.548676\pi$
$420$	0	0
$421$	28.6180	1.39476	0.697379	−	0.716703i	$-0.254350\pi$
$421$	28.6180	1.39476	0.697379	+	0.716703i	$0.254350\pi$
$422$	0	0
$423$	3.90983	0.190102
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.7295	0.519236
$428$	0	0
$429$	1.56231	0.0754288
$430$	0	0
$431$	−29.0902	−1.40122	−0.700612	−	0.713542i	$-0.747090\pi$
$431$	−29.0902	−1.40122	−0.700612	+	0.713542i	$0.747090\pi$
$432$	0	0
$433$	−32.9443	−1.58320	−0.791600	−	0.611039i	$-0.790752\pi$
$433$	−32.9443	−1.58320	−0.791600	+	0.611039i	$0.790752\pi$
$434$	0	0
$435$	55.2148	2.64735
$436$	0	0
$437$	30.1246	1.44106
$438$	0	0
$439$	−6.65248	−0.317505	−0.158753	−	0.987318i	$-0.550747\pi$
$439$	−6.65248	−0.317505	−0.158753	+	0.987318i	$0.550747\pi$
$440$	0	0
$441$	1.94427	0.0925844
$442$	0	0
$443$	9.58359	0.455330	0.227665	−	0.973739i	$-0.426891\pi$
$443$	9.58359	0.455330	0.227665	+	0.973739i	$0.426891\pi$
$444$	0	0
$445$	17.2361	0.817068
$446$	0	0
$447$	15.7082	0.742973
$448$	0	0
$449$	14.3820	0.678727	0.339363	−	0.940655i	$-0.389788\pi$
$449$	14.3820	0.678727	0.339363	+	0.940655i	$0.389788\pi$
$450$	0	0
$451$	0.472136	0.0222320
$452$	0	0
$453$	18.8541	0.885843
$454$	0	0
$455$	21.7852	1.02131
$456$	0	0
$457$	−25.8541	−1.20940	−0.604702	−	0.796452i	$-0.706708\pi$
$457$	−25.8541	−1.20940	−0.604702	+	0.796452i	$0.706708\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.8885	−0.740003	−0.370002	−	0.929031i	$-0.620643\pi$
$461$	−15.8885	−0.740003	−0.370002	+	0.929031i	$0.620643\pi$
$462$	0	0
$463$	−37.1246	−1.72533	−0.862664	−	0.505778i	$-0.831205\pi$
$463$	−37.1246	−1.72533	−0.862664	+	0.505778i	$0.831205\pi$
$464$	0	0
$465$	−59.0689	−2.73925
$466$	0	0
$467$	−28.5066	−1.31913	−0.659564	−	0.751649i	$-0.729259\pi$
$467$	−28.5066	−1.31913	−0.659564	+	0.751649i	$0.729259\pi$
$468$	0	0
$469$	−5.12461	−0.236633
$470$	0	0
$471$	−23.6525	−1.08985
$472$	0	0
$473$	0.347524	0.0159792
$474$	0	0
$475$	35.6525	1.63585
$476$	0	0
$477$	1.20163	0.0550187
$478$	0	0
$479$	−0.708204	−0.0323587	−0.0161793	−	0.999869i	$-0.505150\pi$
$479$	−0.708204	−0.0323587	−0.0161793	+	0.999869i	$0.505150\pi$
$480$	0	0
$481$	−37.7771	−1.72249
$482$	0	0
$483$	18.6180	0.847150
$484$	0	0
$485$	14.8541	0.674490
$486$	0	0
$487$	7.32624	0.331984	0.165992	−	0.986127i	$-0.446917\pi$
$487$	7.32624	0.331984	0.165992	+	0.986127i	$0.446917\pi$
$488$	0	0
$489$	8.85410	0.400396
$490$	0	0
$491$	−9.38197	−0.423402	−0.211701	−	0.977334i	$-0.567900\pi$
$491$	−9.38197	−0.423402	−0.211701	+	0.977334i	$0.567900\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−0.347524	−0.0156200
$496$	0	0
$497$	−15.1246	−0.678432
$498$	0	0
$499$	22.7984	1.02060	0.510298	−	0.859998i	$-0.329535\pi$
$499$	22.7984	1.02060	0.510298	+	0.859998i	$0.329535\pi$
$500$	0	0
$501$	−2.32624	−0.103929
$502$	0	0
$503$	2.12461	0.0947318	0.0473659	−	0.998878i	$-0.484917\pi$
$503$	2.12461	0.0947318	0.0473659	+	0.998878i	$0.484917\pi$
$504$	0	0
$505$	−25.8541	−1.15049
$506$	0	0
$507$	−6.03444	−0.267999
$508$	0	0
$509$	9.29180	0.411852	0.205926	−	0.978568i	$-0.433979\pi$
$509$	9.29180	0.411852	0.205926	+	0.978568i	$0.433979\pi$
$510$	0	0
$511$	5.20163	0.230106
$512$	0	0
$513$	19.7984	0.874120
$514$	0	0
$515$	36.2918	1.59921
$516$	0	0
$517$	2.41641	0.106273
$518$	0	0
$519$	2.09017	0.0917483
$520$	0	0
$521$	20.9443	0.917585	0.458793	−	0.888543i	$-0.348282\pi$
$521$	20.9443	0.917585	0.458793	+	0.888543i	$0.348282\pi$
$522$	0	0
$523$	−36.0132	−1.57475	−0.787373	−	0.616477i	$-0.788559\pi$
$523$	−36.0132	−1.57475	−0.787373	+	0.616477i	$0.788559\pi$
$524$	0	0
$525$	22.0344	0.961662
$526$	0	0
$527$	0	0
$528$	0	0
$529$	46.3262	2.01418
$530$	0	0
$531$	0.180340	0.00782608
$532$	0	0
$533$	−8.18034	−0.354330
$534$	0	0
$535$	−25.8541	−1.11777
$536$	0	0
$537$	−11.9443	−0.515433
$538$	0	0
$539$	1.20163	0.0517577
$540$	0	0
$541$	30.2705	1.30143	0.650715	−	0.759322i	$-0.274469\pi$
$541$	30.2705	1.30143	0.650715	+	0.759322i	$0.274469\pi$
$542$	0	0
$543$	11.3262	0.486055
$544$	0	0
$545$	3.85410	0.165092
$546$	0	0
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	0	0
$549$	2.96556	0.126567
$550$	0	0
$551$	32.0344	1.36471
$552$	0	0
$553$	−13.8197	−0.587672
$554$	0	0
$555$	−57.5967	−2.44485
$556$	0	0
$557$	15.4164	0.653214	0.326607	−	0.945160i	$-0.394095\pi$
$557$	15.4164	0.653214	0.326607	+	0.945160i	$0.394095\pi$
$558$	0	0
$559$	−6.02129	−0.254673
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.1803	−0.429050	−0.214525	−	0.976718i	$-0.568820\pi$
$563$	−10.1803	−0.429050	−0.214525	+	0.976718i	$0.568820\pi$
$564$	0	0
$565$	10.0902	0.424497
$566$	0	0
$567$	10.6525	0.447362
$568$	0	0
$569$	36.5967	1.53422	0.767108	−	0.641518i	$-0.221695\pi$
$569$	36.5967	1.53422	0.767108	+	0.641518i	$0.221695\pi$
$570$	0	0
$571$	−11.7984	−0.493747	−0.246873	−	0.969048i	$-0.579403\pi$
$571$	−11.7984	−0.493747	−0.246873	+	0.969048i	$0.579403\pi$
$572$	0	0
$573$	−21.8541	−0.912968
$574$	0	0
$575$	82.0476	3.42162
$576$	0	0
$577$	−21.1803	−0.881749	−0.440875	−	0.897569i	$-0.645332\pi$
$577$	−21.1803	−0.881749	−0.440875	+	0.897569i	$0.645332\pi$
$578$	0	0
$579$	−31.3262	−1.30187
$580$	0	0
$581$	−21.1803	−0.878709
$582$	0	0
$583$	0.742646	0.0307572
$584$	0	0
$585$	6.02129	0.248950
$586$	0	0
$587$	−0.965558	−0.0398528	−0.0199264	−	0.999801i	$-0.506343\pi$
$587$	−0.965558	−0.0398528	−0.0199264	+	0.999801i	$0.506343\pi$
$588$	0	0
$589$	−34.2705	−1.41209
$590$	0	0
$591$	36.1246	1.48597
$592$	0	0
$593$	−13.3607	−0.548657	−0.274329	−	0.961636i	$-0.588456\pi$
$593$	−13.3607	−0.548657	−0.274329	+	0.961636i	$0.588456\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.854102	0.0349561
$598$	0	0
$599$	−1.87539	−0.0766263	−0.0383131	−	0.999266i	$-0.512198\pi$
$599$	−1.87539	−0.0766263	−0.0383131	+	0.999266i	$0.512198\pi$
$600$	0	0
$601$	46.7426	1.90667	0.953336	−	0.301911i	$-0.0976245\pi$
$601$	46.7426	1.90667	0.953336	+	0.301911i	$0.0976245\pi$
$602$	0	0
$603$	−1.41641	−0.0576806
$604$	0	0
$605$	42.1803	1.71487
$606$	0	0
$607$	−7.29180	−0.295965	−0.147982	−	0.988990i	$-0.547278\pi$
$607$	−7.29180	−0.295965	−0.147982	+	0.988990i	$0.547278\pi$
$608$	0	0
$609$	19.7984	0.802271
$610$	0	0
$611$	−41.8673	−1.69377
$612$	0	0
$613$	3.70820	0.149773	0.0748865	−	0.997192i	$-0.476141\pi$
$613$	3.70820	0.149773	0.0748865	+	0.997192i	$0.476141\pi$
$614$	0	0
$615$	−12.4721	−0.502925
$616$	0	0
$617$	17.0000	0.684394	0.342197	−	0.939628i	$-0.388829\pi$
$617$	17.0000	0.684394	0.342197	+	0.939628i	$0.388829\pi$
$618$	0	0
$619$	33.0689	1.32915	0.664575	−	0.747221i	$-0.268612\pi$
$619$	33.0689	1.32915	0.664575	+	0.747221i	$0.268612\pi$
$620$	0	0
$621$	45.5623	1.82835
$622$	0	0
$623$	6.18034	0.247610
$624$	0	0
$625$	22.8328	0.913313
$626$	0	0
$627$	1.38197	0.0551904
$628$	0	0
$629$	0	0
$630$	0	0
$631$	5.21478	0.207597	0.103799	−	0.994598i	$-0.466900\pi$
$631$	5.21478	0.207597	0.103799	+	0.994598i	$0.466900\pi$
$632$	0	0
$633$	11.9443	0.474742
$634$	0	0
$635$	13.3820	0.531047
$636$	0	0
$637$	−20.8197	−0.824905
$638$	0	0
$639$	−4.18034	−0.165372
$640$	0	0
$641$	6.74265	0.266318	0.133159	−	0.991095i	$-0.457488\pi$
$641$	6.74265	0.266318	0.133159	+	0.991095i	$0.457488\pi$
$642$	0	0
$643$	−42.5410	−1.67765	−0.838827	−	0.544398i	$-0.816758\pi$
$643$	−42.5410	−1.67765	−0.838827	+	0.544398i	$0.816758\pi$
$644$	0	0
$645$	−9.18034	−0.361476
$646$	0	0
$647$	32.7082	1.28589	0.642946	−	0.765911i	$-0.277712\pi$
$647$	32.7082	1.28589	0.642946	+	0.765911i	$0.277712\pi$
$648$	0	0
$649$	0.111456	0.00437504
$650$	0	0
$651$	−21.1803	−0.830123
$652$	0	0
$653$	−26.1246	−1.02234	−0.511168	−	0.859481i	$-0.670787\pi$
$653$	−26.1246	−1.02234	−0.511168	+	0.859481i	$0.670787\pi$
$654$	0	0
$655$	−75.9574	−2.96790
$656$	0	0
$657$	1.43769	0.0560898
$658$	0	0
$659$	15.8328	0.616759	0.308379	−	0.951263i	$-0.400213\pi$
$659$	15.8328	0.616759	0.308379	+	0.951263i	$0.400213\pi$
$660$	0	0
$661$	−29.3820	−1.14283	−0.571413	−	0.820663i	$-0.693605\pi$
$661$	−29.3820	−1.14283	−0.571413	+	0.820663i	$0.693605\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19.2705	0.747278
$666$	0	0
$667$	73.7214	2.85450
$668$	0	0
$669$	24.4164	0.943993
$670$	0	0
$671$	1.83282	0.0707551
$672$	0	0
$673$	2.47214	0.0952938	0.0476469	−	0.998864i	$-0.484828\pi$
$673$	2.47214	0.0952938	0.0476469	+	0.998864i	$0.484828\pi$
$674$	0	0
$675$	53.9230	2.07550
$676$	0	0
$677$	−13.6180	−0.523384	−0.261692	−	0.965151i	$-0.584280\pi$
$677$	−13.6180	−0.523384	−0.261692	+	0.965151i	$0.584280\pi$
$678$	0	0
$679$	5.32624	0.204402
$680$	0	0
$681$	−5.61803	−0.215284
$682$	0	0
$683$	−16.7426	−0.640640	−0.320320	−	0.947309i	$-0.603790\pi$
$683$	−16.7426	−0.640640	−0.320320	+	0.947309i	$0.603790\pi$
$684$	0	0
$685$	26.0689	0.996041
$686$	0	0
$687$	8.00000	0.305219
$688$	0	0
$689$	−12.8673	−0.490203
$690$	0	0
$691$	−48.9443	−1.86193	−0.930964	−	0.365111i	$-0.881031\pi$
$691$	−48.9443	−1.86193	−0.930964	+	0.365111i	$0.881031\pi$
$692$	0	0
$693$	−0.124612	−0.00473361
$694$	0	0
$695$	33.2148	1.25991
$696$	0	0
$697$	0	0
$698$	0	0
$699$	13.2361	0.500634
$700$	0	0
$701$	18.8885	0.713410	0.356705	−	0.934217i	$-0.383900\pi$
$701$	18.8885	0.713410	0.356705	+	0.934217i	$0.383900\pi$
$702$	0	0
$703$	−33.4164	−1.26032
$704$	0	0
$705$	−63.8328	−2.40408
$706$	0	0
$707$	−9.27051	−0.348653
$708$	0	0
$709$	−5.61803	−0.210990	−0.105495	−	0.994420i	$-0.533643\pi$
$709$	−5.61803	−0.210990	−0.105495	+	0.994420i	$0.533643\pi$
$710$	0	0
$711$	−3.81966	−0.143248
$712$	0	0
$713$	−78.8673	−2.95360
$714$	0	0
$715$	3.72136	0.139171
$716$	0	0
$717$	−13.3262	−0.497677
$718$	0	0
$719$	−3.94427	−0.147097	−0.0735483	−	0.997292i	$-0.523432\pi$
$719$	−3.94427	−0.147097	−0.0735483	+	0.997292i	$0.523432\pi$
$720$	0	0
$721$	13.0132	0.484635
$722$	0	0
$723$	38.8885	1.44628
$724$	0	0
$725$	87.2492	3.24035
$726$	0	0
$727$	35.8885	1.33103	0.665516	−	0.746383i	$-0.268211\pi$
$727$	35.8885	1.33103	0.665516	+	0.746383i	$0.268211\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−31.7984	−1.17450	−0.587250	−	0.809406i	$-0.699789\pi$
$733$	−31.7984	−1.17450	−0.587250	+	0.809406i	$0.699789\pi$
$734$	0	0
$735$	−31.7426	−1.17085
$736$	0	0
$737$	−0.875388	−0.0322453
$738$	0	0
$739$	−15.5410	−0.571686	−0.285843	−	0.958277i	$-0.592274\pi$
$739$	−15.5410	−0.571686	−0.285843	+	0.958277i	$0.592274\pi$
$740$	0	0
$741$	−23.9443	−0.879615
$742$	0	0
$743$	−35.9443	−1.31867	−0.659334	−	0.751850i	$-0.729162\pi$
$743$	−35.9443	−1.31867	−0.659334	+	0.751850i	$0.729162\pi$
$744$	0	0
$745$	37.4164	1.37083
$746$	0	0
$747$	−5.85410	−0.214190
$748$	0	0
$749$	−9.27051	−0.338737
$750$	0	0
$751$	40.1246	1.46417	0.732084	−	0.681214i	$-0.238548\pi$
$751$	40.1246	1.46417	0.732084	+	0.681214i	$0.238548\pi$
$752$	0	0
$753$	−22.3262	−0.813613
$754$	0	0
$755$	44.9098	1.63444
$756$	0	0
$757$	−38.7771	−1.40938	−0.704689	−	0.709517i	$-0.748913\pi$
$757$	−38.7771	−1.40938	−0.704689	+	0.709517i	$0.748913\pi$
$758$	0	0
$759$	3.18034	0.115439
$760$	0	0
$761$	−4.52786	−0.164135	−0.0820675	−	0.996627i	$-0.526152\pi$
$761$	−4.52786	−0.164135	−0.0820675	+	0.996627i	$0.526152\pi$
$762$	0	0
$763$	1.38197	0.0500305
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.93112	−0.0697286
$768$	0	0
$769$	20.6738	0.745515	0.372757	−	0.927929i	$-0.378412\pi$
$769$	20.6738	0.745515	0.372757	+	0.927929i	$0.378412\pi$
$770$	0	0
$771$	1.94427	0.0700212
$772$	0	0
$773$	−20.3050	−0.730318	−0.365159	−	0.930945i	$-0.618985\pi$
$773$	−20.3050	−0.730318	−0.365159	+	0.930945i	$0.618985\pi$
$774$	0	0
$775$	−93.3394	−3.35285
$776$	0	0
$777$	−20.6525	−0.740903
$778$	0	0
$779$	−7.23607	−0.259259
$780$	0	0
$781$	−2.58359	−0.0924482
$782$	0	0
$783$	48.4508	1.73149
$784$	0	0
$785$	−56.3394	−2.01084
$786$	0	0
$787$	−18.6525	−0.664889	−0.332444	−	0.943123i	$-0.607873\pi$
$787$	−18.6525	−0.664889	−0.332444	+	0.943123i	$0.607873\pi$
$788$	0	0
$789$	46.2148	1.64529
$790$	0	0
$791$	3.61803	0.128642
$792$	0	0
$793$	−31.7558	−1.12768
$794$	0	0
$795$	−19.6180	−0.695780
$796$	0	0
$797$	2.52786	0.0895415	0.0447708	−	0.998997i	$-0.485744\pi$
$797$	2.52786	0.0895415	0.0447708	+	0.998997i	$0.485744\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	1.70820	0.0603564
$802$	0	0
$803$	0.888544	0.0313560
$804$	0	0
$805$	44.3475	1.56304
$806$	0	0
$807$	31.7426	1.11739
$808$	0	0
$809$	39.0132	1.37163	0.685815	−	0.727776i	$-0.259446\pi$
$809$	39.0132	1.37163	0.685815	+	0.727776i	$0.259446\pi$
$810$	0	0
$811$	23.7082	0.832508	0.416254	−	0.909248i	$-0.363343\pi$
$811$	23.7082	0.832508	0.416254	+	0.909248i	$0.363343\pi$
$812$	0	0
$813$	17.4721	0.612775
$814$	0	0
$815$	21.0902	0.738756
$816$	0	0
$817$	−5.32624	−0.186341
$818$	0	0
$819$	2.15905	0.0754434
$820$	0	0
$821$	−7.29180	−0.254485	−0.127243	−	0.991872i	$-0.540613\pi$
$821$	−7.29180	−0.254485	−0.127243	+	0.991872i	$0.540613\pi$
$822$	0	0
$823$	−53.9443	−1.88038	−0.940190	−	0.340652i	$-0.889352\pi$
$823$	−53.9443	−1.88038	−0.940190	+	0.340652i	$0.889352\pi$
$824$	0	0
$825$	3.76393	0.131043
$826$	0	0
$827$	−2.52786	−0.0879024	−0.0439512	−	0.999034i	$-0.513995\pi$
$827$	−2.52786	−0.0879024	−0.0439512	+	0.999034i	$0.513995\pi$
$828$	0	0
$829$	−50.6525	−1.75923	−0.879617	−	0.475683i	$-0.842201\pi$
$829$	−50.6525	−1.75923	−0.879617	+	0.475683i	$0.842201\pi$
$830$	0	0
$831$	43.4164	1.50610
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.54102	−0.191755
$836$	0	0
$837$	−51.8328	−1.79160
$838$	0	0
$839$	−30.6525	−1.05824	−0.529120	−	0.848547i	$-0.677478\pi$
$839$	−30.6525	−1.05824	−0.529120	+	0.848547i	$0.677478\pi$
$840$	0	0
$841$	49.3951	1.70328
$842$	0	0
$843$	−5.00000	−0.172209
$844$	0	0
$845$	−14.3738	−0.494475
$846$	0	0
$847$	15.1246	0.519688
$848$	0	0
$849$	−22.4721	−0.771242
$850$	0	0
$851$	−76.9017	−2.63616
$852$	0	0
$853$	32.5279	1.11373	0.556866	−	0.830602i	$-0.312004\pi$
$853$	32.5279	1.11373	0.556866	+	0.830602i	$0.312004\pi$
$854$	0	0
$855$	5.32624	0.182153
$856$	0	0
$857$	−39.1803	−1.33837	−0.669187	−	0.743094i	$-0.733358\pi$
$857$	−39.1803	−1.33837	−0.669187	+	0.743094i	$0.733358\pi$
$858$	0	0
$859$	22.5410	0.769090	0.384545	−	0.923106i	$-0.374358\pi$
$859$	22.5410	0.769090	0.384545	+	0.923106i	$0.374358\pi$
$860$	0	0
$861$	−4.47214	−0.152410
$862$	0	0
$863$	49.5623	1.68712	0.843560	−	0.537035i	$-0.180456\pi$
$863$	49.5623	1.68712	0.843560	+	0.537035i	$0.180456\pi$
$864$	0	0
$865$	4.97871	0.169281
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.36068	−0.0800806
$870$	0	0
$871$	15.1672	0.513920
$872$	0	0
$873$	1.47214	0.0498243
$874$	0	0
$875$	25.8541	0.874028
$876$	0	0
$877$	33.7426	1.13941	0.569704	−	0.821850i	$-0.307058\pi$
$877$	33.7426	1.13941	0.569704	+	0.821850i	$0.307058\pi$
$878$	0	0
$879$	31.8885	1.07557
$880$	0	0
$881$	−11.9098	−0.401252	−0.200626	−	0.979668i	$-0.564298\pi$
$881$	−11.9098	−0.401252	−0.200626	+	0.979668i	$0.564298\pi$
$882$	0	0
$883$	34.3820	1.15705	0.578523	−	0.815666i	$-0.303629\pi$
$883$	34.3820	1.15705	0.578523	+	0.815666i	$0.303629\pi$
$884$	0	0
$885$	−2.94427	−0.0989706
$886$	0	0
$887$	−17.5623	−0.589685	−0.294842	−	0.955546i	$-0.595267\pi$
$887$	−17.5623	−0.589685	−0.294842	+	0.955546i	$0.595267\pi$
$888$	0	0
$889$	4.79837	0.160932
$890$	0	0
$891$	1.81966	0.0609609
$892$	0	0
$893$	−37.0344	−1.23931
$894$	0	0
$895$	−28.4508	−0.951007
$896$	0	0
$897$	−55.1033	−1.83985
$898$	0	0
$899$	−83.8673	−2.79713
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−3.29180	−0.109544
$904$	0	0
$905$	26.9787	0.896803
$906$	0	0
$907$	−12.2705	−0.407436	−0.203718	−	0.979030i	$-0.565303\pi$
$907$	−12.2705	−0.407436	−0.203718	+	0.979030i	$0.565303\pi$
$908$	0	0
$909$	−2.56231	−0.0849863
$910$	0	0
$911$	−16.1459	−0.534937	−0.267469	−	0.963567i	$-0.586187\pi$
$911$	−16.1459	−0.534937	−0.267469	+	0.963567i	$0.586187\pi$
$912$	0	0
$913$	−3.61803	−0.119739
$914$	0	0
$915$	−48.4164	−1.60060
$916$	0	0
$917$	−27.2361	−0.899414
$918$	0	0
$919$	−42.9230	−1.41590	−0.707949	−	0.706263i	$-0.750380\pi$
$919$	−42.9230	−1.41590	−0.707949	+	0.706263i	$0.750380\pi$
$920$	0	0
$921$	33.3607	1.09927
$922$	0	0
$923$	44.7639	1.47342
$924$	0	0
$925$	−91.0132	−2.99249
$926$	0	0
$927$	3.59675	0.118133
$928$	0	0
$929$	33.8673	1.11115	0.555574	−	0.831467i	$-0.312498\pi$
$929$	33.8673	1.11115	0.555574	+	0.831467i	$0.312498\pi$
$930$	0	0
$931$	−18.4164	−0.603573
$932$	0	0
$933$	37.2705	1.22018
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−18.3951	−0.600942	−0.300471	−	0.953791i	$-0.597144\pi$
$937$	−18.3951	−0.600942	−0.300471	+	0.953791i	$0.597144\pi$
$938$	0	0
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	−39.6738	−1.29333	−0.646664	−	0.762775i	$-0.723836\pi$
$941$	−39.6738	−1.29333	−0.646664	+	0.762775i	$0.723836\pi$
$942$	0	0
$943$	−16.6525	−0.542279
$944$	0	0
$945$	29.1459	0.948116
$946$	0	0
$947$	−16.2361	−0.527601	−0.263801	−	0.964577i	$-0.584976\pi$
$947$	−16.2361	−0.527601	−0.263801	+	0.964577i	$0.584976\pi$
$948$	0	0
$949$	−15.3951	−0.499747
$950$	0	0
$951$	40.7426	1.32117
$952$	0	0
$953$	0.798374	0.0258619	0.0129309	−	0.999916i	$-0.495884\pi$
$953$	0.798374	0.0258619	0.0129309	+	0.999916i	$0.495884\pi$
$954$	0	0
$955$	−52.0557	−1.68448
$956$	0	0
$957$	3.38197	0.109323
$958$	0	0
$959$	9.34752	0.301847
$960$	0	0
$961$	58.7214	1.89424
$962$	0	0
$963$	−2.56231	−0.0825692
$964$	0	0
$965$	−74.6180	−2.40204
$966$	0	0
$967$	9.20163	0.295904	0.147952	−	0.988995i	$-0.452732\pi$
$967$	9.20163	0.295904	0.147952	+	0.988995i	$0.452732\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.909830	0.0291978	0.0145989	−	0.999893i	$-0.495353\pi$
$971$	0.909830	0.0291978	0.0145989	+	0.999893i	$0.495353\pi$
$972$	0	0
$973$	11.9098	0.381812
$974$	0	0
$975$	−65.2148	−2.08854
$976$	0	0
$977$	32.8541	1.05110	0.525548	−	0.850764i	$-0.323860\pi$
$977$	32.8541	1.05110	0.525548	+	0.850764i	$0.323860\pi$
$978$	0	0
$979$	1.05573	0.0337412
$980$	0	0
$981$	0.381966	0.0121952
$982$	0	0
$983$	−33.0557	−1.05431	−0.527157	−	0.849768i	$-0.676742\pi$
$983$	−33.0557	−1.05431	−0.527157	+	0.849768i	$0.676742\pi$
$984$	0	0
$985$	86.0476	2.74170
$986$	0	0
$987$	−22.8885	−0.728550
$988$	0	0
$989$	−12.2574	−0.389761
$990$	0	0
$991$	3.21478	0.102121	0.0510605	−	0.998696i	$-0.483740\pi$
$991$	3.21478	0.102121	0.0510605	+	0.998696i	$0.483740\pi$
$992$	0	0
$993$	−7.76393	−0.246381
$994$	0	0
$995$	2.03444	0.0644961
$996$	0	0
$997$	18.4508	0.584344	0.292172	−	0.956366i	$-0.405622\pi$
$997$	18.4508	0.584344	0.292172	+	0.956366i	$0.405622\pi$
$998$	0	0
$999$	−50.5410	−1.59905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.2.a.f.1.1	✓	2
4.3	odd	2		4624.2.a.w.1.2		2
17.4	even	4		2312.2.b.k.577.4		4
17.13	even	4		2312.2.b.k.577.1		4
17.16	even	2		2312.2.a.l.1.2	yes	2
68.67	odd	2		4624.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2312.2.a.f.1.1	✓	2	1.1	even	1	trivial
2312.2.a.l.1.2	yes	2	17.16	even	2
2312.2.b.k.577.1		4	17.13	even	4
2312.2.b.k.577.4		4	17.4	even	4
4624.2.a.i.1.1		2	68.67	odd	2
4624.2.a.w.1.2		2	4.3	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)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} -3.85410 q^{5} -1.38197 q^{7} -0.381966 q^{9} -0.236068 q^{11} +4.09017 q^{13} +6.23607 q^{15} +3.61803 q^{19} +2.23607 q^{21} +8.32624 q^{23} +9.85410 q^{25} +5.47214 q^{27} +8.85410 q^{29} -9.47214 q^{31} +0.381966 q^{33} +5.32624 q^{35} -9.23607 q^{37} -6.61803 q^{39} -2.00000 q^{41} -1.47214 q^{43} +1.47214 q^{45} -10.2361 q^{47} -5.09017 q^{49} -3.14590 q^{53} +0.909830 q^{55} -5.85410 q^{57} -0.472136 q^{59} -7.76393 q^{61} +0.527864 q^{63} -15.7639 q^{65} +3.70820 q^{67} -13.4721 q^{69} +10.9443 q^{71} -3.76393 q^{73} -15.9443 q^{75} +0.326238 q^{77} +10.0000 q^{79} -7.70820 q^{81} +15.3262 q^{83} -14.3262 q^{87} -4.47214 q^{89} -5.65248 q^{91} +15.3262 q^{93} -13.9443 q^{95} -3.85410 q^{97} +0.0901699 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} - 5 q^{7} - 3 q^{9} + 4 q^{11} - 3 q^{13} + 8 q^{15} + 5 q^{19} + q^{23} + 13 q^{25} + 2 q^{27} + 11 q^{29} - 10 q^{31} + 3 q^{33} - 5 q^{35} - 14 q^{37} - 11 q^{39} - 4 q^{41} + 6 q^{43}+ \cdots - 11 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 - 5 * q^7 - 3 * q^9 + 4 * q^11 - 3 * q^13 + 8 * q^15 + 5 * q^19 + q^23 + 13 * q^25 + 2 * q^27 + 11 * q^29 - 10 * q^31 + 3 * q^33 - 5 * q^35 - 14 * q^37 - 11 * q^39 - 4 * q^41 + 6 * q^43 - 6 * q^45 - 16 * q^47 + q^49 - 13 * q^53 + 13 * q^55 - 5 * q^57 + 8 * q^59 - 20 * q^61 + 10 * q^63 - 36 * q^65 - 6 * q^67 - 18 * q^69 + 4 * q^71 - 12 * q^73 - 14 * q^75 - 15 * q^77 + 20 * q^79 - 2 * q^81 + 15 * q^83 - 13 * q^87 + 20 * q^91 + 15 * q^93 - 10 * q^95 - q^97 - 11 * q^99

Introduction

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