LMFDB - Embedded newform 2320.2.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(1,2320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2320.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-2,0,3,0,-2,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70928 q^{3} +1.00000 q^{5} +0.630898 q^{7} -0.0783777 q^{9} +0.630898 q^{11} +4.34017 q^{13} +1.70928 q^{15} +4.63090 q^{17} +1.70928 q^{19} +1.07838 q^{21} -1.70928 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.00000 q^{29} -1.70928 q^{31} +1.07838 q^{33} +0.630898 q^{35} +3.36910 q^{37} +7.41855 q^{39} +6.49693 q^{41} -6.38962 q^{43} -0.0783777 q^{45} -9.12783 q^{47} -6.60197 q^{49} +7.91548 q^{51} +5.23513 q^{53} +0.630898 q^{55} +2.92162 q^{57} +4.49693 q^{59} +12.4391 q^{61} -0.0494483 q^{63} +4.34017 q^{65} -1.70928 q^{67} -2.92162 q^{69} -0.183417 q^{71} -1.70928 q^{73} +1.70928 q^{75} +0.398032 q^{77} +13.8927 q^{79} -8.75872 q^{81} +13.3112 q^{83} +4.63090 q^{85} +1.70928 q^{87} +1.10504 q^{89} +2.73820 q^{91} -2.92162 q^{93} +1.70928 q^{95} -7.28458 q^{97} -0.0494483 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} + 10 q^{17} - 2 q^{19} + 2 q^{23} + 3 q^{25} - 8 q^{27} + 3 q^{29} + 2 q^{31} - 2 q^{35} + 14 q^{37} + 8 q^{39} + 2 q^{41} + 10 q^{43}+ \cdots + 18 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 + 10 * q^17 - 2 * q^19 + 2 * q^23 + 3 * q^25 - 8 * q^27 + 3 * q^29 + 2 * q^31 - 2 * q^35 + 14 * q^37 + 8 * q^39 + 2 * q^41 + 10 * q^43 + 3 * q^45 - 6 * q^47 - q^49 - 8 * q^51 + 6 * q^53 - 2 * q^55 + 12 * q^57 - 4 * q^59 - 10 * q^61 + 18 * q^63 + 2 * q^65 + 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 20 * q^77 + 30 * q^79 - q^81 + 14 * q^83 + 10 * q^85 - 2 * q^87 + 2 * q^89 + 16 * q^91 - 12 * q^93 - 2 * q^95 + 6 * q^97 + 18 * 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
$3$	1.70928	0.986851	0.493425	−	0.869788i	$-0.335745\pi$
$3$	1.70928	0.986851	0.493425	+	0.869788i	$0.335745\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.630898	0.238457	0.119228	−	0.992867i	$-0.461958\pi$
$7$	0.630898	0.238457	0.119228	+	0.992867i	$0.461958\pi$
$8$	0	0
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	0.630898	0.190223	0.0951114	−	0.995467i	$-0.469679\pi$
$11$	0.630898	0.190223	0.0951114	+	0.995467i	$0.469679\pi$
$12$	0	0
$13$	4.34017	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$13$	4.34017	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$14$	0	0
$15$	1.70928	0.441333
$16$	0	0
$17$	4.63090	1.12316	0.561579	−	0.827423i	$-0.310194\pi$
$17$	4.63090	1.12316	0.561579	+	0.827423i	$0.310194\pi$
$18$	0	0
$19$	1.70928	0.392135	0.196067	−	0.980590i	$-0.437183\pi$
$19$	1.70928	0.392135	0.196067	+	0.980590i	$0.437183\pi$
$20$	0	0
$21$	1.07838	0.235321
$22$	0	0
$23$	−1.70928	−0.356409	−0.178204	−	0.983994i	$-0.557029\pi$
$23$	−1.70928	−0.356409	−0.178204	+	0.983994i	$0.557029\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.26180	−1.01263
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.70928	−0.306995	−0.153497	−	0.988149i	$-0.549054\pi$
$31$	−1.70928	−0.306995	−0.153497	+	0.988149i	$0.549054\pi$
$32$	0	0
$33$	1.07838	0.187721
$34$	0	0
$35$	0.630898	0.106641
$36$	0	0
$37$	3.36910	0.553877	0.276939	−	0.960888i	$-0.410680\pi$
$37$	3.36910	0.553877	0.276939	+	0.960888i	$0.410680\pi$
$38$	0	0
$39$	7.41855	1.18792
$40$	0	0
$41$	6.49693	1.01465	0.507325	−	0.861755i	$-0.330634\pi$
$41$	6.49693	1.01465	0.507325	+	0.861755i	$0.330634\pi$
$42$	0	0
$43$	−6.38962	−0.974408	−0.487204	−	0.873288i	$-0.661983\pi$
$43$	−6.38962	−0.974408	−0.487204	+	0.873288i	$0.661983\pi$
$44$	0	0
$45$	−0.0783777	−0.0116839
$46$	0	0
$47$	−9.12783	−1.33143	−0.665715	−	0.746206i	$-0.731873\pi$
$47$	−9.12783	−1.33143	−0.665715	+	0.746206i	$0.731873\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	7.91548	1.10839
$52$	0	0
$53$	5.23513	0.719101	0.359550	−	0.933126i	$-0.382930\pi$
$53$	5.23513	0.719101	0.359550	+	0.933126i	$0.382930\pi$
$54$	0	0
$55$	0.630898	0.0850702
$56$	0	0
$57$	2.92162	0.386978
$58$	0	0
$59$	4.49693	0.585450	0.292725	−	0.956197i	$-0.405438\pi$
$59$	4.49693	0.585450	0.292725	+	0.956197i	$0.405438\pi$
$60$	0	0
$61$	12.4391	1.59266	0.796330	−	0.604863i	$-0.206772\pi$
$61$	12.4391	1.59266	0.796330	+	0.604863i	$0.206772\pi$
$62$	0	0
$63$	−0.0494483	−0.00622990
$64$	0	0
$65$	4.34017	0.538332
$66$	0	0
$67$	−1.70928	−0.208821	−0.104411	−	0.994534i	$-0.533296\pi$
$67$	−1.70928	−0.208821	−0.104411	+	0.994534i	$0.533296\pi$
$68$	0	0
$69$	−2.92162	−0.351722
$70$	0	0
$71$	−0.183417	−0.0217677	−0.0108838	−	0.999941i	$-0.503464\pi$
$71$	−0.183417	−0.0217677	−0.0108838	+	0.999941i	$0.503464\pi$
$72$	0	0
$73$	−1.70928	−0.200056	−0.100028	−	0.994985i	$-0.531893\pi$
$73$	−1.70928	−0.200056	−0.100028	+	0.994985i	$0.531893\pi$
$74$	0	0
$75$	1.70928	0.197370
$76$	0	0
$77$	0.398032	0.0453599
$78$	0	0
$79$	13.8927	1.56305	0.781525	−	0.623874i	$-0.214442\pi$
$79$	13.8927	1.56305	0.781525	+	0.623874i	$0.214442\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	0	0
$83$	13.3112	1.46110	0.730549	−	0.682860i	$-0.239264\pi$
$83$	13.3112	1.46110	0.730549	+	0.682860i	$0.239264\pi$
$84$	0	0
$85$	4.63090	0.502291
$86$	0	0
$87$	1.70928	0.183254
$88$	0	0
$89$	1.10504	0.117134	0.0585670	−	0.998283i	$-0.481347\pi$
$89$	1.10504	0.117134	0.0585670	+	0.998283i	$0.481347\pi$
$90$	0	0
$91$	2.73820	0.287042
$92$	0	0
$93$	−2.92162	−0.302958
$94$	0	0
$95$	1.70928	0.175368
$96$	0	0
$97$	−7.28458	−0.739637	−0.369819	−	0.929104i	$-0.620580\pi$
$97$	−7.28458	−0.739637	−0.369819	+	0.929104i	$0.620580\pi$
$98$	0	0
$99$	−0.0494483	−0.00496974
$100$	0	0
$101$	3.26180	0.324561	0.162280	−	0.986745i	$-0.448115\pi$
$101$	3.26180	0.324561	0.162280	+	0.986745i	$0.448115\pi$
$102$	0	0
$103$	15.2846	1.50603	0.753017	−	0.658001i	$-0.228598\pi$
$103$	15.2846	1.50603	0.753017	+	0.658001i	$0.228598\pi$
$104$	0	0
$105$	1.07838	0.105239
$106$	0	0
$107$	−2.10731	−0.203721	−0.101861	−	0.994799i	$-0.532480\pi$
$107$	−2.10731	−0.203721	−0.101861	+	0.994799i	$0.532480\pi$
$108$	0	0
$109$	−9.23513	−0.884565	−0.442283	−	0.896876i	$-0.645831\pi$
$109$	−9.23513	−0.884565	−0.442283	+	0.896876i	$0.645831\pi$
$110$	0	0
$111$	5.75872	0.546594
$112$	0	0
$113$	−3.86603	−0.363686	−0.181843	−	0.983328i	$-0.558206\pi$
$113$	−3.86603	−0.363686	−0.181843	+	0.983328i	$0.558206\pi$
$114$	0	0
$115$	−1.70928	−0.159391
$116$	0	0
$117$	−0.340173	−0.0314490
$118$	0	0
$119$	2.92162	0.267825
$120$	0	0
$121$	−10.6020	−0.963815
$122$	0	0
$123$	11.1050	1.00131
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.54638	−0.758368	−0.379184	−	0.925321i	$-0.623795\pi$
$127$	−8.54638	−0.758368	−0.379184	+	0.925321i	$0.623795\pi$
$128$	0	0
$129$	−10.9216	−0.961595
$130$	0	0
$131$	0.133969	0.0117049	0.00585247	−	0.999983i	$-0.498137\pi$
$131$	0.133969	0.0117049	0.00585247	+	0.999983i	$0.498137\pi$
$132$	0	0
$133$	1.07838	0.0935072
$134$	0	0
$135$	−5.26180	−0.452863
$136$	0	0
$137$	4.63090	0.395644	0.197822	−	0.980238i	$-0.436613\pi$
$137$	4.63090	0.395644	0.197822	+	0.980238i	$0.436613\pi$
$138$	0	0
$139$	−10.2557	−0.869873	−0.434937	−	0.900461i	$-0.643229\pi$
$139$	−10.2557	−0.869873	−0.434937	+	0.900461i	$0.643229\pi$
$140$	0	0
$141$	−15.6020	−1.31392
$142$	0	0
$143$	2.73820	0.228980
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−11.2846	−0.930737
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	1.65983	0.135075	0.0675374	−	0.997717i	$-0.478486\pi$
$151$	1.65983	0.135075	0.0675374	+	0.997717i	$0.478486\pi$
$152$	0	0
$153$	−0.362959	−0.0293435
$154$	0	0
$155$	−1.70928	−0.137292
$156$	0	0
$157$	−11.6514	−0.929884	−0.464942	−	0.885341i	$-0.653925\pi$
$157$	−11.6514	−0.929884	−0.464942	+	0.885341i	$0.653925\pi$
$158$	0	0
$159$	8.94828	0.709645
$160$	0	0
$161$	−1.07838	−0.0849881
$162$	0	0
$163$	−8.23287	−0.644848	−0.322424	−	0.946595i	$-0.604498\pi$
$163$	−8.23287	−0.644848	−0.322424	+	0.946595i	$0.604498\pi$
$164$	0	0
$165$	1.07838	0.0839516
$166$	0	0
$167$	−0.133969	−0.0103668	−0.00518342	−	0.999987i	$-0.501650\pi$
$167$	−0.133969	−0.0103668	−0.00518342	+	0.999987i	$0.501650\pi$
$168$	0	0
$169$	5.83710	0.449008
$170$	0	0
$171$	−0.133969	−0.0102449
$172$	0	0
$173$	6.18342	0.470117	0.235058	−	0.971981i	$-0.424472\pi$
$173$	6.18342	0.470117	0.235058	+	0.971981i	$0.424472\pi$
$174$	0	0
$175$	0.630898	0.0476914
$176$	0	0
$177$	7.68649	0.577752
$178$	0	0
$179$	−22.0410	−1.64742	−0.823712	−	0.567008i	$-0.808101\pi$
$179$	−22.0410	−1.64742	−0.823712	+	0.567008i	$0.808101\pi$
$180$	0	0
$181$	10.5958	0.787581	0.393791	−	0.919200i	$-0.371163\pi$
$181$	10.5958	0.787581	0.393791	+	0.919200i	$0.371163\pi$
$182$	0	0
$183$	21.2618	1.57172
$184$	0	0
$185$	3.36910	0.247701
$186$	0	0
$187$	2.92162	0.213650
$188$	0	0
$189$	−3.31965	−0.241469
$190$	0	0
$191$	−4.26406	−0.308537	−0.154268	−	0.988029i	$-0.549302\pi$
$191$	−4.26406	−0.308537	−0.154268	+	0.988029i	$0.549302\pi$
$192$	0	0
$193$	−1.31124	−0.0943854	−0.0471927	−	0.998886i	$-0.515027\pi$
$193$	−1.31124	−0.0943854	−0.0471927	+	0.998886i	$0.515027\pi$
$194$	0	0
$195$	7.41855	0.531253
$196$	0	0
$197$	24.6225	1.75428	0.877140	−	0.480235i	$-0.159449\pi$
$197$	24.6225	1.75428	0.877140	+	0.480235i	$0.159449\pi$
$198$	0	0
$199$	9.97334	0.706991	0.353496	−	0.935436i	$-0.384993\pi$
$199$	9.97334	0.706991	0.353496	+	0.935436i	$0.384993\pi$
$200$	0	0
$201$	−2.92162	−0.206075
$202$	0	0
$203$	0.630898	0.0442803
$204$	0	0
$205$	6.49693	0.453765
$206$	0	0
$207$	0.133969	0.00931150
$208$	0	0
$209$	1.07838	0.0745929
$210$	0	0
$211$	13.4101	0.923192	0.461596	−	0.887090i	$-0.347277\pi$
$211$	13.4101	0.923192	0.461596	+	0.887090i	$0.347277\pi$
$212$	0	0
$213$	−0.313511	−0.0214814
$214$	0	0
$215$	−6.38962	−0.435769
$216$	0	0
$217$	−1.07838	−0.0732051
$218$	0	0
$219$	−2.92162	−0.197425
$220$	0	0
$221$	20.0989	1.35200
$222$	0	0
$223$	−6.07611	−0.406886	−0.203443	−	0.979087i	$-0.565213\pi$
$223$	−6.07611	−0.406886	−0.203443	+	0.979087i	$0.565213\pi$
$224$	0	0
$225$	−0.0783777	−0.00522518
$226$	0	0
$227$	−20.2329	−1.34290	−0.671451	−	0.741049i	$-0.734329\pi$
$227$	−20.2329	−1.34290	−0.671451	+	0.741049i	$0.734329\pi$
$228$	0	0
$229$	−21.8843	−1.44615	−0.723077	−	0.690767i	$-0.757273\pi$
$229$	−21.8843	−1.44615	−0.723077	+	0.690767i	$0.757273\pi$
$230$	0	0
$231$	0.680346	0.0447635
$232$	0	0
$233$	−10.1978	−0.668080	−0.334040	−	0.942559i	$-0.608412\pi$
$233$	−10.1978	−0.668080	−0.334040	+	0.942559i	$0.608412\pi$
$234$	0	0
$235$	−9.12783	−0.595434
$236$	0	0
$237$	23.7464	1.54250
$238$	0	0
$239$	−5.17727	−0.334890	−0.167445	−	0.985881i	$-0.553552\pi$
$239$	−5.17727	−0.334890	−0.167445	+	0.985881i	$0.553552\pi$
$240$	0	0
$241$	−25.5174	−1.64372	−0.821862	−	0.569687i	$-0.807064\pi$
$241$	−25.5174	−1.64372	−0.821862	+	0.569687i	$0.807064\pi$
$242$	0	0
$243$	0.814315	0.0522383
$244$	0	0
$245$	−6.60197	−0.421784
$246$	0	0
$247$	7.41855	0.472031
$248$	0	0
$249$	22.7526	1.44189
$250$	0	0
$251$	−24.0494	−1.51799	−0.758994	−	0.651098i	$-0.774309\pi$
$251$	−24.0494	−1.51799	−0.758994	+	0.651098i	$0.774309\pi$
$252$	0	0
$253$	−1.07838	−0.0677970
$254$	0	0
$255$	7.91548	0.495687
$256$	0	0
$257$	−22.0144	−1.37322	−0.686610	−	0.727026i	$-0.740902\pi$
$257$	−22.0144	−1.37322	−0.686610	+	0.727026i	$0.740902\pi$
$258$	0	0
$259$	2.12556	0.132076
$260$	0	0
$261$	−0.0783777	−0.00485146
$262$	0	0
$263$	−1.12783	−0.0695447	−0.0347724	−	0.999395i	$-0.511071\pi$
$263$	−1.12783	−0.0695447	−0.0347724	+	0.999395i	$0.511071\pi$
$264$	0	0
$265$	5.23513	0.321592
$266$	0	0
$267$	1.88882	0.115594
$268$	0	0
$269$	−7.85762	−0.479088	−0.239544	−	0.970886i	$-0.576998\pi$
$269$	−7.85762	−0.479088	−0.239544	+	0.970886i	$0.576998\pi$
$270$	0	0
$271$	6.47414	0.393276	0.196638	−	0.980476i	$-0.436998\pi$
$271$	6.47414	0.393276	0.196638	+	0.980476i	$0.436998\pi$
$272$	0	0
$273$	4.68035	0.283267
$274$	0	0
$275$	0.630898	0.0380446
$276$	0	0
$277$	−18.5958	−1.11731	−0.558657	−	0.829399i	$-0.688683\pi$
$277$	−18.5958	−1.11731	−0.558657	+	0.829399i	$0.688683\pi$
$278$	0	0
$279$	0.133969	0.00802052
$280$	0	0
$281$	8.24128	0.491633	0.245817	−	0.969316i	$-0.420944\pi$
$281$	8.24128	0.491633	0.245817	+	0.969316i	$0.420944\pi$
$282$	0	0
$283$	21.5936	1.28360	0.641802	−	0.766870i	$-0.278187\pi$
$283$	21.5936	1.28360	0.641802	+	0.766870i	$0.278187\pi$
$284$	0	0
$285$	2.92162	0.173062
$286$	0	0
$287$	4.09890	0.241950
$288$	0	0
$289$	4.44521	0.261483
$290$	0	0
$291$	−12.4514	−0.729911
$292$	0	0
$293$	31.0166	1.81201	0.906006	−	0.423265i	$-0.139116\pi$
$293$	31.0166	1.81201	0.906006	+	0.423265i	$0.139116\pi$
$294$	0	0
$295$	4.49693	0.261821
$296$	0	0
$297$	−3.31965	−0.192626
$298$	0	0
$299$	−7.41855	−0.429026
$300$	0	0
$301$	−4.03120	−0.232354
$302$	0	0
$303$	5.57531	0.320293
$304$	0	0
$305$	12.4391	0.712259
$306$	0	0
$307$	−13.3958	−0.764537	−0.382268	−	0.924051i	$-0.624857\pi$
$307$	−13.3958	−0.764537	−0.382268	+	0.924051i	$0.624857\pi$
$308$	0	0
$309$	26.1256	1.48623
$310$	0	0
$311$	24.2784	1.37670	0.688352	−	0.725377i	$-0.258334\pi$
$311$	24.2784	1.37670	0.688352	+	0.725377i	$0.258334\pi$
$312$	0	0
$313$	−0.0266620	−0.00150702	−0.000753512	−	1.00000i	$-0.500240\pi$
$313$	−0.0266620	−0.00150702	−0.000753512		1.00000i	$0.500240\pi$
$314$	0	0
$315$	−0.0494483	−0.00278610
$316$	0	0
$317$	−22.9854	−1.29099	−0.645496	−	0.763764i	$-0.723349\pi$
$317$	−22.9854	−1.29099	−0.645496	+	0.763764i	$0.723349\pi$
$318$	0	0
$319$	0.630898	0.0353235
$320$	0	0
$321$	−3.60197	−0.201042
$322$	0	0
$323$	7.91548	0.440429
$324$	0	0
$325$	4.34017	0.240749
$326$	0	0
$327$	−15.7854	−0.872934
$328$	0	0
$329$	−5.75872	−0.317489
$330$	0	0
$331$	−25.9071	−1.42398	−0.711991	−	0.702189i	$-0.752206\pi$
$331$	−25.9071	−1.42398	−0.711991	+	0.702189i	$0.752206\pi$
$332$	0	0
$333$	−0.264063	−0.0144705
$334$	0	0
$335$	−1.70928	−0.0933877
$336$	0	0
$337$	12.8599	0.700523	0.350261	−	0.936652i	$-0.386093\pi$
$337$	12.8599	0.700523	0.350261	+	0.936652i	$0.386093\pi$
$338$	0	0
$339$	−6.60811	−0.358903
$340$	0	0
$341$	−1.07838	−0.0583974
$342$	0	0
$343$	−8.58145	−0.463355
$344$	0	0
$345$	−2.92162	−0.157295
$346$	0	0
$347$	−13.2579	−0.711722	−0.355861	−	0.934539i	$-0.615812\pi$
$347$	−13.2579	−0.711722	−0.355861	+	0.934539i	$0.615812\pi$
$348$	0	0
$349$	4.21008	0.225360	0.112680	−	0.993631i	$-0.464056\pi$
$349$	4.21008	0.225360	0.112680	+	0.993631i	$0.464056\pi$
$350$	0	0
$351$	−22.8371	−1.21895
$352$	0	0
$353$	8.24128	0.438639	0.219319	−	0.975653i	$-0.429616\pi$
$353$	8.24128	0.438639	0.219319	+	0.975653i	$0.429616\pi$
$354$	0	0
$355$	−0.183417	−0.00973479
$356$	0	0
$357$	4.99386	0.264303
$358$	0	0
$359$	−12.2329	−0.645626	−0.322813	−	0.946463i	$-0.604628\pi$
$359$	−12.2329	−0.645626	−0.322813	+	0.946463i	$0.604628\pi$
$360$	0	0
$361$	−16.0784	−0.846230
$362$	0	0
$363$	−18.1217	−0.951142
$364$	0	0
$365$	−1.70928	−0.0894676
$366$	0	0
$367$	3.55252	0.185440	0.0927200	−	0.995692i	$-0.470444\pi$
$367$	3.55252	0.185440	0.0927200	+	0.995692i	$0.470444\pi$
$368$	0	0
$369$	−0.509215	−0.0265087
$370$	0	0
$371$	3.30283	0.171475
$372$	0	0
$373$	7.94214	0.411228	0.205614	−	0.978633i	$-0.434081\pi$
$373$	7.94214	0.411228	0.205614	+	0.978633i	$0.434081\pi$
$374$	0	0
$375$	1.70928	0.0882666
$376$	0	0
$377$	4.34017	0.223530
$378$	0	0
$379$	−21.5259	−1.10571	−0.552855	−	0.833278i	$-0.686461\pi$
$379$	−21.5259	−1.10571	−0.552855	+	0.833278i	$0.686461\pi$
$380$	0	0
$381$	−14.6081	−0.748396
$382$	0	0
$383$	26.2062	1.33907	0.669537	−	0.742778i	$-0.266492\pi$
$383$	26.2062	1.33907	0.669537	+	0.742778i	$0.266492\pi$
$384$	0	0
$385$	0.398032	0.0202856
$386$	0	0
$387$	0.500804	0.0254573
$388$	0	0
$389$	−22.6947	−1.15067	−0.575334	−	0.817919i	$-0.695128\pi$
$389$	−22.6947	−1.15067	−0.575334	+	0.817919i	$0.695128\pi$
$390$	0	0
$391$	−7.91548	−0.400303
$392$	0	0
$393$	0.228990	0.0115510
$394$	0	0
$395$	13.8927	0.699017
$396$	0	0
$397$	−14.4124	−0.723338	−0.361669	−	0.932307i	$-0.617793\pi$
$397$	−14.4124	−0.723338	−0.361669	+	0.932307i	$0.617793\pi$
$398$	0	0
$399$	1.84324	0.0922776
$400$	0	0
$401$	33.1194	1.65390	0.826952	−	0.562272i	$-0.190073\pi$
$401$	33.1194	1.65390	0.826952	+	0.562272i	$0.190073\pi$
$402$	0	0
$403$	−7.41855	−0.369544
$404$	0	0
$405$	−8.75872	−0.435224
$406$	0	0
$407$	2.12556	0.105360
$408$	0	0
$409$	17.2351	0.852222	0.426111	−	0.904671i	$-0.359883\pi$
$409$	17.2351	0.852222	0.426111	+	0.904671i	$0.359883\pi$
$410$	0	0
$411$	7.91548	0.390442
$412$	0	0
$413$	2.83710	0.139605
$414$	0	0
$415$	13.3112	0.653423
$416$	0	0
$417$	−17.5297	−0.858435
$418$	0	0
$419$	3.96880	0.193889	0.0969444	−	0.995290i	$-0.469093\pi$
$419$	3.96880	0.193889	0.0969444	+	0.995290i	$0.469093\pi$
$420$	0	0
$421$	−24.9216	−1.21461	−0.607303	−	0.794470i	$-0.707749\pi$
$421$	−24.9216	−1.21461	−0.607303	+	0.794470i	$0.707749\pi$
$422$	0	0
$423$	0.715418	0.0347848
$424$	0	0
$425$	4.63090	0.224632
$426$	0	0
$427$	7.84778	0.379781
$428$	0	0
$429$	4.68035	0.225969
$430$	0	0
$431$	−5.67420	−0.273317	−0.136658	−	0.990618i	$-0.543636\pi$
$431$	−5.67420	−0.273317	−0.136658	+	0.990618i	$0.543636\pi$
$432$	0	0
$433$	12.3174	0.591936	0.295968	−	0.955198i	$-0.404358\pi$
$433$	12.3174	0.591936	0.295968	+	0.955198i	$0.404358\pi$
$434$	0	0
$435$	1.70928	0.0819535
$436$	0	0
$437$	−2.92162	−0.139760
$438$	0	0
$439$	26.3857	1.25932	0.629661	−	0.776870i	$-0.283194\pi$
$439$	26.3857	1.25932	0.629661	+	0.776870i	$0.283194\pi$
$440$	0	0
$441$	0.517447	0.0246404
$442$	0	0
$443$	−38.1750	−1.81375	−0.906875	−	0.421400i	$-0.861539\pi$
$443$	−38.1750	−1.81375	−0.906875	+	0.421400i	$0.861539\pi$
$444$	0	0
$445$	1.10504	0.0523839
$446$	0	0
$447$	3.41855	0.161692
$448$	0	0
$449$	−5.96880	−0.281685	−0.140843	−	0.990032i	$-0.544981\pi$
$449$	−5.96880	−0.281685	−0.140843	+	0.990032i	$0.544981\pi$
$450$	0	0
$451$	4.09890	0.193010
$452$	0	0
$453$	2.83710	0.133299
$454$	0	0
$455$	2.73820	0.128369
$456$	0	0
$457$	−14.7649	−0.690671	−0.345336	−	0.938479i	$-0.612235\pi$
$457$	−14.7649	−0.690671	−0.345336	+	0.938479i	$0.612235\pi$
$458$	0	0
$459$	−24.3668	−1.13735
$460$	0	0
$461$	13.6163	0.634176	0.317088	−	0.948396i	$-0.397295\pi$
$461$	13.6163	0.634176	0.317088	+	0.948396i	$0.397295\pi$
$462$	0	0
$463$	−27.3835	−1.27262	−0.636309	−	0.771434i	$-0.719540\pi$
$463$	−27.3835	−1.27262	−0.636309	+	0.771434i	$0.719540\pi$
$464$	0	0
$465$	−2.92162	−0.135487
$466$	0	0
$467$	27.4824	1.27173	0.635866	−	0.771799i	$-0.280643\pi$
$467$	27.4824	1.27173	0.635866	+	0.771799i	$0.280643\pi$
$468$	0	0
$469$	−1.07838	−0.0497949
$470$	0	0
$471$	−19.9155	−0.917657
$472$	0	0
$473$	−4.03120	−0.185355
$474$	0	0
$475$	1.70928	0.0784269
$476$	0	0
$477$	−0.410318	−0.0187872
$478$	0	0
$479$	23.1012	1.05552	0.527760	−	0.849394i	$-0.323032\pi$
$479$	23.1012	1.05552	0.527760	+	0.849394i	$0.323032\pi$
$480$	0	0
$481$	14.6225	0.666728
$482$	0	0
$483$	−1.84324	−0.0838705
$484$	0	0
$485$	−7.28458	−0.330776
$486$	0	0
$487$	−19.5669	−0.886661	−0.443330	−	0.896358i	$-0.646203\pi$
$487$	−19.5669	−0.886661	−0.443330	+	0.896358i	$0.646203\pi$
$488$	0	0
$489$	−14.0722	−0.636368
$490$	0	0
$491$	−19.3835	−0.874764	−0.437382	−	0.899276i	$-0.644094\pi$
$491$	−19.3835	−0.874764	−0.437382	+	0.899276i	$0.644094\pi$
$492$	0	0
$493$	4.63090	0.208565
$494$	0	0
$495$	−0.0494483	−0.00222254
$496$	0	0
$497$	−0.115718	−0.00519065
$498$	0	0
$499$	14.3545	0.642598	0.321299	−	0.946978i	$-0.395881\pi$
$499$	14.3545	0.642598	0.321299	+	0.946978i	$0.395881\pi$
$500$	0	0
$501$	−0.228990	−0.0102305
$502$	0	0
$503$	33.5936	1.49786	0.748931	−	0.662648i	$-0.230567\pi$
$503$	33.5936	1.49786	0.748931	+	0.662648i	$0.230567\pi$
$504$	0	0
$505$	3.26180	0.145148
$506$	0	0
$507$	9.97721	0.443104
$508$	0	0
$509$	−33.7998	−1.49815	−0.749074	−	0.662486i	$-0.769501\pi$
$509$	−33.7998	−1.49815	−0.749074	+	0.662486i	$0.769501\pi$
$510$	0	0
$511$	−1.07838	−0.0477046
$512$	0	0
$513$	−8.99386	−0.397088
$514$	0	0
$515$	15.2846	0.673519
$516$	0	0
$517$	−5.75872	−0.253268
$518$	0	0
$519$	10.5692	0.463935
$520$	0	0
$521$	−26.6491	−1.16752	−0.583760	−	0.811926i	$-0.698419\pi$
$521$	−26.6491	−1.16752	−0.583760	+	0.811926i	$0.698419\pi$
$522$	0	0
$523$	−10.2907	−0.449982	−0.224991	−	0.974361i	$-0.572235\pi$
$523$	−10.2907	−0.449982	−0.224991	+	0.974361i	$0.572235\pi$
$524$	0	0
$525$	1.07838	0.0470643
$526$	0	0
$527$	−7.91548	−0.344804
$528$	0	0
$529$	−20.0784	−0.872973
$530$	0	0
$531$	−0.352459	−0.0152954
$532$	0	0
$533$	28.1978	1.22138
$534$	0	0
$535$	−2.10731	−0.0911068
$536$	0	0
$537$	−37.6742	−1.62576
$538$	0	0
$539$	−4.16517	−0.179406
$540$	0	0
$541$	−33.3028	−1.43180	−0.715900	−	0.698203i	$-0.753984\pi$
$541$	−33.3028	−1.43180	−0.715900	+	0.698203i	$0.753984\pi$
$542$	0	0
$543$	18.1112	0.777225
$544$	0	0
$545$	−9.23513	−0.395590
$546$	0	0
$547$	38.2062	1.63358	0.816790	−	0.576936i	$-0.195752\pi$
$547$	38.2062	1.63358	0.816790	+	0.576936i	$0.195752\pi$
$548$	0	0
$549$	−0.974946	−0.0416097
$550$	0	0
$551$	1.70928	0.0728176
$552$	0	0
$553$	8.76487	0.372720
$554$	0	0
$555$	5.75872	0.244444
$556$	0	0
$557$	−41.5174	−1.75915	−0.879575	−	0.475760i	$-0.842173\pi$
$557$	−41.5174	−1.75915	−0.879575	+	0.475760i	$0.842173\pi$
$558$	0	0
$559$	−27.7321	−1.17294
$560$	0	0
$561$	4.99386	0.210841
$562$	0	0
$563$	18.6042	0.784075	0.392038	−	0.919949i	$-0.371770\pi$
$563$	18.6042	0.784075	0.392038	+	0.919949i	$0.371770\pi$
$564$	0	0
$565$	−3.86603	−0.162645
$566$	0	0
$567$	−5.52586	−0.232064
$568$	0	0
$569$	19.4452	0.815186	0.407593	−	0.913164i	$-0.366368\pi$
$569$	19.4452	0.815186	0.407593	+	0.913164i	$0.366368\pi$
$570$	0	0
$571$	−32.5113	−1.36056	−0.680278	−	0.732954i	$-0.738141\pi$
$571$	−32.5113	−1.36056	−0.680278	+	0.732954i	$0.738141\pi$
$572$	0	0
$573$	−7.28846	−0.304480
$574$	0	0
$575$	−1.70928	−0.0712817
$576$	0	0
$577$	−16.9444	−0.705405	−0.352702	−	0.935736i	$-0.614737\pi$
$577$	−16.9444	−0.705405	−0.352702	+	0.935736i	$0.614737\pi$
$578$	0	0
$579$	−2.24128	−0.0931443
$580$	0	0
$581$	8.39803	0.348409
$582$	0	0
$583$	3.30283	0.136789
$584$	0	0
$585$	−0.340173	−0.0140644
$586$	0	0
$587$	7.59809	0.313607	0.156803	−	0.987630i	$-0.449881\pi$
$587$	7.59809	0.313607	0.156803	+	0.987630i	$0.449881\pi$
$588$	0	0
$589$	−2.92162	−0.120383
$590$	0	0
$591$	42.0866	1.73121
$592$	0	0
$593$	13.7998	0.566688	0.283344	−	0.959018i	$-0.408556\pi$
$593$	13.7998	0.566688	0.283344	+	0.959018i	$0.408556\pi$
$594$	0	0
$595$	2.92162	0.119775
$596$	0	0
$597$	17.0472	0.697695
$598$	0	0
$599$	−6.47414	−0.264526	−0.132263	−	0.991215i	$-0.542224\pi$
$599$	−6.47414	−0.264526	−0.132263	+	0.991215i	$0.542224\pi$
$600$	0	0
$601$	−38.3812	−1.56560	−0.782801	−	0.622272i	$-0.786210\pi$
$601$	−38.3812	−1.56560	−0.782801	+	0.622272i	$0.786210\pi$
$602$	0	0
$603$	0.133969	0.00545565
$604$	0	0
$605$	−10.6020	−0.431031
$606$	0	0
$607$	3.81271	0.154753	0.0773765	−	0.997002i	$-0.475346\pi$
$607$	3.81271	0.154753	0.0773765	+	0.997002i	$0.475346\pi$
$608$	0	0
$609$	1.07838	0.0436981
$610$	0	0
$611$	−39.6163	−1.60271
$612$	0	0
$613$	13.6332	0.550638	0.275319	−	0.961353i	$-0.411216\pi$
$613$	13.6332	0.550638	0.275319	+	0.961353i	$0.411216\pi$
$614$	0	0
$615$	11.1050	0.447798
$616$	0	0
$617$	9.70928	0.390881	0.195440	−	0.980716i	$-0.437386\pi$
$617$	9.70928	0.390881	0.195440	+	0.980716i	$0.437386\pi$
$618$	0	0
$619$	−13.9916	−0.562369	−0.281185	−	0.959654i	$-0.590727\pi$
$619$	−13.9916	−0.562369	−0.281185	+	0.959654i	$0.590727\pi$
$620$	0	0
$621$	8.99386	0.360911
$622$	0	0
$623$	0.697167	0.0279314
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.84324	0.0736121
$628$	0	0
$629$	15.6020	0.622091
$630$	0	0
$631$	29.8264	1.18737	0.593686	−	0.804697i	$-0.297672\pi$
$631$	29.8264	1.18737	0.593686	+	0.804697i	$0.297672\pi$
$632$	0	0
$633$	22.9216	0.911053
$634$	0	0
$635$	−8.54638	−0.339153
$636$	0	0
$637$	−28.6537	−1.13530
$638$	0	0
$639$	0.0143758	0.000568700 0
$640$	0	0
$641$	41.8453	1.65279	0.826396	−	0.563090i	$-0.190388\pi$
$641$	41.8453	1.65279	0.826396	+	0.563090i	$0.190388\pi$
$642$	0	0
$643$	−29.7548	−1.17342	−0.586708	−	0.809798i	$-0.699576\pi$
$643$	−29.7548	−1.17342	−0.586708	+	0.809798i	$0.699576\pi$
$644$	0	0
$645$	−10.9216	−0.430038
$646$	0	0
$647$	2.27635	0.0894925	0.0447463	−	0.998998i	$-0.485752\pi$
$647$	2.27635	0.0894925	0.0447463	+	0.998998i	$0.485752\pi$
$648$	0	0
$649$	2.83710	0.111366
$650$	0	0
$651$	−1.84324	−0.0722424
$652$	0	0
$653$	−27.2678	−1.06707	−0.533535	−	0.845778i	$-0.679137\pi$
$653$	−27.2678	−1.06707	−0.533535	+	0.845778i	$0.679137\pi$
$654$	0	0
$655$	0.133969	0.00523461
$656$	0	0
$657$	0.133969	0.00522663
$658$	0	0
$659$	−18.1750	−0.707998	−0.353999	−	0.935246i	$-0.615178\pi$
$659$	−18.1750	−0.707998	−0.353999	+	0.935246i	$0.615178\pi$
$660$	0	0
$661$	−49.2639	−1.91614	−0.958072	−	0.286529i	$-0.907498\pi$
$661$	−49.2639	−1.91614	−0.958072	+	0.286529i	$0.907498\pi$
$662$	0	0
$663$	34.3545	1.33422
$664$	0	0
$665$	1.07838	0.0418177
$666$	0	0
$667$	−1.70928	−0.0661834
$668$	0	0
$669$	−10.3857	−0.401536
$670$	0	0
$671$	7.84778	0.302960
$672$	0	0
$673$	−9.78539	−0.377199	−0.188599	−	0.982054i	$-0.560395\pi$
$673$	−9.78539	−0.377199	−0.188599	+	0.982054i	$0.560395\pi$
$674$	0	0
$675$	−5.26180	−0.202527
$676$	0	0
$677$	−26.4352	−1.01599	−0.507994	−	0.861361i	$-0.669613\pi$
$677$	−26.4352	−1.01599	−0.507994	+	0.861361i	$0.669613\pi$
$678$	0	0
$679$	−4.59583	−0.176372
$680$	0	0
$681$	−34.5835	−1.32524
$682$	0	0
$683$	22.7031	0.868711	0.434356	−	0.900741i	$-0.356976\pi$
$683$	22.7031	0.868711	0.434356	+	0.900741i	$0.356976\pi$
$684$	0	0
$685$	4.63090	0.176938
$686$	0	0
$687$	−37.4063	−1.42714
$688$	0	0
$689$	22.7214	0.865616
$690$	0	0
$691$	5.81205	0.221101	0.110550	−	0.993871i	$-0.464739\pi$
$691$	5.81205	0.221101	0.110550	+	0.993871i	$0.464739\pi$
$692$	0	0
$693$	−0.0311968	−0.00118507
$694$	0	0
$695$	−10.2557	−0.389019
$696$	0	0
$697$	30.0866	1.13961
$698$	0	0
$699$	−17.4308	−0.659295
$700$	0	0
$701$	−38.3111	−1.44699	−0.723494	−	0.690330i	$-0.757465\pi$
$701$	−38.3111	−1.44699	−0.723494	+	0.690330i	$0.757465\pi$
$702$	0	0
$703$	5.75872	0.217194
$704$	0	0
$705$	−15.6020	−0.587604
$706$	0	0
$707$	2.05786	0.0773937
$708$	0	0
$709$	−3.75872	−0.141162	−0.0705809	−	0.997506i	$-0.522485\pi$
$709$	−3.75872	−0.141162	−0.0705809	+	0.997506i	$0.522485\pi$
$710$	0	0
$711$	−1.08888	−0.0408361
$712$	0	0
$713$	2.92162	0.109416
$714$	0	0
$715$	2.73820	0.102403
$716$	0	0
$717$	−8.84939	−0.330486
$718$	0	0
$719$	−21.8576	−0.815152	−0.407576	−	0.913171i	$-0.633626\pi$
$719$	−21.8576	−0.815152	−0.407576	+	0.913171i	$0.633626\pi$
$720$	0	0
$721$	9.64301	0.359124
$722$	0	0
$723$	−43.6163	−1.62211
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	18.2739	0.677742	0.338871	−	0.940833i	$-0.389955\pi$
$727$	18.2739	0.677742	0.338871	+	0.940833i	$0.389955\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	0	0
$731$	−29.5897	−1.09441
$732$	0	0
$733$	49.9383	1.84451	0.922256	−	0.386580i	$-0.126344\pi$
$733$	49.9383	1.84451	0.922256	+	0.386580i	$0.126344\pi$
$734$	0	0
$735$	−11.2846	−0.416238
$736$	0	0
$737$	−1.07838	−0.0397226
$738$	0	0
$739$	−32.6453	−1.20088	−0.600438	−	0.799671i	$-0.705007\pi$
$739$	−32.6453	−1.20088	−0.600438	+	0.799671i	$0.705007\pi$
$740$	0	0
$741$	12.6803	0.465824
$742$	0	0
$743$	38.3896	1.40838	0.704189	−	0.710012i	$-0.251311\pi$
$743$	38.3896	1.40838	0.704189	+	0.710012i	$0.251311\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	−1.04331	−0.0381725
$748$	0	0
$749$	−1.32950	−0.0485787
$750$	0	0
$751$	−44.6453	−1.62913	−0.814565	−	0.580073i	$-0.803024\pi$
$751$	−44.6453	−1.62913	−0.814565	+	0.580073i	$0.803024\pi$
$752$	0	0
$753$	−41.1071	−1.49803
$754$	0	0
$755$	1.65983	0.0604073
$756$	0	0
$757$	26.2907	0.955553	0.477776	−	0.878482i	$-0.341443\pi$
$757$	26.2907	0.955553	0.477776	+	0.878482i	$0.341443\pi$
$758$	0	0
$759$	−1.84324	−0.0669055
$760$	0	0
$761$	−9.15061	−0.331709	−0.165855	−	0.986150i	$-0.553038\pi$
$761$	−9.15061	−0.331709	−0.165855	+	0.986150i	$0.553038\pi$
$762$	0	0
$763$	−5.82642	−0.210931
$764$	0	0
$765$	−0.362959	−0.0131228
$766$	0	0
$767$	19.5174	0.704734
$768$	0	0
$769$	−9.90110	−0.357043	−0.178521	−	0.983936i	$-0.557131\pi$
$769$	−9.90110	−0.357043	−0.178521	+	0.983936i	$0.557131\pi$
$770$	0	0
$771$	−37.6286	−1.35516
$772$	0	0
$773$	26.5197	0.953848	0.476924	−	0.878945i	$-0.341752\pi$
$773$	26.5197	0.953848	0.476924	+	0.878945i	$0.341752\pi$
$774$	0	0
$775$	−1.70928	−0.0613990
$776$	0	0
$777$	3.63317	0.130339
$778$	0	0
$779$	11.1050	0.397879
$780$	0	0
$781$	−0.115718	−0.00414070
$782$	0	0
$783$	−5.26180	−0.188041
$784$	0	0
$785$	−11.6514	−0.415857
$786$	0	0
$787$	52.6141	1.87549	0.937745	−	0.347325i	$-0.112910\pi$
$787$	52.6141	1.87549	0.937745	+	0.347325i	$0.112910\pi$
$788$	0	0
$789$	−1.92777	−0.0686303
$790$	0	0
$791$	−2.43907	−0.0867233
$792$	0	0
$793$	53.9877	1.91716
$794$	0	0
$795$	8.94828	0.317363
$796$	0	0
$797$	18.4885	0.654897	0.327448	−	0.944869i	$-0.393811\pi$
$797$	18.4885	0.654897	0.327448	+	0.944869i	$0.393811\pi$
$798$	0	0
$799$	−42.2700	−1.49541
$800$	0	0
$801$	−0.0866105	−0.00306023
$802$	0	0
$803$	−1.07838	−0.0380551
$804$	0	0
$805$	−1.07838	−0.0380078
$806$	0	0
$807$	−13.4308	−0.472788
$808$	0	0
$809$	54.9770	1.93289	0.966445	−	0.256874i	$-0.0826926\pi$
$809$	54.9770	1.93289	0.966445	+	0.256874i	$0.0826926\pi$
$810$	0	0
$811$	−51.5006	−1.80843	−0.904216	−	0.427076i	$-0.859544\pi$
$811$	−51.5006	−1.80843	−0.904216	+	0.427076i	$0.859544\pi$
$812$	0	0
$813$	11.0661	0.388105
$814$	0	0
$815$	−8.23287	−0.288385
$816$	0	0
$817$	−10.9216	−0.382099
$818$	0	0
$819$	−0.214614	−0.00749923
$820$	0	0
$821$	10.7961	0.376785	0.188393	−	0.982094i	$-0.439672\pi$
$821$	10.7961	0.376785	0.188393	+	0.982094i	$0.439672\pi$
$822$	0	0
$823$	−13.2969	−0.463500	−0.231750	−	0.972775i	$-0.574445\pi$
$823$	−13.2969	−0.463500	−0.231750	+	0.972775i	$0.574445\pi$
$824$	0	0
$825$	1.07838	0.0375443
$826$	0	0
$827$	30.3896	1.05675	0.528375	−	0.849011i	$-0.322801\pi$
$827$	30.3896	1.05675	0.528375	+	0.849011i	$0.322801\pi$
$828$	0	0
$829$	−9.64754	−0.335073	−0.167536	−	0.985866i	$-0.553581\pi$
$829$	−9.64754	−0.335073	−0.167536	+	0.985866i	$0.553581\pi$
$830$	0	0
$831$	−31.7854	−1.10262
$832$	0	0
$833$	−30.5730	−1.05929
$834$	0	0
$835$	−0.133969	−0.00463619
$836$	0	0
$837$	8.99386	0.310873
$838$	0	0
$839$	30.3584	1.04809	0.524045	−	0.851691i	$-0.324423\pi$
$839$	30.3584	1.04809	0.524045	+	0.851691i	$0.324423\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	14.0866	0.485169
$844$	0	0
$845$	5.83710	0.200802
$846$	0	0
$847$	−6.68876	−0.229828
$848$	0	0
$849$	36.9093	1.26673
$850$	0	0
$851$	−5.75872	−0.197407
$852$	0	0
$853$	−16.2472	−0.556295	−0.278147	−	0.960538i	$-0.589720\pi$
$853$	−16.2472	−0.556295	−0.278147	+	0.960538i	$0.589720\pi$
$854$	0	0
$855$	−0.133969	−0.00458165
$856$	0	0
$857$	25.2183	0.861441	0.430721	−	0.902485i	$-0.358259\pi$
$857$	25.2183	0.861441	0.430721	+	0.902485i	$0.358259\pi$
$858$	0	0
$859$	10.4040	0.354980	0.177490	−	0.984123i	$-0.443202\pi$
$859$	10.4040	0.354980	0.177490	+	0.984123i	$0.443202\pi$
$860$	0	0
$861$	7.00614	0.238769
$862$	0	0
$863$	−0.515180	−0.0175369	−0.00876846	−	0.999962i	$-0.502791\pi$
$863$	−0.515180	−0.0175369	−0.00876846	+	0.999962i	$0.502791\pi$
$864$	0	0
$865$	6.18342	0.210243
$866$	0	0
$867$	7.59809	0.258045
$868$	0	0
$869$	8.76487	0.297328
$870$	0	0
$871$	−7.41855	−0.251368
$872$	0	0
$873$	0.570949	0.0193237
$874$	0	0
$875$	0.630898	0.0213282
$876$	0	0
$877$	−44.3234	−1.49669	−0.748347	−	0.663308i	$-0.769152\pi$
$877$	−44.3234	−1.49669	−0.748347	+	0.663308i	$0.769152\pi$
$878$	0	0
$879$	53.0160	1.78818
$880$	0	0
$881$	−48.4222	−1.63139	−0.815694	−	0.578484i	$-0.803644\pi$
$881$	−48.4222	−1.63139	−0.815694	+	0.578484i	$0.803644\pi$
$882$	0	0
$883$	−2.85780	−0.0961726	−0.0480863	−	0.998843i	$-0.515312\pi$
$883$	−2.85780	−0.0961726	−0.0480863	+	0.998843i	$0.515312\pi$
$884$	0	0
$885$	7.68649	0.258379
$886$	0	0
$887$	12.6453	0.424587	0.212293	−	0.977206i	$-0.431907\pi$
$887$	12.6453	0.424587	0.212293	+	0.977206i	$0.431907\pi$
$888$	0	0
$889$	−5.39189	−0.180838
$890$	0	0
$891$	−5.52586	−0.185123
$892$	0	0
$893$	−15.6020	−0.522100
$894$	0	0
$895$	−22.0410	−0.736751
$896$	0	0
$897$	−12.6803	−0.423384
$898$	0	0
$899$	−1.70928	−0.0570075
$900$	0	0
$901$	24.2434	0.807664
$902$	0	0
$903$	−6.89043	−0.229299
$904$	0	0
$905$	10.5958	0.352217
$906$	0	0
$907$	−4.39416	−0.145906	−0.0729528	−	0.997335i	$-0.523242\pi$
$907$	−4.39416	−0.145906	−0.0729528	+	0.997335i	$0.523242\pi$
$908$	0	0
$909$	−0.255652	−0.00847945
$910$	0	0
$911$	25.2267	0.835799	0.417899	−	0.908493i	$-0.362766\pi$
$911$	25.2267	0.835799	0.417899	+	0.908493i	$0.362766\pi$
$912$	0	0
$913$	8.39803	0.277934
$914$	0	0
$915$	21.2618	0.702893
$916$	0	0
$917$	0.0845208	0.00279112
$918$	0	0
$919$	−29.9733	−0.988729	−0.494365	−	0.869255i	$-0.664599\pi$
$919$	−29.9733	−0.988729	−0.494365	+	0.869255i	$0.664599\pi$
$920$	0	0
$921$	−22.8970	−0.754483
$922$	0	0
$923$	−0.796064	−0.0262028
$924$	0	0
$925$	3.36910	0.110775
$926$	0	0
$927$	−1.19797	−0.0393465
$928$	0	0
$929$	52.8659	1.73447	0.867236	−	0.497897i	$-0.165894\pi$
$929$	52.8659	1.73447	0.867236	+	0.497897i	$0.165894\pi$
$930$	0	0
$931$	−11.2846	−0.369837
$932$	0	0
$933$	41.4985	1.35860
$934$	0	0
$935$	2.92162	0.0955473
$936$	0	0
$937$	43.2762	1.41377	0.706885	−	0.707328i	$-0.250100\pi$
$937$	43.2762	1.41377	0.706885	+	0.707328i	$0.250100\pi$
$938$	0	0
$939$	−0.0455727	−0.00148721
$940$	0	0
$941$	−53.1506	−1.73266	−0.866330	−	0.499472i	$-0.833527\pi$
$941$	−53.1506	−1.73266	−0.866330	+	0.499472i	$0.833527\pi$
$942$	0	0
$943$	−11.1050	−0.361630
$944$	0	0
$945$	−3.31965	−0.107988
$946$	0	0
$947$	−24.8599	−0.807838	−0.403919	−	0.914795i	$-0.632352\pi$
$947$	−24.8599	−0.807838	−0.403919	+	0.914795i	$0.632352\pi$
$948$	0	0
$949$	−7.41855	−0.240816
$950$	0	0
$951$	−39.2885	−1.27402
$952$	0	0
$953$	8.47027	0.274379	0.137189	−	0.990545i	$-0.456193\pi$
$953$	8.47027	0.274379	0.137189	+	0.990545i	$0.456193\pi$
$954$	0	0
$955$	−4.26406	−0.137982
$956$	0	0
$957$	1.07838	0.0348590
$958$	0	0
$959$	2.92162	0.0943441
$960$	0	0
$961$	−28.0784	−0.905754
$962$	0	0
$963$	0.165166	0.00532240
$964$	0	0
$965$	−1.31124	−0.0422104
$966$	0	0
$967$	15.9116	0.511683	0.255841	−	0.966719i	$-0.417648\pi$
$967$	15.9116	0.511683	0.255841	+	0.966719i	$0.417648\pi$
$968$	0	0
$969$	13.5297	0.434638
$970$	0	0
$971$	33.9071	1.08813	0.544065	−	0.839043i	$-0.316885\pi$
$971$	33.9071	1.08813	0.544065	+	0.839043i	$0.316885\pi$
$972$	0	0
$973$	−6.47027	−0.207427
$974$	0	0
$975$	7.41855	0.237584
$976$	0	0
$977$	13.1362	0.420265	0.210133	−	0.977673i	$-0.432610\pi$
$977$	13.1362	0.420265	0.210133	+	0.977673i	$0.432610\pi$
$978$	0	0
$979$	0.697167	0.0222816
$980$	0	0
$981$	0.723829	0.0231101
$982$	0	0
$983$	43.0700	1.37372	0.686859	−	0.726790i	$-0.258989\pi$
$983$	43.0700	1.37372	0.686859	+	0.726790i	$0.258989\pi$
$984$	0	0
$985$	24.6225	0.784538
$986$	0	0
$987$	−9.84324	−0.313314
$988$	0	0
$989$	10.9216	0.347287
$990$	0	0
$991$	41.3217	1.31263	0.656314	−	0.754488i	$-0.272115\pi$
$991$	41.3217	1.31263	0.656314	+	0.754488i	$0.272115\pi$
$992$	0	0
$993$	−44.2823	−1.40526
$994$	0	0
$995$	9.97334	0.316176
$996$	0	0
$997$	44.3605	1.40491	0.702456	−	0.711727i	$-0.252087\pi$
$997$	44.3605	1.40491	0.702456	+	0.711727i	$0.252087\pi$
$998$	0	0
$999$	−17.7275	−0.560874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.a.o.1.3		3
4.3	odd	2		580.2.a.d.1.1	✓	3
8.3	odd	2		9280.2.a.bh.1.3		3
8.5	even	2		9280.2.a.bt.1.1		3
12.11	even	2		5220.2.a.w.1.2		3
20.3	even	4		2900.2.c.g.349.2		6
20.7	even	4		2900.2.c.g.349.5		6
20.19	odd	2		2900.2.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.d.1.1	✓	3	4.3	odd	2
2320.2.a.o.1.3		3	1.1	even	1	trivial
2900.2.a.f.1.3		3	20.19	odd	2
2900.2.c.g.349.2		6	20.3	even	4
2900.2.c.g.349.5		6	20.7	even	4
5220.2.a.w.1.2		3	12.11	even	2
9280.2.a.bh.1.3		3	8.3	odd	2
9280.2.a.bt.1.1		3	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70928 q^{3} +1.00000 q^{5} +0.630898 q^{7} -0.0783777 q^{9} +0.630898 q^{11} +4.34017 q^{13} +1.70928 q^{15} +4.63090 q^{17} +1.70928 q^{19} +1.07838 q^{21} -1.70928 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.00000 q^{29} -1.70928 q^{31} +1.07838 q^{33} +0.630898 q^{35} +3.36910 q^{37} +7.41855 q^{39} +6.49693 q^{41} -6.38962 q^{43} -0.0783777 q^{45} -9.12783 q^{47} -6.60197 q^{49} +7.91548 q^{51} +5.23513 q^{53} +0.630898 q^{55} +2.92162 q^{57} +4.49693 q^{59} +12.4391 q^{61} -0.0494483 q^{63} +4.34017 q^{65} -1.70928 q^{67} -2.92162 q^{69} -0.183417 q^{71} -1.70928 q^{73} +1.70928 q^{75} +0.398032 q^{77} +13.8927 q^{79} -8.75872 q^{81} +13.3112 q^{83} +4.63090 q^{85} +1.70928 q^{87} +1.10504 q^{89} +2.73820 q^{91} -2.92162 q^{93} +1.70928 q^{95} -7.28458 q^{97} -0.0494483 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} + 10 q^{17} - 2 q^{19} + 2 q^{23} + 3 q^{25} - 8 q^{27} + 3 q^{29} + 2 q^{31} - 2 q^{35} + 14 q^{37} + 8 q^{39} + 2 q^{41} + 10 q^{43}+ \cdots + 18 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 + 10 * q^17 - 2 * q^19 + 2 * q^23 + 3 * q^25 - 8 * q^27 + 3 * q^29 + 2 * q^31 - 2 * q^35 + 14 * q^37 + 8 * q^39 + 2 * q^41 + 10 * q^43 + 3 * q^45 - 6 * q^47 - q^49 - 8 * q^51 + 6 * q^53 - 2 * q^55 + 12 * q^57 - 4 * q^59 - 10 * q^61 + 18 * q^63 + 2 * q^65 + 2 * q^67 - 12 * q^69 + 4 * q^71 + 2 * q^73 - 2 * q^75 + 20 * q^77 + 30 * q^79 - q^81 + 14 * q^83 + 10 * q^85 - 2 * q^87 + 2 * q^89 + 16 * q^91 - 12 * q^93 - 2 * q^95 + 6 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more