LMFDB - Embedded newform 6080.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.539189 q^{3} -1.00000 q^{5} +0.630898 q^{7} -2.70928 q^{9} +O(q^{10})$	$q-0.539189 q^{3} -1.00000 q^{5} +0.630898 q^{7} -2.70928 q^{9} +1.70928 q^{11} +3.17009 q^{13} +0.539189 q^{15} +1.41855 q^{17} -1.00000 q^{19} -0.340173 q^{21} -4.04945 q^{23} +1.00000 q^{25} +3.07838 q^{27} -3.75872 q^{29} +1.41855 q^{31} -0.921622 q^{33} -0.630898 q^{35} +0.986669 q^{37} -1.70928 q^{39} -9.26180 q^{41} -5.70928 q^{43} +2.70928 q^{45} -4.04945 q^{47} -6.60197 q^{49} -0.764867 q^{51} +7.32684 q^{53} -1.70928 q^{55} +0.539189 q^{57} +13.0205 q^{59} +13.1278 q^{61} -1.70928 q^{63} -3.17009 q^{65} -6.14116 q^{67} +2.18342 q^{69} +7.94214 q^{71} -9.91548 q^{73} -0.539189 q^{75} +1.07838 q^{77} +5.02052 q^{79} +6.46800 q^{81} -3.86603 q^{83} -1.41855 q^{85} +2.02666 q^{87} +5.60197 q^{89} +2.00000 q^{91} -0.764867 q^{93} +1.00000 q^{95} -0.275126 q^{97} -4.63090 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 - 2 * q^7 - q^9	$ 3 q - 3 q^{5} - 2 q^{7} - q^{9} - 2 q^{11} + 4 q^{13} - 10 q^{17} - 3 q^{19} + 10 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} + 14 q^{29} - 10 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 20 q^{41} - 10 q^{43} + q^{45} + 6 q^{47} - q^{49} - 12 q^{51} + 10 q^{53} + 2 q^{55} + 6 q^{59} + 18 q^{61} + 2 q^{63} - 4 q^{65} + 2 q^{67} + 2 q^{69} - 6 q^{71} + 2 q^{73} - 18 q^{79} - 13 q^{81} + 2 q^{83} + 10 q^{85} + 8 q^{87} - 2 q^{89} + 6 q^{91} - 12 q^{93} + 3 q^{95} + 6 q^{97} - 10 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 - 2 * q^7 - q^9 - 2 * q^11 + 4 * q^13 - 10 * q^17 - 3 * q^19 + 10 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 + 14 * q^29 - 10 * q^31 - 6 * q^33 + 2 * q^35 + 2 * q^37 + 2 * q^39 - 20 * q^41 - 10 * q^43 + q^45 + 6 * q^47 - q^49 - 12 * q^51 + 10 * q^53 + 2 * q^55 + 6 * q^59 + 18 * q^61 + 2 * q^63 - 4 * q^65 + 2 * q^67 + 2 * q^69 - 6 * q^71 + 2 * q^73 - 18 * q^79 - 13 * q^81 + 2 * q^83 + 10 * q^85 + 8 * q^87 - 2 * q^89 + 6 * q^91 - 12 * q^93 + 3 * q^95 + 6 * q^97 - 10 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.539189	−0.311301	−0.155650	−	0.987812i	$-0.549747\pi$
$3$	−0.539189	−0.311301	−0.155650	+	0.987812i	$0.549747\pi$
$4$	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
$9$	−2.70928	−0.903092
$10$	0	0
$11$	1.70928	0.515366	0.257683	−	0.966230i	$-0.417041\pi$
$11$	1.70928	0.515366	0.257683	+	0.966230i	$0.417041\pi$
$12$	0	0
$13$	3.17009	0.879224	0.439612	−	0.898188i	$-0.355116\pi$
$13$	3.17009	0.879224	0.439612	+	0.898188i	$0.355116\pi$
$14$	0	0
$15$	0.539189	0.139218
$16$	0	0
$17$	1.41855	0.344049	0.172025	−	0.985093i	$-0.444969\pi$
$17$	1.41855	0.344049	0.172025	+	0.985093i	$0.444969\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.340173	−0.0742318
$22$	0	0
$23$	−4.04945	−0.844368	−0.422184	−	0.906510i	$-0.638736\pi$
$23$	−4.04945	−0.844368	−0.422184	+	0.906510i	$0.638736\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.07838	0.592434
$28$	0	0
$29$	−3.75872	−0.697977	−0.348989	−	0.937127i	$-0.613475\pi$
$29$	−3.75872	−0.697977	−0.348989	+	0.937127i	$0.613475\pi$
$30$	0	0
$31$	1.41855	0.254779	0.127390	−	0.991853i	$-0.459340\pi$
$31$	1.41855	0.254779	0.127390	+	0.991853i	$0.459340\pi$
$32$	0	0
$33$	−0.921622	−0.160434
$34$	0	0
$35$	−0.630898	−0.106641
$36$	0	0
$37$	0.986669	0.162207	0.0811037	−	0.996706i	$-0.474156\pi$
$37$	0.986669	0.162207	0.0811037	+	0.996706i	$0.474156\pi$
$38$	0	0
$39$	−1.70928	−0.273703
$40$	0	0
$41$	−9.26180	−1.44645	−0.723225	−	0.690613i	$-0.757341\pi$
$41$	−9.26180	−1.44645	−0.723225	+	0.690613i	$0.757341\pi$
$42$	0	0
$43$	−5.70928	−0.870656	−0.435328	−	0.900272i	$-0.643368\pi$
$43$	−5.70928	−0.870656	−0.435328	+	0.900272i	$0.643368\pi$
$44$	0	0
$45$	2.70928	0.403875
$46$	0	0
$47$	−4.04945	−0.590673	−0.295336	−	0.955393i	$-0.595432\pi$
$47$	−4.04945	−0.590673	−0.295336	+	0.955393i	$0.595432\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	−0.764867	−0.107103
$52$	0	0
$53$	7.32684	1.00642	0.503210	−	0.864164i	$-0.332152\pi$
$53$	7.32684	1.00642	0.503210	+	0.864164i	$0.332152\pi$
$54$	0	0
$55$	−1.70928	−0.230479
$56$	0	0
$57$	0.539189	0.0714173
$58$	0	0
$59$	13.0205	1.69513	0.847564	−	0.530694i	$-0.178069\pi$
$59$	13.0205	1.69513	0.847564	+	0.530694i	$0.178069\pi$
$60$	0	0
$61$	13.1278	1.68085	0.840423	−	0.541931i	$-0.182307\pi$
$61$	13.1278	1.68085	0.840423	+	0.541931i	$0.182307\pi$
$62$	0	0
$63$	−1.70928	−0.215348
$64$	0	0
$65$	−3.17009	−0.393201
$66$	0	0
$67$	−6.14116	−0.750262	−0.375131	−	0.926972i	$-0.622402\pi$
$67$	−6.14116	−0.750262	−0.375131	+	0.926972i	$0.622402\pi$
$68$	0	0
$69$	2.18342	0.262853
$70$	0	0
$71$	7.94214	0.942559	0.471279	−	0.881984i	$-0.343792\pi$
$71$	7.94214	0.942559	0.471279	+	0.881984i	$0.343792\pi$
$72$	0	0
$73$	−9.91548	−1.16052	−0.580260	−	0.814432i	$-0.697049\pi$
$73$	−9.91548	−1.16052	−0.580260	+	0.814432i	$0.697049\pi$
$74$	0	0
$75$	−0.539189	−0.0622602
$76$	0	0
$77$	1.07838	0.122893
$78$	0	0
$79$	5.02052	0.564853	0.282426	−	0.959289i	$-0.408861\pi$
$79$	5.02052	0.564853	0.282426	+	0.959289i	$0.408861\pi$
$80$	0	0
$81$	6.46800	0.718667
$82$	0	0
$83$	−3.86603	−0.424352	−0.212176	−	0.977231i	$-0.568055\pi$
$83$	−3.86603	−0.424352	−0.212176	+	0.977231i	$0.568055\pi$
$84$	0	0
$85$	−1.41855	−0.153863
$86$	0	0
$87$	2.02666	0.217281
$88$	0	0
$89$	5.60197	0.593807	0.296904	−	0.954907i	$-0.404046\pi$
$89$	5.60197	0.593807	0.296904	+	0.954907i	$0.404046\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	−0.764867	−0.0793130
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−0.275126	−0.0279348	−0.0139674	−	0.999902i	$-0.504446\pi$
$97$	−0.275126	−0.0279348	−0.0139674	+	0.999902i	$0.504446\pi$
$98$	0	0
$99$	−4.63090	−0.465423
$100$	0	0
$101$	16.7298	1.66468	0.832338	−	0.554268i	$-0.187002\pi$
$101$	16.7298	1.66468	0.832338	+	0.554268i	$0.187002\pi$
$102$	0	0
$103$	−17.0628	−1.68125	−0.840623	−	0.541621i	$-0.817811\pi$
$103$	−17.0628	−1.68125	−0.840623	+	0.541621i	$0.817811\pi$
$104$	0	0
$105$	0.340173	0.0331975
$106$	0	0
$107$	5.77432	0.558225	0.279112	−	0.960258i	$-0.409960\pi$
$107$	5.77432	0.558225	0.279112	+	0.960258i	$0.409960\pi$
$108$	0	0
$109$	−1.44521	−0.138426	−0.0692131	−	0.997602i	$-0.522049\pi$
$109$	−1.44521	−0.138426	−0.0692131	+	0.997602i	$0.522049\pi$
$110$	0	0
$111$	−0.532001	−0.0504953
$112$	0	0
$113$	−3.93495	−0.370169	−0.185085	−	0.982723i	$-0.559256\pi$
$113$	−3.93495	−0.370169	−0.185085	+	0.982723i	$0.559256\pi$
$114$	0	0
$115$	4.04945	0.377613
$116$	0	0
$117$	−8.58864	−0.794020
$118$	0	0
$119$	0.894960	0.0820409
$120$	0	0
$121$	−8.07838	−0.734398
$122$	0	0
$123$	4.99386	0.450281
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.71646	−0.862197	−0.431098	−	0.902305i	$-0.641874\pi$
$127$	−9.71646	−0.862197	−0.431098	+	0.902305i	$0.641874\pi$
$128$	0	0
$129$	3.07838	0.271036
$130$	0	0
$131$	−14.6537	−1.28030	−0.640149	−	0.768251i	$-0.721127\pi$
$131$	−14.6537	−1.28030	−0.640149	+	0.768251i	$0.721127\pi$
$132$	0	0
$133$	−0.630898	−0.0547058
$134$	0	0
$135$	−3.07838	−0.264945
$136$	0	0
$137$	8.52359	0.728219	0.364110	−	0.931356i	$-0.381373\pi$
$137$	8.52359	0.728219	0.364110	+	0.931356i	$0.381373\pi$
$138$	0	0
$139$	−22.3896	−1.89906	−0.949531	−	0.313672i	$-0.898441\pi$
$139$	−22.3896	−1.89906	−0.949531	+	0.313672i	$0.898441\pi$
$140$	0	0
$141$	2.18342	0.183877
$142$	0	0
$143$	5.41855	0.453122
$144$	0	0
$145$	3.75872	0.312145
$146$	0	0
$147$	3.55971	0.293600
$148$	0	0
$149$	−6.38962	−0.523458	−0.261729	−	0.965141i	$-0.584293\pi$
$149$	−6.38962	−0.523458	−0.261729	+	0.965141i	$0.584293\pi$
$150$	0	0
$151$	−4.18342	−0.340442	−0.170221	−	0.985406i	$-0.554448\pi$
$151$	−4.18342	−0.340442	−0.170221	+	0.985406i	$0.554448\pi$
$152$	0	0
$153$	−3.84324	−0.310708
$154$	0	0
$155$	−1.41855	−0.113941
$156$	0	0
$157$	15.3607	1.22592	0.612958	−	0.790115i	$-0.289979\pi$
$157$	15.3607	1.22592	0.612958	+	0.790115i	$0.289979\pi$
$158$	0	0
$159$	−3.95055	−0.313299
$160$	0	0
$161$	−2.55479	−0.201345
$162$	0	0
$163$	−22.1217	−1.73270	−0.866352	−	0.499434i	$-0.833541\pi$
$163$	−22.1217	−1.73270	−0.866352	+	0.499434i	$0.833541\pi$
$164$	0	0
$165$	0.921622	0.0717482
$166$	0	0
$167$	1.03612	0.0801772	0.0400886	−	0.999196i	$-0.487236\pi$
$167$	1.03612	0.0801772	0.0400886	+	0.999196i	$0.487236\pi$
$168$	0	0
$169$	−2.95055	−0.226966
$170$	0	0
$171$	2.70928	0.207183
$172$	0	0
$173$	12.6153	0.959123	0.479562	−	0.877508i	$-0.340796\pi$
$173$	12.6153	0.959123	0.479562	+	0.877508i	$0.340796\pi$
$174$	0	0
$175$	0.630898	0.0476914
$176$	0	0
$177$	−7.02052	−0.527695
$178$	0	0
$179$	−23.4908	−1.75578	−0.877892	−	0.478859i	$-0.841051\pi$
$179$	−23.4908	−1.75578	−0.877892	+	0.478859i	$0.841051\pi$
$180$	0	0
$181$	−3.36069	−0.249798	−0.124899	−	0.992169i	$-0.539861\pi$
$181$	−3.36069	−0.249798	−0.124899	+	0.992169i	$0.539861\pi$
$182$	0	0
$183$	−7.07838	−0.523249
$184$	0	0
$185$	−0.986669	−0.0725413
$186$	0	0
$187$	2.42469	0.177311
$188$	0	0
$189$	1.94214	0.141270
$190$	0	0
$191$	−21.4596	−1.55276	−0.776381	−	0.630264i	$-0.782947\pi$
$191$	−21.4596	−1.55276	−0.776381	+	0.630264i	$0.782947\pi$
$192$	0	0
$193$	13.0856	0.941920	0.470960	−	0.882155i	$-0.343908\pi$
$193$	13.0856	0.941920	0.470960	+	0.882155i	$0.343908\pi$
$194$	0	0
$195$	1.70928	0.122404
$196$	0	0
$197$	2.49693	0.177899	0.0889494	−	0.996036i	$-0.471649\pi$
$197$	2.49693	0.177899	0.0889494	+	0.996036i	$0.471649\pi$
$198$	0	0
$199$	−12.6803	−0.898886	−0.449443	−	0.893309i	$-0.648377\pi$
$199$	−12.6803	−0.898886	−0.449443	+	0.893309i	$0.648377\pi$
$200$	0	0
$201$	3.31124	0.233557
$202$	0	0
$203$	−2.37137	−0.166438
$204$	0	0
$205$	9.26180	0.646872
$206$	0	0
$207$	10.9711	0.762542
$208$	0	0
$209$	−1.70928	−0.118233
$210$	0	0
$211$	−16.8638	−1.16095	−0.580475	−	0.814278i	$-0.697133\pi$
$211$	−16.8638	−1.16095	−0.580475	+	0.814278i	$0.697133\pi$
$212$	0	0
$213$	−4.28231	−0.293419
$214$	0	0
$215$	5.70928	0.389369
$216$	0	0
$217$	0.894960	0.0607539
$218$	0	0
$219$	5.34632	0.361271
$220$	0	0
$221$	4.49693	0.302496
$222$	0	0
$223$	−8.22568	−0.550832	−0.275416	−	0.961325i	$-0.588816\pi$
$223$	−8.22568	−0.550832	−0.275416	+	0.961325i	$0.588816\pi$
$224$	0	0
$225$	−2.70928	−0.180618
$226$	0	0
$227$	9.19287	0.610152	0.305076	−	0.952328i	$-0.401318\pi$
$227$	9.19287	0.610152	0.305076	+	0.952328i	$0.401318\pi$
$228$	0	0
$229$	−3.68261	−0.243354	−0.121677	−	0.992570i	$-0.538827\pi$
$229$	−3.68261	−0.243354	−0.121677	+	0.992570i	$0.538827\pi$
$230$	0	0
$231$	−0.581449	−0.0382566
$232$	0	0
$233$	7.17727	0.470199	0.235099	−	0.971971i	$-0.424458\pi$
$233$	7.17727	0.470199	0.235099	+	0.971971i	$0.424458\pi$
$234$	0	0
$235$	4.04945	0.264157
$236$	0	0
$237$	−2.70701	−0.175839
$238$	0	0
$239$	−10.1256	−0.654968	−0.327484	−	0.944857i	$-0.606201\pi$
$239$	−10.1256	−0.654968	−0.327484	+	0.944857i	$0.606201\pi$
$240$	0	0
$241$	−25.3607	−1.63363	−0.816813	−	0.576903i	$-0.804261\pi$
$241$	−25.3607	−1.63363	−0.816813	+	0.576903i	$0.804261\pi$
$242$	0	0
$243$	−12.7226	−0.816156
$244$	0	0
$245$	6.60197	0.421784
$246$	0	0
$247$	−3.17009	−0.201708
$248$	0	0
$249$	2.08452	0.132101
$250$	0	0
$251$	9.26180	0.584599	0.292300	−	0.956327i	$-0.405580\pi$
$251$	9.26180	0.584599	0.292300	+	0.956327i	$0.405580\pi$
$252$	0	0
$253$	−6.92162	−0.435159
$254$	0	0
$255$	0.764867	0.0478978
$256$	0	0
$257$	−6.34736	−0.395937	−0.197969	−	0.980208i	$-0.563434\pi$
$257$	−6.34736	−0.395937	−0.197969	+	0.980208i	$0.563434\pi$
$258$	0	0
$259$	0.622487	0.0386795
$260$	0	0
$261$	10.1834	0.630338
$262$	0	0
$263$	−1.79380	−0.110610	−0.0553051	−	0.998470i	$-0.517613\pi$
$263$	−1.79380	−0.110610	−0.0553051	+	0.998470i	$0.517613\pi$
$264$	0	0
$265$	−7.32684	−0.450084
$266$	0	0
$267$	−3.02052	−0.184853
$268$	0	0
$269$	6.70701	0.408933	0.204467	−	0.978874i	$-0.434454\pi$
$269$	6.70701	0.408933	0.204467	+	0.978874i	$0.434454\pi$
$270$	0	0
$271$	12.8143	0.778414	0.389207	−	0.921150i	$-0.372749\pi$
$271$	12.8143	0.778414	0.389207	+	0.921150i	$0.372749\pi$
$272$	0	0
$273$	−1.07838	−0.0652664
$274$	0	0
$275$	1.70928	0.103073
$276$	0	0
$277$	−4.15676	−0.249755	−0.124878	−	0.992172i	$-0.539854\pi$
$277$	−4.15676	−0.249755	−0.124878	+	0.992172i	$0.539854\pi$
$278$	0	0
$279$	−3.84324	−0.230089
$280$	0	0
$281$	−29.0928	−1.73553	−0.867764	−	0.496976i	$-0.834444\pi$
$281$	−29.0928	−1.73553	−0.867764	+	0.496976i	$0.834444\pi$
$282$	0	0
$283$	−12.1340	−0.721290	−0.360645	−	0.932703i	$-0.617443\pi$
$283$	−12.1340	−0.721290	−0.360645	+	0.932703i	$0.617443\pi$
$284$	0	0
$285$	−0.539189	−0.0319388
$286$	0	0
$287$	−5.84324	−0.344916
$288$	0	0
$289$	−14.9877	−0.881630
$290$	0	0
$291$	0.148345	0.00869614
$292$	0	0
$293$	−11.9227	−0.696530	−0.348265	−	0.937396i	$-0.613229\pi$
$293$	−11.9227	−0.696530	−0.348265	+	0.937396i	$0.613229\pi$
$294$	0	0
$295$	−13.0205	−0.758084
$296$	0	0
$297$	5.26180	0.305320
$298$	0	0
$299$	−12.8371	−0.742389
$300$	0	0
$301$	−3.60197	−0.207614
$302$	0	0
$303$	−9.02052	−0.518215
$304$	0	0
$305$	−13.1278	−0.751697
$306$	0	0
$307$	−3.02174	−0.172460	−0.0862299	−	0.996275i	$-0.527482\pi$
$307$	−3.02174	−0.172460	−0.0862299	+	0.996275i	$0.527482\pi$
$308$	0	0
$309$	9.20006	0.523373
$310$	0	0
$311$	−27.3835	−1.55277	−0.776387	−	0.630256i	$-0.782950\pi$
$311$	−27.3835	−1.55277	−0.776387	+	0.630256i	$0.782950\pi$
$312$	0	0
$313$	15.1773	0.857870	0.428935	−	0.903335i	$-0.358889\pi$
$313$	15.1773	0.857870	0.428935	+	0.903335i	$0.358889\pi$
$314$	0	0
$315$	1.70928	0.0963068
$316$	0	0
$317$	−13.6092	−0.764366	−0.382183	−	0.924087i	$-0.624828\pi$
$317$	−13.6092	−0.764366	−0.382183	+	0.924087i	$0.624828\pi$
$318$	0	0
$319$	−6.42469	−0.359714
$320$	0	0
$321$	−3.11345	−0.173776
$322$	0	0
$323$	−1.41855	−0.0789303
$324$	0	0
$325$	3.17009	0.175845
$326$	0	0
$327$	0.779243	0.0430922
$328$	0	0
$329$	−2.55479	−0.140850
$330$	0	0
$331$	6.92162	0.380447	0.190223	−	0.981741i	$-0.439079\pi$
$331$	6.92162	0.380447	0.190223	+	0.981741i	$0.439079\pi$
$332$	0	0
$333$	−2.67316	−0.146488
$334$	0	0
$335$	6.14116	0.335527
$336$	0	0
$337$	22.6875	1.23587	0.617934	−	0.786230i	$-0.287970\pi$
$337$	22.6875	1.23587	0.617934	+	0.786230i	$0.287970\pi$
$338$	0	0
$339$	2.12168	0.115234
$340$	0	0
$341$	2.42469	0.131305
$342$	0	0
$343$	−8.58145	−0.463355
$344$	0	0
$345$	−2.18342	−0.117551
$346$	0	0
$347$	−28.8287	−1.54761	−0.773803	−	0.633427i	$-0.781648\pi$
$347$	−28.8287	−1.54761	−0.773803	+	0.633427i	$0.781648\pi$
$348$	0	0
$349$	−10.2823	−0.550400	−0.275200	−	0.961387i	$-0.588744\pi$
$349$	−10.2823	−0.550400	−0.275200	+	0.961387i	$0.588744\pi$
$350$	0	0
$351$	9.75872	0.520882
$352$	0	0
$353$	−21.6163	−1.15052	−0.575261	−	0.817970i	$-0.695099\pi$
$353$	−21.6163	−1.15052	−0.575261	+	0.817970i	$0.695099\pi$
$354$	0	0
$355$	−7.94214	−0.421525
$356$	0	0
$357$	−0.482553	−0.0255394
$358$	0	0
$359$	15.0700	0.795362	0.397681	−	0.917524i	$-0.369815\pi$
$359$	15.0700	0.795362	0.397681	+	0.917524i	$0.369815\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.35577	0.228619
$364$	0	0
$365$	9.91548	0.519000
$366$	0	0
$367$	29.3256	1.53078	0.765392	−	0.643564i	$-0.222545\pi$
$367$	29.3256	1.53078	0.765392	+	0.643564i	$0.222545\pi$
$368$	0	0
$369$	25.0928	1.30628
$370$	0	0
$371$	4.62249	0.239988
$372$	0	0
$373$	−0.0917087	−0.00474850	−0.00237425	−	0.999997i	$-0.500756\pi$
$373$	−0.0917087	−0.00474850	−0.00237425	+	0.999997i	$0.500756\pi$
$374$	0	0
$375$	0.539189	0.0278436
$376$	0	0
$377$	−11.9155	−0.613678
$378$	0	0
$379$	11.1506	0.572768	0.286384	−	0.958115i	$-0.407547\pi$
$379$	11.1506	0.572768	0.286384	+	0.958115i	$0.407547\pi$
$380$	0	0
$381$	5.23901	0.268403
$382$	0	0
$383$	14.9060	0.761662	0.380831	−	0.924645i	$-0.375638\pi$
$383$	14.9060	0.761662	0.380831	+	0.924645i	$0.375638\pi$
$384$	0	0
$385$	−1.07838	−0.0549592
$386$	0	0
$387$	15.4680	0.786283
$388$	0	0
$389$	−2.71154	−0.137481	−0.0687403	−	0.997635i	$-0.521898\pi$
$389$	−2.71154	−0.137481	−0.0687403	+	0.997635i	$0.521898\pi$
$390$	0	0
$391$	−5.74435	−0.290504
$392$	0	0
$393$	7.90110	0.398558
$394$	0	0
$395$	−5.02052	−0.252610
$396$	0	0
$397$	−33.3340	−1.67299	−0.836494	−	0.547977i	$-0.815398\pi$
$397$	−33.3340	−1.67299	−0.836494	+	0.547977i	$0.815398\pi$
$398$	0	0
$399$	0.340173	0.0170299
$400$	0	0
$401$	7.88882	0.393949	0.196974	−	0.980409i	$-0.436888\pi$
$401$	7.88882	0.393949	0.196974	+	0.980409i	$0.436888\pi$
$402$	0	0
$403$	4.49693	0.224008
$404$	0	0
$405$	−6.46800	−0.321397
$406$	0	0
$407$	1.68649	0.0835962
$408$	0	0
$409$	−1.97334	−0.0975753	−0.0487876	−	0.998809i	$-0.515536\pi$
$409$	−1.97334	−0.0975753	−0.0487876	+	0.998809i	$0.515536\pi$
$410$	0	0
$411$	−4.59583	−0.226695
$412$	0	0
$413$	8.21461	0.404215
$414$	0	0
$415$	3.86603	0.189776
$416$	0	0
$417$	12.0722	0.591180
$418$	0	0
$419$	−16.9627	−0.828680	−0.414340	−	0.910122i	$-0.635988\pi$
$419$	−16.9627	−0.828680	−0.414340	+	0.910122i	$0.635988\pi$
$420$	0	0
$421$	0.366835	0.0178784	0.00893922	−	0.999960i	$-0.497155\pi$
$421$	0.366835	0.0178784	0.00893922	+	0.999960i	$0.497155\pi$
$422$	0	0
$423$	10.9711	0.533432
$424$	0	0
$425$	1.41855	0.0688098
$426$	0	0
$427$	8.28231	0.400809
$428$	0	0
$429$	−2.92162	−0.141057
$430$	0	0
$431$	8.39803	0.404519	0.202259	−	0.979332i	$-0.435172\pi$
$431$	8.39803	0.404519	0.202259	+	0.979332i	$0.435172\pi$
$432$	0	0
$433$	31.2690	1.50269	0.751346	−	0.659909i	$-0.229405\pi$
$433$	31.2690	1.50269	0.751346	+	0.659909i	$0.229405\pi$
$434$	0	0
$435$	−2.02666	−0.0971710
$436$	0	0
$437$	4.04945	0.193711
$438$	0	0
$439$	−19.0784	−0.910561	−0.455281	−	0.890348i	$-0.650461\pi$
$439$	−19.0784	−0.910561	−0.455281	+	0.890348i	$0.650461\pi$
$440$	0	0
$441$	17.8865	0.851740
$442$	0	0
$443$	8.04945	0.382441	0.191220	−	0.981547i	$-0.438755\pi$
$443$	8.04945	0.382441	0.191220	+	0.981547i	$0.438755\pi$
$444$	0	0
$445$	−5.60197	−0.265559
$446$	0	0
$447$	3.44521	0.162953
$448$	0	0
$449$	−6.92162	−0.326652	−0.163326	−	0.986572i	$-0.552222\pi$
$449$	−6.92162	−0.326652	−0.163326	+	0.986572i	$0.552222\pi$
$450$	0	0
$451$	−15.8310	−0.745451
$452$	0	0
$453$	2.25565	0.105980
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	14.6803	0.686718	0.343359	−	0.939204i	$-0.388435\pi$
$457$	14.6803	0.686718	0.343359	+	0.939204i	$0.388435\pi$
$458$	0	0
$459$	4.36683	0.203826
$460$	0	0
$461$	33.1506	1.54398	0.771989	−	0.635636i	$-0.219262\pi$
$461$	33.1506	1.54398	0.771989	+	0.635636i	$0.219262\pi$
$462$	0	0
$463$	−18.5197	−0.860684	−0.430342	−	0.902666i	$-0.641607\pi$
$463$	−18.5197	−0.860684	−0.430342	+	0.902666i	$0.641607\pi$
$464$	0	0
$465$	0.764867	0.0354698
$466$	0	0
$467$	23.5669	1.09055	0.545273	−	0.838259i	$-0.316426\pi$
$467$	23.5669	1.09055	0.545273	+	0.838259i	$0.316426\pi$
$468$	0	0
$469$	−3.87444	−0.178905
$470$	0	0
$471$	−8.28231	−0.381629
$472$	0	0
$473$	−9.75872	−0.448707
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−19.8504	−0.908889
$478$	0	0
$479$	−10.2907	−0.470195	−0.235098	−	0.971972i	$-0.575541\pi$
$479$	−10.2907	−0.470195	−0.235098	+	0.971972i	$0.575541\pi$
$480$	0	0
$481$	3.12783	0.142617
$482$	0	0
$483$	1.37751	0.0626790
$484$	0	0
$485$	0.275126	0.0124928
$486$	0	0
$487$	−28.9926	−1.31378	−0.656891	−	0.753986i	$-0.728129\pi$
$487$	−28.9926	−1.31378	−0.656891	+	0.753986i	$0.728129\pi$
$488$	0	0
$489$	11.9278	0.539392
$490$	0	0
$491$	18.5692	0.838015	0.419007	−	0.907983i	$-0.362378\pi$
$491$	18.5692	0.838015	0.419007	+	0.907983i	$0.362378\pi$
$492$	0	0
$493$	−5.33194	−0.240139
$494$	0	0
$495$	4.63090	0.208143
$496$	0	0
$497$	5.01068	0.224760
$498$	0	0
$499$	−19.7671	−0.884898	−0.442449	−	0.896794i	$-0.645890\pi$
$499$	−19.7671	−0.884898	−0.442449	+	0.896794i	$0.645890\pi$
$500$	0	0
$501$	−0.558663	−0.0249592
$502$	0	0
$503$	20.8287	0.928705	0.464353	−	0.885650i	$-0.346287\pi$
$503$	20.8287	0.928705	0.464353	+	0.885650i	$0.346287\pi$
$504$	0	0
$505$	−16.7298	−0.744466
$506$	0	0
$507$	1.59090	0.0706546
$508$	0	0
$509$	−24.5380	−1.08763	−0.543813	−	0.839206i	$-0.683020\pi$
$509$	−24.5380	−1.08763	−0.543813	+	0.839206i	$0.683020\pi$
$510$	0	0
$511$	−6.25565	−0.276734
$512$	0	0
$513$	−3.07838	−0.135914
$514$	0	0
$515$	17.0628	0.751876
$516$	0	0
$517$	−6.92162	−0.304413
$518$	0	0
$519$	−6.80203	−0.298576
$520$	0	0
$521$	−6.58145	−0.288339	−0.144169	−	0.989553i	$-0.546051\pi$
$521$	−6.58145	−0.288339	−0.144169	+	0.989553i	$0.546051\pi$
$522$	0	0
$523$	−27.6586	−1.20943	−0.604713	−	0.796443i	$-0.706712\pi$
$523$	−27.6586	−1.20943	−0.604713	+	0.796443i	$0.706712\pi$
$524$	0	0
$525$	−0.340173	−0.0148464
$526$	0	0
$527$	2.01229	0.0876566
$528$	0	0
$529$	−6.60197	−0.287042
$530$	0	0
$531$	−35.2762	−1.53086
$532$	0	0
$533$	−29.3607	−1.27175
$534$	0	0
$535$	−5.77432	−0.249646
$536$	0	0
$537$	12.6660	0.546577
$538$	0	0
$539$	−11.2846	−0.486061
$540$	0	0
$541$	16.9444	0.728497	0.364249	−	0.931302i	$-0.381326\pi$
$541$	16.9444	0.728497	0.364249	+	0.931302i	$0.381326\pi$
$542$	0	0
$543$	1.81205	0.0777624
$544$	0	0
$545$	1.44521	0.0619061
$546$	0	0
$547$	15.4608	0.661057	0.330528	−	0.943796i	$-0.392773\pi$
$547$	15.4608	0.661057	0.330528	+	0.943796i	$0.392773\pi$
$548$	0	0
$549$	−35.5669	−1.51796
$550$	0	0
$551$	3.75872	0.160127
$552$	0	0
$553$	3.16743	0.134693
$554$	0	0
$555$	0.532001	0.0225822
$556$	0	0
$557$	−40.3012	−1.70762	−0.853809	−	0.520587i	$-0.825713\pi$
$557$	−40.3012	−1.70762	−0.853809	+	0.520587i	$0.825713\pi$
$558$	0	0
$559$	−18.0989	−0.765502
$560$	0	0
$561$	−1.30737	−0.0551971
$562$	0	0
$563$	18.2134	0.767603	0.383801	−	0.923416i	$-0.374615\pi$
$563$	18.2134	0.767603	0.383801	+	0.923416i	$0.374615\pi$
$564$	0	0
$565$	3.93495	0.165545
$566$	0	0
$567$	4.08065	0.171371
$568$	0	0
$569$	−4.13009	−0.173143	−0.0865713	−	0.996246i	$-0.527591\pi$
$569$	−4.13009	−0.173143	−0.0865713	+	0.996246i	$0.527591\pi$
$570$	0	0
$571$	25.2267	1.05571	0.527853	−	0.849336i	$-0.322997\pi$
$571$	25.2267	1.05571	0.527853	+	0.849336i	$0.322997\pi$
$572$	0	0
$573$	11.5708	0.483376
$574$	0	0
$575$	−4.04945	−0.168874
$576$	0	0
$577$	34.7480	1.44658	0.723290	−	0.690544i	$-0.242629\pi$
$577$	34.7480	1.44658	0.723290	+	0.690544i	$0.242629\pi$
$578$	0	0
$579$	−7.05559	−0.293220
$580$	0	0
$581$	−2.43907	−0.101190
$582$	0	0
$583$	12.5236	0.518674
$584$	0	0
$585$	8.58864	0.355096
$586$	0	0
$587$	7.91935	0.326867	0.163433	−	0.986554i	$-0.447743\pi$
$587$	7.91935	0.326867	0.163433	+	0.986554i	$0.447743\pi$
$588$	0	0
$589$	−1.41855	−0.0584504
$590$	0	0
$591$	−1.34632	−0.0553800
$592$	0	0
$593$	0.837101	0.0343756	0.0171878	−	0.999852i	$-0.494529\pi$
$593$	0.837101	0.0343756	0.0171878	+	0.999852i	$0.494529\pi$
$594$	0	0
$595$	−0.894960	−0.0366898
$596$	0	0
$597$	6.83710	0.279824
$598$	0	0
$599$	−24.9216	−1.01827	−0.509135	−	0.860687i	$-0.670035\pi$
$599$	−24.9216	−1.01827	−0.509135	+	0.860687i	$0.670035\pi$
$600$	0	0
$601$	−17.7275	−0.723121	−0.361560	−	0.932349i	$-0.617756\pi$
$601$	−17.7275	−0.723121	−0.361560	+	0.932349i	$0.617756\pi$
$602$	0	0
$603$	16.6381	0.677555
$604$	0	0
$605$	8.07838	0.328433
$606$	0	0
$607$	9.40749	0.381838	0.190919	−	0.981606i	$-0.438853\pi$
$607$	9.40749	0.381838	0.190919	+	0.981606i	$0.438853\pi$
$608$	0	0
$609$	1.27862	0.0518121
$610$	0	0
$611$	−12.8371	−0.519334
$612$	0	0
$613$	−15.3919	−0.621673	−0.310836	−	0.950463i	$-0.600609\pi$
$613$	−15.3919	−0.621673	−0.310836	+	0.950463i	$0.600609\pi$
$614$	0	0
$615$	−4.99386	−0.201372
$616$	0	0
$617$	5.28846	0.212905	0.106453	−	0.994318i	$-0.466051\pi$
$617$	5.28846	0.212905	0.106453	+	0.994318i	$0.466051\pi$
$618$	0	0
$619$	−28.9132	−1.16212	−0.581060	−	0.813861i	$-0.697362\pi$
$619$	−28.9132	−1.16212	−0.581060	+	0.813861i	$0.697362\pi$
$620$	0	0
$621$	−12.4657	−0.500233
$622$	0	0
$623$	3.53427	0.141597
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.921622	0.0368060
$628$	0	0
$629$	1.39964	0.0558073
$630$	0	0
$631$	−6.38962	−0.254367	−0.127183	−	0.991879i	$-0.540594\pi$
$631$	−6.38962	−0.254367	−0.127183	+	0.991879i	$0.540594\pi$
$632$	0	0
$633$	9.09275	0.361405
$634$	0	0
$635$	9.71646	0.385586
$636$	0	0
$637$	−20.9288	−0.829230
$638$	0	0
$639$	−21.5174	−0.851217
$640$	0	0
$641$	1.04718	0.0413612	0.0206806	−	0.999786i	$-0.493417\pi$
$641$	1.04718	0.0413612	0.0206806	+	0.999786i	$0.493417\pi$
$642$	0	0
$643$	7.46800	0.294509	0.147255	−	0.989099i	$-0.452956\pi$
$643$	7.46800	0.294509	0.147255	+	0.989099i	$0.452956\pi$
$644$	0	0
$645$	−3.07838	−0.121211
$646$	0	0
$647$	2.70313	0.106271	0.0531355	−	0.998587i	$-0.483078\pi$
$647$	2.70313	0.106271	0.0531355	+	0.998587i	$0.483078\pi$
$648$	0	0
$649$	22.2557	0.873611
$650$	0	0
$651$	−0.482553	−0.0189127
$652$	0	0
$653$	21.4329	0.838735	0.419368	−	0.907817i	$-0.362252\pi$
$653$	21.4329	0.838735	0.419368	+	0.907817i	$0.362252\pi$
$654$	0	0
$655$	14.6537	0.572567
$656$	0	0
$657$	26.8638	1.04806
$658$	0	0
$659$	−19.8576	−0.773543	−0.386772	−	0.922176i	$-0.626410\pi$
$659$	−19.8576	−0.773543	−0.386772	+	0.922176i	$0.626410\pi$
$660$	0	0
$661$	2.83710	0.110350	0.0551752	−	0.998477i	$-0.482428\pi$
$661$	2.83710	0.110350	0.0551752	+	0.998477i	$0.482428\pi$
$662$	0	0
$663$	−2.42469	−0.0941673
$664$	0	0
$665$	0.630898	0.0244652
$666$	0	0
$667$	15.2208	0.589350
$668$	0	0
$669$	4.43519	0.171475
$670$	0	0
$671$	22.4391	0.866251
$672$	0	0
$673$	24.0338	0.926437	0.463218	−	0.886244i	$-0.346695\pi$
$673$	24.0338	0.926437	0.463218	+	0.886244i	$0.346695\pi$
$674$	0	0
$675$	3.07838	0.118487
$676$	0	0
$677$	−19.4114	−0.746039	−0.373020	−	0.927823i	$-0.621678\pi$
$677$	−19.4114	−0.746039	−0.373020	+	0.927823i	$0.621678\pi$
$678$	0	0
$679$	−0.173576	−0.00666125
$680$	0	0
$681$	−4.95669	−0.189941
$682$	0	0
$683$	−0.821503	−0.0314339	−0.0157170	−	0.999876i	$-0.505003\pi$
$683$	−0.821503	−0.0314339	−0.0157170	+	0.999876i	$0.505003\pi$
$684$	0	0
$685$	−8.52359	−0.325670
$686$	0	0
$687$	1.98562	0.0757563
$688$	0	0
$689$	23.2267	0.884868
$690$	0	0
$691$	4.54638	0.172952	0.0864762	−	0.996254i	$-0.472439\pi$
$691$	4.54638	0.172952	0.0864762	+	0.996254i	$0.472439\pi$
$692$	0	0
$693$	−2.92162	−0.110983
$694$	0	0
$695$	22.3896	0.849287
$696$	0	0
$697$	−13.1383	−0.497650
$698$	0	0
$699$	−3.86991	−0.146373
$700$	0	0
$701$	−12.6453	−0.477605	−0.238803	−	0.971068i	$-0.576755\pi$
$701$	−12.6453	−0.477605	−0.238803	+	0.971068i	$0.576755\pi$
$702$	0	0
$703$	−0.986669	−0.0372129
$704$	0	0
$705$	−2.18342	−0.0822323
$706$	0	0
$707$	10.5548	0.396954
$708$	0	0
$709$	12.8059	0.480936	0.240468	−	0.970657i	$-0.422699\pi$
$709$	12.8059	0.480936	0.240468	+	0.970657i	$0.422699\pi$
$710$	0	0
$711$	−13.6020	−0.510114
$712$	0	0
$713$	−5.74435	−0.215128
$714$	0	0
$715$	−5.41855	−0.202642
$716$	0	0
$717$	5.45959	0.203892
$718$	0	0
$719$	−1.90707	−0.0711217	−0.0355608	−	0.999368i	$-0.511322\pi$
$719$	−1.90707	−0.0711217	−0.0355608	+	0.999368i	$0.511322\pi$
$720$	0	0
$721$	−10.7649	−0.400905
$722$	0	0
$723$	13.6742	0.508549
$724$	0	0
$725$	−3.75872	−0.139595
$726$	0	0
$727$	25.9239	0.961464	0.480732	−	0.876868i	$-0.340371\pi$
$727$	25.9239	0.961464	0.480732	+	0.876868i	$0.340371\pi$
$728$	0	0
$729$	−12.5441	−0.464597
$730$	0	0
$731$	−8.09890	−0.299549
$732$	0	0
$733$	−39.1605	−1.44642	−0.723212	−	0.690626i	$-0.757335\pi$
$733$	−39.1605	−1.44642	−0.723212	+	0.690626i	$0.757335\pi$
$734$	0	0
$735$	−3.55971	−0.131302
$736$	0	0
$737$	−10.4969	−0.386659
$738$	0	0
$739$	−8.31351	−0.305817	−0.152909	−	0.988240i	$-0.548864\pi$
$739$	−8.31351	−0.305817	−0.152909	+	0.988240i	$0.548864\pi$
$740$	0	0
$741$	1.70928	0.0627918
$742$	0	0
$743$	−47.5597	−1.74480	−0.872398	−	0.488796i	$-0.837436\pi$
$743$	−47.5597	−1.74480	−0.872398	+	0.488796i	$0.837436\pi$
$744$	0	0
$745$	6.38962	0.234098
$746$	0	0
$747$	10.4741	0.383229
$748$	0	0
$749$	3.64301	0.133113
$750$	0	0
$751$	−23.8622	−0.870742	−0.435371	−	0.900251i	$-0.643383\pi$
$751$	−23.8622	−0.870742	−0.435371	+	0.900251i	$0.643383\pi$
$752$	0	0
$753$	−4.99386	−0.181986
$754$	0	0
$755$	4.18342	0.152250
$756$	0	0
$757$	−44.3402	−1.61157	−0.805785	−	0.592208i	$-0.798257\pi$
$757$	−44.3402	−1.61157	−0.805785	+	0.592208i	$0.798257\pi$
$758$	0	0
$759$	3.73206	0.135465
$760$	0	0
$761$	0.577574	0.0209370	0.0104685	−	0.999945i	$-0.496668\pi$
$761$	0.577574	0.0209370	0.0104685	+	0.999945i	$0.496668\pi$
$762$	0	0
$763$	−0.911781	−0.0330087
$764$	0	0
$765$	3.84324	0.138953
$766$	0	0
$767$	41.2762	1.49040
$768$	0	0
$769$	21.7938	0.785904	0.392952	−	0.919559i	$-0.371454\pi$
$769$	21.7938	0.785904	0.392952	+	0.919559i	$0.371454\pi$
$770$	0	0
$771$	3.42243	0.123256
$772$	0	0
$773$	43.4135	1.56147	0.780737	−	0.624860i	$-0.214844\pi$
$773$	43.4135	1.56147	0.780737	+	0.624860i	$0.214844\pi$
$774$	0	0
$775$	1.41855	0.0509558
$776$	0	0
$777$	−0.335638	−0.0120410
$778$	0	0
$779$	9.26180	0.331838
$780$	0	0
$781$	13.5753	0.485763
$782$	0	0
$783$	−11.5708	−0.413506
$784$	0	0
$785$	−15.3607	−0.548247
$786$	0	0
$787$	−14.2134	−0.506653	−0.253326	−	0.967381i	$-0.581525\pi$
$787$	−14.2134	−0.506653	−0.253326	+	0.967381i	$0.581525\pi$
$788$	0	0
$789$	0.967195	0.0344331
$790$	0	0
$791$	−2.48255	−0.0882694
$792$	0	0
$793$	41.6163	1.47784
$794$	0	0
$795$	3.95055	0.140112
$796$	0	0
$797$	22.0482	0.780988	0.390494	−	0.920605i	$-0.372304\pi$
$797$	22.0482	0.780988	0.390494	+	0.920605i	$0.372304\pi$
$798$	0	0
$799$	−5.74435	−0.203220
$800$	0	0
$801$	−15.1773	−0.536263
$802$	0	0
$803$	−16.9483	−0.598092
$804$	0	0
$805$	2.55479	0.0900444
$806$	0	0
$807$	−3.61634	−0.127301
$808$	0	0
$809$	7.49854	0.263635	0.131817	−	0.991274i	$-0.457919\pi$
$809$	7.49854	0.263635	0.131817	+	0.991274i	$0.457919\pi$
$810$	0	0
$811$	−43.4063	−1.52420	−0.762100	−	0.647459i	$-0.775832\pi$
$811$	−43.4063	−1.52420	−0.762100	+	0.647459i	$0.775832\pi$
$812$	0	0
$813$	−6.90934	−0.242321
$814$	0	0
$815$	22.1217	0.774889
$816$	0	0
$817$	5.70928	0.199742
$818$	0	0
$819$	−5.41855	−0.189339
$820$	0	0
$821$	26.7115	0.932239	0.466120	−	0.884722i	$-0.345652\pi$
$821$	26.7115	0.932239	0.466120	+	0.884722i	$0.345652\pi$
$822$	0	0
$823$	22.2206	0.774561	0.387280	−	0.921962i	$-0.373415\pi$
$823$	22.2206	0.774561	0.387280	+	0.921962i	$0.373415\pi$
$824$	0	0
$825$	−0.921622	−0.0320868
$826$	0	0
$827$	31.7431	1.10382	0.551908	−	0.833905i	$-0.313900\pi$
$827$	31.7431	1.10382	0.551908	+	0.833905i	$0.313900\pi$
$828$	0	0
$829$	−40.9093	−1.42084	−0.710420	−	0.703778i	$-0.751495\pi$
$829$	−40.9093	−1.42084	−0.710420	+	0.703778i	$0.751495\pi$
$830$	0	0
$831$	2.24128	0.0777490
$832$	0	0
$833$	−9.36523	−0.324486
$834$	0	0
$835$	−1.03612	−0.0358563
$836$	0	0
$837$	4.36683	0.150940
$838$	0	0
$839$	−14.3980	−0.497075	−0.248538	−	0.968622i	$-0.579950\pi$
$839$	−14.3980	−0.497075	−0.248538	+	0.968622i	$0.579950\pi$
$840$	0	0
$841$	−14.8720	−0.512827
$842$	0	0
$843$	15.6865	0.540271
$844$	0	0
$845$	2.95055	0.101502
$846$	0	0
$847$	−5.09663	−0.175122
$848$	0	0
$849$	6.54250	0.224538
$850$	0	0
$851$	−3.99547	−0.136963
$852$	0	0
$853$	24.1399	0.826536	0.413268	−	0.910610i	$-0.364387\pi$
$853$	24.1399	0.826536	0.413268	+	0.910610i	$0.364387\pi$
$854$	0	0
$855$	−2.70928	−0.0926553
$856$	0	0
$857$	47.4089	1.61946	0.809729	−	0.586804i	$-0.199614\pi$
$857$	47.4089	1.61946	0.809729	+	0.586804i	$0.199614\pi$
$858$	0	0
$859$	38.9504	1.32897	0.664485	−	0.747302i	$-0.268651\pi$
$859$	38.9504	1.32897	0.664485	+	0.747302i	$0.268651\pi$
$860$	0	0
$861$	3.15061	0.107373
$862$	0	0
$863$	20.5392	0.699162	0.349581	−	0.936906i	$-0.386324\pi$
$863$	20.5392	0.699162	0.349581	+	0.936906i	$0.386324\pi$
$864$	0	0
$865$	−12.6153	−0.428933
$866$	0	0
$867$	8.08121	0.274452
$868$	0	0
$869$	8.58145	0.291106
$870$	0	0
$871$	−19.4680	−0.659648
$872$	0	0
$873$	0.745393	0.0252277
$874$	0	0
$875$	−0.630898	−0.0213282
$876$	0	0
$877$	−2.56198	−0.0865118	−0.0432559	−	0.999064i	$-0.513773\pi$
$877$	−2.56198	−0.0865118	−0.0432559	+	0.999064i	$0.513773\pi$
$878$	0	0
$879$	6.42857	0.216830
$880$	0	0
$881$	−25.0289	−0.843246	−0.421623	−	0.906771i	$-0.638539\pi$
$881$	−25.0289	−0.843246	−0.421623	+	0.906771i	$0.638539\pi$
$882$	0	0
$883$	−57.2411	−1.92632	−0.963158	−	0.268936i	$-0.913328\pi$
$883$	−57.2411	−1.92632	−0.963158	+	0.268936i	$0.913328\pi$
$884$	0	0
$885$	7.02052	0.235992
$886$	0	0
$887$	31.0049	1.04104	0.520522	−	0.853848i	$-0.325737\pi$
$887$	31.0049	1.04104	0.520522	+	0.853848i	$0.325737\pi$
$888$	0	0
$889$	−6.13009	−0.205597
$890$	0	0
$891$	11.0556	0.370376
$892$	0	0
$893$	4.04945	0.135510
$894$	0	0
$895$	23.4908	0.785210
$896$	0	0
$897$	6.92162	0.231106
$898$	0	0
$899$	−5.33194	−0.177830
$900$	0	0
$901$	10.3935	0.346258
$902$	0	0
$903$	1.94214	0.0646304
$904$	0	0
$905$	3.36069	0.111713
$906$	0	0
$907$	24.9783	0.829389	0.414695	−	0.909961i	$-0.363888\pi$
$907$	24.9783	0.829389	0.414695	+	0.909961i	$0.363888\pi$
$908$	0	0
$909$	−45.3256	−1.50336
$910$	0	0
$911$	54.4678	1.80460	0.902300	−	0.431109i	$-0.141878\pi$
$911$	54.4678	1.80460	0.902300	+	0.431109i	$0.141878\pi$
$912$	0	0
$913$	−6.60811	−0.218697
$914$	0	0
$915$	7.07838	0.234004
$916$	0	0
$917$	−9.24497	−0.305296
$918$	0	0
$919$	19.7998	0.653134	0.326567	−	0.945174i	$-0.394108\pi$
$919$	19.7998	0.653134	0.326567	+	0.945174i	$0.394108\pi$
$920$	0	0
$921$	1.62929	0.0536869
$922$	0	0
$923$	25.1773	0.828720
$924$	0	0
$925$	0.986669	0.0324415
$926$	0	0
$927$	46.2278	1.51832
$928$	0	0
$929$	−7.78992	−0.255579	−0.127790	−	0.991801i	$-0.540788\pi$
$929$	−7.78992	−0.255579	−0.127790	+	0.991801i	$0.540788\pi$
$930$	0	0
$931$	6.60197	0.216371
$932$	0	0
$933$	14.7649	0.483380
$934$	0	0
$935$	−2.42469	−0.0792960
$936$	0	0
$937$	2.88058	0.0941046	0.0470523	−	0.998892i	$-0.485017\pi$
$937$	2.88058	0.0941046	0.0470523	+	0.998892i	$0.485017\pi$
$938$	0	0
$939$	−8.18342	−0.267056
$940$	0	0
$941$	53.6163	1.74784	0.873921	−	0.486067i	$-0.161569\pi$
$941$	53.6163	1.74784	0.873921	+	0.486067i	$0.161569\pi$
$942$	0	0
$943$	37.5052	1.22134
$944$	0	0
$945$	−1.94214	−0.0631779
$946$	0	0
$947$	−26.7708	−0.869935	−0.434968	−	0.900446i	$-0.643240\pi$
$947$	−26.7708	−0.869935	−0.434968	+	0.900446i	$0.643240\pi$
$948$	0	0
$949$	−31.4329	−1.02036
$950$	0	0
$951$	7.33791	0.237948
$952$	0	0
$953$	16.1061	0.521727	0.260864	−	0.965376i	$-0.415993\pi$
$953$	16.1061	0.521727	0.260864	+	0.965376i	$0.415993\pi$
$954$	0	0
$955$	21.4596	0.694416
$956$	0	0
$957$	3.46412	0.111979
$958$	0	0
$959$	5.37751	0.173649
$960$	0	0
$961$	−28.9877	−0.935088
$962$	0	0
$963$	−15.6442	−0.504128
$964$	0	0
$965$	−13.0856	−0.421239
$966$	0	0
$967$	−17.8537	−0.574138	−0.287069	−	0.957910i	$-0.592681\pi$
$967$	−17.8537	−0.574138	−0.287069	+	0.957910i	$0.592681\pi$
$968$	0	0
$969$	0.764867	0.0245711
$970$	0	0
$971$	58.1978	1.86766	0.933828	−	0.357722i	$-0.116447\pi$
$971$	58.1978	1.86766	0.933828	+	0.357722i	$0.116447\pi$
$972$	0	0
$973$	−14.1256	−0.452845
$974$	0	0
$975$	−1.70928	−0.0547406
$976$	0	0
$977$	12.7766	0.408759	0.204380	−	0.978892i	$-0.434482\pi$
$977$	12.7766	0.408759	0.204380	+	0.978892i	$0.434482\pi$
$978$	0	0
$979$	9.57531	0.306028
$980$	0	0
$981$	3.91548	0.125012
$982$	0	0
$983$	−40.0833	−1.27846	−0.639229	−	0.769016i	$-0.720747\pi$
$983$	−40.0833	−1.27846	−0.639229	+	0.769016i	$0.720747\pi$
$984$	0	0
$985$	−2.49693	−0.0795588
$986$	0	0
$987$	1.37751	0.0438467
$988$	0	0
$989$	23.1194	0.735155
$990$	0	0
$991$	−27.5318	−0.874577	−0.437289	−	0.899321i	$-0.644061\pi$
$991$	−27.5318	−0.874577	−0.437289	+	0.899321i	$0.644061\pi$
$992$	0	0
$993$	−3.73206	−0.118433
$994$	0	0
$995$	12.6803	0.401994
$996$	0	0
$997$	−20.1399	−0.637838	−0.318919	−	0.947782i	$-0.603320\pi$
$997$	−20.1399	−0.637838	−0.318919	+	0.947782i	$0.603320\pi$
$998$	0	0
$999$	3.03734	0.0960972

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bt.1.2		3
4.3	odd	2		6080.2.a.bu.1.2		3
8.3	odd	2		3040.2.a.m.1.2	yes	3
8.5	even	2		3040.2.a.l.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.l.1.2	✓	3	8.5	even	2
3040.2.a.m.1.2	yes	3	8.3	odd	2
6080.2.a.bt.1.2		3	1.1	even	1	trivial
6080.2.a.bu.1.2		3	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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q-0.539189 q^{3} -1.00000 q^{5} +0.630898 q^{7} -2.70928 q^{9} +O(q^{10})\)	\(q-0.539189 q^{3} -1.00000 q^{5} +0.630898 q^{7} -2.70928 q^{9} +1.70928 q^{11} +3.17009 q^{13} +0.539189 q^{15} +1.41855 q^{17} -1.00000 q^{19} -0.340173 q^{21} -4.04945 q^{23} +1.00000 q^{25} +3.07838 q^{27} -3.75872 q^{29} +1.41855 q^{31} -0.921622 q^{33} -0.630898 q^{35} +0.986669 q^{37} -1.70928 q^{39} -9.26180 q^{41} -5.70928 q^{43} +2.70928 q^{45} -4.04945 q^{47} -6.60197 q^{49} -0.764867 q^{51} +7.32684 q^{53} -1.70928 q^{55} +0.539189 q^{57} +13.0205 q^{59} +13.1278 q^{61} -1.70928 q^{63} -3.17009 q^{65} -6.14116 q^{67} +2.18342 q^{69} +7.94214 q^{71} -9.91548 q^{73} -0.539189 q^{75} +1.07838 q^{77} +5.02052 q^{79} +6.46800 q^{81} -3.86603 q^{83} -1.41855 q^{85} +2.02666 q^{87} +5.60197 q^{89} +2.00000 q^{91} -0.764867 q^{93} +1.00000 q^{95} -0.275126 q^{97} -4.63090 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 - 2 * q^7 - q^9	\( 3 q - 3 q^{5} - 2 q^{7} - q^{9} - 2 q^{11} + 4 q^{13} - 10 q^{17} - 3 q^{19} + 10 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} + 14 q^{29} - 10 q^{31} - 6 q^{33} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 20 q^{41} - 10 q^{43} + q^{45} + 6 q^{47} - q^{49} - 12 q^{51} + 10 q^{53} + 2 q^{55} + 6 q^{59} + 18 q^{61} + 2 q^{63} - 4 q^{65} + 2 q^{67} + 2 q^{69} - 6 q^{71} + 2 q^{73} - 18 q^{79} - 13 q^{81} + 2 q^{83} + 10 q^{85} + 8 q^{87} - 2 q^{89} + 6 q^{91} - 12 q^{93} + 3 q^{95} + 6 q^{97} - 10 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 - 2 * q^7 - q^9 - 2 * q^11 + 4 * q^13 - 10 * q^17 - 3 * q^19 + 10 * q^21 + 6 * q^23 + 3 * q^25 + 6 * q^27 + 14 * q^29 - 10 * q^31 - 6 * q^33 + 2 * q^35 + 2 * q^37 + 2 * q^39 - 20 * q^41 - 10 * q^43 + q^45 + 6 * q^47 - q^49 - 12 * q^51 + 10 * q^53 + 2 * q^55 + 6 * q^59 + 18 * q^61 + 2 * q^63 - 4 * q^65 + 2 * q^67 + 2 * q^69 - 6 * q^71 + 2 * q^73 - 18 * q^79 - 13 * q^81 + 2 * q^83 + 10 * q^85 + 8 * q^87 - 2 * q^89 + 6 * q^91 - 12 * q^93 + 3 * q^95 + 6 * q^97 - 10 * q^99

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