LMFDB - Embedded newform 6080.2.a.ck.1.4

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.2363492.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} - 6x^{2} + 14x + 10 $ x^5 - x^4 - 11x^3 - 6x^2 + 14*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.50919$ 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)

$f(q)$	$=$	$q+2.73033 q^{3} -1.00000 q^{5} +4.47310 q^{7} +4.45473 q^{9} +O(q^{10})$	$q+2.73033 q^{3} -1.00000 q^{5} +4.47310 q^{7} +4.45473 q^{9} +4.94350 q^{11} +2.80521 q^{13} -2.73033 q^{15} +4.39823 q^{17} +1.00000 q^{19} +12.2131 q^{21} +4.47310 q^{23} +1.00000 q^{25} +3.97190 q^{27} -10.3261 q^{29} -3.88701 q^{31} +13.4974 q^{33} -4.47310 q^{35} -4.61140 q^{37} +7.65915 q^{39} +1.09054 q^{41} +3.85296 q^{43} -4.45473 q^{45} -11.3504 q^{47} +13.0086 q^{49} +12.0086 q^{51} +13.6017 q^{53} -4.94350 q^{55} +2.73033 q^{57} -7.09919 q^{59} -5.09325 q^{61} +19.9265 q^{63} -2.80521 q^{65} -3.40192 q^{67} +12.2131 q^{69} -0.909457 q^{71} -0.0281047 q^{73} +2.73033 q^{75} +22.1128 q^{77} -15.0785 q^{79} -2.51958 q^{81} -15.5589 q^{83} -4.39823 q^{85} -28.1936 q^{87} -2.51987 q^{89} +12.5480 q^{91} -10.6128 q^{93} -1.00000 q^{95} +9.13127 q^{97} +22.0220 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9}+O(q^{10}) $ 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9	$ 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9} + 10 q^{11} - 2 q^{15} + 4 q^{17} + 5 q^{19} - 10 q^{21} + 2 q^{23} + 5 q^{25} + 8 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{33} - 2 q^{35} - 2 q^{37} + 10 q^{39} + 12 q^{41} - 2 q^{43} - 19 q^{45} + 22 q^{47} + 19 q^{49} + 14 q^{51} + 18 q^{53} - 10 q^{55} + 2 q^{57} + 4 q^{59} - 6 q^{61} + 10 q^{63} + 20 q^{67} - 10 q^{69} + 2 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 14 q^{79} + 13 q^{81} - 14 q^{83} - 4 q^{85} - 12 q^{87} + 22 q^{89} + 40 q^{91} - 8 q^{93} - 5 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100}) $ 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9 + 10 * q^11 - 2 * q^15 + 4 * q^17 + 5 * q^19 - 10 * q^21 + 2 * q^23 + 5 * q^25 + 8 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^33 - 2 * q^35 - 2 * q^37 + 10 * q^39 + 12 * q^41 - 2 * q^43 - 19 * q^45 + 22 * q^47 + 19 * q^49 + 14 * q^51 + 18 * q^53 - 10 * q^55 + 2 * q^57 + 4 * q^59 - 6 * q^61 + 10 * q^63 + 20 * q^67 - 10 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 + 13 * q^81 - 14 * q^83 - 4 * q^85 - 12 * q^87 + 22 * q^89 + 40 * q^91 - 8 * q^93 - 5 * q^95 - 10 * q^97 + 6 * 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
$3$	2.73033	1.57636	0.788180	−	0.615445i	$-0.211024\pi$
$3$	2.73033	1.57636	0.788180	+	0.615445i	$0.211024\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.47310	1.69067	0.845337	−	0.534234i	$-0.179400\pi$
$7$	4.47310	1.69067	0.845337	+	0.534234i	$0.179400\pi$
$8$	0	0
$9$	4.45473	1.48491
$10$	0	0
$11$	4.94350	1.49052	0.745261	−	0.666773i	$-0.232325\pi$
$11$	4.94350	1.49052	0.745261	+	0.666773i	$0.232325\pi$
$12$	0	0
$13$	2.80521	0.778024	0.389012	−	0.921233i	$-0.372816\pi$
$13$	2.80521	0.778024	0.389012	+	0.921233i	$0.372816\pi$
$14$	0	0
$15$	−2.73033	−0.704969
$16$	0	0
$17$	4.39823	1.06673	0.533364	−	0.845886i	$-0.320928\pi$
$17$	4.39823	1.06673	0.533364	+	0.845886i	$0.320928\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	12.2131	2.66511
$22$	0	0
$23$	4.47310	0.932706	0.466353	−	0.884599i	$-0.345568\pi$
$23$	4.47310	0.932706	0.466353	+	0.884599i	$0.345568\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.97190	0.764392
$28$	0	0
$29$	−10.3261	−1.91750	−0.958751	−	0.284248i	$-0.908256\pi$
$29$	−10.3261	−1.91750	−0.958751	+	0.284248i	$0.908256\pi$
$30$	0	0
$31$	−3.88701	−0.698127	−0.349063	−	0.937099i	$-0.613500\pi$
$31$	−3.88701	−0.698127	−0.349063	+	0.937099i	$0.613500\pi$
$32$	0	0
$33$	13.4974	2.34960
$34$	0	0
$35$	−4.47310	−0.756092
$36$	0	0
$37$	−4.61140	−0.758109	−0.379055	−	0.925374i	$-0.623751\pi$
$37$	−4.61140	−0.758109	−0.379055	+	0.925374i	$0.623751\pi$
$38$	0	0
$39$	7.65915	1.22645
$40$	0	0
$41$	1.09054	0.170314	0.0851571	−	0.996368i	$-0.472861\pi$
$41$	1.09054	0.170314	0.0851571	+	0.996368i	$0.472861\pi$
$42$	0	0
$43$	3.85296	0.587571	0.293785	−	0.955871i	$-0.405085\pi$
$43$	3.85296	0.587571	0.293785	+	0.955871i	$0.405085\pi$
$44$	0	0
$45$	−4.45473	−0.664072
$46$	0	0
$47$	−11.3504	−1.65562	−0.827811	−	0.561007i	$-0.810414\pi$
$47$	−11.3504	−1.65562	−0.827811	+	0.561007i	$0.810414\pi$
$48$	0	0
$49$	13.0086	1.85838
$50$	0	0
$51$	12.0086	1.68155
$52$	0	0
$53$	13.6017	1.86833	0.934166	−	0.356838i	$-0.116145\pi$
$53$	13.6017	1.86833	0.934166	+	0.356838i	$0.116145\pi$
$54$	0	0
$55$	−4.94350	−0.666582
$56$	0	0
$57$	2.73033	0.361642
$58$	0	0
$59$	−7.09919	−0.924235	−0.462118	−	0.886819i	$-0.652910\pi$
$59$	−7.09919	−0.924235	−0.462118	+	0.886819i	$0.652910\pi$
$60$	0	0
$61$	−5.09325	−0.652123	−0.326062	−	0.945349i	$-0.605722\pi$
$61$	−5.09325	−0.652123	−0.326062	+	0.945349i	$0.605722\pi$
$62$	0	0
$63$	19.9265	2.51050
$64$	0	0
$65$	−2.80521	−0.347943
$66$	0	0
$67$	−3.40192	−0.415611	−0.207805	−	0.978170i	$-0.566632\pi$
$67$	−3.40192	−0.415611	−0.207805	+	0.978170i	$0.566632\pi$
$68$	0	0
$69$	12.2131	1.47028
$70$	0	0
$71$	−0.909457	−0.107933	−0.0539663	−	0.998543i	$-0.517186\pi$
$71$	−0.909457	−0.107933	−0.0539663	+	0.998543i	$0.517186\pi$
$72$	0	0
$73$	−0.0281047	−0.00328941	−0.00164470	−	0.999999i	$-0.500524\pi$
$73$	−0.0281047	−0.00328941	−0.00164470	+	0.999999i	$0.500524\pi$
$74$	0	0
$75$	2.73033	0.315272
$76$	0	0
$77$	22.1128	2.51999
$78$	0	0
$79$	−15.0785	−1.69646	−0.848230	−	0.529629i	$-0.822331\pi$
$79$	−15.0785	−1.69646	−0.848230	+	0.529629i	$0.822331\pi$
$80$	0	0
$81$	−2.51958	−0.279953
$82$	0	0
$83$	−15.5589	−1.70781	−0.853904	−	0.520430i	$-0.825772\pi$
$83$	−15.5589	−1.70781	−0.853904	+	0.520430i	$0.825772\pi$
$84$	0	0
$85$	−4.39823	−0.477055
$86$	0	0
$87$	−28.1936	−3.02267
$88$	0	0
$89$	−2.51987	−0.267106	−0.133553	−	0.991042i	$-0.542639\pi$
$89$	−2.51987	−0.267106	−0.133553	+	0.991042i	$0.542639\pi$
$90$	0	0
$91$	12.5480	1.31539
$92$	0	0
$93$	−10.6128	−1.10050
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	9.13127	0.927140	0.463570	−	0.886060i	$-0.346568\pi$
$97$	9.13127	0.927140	0.463570	+	0.886060i	$0.346568\pi$
$98$	0	0
$99$	22.0220	2.21329
$100$	0	0
$101$	−15.9852	−1.59059	−0.795294	−	0.606224i	$-0.792684\pi$
$101$	−15.9852	−1.59059	−0.795294	+	0.606224i	$0.792684\pi$
$102$	0	0
$103$	−5.81764	−0.573229	−0.286614	−	0.958046i	$-0.592530\pi$
$103$	−5.81764	−0.573229	−0.286614	+	0.958046i	$0.592530\pi$
$104$	0	0
$105$	−12.2131	−1.19187
$106$	0	0
$107$	−3.26967	−0.316090	−0.158045	−	0.987432i	$-0.550519\pi$
$107$	−3.26967	−0.316090	−0.158045	+	0.987432i	$0.550519\pi$
$108$	0	0
$109$	−4.47040	−0.428187	−0.214093	−	0.976813i	$-0.568680\pi$
$109$	−4.47040	−0.428187	−0.214093	+	0.976813i	$0.568680\pi$
$110$	0	0
$111$	−12.5907	−1.19505
$112$	0	0
$113$	−0.209928	−0.0197484	−0.00987419	−	0.999951i	$-0.503143\pi$
$113$	−0.209928	−0.0197484	−0.00987419	+	0.999951i	$0.503143\pi$
$114$	0	0
$115$	−4.47310	−0.417119
$116$	0	0
$117$	12.4964	1.15530
$118$	0	0
$119$	19.6737	1.80349
$120$	0	0
$121$	13.4382	1.22166
$122$	0	0
$123$	2.97755	0.268477
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.5236	1.37749	0.688746	−	0.725002i	$-0.258161\pi$
$127$	15.5236	1.37749	0.688746	+	0.725002i	$0.258161\pi$
$128$	0	0
$129$	10.5199	0.926223
$130$	0	0
$131$	−8.58796	−0.750334	−0.375167	−	0.926957i	$-0.622415\pi$
$131$	−8.58796	−0.750334	−0.375167	+	0.926957i	$0.622415\pi$
$132$	0	0
$133$	4.47310	0.387867
$134$	0	0
$135$	−3.97190	−0.341846
$136$	0	0
$137$	−5.48877	−0.468938	−0.234469	−	0.972124i	$-0.575335\pi$
$137$	−5.48877	−0.468938	−0.234469	+	0.972124i	$0.575335\pi$
$138$	0	0
$139$	−13.6838	−1.16064	−0.580320	−	0.814388i	$-0.697073\pi$
$139$	−13.6838	−1.16064	−0.580320	+	0.814388i	$0.697073\pi$
$140$	0	0
$141$	−30.9903	−2.60986
$142$	0	0
$143$	13.8675	1.15966
$144$	0	0
$145$	10.3261	0.857533
$146$	0	0
$147$	35.5180	2.92947
$148$	0	0
$149$	−16.0146	−1.31197	−0.655983	−	0.754776i	$-0.727746\pi$
$149$	−16.0146	−1.31197	−0.655983	+	0.754776i	$0.727746\pi$
$150$	0	0
$151$	−6.24525	−0.508231	−0.254116	−	0.967174i	$-0.581784\pi$
$151$	−6.24525	−0.508231	−0.254116	+	0.967174i	$0.581784\pi$
$152$	0	0
$153$	19.5929	1.58399
$154$	0	0
$155$	3.88701	0.312212
$156$	0	0
$157$	4.03675	0.322168	0.161084	−	0.986941i	$-0.448501\pi$
$157$	4.03675	0.322168	0.161084	+	0.986941i	$0.448501\pi$
$158$	0	0
$159$	37.1371	2.94516
$160$	0	0
$161$	20.0086	1.57690
$162$	0	0
$163$	−1.18137	−0.0925324	−0.0462662	−	0.998929i	$-0.514732\pi$
$163$	−1.18137	−0.0925324	−0.0462662	+	0.998929i	$0.514732\pi$
$164$	0	0
$165$	−13.4974	−1.05077
$166$	0	0
$167$	−12.8520	−0.994516	−0.497258	−	0.867603i	$-0.665660\pi$
$167$	−12.8520	−0.994516	−0.497258	+	0.867603i	$0.665660\pi$
$168$	0	0
$169$	−5.13082	−0.394678
$170$	0	0
$171$	4.45473	0.340662
$172$	0	0
$173$	13.1745	1.00164	0.500819	−	0.865552i	$-0.333032\pi$
$173$	13.1745	1.00164	0.500819	+	0.865552i	$0.333032\pi$
$174$	0	0
$175$	4.47310	0.338135
$176$	0	0
$177$	−19.3832	−1.45693
$178$	0	0
$179$	−7.16659	−0.535656	−0.267828	−	0.963467i	$-0.586306\pi$
$179$	−7.16659	−0.535656	−0.267828	+	0.963467i	$0.586306\pi$
$180$	0	0
$181$	−9.36516	−0.696107	−0.348054	−	0.937475i	$-0.613157\pi$
$181$	−9.36516	−0.696107	−0.348054	+	0.937475i	$0.613157\pi$
$182$	0	0
$183$	−13.9063	−1.02798
$184$	0	0
$185$	4.61140	0.339037
$186$	0	0
$187$	21.7427	1.58998
$188$	0	0
$189$	17.7667	1.29234
$190$	0	0
$191$	12.8405	0.929108	0.464554	−	0.885545i	$-0.346215\pi$
$191$	12.8405	0.929108	0.464554	+	0.885545i	$0.346215\pi$
$192$	0	0
$193$	−19.5381	−1.40639	−0.703193	−	0.710999i	$-0.748243\pi$
$193$	−19.5381	−1.40639	−0.703193	+	0.710999i	$0.748243\pi$
$194$	0	0
$195$	−7.65915	−0.548483
$196$	0	0
$197$	−17.7353	−1.26359	−0.631794	−	0.775137i	$-0.717681\pi$
$197$	−17.7353	−1.26359	−0.631794	+	0.775137i	$0.717681\pi$
$198$	0	0
$199$	26.7275	1.89466	0.947332	−	0.320252i	$-0.103768\pi$
$199$	26.7275	1.89466	0.947332	+	0.320252i	$0.103768\pi$
$200$	0	0
$201$	−9.28838	−0.655152
$202$	0	0
$203$	−46.1895	−3.24187
$204$	0	0
$205$	−1.09054	−0.0761669
$206$	0	0
$207$	19.9265	1.38498
$208$	0	0
$209$	4.94350	0.341949
$210$	0	0
$211$	6.17785	0.425301	0.212650	−	0.977128i	$-0.431790\pi$
$211$	6.17785	0.425301	0.212650	+	0.977128i	$0.431790\pi$
$212$	0	0
$213$	−2.48312	−0.170141
$214$	0	0
$215$	−3.85296	−0.262770
$216$	0	0
$217$	−17.3870	−1.18030
$218$	0	0
$219$	−0.0767352	−0.00518529
$220$	0	0
$221$	12.3379	0.829940
$222$	0	0
$223$	13.1404	0.879950	0.439975	−	0.898010i	$-0.354987\pi$
$223$	13.1404	0.879950	0.439975	+	0.898010i	$0.354987\pi$
$224$	0	0
$225$	4.45473	0.296982
$226$	0	0
$227$	1.82088	0.120856	0.0604280	−	0.998173i	$-0.480753\pi$
$227$	1.82088	0.120856	0.0604280	+	0.998173i	$0.480753\pi$
$228$	0	0
$229$	−4.22309	−0.279069	−0.139535	−	0.990217i	$-0.544561\pi$
$229$	−4.22309	−0.279069	−0.139535	+	0.990217i	$0.544561\pi$
$230$	0	0
$231$	60.3753	3.97240
$232$	0	0
$233$	8.85070	0.579829	0.289914	−	0.957053i	$-0.406373\pi$
$233$	8.85070	0.579829	0.289914	+	0.957053i	$0.406373\pi$
$234$	0	0
$235$	11.3504	0.740417
$236$	0	0
$237$	−41.1692	−2.67423
$238$	0	0
$239$	25.2841	1.63550	0.817748	−	0.575577i	$-0.195222\pi$
$239$	25.2841	1.63550	0.817748	+	0.575577i	$0.195222\pi$
$240$	0	0
$241$	−2.67158	−0.172092	−0.0860459	−	0.996291i	$-0.527423\pi$
$241$	−2.67158	−0.172092	−0.0860459	+	0.996291i	$0.527423\pi$
$242$	0	0
$243$	−18.7950	−1.20570
$244$	0	0
$245$	−13.0086	−0.831092
$246$	0	0
$247$	2.80521	0.178491
$248$	0	0
$249$	−42.4809	−2.69212
$250$	0	0
$251$	−30.7008	−1.93781	−0.968907	−	0.247424i	$-0.920416\pi$
$251$	−30.7008	−1.93781	−0.968907	+	0.247424i	$0.920416\pi$
$252$	0	0
$253$	22.1128	1.39022
$254$	0	0
$255$	−12.0086	−0.752010
$256$	0	0
$257$	26.2043	1.63458	0.817290	−	0.576226i	$-0.195475\pi$
$257$	26.2043	1.63458	0.817290	+	0.576226i	$0.195475\pi$
$258$	0	0
$259$	−20.6273	−1.28172
$260$	0	0
$261$	−45.9998	−2.84732
$262$	0	0
$263$	3.93854	0.242861	0.121430	−	0.992600i	$-0.461252\pi$
$263$	3.93854	0.242861	0.121430	+	0.992600i	$0.461252\pi$
$264$	0	0
$265$	−13.6017	−0.835544
$266$	0	0
$267$	−6.88009	−0.421054
$268$	0	0
$269$	15.4293	0.940743	0.470371	−	0.882469i	$-0.344120\pi$
$269$	15.4293	0.940743	0.470371	+	0.882469i	$0.344120\pi$
$270$	0	0
$271$	14.2279	0.864281	0.432140	−	0.901806i	$-0.357759\pi$
$271$	14.2279	0.864281	0.432140	+	0.901806i	$0.357759\pi$
$272$	0	0
$273$	34.2602	2.07352
$274$	0	0
$275$	4.94350	0.298104
$276$	0	0
$277$	19.7308	1.18551	0.592754	−	0.805384i	$-0.298041\pi$
$277$	19.7308	1.18551	0.592754	+	0.805384i	$0.298041\pi$
$278$	0	0
$279$	−17.3156	−1.03666
$280$	0	0
$281$	21.8308	1.30232	0.651158	−	0.758942i	$-0.274283\pi$
$281$	21.8308	1.30232	0.651158	+	0.758942i	$0.274283\pi$
$282$	0	0
$283$	26.0394	1.54788	0.773942	−	0.633256i	$-0.218282\pi$
$283$	26.0394	1.54788	0.773942	+	0.633256i	$0.218282\pi$
$284$	0	0
$285$	−2.73033	−0.161731
$286$	0	0
$287$	4.87811	0.287946
$288$	0	0
$289$	2.34444	0.137908
$290$	0	0
$291$	24.9314	1.46151
$292$	0	0
$293$	26.3219	1.53774	0.768870	−	0.639405i	$-0.220819\pi$
$293$	26.3219	1.53774	0.768870	+	0.639405i	$0.220819\pi$
$294$	0	0
$295$	7.09919	0.413331
$296$	0	0
$297$	19.6351	1.13934
$298$	0	0
$299$	12.5480	0.725668
$300$	0	0
$301$	17.2347	0.993390
$302$	0	0
$303$	−43.6450	−2.50734
$304$	0	0
$305$	5.09325	0.291638
$306$	0	0
$307$	31.6309	1.80527	0.902637	−	0.430403i	$-0.141629\pi$
$307$	31.6309	1.80527	0.902637	+	0.430403i	$0.141629\pi$
$308$	0	0
$309$	−15.8841	−0.903615
$310$	0	0
$311$	20.0468	1.13675	0.568374	−	0.822770i	$-0.307573\pi$
$311$	20.0468	1.13675	0.568374	+	0.822770i	$0.307573\pi$
$312$	0	0
$313$	−4.83645	−0.273372	−0.136686	−	0.990614i	$-0.543645\pi$
$313$	−4.83645	−0.273372	−0.136686	+	0.990614i	$0.543645\pi$
$314$	0	0
$315$	−19.9265	−1.12273
$316$	0	0
$317$	−22.6553	−1.27245	−0.636223	−	0.771505i	$-0.719504\pi$
$317$	−22.6553	−1.27245	−0.636223	+	0.771505i	$0.719504\pi$
$318$	0	0
$319$	−51.0469	−2.85808
$320$	0	0
$321$	−8.92728	−0.498272
$322$	0	0
$323$	4.39823	0.244724
$324$	0	0
$325$	2.80521	0.155605
$326$	0	0
$327$	−12.2057	−0.674976
$328$	0	0
$329$	−50.7714	−2.79912
$330$	0	0
$331$	−21.6578	−1.19042	−0.595210	−	0.803571i	$-0.702931\pi$
$331$	−21.6578	−1.19042	−0.595210	+	0.803571i	$0.702931\pi$
$332$	0	0
$333$	−20.5425	−1.12572
$334$	0	0
$335$	3.40192	0.185867
$336$	0	0
$337$	18.3292	0.998455	0.499228	−	0.866471i	$-0.333617\pi$
$337$	18.3292	0.998455	0.499228	+	0.866471i	$0.333617\pi$
$338$	0	0
$339$	−0.573174	−0.0311305
$340$	0	0
$341$	−19.2154	−1.04057
$342$	0	0
$343$	26.8773	1.45124
$344$	0	0
$345$	−12.2131	−0.657529
$346$	0	0
$347$	−9.51958	−0.511038	−0.255519	−	0.966804i	$-0.582246\pi$
$347$	−9.51958	−0.511038	−0.255519	+	0.966804i	$0.582246\pi$
$348$	0	0
$349$	−11.3228	−0.606096	−0.303048	−	0.952975i	$-0.598004\pi$
$349$	−11.3228	−0.606096	−0.303048	+	0.952975i	$0.598004\pi$
$350$	0	0
$351$	11.1420	0.594715
$352$	0	0
$353$	19.3758	1.03127	0.515634	−	0.856809i	$-0.327556\pi$
$353$	19.3758	1.03127	0.515634	+	0.856809i	$0.327556\pi$
$354$	0	0
$355$	0.909457	0.0482689
$356$	0	0
$357$	53.7159	2.84295
$358$	0	0
$359$	14.3214	0.755854	0.377927	−	0.925835i	$-0.376637\pi$
$359$	14.3214	0.755854	0.377927	+	0.925835i	$0.376637\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	36.6908	1.92577
$364$	0	0
$365$	0.0281047	0.00147107
$366$	0	0
$367$	3.99082	0.208319	0.104160	−	0.994561i	$-0.466785\pi$
$367$	3.99082	0.208319	0.104160	+	0.994561i	$0.466785\pi$
$368$	0	0
$369$	4.85807	0.252901
$370$	0	0
$371$	60.8417	3.15874
$372$	0	0
$373$	8.10228	0.419520	0.209760	−	0.977753i	$-0.432732\pi$
$373$	8.10228	0.419520	0.209760	+	0.977753i	$0.432732\pi$
$374$	0	0
$375$	−2.73033	−0.140994
$376$	0	0
$377$	−28.9667	−1.49186
$378$	0	0
$379$	10.5480	0.541813	0.270907	−	0.962606i	$-0.412677\pi$
$379$	10.5480	0.541813	0.270907	+	0.962606i	$0.412677\pi$
$380$	0	0
$381$	42.3845	2.17142
$382$	0	0
$383$	38.1059	1.94712	0.973561	−	0.228429i	$-0.0733589\pi$
$383$	38.1059	1.94712	0.973561	+	0.228429i	$0.0733589\pi$
$384$	0	0
$385$	−22.1128	−1.12697
$386$	0	0
$387$	17.1639	0.872489
$388$	0	0
$389$	−10.5998	−0.537433	−0.268717	−	0.963219i	$-0.586600\pi$
$389$	−10.5998	−0.537433	−0.268717	+	0.963219i	$0.586600\pi$
$390$	0	0
$391$	19.6737	0.994944
$392$	0	0
$393$	−23.4480	−1.18280
$394$	0	0
$395$	15.0785	0.758680
$396$	0	0
$397$	35.1584	1.76455	0.882276	−	0.470733i	$-0.156011\pi$
$397$	35.1584	1.76455	0.882276	+	0.470733i	$0.156011\pi$
$398$	0	0
$399$	12.2131	0.611418
$400$	0	0
$401$	−12.9267	−0.645531	−0.322765	−	0.946479i	$-0.604612\pi$
$401$	−12.9267	−0.645531	−0.322765	+	0.946479i	$0.604612\pi$
$402$	0	0
$403$	−10.9039	−0.543159
$404$	0	0
$405$	2.51958	0.125199
$406$	0	0
$407$	−22.7965	−1.12998
$408$	0	0
$409$	4.58148	0.226540	0.113270	−	0.993564i	$-0.463868\pi$
$409$	4.58148	0.226540	0.113270	+	0.993564i	$0.463868\pi$
$410$	0	0
$411$	−14.9862	−0.739214
$412$	0	0
$413$	−31.7554	−1.56258
$414$	0	0
$415$	15.5589	0.763755
$416$	0	0
$417$	−37.3612	−1.82959
$418$	0	0
$419$	5.12926	0.250581	0.125290	−	0.992120i	$-0.460014\pi$
$419$	5.12926	0.250581	0.125290	+	0.992120i	$0.460014\pi$
$420$	0	0
$421$	13.3991	0.653033	0.326516	−	0.945192i	$-0.394125\pi$
$421$	13.3991	0.653033	0.326516	+	0.945192i	$0.394125\pi$
$422$	0	0
$423$	−50.5628	−2.45845
$424$	0	0
$425$	4.39823	0.213346
$426$	0	0
$427$	−22.7826	−1.10253
$428$	0	0
$429$	37.8630	1.82804
$430$	0	0
$431$	−33.4780	−1.61258	−0.806288	−	0.591523i	$-0.798527\pi$
$431$	−33.4780	−1.61258	−0.806288	+	0.591523i	$0.798527\pi$
$432$	0	0
$433$	5.46465	0.262614	0.131307	−	0.991342i	$-0.458083\pi$
$433$	5.46465	0.262614	0.131307	+	0.991342i	$0.458083\pi$
$434$	0	0
$435$	28.1936	1.35178
$436$	0	0
$437$	4.47310	0.213977
$438$	0	0
$439$	17.1915	0.820503	0.410252	−	0.911972i	$-0.365441\pi$
$439$	17.1915	0.820503	0.410252	+	0.911972i	$0.365441\pi$
$440$	0	0
$441$	57.9500	2.75952
$442$	0	0
$443$	32.5492	1.54646	0.773230	−	0.634126i	$-0.218640\pi$
$443$	32.5492	1.54646	0.773230	+	0.634126i	$0.218640\pi$
$444$	0	0
$445$	2.51987	0.119453
$446$	0	0
$447$	−43.7252	−2.06813
$448$	0	0
$449$	−18.4188	−0.869235	−0.434617	−	0.900615i	$-0.643116\pi$
$449$	−18.4188	−0.869235	−0.434617	+	0.900615i	$0.643116\pi$
$450$	0	0
$451$	5.39110	0.253857
$452$	0	0
$453$	−17.0516	−0.801155
$454$	0	0
$455$	−12.5480	−0.588258
$456$	0	0
$457$	16.9503	0.792904	0.396452	−	0.918056i	$-0.370241\pi$
$457$	16.9503	0.792904	0.396452	+	0.918056i	$0.370241\pi$
$458$	0	0
$459$	17.4693	0.815398
$460$	0	0
$461$	2.40885	0.112191	0.0560956	−	0.998425i	$-0.482135\pi$
$461$	2.40885	0.112191	0.0560956	+	0.998425i	$0.482135\pi$
$462$	0	0
$463$	−19.6499	−0.913207	−0.456603	−	0.889670i	$-0.650934\pi$
$463$	−19.6499	−0.913207	−0.456603	+	0.889670i	$0.650934\pi$
$464$	0	0
$465$	10.6128	0.492158
$466$	0	0
$467$	7.57637	0.350592	0.175296	−	0.984516i	$-0.443912\pi$
$467$	7.57637	0.350592	0.175296	+	0.984516i	$0.443912\pi$
$468$	0	0
$469$	−15.2171	−0.702662
$470$	0	0
$471$	11.0217	0.507852
$472$	0	0
$473$	19.0471	0.875787
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	60.5917	2.77430
$478$	0	0
$479$	−27.4891	−1.25601	−0.628005	−	0.778210i	$-0.716128\pi$
$479$	−27.4891	−1.25601	−0.628005	+	0.778210i	$0.716128\pi$
$480$	0	0
$481$	−12.9359	−0.589827
$482$	0	0
$483$	54.6303	2.48576
$484$	0	0
$485$	−9.13127	−0.414629
$486$	0	0
$487$	13.8002	0.625344	0.312672	−	0.949861i	$-0.398776\pi$
$487$	13.8002	0.625344	0.312672	+	0.949861i	$0.398776\pi$
$488$	0	0
$489$	−3.22555	−0.145864
$490$	0	0
$491$	−38.6208	−1.74293	−0.871466	−	0.490457i	$-0.836830\pi$
$491$	−38.6208	−1.74293	−0.871466	+	0.490457i	$0.836830\pi$
$492$	0	0
$493$	−45.4164	−2.04545
$494$	0	0
$495$	−22.0220	−0.989814
$496$	0	0
$497$	−4.06809	−0.182479
$498$	0	0
$499$	−5.80806	−0.260004	−0.130002	−	0.991514i	$-0.541498\pi$
$499$	−5.80806	−0.260004	−0.130002	+	0.991514i	$0.541498\pi$
$500$	0	0
$501$	−35.0902	−1.56771
$502$	0	0
$503$	35.3944	1.57816	0.789080	−	0.614290i	$-0.210558\pi$
$503$	35.3944	1.57816	0.789080	+	0.614290i	$0.210558\pi$
$504$	0	0
$505$	15.9852	0.711333
$506$	0	0
$507$	−14.0089	−0.622155
$508$	0	0
$509$	−32.0685	−1.42141	−0.710705	−	0.703490i	$-0.751624\pi$
$509$	−32.0685	−1.42141	−0.710705	+	0.703490i	$0.751624\pi$
$510$	0	0
$511$	−0.125715	−0.00556131
$512$	0	0
$513$	3.97190	0.175363
$514$	0	0
$515$	5.81764	0.256356
$516$	0	0
$517$	−56.1106	−2.46774
$518$	0	0
$519$	35.9708	1.57894
$520$	0	0
$521$	−42.5996	−1.86632	−0.933162	−	0.359456i	$-0.882962\pi$
$521$	−42.5996	−1.86632	−0.933162	+	0.359456i	$0.882962\pi$
$522$	0	0
$523$	28.4555	1.24427	0.622136	−	0.782909i	$-0.286265\pi$
$523$	28.4555	1.24427	0.622136	+	0.782909i	$0.286265\pi$
$524$	0	0
$525$	12.2131	0.533022
$526$	0	0
$527$	−17.0959	−0.744711
$528$	0	0
$529$	−2.99136	−0.130059
$530$	0	0
$531$	−31.6249	−1.37241
$532$	0	0
$533$	3.05920	0.132509
$534$	0	0
$535$	3.26967	0.141360
$536$	0	0
$537$	−19.5672	−0.844386
$538$	0	0
$539$	64.3083	2.76995
$540$	0	0
$541$	9.12262	0.392212	0.196106	−	0.980583i	$-0.437170\pi$
$541$	9.12262	0.392212	0.196106	+	0.980583i	$0.437170\pi$
$542$	0	0
$543$	−25.5700	−1.09732
$544$	0	0
$545$	4.47040	0.191491
$546$	0	0
$547$	−39.4646	−1.68738	−0.843692	−	0.536827i	$-0.819623\pi$
$547$	−39.4646	−1.68738	−0.843692	+	0.536827i	$0.819623\pi$
$548$	0	0
$549$	−22.6890	−0.968344
$550$	0	0
$551$	−10.3261	−0.439905
$552$	0	0
$553$	−67.4475	−2.86816
$554$	0	0
$555$	12.5907	0.534444
$556$	0	0
$557$	−20.2450	−0.857810	−0.428905	−	0.903350i	$-0.641101\pi$
$557$	−20.2450	−0.857810	−0.428905	+	0.903350i	$0.641101\pi$
$558$	0	0
$559$	10.8083	0.457144
$560$	0	0
$561$	59.3648	2.50638
$562$	0	0
$563$	21.7982	0.918684	0.459342	−	0.888260i	$-0.348085\pi$
$563$	21.7982	0.918684	0.459342	+	0.888260i	$0.348085\pi$
$564$	0	0
$565$	0.209928	0.00883174
$566$	0	0
$567$	−11.2703	−0.473310
$568$	0	0
$569$	−38.3987	−1.60976	−0.804879	−	0.593439i	$-0.797770\pi$
$569$	−38.3987	−1.60976	−0.804879	+	0.593439i	$0.797770\pi$
$570$	0	0
$571$	31.4383	1.31565	0.657826	−	0.753170i	$-0.271476\pi$
$571$	31.4383	1.31565	0.657826	+	0.753170i	$0.271476\pi$
$572$	0	0
$573$	35.0589	1.46461
$574$	0	0
$575$	4.47310	0.186541
$576$	0	0
$577$	−23.7902	−0.990400	−0.495200	−	0.868779i	$-0.664905\pi$
$577$	−23.7902	−0.990400	−0.495200	+	0.868779i	$0.664905\pi$
$578$	0	0
$579$	−53.3457	−2.21697
$580$	0	0
$581$	−69.5964	−2.88735
$582$	0	0
$583$	67.2399	2.78479
$584$	0	0
$585$	−12.4964	−0.516664
$586$	0	0
$587$	−17.8162	−0.735354	−0.367677	−	0.929954i	$-0.619847\pi$
$587$	−17.8162	−0.735354	−0.367677	+	0.929954i	$0.619847\pi$
$588$	0	0
$589$	−3.88701	−0.160161
$590$	0	0
$591$	−48.4233	−1.99187
$592$	0	0
$593$	34.0238	1.39719	0.698594	−	0.715518i	$-0.253809\pi$
$593$	34.0238	1.39719	0.698594	+	0.715518i	$0.253809\pi$
$594$	0	0
$595$	−19.6737	−0.806545
$596$	0	0
$597$	72.9751	2.98667
$598$	0	0
$599$	−2.32886	−0.0951545	−0.0475772	−	0.998868i	$-0.515150\pi$
$599$	−2.32886	−0.0951545	−0.0475772	+	0.998868i	$0.515150\pi$
$600$	0	0
$601$	20.2351	0.825408	0.412704	−	0.910865i	$-0.364584\pi$
$601$	20.2351	0.825408	0.412704	+	0.910865i	$0.364584\pi$
$602$	0	0
$603$	−15.1546	−0.617144
$604$	0	0
$605$	−13.4382	−0.546341
$606$	0	0
$607$	−1.53544	−0.0623216	−0.0311608	−	0.999514i	$-0.509920\pi$
$607$	−1.53544	−0.0623216	−0.0311608	+	0.999514i	$0.509920\pi$
$608$	0	0
$609$	−126.113	−5.11035
$610$	0	0
$611$	−31.8401	−1.28811
$612$	0	0
$613$	7.90449	0.319259	0.159630	−	0.987177i	$-0.448970\pi$
$613$	7.90449	0.319259	0.159630	+	0.987177i	$0.448970\pi$
$614$	0	0
$615$	−2.97755	−0.120066
$616$	0	0
$617$	−5.58555	−0.224866	−0.112433	−	0.993659i	$-0.535864\pi$
$617$	−5.58555	−0.224866	−0.112433	+	0.993659i	$0.535864\pi$
$618$	0	0
$619$	−14.2050	−0.570948	−0.285474	−	0.958386i	$-0.592151\pi$
$619$	−14.2050	−0.570948	−0.285474	+	0.958386i	$0.592151\pi$
$620$	0	0
$621$	17.7667	0.712953
$622$	0	0
$623$	−11.2716	−0.451588
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	13.4974	0.539035
$628$	0	0
$629$	−20.2820	−0.808696
$630$	0	0
$631$	1.39273	0.0554437	0.0277219	−	0.999616i	$-0.491175\pi$
$631$	1.39273	0.0554437	0.0277219	+	0.999616i	$0.491175\pi$
$632$	0	0
$633$	16.8676	0.670427
$634$	0	0
$635$	−15.5236	−0.616034
$636$	0	0
$637$	36.4919	1.44586
$638$	0	0
$639$	−4.05138	−0.160270
$640$	0	0
$641$	−29.1491	−1.15132	−0.575660	−	0.817689i	$-0.695255\pi$
$641$	−29.1491	−1.15132	−0.575660	+	0.817689i	$0.695255\pi$
$642$	0	0
$643$	18.8627	0.743874	0.371937	−	0.928258i	$-0.378694\pi$
$643$	18.8627	0.743874	0.371937	+	0.928258i	$0.378694\pi$
$644$	0	0
$645$	−10.5199	−0.414219
$646$	0	0
$647$	−38.0485	−1.49584	−0.747921	−	0.663788i	$-0.768948\pi$
$647$	−38.0485	−1.49584	−0.747921	+	0.663788i	$0.768948\pi$
$648$	0	0
$649$	−35.0948	−1.37759
$650$	0	0
$651$	−47.4723	−1.86058
$652$	0	0
$653$	34.2524	1.34040	0.670200	−	0.742180i	$-0.266208\pi$
$653$	34.2524	1.34040	0.670200	+	0.742180i	$0.266208\pi$
$654$	0	0
$655$	8.58796	0.335559
$656$	0	0
$657$	−0.125199	−0.00488447
$658$	0	0
$659$	−11.0927	−0.432111	−0.216055	−	0.976381i	$-0.569319\pi$
$659$	−11.0927	−0.432111	−0.216055	+	0.976381i	$0.569319\pi$
$660$	0	0
$661$	−36.6437	−1.42527	−0.712637	−	0.701532i	$-0.752500\pi$
$661$	−36.6437	−1.42527	−0.712637	+	0.701532i	$0.752500\pi$
$662$	0	0
$663$	33.6867	1.30828
$664$	0	0
$665$	−4.47310	−0.173459
$666$	0	0
$667$	−46.1895	−1.78847
$668$	0	0
$669$	35.8778	1.38712
$670$	0	0
$671$	−25.1785	−0.972004
$672$	0	0
$673$	28.5036	1.09873	0.549367	−	0.835581i	$-0.314869\pi$
$673$	28.5036	1.09873	0.549367	+	0.835581i	$0.314869\pi$
$674$	0	0
$675$	3.97190	0.152878
$676$	0	0
$677$	−17.2369	−0.662470	−0.331235	−	0.943548i	$-0.607465\pi$
$677$	−17.2369	−0.662470	−0.331235	+	0.943548i	$0.607465\pi$
$678$	0	0
$679$	40.8451	1.56749
$680$	0	0
$681$	4.97161	0.190512
$682$	0	0
$683$	−33.8742	−1.29616	−0.648081	−	0.761572i	$-0.724428\pi$
$683$	−33.8742	−1.29616	−0.648081	+	0.761572i	$0.724428\pi$
$684$	0	0
$685$	5.48877	0.209715
$686$	0	0
$687$	−11.5304	−0.439914
$688$	0	0
$689$	38.1555	1.45361
$690$	0	0
$691$	2.73697	0.104119	0.0520597	−	0.998644i	$-0.483421\pi$
$691$	2.73697	0.104119	0.0520597	+	0.998644i	$0.483421\pi$
$692$	0	0
$693$	98.5065	3.74195
$694$	0	0
$695$	13.6838	0.519054
$696$	0	0
$697$	4.79646	0.181679
$698$	0	0
$699$	24.1654	0.914018
$700$	0	0
$701$	13.7838	0.520606	0.260303	−	0.965527i	$-0.416178\pi$
$701$	13.7838	0.520606	0.260303	+	0.965527i	$0.416178\pi$
$702$	0	0
$703$	−4.61140	−0.173922
$704$	0	0
$705$	30.9903	1.16716
$706$	0	0
$707$	−71.5035	−2.68917
$708$	0	0
$709$	3.62229	0.136038	0.0680191	−	0.997684i	$-0.478332\pi$
$709$	3.62229	0.136038	0.0680191	+	0.997684i	$0.478332\pi$
$710$	0	0
$711$	−67.1704	−2.51909
$712$	0	0
$713$	−17.3870	−0.651147
$714$	0	0
$715$	−13.8675	−0.518617
$716$	0	0
$717$	69.0342	2.57813
$718$	0	0
$719$	−13.7721	−0.513614	−0.256807	−	0.966463i	$-0.582671\pi$
$719$	−13.7721	−0.513614	−0.256807	+	0.966463i	$0.582671\pi$
$720$	0	0
$721$	−26.0229	−0.969143
$722$	0	0
$723$	−7.29432	−0.271279
$724$	0	0
$725$	−10.3261	−0.383500
$726$	0	0
$727$	8.42623	0.312511	0.156256	−	0.987717i	$-0.450058\pi$
$727$	8.42623	0.312511	0.156256	+	0.987717i	$0.450058\pi$
$728$	0	0
$729$	−43.7579	−1.62066
$730$	0	0
$731$	16.9462	0.626778
$732$	0	0
$733$	32.1690	1.18819	0.594095	−	0.804395i	$-0.297510\pi$
$733$	32.1690	1.18819	0.594095	+	0.804395i	$0.297510\pi$
$734$	0	0
$735$	−35.5180	−1.31010
$736$	0	0
$737$	−16.8174	−0.619477
$738$	0	0
$739$	50.1245	1.84386	0.921929	−	0.387358i	$-0.126612\pi$
$739$	50.1245	1.84386	0.921929	+	0.387358i	$0.126612\pi$
$740$	0	0
$741$	7.65915	0.281366
$742$	0	0
$743$	−30.4116	−1.11569	−0.557846	−	0.829944i	$-0.688372\pi$
$743$	−30.4116	−1.11569	−0.557846	+	0.829944i	$0.688372\pi$
$744$	0	0
$745$	16.0146	0.586729
$746$	0	0
$747$	−69.3106	−2.53594
$748$	0	0
$749$	−14.6255	−0.534406
$750$	0	0
$751$	1.24284	0.0453517	0.0226758	−	0.999743i	$-0.492781\pi$
$751$	1.24284	0.0453517	0.0226758	+	0.999743i	$0.492781\pi$
$752$	0	0
$753$	−83.8233	−3.05469
$754$	0	0
$755$	6.24525	0.227288
$756$	0	0
$757$	33.9263	1.23307	0.616536	−	0.787327i	$-0.288535\pi$
$757$	33.9263	1.23307	0.616536	+	0.787327i	$0.288535\pi$
$758$	0	0
$759$	60.3753	2.19149
$760$	0	0
$761$	44.5699	1.61566	0.807829	−	0.589416i	$-0.200642\pi$
$761$	44.5699	1.61566	0.807829	+	0.589416i	$0.200642\pi$
$762$	0	0
$763$	−19.9966	−0.723924
$764$	0	0
$765$	−19.5929	−0.708384
$766$	0	0
$767$	−19.9147	−0.719077
$768$	0	0
$769$	0.479153	0.0172787	0.00863934	−	0.999963i	$-0.497250\pi$
$769$	0.479153	0.0172787	0.00863934	+	0.999963i	$0.497250\pi$
$770$	0	0
$771$	71.5466	2.57669
$772$	0	0
$773$	−46.9785	−1.68970	−0.844850	−	0.535004i	$-0.820310\pi$
$773$	−46.9785	−1.68970	−0.844850	+	0.535004i	$0.820310\pi$
$774$	0	0
$775$	−3.88701	−0.139625
$776$	0	0
$777$	−56.3193	−2.02044
$778$	0	0
$779$	1.09054	0.0390728
$780$	0	0
$781$	−4.49590	−0.160876
$782$	0	0
$783$	−41.0140	−1.46572
$784$	0	0
$785$	−4.03675	−0.144078
$786$	0	0
$787$	−21.1172	−0.752746	−0.376373	−	0.926468i	$-0.622829\pi$
$787$	−21.1172	−0.752746	−0.376373	+	0.926468i	$0.622829\pi$
$788$	0	0
$789$	10.7535	0.382836
$790$	0	0
$791$	−0.939030	−0.0333881
$792$	0	0
$793$	−14.2876	−0.507368
$794$	0	0
$795$	−37.1371	−1.31712
$796$	0	0
$797$	−11.9178	−0.422149	−0.211074	−	0.977470i	$-0.567696\pi$
$797$	−11.9178	−0.422149	−0.211074	+	0.977470i	$0.567696\pi$
$798$	0	0
$799$	−49.9216	−1.76610
$800$	0	0
$801$	−11.2253	−0.396628
$802$	0	0
$803$	−0.138936	−0.00490293
$804$	0	0
$805$	−20.0086	−0.705212
$806$	0	0
$807$	42.1272	1.48295
$808$	0	0
$809$	−28.2632	−0.993682	−0.496841	−	0.867842i	$-0.665507\pi$
$809$	−28.2632	−0.993682	−0.496841	+	0.867842i	$0.665507\pi$
$810$	0	0
$811$	−9.58879	−0.336708	−0.168354	−	0.985727i	$-0.553845\pi$
$811$	−9.58879	−0.336708	−0.168354	+	0.985727i	$0.553845\pi$
$812$	0	0
$813$	38.8468	1.36242
$814$	0	0
$815$	1.18137	0.0413818
$816$	0	0
$817$	3.85296	0.134798
$818$	0	0
$819$	55.8978	1.95323
$820$	0	0
$821$	44.5031	1.55317	0.776584	−	0.630014i	$-0.216951\pi$
$821$	44.5031	1.55317	0.776584	+	0.630014i	$0.216951\pi$
$822$	0	0
$823$	14.4951	0.505268	0.252634	−	0.967562i	$-0.418703\pi$
$823$	14.4951	0.505268	0.252634	+	0.967562i	$0.418703\pi$
$824$	0	0
$825$	13.4974	0.469920
$826$	0	0
$827$	5.28695	0.183845	0.0919227	−	0.995766i	$-0.470699\pi$
$827$	5.28695	0.183845	0.0919227	+	0.995766i	$0.470699\pi$
$828$	0	0
$829$	−16.2444	−0.564192	−0.282096	−	0.959386i	$-0.591030\pi$
$829$	−16.2444	−0.564192	−0.282096	+	0.959386i	$0.591030\pi$
$830$	0	0
$831$	53.8716	1.86879
$832$	0	0
$833$	57.2150	1.98238
$834$	0	0
$835$	12.8520	0.444761
$836$	0	0
$837$	−15.4388	−0.533642
$838$	0	0
$839$	−17.6884	−0.610672	−0.305336	−	0.952245i	$-0.598769\pi$
$839$	−17.6884	−0.610672	−0.305336	+	0.952245i	$0.598769\pi$
$840$	0	0
$841$	77.6275	2.67681
$842$	0	0
$843$	59.6054	2.05292
$844$	0	0
$845$	5.13082	0.176506
$846$	0	0
$847$	60.1105	2.06542
$848$	0	0
$849$	71.0964	2.44002
$850$	0	0
$851$	−20.6273	−0.707093
$852$	0	0
$853$	22.0681	0.755597	0.377799	−	0.925888i	$-0.376681\pi$
$853$	22.0681	0.755597	0.377799	+	0.925888i	$0.376681\pi$
$854$	0	0
$855$	−4.45473	−0.152348
$856$	0	0
$857$	−38.5101	−1.31548	−0.657740	−	0.753245i	$-0.728487\pi$
$857$	−38.5101	−1.31548	−0.657740	+	0.753245i	$0.728487\pi$
$858$	0	0
$859$	27.0152	0.921747	0.460873	−	0.887466i	$-0.347536\pi$
$859$	27.0152	0.921747	0.460873	+	0.887466i	$0.347536\pi$
$860$	0	0
$861$	13.3189	0.453906
$862$	0	0
$863$	31.7035	1.07920	0.539601	−	0.841921i	$-0.318575\pi$
$863$	31.7035	1.07920	0.539601	+	0.841921i	$0.318575\pi$
$864$	0	0
$865$	−13.1745	−0.447946
$866$	0	0
$867$	6.40109	0.217393
$868$	0	0
$869$	−74.5404	−2.52861
$870$	0	0
$871$	−9.54308	−0.323355
$872$	0	0
$873$	40.6773	1.37672
$874$	0	0
$875$	−4.47310	−0.151218
$876$	0	0
$877$	30.2484	1.02142	0.510708	−	0.859754i	$-0.329383\pi$
$877$	30.2484	1.02142	0.510708	+	0.859754i	$0.329383\pi$
$878$	0	0
$879$	71.8676	2.42403
$880$	0	0
$881$	27.1738	0.915508	0.457754	−	0.889079i	$-0.348654\pi$
$881$	27.1738	0.915508	0.457754	+	0.889079i	$0.348654\pi$
$882$	0	0
$883$	45.4457	1.52937	0.764684	−	0.644405i	$-0.222895\pi$
$883$	45.4457	1.52937	0.764684	+	0.644405i	$0.222895\pi$
$884$	0	0
$885$	19.3832	0.651558
$886$	0	0
$887$	18.7206	0.628576	0.314288	−	0.949328i	$-0.398234\pi$
$887$	18.7206	0.628576	0.314288	+	0.949328i	$0.398234\pi$
$888$	0	0
$889$	69.4385	2.32889
$890$	0	0
$891$	−12.4556	−0.417277
$892$	0	0
$893$	−11.3504	−0.379826
$894$	0	0
$895$	7.16659	0.239553
$896$	0	0
$897$	34.2602	1.14391
$898$	0	0
$899$	40.1375	1.33866
$900$	0	0
$901$	59.8233	1.99300
$902$	0	0
$903$	47.0564	1.56594
$904$	0	0
$905$	9.36516	0.311309
$906$	0	0
$907$	−6.71648	−0.223017	−0.111509	−	0.993763i	$-0.535568\pi$
$907$	−6.71648	−0.223017	−0.111509	+	0.993763i	$0.535568\pi$
$908$	0	0
$909$	−71.2098	−2.36188
$910$	0	0
$911$	−8.19662	−0.271566	−0.135783	−	0.990739i	$-0.543355\pi$
$911$	−8.19662	−0.271566	−0.135783	+	0.990739i	$0.543355\pi$
$912$	0	0
$913$	−76.9154	−2.54553
$914$	0	0
$915$	13.9063	0.459727
$916$	0	0
$917$	−38.4148	−1.26857
$918$	0	0
$919$	4.63203	0.152796	0.0763982	−	0.997077i	$-0.475658\pi$
$919$	4.63203	0.152796	0.0763982	+	0.997077i	$0.475658\pi$
$920$	0	0
$921$	86.3631	2.84576
$922$	0	0
$923$	−2.55121	−0.0839742
$924$	0	0
$925$	−4.61140	−0.151622
$926$	0	0
$927$	−25.9160	−0.851193
$928$	0	0
$929$	28.3717	0.930846	0.465423	−	0.885088i	$-0.345902\pi$
$929$	28.3717	0.930846	0.465423	+	0.885088i	$0.345902\pi$
$930$	0	0
$931$	13.0086	0.426341
$932$	0	0
$933$	54.7344	1.79192
$934$	0	0
$935$	−21.7427	−0.711061
$936$	0	0
$937$	−25.3176	−0.827090	−0.413545	−	0.910484i	$-0.635710\pi$
$937$	−25.3176	−0.827090	−0.413545	+	0.910484i	$0.635710\pi$
$938$	0	0
$939$	−13.2051	−0.430933
$940$	0	0
$941$	6.90754	0.225179	0.112590	−	0.993642i	$-0.464085\pi$
$941$	6.90754	0.225179	0.112590	+	0.993642i	$0.464085\pi$
$942$	0	0
$943$	4.87811	0.158853
$944$	0	0
$945$	−17.7667	−0.577951
$946$	0	0
$947$	37.9219	1.23230	0.616149	−	0.787630i	$-0.288692\pi$
$947$	37.9219	1.23230	0.616149	+	0.787630i	$0.288692\pi$
$948$	0	0
$949$	−0.0788395	−0.00255924
$950$	0	0
$951$	−61.8564	−2.00583
$952$	0	0
$953$	−23.9147	−0.774675	−0.387337	−	0.921938i	$-0.626605\pi$
$953$	−23.9147	−0.774675	−0.387337	+	0.921938i	$0.626605\pi$
$954$	0	0
$955$	−12.8405	−0.415510
$956$	0	0
$957$	−139.375	−4.50536
$958$	0	0
$959$	−24.5518	−0.792821
$960$	0	0
$961$	−15.8912	−0.512619
$962$	0	0
$963$	−14.5655	−0.469366
$964$	0	0
$965$	19.5381	0.628955
$966$	0	0
$967$	−33.7253	−1.08453	−0.542267	−	0.840206i	$-0.682434\pi$
$967$	−33.7253	−1.08453	−0.542267	+	0.840206i	$0.682434\pi$
$968$	0	0
$969$	12.0086	0.385773
$970$	0	0
$971$	−5.30080	−0.170111	−0.0850554	−	0.996376i	$-0.527107\pi$
$971$	−5.30080	−0.170111	−0.0850554	+	0.996376i	$0.527107\pi$
$972$	0	0
$973$	−61.2088	−1.96227
$974$	0	0
$975$	7.65915	0.245289
$976$	0	0
$977$	−29.0358	−0.928936	−0.464468	−	0.885590i	$-0.653754\pi$
$977$	−29.0358	−0.928936	−0.464468	+	0.885590i	$0.653754\pi$
$978$	0	0
$979$	−12.4570	−0.398127
$980$	0	0
$981$	−19.9144	−0.635818
$982$	0	0
$983$	21.1679	0.675150	0.337575	−	0.941299i	$-0.390393\pi$
$983$	21.1679	0.675150	0.337575	+	0.941299i	$0.390393\pi$
$984$	0	0
$985$	17.7353	0.565093
$986$	0	0
$987$	−138.623	−4.41242
$988$	0	0
$989$	17.2347	0.548031
$990$	0	0
$991$	25.3926	0.806622	0.403311	−	0.915063i	$-0.367859\pi$
$991$	25.3926	0.806622	0.403311	+	0.915063i	$0.367859\pi$
$992$	0	0
$993$	−59.1330	−1.87653
$994$	0	0
$995$	−26.7275	−0.847320
$996$	0	0
$997$	12.4402	0.393985	0.196992	−	0.980405i	$-0.436883\pi$
$997$	12.4402	0.393985	0.196992	+	0.980405i	$0.436883\pi$
$998$	0	0
$999$	−18.3160	−0.579492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.ck.1.4		5
4.3	odd	2		6080.2.a.cj.1.2		5
8.3	odd	2		3040.2.a.w.1.4	yes	5
8.5	even	2		3040.2.a.v.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.v.1.2	✓	5	8.5	even	2
3040.2.a.w.1.4	yes	5	8.3	odd	2
6080.2.a.cj.1.2		5	4.3	odd	2
6080.2.a.ck.1.4		5	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.73033 q^{3} -1.00000 q^{5} +4.47310 q^{7} +4.45473 q^{9} +O(q^{10})\)	\(q+2.73033 q^{3} -1.00000 q^{5} +4.47310 q^{7} +4.45473 q^{9} +4.94350 q^{11} +2.80521 q^{13} -2.73033 q^{15} +4.39823 q^{17} +1.00000 q^{19} +12.2131 q^{21} +4.47310 q^{23} +1.00000 q^{25} +3.97190 q^{27} -10.3261 q^{29} -3.88701 q^{31} +13.4974 q^{33} -4.47310 q^{35} -4.61140 q^{37} +7.65915 q^{39} +1.09054 q^{41} +3.85296 q^{43} -4.45473 q^{45} -11.3504 q^{47} +13.0086 q^{49} +12.0086 q^{51} +13.6017 q^{53} -4.94350 q^{55} +2.73033 q^{57} -7.09919 q^{59} -5.09325 q^{61} +19.9265 q^{63} -2.80521 q^{65} -3.40192 q^{67} +12.2131 q^{69} -0.909457 q^{71} -0.0281047 q^{73} +2.73033 q^{75} +22.1128 q^{77} -15.0785 q^{79} -2.51958 q^{81} -15.5589 q^{83} -4.39823 q^{85} -28.1936 q^{87} -2.51987 q^{89} +12.5480 q^{91} -10.6128 q^{93} -1.00000 q^{95} +9.13127 q^{97} +22.0220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9}+O(q^{10}) \) 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9	\( 5 q + 2 q^{3} - 5 q^{5} + 2 q^{7} + 19 q^{9} + 10 q^{11} - 2 q^{15} + 4 q^{17} + 5 q^{19} - 10 q^{21} + 2 q^{23} + 5 q^{25} + 8 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{33} - 2 q^{35} - 2 q^{37} + 10 q^{39} + 12 q^{41} - 2 q^{43} - 19 q^{45} + 22 q^{47} + 19 q^{49} + 14 q^{51} + 18 q^{53} - 10 q^{55} + 2 q^{57} + 4 q^{59} - 6 q^{61} + 10 q^{63} + 20 q^{67} - 10 q^{69} + 2 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 14 q^{79} + 13 q^{81} - 14 q^{83} - 4 q^{85} - 12 q^{87} + 22 q^{89} + 40 q^{91} - 8 q^{93} - 5 q^{95} - 10 q^{97} + 6 q^{99}+O(q^{100}) \) 5 * q + 2 * q^3 - 5 * q^5 + 2 * q^7 + 19 * q^9 + 10 * q^11 - 2 * q^15 + 4 * q^17 + 5 * q^19 - 10 * q^21 + 2 * q^23 + 5 * q^25 + 8 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^33 - 2 * q^35 - 2 * q^37 + 10 * q^39 + 12 * q^41 - 2 * q^43 - 19 * q^45 + 22 * q^47 + 19 * q^49 + 14 * q^51 + 18 * q^53 - 10 * q^55 + 2 * q^57 + 4 * q^59 - 6 * q^61 + 10 * q^63 + 20 * q^67 - 10 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 + 13 * q^81 - 14 * q^83 - 4 * q^85 - 12 * q^87 + 22 * q^89 + 40 * q^91 - 8 * q^93 - 5 * q^95 - 10 * q^97 + 6 * q^99

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