LMFDB - Embedded newform 6160.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
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 3080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.67513 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} +O(q^{10})$	$q-1.67513 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} -1.00000 q^{11} -6.63752 q^{13} +1.67513 q^{15} +4.63752 q^{17} +2.00000 q^{19} +1.67513 q^{21} +4.15633 q^{23} +1.00000 q^{25} +5.35026 q^{27} -7.92478 q^{29} +3.51881 q^{31} +1.67513 q^{33} +1.00000 q^{35} +4.54420 q^{37} +11.1187 q^{39} +2.48119 q^{41} -0.0303172 q^{43} +0.193937 q^{45} +12.9502 q^{47} +1.00000 q^{49} -7.76845 q^{51} +1.38058 q^{53} +1.00000 q^{55} -3.35026 q^{57} -7.05571 q^{59} -1.25694 q^{61} +0.193937 q^{63} +6.63752 q^{65} -4.80606 q^{67} -6.96239 q^{69} +9.92478 q^{71} -5.93700 q^{73} -1.67513 q^{75} +1.00000 q^{77} +9.38058 q^{79} -8.38058 q^{81} +5.35026 q^{83} -4.63752 q^{85} +13.2750 q^{87} -12.1866 q^{89} +6.63752 q^{91} -5.89446 q^{93} -2.00000 q^{95} -6.26187 q^{97} +0.193937 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 3 * q^5 - 3 * q^7 - q^9	$ 3 q - 3 q^{5} - 3 q^{7} - q^{9} - 3 q^{11} - 4 q^{13} - 2 q^{17} + 6 q^{19} + 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 16 q^{31} + 3 q^{35} + 4 q^{37} + 12 q^{39} + 2 q^{41} + 2 q^{43} + q^{45} + 2 q^{47} + 3 q^{49} - 12 q^{51} - 8 q^{53} + 3 q^{55} - 4 q^{59} + q^{63} + 4 q^{65} - 14 q^{67} - 10 q^{69} + 8 q^{71} - 22 q^{73} + 3 q^{77} + 16 q^{79} - 13 q^{81} + 6 q^{83} + 2 q^{85} + 8 q^{87} - 24 q^{89} + 4 q^{91} + 2 q^{93} - 6 q^{95} - 28 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 - q^9 - 3 * q^11 - 4 * q^13 - 2 * q^17 + 6 * q^19 + 2 * q^23 + 3 * q^25 + 6 * q^27 - 2 * q^29 + 16 * q^31 + 3 * q^35 + 4 * q^37 + 12 * q^39 + 2 * q^41 + 2 * q^43 + q^45 + 2 * q^47 + 3 * q^49 - 12 * q^51 - 8 * q^53 + 3 * q^55 - 4 * q^59 + q^63 + 4 * q^65 - 14 * q^67 - 10 * q^69 + 8 * q^71 - 22 * q^73 + 3 * q^77 + 16 * q^79 - 13 * q^81 + 6 * q^83 + 2 * q^85 + 8 * q^87 - 24 * q^89 + 4 * q^91 + 2 * q^93 - 6 * q^95 - 28 * q^97 + 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
$3$	−1.67513	−0.967137	−0.483569	−	0.875306i	$-0.660660\pi$
$3$	−1.67513	−0.967137	−0.483569	+	0.875306i	$0.660660\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.193937	−0.0646455
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.63752	−1.84092	−0.920458	−	0.390841i	$-0.872184\pi$
$13$	−6.63752	−1.84092	−0.920458	+	0.390841i	$0.872184\pi$
$14$	0	0
$15$	1.67513	0.432517
$16$	0	0
$17$	4.63752	1.12476	0.562382	−	0.826878i	$-0.309885\pi$
$17$	4.63752	1.12476	0.562382	+	0.826878i	$0.309885\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	1.67513	0.365544
$22$	0	0
$23$	4.15633	0.866654	0.433327	−	0.901237i	$-0.357340\pi$
$23$	4.15633	0.866654	0.433327	+	0.901237i	$0.357340\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.35026	1.02966
$28$	0	0
$29$	−7.92478	−1.47159	−0.735797	−	0.677202i	$-0.763192\pi$
$29$	−7.92478	−1.47159	−0.735797	+	0.677202i	$0.763192\pi$
$30$	0	0
$31$	3.51881	0.631996	0.315998	−	0.948760i	$-0.397661\pi$
$31$	3.51881	0.631996	0.315998	+	0.948760i	$0.397661\pi$
$32$	0	0
$33$	1.67513	0.291603
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	4.54420	0.747062	0.373531	−	0.927618i	$-0.378147\pi$
$37$	4.54420	0.747062	0.373531	+	0.927618i	$0.378147\pi$
$38$	0	0
$39$	11.1187	1.78042
$40$	0	0
$41$	2.48119	0.387497	0.193749	−	0.981051i	$-0.437935\pi$
$41$	2.48119	0.387497	0.193749	+	0.981051i	$0.437935\pi$
$42$	0	0
$43$	−0.0303172	−0.00462332	−0.00231166	−	0.999997i	$-0.500736\pi$
$43$	−0.0303172	−0.00462332	−0.00231166	+	0.999997i	$0.500736\pi$
$44$	0	0
$45$	0.193937	0.0289104
$46$	0	0
$47$	12.9502	1.88898	0.944488	−	0.328545i	$-0.106558\pi$
$47$	12.9502	1.88898	0.944488	+	0.328545i	$0.106558\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.76845	−1.08780
$52$	0	0
$53$	1.38058	0.189637	0.0948185	−	0.995495i	$-0.469773\pi$
$53$	1.38058	0.189637	0.0948185	+	0.995495i	$0.469773\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−3.35026	−0.443753
$58$	0	0
$59$	−7.05571	−0.918575	−0.459288	−	0.888288i	$-0.651895\pi$
$59$	−7.05571	−0.918575	−0.459288	+	0.888288i	$0.651895\pi$
$60$	0	0
$61$	−1.25694	−0.160935	−0.0804674	−	0.996757i	$-0.525641\pi$
$61$	−1.25694	−0.160935	−0.0804674	+	0.996757i	$0.525641\pi$
$62$	0	0
$63$	0.193937	0.0244337
$64$	0	0
$65$	6.63752	0.823283
$66$	0	0
$67$	−4.80606	−0.587154	−0.293577	−	0.955935i	$-0.594846\pi$
$67$	−4.80606	−0.587154	−0.293577	+	0.955935i	$0.594846\pi$
$68$	0	0
$69$	−6.96239	−0.838173
$70$	0	0
$71$	9.92478	1.17785	0.588927	−	0.808186i	$-0.299550\pi$
$71$	9.92478	1.17785	0.588927	+	0.808186i	$0.299550\pi$
$72$	0	0
$73$	−5.93700	−0.694873	−0.347436	−	0.937704i	$-0.612948\pi$
$73$	−5.93700	−0.694873	−0.347436	+	0.937704i	$0.612948\pi$
$74$	0	0
$75$	−1.67513	−0.193427
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	9.38058	1.05540	0.527699	−	0.849432i	$-0.323055\pi$
$79$	9.38058	1.05540	0.527699	+	0.849432i	$0.323055\pi$
$80$	0	0
$81$	−8.38058	−0.931175
$82$	0	0
$83$	5.35026	0.587268	0.293634	−	0.955918i	$-0.405135\pi$
$83$	5.35026	0.587268	0.293634	+	0.955918i	$0.405135\pi$
$84$	0	0
$85$	−4.63752	−0.503010
$86$	0	0
$87$	13.2750	1.42323
$88$	0	0
$89$	−12.1866	−1.29178	−0.645891	−	0.763430i	$-0.723514\pi$
$89$	−12.1866	−1.29178	−0.645891	+	0.763430i	$0.723514\pi$
$90$	0	0
$91$	6.63752	0.695801
$92$	0	0
$93$	−5.89446	−0.611227
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−6.26187	−0.635796	−0.317898	−	0.948125i	$-0.602977\pi$
$97$	−6.26187	−0.635796	−0.317898	+	0.948125i	$0.602977\pi$
$98$	0	0
$99$	0.193937	0.0194914
$100$	0	0
$101$	17.6810	1.75933	0.879663	−	0.475597i	$-0.157768\pi$
$101$	17.6810	1.75933	0.879663	+	0.475597i	$0.157768\pi$
$102$	0	0
$103$	−18.3004	−1.80320	−0.901598	−	0.432576i	$-0.857605\pi$
$103$	−18.3004	−1.80320	−0.901598	+	0.432576i	$0.857605\pi$
$104$	0	0
$105$	−1.67513	−0.163476
$106$	0	0
$107$	−16.3127	−1.57700	−0.788502	−	0.615033i	$-0.789143\pi$
$107$	−16.3127	−1.57700	−0.788502	+	0.615033i	$0.789143\pi$
$108$	0	0
$109$	6.49929	0.622519	0.311260	−	0.950325i	$-0.399249\pi$
$109$	6.49929	0.622519	0.311260	+	0.950325i	$0.399249\pi$
$110$	0	0
$111$	−7.61213	−0.722511
$112$	0	0
$113$	−16.8265	−1.58291	−0.791453	−	0.611229i	$-0.790675\pi$
$113$	−16.8265	−1.58291	−0.791453	+	0.611229i	$0.790675\pi$
$114$	0	0
$115$	−4.15633	−0.387579
$116$	0	0
$117$	1.28726	0.119007
$118$	0	0
$119$	−4.63752	−0.425121
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.15633	−0.374763
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.73813	0.154235	0.0771173	−	0.997022i	$-0.475428\pi$
$127$	1.73813	0.154235	0.0771173	+	0.997022i	$0.475428\pi$
$128$	0	0
$129$	0.0507852	0.00447139
$130$	0	0
$131$	21.7743	1.90243	0.951216	−	0.308525i	$-0.0998355\pi$
$131$	21.7743	1.90243	0.951216	+	0.308525i	$0.0998355\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	0	0
$135$	−5.35026	−0.460477
$136$	0	0
$137$	−9.38058	−0.801437	−0.400718	−	0.916201i	$-0.631239\pi$
$137$	−9.38058	−0.801437	−0.400718	+	0.916201i	$0.631239\pi$
$138$	0	0
$139$	8.64974	0.733661	0.366831	−	0.930288i	$-0.380443\pi$
$139$	8.64974	0.733661	0.366831	+	0.930288i	$0.380443\pi$
$140$	0	0
$141$	−21.6932	−1.82690
$142$	0	0
$143$	6.63752	0.555057
$144$	0	0
$145$	7.92478	0.658117
$146$	0	0
$147$	−1.67513	−0.138162
$148$	0	0
$149$	10.4387	0.855168	0.427584	−	0.903976i	$-0.359365\pi$
$149$	10.4387	0.855168	0.427584	+	0.903976i	$0.359365\pi$
$150$	0	0
$151$	5.96968	0.485806	0.242903	−	0.970051i	$-0.421900\pi$
$151$	5.96968	0.485806	0.242903	+	0.970051i	$0.421900\pi$
$152$	0	0
$153$	−0.899385	−0.0727109
$154$	0	0
$155$	−3.51881	−0.282637
$156$	0	0
$157$	−8.31265	−0.663422	−0.331711	−	0.943381i	$-0.607626\pi$
$157$	−8.31265	−0.663422	−0.331711	+	0.943381i	$0.607626\pi$
$158$	0	0
$159$	−2.31265	−0.183405
$160$	0	0
$161$	−4.15633	−0.327564
$162$	0	0
$163$	−15.8192	−1.23906	−0.619529	−	0.784974i	$-0.712676\pi$
$163$	−15.8192	−1.23906	−0.619529	+	0.784974i	$0.712676\pi$
$164$	0	0
$165$	−1.67513	−0.130409
$166$	0	0
$167$	−9.53690	−0.737988	−0.368994	−	0.929432i	$-0.620298\pi$
$167$	−9.53690	−0.737988	−0.368994	+	0.929432i	$0.620298\pi$
$168$	0	0
$169$	31.0567	2.38897
$170$	0	0
$171$	−0.387873	−0.0296614
$172$	0	0
$173$	−14.2496	−1.08338	−0.541690	−	0.840578i	$-0.682215\pi$
$173$	−14.2496	−1.08338	−0.541690	+	0.840578i	$0.682215\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	11.8192	0.888388
$178$	0	0
$179$	−8.31265	−0.621317	−0.310658	−	0.950522i	$-0.600549\pi$
$179$	−8.31265	−0.621317	−0.310658	+	0.950522i	$0.600549\pi$
$180$	0	0
$181$	−7.22425	−0.536975	−0.268487	−	0.963283i	$-0.586524\pi$
$181$	−7.22425	−0.536975	−0.268487	+	0.963283i	$0.586524\pi$
$182$	0	0
$183$	2.10554	0.155646
$184$	0	0
$185$	−4.54420	−0.334096
$186$	0	0
$187$	−4.63752	−0.339129
$188$	0	0
$189$	−5.35026	−0.389174
$190$	0	0
$191$	5.27504	0.381688	0.190844	−	0.981620i	$-0.438877\pi$
$191$	5.27504	0.381688	0.190844	+	0.981620i	$0.438877\pi$
$192$	0	0
$193$	−24.2071	−1.74247	−0.871233	−	0.490870i	$-0.836679\pi$
$193$	−24.2071	−1.74247	−0.871233	+	0.490870i	$0.836679\pi$
$194$	0	0
$195$	−11.1187	−0.796228
$196$	0	0
$197$	17.0435	1.21430	0.607149	−	0.794588i	$-0.292313\pi$
$197$	17.0435	1.21430	0.607149	+	0.794588i	$0.292313\pi$
$198$	0	0
$199$	4.43041	0.314063	0.157032	−	0.987594i	$-0.449808\pi$
$199$	4.43041	0.314063	0.157032	+	0.987594i	$0.449808\pi$
$200$	0	0
$201$	8.05079	0.567859
$202$	0	0
$203$	7.92478	0.556210
$204$	0	0
$205$	−2.48119	−0.173294
$206$	0	0
$207$	−0.806063	−0.0560253
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−12.4387	−0.856313	−0.428156	−	0.903705i	$-0.640837\pi$
$211$	−12.4387	−0.856313	−0.428156	+	0.903705i	$0.640837\pi$
$212$	0	0
$213$	−16.6253	−1.13915
$214$	0	0
$215$	0.0303172	0.00206761
$216$	0	0
$217$	−3.51881	−0.238872
$218$	0	0
$219$	9.94525	0.672037
$220$	0	0
$221$	−30.7816	−2.07060
$222$	0	0
$223$	21.6507	1.44984	0.724919	−	0.688834i	$-0.241877\pi$
$223$	21.6507	1.44984	0.724919	+	0.688834i	$0.241877\pi$
$224$	0	0
$225$	−0.193937	−0.0129291
$226$	0	0
$227$	−12.2520	−0.813195	−0.406597	−	0.913607i	$-0.633285\pi$
$227$	−12.2520	−0.813195	−0.406597	+	0.913607i	$0.633285\pi$
$228$	0	0
$229$	−21.6629	−1.43153	−0.715763	−	0.698344i	$-0.753921\pi$
$229$	−21.6629	−1.43153	−0.715763	+	0.698344i	$0.753921\pi$
$230$	0	0
$231$	−1.67513	−0.110216
$232$	0	0
$233$	−15.3054	−1.00269	−0.501344	−	0.865248i	$-0.667161\pi$
$233$	−15.3054	−1.00269	−0.501344	+	0.865248i	$0.667161\pi$
$234$	0	0
$235$	−12.9502	−0.844776
$236$	0	0
$237$	−15.7137	−1.02071
$238$	0	0
$239$	−21.5877	−1.39639	−0.698196	−	0.715907i	$-0.746013\pi$
$239$	−21.5877	−1.39639	−0.698196	+	0.715907i	$0.746013\pi$
$240$	0	0
$241$	20.7186	1.33460	0.667302	−	0.744787i	$-0.267449\pi$
$241$	20.7186	1.33460	0.667302	+	0.744787i	$0.267449\pi$
$242$	0	0
$243$	−2.01222	−0.129084
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−13.2750	−0.844671
$248$	0	0
$249$	−8.96239	−0.567968
$250$	0	0
$251$	7.30773	0.461260	0.230630	−	0.973042i	$-0.425921\pi$
$251$	7.30773	0.461260	0.230630	+	0.973042i	$0.425921\pi$
$252$	0	0
$253$	−4.15633	−0.261306
$254$	0	0
$255$	7.76845	0.486479
$256$	0	0
$257$	26.5501	1.65615	0.828074	−	0.560619i	$-0.189437\pi$
$257$	26.5501	1.65615	0.828074	+	0.560619i	$0.189437\pi$
$258$	0	0
$259$	−4.54420	−0.282363
$260$	0	0
$261$	1.53690	0.0951320
$262$	0	0
$263$	4.96239	0.305994	0.152997	−	0.988227i	$-0.451108\pi$
$263$	4.96239	0.305994	0.152997	+	0.988227i	$0.451108\pi$
$264$	0	0
$265$	−1.38058	−0.0848083
$266$	0	0
$267$	20.4142	1.24933
$268$	0	0
$269$	30.4894	1.85897	0.929487	−	0.368855i	$-0.120250\pi$
$269$	30.4894	1.85897	0.929487	+	0.368855i	$0.120250\pi$
$270$	0	0
$271$	22.5745	1.37130	0.685652	−	0.727929i	$-0.259517\pi$
$271$	22.5745	1.37130	0.685652	+	0.727929i	$0.259517\pi$
$272$	0	0
$273$	−11.1187	−0.672935
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−14.9927	−0.900824	−0.450412	−	0.892821i	$-0.648723\pi$
$277$	−14.9927	−0.900824	−0.450412	+	0.892821i	$0.648723\pi$
$278$	0	0
$279$	−0.682425	−0.0408557
$280$	0	0
$281$	6.83638	0.407824	0.203912	−	0.978989i	$-0.434634\pi$
$281$	6.83638	0.407824	0.203912	+	0.978989i	$0.434634\pi$
$282$	0	0
$283$	23.3865	1.39018	0.695090	−	0.718922i	$-0.255364\pi$
$283$	23.3865	1.39018	0.695090	+	0.718922i	$0.255364\pi$
$284$	0	0
$285$	3.35026	0.198452
$286$	0	0
$287$	−2.48119	−0.146460
$288$	0	0
$289$	4.50659	0.265093
$290$	0	0
$291$	10.4894	0.614902
$292$	0	0
$293$	26.8749	1.57005	0.785026	−	0.619463i	$-0.212650\pi$
$293$	26.8749	1.57005	0.785026	+	0.619463i	$0.212650\pi$
$294$	0	0
$295$	7.05571	0.410799
$296$	0	0
$297$	−5.35026	−0.310454
$298$	0	0
$299$	−27.5877	−1.59544
$300$	0	0
$301$	0.0303172	0.00174745
$302$	0	0
$303$	−29.6180	−1.70151
$304$	0	0
$305$	1.25694	0.0719722
$306$	0	0
$307$	−21.8496	−1.24702	−0.623510	−	0.781815i	$-0.714294\pi$
$307$	−21.8496	−1.24702	−0.623510	+	0.781815i	$0.714294\pi$
$308$	0	0
$309$	30.6556	1.74394
$310$	0	0
$311$	−9.24235	−0.524086	−0.262043	−	0.965056i	$-0.584396\pi$
$311$	−9.24235	−0.524086	−0.262043	+	0.965056i	$0.584396\pi$
$312$	0	0
$313$	−13.1636	−0.744052	−0.372026	−	0.928222i	$-0.621337\pi$
$313$	−13.1636	−0.744052	−0.372026	+	0.928222i	$0.621337\pi$
$314$	0	0
$315$	−0.193937	−0.0109271
$316$	0	0
$317$	−24.5501	−1.37887	−0.689435	−	0.724348i	$-0.742141\pi$
$317$	−24.5501	−1.37887	−0.689435	+	0.724348i	$0.742141\pi$
$318$	0	0
$319$	7.92478	0.443702
$320$	0	0
$321$	27.3258	1.52518
$322$	0	0
$323$	9.27504	0.516077
$324$	0	0
$325$	−6.63752	−0.368183
$326$	0	0
$327$	−10.8872	−0.602062
$328$	0	0
$329$	−12.9502	−0.713966
$330$	0	0
$331$	−11.8134	−0.649321	−0.324660	−	0.945831i	$-0.605250\pi$
$331$	−11.8134	−0.649321	−0.324660	+	0.945831i	$0.605250\pi$
$332$	0	0
$333$	−0.881286	−0.0482942
$334$	0	0
$335$	4.80606	0.262583
$336$	0	0
$337$	4.35756	0.237371	0.118686	−	0.992932i	$-0.462132\pi$
$337$	4.35756	0.237371	0.118686	+	0.992932i	$0.462132\pi$
$338$	0	0
$339$	28.1866	1.53089
$340$	0	0
$341$	−3.51881	−0.190554
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.96239	0.374842
$346$	0	0
$347$	−20.2823	−1.08881	−0.544406	−	0.838822i	$-0.683245\pi$
$347$	−20.2823	−1.08881	−0.544406	+	0.838822i	$0.683245\pi$
$348$	0	0
$349$	11.6810	0.625270	0.312635	−	0.949873i	$-0.398788\pi$
$349$	11.6810	0.625270	0.312635	+	0.949873i	$0.398788\pi$
$350$	0	0
$351$	−35.5125	−1.89552
$352$	0	0
$353$	24.0870	1.28202	0.641010	−	0.767532i	$-0.278516\pi$
$353$	24.0870	1.28202	0.641010	+	0.767532i	$0.278516\pi$
$354$	0	0
$355$	−9.92478	−0.526752
$356$	0	0
$357$	7.76845	0.411150
$358$	0	0
$359$	−18.7816	−0.991256	−0.495628	−	0.868535i	$-0.665062\pi$
$359$	−18.7816	−0.991256	−0.495628	+	0.868535i	$0.665062\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−1.67513	−0.0879216
$364$	0	0
$365$	5.93700	0.310757
$366$	0	0
$367$	3.86177	0.201583	0.100791	−	0.994908i	$-0.467863\pi$
$367$	3.86177	0.201583	0.100791	+	0.994908i	$0.467863\pi$
$368$	0	0
$369$	−0.481194	−0.0250500
$370$	0	0
$371$	−1.38058	−0.0716761
$372$	0	0
$373$	−10.9683	−0.567915	−0.283958	−	0.958837i	$-0.591648\pi$
$373$	−10.9683	−0.567915	−0.283958	+	0.958837i	$0.591648\pi$
$374$	0	0
$375$	1.67513	0.0865034
$376$	0	0
$377$	52.6009	2.70908
$378$	0	0
$379$	−0.625301	−0.0321195	−0.0160598	−	0.999871i	$-0.505112\pi$
$379$	−0.625301	−0.0321195	−0.0160598	+	0.999871i	$0.505112\pi$
$380$	0	0
$381$	−2.91160	−0.149166
$382$	0	0
$383$	−33.1876	−1.69581	−0.847904	−	0.530151i	$-0.822135\pi$
$383$	−33.1876	−1.69581	−0.847904	+	0.530151i	$0.822135\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	0.00587961	0.000298877 0
$388$	0	0
$389$	−8.57452	−0.434745	−0.217373	−	0.976089i	$-0.569749\pi$
$389$	−8.57452	−0.434745	−0.217373	+	0.976089i	$0.569749\pi$
$390$	0	0
$391$	19.2750	0.974781
$392$	0	0
$393$	−36.4749	−1.83991
$394$	0	0
$395$	−9.38058	−0.471988
$396$	0	0
$397$	−31.9511	−1.60358	−0.801790	−	0.597605i	$-0.796119\pi$
$397$	−31.9511	−1.60358	−0.801790	+	0.597605i	$0.796119\pi$
$398$	0	0
$399$	3.35026	0.167723
$400$	0	0
$401$	−10.6253	−0.530602	−0.265301	−	0.964166i	$-0.585471\pi$
$401$	−10.6253	−0.530602	−0.265301	+	0.964166i	$0.585471\pi$
$402$	0	0
$403$	−23.3561	−1.16345
$404$	0	0
$405$	8.38058	0.416434
$406$	0	0
$407$	−4.54420	−0.225248
$408$	0	0
$409$	−30.0689	−1.48681	−0.743405	−	0.668841i	$-0.766791\pi$
$409$	−30.0689	−1.48681	−0.743405	+	0.668841i	$0.766791\pi$
$410$	0	0
$411$	15.7137	0.775100
$412$	0	0
$413$	7.05571	0.347189
$414$	0	0
$415$	−5.35026	−0.262634
$416$	0	0
$417$	−14.4894	−0.709551
$418$	0	0
$419$	22.7694	1.11236	0.556179	−	0.831062i	$-0.312267\pi$
$419$	22.7694	1.11236	0.556179	+	0.831062i	$0.312267\pi$
$420$	0	0
$421$	−23.1695	−1.12921	−0.564606	−	0.825360i	$-0.690972\pi$
$421$	−23.1695	−1.12921	−0.564606	+	0.825360i	$0.690972\pi$
$422$	0	0
$423$	−2.51151	−0.122114
$424$	0	0
$425$	4.63752	0.224953
$426$	0	0
$427$	1.25694	0.0608276
$428$	0	0
$429$	−11.1187	−0.536817
$430$	0	0
$431$	−9.31994	−0.448926	−0.224463	−	0.974483i	$-0.572063\pi$
$431$	−9.31994	−0.448926	−0.224463	+	0.974483i	$0.572063\pi$
$432$	0	0
$433$	−10.7513	−0.516675	−0.258337	−	0.966055i	$-0.583175\pi$
$433$	−10.7513	−0.516675	−0.258337	+	0.966055i	$0.583175\pi$
$434$	0	0
$435$	−13.2750	−0.636489
$436$	0	0
$437$	8.31265	0.397648
$438$	0	0
$439$	1.88858	0.0901370	0.0450685	−	0.998984i	$-0.485649\pi$
$439$	1.88858	0.0901370	0.0450685	+	0.998984i	$0.485649\pi$
$440$	0	0
$441$	−0.193937	−0.00923507
$442$	0	0
$443$	−22.7962	−1.08308	−0.541540	−	0.840675i	$-0.682159\pi$
$443$	−22.7962	−1.08308	−0.541540	+	0.840675i	$0.682159\pi$
$444$	0	0
$445$	12.1866	0.577702
$446$	0	0
$447$	−17.4861	−0.827065
$448$	0	0
$449$	−37.7196	−1.78010	−0.890048	−	0.455866i	$-0.849330\pi$
$449$	−37.7196	−1.78010	−0.890048	+	0.455866i	$0.849330\pi$
$450$	0	0
$451$	−2.48119	−0.116835
$452$	0	0
$453$	−10.0000	−0.469841
$454$	0	0
$455$	−6.63752	−0.311172
$456$	0	0
$457$	23.8945	1.11774	0.558868	−	0.829257i	$-0.311236\pi$
$457$	23.8945	1.11774	0.558868	+	0.829257i	$0.311236\pi$
$458$	0	0
$459$	24.8119	1.15812
$460$	0	0
$461$	−20.0689	−0.934701	−0.467351	−	0.884072i	$-0.654791\pi$
$461$	−20.0689	−0.934701	−0.467351	+	0.884072i	$0.654791\pi$
$462$	0	0
$463$	11.0435	0.513235	0.256617	−	0.966513i	$-0.417392\pi$
$463$	11.0435	0.513235	0.256617	+	0.966513i	$0.417392\pi$
$464$	0	0
$465$	5.89446	0.273349
$466$	0	0
$467$	−35.9488	−1.66351	−0.831755	−	0.555143i	$-0.812664\pi$
$467$	−35.9488	−1.66351	−0.831755	+	0.555143i	$0.812664\pi$
$468$	0	0
$469$	4.80606	0.221923
$470$	0	0
$471$	13.9248	0.641620
$472$	0	0
$473$	0.0303172	0.00139398
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	−0.267745	−0.0122592
$478$	0	0
$479$	9.68735	0.442626	0.221313	−	0.975203i	$-0.428966\pi$
$479$	9.68735	0.442626	0.221313	+	0.975203i	$0.428966\pi$
$480$	0	0
$481$	−30.1622	−1.37528
$482$	0	0
$483$	6.96239	0.316800
$484$	0	0
$485$	6.26187	0.284337
$486$	0	0
$487$	0.781626	0.0354188	0.0177094	−	0.999843i	$-0.494363\pi$
$487$	0.781626	0.0354188	0.0177094	+	0.999843i	$0.494363\pi$
$488$	0	0
$489$	26.4993	1.19834
$490$	0	0
$491$	−36.7064	−1.65654	−0.828268	−	0.560332i	$-0.810674\pi$
$491$	−36.7064	−1.65654	−0.828268	+	0.560332i	$0.810674\pi$
$492$	0	0
$493$	−36.7513	−1.65520
$494$	0	0
$495$	−0.193937	−0.00871680
$496$	0	0
$497$	−9.92478	−0.445187
$498$	0	0
$499$	−17.0884	−0.764982	−0.382491	−	0.923959i	$-0.624934\pi$
$499$	−17.0884	−0.764982	−0.382491	+	0.923959i	$0.624934\pi$
$500$	0	0
$501$	15.9756	0.713735
$502$	0	0
$503$	15.7988	0.704432	0.352216	−	0.935919i	$-0.385428\pi$
$503$	15.7988	0.704432	0.352216	+	0.935919i	$0.385428\pi$
$504$	0	0
$505$	−17.6810	−0.786795
$506$	0	0
$507$	−52.0240	−2.31047
$508$	0	0
$509$	−17.4518	−0.773539	−0.386769	−	0.922176i	$-0.626409\pi$
$509$	−17.4518	−0.773539	−0.386769	+	0.922176i	$0.626409\pi$
$510$	0	0
$511$	5.93700	0.262637
$512$	0	0
$513$	10.7005	0.472440
$514$	0	0
$515$	18.3004	0.806413
$516$	0	0
$517$	−12.9502	−0.569548
$518$	0	0
$519$	23.8700	1.04778
$520$	0	0
$521$	−19.5975	−0.858584	−0.429292	−	0.903166i	$-0.641237\pi$
$521$	−19.5975	−0.858584	−0.429292	+	0.903166i	$0.641237\pi$
$522$	0	0
$523$	−27.6531	−1.20918	−0.604592	−	0.796535i	$-0.706664\pi$
$523$	−27.6531	−1.20918	−0.604592	+	0.796535i	$0.706664\pi$
$524$	0	0
$525$	1.67513	0.0731087
$526$	0	0
$527$	16.3185	0.710846
$528$	0	0
$529$	−5.72496	−0.248911
$530$	0	0
$531$	1.36836	0.0593818
$532$	0	0
$533$	−16.4690	−0.713351
$534$	0	0
$535$	16.3127	0.705257
$536$	0	0
$537$	13.9248	0.600898
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	15.3357	0.659332	0.329666	−	0.944098i	$-0.393064\pi$
$541$	15.3357	0.659332	0.329666	+	0.944098i	$0.393064\pi$
$542$	0	0
$543$	12.1016	0.519328
$544$	0	0
$545$	−6.49929	−0.278399
$546$	0	0
$547$	11.7235	0.501263	0.250631	−	0.968083i	$-0.419362\pi$
$547$	11.7235	0.501263	0.250631	+	0.968083i	$0.419362\pi$
$548$	0	0
$549$	0.243767	0.0104037
$550$	0	0
$551$	−15.8496	−0.675214
$552$	0	0
$553$	−9.38058	−0.398903
$554$	0	0
$555$	7.61213	0.323117
$556$	0	0
$557$	31.6180	1.33970	0.669849	−	0.742497i	$-0.266359\pi$
$557$	31.6180	1.33970	0.669849	+	0.742497i	$0.266359\pi$
$558$	0	0
$559$	0.201231	0.00851116
$560$	0	0
$561$	7.76845	0.327984
$562$	0	0
$563$	−41.5877	−1.75271	−0.876356	−	0.481664i	$-0.840033\pi$
$563$	−41.5877	−1.75271	−0.876356	+	0.481664i	$0.840033\pi$
$564$	0	0
$565$	16.8265	0.707898
$566$	0	0
$567$	8.38058	0.351951
$568$	0	0
$569$	14.4993	0.607842	0.303921	−	0.952697i	$-0.401704\pi$
$569$	14.4993	0.607842	0.303921	+	0.952697i	$0.401704\pi$
$570$	0	0
$571$	29.6893	1.24246	0.621228	−	0.783630i	$-0.286634\pi$
$571$	29.6893	1.24246	0.621228	+	0.783630i	$0.286634\pi$
$572$	0	0
$573$	−8.83638	−0.369145
$574$	0	0
$575$	4.15633	0.173331
$576$	0	0
$577$	−5.79877	−0.241406	−0.120703	−	0.992689i	$-0.538515\pi$
$577$	−5.79877	−0.241406	−0.120703	+	0.992689i	$0.538515\pi$
$578$	0	0
$579$	40.5501	1.68520
$580$	0	0
$581$	−5.35026	−0.221966
$582$	0	0
$583$	−1.38058	−0.0571777
$584$	0	0
$585$	−1.28726	−0.0532216
$586$	0	0
$587$	31.2384	1.28935	0.644673	−	0.764458i	$-0.276994\pi$
$587$	31.2384	1.28935	0.644673	+	0.764458i	$0.276994\pi$
$588$	0	0
$589$	7.03761	0.289980
$590$	0	0
$591$	−28.5501	−1.17439
$592$	0	0
$593$	7.91256	0.324930	0.162465	−	0.986714i	$-0.448056\pi$
$593$	7.91256	0.324930	0.162465	+	0.986714i	$0.448056\pi$
$594$	0	0
$595$	4.63752	0.190120
$596$	0	0
$597$	−7.42152	−0.303742
$598$	0	0
$599$	−0.312650	−0.0127745	−0.00638727	−	0.999980i	$-0.502033\pi$
$599$	−0.312650	−0.0127745	−0.00638727	+	0.999980i	$0.502033\pi$
$600$	0	0
$601$	26.7938	1.09294	0.546472	−	0.837477i	$-0.315971\pi$
$601$	26.7938	1.09294	0.546472	+	0.837477i	$0.315971\pi$
$602$	0	0
$603$	0.932071	0.0379569
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	11.9854	0.486473	0.243236	−	0.969967i	$-0.421791\pi$
$607$	11.9854	0.486473	0.243236	+	0.969967i	$0.421791\pi$
$608$	0	0
$609$	−13.2750	−0.537932
$610$	0	0
$611$	−85.9570	−3.47745
$612$	0	0
$613$	3.87002	0.156309	0.0781544	−	0.996941i	$-0.475097\pi$
$613$	3.87002	0.156309	0.0781544	+	0.996941i	$0.475097\pi$
$614$	0	0
$615$	4.15633	0.167599
$616$	0	0
$617$	23.7685	0.956882	0.478441	−	0.878120i	$-0.341202\pi$
$617$	23.7685	0.956882	0.478441	+	0.878120i	$0.341202\pi$
$618$	0	0
$619$	36.8691	1.48189	0.740946	−	0.671564i	$-0.234377\pi$
$619$	36.8691	1.48189	0.740946	+	0.671564i	$0.234377\pi$
$620$	0	0
$621$	22.2374	0.892357
$622$	0	0
$623$	12.1866	0.488248
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.35026	0.133797
$628$	0	0
$629$	21.0738	0.840268
$630$	0	0
$631$	−23.1754	−0.922597	−0.461299	−	0.887245i	$-0.652616\pi$
$631$	−23.1754	−0.922597	−0.461299	+	0.887245i	$0.652616\pi$
$632$	0	0
$633$	20.8364	0.828172
$634$	0	0
$635$	−1.73813	−0.0689758
$636$	0	0
$637$	−6.63752	−0.262988
$638$	0	0
$639$	−1.92478	−0.0761430
$640$	0	0
$641$	−36.9321	−1.45873	−0.729365	−	0.684125i	$-0.760184\pi$
$641$	−36.9321	−1.45873	−0.729365	+	0.684125i	$0.760184\pi$
$642$	0	0
$643$	15.9878	0.630497	0.315248	−	0.949009i	$-0.397912\pi$
$643$	15.9878	0.630497	0.315248	+	0.949009i	$0.397912\pi$
$644$	0	0
$645$	−0.0507852	−0.00199967
$646$	0	0
$647$	−31.9633	−1.25661	−0.628304	−	0.777968i	$-0.716251\pi$
$647$	−31.9633	−1.25661	−0.628304	+	0.777968i	$0.716251\pi$
$648$	0	0
$649$	7.05571	0.276961
$650$	0	0
$651$	5.89446	0.231022
$652$	0	0
$653$	27.6483	1.08196	0.540981	−	0.841035i	$-0.318053\pi$
$653$	27.6483	1.08196	0.540981	+	0.841035i	$0.318053\pi$
$654$	0	0
$655$	−21.7743	−0.850794
$656$	0	0
$657$	1.15140	0.0449204
$658$	0	0
$659$	17.1041	0.666282	0.333141	−	0.942877i	$-0.391891\pi$
$659$	17.1041	0.666282	0.333141	+	0.942877i	$0.391891\pi$
$660$	0	0
$661$	−13.6483	−0.530858	−0.265429	−	0.964130i	$-0.585514\pi$
$661$	−13.6483	−0.530858	−0.265429	+	0.964130i	$0.585514\pi$
$662$	0	0
$663$	51.5633	2.00255
$664$	0	0
$665$	2.00000	0.0775567
$666$	0	0
$667$	−32.9380	−1.27536
$668$	0	0
$669$	−36.2677	−1.40219
$670$	0	0
$671$	1.25694	0.0485237
$672$	0	0
$673$	16.3284	0.629413	0.314706	−	0.949189i	$-0.398094\pi$
$673$	16.3284	0.629413	0.314706	+	0.949189i	$0.398094\pi$
$674$	0	0
$675$	5.35026	0.205932
$676$	0	0
$677$	−36.6032	−1.40678	−0.703388	−	0.710806i	$-0.748330\pi$
$677$	−36.6032	−1.40678	−0.703388	+	0.710806i	$0.748330\pi$
$678$	0	0
$679$	6.26187	0.240308
$680$	0	0
$681$	20.5237	0.786471
$682$	0	0
$683$	−14.2560	−0.545490	−0.272745	−	0.962086i	$-0.587932\pi$
$683$	−14.2560	−0.545490	−0.272745	+	0.962086i	$0.587932\pi$
$684$	0	0
$685$	9.38058	0.358413
$686$	0	0
$687$	36.2882	1.38448
$688$	0	0
$689$	−9.16362	−0.349106
$690$	0	0
$691$	13.0557	0.496663	0.248331	−	0.968675i	$-0.420118\pi$
$691$	13.0557	0.496663	0.248331	+	0.968675i	$0.420118\pi$
$692$	0	0
$693$	−0.193937	−0.00736704
$694$	0	0
$695$	−8.64974	−0.328103
$696$	0	0
$697$	11.5066	0.435843
$698$	0	0
$699$	25.6385	0.969736
$700$	0	0
$701$	−4.63515	−0.175067	−0.0875336	−	0.996162i	$-0.527899\pi$
$701$	−4.63515	−0.175067	−0.0875336	+	0.996162i	$0.527899\pi$
$702$	0	0
$703$	9.08840	0.342775
$704$	0	0
$705$	21.6932	0.817014
$706$	0	0
$707$	−17.6810	−0.664963
$708$	0	0
$709$	8.32724	0.312736	0.156368	−	0.987699i	$-0.450021\pi$
$709$	8.32724	0.312736	0.156368	+	0.987699i	$0.450021\pi$
$710$	0	0
$711$	−1.81924	−0.0682267
$712$	0	0
$713$	14.6253	0.547722
$714$	0	0
$715$	−6.63752	−0.248229
$716$	0	0
$717$	36.1622	1.35050
$718$	0	0
$719$	−4.47135	−0.166753	−0.0833765	−	0.996518i	$-0.526570\pi$
$719$	−4.47135	−0.166753	−0.0833765	+	0.996518i	$0.526570\pi$
$720$	0	0
$721$	18.3004	0.681544
$722$	0	0
$723$	−34.7064	−1.29075
$724$	0	0
$725$	−7.92478	−0.294319
$726$	0	0
$727$	−24.6375	−0.913755	−0.456878	−	0.889530i	$-0.651032\pi$
$727$	−24.6375	−0.913755	−0.456878	+	0.889530i	$0.651032\pi$
$728$	0	0
$729$	28.5125	1.05602
$730$	0	0
$731$	−0.140596	−0.00520015
$732$	0	0
$733$	26.4119	0.975544	0.487772	−	0.872971i	$-0.337810\pi$
$733$	26.4119	0.975544	0.487772	+	0.872971i	$0.337810\pi$
$734$	0	0
$735$	1.67513	0.0617881
$736$	0	0
$737$	4.80606	0.177034
$738$	0	0
$739$	−21.2995	−0.783514	−0.391757	−	0.920069i	$-0.628133\pi$
$739$	−21.2995	−0.783514	−0.391757	+	0.920069i	$0.628133\pi$
$740$	0	0
$741$	22.2374	0.816912
$742$	0	0
$743$	−23.1636	−0.849791	−0.424895	−	0.905242i	$-0.639689\pi$
$743$	−23.1636	−0.849791	−0.424895	+	0.905242i	$0.639689\pi$
$744$	0	0
$745$	−10.4387	−0.382443
$746$	0	0
$747$	−1.03761	−0.0379642
$748$	0	0
$749$	16.3127	0.596051
$750$	0	0
$751$	12.3780	0.451681	0.225840	−	0.974164i	$-0.427487\pi$
$751$	12.3780	0.451681	0.225840	+	0.974164i	$0.427487\pi$
$752$	0	0
$753$	−12.2414	−0.446101
$754$	0	0
$755$	−5.96968	−0.217259
$756$	0	0
$757$	53.7353	1.95304	0.976521	−	0.215420i	$-0.0691121\pi$
$757$	53.7353	1.95304	0.976521	+	0.215420i	$0.0691121\pi$
$758$	0	0
$759$	6.96239	0.252719
$760$	0	0
$761$	−14.8037	−0.536633	−0.268317	−	0.963331i	$-0.586467\pi$
$761$	−14.8037	−0.536633	−0.268317	+	0.963331i	$0.586467\pi$
$762$	0	0
$763$	−6.49929	−0.235290
$764$	0	0
$765$	0.899385	0.0325173
$766$	0	0
$767$	46.8324	1.69102
$768$	0	0
$769$	29.9819	1.08118	0.540588	−	0.841288i	$-0.318202\pi$
$769$	29.9819	1.08118	0.540588	+	0.841288i	$0.318202\pi$
$770$	0	0
$771$	−44.4749	−1.60172
$772$	0	0
$773$	−30.0625	−1.08127	−0.540637	−	0.841256i	$-0.681817\pi$
$773$	−30.0625	−1.08127	−0.540637	+	0.841256i	$0.681817\pi$
$774$	0	0
$775$	3.51881	0.126399
$776$	0	0
$777$	7.61213	0.273084
$778$	0	0
$779$	4.96239	0.177796
$780$	0	0
$781$	−9.92478	−0.355136
$782$	0	0
$783$	−42.3996	−1.51524
$784$	0	0
$785$	8.31265	0.296691
$786$	0	0
$787$	0.337088	0.0120159	0.00600794	−	0.999982i	$-0.498088\pi$
$787$	0.337088	0.0120159	0.00600794	+	0.999982i	$0.498088\pi$
$788$	0	0
$789$	−8.31265	−0.295938
$790$	0	0
$791$	16.8265	0.598283
$792$	0	0
$793$	8.34297	0.296267
$794$	0	0
$795$	2.31265	0.0820213
$796$	0	0
$797$	34.4894	1.22168	0.610839	−	0.791755i	$-0.290832\pi$
$797$	34.4894	1.22168	0.610839	+	0.791755i	$0.290832\pi$
$798$	0	0
$799$	60.0567	2.12465
$800$	0	0
$801$	2.36344	0.0835079
$802$	0	0
$803$	5.93700	0.209512
$804$	0	0
$805$	4.15633	0.146491
$806$	0	0
$807$	−51.0738	−1.79788
$808$	0	0
$809$	24.9116	0.875845	0.437923	−	0.899013i	$-0.355714\pi$
$809$	24.9116	0.875845	0.437923	+	0.899013i	$0.355714\pi$
$810$	0	0
$811$	13.9149	0.488619	0.244310	−	0.969697i	$-0.421439\pi$
$811$	13.9149	0.488619	0.244310	+	0.969697i	$0.421439\pi$
$812$	0	0
$813$	−37.8153	−1.32624
$814$	0	0
$815$	15.8192	0.554124
$816$	0	0
$817$	−0.0606343	−0.00212133
$818$	0	0
$819$	−1.28726	−0.0449804
$820$	0	0
$821$	−34.9741	−1.22061	−0.610303	−	0.792168i	$-0.708952\pi$
$821$	−34.9741	−1.22061	−0.610303	+	0.792168i	$0.708952\pi$
$822$	0	0
$823$	9.70308	0.338228	0.169114	−	0.985596i	$-0.445909\pi$
$823$	9.70308	0.338228	0.169114	+	0.985596i	$0.445909\pi$
$824$	0	0
$825$	1.67513	0.0583206
$826$	0	0
$827$	−29.8945	−1.03953	−0.519766	−	0.854309i	$-0.673981\pi$
$827$	−29.8945	−1.03953	−0.519766	+	0.854309i	$0.673981\pi$
$828$	0	0
$829$	21.8007	0.757169	0.378584	−	0.925567i	$-0.376411\pi$
$829$	21.8007	0.757169	0.378584	+	0.925567i	$0.376411\pi$
$830$	0	0
$831$	25.1147	0.871221
$832$	0	0
$833$	4.63752	0.160681
$834$	0	0
$835$	9.53690	0.330038
$836$	0	0
$837$	18.8265	0.650740
$838$	0	0
$839$	15.5090	0.535429	0.267714	−	0.963498i	$-0.413732\pi$
$839$	15.5090	0.535429	0.267714	+	0.963498i	$0.413732\pi$
$840$	0	0
$841$	33.8021	1.16559
$842$	0	0
$843$	−11.4518	−0.394422
$844$	0	0
$845$	−31.0567	−1.06838
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−39.1754	−1.34450
$850$	0	0
$851$	18.8872	0.647444
$852$	0	0
$853$	−5.71133	−0.195552	−0.0977761	−	0.995208i	$-0.531173\pi$
$853$	−5.71133	−0.195552	−0.0977761	+	0.995208i	$0.531173\pi$
$854$	0	0
$855$	0.387873	0.0132650
$856$	0	0
$857$	25.8519	0.883085	0.441542	−	0.897240i	$-0.354431\pi$
$857$	25.8519	0.883085	0.441542	+	0.897240i	$0.354431\pi$
$858$	0	0
$859$	3.14552	0.107324	0.0536619	−	0.998559i	$-0.482911\pi$
$859$	3.14552	0.107324	0.0536619	+	0.998559i	$0.482911\pi$
$860$	0	0
$861$	4.15633	0.141647
$862$	0	0
$863$	−44.3185	−1.50862	−0.754310	−	0.656518i	$-0.772028\pi$
$863$	−44.3185	−1.50862	−0.754310	+	0.656518i	$0.772028\pi$
$864$	0	0
$865$	14.2496	0.484503
$866$	0	0
$867$	−7.54912	−0.256382
$868$	0	0
$869$	−9.38058	−0.318214
$870$	0	0
$871$	31.9003	1.08090
$872$	0	0
$873$	1.21440	0.0411014
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−5.53102	−0.186769	−0.0933847	−	0.995630i	$-0.529769\pi$
$877$	−5.53102	−0.186769	−0.0933847	+	0.995630i	$0.529769\pi$
$878$	0	0
$879$	−45.0191	−1.51845
$880$	0	0
$881$	−16.5040	−0.556035	−0.278017	−	0.960576i	$-0.589677\pi$
$881$	−16.5040	−0.556035	−0.278017	+	0.960576i	$0.589677\pi$
$882$	0	0
$883$	25.4314	0.855834	0.427917	−	0.903818i	$-0.359248\pi$
$883$	25.4314	0.855834	0.427917	+	0.903818i	$0.359248\pi$
$884$	0	0
$885$	−11.8192	−0.397299
$886$	0	0
$887$	7.19982	0.241746	0.120873	−	0.992668i	$-0.461431\pi$
$887$	7.19982	0.241746	0.120873	+	0.992668i	$0.461431\pi$
$888$	0	0
$889$	−1.73813	−0.0582952
$890$	0	0
$891$	8.38058	0.280760
$892$	0	0
$893$	25.9003	0.866722
$894$	0	0
$895$	8.31265	0.277861
$896$	0	0
$897$	46.2130	1.54301
$898$	0	0
$899$	−27.8858	−0.930042
$900$	0	0
$901$	6.40246	0.213297
$902$	0	0
$903$	−0.0507852	−0.00169003
$904$	0	0
$905$	7.22425	0.240142
$906$	0	0
$907$	37.9307	1.25947	0.629733	−	0.776811i	$-0.283164\pi$
$907$	37.9307	1.25947	0.629733	+	0.776811i	$0.283164\pi$
$908$	0	0
$909$	−3.42899	−0.113733
$910$	0	0
$911$	24.3733	0.807523	0.403761	−	0.914864i	$-0.367703\pi$
$911$	24.3733	0.807523	0.403761	+	0.914864i	$0.367703\pi$
$912$	0	0
$913$	−5.35026	−0.177068
$914$	0	0
$915$	−2.10554	−0.0696070
$916$	0	0
$917$	−21.7743	−0.719052
$918$	0	0
$919$	10.1114	0.333545	0.166772	−	0.985995i	$-0.446665\pi$
$919$	10.1114	0.333545	0.166772	+	0.985995i	$0.446665\pi$
$920$	0	0
$921$	36.6009	1.20604
$922$	0	0
$923$	−65.8759	−2.16833
$924$	0	0
$925$	4.54420	0.149412
$926$	0	0
$927$	3.54912	0.116568
$928$	0	0
$929$	−40.8773	−1.34114	−0.670571	−	0.741846i	$-0.733951\pi$
$929$	−40.8773	−1.34114	−0.670571	+	0.741846i	$0.733951\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	15.4821	0.506863
$934$	0	0
$935$	4.63752	0.151663
$936$	0	0
$937$	25.3625	0.828556	0.414278	−	0.910150i	$-0.364034\pi$
$937$	25.3625	0.828556	0.414278	+	0.910150i	$0.364034\pi$
$938$	0	0
$939$	22.0508	0.719600
$940$	0	0
$941$	−10.4060	−0.339225	−0.169612	−	0.985511i	$-0.554252\pi$
$941$	−10.4060	−0.339225	−0.169612	+	0.985511i	$0.554252\pi$
$942$	0	0
$943$	10.3127	0.335826
$944$	0	0
$945$	5.35026	0.174044
$946$	0	0
$947$	−26.8568	−0.872730	−0.436365	−	0.899770i	$-0.643734\pi$
$947$	−26.8568	−0.872730	−0.436365	+	0.899770i	$0.643734\pi$
$948$	0	0
$949$	39.4069	1.27920
$950$	0	0
$951$	41.1246	1.33356
$952$	0	0
$953$	−31.4676	−1.01933	−0.509667	−	0.860372i	$-0.670231\pi$
$953$	−31.4676	−1.01933	−0.509667	+	0.860372i	$0.670231\pi$
$954$	0	0
$955$	−5.27504	−0.170696
$956$	0	0
$957$	−13.2750	−0.429121
$958$	0	0
$959$	9.38058	0.302915
$960$	0	0
$961$	−18.6180	−0.600581
$962$	0	0
$963$	3.16362	0.101946
$964$	0	0
$965$	24.2071	0.779254
$966$	0	0
$967$	−45.7850	−1.47234	−0.736172	−	0.676794i	$-0.763369\pi$
$967$	−45.7850	−1.47234	−0.736172	+	0.676794i	$0.763369\pi$
$968$	0	0
$969$	−15.5369	−0.499117
$970$	0	0
$971$	3.50896	0.112608	0.0563039	−	0.998414i	$-0.482068\pi$
$971$	3.50896	0.112608	0.0563039	+	0.998414i	$0.482068\pi$
$972$	0	0
$973$	−8.64974	−0.277298
$974$	0	0
$975$	11.1187	0.356084
$976$	0	0
$977$	10.2276	0.327209	0.163605	−	0.986526i	$-0.447688\pi$
$977$	10.2276	0.327209	0.163605	+	0.986526i	$0.447688\pi$
$978$	0	0
$979$	12.1866	0.389487
$980$	0	0
$981$	−1.26045	−0.0402431
$982$	0	0
$983$	27.1270	0.865216	0.432608	−	0.901582i	$-0.357593\pi$
$983$	27.1270	0.865216	0.432608	+	0.901582i	$0.357593\pi$
$984$	0	0
$985$	−17.0435	−0.543051
$986$	0	0
$987$	21.6932	0.690503
$988$	0	0
$989$	−0.126008	−0.00400682
$990$	0	0
$991$	−55.1392	−1.75155	−0.875777	−	0.482716i	$-0.839650\pi$
$991$	−55.1392	−1.75155	−0.875777	+	0.482716i	$0.839650\pi$
$992$	0	0
$993$	19.7889	0.627982
$994$	0	0
$995$	−4.43041	−0.140453
$996$	0	0
$997$	35.4349	1.12223	0.561117	−	0.827737i	$-0.310372\pi$
$997$	35.4349	1.12223	0.561117	+	0.827737i	$0.310372\pi$
$998$	0	0
$999$	24.3127	0.769218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bg.1.1		3
4.3	odd	2		3080.2.a.l.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3080.2.a.l.1.3	✓	3	4.3	odd	2
6160.2.a.bg.1.1		3	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.67513 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} +O(q^{10})\)	\(q-1.67513 q^{3} -1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} -1.00000 q^{11} -6.63752 q^{13} +1.67513 q^{15} +4.63752 q^{17} +2.00000 q^{19} +1.67513 q^{21} +4.15633 q^{23} +1.00000 q^{25} +5.35026 q^{27} -7.92478 q^{29} +3.51881 q^{31} +1.67513 q^{33} +1.00000 q^{35} +4.54420 q^{37} +11.1187 q^{39} +2.48119 q^{41} -0.0303172 q^{43} +0.193937 q^{45} +12.9502 q^{47} +1.00000 q^{49} -7.76845 q^{51} +1.38058 q^{53} +1.00000 q^{55} -3.35026 q^{57} -7.05571 q^{59} -1.25694 q^{61} +0.193937 q^{63} +6.63752 q^{65} -4.80606 q^{67} -6.96239 q^{69} +9.92478 q^{71} -5.93700 q^{73} -1.67513 q^{75} +1.00000 q^{77} +9.38058 q^{79} -8.38058 q^{81} +5.35026 q^{83} -4.63752 q^{85} +13.2750 q^{87} -12.1866 q^{89} +6.63752 q^{91} -5.89446 q^{93} -2.00000 q^{95} -6.26187 q^{97} +0.193937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^5 - 3 * q^7 - q^9	\( 3 q - 3 q^{5} - 3 q^{7} - q^{9} - 3 q^{11} - 4 q^{13} - 2 q^{17} + 6 q^{19} + 2 q^{23} + 3 q^{25} + 6 q^{27} - 2 q^{29} + 16 q^{31} + 3 q^{35} + 4 q^{37} + 12 q^{39} + 2 q^{41} + 2 q^{43} + q^{45} + 2 q^{47} + 3 q^{49} - 12 q^{51} - 8 q^{53} + 3 q^{55} - 4 q^{59} + q^{63} + 4 q^{65} - 14 q^{67} - 10 q^{69} + 8 q^{71} - 22 q^{73} + 3 q^{77} + 16 q^{79} - 13 q^{81} + 6 q^{83} + 2 q^{85} + 8 q^{87} - 24 q^{89} + 4 q^{91} + 2 q^{93} - 6 q^{95} - 28 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 - q^9 - 3 * q^11 - 4 * q^13 - 2 * q^17 + 6 * q^19 + 2 * q^23 + 3 * q^25 + 6 * q^27 - 2 * q^29 + 16 * q^31 + 3 * q^35 + 4 * q^37 + 12 * q^39 + 2 * q^41 + 2 * q^43 + q^45 + 2 * q^47 + 3 * q^49 - 12 * q^51 - 8 * q^53 + 3 * q^55 - 4 * q^59 + q^63 + 4 * q^65 - 14 * q^67 - 10 * q^69 + 8 * q^71 - 22 * q^73 + 3 * q^77 + 16 * q^79 - 13 * q^81 + 6 * q^83 + 2 * q^85 + 8 * q^87 - 24 * q^89 + 4 * q^91 + 2 * q^93 - 6 * q^95 - 28 * q^97 + q^99

Introduction

L-functions

Modular forms

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Database

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