LMFDB - Embedded newform 6160.2.a.bj.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 385)
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)

$f(q)$	$=$	$q-1.67513 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} +O(q^{10})$	$q-1.67513 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} -1.00000 q^{11} +1.67513 q^{13} -1.67513 q^{15} +0.324869 q^{17} -3.61213 q^{19} +1.67513 q^{21} -0.806063 q^{23} +1.00000 q^{25} +5.35026 q^{27} +7.92478 q^{29} -1.13093 q^{31} +1.67513 q^{33} -1.00000 q^{35} -7.76845 q^{37} -2.80606 q^{39} +8.21933 q^{41} -5.11871 q^{43} -0.193937 q^{45} +2.24965 q^{47} +1.00000 q^{49} -0.544198 q^{51} +10.4690 q^{53} -1.00000 q^{55} +6.05079 q^{57} -4.48119 q^{59} -12.1441 q^{61} +0.193937 q^{63} +1.67513 q^{65} +4.15633 q^{67} +1.35026 q^{69} +0.775746 q^{71} -8.32487 q^{73} -1.67513 q^{75} +1.00000 q^{77} +2.15633 q^{79} -8.38058 q^{81} +12.8872 q^{83} +0.324869 q^{85} -13.2750 q^{87} -6.05079 q^{89} -1.67513 q^{91} +1.89446 q^{93} -3.61213 q^{95} -18.0508 q^{97} +0.193937 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 3 q^{25} + 6 q^{27} + 2 q^{29} - 8 q^{31} - 3 q^{35} - 12 q^{37} - 8 q^{39} + 10 q^{41} + 6 q^{43} - q^{45} - 10 q^{47} + 3 q^{49} + 8 q^{51} - 3 q^{55} - 12 q^{57} - 8 q^{59} + q^{63} + 2 q^{67} - 6 q^{69} + 4 q^{71} - 30 q^{73} + 3 q^{77} - 4 q^{79} - 13 q^{81} + 6 q^{83} + 6 q^{85} - 8 q^{87} + 12 q^{89} - 14 q^{93} - 10 q^{95} - 24 q^{97} + q^{99}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^7 - q^9 - 3 * q^11 + 6 * q^17 - 10 * q^19 - 2 * q^23 + 3 * q^25 + 6 * q^27 + 2 * q^29 - 8 * q^31 - 3 * q^35 - 12 * q^37 - 8 * q^39 + 10 * q^41 + 6 * q^43 - q^45 - 10 * q^47 + 3 * q^49 + 8 * q^51 - 3 * q^55 - 12 * q^57 - 8 * q^59 + q^63 + 2 * q^67 - 6 * q^69 + 4 * q^71 - 30 * q^73 + 3 * q^77 - 4 * q^79 - 13 * q^81 + 6 * q^83 + 6 * q^85 - 8 * q^87 + 12 * q^89 - 14 * q^93 - 10 * q^95 - 24 * 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
$13$	1.67513	0.464598	0.232299	−	0.972644i	$-0.425375\pi$
$13$	1.67513	0.464598	0.232299	+	0.972644i	$0.425375\pi$
$14$	0	0
$15$	−1.67513	−0.432517
$16$	0	0
$17$	0.324869	0.0787923	0.0393962	−	0.999224i	$-0.487457\pi$
$17$	0.324869	0.0787923	0.0393962	+	0.999224i	$0.487457\pi$
$18$	0	0
$19$	−3.61213	−0.828679	−0.414339	−	0.910122i	$-0.635987\pi$
$19$	−3.61213	−0.828679	−0.414339	+	0.910122i	$0.635987\pi$
$20$	0	0
$21$	1.67513	0.365544
$22$	0	0
$23$	−0.806063	−0.168076	−0.0840379	−	0.996463i	$-0.526782\pi$
$23$	−0.806063	−0.168076	−0.0840379	+	0.996463i	$0.526782\pi$
$24$	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	$-0.236808\pi$
$29$	7.92478	1.47159	0.735797	+	0.677202i	$0.236808\pi$
$30$	0	0
$31$	−1.13093	−0.203121	−0.101561	−	0.994829i	$-0.532384\pi$
$31$	−1.13093	−0.203121	−0.101561	+	0.994829i	$0.532384\pi$
$32$	0	0
$33$	1.67513	0.291603
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−7.76845	−1.27713	−0.638563	−	0.769570i	$-0.720471\pi$
$37$	−7.76845	−1.27713	−0.638563	+	0.769570i	$0.720471\pi$
$38$	0	0
$39$	−2.80606	−0.449330
$40$	0	0
$41$	8.21933	1.28364	0.641822	−	0.766854i	$-0.278179\pi$
$41$	8.21933	1.28364	0.641822	+	0.766854i	$0.278179\pi$
$42$	0	0
$43$	−5.11871	−0.780597	−0.390298	−	0.920688i	$-0.627628\pi$
$43$	−5.11871	−0.780597	−0.390298	+	0.920688i	$0.627628\pi$
$44$	0	0
$45$	−0.193937	−0.0289104
$46$	0	0
$47$	2.24965	0.328145	0.164072	−	0.986448i	$-0.447537\pi$
$47$	2.24965	0.328145	0.164072	+	0.986448i	$0.447537\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.544198	−0.0762030
$52$	0	0
$53$	10.4690	1.43802	0.719012	−	0.694997i	$-0.244595\pi$
$53$	10.4690	1.43802	0.719012	+	0.694997i	$0.244595\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	6.05079	0.801446
$58$	0	0
$59$	−4.48119	−0.583402	−0.291701	−	0.956510i	$-0.594221\pi$
$59$	−4.48119	−0.583402	−0.291701	+	0.956510i	$0.594221\pi$
$60$	0	0
$61$	−12.1441	−1.55489	−0.777447	−	0.628949i	$-0.783486\pi$
$61$	−12.1441	−1.55489	−0.777447	+	0.628949i	$0.783486\pi$
$62$	0	0
$63$	0.193937	0.0244337
$64$	0	0
$65$	1.67513	0.207774
$66$	0	0
$67$	4.15633	0.507776	0.253888	−	0.967234i	$-0.418291\pi$
$67$	4.15633	0.507776	0.253888	+	0.967234i	$0.418291\pi$
$68$	0	0
$69$	1.35026	0.162552
$70$	0	0
$71$	0.775746	0.0920641	0.0460321	−	0.998940i	$-0.485342\pi$
$71$	0.775746	0.0920641	0.0460321	+	0.998940i	$0.485342\pi$
$72$	0	0
$73$	−8.32487	−0.974352	−0.487176	−	0.873304i	$-0.661973\pi$
$73$	−8.32487	−0.974352	−0.487176	+	0.873304i	$0.661973\pi$
$74$	0	0
$75$	−1.67513	−0.193427
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	2.15633	0.242606	0.121303	−	0.992616i	$-0.461293\pi$
$79$	2.15633	0.242606	0.121303	+	0.992616i	$0.461293\pi$
$80$	0	0
$81$	−8.38058	−0.931175
$82$	0	0
$83$	12.8872	1.41455	0.707275	−	0.706938i	$-0.249924\pi$
$83$	12.8872	1.41455	0.707275	+	0.706938i	$0.249924\pi$
$84$	0	0
$85$	0.324869	0.0352370
$86$	0	0
$87$	−13.2750	−1.42323
$88$	0	0
$89$	−6.05079	−0.641382	−0.320691	−	0.947184i	$-0.603915\pi$
$89$	−6.05079	−0.641382	−0.320691	+	0.947184i	$0.603915\pi$
$90$	0	0
$91$	−1.67513	−0.175601
$92$	0	0
$93$	1.89446	0.196446
$94$	0	0
$95$	−3.61213	−0.370596
$96$	0	0
$97$	−18.0508	−1.83278	−0.916390	−	0.400287i	$-0.868910\pi$
$97$	−18.0508	−1.83278	−0.916390	+	0.400287i	$0.868910\pi$
$98$	0	0
$99$	0.193937	0.0194914
$100$	0	0
$101$	7.31757	0.728126	0.364063	−	0.931374i	$-0.381389\pi$
$101$	7.31757	0.728126	0.364063	+	0.931374i	$0.381389\pi$
$102$	0	0
$103$	17.6507	1.73917	0.869587	−	0.493779i	$-0.164385\pi$
$103$	17.6507	1.73917	0.869587	+	0.493779i	$0.164385\pi$
$104$	0	0
$105$	1.67513	0.163476
$106$	0	0
$107$	3.53690	0.341925	0.170963	−	0.985277i	$-0.445312\pi$
$107$	3.53690	0.341925	0.170963	+	0.985277i	$0.445312\pi$
$108$	0	0
$109$	−10.4993	−1.00565	−0.502825	−	0.864388i	$-0.667706\pi$
$109$	−10.4993	−1.00565	−0.502825	+	0.864388i	$0.667706\pi$
$110$	0	0
$111$	13.0132	1.23516
$112$	0	0
$113$	−12.0508	−1.13364	−0.566821	−	0.823841i	$-0.691827\pi$
$113$	−12.0508	−1.13364	−0.566821	+	0.823841i	$0.691827\pi$
$114$	0	0
$115$	−0.806063	−0.0751658
$116$	0	0
$117$	−0.324869	−0.0300342
$118$	0	0
$119$	−0.324869	−0.0297807
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−13.7685	−1.24146
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.18664	0.726447	0.363224	−	0.931702i	$-0.381676\pi$
$127$	8.18664	0.726447	0.363224	+	0.931702i	$0.381676\pi$
$128$	0	0
$129$	8.57452	0.754944
$130$	0	0
$131$	−15.4763	−1.35217	−0.676084	−	0.736825i	$-0.736324\pi$
$131$	−15.4763	−1.35217	−0.676084	+	0.736825i	$0.736324\pi$
$132$	0	0
$133$	3.61213	0.313211
$134$	0	0
$135$	5.35026	0.460477
$136$	0	0
$137$	2.93207	0.250504	0.125252	−	0.992125i	$-0.460026\pi$
$137$	2.93207	0.250504	0.125252	+	0.992125i	$0.460026\pi$
$138$	0	0
$139$	−8.12601	−0.689239	−0.344620	−	0.938742i	$-0.611992\pi$
$139$	−8.12601	−0.689239	−0.344620	+	0.938742i	$0.611992\pi$
$140$	0	0
$141$	−3.76845	−0.317361
$142$	0	0
$143$	−1.67513	−0.140081
$144$	0	0
$145$	7.92478	0.658117
$146$	0	0
$147$	−1.67513	−0.138162
$148$	0	0
$149$	−16.8872	−1.38345	−0.691725	−	0.722161i	$-0.743149\pi$
$149$	−16.8872	−1.38345	−0.691725	+	0.722161i	$0.743149\pi$
$150$	0	0
$151$	−0.881286	−0.0717181	−0.0358590	−	0.999357i	$-0.511417\pi$
$151$	−0.881286	−0.0717181	−0.0358590	+	0.999357i	$0.511417\pi$
$152$	0	0
$153$	−0.0630040	−0.00509357
$154$	0	0
$155$	−1.13093	−0.0908387
$156$	0	0
$157$	9.40105	0.750285	0.375143	−	0.926967i	$-0.377594\pi$
$157$	9.40105	0.750285	0.375143	+	0.926967i	$0.377594\pi$
$158$	0	0
$159$	−17.5369	−1.39077
$160$	0	0
$161$	0.806063	0.0635267
$162$	0	0
$163$	−12.0059	−0.940373	−0.470187	−	0.882567i	$-0.655813\pi$
$163$	−12.0059	−0.940373	−0.470187	+	0.882567i	$0.655813\pi$
$164$	0	0
$165$	1.67513	0.130409
$166$	0	0
$167$	−21.0738	−1.63074	−0.815370	−	0.578940i	$-0.803467\pi$
$167$	−21.0738	−1.63074	−0.815370	+	0.578940i	$0.803467\pi$
$168$	0	0
$169$	−10.1939	−0.784149
$170$	0	0
$171$	0.700523	0.0535704
$172$	0	0
$173$	−3.33804	−0.253787	−0.126893	−	0.991916i	$-0.540501\pi$
$173$	−3.33804	−0.253787	−0.126893	+	0.991916i	$0.540501\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	7.50659	0.564230
$178$	0	0
$179$	20.1622	1.50699	0.753497	−	0.657451i	$-0.228365\pi$
$179$	20.1622	1.50699	0.753497	+	0.657451i	$0.228365\pi$
$180$	0	0
$181$	7.01317	0.521285	0.260643	−	0.965435i	$-0.416066\pi$
$181$	7.01317	0.521285	0.260643	+	0.965435i	$0.416066\pi$
$182$	0	0
$183$	20.3430	1.50380
$184$	0	0
$185$	−7.76845	−0.571148
$186$	0	0
$187$	−0.324869	−0.0237568
$188$	0	0
$189$	−5.35026	−0.389174
$190$	0	0
$191$	−10.8265	−0.783380	−0.391690	−	0.920097i	$-0.628109\pi$
$191$	−10.8265	−0.783380	−0.391690	+	0.920097i	$0.628109\pi$
$192$	0	0
$193$	14.7816	1.06400	0.532002	−	0.846743i	$-0.321440\pi$
$193$	14.7816	1.06400	0.532002	+	0.846743i	$0.321440\pi$
$194$	0	0
$195$	−2.80606	−0.200946
$196$	0	0
$197$	4.60483	0.328081	0.164040	−	0.986454i	$-0.447547\pi$
$197$	4.60483	0.328081	0.164040	+	0.986454i	$0.447547\pi$
$198$	0	0
$199$	8.92970	0.633010	0.316505	−	0.948591i	$-0.397491\pi$
$199$	8.92970	0.633010	0.316505	+	0.948591i	$0.397491\pi$
$200$	0	0
$201$	−6.96239	−0.491089
$202$	0	0
$203$	−7.92478	−0.556210
$204$	0	0
$205$	8.21933	0.574063
$206$	0	0
$207$	0.156325	0.0108654
$208$	0	0
$209$	3.61213	0.249856
$210$	0	0
$211$	7.66291	0.527537	0.263768	−	0.964586i	$-0.415035\pi$
$211$	7.66291	0.527537	0.263768	+	0.964586i	$0.415035\pi$
$212$	0	0
$213$	−1.29948	−0.0890387
$214$	0	0
$215$	−5.11871	−0.349093
$216$	0	0
$217$	1.13093	0.0767727
$218$	0	0
$219$	13.9452	0.942332
$220$	0	0
$221$	0.544198	0.0366067
$222$	0	0
$223$	−14.0484	−0.940751	−0.470376	−	0.882466i	$-0.655882\pi$
$223$	−14.0484	−0.940751	−0.470376	+	0.882466i	$0.655882\pi$
$224$	0	0
$225$	−0.193937	−0.0129291
$226$	0	0
$227$	4.77575	0.316977	0.158489	−	0.987361i	$-0.449338\pi$
$227$	4.77575	0.316977	0.158489	+	0.987361i	$0.449338\pi$
$228$	0	0
$229$	7.58769	0.501409	0.250704	−	0.968064i	$-0.419338\pi$
$229$	7.58769	0.501409	0.250704	+	0.968064i	$0.419338\pi$
$230$	0	0
$231$	−1.67513	−0.110216
$232$	0	0
$233$	−17.8192	−1.16738	−0.583689	−	0.811978i	$-0.698391\pi$
$233$	−17.8192	−1.16738	−0.583689	+	0.811978i	$0.698391\pi$
$234$	0	0
$235$	2.24965	0.146751
$236$	0	0
$237$	−3.61213	−0.234633
$238$	0	0
$239$	−16.1866	−1.04703	−0.523513	−	0.852017i	$-0.675379\pi$
$239$	−16.1866	−1.04703	−0.523513	+	0.852017i	$0.675379\pi$
$240$	0	0
$241$	5.98190	0.385328	0.192664	−	0.981265i	$-0.438287\pi$
$241$	5.98190	0.385328	0.192664	+	0.981265i	$0.438287\pi$
$242$	0	0
$243$	−2.01222	−0.129084
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−6.05079	−0.385002
$248$	0	0
$249$	−21.5877	−1.36806
$250$	0	0
$251$	−0.294552	−0.0185920	−0.00929598	−	0.999957i	$-0.502959\pi$
$251$	−0.294552	−0.0185920	−0.00929598	+	0.999957i	$0.502959\pi$
$252$	0	0
$253$	0.806063	0.0506768
$254$	0	0
$255$	−0.544198	−0.0340790
$256$	0	0
$257$	−1.86414	−0.116282	−0.0581410	−	0.998308i	$-0.518517\pi$
$257$	−1.86414	−0.116282	−0.0581410	+	0.998308i	$0.518517\pi$
$258$	0	0
$259$	7.76845	0.482708
$260$	0	0
$261$	−1.53690	−0.0951320
$262$	0	0
$263$	13.5877	0.837853	0.418926	−	0.908020i	$-0.362407\pi$
$263$	13.5877	0.837853	0.418926	+	0.908020i	$0.362407\pi$
$264$	0	0
$265$	10.4690	0.643104
$266$	0	0
$267$	10.1359	0.620304
$268$	0	0
$269$	8.62530	0.525894	0.262947	−	0.964810i	$-0.415306\pi$
$269$	8.62530	0.525894	0.262947	+	0.964810i	$0.415306\pi$
$270$	0	0
$271$	−18.0508	−1.09651	−0.548254	−	0.836312i	$-0.684707\pi$
$271$	−18.0508	−1.09651	−0.548254	+	0.836312i	$0.684707\pi$
$272$	0	0
$273$	2.80606	0.169831
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−15.0581	−0.904752	−0.452376	−	0.891827i	$-0.649424\pi$
$277$	−15.0581	−0.904752	−0.452376	+	0.891827i	$0.649424\pi$
$278$	0	0
$279$	0.219329	0.0131309
$280$	0	0
$281$	−32.3390	−1.92918	−0.964591	−	0.263749i	$-0.915041\pi$
$281$	−32.3390	−1.92918	−0.964591	+	0.263749i	$0.915041\pi$
$282$	0	0
$283$	20.3127	1.20746	0.603731	−	0.797188i	$-0.293680\pi$
$283$	20.3127	1.20746	0.603731	+	0.797188i	$0.293680\pi$
$284$	0	0
$285$	6.05079	0.358418
$286$	0	0
$287$	−8.21933	−0.485172
$288$	0	0
$289$	−16.8945	−0.993792
$290$	0	0
$291$	30.2374	1.77255
$292$	0	0
$293$	16.8895	0.986697	0.493349	−	0.869832i	$-0.335773\pi$
$293$	16.8895	0.986697	0.493349	+	0.869832i	$0.335773\pi$
$294$	0	0
$295$	−4.48119	−0.260905
$296$	0	0
$297$	−5.35026	−0.310454
$298$	0	0
$299$	−1.35026	−0.0780877
$300$	0	0
$301$	5.11871	0.295038
$302$	0	0
$303$	−12.2579	−0.704198
$304$	0	0
$305$	−12.1441	−0.695370
$306$	0	0
$307$	−0.237428	−0.0135507	−0.00677535	−	0.999977i	$-0.502157\pi$
$307$	−0.237428	−0.0135507	−0.00677535	+	0.999977i	$0.502157\pi$
$308$	0	0
$309$	−29.5672	−1.68202
$310$	0	0
$311$	−8.21933	−0.466075	−0.233038	−	0.972468i	$-0.574867\pi$
$311$	−8.21933	−0.466075	−0.233038	+	0.972468i	$0.574867\pi$
$312$	0	0
$313$	−19.6121	−1.10854	−0.554271	−	0.832336i	$-0.687003\pi$
$313$	−19.6121	−1.10854	−0.554271	+	0.832336i	$0.687003\pi$
$314$	0	0
$315$	0.193937	0.0109271
$316$	0	0
$317$	−26.4749	−1.48698	−0.743488	−	0.668749i	$-0.766830\pi$
$317$	−26.4749	−1.48698	−0.743488	+	0.668749i	$0.766830\pi$
$318$	0	0
$319$	−7.92478	−0.443702
$320$	0	0
$321$	−5.92478	−0.330689
$322$	0	0
$323$	−1.17347	−0.0652935
$324$	0	0
$325$	1.67513	0.0929195
$326$	0	0
$327$	17.5877	0.972601
$328$	0	0
$329$	−2.24965	−0.124027
$330$	0	0
$331$	−31.2605	−1.71823	−0.859115	−	0.511783i	$-0.828985\pi$
$331$	−31.2605	−1.71823	−0.859115	+	0.511783i	$0.828985\pi$
$332$	0	0
$333$	1.50659	0.0825605
$334$	0	0
$335$	4.15633	0.227084
$336$	0	0
$337$	−26.5296	−1.44516	−0.722580	−	0.691287i	$-0.757044\pi$
$337$	−26.5296	−1.44516	−0.722580	+	0.691287i	$0.757044\pi$
$338$	0	0
$339$	20.1866	1.09639
$340$	0	0
$341$	1.13093	0.0612434
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.35026	0.0726956
$346$	0	0
$347$	−26.3693	−1.41558	−0.707790	−	0.706423i	$-0.750307\pi$
$347$	−26.3693	−1.41558	−0.707790	+	0.706423i	$0.750307\pi$
$348$	0	0
$349$	26.0444	1.39413	0.697064	−	0.717009i	$-0.254490\pi$
$349$	26.0444	1.39413	0.697064	+	0.717009i	$0.254490\pi$
$350$	0	0
$351$	8.96239	0.478377
$352$	0	0
$353$	3.86414	0.205668	0.102834	−	0.994699i	$-0.467209\pi$
$353$	3.86414	0.205668	0.102834	+	0.994699i	$0.467209\pi$
$354$	0	0
$355$	0.775746	0.0411723
$356$	0	0
$357$	0.544198	0.0288020
$358$	0	0
$359$	−22.1563	−1.16937	−0.584683	−	0.811262i	$-0.698781\pi$
$359$	−22.1563	−1.16937	−0.584683	+	0.811262i	$0.698781\pi$
$360$	0	0
$361$	−5.95254	−0.313292
$362$	0	0
$363$	−1.67513	−0.0879216
$364$	0	0
$365$	−8.32487	−0.435744
$366$	0	0
$367$	−12.3611	−0.645242	−0.322621	−	0.946528i	$-0.604564\pi$
$367$	−12.3611	−0.645242	−0.322621	+	0.946528i	$0.604564\pi$
$368$	0	0
$369$	−1.59403	−0.0829818
$370$	0	0
$371$	−10.4690	−0.543522
$372$	0	0
$373$	33.5428	1.73678	0.868390	−	0.495882i	$-0.165155\pi$
$373$	33.5428	1.73678	0.868390	+	0.495882i	$0.165155\pi$
$374$	0	0
$375$	−1.67513	−0.0865034
$376$	0	0
$377$	13.2750	0.683699
$378$	0	0
$379$	−27.9511	−1.43575	−0.717876	−	0.696171i	$-0.754886\pi$
$379$	−27.9511	−1.43575	−0.717876	+	0.696171i	$0.754886\pi$
$380$	0	0
$381$	−13.7137	−0.702574
$382$	0	0
$383$	−14.1138	−0.721181	−0.360590	−	0.932724i	$-0.617425\pi$
$383$	−14.1138	−0.721181	−0.360590	+	0.932724i	$0.617425\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	0.992706	0.0504621
$388$	0	0
$389$	−32.5745	−1.65159	−0.825797	−	0.563968i	$-0.809274\pi$
$389$	−32.5745	−1.65159	−0.825797	+	0.563968i	$0.809274\pi$
$390$	0	0
$391$	−0.261865	−0.0132431
$392$	0	0
$393$	25.9248	1.30773
$394$	0	0
$395$	2.15633	0.108496
$396$	0	0
$397$	−9.08840	−0.456134	−0.228067	−	0.973645i	$-0.573240\pi$
$397$	−9.08840	−0.456134	−0.228067	+	0.973645i	$0.573240\pi$
$398$	0	0
$399$	−6.05079	−0.302918
$400$	0	0
$401$	26.4749	1.32209	0.661046	−	0.750346i	$-0.270113\pi$
$401$	26.4749	1.32209	0.661046	+	0.750346i	$0.270113\pi$
$402$	0	0
$403$	−1.89446	−0.0943698
$404$	0	0
$405$	−8.38058	−0.416434
$406$	0	0
$407$	7.76845	0.385068
$408$	0	0
$409$	−36.5828	−1.80890	−0.904451	−	0.426578i	$-0.859719\pi$
$409$	−36.5828	−1.80890	−0.904451	+	0.426578i	$0.859719\pi$
$410$	0	0
$411$	−4.91160	−0.242272
$412$	0	0
$413$	4.48119	0.220505
$414$	0	0
$415$	12.8872	0.632606
$416$	0	0
$417$	13.6121	0.666589
$418$	0	0
$419$	−12.9053	−0.630463	−0.315232	−	0.949015i	$-0.602082\pi$
$419$	−12.9053	−0.630463	−0.315232	+	0.949015i	$0.602082\pi$
$420$	0	0
$421$	−15.8437	−0.772173	−0.386087	−	0.922462i	$-0.626173\pi$
$421$	−15.8437	−0.772173	−0.386087	+	0.922462i	$0.626173\pi$
$422$	0	0
$423$	−0.436289	−0.0212131
$424$	0	0
$425$	0.324869	0.0157585
$426$	0	0
$427$	12.1441	0.587694
$428$	0	0
$429$	2.80606	0.135478
$430$	0	0
$431$	−21.9452	−1.05707	−0.528533	−	0.848913i	$-0.677258\pi$
$431$	−21.9452	−1.05707	−0.528533	+	0.848913i	$0.677258\pi$
$432$	0	0
$433$	32.4241	1.55820	0.779101	−	0.626899i	$-0.215676\pi$
$433$	32.4241	1.55820	0.779101	+	0.626899i	$0.215676\pi$
$434$	0	0
$435$	−13.2750	−0.636489
$436$	0	0
$437$	2.91160	0.139281
$438$	0	0
$439$	−3.81336	−0.182002	−0.0910008	−	0.995851i	$-0.529007\pi$
$439$	−3.81336	−0.182002	−0.0910008	+	0.995851i	$0.529007\pi$
$440$	0	0
$441$	−0.193937	−0.00923507
$442$	0	0
$443$	34.3693	1.63294	0.816468	−	0.577391i	$-0.195929\pi$
$443$	34.3693	1.63294	0.816468	+	0.577391i	$0.195929\pi$
$444$	0	0
$445$	−6.05079	−0.286835
$446$	0	0
$447$	28.2882	1.33799
$448$	0	0
$449$	−15.0679	−0.711100	−0.355550	−	0.934657i	$-0.615706\pi$
$449$	−15.0679	−0.711100	−0.355550	+	0.934657i	$0.615706\pi$
$450$	0	0
$451$	−8.21933	−0.387033
$452$	0	0
$453$	1.47627	0.0693612
$454$	0	0
$455$	−1.67513	−0.0785313
$456$	0	0
$457$	11.8291	0.553341	0.276671	−	0.960965i	$-0.410769\pi$
$457$	11.8291	0.553341	0.276671	+	0.960965i	$0.410769\pi$
$458$	0	0
$459$	1.73813	0.0811292
$460$	0	0
$461$	−26.9560	−1.25547	−0.627734	−	0.778428i	$-0.716018\pi$
$461$	−26.9560	−1.25547	−0.627734	+	0.778428i	$0.716018\pi$
$462$	0	0
$463$	3.00729	0.139761	0.0698804	−	0.997555i	$-0.477738\pi$
$463$	3.00729	0.139761	0.0698804	+	0.997555i	$0.477738\pi$
$464$	0	0
$465$	1.89446	0.0878535
$466$	0	0
$467$	16.8994	0.782010	0.391005	−	0.920388i	$-0.372127\pi$
$467$	16.8994	0.782010	0.391005	+	0.920388i	$0.372127\pi$
$468$	0	0
$469$	−4.15633	−0.191921
$470$	0	0
$471$	−15.7480	−0.725629
$472$	0	0
$473$	5.11871	0.235359
$474$	0	0
$475$	−3.61213	−0.165736
$476$	0	0
$477$	−2.03032	−0.0929618
$478$	0	0
$479$	2.37328	0.108438	0.0542191	−	0.998529i	$-0.482733\pi$
$479$	2.37328	0.108438	0.0542191	+	0.998529i	$0.482733\pi$
$480$	0	0
$481$	−13.0132	−0.593350
$482$	0	0
$483$	−1.35026	−0.0614390
$484$	0	0
$485$	−18.0508	−0.819644
$486$	0	0
$487$	23.4168	1.06112	0.530558	−	0.847649i	$-0.321983\pi$
$487$	23.4168	1.06112	0.530558	+	0.847649i	$0.321983\pi$
$488$	0	0
$489$	20.1114	0.909470
$490$	0	0
$491$	4.44263	0.200493	0.100246	−	0.994963i	$-0.468037\pi$
$491$	4.44263	0.200493	0.100246	+	0.994963i	$0.468037\pi$
$492$	0	0
$493$	2.57452	0.115950
$494$	0	0
$495$	0.193937	0.00871680
$496$	0	0
$497$	−0.775746	−0.0347970
$498$	0	0
$499$	−18.4894	−0.827701	−0.413851	−	0.910345i	$-0.635816\pi$
$499$	−18.4894	−0.827701	−0.413851	+	0.910345i	$0.635816\pi$
$500$	0	0
$501$	35.3014	1.57715
$502$	0	0
$503$	−14.3371	−0.639259	−0.319630	−	0.947543i	$-0.603558\pi$
$503$	−14.3371	−0.639259	−0.319630	+	0.947543i	$0.603558\pi$
$504$	0	0
$505$	7.31757	0.325628
$506$	0	0
$507$	17.0762	0.758380
$508$	0	0
$509$	0.201231	0.00891940	0.00445970	−	0.999990i	$-0.498580\pi$
$509$	0.201231	0.00891940	0.00445970	+	0.999990i	$0.498580\pi$
$510$	0	0
$511$	8.32487	0.368271
$512$	0	0
$513$	−19.3258	−0.853256
$514$	0	0
$515$	17.6507	0.777782
$516$	0	0
$517$	−2.24965	−0.0989393
$518$	0	0
$519$	5.59166	0.245447
$520$	0	0
$521$	44.5355	1.95114	0.975568	−	0.219699i	$-0.0705077\pi$
$521$	44.5355	1.95114	0.975568	+	0.219699i	$0.0705077\pi$
$522$	0	0
$523$	−0.387873	−0.0169605	−0.00848025	−	0.999964i	$-0.502699\pi$
$523$	−0.387873	−0.0169605	−0.00848025	+	0.999964i	$0.502699\pi$
$524$	0	0
$525$	1.67513	0.0731087
$526$	0	0
$527$	−0.367405	−0.0160044
$528$	0	0
$529$	−22.3503	−0.971751
$530$	0	0
$531$	0.869067	0.0377143
$532$	0	0
$533$	13.7685	0.596378
$534$	0	0
$535$	3.53690	0.152914
$536$	0	0
$537$	−33.7743	−1.45747
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	20.2619	0.871126	0.435563	−	0.900158i	$-0.356549\pi$
$541$	20.2619	0.871126	0.435563	+	0.900158i	$0.356549\pi$
$542$	0	0
$543$	−11.7480	−0.504154
$544$	0	0
$545$	−10.4993	−0.449740
$546$	0	0
$547$	26.6761	1.14059	0.570294	−	0.821441i	$-0.306829\pi$
$547$	26.6761	1.14059	0.570294	+	0.821441i	$0.306829\pi$
$548$	0	0
$549$	2.35519	0.100517
$550$	0	0
$551$	−28.6253	−1.21948
$552$	0	0
$553$	−2.15633	−0.0916963
$554$	0	0
$555$	13.0132	0.552378
$556$	0	0
$557$	10.6556	0.451493	0.225747	−	0.974186i	$-0.427518\pi$
$557$	10.6556	0.451493	0.225747	+	0.974186i	$0.427518\pi$
$558$	0	0
$559$	−8.57452	−0.362663
$560$	0	0
$561$	0.544198	0.0229761
$562$	0	0
$563$	29.8397	1.25759	0.628797	−	0.777570i	$-0.283548\pi$
$563$	29.8397	1.25759	0.628797	+	0.777570i	$0.283548\pi$
$564$	0	0
$565$	−12.0508	−0.506980
$566$	0	0
$567$	8.38058	0.351951
$568$	0	0
$569$	−28.4241	−1.19160	−0.595799	−	0.803133i	$-0.703165\pi$
$569$	−28.4241	−1.19160	−0.595799	+	0.803133i	$0.703165\pi$
$570$	0	0
$571$	−22.3634	−0.935881	−0.467940	−	0.883760i	$-0.655004\pi$
$571$	−22.3634	−0.935881	−0.467940	+	0.883760i	$0.655004\pi$
$572$	0	0
$573$	18.1359	0.757636
$574$	0	0
$575$	−0.806063	−0.0336152
$576$	0	0
$577$	−18.8872	−0.786283	−0.393142	−	0.919478i	$-0.628612\pi$
$577$	−18.8872	−0.786283	−0.393142	+	0.919478i	$0.628612\pi$
$578$	0	0
$579$	−24.7612	−1.02904
$580$	0	0
$581$	−12.8872	−0.534650
$582$	0	0
$583$	−10.4690	−0.433581
$584$	0	0
$585$	−0.324869	−0.0134317
$586$	0	0
$587$	−25.3380	−1.04581	−0.522906	−	0.852390i	$-0.675152\pi$
$587$	−25.3380	−1.04581	−0.522906	+	0.852390i	$0.675152\pi$
$588$	0	0
$589$	4.08507	0.168322
$590$	0	0
$591$	−7.71370	−0.317299
$592$	0	0
$593$	−26.5477	−1.09018	−0.545092	−	0.838376i	$-0.683505\pi$
$593$	−26.5477	−1.09018	−0.545092	+	0.838376i	$0.683505\pi$
$594$	0	0
$595$	−0.324869	−0.0133183
$596$	0	0
$597$	−14.9584	−0.612207
$598$	0	0
$599$	7.78892	0.318247	0.159123	−	0.987259i	$-0.449133\pi$
$599$	7.78892	0.318247	0.159123	+	0.987259i	$0.449133\pi$
$600$	0	0
$601$	11.1309	0.454040	0.227020	−	0.973890i	$-0.427102\pi$
$601$	11.1309	0.454040	0.227020	+	0.973890i	$0.427102\pi$
$602$	0	0
$603$	−0.806063	−0.0328254
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−40.8021	−1.65611	−0.828053	−	0.560650i	$-0.810551\pi$
$607$	−40.8021	−1.65611	−0.828053	+	0.560650i	$0.810551\pi$
$608$	0	0
$609$	13.2750	0.537932
$610$	0	0
$611$	3.76845	0.152455
$612$	0	0
$613$	−16.0157	−0.646869	−0.323435	−	0.946251i	$-0.604838\pi$
$613$	−16.0157	−0.646869	−0.323435	+	0.946251i	$0.604838\pi$
$614$	0	0
$615$	−13.7685	−0.555198
$616$	0	0
$617$	−33.3199	−1.34141	−0.670705	−	0.741724i	$-0.734008\pi$
$617$	−33.3199	−1.34141	−0.670705	+	0.741724i	$0.734008\pi$
$618$	0	0
$619$	−6.85448	−0.275505	−0.137752	−	0.990467i	$-0.543988\pi$
$619$	−6.85448	−0.275505	−0.137752	+	0.990467i	$0.543988\pi$
$620$	0	0
$621$	−4.31265	−0.173061
$622$	0	0
$623$	6.05079	0.242420
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.05079	−0.241645
$628$	0	0
$629$	−2.52373	−0.100628
$630$	0	0
$631$	−29.2506	−1.16445	−0.582224	−	0.813028i	$-0.697817\pi$
$631$	−29.2506	−1.16445	−0.582224	+	0.813028i	$0.697817\pi$
$632$	0	0
$633$	−12.8364	−0.510200
$634$	0	0
$635$	8.18664	0.324877
$636$	0	0
$637$	1.67513	0.0663711
$638$	0	0
$639$	−0.150446	−0.00595153
$640$	0	0
$641$	−46.3039	−1.82890	−0.914448	−	0.404704i	$-0.867375\pi$
$641$	−46.3039	−1.82890	−0.914448	+	0.404704i	$0.867375\pi$
$642$	0	0
$643$	38.6883	1.52572	0.762859	−	0.646565i	$-0.223795\pi$
$643$	38.6883	1.52572	0.762859	+	0.646565i	$0.223795\pi$
$644$	0	0
$645$	8.57452	0.337621
$646$	0	0
$647$	−0.162664	−0.00639500	−0.00319750	−	0.999995i	$-0.501018\pi$
$647$	−0.162664	−0.00639500	−0.00319750	+	0.999995i	$0.501018\pi$
$648$	0	0
$649$	4.48119	0.175902
$650$	0	0
$651$	−1.89446	−0.0742497
$652$	0	0
$653$	40.0508	1.56731	0.783654	−	0.621197i	$-0.213353\pi$
$653$	40.0508	1.56731	0.783654	+	0.621197i	$0.213353\pi$
$654$	0	0
$655$	−15.4763	−0.604708
$656$	0	0
$657$	1.61450	0.0629875
$658$	0	0
$659$	−34.5950	−1.34763	−0.673815	−	0.738900i	$-0.735345\pi$
$659$	−34.5950	−1.34763	−0.673815	+	0.738900i	$0.735345\pi$
$660$	0	0
$661$	−25.5877	−0.995246	−0.497623	−	0.867393i	$-0.665794\pi$
$661$	−25.5877	−0.995246	−0.497623	+	0.867393i	$0.665794\pi$
$662$	0	0
$663$	−0.911603	−0.0354037
$664$	0	0
$665$	3.61213	0.140072
$666$	0	0
$667$	−6.38787	−0.247339
$668$	0	0
$669$	23.5329	0.909836
$670$	0	0
$671$	12.1441	0.468818
$672$	0	0
$673$	21.8437	0.842012	0.421006	−	0.907058i	$-0.361677\pi$
$673$	21.8437	0.842012	0.421006	+	0.907058i	$0.361677\pi$
$674$	0	0
$675$	5.35026	0.205932
$676$	0	0
$677$	−20.7127	−0.796055	−0.398028	−	0.917373i	$-0.630305\pi$
$677$	−20.7127	−0.796055	−0.398028	+	0.917373i	$0.630305\pi$
$678$	0	0
$679$	18.0508	0.692726
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	11.9043	0.455506	0.227753	−	0.973719i	$-0.426862\pi$
$683$	11.9043	0.455506	0.227753	+	0.973719i	$0.426862\pi$
$684$	0	0
$685$	2.93207	0.112029
$686$	0	0
$687$	−12.7104	−0.484931
$688$	0	0
$689$	17.5369	0.668103
$690$	0	0
$691$	−38.9199	−1.48058	−0.740290	−	0.672287i	$-0.765312\pi$
$691$	−38.9199	−1.48058	−0.740290	+	0.672287i	$0.765312\pi$
$692$	0	0
$693$	−0.193937	−0.00736704
$694$	0	0
$695$	−8.12601	−0.308237
$696$	0	0
$697$	2.67021	0.101141
$698$	0	0
$699$	29.8496	1.12901
$700$	0	0
$701$	26.1378	0.987210	0.493605	−	0.869686i	$-0.335679\pi$
$701$	26.1378	0.987210	0.493605	+	0.869686i	$0.335679\pi$
$702$	0	0
$703$	28.0606	1.05833
$704$	0	0
$705$	−3.76845	−0.141928
$706$	0	0
$707$	−7.31757	−0.275206
$708$	0	0
$709$	18.6253	0.699488	0.349744	−	0.936845i	$-0.386269\pi$
$709$	18.6253	0.699488	0.349744	+	0.936845i	$0.386269\pi$
$710$	0	0
$711$	−0.418190	−0.0156834
$712$	0	0
$713$	0.911603	0.0341398
$714$	0	0
$715$	−1.67513	−0.0626463
$716$	0	0
$717$	27.1147	1.01262
$718$	0	0
$719$	−33.6448	−1.25474	−0.627370	−	0.778721i	$-0.715869\pi$
$719$	−33.6448	−1.25474	−0.627370	+	0.778721i	$0.715869\pi$
$720$	0	0
$721$	−17.6507	−0.657346
$722$	0	0
$723$	−10.0205	−0.372665
$724$	0	0
$725$	7.92478	0.294319
$726$	0	0
$727$	−16.7391	−0.620818	−0.310409	−	0.950603i	$-0.600466\pi$
$727$	−16.7391	−0.620818	−0.310409	+	0.950603i	$0.600466\pi$
$728$	0	0
$729$	28.5125	1.05602
$730$	0	0
$731$	−1.66291	−0.0615050
$732$	0	0
$733$	9.10062	0.336139	0.168069	−	0.985775i	$-0.446247\pi$
$733$	9.10062	0.336139	0.168069	+	0.985775i	$0.446247\pi$
$734$	0	0
$735$	−1.67513	−0.0617881
$736$	0	0
$737$	−4.15633	−0.153100
$738$	0	0
$739$	35.5778	1.30875	0.654376	−	0.756169i	$-0.272931\pi$
$739$	35.5778	1.30875	0.654376	+	0.756169i	$0.272931\pi$
$740$	0	0
$741$	10.1359	0.372350
$742$	0	0
$743$	1.71370	0.0628695	0.0314347	−	0.999506i	$-0.489992\pi$
$743$	1.71370	0.0628695	0.0314347	+	0.999506i	$0.489992\pi$
$744$	0	0
$745$	−16.8872	−0.618698
$746$	0	0
$747$	−2.49929	−0.0914443
$748$	0	0
$749$	−3.53690	−0.129236
$750$	0	0
$751$	−16.6497	−0.607558	−0.303779	−	0.952743i	$-0.598248\pi$
$751$	−16.6497	−0.607558	−0.303779	+	0.952743i	$0.598248\pi$
$752$	0	0
$753$	0.493413	0.0179810
$754$	0	0
$755$	−0.881286	−0.0320733
$756$	0	0
$757$	−44.4650	−1.61611	−0.808054	−	0.589108i	$-0.799479\pi$
$757$	−44.4650	−1.61611	−0.808054	+	0.589108i	$0.799479\pi$
$758$	0	0
$759$	−1.35026	−0.0490114
$760$	0	0
$761$	41.4553	1.50275	0.751377	−	0.659873i	$-0.229390\pi$
$761$	41.4553	1.50275	0.751377	+	0.659873i	$0.229390\pi$
$762$	0	0
$763$	10.4993	0.380100
$764$	0	0
$765$	−0.0630040	−0.00227791
$766$	0	0
$767$	−7.50659	−0.271047
$768$	0	0
$769$	45.3439	1.63514	0.817572	−	0.575827i	$-0.195320\pi$
$769$	45.3439	1.63514	0.817572	+	0.575827i	$0.195320\pi$
$770$	0	0
$771$	3.12268	0.112461
$772$	0	0
$773$	2.42407	0.0871877	0.0435939	−	0.999049i	$-0.486119\pi$
$773$	2.42407	0.0871877	0.0435939	+	0.999049i	$0.486119\pi$
$774$	0	0
$775$	−1.13093	−0.0406243
$776$	0	0
$777$	−13.0132	−0.466845
$778$	0	0
$779$	−29.6893	−1.06373
$780$	0	0
$781$	−0.775746	−0.0277584
$782$	0	0
$783$	42.3996	1.51524
$784$	0	0
$785$	9.40105	0.335538
$786$	0	0
$787$	1.36485	0.0486517	0.0243258	−	0.999704i	$-0.492256\pi$
$787$	1.36485	0.0486517	0.0243258	+	0.999704i	$0.492256\pi$
$788$	0	0
$789$	−22.7612	−0.810319
$790$	0	0
$791$	12.0508	0.428477
$792$	0	0
$793$	−20.3430	−0.722400
$794$	0	0
$795$	−17.5369	−0.621970
$796$	0	0
$797$	16.8970	0.598523	0.299261	−	0.954171i	$-0.403260\pi$
$797$	16.8970	0.598523	0.299261	+	0.954171i	$0.403260\pi$
$798$	0	0
$799$	0.730841	0.0258553
$800$	0	0
$801$	1.17347	0.0414625
$802$	0	0
$803$	8.32487	0.293778
$804$	0	0
$805$	0.806063	0.0284100
$806$	0	0
$807$	−14.4485	−0.508612
$808$	0	0
$809$	−41.2652	−1.45081	−0.725403	−	0.688324i	$-0.758347\pi$
$809$	−41.2652	−1.45081	−0.725403	+	0.688324i	$0.758347\pi$
$810$	0	0
$811$	43.6483	1.53270	0.766350	−	0.642423i	$-0.222071\pi$
$811$	43.6483	1.53270	0.766350	+	0.642423i	$0.222071\pi$
$812$	0	0
$813$	30.2374	1.06047
$814$	0	0
$815$	−12.0059	−0.420548
$816$	0	0
$817$	18.4894	0.646864
$818$	0	0
$819$	0.324869	0.0113518
$820$	0	0
$821$	27.6483	0.964933	0.482467	−	0.875914i	$-0.339741\pi$
$821$	27.6483	0.964933	0.482467	+	0.875914i	$0.339741\pi$
$822$	0	0
$823$	−57.1549	−1.99229	−0.996147	−	0.0876939i	$-0.972050\pi$
$823$	−57.1549	−1.99229	−0.996147	+	0.0876939i	$0.972050\pi$
$824$	0	0
$825$	1.67513	0.0583206
$826$	0	0
$827$	−19.9814	−0.694823	−0.347411	−	0.937713i	$-0.612939\pi$
$827$	−19.9814	−0.694823	−0.347411	+	0.937713i	$0.612939\pi$
$828$	0	0
$829$	4.57925	0.159044	0.0795220	−	0.996833i	$-0.474661\pi$
$829$	4.57925	0.159044	0.0795220	+	0.996833i	$0.474661\pi$
$830$	0	0
$831$	25.2243	0.875020
$832$	0	0
$833$	0.324869	0.0112560
$834$	0	0
$835$	−21.0738	−0.729289
$836$	0	0
$837$	−6.05079	−0.209146
$838$	0	0
$839$	9.15728	0.316144	0.158072	−	0.987428i	$-0.449472\pi$
$839$	9.15728	0.316144	0.158072	+	0.987428i	$0.449472\pi$
$840$	0	0
$841$	33.8021	1.16559
$842$	0	0
$843$	54.1721	1.86578
$844$	0	0
$845$	−10.1939	−0.350682
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−34.0263	−1.16778
$850$	0	0
$851$	6.26187	0.214654
$852$	0	0
$853$	−42.1255	−1.44235	−0.721176	−	0.692752i	$-0.756398\pi$
$853$	−42.1255	−1.44235	−0.721176	+	0.692752i	$0.756398\pi$
$854$	0	0
$855$	0.700523	0.0239574
$856$	0	0
$857$	57.6408	1.96897	0.984487	−	0.175458i	$-0.0561407\pi$
$857$	57.6408	1.96897	0.984487	+	0.175458i	$0.0561407\pi$
$858$	0	0
$859$	33.2687	1.13511	0.567557	−	0.823334i	$-0.307889\pi$
$859$	33.2687	1.13511	0.567557	+	0.823334i	$0.307889\pi$
$860$	0	0
$861$	13.7685	0.469228
$862$	0	0
$863$	49.4920	1.68473	0.842364	−	0.538910i	$-0.181164\pi$
$863$	49.4920	1.68473	0.842364	+	0.538910i	$0.181164\pi$
$864$	0	0
$865$	−3.33804	−0.113497
$866$	0	0
$867$	28.3004	0.961133
$868$	0	0
$869$	−2.15633	−0.0731483
$870$	0	0
$871$	6.96239	0.235912
$872$	0	0
$873$	3.50071	0.118481
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	7.58181	0.256020	0.128010	−	0.991773i	$-0.459141\pi$
$877$	7.58181	0.256020	0.128010	+	0.991773i	$0.459141\pi$
$878$	0	0
$879$	−28.2922	−0.954272
$880$	0	0
$881$	20.4631	0.689419	0.344710	−	0.938709i	$-0.387977\pi$
$881$	20.4631	0.689419	0.344710	+	0.938709i	$0.387977\pi$
$882$	0	0
$883$	39.6932	1.33578	0.667892	−	0.744258i	$-0.267197\pi$
$883$	39.6932	1.33578	0.667892	+	0.744258i	$0.267197\pi$
$884$	0	0
$885$	7.50659	0.252331
$886$	0	0
$887$	−35.0494	−1.17684	−0.588421	−	0.808554i	$-0.700250\pi$
$887$	−35.0494	−1.17684	−0.588421	+	0.808554i	$0.700250\pi$
$888$	0	0
$889$	−8.18664	−0.274571
$890$	0	0
$891$	8.38058	0.280760
$892$	0	0
$893$	−8.12601	−0.271926
$894$	0	0
$895$	20.1622	0.673948
$896$	0	0
$897$	2.26187	0.0755215
$898$	0	0
$899$	−8.96239	−0.298912
$900$	0	0
$901$	3.40105	0.113305
$902$	0	0
$903$	−8.57452	−0.285342
$904$	0	0
$905$	7.01317	0.233126
$906$	0	0
$907$	34.5198	1.14621	0.573105	−	0.819482i	$-0.305739\pi$
$907$	34.5198	1.14621	0.573105	+	0.819482i	$0.305739\pi$
$908$	0	0
$909$	−1.41915	−0.0470701
$910$	0	0
$911$	2.44851	0.0811227	0.0405613	−	0.999177i	$-0.487085\pi$
$911$	2.44851	0.0811227	0.0405613	+	0.999177i	$0.487085\pi$
$912$	0	0
$913$	−12.8872	−0.426503
$914$	0	0
$915$	20.3430	0.672518
$916$	0	0
$917$	15.4763	0.511071
$918$	0	0
$919$	−19.6629	−0.648620	−0.324310	−	0.945951i	$-0.605132\pi$
$919$	−19.6629	−0.648620	−0.324310	+	0.945951i	$0.605132\pi$
$920$	0	0
$921$	0.397722	0.0131054
$922$	0	0
$923$	1.29948	0.0427728
$924$	0	0
$925$	−7.76845	−0.255425
$926$	0	0
$927$	−3.42311	−0.112430
$928$	0	0
$929$	−28.7875	−0.944487	−0.472244	−	0.881468i	$-0.656556\pi$
$929$	−28.7875	−0.944487	−0.472244	+	0.881468i	$0.656556\pi$
$930$	0	0
$931$	−3.61213	−0.118383
$932$	0	0
$933$	13.7685	0.450759
$934$	0	0
$935$	−0.324869	−0.0106244
$936$	0	0
$937$	−18.0729	−0.590414	−0.295207	−	0.955433i	$-0.595389\pi$
$937$	−18.0729	−0.590414	−0.295207	+	0.955433i	$0.595389\pi$
$938$	0	0
$939$	32.8529	1.07211
$940$	0	0
$941$	6.18313	0.201564	0.100782	−	0.994909i	$-0.467865\pi$
$941$	6.18313	0.201564	0.100782	+	0.994909i	$0.467865\pi$
$942$	0	0
$943$	−6.62530	−0.215749
$944$	0	0
$945$	−5.35026	−0.174044
$946$	0	0
$947$	−7.04349	−0.228883	−0.114441	−	0.993430i	$-0.536508\pi$
$947$	−7.04349	−0.228883	−0.114441	+	0.993430i	$0.536508\pi$
$948$	0	0
$949$	−13.9452	−0.452682
$950$	0	0
$951$	44.3488	1.43811
$952$	0	0
$953$	−7.95509	−0.257691	−0.128845	−	0.991665i	$-0.541127\pi$
$953$	−7.95509	−0.257691	−0.128845	+	0.991665i	$0.541127\pi$
$954$	0	0
$955$	−10.8265	−0.350338
$956$	0	0
$957$	13.2750	0.429121
$958$	0	0
$959$	−2.93207	−0.0946815
$960$	0	0
$961$	−29.7210	−0.958742
$962$	0	0
$963$	−0.685935	−0.0221039
$964$	0	0
$965$	14.7816	0.475837
$966$	0	0
$967$	26.0567	0.837926	0.418963	−	0.908003i	$-0.362394\pi$
$967$	26.0567	0.837926	0.418963	+	0.908003i	$0.362394\pi$
$968$	0	0
$969$	1.96571	0.0631478
$970$	0	0
$971$	37.5597	1.20535	0.602675	−	0.797987i	$-0.294102\pi$
$971$	37.5597	1.20535	0.602675	+	0.797987i	$0.294102\pi$
$972$	0	0
$973$	8.12601	0.260508
$974$	0	0
$975$	−2.80606	−0.0898660
$976$	0	0
$977$	37.1509	1.18856	0.594282	−	0.804257i	$-0.297436\pi$
$977$	37.1509	1.18856	0.594282	+	0.804257i	$0.297436\pi$
$978$	0	0
$979$	6.05079	0.193384
$980$	0	0
$981$	2.03620	0.0650108
$982$	0	0
$983$	−6.52610	−0.208150	−0.104075	−	0.994569i	$-0.533188\pi$
$983$	−6.52610	−0.208150	−0.104075	+	0.994569i	$0.533188\pi$
$984$	0	0
$985$	4.60483	0.146722
$986$	0	0
$987$	3.76845	0.119951
$988$	0	0
$989$	4.12601	0.131199
$990$	0	0
$991$	24.1866	0.768314	0.384157	−	0.923268i	$-0.374492\pi$
$991$	24.1866	0.768314	0.384157	+	0.923268i	$0.374492\pi$
$992$	0	0
$993$	52.3653	1.66176
$994$	0	0
$995$	8.92970	0.283091
$996$	0	0
$997$	16.0776	0.509182	0.254591	−	0.967049i	$-0.418059\pi$
$997$	16.0776	0.509182	0.254591	+	0.967049i	$0.418059\pi$
$998$	0	0
$999$	−41.5633	−1.31500

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bj.1.1		3
4.3	odd	2		385.2.a.g.1.1	✓	3
12.11	even	2		3465.2.a.ba.1.3		3
20.3	even	4		1925.2.b.o.1849.5		6
20.7	even	4		1925.2.b.o.1849.2		6
20.19	odd	2		1925.2.a.u.1.3		3
28.27	even	2		2695.2.a.i.1.1		3
44.43	even	2		4235.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.g.1.1	✓	3	4.3	odd	2
1925.2.a.u.1.3		3	20.19	odd	2
1925.2.b.o.1849.2		6	20.7	even	4
1925.2.b.o.1849.5		6	20.3	even	4
2695.2.a.i.1.1		3	28.27	even	2
3465.2.a.ba.1.3		3	12.11	even	2
4235.2.a.o.1.3		3	44.43	even	2
6160.2.a.bj.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)

\(f(q)\)	\(=\)	\(q-1.67513 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} +O(q^{10})\)	\(q-1.67513 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.193937 q^{9} -1.00000 q^{11} +1.67513 q^{13} -1.67513 q^{15} +0.324869 q^{17} -3.61213 q^{19} +1.67513 q^{21} -0.806063 q^{23} +1.00000 q^{25} +5.35026 q^{27} +7.92478 q^{29} -1.13093 q^{31} +1.67513 q^{33} -1.00000 q^{35} -7.76845 q^{37} -2.80606 q^{39} +8.21933 q^{41} -5.11871 q^{43} -0.193937 q^{45} +2.24965 q^{47} +1.00000 q^{49} -0.544198 q^{51} +10.4690 q^{53} -1.00000 q^{55} +6.05079 q^{57} -4.48119 q^{59} -12.1441 q^{61} +0.193937 q^{63} +1.67513 q^{65} +4.15633 q^{67} +1.35026 q^{69} +0.775746 q^{71} -8.32487 q^{73} -1.67513 q^{75} +1.00000 q^{77} +2.15633 q^{79} -8.38058 q^{81} +12.8872 q^{83} +0.324869 q^{85} -13.2750 q^{87} -6.05079 q^{89} -1.67513 q^{91} +1.89446 q^{93} -3.61213 q^{95} -18.0508 q^{97} +0.193937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q + 3 * q^5 - 3 * q^7 - q^9	\( 3 q + 3 q^{5} - 3 q^{7} - q^{9} - 3 q^{11} + 6 q^{17} - 10 q^{19} - 2 q^{23} + 3 q^{25} + 6 q^{27} + 2 q^{29} - 8 q^{31} - 3 q^{35} - 12 q^{37} - 8 q^{39} + 10 q^{41} + 6 q^{43} - q^{45} - 10 q^{47} + 3 q^{49} + 8 q^{51} - 3 q^{55} - 12 q^{57} - 8 q^{59} + q^{63} + 2 q^{67} - 6 q^{69} + 4 q^{71} - 30 q^{73} + 3 q^{77} - 4 q^{79} - 13 q^{81} + 6 q^{83} + 6 q^{85} - 8 q^{87} + 12 q^{89} - 14 q^{93} - 10 q^{95} - 24 q^{97} + q^{99}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^7 - q^9 - 3 * q^11 + 6 * q^17 - 10 * q^19 - 2 * q^23 + 3 * q^25 + 6 * q^27 + 2 * q^29 - 8 * q^31 - 3 * q^35 - 12 * q^37 - 8 * q^39 + 10 * q^41 + 6 * q^43 - q^45 - 10 * q^47 + 3 * q^49 + 8 * q^51 - 3 * q^55 - 12 * q^57 - 8 * q^59 + q^63 + 2 * q^67 - 6 * q^69 + 4 * q^71 - 30 * q^73 + 3 * q^77 - 4 * q^79 - 13 * q^81 + 6 * q^83 + 6 * q^85 - 8 * q^87 + 12 * q^89 - 14 * q^93 - 10 * q^95 - 24 * q^97 + q^99

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