LMFDB - Embedded newform 5850.2.a.ct.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5850,2,Mod(1,5850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.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:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1170)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	5850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.19394 q^{7} +1.00000 q^{8} +2.00000 q^{11} +1.00000 q^{13} +1.19394 q^{14} +1.00000 q^{16} +4.54420 q^{17} +4.15633 q^{19} +2.00000 q^{22} -7.11871 q^{23} +1.00000 q^{26} +1.19394 q^{28} +10.7308 q^{29} -5.35026 q^{31} +1.00000 q^{32} +4.54420 q^{34} -3.92478 q^{37} +4.15633 q^{38} +1.03761 q^{41} +10.8872 q^{43} +2.00000 q^{44} -7.11871 q^{46} -1.61213 q^{47} -5.57452 q^{49} +1.00000 q^{52} +4.18664 q^{53} +1.19394 q^{56} +10.7308 q^{58} -2.31265 q^{59} -7.08840 q^{61} -5.35026 q^{62} +1.00000 q^{64} -4.70052 q^{67} +4.54420 q^{68} -9.27504 q^{71} -3.58181 q^{73} -3.92478 q^{74} +4.15633 q^{76} +2.38787 q^{77} +15.1998 q^{79} +1.03761 q^{82} +1.73813 q^{83} +10.8872 q^{86} +2.00000 q^{88} +14.3127 q^{89} +1.19394 q^{91} -7.11871 q^{92} -1.61213 q^{94} +9.19394 q^{97} -5.57452 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 6 q^{11} + 3 q^{13} + 4 q^{14} + 3 q^{16} + 4 q^{17} + 2 q^{19} + 6 q^{22} + 3 q^{26} + 4 q^{28} + 10 q^{29} - 6 q^{31} + 3 q^{32} + 4 q^{34} + 10 q^{37} + 2 q^{38}+ \cdots - 5 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 6 * q^11 + 3 * q^13 + 4 * q^14 + 3 * q^16 + 4 * q^17 + 2 * q^19 + 6 * q^22 + 3 * q^26 + 4 * q^28 + 10 * q^29 - 6 * q^31 + 3 * q^32 + 4 * q^34 + 10 * q^37 + 2 * q^38 + 14 * q^41 + 6 * q^44 - 4 * q^47 - 5 * q^49 + 3 * q^52 + 4 * q^56 + 10 * q^58 + 14 * q^59 - 2 * q^61 - 6 * q^62 + 3 * q^64 + 6 * q^67 + 4 * q^68 + 4 * q^71 - 12 * q^73 + 10 * q^74 + 2 * q^76 + 8 * q^77 - 8 * q^79 + 14 * q^82 - 4 * q^83 + 6 * q^88 + 22 * q^89 + 4 * q^91 - 4 * q^94 + 28 * q^97 - 5 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.19394	0.451266	0.225633	−	0.974212i	$-0.427555\pi$
$7$	1.19394	0.451266	0.225633	+	0.974212i	$0.427555\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.19394	0.319093
$15$	0	0
$16$	1.00000	0.250000
$17$	4.54420	1.10213	0.551065	−	0.834462i	$-0.314222\pi$
$17$	4.54420	1.10213	0.551065	+	0.834462i	$0.314222\pi$
$18$	0	0
$19$	4.15633	0.953526	0.476763	−	0.879032i	$-0.341810\pi$
$19$	4.15633	0.953526	0.476763	+	0.879032i	$0.341810\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	−7.11871	−1.48435	−0.742177	−	0.670204i	$-0.766207\pi$
$23$	−7.11871	−1.48435	−0.742177	+	0.670204i	$0.766207\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	1.19394	0.225633
$29$	10.7308	1.99267	0.996334	−	0.0855539i	$-0.0272660\pi$
$29$	10.7308	1.99267	0.996334	+	0.0855539i	$0.0272660\pi$
$30$	0	0
$31$	−5.35026	−0.960935	−0.480468	−	0.877012i	$-0.659533\pi$
$31$	−5.35026	−0.960935	−0.480468	+	0.877012i	$0.659533\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.54420	0.779324
$35$	0	0
$36$	0	0
$37$	−3.92478	−0.645229	−0.322615	−	0.946530i	$-0.604562\pi$
$37$	−3.92478	−0.645229	−0.322615	+	0.946530i	$0.604562\pi$
$38$	4.15633	0.674245
$39$	0	0
$40$	0	0
$41$	1.03761	0.162048	0.0810238	−	0.996712i	$-0.474181\pi$
$41$	1.03761	0.162048	0.0810238	+	0.996712i	$0.474181\pi$
$42$	0	0
$43$	10.8872	1.66028	0.830139	−	0.557557i	$-0.188261\pi$
$43$	10.8872	1.66028	0.830139	+	0.557557i	$0.188261\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−7.11871	−1.04960
$47$	−1.61213	−0.235153	−0.117576	−	0.993064i	$-0.537513\pi$
$47$	−1.61213	−0.235153	−0.117576	+	0.993064i	$0.537513\pi$
$48$	0	0
$49$	−5.57452	−0.796359
$50$	0	0
$51$	0	0
$52$	1.00000	0.138675
$53$	4.18664	0.575080	0.287540	−	0.957769i	$-0.407163\pi$
$53$	4.18664	0.575080	0.287540	+	0.957769i	$0.407163\pi$
$54$	0	0
$55$	0	0
$56$	1.19394	0.159546
$57$	0	0
$58$	10.7308	1.40903
$59$	−2.31265	−0.301081	−0.150541	−	0.988604i	$-0.548101\pi$
$59$	−2.31265	−0.301081	−0.150541	+	0.988604i	$0.548101\pi$
$60$	0	0
$61$	−7.08840	−0.907576	−0.453788	−	0.891110i	$-0.649928\pi$
$61$	−7.08840	−0.907576	−0.453788	+	0.891110i	$0.649928\pi$
$62$	−5.35026	−0.679484
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.70052	−0.574260	−0.287130	−	0.957892i	$-0.592701\pi$
$67$	−4.70052	−0.574260	−0.287130	+	0.957892i	$0.592701\pi$
$68$	4.54420	0.551065
$69$	0	0
$70$	0	0
$71$	−9.27504	−1.10074	−0.550372	−	0.834919i	$-0.685514\pi$
$71$	−9.27504	−1.10074	−0.550372	+	0.834919i	$0.685514\pi$
$72$	0	0
$73$	−3.58181	−0.419219	−0.209610	−	0.977785i	$-0.567219\pi$
$73$	−3.58181	−0.419219	−0.209610	+	0.977785i	$0.567219\pi$
$74$	−3.92478	−0.456246
$75$	0	0
$76$	4.15633	0.476763
$77$	2.38787	0.272123
$78$	0	0
$79$	15.1998	1.71011	0.855056	−	0.518535i	$-0.173522\pi$
$79$	15.1998	1.71011	0.855056	+	0.518535i	$0.173522\pi$
$80$	0	0
$81$	0	0
$82$	1.03761	0.114585
$83$	1.73813	0.190785	0.0953925	−	0.995440i	$-0.469589\pi$
$83$	1.73813	0.190785	0.0953925	+	0.995440i	$0.469589\pi$
$84$	0	0
$85$	0	0
$86$	10.8872	1.17399
$87$	0	0
$88$	2.00000	0.213201
$89$	14.3127	1.51714	0.758569	−	0.651593i	$-0.225899\pi$
$89$	14.3127	1.51714	0.758569	+	0.651593i	$0.225899\pi$
$90$	0	0
$91$	1.19394	0.125159
$92$	−7.11871	−0.742177
$93$	0	0
$94$	−1.61213	−0.166278
$95$	0	0
$96$	0	0
$97$	9.19394	0.933503	0.466751	−	0.884389i	$-0.345424\pi$
$97$	9.19394	0.933503	0.466751	+	0.884389i	$0.345424\pi$
$98$	−5.57452	−0.563111
$99$	0	0
$100$	0	0
$101$	13.8945	1.38255	0.691275	−	0.722592i	$-0.257049\pi$
$101$	13.8945	1.38255	0.691275	+	0.722592i	$0.257049\pi$
$102$	0	0
$103$	−7.79877	−0.768436	−0.384218	−	0.923242i	$-0.625529\pi$
$103$	−7.79877	−0.768436	−0.384218	+	0.923242i	$0.625529\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	4.18664	0.406643
$107$	11.5369	1.11531	0.557657	−	0.830071i	$-0.311700\pi$
$107$	11.5369	1.11531	0.557657	+	0.830071i	$0.311700\pi$
$108$	0	0
$109$	−1.58181	−0.151510	−0.0757549	−	0.997126i	$-0.524137\pi$
$109$	−1.58181	−0.151510	−0.0757549	+	0.997126i	$0.524137\pi$
$110$	0	0
$111$	0	0
$112$	1.19394	0.112816
$113$	−14.4690	−1.36113	−0.680563	−	0.732689i	$-0.738265\pi$
$113$	−14.4690	−1.36113	−0.680563	+	0.732689i	$0.738265\pi$
$114$	0	0
$115$	0	0
$116$	10.7308	0.996334
$117$	0	0
$118$	−2.31265	−0.212897
$119$	5.42548	0.497353
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−7.08840	−0.641753
$123$	0	0
$124$	−5.35026	−0.480468
$125$	0	0
$126$	0	0
$127$	2.18664	0.194033	0.0970166	−	0.995283i	$-0.469070\pi$
$127$	2.18664	0.194033	0.0970166	+	0.995283i	$0.469070\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	5.64244	0.492983	0.246491	−	0.969145i	$-0.420722\pi$
$131$	5.64244	0.492983	0.246491	+	0.969145i	$0.420722\pi$
$132$	0	0
$133$	4.96239	0.430294
$134$	−4.70052	−0.406063
$135$	0	0
$136$	4.54420	0.389662
$137$	−2.46310	−0.210436	−0.105218	−	0.994449i	$-0.533554\pi$
$137$	−2.46310	−0.210436	−0.105218	+	0.994449i	$0.533554\pi$
$138$	0	0
$139$	−16.6253	−1.41014	−0.705070	−	0.709138i	$-0.749084\pi$
$139$	−16.6253	−1.41014	−0.705070	+	0.709138i	$0.749084\pi$
$140$	0	0
$141$	0	0
$142$	−9.27504	−0.778344
$143$	2.00000	0.167248
$144$	0	0
$145$	0	0
$146$	−3.58181	−0.296433
$147$	0	0
$148$	−3.92478	−0.322615
$149$	22.0508	1.80647	0.903235	−	0.429146i	$-0.141185\pi$
$149$	22.0508	1.80647	0.903235	+	0.429146i	$0.141185\pi$
$150$	0	0
$151$	−2.96239	−0.241076	−0.120538	−	0.992709i	$-0.538462\pi$
$151$	−2.96239	−0.241076	−0.120538	+	0.992709i	$0.538462\pi$
$152$	4.15633	0.337122
$153$	0	0
$154$	2.38787	0.192420
$155$	0	0
$156$	0	0
$157$	20.3634	1.62518	0.812590	−	0.582836i	$-0.198057\pi$
$157$	20.3634	1.62518	0.812590	+	0.582836i	$0.198057\pi$
$158$	15.1998	1.20923
$159$	0	0
$160$	0	0
$161$	−8.49929	−0.669838
$162$	0	0
$163$	−25.3258	−1.98367	−0.991836	−	0.127521i	$-0.959298\pi$
$163$	−25.3258	−1.98367	−0.991836	+	0.127521i	$0.959298\pi$
$164$	1.03761	0.0810238
$165$	0	0
$166$	1.73813	0.134905
$167$	−6.70052	−0.518502	−0.259251	−	0.965810i	$-0.583476\pi$
$167$	−6.70052	−0.518502	−0.259251	+	0.965810i	$0.583476\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	10.8872	0.830139
$173$	0.649738	0.0493987	0.0246993	−	0.999695i	$-0.492137\pi$
$173$	0.649738	0.0493987	0.0246993	+	0.999695i	$0.492137\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	14.3127	1.07278
$179$	6.10554	0.456349	0.228175	−	0.973620i	$-0.426724\pi$
$179$	6.10554	0.456349	0.228175	+	0.973620i	$0.426724\pi$
$180$	0	0
$181$	22.9380	1.70496	0.852482	−	0.522756i	$-0.175096\pi$
$181$	22.9380	1.70496	0.852482	+	0.522756i	$0.175096\pi$
$182$	1.19394	0.0885005
$183$	0	0
$184$	−7.11871	−0.524799
$185$	0	0
$186$	0	0
$187$	9.08840	0.664609
$188$	−1.61213	−0.117576
$189$	0	0
$190$	0	0
$191$	19.0132	1.37574	0.687872	−	0.725832i	$-0.258545\pi$
$191$	19.0132	1.37574	0.687872	+	0.725832i	$0.258545\pi$
$192$	0	0
$193$	−4.41819	−0.318028	−0.159014	−	0.987276i	$-0.550832\pi$
$193$	−4.41819	−0.318028	−0.159014	+	0.987276i	$0.550832\pi$
$194$	9.19394	0.660086
$195$	0	0
$196$	−5.57452	−0.398180
$197$	−5.22425	−0.372213	−0.186106	−	0.982530i	$-0.559587\pi$
$197$	−5.22425	−0.372213	−0.186106	+	0.982530i	$0.559587\pi$
$198$	0	0
$199$	−19.1998	−1.36104	−0.680519	−	0.732730i	$-0.738246\pi$
$199$	−19.1998	−1.36104	−0.680519	+	0.732730i	$0.738246\pi$
$200$	0	0
$201$	0	0
$202$	13.8945	0.977611
$203$	12.8119	0.899222
$204$	0	0
$205$	0	0
$206$	−7.79877	−0.543366
$207$	0	0
$208$	1.00000	0.0693375
$209$	8.31265	0.574998
$210$	0	0
$211$	9.86414	0.679076	0.339538	−	0.940592i	$-0.389729\pi$
$211$	9.86414	0.679076	0.339538	+	0.940592i	$0.389729\pi$
$212$	4.18664	0.287540
$213$	0	0
$214$	11.5369	0.788647
$215$	0	0
$216$	0	0
$217$	−6.38787	−0.433637
$218$	−1.58181	−0.107134
$219$	0	0
$220$	0	0
$221$	4.54420	0.305676
$222$	0	0
$223$	18.9076	1.26615	0.633074	−	0.774091i	$-0.281793\pi$
$223$	18.9076	1.26615	0.633074	+	0.774091i	$0.281793\pi$
$224$	1.19394	0.0797732
$225$	0	0
$226$	−14.4690	−0.962462
$227$	−19.3258	−1.28270	−0.641350	−	0.767248i	$-0.721625\pi$
$227$	−19.3258	−1.28270	−0.641350	+	0.767248i	$0.721625\pi$
$228$	0	0
$229$	11.5672	0.764383	0.382192	−	0.924083i	$-0.375169\pi$
$229$	11.5672	0.764383	0.382192	+	0.924083i	$0.375169\pi$
$230$	0	0
$231$	0	0
$232$	10.7308	0.704514
$233$	−10.7816	−0.706328	−0.353164	−	0.935561i	$-0.614894\pi$
$233$	−10.7816	−0.706328	−0.353164	+	0.935561i	$0.614894\pi$
$234$	0	0
$235$	0	0
$236$	−2.31265	−0.150541
$237$	0	0
$238$	5.42548	0.351682
$239$	21.9248	1.41820	0.709098	−	0.705110i	$-0.249102\pi$
$239$	21.9248	1.41820	0.709098	+	0.705110i	$0.249102\pi$
$240$	0	0
$241$	−4.23743	−0.272957	−0.136478	−	0.990643i	$-0.543578\pi$
$241$	−4.23743	−0.272957	−0.136478	+	0.990643i	$0.543578\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−7.08840	−0.453788
$245$	0	0
$246$	0	0
$247$	4.15633	0.264461
$248$	−5.35026	−0.339742
$249$	0	0
$250$	0	0
$251$	1.95509	0.123404	0.0617022	−	0.998095i	$-0.480347\pi$
$251$	1.95509	0.123404	0.0617022	+	0.998095i	$0.480347\pi$
$252$	0	0
$253$	−14.2374	−0.895099
$254$	2.18664	0.137202
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.78163	0.423026	0.211513	−	0.977375i	$-0.432161\pi$
$257$	6.78163	0.423026	0.211513	+	0.977375i	$0.432161\pi$
$258$	0	0
$259$	−4.68594	−0.291170
$260$	0	0
$261$	0	0
$262$	5.64244	0.348591
$263$	9.96968	0.614757	0.307378	−	0.951587i	$-0.400548\pi$
$263$	9.96968	0.614757	0.307378	+	0.951587i	$0.400548\pi$
$264$	0	0
$265$	0	0
$266$	4.96239	0.304264
$267$	0	0
$268$	−4.70052	−0.287130
$269$	9.64244	0.587910	0.293955	−	0.955819i	$-0.405028\pi$
$269$	9.64244	0.587910	0.293955	+	0.955819i	$0.405028\pi$
$270$	0	0
$271$	21.1392	1.28411	0.642057	−	0.766657i	$-0.278081\pi$
$271$	21.1392	1.28411	0.642057	+	0.766657i	$0.278081\pi$
$272$	4.54420	0.275532
$273$	0	0
$274$	−2.46310	−0.148801
$275$	0	0
$276$	0	0
$277$	7.48612	0.449797	0.224899	−	0.974382i	$-0.427795\pi$
$277$	7.48612	0.449797	0.224899	+	0.974382i	$0.427795\pi$
$278$	−16.6253	−0.997119
$279$	0	0
$280$	0	0
$281$	−18.1622	−1.08347	−0.541733	−	0.840551i	$-0.682232\pi$
$281$	−18.1622	−1.08347	−0.541733	+	0.840551i	$0.682232\pi$
$282$	0	0
$283$	−11.8134	−0.702231	−0.351116	−	0.936332i	$-0.614198\pi$
$283$	−11.8134	−0.702231	−0.351116	+	0.936332i	$0.614198\pi$
$284$	−9.27504	−0.550372
$285$	0	0
$286$	2.00000	0.118262
$287$	1.23884	0.0731265
$288$	0	0
$289$	3.64974	0.214690
$290$	0	0
$291$	0	0
$292$	−3.58181	−0.209610
$293$	−12.7005	−0.741973	−0.370986	−	0.928638i	$-0.620980\pi$
$293$	−12.7005	−0.741973	−0.370986	+	0.928638i	$0.620980\pi$
$294$	0	0
$295$	0	0
$296$	−3.92478	−0.228123
$297$	0	0
$298$	22.0508	1.27737
$299$	−7.11871	−0.411686
$300$	0	0
$301$	12.9986	0.749226
$302$	−2.96239	−0.170466
$303$	0	0
$304$	4.15633	0.238382
$305$	0	0
$306$	0	0
$307$	−6.98683	−0.398759	−0.199380	−	0.979922i	$-0.563893\pi$
$307$	−6.98683	−0.398759	−0.199380	+	0.979922i	$0.563893\pi$
$308$	2.38787	0.136062
$309$	0	0
$310$	0	0
$311$	−7.53690	−0.427379	−0.213689	−	0.976902i	$-0.568548\pi$
$311$	−7.53690	−0.427379	−0.213689	+	0.976902i	$0.568548\pi$
$312$	0	0
$313$	−21.7137	−1.22733	−0.613665	−	0.789566i	$-0.710306\pi$
$313$	−21.7137	−1.22733	−0.613665	+	0.789566i	$0.710306\pi$
$314$	20.3634	1.14918
$315$	0	0
$316$	15.1998	0.855056
$317$	28.3634	1.59305	0.796525	−	0.604606i	$-0.206669\pi$
$317$	28.3634	1.59305	0.796525	+	0.604606i	$0.206669\pi$
$318$	0	0
$319$	21.4617	1.20162
$320$	0	0
$321$	0	0
$322$	−8.49929	−0.473647
$323$	18.8872	1.05091
$324$	0	0
$325$	0	0
$326$	−25.3258	−1.40267
$327$	0	0
$328$	1.03761	0.0572925
$329$	−1.92478	−0.106116
$330$	0	0
$331$	12.6801	0.696959	0.348479	−	0.937316i	$-0.386698\pi$
$331$	12.6801	0.696959	0.348479	+	0.937316i	$0.386698\pi$
$332$	1.73813	0.0953925
$333$	0	0
$334$	−6.70052	−0.366636
$335$	0	0
$336$	0	0
$337$	−18.1768	−0.990153	−0.495077	−	0.868849i	$-0.664860\pi$
$337$	−18.1768	−0.990153	−0.495077	+	0.868849i	$0.664860\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	0	0
$341$	−10.7005	−0.579466
$342$	0	0
$343$	−15.0132	−0.810635
$344$	10.8872	0.586997
$345$	0	0
$346$	0.649738	0.0349301
$347$	10.7612	0.577689	0.288845	−	0.957376i	$-0.406729\pi$
$347$	10.7612	0.577689	0.288845	+	0.957376i	$0.406729\pi$
$348$	0	0
$349$	−12.8061	−0.685493	−0.342746	−	0.939428i	$-0.611357\pi$
$349$	−12.8061	−0.685493	−0.342746	+	0.939428i	$0.611357\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	−21.5369	−1.14629	−0.573147	−	0.819453i	$-0.694278\pi$
$353$	−21.5369	−1.14629	−0.573147	+	0.819453i	$0.694278\pi$
$354$	0	0
$355$	0	0
$356$	14.3127	0.758569
$357$	0	0
$358$	6.10554	0.322688
$359$	14.8265	0.782514	0.391257	−	0.920281i	$-0.372040\pi$
$359$	14.8265	0.782514	0.391257	+	0.920281i	$0.372040\pi$
$360$	0	0
$361$	−1.72496	−0.0907874
$362$	22.9380	1.20559
$363$	0	0
$364$	1.19394	0.0625793
$365$	0	0
$366$	0	0
$367$	−37.5125	−1.95813	−0.979067	−	0.203536i	$-0.934757\pi$
$367$	−37.5125	−1.95813	−0.979067	+	0.203536i	$0.934757\pi$
$368$	−7.11871	−0.371089
$369$	0	0
$370$	0	0
$371$	4.99859	0.259514
$372$	0	0
$373$	−18.1866	−0.941669	−0.470834	−	0.882222i	$-0.656047\pi$
$373$	−18.1866	−0.941669	−0.470834	+	0.882222i	$0.656047\pi$
$374$	9.08840	0.469950
$375$	0	0
$376$	−1.61213	−0.0831391
$377$	10.7308	0.552666
$378$	0	0
$379$	−24.6312	−1.26522	−0.632609	−	0.774471i	$-0.718016\pi$
$379$	−24.6312	−1.26522	−0.632609	+	0.774471i	$0.718016\pi$
$380$	0	0
$381$	0	0
$382$	19.0132	0.972799
$383$	1.14903	0.0587127	0.0293564	−	0.999569i	$-0.490654\pi$
$383$	1.14903	0.0587127	0.0293564	+	0.999569i	$0.490654\pi$
$384$	0	0
$385$	0	0
$386$	−4.41819	−0.224880
$387$	0	0
$388$	9.19394	0.466751
$389$	−3.50659	−0.177791	−0.0888955	−	0.996041i	$-0.528334\pi$
$389$	−3.50659	−0.177791	−0.0888955	+	0.996041i	$0.528334\pi$
$390$	0	0
$391$	−32.3488	−1.63595
$392$	−5.57452	−0.281556
$393$	0	0
$394$	−5.22425	−0.263194
$395$	0	0
$396$	0	0
$397$	28.8627	1.44858	0.724289	−	0.689496i	$-0.242168\pi$
$397$	28.8627	1.44858	0.724289	+	0.689496i	$0.242168\pi$
$398$	−19.1998	−0.962400
$399$	0	0
$400$	0	0
$401$	17.0376	0.850818	0.425409	−	0.905001i	$-0.360130\pi$
$401$	17.0376	0.850818	0.425409	+	0.905001i	$0.360130\pi$
$402$	0	0
$403$	−5.35026	−0.266516
$404$	13.8945	0.691275
$405$	0	0
$406$	12.8119	0.635846
$407$	−7.84955	−0.389088
$408$	0	0
$409$	6.77575	0.335039	0.167520	−	0.985869i	$-0.446424\pi$
$409$	6.77575	0.335039	0.167520	+	0.985869i	$0.446424\pi$
$410$	0	0
$411$	0	0
$412$	−7.79877	−0.384218
$413$	−2.76116	−0.135868
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	8.31265	0.406585
$419$	−20.7572	−1.01406	−0.507028	−	0.861930i	$-0.669256\pi$
$419$	−20.7572	−1.01406	−0.507028	+	0.861930i	$0.669256\pi$
$420$	0	0
$421$	−36.9683	−1.80172	−0.900862	−	0.434106i	$-0.857064\pi$
$421$	−36.9683	−1.80172	−0.900862	+	0.434106i	$0.857064\pi$
$422$	9.86414	0.480179
$423$	0	0
$424$	4.18664	0.203321
$425$	0	0
$426$	0	0
$427$	−8.46310	−0.409558
$428$	11.5369	0.557657
$429$	0	0
$430$	0	0
$431$	16.6253	0.800813	0.400406	−	0.916338i	$-0.368869\pi$
$431$	16.6253	0.800813	0.400406	+	0.916338i	$0.368869\pi$
$432$	0	0
$433$	−4.89701	−0.235336	−0.117668	−	0.993053i	$-0.537542\pi$
$433$	−4.89701	−0.235336	−0.117668	+	0.993053i	$0.537542\pi$
$434$	−6.38787	−0.306628
$435$	0	0
$436$	−1.58181	−0.0757549
$437$	−29.5877	−1.41537
$438$	0	0
$439$	4.62530	0.220754	0.110377	−	0.993890i	$-0.464794\pi$
$439$	4.62530	0.220754	0.110377	+	0.993890i	$0.464794\pi$
$440$	0	0
$441$	0	0
$442$	4.54420	0.216145
$443$	33.1900	1.57690	0.788451	−	0.615097i	$-0.210883\pi$
$443$	33.1900	1.57690	0.788451	+	0.615097i	$0.210883\pi$
$444$	0	0
$445$	0	0
$446$	18.9076	0.895302
$447$	0	0
$448$	1.19394	0.0564082
$449$	6.06063	0.286019	0.143010	−	0.989721i	$-0.454322\pi$
$449$	6.06063	0.286019	0.143010	+	0.989721i	$0.454322\pi$
$450$	0	0
$451$	2.07522	0.0977184
$452$	−14.4690	−0.680563
$453$	0	0
$454$	−19.3258	−0.907006
$455$	0	0
$456$	0	0
$457$	−36.6820	−1.71591	−0.857955	−	0.513725i	$-0.828265\pi$
$457$	−36.6820	−1.71591	−0.857955	+	0.513725i	$0.828265\pi$
$458$	11.5672	0.540501
$459$	0	0
$460$	0	0
$461$	−7.28963	−0.339512	−0.169756	−	0.985486i	$-0.554298\pi$
$461$	−7.28963	−0.339512	−0.169756	+	0.985486i	$0.554298\pi$
$462$	0	0
$463$	−30.5950	−1.42187	−0.710935	−	0.703258i	$-0.751728\pi$
$463$	−30.5950	−1.42187	−0.710935	+	0.703258i	$0.751728\pi$
$464$	10.7308	0.498167
$465$	0	0
$466$	−10.7816	−0.499449
$467$	10.1359	0.469032	0.234516	−	0.972112i	$-0.424650\pi$
$467$	10.1359	0.469032	0.234516	+	0.972112i	$0.424650\pi$
$468$	0	0
$469$	−5.61213	−0.259144
$470$	0	0
$471$	0	0
$472$	−2.31265	−0.106448
$473$	21.7743	1.00118
$474$	0	0
$475$	0	0
$476$	5.42548	0.248677
$477$	0	0
$478$	21.9248	1.00282
$479$	10.3272	0.471864	0.235932	−	0.971770i	$-0.424186\pi$
$479$	10.3272	0.471864	0.235932	+	0.971770i	$0.424186\pi$
$480$	0	0
$481$	−3.92478	−0.178954
$482$	−4.23743	−0.193010
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	12.5198	0.567325	0.283662	−	0.958924i	$-0.408451\pi$
$487$	12.5198	0.567325	0.283662	+	0.958924i	$0.408451\pi$
$488$	−7.08840	−0.320877
$489$	0	0
$490$	0	0
$491$	26.1465	1.17997	0.589987	−	0.807413i	$-0.299133\pi$
$491$	26.1465	1.17997	0.589987	+	0.807413i	$0.299133\pi$
$492$	0	0
$493$	48.7631	2.19618
$494$	4.15633	0.187002
$495$	0	0
$496$	−5.35026	−0.240234
$497$	−11.0738	−0.496728
$498$	0	0
$499$	24.9438	1.11664	0.558320	−	0.829626i	$-0.311446\pi$
$499$	24.9438	1.11664	0.558320	+	0.829626i	$0.311446\pi$
$500$	0	0
$501$	0	0
$502$	1.95509	0.0872601
$503$	−10.0303	−0.447230	−0.223615	−	0.974678i	$-0.571786\pi$
$503$	−10.0303	−0.447230	−0.223615	+	0.974678i	$0.571786\pi$
$504$	0	0
$505$	0	0
$506$	−14.2374	−0.632931
$507$	0	0
$508$	2.18664	0.0970166
$509$	16.8119	0.745176	0.372588	−	0.927997i	$-0.378470\pi$
$509$	16.8119	0.745176	0.372588	+	0.927997i	$0.378470\pi$
$510$	0	0
$511$	−4.27645	−0.189179
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.78163	0.299125
$515$	0	0
$516$	0	0
$517$	−3.22425	−0.141803
$518$	−4.68594	−0.205888
$519$	0	0
$520$	0	0
$521$	−14.8510	−0.650633	−0.325316	−	0.945605i	$-0.605471\pi$
$521$	−14.8510	−0.650633	−0.325316	+	0.945605i	$0.605471\pi$
$522$	0	0
$523$	−10.5990	−0.463460	−0.231730	−	0.972780i	$-0.574439\pi$
$523$	−10.5990	−0.463460	−0.231730	+	0.972780i	$0.574439\pi$
$524$	5.64244	0.246491
$525$	0	0
$526$	9.96968	0.434699
$527$	−24.3127	−1.05908
$528$	0	0
$529$	27.6761	1.20331
$530$	0	0
$531$	0	0
$532$	4.96239	0.215147
$533$	1.03761	0.0449439
$534$	0	0
$535$	0	0
$536$	−4.70052	−0.203032
$537$	0	0
$538$	9.64244	0.415715
$539$	−11.1490	−0.480223
$540$	0	0
$541$	−12.1925	−0.524197	−0.262099	−	0.965041i	$-0.584415\pi$
$541$	−12.1925	−0.524197	−0.262099	+	0.965041i	$0.584415\pi$
$542$	21.1392	0.908006
$543$	0	0
$544$	4.54420	0.194831
$545$	0	0
$546$	0	0
$547$	−4.12127	−0.176213	−0.0881064	−	0.996111i	$-0.528082\pi$
$547$	−4.12127	−0.176213	−0.0881064	+	0.996111i	$0.528082\pi$
$548$	−2.46310	−0.105218
$549$	0	0
$550$	0	0
$551$	44.6009	1.90006
$552$	0	0
$553$	18.1476	0.771715
$554$	7.48612	0.318055
$555$	0	0
$556$	−16.6253	−0.705070
$557$	−0.111420	−0.00472100	−0.00236050	−	0.999997i	$-0.500751\pi$
$557$	−0.111420	−0.00472100	−0.00236050	+	0.999997i	$0.500751\pi$
$558$	0	0
$559$	10.8872	0.460478
$560$	0	0
$561$	0	0
$562$	−18.1622	−0.766126
$563$	11.7889	0.496844	0.248422	−	0.968652i	$-0.420088\pi$
$563$	11.7889	0.496844	0.248422	+	0.968652i	$0.420088\pi$
$564$	0	0
$565$	0	0
$566$	−11.8134	−0.496552
$567$	0	0
$568$	−9.27504	−0.389172
$569$	−27.6991	−1.16121	−0.580604	−	0.814186i	$-0.697183\pi$
$569$	−27.6991	−1.16121	−0.580604	+	0.814186i	$0.697183\pi$
$570$	0	0
$571$	27.4471	1.14863	0.574313	−	0.818636i	$-0.305269\pi$
$571$	27.4471	1.14863	0.574313	+	0.818636i	$0.305269\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	1.23884	0.0517083
$575$	0	0
$576$	0	0
$577$	8.98286	0.373961	0.186981	−	0.982364i	$-0.440130\pi$
$577$	8.98286	0.373961	0.186981	+	0.982364i	$0.440130\pi$
$578$	3.64974	0.151809
$579$	0	0
$580$	0	0
$581$	2.07522	0.0860947
$582$	0	0
$583$	8.37328	0.346786
$584$	−3.58181	−0.148216
$585$	0	0
$586$	−12.7005	−0.524654
$587$	−41.6531	−1.71921	−0.859603	−	0.510963i	$-0.829289\pi$
$587$	−41.6531	−1.71921	−0.859603	+	0.510963i	$0.829289\pi$
$588$	0	0
$589$	−22.2374	−0.916277
$590$	0	0
$591$	0	0
$592$	−3.92478	−0.161307
$593$	−15.6121	−0.641113	−0.320557	−	0.947229i	$-0.603870\pi$
$593$	−15.6121	−0.641113	−0.320557	+	0.947229i	$0.603870\pi$
$594$	0	0
$595$	0	0
$596$	22.0508	0.903235
$597$	0	0
$598$	−7.11871	−0.291106
$599$	9.92478	0.405515	0.202758	−	0.979229i	$-0.435010\pi$
$599$	9.92478	0.405515	0.202758	+	0.979229i	$0.435010\pi$
$600$	0	0
$601$	−15.6775	−0.639499	−0.319749	−	0.947502i	$-0.603599\pi$
$601$	−15.6775	−0.639499	−0.319749	+	0.947502i	$0.603599\pi$
$602$	12.9986	0.529783
$603$	0	0
$604$	−2.96239	−0.120538
$605$	0	0
$606$	0	0
$607$	−32.5139	−1.31970	−0.659849	−	0.751398i	$-0.729380\pi$
$607$	−32.5139	−1.31970	−0.659849	+	0.751398i	$0.729380\pi$
$608$	4.15633	0.168561
$609$	0	0
$610$	0	0
$611$	−1.61213	−0.0652197
$612$	0	0
$613$	29.9511	1.20971	0.604857	−	0.796334i	$-0.293230\pi$
$613$	29.9511	1.20971	0.604857	+	0.796334i	$0.293230\pi$
$614$	−6.98683	−0.281965
$615$	0	0
$616$	2.38787	0.0962102
$617$	−15.1490	−0.609877	−0.304939	−	0.952372i	$-0.598636\pi$
$617$	−15.1490	−0.609877	−0.304939	+	0.952372i	$0.598636\pi$
$618$	0	0
$619$	2.79621	0.112389	0.0561947	−	0.998420i	$-0.482103\pi$
$619$	2.79621	0.112389	0.0561947	+	0.998420i	$0.482103\pi$
$620$	0	0
$621$	0	0
$622$	−7.53690	−0.302202
$623$	17.0884	0.684632
$624$	0	0
$625$	0	0
$626$	−21.7137	−0.867854
$627$	0	0
$628$	20.3634	0.812590
$629$	−17.8350	−0.711127
$630$	0	0
$631$	13.8134	0.549901	0.274951	−	0.961458i	$-0.411339\pi$
$631$	13.8134	0.549901	0.274951	+	0.961458i	$0.411339\pi$
$632$	15.1998	0.604616
$633$	0	0
$634$	28.3634	1.12646
$635$	0	0
$636$	0	0
$637$	−5.57452	−0.220870
$638$	21.4617	0.849676
$639$	0	0
$640$	0	0
$641$	−26.8627	−1.06101	−0.530507	−	0.847681i	$-0.677998\pi$
$641$	−26.8627	−1.06101	−0.530507	+	0.847681i	$0.677998\pi$
$642$	0	0
$643$	−25.2243	−0.994747	−0.497374	−	0.867536i	$-0.665702\pi$
$643$	−25.2243	−0.994747	−0.497374	+	0.867536i	$0.665702\pi$
$644$	−8.49929	−0.334919
$645$	0	0
$646$	18.8872	0.743106
$647$	−11.7323	−0.461243	−0.230621	−	0.973044i	$-0.574076\pi$
$647$	−11.7323	−0.461243	−0.230621	+	0.973044i	$0.574076\pi$
$648$	0	0
$649$	−4.62530	−0.181559
$650$	0	0
$651$	0	0
$652$	−25.3258	−0.991836
$653$	−23.7645	−0.929976	−0.464988	−	0.885317i	$-0.653941\pi$
$653$	−23.7645	−0.929976	−0.464988	+	0.885317i	$0.653941\pi$
$654$	0	0
$655$	0	0
$656$	1.03761	0.0405119
$657$	0	0
$658$	−1.92478	−0.0750356
$659$	−33.2809	−1.29644	−0.648220	−	0.761453i	$-0.724486\pi$
$659$	−33.2809	−1.29644	−0.648220	+	0.761453i	$0.724486\pi$
$660$	0	0
$661$	−12.0713	−0.469517	−0.234759	−	0.972054i	$-0.575430\pi$
$661$	−12.0713	−0.469517	−0.234759	+	0.972054i	$0.575430\pi$
$662$	12.6801	0.492824
$663$	0	0
$664$	1.73813	0.0674527
$665$	0	0
$666$	0	0
$667$	−76.3898	−2.95782
$668$	−6.70052	−0.259251
$669$	0	0
$670$	0	0
$671$	−14.1768	−0.547289
$672$	0	0
$673$	−38.8627	−1.49805	−0.749024	−	0.662543i	$-0.769477\pi$
$673$	−38.8627	−1.49805	−0.749024	+	0.662543i	$0.769477\pi$
$674$	−18.1768	−0.700144
$675$	0	0
$676$	1.00000	0.0384615
$677$	45.1852	1.73661	0.868305	−	0.496031i	$-0.165210\pi$
$677$	45.1852	1.73661	0.868305	+	0.496031i	$0.165210\pi$
$678$	0	0
$679$	10.9770	0.421258
$680$	0	0
$681$	0	0
$682$	−10.7005	−0.409744
$683$	−7.93463	−0.303610	−0.151805	−	0.988410i	$-0.548509\pi$
$683$	−7.93463	−0.303610	−0.151805	+	0.988410i	$0.548509\pi$
$684$	0	0
$685$	0	0
$686$	−15.0132	−0.573206
$687$	0	0
$688$	10.8872	0.415069
$689$	4.18664	0.159498
$690$	0	0
$691$	37.4069	1.42303	0.711513	−	0.702673i	$-0.248010\pi$
$691$	37.4069	1.42303	0.711513	+	0.702673i	$0.248010\pi$
$692$	0.649738	0.0246993
$693$	0	0
$694$	10.7612	0.408488
$695$	0	0
$696$	0	0
$697$	4.71511	0.178598
$698$	−12.8061	−0.484717
$699$	0	0
$700$	0	0
$701$	11.0435	0.417107	0.208553	−	0.978011i	$-0.433124\pi$
$701$	11.0435	0.417107	0.208553	+	0.978011i	$0.433124\pi$
$702$	0	0
$703$	−16.3127	−0.615243
$704$	2.00000	0.0753778
$705$	0	0
$706$	−21.5369	−0.810552
$707$	16.5891	0.623897
$708$	0	0
$709$	37.7440	1.41751	0.708753	−	0.705457i	$-0.249258\pi$
$709$	37.7440	1.41751	0.708753	+	0.705457i	$0.249258\pi$
$710$	0	0
$711$	0	0
$712$	14.3127	0.536389
$713$	38.0870	1.42637
$714$	0	0
$715$	0	0
$716$	6.10554	0.228175
$717$	0	0
$718$	14.8265	0.553321
$719$	47.8007	1.78266	0.891332	−	0.453351i	$-0.149771\pi$
$719$	47.8007	1.78266	0.891332	+	0.453351i	$0.149771\pi$
$720$	0	0
$721$	−9.31124	−0.346769
$722$	−1.72496	−0.0641964
$723$	0	0
$724$	22.9380	0.852482
$725$	0	0
$726$	0	0
$727$	−33.7645	−1.25226	−0.626128	−	0.779721i	$-0.715361\pi$
$727$	−33.7645	−1.25226	−0.626128	+	0.779721i	$0.715361\pi$
$728$	1.19394	0.0442502
$729$	0	0
$730$	0	0
$731$	49.4734	1.82984
$732$	0	0
$733$	−41.8905	−1.54726	−0.773630	−	0.633637i	$-0.781561\pi$
$733$	−41.8905	−1.54726	−0.773630	+	0.633637i	$0.781561\pi$
$734$	−37.5125	−1.38461
$735$	0	0
$736$	−7.11871	−0.262399
$737$	−9.40105	−0.346292
$738$	0	0
$739$	6.14174	0.225927	0.112964	−	0.993599i	$-0.463966\pi$
$739$	6.14174	0.225927	0.112964	+	0.993599i	$0.463966\pi$
$740$	0	0
$741$	0	0
$742$	4.99859	0.183504
$743$	−38.4894	−1.41204	−0.706020	−	0.708192i	$-0.749511\pi$
$743$	−38.4894	−1.41204	−0.706020	+	0.708192i	$0.749511\pi$
$744$	0	0
$745$	0	0
$746$	−18.1866	−0.665860
$747$	0	0
$748$	9.08840	0.332305
$749$	13.7743	0.503303
$750$	0	0
$751$	−53.2506	−1.94314	−0.971571	−	0.236748i	$-0.923918\pi$
$751$	−53.2506	−1.94314	−0.971571	+	0.236748i	$0.923918\pi$
$752$	−1.61213	−0.0587882
$753$	0	0
$754$	10.7308	0.390794
$755$	0	0
$756$	0	0
$757$	1.59754	0.0580635	0.0290318	−	0.999578i	$-0.490758\pi$
$757$	1.59754	0.0580635	0.0290318	+	0.999578i	$0.490758\pi$
$758$	−24.6312	−0.894645
$759$	0	0
$760$	0	0
$761$	1.56134	0.0565986	0.0282993	−	0.999599i	$-0.490991\pi$
$761$	1.56134	0.0565986	0.0282993	+	0.999599i	$0.490991\pi$
$762$	0	0
$763$	−1.88858	−0.0683712
$764$	19.0132	0.687872
$765$	0	0
$766$	1.14903	0.0415162
$767$	−2.31265	−0.0835050
$768$	0	0
$769$	−32.9234	−1.18725	−0.593623	−	0.804743i	$-0.702303\pi$
$769$	−32.9234	−1.18725	−0.593623	+	0.804743i	$0.702303\pi$
$770$	0	0
$771$	0	0
$772$	−4.41819	−0.159014
$773$	42.8481	1.54114	0.770570	−	0.637355i	$-0.219972\pi$
$773$	42.8481	1.54114	0.770570	+	0.637355i	$0.219972\pi$
$774$	0	0
$775$	0	0
$776$	9.19394	0.330043
$777$	0	0
$778$	−3.50659	−0.125717
$779$	4.31265	0.154517
$780$	0	0
$781$	−18.5501	−0.663774
$782$	−32.3488	−1.15679
$783$	0	0
$784$	−5.57452	−0.199090
$785$	0	0
$786$	0	0
$787$	−38.6253	−1.37684	−0.688422	−	0.725311i	$-0.741696\pi$
$787$	−38.6253	−1.37684	−0.688422	+	0.725311i	$0.741696\pi$
$788$	−5.22425	−0.186106
$789$	0	0
$790$	0	0
$791$	−17.2750	−0.614230
$792$	0	0
$793$	−7.08840	−0.251716
$794$	28.8627	1.02430
$795$	0	0
$796$	−19.1998	−0.680519
$797$	18.0508	0.639392	0.319696	−	0.947520i	$-0.396419\pi$
$797$	18.0508	0.639392	0.319696	+	0.947520i	$0.396419\pi$
$798$	0	0
$799$	−7.32582	−0.259169
$800$	0	0
$801$	0	0
$802$	17.0376	0.601619
$803$	−7.16362	−0.252799
$804$	0	0
$805$	0	0
$806$	−5.35026	−0.188455
$807$	0	0
$808$	13.8945	0.488805
$809$	−1.76257	−0.0619687	−0.0309844	−	0.999520i	$-0.509864\pi$
$809$	−1.76257	−0.0619687	−0.0309844	+	0.999520i	$0.509864\pi$
$810$	0	0
$811$	−44.0665	−1.54738	−0.773692	−	0.633562i	$-0.781592\pi$
$811$	−44.0665	−1.54738	−0.773692	+	0.633562i	$0.781592\pi$
$812$	12.8119	0.449611
$813$	0	0
$814$	−7.84955	−0.275127
$815$	0	0
$816$	0	0
$817$	45.2506	1.58312
$818$	6.77575	0.236908
$819$	0	0
$820$	0	0
$821$	−1.54675	−0.0539821	−0.0269910	−	0.999636i	$-0.508593\pi$
$821$	−1.54675	−0.0539821	−0.0269910	+	0.999636i	$0.508593\pi$
$822$	0	0
$823$	−20.7250	−0.722427	−0.361213	−	0.932483i	$-0.617637\pi$
$823$	−20.7250	−0.722427	−0.361213	+	0.932483i	$0.617637\pi$
$824$	−7.79877	−0.271683
$825$	0	0
$826$	−2.76116	−0.0960730
$827$	−41.7381	−1.45138	−0.725689	−	0.688023i	$-0.758479\pi$
$827$	−41.7381	−1.45138	−0.725689	+	0.688023i	$0.758479\pi$
$828$	0	0
$829$	−26.7757	−0.929960	−0.464980	−	0.885321i	$-0.653939\pi$
$829$	−26.7757	−0.929960	−0.464980	+	0.885321i	$0.653939\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	−25.3317	−0.877692
$834$	0	0
$835$	0	0
$836$	8.31265	0.287499
$837$	0	0
$838$	−20.7572	−0.717045
$839$	−48.6253	−1.67873	−0.839366	−	0.543567i	$-0.817073\pi$
$839$	−48.6253	−1.67873	−0.839366	+	0.543567i	$0.817073\pi$
$840$	0	0
$841$	86.1509	2.97072
$842$	−36.9683	−1.27401
$843$	0	0
$844$	9.86414	0.339538
$845$	0	0
$846$	0	0
$847$	−8.35756	−0.287169
$848$	4.18664	0.143770
$849$	0	0
$850$	0	0
$851$	27.9394	0.957749
$852$	0	0
$853$	4.53832	0.155389	0.0776945	−	0.996977i	$-0.475244\pi$
$853$	4.53832	0.155389	0.0776945	+	0.996977i	$0.475244\pi$
$854$	−8.46310	−0.289601
$855$	0	0
$856$	11.5369	0.394323
$857$	28.2315	0.964371	0.482186	−	0.876069i	$-0.339843\pi$
$857$	28.2315	0.964371	0.482186	+	0.876069i	$0.339843\pi$
$858$	0	0
$859$	−18.1162	−0.618115	−0.309058	−	0.951043i	$-0.600014\pi$
$859$	−18.1162	−0.618115	−0.309058	+	0.951043i	$0.600014\pi$
$860$	0	0
$861$	0	0
$862$	16.6253	0.566260
$863$	−30.0263	−1.02211	−0.511054	−	0.859548i	$-0.670745\pi$
$863$	−30.0263	−1.02211	−0.511054	+	0.859548i	$0.670745\pi$
$864$	0	0
$865$	0	0
$866$	−4.89701	−0.166407
$867$	0	0
$868$	−6.38787	−0.216819
$869$	30.3996	1.03124
$870$	0	0
$871$	−4.70052	−0.159271
$872$	−1.58181	−0.0535668
$873$	0	0
$874$	−29.5877	−1.00082
$875$	0	0
$876$	0	0
$877$	−22.5355	−0.760969	−0.380485	−	0.924787i	$-0.624243\pi$
$877$	−22.5355	−0.760969	−0.380485	+	0.924787i	$0.624243\pi$
$878$	4.62530	0.156096
$879$	0	0
$880$	0	0
$881$	−8.21108	−0.276638	−0.138319	−	0.990388i	$-0.544170\pi$
$881$	−8.21108	−0.276638	−0.138319	+	0.990388i	$0.544170\pi$
$882$	0	0
$883$	−6.44851	−0.217010	−0.108505	−	0.994096i	$-0.534606\pi$
$883$	−6.44851	−0.217010	−0.108505	+	0.994096i	$0.534606\pi$
$884$	4.54420	0.152838
$885$	0	0
$886$	33.1900	1.11504
$887$	34.6067	1.16198	0.580990	−	0.813910i	$-0.302665\pi$
$887$	34.6067	1.16198	0.580990	+	0.813910i	$0.302665\pi$
$888$	0	0
$889$	2.61071	0.0875605
$890$	0	0
$891$	0	0
$892$	18.9076	0.633074
$893$	−6.70052	−0.224224
$894$	0	0
$895$	0	0
$896$	1.19394	0.0398866
$897$	0	0
$898$	6.06063	0.202246
$899$	−57.4128	−1.91482
$900$	0	0
$901$	19.0249	0.633812
$902$	2.07522	0.0690974
$903$	0	0
$904$	−14.4690	−0.481231
$905$	0	0
$906$	0	0
$907$	−20.3371	−0.675282	−0.337641	−	0.941275i	$-0.609629\pi$
$907$	−20.3371	−0.675282	−0.337641	+	0.941275i	$0.609629\pi$
$908$	−19.3258	−0.641350
$909$	0	0
$910$	0	0
$911$	−38.9234	−1.28959	−0.644794	−	0.764356i	$-0.723057\pi$
$911$	−38.9234	−1.28959	−0.644794	+	0.764356i	$0.723057\pi$
$912$	0	0
$913$	3.47627	0.115048
$914$	−36.6820	−1.21333
$915$	0	0
$916$	11.5672	0.382192
$917$	6.73672	0.222466
$918$	0	0
$919$	7.22425	0.238306	0.119153	−	0.992876i	$-0.461982\pi$
$919$	7.22425	0.238306	0.119153	+	0.992876i	$0.461982\pi$
$920$	0	0
$921$	0	0
$922$	−7.28963	−0.240071
$923$	−9.27504	−0.305292
$924$	0	0
$925$	0	0
$926$	−30.5950	−1.00541
$927$	0	0
$928$	10.7308	0.352257
$929$	24.0870	0.790268	0.395134	−	0.918623i	$-0.370698\pi$
$929$	24.0870	0.790268	0.395134	+	0.918623i	$0.370698\pi$
$930$	0	0
$931$	−23.1695	−0.759350
$932$	−10.7816	−0.353164
$933$	0	0
$934$	10.1359	0.331655
$935$	0	0
$936$	0	0
$937$	46.0870	1.50560	0.752798	−	0.658252i	$-0.228704\pi$
$937$	46.0870	1.50560	0.752798	+	0.658252i	$0.228704\pi$
$938$	−5.61213	−0.183242
$939$	0	0
$940$	0	0
$941$	−10.9887	−0.358223	−0.179111	−	0.983829i	$-0.557322\pi$
$941$	−10.9887	−0.358223	−0.179111	+	0.983829i	$0.557322\pi$
$942$	0	0
$943$	−7.38646	−0.240536
$944$	−2.31265	−0.0752704
$945$	0	0
$946$	21.7743	0.707945
$947$	−32.1378	−1.04434	−0.522169	−	0.852842i	$-0.674877\pi$
$947$	−32.1378	−1.04434	−0.522169	+	0.852842i	$0.674877\pi$
$948$	0	0
$949$	−3.58181	−0.116270
$950$	0	0
$951$	0	0
$952$	5.42548	0.175841
$953$	3.61801	0.117199	0.0585994	−	0.998282i	$-0.481337\pi$
$953$	3.61801	0.117199	0.0585994	+	0.998282i	$0.481337\pi$
$954$	0	0
$955$	0	0
$956$	21.9248	0.709098
$957$	0	0
$958$	10.3272	0.333658
$959$	−2.94078	−0.0949627
$960$	0	0
$961$	−2.37470	−0.0766032
$962$	−3.92478	−0.126540
$963$	0	0
$964$	−4.23743	−0.136478
$965$	0	0
$966$	0	0
$967$	26.9194	0.865669	0.432835	−	0.901473i	$-0.357513\pi$
$967$	26.9194	0.865669	0.432835	+	0.901473i	$0.357513\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	−47.8192	−1.53459	−0.767296	−	0.641293i	$-0.778398\pi$
$971$	−47.8192	−1.53459	−0.767296	+	0.641293i	$0.778398\pi$
$972$	0	0
$973$	−19.8496	−0.636348
$974$	12.5198	0.401159
$975$	0	0
$976$	−7.08840	−0.226894
$977$	40.1886	1.28575	0.642873	−	0.765973i	$-0.277742\pi$
$977$	40.1886	1.28575	0.642873	+	0.765973i	$0.277742\pi$
$978$	0	0
$979$	28.6253	0.914869
$980$	0	0
$981$	0	0
$982$	26.1465	0.834368
$983$	6.23743	0.198943	0.0994715	−	0.995040i	$-0.468285\pi$
$983$	6.23743	0.198943	0.0994715	+	0.995040i	$0.468285\pi$
$984$	0	0
$985$	0	0
$986$	48.7631	1.55293
$987$	0	0
$988$	4.15633	0.132230
$989$	−77.5026	−2.46444
$990$	0	0
$991$	40.7221	1.29358	0.646791	−	0.762668i	$-0.276111\pi$
$991$	40.7221	1.29358	0.646791	+	0.762668i	$0.276111\pi$
$992$	−5.35026	−0.169871
$993$	0	0
$994$	−11.0738	−0.351240
$995$	0	0
$996$	0	0
$997$	30.1866	0.956021	0.478010	−	0.878354i	$-0.341358\pi$
$997$	30.1866	0.956021	0.478010	+	0.878354i	$0.341358\pi$
$998$	24.9438	0.789583
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.a.ct.1.2		3
3.2	odd	2		5850.2.a.cq.1.2		3
5.2	odd	4		1170.2.e.h.469.4	yes	6
5.3	odd	4		1170.2.e.h.469.1	yes	6
5.4	even	2		5850.2.a.co.1.2		3
15.2	even	4		1170.2.e.g.469.3	✓	6
15.8	even	4		1170.2.e.g.469.6	yes	6
15.14	odd	2		5850.2.a.cr.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.e.g.469.3	✓	6	15.2	even	4
1170.2.e.g.469.6	yes	6	15.8	even	4
1170.2.e.h.469.1	yes	6	5.3	odd	4
1170.2.e.h.469.4	yes	6	5.2	odd	4
5850.2.a.co.1.2		3	5.4	even	2
5850.2.a.cq.1.2		3	3.2	odd	2
5850.2.a.cr.1.2		3	15.14	odd	2
5850.2.a.ct.1.2		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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.19394 q^{7} +1.00000 q^{8} +2.00000 q^{11} +1.00000 q^{13} +1.19394 q^{14} +1.00000 q^{16} +4.54420 q^{17} +4.15633 q^{19} +2.00000 q^{22} -7.11871 q^{23} +1.00000 q^{26} +1.19394 q^{28} +10.7308 q^{29} -5.35026 q^{31} +1.00000 q^{32} +4.54420 q^{34} -3.92478 q^{37} +4.15633 q^{38} +1.03761 q^{41} +10.8872 q^{43} +2.00000 q^{44} -7.11871 q^{46} -1.61213 q^{47} -5.57452 q^{49} +1.00000 q^{52} +4.18664 q^{53} +1.19394 q^{56} +10.7308 q^{58} -2.31265 q^{59} -7.08840 q^{61} -5.35026 q^{62} +1.00000 q^{64} -4.70052 q^{67} +4.54420 q^{68} -9.27504 q^{71} -3.58181 q^{73} -3.92478 q^{74} +4.15633 q^{76} +2.38787 q^{77} +15.1998 q^{79} +1.03761 q^{82} +1.73813 q^{83} +10.8872 q^{86} +2.00000 q^{88} +14.3127 q^{89} +1.19394 q^{91} -7.11871 q^{92} -1.61213 q^{94} +9.19394 q^{97} -5.57452 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 4 q^{7} + 3 q^{8} + 6 q^{11} + 3 q^{13} + 4 q^{14} + 3 q^{16} + 4 q^{17} + 2 q^{19} + 6 q^{22} + 3 q^{26} + 4 q^{28} + 10 q^{29} - 6 q^{31} + 3 q^{32} + 4 q^{34} + 10 q^{37} + 2 q^{38}+ \cdots - 5 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 4 * q^7 + 3 * q^8 + 6 * q^11 + 3 * q^13 + 4 * q^14 + 3 * q^16 + 4 * q^17 + 2 * q^19 + 6 * q^22 + 3 * q^26 + 4 * q^28 + 10 * q^29 - 6 * q^31 + 3 * q^32 + 4 * q^34 + 10 * q^37 + 2 * q^38 + 14 * q^41 + 6 * q^44 - 4 * q^47 - 5 * q^49 + 3 * q^52 + 4 * q^56 + 10 * q^58 + 14 * q^59 - 2 * q^61 - 6 * q^62 + 3 * q^64 + 6 * q^67 + 4 * q^68 + 4 * q^71 - 12 * q^73 + 10 * q^74 + 2 * q^76 + 8 * q^77 - 8 * q^79 + 14 * q^82 - 4 * q^83 + 6 * q^88 + 22 * q^89 + 4 * q^91 - 4 * q^94 + 28 * q^97 - 5 * q^98

Introduction

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