LMFDB - Embedded newform 6930.2.a.bw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} -1.00000 q^{20} -1.00000 q^{22} -5.46410 q^{23} +1.00000 q^{25} +3.46410 q^{26} -1.00000 q^{28} +4.92820 q^{29} -2.92820 q^{31} +1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} -3.46410 q^{37} -1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -5.46410 q^{46} +6.92820 q^{47} +1.00000 q^{49} +1.00000 q^{50} +3.46410 q^{52} -6.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} +4.92820 q^{58} -8.00000 q^{59} +8.92820 q^{61} -2.92820 q^{62} +1.00000 q^{64} -3.46410 q^{65} +1.46410 q^{67} -3.46410 q^{68} +1.00000 q^{70} -14.9282 q^{71} -11.8564 q^{73} -3.46410 q^{74} +1.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} +13.8564 q^{83} +3.46410 q^{85} -1.00000 q^{88} -15.4641 q^{89} -3.46410 q^{91} -5.46410 q^{92} +6.92820 q^{94} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{35} - 2 q^{40} - 4 q^{41} - 2 q^{44} - 4 q^{46} + 2 q^{49} + 2 q^{50} - 12 q^{53} + 2 q^{55} - 2 q^{56} - 4 q^{58} - 16 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{70} - 16 q^{71} + 4 q^{73} + 2 q^{77} - 8 q^{79} - 2 q^{80} - 4 q^{82} - 2 q^{88} - 24 q^{89} - 4 q^{92} - 20 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^14 + 2 * q^16 - 2 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^25 - 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^35 - 2 * q^40 - 4 * q^41 - 2 * q^44 - 4 * q^46 + 2 * q^49 + 2 * q^50 - 12 * q^53 + 2 * q^55 - 2 * q^56 - 4 * q^58 - 16 * q^59 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^67 + 2 * q^70 - 16 * q^71 + 4 * q^73 + 2 * q^77 - 8 * q^79 - 2 * q^80 - 4 * q^82 - 2 * q^88 - 24 * q^89 - 4 * q^92 - 20 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−5.46410	−1.13934	−0.569672	−	0.821872i	$-0.692930\pi$
$23$	−5.46410	−1.13934	−0.569672	+	0.821872i	$0.692930\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.46410	0.679366
$27$	0	0
$28$	−1.00000	−0.188982
$29$	4.92820	0.915144	0.457572	−	0.889172i	$-0.348719\pi$
$29$	4.92820	0.915144	0.457572	+	0.889172i	$0.348719\pi$
$30$	0	0
$31$	−2.92820	−0.525921	−0.262960	−	0.964807i	$-0.584699\pi$
$31$	−2.92820	−0.525921	−0.262960	+	0.964807i	$0.584699\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	1.00000	0.169031
$36$	0	0
$37$	−3.46410	−0.569495	−0.284747	−	0.958603i	$-0.591910\pi$
$37$	−3.46410	−0.569495	−0.284747	+	0.958603i	$0.591910\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−5.46410	−0.805638
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	3.46410	0.480384
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	4.92820	0.647105
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	−2.92820	−0.371882
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.46410	−0.429669
$66$	0	0
$67$	1.46410	0.178868	0.0894342	−	0.995993i	$-0.471494\pi$
$67$	1.46410	0.178868	0.0894342	+	0.995993i	$0.471494\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	1.00000	0.119523
$71$	−14.9282	−1.77165	−0.885826	−	0.464018i	$-0.846407\pi$
$71$	−14.9282	−1.77165	−0.885826	+	0.464018i	$0.846407\pi$
$72$	0	0
$73$	−11.8564	−1.38769	−0.693844	−	0.720126i	$-0.744084\pi$
$73$	−11.8564	−1.38769	−0.693844	+	0.720126i	$0.744084\pi$
$74$	−3.46410	−0.402694
$75$	0	0
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−2.00000	−0.220863
$83$	13.8564	1.52094	0.760469	−	0.649374i	$-0.224969\pi$
$83$	13.8564	1.52094	0.760469	+	0.649374i	$0.224969\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−15.4641	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$89$	−15.4641	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	−5.46410	−0.569672
$93$	0	0
$94$	6.92820	0.714590
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−19.8564	−1.97579	−0.987893	−	0.155136i	$-0.950418\pi$
$101$	−19.8564	−1.97579	−0.987893	+	0.155136i	$0.950418\pi$
$102$	0	0
$103$	10.9282	1.07679	0.538394	−	0.842693i	$-0.319031\pi$
$103$	10.9282	1.07679	0.538394	+	0.842693i	$0.319031\pi$
$104$	3.46410	0.339683
$105$	0	0
$106$	−6.00000	−0.582772
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	3.46410	0.331801	0.165900	−	0.986143i	$-0.446947\pi$
$109$	3.46410	0.331801	0.165900	+	0.986143i	$0.446947\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−18.3923	−1.73020	−0.865101	−	0.501597i	$-0.832746\pi$
$113$	−18.3923	−1.73020	−0.865101	+	0.501597i	$0.832746\pi$
$114$	0	0
$115$	5.46410	0.509530
$116$	4.92820	0.457572
$117$	0	0
$118$	−8.00000	−0.736460
$119$	3.46410	0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.92820	0.808322
$123$	0	0
$124$	−2.92820	−0.262960
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.92820	0.259836	0.129918	−	0.991525i	$-0.458529\pi$
$127$	2.92820	0.259836	0.129918	+	0.991525i	$0.458529\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.46410	−0.303822
$131$	9.85641	0.861158	0.430579	−	0.902553i	$-0.358309\pi$
$131$	9.85641	0.861158	0.430579	+	0.902553i	$0.358309\pi$
$132$	0	0
$133$	0	0
$134$	1.46410	0.126479
$135$	0	0
$136$	−3.46410	−0.297044
$137$	3.46410	0.295958	0.147979	−	0.988990i	$-0.452723\pi$
$137$	3.46410	0.295958	0.147979	+	0.988990i	$0.452723\pi$
$138$	0	0
$139$	2.92820	0.248367	0.124183	−	0.992259i	$-0.460369\pi$
$139$	2.92820	0.248367	0.124183	+	0.992259i	$0.460369\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−14.9282	−1.25275
$143$	−3.46410	−0.289683
$144$	0	0
$145$	−4.92820	−0.409265
$146$	−11.8564	−0.981243
$147$	0	0
$148$	−3.46410	−0.284747
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	0	0
$153$	0	0
$154$	1.00000	0.0805823
$155$	2.92820	0.235199
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	5.46410	0.430632
$162$	0	0
$163$	15.3205	1.19999	0.599997	−	0.800002i	$-0.295168\pi$
$163$	15.3205	1.19999	0.599997	+	0.800002i	$0.295168\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	13.8564	1.07547
$167$	10.5359	0.815292	0.407646	−	0.913140i	$-0.366350\pi$
$167$	10.5359	0.815292	0.407646	+	0.913140i	$0.366350\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	3.46410	0.265684
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−15.4641	−1.15908
$179$	−6.92820	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$179$	−6.92820	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$180$	0	0
$181$	−3.46410	−0.257485	−0.128742	−	0.991678i	$-0.541094\pi$
$181$	−3.46410	−0.257485	−0.128742	+	0.991678i	$0.541094\pi$
$182$	−3.46410	−0.256776
$183$	0	0
$184$	−5.46410	−0.402819
$185$	3.46410	0.254686
$186$	0	0
$187$	3.46410	0.253320
$188$	6.92820	0.505291
$189$	0	0
$190$	0	0
$191$	−17.8564	−1.29204	−0.646022	−	0.763319i	$-0.723569\pi$
$191$	−17.8564	−1.29204	−0.646022	+	0.763319i	$0.723569\pi$
$192$	0	0
$193$	−11.8564	−0.853443	−0.426721	−	0.904383i	$-0.640332\pi$
$193$	−11.8564	−0.853443	−0.426721	+	0.904383i	$0.640332\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	5.07180	0.359530	0.179765	−	0.983710i	$-0.442466\pi$
$199$	5.07180	0.359530	0.179765	+	0.983710i	$0.442466\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−19.8564	−1.39709
$203$	−4.92820	−0.345892
$204$	0	0
$205$	2.00000	0.139686
$206$	10.9282	0.761404
$207$	0	0
$208$	3.46410	0.240192
$209$	0	0
$210$	0	0
$211$	−16.3923	−1.12849	−0.564246	−	0.825606i	$-0.690833\pi$
$211$	−16.3923	−1.12849	−0.564246	+	0.825606i	$0.690833\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	6.92820	0.473602
$215$	0	0
$216$	0	0
$217$	2.92820	0.198779
$218$	3.46410	0.234619
$219$	0	0
$220$	1.00000	0.0674200
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−27.7128	−1.85579	−0.927894	−	0.372845i	$-0.878382\pi$
$223$	−27.7128	−1.85579	−0.927894	+	0.372845i	$0.878382\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−18.3923	−1.22344
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	23.4641	1.55055	0.775276	−	0.631623i	$-0.217611\pi$
$229$	23.4641	1.55055	0.775276	+	0.631623i	$0.217611\pi$
$230$	5.46410	0.360292
$231$	0	0
$232$	4.92820	0.323552
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	0	0
$235$	−6.92820	−0.451946
$236$	−8.00000	−0.520756
$237$	0	0
$238$	3.46410	0.224544
$239$	−14.5359	−0.940249	−0.470125	−	0.882600i	$-0.655791\pi$
$239$	−14.5359	−0.940249	−0.470125	+	0.882600i	$0.655791\pi$
$240$	0	0
$241$	22.7846	1.46769	0.733843	−	0.679319i	$-0.237725\pi$
$241$	22.7846	1.46769	0.733843	+	0.679319i	$0.237725\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	8.92820	0.571570
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	−2.92820	−0.185941
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	5.46410	0.343525
$254$	2.92820	0.183732
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.9282	−1.05595	−0.527976	−	0.849259i	$-0.677049\pi$
$257$	−16.9282	−1.05595	−0.527976	+	0.849259i	$0.677049\pi$
$258$	0	0
$259$	3.46410	0.215249
$260$	−3.46410	−0.214834
$261$	0	0
$262$	9.85641	0.608931
$263$	−18.9282	−1.16716	−0.583582	−	0.812055i	$-0.698349\pi$
$263$	−18.9282	−1.16716	−0.583582	+	0.812055i	$0.698349\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	1.46410	0.0894342
$269$	−3.07180	−0.187291	−0.0936454	−	0.995606i	$-0.529852\pi$
$269$	−3.07180	−0.187291	−0.0936454	+	0.995606i	$0.529852\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−3.46410	−0.210042
$273$	0	0
$274$	3.46410	0.209274
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	2.92820	0.175622
$279$	0	0
$280$	1.00000	0.0597614
$281$	13.3205	0.794635	0.397317	−	0.917681i	$-0.369941\pi$
$281$	13.3205	0.794635	0.397317	+	0.917681i	$0.369941\pi$
$282$	0	0
$283$	−3.60770	−0.214455	−0.107228	−	0.994234i	$-0.534197\pi$
$283$	−3.60770	−0.214455	−0.107228	+	0.994234i	$0.534197\pi$
$284$	−14.9282	−0.885826
$285$	0	0
$286$	−3.46410	−0.204837
$287$	2.00000	0.118056
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−4.92820	−0.289394
$291$	0	0
$292$	−11.8564	−0.693844
$293$	−22.7846	−1.33109	−0.665546	−	0.746357i	$-0.731801\pi$
$293$	−22.7846	−1.33109	−0.665546	+	0.746357i	$0.731801\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−3.46410	−0.201347
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−18.9282	−1.09465
$300$	0	0
$301$	0	0
$302$	−20.0000	−1.15087
$303$	0	0
$304$	0	0
$305$	−8.92820	−0.511227
$306$	0	0
$307$	−3.60770	−0.205902	−0.102951	−	0.994686i	$-0.532828\pi$
$307$	−3.60770	−0.205902	−0.102951	+	0.994686i	$0.532828\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	2.92820	0.166311
$311$	14.5359	0.824255	0.412128	−	0.911126i	$-0.364786\pi$
$311$	14.5359	0.824255	0.412128	+	0.911126i	$0.364786\pi$
$312$	0	0
$313$	0.143594	0.00811639	0.00405819	−	0.999992i	$-0.498708\pi$
$313$	0.143594	0.00811639	0.00405819	+	0.999992i	$0.498708\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−4.00000	−0.225018
$317$	15.8564	0.890585	0.445292	−	0.895385i	$-0.353100\pi$
$317$	15.8564	0.890585	0.445292	+	0.895385i	$0.353100\pi$
$318$	0	0
$319$	−4.92820	−0.275926
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	5.46410	0.304502
$323$	0	0
$324$	0	0
$325$	3.46410	0.192154
$326$	15.3205	0.848524
$327$	0	0
$328$	−2.00000	−0.110432
$329$	−6.92820	−0.381964
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	13.8564	0.760469
$333$	0	0
$334$	10.5359	0.576499
$335$	−1.46410	−0.0799924
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	3.46410	0.187867
$341$	2.92820	0.158571
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	−6.00000	−0.322562
$347$	1.07180	0.0575371	0.0287685	−	0.999586i	$-0.490841\pi$
$347$	1.07180	0.0575371	0.0287685	+	0.999586i	$0.490841\pi$
$348$	0	0
$349$	−20.9282	−1.12026	−0.560131	−	0.828404i	$-0.689249\pi$
$349$	−20.9282	−1.12026	−0.560131	+	0.828404i	$0.689249\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	14.9282	0.792307
$356$	−15.4641	−0.819596
$357$	0	0
$358$	−6.92820	−0.366167
$359$	−1.46410	−0.0772723	−0.0386362	−	0.999253i	$-0.512301\pi$
$359$	−1.46410	−0.0772723	−0.0386362	+	0.999253i	$0.512301\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−3.46410	−0.182069
$363$	0	0
$364$	−3.46410	−0.181568
$365$	11.8564	0.620593
$366$	0	0
$367$	−24.7846	−1.29375	−0.646873	−	0.762598i	$-0.723924\pi$
$367$	−24.7846	−1.29375	−0.646873	+	0.762598i	$0.723924\pi$
$368$	−5.46410	−0.284836
$369$	0	0
$370$	3.46410	0.180090
$371$	6.00000	0.311504
$372$	0	0
$373$	−6.78461	−0.351294	−0.175647	−	0.984453i	$-0.556202\pi$
$373$	−6.78461	−0.351294	−0.175647	+	0.984453i	$0.556202\pi$
$374$	3.46410	0.179124
$375$	0	0
$376$	6.92820	0.357295
$377$	17.0718	0.879242
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	−17.8564	−0.913613
$383$	14.9282	0.762796	0.381398	−	0.924411i	$-0.375443\pi$
$383$	14.9282	0.762796	0.381398	+	0.924411i	$0.375443\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−11.8564	−0.603475
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−15.0718	−0.764170	−0.382085	−	0.924127i	$-0.624794\pi$
$389$	−15.0718	−0.764170	−0.382085	+	0.924127i	$0.624794\pi$
$390$	0	0
$391$	18.9282	0.957240
$392$	1.00000	0.0505076
$393$	0	0
$394$	−2.00000	−0.100759
$395$	4.00000	0.201262
$396$	0	0
$397$	−6.78461	−0.340510	−0.170255	−	0.985400i	$-0.554459\pi$
$397$	−6.78461	−0.340510	−0.170255	+	0.985400i	$0.554459\pi$
$398$	5.07180	0.254226
$399$	0	0
$400$	1.00000	0.0500000
$401$	−7.85641	−0.392330	−0.196165	−	0.980571i	$-0.562849\pi$
$401$	−7.85641	−0.392330	−0.196165	+	0.980571i	$0.562849\pi$
$402$	0	0
$403$	−10.1436	−0.505288
$404$	−19.8564	−0.987893
$405$	0	0
$406$	−4.92820	−0.244583
$407$	3.46410	0.171709
$408$	0	0
$409$	−20.9282	−1.03483	−0.517417	−	0.855734i	$-0.673106\pi$
$409$	−20.9282	−1.03483	−0.517417	+	0.855734i	$0.673106\pi$
$410$	2.00000	0.0987730
$411$	0	0
$412$	10.9282	0.538394
$413$	8.00000	0.393654
$414$	0	0
$415$	−13.8564	−0.680184
$416$	3.46410	0.169842
$417$	0	0
$418$	0	0
$419$	−10.9282	−0.533878	−0.266939	−	0.963713i	$-0.586012\pi$
$419$	−10.9282	−0.533878	−0.266939	+	0.963713i	$0.586012\pi$
$420$	0	0
$421$	−4.14359	−0.201946	−0.100973	−	0.994889i	$-0.532196\pi$
$421$	−4.14359	−0.201946	−0.100973	+	0.994889i	$0.532196\pi$
$422$	−16.3923	−0.797965
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−3.46410	−0.168034
$426$	0	0
$427$	−8.92820	−0.432066
$428$	6.92820	0.334887
$429$	0	0
$430$	0	0
$431$	11.6077	0.559123	0.279562	−	0.960128i	$-0.409811\pi$
$431$	11.6077	0.559123	0.279562	+	0.960128i	$0.409811\pi$
$432$	0	0
$433$	3.85641	0.185327	0.0926635	−	0.995697i	$-0.470462\pi$
$433$	3.85641	0.185327	0.0926635	+	0.995697i	$0.470462\pi$
$434$	2.92820	0.140558
$435$	0	0
$436$	3.46410	0.165900
$437$	0	0
$438$	0	0
$439$	21.0718	1.00570	0.502851	−	0.864373i	$-0.332284\pi$
$439$	21.0718	1.00570	0.502851	+	0.864373i	$0.332284\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−12.0000	−0.570782
$443$	33.8564	1.60857	0.804283	−	0.594246i	$-0.202550\pi$
$443$	33.8564	1.60857	0.804283	+	0.594246i	$0.202550\pi$
$444$	0	0
$445$	15.4641	0.733069
$446$	−27.7128	−1.31224
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−15.8564	−0.748310	−0.374155	−	0.927366i	$-0.622067\pi$
$449$	−15.8564	−0.748310	−0.374155	+	0.927366i	$0.622067\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	−18.3923	−0.865101
$453$	0	0
$454$	8.00000	0.375459
$455$	3.46410	0.162400
$456$	0	0
$457$	15.8564	0.741731	0.370866	−	0.928687i	$-0.379061\pi$
$457$	15.8564	0.741731	0.370866	+	0.928687i	$0.379061\pi$
$458$	23.4641	1.09641
$459$	0	0
$460$	5.46410	0.254765
$461$	29.7128	1.38386	0.691932	−	0.721963i	$-0.256760\pi$
$461$	29.7128	1.38386	0.691932	+	0.721963i	$0.256760\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	4.92820	0.228786
$465$	0	0
$466$	7.85641	0.363941
$467$	−20.7846	−0.961797	−0.480899	−	0.876776i	$-0.659689\pi$
$467$	−20.7846	−0.961797	−0.480899	+	0.876776i	$0.659689\pi$
$468$	0	0
$469$	−1.46410	−0.0676059
$470$	−6.92820	−0.319574
$471$	0	0
$472$	−8.00000	−0.368230
$473$	0	0
$474$	0	0
$475$	0	0
$476$	3.46410	0.158777
$477$	0	0
$478$	−14.5359	−0.664857
$479$	5.85641	0.267586	0.133793	−	0.991009i	$-0.457284\pi$
$479$	5.85641	0.267586	0.133793	+	0.991009i	$0.457284\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	22.7846	1.03781
$483$	0	0
$484$	1.00000	0.0454545
$485$	10.0000	0.454077
$486$	0	0
$487$	−14.1436	−0.640907	−0.320454	−	0.947264i	$-0.603835\pi$
$487$	−14.1436	−0.640907	−0.320454	+	0.947264i	$0.603835\pi$
$488$	8.92820	0.404161
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	25.8564	1.16688	0.583442	−	0.812155i	$-0.301706\pi$
$491$	25.8564	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$492$	0	0
$493$	−17.0718	−0.768875
$494$	0	0
$495$	0	0
$496$	−2.92820	−0.131480
$497$	14.9282	0.669621
$498$	0	0
$499$	28.7846	1.28858	0.644288	−	0.764783i	$-0.277154\pi$
$499$	28.7846	1.28858	0.644288	+	0.764783i	$0.277154\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	8.00000	0.357057
$503$	40.3923	1.80100	0.900502	−	0.434852i	$-0.143199\pi$
$503$	40.3923	1.80100	0.900502	+	0.434852i	$0.143199\pi$
$504$	0	0
$505$	19.8564	0.883598
$506$	5.46410	0.242909
$507$	0	0
$508$	2.92820	0.129918
$509$	12.9282	0.573033	0.286516	−	0.958075i	$-0.407503\pi$
$509$	12.9282	0.573033	0.286516	+	0.958075i	$0.407503\pi$
$510$	0	0
$511$	11.8564	0.524497
$512$	1.00000	0.0441942
$513$	0	0
$514$	−16.9282	−0.746671
$515$	−10.9282	−0.481554
$516$	0	0
$517$	−6.92820	−0.304702
$518$	3.46410	0.152204
$519$	0	0
$520$	−3.46410	−0.151911
$521$	42.1051	1.84466	0.922329	−	0.386405i	$-0.126283\pi$
$521$	42.1051	1.84466	0.922329	+	0.386405i	$0.126283\pi$
$522$	0	0
$523$	15.3205	0.669919	0.334960	−	0.942233i	$-0.391277\pi$
$523$	15.3205	0.669919	0.334960	+	0.942233i	$0.391277\pi$
$524$	9.85641	0.430579
$525$	0	0
$526$	−18.9282	−0.825309
$527$	10.1436	0.441862
$528$	0	0
$529$	6.85641	0.298105
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	−6.92820	−0.300094
$534$	0	0
$535$	−6.92820	−0.299532
$536$	1.46410	0.0632396
$537$	0	0
$538$	−3.07180	−0.132435
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−7.46410	−0.320907	−0.160453	−	0.987043i	$-0.551296\pi$
$541$	−7.46410	−0.320907	−0.160453	+	0.987043i	$0.551296\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−3.46410	−0.148522
$545$	−3.46410	−0.148386
$546$	0	0
$547$	−30.6410	−1.31012	−0.655058	−	0.755579i	$-0.727356\pi$
$547$	−30.6410	−1.31012	−0.655058	+	0.755579i	$0.727356\pi$
$548$	3.46410	0.147979
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	0	0
$553$	4.00000	0.170097
$554$	−14.0000	−0.594803
$555$	0	0
$556$	2.92820	0.124183
$557$	43.8564	1.85826	0.929128	−	0.369759i	$-0.120560\pi$
$557$	43.8564	1.85826	0.929128	+	0.369759i	$0.120560\pi$
$558$	0	0
$559$	0	0
$560$	1.00000	0.0422577
$561$	0	0
$562$	13.3205	0.561892
$563$	34.9282	1.47205	0.736024	−	0.676955i	$-0.236701\pi$
$563$	34.9282	1.47205	0.736024	+	0.676955i	$0.236701\pi$
$564$	0	0
$565$	18.3923	0.773770
$566$	−3.60770	−0.151643
$567$	0	0
$568$	−14.9282	−0.626373
$569$	−41.3205	−1.73225	−0.866123	−	0.499831i	$-0.833395\pi$
$569$	−41.3205	−1.73225	−0.866123	+	0.499831i	$0.833395\pi$
$570$	0	0
$571$	−41.1769	−1.72320	−0.861600	−	0.507588i	$-0.830537\pi$
$571$	−41.1769	−1.72320	−0.861600	+	0.507588i	$0.830537\pi$
$572$	−3.46410	−0.144841
$573$	0	0
$574$	2.00000	0.0834784
$575$	−5.46410	−0.227869
$576$	0	0
$577$	11.8564	0.493589	0.246794	−	0.969068i	$-0.420623\pi$
$577$	11.8564	0.493589	0.246794	+	0.969068i	$0.420623\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	−4.92820	−0.204633
$581$	−13.8564	−0.574861
$582$	0	0
$583$	6.00000	0.248495
$584$	−11.8564	−0.490622
$585$	0	0
$586$	−22.7846	−0.941224
$587$	9.07180	0.374433	0.187217	−	0.982319i	$-0.440053\pi$
$587$	9.07180	0.374433	0.187217	+	0.982319i	$0.440053\pi$
$588$	0	0
$589$	0	0
$590$	8.00000	0.329355
$591$	0	0
$592$	−3.46410	−0.142374
$593$	−36.2487	−1.48856	−0.744278	−	0.667870i	$-0.767206\pi$
$593$	−36.2487	−1.48856	−0.744278	+	0.667870i	$0.767206\pi$
$594$	0	0
$595$	−3.46410	−0.142014
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−18.9282	−0.774032
$599$	31.7128	1.29575	0.647875	−	0.761746i	$-0.275658\pi$
$599$	31.7128	1.29575	0.647875	+	0.761746i	$0.275658\pi$
$600$	0	0
$601$	16.9282	0.690516	0.345258	−	0.938508i	$-0.387791\pi$
$601$	16.9282	0.690516	0.345258	+	0.938508i	$0.387791\pi$
$602$	0	0
$603$	0	0
$604$	−20.0000	−0.813788
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	−8.92820	−0.361492
$611$	24.0000	0.970936
$612$	0	0
$613$	−16.1436	−0.652034	−0.326017	−	0.945364i	$-0.605707\pi$
$613$	−16.1436	−0.652034	−0.326017	+	0.945364i	$0.605707\pi$
$614$	−3.60770	−0.145595
$615$	0	0
$616$	1.00000	0.0402911
$617$	3.46410	0.139459	0.0697297	−	0.997566i	$-0.477786\pi$
$617$	3.46410	0.139459	0.0697297	+	0.997566i	$0.477786\pi$
$618$	0	0
$619$	−37.4641	−1.50581	−0.752905	−	0.658130i	$-0.771348\pi$
$619$	−37.4641	−1.50581	−0.752905	+	0.658130i	$0.771348\pi$
$620$	2.92820	0.117599
$621$	0	0
$622$	14.5359	0.582836
$623$	15.4641	0.619556
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.143594	0.00573915
$627$	0	0
$628$	−6.00000	−0.239426
$629$	12.0000	0.478471
$630$	0	0
$631$	10.9282	0.435045	0.217522	−	0.976055i	$-0.430202\pi$
$631$	10.9282	0.435045	0.217522	+	0.976055i	$0.430202\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	15.8564	0.629738
$635$	−2.92820	−0.116202
$636$	0	0
$637$	3.46410	0.137253
$638$	−4.92820	−0.195109
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	0	0
$643$	−6.14359	−0.242280	−0.121140	−	0.992635i	$-0.538655\pi$
$643$	−6.14359	−0.242280	−0.121140	+	0.992635i	$0.538655\pi$
$644$	5.46410	0.215316
$645$	0	0
$646$	0	0
$647$	17.0718	0.671162	0.335581	−	0.942011i	$-0.391067\pi$
$647$	17.0718	0.671162	0.335581	+	0.942011i	$0.391067\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	3.46410	0.135873
$651$	0	0
$652$	15.3205	0.599997
$653$	−3.07180	−0.120209	−0.0601043	−	0.998192i	$-0.519143\pi$
$653$	−3.07180	−0.120209	−0.0601043	+	0.998192i	$0.519143\pi$
$654$	0	0
$655$	−9.85641	−0.385122
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	−6.92820	−0.270089
$659$	−17.8564	−0.695587	−0.347793	−	0.937571i	$-0.613069\pi$
$659$	−17.8564	−0.695587	−0.347793	+	0.937571i	$0.613069\pi$
$660$	0	0
$661$	36.5359	1.42108	0.710541	−	0.703656i	$-0.248450\pi$
$661$	36.5359	1.42108	0.710541	+	0.703656i	$0.248450\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	13.8564	0.537733
$665$	0	0
$666$	0	0
$667$	−26.9282	−1.04266
$668$	10.5359	0.407646
$669$	0	0
$670$	−1.46410	−0.0565632
$671$	−8.92820	−0.344669
$672$	0	0
$673$	−8.14359	−0.313912	−0.156956	−	0.987606i	$-0.550168\pi$
$673$	−8.14359	−0.313912	−0.156956	+	0.987606i	$0.550168\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	7.85641	0.301946	0.150973	−	0.988538i	$-0.451759\pi$
$677$	7.85641	0.301946	0.150973	+	0.988538i	$0.451759\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	3.46410	0.132842
$681$	0	0
$682$	2.92820	0.112127
$683$	45.5692	1.74366	0.871829	−	0.489811i	$-0.162934\pi$
$683$	45.5692	1.74366	0.871829	+	0.489811i	$0.162934\pi$
$684$	0	0
$685$	−3.46410	−0.132357
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	0	0
$689$	−20.7846	−0.791831
$690$	0	0
$691$	−21.4641	−0.816533	−0.408266	−	0.912863i	$-0.633867\pi$
$691$	−21.4641	−0.816533	−0.408266	+	0.912863i	$0.633867\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	1.07180	0.0406848
$695$	−2.92820	−0.111073
$696$	0	0
$697$	6.92820	0.262424
$698$	−20.9282	−0.792144
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−30.7846	−1.16272	−0.581359	−	0.813647i	$-0.697479\pi$
$701$	−30.7846	−1.16272	−0.581359	+	0.813647i	$0.697479\pi$
$702$	0	0
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−0.928203	−0.0349334
$707$	19.8564	0.746777
$708$	0	0
$709$	−45.7128	−1.71678	−0.858390	−	0.512997i	$-0.828535\pi$
$709$	−45.7128	−1.71678	−0.858390	+	0.512997i	$0.828535\pi$
$710$	14.9282	0.560245
$711$	0	0
$712$	−15.4641	−0.579542
$713$	16.0000	0.599205
$714$	0	0
$715$	3.46410	0.129550
$716$	−6.92820	−0.258919
$717$	0	0
$718$	−1.46410	−0.0546398
$719$	−22.5359	−0.840447	−0.420224	−	0.907421i	$-0.638048\pi$
$719$	−22.5359	−0.840447	−0.420224	+	0.907421i	$0.638048\pi$
$720$	0	0
$721$	−10.9282	−0.406988
$722$	−19.0000	−0.707107
$723$	0	0
$724$	−3.46410	−0.128742
$725$	4.92820	0.183029
$726$	0	0
$727$	45.8564	1.70072	0.850360	−	0.526201i	$-0.176384\pi$
$727$	45.8564	1.70072	0.850360	+	0.526201i	$0.176384\pi$
$728$	−3.46410	−0.128388
$729$	0	0
$730$	11.8564	0.438825
$731$	0	0
$732$	0	0
$733$	43.4641	1.60538	0.802692	−	0.596394i	$-0.203401\pi$
$733$	43.4641	1.60538	0.802692	+	0.596394i	$0.203401\pi$
$734$	−24.7846	−0.914817
$735$	0	0
$736$	−5.46410	−0.201409
$737$	−1.46410	−0.0539309
$738$	0	0
$739$	−21.4641	−0.789570	−0.394785	−	0.918774i	$-0.629181\pi$
$739$	−21.4641	−0.789570	−0.394785	+	0.918774i	$0.629181\pi$
$740$	3.46410	0.127343
$741$	0	0
$742$	6.00000	0.220267
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−6.78461	−0.248402
$747$	0	0
$748$	3.46410	0.126660
$749$	−6.92820	−0.253151
$750$	0	0
$751$	−13.0718	−0.476997	−0.238498	−	0.971143i	$-0.576655\pi$
$751$	−13.0718	−0.476997	−0.238498	+	0.971143i	$0.576655\pi$
$752$	6.92820	0.252646
$753$	0	0
$754$	17.0718	0.621718
$755$	20.0000	0.727875
$756$	0	0
$757$	−11.4641	−0.416670	−0.208335	−	0.978058i	$-0.566804\pi$
$757$	−11.4641	−0.416670	−0.208335	+	0.978058i	$0.566804\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	33.7128	1.22209	0.611044	−	0.791596i	$-0.290750\pi$
$761$	33.7128	1.22209	0.611044	+	0.791596i	$0.290750\pi$
$762$	0	0
$763$	−3.46410	−0.125409
$764$	−17.8564	−0.646022
$765$	0	0
$766$	14.9282	0.539378
$767$	−27.7128	−1.00065
$768$	0	0
$769$	−40.6410	−1.46555	−0.732776	−	0.680470i	$-0.761776\pi$
$769$	−40.6410	−1.46555	−0.732776	+	0.680470i	$0.761776\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	−11.8564	−0.426721
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	0	0
$775$	−2.92820	−0.105184
$776$	−10.0000	−0.358979
$777$	0	0
$778$	−15.0718	−0.540350
$779$	0	0
$780$	0	0
$781$	14.9282	0.534173
$782$	18.9282	0.676871
$783$	0	0
$784$	1.00000	0.0357143
$785$	6.00000	0.214149
$786$	0	0
$787$	14.5359	0.518149	0.259074	−	0.965857i	$-0.416583\pi$
$787$	14.5359	0.518149	0.259074	+	0.965857i	$0.416583\pi$
$788$	−2.00000	−0.0712470
$789$	0	0
$790$	4.00000	0.142314
$791$	18.3923	0.653955
$792$	0	0
$793$	30.9282	1.09829
$794$	−6.78461	−0.240777
$795$	0	0
$796$	5.07180	0.179765
$797$	21.7128	0.769107	0.384554	−	0.923103i	$-0.374355\pi$
$797$	21.7128	0.769107	0.384554	+	0.923103i	$0.374355\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	0	0
$802$	−7.85641	−0.277419
$803$	11.8564	0.418403
$804$	0	0
$805$	−5.46410	−0.192584
$806$	−10.1436	−0.357293
$807$	0	0
$808$	−19.8564	−0.698546
$809$	−55.1769	−1.93992	−0.969959	−	0.243270i	$-0.921780\pi$
$809$	−55.1769	−1.93992	−0.969959	+	0.243270i	$0.921780\pi$
$810$	0	0
$811$	19.7128	0.692210	0.346105	−	0.938196i	$-0.387504\pi$
$811$	19.7128	0.692210	0.346105	+	0.938196i	$0.387504\pi$
$812$	−4.92820	−0.172946
$813$	0	0
$814$	3.46410	0.121417
$815$	−15.3205	−0.536654
$816$	0	0
$817$	0	0
$818$	−20.9282	−0.731737
$819$	0	0
$820$	2.00000	0.0698430
$821$	1.21539	0.0424174	0.0212087	−	0.999775i	$-0.493249\pi$
$821$	1.21539	0.0424174	0.0212087	+	0.999775i	$0.493249\pi$
$822$	0	0
$823$	−12.7846	−0.445643	−0.222822	−	0.974859i	$-0.571527\pi$
$823$	−12.7846	−0.445643	−0.222822	+	0.974859i	$0.571527\pi$
$824$	10.9282	0.380702
$825$	0	0
$826$	8.00000	0.278356
$827$	17.0718	0.593645	0.296822	−	0.954933i	$-0.404073\pi$
$827$	17.0718	0.593645	0.296822	+	0.954933i	$0.404073\pi$
$828$	0	0
$829$	32.2487	1.12004	0.560022	−	0.828478i	$-0.310793\pi$
$829$	32.2487	1.12004	0.560022	+	0.828478i	$0.310793\pi$
$830$	−13.8564	−0.480963
$831$	0	0
$832$	3.46410	0.120096
$833$	−3.46410	−0.120024
$834$	0	0
$835$	−10.5359	−0.364610
$836$	0	0
$837$	0	0
$838$	−10.9282	−0.377509
$839$	−7.32051	−0.252732	−0.126366	−	0.991984i	$-0.540331\pi$
$839$	−7.32051	−0.252732	−0.126366	+	0.991984i	$0.540331\pi$
$840$	0	0
$841$	−4.71281	−0.162511
$842$	−4.14359	−0.142798
$843$	0	0
$844$	−16.3923	−0.564246
$845$	1.00000	0.0344010
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−3.46410	−0.118818
$851$	18.9282	0.648850
$852$	0	0
$853$	5.60770	0.192004	0.0960019	−	0.995381i	$-0.469395\pi$
$853$	5.60770	0.192004	0.0960019	+	0.995381i	$0.469395\pi$
$854$	−8.92820	−0.305517
$855$	0	0
$856$	6.92820	0.236801
$857$	15.4641	0.528244	0.264122	−	0.964489i	$-0.414918\pi$
$857$	15.4641	0.528244	0.264122	+	0.964489i	$0.414918\pi$
$858$	0	0
$859$	15.6077	0.532528	0.266264	−	0.963900i	$-0.414211\pi$
$859$	15.6077	0.532528	0.266264	+	0.963900i	$0.414211\pi$
$860$	0	0
$861$	0	0
$862$	11.6077	0.395360
$863$	52.1051	1.77368	0.886839	−	0.462078i	$-0.152896\pi$
$863$	52.1051	1.77368	0.886839	+	0.462078i	$0.152896\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	3.85641	0.131046
$867$	0	0
$868$	2.92820	0.0993897
$869$	4.00000	0.135691
$870$	0	0
$871$	5.07180	0.171851
$872$	3.46410	0.117309
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	47.8564	1.61600	0.807998	−	0.589185i	$-0.200551\pi$
$877$	47.8564	1.61600	0.807998	+	0.589185i	$0.200551\pi$
$878$	21.0718	0.711139
$879$	0	0
$880$	1.00000	0.0337100
$881$	−7.46410	−0.251472	−0.125736	−	0.992064i	$-0.540129\pi$
$881$	−7.46410	−0.251472	−0.125736	+	0.992064i	$0.540129\pi$
$882$	0	0
$883$	−12.3923	−0.417034	−0.208517	−	0.978019i	$-0.566864\pi$
$883$	−12.3923	−0.417034	−0.208517	+	0.978019i	$0.566864\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	33.8564	1.13743
$887$	13.4641	0.452080	0.226040	−	0.974118i	$-0.427422\pi$
$887$	13.4641	0.452080	0.226040	+	0.974118i	$0.427422\pi$
$888$	0	0
$889$	−2.92820	−0.0982088
$890$	15.4641	0.518358
$891$	0	0
$892$	−27.7128	−0.927894
$893$	0	0
$894$	0	0
$895$	6.92820	0.231584
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−15.8564	−0.529135
$899$	−14.4308	−0.481293
$900$	0	0
$901$	20.7846	0.692436
$902$	2.00000	0.0665927
$903$	0	0
$904$	−18.3923	−0.611719
$905$	3.46410	0.115151
$906$	0	0
$907$	−33.4641	−1.11116	−0.555579	−	0.831464i	$-0.687503\pi$
$907$	−33.4641	−1.11116	−0.555579	+	0.831464i	$0.687503\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	3.46410	0.114834
$911$	−26.6410	−0.882656	−0.441328	−	0.897346i	$-0.645493\pi$
$911$	−26.6410	−0.882656	−0.441328	+	0.897346i	$0.645493\pi$
$912$	0	0
$913$	−13.8564	−0.458580
$914$	15.8564	0.524483
$915$	0	0
$916$	23.4641	0.775276
$917$	−9.85641	−0.325487
$918$	0	0
$919$	−28.7846	−0.949517	−0.474758	−	0.880116i	$-0.657465\pi$
$919$	−28.7846	−0.949517	−0.474758	+	0.880116i	$0.657465\pi$
$920$	5.46410	0.180146
$921$	0	0
$922$	29.7128	0.978539
$923$	−51.7128	−1.70215
$924$	0	0
$925$	−3.46410	−0.113899
$926$	4.00000	0.131448
$927$	0	0
$928$	4.92820	0.161776
$929$	−20.5359	−0.673761	−0.336880	−	0.941547i	$-0.609372\pi$
$929$	−20.5359	−0.673761	−0.336880	+	0.941547i	$0.609372\pi$
$930$	0	0
$931$	0	0
$932$	7.85641	0.257345
$933$	0	0
$934$	−20.7846	−0.680093
$935$	−3.46410	−0.113288
$936$	0	0
$937$	26.7846	0.875015	0.437508	−	0.899215i	$-0.355861\pi$
$937$	26.7846	0.875015	0.437508	+	0.899215i	$0.355861\pi$
$938$	−1.46410	−0.0478046
$939$	0	0
$940$	−6.92820	−0.225973
$941$	−33.7128	−1.09901	−0.549503	−	0.835492i	$-0.685183\pi$
$941$	−33.7128	−1.09901	−0.549503	+	0.835492i	$0.685183\pi$
$942$	0	0
$943$	10.9282	0.355871
$944$	−8.00000	−0.260378
$945$	0	0
$946$	0	0
$947$	3.21539	0.104486	0.0522431	−	0.998634i	$-0.483363\pi$
$947$	3.21539	0.104486	0.0522431	+	0.998634i	$0.483363\pi$
$948$	0	0
$949$	−41.0718	−1.33325
$950$	0	0
$951$	0	0
$952$	3.46410	0.112272
$953$	36.9282	1.19622	0.598111	−	0.801413i	$-0.295918\pi$
$953$	36.9282	1.19622	0.598111	+	0.801413i	$0.295918\pi$
$954$	0	0
$955$	17.8564	0.577820
$956$	−14.5359	−0.470125
$957$	0	0
$958$	5.85641	0.189212
$959$	−3.46410	−0.111862
$960$	0	0
$961$	−22.4256	−0.723407
$962$	−12.0000	−0.386896
$963$	0	0
$964$	22.7846	0.733843
$965$	11.8564	0.381671
$966$	0	0
$967$	32.7846	1.05428	0.527141	−	0.849778i	$-0.323264\pi$
$967$	32.7846	1.05428	0.527141	+	0.849778i	$0.323264\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	10.0000	0.321081
$971$	45.0718	1.44642	0.723211	−	0.690627i	$-0.242665\pi$
$971$	45.0718	1.44642	0.723211	+	0.690627i	$0.242665\pi$
$972$	0	0
$973$	−2.92820	−0.0938739
$974$	−14.1436	−0.453190
$975$	0	0
$976$	8.92820	0.285785
$977$	22.3923	0.716393	0.358197	−	0.933646i	$-0.383392\pi$
$977$	22.3923	0.716393	0.358197	+	0.933646i	$0.383392\pi$
$978$	0	0
$979$	15.4641	0.494235
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	25.8564	0.825111
$983$	−42.6410	−1.36004	−0.680019	−	0.733195i	$-0.738028\pi$
$983$	−42.6410	−1.36004	−0.680019	+	0.733195i	$0.738028\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	−17.0718	−0.543677
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	2.92820	0.0930174	0.0465087	−	0.998918i	$-0.485190\pi$
$991$	2.92820	0.0930174	0.0465087	+	0.998918i	$0.485190\pi$
$992$	−2.92820	−0.0929705
$993$	0	0
$994$	14.9282	0.473494
$995$	−5.07180	−0.160787
$996$	0	0
$997$	22.3923	0.709171	0.354586	−	0.935024i	$-0.384622\pi$
$997$	22.3923	0.709171	0.354586	+	0.935024i	$0.384622\pi$
$998$	28.7846	0.911161
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.bw.1.2		2
3.2	odd	2		2310.2.a.z.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.z.1.2	✓	2	3.2	odd	2
6930.2.a.bw.1.2		2	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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +3.46410 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} -1.00000 q^{20} -1.00000 q^{22} -5.46410 q^{23} +1.00000 q^{25} +3.46410 q^{26} -1.00000 q^{28} +4.92820 q^{29} -2.92820 q^{31} +1.00000 q^{32} -3.46410 q^{34} +1.00000 q^{35} -3.46410 q^{37} -1.00000 q^{40} -2.00000 q^{41} -1.00000 q^{44} -5.46410 q^{46} +6.92820 q^{47} +1.00000 q^{49} +1.00000 q^{50} +3.46410 q^{52} -6.00000 q^{53} +1.00000 q^{55} -1.00000 q^{56} +4.92820 q^{58} -8.00000 q^{59} +8.92820 q^{61} -2.92820 q^{62} +1.00000 q^{64} -3.46410 q^{65} +1.46410 q^{67} -3.46410 q^{68} +1.00000 q^{70} -14.9282 q^{71} -11.8564 q^{73} -3.46410 q^{74} +1.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} +13.8564 q^{83} +3.46410 q^{85} -1.00000 q^{88} -15.4641 q^{89} -3.46410 q^{91} -5.46410 q^{92} +6.92820 q^{94} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{14} + 2 q^{16} - 2 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{25} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{35} - 2 q^{40} - 4 q^{41} - 2 q^{44} - 4 q^{46} + 2 q^{49} + 2 q^{50} - 12 q^{53} + 2 q^{55} - 2 q^{56} - 4 q^{58} - 16 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} - 4 q^{67} + 2 q^{70} - 16 q^{71} + 4 q^{73} + 2 q^{77} - 8 q^{79} - 2 q^{80} - 4 q^{82} - 2 q^{88} - 24 q^{89} - 4 q^{92} - 20 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^14 + 2 * q^16 - 2 * q^20 - 2 * q^22 - 4 * q^23 + 2 * q^25 - 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^35 - 2 * q^40 - 4 * q^41 - 2 * q^44 - 4 * q^46 + 2 * q^49 + 2 * q^50 - 12 * q^53 + 2 * q^55 - 2 * q^56 - 4 * q^58 - 16 * q^59 + 4 * q^61 + 8 * q^62 + 2 * q^64 - 4 * q^67 + 2 * q^70 - 16 * q^71 + 4 * q^73 + 2 * q^77 - 8 * q^79 - 2 * q^80 - 4 * q^82 - 2 * q^88 - 24 * q^89 - 4 * q^92 - 20 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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