LMFDB - Embedded newform 6930.2.a.bs.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(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.12311 q^{17} -7.12311 q^{19} +1.00000 q^{20} -1.00000 q^{22} +1.12311 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -8.24621 q^{29} +1.12311 q^{31} -1.00000 q^{32} -5.12311 q^{34} -1.00000 q^{35} -1.12311 q^{37} +7.12311 q^{38} -1.00000 q^{40} -0.876894 q^{41} +3.12311 q^{43} +1.00000 q^{44} -1.12311 q^{46} -13.3693 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{55} +1.00000 q^{56} +8.24621 q^{58} -4.00000 q^{59} -5.12311 q^{61} -1.12311 q^{62} +1.00000 q^{64} +2.00000 q^{65} -0.876894 q^{67} +5.12311 q^{68} +1.00000 q^{70} -7.12311 q^{71} +16.2462 q^{73} +1.12311 q^{74} -7.12311 q^{76} -1.00000 q^{77} +5.12311 q^{79} +1.00000 q^{80} +0.876894 q^{82} -2.87689 q^{83} +5.12311 q^{85} -3.12311 q^{86} -1.00000 q^{88} +8.24621 q^{89} -2.00000 q^{91} +1.12311 q^{92} +13.3693 q^{94} -7.12311 q^{95} -11.1231 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 6 q^{19} + 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 6 q^{31} - 2 q^{32} - 2 q^{34} - 2 q^{35} + 6 q^{37} + 6 q^{38} - 2 q^{40} - 10 q^{41} - 2 q^{43} + 2 q^{44} + 6 q^{46} - 2 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} - 20 q^{53} + 2 q^{55} + 2 q^{56} - 8 q^{59} - 2 q^{61} + 6 q^{62} + 2 q^{64} + 4 q^{65} - 10 q^{67} + 2 q^{68} + 2 q^{70} - 6 q^{71} + 16 q^{73} - 6 q^{74} - 6 q^{76} - 2 q^{77} + 2 q^{79} + 2 q^{80} + 10 q^{82} - 14 q^{83} + 2 q^{85} + 2 q^{86} - 2 q^{88} - 4 q^{91} - 6 q^{92} + 2 q^{94} - 6 q^{95} - 14 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 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 6 * q^19 + 2 * q^20 - 2 * q^22 - 6 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 6 * q^31 - 2 * q^32 - 2 * q^34 - 2 * q^35 + 6 * q^37 + 6 * q^38 - 2 * q^40 - 10 * q^41 - 2 * q^43 + 2 * q^44 + 6 * q^46 - 2 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^52 - 20 * q^53 + 2 * q^55 + 2 * q^56 - 8 * q^59 - 2 * q^61 + 6 * q^62 + 2 * q^64 + 4 * q^65 - 10 * q^67 + 2 * q^68 + 2 * q^70 - 6 * q^71 + 16 * q^73 - 6 * q^74 - 6 * q^76 - 2 * q^77 + 2 * q^79 + 2 * q^80 + 10 * q^82 - 14 * q^83 + 2 * q^85 + 2 * q^86 - 2 * q^88 - 4 * q^91 - 6 * q^92 + 2 * q^94 - 6 * q^95 - 14 * 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
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	−7.12311	−1.63415	−0.817076	−	0.576530i	$-0.804407\pi$
$19$	−7.12311	−1.63415	−0.817076	+	0.576530i	$0.804407\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	1.12311	0.234184	0.117092	−	0.993121i	$-0.462643\pi$
$23$	1.12311	0.234184	0.117092	+	0.993121i	$0.462643\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−8.24621	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	−8.24621	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	1.12311	0.201716	0.100858	−	0.994901i	$-0.467841\pi$
$31$	1.12311	0.201716	0.100858	+	0.994901i	$0.467841\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.12311	−0.878605
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−1.12311	−0.184637	−0.0923187	−	0.995730i	$-0.529428\pi$
$37$	−1.12311	−0.184637	−0.0923187	+	0.995730i	$0.529428\pi$
$38$	7.12311	1.15552
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−0.876894	−0.136948	−0.0684739	−	0.997653i	$-0.521813\pi$
$41$	−0.876894	−0.136948	−0.0684739	+	0.997653i	$0.521813\pi$
$42$	0	0
$43$	3.12311	0.476269	0.238135	−	0.971232i	$-0.423464\pi$
$43$	3.12311	0.476269	0.238135	+	0.971232i	$0.423464\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−1.12311	−0.165593
$47$	−13.3693	−1.95012	−0.975058	−	0.221952i	$-0.928757\pi$
$47$	−13.3693	−1.95012	−0.975058	+	0.221952i	$0.928757\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	8.24621	1.08278
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−5.12311	−0.655946	−0.327973	−	0.944687i	$-0.606366\pi$
$61$	−5.12311	−0.655946	−0.327973	+	0.944687i	$0.606366\pi$
$62$	−1.12311	−0.142635
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−0.876894	−0.107130	−0.0535648	−	0.998564i	$-0.517058\pi$
$67$	−0.876894	−0.107130	−0.0535648	+	0.998564i	$0.517058\pi$
$68$	5.12311	0.621268
$69$	0	0
$70$	1.00000	0.119523
$71$	−7.12311	−0.845357	−0.422679	−	0.906280i	$-0.638910\pi$
$71$	−7.12311	−0.845357	−0.422679	+	0.906280i	$0.638910\pi$
$72$	0	0
$73$	16.2462	1.90148	0.950738	−	0.309997i	$-0.100328\pi$
$73$	16.2462	1.90148	0.950738	+	0.309997i	$0.100328\pi$
$74$	1.12311	0.130558
$75$	0	0
$76$	−7.12311	−0.817076
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.12311	0.576394	0.288197	−	0.957571i	$-0.406944\pi$
$79$	5.12311	0.576394	0.288197	+	0.957571i	$0.406944\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	0.876894	0.0968368
$83$	−2.87689	−0.315780	−0.157890	−	0.987457i	$-0.550469\pi$
$83$	−2.87689	−0.315780	−0.157890	+	0.987457i	$0.550469\pi$
$84$	0	0
$85$	5.12311	0.555679
$86$	−3.12311	−0.336773
$87$	0	0
$88$	−1.00000	−0.106600
$89$	8.24621	0.874097	0.437048	−	0.899438i	$-0.356024\pi$
$89$	8.24621	0.874097	0.437048	+	0.899438i	$0.356024\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	1.12311	0.117092
$93$	0	0
$94$	13.3693	1.37894
$95$	−7.12311	−0.730815
$96$	0	0
$97$	−11.1231	−1.12938	−0.564690	−	0.825303i	$-0.691004\pi$
$97$	−11.1231	−1.12938	−0.564690	+	0.825303i	$0.691004\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.3693	−1.33030	−0.665148	−	0.746711i	$-0.731632\pi$
$101$	−13.3693	−1.33030	−0.665148	+	0.746711i	$0.731632\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	10.0000	0.971286
$107$	−1.75379	−0.169545	−0.0847726	−	0.996400i	$-0.527016\pi$
$107$	−1.75379	−0.169545	−0.0847726	+	0.996400i	$0.527016\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	1.12311	0.104730
$116$	−8.24621	−0.765641
$117$	0	0
$118$	4.00000	0.368230
$119$	−5.12311	−0.469634
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.12311	0.463824
$123$	0	0
$124$	1.12311	0.100858
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.2462	−1.61909	−0.809545	−	0.587058i	$-0.800286\pi$
$127$	−18.2462	−1.61909	−0.809545	+	0.587058i	$0.800286\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	22.2462	1.94366	0.971830	−	0.235682i	$-0.0757323\pi$
$131$	22.2462	1.94366	0.971830	+	0.235682i	$0.0757323\pi$
$132$	0	0
$133$	7.12311	0.617652
$134$	0.876894	0.0757521
$135$	0	0
$136$	−5.12311	−0.439303
$137$	−3.75379	−0.320708	−0.160354	−	0.987060i	$-0.551264\pi$
$137$	−3.75379	−0.320708	−0.160354	+	0.987060i	$0.551264\pi$
$138$	0	0
$139$	−0.876894	−0.0743772	−0.0371886	−	0.999308i	$-0.511840\pi$
$139$	−0.876894	−0.0743772	−0.0371886	+	0.999308i	$0.511840\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	7.12311	0.597758
$143$	2.00000	0.167248
$144$	0	0
$145$	−8.24621	−0.684811
$146$	−16.2462	−1.34455
$147$	0	0
$148$	−1.12311	−0.0923187
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	21.1231	1.71897	0.859487	−	0.511158i	$-0.170783\pi$
$151$	21.1231	1.71897	0.859487	+	0.511158i	$0.170783\pi$
$152$	7.12311	0.577760
$153$	0	0
$154$	1.00000	0.0805823
$155$	1.12311	0.0902100
$156$	0	0
$157$	−3.12311	−0.249251	−0.124625	−	0.992204i	$-0.539773\pi$
$157$	−3.12311	−0.249251	−0.124625	+	0.992204i	$0.539773\pi$
$158$	−5.12311	−0.407572
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−1.12311	−0.0885131
$162$	0	0
$163$	−21.3693	−1.67377	−0.836887	−	0.547376i	$-0.815627\pi$
$163$	−21.3693	−1.67377	−0.836887	+	0.547376i	$0.815627\pi$
$164$	−0.876894	−0.0684739
$165$	0	0
$166$	2.87689	0.223290
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−5.12311	−0.392924
$171$	0	0
$172$	3.12311	0.238135
$173$	−12.2462	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$173$	−12.2462	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−8.24621	−0.618080
$179$	2.24621	0.167890	0.0839449	−	0.996470i	$-0.473248\pi$
$179$	2.24621	0.167890	0.0839449	+	0.996470i	$0.473248\pi$
$180$	0	0
$181$	19.3693	1.43971	0.719855	−	0.694124i	$-0.244208\pi$
$181$	19.3693	1.43971	0.719855	+	0.694124i	$0.244208\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	−1.12311	−0.0827964
$185$	−1.12311	−0.0825724
$186$	0	0
$187$	5.12311	0.374639
$188$	−13.3693	−0.975058
$189$	0	0
$190$	7.12311	0.516764
$191$	−15.1231	−1.09427	−0.547135	−	0.837045i	$-0.684281\pi$
$191$	−15.1231	−1.09427	−0.547135	+	0.837045i	$0.684281\pi$
$192$	0	0
$193$	11.1231	0.800659	0.400329	−	0.916371i	$-0.368896\pi$
$193$	11.1231	0.800659	0.400329	+	0.916371i	$0.368896\pi$
$194$	11.1231	0.798592
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.4924	0.747554	0.373777	−	0.927519i	$-0.378063\pi$
$197$	10.4924	0.747554	0.373777	+	0.927519i	$0.378063\pi$
$198$	0	0
$199$	−13.1231	−0.930272	−0.465136	−	0.885239i	$-0.653995\pi$
$199$	−13.1231	−0.930272	−0.465136	+	0.885239i	$0.653995\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	13.3693	0.940662
$203$	8.24621	0.578771
$204$	0	0
$205$	−0.876894	−0.0612450
$206$	2.24621	0.156501
$207$	0	0
$208$	2.00000	0.138675
$209$	−7.12311	−0.492716
$210$	0	0
$211$	17.1231	1.17880	0.589402	−	0.807840i	$-0.299364\pi$
$211$	17.1231	1.17880	0.589402	+	0.807840i	$0.299364\pi$
$212$	−10.0000	−0.686803
$213$	0	0
$214$	1.75379	0.119887
$215$	3.12311	0.212994
$216$	0	0
$217$	−1.12311	−0.0762414
$218$	−18.0000	−1.21911
$219$	0	0
$220$	1.00000	0.0674200
$221$	10.2462	0.689235
$222$	0	0
$223$	−28.4924	−1.90799	−0.953997	−	0.299817i	$-0.903075\pi$
$223$	−28.4924	−1.90799	−0.953997	+	0.299817i	$0.903075\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	3.36932	0.223629	0.111815	−	0.993729i	$-0.464334\pi$
$227$	3.36932	0.223629	0.111815	+	0.993729i	$0.464334\pi$
$228$	0	0
$229$	2.87689	0.190111	0.0950553	−	0.995472i	$-0.469697\pi$
$229$	2.87689	0.190111	0.0950553	+	0.995472i	$0.469697\pi$
$230$	−1.12311	−0.0740554
$231$	0	0
$232$	8.24621	0.541390
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−13.3693	−0.872118
$236$	−4.00000	−0.260378
$237$	0	0
$238$	5.12311	0.332082
$239$	−17.3693	−1.12353	−0.561764	−	0.827298i	$-0.689877\pi$
$239$	−17.3693	−1.12353	−0.561764	+	0.827298i	$0.689877\pi$
$240$	0	0
$241$	−10.4924	−0.675876	−0.337938	−	0.941168i	$-0.609729\pi$
$241$	−10.4924	−0.675876	−0.337938	+	0.941168i	$0.609729\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−5.12311	−0.327973
$245$	1.00000	0.0638877
$246$	0	0
$247$	−14.2462	−0.906465
$248$	−1.12311	−0.0713173
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	16.4924	1.04099	0.520496	−	0.853864i	$-0.325747\pi$
$251$	16.4924	1.04099	0.520496	+	0.853864i	$0.325747\pi$
$252$	0	0
$253$	1.12311	0.0706090
$254$	18.2462	1.14487
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.6155	−1.59785	−0.798926	−	0.601429i	$-0.794598\pi$
$257$	−25.6155	−1.59785	−0.798926	+	0.601429i	$0.794598\pi$
$258$	0	0
$259$	1.12311	0.0697864
$260$	2.00000	0.124035
$261$	0	0
$262$	−22.2462	−1.37438
$263$	−24.4924	−1.51027	−0.755134	−	0.655571i	$-0.772428\pi$
$263$	−24.4924	−1.51027	−0.755134	+	0.655571i	$0.772428\pi$
$264$	0	0
$265$	−10.0000	−0.614295
$266$	−7.12311	−0.436746
$267$	0	0
$268$	−0.876894	−0.0535648
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	5.12311	0.310634
$273$	0	0
$274$	3.75379	0.226775
$275$	1.00000	0.0603023
$276$	0	0
$277$	6.24621	0.375298	0.187649	−	0.982236i	$-0.439913\pi$
$277$	6.24621	0.375298	0.187649	+	0.982236i	$0.439913\pi$
$278$	0.876894	0.0525926
$279$	0	0
$280$	1.00000	0.0597614
$281$	11.3693	0.678237	0.339118	−	0.940744i	$-0.389871\pi$
$281$	11.3693	0.678237	0.339118	+	0.940744i	$0.389871\pi$
$282$	0	0
$283$	−2.24621	−0.133523	−0.0667617	−	0.997769i	$-0.521267\pi$
$283$	−2.24621	−0.133523	−0.0667617	+	0.997769i	$0.521267\pi$
$284$	−7.12311	−0.422679
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0.876894	0.0517614
$288$	0	0
$289$	9.24621	0.543895
$290$	8.24621	0.484234
$291$	0	0
$292$	16.2462	0.950738
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	1.12311	0.0652792
$297$	0	0
$298$	2.00000	0.115857
$299$	2.24621	0.129902
$300$	0	0
$301$	−3.12311	−0.180013
$302$	−21.1231	−1.21550
$303$	0	0
$304$	−7.12311	−0.408538
$305$	−5.12311	−0.293348
$306$	0	0
$307$	14.7386	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$307$	14.7386	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−1.12311	−0.0637881
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	23.6155	1.33483	0.667414	−	0.744687i	$-0.267401\pi$
$313$	23.6155	1.33483	0.667414	+	0.744687i	$0.267401\pi$
$314$	3.12311	0.176247
$315$	0	0
$316$	5.12311	0.288197
$317$	−8.24621	−0.463153	−0.231577	−	0.972817i	$-0.574388\pi$
$317$	−8.24621	−0.463153	−0.231577	+	0.972817i	$0.574388\pi$
$318$	0	0
$319$	−8.24621	−0.461699
$320$	1.00000	0.0559017
$321$	0	0
$322$	1.12311	0.0625882
$323$	−36.4924	−2.03049
$324$	0	0
$325$	2.00000	0.110940
$326$	21.3693	1.18354
$327$	0	0
$328$	0.876894	0.0484184
$329$	13.3693	0.737074
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−2.87689	−0.157890
$333$	0	0
$334$	14.0000	0.766046
$335$	−0.876894	−0.0479099
$336$	0	0
$337$	−11.6155	−0.632738	−0.316369	−	0.948636i	$-0.602464\pi$
$337$	−11.6155	−0.632738	−0.316369	+	0.948636i	$0.602464\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	5.12311	0.277839
$341$	1.12311	0.0608196
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−3.12311	−0.168387
$345$	0	0
$346$	12.2462	0.658360
$347$	−18.7386	−1.00594	−0.502971	−	0.864303i	$-0.667760\pi$
$347$	−18.7386	−1.00594	−0.502971	+	0.864303i	$0.667760\pi$
$348$	0	0
$349$	19.3693	1.03682	0.518408	−	0.855133i	$-0.326525\pi$
$349$	19.3693	1.03682	0.518408	+	0.855133i	$0.326525\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−25.6155	−1.36338	−0.681688	−	0.731643i	$-0.738754\pi$
$353$	−25.6155	−1.36338	−0.681688	+	0.731643i	$0.738754\pi$
$354$	0	0
$355$	−7.12311	−0.378055
$356$	8.24621	0.437048
$357$	0	0
$358$	−2.24621	−0.118716
$359$	−17.3693	−0.916717	−0.458359	−	0.888767i	$-0.651563\pi$
$359$	−17.3693	−0.916717	−0.458359	+	0.888767i	$0.651563\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	−19.3693	−1.01803
$363$	0	0
$364$	−2.00000	−0.104828
$365$	16.2462	0.850366
$366$	0	0
$367$	4.49242	0.234503	0.117251	−	0.993102i	$-0.462592\pi$
$367$	4.49242	0.234503	0.117251	+	0.993102i	$0.462592\pi$
$368$	1.12311	0.0585459
$369$	0	0
$370$	1.12311	0.0583875
$371$	10.0000	0.519174
$372$	0	0
$373$	−5.75379	−0.297920	−0.148960	−	0.988843i	$-0.547593\pi$
$373$	−5.75379	−0.297920	−0.148960	+	0.988843i	$0.547593\pi$
$374$	−5.12311	−0.264909
$375$	0	0
$376$	13.3693	0.689470
$377$	−16.4924	−0.849403
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	−7.12311	−0.365408
$381$	0	0
$382$	15.1231	0.773765
$383$	35.6155	1.81987	0.909934	−	0.414753i	$-0.136132\pi$
$383$	35.6155	1.81987	0.909934	+	0.414753i	$0.136132\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−11.1231	−0.566151
$387$	0	0
$388$	−11.1231	−0.564690
$389$	13.3693	0.677851	0.338926	−	0.940813i	$-0.389936\pi$
$389$	13.3693	0.677851	0.338926	+	0.940813i	$0.389936\pi$
$390$	0	0
$391$	5.75379	0.290982
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−10.4924	−0.528601
$395$	5.12311	0.257771
$396$	0	0
$397$	−2.63068	−0.132030	−0.0660151	−	0.997819i	$-0.521029\pi$
$397$	−2.63068	−0.132030	−0.0660151	+	0.997819i	$0.521029\pi$
$398$	13.1231	0.657802
$399$	0	0
$400$	1.00000	0.0500000
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	0	0
$403$	2.24621	0.111892
$404$	−13.3693	−0.665148
$405$	0	0
$406$	−8.24621	−0.409253
$407$	−1.12311	−0.0556703
$408$	0	0
$409$	−0.246211	−0.0121744	−0.00608718	−	0.999981i	$-0.501938\pi$
$409$	−0.246211	−0.0121744	−0.00608718	+	0.999981i	$0.501938\pi$
$410$	0.876894	0.0433067
$411$	0	0
$412$	−2.24621	−0.110663
$413$	4.00000	0.196827
$414$	0	0
$415$	−2.87689	−0.141221
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	7.12311	0.348402
$419$	−32.4924	−1.58736	−0.793679	−	0.608336i	$-0.791837\pi$
$419$	−32.4924	−1.58736	−0.793679	+	0.608336i	$0.791837\pi$
$420$	0	0
$421$	8.24621	0.401896	0.200948	−	0.979602i	$-0.435598\pi$
$421$	8.24621	0.401896	0.200948	+	0.979602i	$0.435598\pi$
$422$	−17.1231	−0.833540
$423$	0	0
$424$	10.0000	0.485643
$425$	5.12311	0.248507
$426$	0	0
$427$	5.12311	0.247924
$428$	−1.75379	−0.0847726
$429$	0	0
$430$	−3.12311	−0.150610
$431$	−23.6155	−1.13752	−0.568760	−	0.822504i	$-0.692577\pi$
$431$	−23.6155	−1.13752	−0.568760	+	0.822504i	$0.692577\pi$
$432$	0	0
$433$	7.12311	0.342315	0.171157	−	0.985244i	$-0.445249\pi$
$433$	7.12311	0.342315	0.171157	+	0.985244i	$0.445249\pi$
$434$	1.12311	0.0539108
$435$	0	0
$436$	18.0000	0.862044
$437$	−8.00000	−0.382692
$438$	0	0
$439$	26.7386	1.27617	0.638083	−	0.769968i	$-0.279728\pi$
$439$	26.7386	1.27617	0.638083	+	0.769968i	$0.279728\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−10.2462	−0.487363
$443$	13.6155	0.646893	0.323447	−	0.946246i	$-0.395158\pi$
$443$	13.6155	0.646893	0.323447	+	0.946246i	$0.395158\pi$
$444$	0	0
$445$	8.24621	0.390908
$446$	28.4924	1.34916
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	18.7386	0.884331	0.442165	−	0.896934i	$-0.354210\pi$
$449$	18.7386	0.884331	0.442165	+	0.896934i	$0.354210\pi$
$450$	0	0
$451$	−0.876894	−0.0412913
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	−3.36932	−0.158130
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−13.3693	−0.625390	−0.312695	−	0.949854i	$-0.601232\pi$
$457$	−13.3693	−0.625390	−0.312695	+	0.949854i	$0.601232\pi$
$458$	−2.87689	−0.134428
$459$	0	0
$460$	1.12311	0.0523651
$461$	4.87689	0.227140	0.113570	−	0.993530i	$-0.463771\pi$
$461$	4.87689	0.227140	0.113570	+	0.993530i	$0.463771\pi$
$462$	0	0
$463$	−26.7386	−1.24265	−0.621325	−	0.783553i	$-0.713405\pi$
$463$	−26.7386	−1.24265	−0.621325	+	0.783553i	$0.713405\pi$
$464$	−8.24621	−0.382821
$465$	0	0
$466$	6.00000	0.277945
$467$	−20.4924	−0.948276	−0.474138	−	0.880450i	$-0.657240\pi$
$467$	−20.4924	−0.948276	−0.474138	+	0.880450i	$0.657240\pi$
$468$	0	0
$469$	0.876894	0.0404912
$470$	13.3693	0.616681
$471$	0	0
$472$	4.00000	0.184115
$473$	3.12311	0.143601
$474$	0	0
$475$	−7.12311	−0.326831
$476$	−5.12311	−0.234817
$477$	0	0
$478$	17.3693	0.794454
$479$	−16.4924	−0.753558	−0.376779	−	0.926303i	$-0.622968\pi$
$479$	−16.4924	−0.753558	−0.376779	+	0.926303i	$0.622968\pi$
$480$	0	0
$481$	−2.24621	−0.102418
$482$	10.4924	0.477917
$483$	0	0
$484$	1.00000	0.0454545
$485$	−11.1231	−0.505074
$486$	0	0
$487$	−28.4924	−1.29111	−0.645557	−	0.763712i	$-0.723375\pi$
$487$	−28.4924	−1.29111	−0.645557	+	0.763712i	$0.723375\pi$
$488$	5.12311	0.231912
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	10.2462	0.462405	0.231203	−	0.972906i	$-0.425734\pi$
$491$	10.2462	0.462405	0.231203	+	0.972906i	$0.425734\pi$
$492$	0	0
$493$	−42.2462	−1.90267
$494$	14.2462	0.640967
$495$	0	0
$496$	1.12311	0.0504289
$497$	7.12311	0.319515
$498$	0	0
$499$	30.7386	1.37605	0.688025	−	0.725687i	$-0.258478\pi$
$499$	30.7386	1.37605	0.688025	+	0.725687i	$0.258478\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−16.4924	−0.736093
$503$	−16.2462	−0.724383	−0.362191	−	0.932104i	$-0.617971\pi$
$503$	−16.2462	−0.724383	−0.362191	+	0.932104i	$0.617971\pi$
$504$	0	0
$505$	−13.3693	−0.594927
$506$	−1.12311	−0.0499281
$507$	0	0
$508$	−18.2462	−0.809545
$509$	−23.7538	−1.05287	−0.526434	−	0.850216i	$-0.676471\pi$
$509$	−23.7538	−1.05287	−0.526434	+	0.850216i	$0.676471\pi$
$510$	0	0
$511$	−16.2462	−0.718690
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	25.6155	1.12985
$515$	−2.24621	−0.0989799
$516$	0	0
$517$	−13.3693	−0.587982
$518$	−1.12311	−0.0493464
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−4.24621	−0.186030	−0.0930149	−	0.995665i	$-0.529650\pi$
$521$	−4.24621	−0.186030	−0.0930149	+	0.995665i	$0.529650\pi$
$522$	0	0
$523$	−32.4924	−1.42079	−0.710397	−	0.703801i	$-0.751485\pi$
$523$	−32.4924	−1.42079	−0.710397	+	0.703801i	$0.751485\pi$
$524$	22.2462	0.971830
$525$	0	0
$526$	24.4924	1.06792
$527$	5.75379	0.250639
$528$	0	0
$529$	−21.7386	−0.945158
$530$	10.0000	0.434372
$531$	0	0
$532$	7.12311	0.308826
$533$	−1.75379	−0.0759650
$534$	0	0
$535$	−1.75379	−0.0758229
$536$	0.876894	0.0378761
$537$	0	0
$538$	14.0000	0.603583
$539$	1.00000	0.0430730
$540$	0	0
$541$	−28.2462	−1.21440	−0.607200	−	0.794549i	$-0.707707\pi$
$541$	−28.2462	−1.21440	−0.607200	+	0.794549i	$0.707707\pi$
$542$	12.0000	0.515444
$543$	0	0
$544$	−5.12311	−0.219651
$545$	18.0000	0.771035
$546$	0	0
$547$	−35.6155	−1.52281	−0.761405	−	0.648276i	$-0.775490\pi$
$547$	−35.6155	−1.52281	−0.761405	+	0.648276i	$0.775490\pi$
$548$	−3.75379	−0.160354
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	58.7386	2.50235
$552$	0	0
$553$	−5.12311	−0.217857
$554$	−6.24621	−0.265376
$555$	0	0
$556$	−0.876894	−0.0371886
$557$	16.7386	0.709239	0.354619	−	0.935011i	$-0.384610\pi$
$557$	16.7386	0.709239	0.354619	+	0.935011i	$0.384610\pi$
$558$	0	0
$559$	6.24621	0.264187
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−11.3693	−0.479586
$563$	22.8769	0.964146	0.482073	−	0.876131i	$-0.339884\pi$
$563$	22.8769	0.964146	0.482073	+	0.876131i	$0.339884\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	2.24621	0.0944153
$567$	0	0
$568$	7.12311	0.298879
$569$	−22.8769	−0.959049	−0.479525	−	0.877528i	$-0.659191\pi$
$569$	−22.8769	−0.959049	−0.479525	+	0.877528i	$0.659191\pi$
$570$	0	0
$571$	−25.6155	−1.07198	−0.535988	−	0.844225i	$-0.680061\pi$
$571$	−25.6155	−1.07198	−0.535988	+	0.844225i	$0.680061\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−0.876894	−0.0366009
$575$	1.12311	0.0468367
$576$	0	0
$577$	−31.1231	−1.29567	−0.647836	−	0.761780i	$-0.724326\pi$
$577$	−31.1231	−1.29567	−0.647836	+	0.761780i	$0.724326\pi$
$578$	−9.24621	−0.384592
$579$	0	0
$580$	−8.24621	−0.342405
$581$	2.87689	0.119354
$582$	0	0
$583$	−10.0000	−0.414158
$584$	−16.2462	−0.672273
$585$	0	0
$586$	−10.0000	−0.413096
$587$	44.9848	1.85672	0.928362	−	0.371678i	$-0.121218\pi$
$587$	44.9848	1.85672	0.928362	+	0.371678i	$0.121218\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	4.00000	0.164677
$591$	0	0
$592$	−1.12311	−0.0461594
$593$	26.8769	1.10370	0.551851	−	0.833943i	$-0.313922\pi$
$593$	26.8769	1.10370	0.551851	+	0.833943i	$0.313922\pi$
$594$	0	0
$595$	−5.12311	−0.210027
$596$	−2.00000	−0.0819232
$597$	0	0
$598$	−2.24621	−0.0918544
$599$	3.12311	0.127607	0.0638033	−	0.997962i	$-0.479677\pi$
$599$	3.12311	0.127607	0.0638033	+	0.997962i	$0.479677\pi$
$600$	0	0
$601$	−46.9848	−1.91655	−0.958276	−	0.285845i	$-0.907726\pi$
$601$	−46.9848	−1.91655	−0.958276	+	0.285845i	$0.907726\pi$
$602$	3.12311	0.127288
$603$	0	0
$604$	21.1231	0.859487
$605$	1.00000	0.0406558
$606$	0	0
$607$	32.9848	1.33881	0.669407	−	0.742896i	$-0.266548\pi$
$607$	32.9848	1.33881	0.669407	+	0.742896i	$0.266548\pi$
$608$	7.12311	0.288880
$609$	0	0
$610$	5.12311	0.207428
$611$	−26.7386	−1.08173
$612$	0	0
$613$	24.0000	0.969351	0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000	0.969351	0.484675	+	0.874694i	$0.338938\pi$
$614$	−14.7386	−0.594803
$615$	0	0
$616$	1.00000	0.0402911
$617$	−36.2462	−1.45922	−0.729609	−	0.683865i	$-0.760298\pi$
$617$	−36.2462	−1.45922	−0.729609	+	0.683865i	$0.760298\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	1.12311	0.0451050
$621$	0	0
$622$	8.00000	0.320771
$623$	−8.24621	−0.330377
$624$	0	0
$625$	1.00000	0.0400000
$626$	−23.6155	−0.943866
$627$	0	0
$628$	−3.12311	−0.124625
$629$	−5.75379	−0.229419
$630$	0	0
$631$	4.49242	0.178841	0.0894203	−	0.995994i	$-0.471499\pi$
$631$	4.49242	0.178841	0.0894203	+	0.995994i	$0.471499\pi$
$632$	−5.12311	−0.203786
$633$	0	0
$634$	8.24621	0.327499
$635$	−18.2462	−0.724079
$636$	0	0
$637$	2.00000	0.0792429
$638$	8.24621	0.326471
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	18.7386	0.740132	0.370066	−	0.929006i	$-0.379335\pi$
$641$	18.7386	0.740132	0.370066	+	0.929006i	$0.379335\pi$
$642$	0	0
$643$	−21.7538	−0.857886	−0.428943	−	0.903332i	$-0.641114\pi$
$643$	−21.7538	−0.857886	−0.428943	+	0.903332i	$0.641114\pi$
$644$	−1.12311	−0.0442566
$645$	0	0
$646$	36.4924	1.43578
$647$	−27.1231	−1.06632	−0.533160	−	0.846015i	$-0.678995\pi$
$647$	−27.1231	−1.06632	−0.533160	+	0.846015i	$0.678995\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−21.3693	−0.836887
$653$	0.738634	0.0289050	0.0144525	−	0.999896i	$-0.495399\pi$
$653$	0.738634	0.0289050	0.0144525	+	0.999896i	$0.495399\pi$
$654$	0	0
$655$	22.2462	0.869231
$656$	−0.876894	−0.0342370
$657$	0	0
$658$	−13.3693	−0.521190
$659$	−48.4924	−1.88900	−0.944498	−	0.328516i	$-0.893451\pi$
$659$	−48.4924	−1.88900	−0.944498	+	0.328516i	$0.893451\pi$
$660$	0	0
$661$	3.36932	0.131051	0.0655256	−	0.997851i	$-0.479128\pi$
$661$	3.36932	0.131051	0.0655256	+	0.997851i	$0.479128\pi$
$662$	0	0
$663$	0	0
$664$	2.87689	0.111645
$665$	7.12311	0.276222
$666$	0	0
$667$	−9.26137	−0.358602
$668$	−14.0000	−0.541676
$669$	0	0
$670$	0.876894	0.0338774
$671$	−5.12311	−0.197775
$672$	0	0
$673$	17.8617	0.688519	0.344260	−	0.938874i	$-0.388130\pi$
$673$	17.8617	0.688519	0.344260	+	0.938874i	$0.388130\pi$
$674$	11.6155	0.447413
$675$	0	0
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	11.1231	0.426866
$680$	−5.12311	−0.196462
$681$	0	0
$682$	−1.12311	−0.0430059
$683$	13.6155	0.520984	0.260492	−	0.965476i	$-0.416115\pi$
$683$	13.6155	0.520984	0.260492	+	0.965476i	$0.416115\pi$
$684$	0	0
$685$	−3.75379	−0.143425
$686$	1.00000	0.0381802
$687$	0	0
$688$	3.12311	0.119067
$689$	−20.0000	−0.761939
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−12.2462	−0.465531
$693$	0	0
$694$	18.7386	0.711309
$695$	−0.876894	−0.0332625
$696$	0	0
$697$	−4.49242	−0.170163
$698$	−19.3693	−0.733139
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−42.4924	−1.60492	−0.802458	−	0.596708i	$-0.796475\pi$
$701$	−42.4924	−1.60492	−0.802458	+	0.596708i	$0.796475\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	1.00000	0.0376889
$705$	0	0
$706$	25.6155	0.964053
$707$	13.3693	0.502805
$708$	0	0
$709$	−22.9848	−0.863214	−0.431607	−	0.902062i	$-0.642053\pi$
$709$	−22.9848	−0.863214	−0.431607	+	0.902062i	$0.642053\pi$
$710$	7.12311	0.267325
$711$	0	0
$712$	−8.24621	−0.309040
$713$	1.26137	0.0472385
$714$	0	0
$715$	2.00000	0.0747958
$716$	2.24621	0.0839449
$717$	0	0
$718$	17.3693	0.648217
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	2.24621	0.0836533
$722$	−31.7386	−1.18119
$723$	0	0
$724$	19.3693	0.719855
$725$	−8.24621	−0.306257
$726$	0	0
$727$	−29.7538	−1.10351	−0.551753	−	0.834007i	$-0.686041\pi$
$727$	−29.7538	−1.10351	−0.551753	+	0.834007i	$0.686041\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	−16.2462	−0.601299
$731$	16.0000	0.591781
$732$	0	0
$733$	32.7386	1.20923	0.604615	−	0.796518i	$-0.293327\pi$
$733$	32.7386	1.20923	0.604615	+	0.796518i	$0.293327\pi$
$734$	−4.49242	−0.165818
$735$	0	0
$736$	−1.12311	−0.0413982
$737$	−0.876894	−0.0323008
$738$	0	0
$739$	33.1231	1.21845	0.609227	−	0.792996i	$-0.291480\pi$
$739$	33.1231	1.21845	0.609227	+	0.792996i	$0.291480\pi$
$740$	−1.12311	−0.0412862
$741$	0	0
$742$	−10.0000	−0.367112
$743$	46.7386	1.71467	0.857337	−	0.514755i	$-0.172117\pi$
$743$	46.7386	1.71467	0.857337	+	0.514755i	$0.172117\pi$
$744$	0	0
$745$	−2.00000	−0.0732743
$746$	5.75379	0.210661
$747$	0	0
$748$	5.12311	0.187319
$749$	1.75379	0.0640820
$750$	0	0
$751$	−21.7538	−0.793807	−0.396904	−	0.917860i	$-0.629915\pi$
$751$	−21.7538	−0.793807	−0.396904	+	0.917860i	$0.629915\pi$
$752$	−13.3693	−0.487529
$753$	0	0
$754$	16.4924	0.600619
$755$	21.1231	0.768749
$756$	0	0
$757$	7.86174	0.285740	0.142870	−	0.989741i	$-0.454367\pi$
$757$	7.86174	0.285740	0.142870	+	0.989741i	$0.454367\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	7.12311	0.258382
$761$	−40.1080	−1.45391	−0.726956	−	0.686684i	$-0.759066\pi$
$761$	−40.1080	−1.45391	−0.726956	+	0.686684i	$0.759066\pi$
$762$	0	0
$763$	−18.0000	−0.651644
$764$	−15.1231	−0.547135
$765$	0	0
$766$	−35.6155	−1.28684
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	11.1231	0.400329
$773$	−35.7538	−1.28597	−0.642987	−	0.765877i	$-0.722305\pi$
$773$	−35.7538	−1.28597	−0.642987	+	0.765877i	$0.722305\pi$
$774$	0	0
$775$	1.12311	0.0403431
$776$	11.1231	0.399296
$777$	0	0
$778$	−13.3693	−0.479313
$779$	6.24621	0.223794
$780$	0	0
$781$	−7.12311	−0.254885
$782$	−5.75379	−0.205755
$783$	0	0
$784$	1.00000	0.0357143
$785$	−3.12311	−0.111468
$786$	0	0
$787$	10.2462	0.365238	0.182619	−	0.983184i	$-0.441543\pi$
$787$	10.2462	0.365238	0.182619	+	0.983184i	$0.441543\pi$
$788$	10.4924	0.373777
$789$	0	0
$790$	−5.12311	−0.182272
$791$	2.00000	0.0711118
$792$	0	0
$793$	−10.2462	−0.363854
$794$	2.63068	0.0933595
$795$	0	0
$796$	−13.1231	−0.465136
$797$	24.2462	0.858845	0.429422	−	0.903104i	$-0.358717\pi$
$797$	24.2462	0.858845	0.429422	+	0.903104i	$0.358717\pi$
$798$	0	0
$799$	−68.4924	−2.42309
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−32.0000	−1.12996
$803$	16.2462	0.573316
$804$	0	0
$805$	−1.12311	−0.0395843
$806$	−2.24621	−0.0791194
$807$	0	0
$808$	13.3693	0.470331
$809$	−1.12311	−0.0394863	−0.0197431	−	0.999805i	$-0.506285\pi$
$809$	−1.12311	−0.0394863	−0.0197431	+	0.999805i	$0.506285\pi$
$810$	0	0
$811$	−17.8617	−0.627211	−0.313605	−	0.949553i	$-0.601537\pi$
$811$	−17.8617	−0.627211	−0.313605	+	0.949553i	$0.601537\pi$
$812$	8.24621	0.289385
$813$	0	0
$814$	1.12311	0.0393648
$815$	−21.3693	−0.748535
$816$	0	0
$817$	−22.2462	−0.778296
$818$	0.246211	0.00860857
$819$	0	0
$820$	−0.876894	−0.0306225
$821$	−22.9848	−0.802177	−0.401088	−	0.916039i	$-0.631368\pi$
$821$	−22.9848	−0.802177	−0.401088	+	0.916039i	$0.631368\pi$
$822$	0	0
$823$	38.2462	1.33318	0.666590	−	0.745425i	$-0.267753\pi$
$823$	38.2462	1.33318	0.666590	+	0.745425i	$0.267753\pi$
$824$	2.24621	0.0782505
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−46.2462	−1.60814	−0.804069	−	0.594536i	$-0.797336\pi$
$827$	−46.2462	−1.60814	−0.804069	+	0.594536i	$0.797336\pi$
$828$	0	0
$829$	−48.3542	−1.67941	−0.839705	−	0.543043i	$-0.817272\pi$
$829$	−48.3542	−1.67941	−0.839705	+	0.543043i	$0.817272\pi$
$830$	2.87689	0.0998585
$831$	0	0
$832$	2.00000	0.0693375
$833$	5.12311	0.177505
$834$	0	0
$835$	−14.0000	−0.484490
$836$	−7.12311	−0.246358
$837$	0	0
$838$	32.4924	1.12243
$839$	−22.2462	−0.768025	−0.384012	−	0.923328i	$-0.625458\pi$
$839$	−22.2462	−0.768025	−0.384012	+	0.923328i	$0.625458\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	−8.24621	−0.284183
$843$	0	0
$844$	17.1231	0.589402
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−10.0000	−0.343401
$849$	0	0
$850$	−5.12311	−0.175721
$851$	−1.26137	−0.0432391
$852$	0	0
$853$	12.2462	0.419302	0.209651	−	0.977776i	$-0.432767\pi$
$853$	12.2462	0.419302	0.209651	+	0.977776i	$0.432767\pi$
$854$	−5.12311	−0.175309
$855$	0	0
$856$	1.75379	0.0599433
$857$	35.3693	1.20819	0.604096	−	0.796911i	$-0.293534\pi$
$857$	35.3693	1.20819	0.604096	+	0.796911i	$0.293534\pi$
$858$	0	0
$859$	−18.4924	−0.630953	−0.315477	−	0.948933i	$-0.602164\pi$
$859$	−18.4924	−0.630953	−0.315477	+	0.948933i	$0.602164\pi$
$860$	3.12311	0.106497
$861$	0	0
$862$	23.6155	0.804348
$863$	−48.3542	−1.64599	−0.822997	−	0.568045i	$-0.807700\pi$
$863$	−48.3542	−1.64599	−0.822997	+	0.568045i	$0.807700\pi$
$864$	0	0
$865$	−12.2462	−0.416384
$866$	−7.12311	−0.242053
$867$	0	0
$868$	−1.12311	−0.0381207
$869$	5.12311	0.173789
$870$	0	0
$871$	−1.75379	−0.0594249
$872$	−18.0000	−0.609557
$873$	0	0
$874$	8.00000	0.270604
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	26.7386	0.902900	0.451450	−	0.892297i	$-0.350907\pi$
$877$	26.7386	0.902900	0.451450	+	0.892297i	$0.350907\pi$
$878$	−26.7386	−0.902385
$879$	0	0
$880$	1.00000	0.0337100
$881$	−2.49242	−0.0839718	−0.0419859	−	0.999118i	$-0.513368\pi$
$881$	−2.49242	−0.0839718	−0.0419859	+	0.999118i	$0.513368\pi$
$882$	0	0
$883$	−12.3845	−0.416771	−0.208385	−	0.978047i	$-0.566821\pi$
$883$	−12.3845	−0.416771	−0.208385	+	0.978047i	$0.566821\pi$
$884$	10.2462	0.344617
$885$	0	0
$886$	−13.6155	−0.457423
$887$	26.9848	0.906062	0.453031	−	0.891495i	$-0.350343\pi$
$887$	26.9848	0.906062	0.453031	+	0.891495i	$0.350343\pi$
$888$	0	0
$889$	18.2462	0.611958
$890$	−8.24621	−0.276414
$891$	0	0
$892$	−28.4924	−0.953997
$893$	95.2311	3.18679
$894$	0	0
$895$	2.24621	0.0750826
$896$	1.00000	0.0334077
$897$	0	0
$898$	−18.7386	−0.625316
$899$	−9.26137	−0.308884
$900$	0	0
$901$	−51.2311	−1.70675
$902$	0.876894	0.0291974
$903$	0	0
$904$	2.00000	0.0665190
$905$	19.3693	0.643858
$906$	0	0
$907$	−49.8617	−1.65563	−0.827816	−	0.561000i	$-0.810417\pi$
$907$	−49.8617	−1.65563	−0.827816	+	0.561000i	$0.810417\pi$
$908$	3.36932	0.111815
$909$	0	0
$910$	2.00000	0.0662994
$911$	−39.1231	−1.29621	−0.648103	−	0.761553i	$-0.724437\pi$
$911$	−39.1231	−1.29621	−0.648103	+	0.761553i	$0.724437\pi$
$912$	0	0
$913$	−2.87689	−0.0952113
$914$	13.3693	0.442218
$915$	0	0
$916$	2.87689	0.0950553
$917$	−22.2462	−0.734635
$918$	0	0
$919$	14.3845	0.474500	0.237250	−	0.971449i	$-0.423754\pi$
$919$	14.3845	0.474500	0.237250	+	0.971449i	$0.423754\pi$
$920$	−1.12311	−0.0370277
$921$	0	0
$922$	−4.87689	−0.160612
$923$	−14.2462	−0.468920
$924$	0	0
$925$	−1.12311	−0.0369275
$926$	26.7386	0.878686
$927$	0	0
$928$	8.24621	0.270695
$929$	3.26137	0.107002	0.0535010	−	0.998568i	$-0.482962\pi$
$929$	3.26137	0.107002	0.0535010	+	0.998568i	$0.482962\pi$
$930$	0	0
$931$	−7.12311	−0.233450
$932$	−6.00000	−0.196537
$933$	0	0
$934$	20.4924	0.670533
$935$	5.12311	0.167543
$936$	0	0
$937$	−57.2311	−1.86966	−0.934829	−	0.355099i	$-0.884447\pi$
$937$	−57.2311	−1.86966	−0.934829	+	0.355099i	$0.884447\pi$
$938$	−0.876894	−0.0286316
$939$	0	0
$940$	−13.3693	−0.436059
$941$	58.3542	1.90229	0.951146	−	0.308743i	$-0.0999081\pi$
$941$	58.3542	1.90229	0.951146	+	0.308743i	$0.0999081\pi$
$942$	0	0
$943$	−0.984845	−0.0320710
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−3.12311	−0.101541
$947$	47.8617	1.55530	0.777649	−	0.628699i	$-0.216412\pi$
$947$	47.8617	1.55530	0.777649	+	0.628699i	$0.216412\pi$
$948$	0	0
$949$	32.4924	1.05475
$950$	7.12311	0.231104
$951$	0	0
$952$	5.12311	0.166041
$953$	−46.9848	−1.52199	−0.760994	−	0.648759i	$-0.775288\pi$
$953$	−46.9848	−1.52199	−0.760994	+	0.648759i	$0.775288\pi$
$954$	0	0
$955$	−15.1231	−0.489372
$956$	−17.3693	−0.561764
$957$	0	0
$958$	16.4924	0.532846
$959$	3.75379	0.121216
$960$	0	0
$961$	−29.7386	−0.959311
$962$	2.24621	0.0724208
$963$	0	0
$964$	−10.4924	−0.337938
$965$	11.1231	0.358065
$966$	0	0
$967$	−19.5076	−0.627321	−0.313661	−	0.949535i	$-0.601555\pi$
$967$	−19.5076	−0.627321	−0.313661	+	0.949535i	$0.601555\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	11.1231	0.357141
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	0.876894	0.0281119
$974$	28.4924	0.912956
$975$	0	0
$976$	−5.12311	−0.163987
$977$	−37.2311	−1.19113	−0.595564	−	0.803308i	$-0.703071\pi$
$977$	−37.2311	−1.19113	−0.595564	+	0.803308i	$0.703071\pi$
$978$	0	0
$979$	8.24621	0.263550
$980$	1.00000	0.0319438
$981$	0	0
$982$	−10.2462	−0.326970
$983$	37.8617	1.20760	0.603801	−	0.797135i	$-0.293652\pi$
$983$	37.8617	1.20760	0.603801	+	0.797135i	$0.293652\pi$
$984$	0	0
$985$	10.4924	0.334316
$986$	42.2462	1.34539
$987$	0	0
$988$	−14.2462	−0.453232
$989$	3.50758	0.111534
$990$	0	0
$991$	45.4773	1.44463	0.722317	−	0.691563i	$-0.243077\pi$
$991$	45.4773	1.44463	0.722317	+	0.691563i	$0.243077\pi$
$992$	−1.12311	−0.0356586
$993$	0	0
$994$	−7.12311	−0.225931
$995$	−13.1231	−0.416030
$996$	0	0
$997$	0.738634	0.0233928	0.0116964	−	0.999932i	$-0.496277\pi$
$997$	0.738634	0.0233928	0.0116964	+	0.999932i	$0.496277\pi$
$998$	−30.7386	−0.973014
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.bs.1.2	✓	2
3.2	odd	2		6930.2.a.bx.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6930.2.a.bs.1.2	✓	2	1.1	even	1	trivial
6930.2.a.bx.1.1	yes	2	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.12311 q^{17} -7.12311 q^{19} +1.00000 q^{20} -1.00000 q^{22} +1.12311 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -8.24621 q^{29} +1.12311 q^{31} -1.00000 q^{32} -5.12311 q^{34} -1.00000 q^{35} -1.12311 q^{37} +7.12311 q^{38} -1.00000 q^{40} -0.876894 q^{41} +3.12311 q^{43} +1.00000 q^{44} -1.12311 q^{46} -13.3693 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -10.0000 q^{53} +1.00000 q^{55} +1.00000 q^{56} +8.24621 q^{58} -4.00000 q^{59} -5.12311 q^{61} -1.12311 q^{62} +1.00000 q^{64} +2.00000 q^{65} -0.876894 q^{67} +5.12311 q^{68} +1.00000 q^{70} -7.12311 q^{71} +16.2462 q^{73} +1.12311 q^{74} -7.12311 q^{76} -1.00000 q^{77} +5.12311 q^{79} +1.00000 q^{80} +0.876894 q^{82} -2.87689 q^{83} +5.12311 q^{85} -3.12311 q^{86} -1.00000 q^{88} +8.24621 q^{89} -2.00000 q^{91} +1.12311 q^{92} +13.3693 q^{94} -7.12311 q^{95} -11.1231 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 6 q^{19} + 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 6 q^{31} - 2 q^{32} - 2 q^{34} - 2 q^{35} + 6 q^{37} + 6 q^{38} - 2 q^{40} - 10 q^{41} - 2 q^{43} + 2 q^{44} + 6 q^{46} - 2 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} - 20 q^{53} + 2 q^{55} + 2 q^{56} - 8 q^{59} - 2 q^{61} + 6 q^{62} + 2 q^{64} + 4 q^{65} - 10 q^{67} + 2 q^{68} + 2 q^{70} - 6 q^{71} + 16 q^{73} - 6 q^{74} - 6 q^{76} - 2 q^{77} + 2 q^{79} + 2 q^{80} + 10 q^{82} - 14 q^{83} + 2 q^{85} + 2 q^{86} - 2 q^{88} - 4 q^{91} - 6 q^{92} + 2 q^{94} - 6 q^{95} - 14 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 6 * q^19 + 2 * q^20 - 2 * q^22 - 6 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 6 * q^31 - 2 * q^32 - 2 * q^34 - 2 * q^35 + 6 * q^37 + 6 * q^38 - 2 * q^40 - 10 * q^41 - 2 * q^43 + 2 * q^44 + 6 * q^46 - 2 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^52 - 20 * q^53 + 2 * q^55 + 2 * q^56 - 8 * q^59 - 2 * q^61 + 6 * q^62 + 2 * q^64 + 4 * q^65 - 10 * q^67 + 2 * q^68 + 2 * q^70 - 6 * q^71 + 16 * q^73 - 6 * q^74 - 6 * q^76 - 2 * q^77 + 2 * q^79 + 2 * q^80 + 10 * q^82 - 14 * q^83 + 2 * q^85 + 2 * q^86 - 2 * q^88 - 4 * q^91 - 6 * q^92 + 2 * q^94 - 6 * q^95 - 14 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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