LMFDB - Embedded newform 6930.2.a.cl.1.3

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.892.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 10 $ x^3 - x^2 - 8*x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.59774$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +6.69193 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.19547 q^{17} -1.84951 q^{19} +1.00000 q^{20} -1.00000 q^{22} -1.84951 q^{23} +1.00000 q^{25} +6.69193 q^{26} -1.00000 q^{28} -6.84242 q^{29} +6.00000 q^{31} +1.00000 q^{32} +7.19547 q^{34} -1.00000 q^{35} -6.54143 q^{37} -1.84951 q^{38} +1.00000 q^{40} +9.34596 q^{41} +0.503544 q^{43} -1.00000 q^{44} -1.84951 q^{46} +1.49646 q^{47} +1.00000 q^{49} +1.00000 q^{50} +6.69193 q^{52} -6.84242 q^{53} -1.00000 q^{55} -1.00000 q^{56} -6.84242 q^{58} +7.88740 q^{59} +4.50354 q^{61} +6.00000 q^{62} +1.00000 q^{64} +6.69193 q^{65} +8.00000 q^{67} +7.19547 q^{68} -1.00000 q^{70} -0.300986 q^{71} -10.1884 q^{73} -6.54143 q^{74} -1.84951 q^{76} +1.00000 q^{77} -12.2404 q^{79} +1.00000 q^{80} +9.34596 q^{82} +1.30807 q^{83} +7.19547 q^{85} +0.503544 q^{86} -1.00000 q^{88} +8.69193 q^{89} -6.69193 q^{91} -1.84951 q^{92} +1.49646 q^{94} -1.84951 q^{95} +3.84951 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} + 8 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} - 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 4 q^{37} - 2 q^{38} + 3 q^{40} + 18 q^{41} + 8 q^{43} - 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} + 3 q^{50} - 4 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 18 q^{62} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 3 q^{70} - 8 q^{71} - 4 q^{73} + 4 q^{74} - 2 q^{76} + 3 q^{77} - 6 q^{79} + 3 q^{80} + 18 q^{82} + 24 q^{83} + 8 q^{85} + 8 q^{86} - 3 q^{88} + 6 q^{89} - 2 q^{92} - 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8 + 3 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 + 8 * q^17 - 2 * q^19 + 3 * q^20 - 3 * q^22 - 2 * q^23 + 3 * q^25 - 3 * q^28 - 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 - 3 * q^35 + 4 * q^37 - 2 * q^38 + 3 * q^40 + 18 * q^41 + 8 * q^43 - 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 + 3 * q^50 - 4 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 10 * q^59 + 20 * q^61 + 18 * q^62 + 3 * q^64 + 24 * q^67 + 8 * q^68 - 3 * q^70 - 8 * q^71 - 4 * q^73 + 4 * q^74 - 2 * q^76 + 3 * q^77 - 6 * q^79 + 3 * q^80 + 18 * q^82 + 24 * q^83 + 8 * q^85 + 8 * q^86 - 3 * q^88 + 6 * q^89 - 2 * q^92 - 2 * q^94 - 2 * q^95 + 8 * q^97 + 3 * 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
$13$	6.69193	1.85601	0.928003	−	0.372572i	$-0.121524\pi$
$13$	6.69193	1.85601	0.928003	+	0.372572i	$0.121524\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.19547	1.74516	0.872579	−	0.488473i	$-0.162446\pi$
$17$	7.19547	1.74516	0.872579	+	0.488473i	$0.162446\pi$
$18$	0	0
$19$	−1.84951	−0.424306	−0.212153	−	0.977236i	$-0.568048\pi$
$19$	−1.84951	−0.424306	−0.212153	+	0.977236i	$0.568048\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−1.84951	−0.385649	−0.192824	−	0.981233i	$-0.561765\pi$
$23$	−1.84951	−0.385649	−0.192824	+	0.981233i	$0.561765\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.69193	1.31239
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−6.84242	−1.27061	−0.635303	−	0.772263i	$-0.719125\pi$
$29$	−6.84242	−1.27061	−0.635303	+	0.772263i	$0.719125\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.19547	1.23401
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−6.54143	−1.07541	−0.537703	−	0.843135i	$-0.680708\pi$
$37$	−6.54143	−1.07541	−0.537703	+	0.843135i	$0.680708\pi$
$38$	−1.84951	−0.300030
$39$	0	0
$40$	1.00000	0.158114
$41$	9.34596	1.45959	0.729797	−	0.683664i	$-0.239615\pi$
$41$	9.34596	1.45959	0.729797	+	0.683664i	$0.239615\pi$
$42$	0	0
$43$	0.503544	0.0767897	0.0383949	−	0.999263i	$-0.487776\pi$
$43$	0.503544	0.0767897	0.0383949	+	0.999263i	$0.487776\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.84951	−0.272695
$47$	1.49646	0.218281	0.109140	−	0.994026i	$-0.465190\pi$
$47$	1.49646	0.218281	0.109140	+	0.994026i	$0.465190\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	6.69193	0.928003
$53$	−6.84242	−0.939879	−0.469939	−	0.882699i	$-0.655724\pi$
$53$	−6.84242	−0.939879	−0.469939	+	0.882699i	$0.655724\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.84242	−0.898454
$59$	7.88740	1.02685	0.513426	−	0.858134i	$-0.328376\pi$
$59$	7.88740	1.02685	0.513426	+	0.858134i	$0.328376\pi$
$60$	0	0
$61$	4.50354	0.576620	0.288310	−	0.957537i	$-0.406907\pi$
$61$	4.50354	0.576620	0.288310	+	0.957537i	$0.406907\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	6.69193	0.830031
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	7.19547	0.872579
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−0.300986	−0.0357204	−0.0178602	−	0.999840i	$-0.505685\pi$
$71$	−0.300986	−0.0357204	−0.0178602	+	0.999840i	$0.505685\pi$
$72$	0	0
$73$	−10.1884	−1.19246	−0.596230	−	0.802814i	$-0.703335\pi$
$73$	−10.1884	−1.19246	−0.596230	+	0.802814i	$0.703335\pi$
$74$	−6.54143	−0.760426
$75$	0	0
$76$	−1.84951	−0.212153
$77$	1.00000	0.113961
$78$	0	0
$79$	−12.2404	−1.37716	−0.688579	−	0.725161i	$-0.741765\pi$
$79$	−12.2404	−1.37716	−0.688579	+	0.725161i	$0.741765\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	9.34596	1.03209
$83$	1.30807	0.143580	0.0717899	−	0.997420i	$-0.477129\pi$
$83$	1.30807	0.143580	0.0717899	+	0.997420i	$0.477129\pi$
$84$	0	0
$85$	7.19547	0.780458
$86$	0.503544	0.0542985
$87$	0	0
$88$	−1.00000	−0.106600
$89$	8.69193	0.921342	0.460671	−	0.887571i	$-0.347609\pi$
$89$	8.69193	0.921342	0.460671	+	0.887571i	$0.347609\pi$
$90$	0	0
$91$	−6.69193	−0.701505
$92$	−1.84951	−0.192824
$93$	0	0
$94$	1.49646	0.154348
$95$	−1.84951	−0.189755
$96$	0	0
$97$	3.84951	0.390858	0.195429	−	0.980718i	$-0.437390\pi$
$97$	3.84951	0.390858	0.195429	+	0.980718i	$0.437390\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.89448	−0.686027	−0.343013	−	0.939331i	$-0.611448\pi$
$101$	−6.89448	−0.686027	−0.343013	+	0.939331i	$0.611448\pi$
$102$	0	0
$103$	−10.5793	−1.04241	−0.521206	−	0.853431i	$-0.674518\pi$
$103$	−10.5793	−1.04241	−0.521206	+	0.853431i	$0.674518\pi$
$104$	6.69193	0.656197
$105$	0	0
$106$	−6.84242	−0.664595
$107$	7.49646	0.724710	0.362355	−	0.932040i	$-0.381973\pi$
$107$	7.49646	0.724710	0.362355	+	0.932040i	$0.381973\pi$
$108$	0	0
$109$	1.45857	0.139705	0.0698527	−	0.997557i	$-0.477747\pi$
$109$	1.45857	0.139705	0.0698527	+	0.997557i	$0.477747\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−1.84951	−0.172467
$116$	−6.84242	−0.635303
$117$	0	0
$118$	7.88740	0.726094
$119$	−7.19547	−0.659608
$120$	0	0
$121$	1.00000	0.0909091
$122$	4.50354	0.407732
$123$	0	0
$124$	6.00000	0.538816
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.7748	−1.75473	−0.877365	−	0.479824i	$-0.840700\pi$
$127$	−19.7748	−1.75473	−0.877365	+	0.479824i	$0.840700\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	6.69193	0.586921
$131$	−2.15049	−0.187889	−0.0939447	−	0.995577i	$-0.529948\pi$
$131$	−2.15049	−0.187889	−0.0939447	+	0.995577i	$0.529948\pi$
$132$	0	0
$133$	1.84951	0.160373
$134$	8.00000	0.691095
$135$	0	0
$136$	7.19547	0.617006
$137$	8.39094	0.716886	0.358443	−	0.933552i	$-0.383308\pi$
$137$	8.39094	0.716886	0.358443	+	0.933552i	$0.383308\pi$
$138$	0	0
$139$	17.9253	1.52040	0.760201	−	0.649687i	$-0.225100\pi$
$139$	17.9253	1.52040	0.760201	+	0.649687i	$0.225100\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−0.300986	−0.0252582
$143$	−6.69193	−0.559607
$144$	0	0
$145$	−6.84242	−0.568232
$146$	−10.1884	−0.843197
$147$	0	0
$148$	−6.54143	−0.537703
$149$	0.451479	0.0369866	0.0184933	−	0.999829i	$-0.494113\pi$
$149$	0.451479	0.0369866	0.0184933	+	0.999829i	$0.494113\pi$
$150$	0	0
$151$	19.2334	1.56519	0.782594	−	0.622532i	$-0.213896\pi$
$151$	19.2334	1.56519	0.782594	+	0.622532i	$0.213896\pi$
$152$	−1.84951	−0.150015
$153$	0	0
$154$	1.00000	0.0805823
$155$	6.00000	0.481932
$156$	0	0
$157$	−9.69901	−0.774066	−0.387033	−	0.922066i	$-0.626500\pi$
$157$	−9.69901	−0.774066	−0.387033	+	0.922066i	$0.626500\pi$
$158$	−12.2404	−0.973798
$159$	0	0
$160$	1.00000	0.0790569
$161$	1.84951	0.145762
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	9.34596	0.729797
$165$	0	0
$166$	1.30807	0.101526
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	31.7819	2.44476
$170$	7.19547	0.551867
$171$	0	0
$172$	0.503544	0.0383949
$173$	15.4965	1.17817	0.589087	−	0.808070i	$-0.299488\pi$
$173$	15.4965	1.17817	0.589087	+	0.808070i	$0.299488\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	8.69193	0.651487
$179$	13.8874	1.03799	0.518996	−	0.854776i	$-0.326306\pi$
$179$	13.8874	1.03799	0.518996	+	0.854776i	$0.326306\pi$
$180$	0	0
$181$	18.7819	1.39605	0.698023	−	0.716075i	$-0.254063\pi$
$181$	18.7819	1.39605	0.698023	+	0.716075i	$0.254063\pi$
$182$	−6.69193	−0.496039
$183$	0	0
$184$	−1.84951	−0.136347
$185$	−6.54143	−0.480936
$186$	0	0
$187$	−7.19547	−0.526185
$188$	1.49646	0.109140
$189$	0	0
$190$	−1.84951	−0.134177
$191$	6.39094	0.462432	0.231216	−	0.972902i	$-0.425730\pi$
$191$	6.39094	0.462432	0.231216	+	0.972902i	$0.425730\pi$
$192$	0	0
$193$	−13.1955	−0.949831	−0.474915	−	0.880031i	$-0.657521\pi$
$193$	−13.1955	−0.949831	−0.474915	+	0.880031i	$0.657521\pi$
$194$	3.84951	0.276379
$195$	0	0
$196$	1.00000	0.0714286
$197$	4.69193	0.334286	0.167143	−	0.985933i	$-0.446546\pi$
$197$	4.69193	0.334286	0.167143	+	0.985933i	$0.446546\pi$
$198$	0	0
$199$	−9.49646	−0.673186	−0.336593	−	0.941650i	$-0.609275\pi$
$199$	−9.49646	−0.673186	−0.336593	+	0.941650i	$0.609275\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.89448	−0.485094
$203$	6.84242	0.480244
$204$	0	0
$205$	9.34596	0.652750
$206$	−10.5793	−0.737096
$207$	0	0
$208$	6.69193	0.464002
$209$	1.84951	0.127933
$210$	0	0
$211$	−2.39094	−0.164599	−0.0822996	−	0.996608i	$-0.526226\pi$
$211$	−2.39094	−0.164599	−0.0822996	+	0.996608i	$0.526226\pi$
$212$	−6.84242	−0.469939
$213$	0	0
$214$	7.49646	0.512447
$215$	0.503544	0.0343414
$216$	0	0
$217$	−6.00000	−0.407307
$218$	1.45857	0.0987866
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	48.1516	3.23902
$222$	0	0
$223$	0.188383	0.0126150	0.00630752	−	0.999980i	$-0.497992\pi$
$223$	0.188383	0.0126150	0.00630752	+	0.999980i	$0.497992\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	6.99291	0.464136	0.232068	−	0.972700i	$-0.425451\pi$
$227$	6.99291	0.464136	0.232068	+	0.972700i	$0.425451\pi$
$228$	0	0
$229$	−9.19547	−0.607654	−0.303827	−	0.952727i	$-0.598264\pi$
$229$	−9.19547	−0.607654	−0.303827	+	0.952727i	$0.598264\pi$
$230$	−1.84951	−0.121953
$231$	0	0
$232$	−6.84242	−0.449227
$233$	0.992912	0.0650479	0.0325239	−	0.999471i	$-0.489645\pi$
$233$	0.992912	0.0650479	0.0325239	+	0.999471i	$0.489645\pi$
$234$	0	0
$235$	1.49646	0.0976180
$236$	7.88740	0.513426
$237$	0	0
$238$	−7.19547	−0.466413
$239$	1.14341	0.0739607	0.0369804	−	0.999316i	$-0.488226\pi$
$239$	1.14341	0.0739607	0.0369804	+	0.999316i	$0.488226\pi$
$240$	0	0
$241$	8.33888	0.537154	0.268577	−	0.963258i	$-0.413447\pi$
$241$	8.33888	0.537154	0.268577	+	0.963258i	$0.413447\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	4.50354	0.288310
$245$	1.00000	0.0638877
$246$	0	0
$247$	−12.3768	−0.787515
$248$	6.00000	0.381000
$249$	0	0
$250$	1.00000	0.0632456
$251$	−23.8874	−1.50776	−0.753880	−	0.657013i	$-0.771820\pi$
$251$	−23.8874	−1.50776	−0.753880	+	0.657013i	$0.771820\pi$
$252$	0	0
$253$	1.84951	0.116278
$254$	−19.7748	−1.24078
$255$	0	0
$256$	1.00000	0.0625000
$257$	31.6243	1.97267	0.986335	−	0.164753i	$-0.0526827\pi$
$257$	31.6243	1.97267	0.986335	+	0.164753i	$0.0526827\pi$
$258$	0	0
$259$	6.54143	0.406465
$260$	6.69193	0.415016
$261$	0	0
$262$	−2.15049	−0.132858
$263$	−13.3839	−0.825284	−0.412642	−	0.910893i	$-0.635394\pi$
$263$	−13.3839	−0.825284	−0.412642	+	0.910893i	$0.635394\pi$
$264$	0	0
$265$	−6.84242	−0.420326
$266$	1.84951	0.113401
$267$	0	0
$268$	8.00000	0.488678
$269$	−5.79744	−0.353476	−0.176738	−	0.984258i	$-0.556555\pi$
$269$	−5.79744	−0.353476	−0.176738	+	0.984258i	$0.556555\pi$
$270$	0	0
$271$	−20.0758	−1.21952	−0.609758	−	0.792587i	$-0.708734\pi$
$271$	−20.0758	−1.21952	−0.609758	+	0.792587i	$0.708734\pi$
$272$	7.19547	0.436289
$273$	0	0
$274$	8.39094	0.506915
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	17.9253	1.07509
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	6.30099	0.375885	0.187943	−	0.982180i	$-0.439818\pi$
$281$	6.30099	0.375885	0.187943	+	0.982180i	$0.439818\pi$
$282$	0	0
$283$	22.3909	1.33100	0.665502	−	0.746396i	$-0.268218\pi$
$283$	22.3909	1.33100	0.665502	+	0.746396i	$0.268218\pi$
$284$	−0.300986	−0.0178602
$285$	0	0
$286$	−6.69193	−0.395702
$287$	−9.34596	−0.551675
$288$	0	0
$289$	34.7748	2.04558
$290$	−6.84242	−0.401801
$291$	0	0
$292$	−10.1884	−0.596230
$293$	10.6919	0.624629	0.312315	−	0.949979i	$-0.398896\pi$
$293$	10.6919	0.624629	0.312315	+	0.949979i	$0.398896\pi$
$294$	0	0
$295$	7.88740	0.459222
$296$	−6.54143	−0.380213
$297$	0	0
$298$	0.451479	0.0261535
$299$	−12.3768	−0.715767
$300$	0	0
$301$	−0.503544	−0.0290238
$302$	19.2334	1.10676
$303$	0	0
$304$	−1.84951	−0.106077
$305$	4.50354	0.257872
$306$	0	0
$307$	−9.08287	−0.518387	−0.259193	−	0.965825i	$-0.583457\pi$
$307$	−9.08287	−0.518387	−0.259193	+	0.965825i	$0.583457\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	6.00000	0.340777
$311$	11.3839	0.645519	0.322760	−	0.946481i	$-0.395389\pi$
$311$	11.3839	0.645519	0.322760	+	0.946481i	$0.395389\pi$
$312$	0	0
$313$	−23.6243	−1.33532	−0.667662	−	0.744464i	$-0.732705\pi$
$313$	−23.6243	−1.33532	−0.667662	+	0.744464i	$0.732705\pi$
$314$	−9.69901	−0.547347
$315$	0	0
$316$	−12.2404	−0.688579
$317$	9.53435	0.535502	0.267751	−	0.963488i	$-0.413720\pi$
$317$	9.53435	0.535502	0.267751	+	0.963488i	$0.413720\pi$
$318$	0	0
$319$	6.84242	0.383102
$320$	1.00000	0.0559017
$321$	0	0
$322$	1.84951	0.103069
$323$	−13.3081	−0.740481
$324$	0	0
$325$	6.69193	0.371201
$326$	−8.00000	−0.443079
$327$	0	0
$328$	9.34596	0.516044
$329$	−1.49646	−0.0825023
$330$	0	0
$331$	0.503544	0.0276773	0.0138386	−	0.999904i	$-0.495595\pi$
$331$	0.503544	0.0276773	0.0138386	+	0.999904i	$0.495595\pi$
$332$	1.30807	0.0717899
$333$	0	0
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	12.3909	0.674978	0.337489	−	0.941330i	$-0.390422\pi$
$337$	12.3909	0.674978	0.337489	+	0.941330i	$0.390422\pi$
$338$	31.7819	1.72871
$339$	0	0
$340$	7.19547	0.390229
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0.503544	0.0271493
$345$	0	0
$346$	15.4965	0.833095
$347$	−20.8803	−1.12091	−0.560457	−	0.828184i	$-0.689374\pi$
$347$	−20.8803	−1.12091	−0.560457	+	0.828184i	$0.689374\pi$
$348$	0	0
$349$	−22.1884	−1.18772	−0.593858	−	0.804570i	$-0.702396\pi$
$349$	−22.1884	−1.18772	−0.593858	+	0.804570i	$0.702396\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	31.6243	1.68319	0.841596	−	0.540108i	$-0.181617\pi$
$353$	31.6243	1.68319	0.841596	+	0.540108i	$0.181617\pi$
$354$	0	0
$355$	−0.300986	−0.0159747
$356$	8.69193	0.460671
$357$	0	0
$358$	13.8874	0.733972
$359$	−30.5273	−1.61117	−0.805584	−	0.592482i	$-0.798148\pi$
$359$	−30.5273	−1.61117	−0.805584	+	0.592482i	$0.798148\pi$
$360$	0	0
$361$	−15.5793	−0.819964
$362$	18.7819	0.987154
$363$	0	0
$364$	−6.69193	−0.350752
$365$	−10.1884	−0.533284
$366$	0	0
$367$	−10.5793	−0.552236	−0.276118	−	0.961124i	$-0.589048\pi$
$367$	−10.5793	−0.552236	−0.276118	+	0.961124i	$0.589048\pi$
$368$	−1.84951	−0.0964122
$369$	0	0
$370$	−6.54143	−0.340073
$371$	6.84242	0.355241
$372$	0	0
$373$	36.4667	1.88818	0.944088	−	0.329695i	$-0.106946\pi$
$373$	36.4667	1.88818	0.944088	+	0.329695i	$0.106946\pi$
$374$	−7.19547	−0.372069
$375$	0	0
$376$	1.49646	0.0771738
$377$	−45.7890	−2.35825
$378$	0	0
$379$	14.9929	0.770134	0.385067	−	0.922889i	$-0.374178\pi$
$379$	14.9929	0.770134	0.385067	+	0.922889i	$0.374178\pi$
$380$	−1.84951	−0.0948777
$381$	0	0
$382$	6.39094	0.326989
$383$	−35.1813	−1.79768	−0.898840	−	0.438277i	$-0.855589\pi$
$383$	−35.1813	−1.79768	−0.898840	+	0.438277i	$0.855589\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	−13.1955	−0.671632
$387$	0	0
$388$	3.84951	0.195429
$389$	−2.30099	−0.116665	−0.0583323	−	0.998297i	$-0.518578\pi$
$389$	−2.30099	−0.116665	−0.0583323	+	0.998297i	$0.518578\pi$
$390$	0	0
$391$	−13.3081	−0.673018
$392$	1.00000	0.0505076
$393$	0	0
$394$	4.69193	0.236376
$395$	−12.2404	−0.615884
$396$	0	0
$397$	18.0758	0.907197	0.453599	−	0.891206i	$-0.350140\pi$
$397$	18.0758	0.907197	0.453599	+	0.891206i	$0.350140\pi$
$398$	−9.49646	−0.476014
$399$	0	0
$400$	1.00000	0.0500000
$401$	−22.5793	−1.12756	−0.563779	−	0.825926i	$-0.690653\pi$
$401$	−22.5793	−1.12756	−0.563779	+	0.825926i	$0.690653\pi$
$402$	0	0
$403$	40.1516	2.00009
$404$	−6.89448	−0.343013
$405$	0	0
$406$	6.84242	0.339584
$407$	6.54143	0.324247
$408$	0	0
$409$	1.04498	0.0516708	0.0258354	−	0.999666i	$-0.491775\pi$
$409$	1.04498	0.0516708	0.0258354	+	0.999666i	$0.491775\pi$
$410$	9.34596	0.461564
$411$	0	0
$412$	−10.5793	−0.521206
$413$	−7.88740	−0.388113
$414$	0	0
$415$	1.30807	0.0642108
$416$	6.69193	0.328099
$417$	0	0
$418$	1.84951	0.0904623
$419$	−5.49646	−0.268519	−0.134260	−	0.990946i	$-0.542866\pi$
$419$	−5.49646	−0.268519	−0.134260	+	0.990946i	$0.542866\pi$
$420$	0	0
$421$	2.60197	0.126812	0.0634062	−	0.997988i	$-0.479804\pi$
$421$	2.60197	0.126812	0.0634062	+	0.997988i	$0.479804\pi$
$422$	−2.39094	−0.116389
$423$	0	0
$424$	−6.84242	−0.332297
$425$	7.19547	0.349032
$426$	0	0
$427$	−4.50354	−0.217942
$428$	7.49646	0.362355
$429$	0	0
$430$	0.503544	0.0242830
$431$	−19.2334	−0.926438	−0.463219	−	0.886244i	$-0.653306\pi$
$431$	−19.2334	−0.926438	−0.463219	+	0.886244i	$0.653306\pi$
$432$	0	0
$433$	−8.93237	−0.429263	−0.214631	−	0.976695i	$-0.568855\pi$
$433$	−8.93237	−0.429263	−0.214631	+	0.976695i	$0.568855\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	1.45857	0.0698527
$437$	3.42068	0.163633
$438$	0	0
$439$	−0.300986	−0.0143653	−0.00718264	−	0.999974i	$-0.502286\pi$
$439$	−0.300986	−0.0143653	−0.00718264	+	0.999974i	$0.502286\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	48.1516	2.29034
$443$	−30.7677	−1.46182	−0.730909	−	0.682475i	$-0.760904\pi$
$443$	−30.7677	−1.46182	−0.730909	+	0.682475i	$0.760904\pi$
$444$	0	0
$445$	8.69193	0.412037
$446$	0.188383	0.00892018
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	1.19547	0.0564177	0.0282089	−	0.999602i	$-0.491020\pi$
$449$	1.19547	0.0564177	0.0282089	+	0.999602i	$0.491020\pi$
$450$	0	0
$451$	−9.34596	−0.440084
$452$	14.0000	0.658505
$453$	0	0
$454$	6.99291	0.328194
$455$	−6.69193	−0.313722
$456$	0	0
$457$	−25.7748	−1.20569	−0.602847	−	0.797857i	$-0.705967\pi$
$457$	−25.7748	−1.20569	−0.602847	+	0.797857i	$0.705967\pi$
$458$	−9.19547	−0.429676
$459$	0	0
$460$	−1.84951	−0.0862337
$461$	−11.5722	−0.538973	−0.269486	−	0.963004i	$-0.586854\pi$
$461$	−11.5722	−0.538973	−0.269486	+	0.963004i	$0.586854\pi$
$462$	0	0
$463$	−32.9182	−1.52984	−0.764919	−	0.644126i	$-0.777221\pi$
$463$	−32.9182	−1.52984	−0.764919	+	0.644126i	$0.777221\pi$
$464$	−6.84242	−0.317651
$465$	0	0
$466$	0.992912	0.0459958
$467$	14.6399	0.677452	0.338726	−	0.940885i	$-0.390004\pi$
$467$	14.6399	0.677452	0.338726	+	0.940885i	$0.390004\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	1.49646	0.0690264
$471$	0	0
$472$	7.88740	0.363047
$473$	−0.503544	−0.0231530
$474$	0	0
$475$	−1.84951	−0.0848612
$476$	−7.19547	−0.329804
$477$	0	0
$478$	1.14341	0.0522981
$479$	−10.3909	−0.474774	−0.237387	−	0.971415i	$-0.576291\pi$
$479$	−10.3909	−0.474774	−0.237387	+	0.971415i	$0.576291\pi$
$480$	0	0
$481$	−43.7748	−1.99596
$482$	8.33888	0.379825
$483$	0	0
$484$	1.00000	0.0454545
$485$	3.84951	0.174797
$486$	0	0
$487$	39.3091	1.78127	0.890634	−	0.454722i	$-0.150261\pi$
$487$	39.3091	1.78127	0.890634	+	0.454722i	$0.150261\pi$
$488$	4.50354	0.203866
$489$	0	0
$490$	1.00000	0.0451754
$491$	−23.7748	−1.07294	−0.536471	−	0.843919i	$-0.680243\pi$
$491$	−23.7748	−1.07294	−0.536471	+	0.843919i	$0.680243\pi$
$492$	0	0
$493$	−49.2344	−2.21741
$494$	−12.3768	−0.556857
$495$	0	0
$496$	6.00000	0.269408
$497$	0.300986	0.0135011
$498$	0	0
$499$	−6.11260	−0.273638	−0.136819	−	0.990596i	$-0.543688\pi$
$499$	−6.11260	−0.273638	−0.136819	+	0.990596i	$0.543688\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−23.8874	−1.06615
$503$	13.1586	0.586715	0.293358	−	0.956003i	$-0.405227\pi$
$503$	13.1586	0.586715	0.293358	+	0.956003i	$0.405227\pi$
$504$	0	0
$505$	−6.89448	−0.306801
$506$	1.84951	0.0822206
$507$	0	0
$508$	−19.7748	−0.877365
$509$	−34.1799	−1.51500	−0.757499	−	0.652836i	$-0.773579\pi$
$509$	−34.1799	−1.51500	−0.757499	+	0.652836i	$0.773579\pi$
$510$	0	0
$511$	10.1884	0.450708
$512$	1.00000	0.0441942
$513$	0	0
$514$	31.6243	1.39489
$515$	−10.5793	−0.466181
$516$	0	0
$517$	−1.49646	−0.0658141
$518$	6.54143	0.287414
$519$	0	0
$520$	6.69193	0.293460
$521$	26.6778	1.16877	0.584387	−	0.811475i	$-0.301335\pi$
$521$	26.6778	1.16877	0.584387	+	0.811475i	$0.301335\pi$
$522$	0	0
$523$	25.4880	1.11451	0.557256	−	0.830341i	$-0.311854\pi$
$523$	25.4880	1.11451	0.557256	+	0.830341i	$0.311854\pi$
$524$	−2.15049	−0.0939447
$525$	0	0
$526$	−13.3839	−0.583564
$527$	43.1728	1.88064
$528$	0	0
$529$	−19.5793	−0.851275
$530$	−6.84242	−0.297216
$531$	0	0
$532$	1.84951	0.0801863
$533$	62.5425	2.70902
$534$	0	0
$535$	7.49646	0.324100
$536$	8.00000	0.345547
$537$	0	0
$538$	−5.79744	−0.249945
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	4.45148	0.191384	0.0956920	−	0.995411i	$-0.469494\pi$
$541$	4.45148	0.191384	0.0956920	+	0.995411i	$0.469494\pi$
$542$	−20.0758	−0.862329
$543$	0	0
$544$	7.19547	0.308503
$545$	1.45857	0.0624781
$546$	0	0
$547$	−1.38385	−0.0591693	−0.0295846	−	0.999562i	$-0.509418\pi$
$547$	−1.38385	−0.0591693	−0.0295846	+	0.999562i	$0.509418\pi$
$548$	8.39094	0.358443
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	12.6551	0.539126
$552$	0	0
$553$	12.2404	0.520517
$554$	22.0000	0.934690
$555$	0	0
$556$	17.9253	0.760201
$557$	6.70610	0.284147	0.142073	−	0.989856i	$-0.454623\pi$
$557$	6.70610	0.284147	0.142073	+	0.989856i	$0.454623\pi$
$558$	0	0
$559$	3.36968	0.142522
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	6.30099	0.265791
$563$	−3.59488	−0.151506	−0.0757532	−	0.997127i	$-0.524136\pi$
$563$	−3.59488	−0.151506	−0.0757532	+	0.997127i	$0.524136\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	22.3909	0.941161
$567$	0	0
$568$	−0.300986	−0.0126291
$569$	−31.7606	−1.33147	−0.665737	−	0.746186i	$-0.731883\pi$
$569$	−31.7606	−1.33147	−0.665737	+	0.746186i	$0.731883\pi$
$570$	0	0
$571$	−5.68484	−0.237903	−0.118952	−	0.992900i	$-0.537953\pi$
$571$	−5.68484	−0.237903	−0.118952	+	0.992900i	$0.537953\pi$
$572$	−6.69193	−0.279804
$573$	0	0
$574$	−9.34596	−0.390093
$575$	−1.84951	−0.0771298
$576$	0	0
$577$	13.5343	0.563442	0.281721	−	0.959496i	$-0.409095\pi$
$577$	13.5343	0.563442	0.281721	+	0.959496i	$0.409095\pi$
$578$	34.7748	1.44644
$579$	0	0
$580$	−6.84242	−0.284116
$581$	−1.30807	−0.0542680
$582$	0	0
$583$	6.84242	0.283384
$584$	−10.1884	−0.421598
$585$	0	0
$586$	10.6919	0.441679
$587$	0.0520650	0.00214895	0.00107448	−	0.999999i	$-0.499658\pi$
$587$	0.0520650	0.00214895	0.00107448	+	0.999999i	$0.499658\pi$
$588$	0	0
$589$	−11.0970	−0.457246
$590$	7.88740	0.324719
$591$	0	0
$592$	−6.54143	−0.268851
$593$	−31.9774	−1.31315	−0.656576	−	0.754260i	$-0.727996\pi$
$593$	−31.9774	−1.31315	−0.656576	+	0.754260i	$0.727996\pi$
$594$	0	0
$595$	−7.19547	−0.294986
$596$	0.451479	0.0184933
$597$	0	0
$598$	−12.3768	−0.506124
$599$	−21.0829	−0.861423	−0.430711	−	0.902490i	$-0.641737\pi$
$599$	−21.0829	−0.861423	−0.430711	+	0.902490i	$0.641737\pi$
$600$	0	0
$601$	−21.6469	−0.882997	−0.441499	−	0.897262i	$-0.645553\pi$
$601$	−21.6469	−0.882997	−0.441499	+	0.897262i	$0.645553\pi$
$602$	−0.503544	−0.0205229
$603$	0	0
$604$	19.2334	0.782594
$605$	1.00000	0.0406558
$606$	0	0
$607$	21.6622	0.879241	0.439621	−	0.898184i	$-0.355113\pi$
$607$	21.6622	0.879241	0.439621	+	0.898184i	$0.355113\pi$
$608$	−1.84951	−0.0750074
$609$	0	0
$610$	4.50354	0.182343
$611$	10.0142	0.405130
$612$	0	0
$613$	8.31516	0.335846	0.167923	−	0.985800i	$-0.446294\pi$
$613$	8.31516	0.335846	0.167923	+	0.985800i	$0.446294\pi$
$614$	−9.08287	−0.366555
$615$	0	0
$616$	1.00000	0.0402911
$617$	−12.4667	−0.501891	−0.250946	−	0.968001i	$-0.580742\pi$
$617$	−12.4667	−0.501891	−0.250946	+	0.968001i	$0.580742\pi$
$618$	0	0
$619$	5.27125	0.211869	0.105935	−	0.994373i	$-0.466217\pi$
$619$	5.27125	0.211869	0.105935	+	0.994373i	$0.466217\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	11.3839	0.456451
$623$	−8.69193	−0.348235
$624$	0	0
$625$	1.00000	0.0400000
$626$	−23.6243	−0.944217
$627$	0	0
$628$	−9.69901	−0.387033
$629$	−47.0687	−1.87675
$630$	0	0
$631$	−17.1586	−0.683075	−0.341537	−	0.939868i	$-0.610948\pi$
$631$	−17.1586	−0.683075	−0.341537	+	0.939868i	$0.610948\pi$
$632$	−12.2404	−0.486899
$633$	0	0
$634$	9.53435	0.378657
$635$	−19.7748	−0.784739
$636$	0	0
$637$	6.69193	0.265144
$638$	6.84242	0.270894
$639$	0	0
$640$	1.00000	0.0395285
$641$	−24.7677	−0.978266	−0.489133	−	0.872209i	$-0.662687\pi$
$641$	−24.7677	−0.978266	−0.489133	+	0.872209i	$0.662687\pi$
$642$	0	0
$643$	1.03080	0.0406509	0.0203254	−	0.999793i	$-0.493530\pi$
$643$	1.03080	0.0406509	0.0203254	+	0.999793i	$0.493530\pi$
$644$	1.84951	0.0728808
$645$	0	0
$646$	−13.3081	−0.523599
$647$	−12.7904	−0.502841	−0.251420	−	0.967878i	$-0.580898\pi$
$647$	−12.7904	−0.502841	−0.251420	+	0.967878i	$0.580898\pi$
$648$	0	0
$649$	−7.88740	−0.309607
$650$	6.69193	0.262479
$651$	0	0
$652$	−8.00000	−0.313304
$653$	22.3162	0.873301	0.436651	−	0.899631i	$-0.356165\pi$
$653$	22.3162	0.873301	0.436651	+	0.899631i	$0.356165\pi$
$654$	0	0
$655$	−2.15049	−0.0840267
$656$	9.34596	0.364899
$657$	0	0
$658$	−1.49646	−0.0583379
$659$	−22.6919	−0.883952	−0.441976	−	0.897027i	$-0.645722\pi$
$659$	−22.6919	−0.883952	−0.441976	+	0.897027i	$0.645722\pi$
$660$	0	0
$661$	43.5638	1.69443	0.847217	−	0.531247i	$-0.178276\pi$
$661$	43.5638	1.69443	0.847217	+	0.531247i	$0.178276\pi$
$662$	0.503544	0.0195708
$663$	0	0
$664$	1.30807	0.0507631
$665$	1.84951	0.0717208
$666$	0	0
$667$	12.6551	0.490008
$668$	−8.00000	−0.309529
$669$	0	0
$670$	8.00000	0.309067
$671$	−4.50354	−0.173857
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	12.3909	0.477281
$675$	0	0
$676$	31.7819	1.22238
$677$	16.0984	0.618713	0.309356	−	0.950946i	$-0.399886\pi$
$677$	16.0984	0.618713	0.309356	+	0.950946i	$0.399886\pi$
$678$	0	0
$679$	−3.84951	−0.147731
$680$	7.19547	0.275934
$681$	0	0
$682$	−6.00000	−0.229752
$683$	−29.0829	−1.11282	−0.556412	−	0.830906i	$-0.687823\pi$
$683$	−29.0829	−1.11282	−0.556412	+	0.830906i	$0.687823\pi$
$684$	0	0
$685$	8.39094	0.320601
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	0.503544	0.0191974
$689$	−45.7890	−1.74442
$690$	0	0
$691$	7.58641	0.288601	0.144300	−	0.989534i	$-0.453907\pi$
$691$	7.58641	0.288601	0.144300	+	0.989534i	$0.453907\pi$
$692$	15.4965	0.589087
$693$	0	0
$694$	−20.8803	−0.792606
$695$	17.9253	0.679945
$696$	0	0
$697$	67.2486	2.54722
$698$	−22.1884	−0.839843
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−4.07471	−0.153900	−0.0769499	−	0.997035i	$-0.524518\pi$
$701$	−4.07471	−0.153900	−0.0769499	+	0.997035i	$0.524518\pi$
$702$	0	0
$703$	12.0984	0.456301
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	31.6243	1.19020
$707$	6.89448	0.259294
$708$	0	0
$709$	−46.4525	−1.74456	−0.872281	−	0.489005i	$-0.837360\pi$
$709$	−46.4525	−1.74456	−0.872281	+	0.489005i	$0.837360\pi$
$710$	−0.300986	−0.0112958
$711$	0	0
$712$	8.69193	0.325744
$713$	−11.0970	−0.415588
$714$	0	0
$715$	−6.69193	−0.250264
$716$	13.8874	0.518996
$717$	0	0
$718$	−30.5273	−1.13927
$719$	17.8732	0.666559	0.333279	−	0.942828i	$-0.391845\pi$
$719$	17.8732	0.666559	0.333279	+	0.942828i	$0.391845\pi$
$720$	0	0
$721$	10.5793	0.393995
$722$	−15.5793	−0.579802
$723$	0	0
$724$	18.7819	0.698023
$725$	−6.84242	−0.254121
$726$	0	0
$727$	−11.8874	−0.440879	−0.220440	−	0.975401i	$-0.570749\pi$
$727$	−11.8874	−0.440879	−0.220440	+	0.975401i	$0.570749\pi$
$728$	−6.69193	−0.248019
$729$	0	0
$730$	−10.1884	−0.377089
$731$	3.62323	0.134010
$732$	0	0
$733$	−31.6764	−1.16999	−0.584997	−	0.811036i	$-0.698904\pi$
$733$	−31.6764	−1.16999	−0.584997	+	0.811036i	$0.698904\pi$
$734$	−10.5793	−0.390490
$735$	0	0
$736$	−1.84951	−0.0681737
$737$	−8.00000	−0.294684
$738$	0	0
$739$	28.5567	1.05047	0.525237	−	0.850956i	$-0.323977\pi$
$739$	28.5567	1.05047	0.525237	+	0.850956i	$0.323977\pi$
$740$	−6.54143	−0.240468
$741$	0	0
$742$	6.84242	0.251193
$743$	−13.9858	−0.513090	−0.256545	−	0.966532i	$-0.582584\pi$
$743$	−13.9858	−0.513090	−0.256545	+	0.966532i	$0.582584\pi$
$744$	0	0
$745$	0.451479	0.0165409
$746$	36.4667	1.33514
$747$	0	0
$748$	−7.19547	−0.263092
$749$	−7.49646	−0.273915
$750$	0	0
$751$	18.9171	0.690296	0.345148	−	0.938548i	$-0.387829\pi$
$751$	18.9171	0.690296	0.345148	+	0.938548i	$0.387829\pi$
$752$	1.49646	0.0545701
$753$	0	0
$754$	−45.7890	−1.66754
$755$	19.2334	0.699974
$756$	0	0
$757$	10.9182	0.396829	0.198414	−	0.980118i	$-0.436421\pi$
$757$	10.9182	0.396829	0.198414	+	0.980118i	$0.436421\pi$
$758$	14.9929	0.544567
$759$	0	0
$760$	−1.84951	−0.0670887
$761$	30.3531	1.10030	0.550149	−	0.835067i	$-0.314571\pi$
$761$	30.3531	1.10030	0.550149	+	0.835067i	$0.314571\pi$
$762$	0	0
$763$	−1.45857	−0.0528036
$764$	6.39094	0.231216
$765$	0	0
$766$	−35.1813	−1.27115
$767$	52.7819	1.90584
$768$	0	0
$769$	−19.7369	−0.711731	−0.355865	−	0.934537i	$-0.615814\pi$
$769$	−19.7369	−0.711731	−0.355865	+	0.934537i	$0.615814\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−13.1955	−0.474915
$773$	24.3909	0.877281	0.438641	−	0.898663i	$-0.355460\pi$
$773$	24.3909	0.877281	0.438641	+	0.898663i	$0.355460\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	3.84951	0.138189
$777$	0	0
$778$	−2.30099	−0.0824943
$779$	−17.2854	−0.619315
$780$	0	0
$781$	0.300986	0.0107701
$782$	−13.3081	−0.475896
$783$	0	0
$784$	1.00000	0.0357143
$785$	−9.69901	−0.346173
$786$	0	0
$787$	33.7890	1.20445	0.602223	−	0.798328i	$-0.294282\pi$
$787$	33.7890	1.20445	0.602223	+	0.798328i	$0.294282\pi$
$788$	4.69193	0.167143
$789$	0	0
$790$	−12.2404	−0.435496
$791$	−14.0000	−0.497783
$792$	0	0
$793$	30.1374	1.07021
$794$	18.0758	0.641485
$795$	0	0
$796$	−9.49646	−0.336593
$797$	−36.8435	−1.30506	−0.652532	−	0.757761i	$-0.726293\pi$
$797$	−36.8435	−1.30506	−0.652532	+	0.757761i	$0.726293\pi$
$798$	0	0
$799$	10.7677	0.380934
$800$	1.00000	0.0353553
$801$	0	0
$802$	−22.5793	−0.797304
$803$	10.1884	0.359540
$804$	0	0
$805$	1.84951	0.0651866
$806$	40.1516	1.41428
$807$	0	0
$808$	−6.89448	−0.242547
$809$	−37.6990	−1.32543	−0.662713	−	0.748873i	$-0.730595\pi$
$809$	−37.6990	−1.32543	−0.662713	+	0.748873i	$0.730595\pi$
$810$	0	0
$811$	38.3304	1.34596	0.672981	−	0.739660i	$-0.265013\pi$
$811$	38.3304	1.34596	0.672981	+	0.739660i	$0.265013\pi$
$812$	6.84242	0.240122
$813$	0	0
$814$	6.54143	0.229277
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−0.931308	−0.0325823
$818$	1.04498	0.0365368
$819$	0	0
$820$	9.34596	0.326375
$821$	18.1363	0.632962	0.316481	−	0.948599i	$-0.397499\pi$
$821$	18.1363	0.632962	0.316481	+	0.948599i	$0.397499\pi$
$822$	0	0
$823$	−7.86368	−0.274111	−0.137055	−	0.990563i	$-0.543764\pi$
$823$	−7.86368	−0.274111	−0.137055	+	0.990563i	$0.543764\pi$
$824$	−10.5793	−0.368548
$825$	0	0
$826$	−7.88740	−0.274438
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	2.80453	0.0974053	0.0487027	−	0.998813i	$-0.484491\pi$
$829$	2.80453	0.0974053	0.0487027	+	0.998813i	$0.484491\pi$
$830$	1.30807	0.0454039
$831$	0	0
$832$	6.69193	0.232001
$833$	7.19547	0.249308
$834$	0	0
$835$	−8.00000	−0.276851
$836$	1.84951	0.0639665
$837$	0	0
$838$	−5.49646	−0.189872
$839$	−36.9929	−1.27714	−0.638569	−	0.769565i	$-0.720473\pi$
$839$	−36.9929	−1.27714	−0.638569	+	0.769565i	$0.720473\pi$
$840$	0	0
$841$	17.8187	0.614438
$842$	2.60197	0.0896699
$843$	0	0
$844$	−2.39094	−0.0822996
$845$	31.7819	1.09333
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−6.84242	−0.234970
$849$	0	0
$850$	7.19547	0.246803
$851$	12.0984	0.414729
$852$	0	0
$853$	−49.2571	−1.68653	−0.843265	−	0.537498i	$-0.819370\pi$
$853$	−49.2571	−1.68653	−0.843265	+	0.537498i	$0.819370\pi$
$854$	−4.50354	−0.154108
$855$	0	0
$856$	7.49646	0.256224
$857$	20.3541	0.695283	0.347642	−	0.937627i	$-0.386983\pi$
$857$	20.3541	0.695283	0.347642	+	0.937627i	$0.386983\pi$
$858$	0	0
$859$	−34.2783	−1.16956	−0.584781	−	0.811191i	$-0.698820\pi$
$859$	−34.2783	−1.16956	−0.584781	+	0.811191i	$0.698820\pi$
$860$	0.503544	0.0171707
$861$	0	0
$862$	−19.2334	−0.655091
$863$	12.4373	0.423371	0.211685	−	0.977338i	$-0.432105\pi$
$863$	12.4373	0.423371	0.211685	+	0.977338i	$0.432105\pi$
$864$	0	0
$865$	15.4965	0.526895
$866$	−8.93237	−0.303534
$867$	0	0
$868$	−6.00000	−0.203653
$869$	12.2404	0.415229
$870$	0	0
$871$	53.5354	1.81398
$872$	1.45857	0.0493933
$873$	0	0
$874$	3.42068	0.115706
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	34.4809	1.16434	0.582169	−	0.813068i	$-0.302204\pi$
$877$	34.4809	1.16434	0.582169	+	0.813068i	$0.302204\pi$
$878$	−0.300986	−0.0101578
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−27.3839	−0.922585	−0.461293	−	0.887248i	$-0.652614\pi$
$881$	−27.3839	−0.922585	−0.461293	+	0.887248i	$0.652614\pi$
$882$	0	0
$883$	22.0900	0.743386	0.371693	−	0.928356i	$-0.378777\pi$
$883$	22.0900	0.743386	0.371693	+	0.928356i	$0.378777\pi$
$884$	48.1516	1.61951
$885$	0	0
$886$	−30.7677	−1.03366
$887$	−25.7890	−0.865909	−0.432954	−	0.901416i	$-0.642529\pi$
$887$	−25.7890	−0.865909	−0.432954	+	0.901416i	$0.642529\pi$
$888$	0	0
$889$	19.7748	0.663225
$890$	8.69193	0.291354
$891$	0	0
$892$	0.188383	0.00630752
$893$	−2.76771	−0.0926178
$894$	0	0
$895$	13.8874	0.464204
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	1.19547	0.0398934
$899$	−41.0545	−1.36924
$900$	0	0
$901$	−49.2344	−1.64024
$902$	−9.34596	−0.311187
$903$	0	0
$904$	14.0000	0.465633
$905$	18.7819	0.624331
$906$	0	0
$907$	−45.4596	−1.50946	−0.754731	−	0.656034i	$-0.772233\pi$
$907$	−45.4596	−1.50946	−0.754731	+	0.656034i	$0.772233\pi$
$908$	6.99291	0.232068
$909$	0	0
$910$	−6.69193	−0.221835
$911$	15.8506	0.525153	0.262576	−	0.964911i	$-0.415428\pi$
$911$	15.8506	0.525153	0.262576	+	0.964911i	$0.415428\pi$
$912$	0	0
$913$	−1.30807	−0.0432909
$914$	−25.7748	−0.852554
$915$	0	0
$916$	−9.19547	−0.303827
$917$	2.15049	0.0710155
$918$	0	0
$919$	18.6314	0.614593	0.307296	−	0.951614i	$-0.400576\pi$
$919$	18.6314	0.614593	0.307296	+	0.951614i	$0.400576\pi$
$920$	−1.84951	−0.0609764
$921$	0	0
$922$	−11.5722	−0.381111
$923$	−2.01418	−0.0662974
$924$	0	0
$925$	−6.54143	−0.215081
$926$	−32.9182	−1.08176
$927$	0	0
$928$	−6.84242	−0.224613
$929$	29.2486	0.959616	0.479808	−	0.877374i	$-0.340706\pi$
$929$	29.2486	0.959616	0.479808	+	0.877374i	$0.340706\pi$
$930$	0	0
$931$	−1.84951	−0.0606151
$932$	0.992912	0.0325239
$933$	0	0
$934$	14.6399	0.479031
$935$	−7.19547	−0.235317
$936$	0	0
$937$	−46.4441	−1.51726	−0.758631	−	0.651521i	$-0.774131\pi$
$937$	−46.4441	−1.51726	−0.758631	+	0.651521i	$0.774131\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	1.49646	0.0488090
$941$	39.2713	1.28021	0.640103	−	0.768289i	$-0.278892\pi$
$941$	39.2713	1.28021	0.640103	+	0.768289i	$0.278892\pi$
$942$	0	0
$943$	−17.2854	−0.562891
$944$	7.88740	0.256713
$945$	0	0
$946$	−0.503544	−0.0163716
$947$	−54.2415	−1.76261	−0.881306	−	0.472546i	$-0.843335\pi$
$947$	−54.2415	−1.76261	−0.881306	+	0.472546i	$0.843335\pi$
$948$	0	0
$949$	−68.1799	−2.21321
$950$	−1.84951	−0.0600059
$951$	0	0
$952$	−7.19547	−0.233207
$953$	−59.3612	−1.92290	−0.961449	−	0.274983i	$-0.911328\pi$
$953$	−59.3612	−1.92290	−0.961449	+	0.274983i	$0.911328\pi$
$954$	0	0
$955$	6.39094	0.206806
$956$	1.14341	0.0369804
$957$	0	0
$958$	−10.3909	−0.335716
$959$	−8.39094	−0.270958
$960$	0	0
$961$	5.00000	0.161290
$962$	−43.7748	−1.41136
$963$	0	0
$964$	8.33888	0.268577
$965$	−13.1955	−0.424777
$966$	0	0
$967$	−18.7677	−0.603529	−0.301764	−	0.953383i	$-0.597576\pi$
$967$	−18.7677	−0.603529	−0.301764	+	0.953383i	$0.597576\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	3.84951	0.123600
$971$	59.4370	1.90742	0.953712	−	0.300722i	$-0.0972277\pi$
$971$	59.4370	1.90742	0.953712	+	0.300722i	$0.0972277\pi$
$972$	0	0
$973$	−17.9253	−0.574658
$974$	39.3091	1.25955
$975$	0	0
$976$	4.50354	0.144155
$977$	32.7677	1.04833	0.524166	−	0.851616i	$-0.324377\pi$
$977$	32.7677	1.04833	0.524166	+	0.851616i	$0.324377\pi$
$978$	0	0
$979$	−8.69193	−0.277795
$980$	1.00000	0.0319438
$981$	0	0
$982$	−23.7748	−0.758684
$983$	−13.4207	−0.428053	−0.214027	−	0.976828i	$-0.568658\pi$
$983$	−13.4207	−0.428053	−0.214027	+	0.976828i	$0.568658\pi$
$984$	0	0
$985$	4.69193	0.149497
$986$	−49.2344	−1.56794
$987$	0	0
$988$	−12.3768	−0.393757
$989$	−0.931308	−0.0296139
$990$	0	0
$991$	−38.1657	−1.21237	−0.606187	−	0.795322i	$-0.707302\pi$
$991$	−38.1657	−1.21237	−0.606187	+	0.795322i	$0.707302\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	0.300986	0.00954669
$995$	−9.49646	−0.301058
$996$	0	0
$997$	−33.6622	−1.06609	−0.533046	−	0.846086i	$-0.678953\pi$
$997$	−33.6622	−1.06609	−0.533046	+	0.846086i	$0.678953\pi$
$998$	−6.11260	−0.193491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.cl.1.3		3
3.2	odd	2		770.2.a.l.1.3	✓	3
12.11	even	2		6160.2.a.bi.1.1		3
15.2	even	4		3850.2.c.z.1849.1		6
15.8	even	4		3850.2.c.z.1849.6		6
15.14	odd	2		3850.2.a.bu.1.1		3
21.20	even	2		5390.2.a.bz.1.1		3
33.32	even	2		8470.2.a.cl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.l.1.3	✓	3	3.2	odd	2
3850.2.a.bu.1.1		3	15.14	odd	2
3850.2.c.z.1849.1		6	15.2	even	4
3850.2.c.z.1849.6		6	15.8	even	4
5390.2.a.bz.1.1		3	21.20	even	2
6160.2.a.bi.1.1		3	12.11	even	2
6930.2.a.cl.1.3		3	1.1	even	1	trivial
8470.2.a.cl.1.3		3	33.32	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +6.69193 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.19547 q^{17} -1.84951 q^{19} +1.00000 q^{20} -1.00000 q^{22} -1.84951 q^{23} +1.00000 q^{25} +6.69193 q^{26} -1.00000 q^{28} -6.84242 q^{29} +6.00000 q^{31} +1.00000 q^{32} +7.19547 q^{34} -1.00000 q^{35} -6.54143 q^{37} -1.84951 q^{38} +1.00000 q^{40} +9.34596 q^{41} +0.503544 q^{43} -1.00000 q^{44} -1.84951 q^{46} +1.49646 q^{47} +1.00000 q^{49} +1.00000 q^{50} +6.69193 q^{52} -6.84242 q^{53} -1.00000 q^{55} -1.00000 q^{56} -6.84242 q^{58} +7.88740 q^{59} +4.50354 q^{61} +6.00000 q^{62} +1.00000 q^{64} +6.69193 q^{65} +8.00000 q^{67} +7.19547 q^{68} -1.00000 q^{70} -0.300986 q^{71} -10.1884 q^{73} -6.54143 q^{74} -1.84951 q^{76} +1.00000 q^{77} -12.2404 q^{79} +1.00000 q^{80} +9.34596 q^{82} +1.30807 q^{83} +7.19547 q^{85} +0.503544 q^{86} -1.00000 q^{88} +8.69193 q^{89} -6.69193 q^{91} -1.84951 q^{92} +1.49646 q^{94} -1.84951 q^{95} +3.84951 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 3 q^{14} + 3 q^{16} + 8 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} - 2 q^{23} + 3 q^{25} - 3 q^{28} - 4 q^{29} + 18 q^{31} + 3 q^{32} + 8 q^{34} - 3 q^{35} + 4 q^{37} - 2 q^{38} + 3 q^{40} + 18 q^{41} + 8 q^{43} - 3 q^{44} - 2 q^{46} - 2 q^{47} + 3 q^{49} + 3 q^{50} - 4 q^{53} - 3 q^{55} - 3 q^{56} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 18 q^{62} + 3 q^{64} + 24 q^{67} + 8 q^{68} - 3 q^{70} - 8 q^{71} - 4 q^{73} + 4 q^{74} - 2 q^{76} + 3 q^{77} - 6 q^{79} + 3 q^{80} + 18 q^{82} + 24 q^{83} + 8 q^{85} + 8 q^{86} - 3 q^{88} + 6 q^{89} - 2 q^{92} - 2 q^{94} - 2 q^{95} + 8 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8 + 3 * q^10 - 3 * q^11 - 3 * q^14 + 3 * q^16 + 8 * q^17 - 2 * q^19 + 3 * q^20 - 3 * q^22 - 2 * q^23 + 3 * q^25 - 3 * q^28 - 4 * q^29 + 18 * q^31 + 3 * q^32 + 8 * q^34 - 3 * q^35 + 4 * q^37 - 2 * q^38 + 3 * q^40 + 18 * q^41 + 8 * q^43 - 3 * q^44 - 2 * q^46 - 2 * q^47 + 3 * q^49 + 3 * q^50 - 4 * q^53 - 3 * q^55 - 3 * q^56 - 4 * q^58 - 10 * q^59 + 20 * q^61 + 18 * q^62 + 3 * q^64 + 24 * q^67 + 8 * q^68 - 3 * q^70 - 8 * q^71 - 4 * q^73 + 4 * q^74 - 2 * q^76 + 3 * q^77 - 6 * q^79 + 3 * q^80 + 18 * q^82 + 24 * q^83 + 8 * q^85 + 8 * q^86 - 3 * q^88 + 6 * q^89 - 2 * q^92 - 2 * q^94 - 2 * q^95 + 8 * q^97 + 3 * q^98

Introduction

L-functions

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Database

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