LMFDB - Embedded newform 6930.2.a.ck.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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +4.09783 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.87800 q^{17} -0.219833 q^{19} +1.00000 q^{20} -1.00000 q^{22} -2.89008 q^{23} +1.00000 q^{25} +4.09783 q^{26} -1.00000 q^{28} -9.20775 q^{29} +1.00000 q^{32} -7.87800 q^{34} -1.00000 q^{35} -8.31767 q^{37} -0.219833 q^{38} +1.00000 q^{40} -7.42758 q^{41} +2.98792 q^{43} -1.00000 q^{44} -2.89008 q^{46} -6.98792 q^{47} +1.00000 q^{49} +1.00000 q^{50} +4.09783 q^{52} +12.4155 q^{53} -1.00000 q^{55} -1.00000 q^{56} -9.20775 q^{58} +2.21983 q^{59} +0.987918 q^{61} +1.00000 q^{64} +4.09783 q^{65} -6.67025 q^{67} -7.87800 q^{68} -1.00000 q^{70} -4.21983 q^{71} -7.78017 q^{73} -8.31767 q^{74} -0.219833 q^{76} +1.00000 q^{77} -6.76809 q^{79} +1.00000 q^{80} -7.42758 q^{82} -5.42758 q^{83} -7.87800 q^{85} +2.98792 q^{86} -1.00000 q^{88} +0.0978347 q^{89} -4.09783 q^{91} -2.89008 q^{92} -6.98792 q^{94} -0.219833 q^{95} -3.78017 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{13} - 3 q^{14} + 3 q^{16} - 4 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} - 8 q^{23} + 3 q^{25} - 6 q^{26} - 3 q^{28} - 10 q^{29} + 3 q^{32} - 4 q^{34} - 3 q^{35} - 8 q^{37} - 2 q^{38} + 3 q^{40} - 6 q^{41} - 10 q^{43} - 3 q^{44} - 8 q^{46} - 2 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{52} + 2 q^{53} - 3 q^{55} - 3 q^{56} - 10 q^{58} + 8 q^{59} - 16 q^{61} + 3 q^{64} - 6 q^{65} - 18 q^{67} - 4 q^{68} - 3 q^{70} - 14 q^{71} - 22 q^{73} - 8 q^{74} - 2 q^{76} + 3 q^{77} + 3 q^{80} - 6 q^{82} - 4 q^{85} - 10 q^{86} - 3 q^{88} - 18 q^{89} + 6 q^{91} - 8 q^{92} - 2 q^{94} - 2 q^{95} - 10 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 - 6 * q^13 - 3 * q^14 + 3 * q^16 - 4 * q^17 - 2 * q^19 + 3 * q^20 - 3 * q^22 - 8 * q^23 + 3 * q^25 - 6 * q^26 - 3 * q^28 - 10 * q^29 + 3 * q^32 - 4 * q^34 - 3 * q^35 - 8 * q^37 - 2 * q^38 + 3 * q^40 - 6 * q^41 - 10 * q^43 - 3 * q^44 - 8 * q^46 - 2 * q^47 + 3 * q^49 + 3 * q^50 - 6 * q^52 + 2 * q^53 - 3 * q^55 - 3 * q^56 - 10 * q^58 + 8 * q^59 - 16 * q^61 + 3 * q^64 - 6 * q^65 - 18 * q^67 - 4 * q^68 - 3 * q^70 - 14 * q^71 - 22 * q^73 - 8 * q^74 - 2 * q^76 + 3 * q^77 + 3 * q^80 - 6 * q^82 - 4 * q^85 - 10 * q^86 - 3 * q^88 - 18 * q^89 + 6 * q^91 - 8 * q^92 - 2 * q^94 - 2 * q^95 - 10 * 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
$13$	4.09783	1.13653	0.568267	−	0.822844i	$-0.307614\pi$
$13$	4.09783	1.13653	0.568267	+	0.822844i	$0.307614\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.87800	−1.91070	−0.955348	−	0.295483i	$-0.904520\pi$
$17$	−7.87800	−1.91070	−0.955348	+	0.295483i	$0.904520\pi$
$18$	0	0
$19$	−0.219833	−0.0504330	−0.0252165	−	0.999682i	$-0.508028\pi$
$19$	−0.219833	−0.0504330	−0.0252165	+	0.999682i	$0.508028\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−2.89008	−0.602624	−0.301312	−	0.953526i	$-0.597425\pi$
$23$	−2.89008	−0.602624	−0.301312	+	0.953526i	$0.597425\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.09783	0.803651
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−9.20775	−1.70984	−0.854918	−	0.518763i	$-0.826393\pi$
$29$	−9.20775	−1.70984	−0.854918	+	0.518763i	$0.826393\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.87800	−1.35107
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.31767	−1.36742	−0.683708	−	0.729756i	$-0.739634\pi$
$37$	−8.31767	−1.36742	−0.683708	+	0.729756i	$0.739634\pi$
$38$	−0.219833	−0.0356615
$39$	0	0
$40$	1.00000	0.158114
$41$	−7.42758	−1.15999	−0.579997	−	0.814619i	$-0.696946\pi$
$41$	−7.42758	−1.15999	−0.579997	+	0.814619i	$0.696946\pi$
$42$	0	0
$43$	2.98792	0.455653	0.227827	−	0.973702i	$-0.426838\pi$
$43$	2.98792	0.455653	0.227827	+	0.973702i	$0.426838\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−2.89008	−0.426120
$47$	−6.98792	−1.01929	−0.509646	−	0.860384i	$-0.670224\pi$
$47$	−6.98792	−1.01929	−0.509646	+	0.860384i	$0.670224\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	4.09783	0.568267
$53$	12.4155	1.70540	0.852700	−	0.522401i	$-0.174963\pi$
$53$	12.4155	1.70540	0.852700	+	0.522401i	$0.174963\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−9.20775	−1.20904
$59$	2.21983	0.288998	0.144499	−	0.989505i	$-0.453843\pi$
$59$	2.21983	0.288998	0.144499	+	0.989505i	$0.453843\pi$
$60$	0	0
$61$	0.987918	0.126490	0.0632450	−	0.997998i	$-0.479855\pi$
$61$	0.987918	0.126490	0.0632450	+	0.997998i	$0.479855\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	4.09783	0.508274
$66$	0	0
$67$	−6.67025	−0.814901	−0.407450	−	0.913227i	$-0.633582\pi$
$67$	−6.67025	−0.814901	−0.407450	+	0.913227i	$0.633582\pi$
$68$	−7.87800	−0.955348
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−4.21983	−0.500802	−0.250401	−	0.968142i	$-0.580562\pi$
$71$	−4.21983	−0.500802	−0.250401	+	0.968142i	$0.580562\pi$
$72$	0	0
$73$	−7.78017	−0.910600	−0.455300	−	0.890338i	$-0.650468\pi$
$73$	−7.78017	−0.910600	−0.455300	+	0.890338i	$0.650468\pi$
$74$	−8.31767	−0.966909
$75$	0	0
$76$	−0.219833	−0.0252165
$77$	1.00000	0.113961
$78$	0	0
$79$	−6.76809	−0.761469	−0.380735	−	0.924684i	$-0.624329\pi$
$79$	−6.76809	−0.761469	−0.380735	+	0.924684i	$0.624329\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−7.42758	−0.820239
$83$	−5.42758	−0.595755	−0.297877	−	0.954604i	$-0.596279\pi$
$83$	−5.42758	−0.595755	−0.297877	+	0.954604i	$0.596279\pi$
$84$	0	0
$85$	−7.87800	−0.854489
$86$	2.98792	0.322196
$87$	0	0
$88$	−1.00000	−0.106600
$89$	0.0978347	0.0103705	0.00518523	−	0.999987i	$-0.498349\pi$
$89$	0.0978347	0.0103705	0.00518523	+	0.999987i	$0.498349\pi$
$90$	0	0
$91$	−4.09783	−0.429570
$92$	−2.89008	−0.301312
$93$	0	0
$94$	−6.98792	−0.720749
$95$	−0.219833	−0.0225543
$96$	0	0
$97$	−3.78017	−0.383818	−0.191909	−	0.981413i	$-0.561468\pi$
$97$	−3.78017	−0.383818	−0.191909	+	0.981413i	$0.561468\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	3.42758	0.341057	0.170529	−	0.985353i	$-0.445452\pi$
$101$	3.42758	0.341057	0.170529	+	0.985353i	$0.445452\pi$
$102$	0	0
$103$	16.6353	1.63913	0.819564	−	0.572988i	$-0.194216\pi$
$103$	16.6353	1.63913	0.819564	+	0.572988i	$0.194216\pi$
$104$	4.09783	0.401826
$105$	0	0
$106$	12.4155	1.20590
$107$	−12.9879	−1.25559	−0.627795	−	0.778379i	$-0.716042\pi$
$107$	−12.9879	−1.25559	−0.627795	+	0.778379i	$0.716042\pi$
$108$	0	0
$109$	−6.31767	−0.605123	−0.302561	−	0.953130i	$-0.597842\pi$
$109$	−6.31767	−0.605123	−0.302561	+	0.953130i	$0.597842\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	8.29350	0.780187	0.390094	−	0.920775i	$-0.372443\pi$
$113$	8.29350	0.780187	0.390094	+	0.920775i	$0.372443\pi$
$114$	0	0
$115$	−2.89008	−0.269502
$116$	−9.20775	−0.854918
$117$	0	0
$118$	2.21983	0.204352
$119$	7.87800	0.722175
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.987918	0.0894419
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.1836	1.16985	0.584927	−	0.811086i	$-0.301123\pi$
$127$	13.1836	1.16985	0.584927	+	0.811086i	$0.301123\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.09783	0.359404
$131$	17.1836	1.50134	0.750669	−	0.660679i	$-0.229732\pi$
$131$	17.1836	1.50134	0.750669	+	0.660679i	$0.229732\pi$
$132$	0	0
$133$	0.219833	0.0190619
$134$	−6.67025	−0.576222
$135$	0	0
$136$	−7.87800	−0.675533
$137$	13.8780	1.18568	0.592839	−	0.805321i	$-0.298007\pi$
$137$	13.8780	1.18568	0.592839	+	0.805321i	$0.298007\pi$
$138$	0	0
$139$	1.56033	0.132346	0.0661729	−	0.997808i	$-0.478921\pi$
$139$	1.56033	0.132346	0.0661729	+	0.997808i	$0.478921\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−4.21983	−0.354120
$143$	−4.09783	−0.342678
$144$	0	0
$145$	−9.20775	−0.764662
$146$	−7.78017	−0.643891
$147$	0	0
$148$	−8.31767	−0.683708
$149$	15.6233	1.27991	0.639953	−	0.768414i	$-0.278954\pi$
$149$	15.6233	1.27991	0.639953	+	0.768414i	$0.278954\pi$
$150$	0	0
$151$	−3.20775	−0.261043	−0.130522	−	0.991445i	$-0.541665\pi$
$151$	−3.20775	−0.261043	−0.130522	+	0.991445i	$0.541665\pi$
$152$	−0.219833	−0.0178308
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	−21.4034	−1.70818	−0.854089	−	0.520126i	$-0.825885\pi$
$157$	−21.4034	−1.70818	−0.854089	+	0.520126i	$0.825885\pi$
$158$	−6.76809	−0.538440
$159$	0	0
$160$	1.00000	0.0790569
$161$	2.89008	0.227771
$162$	0	0
$163$	1.32975	0.104154	0.0520770	−	0.998643i	$-0.483416\pi$
$163$	1.32975	0.104154	0.0520770	+	0.998643i	$0.483416\pi$
$164$	−7.42758	−0.579997
$165$	0	0
$166$	−5.42758	−0.421262
$167$	−20.2935	−1.57036	−0.785179	−	0.619269i	$-0.787429\pi$
$167$	−20.2935	−1.57036	−0.785179	+	0.619269i	$0.787429\pi$
$168$	0	0
$169$	3.79225	0.291711
$170$	−7.87800	−0.604215
$171$	0	0
$172$	2.98792	0.227827
$173$	−10.9879	−0.835396	−0.417698	−	0.908586i	$-0.637163\pi$
$173$	−10.9879	−0.835396	−0.417698	+	0.908586i	$0.637163\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	0.0978347	0.00733302
$179$	−24.7439	−1.84945	−0.924724	−	0.380639i	$-0.875704\pi$
$179$	−24.7439	−1.84945	−0.924724	+	0.380639i	$0.875704\pi$
$180$	0	0
$181$	20.4263	1.51827	0.759136	−	0.650932i	$-0.225622\pi$
$181$	20.4263	1.51827	0.759136	+	0.650932i	$0.225622\pi$
$182$	−4.09783	−0.303752
$183$	0	0
$184$	−2.89008	−0.213060
$185$	−8.31767	−0.611527
$186$	0	0
$187$	7.87800	0.576097
$188$	−6.98792	−0.509646
$189$	0	0
$190$	−0.219833	−0.0159483
$191$	−24.4155	−1.76664	−0.883322	−	0.468767i	$-0.844698\pi$
$191$	−24.4155	−1.76664	−0.883322	+	0.468767i	$0.844698\pi$
$192$	0	0
$193$	−1.45175	−0.104499	−0.0522495	−	0.998634i	$-0.516639\pi$
$193$	−1.45175	−0.104499	−0.0522495	+	0.998634i	$0.516639\pi$
$194$	−3.78017	−0.271400
$195$	0	0
$196$	1.00000	0.0714286
$197$	−7.23191	−0.515253	−0.257626	−	0.966245i	$-0.582940\pi$
$197$	−7.23191	−0.515253	−0.257626	+	0.966245i	$0.582940\pi$
$198$	0	0
$199$	−4.63533	−0.328590	−0.164295	−	0.986411i	$-0.552535\pi$
$199$	−4.63533	−0.328590	−0.164295	+	0.986411i	$0.552535\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	3.42758	0.241164
$203$	9.20775	0.646257
$204$	0	0
$205$	−7.42758	−0.518765
$206$	16.6353	1.15904
$207$	0	0
$208$	4.09783	0.284134
$209$	0.219833	0.0152061
$210$	0	0
$211$	19.0858	1.31392	0.656959	−	0.753927i	$-0.271843\pi$
$211$	19.0858	1.31392	0.656959	+	0.753927i	$0.271843\pi$
$212$	12.4155	0.852700
$213$	0	0
$214$	−12.9879	−0.887836
$215$	2.98792	0.203774
$216$	0	0
$217$	0	0
$218$	−6.31767	−0.427886
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	−32.2828	−2.17157
$222$	0	0
$223$	−5.18359	−0.347119	−0.173559	−	0.984823i	$-0.555527\pi$
$223$	−5.18359	−0.347119	−0.173559	+	0.984823i	$0.555527\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	8.29350	0.551676
$227$	11.6474	0.773066	0.386533	−	0.922276i	$-0.373673\pi$
$227$	11.6474	0.773066	0.386533	+	0.922276i	$0.373673\pi$
$228$	0	0
$229$	21.5013	1.42084	0.710421	−	0.703776i	$-0.248504\pi$
$229$	21.5013	1.42084	0.710421	+	0.703776i	$0.248504\pi$
$230$	−2.89008	−0.190566
$231$	0	0
$232$	−9.20775	−0.604518
$233$	−27.9758	−1.83276	−0.916379	−	0.400312i	$-0.868902\pi$
$233$	−27.9758	−1.83276	−0.916379	+	0.400312i	$0.868902\pi$
$234$	0	0
$235$	−6.98792	−0.455842
$236$	2.21983	0.144499
$237$	0	0
$238$	7.87800	0.510655
$239$	4.09783	0.265067	0.132533	−	0.991179i	$-0.457689\pi$
$239$	4.09783	0.265067	0.132533	+	0.991179i	$0.457689\pi$
$240$	0	0
$241$	−18.7439	−1.20740	−0.603701	−	0.797211i	$-0.706308\pi$
$241$	−18.7439	−1.20740	−0.603701	+	0.797211i	$0.706308\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	0.987918	0.0632450
$245$	1.00000	0.0638877
$246$	0	0
$247$	−0.900837	−0.0573189
$248$	0	0
$249$	0	0
$250$	1.00000	0.0632456
$251$	12.4397	0.785185	0.392592	−	0.919713i	$-0.371578\pi$
$251$	12.4397	0.785185	0.392592	+	0.919713i	$0.371578\pi$
$252$	0	0
$253$	2.89008	0.181698
$254$	13.1836	0.827212
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	8.31767	0.516835
$260$	4.09783	0.254137
$261$	0	0
$262$	17.1836	1.06161
$263$	−8.19567	−0.505367	−0.252683	−	0.967549i	$-0.581313\pi$
$263$	−8.19567	−0.505367	−0.252683	+	0.967549i	$0.581313\pi$
$264$	0	0
$265$	12.4155	0.762678
$266$	0.219833	0.0134788
$267$	0	0
$268$	−6.67025	−0.407450
$269$	−16.8552	−1.02768	−0.513839	−	0.857887i	$-0.671777\pi$
$269$	−16.8552	−1.02768	−0.513839	+	0.857887i	$0.671777\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−7.87800	−0.477674
$273$	0	0
$274$	13.8780	0.838401
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−22.9638	−1.37976	−0.689879	−	0.723925i	$-0.742336\pi$
$277$	−22.9638	−1.37976	−0.689879	+	0.723925i	$0.742336\pi$
$278$	1.56033	0.0935827
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−10.4504	−0.623420	−0.311710	−	0.950177i	$-0.600902\pi$
$281$	−10.4504	−0.623420	−0.311710	+	0.950177i	$0.600902\pi$
$282$	0	0
$283$	−8.12200	−0.482803	−0.241401	−	0.970425i	$-0.577607\pi$
$283$	−8.12200	−0.482803	−0.241401	+	0.970425i	$0.577607\pi$
$284$	−4.21983	−0.250401
$285$	0	0
$286$	−4.09783	−0.242310
$287$	7.42758	0.438436
$288$	0	0
$289$	45.0629	2.65076
$290$	−9.20775	−0.540698
$291$	0	0
$292$	−7.78017	−0.455300
$293$	16.7681	0.979602	0.489801	−	0.871834i	$-0.337069\pi$
$293$	16.7681	0.979602	0.489801	+	0.871834i	$0.337069\pi$
$294$	0	0
$295$	2.21983	0.129244
$296$	−8.31767	−0.483455
$297$	0	0
$298$	15.6233	0.905031
$299$	−11.8431	−0.684903
$300$	0	0
$301$	−2.98792	−0.172221
$302$	−3.20775	−0.184585
$303$	0	0
$304$	−0.219833	−0.0126083
$305$	0.987918	0.0565680
$306$	0	0
$307$	12.3177	0.703006	0.351503	−	0.936187i	$-0.385671\pi$
$307$	12.3177	0.703006	0.351503	+	0.936187i	$0.385671\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	0	0
$311$	−23.4383	−1.32907	−0.664533	−	0.747259i	$-0.731370\pi$
$311$	−23.4383	−1.32907	−0.664533	+	0.747259i	$0.731370\pi$
$312$	0	0
$313$	−26.2828	−1.48559	−0.742794	−	0.669520i	$-0.766500\pi$
$313$	−26.2828	−1.48559	−0.742794	+	0.669520i	$0.766500\pi$
$314$	−21.4034	−1.20786
$315$	0	0
$316$	−6.76809	−0.380735
$317$	29.3163	1.64657	0.823285	−	0.567628i	$-0.192139\pi$
$317$	29.3163	1.64657	0.823285	+	0.567628i	$0.192139\pi$
$318$	0	0
$319$	9.20775	0.515535
$320$	1.00000	0.0559017
$321$	0	0
$322$	2.89008	0.161058
$323$	1.73184	0.0963622
$324$	0	0
$325$	4.09783	0.227307
$326$	1.32975	0.0736480
$327$	0	0
$328$	−7.42758	−0.410120
$329$	6.98792	0.385256
$330$	0	0
$331$	35.9517	1.97608	0.988041	−	0.154189i	$-0.0492765\pi$
$331$	35.9517	1.97608	0.988041	+	0.154189i	$0.0492765\pi$
$332$	−5.42758	−0.297877
$333$	0	0
$334$	−20.2935	−1.11041
$335$	−6.67025	−0.364435
$336$	0	0
$337$	−12.4155	−0.676315	−0.338158	−	0.941089i	$-0.609804\pi$
$337$	−12.4155	−0.676315	−0.338158	+	0.941089i	$0.609804\pi$
$338$	3.79225	0.206271
$339$	0	0
$340$	−7.87800	−0.427245
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.98792	0.161098
$345$	0	0
$346$	−10.9879	−0.590714
$347$	29.3793	1.57716	0.788581	−	0.614931i	$-0.210816\pi$
$347$	29.3793	1.57716	0.788581	+	0.614931i	$0.210816\pi$
$348$	0	0
$349$	−11.4034	−0.610411	−0.305206	−	0.952286i	$-0.598725\pi$
$349$	−11.4034	−0.610411	−0.305206	+	0.952286i	$0.598725\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	−4.21983	−0.223965
$356$	0.0978347	0.00518523
$357$	0	0
$358$	−24.7439	−1.30776
$359$	−21.6340	−1.14180	−0.570900	−	0.821020i	$-0.693405\pi$
$359$	−21.6340	−1.14180	−0.570900	+	0.821020i	$0.693405\pi$
$360$	0	0
$361$	−18.9517	−0.997457
$362$	20.4263	1.07358
$363$	0	0
$364$	−4.09783	−0.214785
$365$	−7.78017	−0.407233
$366$	0	0
$367$	27.3163	1.42590	0.712951	−	0.701214i	$-0.247358\pi$
$367$	27.3163	1.42590	0.712951	+	0.701214i	$0.247358\pi$
$368$	−2.89008	−0.150656
$369$	0	0
$370$	−8.31767	−0.432415
$371$	−12.4155	−0.644581
$372$	0	0
$373$	−24.0388	−1.24468	−0.622340	−	0.782747i	$-0.713818\pi$
$373$	−24.0388	−1.24468	−0.622340	+	0.782747i	$0.713818\pi$
$374$	7.87800	0.407362
$375$	0	0
$376$	−6.98792	−0.360374
$377$	−37.7318	−1.94329
$378$	0	0
$379$	−25.9758	−1.33429	−0.667145	−	0.744928i	$-0.732484\pi$
$379$	−25.9758	−1.33429	−0.667145	+	0.744928i	$0.732484\pi$
$380$	−0.219833	−0.0112772
$381$	0	0
$382$	−24.4155	−1.24921
$383$	−6.10859	−0.312134	−0.156067	−	0.987746i	$-0.549882\pi$
$383$	−6.10859	−0.312134	−0.156067	+	0.987746i	$0.549882\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	−1.45175	−0.0738920
$387$	0	0
$388$	−3.78017	−0.191909
$389$	−15.1836	−0.769838	−0.384919	−	0.922950i	$-0.625771\pi$
$389$	−15.1836	−0.769838	−0.384919	+	0.922950i	$0.625771\pi$
$390$	0	0
$391$	22.7681	1.15143
$392$	1.00000	0.0505076
$393$	0	0
$394$	−7.23191	−0.364339
$395$	−6.76809	−0.340539
$396$	0	0
$397$	7.62325	0.382600	0.191300	−	0.981532i	$-0.438730\pi$
$397$	7.62325	0.382600	0.191300	+	0.981532i	$0.438730\pi$
$398$	−4.63533	−0.232348
$399$	0	0
$400$	1.00000	0.0500000
$401$	9.97584	0.498170	0.249085	−	0.968482i	$-0.419870\pi$
$401$	9.97584	0.498170	0.249085	+	0.968482i	$0.419870\pi$
$402$	0	0
$403$	0	0
$404$	3.42758	0.170529
$405$	0	0
$406$	9.20775	0.456973
$407$	8.31767	0.412291
$408$	0	0
$409$	14.6353	0.723671	0.361835	−	0.932242i	$-0.382150\pi$
$409$	14.6353	0.723671	0.361835	+	0.932242i	$0.382150\pi$
$410$	−7.42758	−0.366822
$411$	0	0
$412$	16.6353	0.819564
$413$	−2.21983	−0.109231
$414$	0	0
$415$	−5.42758	−0.266430
$416$	4.09783	0.200913
$417$	0	0
$418$	0.219833	0.0107524
$419$	35.9517	1.75635	0.878177	−	0.478336i	$-0.158760\pi$
$419$	35.9517	1.75635	0.878177	+	0.478336i	$0.158760\pi$
$420$	0	0
$421$	10.4397	0.508798	0.254399	−	0.967099i	$-0.418122\pi$
$421$	10.4397	0.508798	0.254399	+	0.967099i	$0.418122\pi$
$422$	19.0858	0.929080
$423$	0	0
$424$	12.4155	0.602950
$425$	−7.87800	−0.382139
$426$	0	0
$427$	−0.987918	−0.0478087
$428$	−12.9879	−0.627795
$429$	0	0
$430$	2.98792	0.144090
$431$	−11.4625	−0.552129	−0.276065	−	0.961139i	$-0.589030\pi$
$431$	−11.4625	−0.552129	−0.276065	+	0.961139i	$0.589030\pi$
$432$	0	0
$433$	13.8043	0.663394	0.331697	−	0.943386i	$-0.392379\pi$
$433$	13.8043	0.663394	0.331697	+	0.943386i	$0.392379\pi$
$434$	0	0
$435$	0	0
$436$	−6.31767	−0.302561
$437$	0.635334	0.0303922
$438$	0	0
$439$	18.0629	0.862096	0.431048	−	0.902329i	$-0.358144\pi$
$439$	18.0629	0.862096	0.431048	+	0.902329i	$0.358144\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	−32.2828	−1.53553
$443$	−12.0871	−0.574275	−0.287137	−	0.957889i	$-0.592704\pi$
$443$	−12.0871	−0.574275	−0.287137	+	0.957889i	$0.592704\pi$
$444$	0	0
$445$	0.0978347	0.00463781
$446$	−5.18359	−0.245450
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−32.8310	−1.54939	−0.774695	−	0.632335i	$-0.782097\pi$
$449$	−32.8310	−1.54939	−0.774695	+	0.632335i	$0.782097\pi$
$450$	0	0
$451$	7.42758	0.349751
$452$	8.29350	0.390094
$453$	0	0
$454$	11.6474	0.546640
$455$	−4.09783	−0.192109
$456$	0	0
$457$	−41.4034	−1.93677	−0.968385	−	0.249460i	$-0.919747\pi$
$457$	−41.4034	−1.93677	−0.968385	+	0.249460i	$0.919747\pi$
$458$	21.5013	1.00469
$459$	0	0
$460$	−2.89008	−0.134751
$461$	−21.3163	−0.992801	−0.496400	−	0.868094i	$-0.665345\pi$
$461$	−21.3163	−0.992801	−0.496400	+	0.868094i	$0.665345\pi$
$462$	0	0
$463$	20.7922	0.966298	0.483149	−	0.875538i	$-0.339493\pi$
$463$	20.7922	0.966298	0.483149	+	0.875538i	$0.339493\pi$
$464$	−9.20775	−0.427459
$465$	0	0
$466$	−27.9758	−1.29596
$467$	32.7439	1.51521	0.757604	−	0.652714i	$-0.226370\pi$
$467$	32.7439	1.51521	0.757604	+	0.652714i	$0.226370\pi$
$468$	0	0
$469$	6.67025	0.308004
$470$	−6.98792	−0.322329
$471$	0	0
$472$	2.21983	0.102176
$473$	−2.98792	−0.137385
$474$	0	0
$475$	−0.219833	−0.0100866
$476$	7.87800	0.361088
$477$	0	0
$478$	4.09783	0.187431
$479$	26.4155	1.20696	0.603478	−	0.797380i	$-0.293781\pi$
$479$	26.4155	1.20696	0.603478	+	0.797380i	$0.293781\pi$
$480$	0	0
$481$	−34.0844	−1.55412
$482$	−18.7439	−0.853762
$483$	0	0
$484$	1.00000	0.0454545
$485$	−3.78017	−0.171649
$486$	0	0
$487$	2.13275	0.0966442	0.0483221	−	0.998832i	$-0.484613\pi$
$487$	2.13275	0.0966442	0.0483221	+	0.998832i	$0.484613\pi$
$488$	0.987918	0.0447210
$489$	0	0
$490$	1.00000	0.0451754
$491$	−5.78017	−0.260855	−0.130428	−	0.991458i	$-0.541635\pi$
$491$	−5.78017	−0.260855	−0.130428	+	0.991458i	$0.541635\pi$
$492$	0	0
$493$	72.5387	3.26698
$494$	−0.900837	−0.0405306
$495$	0	0
$496$	0	0
$497$	4.21983	0.189285
$498$	0	0
$499$	4.90084	0.219392	0.109696	−	0.993965i	$-0.465012\pi$
$499$	4.90084	0.219392	0.109696	+	0.993965i	$0.465012\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	12.4397	0.555209
$503$	43.6098	1.94447	0.972233	−	0.234013i	$-0.0751859\pi$
$503$	43.6098	1.94447	0.972233	+	0.234013i	$0.0751859\pi$
$504$	0	0
$505$	3.42758	0.152525
$506$	2.89008	0.128480
$507$	0	0
$508$	13.1836	0.584927
$509$	−2.63533	−0.116809	−0.0584046	−	0.998293i	$-0.518601\pi$
$509$	−2.63533	−0.116809	−0.0584046	+	0.998293i	$0.518601\pi$
$510$	0	0
$511$	7.78017	0.344174
$512$	1.00000	0.0441942
$513$	0	0
$514$	0	0
$515$	16.6353	0.733040
$516$	0	0
$517$	6.98792	0.307328
$518$	8.31767	0.365457
$519$	0	0
$520$	4.09783	0.179702
$521$	−41.1728	−1.80381	−0.901907	−	0.431930i	$-0.857833\pi$
$521$	−41.1728	−1.80381	−0.901907	+	0.431930i	$0.857833\pi$
$522$	0	0
$523$	2.53750	0.110957	0.0554786	−	0.998460i	$-0.482332\pi$
$523$	2.53750	0.110957	0.0554786	+	0.998460i	$0.482332\pi$
$524$	17.1836	0.750669
$525$	0	0
$526$	−8.19567	−0.357348
$527$	0	0
$528$	0	0
$529$	−14.6474	−0.636844
$530$	12.4155	0.539295
$531$	0	0
$532$	0.219833	0.00953095
$533$	−30.4370	−1.31837
$534$	0	0
$535$	−12.9879	−0.561517
$536$	−6.67025	−0.288111
$537$	0	0
$538$	−16.8552	−0.726678
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	16.2935	0.700512	0.350256	−	0.936654i	$-0.386095\pi$
$541$	16.2935	0.700512	0.350256	+	0.936654i	$0.386095\pi$
$542$	12.0000	0.515444
$543$	0	0
$544$	−7.87800	−0.337767
$545$	−6.31767	−0.270619
$546$	0	0
$547$	0.768086	0.0328410	0.0164205	−	0.999865i	$-0.494773\pi$
$547$	0.768086	0.0328410	0.0164205	+	0.999865i	$0.494773\pi$
$548$	13.8780	0.592839
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	2.02416	0.0862323
$552$	0	0
$553$	6.76809	0.287808
$554$	−22.9638	−0.975636
$555$	0	0
$556$	1.56033	0.0661729
$557$	−2.52675	−0.107062	−0.0535308	−	0.998566i	$-0.517048\pi$
$557$	−2.52675	−0.107062	−0.0535308	+	0.998566i	$0.517048\pi$
$558$	0	0
$559$	12.2440	0.517866
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−10.4504	−0.440824
$563$	−16.0871	−0.677990	−0.338995	−	0.940788i	$-0.610087\pi$
$563$	−16.0871	−0.677990	−0.338995	+	0.940788i	$0.610087\pi$
$564$	0	0
$565$	8.29350	0.348910
$566$	−8.12200	−0.341393
$567$	0	0
$568$	−4.21983	−0.177060
$569$	24.1608	1.01287	0.506436	−	0.862278i	$-0.330963\pi$
$569$	24.1608	1.01287	0.506436	+	0.862278i	$0.330963\pi$
$570$	0	0
$571$	36.1608	1.51328	0.756640	−	0.653831i	$-0.226839\pi$
$571$	36.1608	1.51328	0.756640	+	0.653831i	$0.226839\pi$
$572$	−4.09783	−0.171339
$573$	0	0
$574$	7.42758	0.310021
$575$	−2.89008	−0.120525
$576$	0	0
$577$	−34.9879	−1.45657	−0.728283	−	0.685276i	$-0.759681\pi$
$577$	−34.9879	−1.45657	−0.728283	+	0.685276i	$0.759681\pi$
$578$	45.0629	1.87437
$579$	0	0
$580$	−9.20775	−0.382331
$581$	5.42758	0.225174
$582$	0	0
$583$	−12.4155	−0.514198
$584$	−7.78017	−0.321946
$585$	0	0
$586$	16.7681	0.692683
$587$	33.5749	1.38579	0.692893	−	0.721041i	$-0.256336\pi$
$587$	33.5749	1.38579	0.692893	+	0.721041i	$0.256336\pi$
$588$	0	0
$589$	0	0
$590$	2.21983	0.0913891
$591$	0	0
$592$	−8.31767	−0.341854
$593$	20.7090	0.850417	0.425208	−	0.905095i	$-0.360201\pi$
$593$	20.7090	0.850417	0.425208	+	0.905095i	$0.360201\pi$
$594$	0	0
$595$	7.87800	0.322967
$596$	15.6233	0.639953
$597$	0	0
$598$	−11.8431	−0.484300
$599$	31.2707	1.27768	0.638842	−	0.769338i	$-0.279414\pi$
$599$	31.2707	1.27768	0.638842	+	0.769338i	$0.279414\pi$
$600$	0	0
$601$	25.4905	1.03978	0.519890	−	0.854233i	$-0.325973\pi$
$601$	25.4905	1.03978	0.519890	+	0.854233i	$0.325973\pi$
$602$	−2.98792	−0.121778
$603$	0	0
$604$	−3.20775	−0.130522
$605$	1.00000	0.0406558
$606$	0	0
$607$	−44.5870	−1.80973	−0.904865	−	0.425698i	$-0.860029\pi$
$607$	−44.5870	−1.80973	−0.904865	+	0.425698i	$0.860029\pi$
$608$	−0.219833	−0.00891539
$609$	0	0
$610$	0.987918	0.0399996
$611$	−28.6353	−1.15846
$612$	0	0
$613$	−11.4517	−0.462532	−0.231266	−	0.972891i	$-0.574287\pi$
$613$	−11.4517	−0.462532	−0.231266	+	0.972891i	$0.574287\pi$
$614$	12.3177	0.497101
$615$	0	0
$616$	1.00000	0.0402911
$617$	−34.4650	−1.38751	−0.693755	−	0.720212i	$-0.744045\pi$
$617$	−34.4650	−1.38751	−0.693755	+	0.720212i	$0.744045\pi$
$618$	0	0
$619$	38.4263	1.54448	0.772241	−	0.635330i	$-0.219136\pi$
$619$	38.4263	1.54448	0.772241	+	0.635330i	$0.219136\pi$
$620$	0	0
$621$	0	0
$622$	−23.4383	−0.939792
$623$	−0.0978347	−0.00391966
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.2828	−1.05047
$627$	0	0
$628$	−21.4034	−0.854089
$629$	65.5266	2.61272
$630$	0	0
$631$	−20.9396	−0.833592	−0.416796	−	0.909000i	$-0.636847\pi$
$631$	−20.9396	−0.833592	−0.416796	+	0.909000i	$0.636847\pi$
$632$	−6.76809	−0.269220
$633$	0	0
$634$	29.3163	1.16430
$635$	13.1836	0.523175
$636$	0	0
$637$	4.09783	0.162362
$638$	9.20775	0.364538
$639$	0	0
$640$	1.00000	0.0395285
$641$	−1.14483	−0.0452182	−0.0226091	−	0.999744i	$-0.507197\pi$
$641$	−1.14483	−0.0452182	−0.0226091	+	0.999744i	$0.507197\pi$
$642$	0	0
$643$	−1.97584	−0.0779194	−0.0389597	−	0.999241i	$-0.512404\pi$
$643$	−1.97584	−0.0779194	−0.0389597	+	0.999241i	$0.512404\pi$
$644$	2.89008	0.113885
$645$	0	0
$646$	1.73184	0.0681384
$647$	−33.8431	−1.33051	−0.665254	−	0.746617i	$-0.731677\pi$
$647$	−33.8431	−1.33051	−0.665254	+	0.746617i	$0.731677\pi$
$648$	0	0
$649$	−2.21983	−0.0871360
$650$	4.09783	0.160730
$651$	0	0
$652$	1.32975	0.0520770
$653$	−11.5362	−0.451445	−0.225723	−	0.974192i	$-0.572474\pi$
$653$	−11.5362	−0.451445	−0.225723	+	0.974192i	$0.572474\pi$
$654$	0	0
$655$	17.1836	0.671418
$656$	−7.42758	−0.289998
$657$	0	0
$658$	6.98792	0.272417
$659$	9.53617	0.371477	0.185738	−	0.982599i	$-0.440532\pi$
$659$	9.53617	0.371477	0.185738	+	0.982599i	$0.440532\pi$
$660$	0	0
$661$	−3.08575	−0.120022	−0.0600109	−	0.998198i	$-0.519114\pi$
$661$	−3.08575	−0.120022	−0.0600109	+	0.998198i	$0.519114\pi$
$662$	35.9517	1.39730
$663$	0	0
$664$	−5.42758	−0.210631
$665$	0.219833	0.00852474
$666$	0	0
$667$	26.6112	1.03039
$668$	−20.2935	−0.785179
$669$	0	0
$670$	−6.67025	−0.257694
$671$	−0.987918	−0.0381382
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−12.4155	−0.478227
$675$	0	0
$676$	3.79225	0.145856
$677$	−27.1353	−1.04289	−0.521446	−	0.853284i	$-0.674607\pi$
$677$	−27.1353	−1.04289	−0.521446	+	0.853284i	$0.674607\pi$
$678$	0	0
$679$	3.78017	0.145070
$680$	−7.87800	−0.302108
$681$	0	0
$682$	0	0
$683$	−28.9396	−1.10734	−0.553671	−	0.832735i	$-0.686774\pi$
$683$	−28.9396	−1.10734	−0.553671	+	0.832735i	$0.686774\pi$
$684$	0	0
$685$	13.8780	0.530251
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	2.98792	0.113913
$689$	50.8767	1.93825
$690$	0	0
$691$	0.0107536	0.000409087 0	0.000204544	−	1.00000i	$-0.499935\pi$
$691$	0.0107536	0.000409087 0	0.000204544		1.00000i	$0.499935\pi$
$692$	−10.9879	−0.417698
$693$	0	0
$694$	29.3793	1.11522
$695$	1.56033	0.0591869
$696$	0	0
$697$	58.5145	2.21640
$698$	−11.4034	−0.431626
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−46.7439	−1.76549	−0.882747	−	0.469849i	$-0.844308\pi$
$701$	−46.7439	−1.76549	−0.882747	+	0.469849i	$0.844308\pi$
$702$	0	0
$703$	1.82849	0.0689630
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	0	0
$707$	−3.42758	−0.128908
$708$	0	0
$709$	14.4397	0.542293	0.271147	−	0.962538i	$-0.412597\pi$
$709$	14.4397	0.542293	0.271147	+	0.962538i	$0.412597\pi$
$710$	−4.21983	−0.158367
$711$	0	0
$712$	0.0978347	0.00366651
$713$	0	0
$714$	0	0
$715$	−4.09783	−0.153250
$716$	−24.7439	−0.924724
$717$	0	0
$718$	−21.6340	−0.807374
$719$	−22.0978	−0.824110	−0.412055	−	0.911159i	$-0.635189\pi$
$719$	−22.0978	−0.824110	−0.412055	+	0.911159i	$0.635189\pi$
$720$	0	0
$721$	−16.6353	−0.619532
$722$	−18.9517	−0.705308
$723$	0	0
$724$	20.4263	0.759136
$725$	−9.20775	−0.341967
$726$	0	0
$727$	2.81641	0.104455	0.0522275	−	0.998635i	$-0.483368\pi$
$727$	2.81641	0.104455	0.0522275	+	0.998635i	$0.483368\pi$
$728$	−4.09783	−0.151876
$729$	0	0
$730$	−7.78017	−0.287957
$731$	−23.5388	−0.870615
$732$	0	0
$733$	−4.99867	−0.184630	−0.0923151	−	0.995730i	$-0.529427\pi$
$733$	−4.99867	−0.184630	−0.0923151	+	0.995730i	$0.529427\pi$
$734$	27.3163	1.00826
$735$	0	0
$736$	−2.89008	−0.106530
$737$	6.67025	0.245702
$738$	0	0
$739$	14.1849	0.521801	0.260900	−	0.965366i	$-0.415981\pi$
$739$	14.1849	0.521801	0.260900	+	0.965366i	$0.415981\pi$
$740$	−8.31767	−0.305764
$741$	0	0
$742$	−12.4155	−0.455787
$743$	−23.2948	−0.854605	−0.427302	−	0.904109i	$-0.640536\pi$
$743$	−23.2948	−0.854605	−0.427302	+	0.904109i	$0.640536\pi$
$744$	0	0
$745$	15.6233	0.572392
$746$	−24.0388	−0.880121
$747$	0	0
$748$	7.87800	0.288048
$749$	12.9879	0.474568
$750$	0	0
$751$	11.6689	0.425805	0.212903	−	0.977073i	$-0.431708\pi$
$751$	11.6689	0.425805	0.212903	+	0.977073i	$0.431708\pi$
$752$	−6.98792	−0.254823
$753$	0	0
$754$	−37.7318	−1.37411
$755$	−3.20775	−0.116742
$756$	0	0
$757$	−16.7090	−0.607299	−0.303650	−	0.952784i	$-0.598205\pi$
$757$	−16.7090	−0.607299	−0.303650	+	0.952784i	$0.598205\pi$
$758$	−25.9758	−0.943485
$759$	0	0
$760$	−0.219833	−0.00797416
$761$	19.8672	0.720187	0.360094	−	0.932916i	$-0.382745\pi$
$761$	19.8672	0.720187	0.360094	+	0.932916i	$0.382745\pi$
$762$	0	0
$763$	6.31767	0.228715
$764$	−24.4155	−0.883322
$765$	0	0
$766$	−6.10859	−0.220712
$767$	9.09651	0.328456
$768$	0	0
$769$	18.1473	0.654410	0.327205	−	0.944953i	$-0.393893\pi$
$769$	18.1473	0.654410	0.327205	+	0.944953i	$0.393893\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−1.45175	−0.0522495
$773$	33.7948	1.21551	0.607756	−	0.794123i	$-0.292070\pi$
$773$	33.7948	1.21551	0.607756	+	0.794123i	$0.292070\pi$
$774$	0	0
$775$	0	0
$776$	−3.78017	−0.135700
$777$	0	0
$778$	−15.1836	−0.544358
$779$	1.63282	0.0585020
$780$	0	0
$781$	4.21983	0.150997
$782$	22.7681	0.814185
$783$	0	0
$784$	1.00000	0.0357143
$785$	−21.4034	−0.763921
$786$	0	0
$787$	−34.4892	−1.22941	−0.614703	−	0.788759i	$-0.710724\pi$
$787$	−34.4892	−1.22941	−0.614703	+	0.788759i	$0.710724\pi$
$788$	−7.23191	−0.257626
$789$	0	0
$790$	−6.76809	−0.240798
$791$	−8.29350	−0.294883
$792$	0	0
$793$	4.04833	0.143760
$794$	7.62325	0.270539
$795$	0	0
$796$	−4.63533	−0.164295
$797$	8.57242	0.303651	0.151825	−	0.988407i	$-0.451485\pi$
$797$	8.57242	0.303651	0.151825	+	0.988407i	$0.451485\pi$
$798$	0	0
$799$	55.0508	1.94756
$800$	1.00000	0.0353553
$801$	0	0
$802$	9.97584	0.352259
$803$	7.78017	0.274556
$804$	0	0
$805$	2.89008	0.101862
$806$	0	0
$807$	0	0
$808$	3.42758	0.120582
$809$	−1.10992	−0.0390226	−0.0195113	−	0.999810i	$-0.506211\pi$
$809$	−1.10992	−0.0390226	−0.0195113	+	0.999810i	$0.506211\pi$
$810$	0	0
$811$	7.82849	0.274896	0.137448	−	0.990509i	$-0.456110\pi$
$811$	7.82849	0.274896	0.137448	+	0.990509i	$0.456110\pi$
$812$	9.20775	0.323129
$813$	0	0
$814$	8.31767	0.291534
$815$	1.32975	0.0465791
$816$	0	0
$817$	−0.656842	−0.0229800
$818$	14.6353	0.511712
$819$	0	0
$820$	−7.42758	−0.259382
$821$	−34.5483	−1.20574	−0.602871	−	0.797839i	$-0.705977\pi$
$821$	−34.5483	−1.20574	−0.602871	+	0.797839i	$0.705977\pi$
$822$	0	0
$823$	−49.1353	−1.71275	−0.856374	−	0.516356i	$-0.827288\pi$
$823$	−49.1353	−1.71275	−0.856374	+	0.516356i	$0.827288\pi$
$824$	16.6353	0.579519
$825$	0	0
$826$	−2.21983	−0.0772379
$827$	4.28275	0.148926	0.0744629	−	0.997224i	$-0.476276\pi$
$827$	4.28275	0.148926	0.0744629	+	0.997224i	$0.476276\pi$
$828$	0	0
$829$	52.1608	1.81162	0.905809	−	0.423686i	$-0.139264\pi$
$829$	52.1608	1.81162	0.905809	+	0.423686i	$0.139264\pi$
$830$	−5.42758	−0.188394
$831$	0	0
$832$	4.09783	0.142067
$833$	−7.87800	−0.272957
$834$	0	0
$835$	−20.2935	−0.702286
$836$	0.219833	0.00760307
$837$	0	0
$838$	35.9517	1.24193
$839$	−26.2452	−0.906084	−0.453042	−	0.891489i	$-0.649661\pi$
$839$	−26.2452	−0.906084	−0.453042	+	0.891489i	$0.649661\pi$
$840$	0	0
$841$	55.7827	1.92354
$842$	10.4397	0.359775
$843$	0	0
$844$	19.0858	0.656959
$845$	3.79225	0.130457
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	12.4155	0.426350
$849$	0	0
$850$	−7.87800	−0.270213
$851$	24.0388	0.824038
$852$	0	0
$853$	−9.43834	−0.323163	−0.161581	−	0.986859i	$-0.551659\pi$
$853$	−9.43834	−0.323163	−0.161581	+	0.986859i	$0.551659\pi$
$854$	−0.987918	−0.0338059
$855$	0	0
$856$	−12.9879	−0.443918
$857$	−41.1702	−1.40635	−0.703173	−	0.711019i	$-0.748234\pi$
$857$	−41.1702	−1.40635	−0.703173	+	0.711019i	$0.748234\pi$
$858$	0	0
$859$	−1.72109	−0.0587227	−0.0293614	−	0.999569i	$-0.509347\pi$
$859$	−1.72109	−0.0587227	−0.0293614	+	0.999569i	$0.509347\pi$
$860$	2.98792	0.101887
$861$	0	0
$862$	−11.4625	−0.390414
$863$	−12.0349	−0.409673	−0.204837	−	0.978796i	$-0.565666\pi$
$863$	−12.0349	−0.409673	−0.204837	+	0.978796i	$0.565666\pi$
$864$	0	0
$865$	−10.9879	−0.373600
$866$	13.8043	0.469090
$867$	0	0
$868$	0	0
$869$	6.76809	0.229592
$870$	0	0
$871$	−27.3336	−0.926163
$872$	−6.31767	−0.213943
$873$	0	0
$874$	0.635334	0.0214905
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	9.81892	0.331561	0.165781	−	0.986163i	$-0.446986\pi$
$877$	9.81892	0.331561	0.165781	+	0.986163i	$0.446986\pi$
$878$	18.0629	0.609594
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−3.90217	−0.131467	−0.0657336	−	0.997837i	$-0.520939\pi$
$881$	−3.90217	−0.131467	−0.0657336	+	0.997837i	$0.520939\pi$
$882$	0	0
$883$	34.6004	1.16440	0.582198	−	0.813047i	$-0.302193\pi$
$883$	34.6004	1.16440	0.582198	+	0.813047i	$0.302193\pi$
$884$	−32.2828	−1.08579
$885$	0	0
$886$	−12.0871	−0.406073
$887$	10.5832	0.355348	0.177674	−	0.984089i	$-0.443143\pi$
$887$	10.5832	0.355348	0.177674	+	0.984089i	$0.443143\pi$
$888$	0	0
$889$	−13.1836	−0.442163
$890$	0.0978347	0.00327943
$891$	0	0
$892$	−5.18359	−0.173559
$893$	1.53617	0.0514060
$894$	0	0
$895$	−24.7439	−0.827098
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−32.8310	−1.09558
$899$	0	0
$900$	0	0
$901$	−97.8094	−3.25850
$902$	7.42758	0.247311
$903$	0	0
$904$	8.29350	0.275838
$905$	20.4263	0.678992
$906$	0	0
$907$	−34.8659	−1.15770	−0.578852	−	0.815433i	$-0.696499\pi$
$907$	−34.8659	−1.15770	−0.578852	+	0.815433i	$0.696499\pi$
$908$	11.6474	0.386533
$909$	0	0
$910$	−4.09783	−0.135842
$911$	49.7077	1.64689	0.823444	−	0.567397i	$-0.192049\pi$
$911$	49.7077	1.64689	0.823444	+	0.567397i	$0.192049\pi$
$912$	0	0
$913$	5.42758	0.179627
$914$	−41.4034	−1.36950
$915$	0	0
$916$	21.5013	0.710421
$917$	−17.1836	−0.567452
$918$	0	0
$919$	24.3526	0.803318	0.401659	−	0.915789i	$-0.368434\pi$
$919$	24.3526	0.803318	0.401659	+	0.915789i	$0.368434\pi$
$920$	−2.89008	−0.0952832
$921$	0	0
$922$	−21.3163	−0.702016
$923$	−17.2922	−0.569179
$924$	0	0
$925$	−8.31767	−0.273483
$926$	20.7922	0.683276
$927$	0	0
$928$	−9.20775	−0.302259
$929$	2.36599	0.0776257	0.0388129	−	0.999246i	$-0.487642\pi$
$929$	2.36599	0.0776257	0.0388129	+	0.999246i	$0.487642\pi$
$930$	0	0
$931$	−0.219833	−0.00720472
$932$	−27.9758	−0.916379
$933$	0	0
$934$	32.7439	1.07141
$935$	7.87800	0.257638
$936$	0	0
$937$	8.17151	0.266951	0.133476	−	0.991052i	$-0.457386\pi$
$937$	8.17151	0.266951	0.133476	+	0.991052i	$0.457386\pi$
$938$	6.67025	0.217791
$939$	0	0
$940$	−6.98792	−0.227921
$941$	31.3793	1.02293	0.511467	−	0.859303i	$-0.329102\pi$
$941$	31.3793	1.02293	0.511467	+	0.859303i	$0.329102\pi$
$942$	0	0
$943$	21.4663	0.699040
$944$	2.21983	0.0722494
$945$	0	0
$946$	−2.98792	−0.0971456
$947$	−7.64742	−0.248508	−0.124254	−	0.992250i	$-0.539654\pi$
$947$	−7.64742	−0.248508	−0.124254	+	0.992250i	$0.539654\pi$
$948$	0	0
$949$	−31.8818	−1.03493
$950$	−0.219833	−0.00713231
$951$	0	0
$952$	7.87800	0.255328
$953$	−31.7802	−1.02946	−0.514730	−	0.857352i	$-0.672108\pi$
$953$	−31.7802	−1.02946	−0.514730	+	0.857352i	$0.672108\pi$
$954$	0	0
$955$	−24.4155	−0.790067
$956$	4.09783	0.132533
$957$	0	0
$958$	26.4155	0.853446
$959$	−13.8780	−0.448144
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−34.0844	−1.09893
$963$	0	0
$964$	−18.7439	−0.603701
$965$	−1.45175	−0.0467334
$966$	0	0
$967$	41.8404	1.34550	0.672749	−	0.739871i	$-0.265113\pi$
$967$	41.8404	1.34550	0.672749	+	0.739871i	$0.265113\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−3.78017	−0.121374
$971$	35.7077	1.14591	0.572957	−	0.819585i	$-0.305796\pi$
$971$	35.7077	1.14591	0.572957	+	0.819585i	$0.305796\pi$
$972$	0	0
$973$	−1.56033	−0.0500220
$974$	2.13275	0.0683378
$975$	0	0
$976$	0.987918	0.0316225
$977$	17.4168	0.557214	0.278607	−	0.960405i	$-0.410127\pi$
$977$	17.4168	0.557214	0.278607	+	0.960405i	$0.410127\pi$
$978$	0	0
$979$	−0.0978347	−0.00312681
$980$	1.00000	0.0319438
$981$	0	0
$982$	−5.78017	−0.184453
$983$	54.2344	1.72981	0.864905	−	0.501936i	$-0.167379\pi$
$983$	54.2344	1.72981	0.864905	+	0.501936i	$0.167379\pi$
$984$	0	0
$985$	−7.23191	−0.230428
$986$	72.5387	2.31010
$987$	0	0
$988$	−0.900837	−0.0286595
$989$	−8.63533	−0.274588
$990$	0	0
$991$	17.6931	0.562039	0.281020	−	0.959702i	$-0.409327\pi$
$991$	17.6931	0.562039	0.281020	+	0.959702i	$0.409327\pi$
$992$	0	0
$993$	0	0
$994$	4.21983	0.133845
$995$	−4.63533	−0.146950
$996$	0	0
$997$	2.36599	0.0749318	0.0374659	−	0.999298i	$-0.488071\pi$
$997$	2.36599	0.0749318	0.0374659	+	0.999298i	$0.488071\pi$
$998$	4.90084	0.155133
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.ck.1.3	yes	3
3.2	odd	2		6930.2.a.cg.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6930.2.a.cg.1.3	✓	3	3.2	odd	2
6930.2.a.ck.1.3	yes	3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +4.09783 q^{13} -1.00000 q^{14} +1.00000 q^{16} -7.87800 q^{17} -0.219833 q^{19} +1.00000 q^{20} -1.00000 q^{22} -2.89008 q^{23} +1.00000 q^{25} +4.09783 q^{26} -1.00000 q^{28} -9.20775 q^{29} +1.00000 q^{32} -7.87800 q^{34} -1.00000 q^{35} -8.31767 q^{37} -0.219833 q^{38} +1.00000 q^{40} -7.42758 q^{41} +2.98792 q^{43} -1.00000 q^{44} -2.89008 q^{46} -6.98792 q^{47} +1.00000 q^{49} +1.00000 q^{50} +4.09783 q^{52} +12.4155 q^{53} -1.00000 q^{55} -1.00000 q^{56} -9.20775 q^{58} +2.21983 q^{59} +0.987918 q^{61} +1.00000 q^{64} +4.09783 q^{65} -6.67025 q^{67} -7.87800 q^{68} -1.00000 q^{70} -4.21983 q^{71} -7.78017 q^{73} -8.31767 q^{74} -0.219833 q^{76} +1.00000 q^{77} -6.76809 q^{79} +1.00000 q^{80} -7.42758 q^{82} -5.42758 q^{83} -7.87800 q^{85} +2.98792 q^{86} -1.00000 q^{88} +0.0978347 q^{89} -4.09783 q^{91} -2.89008 q^{92} -6.98792 q^{94} -0.219833 q^{95} -3.78017 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8} + 3 q^{10} - 3 q^{11} - 6 q^{13} - 3 q^{14} + 3 q^{16} - 4 q^{17} - 2 q^{19} + 3 q^{20} - 3 q^{22} - 8 q^{23} + 3 q^{25} - 6 q^{26} - 3 q^{28} - 10 q^{29} + 3 q^{32} - 4 q^{34} - 3 q^{35} - 8 q^{37} - 2 q^{38} + 3 q^{40} - 6 q^{41} - 10 q^{43} - 3 q^{44} - 8 q^{46} - 2 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{52} + 2 q^{53} - 3 q^{55} - 3 q^{56} - 10 q^{58} + 8 q^{59} - 16 q^{61} + 3 q^{64} - 6 q^{65} - 18 q^{67} - 4 q^{68} - 3 q^{70} - 14 q^{71} - 22 q^{73} - 8 q^{74} - 2 q^{76} + 3 q^{77} + 3 q^{80} - 6 q^{82} - 4 q^{85} - 10 q^{86} - 3 q^{88} - 18 q^{89} + 6 q^{91} - 8 q^{92} - 2 q^{94} - 2 q^{95} - 10 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 3 * q^8 + 3 * q^10 - 3 * q^11 - 6 * q^13 - 3 * q^14 + 3 * q^16 - 4 * q^17 - 2 * q^19 + 3 * q^20 - 3 * q^22 - 8 * q^23 + 3 * q^25 - 6 * q^26 - 3 * q^28 - 10 * q^29 + 3 * q^32 - 4 * q^34 - 3 * q^35 - 8 * q^37 - 2 * q^38 + 3 * q^40 - 6 * q^41 - 10 * q^43 - 3 * q^44 - 8 * q^46 - 2 * q^47 + 3 * q^49 + 3 * q^50 - 6 * q^52 + 2 * q^53 - 3 * q^55 - 3 * q^56 - 10 * q^58 + 8 * q^59 - 16 * q^61 + 3 * q^64 - 6 * q^65 - 18 * q^67 - 4 * q^68 - 3 * q^70 - 14 * q^71 - 22 * q^73 - 8 * q^74 - 2 * q^76 + 3 * q^77 + 3 * q^80 - 6 * q^82 - 4 * q^85 - 10 * q^86 - 3 * q^88 - 18 * q^89 + 6 * q^91 - 8 * q^92 - 2 * q^94 - 2 * q^95 - 10 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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