LMFDB - Embedded newform 6930.2.a.cg.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:	$0$
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			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -4.71379 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.70171 q^{17} -6.98792 q^{19} -1.00000 q^{20} -1.00000 q^{22} -0.493959 q^{23} +1.00000 q^{25} +4.71379 q^{26} -1.00000 q^{28} +3.78017 q^{29} -1.00000 q^{32} +7.70171 q^{34} +1.00000 q^{35} -6.27413 q^{37} +6.98792 q^{38} +1.00000 q^{40} +8.76809 q^{41} -9.20775 q^{43} +1.00000 q^{44} +0.493959 q^{46} -5.20775 q^{47} +1.00000 q^{49} -1.00000 q^{50} -4.71379 q^{52} -1.56033 q^{53} -1.00000 q^{55} +1.00000 q^{56} -3.78017 q^{58} -8.98792 q^{59} -11.2078 q^{61} +1.00000 q^{64} +4.71379 q^{65} +3.48188 q^{67} -7.70171 q^{68} -1.00000 q^{70} +10.9879 q^{71} -1.01208 q^{73} +6.27413 q^{74} -6.98792 q^{76} -1.00000 q^{77} +12.1957 q^{79} -1.00000 q^{80} -8.76809 q^{82} +6.76809 q^{83} +7.70171 q^{85} +9.20775 q^{86} -1.00000 q^{88} +8.71379 q^{89} +4.71379 q^{91} -0.493959 q^{92} +5.20775 q^{94} +6.98792 q^{95} +2.98792 q^{97} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−4.71379	−1.30737	−0.653685	−	0.756766i	$-0.726778\pi$
$13$	−4.71379	−1.30737	−0.653685	+	0.756766i	$0.726778\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.70171	−1.86794	−0.933970	−	0.357353i	$-0.883679\pi$
$17$	−7.70171	−1.86794	−0.933970	+	0.357353i	$0.883679\pi$
$18$	0	0
$19$	−6.98792	−1.60314	−0.801569	−	0.597902i	$-0.796001\pi$
$19$	−6.98792	−1.60314	−0.801569	+	0.597902i	$0.796001\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−0.493959	−0.102998	−0.0514988	−	0.998673i	$-0.516400\pi$
$23$	−0.493959	−0.102998	−0.0514988	+	0.998673i	$0.516400\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.71379	0.924451
$27$	0	0
$28$	−1.00000	−0.188982
$29$	3.78017	0.701959	0.350980	−	0.936383i	$-0.385849\pi$
$29$	3.78017	0.701959	0.350980	+	0.936383i	$0.385849\pi$
$30$	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
$34$	7.70171	1.32083
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.27413	−1.03146	−0.515730	−	0.856751i	$-0.672479\pi$
$37$	−6.27413	−1.03146	−0.515730	+	0.856751i	$0.672479\pi$
$38$	6.98792	1.13359
$39$	0	0
$40$	1.00000	0.158114
$41$	8.76809	1.36934	0.684672	−	0.728851i	$-0.259945\pi$
$41$	8.76809	1.36934	0.684672	+	0.728851i	$0.259945\pi$
$42$	0	0
$43$	−9.20775	−1.40417	−0.702084	−	0.712094i	$-0.747747\pi$
$43$	−9.20775	−1.40417	−0.702084	+	0.712094i	$0.747747\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0.493959	0.0728303
$47$	−5.20775	−0.759629	−0.379814	−	0.925063i	$-0.624012\pi$
$47$	−5.20775	−0.759629	−0.379814	+	0.925063i	$0.624012\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−4.71379	−0.653685
$53$	−1.56033	−0.214328	−0.107164	−	0.994241i	$-0.534177\pi$
$53$	−1.56033	−0.214328	−0.107164	+	0.994241i	$0.534177\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	−3.78017	−0.496360
$59$	−8.98792	−1.17013	−0.585064	−	0.810987i	$-0.698930\pi$
$59$	−8.98792	−1.17013	−0.585064	+	0.810987i	$0.698930\pi$
$60$	0	0
$61$	−11.2078	−1.43501	−0.717503	−	0.696556i	$-0.754715\pi$
$61$	−11.2078	−1.43501	−0.717503	+	0.696556i	$0.754715\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	4.71379	0.584674
$66$	0	0
$67$	3.48188	0.425379	0.212690	−	0.977120i	$-0.431778\pi$
$67$	3.48188	0.425379	0.212690	+	0.977120i	$0.431778\pi$
$68$	−7.70171	−0.933970
$69$	0	0
$70$	−1.00000	−0.119523
$71$	10.9879	1.30403	0.652013	−	0.758208i	$-0.273925\pi$
$71$	10.9879	1.30403	0.652013	+	0.758208i	$0.273925\pi$
$72$	0	0
$73$	−1.01208	−0.118455	−0.0592276	−	0.998245i	$-0.518864\pi$
$73$	−1.01208	−0.118455	−0.0592276	+	0.998245i	$0.518864\pi$
$74$	6.27413	0.729352
$75$	0	0
$76$	−6.98792	−0.801569
$77$	−1.00000	−0.113961
$78$	0	0
$79$	12.1957	1.37212	0.686060	−	0.727545i	$-0.259339\pi$
$79$	12.1957	1.37212	0.686060	+	0.727545i	$0.259339\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−8.76809	−0.968273
$83$	6.76809	0.742894	0.371447	−	0.928454i	$-0.378862\pi$
$83$	6.76809	0.742894	0.371447	+	0.928454i	$0.378862\pi$
$84$	0	0
$85$	7.70171	0.835368
$86$	9.20775	0.992897
$87$	0	0
$88$	−1.00000	−0.106600
$89$	8.71379	0.923660	0.461830	−	0.886968i	$-0.347193\pi$
$89$	8.71379	0.923660	0.461830	+	0.886968i	$0.347193\pi$
$90$	0	0
$91$	4.71379	0.494140
$92$	−0.493959	−0.0514988
$93$	0	0
$94$	5.20775	0.537138
$95$	6.98792	0.716945
$96$	0	0
$97$	2.98792	0.303377	0.151689	−	0.988428i	$-0.451529\pi$
$97$	2.98792	0.303377	0.151689	+	0.988428i	$0.451529\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−4.76809	−0.474442	−0.237221	−	0.971456i	$-0.576237\pi$
$101$	−4.76809	−0.474442	−0.237221	+	0.971456i	$0.576237\pi$
$102$	0	0
$103$	12.5483	1.23642	0.618208	−	0.786014i	$-0.287859\pi$
$103$	12.5483	1.23642	0.618208	+	0.786014i	$0.287859\pi$
$104$	4.71379	0.462225
$105$	0	0
$106$	1.56033	0.151553
$107$	0.792249	0.0765896	0.0382948	−	0.999266i	$-0.487807\pi$
$107$	0.792249	0.0765896	0.0382948	+	0.999266i	$0.487807\pi$
$108$	0	0
$109$	−4.27413	−0.409387	−0.204694	−	0.978826i	$-0.565620\pi$
$109$	−4.27413	−0.409387	−0.204694	+	0.978826i	$0.565620\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	18.1414	1.70660	0.853299	−	0.521423i	$-0.174599\pi$
$113$	18.1414	1.70660	0.853299	+	0.521423i	$0.174599\pi$
$114$	0	0
$115$	0.493959	0.0460619
$116$	3.78017	0.350980
$117$	0	0
$118$	8.98792	0.827405
$119$	7.70171	0.706015
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.2078	1.01470
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.6353	−1.47615	−0.738074	−	0.674720i	$-0.764264\pi$
$127$	−16.6353	−1.47615	−0.738074	+	0.674720i	$0.764264\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−4.71379	−0.413427
$131$	12.6353	1.10395	0.551977	−	0.833859i	$-0.313874\pi$
$131$	12.6353	1.10395	0.551977	+	0.833859i	$0.313874\pi$
$132$	0	0
$133$	6.98792	0.605929
$134$	−3.48188	−0.300788
$135$	0	0
$136$	7.70171	0.660416
$137$	1.70171	0.145387	0.0726935	−	0.997354i	$-0.476841\pi$
$137$	1.70171	0.145387	0.0726935	+	0.997354i	$0.476841\pi$
$138$	0	0
$139$	−11.9758	−1.01578	−0.507889	−	0.861423i	$-0.669574\pi$
$139$	−11.9758	−1.01578	−0.507889	+	0.861423i	$0.669574\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−10.9879	−0.922086
$143$	−4.71379	−0.394187
$144$	0	0
$145$	−3.78017	−0.313926
$146$	1.01208	0.0837605
$147$	0	0
$148$	−6.27413	−0.515730
$149$	0.659498	0.0540281	0.0270141	−	0.999635i	$-0.491400\pi$
$149$	0.659498	0.0540281	0.0270141	+	0.999635i	$0.491400\pi$
$150$	0	0
$151$	2.21983	0.180647	0.0903237	−	0.995912i	$-0.471210\pi$
$151$	2.21983	0.180647	0.0903237	+	0.995912i	$0.471210\pi$
$152$	6.98792	0.566795
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	1.64742	0.131478	0.0657391	−	0.997837i	$-0.479060\pi$
$157$	1.64742	0.131478	0.0657391	+	0.997837i	$0.479060\pi$
$158$	−12.1957	−0.970235
$159$	0	0
$160$	1.00000	0.0790569
$161$	0.493959	0.0389294
$162$	0	0
$163$	11.4819	0.899330	0.449665	−	0.893197i	$-0.351543\pi$
$163$	11.4819	0.899330	0.449665	+	0.893197i	$0.351543\pi$
$164$	8.76809	0.684672
$165$	0	0
$166$	−6.76809	−0.525305
$167$	−6.14138	−0.475234	−0.237617	−	0.971359i	$-0.576366\pi$
$167$	−6.14138	−0.475234	−0.237617	+	0.971359i	$0.576366\pi$
$168$	0	0
$169$	9.21983	0.709218
$170$	−7.70171	−0.590694
$171$	0	0
$172$	−9.20775	−0.702084
$173$	−1.20775	−0.0918236	−0.0459118	−	0.998945i	$-0.514619\pi$
$173$	−1.20775	−0.0918236	−0.0459118	+	0.998945i	$0.514619\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−8.71379	−0.653126
$179$	−18.6112	−1.39106	−0.695532	−	0.718495i	$-0.744831\pi$
$179$	−18.6112	−1.39106	−0.695532	+	0.718495i	$0.744831\pi$
$180$	0	0
$181$	−20.8853	−1.55239	−0.776196	−	0.630492i	$-0.782853\pi$
$181$	−20.8853	−1.55239	−0.776196	+	0.630492i	$0.782853\pi$
$182$	−4.71379	−0.349409
$183$	0	0
$184$	0.493959	0.0364152
$185$	6.27413	0.461283
$186$	0	0
$187$	−7.70171	−0.563205
$188$	−5.20775	−0.379814
$189$	0	0
$190$	−6.98792	−0.506957
$191$	13.5603	0.981191	0.490596	−	0.871387i	$-0.336779\pi$
$191$	13.5603	0.981191	0.490596	+	0.871387i	$0.336779\pi$
$192$	0	0
$193$	−27.1836	−1.95672	−0.978359	−	0.206916i	$-0.933657\pi$
$193$	−27.1836	−1.95672	−0.978359	+	0.206916i	$0.933657\pi$
$194$	−2.98792	−0.214520
$195$	0	0
$196$	1.00000	0.0714286
$197$	26.1957	1.86636	0.933182	−	0.359404i	$-0.117020\pi$
$197$	26.1957	1.86636	0.933182	+	0.359404i	$0.117020\pi$
$198$	0	0
$199$	−0.548253	−0.0388647	−0.0194323	−	0.999811i	$-0.506186\pi$
$199$	−0.548253	−0.0388647	−0.0194323	+	0.999811i	$0.506186\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	4.76809	0.335481
$203$	−3.78017	−0.265316
$204$	0	0
$205$	−8.76809	−0.612390
$206$	−12.5483	−0.874278
$207$	0	0
$208$	−4.71379	−0.326843
$209$	−6.98792	−0.483364
$210$	0	0
$211$	−1.92154	−0.132284	−0.0661422	−	0.997810i	$-0.521069\pi$
$211$	−1.92154	−0.132284	−0.0661422	+	0.997810i	$0.521069\pi$
$212$	−1.56033	−0.107164
$213$	0	0
$214$	−0.792249	−0.0541570
$215$	9.20775	0.627963
$216$	0	0
$217$	0	0
$218$	4.27413	0.289480
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	36.3043	2.44209
$222$	0	0
$223$	24.6353	1.64970	0.824852	−	0.565349i	$-0.191258\pi$
$223$	24.6353	1.64970	0.824852	+	0.565349i	$0.191258\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−18.1414	−1.20675
$227$	−19.7560	−1.31125	−0.655626	−	0.755086i	$-0.727595\pi$
$227$	−19.7560	−1.31125	−0.655626	+	0.755086i	$0.727595\pi$
$228$	0	0
$229$	−10.3612	−0.684688	−0.342344	−	0.939575i	$-0.611221\pi$
$229$	−10.3612	−0.684688	−0.342344	+	0.939575i	$0.611221\pi$
$230$	−0.493959	−0.0325707
$231$	0	0
$232$	−3.78017	−0.248180
$233$	3.58450	0.234828	0.117414	−	0.993083i	$-0.462540\pi$
$233$	3.58450	0.234828	0.117414	+	0.993083i	$0.462540\pi$
$234$	0	0
$235$	5.20775	0.339716
$236$	−8.98792	−0.585064
$237$	0	0
$238$	−7.70171	−0.499228
$239$	4.71379	0.304910	0.152455	−	0.988310i	$-0.451282\pi$
$239$	4.71379	0.304910	0.152455	+	0.988310i	$0.451282\pi$
$240$	0	0
$241$	24.6112	1.58534	0.792672	−	0.609648i	$-0.208689\pi$
$241$	24.6112	1.58534	0.792672	+	0.609648i	$0.208689\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−11.2078	−0.717503
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	32.9396	2.09590
$248$	0	0
$249$	0	0
$250$	1.00000	0.0632456
$251$	−25.9758	−1.63958	−0.819790	−	0.572664i	$-0.805910\pi$
$251$	−25.9758	−1.63958	−0.819790	+	0.572664i	$0.805910\pi$
$252$	0	0
$253$	−0.493959	−0.0310549
$254$	16.6353	1.04379
$255$	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
$259$	6.27413	0.389855
$260$	4.71379	0.292337
$261$	0	0
$262$	−12.6353	−0.780614
$263$	−9.42758	−0.581330	−0.290665	−	0.956825i	$-0.593876\pi$
$263$	−9.42758	−0.581330	−0.290665	+	0.956825i	$0.593876\pi$
$264$	0	0
$265$	1.56033	0.0958506
$266$	−6.98792	−0.428457
$267$	0	0
$268$	3.48188	0.212690
$269$	19.5362	1.19114	0.595571	−	0.803303i	$-0.296926\pi$
$269$	19.5362	1.19114	0.595571	+	0.803303i	$0.296926\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	−7.70171	−0.466985
$273$	0	0
$274$	−1.70171	−0.102804
$275$	1.00000	0.0603023
$276$	0	0
$277$	13.6233	0.818542	0.409271	−	0.912413i	$-0.365783\pi$
$277$	13.6233	0.818542	0.409271	+	0.912413i	$0.365783\pi$
$278$	11.9758	0.718263
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−6.46980	−0.385956	−0.192978	−	0.981203i	$-0.561815\pi$
$281$	−6.46980	−0.385956	−0.192978	+	0.981203i	$0.561815\pi$
$282$	0	0
$283$	−23.7017	−1.40892	−0.704460	−	0.709743i	$-0.748811\pi$
$283$	−23.7017	−1.40892	−0.704460	+	0.709743i	$0.748811\pi$
$284$	10.9879	0.652013
$285$	0	0
$286$	4.71379	0.278732
$287$	−8.76809	−0.517564
$288$	0	0
$289$	42.3163	2.48920
$290$	3.78017	0.221979
$291$	0	0
$292$	−1.01208	−0.0592276
$293$	2.19567	0.128272	0.0641362	−	0.997941i	$-0.479571\pi$
$293$	2.19567	0.128272	0.0641362	+	0.997941i	$0.479571\pi$
$294$	0	0
$295$	8.98792	0.523297
$296$	6.27413	0.364676
$297$	0	0
$298$	−0.659498	−0.0382037
$299$	2.32842	0.134656
$300$	0	0
$301$	9.20775	0.530726
$302$	−2.21983	−0.127737
$303$	0	0
$304$	−6.98792	−0.400785
$305$	11.2078	0.641754
$306$	0	0
$307$	10.2741	0.586375	0.293188	−	0.956055i	$-0.405284\pi$
$307$	10.2741	0.586375	0.293188	+	0.956055i	$0.405284\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	0	0
$311$	−5.67755	−0.321944	−0.160972	−	0.986959i	$-0.551463\pi$
$311$	−5.67755	−0.321944	−0.160972	+	0.986959i	$0.551463\pi$
$312$	0	0
$313$	−30.3043	−1.71290	−0.856449	−	0.516232i	$-0.827334\pi$
$313$	−30.3043	−1.71290	−0.856449	+	0.516232i	$0.827334\pi$
$314$	−1.64742	−0.0929691
$315$	0	0
$316$	12.1957	0.686060
$317$	15.3793	0.863785	0.431893	−	0.901925i	$-0.357846\pi$
$317$	15.3793	0.863785	0.431893	+	0.901925i	$0.357846\pi$
$318$	0	0
$319$	3.78017	0.211649
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−0.493959	−0.0275273
$323$	53.8189	2.99456
$324$	0	0
$325$	−4.71379	−0.261474
$326$	−11.4819	−0.635922
$327$	0	0
$328$	−8.76809	−0.484137
$329$	5.20775	0.287113
$330$	0	0
$331$	−12.8310	−0.705256	−0.352628	−	0.935764i	$-0.614712\pi$
$331$	−12.8310	−0.705256	−0.352628	+	0.935764i	$0.614712\pi$
$332$	6.76809	0.371447
$333$	0	0
$334$	6.14138	0.336041
$335$	−3.48188	−0.190235
$336$	0	0
$337$	−1.56033	−0.0849969	−0.0424984	−	0.999097i	$-0.513532\pi$
$337$	−1.56033	−0.0849969	−0.0424984	+	0.999097i	$0.513532\pi$
$338$	−9.21983	−0.501493
$339$	0	0
$340$	7.70171	0.417684
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	9.20775	0.496449
$345$	0	0
$346$	1.20775	0.0649291
$347$	18.0629	0.969668	0.484834	−	0.874606i	$-0.338880\pi$
$347$	18.0629	0.969668	0.484834	+	0.874606i	$0.338880\pi$
$348$	0	0
$349$	11.6474	0.623472	0.311736	−	0.950169i	$-0.399090\pi$
$349$	11.6474	0.623472	0.311736	+	0.950169i	$0.399090\pi$
$350$	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
$355$	−10.9879	−0.583178
$356$	8.71379	0.461830
$357$	0	0
$358$	18.6112	0.983631
$359$	−25.1051	−1.32500	−0.662499	−	0.749063i	$-0.730504\pi$
$359$	−25.1051	−1.32500	−0.662499	+	0.749063i	$0.730504\pi$
$360$	0	0
$361$	29.8310	1.57005
$362$	20.8853	1.09771
$363$	0	0
$364$	4.71379	0.247070
$365$	1.01208	0.0529748
$366$	0	0
$367$	−17.3793	−0.907190	−0.453595	−	0.891208i	$-0.649859\pi$
$367$	−17.3793	−0.907190	−0.453595	+	0.891208i	$0.649859\pi$
$368$	−0.493959	−0.0257494
$369$	0	0
$370$	−6.27413	−0.326176
$371$	1.56033	0.0810086
$372$	0	0
$373$	3.09916	0.160469	0.0802343	−	0.996776i	$-0.474433\pi$
$373$	3.09916	0.160469	0.0802343	+	0.996776i	$0.474433\pi$
$374$	7.70171	0.398246
$375$	0	0
$376$	5.20775	0.268569
$377$	−17.8189	−0.917721
$378$	0	0
$379$	−1.58450	−0.0813902	−0.0406951	−	0.999172i	$-0.512957\pi$
$379$	−1.58450	−0.0813902	−0.0406951	+	0.999172i	$0.512957\pi$
$380$	6.98792	0.358473
$381$	0	0
$382$	−13.5603	−0.693807
$383$	−33.1594	−1.69437	−0.847184	−	0.531300i	$-0.821704\pi$
$383$	−33.1594	−1.69437	−0.847184	+	0.531300i	$0.821704\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	27.1836	1.38361
$387$	0	0
$388$	2.98792	0.151689
$389$	−14.6353	−0.742041	−0.371020	−	0.928625i	$-0.620992\pi$
$389$	−14.6353	−0.742041	−0.371020	+	0.928625i	$0.620992\pi$
$390$	0	0
$391$	3.80433	0.192393
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−26.1957	−1.31972
$395$	−12.1957	−0.613631
$396$	0	0
$397$	−8.65950	−0.434608	−0.217304	−	0.976104i	$-0.569726\pi$
$397$	−8.65950	−0.434608	−0.217304	+	0.976104i	$0.569726\pi$
$398$	0.548253	0.0274815
$399$	0	0
$400$	1.00000	0.0500000
$401$	14.4155	0.719876	0.359938	−	0.932976i	$-0.382798\pi$
$401$	14.4155	0.719876	0.359938	+	0.932976i	$0.382798\pi$
$402$	0	0
$403$	0	0
$404$	−4.76809	−0.237221
$405$	0	0
$406$	3.78017	0.187607
$407$	−6.27413	−0.310997
$408$	0	0
$409$	10.5483	0.521578	0.260789	−	0.965396i	$-0.416017\pi$
$409$	10.5483	0.521578	0.260789	+	0.965396i	$0.416017\pi$
$410$	8.76809	0.433025
$411$	0	0
$412$	12.5483	0.618208
$413$	8.98792	0.442267
$414$	0	0
$415$	−6.76809	−0.332232
$416$	4.71379	0.231113
$417$	0	0
$418$	6.98792	0.341790
$419$	12.8310	0.626836	0.313418	−	0.949615i	$-0.398526\pi$
$419$	12.8310	0.626836	0.313418	+	0.949615i	$0.398526\pi$
$420$	0	0
$421$	23.9758	1.16851	0.584255	−	0.811570i	$-0.301387\pi$
$421$	23.9758	1.16851	0.584255	+	0.811570i	$0.301387\pi$
$422$	1.92154	0.0935392
$423$	0	0
$424$	1.56033	0.0757766
$425$	−7.70171	−0.373588
$426$	0	0
$427$	11.2078	0.542381
$428$	0.792249	0.0382948
$429$	0	0
$430$	−9.20775	−0.444037
$431$	6.73795	0.324556	0.162278	−	0.986745i	$-0.448116\pi$
$431$	6.73795	0.324556	0.162278	+	0.986745i	$0.448116\pi$
$432$	0	0
$433$	31.4276	1.51031	0.755157	−	0.655544i	$-0.227561\pi$
$433$	31.4276	1.51031	0.755157	+	0.655544i	$0.227561\pi$
$434$	0	0
$435$	0	0
$436$	−4.27413	−0.204694
$437$	3.45175	0.165119
$438$	0	0
$439$	15.3163	0.731009	0.365504	−	0.930810i	$-0.380896\pi$
$439$	15.3163	0.731009	0.365504	+	0.930810i	$0.380896\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−36.3043	−1.72682
$443$	33.7318	1.60265	0.801324	−	0.598231i	$-0.204129\pi$
$443$	33.7318	1.60265	0.801324	+	0.598231i	$0.204129\pi$
$444$	0	0
$445$	−8.71379	−0.413073
$446$	−24.6353	−1.16652
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	11.1207	0.524817	0.262408	−	0.964957i	$-0.415483\pi$
$449$	11.1207	0.524817	0.262408	+	0.964957i	$0.415483\pi$
$450$	0	0
$451$	8.76809	0.412873
$452$	18.1414	0.853299
$453$	0	0
$454$	19.7560	0.927195
$455$	−4.71379	−0.220986
$456$	0	0
$457$	−18.3526	−0.858498	−0.429249	−	0.903186i	$-0.641222\pi$
$457$	−18.3526	−0.858498	−0.429249	+	0.903186i	$0.641222\pi$
$458$	10.3612	0.484147
$459$	0	0
$460$	0.493959	0.0230310
$461$	−23.3793	−1.08888	−0.544440	−	0.838800i	$-0.683258\pi$
$461$	−23.3793	−1.08888	−0.544440	+	0.838800i	$0.683258\pi$
$462$	0	0
$463$	26.2198	1.21854	0.609269	−	0.792963i	$-0.291463\pi$
$463$	26.2198	1.21854	0.609269	+	0.792963i	$0.291463\pi$
$464$	3.78017	0.175490
$465$	0	0
$466$	−3.58450	−0.166049
$467$	10.6112	0.491026	0.245513	−	0.969393i	$-0.421044\pi$
$467$	10.6112	0.491026	0.245513	+	0.969393i	$0.421044\pi$
$468$	0	0
$469$	−3.48188	−0.160778
$470$	−5.20775	−0.240216
$471$	0	0
$472$	8.98792	0.413702
$473$	−9.20775	−0.423373
$474$	0	0
$475$	−6.98792	−0.320628
$476$	7.70171	0.353007
$477$	0	0
$478$	−4.71379	−0.215604
$479$	−15.5603	−0.710970	−0.355485	−	0.934682i	$-0.615684\pi$
$479$	−15.5603	−0.710970	−0.355485	+	0.934682i	$0.615684\pi$
$480$	0	0
$481$	29.5749	1.34850
$482$	−24.6112	−1.12101
$483$	0	0
$484$	1.00000	0.0454545
$485$	−2.98792	−0.135674
$486$	0	0
$487$	−12.7439	−0.577482	−0.288741	−	0.957407i	$-0.593237\pi$
$487$	−12.7439	−0.577482	−0.288741	+	0.957407i	$0.593237\pi$
$488$	11.2078	0.507351
$489$	0	0
$490$	1.00000	0.0451754
$491$	−0.987918	−0.0445841	−0.0222921	−	0.999752i	$-0.507096\pi$
$491$	−0.987918	−0.0445841	−0.0222921	+	0.999752i	$0.507096\pi$
$492$	0	0
$493$	−29.1138	−1.31122
$494$	−32.9396	−1.48202
$495$	0	0
$496$	0	0
$497$	−10.9879	−0.492876
$498$	0	0
$499$	−28.9396	−1.29551	−0.647757	−	0.761847i	$-0.724293\pi$
$499$	−28.9396	−1.29551	−0.647757	+	0.761847i	$0.724293\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	25.9758	1.15936
$503$	27.5206	1.22708	0.613542	−	0.789662i	$-0.289744\pi$
$503$	27.5206	1.22708	0.613542	+	0.789662i	$0.289744\pi$
$504$	0	0
$505$	4.76809	0.212177
$506$	0.493959	0.0219592
$507$	0	0
$508$	−16.6353	−0.738074
$509$	−1.45175	−0.0643475	−0.0321738	−	0.999482i	$-0.510243\pi$
$509$	−1.45175	−0.0643475	−0.0321738	+	0.999482i	$0.510243\pi$
$510$	0	0
$511$	1.01208	0.0447719
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0	0
$515$	−12.5483	−0.552942
$516$	0	0
$517$	−5.20775	−0.229037
$518$	−6.27413	−0.275669
$519$	0	0
$520$	−4.71379	−0.206713
$521$	41.8103	1.83174	0.915871	−	0.401472i	$-0.131501\pi$
$521$	41.8103	1.83174	0.915871	+	0.401472i	$0.131501\pi$
$522$	0	0
$523$	7.26205	0.317547	0.158774	−	0.987315i	$-0.449246\pi$
$523$	7.26205	0.317547	0.158774	+	0.987315i	$0.449246\pi$
$524$	12.6353	0.551977
$525$	0	0
$526$	9.42758	0.411062
$527$	0	0
$528$	0	0
$529$	−22.7560	−0.989391
$530$	−1.56033	−0.0677766
$531$	0	0
$532$	6.98792	0.302965
$533$	−41.3309	−1.79024
$534$	0	0
$535$	−0.792249	−0.0342519
$536$	−3.48188	−0.150394
$537$	0	0
$538$	−19.5362	−0.842264
$539$	1.00000	0.0430730
$540$	0	0
$541$	−10.1414	−0.436012	−0.218006	−	0.975947i	$-0.569955\pi$
$541$	−10.1414	−0.436012	−0.218006	+	0.975947i	$0.569955\pi$
$542$	−12.0000	−0.515444
$543$	0	0
$544$	7.70171	0.330208
$545$	4.27413	0.183083
$546$	0	0
$547$	−18.1957	−0.777991	−0.388995	−	0.921240i	$-0.627178\pi$
$547$	−18.1957	−0.777991	−0.388995	+	0.921240i	$0.627178\pi$
$548$	1.70171	0.0726935
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−26.4155	−1.12534
$552$	0	0
$553$	−12.1957	−0.518613
$554$	−13.6233	−0.578797
$555$	0	0
$556$	−11.9758	−0.507889
$557$	37.7077	1.59773	0.798863	−	0.601513i	$-0.205435\pi$
$557$	37.7077	1.59773	0.798863	+	0.601513i	$0.205435\pi$
$558$	0	0
$559$	43.4034	1.83577
$560$	1.00000	0.0422577
$561$	0	0
$562$	6.46980	0.272912
$563$	37.7318	1.59021	0.795104	−	0.606473i	$-0.207416\pi$
$563$	37.7318	1.59021	0.795104	+	0.606473i	$0.207416\pi$
$564$	0	0
$565$	−18.1414	−0.763213
$566$	23.7017	0.996257
$567$	0	0
$568$	−10.9879	−0.461043
$569$	−12.6025	−0.528326	−0.264163	−	0.964478i	$-0.585096\pi$
$569$	−12.6025	−0.528326	−0.264163	+	0.964478i	$0.585096\pi$
$570$	0	0
$571$	24.6025	1.02958	0.514792	−	0.857315i	$-0.327869\pi$
$571$	24.6025	1.02958	0.514792	+	0.857315i	$0.327869\pi$
$572$	−4.71379	−0.197094
$573$	0	0
$574$	8.76809	0.365973
$575$	−0.493959	−0.0205995
$576$	0	0
$577$	−22.7922	−0.948854	−0.474427	−	0.880295i	$-0.657345\pi$
$577$	−22.7922	−0.948854	−0.474427	+	0.880295i	$0.657345\pi$
$578$	−42.3163	−1.76013
$579$	0	0
$580$	−3.78017	−0.156963
$581$	−6.76809	−0.280788
$582$	0	0
$583$	−1.56033	−0.0646225
$584$	1.01208	0.0418802
$585$	0	0
$586$	−2.19567	−0.0907023
$587$	31.4905	1.29975	0.649876	−	0.760040i	$-0.274821\pi$
$587$	31.4905	1.29975	0.649876	+	0.760040i	$0.274821\pi$
$588$	0	0
$589$	0	0
$590$	−8.98792	−0.370027
$591$	0	0
$592$	−6.27413	−0.257865
$593$	16.5810	0.680902	0.340451	−	0.940262i	$-0.389420\pi$
$593$	16.5810	0.680902	0.340451	+	0.940262i	$0.389420\pi$
$594$	0	0
$595$	−7.70171	−0.315739
$596$	0.659498	0.0270141
$597$	0	0
$598$	−2.32842	−0.0952162
$599$	−23.0965	−0.943698	−0.471849	−	0.881679i	$-0.656413\pi$
$599$	−23.0965	−0.943698	−0.471849	+	0.881679i	$0.656413\pi$
$600$	0	0
$601$	24.0844	0.982424	0.491212	−	0.871040i	$-0.336554\pi$
$601$	24.0844	0.982424	0.491212	+	0.871040i	$0.336554\pi$
$602$	−9.20775	−0.375280
$603$	0	0
$604$	2.21983	0.0903237
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	8.28275	0.336186	0.168093	−	0.985771i	$-0.446239\pi$
$607$	8.28275	0.336186	0.168093	+	0.985771i	$0.446239\pi$
$608$	6.98792	0.283398
$609$	0	0
$610$	−11.2078	−0.453789
$611$	24.5483	0.993116
$612$	0	0
$613$	−37.1836	−1.50183	−0.750915	−	0.660398i	$-0.770387\pi$
$613$	−37.1836	−1.50183	−0.750915	+	0.660398i	$0.770387\pi$
$614$	−10.2741	−0.414630
$615$	0	0
$616$	1.00000	0.0402911
$617$	−33.9845	−1.36816	−0.684081	−	0.729406i	$-0.739797\pi$
$617$	−33.9845	−1.36816	−0.684081	+	0.729406i	$0.739797\pi$
$618$	0	0
$619$	−2.88530	−0.115970	−0.0579850	−	0.998317i	$-0.518468\pi$
$619$	−2.88530	−0.115970	−0.0579850	+	0.998317i	$0.518468\pi$
$620$	0	0
$621$	0	0
$622$	5.67755	0.227649
$623$	−8.71379	−0.349111
$624$	0	0
$625$	1.00000	0.0400000
$626$	30.3043	1.21120
$627$	0	0
$628$	1.64742	0.0657391
$629$	48.3215	1.92670
$630$	0	0
$631$	40.0388	1.59392	0.796959	−	0.604034i	$-0.206441\pi$
$631$	40.0388	1.59392	0.796959	+	0.604034i	$0.206441\pi$
$632$	−12.1957	−0.485118
$633$	0	0
$634$	−15.3793	−0.610788
$635$	16.6353	0.660153
$636$	0	0
$637$	−4.71379	−0.186767
$638$	−3.78017	−0.149658
$639$	0	0
$640$	1.00000	0.0395285
$641$	−1.53617	−0.0606751	−0.0303376	−	0.999540i	$-0.509658\pi$
$641$	−1.53617	−0.0606751	−0.0303376	+	0.999540i	$0.509658\pi$
$642$	0	0
$643$	22.4155	0.883981	0.441991	−	0.897020i	$-0.354272\pi$
$643$	22.4155	0.883981	0.441991	+	0.897020i	$0.354272\pi$
$644$	0.493959	0.0194647
$645$	0	0
$646$	−53.8189	−2.11748
$647$	24.3284	0.956449	0.478224	−	0.878238i	$-0.341281\pi$
$647$	24.3284	0.956449	0.478224	+	0.878238i	$0.341281\pi$
$648$	0	0
$649$	−8.98792	−0.352807
$650$	4.71379	0.184890
$651$	0	0
$652$	11.4819	0.449665
$653$	−26.3913	−1.03277	−0.516386	−	0.856356i	$-0.672723\pi$
$653$	−26.3913	−1.03277	−0.516386	+	0.856356i	$0.672723\pi$
$654$	0	0
$655$	−12.6353	−0.493703
$656$	8.76809	0.342336
$657$	0	0
$658$	−5.20775	−0.203019
$659$	28.3913	1.10597	0.552985	−	0.833191i	$-0.313489\pi$
$659$	28.3913	1.10597	0.552985	+	0.833191i	$0.313489\pi$
$660$	0	0
$661$	17.9215	0.697067	0.348534	−	0.937296i	$-0.386680\pi$
$661$	17.9215	0.697067	0.348534	+	0.937296i	$0.386680\pi$
$662$	12.8310	0.498691
$663$	0	0
$664$	−6.76809	−0.262653
$665$	−6.98792	−0.270980
$666$	0	0
$667$	−1.86725	−0.0723002
$668$	−6.14138	−0.237617
$669$	0	0
$670$	3.48188	0.134517
$671$	−11.2078	−0.432670
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	1.56033	0.0601019
$675$	0	0
$676$	9.21983	0.354609
$677$	−51.4663	−1.97801	−0.989006	−	0.147875i	$-0.952757\pi$
$677$	−51.4663	−1.97801	−0.989006	+	0.147875i	$0.952757\pi$
$678$	0	0
$679$	−2.98792	−0.114666
$680$	−7.70171	−0.295347
$681$	0	0
$682$	0	0
$683$	−32.0388	−1.22593	−0.612964	−	0.790110i	$-0.710023\pi$
$683$	−32.0388	−1.22593	−0.612964	+	0.790110i	$0.710023\pi$
$684$	0	0
$685$	−1.70171	−0.0650190
$686$	1.00000	0.0381802
$687$	0	0
$688$	−9.20775	−0.351042
$689$	7.35509	0.280207
$690$	0	0
$691$	−30.4456	−1.15821	−0.579103	−	0.815254i	$-0.696597\pi$
$691$	−30.4456	−1.15821	−0.579103	+	0.815254i	$0.696597\pi$
$692$	−1.20775	−0.0459118
$693$	0	0
$694$	−18.0629	−0.685659
$695$	11.9758	0.454269
$696$	0	0
$697$	−67.5293	−2.55785
$698$	−11.6474	−0.440861
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	3.38883	0.127994	0.0639971	−	0.997950i	$-0.479615\pi$
$701$	3.38883	0.127994	0.0639971	+	0.997950i	$0.479615\pi$
$702$	0	0
$703$	43.8431	1.65357
$704$	1.00000	0.0376889
$705$	0	0
$706$	0	0
$707$	4.76809	0.179322
$708$	0	0
$709$	27.9758	1.05065	0.525327	−	0.850900i	$-0.323943\pi$
$709$	27.9758	1.05065	0.525327	+	0.850900i	$0.323943\pi$
$710$	10.9879	0.412369
$711$	0	0
$712$	−8.71379	−0.326563
$713$	0	0
$714$	0	0
$715$	4.71379	0.176286
$716$	−18.6112	−0.695532
$717$	0	0
$718$	25.1051	0.936915
$719$	13.2862	0.495492	0.247746	−	0.968825i	$-0.420310\pi$
$719$	13.2862	0.495492	0.247746	+	0.968825i	$0.420310\pi$
$720$	0	0
$721$	−12.5483	−0.467321
$722$	−29.8310	−1.11020
$723$	0	0
$724$	−20.8853	−0.776196
$725$	3.78017	0.140392
$726$	0	0
$727$	32.6353	1.21038	0.605189	−	0.796082i	$-0.293098\pi$
$727$	32.6353	1.21038	0.605189	+	0.796082i	$0.293098\pi$
$728$	−4.71379	−0.174705
$729$	0	0
$730$	−1.01208	−0.0374588
$731$	70.9154	2.62290
$732$	0	0
$733$	37.6534	1.39076	0.695380	−	0.718642i	$-0.255236\pi$
$733$	37.6534	1.39076	0.695380	+	0.718642i	$0.255236\pi$
$734$	17.3793	0.641480
$735$	0	0
$736$	0.493959	0.0182076
$737$	3.48188	0.128257
$738$	0	0
$739$	27.0180	0.993875	0.496938	−	0.867786i	$-0.334458\pi$
$739$	27.0180	0.993875	0.496938	+	0.867786i	$0.334458\pi$
$740$	6.27413	0.230641
$741$	0	0
$742$	−1.56033	−0.0572817
$743$	39.5120	1.44956	0.724778	−	0.688983i	$-0.241943\pi$
$743$	39.5120	1.44956	0.724778	+	0.688983i	$0.241943\pi$
$744$	0	0
$745$	−0.659498	−0.0241621
$746$	−3.09916	−0.113468
$747$	0	0
$748$	−7.70171	−0.281602
$749$	−0.792249	−0.0289482
$750$	0	0
$751$	−41.1353	−1.50105	−0.750524	−	0.660844i	$-0.770199\pi$
$751$	−41.1353	−1.50105	−0.750524	+	0.660844i	$0.770199\pi$
$752$	−5.20775	−0.189907
$753$	0	0
$754$	17.8189	0.648927
$755$	−2.21983	−0.0807880
$756$	0	0
$757$	20.5810	0.748031	0.374015	−	0.927423i	$-0.377981\pi$
$757$	20.5810	0.748031	0.374015	+	0.927423i	$0.377981\pi$
$758$	1.58450	0.0575516
$759$	0	0
$760$	−6.98792	−0.253478
$761$	−34.7439	−1.25947	−0.629733	−	0.776812i	$-0.716836\pi$
$761$	−34.7439	−1.25947	−0.629733	+	0.776812i	$0.716836\pi$
$762$	0	0
$763$	4.27413	0.154734
$764$	13.5603	0.490596
$765$	0	0
$766$	33.1594	1.19810
$767$	42.3672	1.52979
$768$	0	0
$769$	−48.2586	−1.74025	−0.870125	−	0.492832i	$-0.835962\pi$
$769$	−48.2586	−1.74025	−0.870125	+	0.492832i	$0.835962\pi$
$770$	−1.00000	−0.0360375
$771$	0	0
$772$	−27.1836	−0.978359
$773$	24.5026	0.881297	0.440648	−	0.897680i	$-0.354749\pi$
$773$	24.5026	0.881297	0.440648	+	0.897680i	$0.354749\pi$
$774$	0	0
$775$	0	0
$776$	−2.98792	−0.107260
$777$	0	0
$778$	14.6353	0.524702
$779$	−61.2707	−2.19525
$780$	0	0
$781$	10.9879	0.393179
$782$	−3.80433	−0.136043
$783$	0	0
$784$	1.00000	0.0357143
$785$	−1.64742	−0.0587988
$786$	0	0
$787$	9.56896	0.341097	0.170548	−	0.985349i	$-0.445446\pi$
$787$	9.56896	0.341097	0.170548	+	0.985349i	$0.445446\pi$
$788$	26.1957	0.933182
$789$	0	0
$790$	12.1957	0.433902
$791$	−18.1414	−0.645033
$792$	0	0
$793$	52.8310	1.87608
$794$	8.65950	0.307314
$795$	0	0
$796$	−0.548253	−0.0194323
$797$	−7.23191	−0.256168	−0.128084	−	0.991763i	$-0.540883\pi$
$797$	−7.23191	−0.256168	−0.128084	+	0.991763i	$0.540883\pi$
$798$	0	0
$799$	40.1086	1.41894
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−14.4155	−0.509029
$803$	−1.01208	−0.0357156
$804$	0	0
$805$	−0.493959	−0.0174098
$806$	0	0
$807$	0	0
$808$	4.76809	0.167741
$809$	4.49396	0.157999	0.0789996	−	0.996875i	$-0.474827\pi$
$809$	4.49396	0.157999	0.0789996	+	0.996875i	$0.474827\pi$
$810$	0	0
$811$	49.8431	1.75023	0.875114	−	0.483917i	$-0.160786\pi$
$811$	49.8431	1.75023	0.875114	+	0.483917i	$0.160786\pi$
$812$	−3.78017	−0.132658
$813$	0	0
$814$	6.27413	0.219908
$815$	−11.4819	−0.402193
$816$	0	0
$817$	64.3430	2.25108
$818$	−10.5483	−0.368811
$819$	0	0
$820$	−8.76809	−0.306195
$821$	8.81641	0.307695	0.153847	−	0.988095i	$-0.450834\pi$
$821$	8.81641	0.307695	0.153847	+	0.988095i	$0.450834\pi$
$822$	0	0
$823$	29.4663	1.02713	0.513566	−	0.858050i	$-0.328324\pi$
$823$	29.4663	1.02713	0.513566	+	0.858050i	$0.328324\pi$
$824$	−12.5483	−0.437139
$825$	0	0
$826$	−8.98792	−0.312730
$827$	−8.30426	−0.288767	−0.144384	−	0.989522i	$-0.546120\pi$
$827$	−8.30426	−0.288767	−0.144384	+	0.989522i	$0.546120\pi$
$828$	0	0
$829$	40.6025	1.41018	0.705092	−	0.709115i	$-0.250906\pi$
$829$	40.6025	1.41018	0.705092	+	0.709115i	$0.250906\pi$
$830$	6.76809	0.234924
$831$	0	0
$832$	−4.71379	−0.163421
$833$	−7.70171	−0.266848
$834$	0	0
$835$	6.14138	0.212531
$836$	−6.98792	−0.241682
$837$	0	0
$838$	−12.8310	−0.443240
$839$	−48.9724	−1.69071	−0.845357	−	0.534202i	$-0.820612\pi$
$839$	−48.9724	−1.69071	−0.845357	+	0.534202i	$0.820612\pi$
$840$	0	0
$841$	−14.7103	−0.507253
$842$	−23.9758	−0.826262
$843$	0	0
$844$	−1.92154	−0.0661422
$845$	−9.21983	−0.317172
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−1.56033	−0.0535821
$849$	0	0
$850$	7.70171	0.264166
$851$	3.09916	0.106238
$852$	0	0
$853$	19.6775	0.673746	0.336873	−	0.941550i	$-0.390631\pi$
$853$	19.6775	0.673746	0.336873	+	0.941550i	$0.390631\pi$
$854$	−11.2078	−0.383521
$855$	0	0
$856$	−0.792249	−0.0270785
$857$	−43.4965	−1.48581	−0.742906	−	0.669396i	$-0.766553\pi$
$857$	−43.4965	−1.48581	−0.742906	+	0.669396i	$0.766553\pi$
$858$	0	0
$859$	23.3733	0.797486	0.398743	−	0.917063i	$-0.369446\pi$
$859$	23.3733	0.797486	0.398743	+	0.917063i	$0.369446\pi$
$860$	9.20775	0.313982
$861$	0	0
$862$	−6.73795	−0.229496
$863$	5.96987	0.203217	0.101608	−	0.994824i	$-0.467601\pi$
$863$	5.96987	0.203217	0.101608	+	0.994824i	$0.467601\pi$
$864$	0	0
$865$	1.20775	0.0410648
$866$	−31.4276	−1.06795
$867$	0	0
$868$	0	0
$869$	12.1957	0.413710
$870$	0	0
$871$	−16.4128	−0.556128
$872$	4.27413	0.144740
$873$	0	0
$874$	−3.45175	−0.116757
$875$	1.00000	0.0338062
$876$	0	0
$877$	−24.0871	−0.813363	−0.406681	−	0.913570i	$-0.633314\pi$
$877$	−24.0871	−0.813363	−0.406681	+	0.913570i	$0.633314\pi$
$878$	−15.3163	−0.516901
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	12.7138	0.428339	0.214169	−	0.976797i	$-0.431296\pi$
$881$	12.7138	0.428339	0.214169	+	0.976797i	$0.431296\pi$
$882$	0	0
$883$	36.5784	1.23096	0.615480	−	0.788152i	$-0.288962\pi$
$883$	36.5784	1.23096	0.615480	+	0.788152i	$0.288962\pi$
$884$	36.3043	1.22104
$885$	0	0
$886$	−33.7318	−1.13324
$887$	21.2137	0.712287	0.356143	−	0.934431i	$-0.384091\pi$
$887$	21.2137	0.712287	0.356143	+	0.934431i	$0.384091\pi$
$888$	0	0
$889$	16.6353	0.557931
$890$	8.71379	0.292087
$891$	0	0
$892$	24.6353	0.824852
$893$	36.3913	1.21779
$894$	0	0
$895$	18.6112	0.622103
$896$	1.00000	0.0334077
$897$	0	0
$898$	−11.1207	−0.371102
$899$	0	0
$900$	0	0
$901$	12.0172	0.400353
$902$	−8.76809	−0.291945
$903$	0	0
$904$	−18.1414	−0.603373
$905$	20.8853	0.694251
$906$	0	0
$907$	−7.09054	−0.235437	−0.117719	−	0.993047i	$-0.537558\pi$
$907$	−7.09054	−0.235437	−0.117719	+	0.993047i	$0.537558\pi$
$908$	−19.7560	−0.655626
$909$	0	0
$910$	4.71379	0.156261
$911$	30.2344	1.00171	0.500856	−	0.865531i	$-0.333019\pi$
$911$	30.2344	1.00171	0.500856	+	0.865531i	$0.333019\pi$
$912$	0	0
$913$	6.76809	0.223991
$914$	18.3526	0.607050
$915$	0	0
$916$	−10.3612	−0.342344
$917$	−12.6353	−0.417256
$918$	0	0
$919$	16.2440	0.535840	0.267920	−	0.963441i	$-0.413664\pi$
$919$	16.2440	0.535840	0.267920	+	0.963441i	$0.413664\pi$
$920$	−0.493959	−0.0162854
$921$	0	0
$922$	23.3793	0.769955
$923$	−51.7948	−1.70485
$924$	0	0
$925$	−6.27413	−0.206292
$926$	−26.2198	−0.861637
$927$	0	0
$928$	−3.78017	−0.124090
$929$	−49.1051	−1.61109	−0.805544	−	0.592537i	$-0.798127\pi$
$929$	−49.1051	−1.61109	−0.805544	+	0.592537i	$0.798127\pi$
$930$	0	0
$931$	−6.98792	−0.229020
$932$	3.58450	0.117414
$933$	0	0
$934$	−10.6112	−0.347208
$935$	7.70171	0.251873
$936$	0	0
$937$	−33.8431	−1.10561	−0.552803	−	0.833312i	$-0.686442\pi$
$937$	−33.8431	−1.10561	−0.552803	+	0.833312i	$0.686442\pi$
$938$	3.48188	0.113687
$939$	0	0
$940$	5.20775	0.169858
$941$	16.0629	0.523636	0.261818	−	0.965117i	$-0.415678\pi$
$941$	16.0629	0.523636	0.261818	+	0.965117i	$0.415678\pi$
$942$	0	0
$943$	−4.33108	−0.141039
$944$	−8.98792	−0.292532
$945$	0	0
$946$	9.20775	0.299370
$947$	15.7560	0.512001	0.256001	−	0.966677i	$-0.417595\pi$
$947$	15.7560	0.512001	0.256001	+	0.966677i	$0.417595\pi$
$948$	0	0
$949$	4.77074	0.154865
$950$	6.98792	0.226718
$951$	0	0
$952$	−7.70171	−0.249614
$953$	25.0121	0.810221	0.405110	−	0.914268i	$-0.367233\pi$
$953$	25.0121	0.810221	0.405110	+	0.914268i	$0.367233\pi$
$954$	0	0
$955$	−13.5603	−0.438802
$956$	4.71379	0.152455
$957$	0	0
$958$	15.5603	0.502732
$959$	−1.70171	−0.0549511
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−29.5749	−0.953534
$963$	0	0
$964$	24.6112	0.792672
$965$	27.1836	0.875071
$966$	0	0
$967$	−52.9783	−1.70367	−0.851834	−	0.523811i	$-0.824510\pi$
$967$	−52.9783	−1.70367	−0.851834	+	0.523811i	$0.824510\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	2.98792	0.0959363
$971$	44.2344	1.41955	0.709775	−	0.704428i	$-0.248797\pi$
$971$	44.2344	1.41955	0.709775	+	0.704428i	$0.248797\pi$
$972$	0	0
$973$	11.9758	0.383928
$974$	12.7439	0.408342
$975$	0	0
$976$	−11.2078	−0.358751
$977$	−49.2137	−1.57449	−0.787243	−	0.616643i	$-0.788492\pi$
$977$	−49.2137	−1.57449	−0.787243	+	0.616643i	$0.788492\pi$
$978$	0	0
$979$	8.71379	0.278494
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	0.987918	0.0315257
$983$	−9.47325	−0.302150	−0.151075	−	0.988522i	$-0.548273\pi$
$983$	−9.47325	−0.302150	−0.151075	+	0.988522i	$0.548273\pi$
$984$	0	0
$985$	−26.1957	−0.834663
$986$	29.1138	0.927171
$987$	0	0
$988$	32.9396	1.04795
$989$	4.54825	0.144626
$990$	0	0
$991$	−10.7198	−0.340524	−0.170262	−	0.985399i	$-0.554461\pi$
$991$	−10.7198	−0.340524	−0.170262	+	0.985399i	$0.554461\pi$
$992$	0	0
$993$	0	0
$994$	10.9879	0.348516
$995$	0.548253	0.0173808
$996$	0	0
$997$	49.1051	1.55518	0.777588	−	0.628775i	$-0.216443\pi$
$997$	49.1051	1.55518	0.777588	+	0.628775i	$0.216443\pi$
$998$	28.9396	0.916067
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.cg.1.2	✓	3
3.2	odd	2		6930.2.a.ck.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6930.2.a.cg.1.2	✓	3	1.1	even	1	trivial
6930.2.a.ck.1.2	yes	3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -4.71379 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.70171 q^{17} -6.98792 q^{19} -1.00000 q^{20} -1.00000 q^{22} -0.493959 q^{23} +1.00000 q^{25} +4.71379 q^{26} -1.00000 q^{28} +3.78017 q^{29} -1.00000 q^{32} +7.70171 q^{34} +1.00000 q^{35} -6.27413 q^{37} +6.98792 q^{38} +1.00000 q^{40} +8.76809 q^{41} -9.20775 q^{43} +1.00000 q^{44} +0.493959 q^{46} -5.20775 q^{47} +1.00000 q^{49} -1.00000 q^{50} -4.71379 q^{52} -1.56033 q^{53} -1.00000 q^{55} +1.00000 q^{56} -3.78017 q^{58} -8.98792 q^{59} -11.2078 q^{61} +1.00000 q^{64} +4.71379 q^{65} +3.48188 q^{67} -7.70171 q^{68} -1.00000 q^{70} +10.9879 q^{71} -1.01208 q^{73} +6.27413 q^{74} -6.98792 q^{76} -1.00000 q^{77} +12.1957 q^{79} -1.00000 q^{80} -8.76809 q^{82} +6.76809 q^{83} +7.70171 q^{85} +9.20775 q^{86} -1.00000 q^{88} +8.71379 q^{89} +4.71379 q^{91} -0.493959 q^{92} +5.20775 q^{94} +6.98792 q^{95} +2.98792 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 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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