LMFDB - Embedded newform 6930.2.a.cl.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:	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			1.2
Root			$-2.91729$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -0.813457 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.83457 q^{17} +5.42784 q^{19} +1.00000 q^{20} -1.00000 q^{22} +5.42784 q^{23} +1.00000 q^{25} -0.813457 q^{26} -1.00000 q^{28} -6.61439 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.83457 q^{34} -1.00000 q^{35} +8.24130 q^{37} +5.42784 q^{38} +1.00000 q^{40} +5.59327 q^{41} -3.02112 q^{43} -1.00000 q^{44} +5.42784 q^{46} +5.02112 q^{47} +1.00000 q^{49} +1.00000 q^{50} -0.813457 q^{52} -6.61439 q^{53} -1.00000 q^{55} -1.00000 q^{56} -6.61439 q^{58} -10.6480 q^{59} +0.978885 q^{61} +6.00000 q^{62} +1.00000 q^{64} -0.813457 q^{65} +8.00000 q^{67} -3.83457 q^{68} -1.00000 q^{70} -14.8557 q^{71} -6.20766 q^{73} +8.24130 q^{74} +5.42784 q^{76} +1.00000 q^{77} +17.0970 q^{79} +1.00000 q^{80} +5.59327 q^{82} +8.81346 q^{83} -3.83457 q^{85} -3.02112 q^{86} -1.00000 q^{88} +1.18654 q^{89} +0.813457 q^{91} +5.42784 q^{92} +5.02112 q^{94} +5.42784 q^{95} -3.42784 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−0.813457	−0.225612	−0.112806	−	0.993617i	$-0.535984\pi$
$13$	−0.813457	−0.225612	−0.112806	+	0.993617i	$0.535984\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.83457	−0.930020	−0.465010	−	0.885305i	$-0.653949\pi$
$17$	−3.83457	−0.930020	−0.465010	+	0.885305i	$0.653949\pi$
$18$	0	0
$19$	5.42784	1.24523	0.622616	−	0.782527i	$-0.286070\pi$
$19$	5.42784	1.24523	0.622616	+	0.782527i	$0.286070\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	5.42784	1.13178	0.565892	−	0.824480i	$-0.308532\pi$
$23$	5.42784	1.13178	0.565892	+	0.824480i	$0.308532\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.813457	−0.159532
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−6.61439	−1.22826	−0.614130	−	0.789205i	$-0.710493\pi$
$29$	−6.61439	−1.22826	−0.614130	+	0.789205i	$0.710493\pi$
$30$	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
$34$	−3.83457	−0.657624
$35$	−1.00000	−0.169031
$36$	0	0
$37$	8.24130	1.35486	0.677431	−	0.735587i	$-0.263093\pi$
$37$	8.24130	1.35486	0.677431	+	0.735587i	$0.263093\pi$
$38$	5.42784	0.880512
$39$	0	0
$40$	1.00000	0.158114
$41$	5.59327	0.873522	0.436761	−	0.899578i	$-0.356125\pi$
$41$	5.59327	0.873522	0.436761	+	0.899578i	$0.356125\pi$
$42$	0	0
$43$	−3.02112	−0.460716	−0.230358	−	0.973106i	$-0.573990\pi$
$43$	−3.02112	−0.460716	−0.230358	+	0.973106i	$0.573990\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	5.42784	0.800292
$47$	5.02112	0.732405	0.366202	−	0.930535i	$-0.380658\pi$
$47$	5.02112	0.732405	0.366202	+	0.930535i	$0.380658\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−0.813457	−0.112806
$53$	−6.61439	−0.908556	−0.454278	−	0.890860i	$-0.650103\pi$
$53$	−6.61439	−0.908556	−0.454278	+	0.890860i	$0.650103\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−6.61439	−0.868512
$59$	−10.6480	−1.38626	−0.693128	−	0.720815i	$-0.743768\pi$
$59$	−10.6480	−1.38626	−0.693128	+	0.720815i	$0.743768\pi$
$60$	0	0
$61$	0.978885	0.125333	0.0626667	−	0.998035i	$-0.480039\pi$
$61$	0.978885	0.125333	0.0626667	+	0.998035i	$0.480039\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.813457	−0.100897
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−3.83457	−0.465010
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−14.8557	−1.76305	−0.881523	−	0.472141i	$-0.843481\pi$
$71$	−14.8557	−1.76305	−0.881523	+	0.472141i	$0.843481\pi$
$72$	0	0
$73$	−6.20766	−0.726551	−0.363276	−	0.931682i	$-0.618342\pi$
$73$	−6.20766	−0.726551	−0.363276	+	0.931682i	$0.618342\pi$
$74$	8.24130	0.958032
$75$	0	0
$76$	5.42784	0.622616
$77$	1.00000	0.113961
$78$	0	0
$79$	17.0970	1.92356	0.961781	−	0.273821i	$-0.0882876\pi$
$79$	17.0970	1.92356	0.961781	+	0.273821i	$0.0882876\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	5.59327	0.617674
$83$	8.81346	0.967403	0.483701	−	0.875233i	$-0.339292\pi$
$83$	8.81346	0.967403	0.483701	+	0.875233i	$0.339292\pi$
$84$	0	0
$85$	−3.83457	−0.415918
$86$	−3.02112	−0.325775
$87$	0	0
$88$	−1.00000	−0.106600
$89$	1.18654	0.125773	0.0628867	−	0.998021i	$-0.479969\pi$
$89$	1.18654	0.125773	0.0628867	+	0.998021i	$0.479969\pi$
$90$	0	0
$91$	0.813457	0.0852734
$92$	5.42784	0.565892
$93$	0	0
$94$	5.02112	0.517888
$95$	5.42784	0.556885
$96$	0	0
$97$	−3.42784	−0.348045	−0.174022	−	0.984742i	$-0.555677\pi$
$97$	−3.42784	−0.348045	−0.174022	+	0.984742i	$0.555677\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	18.6903	1.85975	0.929875	−	0.367875i	$-0.119915\pi$
$101$	18.6903	1.85975	0.929875	+	0.367875i	$0.119915\pi$
$102$	0	0
$103$	15.4615	1.52347	0.761733	−	0.647891i	$-0.224349\pi$
$103$	15.4615	1.52347	0.761733	+	0.647891i	$0.224349\pi$
$104$	−0.813457	−0.0797660
$105$	0	0
$106$	−6.61439	−0.642446
$107$	11.0211	1.06545	0.532726	−	0.846288i	$-0.321168\pi$
$107$	11.0211	1.06545	0.532726	+	0.846288i	$0.321168\pi$
$108$	0	0
$109$	16.2413	1.55563	0.777817	−	0.628491i	$-0.216327\pi$
$109$	16.2413	1.55563	0.777817	+	0.628491i	$0.216327\pi$
$110$	−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
$115$	5.42784	0.506149
$116$	−6.61439	−0.614130
$117$	0	0
$118$	−10.6480	−0.980231
$119$	3.83457	0.351515
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.978885	0.0886241
$123$	0	0
$124$	6.00000	0.538816
$125$	1.00000	0.0894427
$126$	0	0
$127$	17.2961	1.53478	0.767388	−	0.641182i	$-0.221556\pi$
$127$	17.2961	1.53478	0.767388	+	0.641182i	$0.221556\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.813457	−0.0713449
$131$	−9.42784	−0.823715	−0.411857	−	0.911248i	$-0.635120\pi$
$131$	−9.42784	−0.823715	−0.411857	+	0.911248i	$0.635120\pi$
$132$	0	0
$133$	−5.42784	−0.470654
$134$	8.00000	0.691095
$135$	0	0
$136$	−3.83457	−0.328812
$137$	−13.6691	−1.16783	−0.583917	−	0.811813i	$-0.698481\pi$
$137$	−13.6691	−1.16783	−0.583917	+	0.811813i	$0.698481\pi$
$138$	0	0
$139$	−11.8682	−1.00665	−0.503324	−	0.864098i	$-0.667890\pi$
$139$	−11.8682	−1.00665	−0.503324	+	0.864098i	$0.667890\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−14.8557	−1.24666
$143$	0.813457	0.0680247
$144$	0	0
$145$	−6.61439	−0.549295
$146$	−6.20766	−0.513749
$147$	0	0
$148$	8.24130	0.677431
$149$	22.2835	1.82554	0.912769	−	0.408476i	$-0.133940\pi$
$149$	22.2835	1.82554	0.912769	+	0.408476i	$0.133940\pi$
$150$	0	0
$151$	−3.05476	−0.248593	−0.124296	−	0.992245i	$-0.539667\pi$
$151$	−3.05476	−0.248593	−0.124296	+	0.992245i	$0.539667\pi$
$152$	5.42784	0.440256
$153$	0	0
$154$	1.00000	0.0805823
$155$	6.00000	0.481932
$156$	0	0
$157$	4.85569	0.387526	0.193763	−	0.981048i	$-0.437931\pi$
$157$	4.85569	0.387526	0.193763	+	0.981048i	$0.437931\pi$
$158$	17.0970	1.36016
$159$	0	0
$160$	1.00000	0.0790569
$161$	−5.42784	−0.427774
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	5.59327	0.436761
$165$	0	0
$166$	8.81346	0.684057
$167$	−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
$169$	−12.3383	−0.949099
$170$	−3.83457	−0.294098
$171$	0	0
$172$	−3.02112	−0.230358
$173$	19.0211	1.44615	0.723074	−	0.690770i	$-0.242728\pi$
$173$	19.0211	1.44615	0.723074	+	0.690770i	$0.242728\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	1.18654	0.0889352
$179$	−4.64803	−0.347410	−0.173705	−	0.984798i	$-0.555574\pi$
$179$	−4.64803	−0.347410	−0.173705	+	0.984798i	$0.555574\pi$
$180$	0	0
$181$	−25.3383	−1.88338	−0.941690	−	0.336482i	$-0.890763\pi$
$181$	−25.3383	−1.88338	−0.941690	+	0.336482i	$0.890763\pi$
$182$	0.813457	0.0602974
$183$	0	0
$184$	5.42784	0.400146
$185$	8.24130	0.605912
$186$	0	0
$187$	3.83457	0.280412
$188$	5.02112	0.366202
$189$	0	0
$190$	5.42784	0.393777
$191$	−15.6691	−1.13378	−0.566890	−	0.823794i	$-0.691853\pi$
$191$	−15.6691	−1.13378	−0.566890	+	0.823794i	$0.691853\pi$
$192$	0	0
$193$	−2.16543	−0.155871	−0.0779355	−	0.996958i	$-0.524833\pi$
$193$	−2.16543	−0.155871	−0.0779355	+	0.996958i	$0.524833\pi$
$194$	−3.42784	−0.246105
$195$	0	0
$196$	1.00000	0.0714286
$197$	−2.81346	−0.200451	−0.100225	−	0.994965i	$-0.531956\pi$
$197$	−2.81346	−0.200451	−0.100225	+	0.994965i	$0.531956\pi$
$198$	0	0
$199$	−13.0211	−0.923042	−0.461521	−	0.887129i	$-0.652696\pi$
$199$	−13.0211	−0.923042	−0.461521	+	0.887129i	$0.652696\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	18.6903	1.31504
$203$	6.61439	0.464239
$204$	0	0
$205$	5.59327	0.390651
$206$	15.4615	1.07725
$207$	0	0
$208$	−0.813457	−0.0564031
$209$	−5.42784	−0.375452
$210$	0	0
$211$	19.6691	1.35408	0.677040	−	0.735946i	$-0.263262\pi$
$211$	19.6691	1.35408	0.677040	+	0.735946i	$0.263262\pi$
$212$	−6.61439	−0.454278
$213$	0	0
$214$	11.0211	0.753388
$215$	−3.02112	−0.206038
$216$	0	0
$217$	−6.00000	−0.407307
$218$	16.2413	1.10000
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	3.11926	0.209824
$222$	0	0
$223$	−3.79234	−0.253954	−0.126977	−	0.991906i	$-0.540527\pi$
$223$	−3.79234	−0.253954	−0.126977	+	0.991906i	$0.540527\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	14.0422	0.932016	0.466008	−	0.884781i	$-0.345692\pi$
$227$	14.0422	0.932016	0.466008	+	0.884781i	$0.345692\pi$
$228$	0	0
$229$	1.83457	0.121232	0.0606160	−	0.998161i	$-0.480694\pi$
$229$	1.83457	0.121232	0.0606160	+	0.998161i	$0.480694\pi$
$230$	5.42784	0.357901
$231$	0	0
$232$	−6.61439	−0.434256
$233$	8.04223	0.526864	0.263432	−	0.964678i	$-0.415146\pi$
$233$	8.04223	0.526864	0.263432	+	0.964678i	$0.415146\pi$
$234$	0	0
$235$	5.02112	0.327541
$236$	−10.6480	−0.693128
$237$	0	0
$238$	3.83457	0.248558
$239$	15.4701	1.00068	0.500338	−	0.865830i	$-0.333209\pi$
$239$	15.4701	1.00068	0.500338	+	0.865830i	$0.333209\pi$
$240$	0	0
$241$	11.6355	0.749509	0.374754	−	0.927124i	$-0.377727\pi$
$241$	11.6355	0.749509	0.374754	+	0.927124i	$0.377727\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	0.978885	0.0626667
$245$	1.00000	0.0638877
$246$	0	0
$247$	−4.41532	−0.280940
$248$	6.00000	0.381000
$249$	0	0
$250$	1.00000	0.0632456
$251$	−5.35197	−0.337813	−0.168907	−	0.985632i	$-0.554024\pi$
$251$	−5.35197	−0.337813	−0.168907	+	0.985632i	$0.554024\pi$
$252$	0	0
$253$	−5.42784	−0.341246
$254$	17.2961	1.08525
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.7239	−0.793695	−0.396848	−	0.917885i	$-0.629896\pi$
$257$	−12.7239	−0.793695	−0.396848	+	0.917885i	$0.629896\pi$
$258$	0	0
$259$	−8.24130	−0.512089
$260$	−0.813457	−0.0504485
$261$	0	0
$262$	−9.42784	−0.582454
$263$	1.62691	0.100320	0.0501599	−	0.998741i	$-0.484027\pi$
$263$	1.62691	0.100320	0.0501599	+	0.998741i	$0.484027\pi$
$264$	0	0
$265$	−6.61439	−0.406319
$266$	−5.42784	−0.332802
$267$	0	0
$268$	8.00000	0.488678
$269$	−23.8768	−1.45579	−0.727897	−	0.685686i	$-0.759502\pi$
$269$	−23.8768	−1.45579	−0.727897	+	0.685686i	$0.759502\pi$
$270$	0	0
$271$	2.44037	0.148242	0.0741210	−	0.997249i	$-0.476385\pi$
$271$	2.44037	0.148242	0.0741210	+	0.997249i	$0.476385\pi$
$272$	−3.83457	−0.232505
$273$	0	0
$274$	−13.6691	−0.825783
$275$	−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	$0.270164\pi$
$278$	−11.8682	−0.711808
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	20.8557	1.24415	0.622073	−	0.782959i	$-0.286291\pi$
$281$	20.8557	1.24415	0.622073	+	0.782959i	$0.286291\pi$
$282$	0	0
$283$	0.330856	0.0196673	0.00983367	−	0.999952i	$-0.496870\pi$
$283$	0.330856	0.0196673	0.00983367	+	0.999952i	$0.496870\pi$
$284$	−14.8557	−0.881523
$285$	0	0
$286$	0.813457	0.0481007
$287$	−5.59327	−0.330160
$288$	0	0
$289$	−2.29606	−0.135062
$290$	−6.61439	−0.388410
$291$	0	0
$292$	−6.20766	−0.363276
$293$	3.18654	0.186160	0.0930799	−	0.995659i	$-0.470329\pi$
$293$	3.18654	0.186160	0.0930799	+	0.995659i	$0.470329\pi$
$294$	0	0
$295$	−10.6480	−0.619952
$296$	8.24130	0.479016
$297$	0	0
$298$	22.2835	1.29085
$299$	−4.41532	−0.255344
$300$	0	0
$301$	3.02112	0.174134
$302$	−3.05476	−0.175782
$303$	0	0
$304$	5.42784	0.311308
$305$	0.978885	0.0560508
$306$	0	0
$307$	20.4826	1.16900	0.584502	−	0.811392i	$-0.301290\pi$
$307$	20.4826	1.16900	0.584502	+	0.811392i	$0.301290\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	6.00000	0.340777
$311$	−3.62691	−0.205663	−0.102832	−	0.994699i	$-0.532790\pi$
$311$	−3.62691	−0.205663	−0.102832	+	0.994699i	$0.532790\pi$
$312$	0	0
$313$	20.7239	1.17138	0.585692	−	0.810534i	$-0.300823\pi$
$313$	20.7239	1.17138	0.585692	+	0.810534i	$0.300823\pi$
$314$	4.85569	0.274022
$315$	0	0
$316$	17.0970	0.961781
$317$	1.80093	0.101150	0.0505751	−	0.998720i	$-0.483895\pi$
$317$	1.80093	0.101150	0.0505751	+	0.998720i	$0.483895\pi$
$318$	0	0
$319$	6.61439	0.370335
$320$	1.00000	0.0559017
$321$	0	0
$322$	−5.42784	−0.302482
$323$	−20.8135	−1.15809
$324$	0	0
$325$	−0.813457	−0.0451225
$326$	−8.00000	−0.443079
$327$	0	0
$328$	5.59327	0.308837
$329$	−5.02112	−0.276823
$330$	0	0
$331$	−3.02112	−0.166056	−0.0830278	−	0.996547i	$-0.526459\pi$
$331$	−3.02112	−0.166056	−0.0830278	+	0.996547i	$0.526459\pi$
$332$	8.81346	0.483701
$333$	0	0
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	−9.66914	−0.526712	−0.263356	−	0.964699i	$-0.584829\pi$
$337$	−9.66914	−0.526712	−0.263356	+	0.964699i	$0.584829\pi$
$338$	−12.3383	−0.671114
$339$	0	0
$340$	−3.83457	−0.207959
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−3.02112	−0.162888
$345$	0	0
$346$	19.0211	1.02258
$347$	−9.39420	−0.504307	−0.252154	−	0.967687i	$-0.581139\pi$
$347$	−9.39420	−0.504307	−0.252154	+	0.967687i	$0.581139\pi$
$348$	0	0
$349$	−18.2077	−0.974634	−0.487317	−	0.873225i	$-0.662024\pi$
$349$	−18.2077	−0.974634	−0.487317	+	0.873225i	$0.662024\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−12.7239	−0.677225	−0.338612	−	0.940926i	$-0.609958\pi$
$353$	−12.7239	−0.677225	−0.338612	+	0.940926i	$0.609958\pi$
$354$	0	0
$355$	−14.8557	−0.788458
$356$	1.18654	0.0628867
$357$	0	0
$358$	−4.64803	−0.245656
$359$	−29.8432	−1.57506	−0.787531	−	0.616275i	$-0.788641\pi$
$359$	−29.8432	−1.57506	−0.787531	+	0.616275i	$0.788641\pi$
$360$	0	0
$361$	10.4615	0.550605
$362$	−25.3383	−1.33175
$363$	0	0
$364$	0.813457	0.0426367
$365$	−6.20766	−0.324924
$366$	0	0
$367$	15.4615	0.807083	0.403541	−	0.914961i	$-0.367779\pi$
$367$	15.4615	0.807083	0.403541	+	0.914961i	$0.367779\pi$
$368$	5.42784	0.282946
$369$	0	0
$370$	8.24130	0.428445
$371$	6.61439	0.343402
$372$	0	0
$373$	−8.10951	−0.419895	−0.209947	−	0.977713i	$-0.567329\pi$
$373$	−8.10951	−0.419895	−0.209947	+	0.977713i	$0.567329\pi$
$374$	3.83457	0.198281
$375$	0	0
$376$	5.02112	0.258944
$377$	5.38052	0.277111
$378$	0	0
$379$	22.0422	1.13223	0.566117	−	0.824325i	$-0.308445\pi$
$379$	22.0422	1.13223	0.566117	+	0.824325i	$0.308445\pi$
$380$	5.42784	0.278442
$381$	0	0
$382$	−15.6691	−0.801703
$383$	−38.2499	−1.95448	−0.977239	−	0.212141i	$-0.931956\pi$
$383$	−38.2499	−1.95448	−0.977239	+	0.212141i	$0.931956\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	−2.16543	−0.110217
$387$	0	0
$388$	−3.42784	−0.174022
$389$	−16.8557	−0.854617	−0.427309	−	0.904106i	$-0.640538\pi$
$389$	−16.8557	−0.854617	−0.427309	+	0.904106i	$0.640538\pi$
$390$	0	0
$391$	−20.8135	−1.05258
$392$	1.00000	0.0505076
$393$	0	0
$394$	−2.81346	−0.141740
$395$	17.0970	0.860243
$396$	0	0
$397$	−4.44037	−0.222856	−0.111428	−	0.993773i	$-0.535542\pi$
$397$	−4.44037	−0.222856	−0.111428	+	0.993773i	$0.535542\pi$
$398$	−13.0211	−0.652690
$399$	0	0
$400$	1.00000	0.0500000
$401$	3.46149	0.172858	0.0864292	−	0.996258i	$-0.472454\pi$
$401$	3.46149	0.172858	0.0864292	+	0.996258i	$0.472454\pi$
$402$	0	0
$403$	−4.88074	−0.243127
$404$	18.6903	0.929875
$405$	0	0
$406$	6.61439	0.328267
$407$	−8.24130	−0.408506
$408$	0	0
$409$	−17.2624	−0.853572	−0.426786	−	0.904353i	$-0.640354\pi$
$409$	−17.2624	−0.853572	−0.426786	+	0.904353i	$0.640354\pi$
$410$	5.59327	0.276232
$411$	0	0
$412$	15.4615	0.761733
$413$	10.6480	0.523955
$414$	0	0
$415$	8.81346	0.432636
$416$	−0.813457	−0.0398830
$417$	0	0
$418$	−5.42784	−0.265485
$419$	−9.02112	−0.440710	−0.220355	−	0.975420i	$-0.570722\pi$
$419$	−9.02112	−0.440710	−0.220355	+	0.975420i	$0.570722\pi$
$420$	0	0
$421$	31.7114	1.54552	0.772759	−	0.634700i	$-0.218876\pi$
$421$	31.7114	1.54552	0.772759	+	0.634700i	$0.218876\pi$
$422$	19.6691	0.957479
$423$	0	0
$424$	−6.61439	−0.321223
$425$	−3.83457	−0.186004
$426$	0	0
$427$	−0.978885	−0.0473716
$428$	11.0211	0.532726
$429$	0	0
$430$	−3.02112	−0.145691
$431$	3.05476	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$431$	3.05476	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$432$	0	0
$433$	27.9104	1.34129	0.670645	−	0.741778i	$-0.266017\pi$
$433$	27.9104	1.34129	0.670645	+	0.741778i	$0.266017\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	16.2413	0.777817
$437$	29.4615	1.40933
$438$	0	0
$439$	−14.8557	−0.709023	−0.354512	−	0.935052i	$-0.615353\pi$
$439$	−14.8557	−0.709023	−0.354512	+	0.935052i	$0.615353\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	3.11926	0.148368
$443$	−0.746173	−0.0354517	−0.0177259	−	0.999843i	$-0.505643\pi$
$443$	−0.746173	−0.0354517	−0.0177259	+	0.999843i	$0.505643\pi$
$444$	0	0
$445$	1.18654	0.0562475
$446$	−3.79234	−0.179573
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−9.83457	−0.464122	−0.232061	−	0.972701i	$-0.574547\pi$
$449$	−9.83457	−0.464122	−0.232061	+	0.972701i	$0.574547\pi$
$450$	0	0
$451$	−5.59327	−0.263377
$452$	14.0000	0.658505
$453$	0	0
$454$	14.0422	0.659035
$455$	0.813457	0.0381354
$456$	0	0
$457$	11.2961	0.528407	0.264204	−	0.964467i	$-0.414891\pi$
$457$	11.2961	0.528407	0.264204	+	0.964467i	$0.414891\pi$
$458$	1.83457	0.0857239
$459$	0	0
$460$	5.42784	0.253075
$461$	7.41926	0.345549	0.172775	−	0.984961i	$-0.444727\pi$
$461$	7.41926	0.345549	0.172775	+	0.984961i	$0.444727\pi$
$462$	0	0
$463$	−10.1740	−0.472827	−0.236413	−	0.971653i	$-0.575972\pi$
$463$	−10.1740	−0.472827	−0.236413	+	0.971653i	$0.575972\pi$
$464$	−6.61439	−0.307065
$465$	0	0
$466$	8.04223	0.372549
$467$	32.4912	1.50351	0.751756	−	0.659441i	$-0.229207\pi$
$467$	32.4912	1.50351	0.751756	+	0.659441i	$0.229207\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	5.02112	0.231607
$471$	0	0
$472$	−10.6480	−0.490115
$473$	3.02112	0.138911
$474$	0	0
$475$	5.42784	0.249047
$476$	3.83457	0.175757
$477$	0	0
$478$	15.4701	0.707585
$479$	11.6691	0.533177	0.266588	−	0.963810i	$-0.414104\pi$
$479$	11.6691	0.533177	0.266588	+	0.963810i	$0.414104\pi$
$480$	0	0
$481$	−6.70394	−0.305673
$482$	11.6355	0.529983
$483$	0	0
$484$	1.00000	0.0454545
$485$	−3.42784	−0.155650
$486$	0	0
$487$	−5.49513	−0.249008	−0.124504	−	0.992219i	$-0.539734\pi$
$487$	−5.49513	−0.249008	−0.124504	+	0.992219i	$0.539734\pi$
$488$	0.978885	0.0443120
$489$	0	0
$490$	1.00000	0.0451754
$491$	13.2961	0.600043	0.300021	−	0.953932i	$-0.403006\pi$
$491$	13.2961	0.600043	0.300021	+	0.953932i	$0.403006\pi$
$492$	0	0
$493$	25.3633	1.14231
$494$	−4.41532	−0.198654
$495$	0	0
$496$	6.00000	0.269408
$497$	14.8557	0.666369
$498$	0	0
$499$	−24.6480	−1.10340	−0.551699	−	0.834044i	$-0.686020\pi$
$499$	−24.6480	−1.10340	−0.551699	+	0.834044i	$0.686020\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−5.35197	−0.238870
$503$	−38.9230	−1.73549	−0.867745	−	0.497010i	$-0.834431\pi$
$503$	−38.9230	−1.73549	−0.867745	+	0.497010i	$0.834431\pi$
$504$	0	0
$505$	18.6903	0.831706
$506$	−5.42784	−0.241297
$507$	0	0
$508$	17.2961	0.767388
$509$	39.0497	1.73085	0.865423	−	0.501042i	$-0.167050\pi$
$509$	39.0497	1.73085	0.865423	+	0.501042i	$0.167050\pi$
$510$	0	0
$511$	6.20766	0.274611
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.7239	−0.561227
$515$	15.4615	0.681314
$516$	0	0
$517$	−5.02112	−0.220828
$518$	−8.24130	−0.362102
$519$	0	0
$520$	−0.813457	−0.0356724
$521$	33.2710	1.45763	0.728815	−	0.684711i	$-0.240072\pi$
$521$	33.2710	1.45763	0.728815	+	0.684711i	$0.240072\pi$
$522$	0	0
$523$	−40.2362	−1.75941	−0.879703	−	0.475523i	$-0.842259\pi$
$523$	−40.2362	−1.75941	−0.879703	+	0.475523i	$0.842259\pi$
$524$	−9.42784	−0.411857
$525$	0	0
$526$	1.62691	0.0709368
$527$	−23.0074	−1.00222
$528$	0	0
$529$	6.46149	0.280934
$530$	−6.61439	−0.287311
$531$	0	0
$532$	−5.42784	−0.235327
$533$	−4.54989	−0.197077
$534$	0	0
$535$	11.0211	0.476484
$536$	8.00000	0.345547
$537$	0	0
$538$	−23.8768	−1.02940
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	26.2835	1.13002	0.565009	−	0.825085i	$-0.308873\pi$
$541$	26.2835	1.13002	0.565009	+	0.825085i	$0.308873\pi$
$542$	2.44037	0.104823
$543$	0	0
$544$	−3.83457	−0.164406
$545$	16.2413	0.695701
$546$	0	0
$547$	13.6269	0.582645	0.291322	−	0.956625i	$-0.405905\pi$
$547$	13.6269	0.582645	0.291322	+	0.956625i	$0.405905\pi$
$548$	−13.6691	−0.583917
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−35.9019	−1.52947
$552$	0	0
$553$	−17.0970	−0.727038
$554$	22.0000	0.934690
$555$	0	0
$556$	−11.8682	−0.503324
$557$	−14.8979	−0.631245	−0.315623	−	0.948885i	$-0.602213\pi$
$557$	−14.8979	−0.631245	−0.315623	+	0.948885i	$0.602213\pi$
$558$	0	0
$559$	2.45755	0.103943
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	20.8557	0.879744
$563$	−39.7536	−1.67541	−0.837707	−	0.546119i	$-0.816104\pi$
$563$	−39.7536	−1.67541	−0.837707	+	0.546119i	$0.816104\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0.330856	0.0139069
$567$	0	0
$568$	−14.8557	−0.623331
$569$	−8.78840	−0.368429	−0.184215	−	0.982886i	$-0.558974\pi$
$569$	−8.78840	−0.368429	−0.184215	+	0.982886i	$0.558974\pi$
$570$	0	0
$571$	−5.22877	−0.218817	−0.109409	−	0.993997i	$-0.534896\pi$
$571$	−5.22877	−0.218817	−0.109409	+	0.993997i	$0.534896\pi$
$572$	0.813457	0.0340123
$573$	0	0
$574$	−5.59327	−0.233459
$575$	5.42784	0.226357
$576$	0	0
$577$	5.80093	0.241496	0.120748	−	0.992683i	$-0.461471\pi$
$577$	5.80093	0.241496	0.120748	+	0.992683i	$0.461471\pi$
$578$	−2.29606	−0.0955034
$579$	0	0
$580$	−6.61439	−0.274647
$581$	−8.81346	−0.365644
$582$	0	0
$583$	6.61439	0.273940
$584$	−6.20766	−0.256875
$585$	0	0
$586$	3.18654	0.131635
$587$	−25.3046	−1.04443	−0.522217	−	0.852812i	$-0.674895\pi$
$587$	−25.3046	−1.04443	−0.522217	+	0.852812i	$0.674895\pi$
$588$	0	0
$589$	32.5671	1.34190
$590$	−10.6480	−0.438372
$591$	0	0
$592$	8.24130	0.338715
$593$	23.1729	0.951595	0.475798	−	0.879555i	$-0.342159\pi$
$593$	23.1729	0.951595	0.475798	+	0.879555i	$0.342159\pi$
$594$	0	0
$595$	3.83457	0.157202
$596$	22.2835	0.912769
$597$	0	0
$598$	−4.41532	−0.180556
$599$	8.48260	0.346590	0.173295	−	0.984870i	$-0.444559\pi$
$599$	8.48260	0.346590	0.173295	+	0.984870i	$0.444559\pi$
$600$	0	0
$601$	−32.4490	−1.32362	−0.661810	−	0.749671i	$-0.730212\pi$
$601$	−32.4490	−1.32362	−0.661810	+	0.749671i	$0.730212\pi$
$602$	3.02112	0.123131
$603$	0	0
$604$	−3.05476	−0.124296
$605$	1.00000	0.0406558
$606$	0	0
$607$	−33.9441	−1.37775	−0.688874	−	0.724881i	$-0.741895\pi$
$607$	−33.9441	−1.37775	−0.688874	+	0.724881i	$0.741895\pi$
$608$	5.42784	0.220128
$609$	0	0
$610$	0.978885	0.0396339
$611$	−4.08446	−0.165240
$612$	0	0
$613$	8.77123	0.354267	0.177133	−	0.984187i	$-0.443318\pi$
$613$	8.77123	0.354267	0.177133	+	0.984187i	$0.443318\pi$
$614$	20.4826	0.826610
$615$	0	0
$616$	1.00000	0.0402911
$617$	32.1095	1.29268	0.646340	−	0.763049i	$-0.276299\pi$
$617$	32.1095	1.29268	0.646340	+	0.763049i	$0.276299\pi$
$618$	0	0
$619$	−28.2749	−1.13647	−0.568233	−	0.822868i	$-0.692373\pi$
$619$	−28.2749	−1.13647	−0.568233	+	0.822868i	$0.692373\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	−3.62691	−0.145426
$623$	−1.18654	−0.0475378
$624$	0	0
$625$	1.00000	0.0400000
$626$	20.7239	0.828294
$627$	0	0
$628$	4.85569	0.193763
$629$	−31.6019	−1.26005
$630$	0	0
$631$	34.9230	1.39026	0.695131	−	0.718883i	$-0.255346\pi$
$631$	34.9230	1.39026	0.695131	+	0.718883i	$0.255346\pi$
$632$	17.0970	0.680082
$633$	0	0
$634$	1.80093	0.0715241
$635$	17.2961	0.686373
$636$	0	0
$637$	−0.813457	−0.0322303
$638$	6.61439	0.261866
$639$	0	0
$640$	1.00000	0.0395285
$641$	5.25383	0.207514	0.103757	−	0.994603i	$-0.466914\pi$
$641$	5.25383	0.207514	0.103757	+	0.994603i	$0.466914\pi$
$642$	0	0
$643$	−3.17796	−0.125326	−0.0626632	−	0.998035i	$-0.519959\pi$
$643$	−3.17796	−0.125326	−0.0626632	+	0.998035i	$0.519959\pi$
$644$	−5.42784	−0.213887
$645$	0	0
$646$	−20.8135	−0.818895
$647$	−37.9190	−1.49075	−0.745375	−	0.666645i	$-0.767730\pi$
$647$	−37.9190	−1.49075	−0.745375	+	0.666645i	$0.767730\pi$
$648$	0	0
$649$	10.6480	0.417972
$650$	−0.813457	−0.0319064
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−29.5374	−1.15589	−0.577943	−	0.816077i	$-0.696144\pi$
$653$	−29.5374	−1.15589	−0.577943	+	0.816077i	$0.696144\pi$
$654$	0	0
$655$	−9.42784	−0.368376
$656$	5.59327	0.218381
$657$	0	0
$658$	−5.02112	−0.195743
$659$	−15.1865	−0.591584	−0.295792	−	0.955252i	$-0.595583\pi$
$659$	−15.1865	−0.591584	−0.295792	+	0.955252i	$0.595583\pi$
$660$	0	0
$661$	−44.6766	−1.73772	−0.868859	−	0.495060i	$-0.835146\pi$
$661$	−44.6766	−1.73772	−0.868859	+	0.495060i	$0.835146\pi$
$662$	−3.02112	−0.117419
$663$	0	0
$664$	8.81346	0.342028
$665$	−5.42784	−0.210483
$666$	0	0
$667$	−35.9019	−1.39013
$668$	−8.00000	−0.309529
$669$	0	0
$670$	8.00000	0.309067
$671$	−0.978885	−0.0377894
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	−9.66914	−0.372442
$675$	0	0
$676$	−12.3383	−0.474550
$677$	48.7325	1.87294	0.936471	−	0.350745i	$-0.114072\pi$
$677$	48.7325	1.87294	0.936471	+	0.350745i	$0.114072\pi$
$678$	0	0
$679$	3.42784	0.131549
$680$	−3.83457	−0.147049
$681$	0	0
$682$	−6.00000	−0.229752
$683$	0.482601	0.0184662	0.00923310	−	0.999957i	$-0.497061\pi$
$683$	0.482601	0.0184662	0.00923310	+	0.999957i	$0.497061\pi$
$684$	0	0
$685$	−13.6691	−0.522271
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−3.02112	−0.115179
$689$	5.38052	0.204981
$690$	0	0
$691$	−25.5037	−0.970207	−0.485104	−	0.874457i	$-0.661218\pi$
$691$	−25.5037	−0.970207	−0.485104	+	0.874457i	$0.661218\pi$
$692$	19.0211	0.723074
$693$	0	0
$694$	−9.39420	−0.356599
$695$	−11.8682	−0.450187
$696$	0	0
$697$	−21.4478	−0.812393
$698$	−18.2077	−0.689170
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−33.8682	−1.27918	−0.639592	−	0.768714i	$-0.720897\pi$
$701$	−33.8682	−1.27918	−0.639592	+	0.768714i	$0.720897\pi$
$702$	0	0
$703$	44.7325	1.68712
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−12.7239	−0.478870
$707$	−18.6903	−0.702920
$708$	0	0
$709$	−15.9749	−0.599952	−0.299976	−	0.953947i	$-0.596979\pi$
$709$	−15.9749	−0.599952	−0.299976	+	0.953947i	$0.596979\pi$
$710$	−14.8557	−0.557524
$711$	0	0
$712$	1.18654	0.0444676
$713$	32.5671	1.21965
$714$	0	0
$715$	0.813457	0.0304216
$716$	−4.64803	−0.173705
$717$	0	0
$718$	−29.8432	−1.11374
$719$	13.4364	0.501094	0.250547	−	0.968104i	$-0.419389\pi$
$719$	13.4364	0.501094	0.250547	+	0.968104i	$0.419389\pi$
$720$	0	0
$721$	−15.4615	−0.575816
$722$	10.4615	0.389336
$723$	0	0
$724$	−25.3383	−0.941690
$725$	−6.61439	−0.245652
$726$	0	0
$727$	6.64803	0.246562	0.123281	−	0.992372i	$-0.460658\pi$
$727$	6.64803	0.246562	0.123281	+	0.992372i	$0.460658\pi$
$728$	0.813457	0.0301487
$729$	0	0
$730$	−6.20766	−0.229756
$731$	11.5847	0.428475
$732$	0	0
$733$	38.0285	1.40462	0.702308	−	0.711873i	$-0.252153\pi$
$733$	38.0285	1.40462	0.702308	+	0.711873i	$0.252153\pi$
$734$	15.4615	0.570694
$735$	0	0
$736$	5.42784	0.200073
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−52.6343	−1.93619	−0.968093	−	0.250592i	$-0.919375\pi$
$739$	−52.6343	−1.93619	−0.968093	+	0.250592i	$0.919375\pi$
$740$	8.24130	0.302956
$741$	0	0
$742$	6.61439	0.242822
$743$	−28.0845	−1.03032	−0.515159	−	0.857094i	$-0.672267\pi$
$743$	−28.0845	−1.03032	−0.515159	+	0.857094i	$0.672267\pi$
$744$	0	0
$745$	22.2835	0.816405
$746$	−8.10951	−0.296910
$747$	0	0
$748$	3.83457	0.140206
$749$	−11.0211	−0.402703
$750$	0	0
$751$	48.4826	1.76916	0.884578	−	0.466393i	$-0.154447\pi$
$751$	48.4826	1.76916	0.884578	+	0.466393i	$0.154447\pi$
$752$	5.02112	0.183101
$753$	0	0
$754$	5.38052	0.195947
$755$	−3.05476	−0.111174
$756$	0	0
$757$	−11.8260	−0.429823	−0.214911	−	0.976634i	$-0.568946\pi$
$757$	−11.8260	−0.429823	−0.214911	+	0.976634i	$0.568946\pi$
$758$	22.0422	0.800610
$759$	0	0
$760$	5.42784	0.196889
$761$	19.5510	0.708725	0.354362	−	0.935108i	$-0.384698\pi$
$761$	19.5510	0.708725	0.354362	+	0.935108i	$0.384698\pi$
$762$	0	0
$763$	−16.2413	−0.587975
$764$	−15.6691	−0.566890
$765$	0	0
$766$	−38.2499	−1.38202
$767$	8.66171	0.312756
$768$	0	0
$769$	6.07587	0.219102	0.109551	−	0.993981i	$-0.465059\pi$
$769$	6.07587	0.219102	0.109551	+	0.993981i	$0.465059\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−2.16543	−0.0779355
$773$	2.33086	0.0838351	0.0419175	−	0.999121i	$-0.486653\pi$
$773$	2.33086	0.0838351	0.0419175	+	0.999121i	$0.486653\pi$
$774$	0	0
$775$	6.00000	0.215526
$776$	−3.42784	−0.123052
$777$	0	0
$778$	−16.8557	−0.604306
$779$	30.3594	1.08774
$780$	0	0
$781$	14.8557	0.531578
$782$	−20.8135	−0.744288
$783$	0	0
$784$	1.00000	0.0357143
$785$	4.85569	0.173307
$786$	0	0
$787$	−17.3805	−0.619549	−0.309774	−	0.950810i	$-0.600253\pi$
$787$	−17.3805	−0.619549	−0.309774	+	0.950810i	$0.600253\pi$
$788$	−2.81346	−0.100225
$789$	0	0
$790$	17.0970	0.608284
$791$	−14.0000	−0.497783
$792$	0	0
$793$	−0.796281	−0.0282768
$794$	−4.44037	−0.157583
$795$	0	0
$796$	−13.0211	−0.461521
$797$	15.6942	0.555917	0.277959	−	0.960593i	$-0.410342\pi$
$797$	15.6942	0.555917	0.277959	+	0.960593i	$0.410342\pi$
$798$	0	0
$799$	−19.2538	−0.681151
$800$	1.00000	0.0353553
$801$	0	0
$802$	3.46149	0.122229
$803$	6.20766	0.219064
$804$	0	0
$805$	−5.42784	−0.191306
$806$	−4.88074	−0.171917
$807$	0	0
$808$	18.6903	0.657521
$809$	−23.1443	−0.813711	−0.406855	−	0.913493i	$-0.633375\pi$
$809$	−23.1443	−0.813711	−0.406855	+	0.913493i	$0.633375\pi$
$810$	0	0
$811$	−27.6218	−0.969933	−0.484967	−	0.874533i	$-0.661168\pi$
$811$	−27.6218	−0.969933	−0.484967	+	0.874533i	$0.661168\pi$
$812$	6.61439	0.232119
$813$	0	0
$814$	−8.24130	−0.288857
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−16.3981	−0.573698
$818$	−17.2624	−0.603566
$819$	0	0
$820$	5.59327	0.195326
$821$	39.5123	1.37899	0.689494	−	0.724291i	$-0.257833\pi$
$821$	39.5123	1.37899	0.689494	+	0.724291i	$0.257833\pi$
$822$	0	0
$823$	13.5123	0.471009	0.235505	−	0.971873i	$-0.424326\pi$
$823$	13.5123	0.471009	0.235505	+	0.971873i	$0.424326\pi$
$824$	15.4615	0.538626
$825$	0	0
$826$	10.6480	0.370492
$827$	−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
$829$	13.8346	0.480495	0.240247	−	0.970712i	$-0.422771\pi$
$829$	13.8346	0.480495	0.240247	+	0.970712i	$0.422771\pi$
$830$	8.81346	0.305920
$831$	0	0
$832$	−0.813457	−0.0282015
$833$	−3.83457	−0.132860
$834$	0	0
$835$	−8.00000	−0.276851
$836$	−5.42784	−0.187726
$837$	0	0
$838$	−9.02112	−0.311629
$839$	−44.0422	−1.52051	−0.760253	−	0.649627i	$-0.774925\pi$
$839$	−44.0422	−1.52051	−0.760253	+	0.649627i	$0.774925\pi$
$840$	0	0
$841$	14.7501	0.508625
$842$	31.7114	1.09285
$843$	0	0
$844$	19.6691	0.677040
$845$	−12.3383	−0.424450
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−6.61439	−0.227139
$849$	0	0
$850$	−3.83457	−0.131525
$851$	44.7325	1.53341
$852$	0	0
$853$	−29.8095	−1.02066	−0.510329	−	0.859979i	$-0.670476\pi$
$853$	−29.8095	−1.02066	−0.510329	+	0.859979i	$0.670476\pi$
$854$	−0.978885	−0.0334968
$855$	0	0
$856$	11.0211	0.376694
$857$	−42.7575	−1.46057	−0.730285	−	0.683143i	$-0.760613\pi$
$857$	−42.7575	−1.46057	−0.730285	+	0.683143i	$0.760613\pi$
$858$	0	0
$859$	6.31717	0.215539	0.107770	−	0.994176i	$-0.465629\pi$
$859$	6.31717	0.215539	0.107770	+	0.994176i	$0.465629\pi$
$860$	−3.02112	−0.103019
$861$	0	0
$862$	3.05476	0.104045
$863$	48.3680	1.64647	0.823233	−	0.567704i	$-0.192168\pi$
$863$	48.3680	1.64647	0.823233	+	0.567704i	$0.192168\pi$
$864$	0	0
$865$	19.0211	0.646737
$866$	27.9104	0.948436
$867$	0	0
$868$	−6.00000	−0.203653
$869$	−17.0970	−0.579976
$870$	0	0
$871$	−6.50765	−0.220503
$872$	16.2413	0.550000
$873$	0	0
$874$	29.4615	0.996550
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−24.1940	−0.816972	−0.408486	−	0.912764i	$-0.633943\pi$
$877$	−24.1940	−0.816972	−0.408486	+	0.912764i	$0.633943\pi$
$878$	−14.8557	−0.501355
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−12.3731	−0.416860	−0.208430	−	0.978037i	$-0.566835\pi$
$881$	−12.3731	−0.416860	−0.208430	+	0.978037i	$0.566835\pi$
$882$	0	0
$883$	−14.5248	−0.488799	−0.244400	−	0.969675i	$-0.578591\pi$
$883$	−14.5248	−0.488799	−0.244400	+	0.969675i	$0.578591\pi$
$884$	3.11926	0.104912
$885$	0	0
$886$	−0.746173	−0.0250682
$887$	25.3805	0.852194	0.426097	−	0.904677i	$-0.359888\pi$
$887$	25.3805	0.852194	0.426097	+	0.904677i	$0.359888\pi$
$888$	0	0
$889$	−17.2961	−0.580091
$890$	1.18654	0.0397730
$891$	0	0
$892$	−3.79234	−0.126977
$893$	27.2538	0.912015
$894$	0	0
$895$	−4.64803	−0.155366
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−9.83457	−0.328184
$899$	−39.6863	−1.32361
$900$	0	0
$901$	25.3633	0.844975
$902$	−5.59327	−0.186236
$903$	0	0
$904$	14.0000	0.465633
$905$	−25.3383	−0.842273
$906$	0	0
$907$	−7.93272	−0.263402	−0.131701	−	0.991290i	$-0.542044\pi$
$907$	−7.93272	−0.263402	−0.131701	+	0.991290i	$0.542044\pi$
$908$	14.0422	0.466008
$909$	0	0
$910$	0.813457	0.0269658
$911$	−43.7364	−1.44905	−0.724526	−	0.689247i	$-0.757941\pi$
$911$	−43.7364	−1.44905	−0.724526	+	0.689247i	$0.757941\pi$
$912$	0	0
$913$	−8.81346	−0.291683
$914$	11.2961	0.373640
$915$	0	0
$916$	1.83457	0.0606160
$917$	9.42784	0.311335
$918$	0	0
$919$	−32.7661	−1.08085	−0.540427	−	0.841391i	$-0.681737\pi$
$919$	−32.7661	−1.08085	−0.540427	+	0.841391i	$0.681737\pi$
$920$	5.42784	0.178951
$921$	0	0
$922$	7.41926	0.244340
$923$	12.0845	0.397765
$924$	0	0
$925$	8.24130	0.270972
$926$	−10.1740	−0.334339
$927$	0	0
$928$	−6.61439	−0.217128
$929$	−59.4478	−1.95042	−0.975210	−	0.221283i	$-0.928975\pi$
$929$	−59.4478	−1.95042	−0.975210	+	0.221283i	$0.928975\pi$
$930$	0	0
$931$	5.42784	0.177890
$932$	8.04223	0.263432
$933$	0	0
$934$	32.4912	1.06314
$935$	3.83457	0.125404
$936$	0	0
$937$	53.2824	1.74066	0.870330	−	0.492470i	$-0.163906\pi$
$937$	53.2824	1.74066	0.870330	+	0.492470i	$0.163906\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	5.02112	0.163771
$941$	5.72506	0.186632	0.0933158	−	0.995637i	$-0.470253\pi$
$941$	5.72506	0.186632	0.0933158	+	0.995637i	$0.470253\pi$
$942$	0	0
$943$	30.3594	0.988638
$944$	−10.6480	−0.346564
$945$	0	0
$946$	3.02112	0.0982249
$947$	27.4056	0.890561	0.445281	−	0.895391i	$-0.353104\pi$
$947$	27.4056	0.890561	0.445281	+	0.895391i	$0.353104\pi$
$948$	0	0
$949$	5.04966	0.163919
$950$	5.42784	0.176102
$951$	0	0
$952$	3.83457	0.124279
$953$	10.7998	0.349839	0.174919	−	0.984583i	$-0.444033\pi$
$953$	10.7998	0.349839	0.174919	+	0.984583i	$0.444033\pi$
$954$	0	0
$955$	−15.6691	−0.507042
$956$	15.4701	0.500338
$957$	0	0
$958$	11.6691	0.377013
$959$	13.6691	0.441400
$960$	0	0
$961$	5.00000	0.161290
$962$	−6.70394	−0.216144
$963$	0	0
$964$	11.6355	0.374754
$965$	−2.16543	−0.0697076
$966$	0	0
$967$	11.2538	0.361899	0.180949	−	0.983492i	$-0.442083\pi$
$967$	11.2538	0.361899	0.180949	+	0.983492i	$0.442083\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−3.42784	−0.110061
$971$	−33.2401	−1.06673	−0.533363	−	0.845886i	$-0.679072\pi$
$971$	−33.2401	−1.06673	−0.533363	+	0.845886i	$0.679072\pi$
$972$	0	0
$973$	11.8682	0.380477
$974$	−5.49513	−0.176075
$975$	0	0
$976$	0.978885	0.0313333
$977$	2.74617	0.0878578	0.0439289	−	0.999035i	$-0.486012\pi$
$977$	2.74617	0.0878578	0.0439289	+	0.999035i	$0.486012\pi$
$978$	0	0
$979$	−1.18654	−0.0379221
$980$	1.00000	0.0319438
$981$	0	0
$982$	13.2961	0.424294
$983$	−39.4615	−1.25863	−0.629313	−	0.777152i	$-0.716664\pi$
$983$	−39.4615	−1.25863	−0.629313	+	0.777152i	$0.716664\pi$
$984$	0	0
$985$	−2.81346	−0.0896442
$986$	25.3633	0.807733
$987$	0	0
$988$	−4.41532	−0.140470
$989$	−16.3981	−0.521431
$990$	0	0
$991$	20.9652	0.665982	0.332991	−	0.942930i	$-0.391942\pi$
$991$	20.9652	0.665982	0.332991	+	0.942930i	$0.391942\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	14.8557	0.471194
$995$	−13.0211	−0.412797
$996$	0	0
$997$	21.9441	0.694976	0.347488	−	0.937684i	$-0.387035\pi$
$997$	21.9441	0.694976	0.347488	+	0.937684i	$0.387035\pi$
$998$	−24.6480	−0.780220
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.cl.1.2		3
3.2	odd	2		770.2.a.l.1.2	✓	3
12.11	even	2		6160.2.a.bi.1.2		3
15.2	even	4		3850.2.c.z.1849.2		6
15.8	even	4		3850.2.c.z.1849.5		6
15.14	odd	2		3850.2.a.bu.1.2		3
21.20	even	2		5390.2.a.bz.1.2		3
33.32	even	2		8470.2.a.cl.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.l.1.2	✓	3	3.2	odd	2
3850.2.a.bu.1.2		3	15.14	odd	2
3850.2.c.z.1849.2		6	15.2	even	4
3850.2.c.z.1849.5		6	15.8	even	4
5390.2.a.bz.1.2		3	21.20	even	2
6160.2.a.bi.1.2		3	12.11	even	2
6930.2.a.cl.1.2		3	1.1	even	1	trivial
8470.2.a.cl.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -0.813457 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.83457 q^{17} +5.42784 q^{19} +1.00000 q^{20} -1.00000 q^{22} +5.42784 q^{23} +1.00000 q^{25} -0.813457 q^{26} -1.00000 q^{28} -6.61439 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.83457 q^{34} -1.00000 q^{35} +8.24130 q^{37} +5.42784 q^{38} +1.00000 q^{40} +5.59327 q^{41} -3.02112 q^{43} -1.00000 q^{44} +5.42784 q^{46} +5.02112 q^{47} +1.00000 q^{49} +1.00000 q^{50} -0.813457 q^{52} -6.61439 q^{53} -1.00000 q^{55} -1.00000 q^{56} -6.61439 q^{58} -10.6480 q^{59} +0.978885 q^{61} +6.00000 q^{62} +1.00000 q^{64} -0.813457 q^{65} +8.00000 q^{67} -3.83457 q^{68} -1.00000 q^{70} -14.8557 q^{71} -6.20766 q^{73} +8.24130 q^{74} +5.42784 q^{76} +1.00000 q^{77} +17.0970 q^{79} +1.00000 q^{80} +5.59327 q^{82} +8.81346 q^{83} -3.83457 q^{85} -3.02112 q^{86} -1.00000 q^{88} +1.18654 q^{89} +0.813457 q^{91} +5.42784 q^{92} +5.02112 q^{94} +5.42784 q^{95} -3.42784 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 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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Varieties

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Database

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