LMFDB - Embedded newform 6930.2.a.br.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.65685 q^{17} -2.82843 q^{19} +1.00000 q^{20} -1.00000 q^{22} +2.82843 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -3.17157 q^{29} -5.65685 q^{31} -1.00000 q^{32} +7.65685 q^{34} -1.00000 q^{35} +3.17157 q^{37} +2.82843 q^{38} -1.00000 q^{40} +0.828427 q^{41} -9.65685 q^{43} +1.00000 q^{44} -2.82843 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +10.4853 q^{53} +1.00000 q^{55} +1.00000 q^{56} +3.17157 q^{58} +8.00000 q^{59} -9.31371 q^{61} +5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} +15.3137 q^{67} -7.65685 q^{68} +1.00000 q^{70} +5.65685 q^{71} +7.65685 q^{73} -3.17157 q^{74} -2.82843 q^{76} -1.00000 q^{77} -5.17157 q^{79} +1.00000 q^{80} -0.828427 q^{82} +2.34315 q^{83} -7.65685 q^{85} +9.65685 q^{86} -1.00000 q^{88} +11.6569 q^{89} -2.00000 q^{91} +2.82843 q^{92} -8.00000 q^{94} -2.82843 q^{95} -0.828427 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{20} - 2 q^{22} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 12 q^{37} - 2 q^{40} - 4 q^{41} - 8 q^{43} + 2 q^{44} + 16 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} + 2 q^{55} + 2 q^{56} + 12 q^{58} + 16 q^{59} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} + 2 q^{70} + 4 q^{73} - 12 q^{74} - 2 q^{77} - 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} - 4 q^{85} + 8 q^{86} - 2 q^{88} + 12 q^{89} - 4 q^{91} - 16 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^20 - 2 * q^22 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 12 * q^29 - 2 * q^32 + 4 * q^34 - 2 * q^35 + 12 * q^37 - 2 * q^40 - 4 * q^41 - 8 * q^43 + 2 * q^44 + 16 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 2 * q^55 + 2 * q^56 + 12 * q^58 + 16 * q^59 + 4 * q^61 + 2 * q^64 + 4 * q^65 + 8 * q^67 - 4 * q^68 + 2 * q^70 + 4 * q^73 - 12 * q^74 - 2 * q^77 - 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 - 4 * q^85 + 8 * q^86 - 2 * q^88 + 12 * q^89 - 4 * q^91 - 16 * q^94 + 4 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.65685	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$17$	−7.65685	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.17157	−0.588946	−0.294473	−	0.955660i	$-0.595144\pi$
$29$	−3.17157	−0.588946	−0.294473	+	0.955660i	$0.595144\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.65685	1.31314
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.17157	0.521403	0.260702	−	0.965419i	$-0.416046\pi$
$37$	3.17157	0.521403	0.260702	+	0.965419i	$0.416046\pi$
$38$	2.82843	0.458831
$39$	0	0
$40$	−1.00000	−0.158114
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−2.82843	−0.417029
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.00000	0.277350
$53$	10.4853	1.44026	0.720132	−	0.693837i	$-0.244081\pi$
$53$	10.4853	1.44026	0.720132	+	0.693837i	$0.244081\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	3.17157	0.416448
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	5.65685	0.718421
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	15.3137	1.87087	0.935434	−	0.353502i	$-0.115009\pi$
$67$	15.3137	1.87087	0.935434	+	0.353502i	$0.115009\pi$
$68$	−7.65685	−0.928530
$69$	0	0
$70$	1.00000	0.119523
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	7.65685	0.896167	0.448084	−	0.893992i	$-0.352107\pi$
$73$	7.65685	0.896167	0.448084	+	0.893992i	$0.352107\pi$
$74$	−3.17157	−0.368688
$75$	0	0
$76$	−2.82843	−0.324443
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−5.17157	−0.581847	−0.290924	−	0.956746i	$-0.593963\pi$
$79$	−5.17157	−0.581847	−0.290924	+	0.956746i	$0.593963\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−0.828427	−0.0914845
$83$	2.34315	0.257194	0.128597	−	0.991697i	$-0.458953\pi$
$83$	2.34315	0.257194	0.128597	+	0.991697i	$0.458953\pi$
$84$	0	0
$85$	−7.65685	−0.830502
$86$	9.65685	1.04133
$87$	0	0
$88$	−1.00000	−0.106600
$89$	11.6569	1.23562	0.617812	−	0.786326i	$-0.288019\pi$
$89$	11.6569	1.23562	0.617812	+	0.786326i	$0.288019\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	2.82843	0.294884
$93$	0	0
$94$	−8.00000	−0.825137
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−0.828427	−0.0841140	−0.0420570	−	0.999115i	$-0.513391\pi$
$97$	−0.828427	−0.0841140	−0.0420570	+	0.999115i	$0.513391\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−5.65685	−0.557386	−0.278693	−	0.960380i	$-0.589901\pi$
$103$	−5.65685	−0.557386	−0.278693	+	0.960380i	$0.589901\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−10.4853	−1.01842
$107$	1.65685	0.160174	0.0800871	−	0.996788i	$-0.474480\pi$
$107$	1.65685	0.160174	0.0800871	+	0.996788i	$0.474480\pi$
$108$	0	0
$109$	14.4853	1.38744	0.693719	−	0.720246i	$-0.255971\pi$
$109$	14.4853	1.38744	0.693719	+	0.720246i	$0.255971\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	2.82843	0.263752
$116$	−3.17157	−0.294473
$117$	0	0
$118$	−8.00000	−0.736460
$119$	7.65685	0.701903
$120$	0	0
$121$	1.00000	0.0909091
$122$	9.31371	0.843224
$123$	0	0
$124$	−5.65685	−0.508001
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.3137	1.71381	0.856907	−	0.515471i	$-0.172383\pi$
$127$	19.3137	1.71381	0.856907	+	0.515471i	$0.172383\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−18.8284	−1.64505	−0.822524	−	0.568731i	$-0.807435\pi$
$131$	−18.8284	−1.64505	−0.822524	+	0.568731i	$0.807435\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	−15.3137	−1.32290
$135$	0	0
$136$	7.65685	0.656570
$137$	−5.31371	−0.453981	−0.226990	−	0.973897i	$-0.572889\pi$
$137$	−5.31371	−0.453981	−0.226990	+	0.973897i	$0.572889\pi$
$138$	0	0
$139$	18.8284	1.59701	0.798503	−	0.601991i	$-0.205626\pi$
$139$	18.8284	1.59701	0.798503	+	0.601991i	$0.205626\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−5.65685	−0.474713
$143$	2.00000	0.167248
$144$	0	0
$145$	−3.17157	−0.263385
$146$	−7.65685	−0.633686
$147$	0	0
$148$	3.17157	0.260702
$149$	1.51472	0.124091	0.0620453	−	0.998073i	$-0.480238\pi$
$149$	1.51472	0.124091	0.0620453	+	0.998073i	$0.480238\pi$
$150$	0	0
$151$	10.8284	0.881205	0.440602	−	0.897702i	$-0.354765\pi$
$151$	10.8284	0.881205	0.440602	+	0.897702i	$0.354765\pi$
$152$	2.82843	0.229416
$153$	0	0
$154$	1.00000	0.0805823
$155$	−5.65685	−0.454369
$156$	0	0
$157$	15.6569	1.24955	0.624777	−	0.780804i	$-0.285190\pi$
$157$	15.6569	1.24955	0.624777	+	0.780804i	$0.285190\pi$
$158$	5.17157	0.411428
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−2.82843	−0.222911
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0.828427	0.0646893
$165$	0	0
$166$	−2.34315	−0.181863
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	7.65685	0.587254
$171$	0	0
$172$	−9.65685	−0.736328
$173$	1.31371	0.0998794	0.0499397	−	0.998752i	$-0.484097\pi$
$173$	1.31371	0.0998794	0.0499397	+	0.998752i	$0.484097\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−11.6569	−0.873718
$179$	−6.34315	−0.474109	−0.237054	−	0.971496i	$-0.576182\pi$
$179$	−6.34315	−0.474109	−0.237054	+	0.971496i	$0.576182\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	2.00000	0.148250
$183$	0	0
$184$	−2.82843	−0.208514
$185$	3.17157	0.233179
$186$	0	0
$187$	−7.65685	−0.559925
$188$	8.00000	0.583460
$189$	0	0
$190$	2.82843	0.205196
$191$	−19.3137	−1.39749	−0.698745	−	0.715370i	$-0.746258\pi$
$191$	−19.3137	−1.39749	−0.698745	+	0.715370i	$0.746258\pi$
$192$	0	0
$193$	21.3137	1.53419	0.767097	−	0.641531i	$-0.221700\pi$
$193$	21.3137	1.53419	0.767097	+	0.641531i	$0.221700\pi$
$194$	0.828427	0.0594776
$195$	0	0
$196$	1.00000	0.0714286
$197$	12.3431	0.879413	0.439706	−	0.898142i	$-0.355083\pi$
$197$	12.3431	0.879413	0.439706	+	0.898142i	$0.355083\pi$
$198$	0	0
$199$	−11.3137	−0.802008	−0.401004	−	0.916076i	$-0.631339\pi$
$199$	−11.3137	−0.802008	−0.401004	+	0.916076i	$0.631339\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	10.0000	0.703598
$203$	3.17157	0.222601
$204$	0	0
$205$	0.828427	0.0578599
$206$	5.65685	0.394132
$207$	0	0
$208$	2.00000	0.138675
$209$	−2.82843	−0.195646
$210$	0	0
$211$	15.3137	1.05424	0.527120	−	0.849791i	$-0.323272\pi$
$211$	15.3137	1.05424	0.527120	+	0.849791i	$0.323272\pi$
$212$	10.4853	0.720132
$213$	0	0
$214$	−1.65685	−0.113260
$215$	−9.65685	−0.658592
$216$	0	0
$217$	5.65685	0.384012
$218$	−14.4853	−0.981067
$219$	0	0
$220$	1.00000	0.0674200
$221$	−15.3137	−1.03011
$222$	0	0
$223$	−28.2843	−1.89405	−0.947027	−	0.321153i	$-0.895930\pi$
$223$	−28.2843	−1.89405	−0.947027	+	0.321153i	$0.895930\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	−2.82843	−0.186501
$231$	0	0
$232$	3.17157	0.208224
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	8.00000	0.520756
$237$	0	0
$238$	−7.65685	−0.496320
$239$	7.51472	0.486087	0.243043	−	0.970015i	$-0.421854\pi$
$239$	7.51472	0.486087	0.243043	+	0.970015i	$0.421854\pi$
$240$	0	0
$241$	−8.82843	−0.568689	−0.284344	−	0.958722i	$-0.591776\pi$
$241$	−8.82843	−0.568689	−0.284344	+	0.958722i	$0.591776\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	−9.31371	−0.596249
$245$	1.00000	0.0638877
$246$	0	0
$247$	−5.65685	−0.359937
$248$	5.65685	0.359211
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	11.3137	0.714115	0.357057	−	0.934082i	$-0.383780\pi$
$251$	11.3137	0.714115	0.357057	+	0.934082i	$0.383780\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	−19.3137	−1.21185
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.1716	−1.44540	−0.722702	−	0.691160i	$-0.757100\pi$
$257$	−23.1716	−1.44540	−0.722702	+	0.691160i	$0.757100\pi$
$258$	0	0
$259$	−3.17157	−0.197072
$260$	2.00000	0.124035
$261$	0	0
$262$	18.8284	1.16322
$263$	14.6274	0.901965	0.450983	−	0.892533i	$-0.351074\pi$
$263$	14.6274	0.901965	0.450983	+	0.892533i	$0.351074\pi$
$264$	0	0
$265$	10.4853	0.644106
$266$	−2.82843	−0.173422
$267$	0	0
$268$	15.3137	0.935434
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	−7.65685	−0.464265
$273$	0	0
$274$	5.31371	0.321013
$275$	1.00000	0.0603023
$276$	0	0
$277$	−13.3137	−0.799943	−0.399972	−	0.916528i	$-0.630980\pi$
$277$	−13.3137	−0.799943	−0.399972	+	0.916528i	$0.630980\pi$
$278$	−18.8284	−1.12925
$279$	0	0
$280$	1.00000	0.0597614
$281$	6.97056	0.415829	0.207914	−	0.978147i	$-0.433332\pi$
$281$	6.97056	0.415829	0.207914	+	0.978147i	$0.433332\pi$
$282$	0	0
$283$	−22.6274	−1.34506	−0.672530	−	0.740070i	$-0.734792\pi$
$283$	−22.6274	−1.34506	−0.672530	+	0.740070i	$0.734792\pi$
$284$	5.65685	0.335673
$285$	0	0
$286$	−2.00000	−0.118262
$287$	−0.828427	−0.0489005
$288$	0	0
$289$	41.6274	2.44867
$290$	3.17157	0.186241
$291$	0	0
$292$	7.65685	0.448084
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−3.17157	−0.184344
$297$	0	0
$298$	−1.51472	−0.0877453
$299$	5.65685	0.327144
$300$	0	0
$301$	9.65685	0.556612
$302$	−10.8284	−0.623106
$303$	0	0
$304$	−2.82843	−0.162221
$305$	−9.31371	−0.533301
$306$	0	0
$307$	−8.97056	−0.511977	−0.255989	−	0.966680i	$-0.582401\pi$
$307$	−8.97056	−0.511977	−0.255989	+	0.966680i	$0.582401\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	5.65685	0.321288
$311$	−0.970563	−0.0550356	−0.0275178	−	0.999621i	$-0.508760\pi$
$311$	−0.970563	−0.0550356	−0.0275178	+	0.999621i	$0.508760\pi$
$312$	0	0
$313$	−8.82843	−0.499012	−0.249506	−	0.968373i	$-0.580268\pi$
$313$	−8.82843	−0.499012	−0.249506	+	0.968373i	$0.580268\pi$
$314$	−15.6569	−0.883567
$315$	0	0
$316$	−5.17157	−0.290924
$317$	−3.17157	−0.178133	−0.0890666	−	0.996026i	$-0.528388\pi$
$317$	−3.17157	−0.178133	−0.0890666	+	0.996026i	$0.528388\pi$
$318$	0	0
$319$	−3.17157	−0.177574
$320$	1.00000	0.0559017
$321$	0	0
$322$	2.82843	0.157622
$323$	21.6569	1.20502
$324$	0	0
$325$	2.00000	0.110940
$326$	−12.0000	−0.664619
$327$	0	0
$328$	−0.828427	−0.0457422
$329$	−8.00000	−0.441054
$330$	0	0
$331$	28.9706	1.59237	0.796183	−	0.605056i	$-0.206849\pi$
$331$	28.9706	1.59237	0.796183	+	0.605056i	$0.206849\pi$
$332$	2.34315	0.128597
$333$	0	0
$334$	−16.0000	−0.875481
$335$	15.3137	0.836677
$336$	0	0
$337$	29.3137	1.59682	0.798410	−	0.602115i	$-0.205675\pi$
$337$	29.3137	1.59682	0.798410	+	0.602115i	$0.205675\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−7.65685	−0.415251
$341$	−5.65685	−0.306336
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	9.65685	0.520663
$345$	0	0
$346$	−1.31371	−0.0706254
$347$	−28.9706	−1.55522	−0.777611	−	0.628746i	$-0.783568\pi$
$347$	−28.9706	−1.55522	−0.777611	+	0.628746i	$0.783568\pi$
$348$	0	0
$349$	26.9706	1.44370	0.721851	−	0.692049i	$-0.243292\pi$
$349$	26.9706	1.44370	0.721851	+	0.692049i	$0.243292\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	24.8284	1.32148	0.660742	−	0.750613i	$-0.270242\pi$
$353$	24.8284	1.32148	0.660742	+	0.750613i	$0.270242\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	11.6569	0.617812
$357$	0	0
$358$	6.34315	0.335246
$359$	−33.4558	−1.76573	−0.882866	−	0.469625i	$-0.844389\pi$
$359$	−33.4558	−1.76573	−0.882866	+	0.469625i	$0.844389\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	14.0000	0.735824
$363$	0	0
$364$	−2.00000	−0.104828
$365$	7.65685	0.400778
$366$	0	0
$367$	28.2843	1.47643	0.738213	−	0.674567i	$-0.235670\pi$
$367$	28.2843	1.47643	0.738213	+	0.674567i	$0.235670\pi$
$368$	2.82843	0.147442
$369$	0	0
$370$	−3.17157	−0.164882
$371$	−10.4853	−0.544369
$372$	0	0
$373$	3.65685	0.189345	0.0946724	−	0.995508i	$-0.469820\pi$
$373$	3.65685	0.189345	0.0946724	+	0.995508i	$0.469820\pi$
$374$	7.65685	0.395927
$375$	0	0
$376$	−8.00000	−0.412568
$377$	−6.34315	−0.326689
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−2.82843	−0.145095
$381$	0	0
$382$	19.3137	0.988175
$383$	29.6569	1.51539	0.757697	−	0.652606i	$-0.226324\pi$
$383$	29.6569	1.51539	0.757697	+	0.652606i	$0.226324\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−21.3137	−1.08484
$387$	0	0
$388$	−0.828427	−0.0420570
$389$	−30.9706	−1.57027	−0.785135	−	0.619325i	$-0.787406\pi$
$389$	−30.9706	−1.57027	−0.785135	+	0.619325i	$0.787406\pi$
$390$	0	0
$391$	−21.6569	−1.09523
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−12.3431	−0.621839
$395$	−5.17157	−0.260210
$396$	0	0
$397$	−22.9706	−1.15286	−0.576430	−	0.817147i	$-0.695555\pi$
$397$	−22.9706	−1.15286	−0.576430	+	0.817147i	$0.695555\pi$
$398$	11.3137	0.567105
$399$	0	0
$400$	1.00000	0.0500000
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	−11.3137	−0.563576
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−3.17157	−0.157403
$407$	3.17157	0.157209
$408$	0	0
$409$	−12.1421	−0.600390	−0.300195	−	0.953878i	$-0.597052\pi$
$409$	−12.1421	−0.600390	−0.300195	+	0.953878i	$0.597052\pi$
$410$	−0.828427	−0.0409131
$411$	0	0
$412$	−5.65685	−0.278693
$413$	−8.00000	−0.393654
$414$	0	0
$415$	2.34315	0.115021
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	2.82843	0.138343
$419$	30.6274	1.49625	0.748124	−	0.663559i	$-0.230955\pi$
$419$	30.6274	1.49625	0.748124	+	0.663559i	$0.230955\pi$
$420$	0	0
$421$	28.6274	1.39521	0.697607	−	0.716480i	$-0.254248\pi$
$421$	28.6274	1.39521	0.697607	+	0.716480i	$0.254248\pi$
$422$	−15.3137	−0.745460
$423$	0	0
$424$	−10.4853	−0.509210
$425$	−7.65685	−0.371412
$426$	0	0
$427$	9.31371	0.450722
$428$	1.65685	0.0800871
$429$	0	0
$430$	9.65685	0.465695
$431$	19.7990	0.953684	0.476842	−	0.878989i	$-0.341781\pi$
$431$	19.7990	0.953684	0.476842	+	0.878989i	$0.341781\pi$
$432$	0	0
$433$	20.8284	1.00095	0.500475	−	0.865751i	$-0.333159\pi$
$433$	20.8284	1.00095	0.500475	+	0.865751i	$0.333159\pi$
$434$	−5.65685	−0.271538
$435$	0	0
$436$	14.4853	0.693719
$437$	−8.00000	−0.382692
$438$	0	0
$439$	24.9706	1.19178	0.595890	−	0.803066i	$-0.296799\pi$
$439$	24.9706	1.19178	0.595890	+	0.803066i	$0.296799\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	15.3137	0.728399
$443$	8.68629	0.412698	0.206349	−	0.978478i	$-0.433842\pi$
$443$	8.68629	0.412698	0.206349	+	0.978478i	$0.433842\pi$
$444$	0	0
$445$	11.6569	0.552588
$446$	28.2843	1.33930
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0.828427	0.0390091
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	−24.0000	−1.12638
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	1.31371	0.0613856
$459$	0	0
$460$	2.82843	0.131876
$461$	11.6569	0.542914	0.271457	−	0.962451i	$-0.412495\pi$
$461$	11.6569	0.542914	0.271457	+	0.962451i	$0.412495\pi$
$462$	0	0
$463$	−9.45584	−0.439450	−0.219725	−	0.975562i	$-0.570516\pi$
$463$	−9.45584	−0.439450	−0.219725	+	0.975562i	$0.570516\pi$
$464$	−3.17157	−0.147237
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−11.7990	−0.545992	−0.272996	−	0.962015i	$-0.588015\pi$
$467$	−11.7990	−0.545992	−0.272996	+	0.962015i	$0.588015\pi$
$468$	0	0
$469$	−15.3137	−0.707121
$470$	−8.00000	−0.369012
$471$	0	0
$472$	−8.00000	−0.368230
$473$	−9.65685	−0.444023
$474$	0	0
$475$	−2.82843	−0.129777
$476$	7.65685	0.350951
$477$	0	0
$478$	−7.51472	−0.343715
$479$	−35.3137	−1.61352	−0.806762	−	0.590876i	$-0.798782\pi$
$479$	−35.3137	−1.61352	−0.806762	+	0.590876i	$0.798782\pi$
$480$	0	0
$481$	6.34315	0.289223
$482$	8.82843	0.402124
$483$	0	0
$484$	1.00000	0.0454545
$485$	−0.828427	−0.0376169
$486$	0	0
$487$	−36.7696	−1.66619	−0.833094	−	0.553132i	$-0.813433\pi$
$487$	−36.7696	−1.66619	−0.833094	+	0.553132i	$0.813433\pi$
$488$	9.31371	0.421612
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	8.68629	0.392007	0.196003	−	0.980603i	$-0.437204\pi$
$491$	8.68629	0.392007	0.196003	+	0.980603i	$0.437204\pi$
$492$	0	0
$493$	24.2843	1.09371
$494$	5.65685	0.254514
$495$	0	0
$496$	−5.65685	−0.254000
$497$	−5.65685	−0.253745
$498$	0	0
$499$	28.9706	1.29690	0.648450	−	0.761257i	$-0.275417\pi$
$499$	28.9706	1.29690	0.648450	+	0.761257i	$0.275417\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−11.3137	−0.504956
$503$	3.31371	0.147751	0.0738755	−	0.997267i	$-0.476463\pi$
$503$	3.31371	0.147751	0.0738755	+	0.997267i	$0.476463\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	−2.82843	−0.125739
$507$	0	0
$508$	19.3137	0.856907
$509$	10.6863	0.473662	0.236831	−	0.971551i	$-0.423891\pi$
$509$	10.6863	0.473662	0.236831	+	0.971551i	$0.423891\pi$
$510$	0	0
$511$	−7.65685	−0.338719
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	23.1716	1.02205
$515$	−5.65685	−0.249271
$516$	0	0
$517$	8.00000	0.351840
$518$	3.17157	0.139351
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−41.5980	−1.82244	−0.911220	−	0.411919i	$-0.864859\pi$
$521$	−41.5980	−1.82244	−0.911220	+	0.411919i	$0.864859\pi$
$522$	0	0
$523$	−10.3431	−0.452274	−0.226137	−	0.974095i	$-0.572610\pi$
$523$	−10.3431	−0.452274	−0.226137	+	0.974095i	$0.572610\pi$
$524$	−18.8284	−0.822524
$525$	0	0
$526$	−14.6274	−0.637786
$527$	43.3137	1.88677
$528$	0	0
$529$	−15.0000	−0.652174
$530$	−10.4853	−0.455452
$531$	0	0
$532$	2.82843	0.122628
$533$	1.65685	0.0717663
$534$	0	0
$535$	1.65685	0.0716321
$536$	−15.3137	−0.661451
$537$	0	0
$538$	2.00000	0.0862261
$539$	1.00000	0.0430730
$540$	0	0
$541$	−33.1127	−1.42363	−0.711813	−	0.702369i	$-0.752126\pi$
$541$	−33.1127	−1.42363	−0.711813	+	0.702369i	$0.752126\pi$
$542$	−16.9706	−0.728948
$543$	0	0
$544$	7.65685	0.328285
$545$	14.4853	0.620481
$546$	0	0
$547$	23.3137	0.996822	0.498411	−	0.866941i	$-0.333917\pi$
$547$	23.3137	0.996822	0.498411	+	0.866941i	$0.333917\pi$
$548$	−5.31371	−0.226990
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	8.97056	0.382159
$552$	0	0
$553$	5.17157	0.219918
$554$	13.3137	0.565645
$555$	0	0
$556$	18.8284	0.798503
$557$	26.9706	1.14278	0.571390	−	0.820679i	$-0.306404\pi$
$557$	26.9706	1.14278	0.571390	+	0.820679i	$0.306404\pi$
$558$	0	0
$559$	−19.3137	−0.816883
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−6.97056	−0.294035
$563$	22.6274	0.953632	0.476816	−	0.879003i	$-0.341791\pi$
$563$	22.6274	0.953632	0.476816	+	0.879003i	$0.341791\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	22.6274	0.951101
$567$	0	0
$568$	−5.65685	−0.237356
$569$	−16.6274	−0.697058	−0.348529	−	0.937298i	$-0.613319\pi$
$569$	−16.6274	−0.697058	−0.348529	+	0.937298i	$0.613319\pi$
$570$	0	0
$571$	−35.5980	−1.48973	−0.744865	−	0.667216i	$-0.767486\pi$
$571$	−35.5980	−1.48973	−0.744865	+	0.667216i	$0.767486\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0.828427	0.0345779
$575$	2.82843	0.117954
$576$	0	0
$577$	25.1127	1.04546	0.522728	−	0.852500i	$-0.324914\pi$
$577$	25.1127	1.04546	0.522728	+	0.852500i	$0.324914\pi$
$578$	−41.6274	−1.73147
$579$	0	0
$580$	−3.17157	−0.131692
$581$	−2.34315	−0.0972101
$582$	0	0
$583$	10.4853	0.434256
$584$	−7.65685	−0.316843
$585$	0	0
$586$	2.00000	0.0826192
$587$	38.1421	1.57429	0.787147	−	0.616765i	$-0.211557\pi$
$587$	38.1421	1.57429	0.787147	+	0.616765i	$0.211557\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	−8.00000	−0.329355
$591$	0	0
$592$	3.17157	0.130351
$593$	−20.3431	−0.835393	−0.417696	−	0.908587i	$-0.637162\pi$
$593$	−20.3431	−0.835393	−0.417696	+	0.908587i	$0.637162\pi$
$594$	0	0
$595$	7.65685	0.313900
$596$	1.51472	0.0620453
$597$	0	0
$598$	−5.65685	−0.231326
$599$	−21.6569	−0.884875	−0.442438	−	0.896799i	$-0.645886\pi$
$599$	−21.6569	−0.884875	−0.442438	+	0.896799i	$0.645886\pi$
$600$	0	0
$601$	12.8284	0.523282	0.261641	−	0.965165i	$-0.415736\pi$
$601$	12.8284	0.523282	0.261641	+	0.965165i	$0.415736\pi$
$602$	−9.65685	−0.393584
$603$	0	0
$604$	10.8284	0.440602
$605$	1.00000	0.0406558
$606$	0	0
$607$	29.6569	1.20373	0.601867	−	0.798596i	$-0.294424\pi$
$607$	29.6569	1.20373	0.601867	+	0.798596i	$0.294424\pi$
$608$	2.82843	0.114708
$609$	0	0
$610$	9.31371	0.377101
$611$	16.0000	0.647291
$612$	0	0
$613$	22.9706	0.927772	0.463886	−	0.885895i	$-0.346455\pi$
$613$	22.9706	0.927772	0.463886	+	0.885895i	$0.346455\pi$
$614$	8.97056	0.362022
$615$	0	0
$616$	1.00000	0.0402911
$617$	32.3431	1.30209	0.651043	−	0.759041i	$-0.274332\pi$
$617$	32.3431	1.30209	0.651043	+	0.759041i	$0.274332\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	−5.65685	−0.227185
$621$	0	0
$622$	0.970563	0.0389160
$623$	−11.6569	−0.467022
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.82843	0.352855
$627$	0	0
$628$	15.6569	0.624777
$629$	−24.2843	−0.968277
$630$	0	0
$631$	−22.6274	−0.900783	−0.450392	−	0.892831i	$-0.648716\pi$
$631$	−22.6274	−0.900783	−0.450392	+	0.892831i	$0.648716\pi$
$632$	5.17157	0.205714
$633$	0	0
$634$	3.17157	0.125959
$635$	19.3137	0.766441
$636$	0	0
$637$	2.00000	0.0792429
$638$	3.17157	0.125564
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	14.1421	0.557711	0.278856	−	0.960333i	$-0.410045\pi$
$643$	14.1421	0.557711	0.278856	+	0.960333i	$0.410045\pi$
$644$	−2.82843	−0.111456
$645$	0	0
$646$	−21.6569	−0.852078
$647$	−12.2843	−0.482945	−0.241472	−	0.970408i	$-0.577630\pi$
$647$	−12.2843	−0.482945	−0.241472	+	0.970408i	$0.577630\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	12.0000	0.469956
$653$	−7.45584	−0.291770	−0.145885	−	0.989302i	$-0.546603\pi$
$653$	−7.45584	−0.291770	−0.145885	+	0.989302i	$0.546603\pi$
$654$	0	0
$655$	−18.8284	−0.735688
$656$	0.828427	0.0323446
$657$	0	0
$658$	8.00000	0.311872
$659$	9.65685	0.376178	0.188089	−	0.982152i	$-0.439771\pi$
$659$	9.65685	0.376178	0.188089	+	0.982152i	$0.439771\pi$
$660$	0	0
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	−28.9706	−1.12597
$663$	0	0
$664$	−2.34315	−0.0909317
$665$	2.82843	0.109682
$666$	0	0
$667$	−8.97056	−0.347342
$668$	16.0000	0.619059
$669$	0	0
$670$	−15.3137	−0.591620
$671$	−9.31371	−0.359552
$672$	0	0
$673$	−36.6274	−1.41188	−0.705942	−	0.708270i	$-0.749476\pi$
$673$	−36.6274	−1.41188	−0.705942	+	0.708270i	$0.749476\pi$
$674$	−29.3137	−1.12912
$675$	0	0
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	0.828427	0.0317921
$680$	7.65685	0.293627
$681$	0	0
$682$	5.65685	0.216612
$683$	−22.3431	−0.854937	−0.427468	−	0.904030i	$-0.640594\pi$
$683$	−22.3431	−0.854937	−0.427468	+	0.904030i	$0.640594\pi$
$684$	0	0
$685$	−5.31371	−0.203026
$686$	1.00000	0.0381802
$687$	0	0
$688$	−9.65685	−0.368164
$689$	20.9706	0.798915
$690$	0	0
$691$	0.970563	0.0369219	0.0184610	−	0.999830i	$-0.494123\pi$
$691$	0.970563	0.0369219	0.0184610	+	0.999830i	$0.494123\pi$
$692$	1.31371	0.0499397
$693$	0	0
$694$	28.9706	1.09971
$695$	18.8284	0.714203
$696$	0	0
$697$	−6.34315	−0.240264
$698$	−26.9706	−1.02085
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−40.8284	−1.54207	−0.771034	−	0.636794i	$-0.780260\pi$
$701$	−40.8284	−1.54207	−0.771034	+	0.636794i	$0.780260\pi$
$702$	0	0
$703$	−8.97056	−0.338331
$704$	1.00000	0.0376889
$705$	0	0
$706$	−24.8284	−0.934430
$707$	10.0000	0.376089
$708$	0	0
$709$	26.2843	0.987127	0.493563	−	0.869710i	$-0.335694\pi$
$709$	26.2843	0.987127	0.493563	+	0.869710i	$0.335694\pi$
$710$	−5.65685	−0.212298
$711$	0	0
$712$	−11.6569	−0.436859
$713$	−16.0000	−0.599205
$714$	0	0
$715$	2.00000	0.0747958
$716$	−6.34315	−0.237054
$717$	0	0
$718$	33.4558	1.24856
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	5.65685	0.210672
$722$	11.0000	0.409378
$723$	0	0
$724$	−14.0000	−0.520306
$725$	−3.17157	−0.117789
$726$	0	0
$727$	4.68629	0.173805	0.0869025	−	0.996217i	$-0.472303\pi$
$727$	4.68629	0.173805	0.0869025	+	0.996217i	$0.472303\pi$
$728$	2.00000	0.0741249
$729$	0	0
$730$	−7.65685	−0.283393
$731$	73.9411	2.73481
$732$	0	0
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−28.2843	−1.04399
$735$	0	0
$736$	−2.82843	−0.104257
$737$	15.3137	0.564088
$738$	0	0
$739$	41.2548	1.51758	0.758792	−	0.651333i	$-0.225790\pi$
$739$	41.2548	1.51758	0.758792	+	0.651333i	$0.225790\pi$
$740$	3.17157	0.116589
$741$	0	0
$742$	10.4853	0.384927
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	1.51472	0.0554950
$746$	−3.65685	−0.133887
$747$	0	0
$748$	−7.65685	−0.279962
$749$	−1.65685	−0.0605401
$750$	0	0
$751$	39.5980	1.44495	0.722475	−	0.691397i	$-0.243004\pi$
$751$	39.5980	1.44495	0.722475	+	0.691397i	$0.243004\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	6.34315	0.231004
$755$	10.8284	0.394087
$756$	0	0
$757$	−25.1127	−0.912737	−0.456368	−	0.889791i	$-0.650850\pi$
$757$	−25.1127	−0.912737	−0.456368	+	0.889791i	$0.650850\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	2.82843	0.102598
$761$	7.45584	0.270274	0.135137	−	0.990827i	$-0.456853\pi$
$761$	7.45584	0.270274	0.135137	+	0.990827i	$0.456853\pi$
$762$	0	0
$763$	−14.4853	−0.524402
$764$	−19.3137	−0.698745
$765$	0	0
$766$	−29.6569	−1.07155
$767$	16.0000	0.577727
$768$	0	0
$769$	−14.4853	−0.522353	−0.261176	−	0.965291i	$-0.584110\pi$
$769$	−14.4853	−0.522353	−0.261176	+	0.965291i	$0.584110\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	21.3137	0.767097
$773$	10.6863	0.384359	0.192180	−	0.981360i	$-0.438444\pi$
$773$	10.6863	0.384359	0.192180	+	0.981360i	$0.438444\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0.828427	0.0297388
$777$	0	0
$778$	30.9706	1.11035
$779$	−2.34315	−0.0839519
$780$	0	0
$781$	5.65685	0.202418
$782$	21.6569	0.774448
$783$	0	0
$784$	1.00000	0.0357143
$785$	15.6569	0.558817
$786$	0	0
$787$	30.6274	1.09175	0.545875	−	0.837867i	$-0.316197\pi$
$787$	30.6274	1.09175	0.545875	+	0.837867i	$0.316197\pi$
$788$	12.3431	0.439706
$789$	0	0
$790$	5.17157	0.183996
$791$	2.00000	0.0711118
$792$	0	0
$793$	−18.6274	−0.661479
$794$	22.9706	0.815195
$795$	0	0
$796$	−11.3137	−0.401004
$797$	−6.28427	−0.222600	−0.111300	−	0.993787i	$-0.535501\pi$
$797$	−6.28427	−0.222600	−0.111300	+	0.993787i	$0.535501\pi$
$798$	0	0
$799$	−61.2548	−2.16704
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	10.0000	0.353112
$803$	7.65685	0.270205
$804$	0	0
$805$	−2.82843	−0.0996890
$806$	11.3137	0.398508
$807$	0	0
$808$	10.0000	0.351799
$809$	−22.2843	−0.783473	−0.391737	−	0.920077i	$-0.628126\pi$
$809$	−22.2843	−0.783473	−0.391737	+	0.920077i	$0.628126\pi$
$810$	0	0
$811$	22.1421	0.777516	0.388758	−	0.921340i	$-0.372904\pi$
$811$	22.1421	0.777516	0.388758	+	0.921340i	$0.372904\pi$
$812$	3.17157	0.111300
$813$	0	0
$814$	−3.17157	−0.111164
$815$	12.0000	0.420342
$816$	0	0
$817$	27.3137	0.955586
$818$	12.1421	0.424540
$819$	0	0
$820$	0.828427	0.0289299
$821$	29.7990	1.03999	0.519996	−	0.854169i	$-0.325933\pi$
$821$	29.7990	1.03999	0.519996	+	0.854169i	$0.325933\pi$
$822$	0	0
$823$	21.1716	0.737995	0.368997	−	0.929430i	$-0.379701\pi$
$823$	21.1716	0.737995	0.368997	+	0.929430i	$0.379701\pi$
$824$	5.65685	0.197066
$825$	0	0
$826$	8.00000	0.278356
$827$	49.2548	1.71276	0.856379	−	0.516347i	$-0.172709\pi$
$827$	49.2548	1.71276	0.856379	+	0.516347i	$0.172709\pi$
$828$	0	0
$829$	39.2548	1.36338	0.681688	−	0.731643i	$-0.261246\pi$
$829$	39.2548	1.36338	0.681688	+	0.731643i	$0.261246\pi$
$830$	−2.34315	−0.0813318
$831$	0	0
$832$	2.00000	0.0693375
$833$	−7.65685	−0.265294
$834$	0	0
$835$	16.0000	0.553703
$836$	−2.82843	−0.0978232
$837$	0	0
$838$	−30.6274	−1.05801
$839$	34.3431	1.18566	0.592829	−	0.805329i	$-0.298011\pi$
$839$	34.3431	1.18566	0.592829	+	0.805329i	$0.298011\pi$
$840$	0	0
$841$	−18.9411	−0.653142
$842$	−28.6274	−0.986566
$843$	0	0
$844$	15.3137	0.527120
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	10.4853	0.360066
$849$	0	0
$850$	7.65685	0.262628
$851$	8.97056	0.307507
$852$	0	0
$853$	−25.3137	−0.866725	−0.433362	−	0.901220i	$-0.642673\pi$
$853$	−25.3137	−0.866725	−0.433362	+	0.901220i	$0.642673\pi$
$854$	−9.31371	−0.318709
$855$	0	0
$856$	−1.65685	−0.0566301
$857$	6.97056	0.238110	0.119055	−	0.992888i	$-0.462014\pi$
$857$	6.97056	0.238110	0.119055	+	0.992888i	$0.462014\pi$
$858$	0	0
$859$	4.68629	0.159894	0.0799471	−	0.996799i	$-0.474525\pi$
$859$	4.68629	0.159894	0.0799471	+	0.996799i	$0.474525\pi$
$860$	−9.65685	−0.329296
$861$	0	0
$862$	−19.7990	−0.674356
$863$	36.7696	1.25165	0.625825	−	0.779963i	$-0.284762\pi$
$863$	36.7696	1.25165	0.625825	+	0.779963i	$0.284762\pi$
$864$	0	0
$865$	1.31371	0.0446674
$866$	−20.8284	−0.707779
$867$	0	0
$868$	5.65685	0.192006
$869$	−5.17157	−0.175434
$870$	0	0
$871$	30.6274	1.03777
$872$	−14.4853	−0.490534
$873$	0	0
$874$	8.00000	0.270604
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−2.97056	−0.100309	−0.0501544	−	0.998741i	$-0.515971\pi$
$877$	−2.97056	−0.100309	−0.0501544	+	0.998741i	$0.515971\pi$
$878$	−24.9706	−0.842716
$879$	0	0
$880$	1.00000	0.0337100
$881$	2.68629	0.0905035	0.0452517	−	0.998976i	$-0.485591\pi$
$881$	2.68629	0.0905035	0.0452517	+	0.998976i	$0.485591\pi$
$882$	0	0
$883$	41.6569	1.40186	0.700932	−	0.713228i	$-0.252767\pi$
$883$	41.6569	1.40186	0.700932	+	0.713228i	$0.252767\pi$
$884$	−15.3137	−0.515056
$885$	0	0
$886$	−8.68629	−0.291822
$887$	22.6274	0.759754	0.379877	−	0.925037i	$-0.375966\pi$
$887$	22.6274	0.759754	0.379877	+	0.925037i	$0.375966\pi$
$888$	0	0
$889$	−19.3137	−0.647761
$890$	−11.6569	−0.390739
$891$	0	0
$892$	−28.2843	−0.947027
$893$	−22.6274	−0.757198
$894$	0	0
$895$	−6.34315	−0.212028
$896$	1.00000	0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	17.9411	0.598370
$900$	0	0
$901$	−80.2843	−2.67466
$902$	−0.828427	−0.0275836
$903$	0	0
$904$	2.00000	0.0665190
$905$	−14.0000	−0.465376
$906$	0	0
$907$	17.6569	0.586286	0.293143	−	0.956069i	$-0.405299\pi$
$907$	17.6569	0.586286	0.293143	+	0.956069i	$0.405299\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	2.00000	0.0662994
$911$	21.6569	0.717524	0.358762	−	0.933429i	$-0.383199\pi$
$911$	21.6569	0.717524	0.358762	+	0.933429i	$0.383199\pi$
$912$	0	0
$913$	2.34315	0.0775468
$914$	22.0000	0.727695
$915$	0	0
$916$	−1.31371	−0.0434062
$917$	18.8284	0.621769
$918$	0	0
$919$	20.7696	0.685124	0.342562	−	0.939495i	$-0.388705\pi$
$919$	20.7696	0.685124	0.342562	+	0.939495i	$0.388705\pi$
$920$	−2.82843	−0.0932505
$921$	0	0
$922$	−11.6569	−0.383898
$923$	11.3137	0.372395
$924$	0	0
$925$	3.17157	0.104281
$926$	9.45584	0.310738
$927$	0	0
$928$	3.17157	0.104112
$929$	38.9706	1.27858	0.639291	−	0.768965i	$-0.279228\pi$
$929$	38.9706	1.27858	0.639291	+	0.768965i	$0.279228\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	6.00000	0.196537
$933$	0	0
$934$	11.7990	0.386075
$935$	−7.65685	−0.250406
$936$	0	0
$937$	24.6274	0.804543	0.402271	−	0.915520i	$-0.368221\pi$
$937$	24.6274	0.804543	0.402271	+	0.915520i	$0.368221\pi$
$938$	15.3137	0.500010
$939$	0	0
$940$	8.00000	0.260931
$941$	−13.3137	−0.434014	−0.217007	−	0.976170i	$-0.569630\pi$
$941$	−13.3137	−0.434014	−0.217007	+	0.976170i	$0.569630\pi$
$942$	0	0
$943$	2.34315	0.0763033
$944$	8.00000	0.260378
$945$	0	0
$946$	9.65685	0.313971
$947$	−35.5980	−1.15678	−0.578389	−	0.815761i	$-0.696319\pi$
$947$	−35.5980	−1.15678	−0.578389	+	0.815761i	$0.696319\pi$
$948$	0	0
$949$	15.3137	0.497104
$950$	2.82843	0.0917663
$951$	0	0
$952$	−7.65685	−0.248160
$953$	43.2548	1.40116	0.700581	−	0.713573i	$-0.252924\pi$
$953$	43.2548	1.40116	0.700581	+	0.713573i	$0.252924\pi$
$954$	0	0
$955$	−19.3137	−0.624977
$956$	7.51472	0.243043
$957$	0	0
$958$	35.3137	1.14093
$959$	5.31371	0.171589
$960$	0	0
$961$	1.00000	0.0322581
$962$	−6.34315	−0.204511
$963$	0	0
$964$	−8.82843	−0.284344
$965$	21.3137	0.686113
$966$	0	0
$967$	59.3137	1.90740	0.953700	−	0.300759i	$-0.0972400\pi$
$967$	59.3137	1.90740	0.953700	+	0.300759i	$0.0972400\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	0.828427	0.0265992
$971$	41.9411	1.34595	0.672977	−	0.739663i	$-0.265015\pi$
$971$	41.9411	1.34595	0.672977	+	0.739663i	$0.265015\pi$
$972$	0	0
$973$	−18.8284	−0.603612
$974$	36.7696	1.17817
$975$	0	0
$976$	−9.31371	−0.298125
$977$	20.6274	0.659930	0.329965	−	0.943993i	$-0.392963\pi$
$977$	20.6274	0.659930	0.329965	+	0.943993i	$0.392963\pi$
$978$	0	0
$979$	11.6569	0.372555
$980$	1.00000	0.0319438
$981$	0	0
$982$	−8.68629	−0.277191
$983$	−39.5980	−1.26298	−0.631490	−	0.775384i	$-0.717556\pi$
$983$	−39.5980	−1.26298	−0.631490	+	0.775384i	$0.717556\pi$
$984$	0	0
$985$	12.3431	0.393285
$986$	−24.2843	−0.773369
$987$	0	0
$988$	−5.65685	−0.179969
$989$	−27.3137	−0.868525
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	5.65685	0.179605
$993$	0	0
$994$	5.65685	0.179425
$995$	−11.3137	−0.358669
$996$	0	0
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−28.9706	−0.917047
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.br.1.1		2
3.2	odd	2		770.2.a.i.1.2	✓	2
12.11	even	2		6160.2.a.ba.1.1		2
15.2	even	4		3850.2.c.u.1849.3		4
15.8	even	4		3850.2.c.u.1849.2		4
15.14	odd	2		3850.2.a.bi.1.1		2
21.20	even	2		5390.2.a.bt.1.1		2
33.32	even	2		8470.2.a.bo.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.i.1.2	✓	2	3.2	odd	2
3850.2.a.bi.1.1		2	15.14	odd	2
3850.2.c.u.1849.2		4	15.8	even	4
3850.2.c.u.1849.3		4	15.2	even	4
5390.2.a.bt.1.1		2	21.20	even	2
6160.2.a.ba.1.1		2	12.11	even	2
6930.2.a.br.1.1		2	1.1	even	1	trivial
8470.2.a.bo.1.2		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.65685 q^{17} -2.82843 q^{19} +1.00000 q^{20} -1.00000 q^{22} +2.82843 q^{23} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{28} -3.17157 q^{29} -5.65685 q^{31} -1.00000 q^{32} +7.65685 q^{34} -1.00000 q^{35} +3.17157 q^{37} +2.82843 q^{38} -1.00000 q^{40} +0.828427 q^{41} -9.65685 q^{43} +1.00000 q^{44} -2.82843 q^{46} +8.00000 q^{47} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} +10.4853 q^{53} +1.00000 q^{55} +1.00000 q^{56} +3.17157 q^{58} +8.00000 q^{59} -9.31371 q^{61} +5.65685 q^{62} +1.00000 q^{64} +2.00000 q^{65} +15.3137 q^{67} -7.65685 q^{68} +1.00000 q^{70} +5.65685 q^{71} +7.65685 q^{73} -3.17157 q^{74} -2.82843 q^{76} -1.00000 q^{77} -5.17157 q^{79} +1.00000 q^{80} -0.828427 q^{82} +2.34315 q^{83} -7.65685 q^{85} +9.65685 q^{86} -1.00000 q^{88} +11.6569 q^{89} -2.00000 q^{91} +2.82843 q^{92} -8.00000 q^{94} -2.82843 q^{95} -0.828427 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{20} - 2 q^{22} + 2 q^{25} - 4 q^{26} - 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 12 q^{37} - 2 q^{40} - 4 q^{41} - 8 q^{43} + 2 q^{44} + 16 q^{47} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 4 q^{53} + 2 q^{55} + 2 q^{56} + 12 q^{58} + 16 q^{59} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} + 2 q^{70} + 4 q^{73} - 12 q^{74} - 2 q^{77} - 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} - 4 q^{85} + 8 q^{86} - 2 q^{88} + 12 q^{89} - 4 q^{91} - 16 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 - 2 * q^8 - 2 * q^10 + 2 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 4 * q^17 + 2 * q^20 - 2 * q^22 + 2 * q^25 - 4 * q^26 - 2 * q^28 - 12 * q^29 - 2 * q^32 + 4 * q^34 - 2 * q^35 + 12 * q^37 - 2 * q^40 - 4 * q^41 - 8 * q^43 + 2 * q^44 + 16 * q^47 + 2 * q^49 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 2 * q^55 + 2 * q^56 + 12 * q^58 + 16 * q^59 + 4 * q^61 + 2 * q^64 + 4 * q^65 + 8 * q^67 - 4 * q^68 + 2 * q^70 + 4 * q^73 - 12 * q^74 - 2 * q^77 - 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 - 4 * q^85 + 8 * q^86 - 2 * q^88 + 12 * q^89 - 4 * q^91 - 16 * q^94 + 4 * q^97 - 2 * q^98

Introduction

L-functions

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