LMFDB - Embedded newform 6930.2.a.ca.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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -1.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} -2.73205 q^{19} -1.00000 q^{20} -1.00000 q^{22} -1.26795 q^{23} +1.00000 q^{25} -1.46410 q^{26} +1.00000 q^{28} +4.73205 q^{29} +8.92820 q^{31} +1.00000 q^{32} -3.46410 q^{34} -1.00000 q^{35} +3.26795 q^{37} -2.73205 q^{38} -1.00000 q^{40} +4.73205 q^{41} -4.92820 q^{43} -1.00000 q^{44} -1.26795 q^{46} +1.00000 q^{49} +1.00000 q^{50} -1.46410 q^{52} +4.73205 q^{53} +1.00000 q^{55} +1.00000 q^{56} +4.73205 q^{58} +13.8564 q^{59} +2.00000 q^{61} +8.92820 q^{62} +1.00000 q^{64} +1.46410 q^{65} -10.9282 q^{67} -3.46410 q^{68} -1.00000 q^{70} -9.46410 q^{71} +11.4641 q^{73} +3.26795 q^{74} -2.73205 q^{76} -1.00000 q^{77} +6.73205 q^{79} -1.00000 q^{80} +4.73205 q^{82} +4.39230 q^{83} +3.46410 q^{85} -4.92820 q^{86} -1.00000 q^{88} +15.4641 q^{89} -1.46410 q^{91} -1.26795 q^{92} +2.73205 q^{95} +5.80385 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} + 4 q^{52} + 6 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} + 16 q^{73} + 10 q^{74} - 2 q^{76} - 2 q^{77} + 10 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 4 q^{86} - 2 q^{88} + 24 q^{89} + 4 q^{91} - 6 q^{92} + 2 q^{95} + 22 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 - 2 * q^19 - 2 * q^20 - 2 * q^22 - 6 * q^23 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 6 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^35 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 4 * q^43 - 2 * q^44 - 6 * q^46 + 2 * q^49 + 2 * q^50 + 4 * q^52 + 6 * q^53 + 2 * q^55 + 2 * q^56 + 6 * q^58 + 4 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 2 * q^70 - 12 * q^71 + 16 * q^73 + 10 * q^74 - 2 * q^76 - 2 * q^77 + 10 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 4 * q^86 - 2 * q^88 + 24 * q^89 + 4 * q^91 - 6 * q^92 + 2 * q^95 + 22 * 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
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	−2.73205	−0.626775	−0.313388	−	0.949625i	$-0.601464\pi$
$19$	−2.73205	−0.626775	−0.313388	+	0.949625i	$0.601464\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−1.26795	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$23$	−1.26795	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.46410	−0.287134
$27$	0	0
$28$	1.00000	0.188982
$29$	4.73205	0.878720	0.439360	−	0.898311i	$-0.355205\pi$
$29$	4.73205	0.878720	0.439360	+	0.898311i	$0.355205\pi$
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.26795	0.537248	0.268624	−	0.963245i	$-0.413431\pi$
$37$	3.26795	0.537248	0.268624	+	0.963245i	$0.413431\pi$
$38$	−2.73205	−0.443197
$39$	0	0
$40$	−1.00000	−0.158114
$41$	4.73205	0.739022	0.369511	−	0.929226i	$-0.379525\pi$
$41$	4.73205	0.739022	0.369511	+	0.929226i	$0.379525\pi$
$42$	0	0
$43$	−4.92820	−0.751544	−0.375772	−	0.926712i	$-0.622622\pi$
$43$	−4.92820	−0.751544	−0.375772	+	0.926712i	$0.622622\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−1.26795	−0.186949
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	−1.46410	−0.203034
$53$	4.73205	0.649997	0.324999	−	0.945715i	$-0.394636\pi$
$53$	4.73205	0.649997	0.324999	+	0.945715i	$0.394636\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	4.73205	0.621349
$59$	13.8564	1.80395	0.901975	−	0.431788i	$-0.142117\pi$
$59$	13.8564	1.80395	0.901975	+	0.431788i	$0.142117\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	8.92820	1.13388
$63$	0	0
$64$	1.00000	0.125000
$65$	1.46410	0.181599
$66$	0	0
$67$	−10.9282	−1.33509	−0.667546	−	0.744568i	$-0.732655\pi$
$67$	−10.9282	−1.33509	−0.667546	+	0.744568i	$0.732655\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	0	0
$73$	11.4641	1.34177	0.670886	−	0.741561i	$-0.265914\pi$
$73$	11.4641	1.34177	0.670886	+	0.741561i	$0.265914\pi$
$74$	3.26795	0.379891
$75$	0	0
$76$	−2.73205	−0.313388
$77$	−1.00000	−0.113961
$78$	0	0
$79$	6.73205	0.757415	0.378707	−	0.925516i	$-0.376369\pi$
$79$	6.73205	0.757415	0.378707	+	0.925516i	$0.376369\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	4.73205	0.522568
$83$	4.39230	0.482118	0.241059	−	0.970510i	$-0.422505\pi$
$83$	4.39230	0.482118	0.241059	+	0.970510i	$0.422505\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	−4.92820	−0.531422
$87$	0	0
$88$	−1.00000	−0.106600
$89$	15.4641	1.63919	0.819596	−	0.572942i	$-0.194198\pi$
$89$	15.4641	1.63919	0.819596	+	0.572942i	$0.194198\pi$
$90$	0	0
$91$	−1.46410	−0.153480
$92$	−1.26795	−0.132193
$93$	0	0
$94$	0	0
$95$	2.73205	0.280302
$96$	0	0
$97$	5.80385	0.589291	0.294646	−	0.955607i	$-0.404798\pi$
$97$	5.80385	0.589291	0.294646	+	0.955607i	$0.404798\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	10.5359	1.03813	0.519066	−	0.854734i	$-0.326280\pi$
$103$	10.5359	1.03813	0.519066	+	0.854734i	$0.326280\pi$
$104$	−1.46410	−0.143567
$105$	0	0
$106$	4.73205	0.459617
$107$	−0.928203	−0.0897328	−0.0448664	−	0.998993i	$-0.514286\pi$
$107$	−0.928203	−0.0897328	−0.0448664	+	0.998993i	$0.514286\pi$
$108$	0	0
$109$	−1.80385	−0.172777	−0.0863886	−	0.996262i	$-0.527533\pi$
$109$	−1.80385	−0.172777	−0.0863886	+	0.996262i	$0.527533\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	−12.9282	−1.21618	−0.608092	−	0.793867i	$-0.708065\pi$
$113$	−12.9282	−1.21618	−0.608092	+	0.793867i	$0.708065\pi$
$114$	0	0
$115$	1.26795	0.118237
$116$	4.73205	0.439360
$117$	0	0
$118$	13.8564	1.27559
$119$	−3.46410	−0.317554
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	0	0
$124$	8.92820	0.801776
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.85641	0.874615	0.437307	−	0.899312i	$-0.355932\pi$
$127$	9.85641	0.874615	0.437307	+	0.899312i	$0.355932\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.46410	0.128410
$131$	17.6603	1.54298	0.771492	−	0.636239i	$-0.219511\pi$
$131$	17.6603	1.54298	0.771492	+	0.636239i	$0.219511\pi$
$132$	0	0
$133$	−2.73205	−0.236899
$134$	−10.9282	−0.944053
$135$	0	0
$136$	−3.46410	−0.297044
$137$	−7.85641	−0.671218	−0.335609	−	0.942001i	$-0.608942\pi$
$137$	−7.85641	−0.671218	−0.335609	+	0.942001i	$0.608942\pi$
$138$	0	0
$139$	4.19615	0.355913	0.177957	−	0.984038i	$-0.443051\pi$
$139$	4.19615	0.355913	0.177957	+	0.984038i	$0.443051\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−9.46410	−0.794210
$143$	1.46410	0.122434
$144$	0	0
$145$	−4.73205	−0.392975
$146$	11.4641	0.948776
$147$	0	0
$148$	3.26795	0.268624
$149$	7.26795	0.595414	0.297707	−	0.954657i	$-0.403778\pi$
$149$	7.26795	0.595414	0.297707	+	0.954657i	$0.403778\pi$
$150$	0	0
$151$	−7.80385	−0.635068	−0.317534	−	0.948247i	$-0.602855\pi$
$151$	−7.80385	−0.635068	−0.317534	+	0.948247i	$0.602855\pi$
$152$	−2.73205	−0.221599
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−8.92820	−0.717131
$156$	0	0
$157$	6.39230	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$157$	6.39230	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$158$	6.73205	0.535573
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−1.26795	−0.0999284
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	4.73205	0.369511
$165$	0	0
$166$	4.39230	0.340909
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	3.46410	0.265684
$171$	0	0
$172$	−4.92820	−0.375772
$173$	−0.928203	−0.0705700	−0.0352850	−	0.999377i	$-0.511234\pi$
$173$	−0.928203	−0.0705700	−0.0352850	+	0.999377i	$0.511234\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	15.4641	1.15908
$179$	19.8564	1.48414	0.742069	−	0.670324i	$-0.233845\pi$
$179$	19.8564	1.48414	0.742069	+	0.670324i	$0.233845\pi$
$180$	0	0
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	−1.46410	−0.108526
$183$	0	0
$184$	−1.26795	−0.0934745
$185$	−3.26795	−0.240264
$186$	0	0
$187$	3.46410	0.253320
$188$	0	0
$189$	0	0
$190$	2.73205	0.198204
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	−25.4641	−1.83295	−0.916473	−	0.400096i	$-0.868977\pi$
$193$	−25.4641	−1.83295	−0.916473	+	0.400096i	$0.868977\pi$
$194$	5.80385	0.416692
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.3923	1.59539	0.797693	−	0.603064i	$-0.206054\pi$
$197$	22.3923	1.59539	0.797693	+	0.603064i	$0.206054\pi$
$198$	0	0
$199$	−24.7846	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	4.73205	0.332125
$204$	0	0
$205$	−4.73205	−0.330501
$206$	10.5359	0.734071
$207$	0	0
$208$	−1.46410	−0.101517
$209$	2.73205	0.188980
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	4.73205	0.324999
$213$	0	0
$214$	−0.928203	−0.0634507
$215$	4.92820	0.336101
$216$	0	0
$217$	8.92820	0.606086
$218$	−1.80385	−0.122172
$219$	0	0
$220$	1.00000	0.0674200
$221$	5.07180	0.341166
$222$	0	0
$223$	−8.39230	−0.561990	−0.280995	−	0.959709i	$-0.590664\pi$
$223$	−8.39230	−0.561990	−0.280995	+	0.959709i	$0.590664\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−12.9282	−0.859971
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	0	0
$229$	−13.4641	−0.889733	−0.444866	−	0.895597i	$-0.646749\pi$
$229$	−13.4641	−0.889733	−0.444866	+	0.895597i	$0.646749\pi$
$230$	1.26795	0.0836061
$231$	0	0
$232$	4.73205	0.310674
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	0	0
$235$	0	0
$236$	13.8564	0.901975
$237$	0	0
$238$	−3.46410	−0.224544
$239$	3.80385	0.246050	0.123025	−	0.992404i	$-0.460740\pi$
$239$	3.80385	0.246050	0.123025	+	0.992404i	$0.460740\pi$
$240$	0	0
$241$	−18.1962	−1.17212	−0.586059	−	0.810269i	$-0.699321\pi$
$241$	−18.1962	−1.17212	−0.586059	+	0.810269i	$0.699321\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	4.00000	0.254514
$248$	8.92820	0.566941
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	1.26795	0.0797153
$254$	9.85641	0.618446
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.9808	−1.43350	−0.716750	−	0.697330i	$-0.754371\pi$
$257$	−22.9808	−1.43350	−0.716750	+	0.697330i	$0.754371\pi$
$258$	0	0
$259$	3.26795	0.203060
$260$	1.46410	0.0907997
$261$	0	0
$262$	17.6603	1.09105
$263$	−18.9282	−1.16716	−0.583582	−	0.812055i	$-0.698349\pi$
$263$	−18.9282	−1.16716	−0.583582	+	0.812055i	$0.698349\pi$
$264$	0	0
$265$	−4.73205	−0.290688
$266$	−2.73205	−0.167513
$267$	0	0
$268$	−10.9282	−0.667546
$269$	−4.39230	−0.267804	−0.133902	−	0.990995i	$-0.542751\pi$
$269$	−4.39230	−0.267804	−0.133902	+	0.990995i	$0.542751\pi$
$270$	0	0
$271$	24.3923	1.48173	0.740863	−	0.671656i	$-0.234417\pi$
$271$	24.3923	1.48173	0.740863	+	0.671656i	$0.234417\pi$
$272$	−3.46410	−0.210042
$273$	0	0
$274$	−7.85641	−0.474623
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	27.8564	1.67373	0.836865	−	0.547410i	$-0.184386\pi$
$277$	27.8564	1.67373	0.836865	+	0.547410i	$0.184386\pi$
$278$	4.19615	0.251668
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	10.3923	0.619953	0.309976	−	0.950744i	$-0.399679\pi$
$281$	10.3923	0.619953	0.309976	+	0.950744i	$0.399679\pi$
$282$	0	0
$283$	−10.9282	−0.649614	−0.324807	−	0.945780i	$-0.605299\pi$
$283$	−10.9282	−0.649614	−0.324807	+	0.945780i	$0.605299\pi$
$284$	−9.46410	−0.561591
$285$	0	0
$286$	1.46410	0.0865741
$287$	4.73205	0.279324
$288$	0	0
$289$	−5.00000	−0.294118
$290$	−4.73205	−0.277876
$291$	0	0
$292$	11.4641	0.670886
$293$	2.53590	0.148149	0.0740744	−	0.997253i	$-0.476400\pi$
$293$	2.53590	0.148149	0.0740744	+	0.997253i	$0.476400\pi$
$294$	0	0
$295$	−13.8564	−0.806751
$296$	3.26795	0.189946
$297$	0	0
$298$	7.26795	0.421021
$299$	1.85641	0.107359
$300$	0	0
$301$	−4.92820	−0.284057
$302$	−7.80385	−0.449061
$303$	0	0
$304$	−2.73205	−0.156694
$305$	−2.00000	−0.114520
$306$	0	0
$307$	31.3205	1.78756	0.893778	−	0.448510i	$-0.148045\pi$
$307$	31.3205	1.78756	0.893778	+	0.448510i	$0.148045\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−8.92820	−0.507088
$311$	−7.85641	−0.445496	−0.222748	−	0.974876i	$-0.571503\pi$
$311$	−7.85641	−0.445496	−0.222748	+	0.974876i	$0.571503\pi$
$312$	0	0
$313$	27.2679	1.54128	0.770638	−	0.637273i	$-0.219938\pi$
$313$	27.2679	1.54128	0.770638	+	0.637273i	$0.219938\pi$
$314$	6.39230	0.360739
$315$	0	0
$316$	6.73205	0.378707
$317$	32.4449	1.82229	0.911143	−	0.412091i	$-0.135202\pi$
$317$	32.4449	1.82229	0.911143	+	0.412091i	$0.135202\pi$
$318$	0	0
$319$	−4.73205	−0.264944
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−1.26795	−0.0706600
$323$	9.46410	0.526597
$324$	0	0
$325$	−1.46410	−0.0812137
$326$	8.00000	0.443079
$327$	0	0
$328$	4.73205	0.261284
$329$	0	0
$330$	0	0
$331$	−4.92820	−0.270879	−0.135439	−	0.990786i	$-0.543245\pi$
$331$	−4.92820	−0.270879	−0.135439	+	0.990786i	$0.543245\pi$
$332$	4.39230	0.241059
$333$	0	0
$334$	13.8564	0.758189
$335$	10.9282	0.597072
$336$	0	0
$337$	−30.7846	−1.67694	−0.838472	−	0.544944i	$-0.816551\pi$
$337$	−30.7846	−1.67694	−0.838472	+	0.544944i	$0.816551\pi$
$338$	−10.8564	−0.590511
$339$	0	0
$340$	3.46410	0.187867
$341$	−8.92820	−0.483489
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.92820	−0.265711
$345$	0	0
$346$	−0.928203	−0.0499005
$347$	−7.85641	−0.421754	−0.210877	−	0.977513i	$-0.567632\pi$
$347$	−7.85641	−0.421754	−0.210877	+	0.977513i	$0.567632\pi$
$348$	0	0
$349$	30.3923	1.62686	0.813431	−	0.581661i	$-0.197597\pi$
$349$	30.3923	1.62686	0.813431	+	0.581661i	$0.197597\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	32.4449	1.72687	0.863433	−	0.504464i	$-0.168310\pi$
$353$	32.4449	1.72687	0.863433	+	0.504464i	$0.168310\pi$
$354$	0	0
$355$	9.46410	0.502302
$356$	15.4641	0.819596
$357$	0	0
$358$	19.8564	1.04944
$359$	1.26795	0.0669198	0.0334599	−	0.999440i	$-0.489347\pi$
$359$	1.26795	0.0669198	0.0334599	+	0.999440i	$0.489347\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	−11.8564	−0.623159
$363$	0	0
$364$	−1.46410	−0.0767398
$365$	−11.4641	−0.600059
$366$	0	0
$367$	29.4641	1.53801	0.769007	−	0.639241i	$-0.220751\pi$
$367$	29.4641	1.53801	0.769007	+	0.639241i	$0.220751\pi$
$368$	−1.26795	−0.0660964
$369$	0	0
$370$	−3.26795	−0.169893
$371$	4.73205	0.245676
$372$	0	0
$373$	30.3923	1.57365	0.786827	−	0.617174i	$-0.211722\pi$
$373$	30.3923	1.57365	0.786827	+	0.617174i	$0.211722\pi$
$374$	3.46410	0.179124
$375$	0	0
$376$	0	0
$377$	−6.92820	−0.356821
$378$	0	0
$379$	23.7128	1.21805	0.609023	−	0.793153i	$-0.291562\pi$
$379$	23.7128	1.21805	0.609023	+	0.793153i	$0.291562\pi$
$380$	2.73205	0.140151
$381$	0	0
$382$	−6.92820	−0.354478
$383$	−16.3923	−0.837608	−0.418804	−	0.908077i	$-0.637551\pi$
$383$	−16.3923	−0.837608	−0.418804	+	0.908077i	$0.637551\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	−25.4641	−1.29609
$387$	0	0
$388$	5.80385	0.294646
$389$	24.2487	1.22946	0.614729	−	0.788738i	$-0.289265\pi$
$389$	24.2487	1.22946	0.614729	+	0.788738i	$0.289265\pi$
$390$	0	0
$391$	4.39230	0.222128
$392$	1.00000	0.0505076
$393$	0	0
$394$	22.3923	1.12811
$395$	−6.73205	−0.338726
$396$	0	0
$397$	−38.3923	−1.92685	−0.963427	−	0.267970i	$-0.913647\pi$
$397$	−38.3923	−1.92685	−0.963427	+	0.267970i	$0.913647\pi$
$398$	−24.7846	−1.24234
$399$	0	0
$400$	1.00000	0.0500000
$401$	25.1769	1.25728	0.628638	−	0.777698i	$-0.283613\pi$
$401$	25.1769	1.25728	0.628638	+	0.777698i	$0.283613\pi$
$402$	0	0
$403$	−13.0718	−0.651153
$404$	6.00000	0.298511
$405$	0	0
$406$	4.73205	0.234848
$407$	−3.26795	−0.161986
$408$	0	0
$409$	22.1962	1.09753	0.548765	−	0.835977i	$-0.315098\pi$
$409$	22.1962	1.09753	0.548765	+	0.835977i	$0.315098\pi$
$410$	−4.73205	−0.233699
$411$	0	0
$412$	10.5359	0.519066
$413$	13.8564	0.681829
$414$	0	0
$415$	−4.39230	−0.215610
$416$	−1.46410	−0.0717835
$417$	0	0
$418$	2.73205	0.133629
$419$	−32.7846	−1.60163	−0.800816	−	0.598910i	$-0.795601\pi$
$419$	−32.7846	−1.60163	−0.800816	+	0.598910i	$0.795601\pi$
$420$	0	0
$421$	10.7846	0.525610	0.262805	−	0.964849i	$-0.415352\pi$
$421$	10.7846	0.525610	0.262805	+	0.964849i	$0.415352\pi$
$422$	8.00000	0.389434
$423$	0	0
$424$	4.73205	0.229809
$425$	−3.46410	−0.168034
$426$	0	0
$427$	2.00000	0.0967868
$428$	−0.928203	−0.0448664
$429$	0	0
$430$	4.92820	0.237659
$431$	−18.3397	−0.883394	−0.441697	−	0.897164i	$-0.645623\pi$
$431$	−18.3397	−0.883394	−0.441697	+	0.897164i	$0.645623\pi$
$432$	0	0
$433$	0.732051	0.0351801	0.0175901	−	0.999845i	$-0.494401\pi$
$433$	0.732051	0.0351801	0.0175901	+	0.999845i	$0.494401\pi$
$434$	8.92820	0.428567
$435$	0	0
$436$	−1.80385	−0.0863886
$437$	3.46410	0.165710
$438$	0	0
$439$	36.3923	1.73691	0.868455	−	0.495768i	$-0.165113\pi$
$439$	36.3923	1.73691	0.868455	+	0.495768i	$0.165113\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	5.07180	0.241241
$443$	−20.7846	−0.987507	−0.493753	−	0.869602i	$-0.664375\pi$
$443$	−20.7846	−0.987507	−0.493753	+	0.869602i	$0.664375\pi$
$444$	0	0
$445$	−15.4641	−0.733069
$446$	−8.39230	−0.397387
$447$	0	0
$448$	1.00000	0.0472456
$449$	35.3205	1.66688	0.833439	−	0.552612i	$-0.186369\pi$
$449$	35.3205	1.66688	0.833439	+	0.552612i	$0.186369\pi$
$450$	0	0
$451$	−4.73205	−0.222824
$452$	−12.9282	−0.608092
$453$	0	0
$454$	−13.8564	−0.650313
$455$	1.46410	0.0686381
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	−13.4641	−0.629136
$459$	0	0
$460$	1.26795	0.0591184
$461$	−29.3205	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$461$	−29.3205	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$462$	0	0
$463$	12.9808	0.603267	0.301634	−	0.953424i	$-0.402468\pi$
$463$	12.9808	0.603267	0.301634	+	0.953424i	$0.402468\pi$
$464$	4.73205	0.219680
$465$	0	0
$466$	−7.85641	−0.363941
$467$	−17.6603	−0.817219	−0.408610	−	0.912709i	$-0.633986\pi$
$467$	−17.6603	−0.817219	−0.408610	+	0.912709i	$0.633986\pi$
$468$	0	0
$469$	−10.9282	−0.504618
$470$	0	0
$471$	0	0
$472$	13.8564	0.637793
$473$	4.92820	0.226599
$474$	0	0
$475$	−2.73205	−0.125355
$476$	−3.46410	−0.158777
$477$	0	0
$478$	3.80385	0.173984
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	0	0
$481$	−4.78461	−0.218159
$482$	−18.1962	−0.828812
$483$	0	0
$484$	1.00000	0.0454545
$485$	−5.80385	−0.263539
$486$	0	0
$487$	4.87564	0.220937	0.110468	−	0.993880i	$-0.464765\pi$
$487$	4.87564	0.220937	0.110468	+	0.993880i	$0.464765\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−42.9282	−1.93732	−0.968661	−	0.248385i	$-0.920100\pi$
$491$	−42.9282	−1.93732	−0.968661	+	0.248385i	$0.920100\pi$
$492$	0	0
$493$	−16.3923	−0.738272
$494$	4.00000	0.179969
$495$	0	0
$496$	8.92820	0.400888
$497$	−9.46410	−0.424523
$498$	0	0
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	0	0
$503$	18.9282	0.843967	0.421983	−	0.906604i	$-0.361334\pi$
$503$	18.9282	0.843967	0.421983	+	0.906604i	$0.361334\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	1.26795	0.0563672
$507$	0	0
$508$	9.85641	0.437307
$509$	−2.78461	−0.123426	−0.0617128	−	0.998094i	$-0.519656\pi$
$509$	−2.78461	−0.123426	−0.0617128	+	0.998094i	$0.519656\pi$
$510$	0	0
$511$	11.4641	0.507142
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.9808	−1.01364
$515$	−10.5359	−0.464267
$516$	0	0
$517$	0	0
$518$	3.26795	0.143585
$519$	0	0
$520$	1.46410	0.0642051
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	0	0
$523$	19.3205	0.844827	0.422413	−	0.906403i	$-0.361183\pi$
$523$	19.3205	0.844827	0.422413	+	0.906403i	$0.361183\pi$
$524$	17.6603	0.771492
$525$	0	0
$526$	−18.9282	−0.825309
$527$	−30.9282	−1.34725
$528$	0	0
$529$	−21.3923	−0.930100
$530$	−4.73205	−0.205547
$531$	0	0
$532$	−2.73205	−0.118449
$533$	−6.92820	−0.300094
$534$	0	0
$535$	0.928203	0.0401297
$536$	−10.9282	−0.472026
$537$	0	0
$538$	−4.39230	−0.189366
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	17.1244	0.736234	0.368117	−	0.929780i	$-0.380003\pi$
$541$	17.1244	0.736234	0.368117	+	0.929780i	$0.380003\pi$
$542$	24.3923	1.04774
$543$	0	0
$544$	−3.46410	−0.148522
$545$	1.80385	0.0772683
$546$	0	0
$547$	4.78461	0.204575	0.102288	−	0.994755i	$-0.467384\pi$
$547$	4.78461	0.204575	0.102288	+	0.994755i	$0.467384\pi$
$548$	−7.85641	−0.335609
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	−12.9282	−0.550760
$552$	0	0
$553$	6.73205	0.286276
$554$	27.8564	1.18351
$555$	0	0
$556$	4.19615	0.177957
$557$	12.2487	0.518995	0.259497	−	0.965744i	$-0.416443\pi$
$557$	12.2487	0.518995	0.259497	+	0.965744i	$0.416443\pi$
$558$	0	0
$559$	7.21539	0.305178
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	10.3923	0.438373
$563$	20.7846	0.875967	0.437983	−	0.898983i	$-0.355693\pi$
$563$	20.7846	0.875967	0.437983	+	0.898983i	$0.355693\pi$
$564$	0	0
$565$	12.9282	0.543894
$566$	−10.9282	−0.459347
$567$	0	0
$568$	−9.46410	−0.397105
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−20.3923	−0.853391	−0.426696	−	0.904395i	$-0.640322\pi$
$571$	−20.3923	−0.853391	−0.426696	+	0.904395i	$0.640322\pi$
$572$	1.46410	0.0612172
$573$	0	0
$574$	4.73205	0.197512
$575$	−1.26795	−0.0528771
$576$	0	0
$577$	8.33975	0.347188	0.173594	−	0.984817i	$-0.444462\pi$
$577$	8.33975	0.347188	0.173594	+	0.984817i	$0.444462\pi$
$578$	−5.00000	−0.207973
$579$	0	0
$580$	−4.73205	−0.196488
$581$	4.39230	0.182224
$582$	0	0
$583$	−4.73205	−0.195982
$584$	11.4641	0.474388
$585$	0	0
$586$	2.53590	0.104757
$587$	28.9808	1.19616	0.598082	−	0.801435i	$-0.295930\pi$
$587$	28.9808	1.19616	0.598082	+	0.801435i	$0.295930\pi$
$588$	0	0
$589$	−24.3923	−1.00507
$590$	−13.8564	−0.570459
$591$	0	0
$592$	3.26795	0.134312
$593$	−32.5359	−1.33609	−0.668045	−	0.744121i	$-0.732868\pi$
$593$	−32.5359	−1.33609	−0.668045	+	0.744121i	$0.732868\pi$
$594$	0	0
$595$	3.46410	0.142014
$596$	7.26795	0.297707
$597$	0	0
$598$	1.85641	0.0759141
$599$	32.1051	1.31178	0.655890	−	0.754857i	$-0.272294\pi$
$599$	32.1051	1.31178	0.655890	+	0.754857i	$0.272294\pi$
$600$	0	0
$601$	28.4449	1.16029	0.580145	−	0.814513i	$-0.302996\pi$
$601$	28.4449	1.16029	0.580145	+	0.814513i	$0.302996\pi$
$602$	−4.92820	−0.200859
$603$	0	0
$604$	−7.80385	−0.317534
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	34.7846	1.41186	0.705932	−	0.708280i	$-0.250528\pi$
$607$	34.7846	1.41186	0.705932	+	0.708280i	$0.250528\pi$
$608$	−2.73205	−0.110799
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	0	0
$613$	27.1769	1.09767	0.548833	−	0.835932i	$-0.315072\pi$
$613$	27.1769	1.09767	0.548833	+	0.835932i	$0.315072\pi$
$614$	31.3205	1.26399
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−8.53590	−0.343642	−0.171821	−	0.985128i	$-0.554965\pi$
$617$	−8.53590	−0.343642	−0.171821	+	0.985128i	$0.554965\pi$
$618$	0	0
$619$	−10.9282	−0.439242	−0.219621	−	0.975585i	$-0.570482\pi$
$619$	−10.9282	−0.439242	−0.219621	+	0.975585i	$0.570482\pi$
$620$	−8.92820	−0.358565
$621$	0	0
$622$	−7.85641	−0.315013
$623$	15.4641	0.619556
$624$	0	0
$625$	1.00000	0.0400000
$626$	27.2679	1.08985
$627$	0	0
$628$	6.39230	0.255081
$629$	−11.3205	−0.451378
$630$	0	0
$631$	−12.7846	−0.508947	−0.254474	−	0.967080i	$-0.581902\pi$
$631$	−12.7846	−0.508947	−0.254474	+	0.967080i	$0.581902\pi$
$632$	6.73205	0.267787
$633$	0	0
$634$	32.4449	1.28855
$635$	−9.85641	−0.391140
$636$	0	0
$637$	−1.46410	−0.0580098
$638$	−4.73205	−0.187344
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−19.8564	−0.784281	−0.392140	−	0.919905i	$-0.628265\pi$
$641$	−19.8564	−0.784281	−0.392140	+	0.919905i	$0.628265\pi$
$642$	0	0
$643$	−14.7321	−0.580975	−0.290488	−	0.956879i	$-0.593818\pi$
$643$	−14.7321	−0.580975	−0.290488	+	0.956879i	$0.593818\pi$
$644$	−1.26795	−0.0499642
$645$	0	0
$646$	9.46410	0.372360
$647$	−37.1769	−1.46158	−0.730788	−	0.682605i	$-0.760847\pi$
$647$	−37.1769	−1.46158	−0.730788	+	0.682605i	$0.760847\pi$
$648$	0	0
$649$	−13.8564	−0.543912
$650$	−1.46410	−0.0574268
$651$	0	0
$652$	8.00000	0.313304
$653$	9.80385	0.383654	0.191827	−	0.981429i	$-0.438559\pi$
$653$	9.80385	0.383654	0.191827	+	0.981429i	$0.438559\pi$
$654$	0	0
$655$	−17.6603	−0.690043
$656$	4.73205	0.184756
$657$	0	0
$658$	0	0
$659$	−6.24871	−0.243415	−0.121708	−	0.992566i	$-0.538837\pi$
$659$	−6.24871	−0.243415	−0.121708	+	0.992566i	$0.538837\pi$
$660$	0	0
$661$	3.85641	0.149997	0.0749984	−	0.997184i	$-0.476105\pi$
$661$	3.85641	0.149997	0.0749984	+	0.997184i	$0.476105\pi$
$662$	−4.92820	−0.191540
$663$	0	0
$664$	4.39230	0.170454
$665$	2.73205	0.105944
$666$	0	0
$667$	−6.00000	−0.232321
$668$	13.8564	0.536120
$669$	0	0
$670$	10.9282	0.422193
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−20.1436	−0.776478	−0.388239	−	0.921559i	$-0.626917\pi$
$673$	−20.1436	−0.776478	−0.388239	+	0.921559i	$0.626917\pi$
$674$	−30.7846	−1.18578
$675$	0	0
$676$	−10.8564	−0.417554
$677$	−19.8564	−0.763144	−0.381572	−	0.924339i	$-0.624617\pi$
$677$	−19.8564	−0.763144	−0.381572	+	0.924339i	$0.624617\pi$
$678$	0	0
$679$	5.80385	0.222731
$680$	3.46410	0.132842
$681$	0	0
$682$	−8.92820	−0.341879
$683$	−42.2487	−1.61660	−0.808301	−	0.588769i	$-0.799613\pi$
$683$	−42.2487	−1.61660	−0.808301	+	0.588769i	$0.799613\pi$
$684$	0	0
$685$	7.85641	0.300178
$686$	1.00000	0.0381802
$687$	0	0
$688$	−4.92820	−0.187886
$689$	−6.92820	−0.263944
$690$	0	0
$691$	31.3205	1.19149	0.595744	−	0.803174i	$-0.296857\pi$
$691$	31.3205	1.19149	0.595744	+	0.803174i	$0.296857\pi$
$692$	−0.928203	−0.0352850
$693$	0	0
$694$	−7.85641	−0.298225
$695$	−4.19615	−0.159169
$696$	0	0
$697$	−16.3923	−0.620903
$698$	30.3923	1.15037
$699$	0	0
$700$	1.00000	0.0377964
$701$	−14.1962	−0.536181	−0.268091	−	0.963394i	$-0.586393\pi$
$701$	−14.1962	−0.536181	−0.268091	+	0.963394i	$0.586393\pi$
$702$	0	0
$703$	−8.92820	−0.336734
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	32.4449	1.22108
$707$	6.00000	0.225653
$708$	0	0
$709$	−2.39230	−0.0898449	−0.0449224	−	0.998990i	$-0.514304\pi$
$709$	−2.39230	−0.0898449	−0.0449224	+	0.998990i	$0.514304\pi$
$710$	9.46410	0.355181
$711$	0	0
$712$	15.4641	0.579542
$713$	−11.3205	−0.423956
$714$	0	0
$715$	−1.46410	−0.0547543
$716$	19.8564	0.742069
$717$	0	0
$718$	1.26795	0.0473194
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	10.5359	0.392377
$722$	−11.5359	−0.429322
$723$	0	0
$724$	−11.8564	−0.440640
$725$	4.73205	0.175744
$726$	0	0
$727$	−38.6410	−1.43312	−0.716558	−	0.697528i	$-0.754283\pi$
$727$	−38.6410	−1.43312	−0.716558	+	0.697528i	$0.754283\pi$
$728$	−1.46410	−0.0542632
$729$	0	0
$730$	−11.4641	−0.424305
$731$	17.0718	0.631423
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	29.4641	1.08754
$735$	0	0
$736$	−1.26795	−0.0467372
$737$	10.9282	0.402546
$738$	0	0
$739$	18.1436	0.667423	0.333711	−	0.942675i	$-0.391699\pi$
$739$	18.1436	0.667423	0.333711	+	0.942675i	$0.391699\pi$
$740$	−3.26795	−0.120132
$741$	0	0
$742$	4.73205	0.173719
$743$	−3.71281	−0.136210	−0.0681049	−	0.997678i	$-0.521695\pi$
$743$	−3.71281	−0.136210	−0.0681049	+	0.997678i	$0.521695\pi$
$744$	0	0
$745$	−7.26795	−0.266277
$746$	30.3923	1.11274
$747$	0	0
$748$	3.46410	0.126660
$749$	−0.928203	−0.0339158
$750$	0	0
$751$	2.24871	0.0820566	0.0410283	−	0.999158i	$-0.486937\pi$
$751$	2.24871	0.0820566	0.0410283	+	0.999158i	$0.486937\pi$
$752$	0	0
$753$	0	0
$754$	−6.92820	−0.252310
$755$	7.80385	0.284011
$756$	0	0
$757$	23.3731	0.849509	0.424754	−	0.905309i	$-0.360360\pi$
$757$	23.3731	0.849509	0.424754	+	0.905309i	$0.360360\pi$
$758$	23.7128	0.861288
$759$	0	0
$760$	2.73205	0.0991019
$761$	−19.2679	−0.698463	−0.349231	−	0.937037i	$-0.613557\pi$
$761$	−19.2679	−0.698463	−0.349231	+	0.937037i	$0.613557\pi$
$762$	0	0
$763$	−1.80385	−0.0653037
$764$	−6.92820	−0.250654
$765$	0	0
$766$	−16.3923	−0.592278
$767$	−20.2872	−0.732528
$768$	0	0
$769$	−24.4449	−0.881504	−0.440752	−	0.897629i	$-0.645288\pi$
$769$	−24.4449	−0.881504	−0.440752	+	0.897629i	$0.645288\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−25.4641	−0.916473
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	0	0
$775$	8.92820	0.320711
$776$	5.80385	0.208346
$777$	0	0
$778$	24.2487	0.869358
$779$	−12.9282	−0.463201
$780$	0	0
$781$	9.46410	0.338652
$782$	4.39230	0.157069
$783$	0	0
$784$	1.00000	0.0357143
$785$	−6.39230	−0.228151
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	22.3923	0.797693
$789$	0	0
$790$	−6.73205	−0.239516
$791$	−12.9282	−0.459674
$792$	0	0
$793$	−2.92820	−0.103984
$794$	−38.3923	−1.36249
$795$	0	0
$796$	−24.7846	−0.878467
$797$	−6.67949	−0.236600	−0.118300	−	0.992978i	$-0.537744\pi$
$797$	−6.67949	−0.236600	−0.118300	+	0.992978i	$0.537744\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	25.1769	0.889028
$803$	−11.4641	−0.404559
$804$	0	0
$805$	1.26795	0.0446893
$806$	−13.0718	−0.460434
$807$	0	0
$808$	6.00000	0.211079
$809$	−5.32051	−0.187059	−0.0935296	−	0.995617i	$-0.529815\pi$
$809$	−5.32051	−0.187059	−0.0935296	+	0.995617i	$0.529815\pi$
$810$	0	0
$811$	−4.58846	−0.161123	−0.0805613	−	0.996750i	$-0.525671\pi$
$811$	−4.58846	−0.161123	−0.0805613	+	0.996750i	$0.525671\pi$
$812$	4.73205	0.166062
$813$	0	0
$814$	−3.26795	−0.114542
$815$	−8.00000	−0.280228
$816$	0	0
$817$	13.4641	0.471049
$818$	22.1962	0.776070
$819$	0	0
$820$	−4.73205	−0.165250
$821$	−52.0526	−1.81665	−0.908323	−	0.418269i	$-0.862637\pi$
$821$	−52.0526	−1.81665	−0.908323	+	0.418269i	$0.862637\pi$
$822$	0	0
$823$	−24.1962	−0.843425	−0.421712	−	0.906730i	$-0.638571\pi$
$823$	−24.1962	−0.843425	−0.421712	+	0.906730i	$0.638571\pi$
$824$	10.5359	0.367035
$825$	0	0
$826$	13.8564	0.482126
$827$	−53.5692	−1.86278	−0.931392	−	0.364017i	$-0.881405\pi$
$827$	−53.5692	−1.86278	−0.931392	+	0.364017i	$0.881405\pi$
$828$	0	0
$829$	40.1051	1.39291	0.696454	−	0.717601i	$-0.254760\pi$
$829$	40.1051	1.39291	0.696454	+	0.717601i	$0.254760\pi$
$830$	−4.39230	−0.152459
$831$	0	0
$832$	−1.46410	−0.0507586
$833$	−3.46410	−0.120024
$834$	0	0
$835$	−13.8564	−0.479521
$836$	2.73205	0.0944900
$837$	0	0
$838$	−32.7846	−1.13253
$839$	38.7846	1.33899	0.669497	−	0.742815i	$-0.266510\pi$
$839$	38.7846	1.33899	0.669497	+	0.742815i	$0.266510\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	10.7846	0.371662
$843$	0	0
$844$	8.00000	0.275371
$845$	10.8564	0.373472
$846$	0	0
$847$	1.00000	0.0343604
$848$	4.73205	0.162499
$849$	0	0
$850$	−3.46410	−0.118818
$851$	−4.14359	−0.142041
$852$	0	0
$853$	20.9282	0.716568	0.358284	−	0.933613i	$-0.383362\pi$
$853$	20.9282	0.716568	0.358284	+	0.933613i	$0.383362\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−0.928203	−0.0317253
$857$	19.1769	0.655071	0.327535	−	0.944839i	$-0.393782\pi$
$857$	19.1769	0.655071	0.327535	+	0.944839i	$0.393782\pi$
$858$	0	0
$859$	40.7846	1.39155	0.695776	−	0.718258i	$-0.255060\pi$
$859$	40.7846	1.39155	0.695776	+	0.718258i	$0.255060\pi$
$860$	4.92820	0.168050
$861$	0	0
$862$	−18.3397	−0.624654
$863$	−24.5885	−0.837001	−0.418500	−	0.908217i	$-0.637444\pi$
$863$	−24.5885	−0.837001	−0.418500	+	0.908217i	$0.637444\pi$
$864$	0	0
$865$	0.928203	0.0315599
$866$	0.732051	0.0248761
$867$	0	0
$868$	8.92820	0.303043
$869$	−6.73205	−0.228369
$870$	0	0
$871$	16.0000	0.542139
$872$	−1.80385	−0.0610860
$873$	0	0
$874$	3.46410	0.117175
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−23.1769	−0.782629	−0.391314	−	0.920257i	$-0.627979\pi$
$877$	−23.1769	−0.782629	−0.391314	+	0.920257i	$0.627979\pi$
$878$	36.3923	1.22818
$879$	0	0
$880$	1.00000	0.0337100
$881$	−24.9282	−0.839853	−0.419926	−	0.907558i	$-0.637944\pi$
$881$	−24.9282	−0.839853	−0.419926	+	0.907558i	$0.637944\pi$
$882$	0	0
$883$	−41.1769	−1.38571	−0.692857	−	0.721075i	$-0.743648\pi$
$883$	−41.1769	−1.38571	−0.692857	+	0.721075i	$0.743648\pi$
$884$	5.07180	0.170583
$885$	0	0
$886$	−20.7846	−0.698273
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	9.85641	0.330573
$890$	−15.4641	−0.518358
$891$	0	0
$892$	−8.39230	−0.280995
$893$	0	0
$894$	0	0
$895$	−19.8564	−0.663726
$896$	1.00000	0.0334077
$897$	0	0
$898$	35.3205	1.17866
$899$	42.2487	1.40907
$900$	0	0
$901$	−16.3923	−0.546107
$902$	−4.73205	−0.157560
$903$	0	0
$904$	−12.9282	−0.429986
$905$	11.8564	0.394120
$906$	0	0
$907$	−15.3205	−0.508709	−0.254355	−	0.967111i	$-0.581863\pi$
$907$	−15.3205	−0.508709	−0.254355	+	0.967111i	$0.581863\pi$
$908$	−13.8564	−0.459841
$909$	0	0
$910$	1.46410	0.0485345
$911$	−14.5359	−0.481596	−0.240798	−	0.970575i	$-0.577409\pi$
$911$	−14.5359	−0.481596	−0.240798	+	0.970575i	$0.577409\pi$
$912$	0	0
$913$	−4.39230	−0.145364
$914$	−34.0000	−1.12462
$915$	0	0
$916$	−13.4641	−0.444866
$917$	17.6603	0.583193
$918$	0	0
$919$	24.9808	0.824039	0.412020	−	0.911175i	$-0.364824\pi$
$919$	24.9808	0.824039	0.412020	+	0.911175i	$0.364824\pi$
$920$	1.26795	0.0418030
$921$	0	0
$922$	−29.3205	−0.965620
$923$	13.8564	0.456089
$924$	0	0
$925$	3.26795	0.107450
$926$	12.9808	0.426574
$927$	0	0
$928$	4.73205	0.155337
$929$	−26.1051	−0.856481	−0.428241	−	0.903665i	$-0.640866\pi$
$929$	−26.1051	−0.856481	−0.428241	+	0.903665i	$0.640866\pi$
$930$	0	0
$931$	−2.73205	−0.0895393
$932$	−7.85641	−0.257345
$933$	0	0
$934$	−17.6603	−0.577861
$935$	−3.46410	−0.113288
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	−10.9282	−0.356818
$939$	0	0
$940$	0	0
$941$	35.5692	1.15952	0.579762	−	0.814786i	$-0.303146\pi$
$941$	35.5692	1.15952	0.579762	+	0.814786i	$0.303146\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	13.8564	0.450988
$945$	0	0
$946$	4.92820	0.160230
$947$	25.1769	0.818140	0.409070	−	0.912503i	$-0.365853\pi$
$947$	25.1769	0.818140	0.409070	+	0.912503i	$0.365853\pi$
$948$	0	0
$949$	−16.7846	−0.544851
$950$	−2.73205	−0.0886394
$951$	0	0
$952$	−3.46410	−0.112272
$953$	11.3205	0.366707	0.183354	−	0.983047i	$-0.441305\pi$
$953$	11.3205	0.366707	0.183354	+	0.983047i	$0.441305\pi$
$954$	0	0
$955$	6.92820	0.224191
$956$	3.80385	0.123025
$957$	0	0
$958$	−13.8564	−0.447680
$959$	−7.85641	−0.253697
$960$	0	0
$961$	48.7128	1.57138
$962$	−4.78461	−0.154262
$963$	0	0
$964$	−18.1962	−0.586059
$965$	25.4641	0.819718
$966$	0	0
$967$	−26.1436	−0.840721	−0.420361	−	0.907357i	$-0.638096\pi$
$967$	−26.1436	−0.840721	−0.420361	+	0.907357i	$0.638096\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−5.80385	−0.186350
$971$	−49.8564	−1.59997	−0.799984	−	0.600021i	$-0.795159\pi$
$971$	−49.8564	−1.59997	−0.799984	+	0.600021i	$0.795159\pi$
$972$	0	0
$973$	4.19615	0.134522
$974$	4.87564	0.156226
$975$	0	0
$976$	2.00000	0.0640184
$977$	0.928203	0.0296959	0.0148479	−	0.999890i	$-0.495274\pi$
$977$	0.928203	0.0296959	0.0148479	+	0.999890i	$0.495274\pi$
$978$	0	0
$979$	−15.4641	−0.494235
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−42.9282	−1.36989
$983$	7.60770	0.242648	0.121324	−	0.992613i	$-0.461286\pi$
$983$	7.60770	0.242648	0.121324	+	0.992613i	$0.461286\pi$
$984$	0	0
$985$	−22.3923	−0.713478
$986$	−16.3923	−0.522037
$987$	0	0
$988$	4.00000	0.127257
$989$	6.24871	0.198697
$990$	0	0
$991$	−22.9282	−0.728338	−0.364169	−	0.931333i	$-0.618647\pi$
$991$	−22.9282	−0.728338	−0.364169	+	0.931333i	$0.618647\pi$
$992$	8.92820	0.283471
$993$	0	0
$994$	−9.46410	−0.300183
$995$	24.7846	0.785725
$996$	0	0
$997$	−23.8564	−0.755540	−0.377770	−	0.925899i	$-0.623309\pi$
$997$	−23.8564	−0.755540	−0.377770	+	0.925899i	$0.623309\pi$
$998$	2.00000	0.0633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.ca.1.1		2
3.2	odd	2		770.2.a.h.1.2	✓	2
12.11	even	2		6160.2.a.v.1.1		2
15.2	even	4		3850.2.c.s.1849.1		4
15.8	even	4		3850.2.c.s.1849.4		4
15.14	odd	2		3850.2.a.bm.1.1		2
21.20	even	2		5390.2.a.bk.1.1		2
33.32	even	2		8470.2.a.ce.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.h.1.2	✓	2	3.2	odd	2
3850.2.a.bm.1.1		2	15.14	odd	2
3850.2.c.s.1849.1		4	15.2	even	4
3850.2.c.s.1849.4		4	15.8	even	4
5390.2.a.bk.1.1		2	21.20	even	2
6160.2.a.v.1.1		2	12.11	even	2
6930.2.a.ca.1.1		2	1.1	even	1	trivial
8470.2.a.ce.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} -1.46410 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.46410 q^{17} -2.73205 q^{19} -1.00000 q^{20} -1.00000 q^{22} -1.26795 q^{23} +1.00000 q^{25} -1.46410 q^{26} +1.00000 q^{28} +4.73205 q^{29} +8.92820 q^{31} +1.00000 q^{32} -3.46410 q^{34} -1.00000 q^{35} +3.26795 q^{37} -2.73205 q^{38} -1.00000 q^{40} +4.73205 q^{41} -4.92820 q^{43} -1.00000 q^{44} -1.26795 q^{46} +1.00000 q^{49} +1.00000 q^{50} -1.46410 q^{52} +4.73205 q^{53} +1.00000 q^{55} +1.00000 q^{56} +4.73205 q^{58} +13.8564 q^{59} +2.00000 q^{61} +8.92820 q^{62} +1.00000 q^{64} +1.46410 q^{65} -10.9282 q^{67} -3.46410 q^{68} -1.00000 q^{70} -9.46410 q^{71} +11.4641 q^{73} +3.26795 q^{74} -2.73205 q^{76} -1.00000 q^{77} +6.73205 q^{79} -1.00000 q^{80} +4.73205 q^{82} +4.39230 q^{83} +3.46410 q^{85} -4.92820 q^{86} -1.00000 q^{88} +15.4641 q^{89} -1.46410 q^{91} -1.26795 q^{92} +2.73205 q^{95} +5.80385 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 6 q^{29} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 4 q^{43} - 2 q^{44} - 6 q^{46} + 2 q^{49} + 2 q^{50} + 4 q^{52} + 6 q^{53} + 2 q^{55} + 2 q^{56} + 6 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} - 12 q^{71} + 16 q^{73} + 10 q^{74} - 2 q^{76} - 2 q^{77} + 10 q^{79} - 2 q^{80} + 6 q^{82} - 12 q^{83} + 4 q^{86} - 2 q^{88} + 24 q^{89} + 4 q^{91} - 6 q^{92} + 2 q^{95} + 22 q^{97} + 2 q^{98}+O(q^{100}) \) 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 - 2 * q^19 - 2 * q^20 - 2 * q^22 - 6 * q^23 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 6 * q^29 + 4 * q^31 + 2 * q^32 - 2 * q^35 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 4 * q^43 - 2 * q^44 - 6 * q^46 + 2 * q^49 + 2 * q^50 + 4 * q^52 + 6 * q^53 + 2 * q^55 + 2 * q^56 + 6 * q^58 + 4 * q^61 + 4 * q^62 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 2 * q^70 - 12 * q^71 + 16 * q^73 + 10 * q^74 - 2 * q^76 - 2 * q^77 + 10 * q^79 - 2 * q^80 + 6 * q^82 - 12 * q^83 + 4 * q^86 - 2 * q^88 + 24 * q^89 + 4 * q^91 - 6 * q^92 + 2 * q^95 + 22 * q^97 + 2 * q^98

Introduction

L-functions

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Database

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