LMFDB - Embedded newform 6930.2.a.by.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:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2310)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +6.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.74456 q^{17} -1.00000 q^{20} +1.00000 q^{22} -4.74456 q^{23} +1.00000 q^{25} +6.74456 q^{26} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{32} +6.74456 q^{34} +1.00000 q^{35} -2.74456 q^{37} -1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -4.74456 q^{46} +4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +6.74456 q^{52} +6.00000 q^{53} -1.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{64} -6.74456 q^{65} -12.7446 q^{67} +6.74456 q^{68} +1.00000 q^{70} +6.00000 q^{73} -2.74456 q^{74} -1.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} -6.74456 q^{85} +4.00000 q^{86} +1.00000 q^{88} -6.74456 q^{89} -6.74456 q^{91} -4.74456 q^{92} +4.00000 q^{94} +15.4891 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{20} + 2 q^{22} + 2 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} - 4 q^{29} + 2 q^{32} + 2 q^{34} + 2 q^{35} + 6 q^{37} - 2 q^{40} - 12 q^{41} + 8 q^{43} + 2 q^{44} + 2 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} + 2 q^{52} + 12 q^{53} - 2 q^{55} - 2 q^{56} - 4 q^{58} - 8 q^{59} + 12 q^{61} + 2 q^{64} - 2 q^{65} - 14 q^{67} + 2 q^{68} + 2 q^{70} + 12 q^{73} + 6 q^{74} - 2 q^{77} + 24 q^{79} - 2 q^{80} - 12 q^{82} + 8 q^{83} - 2 q^{85} + 8 q^{86} + 2 q^{88} - 2 q^{89} - 2 q^{91} + 2 q^{92} + 8 q^{94} + 8 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 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^20 + 2 * q^22 + 2 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 - 4 * q^29 + 2 * q^32 + 2 * q^34 + 2 * q^35 + 6 * q^37 - 2 * q^40 - 12 * q^41 + 8 * q^43 + 2 * q^44 + 2 * q^46 + 8 * q^47 + 2 * q^49 + 2 * q^50 + 2 * q^52 + 12 * q^53 - 2 * q^55 - 2 * q^56 - 4 * q^58 - 8 * q^59 + 12 * q^61 + 2 * q^64 - 2 * q^65 - 14 * q^67 + 2 * q^68 + 2 * q^70 + 12 * q^73 + 6 * q^74 - 2 * q^77 + 24 * q^79 - 2 * q^80 - 12 * q^82 + 8 * q^83 - 2 * q^85 + 8 * q^86 + 2 * q^88 - 2 * q^89 - 2 * q^91 + 2 * q^92 + 8 * q^94 + 8 * 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
$13$	6.74456	1.87061	0.935303	−	0.353849i	$-0.115127\pi$
$13$	6.74456	1.87061	0.935303	+	0.353849i	$0.115127\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.74456	1.63580	0.817898	−	0.575363i	$-0.195139\pi$
$17$	6.74456	1.63580	0.817898	+	0.575363i	$0.195139\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.00000	0.213201
$23$	−4.74456	−0.989310	−0.494655	−	0.869090i	$-0.664706\pi$
$23$	−4.74456	−0.989310	−0.494655	+	0.869090i	$0.664706\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.74456	1.32272
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.74456	1.15668
$35$	1.00000	0.169031
$36$	0	0
$37$	−2.74456	−0.451203	−0.225602	−	0.974220i	$-0.572435\pi$
$37$	−2.74456	−0.451203	−0.225602	+	0.974220i	$0.572435\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.74456	−0.699548
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	6.74456	0.935303
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.74456	−0.836560
$66$	0	0
$67$	−12.7446	−1.55700	−0.778498	−	0.627647i	$-0.784018\pi$
$67$	−12.7446	−1.55700	−0.778498	+	0.627647i	$0.784018\pi$
$68$	6.74456	0.817898
$69$	0	0
$70$	1.00000	0.119523
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−2.74456	−0.319049
$75$	0	0
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−6.74456	−0.731551
$86$	4.00000	0.431331
$87$	0	0
$88$	1.00000	0.106600
$89$	−6.74456	−0.714922	−0.357461	−	0.933928i	$-0.616358\pi$
$89$	−6.74456	−0.714922	−0.357461	+	0.933928i	$0.616358\pi$
$90$	0	0
$91$	−6.74456	−0.707022
$92$	−4.74456	−0.494655
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	15.4891	1.57268	0.786341	−	0.617792i	$-0.211973\pi$
$97$	15.4891	1.57268	0.786341	+	0.617792i	$0.211973\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−13.4891	−1.32912	−0.664562	−	0.747234i	$-0.731382\pi$
$103$	−13.4891	−1.32912	−0.664562	+	0.747234i	$0.731382\pi$
$104$	6.74456	0.661359
$105$	0	0
$106$	6.00000	0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	14.7446	1.41227	0.706136	−	0.708076i	$-0.250436\pi$
$109$	14.7446	1.41227	0.706136	+	0.708076i	$0.250436\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	10.7446	1.01076	0.505382	−	0.862896i	$-0.331352\pi$
$113$	10.7446	1.01076	0.505382	+	0.862896i	$0.331352\pi$
$114$	0	0
$115$	4.74456	0.442433
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−4.00000	−0.368230
$119$	−6.74456	−0.618273
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.00000	0.543214
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.74456	−0.591537
$131$	−9.48913	−0.829069	−0.414534	−	0.910034i	$-0.636056\pi$
$131$	−9.48913	−0.829069	−0.414534	+	0.910034i	$0.636056\pi$
$132$	0	0
$133$	0	0
$134$	−12.7446	−1.10096
$135$	0	0
$136$	6.74456	0.578341
$137$	−16.2337	−1.38694	−0.693469	−	0.720487i	$-0.743918\pi$
$137$	−16.2337	−1.38694	−0.693469	+	0.720487i	$0.743918\pi$
$138$	0	0
$139$	−1.48913	−0.126306	−0.0631530	−	0.998004i	$-0.520116\pi$
$139$	−1.48913	−0.126306	−0.0631530	+	0.998004i	$0.520116\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	0	0
$143$	6.74456	0.564009
$144$	0	0
$145$	2.00000	0.166091
$146$	6.00000	0.496564
$147$	0	0
$148$	−2.74456	−0.225602
$149$	7.48913	0.613533	0.306767	−	0.951785i	$-0.400753\pi$
$149$	7.48913	0.613533	0.306767	+	0.951785i	$0.400753\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	4.74456	0.373924
$162$	0	0
$163$	3.25544	0.254986	0.127493	−	0.991840i	$-0.459307\pi$
$163$	3.25544	0.254986	0.127493	+	0.991840i	$0.459307\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	14.2337	1.10144	0.550718	−	0.834691i	$-0.314354\pi$
$167$	14.2337	1.10144	0.550718	+	0.834691i	$0.314354\pi$
$168$	0	0
$169$	32.4891	2.49916
$170$	−6.74456	−0.517284
$171$	0	0
$172$	4.00000	0.304997
$173$	−0.510875	−0.0388411	−0.0194205	−	0.999811i	$-0.506182\pi$
$173$	−0.510875	−0.0388411	−0.0194205	+	0.999811i	$0.506182\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−6.74456	−0.505526
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	1.25544	0.0933159	0.0466580	−	0.998911i	$-0.485143\pi$
$181$	1.25544	0.0933159	0.0466580	+	0.998911i	$0.485143\pi$
$182$	−6.74456	−0.499940
$183$	0	0
$184$	−4.74456	−0.349774
$185$	2.74456	0.201784
$186$	0	0
$187$	6.74456	0.493211
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	−17.4891	−1.26547	−0.632734	−	0.774369i	$-0.718067\pi$
$191$	−17.4891	−1.26547	−0.632734	+	0.774369i	$0.718067\pi$
$192$	0	0
$193$	24.9783	1.79797	0.898987	−	0.437975i	$-0.144304\pi$
$193$	24.9783	1.79797	0.898987	+	0.437975i	$0.144304\pi$
$194$	15.4891	1.11205
$195$	0	0
$196$	1.00000	0.0714286
$197$	−0.510875	−0.0363983	−0.0181992	−	0.999834i	$-0.505793\pi$
$197$	−0.510875	−0.0363983	−0.0181992	+	0.999834i	$0.505793\pi$
$198$	0	0
$199$	−18.9783	−1.34533	−0.672666	−	0.739946i	$-0.734851\pi$
$199$	−18.9783	−1.34533	−0.672666	+	0.739946i	$0.734851\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	2.00000	0.140372
$204$	0	0
$205$	6.00000	0.419058
$206$	−13.4891	−0.939832
$207$	0	0
$208$	6.74456	0.467651
$209$	0	0
$210$	0	0
$211$	24.7446	1.70349	0.851743	−	0.523960i	$-0.175546\pi$
$211$	24.7446	1.70349	0.851743	+	0.523960i	$0.175546\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	4.00000	0.273434
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	14.7446	0.998628
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	45.4891	3.05993
$222$	0	0
$223$	21.4891	1.43902	0.719509	−	0.694483i	$-0.244367\pi$
$223$	21.4891	1.43902	0.719509	+	0.694483i	$0.244367\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	10.7446	0.714718
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	26.7446	1.76733	0.883665	−	0.468119i	$-0.155068\pi$
$229$	26.7446	1.76733	0.883665	+	0.468119i	$0.155068\pi$
$230$	4.74456	0.312847
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−6.74456	−0.437185
$239$	−18.2337	−1.17944	−0.589720	−	0.807608i	$-0.700762\pi$
$239$	−18.2337	−1.17944	−0.589720	+	0.807608i	$0.700762\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	6.00000	0.384111
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	−4.74456	−0.298288
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.4891	−1.46521	−0.732606	−	0.680653i	$-0.761696\pi$
$257$	−23.4891	−1.46521	−0.732606	+	0.680653i	$0.761696\pi$
$258$	0	0
$259$	2.74456	0.170539
$260$	−6.74456	−0.418280
$261$	0	0
$262$	−9.48913	−0.586240
$263$	25.4891	1.57173	0.785863	−	0.618400i	$-0.212219\pi$
$263$	25.4891	1.57173	0.785863	+	0.618400i	$0.212219\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	−12.7446	−0.778498
$269$	−23.4891	−1.43216	−0.716079	−	0.698020i	$-0.754065\pi$
$269$	−23.4891	−1.43216	−0.716079	+	0.698020i	$0.754065\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.74456	0.408949
$273$	0	0
$274$	−16.2337	−0.980713
$275$	1.00000	0.0603023
$276$	0	0
$277$	19.4891	1.17099	0.585494	−	0.810677i	$-0.300901\pi$
$277$	19.4891	1.17099	0.585494	+	0.810677i	$0.300901\pi$
$278$	−1.48913	−0.0893118
$279$	0	0
$280$	1.00000	0.0597614
$281$	20.2337	1.20704	0.603520	−	0.797348i	$-0.293764\pi$
$281$	20.2337	1.20704	0.603520	+	0.797348i	$0.293764\pi$
$282$	0	0
$283$	10.2337	0.608330	0.304165	−	0.952619i	$-0.401623\pi$
$283$	10.2337	0.608330	0.304165	+	0.952619i	$0.401623\pi$
$284$	0	0
$285$	0	0
$286$	6.74456	0.398814
$287$	6.00000	0.354169
$288$	0	0
$289$	28.4891	1.67583
$290$	2.00000	0.117444
$291$	0	0
$292$	6.00000	0.351123
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−2.74456	−0.159524
$297$	0	0
$298$	7.48913	0.433833
$299$	−32.0000	−1.85061
$300$	0	0
$301$	−4.00000	−0.230556
$302$	−4.00000	−0.230174
$303$	0	0
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	18.2337	1.04065	0.520326	−	0.853968i	$-0.325811\pi$
$307$	18.2337	1.04065	0.520326	+	0.853968i	$0.325811\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	0	0
$311$	−4.74456	−0.269039	−0.134520	−	0.990911i	$-0.542949\pi$
$311$	−4.74456	−0.269039	−0.134520	+	0.990911i	$0.542949\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	12.0000	0.675053
$317$	16.9783	0.953594	0.476797	−	0.879014i	$-0.341798\pi$
$317$	16.9783	0.953594	0.476797	+	0.879014i	$0.341798\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	4.74456	0.264404
$323$	0	0
$324$	0	0
$325$	6.74456	0.374121
$326$	3.25544	0.180302
$327$	0	0
$328$	−6.00000	−0.331295
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	14.2337	0.778833
$335$	12.7446	0.696310
$336$	0	0
$337$	15.4891	0.843746	0.421873	−	0.906655i	$-0.361373\pi$
$337$	15.4891	0.843746	0.421873	+	0.906655i	$0.361373\pi$
$338$	32.4891	1.76718
$339$	0	0
$340$	−6.74456	−0.365775
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	−0.510875	−0.0274648
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−20.9783	−1.12294	−0.561470	−	0.827497i	$-0.689764\pi$
$349$	−20.9783	−1.12294	−0.561470	+	0.827497i	$0.689764\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	1.00000	0.0533002
$353$	8.51087	0.452988	0.226494	−	0.974013i	$-0.427274\pi$
$353$	8.51087	0.452988	0.226494	+	0.974013i	$0.427274\pi$
$354$	0	0
$355$	0	0
$356$	−6.74456	−0.357461
$357$	0	0
$358$	4.00000	0.211407
$359$	7.25544	0.382927	0.191464	−	0.981500i	$-0.438677\pi$
$359$	7.25544	0.382927	0.191464	+	0.981500i	$0.438677\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	1.25544	0.0659843
$363$	0	0
$364$	−6.74456	−0.353511
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−21.4891	−1.12172	−0.560862	−	0.827910i	$-0.689530\pi$
$367$	−21.4891	−1.12172	−0.560862	+	0.827910i	$0.689530\pi$
$368$	−4.74456	−0.247327
$369$	0	0
$370$	2.74456	0.142683
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−24.9783	−1.29332	−0.646662	−	0.762776i	$-0.723836\pi$
$373$	−24.9783	−1.29332	−0.646662	+	0.762776i	$0.723836\pi$
$374$	6.74456	0.348753
$375$	0	0
$376$	4.00000	0.206284
$377$	−13.4891	−0.694725
$378$	0	0
$379$	−14.9783	−0.769381	−0.384691	−	0.923046i	$-0.625692\pi$
$379$	−14.9783	−0.769381	−0.384691	+	0.923046i	$0.625692\pi$
$380$	0	0
$381$	0	0
$382$	−17.4891	−0.894821
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	24.9783	1.27136
$387$	0	0
$388$	15.4891	0.786341
$389$	27.4891	1.39375	0.696877	−	0.717191i	$-0.254572\pi$
$389$	27.4891	1.39375	0.696877	+	0.717191i	$0.254572\pi$
$390$	0	0
$391$	−32.0000	−1.61831
$392$	1.00000	0.0505076
$393$	0	0
$394$	−0.510875	−0.0257375
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−38.4674	−1.93062	−0.965311	−	0.261102i	$-0.915914\pi$
$397$	−38.4674	−1.93062	−0.965311	+	0.261102i	$0.915914\pi$
$398$	−18.9783	−0.951294
$399$	0	0
$400$	1.00000	0.0500000
$401$	32.9783	1.64686	0.823428	−	0.567421i	$-0.192059\pi$
$401$	32.9783	1.64686	0.823428	+	0.567421i	$0.192059\pi$
$402$	0	0
$403$	0	0
$404$	−6.00000	−0.298511
$405$	0	0
$406$	2.00000	0.0992583
$407$	−2.74456	−0.136043
$408$	0	0
$409$	23.4891	1.16146	0.580731	−	0.814095i	$-0.302767\pi$
$409$	23.4891	1.16146	0.580731	+	0.814095i	$0.302767\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	−13.4891	−0.664562
$413$	4.00000	0.196827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	6.74456	0.330679
$417$	0	0
$418$	0	0
$419$	2.51087	0.122664	0.0613321	−	0.998117i	$-0.480465\pi$
$419$	2.51087	0.122664	0.0613321	+	0.998117i	$0.480465\pi$
$420$	0	0
$421$	−28.9783	−1.41231	−0.706157	−	0.708056i	$-0.749573\pi$
$421$	−28.9783	−1.41231	−0.706157	+	0.708056i	$0.749573\pi$
$422$	24.7446	1.20455
$423$	0	0
$424$	6.00000	0.291386
$425$	6.74456	0.327159
$426$	0	0
$427$	−6.00000	−0.290360
$428$	4.00000	0.193347
$429$	0	0
$430$	−4.00000	−0.192897
$431$	24.7446	1.19190	0.595952	−	0.803020i	$-0.296775\pi$
$431$	24.7446	1.19190	0.595952	+	0.803020i	$0.296775\pi$
$432$	0	0
$433$	−27.4891	−1.32104	−0.660522	−	0.750807i	$-0.729665\pi$
$433$	−27.4891	−1.32104	−0.660522	+	0.750807i	$0.729665\pi$
$434$	0	0
$435$	0	0
$436$	14.7446	0.706136
$437$	0	0
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	45.4891	2.16370
$443$	−34.9783	−1.66187	−0.830933	−	0.556372i	$-0.812193\pi$
$443$	−34.9783	−1.66187	−0.830933	+	0.556372i	$0.812193\pi$
$444$	0	0
$445$	6.74456	0.319723
$446$	21.4891	1.01754
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	10.7446	0.505382
$453$	0	0
$454$	−4.00000	−0.187729
$455$	6.74456	0.316190
$456$	0	0
$457$	−28.9783	−1.35555	−0.677773	−	0.735271i	$-0.737055\pi$
$457$	−28.9783	−1.35555	−0.677773	+	0.735271i	$0.737055\pi$
$458$	26.7446	1.24969
$459$	0	0
$460$	4.74456	0.221216
$461$	0.510875	0.0237938	0.0118969	−	0.999929i	$-0.496213\pi$
$461$	0.510875	0.0237938	0.0118969	+	0.999929i	$0.496213\pi$
$462$	0	0
$463$	−14.5109	−0.674378	−0.337189	−	0.941437i	$-0.609476\pi$
$463$	−14.5109	−0.674378	−0.337189	+	0.941437i	$0.609476\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	12.7446	0.588489
$470$	−4.00000	−0.184506
$471$	0	0
$472$	−4.00000	−0.184115
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	−6.74456	−0.309137
$477$	0	0
$478$	−18.2337	−0.833989
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−18.5109	−0.844023
$482$	14.0000	0.637683
$483$	0	0
$484$	1.00000	0.0454545
$485$	−15.4891	−0.703325
$486$	0	0
$487$	−33.4891	−1.51754	−0.758769	−	0.651360i	$-0.774199\pi$
$487$	−33.4891	−1.51754	−0.758769	+	0.651360i	$0.774199\pi$
$488$	6.00000	0.271607
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	−22.9783	−1.03699	−0.518497	−	0.855079i	$-0.673508\pi$
$491$	−22.9783	−1.03699	−0.518497	+	0.855079i	$0.673508\pi$
$492$	0	0
$493$	−13.4891	−0.607520
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.51087	−0.112402	−0.0562011	−	0.998419i	$-0.517899\pi$
$499$	−2.51087	−0.112402	−0.0562011	+	0.998419i	$0.517899\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	4.00000	0.178529
$503$	−28.7446	−1.28166	−0.640828	−	0.767684i	$-0.721409\pi$
$503$	−28.7446	−1.28166	−0.640828	+	0.767684i	$0.721409\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−4.74456	−0.210922
$507$	0	0
$508$	8.00000	0.354943
$509$	−39.4891	−1.75032	−0.875162	−	0.483829i	$-0.839246\pi$
$509$	−39.4891	−1.75032	−0.875162	+	0.483829i	$0.839246\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	1.00000	0.0441942
$513$	0	0
$514$	−23.4891	−1.03606
$515$	13.4891	0.594402
$516$	0	0
$517$	4.00000	0.175920
$518$	2.74456	0.120589
$519$	0	0
$520$	−6.74456	−0.295769
$521$	−25.7228	−1.12694	−0.563468	−	0.826138i	$-0.690533\pi$
$521$	−25.7228	−1.12694	−0.563468	+	0.826138i	$0.690533\pi$
$522$	0	0
$523$	16.7446	0.732189	0.366094	−	0.930578i	$-0.380695\pi$
$523$	16.7446	0.732189	0.366094	+	0.930578i	$0.380695\pi$
$524$	−9.48913	−0.414534
$525$	0	0
$526$	25.4891	1.11138
$527$	0	0
$528$	0	0
$529$	−0.489125	−0.0212663
$530$	−6.00000	−0.260623
$531$	0	0
$532$	0	0
$533$	−40.4674	−1.75284
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−12.7446	−0.550481
$537$	0	0
$538$	−23.4891	−1.01269
$539$	1.00000	0.0430730
$540$	0	0
$541$	16.2337	0.697941	0.348970	−	0.937134i	$-0.386531\pi$
$541$	16.2337	0.697941	0.348970	+	0.937134i	$0.386531\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	6.74456	0.289171
$545$	−14.7446	−0.631588
$546$	0	0
$547$	32.4674	1.38820	0.694102	−	0.719876i	$-0.255801\pi$
$547$	32.4674	1.38820	0.694102	+	0.719876i	$0.255801\pi$
$548$	−16.2337	−0.693469
$549$	0	0
$550$	1.00000	0.0426401
$551$	0	0
$552$	0	0
$553$	−12.0000	−0.510292
$554$	19.4891	0.828014
$555$	0	0
$556$	−1.48913	−0.0631530
$557$	−11.4891	−0.486810	−0.243405	−	0.969925i	$-0.578264\pi$
$557$	−11.4891	−0.486810	−0.243405	+	0.969925i	$0.578264\pi$
$558$	0	0
$559$	26.9783	1.14106
$560$	1.00000	0.0422577
$561$	0	0
$562$	20.2337	0.853507
$563$	−5.48913	−0.231339	−0.115670	−	0.993288i	$-0.536901\pi$
$563$	−5.48913	−0.231339	−0.115670	+	0.993288i	$0.536901\pi$
$564$	0	0
$565$	−10.7446	−0.452027
$566$	10.2337	0.430154
$567$	0	0
$568$	0	0
$569$	26.7446	1.12119	0.560595	−	0.828090i	$-0.310572\pi$
$569$	26.7446	1.12119	0.560595	+	0.828090i	$0.310572\pi$
$570$	0	0
$571$	−40.7446	−1.70511	−0.852553	−	0.522640i	$-0.824947\pi$
$571$	−40.7446	−1.70511	−0.852553	+	0.522640i	$0.824947\pi$
$572$	6.74456	0.282004
$573$	0	0
$574$	6.00000	0.250435
$575$	−4.74456	−0.197862
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	28.4891	1.18499
$579$	0	0
$580$	2.00000	0.0830455
$581$	−4.00000	−0.165948
$582$	0	0
$583$	6.00000	0.248495
$584$	6.00000	0.248282
$585$	0	0
$586$	6.00000	0.247858
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	0	0
$590$	4.00000	0.164677
$591$	0	0
$592$	−2.74456	−0.112801
$593$	0.233688	0.00959641	0.00479821	−	0.999988i	$-0.498473\pi$
$593$	0.233688	0.00959641	0.00479821	+	0.999988i	$0.498473\pi$
$594$	0	0
$595$	6.74456	0.276500
$596$	7.48913	0.306767
$597$	0	0
$598$	−32.0000	−1.30858
$599$	44.4674	1.81689	0.908444	−	0.418007i	$-0.137271\pi$
$599$	44.4674	1.81689	0.908444	+	0.418007i	$0.137271\pi$
$600$	0	0
$601$	−30.4674	−1.24279	−0.621395	−	0.783497i	$-0.713434\pi$
$601$	−30.4674	−1.24279	−0.621395	+	0.783497i	$0.713434\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−4.00000	−0.162758
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−34.9783	−1.41972	−0.709862	−	0.704341i	$-0.751243\pi$
$607$	−34.9783	−1.41972	−0.709862	+	0.704341i	$0.751243\pi$
$608$	0	0
$609$	0	0
$610$	−6.00000	−0.242933
$611$	26.9783	1.09142
$612$	0	0
$613$	−31.4891	−1.27183	−0.635917	−	0.771758i	$-0.719378\pi$
$613$	−31.4891	−1.27183	−0.635917	+	0.771758i	$0.719378\pi$
$614$	18.2337	0.735852
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	7.76631	0.312660	0.156330	−	0.987705i	$-0.450034\pi$
$617$	7.76631	0.312660	0.156330	+	0.987705i	$0.450034\pi$
$618$	0	0
$619$	−2.23369	−0.0897795	−0.0448897	−	0.998992i	$-0.514294\pi$
$619$	−2.23369	−0.0897795	−0.0448897	+	0.998992i	$0.514294\pi$
$620$	0	0
$621$	0	0
$622$	−4.74456	−0.190240
$623$	6.74456	0.270215
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	0	0
$628$	6.00000	0.239426
$629$	−18.5109	−0.738077
$630$	0	0
$631$	26.9783	1.07399	0.536994	−	0.843586i	$-0.319560\pi$
$631$	26.9783	1.07399	0.536994	+	0.843586i	$0.319560\pi$
$632$	12.0000	0.477334
$633$	0	0
$634$	16.9783	0.674292
$635$	−8.00000	−0.317470
$636$	0	0
$637$	6.74456	0.267229
$638$	−2.00000	−0.0791808
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	48.9783	1.93452	0.967262	−	0.253779i	$-0.0816735\pi$
$641$	48.9783	1.93452	0.967262	+	0.253779i	$0.0816735\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	4.74456	0.186962
$645$	0	0
$646$	0	0
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	6.74456	0.264544
$651$	0	0
$652$	3.25544	0.127493
$653$	−11.4891	−0.449604	−0.224802	−	0.974404i	$-0.572174\pi$
$653$	−11.4891	−0.449604	−0.224802	+	0.974404i	$0.572174\pi$
$654$	0	0
$655$	9.48913	0.370771
$656$	−6.00000	−0.234261
$657$	0	0
$658$	−4.00000	−0.155936
$659$	38.9783	1.51838	0.759189	−	0.650871i	$-0.225596\pi$
$659$	38.9783	1.51838	0.759189	+	0.650871i	$0.225596\pi$
$660$	0	0
$661$	−11.7663	−0.457656	−0.228828	−	0.973467i	$-0.573489\pi$
$661$	−11.7663	−0.457656	−0.228828	+	0.973467i	$0.573489\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	9.48913	0.367420
$668$	14.2337	0.550718
$669$	0	0
$670$	12.7446	0.492365
$671$	6.00000	0.231627
$672$	0	0
$673$	31.4891	1.21382	0.606908	−	0.794772i	$-0.292410\pi$
$673$	31.4891	1.21382	0.606908	+	0.794772i	$0.292410\pi$
$674$	15.4891	0.596619
$675$	0	0
$676$	32.4891	1.24958
$677$	−22.4674	−0.863491	−0.431746	−	0.901995i	$-0.642102\pi$
$677$	−22.4674	−0.863491	−0.431746	+	0.901995i	$0.642102\pi$
$678$	0	0
$679$	−15.4891	−0.594418
$680$	−6.74456	−0.258642
$681$	0	0
$682$	0	0
$683$	34.9783	1.33841	0.669203	−	0.743080i	$-0.266636\pi$
$683$	34.9783	1.33841	0.669203	+	0.743080i	$0.266636\pi$
$684$	0	0
$685$	16.2337	0.620257
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	40.4674	1.54168
$690$	0	0
$691$	−18.2337	−0.693642	−0.346821	−	0.937931i	$-0.612739\pi$
$691$	−18.2337	−0.693642	−0.346821	+	0.937931i	$0.612739\pi$
$692$	−0.510875	−0.0194205
$693$	0	0
$694$	20.0000	0.759190
$695$	1.48913	0.0564857
$696$	0	0
$697$	−40.4674	−1.53281
$698$	−20.9783	−0.794038
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	0	0
$703$	0	0
$704$	1.00000	0.0376889
$705$	0	0
$706$	8.51087	0.320311
$707$	6.00000	0.225653
$708$	0	0
$709$	16.9783	0.637632	0.318816	−	0.947817i	$-0.396715\pi$
$709$	16.9783	0.637632	0.318816	+	0.947817i	$0.396715\pi$
$710$	0	0
$711$	0	0
$712$	−6.74456	−0.252763
$713$	0	0
$714$	0	0
$715$	−6.74456	−0.252232
$716$	4.00000	0.149487
$717$	0	0
$718$	7.25544	0.270771
$719$	23.7228	0.884712	0.442356	−	0.896840i	$-0.354143\pi$
$719$	23.7228	0.884712	0.442356	+	0.896840i	$0.354143\pi$
$720$	0	0
$721$	13.4891	0.502361
$722$	−19.0000	−0.707107
$723$	0	0
$724$	1.25544	0.0466580
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−10.5109	−0.389827	−0.194913	−	0.980820i	$-0.562443\pi$
$727$	−10.5109	−0.389827	−0.194913	+	0.980820i	$0.562443\pi$
$728$	−6.74456	−0.249970
$729$	0	0
$730$	−6.00000	−0.222070
$731$	26.9783	0.997827
$732$	0	0
$733$	−1.25544	−0.0463706	−0.0231853	−	0.999731i	$-0.507381\pi$
$733$	−1.25544	−0.0463706	−0.0231853	+	0.999731i	$0.507381\pi$
$734$	−21.4891	−0.793178
$735$	0	0
$736$	−4.74456	−0.174887
$737$	−12.7446	−0.469452
$738$	0	0
$739$	31.2554	1.14975	0.574875	−	0.818241i	$-0.305051\pi$
$739$	31.2554	1.14975	0.574875	+	0.818241i	$0.305051\pi$
$740$	2.74456	0.100892
$741$	0	0
$742$	−6.00000	−0.220267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−7.48913	−0.274380
$746$	−24.9783	−0.914519
$747$	0	0
$748$	6.74456	0.246606
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−5.02175	−0.183246	−0.0916231	−	0.995794i	$-0.529206\pi$
$751$	−5.02175	−0.183246	−0.0916231	+	0.995794i	$0.529206\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	−13.4891	−0.491245
$755$	4.00000	0.145575
$756$	0	0
$757$	−5.72281	−0.207999	−0.104000	−	0.994577i	$-0.533164\pi$
$757$	−5.72281	−0.207999	−0.104000	+	0.994577i	$0.533164\pi$
$758$	−14.9783	−0.544035
$759$	0	0
$760$	0	0
$761$	−51.9565	−1.88342	−0.941711	−	0.336423i	$-0.890783\pi$
$761$	−51.9565	−1.88342	−0.941711	+	0.336423i	$0.890783\pi$
$762$	0	0
$763$	−14.7446	−0.533789
$764$	−17.4891	−0.632734
$765$	0	0
$766$	−20.0000	−0.722629
$767$	−26.9783	−0.974128
$768$	0	0
$769$	31.4891	1.13553	0.567763	−	0.823192i	$-0.307809\pi$
$769$	31.4891	1.13553	0.567763	+	0.823192i	$0.307809\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	24.9783	0.898987
$773$	14.4674	0.520355	0.260178	−	0.965561i	$-0.416219\pi$
$773$	14.4674	0.520355	0.260178	+	0.965561i	$0.416219\pi$
$774$	0	0
$775$	0	0
$776$	15.4891	0.556027
$777$	0	0
$778$	27.4891	0.985533
$779$	0	0
$780$	0	0
$781$	0	0
$782$	−32.0000	−1.14432
$783$	0	0
$784$	1.00000	0.0357143
$785$	−6.00000	−0.214149
$786$	0	0
$787$	−5.76631	−0.205547	−0.102773	−	0.994705i	$-0.532772\pi$
$787$	−5.76631	−0.205547	−0.102773	+	0.994705i	$0.532772\pi$
$788$	−0.510875	−0.0181992
$789$	0	0
$790$	−12.0000	−0.426941
$791$	−10.7446	−0.382033
$792$	0	0
$793$	40.4674	1.43704
$794$	−38.4674	−1.36516
$795$	0	0
$796$	−18.9783	−0.672666
$797$	−47.4891	−1.68215	−0.841076	−	0.540918i	$-0.818077\pi$
$797$	−47.4891	−1.68215	−0.841076	+	0.540918i	$0.818077\pi$
$798$	0	0
$799$	26.9783	0.954422
$800$	1.00000	0.0353553
$801$	0	0
$802$	32.9783	1.16450
$803$	6.00000	0.211735
$804$	0	0
$805$	−4.74456	−0.167224
$806$	0	0
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−8.23369	−0.289481	−0.144741	−	0.989470i	$-0.546235\pi$
$809$	−8.23369	−0.289481	−0.144741	+	0.989470i	$0.546235\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	2.00000	0.0701862
$813$	0	0
$814$	−2.74456	−0.0961969
$815$	−3.25544	−0.114033
$816$	0	0
$817$	0	0
$818$	23.4891	0.821278
$819$	0	0
$820$	6.00000	0.209529
$821$	0.978251	0.0341412	0.0170706	−	0.999854i	$-0.494566\pi$
$821$	0.978251	0.0341412	0.0170706	+	0.999854i	$0.494566\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−13.4891	−0.469916
$825$	0	0
$826$	4.00000	0.139178
$827$	−1.02175	−0.0355297	−0.0177649	−	0.999842i	$-0.505655\pi$
$827$	−1.02175	−0.0355297	−0.0177649	+	0.999842i	$0.505655\pi$
$828$	0	0
$829$	28.2337	0.980597	0.490298	−	0.871555i	$-0.336888\pi$
$829$	28.2337	0.980597	0.490298	+	0.871555i	$0.336888\pi$
$830$	−4.00000	−0.138842
$831$	0	0
$832$	6.74456	0.233826
$833$	6.74456	0.233685
$834$	0	0
$835$	−14.2337	−0.492577
$836$	0	0
$837$	0	0
$838$	2.51087	0.0867367
$839$	−28.7446	−0.992373	−0.496186	−	0.868216i	$-0.665267\pi$
$839$	−28.7446	−0.992373	−0.496186	+	0.868216i	$0.665267\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−28.9783	−0.998656
$843$	0	0
$844$	24.7446	0.851743
$845$	−32.4891	−1.11766
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	6.00000	0.206041
$849$	0	0
$850$	6.74456	0.231337
$851$	13.0217	0.446380
$852$	0	0
$853$	−44.2337	−1.51453	−0.757266	−	0.653106i	$-0.773466\pi$
$853$	−44.2337	−1.51453	−0.757266	+	0.653106i	$0.773466\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	4.00000	0.136717
$857$	−26.7446	−0.913577	−0.456788	−	0.889575i	$-0.651000\pi$
$857$	−26.7446	−0.913577	−0.456788	+	0.889575i	$0.651000\pi$
$858$	0	0
$859$	58.2337	1.98691	0.993454	−	0.114234i	$-0.0364413\pi$
$859$	58.2337	1.98691	0.993454	+	0.114234i	$0.0364413\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	24.7446	0.842803
$863$	−20.7446	−0.706153	−0.353077	−	0.935594i	$-0.614864\pi$
$863$	−20.7446	−0.706153	−0.353077	+	0.935594i	$0.614864\pi$
$864$	0	0
$865$	0.510875	0.0173703
$866$	−27.4891	−0.934119
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−85.9565	−2.91252
$872$	14.7446	0.499314
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−4.51087	−0.152321	−0.0761607	−	0.997096i	$-0.524266\pi$
$877$	−4.51087	−0.152321	−0.0761607	+	0.997096i	$0.524266\pi$
$878$	16.0000	0.539974
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	9.25544	0.311824	0.155912	−	0.987771i	$-0.450168\pi$
$881$	9.25544	0.311824	0.155912	+	0.987771i	$0.450168\pi$
$882$	0	0
$883$	−58.7011	−1.97545	−0.987724	−	0.156209i	$-0.950073\pi$
$883$	−58.7011	−1.97545	−0.987724	+	0.156209i	$0.950073\pi$
$884$	45.4891	1.52996
$885$	0	0
$886$	−34.9783	−1.17512
$887$	23.7228	0.796534	0.398267	−	0.917270i	$-0.369612\pi$
$887$	23.7228	0.796534	0.398267	+	0.917270i	$0.369612\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	6.74456	0.226078
$891$	0	0
$892$	21.4891	0.719509
$893$	0	0
$894$	0	0
$895$	−4.00000	−0.133705
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	14.0000	0.467186
$899$	0	0
$900$	0	0
$901$	40.4674	1.34816
$902$	−6.00000	−0.199778
$903$	0	0
$904$	10.7446	0.357359
$905$	−1.25544	−0.0417321
$906$	0	0
$907$	12.7446	0.423176	0.211588	−	0.977359i	$-0.432136\pi$
$907$	12.7446	0.423176	0.211588	+	0.977359i	$0.432136\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	6.74456	0.223580
$911$	−34.9783	−1.15888	−0.579441	−	0.815014i	$-0.696729\pi$
$911$	−34.9783	−1.15888	−0.579441	+	0.815014i	$0.696729\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	−28.9783	−0.958515
$915$	0	0
$916$	26.7446	0.883665
$917$	9.48913	0.313359
$918$	0	0
$919$	−29.4891	−0.972756	−0.486378	−	0.873748i	$-0.661682\pi$
$919$	−29.4891	−0.972756	−0.486378	+	0.873748i	$0.661682\pi$
$920$	4.74456	0.156424
$921$	0	0
$922$	0.510875	0.0168248
$923$	0	0
$924$	0	0
$925$	−2.74456	−0.0902407
$926$	−14.5109	−0.476857
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	26.7446	0.877461	0.438730	−	0.898619i	$-0.355428\pi$
$929$	26.7446	0.877461	0.438730	+	0.898619i	$0.355428\pi$
$930$	0	0
$931$	0	0
$932$	−22.0000	−0.720634
$933$	0	0
$934$	−4.00000	−0.130884
$935$	−6.74456	−0.220571
$936$	0	0
$937$	−3.48913	−0.113985	−0.0569924	−	0.998375i	$-0.518151\pi$
$937$	−3.48913	−0.113985	−0.0569924	+	0.998375i	$0.518151\pi$
$938$	12.7446	0.416125
$939$	0	0
$940$	−4.00000	−0.130466
$941$	−51.9565	−1.69373	−0.846867	−	0.531805i	$-0.821514\pi$
$941$	−51.9565	−1.69373	−0.846867	+	0.531805i	$0.821514\pi$
$942$	0	0
$943$	28.4674	0.927025
$944$	−4.00000	−0.130189
$945$	0	0
$946$	4.00000	0.130051
$947$	41.4891	1.34822	0.674108	−	0.738633i	$-0.264528\pi$
$947$	41.4891	1.34822	0.674108	+	0.738633i	$0.264528\pi$
$948$	0	0
$949$	40.4674	1.31363
$950$	0	0
$951$	0	0
$952$	−6.74456	−0.218593
$953$	16.5109	0.534840	0.267420	−	0.963580i	$-0.413829\pi$
$953$	16.5109	0.534840	0.267420	+	0.963580i	$0.413829\pi$
$954$	0	0
$955$	17.4891	0.565935
$956$	−18.2337	−0.589720
$957$	0	0
$958$	8.00000	0.258468
$959$	16.2337	0.524213
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−18.5109	−0.596815
$963$	0	0
$964$	14.0000	0.450910
$965$	−24.9783	−0.804078
$966$	0	0
$967$	−18.9783	−0.610299	−0.305150	−	0.952304i	$-0.598707\pi$
$967$	−18.9783	−0.610299	−0.305150	+	0.952304i	$0.598707\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−15.4891	−0.497326
$971$	5.48913	0.176154	0.0880772	−	0.996114i	$-0.471928\pi$
$971$	5.48913	0.176154	0.0880772	+	0.996114i	$0.471928\pi$
$972$	0	0
$973$	1.48913	0.0477392
$974$	−33.4891	−1.07306
$975$	0	0
$976$	6.00000	0.192055
$977$	−14.7446	−0.471720	−0.235860	−	0.971787i	$-0.575791\pi$
$977$	−14.7446	−0.471720	−0.235860	+	0.971787i	$0.575791\pi$
$978$	0	0
$979$	−6.74456	−0.215557
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−22.9783	−0.733265
$983$	54.9783	1.75353	0.876767	−	0.480916i	$-0.159696\pi$
$983$	54.9783	1.75353	0.876767	+	0.480916i	$0.159696\pi$
$984$	0	0
$985$	0.510875	0.0162778
$986$	−13.4891	−0.429581
$987$	0	0
$988$	0	0
$989$	−18.9783	−0.603473
$990$	0	0
$991$	10.9783	0.348736	0.174368	−	0.984681i	$-0.444212\pi$
$991$	10.9783	0.348736	0.174368	+	0.984681i	$0.444212\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.9783	0.601651
$996$	0	0
$997$	48.2337	1.52758	0.763788	−	0.645467i	$-0.223337\pi$
$997$	48.2337	1.52758	0.763788	+	0.645467i	$0.223337\pi$
$998$	−2.51087	−0.0794804
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.by.1.2		2
3.2	odd	2		2310.2.a.x.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.2.a.x.1.2	✓	2	3.2	odd	2
6930.2.a.by.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +6.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.74456 q^{17} -1.00000 q^{20} +1.00000 q^{22} -4.74456 q^{23} +1.00000 q^{25} +6.74456 q^{26} -1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{32} +6.74456 q^{34} +1.00000 q^{35} -2.74456 q^{37} -1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -4.74456 q^{46} +4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +6.74456 q^{52} +6.00000 q^{53} -1.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{64} -6.74456 q^{65} -12.7446 q^{67} +6.74456 q^{68} +1.00000 q^{70} +6.00000 q^{73} -2.74456 q^{74} -1.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} -6.74456 q^{85} +4.00000 q^{86} +1.00000 q^{88} -6.74456 q^{89} -6.74456 q^{91} -4.74456 q^{92} +4.00000 q^{94} +15.4891 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{20} + 2 q^{22} + 2 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} - 4 q^{29} + 2 q^{32} + 2 q^{34} + 2 q^{35} + 6 q^{37} - 2 q^{40} - 12 q^{41} + 8 q^{43} + 2 q^{44} + 2 q^{46} + 8 q^{47} + 2 q^{49} + 2 q^{50} + 2 q^{52} + 12 q^{53} - 2 q^{55} - 2 q^{56} - 4 q^{58} - 8 q^{59} + 12 q^{61} + 2 q^{64} - 2 q^{65} - 14 q^{67} + 2 q^{68} + 2 q^{70} + 12 q^{73} + 6 q^{74} - 2 q^{77} + 24 q^{79} - 2 q^{80} - 12 q^{82} + 8 q^{83} - 2 q^{85} + 8 q^{86} + 2 q^{88} - 2 q^{89} - 2 q^{91} + 2 q^{92} + 8 q^{94} + 8 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^8 - 2 * q^10 + 2 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^20 + 2 * q^22 + 2 * q^23 + 2 * q^25 + 2 * q^26 - 2 * q^28 - 4 * q^29 + 2 * q^32 + 2 * q^34 + 2 * q^35 + 6 * q^37 - 2 * q^40 - 12 * q^41 + 8 * q^43 + 2 * q^44 + 2 * q^46 + 8 * q^47 + 2 * q^49 + 2 * q^50 + 2 * q^52 + 12 * q^53 - 2 * q^55 - 2 * q^56 - 4 * q^58 - 8 * q^59 + 12 * q^61 + 2 * q^64 - 2 * q^65 - 14 * q^67 + 2 * q^68 + 2 * q^70 + 12 * q^73 + 6 * q^74 - 2 * q^77 + 24 * q^79 - 2 * q^80 - 12 * q^82 + 8 * q^83 - 2 * q^85 + 8 * q^86 + 2 * q^88 - 2 * q^89 - 2 * q^91 + 2 * q^92 + 8 * q^94 + 8 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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