LMFDB - Embedded newform 6930.2.a.bo.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 770)
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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +6.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.74456 q^{17} -6.74456 q^{19} -1.00000 q^{20} -1.00000 q^{22} +6.74456 q^{23} +1.00000 q^{25} -6.74456 q^{26} +1.00000 q^{28} -8.74456 q^{29} +4.74456 q^{31} -1.00000 q^{32} -6.74456 q^{34} -1.00000 q^{35} +0.744563 q^{37} +6.74456 q^{38} +1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -6.74456 q^{46} -4.74456 q^{47} +1.00000 q^{49} -1.00000 q^{50} +6.74456 q^{52} +12.7446 q^{53} -1.00000 q^{55} -1.00000 q^{56} +8.74456 q^{58} +8.74456 q^{59} +1.25544 q^{61} -4.74456 q^{62} +1.00000 q^{64} -6.74456 q^{65} -4.00000 q^{67} +6.74456 q^{68} +1.00000 q^{70} +4.00000 q^{71} +10.7446 q^{73} -0.744563 q^{74} -6.74456 q^{76} +1.00000 q^{77} +6.74456 q^{79} -1.00000 q^{80} -4.00000 q^{82} -8.00000 q^{83} -6.74456 q^{85} -4.00000 q^{86} -1.00000 q^{88} -15.4891 q^{89} +6.74456 q^{91} +6.74456 q^{92} +4.74456 q^{94} +6.74456 q^{95} -16.7446 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} - 6 q^{29} - 2 q^{31} - 2 q^{32} - 2 q^{34} - 2 q^{35} - 10 q^{37} + 2 q^{38} + 2 q^{40} + 8 q^{41} + 8 q^{43} + 2 q^{44} - 2 q^{46} + 2 q^{47} + 2 q^{49} - 2 q^{50} + 2 q^{52} + 14 q^{53} - 2 q^{55} - 2 q^{56} + 6 q^{58} + 6 q^{59} + 14 q^{61} + 2 q^{62} + 2 q^{64} - 2 q^{65} - 8 q^{67} + 2 q^{68} + 2 q^{70} + 8 q^{71} + 10 q^{73} + 10 q^{74} - 2 q^{76} + 2 q^{77} + 2 q^{79} - 2 q^{80} - 8 q^{82} - 16 q^{83} - 2 q^{85} - 8 q^{86} - 2 q^{88} - 8 q^{89} + 2 q^{91} + 2 q^{92} - 2 q^{94} + 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 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 - 2 * q^20 - 2 * q^22 + 2 * q^23 + 2 * q^25 - 2 * q^26 + 2 * q^28 - 6 * q^29 - 2 * q^31 - 2 * q^32 - 2 * q^34 - 2 * q^35 - 10 * q^37 + 2 * q^38 + 2 * q^40 + 8 * q^41 + 8 * q^43 + 2 * q^44 - 2 * q^46 + 2 * q^47 + 2 * q^49 - 2 * q^50 + 2 * q^52 + 14 * q^53 - 2 * q^55 - 2 * q^56 + 6 * q^58 + 6 * q^59 + 14 * q^61 + 2 * q^62 + 2 * q^64 - 2 * q^65 - 8 * q^67 + 2 * q^68 + 2 * q^70 + 8 * q^71 + 10 * q^73 + 10 * q^74 - 2 * q^76 + 2 * q^77 + 2 * q^79 - 2 * q^80 - 8 * q^82 - 16 * q^83 - 2 * q^85 - 8 * q^86 - 2 * q^88 - 8 * q^89 + 2 * q^91 + 2 * q^92 - 2 * q^94 + 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
$13$	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
$19$	−6.74456	−1.54731	−0.773654	−	0.633608i	$-0.781573\pi$
$19$	−6.74456	−1.54731	−0.773654	+	0.633608i	$0.781573\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	6.74456	1.40634	0.703169	−	0.711022i	$-0.251768\pi$
$23$	6.74456	1.40634	0.703169	+	0.711022i	$0.251768\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.74456	−1.32272
$27$	0	0
$28$	1.00000	0.188982
$29$	−8.74456	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$30$	0	0
$31$	4.74456	0.852149	0.426074	−	0.904688i	$-0.359896\pi$
$31$	4.74456	0.852149	0.426074	+	0.904688i	$0.359896\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.74456	−1.15668
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.744563	0.122405	0.0612027	−	0.998125i	$-0.480506\pi$
$37$	0.744563	0.122405	0.0612027	+	0.998125i	$0.480506\pi$
$38$	6.74456	1.09411
$39$	0	0
$40$	1.00000	0.158114
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	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
$46$	−6.74456	−0.994432
$47$	−4.74456	−0.692066	−0.346033	−	0.938222i	$-0.612471\pi$
$47$	−4.74456	−0.692066	−0.346033	+	0.938222i	$0.612471\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	6.74456	0.935303
$53$	12.7446	1.75060	0.875300	−	0.483580i	$-0.160664\pi$
$53$	12.7446	1.75060	0.875300	+	0.483580i	$0.160664\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	8.74456	1.14822
$59$	8.74456	1.13845	0.569223	−	0.822183i	$-0.307244\pi$
$59$	8.74456	1.13845	0.569223	+	0.822183i	$0.307244\pi$
$60$	0	0
$61$	1.25544	0.160742	0.0803711	−	0.996765i	$-0.474389\pi$
$61$	1.25544	0.160742	0.0803711	+	0.996765i	$0.474389\pi$
$62$	−4.74456	−0.602560
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.74456	−0.836560
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.74456	0.817898
$69$	0	0
$70$	1.00000	0.119523
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	10.7446	1.25756	0.628778	−	0.777585i	$-0.283555\pi$
$73$	10.7446	1.25756	0.628778	+	0.777585i	$0.283555\pi$
$74$	−0.744563	−0.0865536
$75$	0	0
$76$	−6.74456	−0.773654
$77$	1.00000	0.113961
$78$	0	0
$79$	6.74456	0.758823	0.379411	−	0.925228i	$-0.376127\pi$
$79$	6.74456	0.758823	0.379411	+	0.925228i	$0.376127\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−4.00000	−0.441726
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−6.74456	−0.731551
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−15.4891	−1.64184	−0.820922	−	0.571040i	$-0.806540\pi$
$89$	−15.4891	−1.64184	−0.820922	+	0.571040i	$0.806540\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	6.74456	0.703169
$93$	0	0
$94$	4.74456	0.489364
$95$	6.74456	0.691978
$96$	0	0
$97$	−16.7446	−1.70015	−0.850076	−	0.526659i	$-0.823444\pi$
$97$	−16.7446	−1.70015	−0.850076	+	0.526659i	$0.823444\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.74456	−0.273094	−0.136547	−	0.990634i	$-0.543601\pi$
$101$	−2.74456	−0.273094	−0.136547	+	0.990634i	$0.543601\pi$
$102$	0	0
$103$	−0.744563	−0.0733639	−0.0366820	−	0.999327i	$-0.511679\pi$
$103$	−0.744563	−0.0733639	−0.0366820	+	0.999327i	$0.511679\pi$
$104$	−6.74456	−0.661359
$105$	0	0
$106$	−12.7446	−1.23786
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−18.2337	−1.74647	−0.873235	−	0.487299i	$-0.837982\pi$
$109$	−18.2337	−1.74647	−0.873235	+	0.487299i	$0.837982\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−6.74456	−0.628934
$116$	−8.74456	−0.811912
$117$	0	0
$118$	−8.74456	−0.805002
$119$	6.74456	0.618273
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.25544	−0.113662
$123$	0	0
$124$	4.74456	0.426074
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.74456	0.591537
$131$	2.74456	0.239794	0.119897	−	0.992786i	$-0.461744\pi$
$131$	2.74456	0.239794	0.119897	+	0.992786i	$0.461744\pi$
$132$	0	0
$133$	−6.74456	−0.584828
$134$	4.00000	0.345547
$135$	0	0
$136$	−6.74456	−0.578341
$137$	−3.48913	−0.298096	−0.149048	−	0.988830i	$-0.547621\pi$
$137$	−3.48913	−0.298096	−0.149048	+	0.988830i	$0.547621\pi$
$138$	0	0
$139$	14.7446	1.25062	0.625309	−	0.780377i	$-0.284973\pi$
$139$	14.7446	1.25062	0.625309	+	0.780377i	$0.284973\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−4.00000	−0.335673
$143$	6.74456	0.564009
$144$	0	0
$145$	8.74456	0.726196
$146$	−10.7446	−0.889226
$147$	0	0
$148$	0.744563	0.0612027
$149$	0.744563	0.0609969	0.0304985	−	0.999535i	$-0.490291\pi$
$149$	0.744563	0.0609969	0.0304985	+	0.999535i	$0.490291\pi$
$150$	0	0
$151$	−9.25544	−0.753197	−0.376598	−	0.926377i	$-0.622906\pi$
$151$	−9.25544	−0.753197	−0.376598	+	0.926377i	$0.622906\pi$
$152$	6.74456	0.547056
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−4.74456	−0.381092
$156$	0	0
$157$	0.510875	0.0407722	0.0203861	−	0.999792i	$-0.493510\pi$
$157$	0.510875	0.0407722	0.0203861	+	0.999792i	$0.493510\pi$
$158$	−6.74456	−0.536569
$159$	0	0
$160$	1.00000	0.0790569
$161$	6.74456	0.531546
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	8.00000	0.620920
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	32.4891	2.49916
$170$	6.74456	0.517284
$171$	0	0
$172$	4.00000	0.304997
$173$	−20.2337	−1.53834	−0.769169	−	0.639045i	$-0.779330\pi$
$173$	−20.2337	−1.53834	−0.769169	+	0.639045i	$0.779330\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	15.4891	1.16096
$179$	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
$181$	−19.4891	−1.44862	−0.724308	−	0.689477i	$-0.757840\pi$
$181$	−19.4891	−1.44862	−0.724308	+	0.689477i	$0.757840\pi$
$182$	−6.74456	−0.499940
$183$	0	0
$184$	−6.74456	−0.497216
$185$	−0.744563	−0.0547413
$186$	0	0
$187$	6.74456	0.493211
$188$	−4.74456	−0.346033
$189$	0	0
$190$	−6.74456	−0.489302
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	16.7446	1.20219
$195$	0	0
$196$	1.00000	0.0714286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	3.25544	0.230772	0.115386	−	0.993321i	$-0.463190\pi$
$199$	3.25544	0.230772	0.115386	+	0.993321i	$0.463190\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	2.74456	0.193107
$203$	−8.74456	−0.613748
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0.744563	0.0518761
$207$	0	0
$208$	6.74456	0.467651
$209$	−6.74456	−0.466531
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	12.7446	0.875300
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	4.74456	0.322082
$218$	18.2337	1.23494
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	45.4891	3.05993
$222$	0	0
$223$	15.2554	1.02158	0.510790	−	0.859706i	$-0.329353\pi$
$223$	15.2554	1.02158	0.510790	+	0.859706i	$0.329353\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	10.0000	0.665190
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	6.74456	0.444723
$231$	0	0
$232$	8.74456	0.574109
$233$	24.9783	1.63638	0.818190	−	0.574948i	$-0.194978\pi$
$233$	24.9783	1.63638	0.818190	+	0.574948i	$0.194978\pi$
$234$	0	0
$235$	4.74456	0.309501
$236$	8.74456	0.569223
$237$	0	0
$238$	−6.74456	−0.437185
$239$	14.7446	0.953746	0.476873	−	0.878972i	$-0.341770\pi$
$239$	14.7446	0.953746	0.476873	+	0.878972i	$0.341770\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	1.25544	0.0803711
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−45.4891	−2.89440
$248$	−4.74456	−0.301280
$249$	0	0
$250$	1.00000	0.0632456
$251$	26.2337	1.65586	0.827928	−	0.560835i	$-0.189520\pi$
$251$	26.2337	1.65586	0.827928	+	0.560835i	$0.189520\pi$
$252$	0	0
$253$	6.74456	0.424027
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.2337	0.638360	0.319180	−	0.947694i	$-0.396593\pi$
$257$	10.2337	0.638360	0.319180	+	0.947694i	$0.396593\pi$
$258$	0	0
$259$	0.744563	0.0462649
$260$	−6.74456	−0.418280
$261$	0	0
$262$	−2.74456	−0.169560
$263$	−26.9783	−1.66355	−0.831775	−	0.555113i	$-0.812675\pi$
$263$	−26.9783	−1.66355	−0.831775	+	0.555113i	$0.812675\pi$
$264$	0	0
$265$	−12.7446	−0.782892
$266$	6.74456	0.413536
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−20.9783	−1.27907	−0.639533	−	0.768763i	$-0.720872\pi$
$269$	−20.9783	−1.27907	−0.639533	+	0.768763i	$0.720872\pi$
$270$	0	0
$271$	14.9783	0.909864	0.454932	−	0.890526i	$-0.349664\pi$
$271$	14.9783	0.909864	0.454932	+	0.890526i	$0.349664\pi$
$272$	6.74456	0.408949
$273$	0	0
$274$	3.48913	0.210786
$275$	1.00000	0.0603023
$276$	0	0
$277$	15.4891	0.930651	0.465326	−	0.885140i	$-0.345937\pi$
$277$	15.4891	0.930651	0.465326	+	0.885140i	$0.345937\pi$
$278$	−14.7446	−0.884320
$279$	0	0
$280$	1.00000	0.0597614
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	−6.74456	−0.398814
$287$	4.00000	0.236113
$288$	0	0
$289$	28.4891	1.67583
$290$	−8.74456	−0.513498
$291$	0	0
$292$	10.7446	0.628778
$293$	−13.2554	−0.774391	−0.387195	−	0.921998i	$-0.626556\pi$
$293$	−13.2554	−0.774391	−0.387195	+	0.921998i	$0.626556\pi$
$294$	0	0
$295$	−8.74456	−0.509128
$296$	−0.744563	−0.0432768
$297$	0	0
$298$	−0.744563	−0.0431314
$299$	45.4891	2.63070
$300$	0	0
$301$	4.00000	0.230556
$302$	9.25544	0.532591
$303$	0	0
$304$	−6.74456	−0.386827
$305$	−1.25544	−0.0718861
$306$	0	0
$307$	−1.48913	−0.0849889	−0.0424944	−	0.999097i	$-0.513530\pi$
$307$	−1.48913	−0.0849889	−0.0424944	+	0.999097i	$0.513530\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	4.74456	0.269473
$311$	20.7446	1.17632	0.588158	−	0.808746i	$-0.299853\pi$
$311$	20.7446	1.17632	0.588158	+	0.808746i	$0.299853\pi$
$312$	0	0
$313$	−2.23369	−0.126256	−0.0631278	−	0.998005i	$-0.520108\pi$
$313$	−2.23369	−0.126256	−0.0631278	+	0.998005i	$0.520108\pi$
$314$	−0.510875	−0.0288303
$315$	0	0
$316$	6.74456	0.379411
$317$	−2.23369	−0.125456	−0.0627282	−	0.998031i	$-0.519980\pi$
$317$	−2.23369	−0.125456	−0.0627282	+	0.998031i	$0.519980\pi$
$318$	0	0
$319$	−8.74456	−0.489602
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−6.74456	−0.375860
$323$	−45.4891	−2.53108
$324$	0	0
$325$	6.74456	0.374121
$326$	4.00000	0.221540
$327$	0	0
$328$	−4.00000	−0.220863
$329$	−4.74456	−0.261576
$330$	0	0
$331$	14.9783	0.823279	0.411640	−	0.911347i	$-0.364956\pi$
$331$	14.9783	0.823279	0.411640	+	0.911347i	$0.364956\pi$
$332$	−8.00000	−0.439057
$333$	0	0
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	−32.4891	−1.76718
$339$	0	0
$340$	−6.74456	−0.365775
$341$	4.74456	0.256932
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	20.2337	1.08777
$347$	22.9783	1.23354	0.616769	−	0.787145i	$-0.288441\pi$
$347$	22.9783	1.23354	0.616769	+	0.787145i	$0.288441\pi$
$348$	0	0
$349$	30.7446	1.64572	0.822859	−	0.568245i	$-0.192377\pi$
$349$	30.7446	1.64572	0.822859	+	0.568245i	$0.192377\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−8.74456	−0.465426	−0.232713	−	0.972545i	$-0.574760\pi$
$353$	−8.74456	−0.465426	−0.232713	+	0.972545i	$0.574760\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	−15.4891	−0.820922
$357$	0	0
$358$	−4.00000	−0.211407
$359$	1.25544	0.0662594	0.0331297	−	0.999451i	$-0.489453\pi$
$359$	1.25544	0.0662594	0.0331297	+	0.999451i	$0.489453\pi$
$360$	0	0
$361$	26.4891	1.39416
$362$	19.4891	1.02433
$363$	0	0
$364$	6.74456	0.353511
$365$	−10.7446	−0.562396
$366$	0	0
$367$	−8.74456	−0.456462	−0.228231	−	0.973607i	$-0.573294\pi$
$367$	−8.74456	−0.456462	−0.228231	+	0.973607i	$0.573294\pi$
$368$	6.74456	0.351585
$369$	0	0
$370$	0.744563	0.0387080
$371$	12.7446	0.661665
$372$	0	0
$373$	−16.9783	−0.879100	−0.439550	−	0.898218i	$-0.644862\pi$
$373$	−16.9783	−0.879100	−0.439550	+	0.898218i	$0.644862\pi$
$374$	−6.74456	−0.348753
$375$	0	0
$376$	4.74456	0.244682
$377$	−58.9783	−3.03753
$378$	0	0
$379$	−37.4891	−1.92569	−0.962844	−	0.270060i	$-0.912956\pi$
$379$	−37.4891	−1.92569	−0.962844	+	0.270060i	$0.912956\pi$
$380$	6.74456	0.345989
$381$	0	0
$382$	−16.0000	−0.818631
$383$	−19.7228	−1.00779	−0.503894	−	0.863765i	$-0.668100\pi$
$383$	−19.7228	−1.00779	−0.503894	+	0.863765i	$0.668100\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	2.00000	0.101797
$387$	0	0
$388$	−16.7446	−0.850076
$389$	7.48913	0.379714	0.189857	−	0.981812i	$-0.439198\pi$
$389$	7.48913	0.379714	0.189857	+	0.981812i	$0.439198\pi$
$390$	0	0
$391$	45.4891	2.30048
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	10.0000	0.503793
$395$	−6.74456	−0.339356
$396$	0	0
$397$	35.4891	1.78115	0.890574	−	0.454838i	$-0.150303\pi$
$397$	35.4891	1.78115	0.890574	+	0.454838i	$0.150303\pi$
$398$	−3.25544	−0.163180
$399$	0	0
$400$	1.00000	0.0500000
$401$	−23.4891	−1.17299	−0.586495	−	0.809953i	$-0.699493\pi$
$401$	−23.4891	−1.17299	−0.586495	+	0.809953i	$0.699493\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	−2.74456	−0.136547
$405$	0	0
$406$	8.74456	0.433985
$407$	0.744563	0.0369066
$408$	0	0
$409$	−21.4891	−1.06257	−0.531284	−	0.847194i	$-0.678290\pi$
$409$	−21.4891	−1.06257	−0.531284	+	0.847194i	$0.678290\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	−0.744563	−0.0366820
$413$	8.74456	0.430292
$414$	0	0
$415$	8.00000	0.392705
$416$	−6.74456	−0.330679
$417$	0	0
$418$	6.74456	0.329887
$419$	0.744563	0.0363743	0.0181871	−	0.999835i	$-0.494211\pi$
$419$	0.744563	0.0363743	0.0181871	+	0.999835i	$0.494211\pi$
$420$	0	0
$421$	4.51087	0.219847	0.109923	−	0.993940i	$-0.464939\pi$
$421$	4.51087	0.219847	0.109923	+	0.993940i	$0.464939\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−12.7446	−0.618931
$425$	6.74456	0.327159
$426$	0	0
$427$	1.25544	0.0607549
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	17.2554	0.831165	0.415583	−	0.909555i	$-0.363578\pi$
$431$	17.2554	0.831165	0.415583	+	0.909555i	$0.363578\pi$
$432$	0	0
$433$	36.7446	1.76583	0.882915	−	0.469532i	$-0.155577\pi$
$433$	36.7446	1.76583	0.882915	+	0.469532i	$0.155577\pi$
$434$	−4.74456	−0.227746
$435$	0	0
$436$	−18.2337	−0.873235
$437$	−45.4891	−2.17604
$438$	0	0
$439$	14.9783	0.714873	0.357436	−	0.933937i	$-0.383651\pi$
$439$	14.9783	0.714873	0.357436	+	0.933937i	$0.383651\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−45.4891	−2.16370
$443$	29.4891	1.40107	0.700535	−	0.713618i	$-0.252945\pi$
$443$	29.4891	1.40107	0.700535	+	0.713618i	$0.252945\pi$
$444$	0	0
$445$	15.4891	0.734255
$446$	−15.2554	−0.722366
$447$	0	0
$448$	1.00000	0.0472456
$449$	4.97825	0.234938	0.117469	−	0.993077i	$-0.462522\pi$
$449$	4.97825	0.234938	0.117469	+	0.993077i	$0.462522\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	−10.0000	−0.470360
$453$	0	0
$454$	20.0000	0.938647
$455$	−6.74456	−0.316190
$456$	0	0
$457$	3.48913	0.163214	0.0816072	−	0.996665i	$-0.473995\pi$
$457$	3.48913	0.163214	0.0816072	+	0.996665i	$0.473995\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	−6.74456	−0.314467
$461$	−33.7228	−1.57063	−0.785314	−	0.619098i	$-0.787498\pi$
$461$	−33.7228	−1.57063	−0.785314	+	0.619098i	$0.787498\pi$
$462$	0	0
$463$	1.25544	0.0583451	0.0291726	−	0.999574i	$-0.490713\pi$
$463$	1.25544	0.0583451	0.0291726	+	0.999574i	$0.490713\pi$
$464$	−8.74456	−0.405956
$465$	0	0
$466$	−24.9783	−1.15710
$467$	−16.9783	−0.785660	−0.392830	−	0.919611i	$-0.628504\pi$
$467$	−16.9783	−0.785660	−0.392830	+	0.919611i	$0.628504\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−4.74456	−0.218850
$471$	0	0
$472$	−8.74456	−0.402501
$473$	4.00000	0.183920
$474$	0	0
$475$	−6.74456	−0.309462
$476$	6.74456	0.309137
$477$	0	0
$478$	−14.7446	−0.674401
$479$	41.4891	1.89569	0.947843	−	0.318737i	$-0.103259\pi$
$479$	41.4891	1.89569	0.947843	+	0.318737i	$0.103259\pi$
$480$	0	0
$481$	5.02175	0.228972
$482$	−20.0000	−0.910975
$483$	0	0
$484$	1.00000	0.0454545
$485$	16.7446	0.760331
$486$	0	0
$487$	9.25544	0.419404	0.209702	−	0.977765i	$-0.432751\pi$
$487$	9.25544	0.419404	0.209702	+	0.977765i	$0.432751\pi$
$488$	−1.25544	−0.0568310
$489$	0	0
$490$	1.00000	0.0451754
$491$	−30.9783	−1.39803	−0.699014	−	0.715108i	$-0.746378\pi$
$491$	−30.9783	−1.39803	−0.699014	+	0.715108i	$0.746378\pi$
$492$	0	0
$493$	−58.9783	−2.65625
$494$	45.4891	2.04665
$495$	0	0
$496$	4.74456	0.213037
$497$	4.00000	0.179425
$498$	0	0
$499$	−13.4891	−0.603856	−0.301928	−	0.953331i	$-0.597630\pi$
$499$	−13.4891	−0.603856	−0.301928	+	0.953331i	$0.597630\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−26.2337	−1.17087
$503$	−6.51087	−0.290306	−0.145153	−	0.989409i	$-0.546367\pi$
$503$	−6.51087	−0.290306	−0.145153	+	0.989409i	$0.546367\pi$
$504$	0	0
$505$	2.74456	0.122131
$506$	−6.74456	−0.299832
$507$	0	0
$508$	0	0
$509$	35.4891	1.57303	0.786514	−	0.617573i	$-0.211884\pi$
$509$	35.4891	1.57303	0.786514	+	0.617573i	$0.211884\pi$
$510$	0	0
$511$	10.7446	0.475311
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.2337	−0.451389
$515$	0.744563	0.0328094
$516$	0	0
$517$	−4.74456	−0.208666
$518$	−0.744563	−0.0327142
$519$	0	0
$520$	6.74456	0.295769
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	0	0
$523$	−17.4891	−0.764746	−0.382373	−	0.924008i	$-0.624893\pi$
$523$	−17.4891	−0.764746	−0.382373	+	0.924008i	$0.624893\pi$
$524$	2.74456	0.119897
$525$	0	0
$526$	26.9783	1.17631
$527$	32.0000	1.39394
$528$	0	0
$529$	22.4891	0.977788
$530$	12.7446	0.553588
$531$	0	0
$532$	−6.74456	−0.292414
$533$	26.9783	1.16856
$534$	0	0
$535$	−12.0000	−0.518805
$536$	4.00000	0.172774
$537$	0	0
$538$	20.9783	0.904437
$539$	1.00000	0.0430730
$540$	0	0
$541$	1.76631	0.0759397	0.0379698	−	0.999279i	$-0.487911\pi$
$541$	1.76631	0.0759397	0.0379698	+	0.999279i	$0.487911\pi$
$542$	−14.9783	−0.643371
$543$	0	0
$544$	−6.74456	−0.289171
$545$	18.2337	0.781045
$546$	0	0
$547$	14.9783	0.640424	0.320212	−	0.947346i	$-0.396246\pi$
$547$	14.9783	0.640424	0.320212	+	0.947346i	$0.396246\pi$
$548$	−3.48913	−0.149048
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	58.9783	2.51256
$552$	0	0
$553$	6.74456	0.286808
$554$	−15.4891	−0.658070
$555$	0	0
$556$	14.7446	0.625309
$557$	0.978251	0.0414498	0.0207249	−	0.999785i	$-0.493403\pi$
$557$	0.978251	0.0414498	0.0207249	+	0.999785i	$0.493403\pi$
$558$	0	0
$559$	26.9783	1.14106
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	14.0000	0.590554
$563$	−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
$565$	10.0000	0.420703
$566$	−28.0000	−1.17693
$567$	0	0
$568$	−4.00000	−0.167836
$569$	−16.5109	−0.692172	−0.346086	−	0.938203i	$-0.612489\pi$
$569$	−16.5109	−0.692172	−0.346086	+	0.938203i	$0.612489\pi$
$570$	0	0
$571$	−17.4891	−0.731897	−0.365949	−	0.930635i	$-0.619255\pi$
$571$	−17.4891	−0.731897	−0.365949	+	0.930635i	$0.619255\pi$
$572$	6.74456	0.282004
$573$	0	0
$574$	−4.00000	−0.166957
$575$	6.74456	0.281268
$576$	0	0
$577$	−8.74456	−0.364041	−0.182020	−	0.983295i	$-0.558264\pi$
$577$	−8.74456	−0.364041	−0.182020	+	0.983295i	$0.558264\pi$
$578$	−28.4891	−1.18499
$579$	0	0
$580$	8.74456	0.363098
$581$	−8.00000	−0.331896
$582$	0	0
$583$	12.7446	0.527826
$584$	−10.7446	−0.444613
$585$	0	0
$586$	13.2554	0.547577
$587$	4.97825	0.205474	0.102737	−	0.994709i	$-0.467240\pi$
$587$	4.97825	0.205474	0.102737	+	0.994709i	$0.467240\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	8.74456	0.360008
$591$	0	0
$592$	0.744563	0.0306013
$593$	40.2337	1.65220	0.826100	−	0.563524i	$-0.190555\pi$
$593$	40.2337	1.65220	0.826100	+	0.563524i	$0.190555\pi$
$594$	0	0
$595$	−6.74456	−0.276500
$596$	0.744563	0.0304985
$597$	0	0
$598$	−45.4891	−1.86019
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−33.4891	−1.36605	−0.683025	−	0.730395i	$-0.739336\pi$
$601$	−33.4891	−1.36605	−0.683025	+	0.730395i	$0.739336\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−9.25544	−0.376598
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	6.74456	0.273528
$609$	0	0
$610$	1.25544	0.0508312
$611$	−32.0000	−1.29458
$612$	0	0
$613$	11.4891	0.464041	0.232021	−	0.972711i	$-0.425466\pi$
$613$	11.4891	0.464041	0.232021	+	0.972711i	$0.425466\pi$
$614$	1.48913	0.0600962
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	0.744563	0.0299265	0.0149632	−	0.999888i	$-0.495237\pi$
$619$	0.744563	0.0299265	0.0149632	+	0.999888i	$0.495237\pi$
$620$	−4.74456	−0.190546
$621$	0	0
$622$	−20.7446	−0.831781
$623$	−15.4891	−0.620559
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.23369	0.0892761
$627$	0	0
$628$	0.510875	0.0203861
$629$	5.02175	0.200230
$630$	0	0
$631$	−2.97825	−0.118562	−0.0592811	−	0.998241i	$-0.518881\pi$
$631$	−2.97825	−0.118562	−0.0592811	+	0.998241i	$0.518881\pi$
$632$	−6.74456	−0.268284
$633$	0	0
$634$	2.23369	0.0887111
$635$	0	0
$636$	0	0
$637$	6.74456	0.267229
$638$	8.74456	0.346201
$639$	0	0
$640$	1.00000	0.0395285
$641$	−11.4891	−0.453793	−0.226897	−	0.973919i	$-0.572858\pi$
$641$	−11.4891	−0.453793	−0.226897	+	0.973919i	$0.572858\pi$
$642$	0	0
$643$	46.4674	1.83249	0.916247	−	0.400613i	$-0.131203\pi$
$643$	46.4674	1.83249	0.916247	+	0.400613i	$0.131203\pi$
$644$	6.74456	0.265773
$645$	0	0
$646$	45.4891	1.78975
$647$	−8.74456	−0.343784	−0.171892	−	0.985116i	$-0.554988\pi$
$647$	−8.74456	−0.343784	−0.171892	+	0.985116i	$0.554988\pi$
$648$	0	0
$649$	8.74456	0.343254
$650$	−6.74456	−0.264544
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−15.7228	−0.615281	−0.307641	−	0.951503i	$-0.599539\pi$
$653$	−15.7228	−0.615281	−0.307641	+	0.951503i	$0.599539\pi$
$654$	0	0
$655$	−2.74456	−0.107239
$656$	4.00000	0.156174
$657$	0	0
$658$	4.74456	0.184962
$659$	41.4891	1.61619	0.808093	−	0.589054i	$-0.200500\pi$
$659$	41.4891	1.61619	0.808093	+	0.589054i	$0.200500\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−14.9783	−0.582146
$663$	0	0
$664$	8.00000	0.310460
$665$	6.74456	0.261543
$666$	0	0
$667$	−58.9783	−2.28365
$668$	0	0
$669$	0	0
$670$	−4.00000	−0.154533
$671$	1.25544	0.0484656
$672$	0	0
$673$	20.9783	0.808652	0.404326	−	0.914615i	$-0.367506\pi$
$673$	20.9783	0.808652	0.404326	+	0.914615i	$0.367506\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	32.4891	1.24958
$677$	−36.2337	−1.39257	−0.696287	−	0.717764i	$-0.745166\pi$
$677$	−36.2337	−1.39257	−0.696287	+	0.717764i	$0.745166\pi$
$678$	0	0
$679$	−16.7446	−0.642597
$680$	6.74456	0.258642
$681$	0	0
$682$	−4.74456	−0.181679
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	3.48913	0.133313
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	85.9565	3.27468
$690$	0	0
$691$	1.76631	0.0671937	0.0335968	−	0.999435i	$-0.489304\pi$
$691$	1.76631	0.0671937	0.0335968	+	0.999435i	$0.489304\pi$
$692$	−20.2337	−0.769169
$693$	0	0
$694$	−22.9783	−0.872242
$695$	−14.7446	−0.559293
$696$	0	0
$697$	26.9783	1.02187
$698$	−30.7446	−1.16370
$699$	0	0
$700$	1.00000	0.0377964
$701$	−22.2337	−0.839755	−0.419877	−	0.907581i	$-0.637927\pi$
$701$	−22.2337	−0.839755	−0.419877	+	0.907581i	$0.637927\pi$
$702$	0	0
$703$	−5.02175	−0.189399
$704$	1.00000	0.0376889
$705$	0	0
$706$	8.74456	0.329106
$707$	−2.74456	−0.103220
$708$	0	0
$709$	0.510875	0.0191863	0.00959315	−	0.999954i	$-0.496946\pi$
$709$	0.510875	0.0191863	0.00959315	+	0.999954i	$0.496946\pi$
$710$	4.00000	0.150117
$711$	0	0
$712$	15.4891	0.580480
$713$	32.0000	1.19841
$714$	0	0
$715$	−6.74456	−0.252232
$716$	4.00000	0.149487
$717$	0	0
$718$	−1.25544	−0.0468525
$719$	−7.72281	−0.288012	−0.144006	−	0.989577i	$-0.545999\pi$
$719$	−7.72281	−0.288012	−0.144006	+	0.989577i	$0.545999\pi$
$720$	0	0
$721$	−0.744563	−0.0277290
$722$	−26.4891	−0.985823
$723$	0	0
$724$	−19.4891	−0.724308
$725$	−8.74456	−0.324765
$726$	0	0
$727$	14.2337	0.527898	0.263949	−	0.964537i	$-0.414975\pi$
$727$	14.2337	0.527898	0.263949	+	0.964537i	$0.414975\pi$
$728$	−6.74456	−0.249970
$729$	0	0
$730$	10.7446	0.397674
$731$	26.9783	0.997827
$732$	0	0
$733$	−16.2337	−0.599605	−0.299802	−	0.954001i	$-0.596921\pi$
$733$	−16.2337	−0.599605	−0.299802	+	0.954001i	$0.596921\pi$
$734$	8.74456	0.322768
$735$	0	0
$736$	−6.74456	−0.248608
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−30.9783	−1.13955	−0.569777	−	0.821800i	$-0.692970\pi$
$739$	−30.9783	−1.13955	−0.569777	+	0.821800i	$0.692970\pi$
$740$	−0.744563	−0.0273707
$741$	0	0
$742$	−12.7446	−0.467868
$743$	−26.9783	−0.989736	−0.494868	−	0.868968i	$-0.664784\pi$
$743$	−26.9783	−0.989736	−0.494868	+	0.868968i	$0.664784\pi$
$744$	0	0
$745$	−0.744563	−0.0272787
$746$	16.9783	0.621618
$747$	0	0
$748$	6.74456	0.246606
$749$	12.0000	0.438470
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	−4.74456	−0.173016
$753$	0	0
$754$	58.9783	2.14786
$755$	9.25544	0.336840
$756$	0	0
$757$	−19.2554	−0.699851	−0.349925	−	0.936778i	$-0.613793\pi$
$757$	−19.2554	−0.699851	−0.349925	+	0.936778i	$0.613793\pi$
$758$	37.4891	1.36167
$759$	0	0
$760$	−6.74456	−0.244651
$761$	29.4891	1.06898	0.534490	−	0.845175i	$-0.320504\pi$
$761$	29.4891	1.06898	0.534490	+	0.845175i	$0.320504\pi$
$762$	0	0
$763$	−18.2337	−0.660104
$764$	16.0000	0.578860
$765$	0	0
$766$	19.7228	0.712614
$767$	58.9783	2.12958
$768$	0	0
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	51.4891	1.85194	0.925968	−	0.377603i	$-0.123252\pi$
$773$	51.4891	1.85194	0.925968	+	0.377603i	$0.123252\pi$
$774$	0	0
$775$	4.74456	0.170430
$776$	16.7446	0.601095
$777$	0	0
$778$	−7.48913	−0.268498
$779$	−26.9783	−0.966596
$780$	0	0
$781$	4.00000	0.143131
$782$	−45.4891	−1.62669
$783$	0	0
$784$	1.00000	0.0357143
$785$	−0.510875	−0.0182339
$786$	0	0
$787$	5.48913	0.195666	0.0978331	−	0.995203i	$-0.468809\pi$
$787$	5.48913	0.195666	0.0978331	+	0.995203i	$0.468809\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	6.74456	0.239961
$791$	−10.0000	−0.355559
$792$	0	0
$793$	8.46738	0.300685
$794$	−35.4891	−1.25946
$795$	0	0
$796$	3.25544	0.115386
$797$	42.4674	1.50427	0.752136	−	0.659008i	$-0.229024\pi$
$797$	42.4674	1.50427	0.752136	+	0.659008i	$0.229024\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	23.4891	0.829430
$803$	10.7446	0.379167
$804$	0	0
$805$	−6.74456	−0.237715
$806$	−32.0000	−1.12715
$807$	0	0
$808$	2.74456	0.0965534
$809$	−34.4674	−1.21181	−0.605904	−	0.795538i	$-0.707189\pi$
$809$	−34.4674	−1.21181	−0.605904	+	0.795538i	$0.707189\pi$
$810$	0	0
$811$	−18.7446	−0.658211	−0.329105	−	0.944293i	$-0.606747\pi$
$811$	−18.7446	−0.658211	−0.329105	+	0.944293i	$0.606747\pi$
$812$	−8.74456	−0.306874
$813$	0	0
$814$	−0.744563	−0.0260969
$815$	4.00000	0.140114
$816$	0	0
$817$	−26.9783	−0.943850
$818$	21.4891	0.751350
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−7.72281	−0.269528	−0.134764	−	0.990878i	$-0.543028\pi$
$821$	−7.72281	−0.269528	−0.134764	+	0.990878i	$0.543028\pi$
$822$	0	0
$823$	7.76631	0.270717	0.135358	−	0.990797i	$-0.456781\pi$
$823$	7.76631	0.270717	0.135358	+	0.990797i	$0.456781\pi$
$824$	0.744563	0.0259381
$825$	0	0
$826$	−8.74456	−0.304262
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	14.4674	0.502473	0.251236	−	0.967926i	$-0.419163\pi$
$829$	14.4674	0.502473	0.251236	+	0.967926i	$0.419163\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	6.74456	0.233826
$833$	6.74456	0.233685
$834$	0	0
$835$	0	0
$836$	−6.74456	−0.233266
$837$	0	0
$838$	−0.744563	−0.0257205
$839$	3.25544	0.112390	0.0561951	−	0.998420i	$-0.482103\pi$
$839$	3.25544	0.112390	0.0561951	+	0.998420i	$0.482103\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	−4.51087	−0.155455
$843$	0	0
$844$	−12.0000	−0.413057
$845$	−32.4891	−1.11766
$846$	0	0
$847$	1.00000	0.0343604
$848$	12.7446	0.437650
$849$	0	0
$850$	−6.74456	−0.231337
$851$	5.02175	0.172143
$852$	0	0
$853$	−22.7446	−0.778759	−0.389379	−	0.921077i	$-0.627311\pi$
$853$	−22.7446	−0.778759	−0.389379	+	0.921077i	$0.627311\pi$
$854$	−1.25544	−0.0429602
$855$	0	0
$856$	−12.0000	−0.410152
$857$	43.2119	1.47609	0.738046	−	0.674751i	$-0.235749\pi$
$857$	43.2119	1.47609	0.738046	+	0.674751i	$0.235749\pi$
$858$	0	0
$859$	−29.2119	−0.996698	−0.498349	−	0.866976i	$-0.666060\pi$
$859$	−29.2119	−0.996698	−0.498349	+	0.866976i	$0.666060\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−17.2554	−0.587723
$863$	40.2337	1.36957	0.684785	−	0.728745i	$-0.259896\pi$
$863$	40.2337	1.36957	0.684785	+	0.728745i	$0.259896\pi$
$864$	0	0
$865$	20.2337	0.687966
$866$	−36.7446	−1.24863
$867$	0	0
$868$	4.74456	0.161041
$869$	6.74456	0.228794
$870$	0	0
$871$	−26.9783	−0.914123
$872$	18.2337	0.617471
$873$	0	0
$874$	45.4891	1.53869
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−26.4674	−0.893740	−0.446870	−	0.894599i	$-0.647461\pi$
$877$	−26.4674	−0.893740	−0.446870	+	0.894599i	$0.647461\pi$
$878$	−14.9783	−0.505491
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	55.4891	1.86948	0.934738	−	0.355338i	$-0.115634\pi$
$881$	55.4891	1.86948	0.934738	+	0.355338i	$0.115634\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	45.4891	1.52996
$885$	0	0
$886$	−29.4891	−0.990707
$887$	41.4891	1.39307	0.696534	−	0.717524i	$-0.254724\pi$
$887$	41.4891	1.39307	0.696534	+	0.717524i	$0.254724\pi$
$888$	0	0
$889$	0	0
$890$	−15.4891	−0.519197
$891$	0	0
$892$	15.2554	0.510790
$893$	32.0000	1.07084
$894$	0	0
$895$	−4.00000	−0.133705
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−4.97825	−0.166126
$899$	−41.4891	−1.38374
$900$	0	0
$901$	85.9565	2.86363
$902$	−4.00000	−0.133185
$903$	0	0
$904$	10.0000	0.332595
$905$	19.4891	0.647840
$906$	0	0
$907$	−14.5109	−0.481826	−0.240913	−	0.970547i	$-0.577447\pi$
$907$	−14.5109	−0.481826	−0.240913	+	0.970547i	$0.577447\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	6.74456	0.223580
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	−3.48913	−0.115410
$915$	0	0
$916$	−6.00000	−0.198246
$917$	2.74456	0.0906334
$918$	0	0
$919$	−55.2119	−1.82127	−0.910637	−	0.413207i	$-0.864408\pi$
$919$	−55.2119	−1.82127	−0.910637	+	0.413207i	$0.864408\pi$
$920$	6.74456	0.222362
$921$	0	0
$922$	33.7228	1.11060
$923$	26.9783	0.888000
$924$	0	0
$925$	0.744563	0.0244811
$926$	−1.25544	−0.0412562
$927$	0	0
$928$	8.74456	0.287054
$929$	28.9783	0.950746	0.475373	−	0.879784i	$-0.342313\pi$
$929$	28.9783	0.950746	0.475373	+	0.879784i	$0.342313\pi$
$930$	0	0
$931$	−6.74456	−0.221044
$932$	24.9783	0.818190
$933$	0	0
$934$	16.9783	0.555545
$935$	−6.74456	−0.220571
$936$	0	0
$937$	−18.7446	−0.612358	−0.306179	−	0.951974i	$-0.599051\pi$
$937$	−18.7446	−0.612358	−0.306179	+	0.951974i	$0.599051\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	4.74456	0.154751
$941$	−12.2337	−0.398807	−0.199403	−	0.979917i	$-0.563900\pi$
$941$	−12.2337	−0.398807	−0.199403	+	0.979917i	$0.563900\pi$
$942$	0	0
$943$	26.9783	0.878533
$944$	8.74456	0.284611
$945$	0	0
$946$	−4.00000	−0.130051
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	72.4674	2.35239
$950$	6.74456	0.218823
$951$	0	0
$952$	−6.74456	−0.218593
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	14.7446	0.476873
$957$	0	0
$958$	−41.4891	−1.34045
$959$	−3.48913	−0.112670
$960$	0	0
$961$	−8.48913	−0.273843
$962$	−5.02175	−0.161908
$963$	0	0
$964$	20.0000	0.644157
$965$	2.00000	0.0643823
$966$	0	0
$967$	−50.9783	−1.63935	−0.819675	−	0.572829i	$-0.805846\pi$
$967$	−50.9783	−1.63935	−0.819675	+	0.572829i	$0.805846\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−16.7446	−0.537636
$971$	−19.7228	−0.632935	−0.316468	−	0.948603i	$-0.602497\pi$
$971$	−19.7228	−0.632935	−0.316468	+	0.948603i	$0.602497\pi$
$972$	0	0
$973$	14.7446	0.472689
$974$	−9.25544	−0.296563
$975$	0	0
$976$	1.25544	0.0401856
$977$	−12.9783	−0.415211	−0.207606	−	0.978213i	$-0.566567\pi$
$977$	−12.9783	−0.415211	−0.207606	+	0.978213i	$0.566567\pi$
$978$	0	0
$979$	−15.4891	−0.495035
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	30.9783	0.988556
$983$	−2.23369	−0.0712436	−0.0356218	−	0.999365i	$-0.511341\pi$
$983$	−2.23369	−0.0712436	−0.0356218	+	0.999365i	$0.511341\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	58.9783	1.87825
$987$	0	0
$988$	−45.4891	−1.44720
$989$	26.9783	0.857858
$990$	0	0
$991$	1.48913	0.0473036	0.0236518	−	0.999720i	$-0.492471\pi$
$991$	1.48913	0.0473036	0.0236518	+	0.999720i	$0.492471\pi$
$992$	−4.74456	−0.150640
$993$	0	0
$994$	−4.00000	−0.126872
$995$	−3.25544	−0.103204
$996$	0	0
$997$	39.2119	1.24185	0.620927	−	0.783868i	$-0.286756\pi$
$997$	39.2119	1.24185	0.620927	+	0.783868i	$0.286756\pi$
$998$	13.4891	0.426991
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.bo.1.2		2
3.2	odd	2		770.2.a.k.1.2	✓	2
12.11	even	2		6160.2.a.r.1.2		2
15.2	even	4		3850.2.c.y.1849.3		4
15.8	even	4		3850.2.c.y.1849.2		4
15.14	odd	2		3850.2.a.bc.1.1		2
21.20	even	2		5390.2.a.bq.1.1		2
33.32	even	2		8470.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.k.1.2	✓	2	3.2	odd	2
3850.2.a.bc.1.1		2	15.14	odd	2
3850.2.c.y.1849.2		4	15.8	even	4
3850.2.c.y.1849.3		4	15.2	even	4
5390.2.a.bq.1.1		2	21.20	even	2
6160.2.a.r.1.2		2	12.11	even	2
6930.2.a.bo.1.2		2	1.1	even	1	trivial
8470.2.a.bu.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +6.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.74456 q^{17} -6.74456 q^{19} -1.00000 q^{20} -1.00000 q^{22} +6.74456 q^{23} +1.00000 q^{25} -6.74456 q^{26} +1.00000 q^{28} -8.74456 q^{29} +4.74456 q^{31} -1.00000 q^{32} -6.74456 q^{34} -1.00000 q^{35} +0.744563 q^{37} +6.74456 q^{38} +1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -6.74456 q^{46} -4.74456 q^{47} +1.00000 q^{49} -1.00000 q^{50} +6.74456 q^{52} +12.7446 q^{53} -1.00000 q^{55} -1.00000 q^{56} +8.74456 q^{58} +8.74456 q^{59} +1.25544 q^{61} -4.74456 q^{62} +1.00000 q^{64} -6.74456 q^{65} -4.00000 q^{67} +6.74456 q^{68} +1.00000 q^{70} +4.00000 q^{71} +10.7446 q^{73} -0.744563 q^{74} -6.74456 q^{76} +1.00000 q^{77} +6.74456 q^{79} -1.00000 q^{80} -4.00000 q^{82} -8.00000 q^{83} -6.74456 q^{85} -4.00000 q^{86} -1.00000 q^{88} -15.4891 q^{89} +6.74456 q^{91} +6.74456 q^{92} +4.74456 q^{94} +6.74456 q^{95} -16.7446 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 2 q^{25} - 2 q^{26} + 2 q^{28} - 6 q^{29} - 2 q^{31} - 2 q^{32} - 2 q^{34} - 2 q^{35} - 10 q^{37} + 2 q^{38} + 2 q^{40} + 8 q^{41} + 8 q^{43} + 2 q^{44} - 2 q^{46} + 2 q^{47} + 2 q^{49} - 2 q^{50} + 2 q^{52} + 14 q^{53} - 2 q^{55} - 2 q^{56} + 6 q^{58} + 6 q^{59} + 14 q^{61} + 2 q^{62} + 2 q^{64} - 2 q^{65} - 8 q^{67} + 2 q^{68} + 2 q^{70} + 8 q^{71} + 10 q^{73} + 10 q^{74} - 2 q^{76} + 2 q^{77} + 2 q^{79} - 2 q^{80} - 8 q^{82} - 16 q^{83} - 2 q^{85} - 8 q^{86} - 2 q^{88} - 8 q^{89} + 2 q^{91} + 2 q^{92} - 2 q^{94} + 2 q^{95} - 22 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 2 * q^10 + 2 * q^11 + 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 2 * q^19 - 2 * q^20 - 2 * q^22 + 2 * q^23 + 2 * q^25 - 2 * q^26 + 2 * q^28 - 6 * q^29 - 2 * q^31 - 2 * q^32 - 2 * q^34 - 2 * q^35 - 10 * q^37 + 2 * q^38 + 2 * q^40 + 8 * q^41 + 8 * q^43 + 2 * q^44 - 2 * q^46 + 2 * q^47 + 2 * q^49 - 2 * q^50 + 2 * q^52 + 14 * q^53 - 2 * q^55 - 2 * q^56 + 6 * q^58 + 6 * q^59 + 14 * q^61 + 2 * q^62 + 2 * q^64 - 2 * q^65 - 8 * q^67 + 2 * q^68 + 2 * q^70 + 8 * q^71 + 10 * q^73 + 10 * q^74 - 2 * q^76 + 2 * q^77 + 2 * q^79 - 2 * q^80 - 8 * q^82 - 16 * q^83 - 2 * q^85 - 8 * q^86 - 2 * q^88 - 8 * q^89 + 2 * q^91 + 2 * q^92 - 2 * q^94 + 2 * q^95 - 22 * q^97 - 2 * q^98

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