LMFDB - Embedded newform 6930.2.a.bo.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(\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			1.1
Root			$-2.37228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -4.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.74456 q^{17} +4.74456 q^{19} -1.00000 q^{20} -1.00000 q^{22} -4.74456 q^{23} +1.00000 q^{25} +4.74456 q^{26} +1.00000 q^{28} +2.74456 q^{29} -6.74456 q^{31} -1.00000 q^{32} +4.74456 q^{34} -1.00000 q^{35} -10.7446 q^{37} -4.74456 q^{38} +1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.74456 q^{46} +6.74456 q^{47} +1.00000 q^{49} -1.00000 q^{50} -4.74456 q^{52} +1.25544 q^{53} -1.00000 q^{55} -1.00000 q^{56} -2.74456 q^{58} -2.74456 q^{59} +12.7446 q^{61} +6.74456 q^{62} +1.00000 q^{64} +4.74456 q^{65} -4.00000 q^{67} -4.74456 q^{68} +1.00000 q^{70} +4.00000 q^{71} -0.744563 q^{73} +10.7446 q^{74} +4.74456 q^{76} +1.00000 q^{77} -4.74456 q^{79} -1.00000 q^{80} -4.00000 q^{82} -8.00000 q^{83} +4.74456 q^{85} -4.00000 q^{86} -1.00000 q^{88} +7.48913 q^{89} -4.74456 q^{91} -4.74456 q^{92} -6.74456 q^{94} -4.74456 q^{95} -5.25544 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$13$	−4.74456	−1.31590	−0.657952	−	0.753059i	$-0.728577\pi$
$13$	−4.74456	−1.31590	−0.657952	+	0.753059i	$0.728577\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.74456	−1.15073	−0.575363	−	0.817898i	$-0.695139\pi$
$17$	−4.74456	−1.15073	−0.575363	+	0.817898i	$0.695139\pi$
$18$	0	0
$19$	4.74456	1.08848	0.544239	−	0.838930i	$-0.316819\pi$
$19$	4.74456	1.08848	0.544239	+	0.838930i	$0.316819\pi$
$20$	−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
$26$	4.74456	0.930485
$27$	0	0
$28$	1.00000	0.188982
$29$	2.74456	0.509652	0.254826	−	0.966987i	$-0.417982\pi$
$29$	2.74456	0.509652	0.254826	+	0.966987i	$0.417982\pi$
$30$	0	0
$31$	−6.74456	−1.21136	−0.605680	−	0.795709i	$-0.707099\pi$
$31$	−6.74456	−1.21136	−0.605680	+	0.795709i	$0.707099\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.74456	0.813686
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−10.7446	−1.76640	−0.883198	−	0.469001i	$-0.844614\pi$
$37$	−10.7446	−1.76640	−0.883198	+	0.469001i	$0.844614\pi$
$38$	−4.74456	−0.769670
$39$	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
$46$	4.74456	0.699548
$47$	6.74456	0.983796	0.491898	−	0.870653i	$-0.336303\pi$
$47$	6.74456	0.983796	0.491898	+	0.870653i	$0.336303\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−4.74456	−0.657952
$53$	1.25544	0.172448	0.0862238	−	0.996276i	$-0.472520\pi$
$53$	1.25544	0.172448	0.0862238	+	0.996276i	$0.472520\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−2.74456	−0.360379
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	12.7446	1.63177	0.815887	−	0.578211i	$-0.196249\pi$
$61$	12.7446	1.63177	0.815887	+	0.578211i	$0.196249\pi$
$62$	6.74456	0.856560
$63$	0	0
$64$	1.00000	0.125000
$65$	4.74456	0.588491
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−4.74456	−0.575363
$69$	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
$73$	−0.744563	−0.0871445	−0.0435722	−	0.999050i	$-0.513874\pi$
$73$	−0.744563	−0.0871445	−0.0435722	+	0.999050i	$0.513874\pi$
$74$	10.7446	1.24903
$75$	0	0
$76$	4.74456	0.544239
$77$	1.00000	0.113961
$78$	0	0
$79$	−4.74456	−0.533805	−0.266903	−	0.963724i	$-0.586000\pi$
$79$	−4.74456	−0.533805	−0.266903	+	0.963724i	$0.586000\pi$
$80$	−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
$85$	4.74456	0.514620
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	7.48913	0.793846	0.396923	−	0.917852i	$-0.370078\pi$
$89$	7.48913	0.793846	0.396923	+	0.917852i	$0.370078\pi$
$90$	0	0
$91$	−4.74456	−0.497365
$92$	−4.74456	−0.494655
$93$	0	0
$94$	−6.74456	−0.695649
$95$	−4.74456	−0.486782
$96$	0	0
$97$	−5.25544	−0.533609	−0.266804	−	0.963751i	$-0.585968\pi$
$97$	−5.25544	−0.533609	−0.266804	+	0.963751i	$0.585968\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	8.74456	0.870117	0.435058	−	0.900402i	$-0.356728\pi$
$101$	8.74456	0.870117	0.435058	+	0.900402i	$0.356728\pi$
$102$	0	0
$103$	10.7446	1.05869	0.529347	−	0.848406i	$-0.322437\pi$
$103$	10.7446	1.05869	0.529347	+	0.848406i	$0.322437\pi$
$104$	4.74456	0.465243
$105$	0	0
$106$	−1.25544	−0.121939
$107$	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
$109$	16.2337	1.55491	0.777453	−	0.628941i	$-0.216511\pi$
$109$	16.2337	1.55491	0.777453	+	0.628941i	$0.216511\pi$
$110$	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
$115$	4.74456	0.442433
$116$	2.74456	0.254826
$117$	0	0
$118$	2.74456	0.252657
$119$	−4.74456	−0.434933
$120$	0	0
$121$	1.00000	0.0909091
$122$	−12.7446	−1.15384
$123$	0	0
$124$	−6.74456	−0.605680
$125$	−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
$130$	−4.74456	−0.416126
$131$	−8.74456	−0.764016	−0.382008	−	0.924159i	$-0.624767\pi$
$131$	−8.74456	−0.764016	−0.382008	+	0.924159i	$0.624767\pi$
$132$	0	0
$133$	4.74456	0.411406
$134$	4.00000	0.345547
$135$	0	0
$136$	4.74456	0.406843
$137$	19.4891	1.66507	0.832534	−	0.553974i	$-0.186889\pi$
$137$	19.4891	1.66507	0.832534	+	0.553974i	$0.186889\pi$
$138$	0	0
$139$	3.25544	0.276123	0.138061	−	0.990424i	$-0.455913\pi$
$139$	3.25544	0.276123	0.138061	+	0.990424i	$0.455913\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−4.00000	−0.335673
$143$	−4.74456	−0.396760
$144$	0	0
$145$	−2.74456	−0.227924
$146$	0.744563	0.0616204
$147$	0	0
$148$	−10.7446	−0.883198
$149$	−10.7446	−0.880229	−0.440114	−	0.897942i	$-0.645062\pi$
$149$	−10.7446	−0.880229	−0.440114	+	0.897942i	$0.645062\pi$
$150$	0	0
$151$	−20.7446	−1.68817	−0.844084	−	0.536211i	$-0.819855\pi$
$151$	−20.7446	−1.68817	−0.844084	+	0.536211i	$0.819855\pi$
$152$	−4.74456	−0.384835
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	6.74456	0.541736
$156$	0	0
$157$	23.4891	1.87464	0.937318	−	0.348475i	$-0.113300\pi$
$157$	23.4891	1.87464	0.937318	+	0.348475i	$0.113300\pi$
$158$	4.74456	0.377457
$159$	0	0
$160$	1.00000	0.0790569
$161$	−4.74456	−0.373924
$162$	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
$169$	9.51087	0.731606
$170$	−4.74456	−0.363891
$171$	0	0
$172$	4.00000	0.304997
$173$	14.2337	1.08217	0.541084	−	0.840969i	$-0.318014\pi$
$173$	14.2337	1.08217	0.541084	+	0.840969i	$0.318014\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	0	0
$178$	−7.48913	−0.561334
$179$	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
$181$	3.48913	0.259345	0.129672	−	0.991557i	$-0.458607\pi$
$181$	3.48913	0.259345	0.129672	+	0.991557i	$0.458607\pi$
$182$	4.74456	0.351690
$183$	0	0
$184$	4.74456	0.349774
$185$	10.7446	0.789956
$186$	0	0
$187$	−4.74456	−0.346957
$188$	6.74456	0.491898
$189$	0	0
$190$	4.74456	0.344207
$191$	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	$0.522932\pi$
$194$	5.25544	0.377318
$195$	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
$199$	14.7446	1.04521	0.522607	−	0.852574i	$-0.324959\pi$
$199$	14.7446	1.04521	0.522607	+	0.852574i	$0.324959\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−8.74456	−0.615265
$203$	2.74456	0.192631
$204$	0	0
$205$	−4.00000	−0.279372
$206$	−10.7446	−0.748609
$207$	0	0
$208$	−4.74456	−0.328976
$209$	4.74456	0.328188
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	1.25544	0.0862238
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	0	0
$217$	−6.74456	−0.457851
$218$	−16.2337	−1.09948
$219$	0	0
$220$	−1.00000	−0.0674200
$221$	22.5109	1.51425
$222$	0	0
$223$	26.7446	1.79095	0.895474	−	0.445113i	$-0.146837\pi$
$223$	26.7446	1.79095	0.895474	+	0.445113i	$0.146837\pi$
$224$	−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	$0.563524\pi$
$230$	−4.74456	−0.312847
$231$	0	0
$232$	−2.74456	−0.180189
$233$	−20.9783	−1.37433	−0.687165	−	0.726501i	$-0.741145\pi$
$233$	−20.9783	−1.37433	−0.687165	+	0.726501i	$0.741145\pi$
$234$	0	0
$235$	−6.74456	−0.439967
$236$	−2.74456	−0.178656
$237$	0	0
$238$	4.74456	0.307544
$239$	3.25544	0.210577	0.105288	−	0.994442i	$-0.466423\pi$
$239$	3.25544	0.210577	0.105288	+	0.994442i	$0.466423\pi$
$240$	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
$244$	12.7446	0.815887
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−22.5109	−1.43233
$248$	6.74456	0.428280
$249$	0	0
$250$	1.00000	0.0632456
$251$	−8.23369	−0.519706	−0.259853	−	0.965648i	$-0.583674\pi$
$251$	−8.23369	−0.519706	−0.259853	+	0.965648i	$0.583674\pi$
$252$	0	0
$253$	−4.74456	−0.298288
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.2337	−1.51166	−0.755828	−	0.654770i	$-0.772765\pi$
$257$	−24.2337	−1.51166	−0.755828	+	0.654770i	$0.772765\pi$
$258$	0	0
$259$	−10.7446	−0.667635
$260$	4.74456	0.294245
$261$	0	0
$262$	8.74456	0.540241
$263$	18.9783	1.17025	0.585125	−	0.810943i	$-0.301046\pi$
$263$	18.9783	1.17025	0.585125	+	0.810943i	$0.301046\pi$
$264$	0	0
$265$	−1.25544	−0.0771209
$266$	−4.74456	−0.290908
$267$	0	0
$268$	−4.00000	−0.244339
$269$	24.9783	1.52295	0.761475	−	0.648194i	$-0.224475\pi$
$269$	24.9783	1.52295	0.761475	+	0.648194i	$0.224475\pi$
$270$	0	0
$271$	−30.9783	−1.88179	−0.940897	−	0.338692i	$-0.890016\pi$
$271$	−30.9783	−1.88179	−0.940897	+	0.338692i	$0.890016\pi$
$272$	−4.74456	−0.287681
$273$	0	0
$274$	−19.4891	−1.17738
$275$	1.00000	0.0603023
$276$	0	0
$277$	−7.48913	−0.449978	−0.224989	−	0.974361i	$-0.572235\pi$
$277$	−7.48913	−0.449978	−0.224989	+	0.974361i	$0.572235\pi$
$278$	−3.25544	−0.195248
$279$	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
$286$	4.74456	0.280552
$287$	4.00000	0.236113
$288$	0	0
$289$	5.51087	0.324169
$290$	2.74456	0.161166
$291$	0	0
$292$	−0.744563	−0.0435722
$293$	−24.7446	−1.44559	−0.722796	−	0.691061i	$-0.757144\pi$
$293$	−24.7446	−1.44559	−0.722796	+	0.691061i	$0.757144\pi$
$294$	0	0
$295$	2.74456	0.159795
$296$	10.7446	0.624515
$297$	0	0
$298$	10.7446	0.622416
$299$	22.5109	1.30184
$300$	0	0
$301$	4.00000	0.230556
$302$	20.7446	1.19372
$303$	0	0
$304$	4.74456	0.272119
$305$	−12.7446	−0.729752
$306$	0	0
$307$	21.4891	1.22645	0.613225	−	0.789909i	$-0.289872\pi$
$307$	21.4891	1.22645	0.613225	+	0.789909i	$0.289872\pi$
$308$	1.00000	0.0569803
$309$	0	0
$310$	−6.74456	−0.383065
$311$	9.25544	0.524828	0.262414	−	0.964955i	$-0.415481\pi$
$311$	9.25544	0.524828	0.262414	+	0.964955i	$0.415481\pi$
$312$	0	0
$313$	32.2337	1.82196	0.910978	−	0.412455i	$-0.135329\pi$
$313$	32.2337	1.82196	0.910978	+	0.412455i	$0.135329\pi$
$314$	−23.4891	−1.32557
$315$	0	0
$316$	−4.74456	−0.266903
$317$	32.2337	1.81042	0.905212	−	0.424960i	$-0.139712\pi$
$317$	32.2337	1.81042	0.905212	+	0.424960i	$0.139712\pi$
$318$	0	0
$319$	2.74456	0.153666
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	4.74456	0.264404
$323$	−22.5109	−1.25254
$324$	0	0
$325$	−4.74456	−0.263181
$326$	4.00000	0.221540
$327$	0	0
$328$	−4.00000	−0.220863
$329$	6.74456	0.371840
$330$	0	0
$331$	−30.9783	−1.70272	−0.851359	−	0.524583i	$-0.824221\pi$
$331$	−30.9783	−1.70272	−0.851359	+	0.524583i	$0.824221\pi$
$332$	−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	$0.249528\pi$
$338$	−9.51087	−0.517323
$339$	0	0
$340$	4.74456	0.257310
$341$	−6.74456	−0.365239
$342$	0	0
$343$	1.00000	0.0539949
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−14.2337	−0.765208
$347$	−22.9783	−1.23354	−0.616769	−	0.787145i	$-0.711559\pi$
$347$	−22.9783	−1.23354	−0.616769	+	0.787145i	$0.711559\pi$
$348$	0	0
$349$	19.2554	1.03072	0.515360	−	0.856974i	$-0.327658\pi$
$349$	19.2554	1.03072	0.515360	+	0.856974i	$0.327658\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	2.74456	0.146078	0.0730392	−	0.997329i	$-0.476730\pi$
$353$	2.74456	0.146078	0.0730392	+	0.997329i	$0.476730\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	7.48913	0.396923
$357$	0	0
$358$	−4.00000	−0.211407
$359$	12.7446	0.672632	0.336316	−	0.941749i	$-0.390819\pi$
$359$	12.7446	0.672632	0.336316	+	0.941749i	$0.390819\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	−3.48913	−0.183384
$363$	0	0
$364$	−4.74456	−0.248683
$365$	0.744563	0.0389722
$366$	0	0
$367$	2.74456	0.143265	0.0716325	−	0.997431i	$-0.477179\pi$
$367$	2.74456	0.143265	0.0716325	+	0.997431i	$0.477179\pi$
$368$	−4.74456	−0.247327
$369$	0	0
$370$	−10.7446	−0.558583
$371$	1.25544	0.0651791
$372$	0	0
$373$	28.9783	1.50044	0.750218	−	0.661190i	$-0.229948\pi$
$373$	28.9783	1.50044	0.750218	+	0.661190i	$0.229948\pi$
$374$	4.74456	0.245335
$375$	0	0
$376$	−6.74456	−0.347824
$377$	−13.0217	−0.670654
$378$	0	0
$379$	−14.5109	−0.745374	−0.372687	−	0.927957i	$-0.621563\pi$
$379$	−14.5109	−0.745374	−0.372687	+	0.927957i	$0.621563\pi$
$380$	−4.74456	−0.243391
$381$	0	0
$382$	−16.0000	−0.818631
$383$	37.7228	1.92755	0.963773	−	0.266724i	$-0.0859413\pi$
$383$	37.7228	1.92755	0.963773	+	0.266724i	$0.0859413\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	2.00000	0.101797
$387$	0	0
$388$	−5.25544	−0.266804
$389$	−15.4891	−0.785330	−0.392665	−	0.919682i	$-0.628447\pi$
$389$	−15.4891	−0.785330	−0.392665	+	0.919682i	$0.628447\pi$
$390$	0	0
$391$	22.5109	1.13842
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	10.0000	0.503793
$395$	4.74456	0.238725
$396$	0	0
$397$	12.5109	0.627903	0.313951	−	0.949439i	$-0.398347\pi$
$397$	12.5109	0.627903	0.313951	+	0.949439i	$0.398347\pi$
$398$	−14.7446	−0.739078
$399$	0	0
$400$	1.00000	0.0500000
$401$	−0.510875	−0.0255119	−0.0127559	−	0.999919i	$-0.504060\pi$
$401$	−0.510875	−0.0255119	−0.0127559	+	0.999919i	$0.504060\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	8.74456	0.435058
$405$	0	0
$406$	−2.74456	−0.136210
$407$	−10.7446	−0.532588
$408$	0	0
$409$	1.48913	0.0736325	0.0368163	−	0.999322i	$-0.488278\pi$
$409$	1.48913	0.0736325	0.0368163	+	0.999322i	$0.488278\pi$
$410$	4.00000	0.197546
$411$	0	0
$412$	10.7446	0.529347
$413$	−2.74456	−0.135051
$414$	0	0
$415$	8.00000	0.392705
$416$	4.74456	0.232621
$417$	0	0
$418$	−4.74456	−0.232064
$419$	−10.7446	−0.524906	−0.262453	−	0.964945i	$-0.584532\pi$
$419$	−10.7446	−0.524906	−0.262453	+	0.964945i	$0.584532\pi$
$420$	0	0
$421$	27.4891	1.33974	0.669869	−	0.742479i	$-0.266350\pi$
$421$	27.4891	1.33974	0.669869	+	0.742479i	$0.266350\pi$
$422$	12.0000	0.584151
$423$	0	0
$424$	−1.25544	−0.0609694
$425$	−4.74456	−0.230145
$426$	0	0
$427$	12.7446	0.616753
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	28.7446	1.38458	0.692288	−	0.721621i	$-0.256603\pi$
$431$	28.7446	1.38458	0.692288	+	0.721621i	$0.256603\pi$
$432$	0	0
$433$	25.2554	1.21370	0.606849	−	0.794817i	$-0.292433\pi$
$433$	25.2554	1.21370	0.606849	+	0.794817i	$0.292433\pi$
$434$	6.74456	0.323749
$435$	0	0
$436$	16.2337	0.777453
$437$	−22.5109	−1.07684
$438$	0	0
$439$	−30.9783	−1.47851	−0.739256	−	0.673425i	$-0.764822\pi$
$439$	−30.9783	−1.47851	−0.739256	+	0.673425i	$0.764822\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	−22.5109	−1.07073
$443$	6.51087	0.309341	0.154670	−	0.987966i	$-0.450568\pi$
$443$	6.51087	0.309341	0.154670	+	0.987966i	$0.450568\pi$
$444$	0	0
$445$	−7.48913	−0.355019
$446$	−26.7446	−1.26639
$447$	0	0
$448$	1.00000	0.0472456
$449$	−40.9783	−1.93388	−0.966942	−	0.254998i	$-0.917925\pi$
$449$	−40.9783	−1.93388	−0.966942	+	0.254998i	$0.917925\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	−10.0000	−0.470360
$453$	0	0
$454$	20.0000	0.938647
$455$	4.74456	0.222429
$456$	0	0
$457$	−19.4891	−0.911663	−0.455831	−	0.890066i	$-0.650658\pi$
$457$	−19.4891	−0.911663	−0.455831	+	0.890066i	$0.650658\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	4.74456	0.221216
$461$	23.7228	1.10488	0.552441	−	0.833552i	$-0.313697\pi$
$461$	23.7228	1.10488	0.552441	+	0.833552i	$0.313697\pi$
$462$	0	0
$463$	12.7446	0.592290	0.296145	−	0.955143i	$-0.404299\pi$
$463$	12.7446	0.592290	0.296145	+	0.955143i	$0.404299\pi$
$464$	2.74456	0.127413
$465$	0	0
$466$	20.9783	0.971799
$467$	28.9783	1.34095	0.670477	−	0.741931i	$-0.266090\pi$
$467$	28.9783	1.34095	0.670477	+	0.741931i	$0.266090\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	6.74456	0.311103
$471$	0	0
$472$	2.74456	0.126329
$473$	4.00000	0.183920
$474$	0	0
$475$	4.74456	0.217695
$476$	−4.74456	−0.217467
$477$	0	0
$478$	−3.25544	−0.148900
$479$	18.5109	0.845783	0.422892	−	0.906180i	$-0.361015\pi$
$479$	18.5109	0.845783	0.422892	+	0.906180i	$0.361015\pi$
$480$	0	0
$481$	50.9783	2.32441
$482$	−20.0000	−0.910975
$483$	0	0
$484$	1.00000	0.0454545
$485$	5.25544	0.238637
$486$	0	0
$487$	20.7446	0.940026	0.470013	−	0.882660i	$-0.344249\pi$
$487$	20.7446	0.940026	0.470013	+	0.882660i	$0.344249\pi$
$488$	−12.7446	−0.576919
$489$	0	0
$490$	1.00000	0.0451754
$491$	14.9783	0.675959	0.337979	−	0.941153i	$-0.390257\pi$
$491$	14.9783	0.675959	0.337979	+	0.941153i	$0.390257\pi$
$492$	0	0
$493$	−13.0217	−0.586470
$494$	22.5109	1.01281
$495$	0	0
$496$	−6.74456	−0.302840
$497$	4.00000	0.179425
$498$	0	0
$499$	9.48913	0.424792	0.212396	−	0.977184i	$-0.431873\pi$
$499$	9.48913	0.424792	0.212396	+	0.977184i	$0.431873\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	8.23369	0.367487
$503$	−29.4891	−1.31486	−0.657428	−	0.753518i	$-0.728355\pi$
$503$	−29.4891	−1.31486	−0.657428	+	0.753518i	$0.728355\pi$
$504$	0	0
$505$	−8.74456	−0.389128
$506$	4.74456	0.210922
$507$	0	0
$508$	0	0
$509$	12.5109	0.554535	0.277267	−	0.960793i	$-0.410571\pi$
$509$	12.5109	0.554535	0.277267	+	0.960793i	$0.410571\pi$
$510$	0	0
$511$	−0.744563	−0.0329375
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	24.2337	1.06890
$515$	−10.7446	−0.473462
$516$	0	0
$517$	6.74456	0.296626
$518$	10.7446	0.472089
$519$	0	0
$520$	−4.74456	−0.208063
$521$	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
$523$	5.48913	0.240023	0.120011	−	0.992773i	$-0.461707\pi$
$523$	5.48913	0.240023	0.120011	+	0.992773i	$0.461707\pi$
$524$	−8.74456	−0.382008
$525$	0	0
$526$	−18.9783	−0.827491
$527$	32.0000	1.39394
$528$	0	0
$529$	−0.489125	−0.0212663
$530$	1.25544	0.0545327
$531$	0	0
$532$	4.74456	0.205703
$533$	−18.9783	−0.822039
$534$	0	0
$535$	−12.0000	−0.518805
$536$	4.00000	0.172774
$537$	0	0
$538$	−24.9783	−1.07689
$539$	1.00000	0.0430730
$540$	0	0
$541$	36.2337	1.55781	0.778904	−	0.627143i	$-0.215776\pi$
$541$	36.2337	1.55781	0.778904	+	0.627143i	$0.215776\pi$
$542$	30.9783	1.33063
$543$	0	0
$544$	4.74456	0.203421
$545$	−16.2337	−0.695375
$546$	0	0
$547$	−30.9783	−1.32453	−0.662267	−	0.749268i	$-0.730406\pi$
$547$	−30.9783	−1.32453	−0.662267	+	0.749268i	$0.730406\pi$
$548$	19.4891	0.832534
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	13.0217	0.554745
$552$	0	0
$553$	−4.74456	−0.201759
$554$	7.48913	0.318182
$555$	0	0
$556$	3.25544	0.138061
$557$	−44.9783	−1.90579	−0.952895	−	0.303301i	$-0.901911\pi$
$557$	−44.9783	−1.90579	−0.952895	+	0.303301i	$0.901911\pi$
$558$	0	0
$559$	−18.9783	−0.802694
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	14.0000	0.590554
$563$	17.4891	0.737079	0.368539	−	0.929612i	$-0.379858\pi$
$563$	17.4891	0.737079	0.368539	+	0.929612i	$0.379858\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	−28.0000	−1.17693
$567$	0	0
$568$	−4.00000	−0.167836
$569$	−39.4891	−1.65547	−0.827735	−	0.561119i	$-0.810371\pi$
$569$	−39.4891	−1.65547	−0.827735	+	0.561119i	$0.810371\pi$
$570$	0	0
$571$	5.48913	0.229713	0.114856	−	0.993382i	$-0.463359\pi$
$571$	5.48913	0.229713	0.114856	+	0.993382i	$0.463359\pi$
$572$	−4.74456	−0.198380
$573$	0	0
$574$	−4.00000	−0.166957
$575$	−4.74456	−0.197862
$576$	0	0
$577$	2.74456	0.114258	0.0571288	−	0.998367i	$-0.481805\pi$
$577$	2.74456	0.114258	0.0571288	+	0.998367i	$0.481805\pi$
$578$	−5.51087	−0.229222
$579$	0	0
$580$	−2.74456	−0.113962
$581$	−8.00000	−0.331896
$582$	0	0
$583$	1.25544	0.0519949
$584$	0.744563	0.0308102
$585$	0	0
$586$	24.7446	1.02219
$587$	−40.9783	−1.69135	−0.845677	−	0.533696i	$-0.820803\pi$
$587$	−40.9783	−1.69135	−0.845677	+	0.533696i	$0.820803\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	−2.74456	−0.112992
$591$	0	0
$592$	−10.7446	−0.441599
$593$	5.76631	0.236794	0.118397	−	0.992966i	$-0.462224\pi$
$593$	5.76631	0.236794	0.118397	+	0.992966i	$0.462224\pi$
$594$	0	0
$595$	4.74456	0.194508
$596$	−10.7446	−0.440114
$597$	0	0
$598$	−22.5109	−0.920538
$599$	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
$601$	−10.5109	−0.428748	−0.214374	−	0.976752i	$-0.568771\pi$
$601$	−10.5109	−0.428748	−0.214374	+	0.976752i	$0.568771\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−20.7446	−0.844084
$605$	−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	$0.448091\pi$
$608$	−4.74456	−0.192417
$609$	0	0
$610$	12.7446	0.516012
$611$	−32.0000	−1.29458
$612$	0	0
$613$	−11.4891	−0.464041	−0.232021	−	0.972711i	$-0.574534\pi$
$613$	−11.4891	−0.464041	−0.232021	+	0.972711i	$0.574534\pi$
$614$	−21.4891	−0.867231
$615$	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
$619$	−10.7446	−0.431860	−0.215930	−	0.976409i	$-0.569278\pi$
$619$	−10.7446	−0.431860	−0.215930	+	0.976409i	$0.569278\pi$
$620$	6.74456	0.270868
$621$	0	0
$622$	−9.25544	−0.371109
$623$	7.48913	0.300045
$624$	0	0
$625$	1.00000	0.0400000
$626$	−32.2337	−1.28832
$627$	0	0
$628$	23.4891	0.937318
$629$	50.9783	2.03264
$630$	0	0
$631$	42.9783	1.71094	0.855469	−	0.517855i	$-0.173269\pi$
$631$	42.9783	1.71094	0.855469	+	0.517855i	$0.173269\pi$
$632$	4.74456	0.188729
$633$	0	0
$634$	−32.2337	−1.28016
$635$	0	0
$636$	0	0
$637$	−4.74456	−0.187986
$638$	−2.74456	−0.108658
$639$	0	0
$640$	1.00000	0.0395285
$641$	11.4891	0.453793	0.226897	−	0.973919i	$-0.427142\pi$
$641$	11.4891	0.453793	0.226897	+	0.973919i	$0.427142\pi$
$642$	0	0
$643$	−22.4674	−0.886027	−0.443013	−	0.896515i	$-0.646091\pi$
$643$	−22.4674	−0.886027	−0.443013	+	0.896515i	$0.646091\pi$
$644$	−4.74456	−0.186962
$645$	0	0
$646$	22.5109	0.885679
$647$	2.74456	0.107900	0.0539499	−	0.998544i	$-0.482819\pi$
$647$	2.74456	0.107900	0.0539499	+	0.998544i	$0.482819\pi$
$648$	0	0
$649$	−2.74456	−0.107734
$650$	4.74456	0.186097
$651$	0	0
$652$	−4.00000	−0.156652
$653$	41.7228	1.63274	0.816370	−	0.577529i	$-0.195983\pi$
$653$	41.7228	1.63274	0.816370	+	0.577529i	$0.195983\pi$
$654$	0	0
$655$	8.74456	0.341678
$656$	4.00000	0.156174
$657$	0	0
$658$	−6.74456	−0.262930
$659$	18.5109	0.721081	0.360541	−	0.932744i	$-0.382592\pi$
$659$	18.5109	0.721081	0.360541	+	0.932744i	$0.382592\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	30.9783	1.20400
$663$	0	0
$664$	8.00000	0.310460
$665$	−4.74456	−0.183986
$666$	0	0
$667$	−13.0217	−0.504204
$668$	0	0
$669$	0	0
$670$	−4.00000	−0.154533
$671$	12.7446	0.491998
$672$	0	0
$673$	−24.9783	−0.962841	−0.481420	−	0.876490i	$-0.659879\pi$
$673$	−24.9783	−0.962841	−0.481420	+	0.876490i	$0.659879\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	9.51087	0.365803
$677$	−1.76631	−0.0678849	−0.0339424	−	0.999424i	$-0.510806\pi$
$677$	−1.76631	−0.0678849	−0.0339424	+	0.999424i	$0.510806\pi$
$678$	0	0
$679$	−5.25544	−0.201685
$680$	−4.74456	−0.181946
$681$	0	0
$682$	6.74456	0.258263
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	−19.4891	−0.744641
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	4.00000	0.152499
$689$	−5.95650	−0.226925
$690$	0	0
$691$	36.2337	1.37839	0.689197	−	0.724574i	$-0.257963\pi$
$691$	36.2337	1.37839	0.689197	+	0.724574i	$0.257963\pi$
$692$	14.2337	0.541084
$693$	0	0
$694$	22.9783	0.872242
$695$	−3.25544	−0.123486
$696$	0	0
$697$	−18.9783	−0.718853
$698$	−19.2554	−0.728829
$699$	0	0
$700$	1.00000	0.0377964
$701$	12.2337	0.462060	0.231030	−	0.972947i	$-0.425790\pi$
$701$	12.2337	0.462060	0.231030	+	0.972947i	$0.425790\pi$
$702$	0	0
$703$	−50.9783	−1.92268
$704$	1.00000	0.0376889
$705$	0	0
$706$	−2.74456	−0.103293
$707$	8.74456	0.328873
$708$	0	0
$709$	23.4891	0.882153	0.441076	−	0.897470i	$-0.354597\pi$
$709$	23.4891	0.882153	0.441076	+	0.897470i	$0.354597\pi$
$710$	4.00000	0.150117
$711$	0	0
$712$	−7.48913	−0.280667
$713$	32.0000	1.19841
$714$	0	0
$715$	4.74456	0.177437
$716$	4.00000	0.149487
$717$	0	0
$718$	−12.7446	−0.475623
$719$	49.7228	1.85435	0.927174	−	0.374631i	$-0.122231\pi$
$719$	49.7228	1.85435	0.927174	+	0.374631i	$0.122231\pi$
$720$	0	0
$721$	10.7446	0.400148
$722$	−3.51087	−0.130661
$723$	0	0
$724$	3.48913	0.129672
$725$	2.74456	0.101930
$726$	0	0
$727$	−20.2337	−0.750426	−0.375213	−	0.926939i	$-0.622430\pi$
$727$	−20.2337	−0.750426	−0.375213	+	0.926939i	$0.622430\pi$
$728$	4.74456	0.175845
$729$	0	0
$730$	−0.744563	−0.0275575
$731$	−18.9783	−0.701936
$732$	0	0
$733$	18.2337	0.673477	0.336738	−	0.941598i	$-0.390676\pi$
$733$	18.2337	0.673477	0.336738	+	0.941598i	$0.390676\pi$
$734$	−2.74456	−0.101304
$735$	0	0
$736$	4.74456	0.174887
$737$	−4.00000	−0.147342
$738$	0	0
$739$	14.9783	0.550984	0.275492	−	0.961303i	$-0.411159\pi$
$739$	14.9783	0.550984	0.275492	+	0.961303i	$0.411159\pi$
$740$	10.7446	0.394978
$741$	0	0
$742$	−1.25544	−0.0460886
$743$	18.9783	0.696244	0.348122	−	0.937449i	$-0.386819\pi$
$743$	18.9783	0.696244	0.348122	+	0.937449i	$0.386819\pi$
$744$	0	0
$745$	10.7446	0.393650
$746$	−28.9783	−1.06097
$747$	0	0
$748$	−4.74456	−0.173478
$749$	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	$0.381101\pi$
$752$	6.74456	0.245949
$753$	0	0
$754$	13.0217	0.474224
$755$	20.7446	0.754972
$756$	0	0
$757$	−30.7446	−1.11743	−0.558715	−	0.829360i	$-0.688705\pi$
$757$	−30.7446	−1.11743	−0.558715	+	0.829360i	$0.688705\pi$
$758$	14.5109	0.527059
$759$	0	0
$760$	4.74456	0.172103
$761$	6.51087	0.236019	0.118010	−	0.993012i	$-0.462349\pi$
$761$	6.51087	0.236019	0.118010	+	0.993012i	$0.462349\pi$
$762$	0	0
$763$	16.2337	0.587699
$764$	16.0000	0.578860
$765$	0	0
$766$	−37.7228	−1.36298
$767$	13.0217	0.470188
$768$	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
$773$	28.5109	1.02546	0.512732	−	0.858548i	$-0.328633\pi$
$773$	28.5109	1.02546	0.512732	+	0.858548i	$0.328633\pi$
$774$	0	0
$775$	−6.74456	−0.242272
$776$	5.25544	0.188659
$777$	0	0
$778$	15.4891	0.555312
$779$	18.9783	0.679966
$780$	0	0
$781$	4.00000	0.143131
$782$	−22.5109	−0.804987
$783$	0	0
$784$	1.00000	0.0357143
$785$	−23.4891	−0.838363
$786$	0	0
$787$	−17.4891	−0.623420	−0.311710	−	0.950177i	$-0.600902\pi$
$787$	−17.4891	−0.623420	−0.311710	+	0.950177i	$0.600902\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	−4.74456	−0.168804
$791$	−10.0000	−0.355559
$792$	0	0
$793$	−60.4674	−2.14726
$794$	−12.5109	−0.443994
$795$	0	0
$796$	14.7446	0.522607
$797$	−26.4674	−0.937523	−0.468761	−	0.883325i	$-0.655300\pi$
$797$	−26.4674	−0.937523	−0.468761	+	0.883325i	$0.655300\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	0.510875	0.0180396
$803$	−0.744563	−0.0262750
$804$	0	0
$805$	4.74456	0.167224
$806$	−32.0000	−1.12715
$807$	0	0
$808$	−8.74456	−0.307633
$809$	34.4674	1.21181	0.605904	−	0.795538i	$-0.292811\pi$
$809$	34.4674	1.21181	0.605904	+	0.795538i	$0.292811\pi$
$810$	0	0
$811$	−7.25544	−0.254773	−0.127386	−	0.991853i	$-0.540659\pi$
$811$	−7.25544	−0.254773	−0.127386	+	0.991853i	$0.540659\pi$
$812$	2.74456	0.0963153
$813$	0	0
$814$	10.7446	0.376597
$815$	4.00000	0.140114
$816$	0	0
$817$	18.9783	0.663965
$818$	−1.48913	−0.0520660
$819$	0	0
$820$	−4.00000	−0.139686
$821$	49.7228	1.73534	0.867669	−	0.497142i	$-0.165617\pi$
$821$	49.7228	1.73534	0.867669	+	0.497142i	$0.165617\pi$
$822$	0	0
$823$	42.2337	1.47217	0.736087	−	0.676887i	$-0.236671\pi$
$823$	42.2337	1.47217	0.736087	+	0.676887i	$0.236671\pi$
$824$	−10.7446	−0.374305
$825$	0	0
$826$	2.74456	0.0954955
$827$	−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
$829$	−54.4674	−1.89173	−0.945865	−	0.324560i	$-0.894784\pi$
$829$	−54.4674	−1.89173	−0.945865	+	0.324560i	$0.894784\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	−4.74456	−0.164488
$833$	−4.74456	−0.164389
$834$	0	0
$835$	0	0
$836$	4.74456	0.164094
$837$	0	0
$838$	10.7446	0.371165
$839$	14.7446	0.509039	0.254519	−	0.967068i	$-0.418083\pi$
$839$	14.7446	0.509039	0.254519	+	0.967068i	$0.418083\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	−27.4891	−0.947338
$843$	0	0
$844$	−12.0000	−0.413057
$845$	−9.51087	−0.327184
$846$	0	0
$847$	1.00000	0.0343604
$848$	1.25544	0.0431119
$849$	0	0
$850$	4.74456	0.162737
$851$	50.9783	1.74751
$852$	0	0
$853$	−11.2554	−0.385379	−0.192689	−	0.981260i	$-0.561721\pi$
$853$	−11.2554	−0.385379	−0.192689	+	0.981260i	$0.561721\pi$
$854$	−12.7446	−0.436110
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−37.2119	−1.27114	−0.635568	−	0.772045i	$-0.719234\pi$
$857$	−37.2119	−1.27114	−0.635568	+	0.772045i	$0.719234\pi$
$858$	0	0
$859$	51.2119	1.74733	0.873664	−	0.486529i	$-0.161737\pi$
$859$	51.2119	1.74733	0.873664	+	0.486529i	$0.161737\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−28.7446	−0.979044
$863$	5.76631	0.196288	0.0981438	−	0.995172i	$-0.468709\pi$
$863$	5.76631	0.196288	0.0981438	+	0.995172i	$0.468709\pi$
$864$	0	0
$865$	−14.2337	−0.483960
$866$	−25.2554	−0.858215
$867$	0	0
$868$	−6.74456	−0.228925
$869$	−4.74456	−0.160948
$870$	0	0
$871$	18.9783	0.643053
$872$	−16.2337	−0.549742
$873$	0	0
$874$	22.5109	0.761442
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	42.4674	1.43402	0.717011	−	0.697062i	$-0.245510\pi$
$877$	42.4674	1.43402	0.717011	+	0.697062i	$0.245510\pi$
$878$	30.9783	1.04547
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	32.5109	1.09532	0.547660	−	0.836701i	$-0.315519\pi$
$881$	32.5109	1.09532	0.547660	+	0.836701i	$0.315519\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	22.5109	0.757123
$885$	0	0
$886$	−6.51087	−0.218737
$887$	18.5109	0.621534	0.310767	−	0.950486i	$-0.399414\pi$
$887$	18.5109	0.621534	0.310767	+	0.950486i	$0.399414\pi$
$888$	0	0
$889$	0	0
$890$	7.48913	0.251036
$891$	0	0
$892$	26.7446	0.895474
$893$	32.0000	1.07084
$894$	0	0
$895$	−4.00000	−0.133705
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	40.9783	1.36746
$899$	−18.5109	−0.617372
$900$	0	0
$901$	−5.95650	−0.198440
$902$	−4.00000	−0.133185
$903$	0	0
$904$	10.0000	0.332595
$905$	−3.48913	−0.115982
$906$	0	0
$907$	−37.4891	−1.24481	−0.622403	−	0.782697i	$-0.713843\pi$
$907$	−37.4891	−1.24481	−0.622403	+	0.782697i	$0.713843\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	−4.74456	−0.157281
$911$	−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
$914$	19.4891	0.644643
$915$	0	0
$916$	−6.00000	−0.198246
$917$	−8.74456	−0.288771
$918$	0	0
$919$	25.2119	0.831665	0.415833	−	0.909441i	$-0.363490\pi$
$919$	25.2119	0.831665	0.415833	+	0.909441i	$0.363490\pi$
$920$	−4.74456	−0.156424
$921$	0	0
$922$	−23.7228	−0.781269
$923$	−18.9783	−0.624677
$924$	0	0
$925$	−10.7446	−0.353279
$926$	−12.7446	−0.418812
$927$	0	0
$928$	−2.74456	−0.0900947
$929$	−16.9783	−0.557038	−0.278519	−	0.960431i	$-0.589844\pi$
$929$	−16.9783	−0.557038	−0.278519	+	0.960431i	$0.589844\pi$
$930$	0	0
$931$	4.74456	0.155497
$932$	−20.9783	−0.687165
$933$	0	0
$934$	−28.9783	−0.948197
$935$	4.74456	0.155164
$936$	0	0
$937$	−7.25544	−0.237025	−0.118512	−	0.992953i	$-0.537813\pi$
$937$	−7.25544	−0.237025	−0.118512	+	0.992953i	$0.537813\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	−6.74456	−0.219983
$941$	22.2337	0.724798	0.362399	−	0.932023i	$-0.381958\pi$
$941$	22.2337	0.724798	0.362399	+	0.932023i	$0.381958\pi$
$942$	0	0
$943$	−18.9783	−0.618017
$944$	−2.74456	−0.0893279
$945$	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
$949$	3.53262	0.114674
$950$	−4.74456	−0.153934
$951$	0	0
$952$	4.74456	0.153772
$953$	−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
$956$	3.25544	0.105288
$957$	0	0
$958$	−18.5109	−0.598059
$959$	19.4891	0.629337
$960$	0	0
$961$	14.4891	0.467391
$962$	−50.9783	−1.64360
$963$	0	0
$964$	20.0000	0.644157
$965$	2.00000	0.0643823
$966$	0	0
$967$	−5.02175	−0.161489	−0.0807443	−	0.996735i	$-0.525730\pi$
$967$	−5.02175	−0.161489	−0.0807443	+	0.996735i	$0.525730\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−5.25544	−0.168742
$971$	37.7228	1.21058	0.605291	−	0.796004i	$-0.293057\pi$
$971$	37.7228	1.21058	0.605291	+	0.796004i	$0.293057\pi$
$972$	0	0
$973$	3.25544	0.104365
$974$	−20.7446	−0.664699
$975$	0	0
$976$	12.7446	0.407944
$977$	32.9783	1.05507	0.527534	−	0.849534i	$-0.323117\pi$
$977$	32.9783	1.05507	0.527534	+	0.849534i	$0.323117\pi$
$978$	0	0
$979$	7.48913	0.239353
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−14.9783	−0.477975
$983$	32.2337	1.02809	0.514047	−	0.857762i	$-0.328146\pi$
$983$	32.2337	1.02809	0.514047	+	0.857762i	$0.328146\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	13.0217	0.414697
$987$	0	0
$988$	−22.5109	−0.716166
$989$	−18.9783	−0.603473
$990$	0	0
$991$	−21.4891	−0.682625	−0.341312	−	0.939950i	$-0.610871\pi$
$991$	−21.4891	−0.682625	−0.341312	+	0.939950i	$0.610871\pi$
$992$	6.74456	0.214140
$993$	0	0
$994$	−4.00000	−0.126872
$995$	−14.7446	−0.467434
$996$	0	0
$997$	−41.2119	−1.30520	−0.652598	−	0.757705i	$-0.726321\pi$
$997$	−41.2119	−1.30520	−0.652598	+	0.757705i	$0.726321\pi$
$998$	−9.48913	−0.300373
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.bo.1.1		2
3.2	odd	2		770.2.a.k.1.1	✓	2
12.11	even	2		6160.2.a.r.1.1		2
15.2	even	4		3850.2.c.y.1849.4		4
15.8	even	4		3850.2.c.y.1849.1		4
15.14	odd	2		3850.2.a.bc.1.2		2
21.20	even	2		5390.2.a.bq.1.2		2
33.32	even	2		8470.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.k.1.1	✓	2	3.2	odd	2
3850.2.a.bc.1.2		2	15.14	odd	2
3850.2.c.y.1849.1		4	15.8	even	4
3850.2.c.y.1849.4		4	15.2	even	4
5390.2.a.bq.1.2		2	21.20	even	2
6160.2.a.r.1.1		2	12.11	even	2
6930.2.a.bo.1.1		2	1.1	even	1	trivial
8470.2.a.bu.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -4.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.74456 q^{17} +4.74456 q^{19} -1.00000 q^{20} -1.00000 q^{22} -4.74456 q^{23} +1.00000 q^{25} +4.74456 q^{26} +1.00000 q^{28} +2.74456 q^{29} -6.74456 q^{31} -1.00000 q^{32} +4.74456 q^{34} -1.00000 q^{35} -10.7446 q^{37} -4.74456 q^{38} +1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.74456 q^{46} +6.74456 q^{47} +1.00000 q^{49} -1.00000 q^{50} -4.74456 q^{52} +1.25544 q^{53} -1.00000 q^{55} -1.00000 q^{56} -2.74456 q^{58} -2.74456 q^{59} +12.7446 q^{61} +6.74456 q^{62} +1.00000 q^{64} +4.74456 q^{65} -4.00000 q^{67} -4.74456 q^{68} +1.00000 q^{70} +4.00000 q^{71} -0.744563 q^{73} +10.7446 q^{74} +4.74456 q^{76} +1.00000 q^{77} -4.74456 q^{79} -1.00000 q^{80} -4.00000 q^{82} -8.00000 q^{83} +4.74456 q^{85} -4.00000 q^{86} -1.00000 q^{88} +7.48913 q^{89} -4.74456 q^{91} -4.74456 q^{92} -6.74456 q^{94} -4.74456 q^{95} -5.25544 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) 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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