LMFDB - Embedded newform 7600.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} -4.82843 q^{11} -0.585786 q^{13} +0.828427 q^{17} +1.00000 q^{19} +4.00000 q^{21} +4.00000 q^{23} -5.65685 q^{27} +4.82843 q^{29} -6.82843 q^{33} -1.75736 q^{37} -0.828427 q^{39} +4.82843 q^{41} -2.82843 q^{43} +8.48528 q^{47} +1.00000 q^{49} +1.17157 q^{51} +1.07107 q^{53} +1.41421 q^{57} +2.82843 q^{59} +9.65685 q^{61} -2.82843 q^{63} -6.58579 q^{67} +5.65685 q^{69} +7.31371 q^{71} +16.1421 q^{73} -13.6569 q^{77} +14.8284 q^{79} -5.00000 q^{81} +8.00000 q^{83} +6.82843 q^{87} -8.82843 q^{89} -1.65685 q^{91} -6.24264 q^{97} +4.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{17} + 2 q^{19} + 8 q^{21} + 8 q^{23} + 4 q^{29} - 8 q^{33} - 12 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{49} + 8 q^{51} - 12 q^{53} + 8 q^{61} - 16 q^{67} - 8 q^{71} + 4 q^{73} - 16 q^{77} + 24 q^{79} - 10 q^{81} + 16 q^{83} + 8 q^{87} - 12 q^{89} + 8 q^{91} - 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^11 - 4 * q^13 - 4 * q^17 + 2 * q^19 + 8 * q^21 + 8 * q^23 + 4 * q^29 - 8 * q^33 - 12 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^49 + 8 * q^51 - 12 * q^53 + 8 * q^61 - 16 * q^67 - 8 * q^71 + 4 * q^73 - 16 * q^77 + 24 * q^79 - 10 * q^81 + 16 * q^83 + 8 * q^87 - 12 * q^89 + 8 * q^91 - 4 * q^97 + 4 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	−0.585786	−0.162468	−0.0812340	−	0.996695i	$-0.525886\pi$
$13$	−0.585786	−0.162468	−0.0812340	+	0.996695i	$0.525886\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−6.82843	−1.18868
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.75736	−0.288908	−0.144454	−	0.989512i	$-0.546143\pi$
$37$	−1.75736	−0.288908	−0.144454	+	0.989512i	$0.546143\pi$
$38$	0	0
$39$	−0.828427	−0.132655
$40$	0	0
$41$	4.82843	0.754074	0.377037	−	0.926198i	$-0.376943\pi$
$41$	4.82843	0.754074	0.377037	+	0.926198i	$0.376943\pi$
$42$	0	0
$43$	−2.82843	−0.431331	−0.215666	−	0.976467i	$-0.569192\pi$
$43$	−2.82843	−0.431331	−0.215666	+	0.976467i	$0.569192\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.48528	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$47$	8.48528	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.17157	0.164053
$52$	0	0
$53$	1.07107	0.147122	0.0735612	−	0.997291i	$-0.476564\pi$
$53$	1.07107	0.147122	0.0735612	+	0.997291i	$0.476564\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.41421	0.187317
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	9.65685	1.23643	0.618217	−	0.786008i	$-0.287855\pi$
$61$	9.65685	1.23643	0.618217	+	0.786008i	$0.287855\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.58579	−0.804582	−0.402291	−	0.915512i	$-0.631786\pi$
$67$	−6.58579	−0.804582	−0.402291	+	0.915512i	$0.631786\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	7.31371	0.867978	0.433989	−	0.900918i	$-0.357106\pi$
$71$	7.31371	0.867978	0.433989	+	0.900918i	$0.357106\pi$
$72$	0	0
$73$	16.1421	1.88929	0.944647	−	0.328088i	$-0.106404\pi$
$73$	16.1421	1.88929	0.944647	+	0.328088i	$0.106404\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.6569	−1.55634
$78$	0	0
$79$	14.8284	1.66833	0.834164	−	0.551516i	$-0.185951\pi$
$79$	14.8284	1.66833	0.834164	+	0.551516i	$0.185951\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.82843	0.732084
$88$	0	0
$89$	−8.82843	−0.935811	−0.467906	−	0.883778i	$-0.654991\pi$
$89$	−8.82843	−0.935811	−0.467906	+	0.883778i	$0.654991\pi$
$90$	0	0
$91$	−1.65685	−0.173686
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.24264	−0.633844	−0.316922	−	0.948452i	$-0.602649\pi$
$97$	−6.24264	−0.633844	−0.316922	+	0.948452i	$0.602649\pi$
$98$	0	0
$99$	4.82843	0.485275
$100$	0	0
$101$	6.34315	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$101$	6.34315	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$102$	0	0
$103$	−0.242641	−0.0239081	−0.0119540	−	0.999929i	$-0.503805\pi$
$103$	−0.242641	−0.0239081	−0.0119540	+	0.999929i	$0.503805\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.41421	−0.136717	−0.0683586	−	0.997661i	$-0.521776\pi$
$107$	−1.41421	−0.136717	−0.0683586	+	0.997661i	$0.521776\pi$
$108$	0	0
$109$	−7.65685	−0.733394	−0.366697	−	0.930341i	$-0.619511\pi$
$109$	−7.65685	−0.733394	−0.366697	+	0.930341i	$0.619511\pi$
$110$	0	0
$111$	−2.48528	−0.235892
$112$	0	0
$113$	−10.7279	−1.00920	−0.504599	−	0.863354i	$-0.668360\pi$
$113$	−10.7279	−1.00920	−0.504599	+	0.863354i	$0.668360\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.585786	0.0541560
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	6.82843	0.615699
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.5858	−0.939337	−0.469668	−	0.882843i	$-0.655626\pi$
$127$	−10.5858	−0.939337	−0.469668	+	0.882843i	$0.655626\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.8284	1.09601	0.548003	−	0.836476i	$-0.315388\pi$
$137$	12.8284	1.09601	0.548003	+	0.836476i	$0.315388\pi$
$138$	0	0
$139$	8.82843	0.748817	0.374409	−	0.927264i	$-0.377846\pi$
$139$	8.82843	0.748817	0.374409	+	0.927264i	$0.377846\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.41421	0.116642
$148$	0	0
$149$	−9.65685	−0.791120	−0.395560	−	0.918440i	$-0.629450\pi$
$149$	−9.65685	−0.791120	−0.395560	+	0.918440i	$0.629450\pi$
$150$	0	0
$151$	22.1421	1.80190	0.900951	−	0.433921i	$-0.142870\pi$
$151$	22.1421	1.80190	0.900951	+	0.433921i	$0.142870\pi$
$152$	0	0
$153$	−0.828427	−0.0669744
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	0	0
$159$	1.51472	0.120125
$160$	0	0
$161$	11.3137	0.891645
$162$	0	0
$163$	9.17157	0.718373	0.359187	−	0.933266i	$-0.383054\pi$
$163$	9.17157	0.718373	0.359187	+	0.933266i	$0.383054\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.8995	1.69463	0.847317	−	0.531088i	$-0.178217\pi$
$167$	21.8995	1.69463	0.847317	+	0.531088i	$0.178217\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−14.7279	−1.11974	−0.559872	−	0.828579i	$-0.689150\pi$
$173$	−14.7279	−1.11974	−0.559872	+	0.828579i	$0.689150\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	8.48528	0.634220	0.317110	−	0.948389i	$-0.397288\pi$
$179$	8.48528	0.634220	0.317110	+	0.948389i	$0.397288\pi$
$180$	0	0
$181$	17.3137	1.28692	0.643459	−	0.765481i	$-0.277499\pi$
$181$	17.3137	1.28692	0.643459	+	0.765481i	$0.277499\pi$
$182$	0	0
$183$	13.6569	1.00954
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	−16.0000	−1.16383
$190$	0	0
$191$	19.3137	1.39749	0.698745	−	0.715370i	$-0.253742\pi$
$191$	19.3137	1.39749	0.698745	+	0.715370i	$0.253742\pi$
$192$	0	0
$193$	21.0711	1.51673	0.758364	−	0.651831i	$-0.225999\pi$
$193$	21.0711	1.51673	0.758364	+	0.651831i	$0.225999\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.68629	−0.191390	−0.0956952	−	0.995411i	$-0.530507\pi$
$197$	−2.68629	−0.191390	−0.0956952	+	0.995411i	$0.530507\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	−9.31371	−0.656938
$202$	0	0
$203$	13.6569	0.958523
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	−4.82843	−0.333989
$210$	0	0
$211$	−0.485281	−0.0334081	−0.0167041	−	0.999860i	$-0.505317\pi$
$211$	−0.485281	−0.0334081	−0.0167041	+	0.999860i	$0.505317\pi$
$212$	0	0
$213$	10.3431	0.708701
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	22.8284	1.54260
$220$	0	0
$221$	−0.485281	−0.0326436
$222$	0	0
$223$	10.5858	0.708877	0.354438	−	0.935079i	$-0.384672\pi$
$223$	10.5858	0.708877	0.354438	+	0.935079i	$0.384672\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.0416	−1.59570	−0.797850	−	0.602857i	$-0.794029\pi$
$227$	−24.0416	−1.59570	−0.797850	+	0.602857i	$0.794029\pi$
$228$	0	0
$229$	−11.3137	−0.747631	−0.373815	−	0.927503i	$-0.621951\pi$
$229$	−11.3137	−0.747631	−0.373815	+	0.927503i	$0.621951\pi$
$230$	0	0
$231$	−19.3137	−1.27075
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	20.9706	1.36218
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−20.1421	−1.29747	−0.648735	−	0.761015i	$-0.724701\pi$
$241$	−20.1421	−1.29747	−0.648735	+	0.761015i	$0.724701\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.585786	−0.0372727
$248$	0	0
$249$	11.3137	0.716977
$250$	0	0
$251$	4.97056	0.313739	0.156870	−	0.987619i	$-0.449860\pi$
$251$	4.97056	0.313739	0.156870	+	0.987619i	$0.449860\pi$
$252$	0	0
$253$	−19.3137	−1.21424
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.8995	1.49081	0.745405	−	0.666612i	$-0.232256\pi$
$257$	23.8995	1.49081	0.745405	+	0.666612i	$0.232256\pi$
$258$	0	0
$259$	−4.97056	−0.308856
$260$	0	0
$261$	−4.82843	−0.298872
$262$	0	0
$263$	4.68629	0.288969	0.144485	−	0.989507i	$-0.453848\pi$
$263$	4.68629	0.288969	0.144485	+	0.989507i	$0.453848\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−12.4853	−0.764087
$268$	0	0
$269$	−6.68629	−0.407670	−0.203835	−	0.979005i	$-0.565341\pi$
$269$	−6.68629	−0.407670	−0.203835	+	0.979005i	$0.565341\pi$
$270$	0	0
$271$	26.4853	1.60887	0.804433	−	0.594043i	$-0.202469\pi$
$271$	26.4853	1.60887	0.804433	+	0.594043i	$0.202469\pi$
$272$	0	0
$273$	−2.34315	−0.141814
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−24.8284	−1.49180	−0.745898	−	0.666060i	$-0.767979\pi$
$277$	−24.8284	−1.49180	−0.745898	+	0.666060i	$0.767979\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.8284	1.00390	0.501950	−	0.864897i	$-0.332616\pi$
$281$	16.8284	1.00390	0.501950	+	0.864897i	$0.332616\pi$
$282$	0	0
$283$	17.6569	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$283$	17.6569	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.6569	0.806139
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	−8.82843	−0.517532
$292$	0	0
$293$	4.38478	0.256161	0.128081	−	0.991764i	$-0.459118\pi$
$293$	4.38478	0.256161	0.128081	+	0.991764i	$0.459118\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	27.3137	1.58490
$298$	0	0
$299$	−2.34315	−0.135508
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	8.97056	0.515345
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.41421	−0.0807134	−0.0403567	−	0.999185i	$-0.512849\pi$
$307$	−1.41421	−0.0807134	−0.0403567	+	0.999185i	$0.512849\pi$
$308$	0	0
$309$	−0.343146	−0.0195209
$310$	0	0
$311$	−24.8284	−1.40789	−0.703945	−	0.710254i	$-0.748580\pi$
$311$	−24.8284	−1.40789	−0.703945	+	0.710254i	$0.748580\pi$
$312$	0	0
$313$	−17.3137	−0.978629	−0.489314	−	0.872107i	$-0.662753\pi$
$313$	−17.3137	−0.978629	−0.489314	+	0.872107i	$0.662753\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.8995	−1.11767	−0.558833	−	0.829280i	$-0.688751\pi$
$317$	−19.8995	−1.11767	−0.558833	+	0.829280i	$0.688751\pi$
$318$	0	0
$319$	−23.3137	−1.30532
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	0.828427	0.0460949
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.8284	−0.598813
$328$	0	0
$329$	24.0000	1.32316
$330$	0	0
$331$	−33.4558	−1.83890	−0.919450	−	0.393208i	$-0.871365\pi$
$331$	−33.4558	−1.83890	−0.919450	+	0.393208i	$0.871365\pi$
$332$	0	0
$333$	1.75736	0.0963027
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.3848	−0.892536	−0.446268	−	0.894899i	$-0.647247\pi$
$337$	−16.3848	−0.892536	−0.446268	+	0.894899i	$0.647247\pi$
$338$	0	0
$339$	−15.1716	−0.824007
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.2843	1.94784	0.973921	−	0.226888i	$-0.0728551\pi$
$347$	36.2843	1.94784	0.973921	+	0.226888i	$0.0728551\pi$
$348$	0	0
$349$	23.6569	1.26632	0.633161	−	0.774020i	$-0.281757\pi$
$349$	23.6569	1.26632	0.633161	+	0.774020i	$0.281757\pi$
$350$	0	0
$351$	3.31371	0.176873
$352$	0	0
$353$	−16.1421	−0.859159	−0.429580	−	0.903029i	$-0.641338\pi$
$353$	−16.1421	−0.859159	−0.429580	+	0.903029i	$0.641338\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.31371	0.175380
$358$	0	0
$359$	−31.4558	−1.66018	−0.830088	−	0.557632i	$-0.811710\pi$
$359$	−31.4558	−1.66018	−0.830088	+	0.557632i	$0.811710\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	17.4142	0.914009
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−31.1127	−1.62407	−0.812035	−	0.583609i	$-0.801640\pi$
$367$	−31.1127	−1.62407	−0.812035	+	0.583609i	$0.801640\pi$
$368$	0	0
$369$	−4.82843	−0.251358
$370$	0	0
$371$	3.02944	0.157281
$372$	0	0
$373$	2.44365	0.126527	0.0632637	−	0.997997i	$-0.479849\pi$
$373$	2.44365	0.126527	0.0632637	+	0.997997i	$0.479849\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.82843	−0.145671
$378$	0	0
$379$	1.65685	0.0851069	0.0425534	−	0.999094i	$-0.486451\pi$
$379$	1.65685	0.0851069	0.0425534	+	0.999094i	$0.486451\pi$
$380$	0	0
$381$	−14.9706	−0.766965
$382$	0	0
$383$	7.27208	0.371586	0.185793	−	0.982589i	$-0.440515\pi$
$383$	7.27208	0.371586	0.185793	+	0.982589i	$0.440515\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.82843	0.143777
$388$	0	0
$389$	−13.3137	−0.675032	−0.337516	−	0.941320i	$-0.609587\pi$
$389$	−13.3137	−0.675032	−0.337516	+	0.941320i	$0.609587\pi$
$390$	0	0
$391$	3.31371	0.167581
$392$	0	0
$393$	21.6569	1.09244
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−31.1716	−1.56446	−0.782228	−	0.622992i	$-0.785917\pi$
$397$	−31.1716	−1.56446	−0.782228	+	0.622992i	$0.785917\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−18.9706	−0.947345	−0.473672	−	0.880701i	$-0.657072\pi$
$401$	−18.9706	−0.947345	−0.473672	+	0.880701i	$0.657072\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.48528	0.420600
$408$	0	0
$409$	26.2843	1.29967	0.649837	−	0.760074i	$-0.274837\pi$
$409$	26.2843	1.29967	0.649837	+	0.760074i	$0.274837\pi$
$410$	0	0
$411$	18.1421	0.894886
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.4853	0.611407
$418$	0	0
$419$	9.65685	0.471768	0.235884	−	0.971781i	$-0.424201\pi$
$419$	9.65685	0.471768	0.235884	+	0.971781i	$0.424201\pi$
$420$	0	0
$421$	−15.1716	−0.739417	−0.369709	−	0.929148i	$-0.620542\pi$
$421$	−15.1716	−0.739417	−0.369709	+	0.929148i	$0.620542\pi$
$422$	0	0
$423$	−8.48528	−0.412568
$424$	0	0
$425$	0	0
$426$	0	0
$427$	27.3137	1.32180
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	9.17157	0.441779	0.220890	−	0.975299i	$-0.429104\pi$
$431$	9.17157	0.441779	0.220890	+	0.975299i	$0.429104\pi$
$432$	0	0
$433$	−30.0416	−1.44371	−0.721854	−	0.692045i	$-0.756710\pi$
$433$	−30.0416	−1.44371	−0.721854	+	0.692045i	$0.756710\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	14.1421	0.674967	0.337484	−	0.941331i	$-0.390424\pi$
$439$	14.1421	0.674967	0.337484	+	0.941331i	$0.390424\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	23.3137	1.10767	0.553834	−	0.832627i	$-0.313164\pi$
$443$	23.3137	1.10767	0.553834	+	0.832627i	$0.313164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.6569	−0.645947
$448$	0	0
$449$	−30.2843	−1.42920	−0.714602	−	0.699532i	$-0.753392\pi$
$449$	−30.2843	−1.42920	−0.714602	+	0.699532i	$0.753392\pi$
$450$	0	0
$451$	−23.3137	−1.09780
$452$	0	0
$453$	31.3137	1.47125
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.2843	−1.04241	−0.521207	−	0.853430i	$-0.674518\pi$
$457$	−22.2843	−1.04241	−0.521207	+	0.853430i	$0.674518\pi$
$458$	0	0
$459$	−4.68629	−0.218737
$460$	0	0
$461$	36.6274	1.70591	0.852954	−	0.521985i	$-0.174808\pi$
$461$	36.6274	1.70591	0.852954	+	0.521985i	$0.174808\pi$
$462$	0	0
$463$	−18.6274	−0.865689	−0.432845	−	0.901468i	$-0.642490\pi$
$463$	−18.6274	−0.865689	−0.432845	+	0.901468i	$0.642490\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	−18.6274	−0.860134
$470$	0	0
$471$	−9.17157	−0.422604
$472$	0	0
$473$	13.6569	0.627943
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−1.07107	−0.0490408
$478$	0	0
$479$	27.4558	1.25449	0.627245	−	0.778822i	$-0.284183\pi$
$479$	27.4558	1.25449	0.627245	+	0.778822i	$0.284183\pi$
$480$	0	0
$481$	1.02944	0.0469383
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.55635	−0.161154	−0.0805768	−	0.996748i	$-0.525676\pi$
$487$	−3.55635	−0.161154	−0.0805768	+	0.996748i	$0.525676\pi$
$488$	0	0
$489$	12.9706	0.586549
$490$	0	0
$491$	−20.9706	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$491$	−20.9706	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.6863	0.927907
$498$	0	0
$499$	24.1421	1.08075	0.540375	−	0.841424i	$-0.318282\pi$
$499$	24.1421	1.08075	0.540375	+	0.841424i	$0.318282\pi$
$500$	0	0
$501$	30.9706	1.38366
$502$	0	0
$503$	9.65685	0.430578	0.215289	−	0.976550i	$-0.430931\pi$
$503$	9.65685	0.430578	0.215289	+	0.976550i	$0.430931\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−17.8995	−0.794944
$508$	0	0
$509$	35.4558	1.57155	0.785776	−	0.618511i	$-0.212264\pi$
$509$	35.4558	1.57155	0.785776	+	0.618511i	$0.212264\pi$
$510$	0	0
$511$	45.6569	2.01974
$512$	0	0
$513$	−5.65685	−0.249756
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−40.9706	−1.80188
$518$	0	0
$519$	−20.8284	−0.914266
$520$	0	0
$521$	37.3137	1.63474	0.817372	−	0.576111i	$-0.195430\pi$
$521$	37.3137	1.63474	0.817372	+	0.576111i	$0.195430\pi$
$522$	0	0
$523$	−27.7574	−1.21374	−0.606872	−	0.794799i	$-0.707576\pi$
$523$	−27.7574	−1.21374	−0.606872	+	0.794799i	$0.707576\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−2.82843	−0.122743
$532$	0	0
$533$	−2.82843	−0.122513
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	−4.82843	−0.207975
$540$	0	0
$541$	−20.2843	−0.872089	−0.436044	−	0.899925i	$-0.643621\pi$
$541$	−20.2843	−0.872089	−0.436044	+	0.899925i	$0.643621\pi$
$542$	0	0
$543$	24.4853	1.05076
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24.9289	−1.06588	−0.532942	−	0.846152i	$-0.678914\pi$
$547$	−24.9289	−1.06588	−0.532942	+	0.846152i	$0.678914\pi$
$548$	0	0
$549$	−9.65685	−0.412144
$550$	0	0
$551$	4.82843	0.205698
$552$	0	0
$553$	41.9411	1.78352
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−33.7990	−1.43211	−0.716055	−	0.698044i	$-0.754054\pi$
$557$	−33.7990	−1.43211	−0.716055	+	0.698044i	$0.754054\pi$
$558$	0	0
$559$	1.65685	0.0700775
$560$	0	0
$561$	−5.65685	−0.238833
$562$	0	0
$563$	4.24264	0.178806	0.0894030	−	0.995996i	$-0.471504\pi$
$563$	4.24264	0.178806	0.0894030	+	0.995996i	$0.471504\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−14.1421	−0.593914
$568$	0	0
$569$	35.9411	1.50673	0.753365	−	0.657602i	$-0.228429\pi$
$569$	35.9411	1.50673	0.753365	+	0.657602i	$0.228429\pi$
$570$	0	0
$571$	13.5147	0.565573	0.282787	−	0.959183i	$-0.408741\pi$
$571$	13.5147	0.565573	0.282787	+	0.959183i	$0.408741\pi$
$572$	0	0
$573$	27.3137	1.14105
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.627417	0.0261197	0.0130599	−	0.999915i	$-0.495843\pi$
$577$	0.627417	0.0261197	0.0130599	+	0.999915i	$0.495843\pi$
$578$	0	0
$579$	29.7990	1.23840
$580$	0	0
$581$	22.6274	0.938743
$582$	0	0
$583$	−5.17157	−0.214185
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.65685	0.233483	0.116742	−	0.993162i	$-0.462755\pi$
$587$	5.65685	0.233483	0.116742	+	0.993162i	$0.462755\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−3.79899	−0.156270
$592$	0	0
$593$	28.6274	1.17559	0.587794	−	0.809011i	$-0.299997\pi$
$593$	28.6274	1.17559	0.587794	+	0.809011i	$0.299997\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	22.1421	0.904703	0.452352	−	0.891840i	$-0.350585\pi$
$599$	22.1421	0.904703	0.452352	+	0.891840i	$0.350585\pi$
$600$	0	0
$601$	2.48528	0.101377	0.0506884	−	0.998715i	$-0.483858\pi$
$601$	2.48528	0.101377	0.0506884	+	0.998715i	$0.483858\pi$
$602$	0	0
$603$	6.58579	0.268194
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.2426	−0.659268	−0.329634	−	0.944109i	$-0.606925\pi$
$607$	−16.2426	−0.659268	−0.329634	+	0.944109i	$0.606925\pi$
$608$	0	0
$609$	19.3137	0.782631
$610$	0	0
$611$	−4.97056	−0.201087
$612$	0	0
$613$	21.7990	0.880453	0.440226	−	0.897887i	$-0.354898\pi$
$613$	21.7990	0.880453	0.440226	+	0.897887i	$0.354898\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.79899	−0.0724246	−0.0362123	−	0.999344i	$-0.511529\pi$
$617$	−1.79899	−0.0724246	−0.0362123	+	0.999344i	$0.511529\pi$
$618$	0	0
$619$	−11.8579	−0.476608	−0.238304	−	0.971191i	$-0.576591\pi$
$619$	−11.8579	−0.476608	−0.238304	+	0.971191i	$0.576591\pi$
$620$	0	0
$621$	−22.6274	−0.908007
$622$	0	0
$623$	−24.9706	−1.00042
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.82843	−0.272701
$628$	0	0
$629$	−1.45584	−0.0580483
$630$	0	0
$631$	17.5147	0.697250	0.348625	−	0.937262i	$-0.386649\pi$
$631$	17.5147	0.697250	0.348625	+	0.937262i	$0.386649\pi$
$632$	0	0
$633$	−0.686292	−0.0272776
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.585786	−0.0232097
$638$	0	0
$639$	−7.31371	−0.289326
$640$	0	0
$641$	−23.4558	−0.926450	−0.463225	−	0.886241i	$-0.653308\pi$
$641$	−23.4558	−0.926450	−0.463225	+	0.886241i	$0.653308\pi$
$642$	0	0
$643$	−12.6863	−0.500298	−0.250149	−	0.968207i	$-0.580480\pi$
$643$	−12.6863	−0.500298	−0.250149	+	0.968207i	$0.580480\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−39.3137	−1.54558	−0.772791	−	0.634661i	$-0.781140\pi$
$647$	−39.3137	−1.54558	−0.772791	+	0.634661i	$0.781140\pi$
$648$	0	0
$649$	−13.6569	−0.536078
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.1421	0.631691	0.315845	−	0.948811i	$-0.397712\pi$
$653$	16.1421	0.631691	0.315845	+	0.948811i	$0.397712\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−16.1421	−0.629765
$658$	0	0
$659$	11.5147	0.448550	0.224275	−	0.974526i	$-0.427999\pi$
$659$	11.5147	0.448550	0.224275	+	0.974526i	$0.427999\pi$
$660$	0	0
$661$	3.17157	0.123360	0.0616799	−	0.998096i	$-0.480354\pi$
$661$	3.17157	0.123360	0.0616799	+	0.998096i	$0.480354\pi$
$662$	0	0
$663$	−0.686292	−0.0266534
$664$	0	0
$665$	0	0
$666$	0	0
$667$	19.3137	0.747830
$668$	0	0
$669$	14.9706	0.578795
$670$	0	0
$671$	−46.6274	−1.80003
$672$	0	0
$673$	28.3848	1.09415	0.547076	−	0.837083i	$-0.315741\pi$
$673$	28.3848	1.09415	0.547076	+	0.837083i	$0.315741\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	44.3848	1.70585	0.852923	−	0.522037i	$-0.174828\pi$
$677$	44.3848	1.70585	0.852923	+	0.522037i	$0.174828\pi$
$678$	0	0
$679$	−17.6569	−0.677608
$680$	0	0
$681$	−34.0000	−1.30288
$682$	0	0
$683$	9.89949	0.378794	0.189397	−	0.981901i	$-0.439347\pi$
$683$	9.89949	0.378794	0.189397	+	0.981901i	$0.439347\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.0000	−0.610438
$688$	0	0
$689$	−0.627417	−0.0239027
$690$	0	0
$691$	40.8284	1.55319	0.776593	−	0.630002i	$-0.216946\pi$
$691$	40.8284	1.55319	0.776593	+	0.630002i	$0.216946\pi$
$692$	0	0
$693$	13.6569	0.518781
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	14.1421	0.534905
$700$	0	0
$701$	−24.0000	−0.906467	−0.453234	−	0.891392i	$-0.649730\pi$
$701$	−24.0000	−0.906467	−0.453234	+	0.891392i	$0.649730\pi$
$702$	0	0
$703$	−1.75736	−0.0662801
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.9411	0.674745
$708$	0	0
$709$	38.2843	1.43780	0.718898	−	0.695116i	$-0.244647\pi$
$709$	38.2843	1.43780	0.718898	+	0.695116i	$0.244647\pi$
$710$	0	0
$711$	−14.8284	−0.556109
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.3137	−0.422518
$718$	0	0
$719$	−19.4558	−0.725581	−0.362790	−	0.931871i	$-0.618176\pi$
$719$	−19.4558	−0.725581	−0.362790	+	0.931871i	$0.618176\pi$
$720$	0	0
$721$	−0.686292	−0.0255588
$722$	0	0
$723$	−28.4853	−1.05938
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.45584	0.350698	0.175349	−	0.984506i	$-0.443895\pi$
$727$	9.45584	0.350698	0.175349	+	0.984506i	$0.443895\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−2.34315	−0.0866644
$732$	0	0
$733$	−15.6569	−0.578299	−0.289150	−	0.957284i	$-0.593372\pi$
$733$	−15.6569	−0.578299	−0.289150	+	0.957284i	$0.593372\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31.7990	1.17133
$738$	0	0
$739$	−23.3137	−0.857609	−0.428804	−	0.903397i	$-0.641065\pi$
$739$	−23.3137	−0.857609	−0.428804	+	0.903397i	$0.641065\pi$
$740$	0	0
$741$	−0.828427	−0.0304330
$742$	0	0
$743$	−24.2426	−0.889376	−0.444688	−	0.895685i	$-0.646685\pi$
$743$	−24.2426	−0.889376	−0.444688	+	0.895685i	$0.646685\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	27.5147	1.00403	0.502013	−	0.864860i	$-0.332593\pi$
$751$	27.5147	1.00403	0.502013	+	0.864860i	$0.332593\pi$
$752$	0	0
$753$	7.02944	0.256167
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.2843	−1.24608	−0.623042	−	0.782189i	$-0.714103\pi$
$757$	−34.2843	−1.24608	−0.623042	+	0.782189i	$0.714103\pi$
$758$	0	0
$759$	−27.3137	−0.991425
$760$	0	0
$761$	46.6274	1.69024	0.845121	−	0.534575i	$-0.179528\pi$
$761$	46.6274	1.69024	0.845121	+	0.534575i	$0.179528\pi$
$762$	0	0
$763$	−21.6569	−0.784031
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.65685	−0.0598255
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	33.7990	1.21724
$772$	0	0
$773$	−4.87006	−0.175164	−0.0875819	−	0.996157i	$-0.527914\pi$
$773$	−4.87006	−0.175164	−0.0875819	+	0.996157i	$0.527914\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.02944	−0.252180
$778$	0	0
$779$	4.82843	0.172996
$780$	0	0
$781$	−35.3137	−1.26362
$782$	0	0
$783$	−27.3137	−0.976112
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.41421	0.0504113	0.0252056	−	0.999682i	$-0.491976\pi$
$787$	1.41421	0.0504113	0.0252056	+	0.999682i	$0.491976\pi$
$788$	0	0
$789$	6.62742	0.235942
$790$	0	0
$791$	−30.3431	−1.07888
$792$	0	0
$793$	−5.65685	−0.200881
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.6985	1.12282	0.561409	−	0.827538i	$-0.310259\pi$
$797$	31.6985	1.12282	0.561409	+	0.827538i	$0.310259\pi$
$798$	0	0
$799$	7.02944	0.248684
$800$	0	0
$801$	8.82843	0.311937
$802$	0	0
$803$	−77.9411	−2.75048
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.45584	−0.332861
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	23.3137	0.818655	0.409328	−	0.912388i	$-0.365763\pi$
$811$	23.3137	0.818655	0.409328	+	0.912388i	$0.365763\pi$
$812$	0	0
$813$	37.4558	1.31363
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.82843	−0.0989541
$818$	0	0
$819$	1.65685	0.0578952
$820$	0	0
$821$	29.5980	1.03298	0.516488	−	0.856294i	$-0.327239\pi$
$821$	29.5980	1.03298	0.516488	+	0.856294i	$0.327239\pi$
$822$	0	0
$823$	0.485281	0.0169158	0.00845792	−	0.999964i	$-0.497308\pi$
$823$	0.485281	0.0169158	0.00845792	+	0.999964i	$0.497308\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.6985	−1.58909	−0.794546	−	0.607204i	$-0.792291\pi$
$827$	−45.6985	−1.58909	−0.794546	+	0.607204i	$0.792291\pi$
$828$	0	0
$829$	29.5980	1.02798	0.513990	−	0.857796i	$-0.328167\pi$
$829$	29.5980	1.02798	0.513990	+	0.857796i	$0.328167\pi$
$830$	0	0
$831$	−35.1127	−1.21805
$832$	0	0
$833$	0.828427	0.0287033
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−39.7990	−1.37401	−0.687007	−	0.726651i	$-0.741076\pi$
$839$	−39.7990	−1.37401	−0.687007	+	0.726651i	$0.741076\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	23.7990	0.819681
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34.8284	1.19672
$848$	0	0
$849$	24.9706	0.856987
$850$	0	0
$851$	−7.02944	−0.240966
$852$	0	0
$853$	−39.9411	−1.36756	−0.683779	−	0.729689i	$-0.739665\pi$
$853$	−39.9411	−1.36756	−0.683779	+	0.729689i	$0.739665\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.2132	0.383036	0.191518	−	0.981489i	$-0.438659\pi$
$857$	11.2132	0.383036	0.191518	+	0.981489i	$0.438659\pi$
$858$	0	0
$859$	−17.6569	−0.602444	−0.301222	−	0.953554i	$-0.597395\pi$
$859$	−17.6569	−0.602444	−0.301222	+	0.953554i	$0.597395\pi$
$860$	0	0
$861$	19.3137	0.658209
$862$	0	0
$863$	4.04163	0.137579	0.0687894	−	0.997631i	$-0.478086\pi$
$863$	4.04163	0.137579	0.0687894	+	0.997631i	$0.478086\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−23.0711	−0.783535
$868$	0	0
$869$	−71.5980	−2.42880
$870$	0	0
$871$	3.85786	0.130719
$872$	0	0
$873$	6.24264	0.211281
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.72792	0.227186	0.113593	−	0.993527i	$-0.463764\pi$
$877$	6.72792	0.227186	0.113593	+	0.993527i	$0.463764\pi$
$878$	0	0
$879$	6.20101	0.209155
$880$	0	0
$881$	9.65685	0.325348	0.162674	−	0.986680i	$-0.447988\pi$
$881$	9.65685	0.325348	0.162674	+	0.986680i	$0.447988\pi$
$882$	0	0
$883$	−5.17157	−0.174037	−0.0870186	−	0.996207i	$-0.527734\pi$
$883$	−5.17157	−0.174037	−0.0870186	+	0.996207i	$0.527734\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−40.7279	−1.36751	−0.683755	−	0.729712i	$-0.739654\pi$
$887$	−40.7279	−1.36751	−0.683755	+	0.729712i	$0.739654\pi$
$888$	0	0
$889$	−29.9411	−1.00419
$890$	0	0
$891$	24.1421	0.808792
$892$	0	0
$893$	8.48528	0.283949
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.31371	−0.110642
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.887302	0.0295603
$902$	0	0
$903$	−11.3137	−0.376497
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−0.0416306	−0.00138232	−0.000691160	−	1.00000i	$-0.500220\pi$
$907$	−0.0416306	−0.00138232	−0.000691160		1.00000i	$0.500220\pi$
$908$	0	0
$909$	−6.34315	−0.210389
$910$	0	0
$911$	14.8284	0.491288	0.245644	−	0.969360i	$-0.421001\pi$
$911$	14.8284	0.491288	0.245644	+	0.969360i	$0.421001\pi$
$912$	0	0
$913$	−38.6274	−1.27838
$914$	0	0
$915$	0	0
$916$	0	0
$917$	43.3137	1.43034
$918$	0	0
$919$	−49.9411	−1.64741	−0.823703	−	0.567022i	$-0.808096\pi$
$919$	−49.9411	−1.64741	−0.823703	+	0.567022i	$0.808096\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	−4.28427	−0.141019
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.242641	0.00796937
$928$	0	0
$929$	18.6863	0.613077	0.306539	−	0.951858i	$-0.400829\pi$
$929$	18.6863	0.613077	0.306539	+	0.951858i	$0.400829\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−35.1127	−1.14954
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.54416	−0.148451	−0.0742256	−	0.997241i	$-0.523648\pi$
$937$	−4.54416	−0.148451	−0.0742256	+	0.997241i	$0.523648\pi$
$938$	0	0
$939$	−24.4853	−0.799047
$940$	0	0
$941$	−57.5980	−1.87764	−0.938820	−	0.344408i	$-0.888080\pi$
$941$	−57.5980	−1.87764	−0.938820	+	0.344408i	$0.888080\pi$
$942$	0	0
$943$	19.3137	0.628941
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.6863	0.802197	0.401098	−	0.916035i	$-0.368629\pi$
$947$	24.6863	0.802197	0.401098	+	0.916035i	$0.368629\pi$
$948$	0	0
$949$	−9.45584	−0.306950
$950$	0	0
$951$	−28.1421	−0.912571
$952$	0	0
$953$	9.27208	0.300352	0.150176	−	0.988659i	$-0.452016\pi$
$953$	9.27208	0.300352	0.150176	+	0.988659i	$0.452016\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−32.9706	−1.06579
$958$	0	0
$959$	36.2843	1.17168
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	1.41421	0.0455724
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.3431	0.847138	0.423569	−	0.905864i	$-0.360777\pi$
$967$	26.3431	0.847138	0.423569	+	0.905864i	$0.360777\pi$
$968$	0	0
$969$	1.17157	0.0376363
$970$	0	0
$971$	41.9411	1.34595	0.672977	−	0.739663i	$-0.265015\pi$
$971$	41.9411	1.34595	0.672977	+	0.739663i	$0.265015\pi$
$972$	0	0
$973$	24.9706	0.800519
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−33.8406	−1.08266	−0.541329	−	0.840811i	$-0.682079\pi$
$977$	−33.8406	−1.08266	−0.541329	+	0.840811i	$0.682079\pi$
$978$	0	0
$979$	42.6274	1.36238
$980$	0	0
$981$	7.65685	0.244465
$982$	0	0
$983$	−44.0416	−1.40471	−0.702355	−	0.711827i	$-0.747868\pi$
$983$	−44.0416	−1.40471	−0.702355	+	0.711827i	$0.747868\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	33.9411	1.08036
$988$	0	0
$989$	−11.3137	−0.359755
$990$	0	0
$991$	28.7696	0.913895	0.456947	−	0.889494i	$-0.348943\pi$
$991$	28.7696	0.913895	0.456947	+	0.889494i	$0.348943\pi$
$992$	0	0
$993$	−47.3137	−1.50146
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−7.17157	−0.227126	−0.113563	−	0.993531i	$-0.536226\pi$
$997$	−7.17157	−0.227126	−0.113563	+	0.993531i	$0.536226\pi$
$998$	0	0
$999$	9.94113	0.314523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.ba.1.2		2
4.3	odd	2		3800.2.a.n.1.1		2
5.4	even	2		1520.2.a.m.1.1		2
20.3	even	4		3800.2.d.i.3649.2		4
20.7	even	4		3800.2.d.i.3649.4		4
20.19	odd	2		760.2.a.f.1.2	✓	2
40.19	odd	2		6080.2.a.bf.1.1		2
40.29	even	2		6080.2.a.bg.1.2		2
60.59	even	2		6840.2.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.f.1.2	✓	2	20.19	odd	2
1520.2.a.m.1.1		2	5.4	even	2
3800.2.a.n.1.1		2	4.3	odd	2
3800.2.d.i.3649.2		4	20.3	even	4
3800.2.d.i.3649.4		4	20.7	even	4
6080.2.a.bf.1.1		2	40.19	odd	2
6080.2.a.bg.1.2		2	40.29	even	2
6840.2.a.z.1.2		2	60.59	even	2
7600.2.a.ba.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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} +2.82843 q^{7} -1.00000 q^{9} -4.82843 q^{11} -0.585786 q^{13} +0.828427 q^{17} +1.00000 q^{19} +4.00000 q^{21} +4.00000 q^{23} -5.65685 q^{27} +4.82843 q^{29} -6.82843 q^{33} -1.75736 q^{37} -0.828427 q^{39} +4.82843 q^{41} -2.82843 q^{43} +8.48528 q^{47} +1.00000 q^{49} +1.17157 q^{51} +1.07107 q^{53} +1.41421 q^{57} +2.82843 q^{59} +9.65685 q^{61} -2.82843 q^{63} -6.58579 q^{67} +5.65685 q^{69} +7.31371 q^{71} +16.1421 q^{73} -13.6569 q^{77} +14.8284 q^{79} -5.00000 q^{81} +8.00000 q^{83} +6.82843 q^{87} -8.82843 q^{89} -1.65685 q^{91} -6.24264 q^{97} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{17} + 2 q^{19} + 8 q^{21} + 8 q^{23} + 4 q^{29} - 8 q^{33} - 12 q^{37} + 4 q^{39} + 4 q^{41} + 2 q^{49} + 8 q^{51} - 12 q^{53} + 8 q^{61} - 16 q^{67} - 8 q^{71} + 4 q^{73} - 16 q^{77} + 24 q^{79} - 10 q^{81} + 16 q^{83} + 8 q^{87} - 12 q^{89} + 8 q^{91} - 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^11 - 4 * q^13 - 4 * q^17 + 2 * q^19 + 8 * q^21 + 8 * q^23 + 4 * q^29 - 8 * q^33 - 12 * q^37 + 4 * q^39 + 4 * q^41 + 2 * q^49 + 8 * q^51 - 12 * q^53 + 8 * q^61 - 16 * q^67 - 8 * q^71 + 4 * q^73 - 16 * q^77 + 24 * q^79 - 10 * q^81 + 16 * q^83 + 8 * q^87 - 12 * q^89 + 8 * q^91 - 4 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more