LMFDB - Embedded newform 6040.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 $ x^19 - 5x^18 - 29x^17 + 165x^16 + 325x^15 - 2208x^14 - 1891x^13 + 15895x^12 + 6652x^11 - 67665x^10 - 17345x^9 + 174105x^8 + 41499x^7 - 262172x^6 - 80919x^5 + 206783x^4 + 91643x^3 - 59750x^2 - 40224x - 5628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.31016$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.31016 q^{3} +1.00000 q^{5} -0.477457 q^{7} +7.95713 q^{9} +O(q^{10})$	$q-3.31016 q^{3} +1.00000 q^{5} -0.477457 q^{7} +7.95713 q^{9} +4.57039 q^{11} +2.86935 q^{13} -3.31016 q^{15} +7.02125 q^{17} -5.71822 q^{19} +1.58046 q^{21} -1.60378 q^{23} +1.00000 q^{25} -16.4089 q^{27} -8.07405 q^{29} -11.0218 q^{31} -15.1287 q^{33} -0.477457 q^{35} -2.92795 q^{37} -9.49800 q^{39} -5.80115 q^{41} -7.00093 q^{43} +7.95713 q^{45} +12.4002 q^{47} -6.77203 q^{49} -23.2414 q^{51} -0.250729 q^{53} +4.57039 q^{55} +18.9282 q^{57} +2.86948 q^{59} +2.93287 q^{61} -3.79919 q^{63} +2.86935 q^{65} +0.359307 q^{67} +5.30876 q^{69} +13.8626 q^{71} +6.69059 q^{73} -3.31016 q^{75} -2.18217 q^{77} -16.4177 q^{79} +30.4446 q^{81} +4.08016 q^{83} +7.02125 q^{85} +26.7264 q^{87} -2.51700 q^{89} -1.36999 q^{91} +36.4840 q^{93} -5.71822 q^{95} -13.1177 q^{97} +36.3672 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * 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
$3$	−3.31016	−1.91112	−0.955560	−	0.294797i	$-0.904748\pi$
$3$	−3.31016	−1.91112	−0.955560	+	0.294797i	$0.904748\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.477457	−0.180462	−0.0902310	−	0.995921i	$-0.528761\pi$
$7$	−0.477457	−0.180462	−0.0902310	+	0.995921i	$0.528761\pi$
$8$	0	0
$9$	7.95713	2.65238
$10$	0	0
$11$	4.57039	1.37803	0.689013	−	0.724749i	$-0.258044\pi$
$11$	4.57039	1.37803	0.689013	+	0.724749i	$0.258044\pi$
$12$	0	0
$13$	2.86935	0.795815	0.397907	−	0.917426i	$-0.369737\pi$
$13$	2.86935	0.795815	0.397907	+	0.917426i	$0.369737\pi$
$14$	0	0
$15$	−3.31016	−0.854679
$16$	0	0
$17$	7.02125	1.70290	0.851451	−	0.524433i	$-0.175723\pi$
$17$	7.02125	1.70290	0.851451	+	0.524433i	$0.175723\pi$
$18$	0	0
$19$	−5.71822	−1.31185	−0.655925	−	0.754826i	$-0.727721\pi$
$19$	−5.71822	−1.31185	−0.655925	+	0.754826i	$0.727721\pi$
$20$	0	0
$21$	1.58046	0.344884
$22$	0	0
$23$	−1.60378	−0.334411	−0.167205	−	0.985922i	$-0.553474\pi$
$23$	−1.60378	−0.334411	−0.167205	+	0.985922i	$0.553474\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−16.4089	−3.15789
$28$	0	0
$29$	−8.07405	−1.49931	−0.749657	−	0.661827i	$-0.769781\pi$
$29$	−8.07405	−1.49931	−0.749657	+	0.661827i	$0.769781\pi$
$30$	0	0
$31$	−11.0218	−1.97958	−0.989789	−	0.142537i	$-0.954474\pi$
$31$	−11.0218	−1.97958	−0.989789	+	0.142537i	$0.954474\pi$
$32$	0	0
$33$	−15.1287	−2.63357
$34$	0	0
$35$	−0.477457	−0.0807050
$36$	0	0
$37$	−2.92795	−0.481352	−0.240676	−	0.970606i	$-0.577369\pi$
$37$	−2.92795	−0.481352	−0.240676	+	0.970606i	$0.577369\pi$
$38$	0	0
$39$	−9.49800	−1.52090
$40$	0	0
$41$	−5.80115	−0.905988	−0.452994	−	0.891514i	$-0.649644\pi$
$41$	−5.80115	−0.905988	−0.452994	+	0.891514i	$0.649644\pi$
$42$	0	0
$43$	−7.00093	−1.06763	−0.533816	−	0.845601i	$-0.679242\pi$
$43$	−7.00093	−1.06763	−0.533816	+	0.845601i	$0.679242\pi$
$44$	0	0
$45$	7.95713	1.18618
$46$	0	0
$47$	12.4002	1.80876	0.904378	−	0.426733i	$-0.140336\pi$
$47$	12.4002	1.80876	0.904378	+	0.426733i	$0.140336\pi$
$48$	0	0
$49$	−6.77203	−0.967433
$50$	0	0
$51$	−23.2414	−3.25445
$52$	0	0
$53$	−0.250729	−0.0344402	−0.0172201	−	0.999852i	$-0.505482\pi$
$53$	−0.250729	−0.0344402	−0.0172201	+	0.999852i	$0.505482\pi$
$54$	0	0
$55$	4.57039	0.616272
$56$	0	0
$57$	18.9282	2.50710
$58$	0	0
$59$	2.86948	0.373574	0.186787	−	0.982400i	$-0.440192\pi$
$59$	2.86948	0.373574	0.186787	+	0.982400i	$0.440192\pi$
$60$	0	0
$61$	2.93287	0.375515	0.187758	−	0.982215i	$-0.439878\pi$
$61$	2.93287	0.375515	0.187758	+	0.982215i	$0.439878\pi$
$62$	0	0
$63$	−3.79919	−0.478653
$64$	0	0
$65$	2.86935	0.355899
$66$	0	0
$67$	0.359307	0.0438964	0.0219482	−	0.999759i	$-0.493013\pi$
$67$	0.359307	0.0438964	0.0219482	+	0.999759i	$0.493013\pi$
$68$	0	0
$69$	5.30876	0.639099
$70$	0	0
$71$	13.8626	1.64518	0.822591	−	0.568633i	$-0.192528\pi$
$71$	13.8626	1.64518	0.822591	+	0.568633i	$0.192528\pi$
$72$	0	0
$73$	6.69059	0.783074	0.391537	−	0.920162i	$-0.371943\pi$
$73$	6.69059	0.783074	0.391537	+	0.920162i	$0.371943\pi$
$74$	0	0
$75$	−3.31016	−0.382224
$76$	0	0
$77$	−2.18217	−0.248681
$78$	0	0
$79$	−16.4177	−1.84713	−0.923567	−	0.383438i	$-0.874740\pi$
$79$	−16.4177	−1.84713	−0.923567	+	0.383438i	$0.874740\pi$
$80$	0	0
$81$	30.4446	3.38273
$82$	0	0
$83$	4.08016	0.447856	0.223928	−	0.974606i	$-0.428112\pi$
$83$	4.08016	0.447856	0.223928	+	0.974606i	$0.428112\pi$
$84$	0	0
$85$	7.02125	0.761561
$86$	0	0
$87$	26.7264	2.86537
$88$	0	0
$89$	−2.51700	−0.266802	−0.133401	−	0.991062i	$-0.542590\pi$
$89$	−2.51700	−0.266802	−0.133401	+	0.991062i	$0.542590\pi$
$90$	0	0
$91$	−1.36999	−0.143614
$92$	0	0
$93$	36.4840	3.78321
$94$	0	0
$95$	−5.71822	−0.586677
$96$	0	0
$97$	−13.1177	−1.33190	−0.665948	−	0.745998i	$-0.731973\pi$
$97$	−13.1177	−1.33190	−0.665948	+	0.745998i	$0.731973\pi$
$98$	0	0
$99$	36.3672	3.65504
$100$	0	0
$101$	−5.13472	−0.510924	−0.255462	−	0.966819i	$-0.582228\pi$
$101$	−5.13472	−0.510924	−0.255462	+	0.966819i	$0.582228\pi$
$102$	0	0
$103$	−8.82746	−0.869796	−0.434898	−	0.900480i	$-0.643216\pi$
$103$	−8.82746	−0.869796	−0.434898	+	0.900480i	$0.643216\pi$
$104$	0	0
$105$	1.58046	0.154237
$106$	0	0
$107$	−9.90023	−0.957092	−0.478546	−	0.878063i	$-0.658836\pi$
$107$	−9.90023	−0.957092	−0.478546	+	0.878063i	$0.658836\pi$
$108$	0	0
$109$	−13.9578	−1.33691	−0.668455	−	0.743752i	$-0.733044\pi$
$109$	−13.9578	−1.33691	−0.668455	+	0.743752i	$0.733044\pi$
$110$	0	0
$111$	9.69197	0.919921
$112$	0	0
$113$	12.2709	1.15435	0.577173	−	0.816622i	$-0.304156\pi$
$113$	12.2709	1.15435	0.577173	+	0.816622i	$0.304156\pi$
$114$	0	0
$115$	−1.60378	−0.149553
$116$	0	0
$117$	22.8318	2.11080
$118$	0	0
$119$	−3.35235	−0.307309
$120$	0	0
$121$	9.88850	0.898954
$122$	0	0
$123$	19.2027	1.73145
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−0.370668	−0.0328915	−0.0164457	−	0.999865i	$-0.505235\pi$
$127$	−0.370668	−0.0328915	−0.0164457	+	0.999865i	$0.505235\pi$
$128$	0	0
$129$	23.1742	2.04037
$130$	0	0
$131$	−1.83804	−0.160591	−0.0802953	−	0.996771i	$-0.525586\pi$
$131$	−1.83804	−0.160591	−0.0802953	+	0.996771i	$0.525586\pi$
$132$	0	0
$133$	2.73021	0.236739
$134$	0	0
$135$	−16.4089	−1.41225
$136$	0	0
$137$	−16.3748	−1.39900	−0.699498	−	0.714635i	$-0.746593\pi$
$137$	−16.3748	−1.39900	−0.699498	+	0.714635i	$0.746593\pi$
$138$	0	0
$139$	−12.2506	−1.03908	−0.519542	−	0.854445i	$-0.673897\pi$
$139$	−12.2506	−1.03908	−0.519542	+	0.854445i	$0.673897\pi$
$140$	0	0
$141$	−41.0466	−3.45675
$142$	0	0
$143$	13.1141	1.09665
$144$	0	0
$145$	−8.07405	−0.670513
$146$	0	0
$147$	22.4165	1.84888
$148$	0	0
$149$	−8.94591	−0.732877	−0.366439	−	0.930442i	$-0.619423\pi$
$149$	−8.94591	−0.732877	−0.366439	+	0.930442i	$0.619423\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	55.8690	4.51674
$154$	0	0
$155$	−11.0218	−0.885295
$156$	0	0
$157$	10.0682	0.803526	0.401763	−	0.915744i	$-0.368398\pi$
$157$	10.0682	0.803526	0.401763	+	0.915744i	$0.368398\pi$
$158$	0	0
$159$	0.829951	0.0658194
$160$	0	0
$161$	0.765736	0.0603484
$162$	0	0
$163$	−7.23556	−0.566733	−0.283366	−	0.959012i	$-0.591451\pi$
$163$	−7.23556	−0.566733	−0.283366	+	0.959012i	$0.591451\pi$
$164$	0	0
$165$	−15.1287	−1.17777
$166$	0	0
$167$	19.6561	1.52103	0.760516	−	0.649320i	$-0.224946\pi$
$167$	19.6561	1.52103	0.760516	+	0.649320i	$0.224946\pi$
$168$	0	0
$169$	−4.76682	−0.366679
$170$	0	0
$171$	−45.5007	−3.47952
$172$	0	0
$173$	−22.8776	−1.73935	−0.869676	−	0.493622i	$-0.835673\pi$
$173$	−22.8776	−1.73935	−0.869676	+	0.493622i	$0.835673\pi$
$174$	0	0
$175$	−0.477457	−0.0360924
$176$	0	0
$177$	−9.49843	−0.713946
$178$	0	0
$179$	8.42748	0.629900	0.314950	−	0.949108i	$-0.398012\pi$
$179$	8.42748	0.629900	0.314950	+	0.949108i	$0.398012\pi$
$180$	0	0
$181$	1.21187	0.0900775	0.0450388	−	0.998985i	$-0.485659\pi$
$181$	1.21187	0.0900775	0.0450388	+	0.998985i	$0.485659\pi$
$182$	0	0
$183$	−9.70825	−0.717654
$184$	0	0
$185$	−2.92795	−0.215267
$186$	0	0
$187$	32.0899	2.34664
$188$	0	0
$189$	7.83455	0.569879
$190$	0	0
$191$	−12.1900	−0.882036	−0.441018	−	0.897498i	$-0.645383\pi$
$191$	−12.1900	−0.882036	−0.441018	+	0.897498i	$0.645383\pi$
$192$	0	0
$193$	−5.35965	−0.385796	−0.192898	−	0.981219i	$-0.561789\pi$
$193$	−5.35965	−0.385796	−0.192898	+	0.981219i	$0.561789\pi$
$194$	0	0
$195$	−9.49800	−0.680166
$196$	0	0
$197$	−13.8689	−0.988117	−0.494059	−	0.869429i	$-0.664487\pi$
$197$	−13.8689	−0.988117	−0.494059	+	0.869429i	$0.664487\pi$
$198$	0	0
$199$	−21.3390	−1.51268	−0.756342	−	0.654176i	$-0.773015\pi$
$199$	−21.3390	−1.51268	−0.756342	+	0.654176i	$0.773015\pi$
$200$	0	0
$201$	−1.18936	−0.0838912
$202$	0	0
$203$	3.85501	0.270569
$204$	0	0
$205$	−5.80115	−0.405170
$206$	0	0
$207$	−12.7615	−0.886984
$208$	0	0
$209$	−26.1345	−1.80776
$210$	0	0
$211$	7.12888	0.490772	0.245386	−	0.969425i	$-0.421085\pi$
$211$	7.12888	0.490772	0.245386	+	0.969425i	$0.421085\pi$
$212$	0	0
$213$	−45.8872	−3.14414
$214$	0	0
$215$	−7.00093	−0.477459
$216$	0	0
$217$	5.26245	0.357239
$218$	0	0
$219$	−22.1469	−1.49655
$220$	0	0
$221$	20.1464	1.35520
$222$	0	0
$223$	−2.28108	−0.152753	−0.0763763	−	0.997079i	$-0.524335\pi$
$223$	−2.28108	−0.152753	−0.0763763	+	0.997079i	$0.524335\pi$
$224$	0	0
$225$	7.95713	0.530476
$226$	0	0
$227$	17.9633	1.19227	0.596134	−	0.802885i	$-0.296703\pi$
$227$	17.9633	1.19227	0.596134	+	0.802885i	$0.296703\pi$
$228$	0	0
$229$	14.1077	0.932266	0.466133	−	0.884715i	$-0.345647\pi$
$229$	14.1077	0.932266	0.466133	+	0.884715i	$0.345647\pi$
$230$	0	0
$231$	7.22332	0.475259
$232$	0	0
$233$	26.2492	1.71964	0.859820	−	0.510598i	$-0.170576\pi$
$233$	26.2492	1.71964	0.859820	+	0.510598i	$0.170576\pi$
$234$	0	0
$235$	12.4002	0.808900
$236$	0	0
$237$	54.3451	3.53009
$238$	0	0
$239$	−15.0090	−0.970849	−0.485425	−	0.874279i	$-0.661335\pi$
$239$	−15.0090	−0.970849	−0.485425	+	0.874279i	$0.661335\pi$
$240$	0	0
$241$	0.442366	0.0284953	0.0142476	−	0.999898i	$-0.495465\pi$
$241$	0.442366	0.0284953	0.0142476	+	0.999898i	$0.495465\pi$
$242$	0	0
$243$	−51.5497	−3.30691
$244$	0	0
$245$	−6.77203	−0.432649
$246$	0	0
$247$	−16.4076	−1.04399
$248$	0	0
$249$	−13.5060	−0.855906
$250$	0	0
$251$	0.341253	0.0215397	0.0107699	−	0.999942i	$-0.496572\pi$
$251$	0.341253	0.0215397	0.0107699	+	0.999942i	$0.496572\pi$
$252$	0	0
$253$	−7.32990	−0.460827
$254$	0	0
$255$	−23.2414	−1.45543
$256$	0	0
$257$	19.4223	1.21153	0.605764	−	0.795644i	$-0.292868\pi$
$257$	19.4223	1.21153	0.605764	+	0.795644i	$0.292868\pi$
$258$	0	0
$259$	1.39797	0.0868657
$260$	0	0
$261$	−64.2463	−3.97675
$262$	0	0
$263$	11.3773	0.701554	0.350777	−	0.936459i	$-0.385918\pi$
$263$	11.3773	0.701554	0.350777	+	0.936459i	$0.385918\pi$
$264$	0	0
$265$	−0.250729	−0.0154021
$266$	0	0
$267$	8.33167	0.509890
$268$	0	0
$269$	2.12272	0.129425	0.0647124	−	0.997904i	$-0.479387\pi$
$269$	2.12272	0.129425	0.0647124	+	0.997904i	$0.479387\pi$
$270$	0	0
$271$	10.8941	0.661770	0.330885	−	0.943671i	$-0.392653\pi$
$271$	10.8941	0.661770	0.330885	+	0.943671i	$0.392653\pi$
$272$	0	0
$273$	4.53489	0.274464
$274$	0	0
$275$	4.57039	0.275605
$276$	0	0
$277$	−8.45475	−0.507997	−0.253998	−	0.967205i	$-0.581746\pi$
$277$	−8.45475	−0.507997	−0.253998	+	0.967205i	$0.581746\pi$
$278$	0	0
$279$	−87.7022	−5.25059
$280$	0	0
$281$	31.7396	1.89343	0.946714	−	0.322075i	$-0.104380\pi$
$281$	31.7396	1.89343	0.946714	+	0.322075i	$0.104380\pi$
$282$	0	0
$283$	−11.1546	−0.663070	−0.331535	−	0.943443i	$-0.607567\pi$
$283$	−11.1546	−0.663070	−0.331535	+	0.943443i	$0.607567\pi$
$284$	0	0
$285$	18.9282	1.12121
$286$	0	0
$287$	2.76980	0.163496
$288$	0	0
$289$	32.2979	1.89988
$290$	0	0
$291$	43.4215	2.54541
$292$	0	0
$293$	17.1225	1.00031	0.500154	−	0.865936i	$-0.333277\pi$
$293$	17.1225	1.00031	0.500154	+	0.865936i	$0.333277\pi$
$294$	0	0
$295$	2.86948	0.167068
$296$	0	0
$297$	−74.9951	−4.35166
$298$	0	0
$299$	−4.60180	−0.266129
$300$	0	0
$301$	3.34264	0.192667
$302$	0	0
$303$	16.9967	0.976437
$304$	0	0
$305$	2.93287	0.167935
$306$	0	0
$307$	−26.2164	−1.49625	−0.748126	−	0.663557i	$-0.769046\pi$
$307$	−26.2164	−1.49625	−0.748126	+	0.663557i	$0.769046\pi$
$308$	0	0
$309$	29.2203	1.66228
$310$	0	0
$311$	7.64048	0.433252	0.216626	−	0.976255i	$-0.430495\pi$
$311$	7.64048	0.433252	0.216626	+	0.976255i	$0.430495\pi$
$312$	0	0
$313$	−20.0333	−1.13235	−0.566175	−	0.824285i	$-0.691577\pi$
$313$	−20.0333	−1.13235	−0.566175	+	0.824285i	$0.691577\pi$
$314$	0	0
$315$	−3.79919	−0.214060
$316$	0	0
$317$	−21.8570	−1.22761	−0.613806	−	0.789457i	$-0.710362\pi$
$317$	−21.8570	−1.22761	−0.613806	+	0.789457i	$0.710362\pi$
$318$	0	0
$319$	−36.9016	−2.06609
$320$	0	0
$321$	32.7713	1.82912
$322$	0	0
$323$	−40.1491	−2.23395
$324$	0	0
$325$	2.86935	0.159163
$326$	0	0
$327$	46.2023	2.55500
$328$	0	0
$329$	−5.92057	−0.326411
$330$	0	0
$331$	2.17211	0.119390	0.0596950	−	0.998217i	$-0.480987\pi$
$331$	2.17211	0.119390	0.0596950	+	0.998217i	$0.480987\pi$
$332$	0	0
$333$	−23.2981	−1.27673
$334$	0	0
$335$	0.359307	0.0196310
$336$	0	0
$337$	17.7985	0.969547	0.484773	−	0.874640i	$-0.338902\pi$
$337$	17.7985	0.969547	0.484773	+	0.874640i	$0.338902\pi$
$338$	0	0
$339$	−40.6185	−2.20609
$340$	0	0
$341$	−50.3741	−2.72791
$342$	0	0
$343$	6.57556	0.355047
$344$	0	0
$345$	5.30876	0.285814
$346$	0	0
$347$	9.81221	0.526747	0.263373	−	0.964694i	$-0.415165\pi$
$347$	9.81221	0.526747	0.263373	+	0.964694i	$0.415165\pi$
$348$	0	0
$349$	10.2946	0.551056	0.275528	−	0.961293i	$-0.411147\pi$
$349$	10.2946	0.551056	0.275528	+	0.961293i	$0.411147\pi$
$350$	0	0
$351$	−47.0829	−2.51310
$352$	0	0
$353$	16.4544	0.875780	0.437890	−	0.899029i	$-0.355726\pi$
$353$	16.4544	0.875780	0.437890	+	0.899029i	$0.355726\pi$
$354$	0	0
$355$	13.8626	0.735748
$356$	0	0
$357$	11.0968	0.587305
$358$	0	0
$359$	−3.15583	−0.166558	−0.0832792	−	0.996526i	$-0.526539\pi$
$359$	−3.15583	−0.166558	−0.0832792	+	0.996526i	$0.526539\pi$
$360$	0	0
$361$	13.6981	0.720950
$362$	0	0
$363$	−32.7325	−1.71801
$364$	0	0
$365$	6.69059	0.350201
$366$	0	0
$367$	10.8928	0.568599	0.284300	−	0.958735i	$-0.408239\pi$
$367$	10.8928	0.568599	0.284300	+	0.958735i	$0.408239\pi$
$368$	0	0
$369$	−46.1606	−2.40302
$370$	0	0
$371$	0.119712	0.00621515
$372$	0	0
$373$	−37.6287	−1.94834	−0.974169	−	0.225820i	$-0.927494\pi$
$373$	−37.6287	−1.94834	−0.974169	+	0.225820i	$0.927494\pi$
$374$	0	0
$375$	−3.31016	−0.170936
$376$	0	0
$377$	−23.1673	−1.19318
$378$	0	0
$379$	−20.4558	−1.05075	−0.525373	−	0.850872i	$-0.676074\pi$
$379$	−20.4558	−1.05075	−0.525373	+	0.850872i	$0.676074\pi$
$380$	0	0
$381$	1.22697	0.0628595
$382$	0	0
$383$	−10.2171	−0.522069	−0.261034	−	0.965330i	$-0.584064\pi$
$383$	−10.2171	−0.522069	−0.261034	+	0.965330i	$0.584064\pi$
$384$	0	0
$385$	−2.18217	−0.111214
$386$	0	0
$387$	−55.7073	−2.83176
$388$	0	0
$389$	4.13479	0.209642	0.104821	−	0.994491i	$-0.466573\pi$
$389$	4.13479	0.209642	0.104821	+	0.994491i	$0.466573\pi$
$390$	0	0
$391$	−11.2605	−0.569469
$392$	0	0
$393$	6.08421	0.306908
$394$	0	0
$395$	−16.4177	−0.826063
$396$	0	0
$397$	2.87821	0.144453	0.0722266	−	0.997388i	$-0.476990\pi$
$397$	2.87821	0.144453	0.0722266	+	0.997388i	$0.476990\pi$
$398$	0	0
$399$	−9.03741	−0.452436
$400$	0	0
$401$	−17.5295	−0.875381	−0.437691	−	0.899126i	$-0.644203\pi$
$401$	−17.5295	−0.875381	−0.437691	+	0.899126i	$0.644203\pi$
$402$	0	0
$403$	−31.6255	−1.57538
$404$	0	0
$405$	30.4446	1.51280
$406$	0	0
$407$	−13.3819	−0.663315
$408$	0	0
$409$	−23.3765	−1.15589	−0.577946	−	0.816075i	$-0.696146\pi$
$409$	−23.3765	−1.15589	−0.577946	+	0.816075i	$0.696146\pi$
$410$	0	0
$411$	54.2032	2.67365
$412$	0	0
$413$	−1.37005	−0.0674160
$414$	0	0
$415$	4.08016	0.200287
$416$	0	0
$417$	40.5515	1.98581
$418$	0	0
$419$	34.1445	1.66807	0.834033	−	0.551715i	$-0.186026\pi$
$419$	34.1445	1.66807	0.834033	+	0.551715i	$0.186026\pi$
$420$	0	0
$421$	−17.1458	−0.835637	−0.417818	−	0.908531i	$-0.637205\pi$
$421$	−17.1458	−0.835637	−0.417818	+	0.908531i	$0.637205\pi$
$422$	0	0
$423$	98.6701	4.79750
$424$	0	0
$425$	7.02125	0.340581
$426$	0	0
$427$	−1.40032	−0.0677662
$428$	0	0
$429$	−43.4096	−2.09584
$430$	0	0
$431$	−27.0734	−1.30408	−0.652040	−	0.758185i	$-0.726086\pi$
$431$	−27.0734	−1.30408	−0.652040	+	0.758185i	$0.726086\pi$
$432$	0	0
$433$	−15.9416	−0.766106	−0.383053	−	0.923726i	$-0.625127\pi$
$433$	−15.9416	−0.766106	−0.383053	+	0.923726i	$0.625127\pi$
$434$	0	0
$435$	26.7264	1.28143
$436$	0	0
$437$	9.17076	0.438697
$438$	0	0
$439$	3.36694	0.160695	0.0803476	−	0.996767i	$-0.474397\pi$
$439$	3.36694	0.160695	0.0803476	+	0.996767i	$0.474397\pi$
$440$	0	0
$441$	−53.8860	−2.56600
$442$	0	0
$443$	−25.2415	−1.19926	−0.599629	−	0.800278i	$-0.704685\pi$
$443$	−25.2415	−1.19926	−0.599629	+	0.800278i	$0.704685\pi$
$444$	0	0
$445$	−2.51700	−0.119317
$446$	0	0
$447$	29.6124	1.40062
$448$	0	0
$449$	19.5496	0.922603	0.461301	−	0.887243i	$-0.347383\pi$
$449$	19.5496	0.922603	0.461301	+	0.887243i	$0.347383\pi$
$450$	0	0
$451$	−26.5135	−1.24847
$452$	0	0
$453$	−3.31016	−0.155525
$454$	0	0
$455$	−1.36999	−0.0642263
$456$	0	0
$457$	32.8315	1.53579	0.767897	−	0.640573i	$-0.221303\pi$
$457$	32.8315	1.53579	0.767897	+	0.640573i	$0.221303\pi$
$458$	0	0
$459$	−115.211	−5.37758
$460$	0	0
$461$	−15.1822	−0.707104	−0.353552	−	0.935415i	$-0.615026\pi$
$461$	−15.1822	−0.707104	−0.353552	+	0.935415i	$0.615026\pi$
$462$	0	0
$463$	3.15228	0.146499	0.0732494	−	0.997314i	$-0.476663\pi$
$463$	3.15228	0.146499	0.0732494	+	0.997314i	$0.476663\pi$
$464$	0	0
$465$	36.4840	1.69190
$466$	0	0
$467$	−24.8132	−1.14822	−0.574108	−	0.818779i	$-0.694651\pi$
$467$	−24.8132	−1.14822	−0.574108	+	0.818779i	$0.694651\pi$
$468$	0	0
$469$	−0.171554	−0.00792162
$470$	0	0
$471$	−33.3272	−1.53563
$472$	0	0
$473$	−31.9970	−1.47122
$474$	0	0
$475$	−5.71822	−0.262370
$476$	0	0
$477$	−1.99508	−0.0913486
$478$	0	0
$479$	−3.81391	−0.174262	−0.0871309	−	0.996197i	$-0.527770\pi$
$479$	−3.81391	−0.174262	−0.0871309	+	0.996197i	$0.527770\pi$
$480$	0	0
$481$	−8.40131	−0.383067
$482$	0	0
$483$	−2.53471	−0.115333
$484$	0	0
$485$	−13.1177	−0.595642
$486$	0	0
$487$	36.6581	1.66113	0.830567	−	0.556918i	$-0.188016\pi$
$487$	36.6581	1.66113	0.830567	+	0.556918i	$0.188016\pi$
$488$	0	0
$489$	23.9508	1.08309
$490$	0	0
$491$	−0.660116	−0.0297906	−0.0148953	−	0.999889i	$-0.504742\pi$
$491$	−0.660116	−0.0297906	−0.0148953	+	0.999889i	$0.504742\pi$
$492$	0	0
$493$	−56.6899	−2.55319
$494$	0	0
$495$	36.3672	1.63459
$496$	0	0
$497$	−6.61878	−0.296893
$498$	0	0
$499$	28.5930	1.28000	0.639999	−	0.768375i	$-0.278935\pi$
$499$	28.5930	1.28000	0.639999	+	0.768375i	$0.278935\pi$
$500$	0	0
$501$	−65.0646	−2.90687
$502$	0	0
$503$	−15.2567	−0.680265	−0.340132	−	0.940378i	$-0.610472\pi$
$503$	−15.2567	−0.680265	−0.340132	+	0.940378i	$0.610472\pi$
$504$	0	0
$505$	−5.13472	−0.228492
$506$	0	0
$507$	15.7789	0.700767
$508$	0	0
$509$	−11.0125	−0.488121	−0.244060	−	0.969760i	$-0.578480\pi$
$509$	−11.0125	−0.488121	−0.244060	+	0.969760i	$0.578480\pi$
$510$	0	0
$511$	−3.19447	−0.141315
$512$	0	0
$513$	93.8297	4.14268
$514$	0	0
$515$	−8.82746	−0.388984
$516$	0	0
$517$	56.6738	2.49251
$518$	0	0
$519$	75.7285	3.32411
$520$	0	0
$521$	13.2241	0.579358	0.289679	−	0.957124i	$-0.406451\pi$
$521$	13.2241	0.579358	0.289679	+	0.957124i	$0.406451\pi$
$522$	0	0
$523$	23.0224	1.00670	0.503350	−	0.864082i	$-0.332101\pi$
$523$	23.0224	1.00670	0.503350	+	0.864082i	$0.332101\pi$
$524$	0	0
$525$	1.58046	0.0689769
$526$	0	0
$527$	−77.3870	−3.37103
$528$	0	0
$529$	−20.4279	−0.888169
$530$	0	0
$531$	22.8328	0.990861
$532$	0	0
$533$	−16.6455	−0.720999
$534$	0	0
$535$	−9.90023	−0.428024
$536$	0	0
$537$	−27.8963	−1.20381
$538$	0	0
$539$	−30.9509	−1.33315
$540$	0	0
$541$	−9.41992	−0.404994	−0.202497	−	0.979283i	$-0.564906\pi$
$541$	−9.41992	−0.404994	−0.202497	+	0.979283i	$0.564906\pi$
$542$	0	0
$543$	−4.01148	−0.172149
$544$	0	0
$545$	−13.9578	−0.597884
$546$	0	0
$547$	−37.4340	−1.60056	−0.800280	−	0.599626i	$-0.795316\pi$
$547$	−37.4340	−1.60056	−0.800280	+	0.599626i	$0.795316\pi$
$548$	0	0
$549$	23.3372	0.996008
$550$	0	0
$551$	46.1692	1.96687
$552$	0	0
$553$	7.83874	0.333337
$554$	0	0
$555$	9.69197	0.411401
$556$	0	0
$557$	−3.10073	−0.131382	−0.0656910	−	0.997840i	$-0.520925\pi$
$557$	−3.10073	−0.131382	−0.0656910	+	0.997840i	$0.520925\pi$
$558$	0	0
$559$	−20.0881	−0.849637
$560$	0	0
$561$	−106.222	−4.48472
$562$	0	0
$563$	21.6687	0.913229	0.456614	−	0.889665i	$-0.349062\pi$
$563$	21.6687	0.913229	0.456614	+	0.889665i	$0.349062\pi$
$564$	0	0
$565$	12.2709	0.516239
$566$	0	0
$567$	−14.5360	−0.610454
$568$	0	0
$569$	34.0389	1.42698	0.713492	−	0.700663i	$-0.247112\pi$
$569$	34.0389	1.42698	0.713492	+	0.700663i	$0.247112\pi$
$570$	0	0
$571$	−34.5849	−1.44733	−0.723666	−	0.690150i	$-0.757545\pi$
$571$	−34.5849	−1.44733	−0.723666	+	0.690150i	$0.757545\pi$
$572$	0	0
$573$	40.3507	1.68568
$574$	0	0
$575$	−1.60378	−0.0668822
$576$	0	0
$577$	14.3184	0.596082	0.298041	−	0.954553i	$-0.403667\pi$
$577$	14.3184	0.596082	0.298041	+	0.954553i	$0.403667\pi$
$578$	0	0
$579$	17.7413	0.737302
$580$	0	0
$581$	−1.94810	−0.0808209
$582$	0	0
$583$	−1.14593	−0.0474595
$584$	0	0
$585$	22.8318	0.943979
$586$	0	0
$587$	47.3341	1.95369	0.976845	−	0.213950i	$-0.0686330\pi$
$587$	47.3341	1.95369	0.976845	+	0.213950i	$0.0686330\pi$
$588$	0	0
$589$	63.0253	2.59691
$590$	0	0
$591$	45.9082	1.88841
$592$	0	0
$593$	1.53057	0.0628529	0.0314264	−	0.999506i	$-0.489995\pi$
$593$	1.53057	0.0628529	0.0314264	+	0.999506i	$0.489995\pi$
$594$	0	0
$595$	−3.35235	−0.137433
$596$	0	0
$597$	70.6356	2.89092
$598$	0	0
$599$	24.7179	1.00995	0.504974	−	0.863135i	$-0.331502\pi$
$599$	24.7179	1.00995	0.504974	+	0.863135i	$0.331502\pi$
$600$	0	0
$601$	−28.5283	−1.16369	−0.581846	−	0.813299i	$-0.697669\pi$
$601$	−28.5283	−1.16369	−0.581846	+	0.813299i	$0.697669\pi$
$602$	0	0
$603$	2.85906	0.116430
$604$	0	0
$605$	9.88850	0.402025
$606$	0	0
$607$	26.5748	1.07864	0.539319	−	0.842102i	$-0.318682\pi$
$607$	26.5748	1.07864	0.539319	+	0.842102i	$0.318682\pi$
$608$	0	0
$609$	−12.7607	−0.517090
$610$	0	0
$611$	35.5805	1.43943
$612$	0	0
$613$	−43.3039	−1.74903	−0.874514	−	0.485000i	$-0.838820\pi$
$613$	−43.3039	−1.74903	−0.874514	+	0.485000i	$0.838820\pi$
$614$	0	0
$615$	19.2027	0.774328
$616$	0	0
$617$	−28.3610	−1.14177	−0.570886	−	0.821030i	$-0.693400\pi$
$617$	−28.3610	−1.14177	−0.570886	+	0.821030i	$0.693400\pi$
$618$	0	0
$619$	−2.00072	−0.0804157	−0.0402078	−	0.999191i	$-0.512802\pi$
$619$	−2.00072	−0.0804157	−0.0402078	+	0.999191i	$0.512802\pi$
$620$	0	0
$621$	26.3162	1.05603
$622$	0	0
$623$	1.20176	0.0481475
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	86.5094	3.45485
$628$	0	0
$629$	−20.5579	−0.819695
$630$	0	0
$631$	3.47057	0.138161	0.0690806	−	0.997611i	$-0.477993\pi$
$631$	3.47057	0.138161	0.0690806	+	0.997611i	$0.477993\pi$
$632$	0	0
$633$	−23.5977	−0.937924
$634$	0	0
$635$	−0.370668	−0.0147095
$636$	0	0
$637$	−19.4313	−0.769898
$638$	0	0
$639$	110.306	4.36365
$640$	0	0
$641$	1.06700	0.0421438	0.0210719	−	0.999778i	$-0.493292\pi$
$641$	1.06700	0.0421438	0.0210719	+	0.999778i	$0.493292\pi$
$642$	0	0
$643$	3.05355	0.120420	0.0602101	−	0.998186i	$-0.480823\pi$
$643$	3.05355	0.120420	0.0602101	+	0.998186i	$0.480823\pi$
$644$	0	0
$645$	23.1742	0.912482
$646$	0	0
$647$	−6.50100	−0.255581	−0.127790	−	0.991801i	$-0.540788\pi$
$647$	−6.50100	−0.255581	−0.127790	+	0.991801i	$0.540788\pi$
$648$	0	0
$649$	13.1147	0.514795
$650$	0	0
$651$	−17.4195	−0.682726
$652$	0	0
$653$	42.7661	1.67357	0.836784	−	0.547534i	$-0.184433\pi$
$653$	42.7661	1.67357	0.836784	+	0.547534i	$0.184433\pi$
$654$	0	0
$655$	−1.83804	−0.0718183
$656$	0	0
$657$	53.2379	2.07701
$658$	0	0
$659$	−27.3064	−1.06371	−0.531853	−	0.846837i	$-0.678504\pi$
$659$	−27.3064	−1.06371	−0.531853	+	0.846837i	$0.678504\pi$
$660$	0	0
$661$	24.0143	0.934047	0.467024	−	0.884245i	$-0.345326\pi$
$661$	24.0143	0.934047	0.467024	+	0.884245i	$0.345326\pi$
$662$	0	0
$663$	−66.6878	−2.58994
$664$	0	0
$665$	2.73021	0.105873
$666$	0	0
$667$	12.9490	0.501387
$668$	0	0
$669$	7.55074	0.291929
$670$	0	0
$671$	13.4044	0.517469
$672$	0	0
$673$	−39.7984	−1.53412	−0.767058	−	0.641578i	$-0.778280\pi$
$673$	−39.7984	−1.53412	−0.767058	+	0.641578i	$0.778280\pi$
$674$	0	0
$675$	−16.4089	−0.631578
$676$	0	0
$677$	−4.22897	−0.162533	−0.0812663	−	0.996692i	$-0.525896\pi$
$677$	−4.22897	−0.162533	−0.0812663	+	0.996692i	$0.525896\pi$
$678$	0	0
$679$	6.26312	0.240357
$680$	0	0
$681$	−59.4615	−2.27857
$682$	0	0
$683$	−43.1761	−1.65209	−0.826044	−	0.563605i	$-0.809414\pi$
$683$	−43.1761	−1.65209	−0.826044	+	0.563605i	$0.809414\pi$
$684$	0	0
$685$	−16.3748	−0.625650
$686$	0	0
$687$	−46.6988	−1.78167
$688$	0	0
$689$	−0.719429	−0.0274081
$690$	0	0
$691$	26.8640	1.02196	0.510978	−	0.859594i	$-0.329283\pi$
$691$	26.8640	1.02196	0.510978	+	0.859594i	$0.329283\pi$
$692$	0	0
$693$	−17.3638	−0.659596
$694$	0	0
$695$	−12.2506	−0.464692
$696$	0	0
$697$	−40.7313	−1.54281
$698$	0	0
$699$	−86.8888	−3.28644
$700$	0	0
$701$	−11.3675	−0.429346	−0.214673	−	0.976686i	$-0.568869\pi$
$701$	−11.3675	−0.429346	−0.214673	+	0.976686i	$0.568869\pi$
$702$	0	0
$703$	16.7427	0.631461
$704$	0	0
$705$	−41.0466	−1.54590
$706$	0	0
$707$	2.45161	0.0922024
$708$	0	0
$709$	6.65280	0.249851	0.124926	−	0.992166i	$-0.460131\pi$
$709$	6.65280	0.249851	0.124926	+	0.992166i	$0.460131\pi$
$710$	0	0
$711$	−130.638	−4.89930
$712$	0	0
$713$	17.6766	0.661993
$714$	0	0
$715$	13.1141	0.490438
$716$	0	0
$717$	49.6820	1.85541
$718$	0	0
$719$	−45.6899	−1.70394	−0.851972	−	0.523587i	$-0.824594\pi$
$719$	−45.6899	−1.70394	−0.851972	+	0.523587i	$0.824594\pi$
$720$	0	0
$721$	4.21474	0.156965
$722$	0	0
$723$	−1.46430	−0.0544579
$724$	0	0
$725$	−8.07405	−0.299863
$726$	0	0
$727$	16.1171	0.597752	0.298876	−	0.954292i	$-0.403388\pi$
$727$	16.1171	0.597752	0.298876	+	0.954292i	$0.403388\pi$
$728$	0	0
$729$	79.3037	2.93717
$730$	0	0
$731$	−49.1553	−1.81807
$732$	0	0
$733$	−32.2680	−1.19185	−0.595924	−	0.803041i	$-0.703214\pi$
$733$	−32.2680	−1.19185	−0.595924	+	0.803041i	$0.703214\pi$
$734$	0	0
$735$	22.4165	0.826845
$736$	0	0
$737$	1.64218	0.0604903
$738$	0	0
$739$	−0.315945	−0.0116222	−0.00581110	−	0.999983i	$-0.501850\pi$
$739$	−0.315945	−0.0116222	−0.00581110	+	0.999983i	$0.501850\pi$
$740$	0	0
$741$	54.3117	1.99519
$742$	0	0
$743$	−12.5495	−0.460398	−0.230199	−	0.973144i	$-0.573938\pi$
$743$	−12.5495	−0.460398	−0.230199	+	0.973144i	$0.573938\pi$
$744$	0	0
$745$	−8.94591	−0.327753
$746$	0	0
$747$	32.4664	1.18788
$748$	0	0
$749$	4.72694	0.172719
$750$	0	0
$751$	−48.2774	−1.76167	−0.880834	−	0.473424i	$-0.843018\pi$
$751$	−48.2774	−1.76167	−0.880834	+	0.473424i	$0.843018\pi$
$752$	0	0
$753$	−1.12960	−0.0411650
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−33.6955	−1.22468	−0.612342	−	0.790593i	$-0.709773\pi$
$757$	−33.6955	−1.22468	−0.612342	+	0.790593i	$0.709773\pi$
$758$	0	0
$759$	24.2631	0.880695
$760$	0	0
$761$	15.9571	0.578445	0.289222	−	0.957262i	$-0.406603\pi$
$761$	15.9571	0.578445	0.289222	+	0.957262i	$0.406603\pi$
$762$	0	0
$763$	6.66423	0.241261
$764$	0	0
$765$	55.8690	2.01995
$766$	0	0
$767$	8.23355	0.297296
$768$	0	0
$769$	21.3174	0.768724	0.384362	−	0.923182i	$-0.374421\pi$
$769$	21.3174	0.768724	0.384362	+	0.923182i	$0.374421\pi$
$770$	0	0
$771$	−64.2908	−2.31538
$772$	0	0
$773$	12.5965	0.453065	0.226532	−	0.974004i	$-0.427261\pi$
$773$	12.5965	0.453065	0.226532	+	0.974004i	$0.427261\pi$
$774$	0	0
$775$	−11.0218	−0.395916
$776$	0	0
$777$	−4.62750	−0.166011
$778$	0	0
$779$	33.1723	1.18852
$780$	0	0
$781$	63.3573	2.26710
$782$	0	0
$783$	132.486	4.73467
$784$	0	0
$785$	10.0682	0.359348
$786$	0	0
$787$	23.3087	0.830867	0.415433	−	0.909624i	$-0.363630\pi$
$787$	23.3087	0.830867	0.415433	+	0.909624i	$0.363630\pi$
$788$	0	0
$789$	−37.6606	−1.34075
$790$	0	0
$791$	−5.85881	−0.208315
$792$	0	0
$793$	8.41543	0.298841
$794$	0	0
$795$	0.829951	0.0294353
$796$	0	0
$797$	−24.2039	−0.857347	−0.428674	−	0.903459i	$-0.641019\pi$
$797$	−24.2039	−0.857347	−0.428674	+	0.903459i	$0.641019\pi$
$798$	0	0
$799$	87.0649	3.08013
$800$	0	0
$801$	−20.0281	−0.707659
$802$	0	0
$803$	30.5786	1.07910
$804$	0	0
$805$	0.765736	0.0269886
$806$	0	0
$807$	−7.02655	−0.247346
$808$	0	0
$809$	−46.2509	−1.62610	−0.813048	−	0.582197i	$-0.802193\pi$
$809$	−46.2509	−1.62610	−0.813048	+	0.582197i	$0.802193\pi$
$810$	0	0
$811$	44.1204	1.54928	0.774639	−	0.632404i	$-0.217932\pi$
$811$	44.1204	1.54928	0.774639	+	0.632404i	$0.217932\pi$
$812$	0	0
$813$	−36.0612	−1.26472
$814$	0	0
$815$	−7.23556	−0.253451
$816$	0	0
$817$	40.0329	1.40057
$818$	0	0
$819$	−10.9012	−0.380919
$820$	0	0
$821$	13.4583	0.469696	0.234848	−	0.972032i	$-0.424541\pi$
$821$	13.4583	0.469696	0.234848	+	0.972032i	$0.424541\pi$
$822$	0	0
$823$	29.8848	1.04172	0.520859	−	0.853643i	$-0.325612\pi$
$823$	29.8848	1.04172	0.520859	+	0.853643i	$0.325612\pi$
$824$	0	0
$825$	−15.1287	−0.526714
$826$	0	0
$827$	−37.7344	−1.31215	−0.656076	−	0.754695i	$-0.727785\pi$
$827$	−37.7344	−1.31215	−0.656076	+	0.754695i	$0.727785\pi$
$828$	0	0
$829$	−9.46552	−0.328751	−0.164375	−	0.986398i	$-0.552561\pi$
$829$	−9.46552	−0.328751	−0.164375	+	0.986398i	$0.552561\pi$
$830$	0	0
$831$	27.9866	0.970843
$832$	0	0
$833$	−47.5481	−1.64745
$834$	0	0
$835$	19.6561	0.680226
$836$	0	0
$837$	180.856	6.25130
$838$	0	0
$839$	7.77972	0.268586	0.134293	−	0.990942i	$-0.457124\pi$
$839$	7.77972	0.268586	0.134293	+	0.990942i	$0.457124\pi$
$840$	0	0
$841$	36.1903	1.24794
$842$	0	0
$843$	−105.063	−3.61857
$844$	0	0
$845$	−4.76682	−0.163984
$846$	0	0
$847$	−4.72134	−0.162227
$848$	0	0
$849$	36.9234	1.26721
$850$	0	0
$851$	4.69578	0.160969
$852$	0	0
$853$	−9.58721	−0.328260	−0.164130	−	0.986439i	$-0.552482\pi$
$853$	−9.58721	−0.328260	−0.164130	+	0.986439i	$0.552482\pi$
$854$	0	0
$855$	−45.5007	−1.55609
$856$	0	0
$857$	−48.6213	−1.66087	−0.830436	−	0.557115i	$-0.811908\pi$
$857$	−48.6213	−1.66087	−0.830436	+	0.557115i	$0.811908\pi$
$858$	0	0
$859$	−42.2174	−1.44044	−0.720219	−	0.693747i	$-0.755959\pi$
$859$	−42.2174	−1.44044	−0.720219	+	0.693747i	$0.755959\pi$
$860$	0	0
$861$	−9.16848	−0.312461
$862$	0	0
$863$	3.78023	0.128680	0.0643402	−	0.997928i	$-0.479506\pi$
$863$	3.78023	0.128680	0.0643402	+	0.997928i	$0.479506\pi$
$864$	0	0
$865$	−22.8776	−0.777862
$866$	0	0
$867$	−106.911	−3.63090
$868$	0	0
$869$	−75.0353	−2.54540
$870$	0	0
$871$	1.03098	0.0349334
$872$	0	0
$873$	−104.379	−3.53269
$874$	0	0
$875$	−0.477457	−0.0161410
$876$	0	0
$877$	−18.2569	−0.616491	−0.308246	−	0.951307i	$-0.599742\pi$
$877$	−18.2569	−0.616491	−0.308246	+	0.951307i	$0.599742\pi$
$878$	0	0
$879$	−56.6782	−1.91171
$880$	0	0
$881$	−16.1548	−0.544269	−0.272135	−	0.962259i	$-0.587730\pi$
$881$	−16.1548	−0.544269	−0.272135	+	0.962259i	$0.587730\pi$
$882$	0	0
$883$	−44.0584	−1.48268	−0.741341	−	0.671128i	$-0.765810\pi$
$883$	−44.0584	−1.48268	−0.741341	+	0.671128i	$0.765810\pi$
$884$	0	0
$885$	−9.49843	−0.319286
$886$	0	0
$887$	42.4081	1.42393	0.711963	−	0.702217i	$-0.247806\pi$
$887$	42.4081	1.42393	0.711963	+	0.702217i	$0.247806\pi$
$888$	0	0
$889$	0.176978	0.00593566
$890$	0	0
$891$	139.144	4.66149
$892$	0	0
$893$	−70.9071	−2.37282
$894$	0	0
$895$	8.42748	0.281700
$896$	0	0
$897$	15.2327	0.508605
$898$	0	0
$899$	88.9908	2.96801
$900$	0	0
$901$	−1.76043	−0.0586484
$902$	0	0
$903$	−11.0647	−0.368209
$904$	0	0
$905$	1.21187	0.0402839
$906$	0	0
$907$	−4.84139	−0.160756	−0.0803779	−	0.996764i	$-0.525613\pi$
$907$	−4.84139	−0.160756	−0.0803779	+	0.996764i	$0.525613\pi$
$908$	0	0
$909$	−40.8577	−1.35516
$910$	0	0
$911$	41.8318	1.38595	0.692974	−	0.720963i	$-0.256300\pi$
$911$	41.8318	1.38595	0.692974	+	0.720963i	$0.256300\pi$
$912$	0	0
$913$	18.6479	0.617157
$914$	0	0
$915$	−9.70825	−0.320945
$916$	0	0
$917$	0.877587	0.0289805
$918$	0	0
$919$	18.0845	0.596552	0.298276	−	0.954480i	$-0.403588\pi$
$919$	18.0845	0.596552	0.298276	+	0.954480i	$0.403588\pi$
$920$	0	0
$921$	86.7805	2.85951
$922$	0	0
$923$	39.7765	1.30926
$924$	0	0
$925$	−2.92795	−0.0962704
$926$	0	0
$927$	−70.2413	−2.30703
$928$	0	0
$929$	12.6937	0.416468	0.208234	−	0.978079i	$-0.433228\pi$
$929$	12.6937	0.416468	0.208234	+	0.978079i	$0.433228\pi$
$930$	0	0
$931$	38.7240	1.26913
$932$	0	0
$933$	−25.2912	−0.827996
$934$	0	0
$935$	32.0899	1.04945
$936$	0	0
$937$	−39.3640	−1.28597	−0.642983	−	0.765880i	$-0.722303\pi$
$937$	−39.3640	−1.28597	−0.642983	+	0.765880i	$0.722303\pi$
$938$	0	0
$939$	66.3134	2.16406
$940$	0	0
$941$	−21.2284	−0.692025	−0.346012	−	0.938230i	$-0.612465\pi$
$941$	−21.2284	−0.692025	−0.346012	+	0.938230i	$0.612465\pi$
$942$	0	0
$943$	9.30376	0.302972
$944$	0	0
$945$	7.83455	0.254858
$946$	0	0
$947$	15.0570	0.489287	0.244644	−	0.969613i	$-0.421329\pi$
$947$	15.0570	0.489287	0.244644	+	0.969613i	$0.421329\pi$
$948$	0	0
$949$	19.1977	0.623182
$950$	0	0
$951$	72.3501	2.34611
$952$	0	0
$953$	−1.98812	−0.0644014	−0.0322007	−	0.999481i	$-0.510252\pi$
$953$	−1.98812	−0.0644014	−0.0322007	+	0.999481i	$0.510252\pi$
$954$	0	0
$955$	−12.1900	−0.394459
$956$	0	0
$957$	122.150	3.94855
$958$	0	0
$959$	7.81828	0.252465
$960$	0	0
$961$	90.4807	2.91873
$962$	0	0
$963$	−78.7775	−2.53857
$964$	0	0
$965$	−5.35965	−0.172533
$966$	0	0
$967$	−33.9801	−1.09273	−0.546364	−	0.837548i	$-0.683988\pi$
$967$	−33.9801	−1.09273	−0.546364	+	0.837548i	$0.683988\pi$
$968$	0	0
$969$	132.900	4.26935
$970$	0	0
$971$	−3.13278	−0.100536	−0.0502678	−	0.998736i	$-0.516007\pi$
$971$	−3.13278	−0.100536	−0.0502678	+	0.998736i	$0.516007\pi$
$972$	0	0
$973$	5.84915	0.187515
$974$	0	0
$975$	−9.49800	−0.304180
$976$	0	0
$977$	34.3665	1.09948	0.549740	−	0.835336i	$-0.314727\pi$
$977$	34.3665	1.09948	0.549740	+	0.835336i	$0.314727\pi$
$978$	0	0
$979$	−11.5037	−0.367659
$980$	0	0
$981$	−111.064	−3.54599
$982$	0	0
$983$	−8.60360	−0.274412	−0.137206	−	0.990543i	$-0.543812\pi$
$983$	−8.60360	−0.274412	−0.137206	+	0.990543i	$0.543812\pi$
$984$	0	0
$985$	−13.8689	−0.441900
$986$	0	0
$987$	19.5980	0.623811
$988$	0	0
$989$	11.2279	0.357028
$990$	0	0
$991$	−17.2975	−0.549473	−0.274736	−	0.961520i	$-0.588591\pi$
$991$	−17.2975	−0.549473	−0.274736	+	0.961520i	$0.588591\pi$
$992$	0	0
$993$	−7.19002	−0.228168
$994$	0	0
$995$	−21.3390	−0.676493
$996$	0	0
$997$	−42.2430	−1.33785	−0.668925	−	0.743330i	$-0.733245\pi$
$997$	−42.2430	−1.33785	−0.668925	+	0.743330i	$0.733245\pi$
$998$	0	0
$999$	48.0444	1.52006

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.2	✓	19	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-3.31016 q^{3} +1.00000 q^{5} -0.477457 q^{7} +7.95713 q^{9} +O(q^{10})\)	\(q-3.31016 q^{3} +1.00000 q^{5} -0.477457 q^{7} +7.95713 q^{9} +4.57039 q^{11} +2.86935 q^{13} -3.31016 q^{15} +7.02125 q^{17} -5.71822 q^{19} +1.58046 q^{21} -1.60378 q^{23} +1.00000 q^{25} -16.4089 q^{27} -8.07405 q^{29} -11.0218 q^{31} -15.1287 q^{33} -0.477457 q^{35} -2.92795 q^{37} -9.49800 q^{39} -5.80115 q^{41} -7.00093 q^{43} +7.95713 q^{45} +12.4002 q^{47} -6.77203 q^{49} -23.2414 q^{51} -0.250729 q^{53} +4.57039 q^{55} +18.9282 q^{57} +2.86948 q^{59} +2.93287 q^{61} -3.79919 q^{63} +2.86935 q^{65} +0.359307 q^{67} +5.30876 q^{69} +13.8626 q^{71} +6.69059 q^{73} -3.31016 q^{75} -2.18217 q^{77} -16.4177 q^{79} +30.4446 q^{81} +4.08016 q^{83} +7.02125 q^{85} +26.7264 q^{87} -2.51700 q^{89} -1.36999 q^{91} +36.4840 q^{93} -5.71822 q^{95} -13.1177 q^{97} +36.3672 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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