LMFDB - Embedded newform 6080.2.a.ce.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.78292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 8x + 18 $ x^4 - x^3 - 10x^2 + 8x + 18
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.78678$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.78678 q^{3} +1.00000 q^{5} -3.76616 q^{7} +4.76616 q^{9} +O(q^{10})$	$q+2.78678 q^{3} +1.00000 q^{5} -3.76616 q^{7} +4.76616 q^{9} -1.15575 q^{11} -1.63103 q^{13} +2.78678 q^{15} -1.38959 q^{17} -1.00000 q^{19} -10.4955 q^{21} -7.76616 q^{23} +1.00000 q^{25} +4.92191 q^{27} -0.651655 q^{29} +4.31150 q^{31} -3.22082 q^{33} -3.76616 q^{35} -3.02062 q^{37} -4.54534 q^{39} +9.84382 q^{41} -9.95014 q^{43} +4.76616 q^{45} -3.62343 q^{47} +7.18398 q^{49} -3.87248 q^{51} -5.20460 q^{53} -1.15575 q^{55} -2.78678 q^{57} -1.65985 q^{59} -1.15575 q^{61} -17.9501 q^{63} -1.63103 q^{65} -1.99239 q^{67} -21.6426 q^{69} -5.53232 q^{71} +1.87248 q^{73} +2.78678 q^{75} +4.35274 q^{77} -0.0412422 q^{79} -0.582183 q^{81} -5.95014 q^{83} -1.38959 q^{85} -1.81602 q^{87} -0.794392 q^{89} +6.14274 q^{91} +12.0152 q^{93} -1.00000 q^{95} -18.9209 q^{97} -5.50849 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9} + 6 q^{11} - 5 q^{13} - q^{15} - 5 q^{17} - 4 q^{19} + 3 q^{21} - 21 q^{23} + 4 q^{25} - q^{27} + q^{29} - 4 q^{31} - 14 q^{33} - 5 q^{35} - 10 q^{37} - 7 q^{39} - 2 q^{41} - 6 q^{43} + 9 q^{45} - 24 q^{47} + 5 q^{49} - 13 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 11 q^{59} + 6 q^{61} - 38 q^{63} - 5 q^{65} - 19 q^{67} + 7 q^{69} - 2 q^{71} + 5 q^{73} - q^{75} - 8 q^{77} + 4 q^{79} - 16 q^{81} + 10 q^{83} - 5 q^{85} - 31 q^{87} + 20 q^{89} + 5 q^{91} + 26 q^{93} - 4 q^{95} - 6 q^{97} + 14 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9 + 6 * q^11 - 5 * q^13 - q^15 - 5 * q^17 - 4 * q^19 + 3 * q^21 - 21 * q^23 + 4 * q^25 - q^27 + q^29 - 4 * q^31 - 14 * q^33 - 5 * q^35 - 10 * q^37 - 7 * q^39 - 2 * q^41 - 6 * q^43 + 9 * q^45 - 24 * q^47 + 5 * q^49 - 13 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 11 * q^59 + 6 * q^61 - 38 * q^63 - 5 * q^65 - 19 * q^67 + 7 * q^69 - 2 * q^71 + 5 * q^73 - q^75 - 8 * q^77 + 4 * q^79 - 16 * q^81 + 10 * q^83 - 5 * q^85 - 31 * q^87 + 20 * q^89 + 5 * q^91 + 26 * q^93 - 4 * q^95 - 6 * q^97 + 14 * 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
$3$	2.78678	1.60895	0.804475	−	0.593986i	$-0.202447\pi$
$3$	2.78678	1.60895	0.804475	+	0.593986i	$0.202447\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.76616	−1.42348	−0.711738	−	0.702445i	$-0.752092\pi$
$7$	−3.76616	−1.42348	−0.711738	+	0.702445i	$0.752092\pi$
$8$	0	0
$9$	4.76616	1.58872
$10$	0	0
$11$	−1.15575	−0.348472	−0.174236	−	0.984704i	$-0.555746\pi$
$11$	−1.15575	−0.348472	−0.174236	+	0.984704i	$0.555746\pi$
$12$	0	0
$13$	−1.63103	−0.452367	−0.226184	−	0.974085i	$-0.572625\pi$
$13$	−1.63103	−0.452367	−0.226184	+	0.974085i	$0.572625\pi$
$14$	0	0
$15$	2.78678	0.719544
$16$	0	0
$17$	−1.38959	−0.337024	−0.168512	−	0.985700i	$-0.553896\pi$
$17$	−1.38959	−0.337024	−0.168512	+	0.985700i	$0.553896\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−10.4955	−2.29030
$22$	0	0
$23$	−7.76616	−1.61936	−0.809678	−	0.586874i	$-0.800358\pi$
$23$	−7.76616	−1.61936	−0.809678	+	0.586874i	$0.800358\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.92191	0.947222
$28$	0	0
$29$	−0.651655	−0.121009	−0.0605046	−	0.998168i	$-0.519271\pi$
$29$	−0.651655	−0.121009	−0.0605046	+	0.998168i	$0.519271\pi$
$30$	0	0
$31$	4.31150	0.774368	0.387184	−	0.922002i	$-0.373448\pi$
$31$	4.31150	0.774368	0.387184	+	0.922002i	$0.373448\pi$
$32$	0	0
$33$	−3.22082	−0.560674
$34$	0	0
$35$	−3.76616	−0.636598
$36$	0	0
$37$	−3.02062	−0.496587	−0.248294	−	0.968685i	$-0.579870\pi$
$37$	−3.02062	−0.496587	−0.248294	+	0.968685i	$0.579870\pi$
$38$	0	0
$39$	−4.54534	−0.727836
$40$	0	0
$41$	9.84382	1.53735	0.768674	−	0.639641i	$-0.220917\pi$
$41$	9.84382	1.53735	0.768674	+	0.639641i	$0.220917\pi$
$42$	0	0
$43$	−9.95014	−1.51738	−0.758691	−	0.651450i	$-0.774161\pi$
$43$	−9.95014	−1.51738	−0.758691	+	0.651450i	$0.774161\pi$
$44$	0	0
$45$	4.76616	0.710498
$46$	0	0
$47$	−3.62343	−0.528531	−0.264265	−	0.964450i	$-0.585129\pi$
$47$	−3.62343	−0.528531	−0.264265	+	0.964450i	$0.585129\pi$
$48$	0	0
$49$	7.18398	1.02628
$50$	0	0
$51$	−3.87248	−0.542256
$52$	0	0
$53$	−5.20460	−0.714907	−0.357453	−	0.933931i	$-0.616355\pi$
$53$	−5.20460	−0.714907	−0.357453	+	0.933931i	$0.616355\pi$
$54$	0	0
$55$	−1.15575	−0.155841
$56$	0	0
$57$	−2.78678	−0.369118
$58$	0	0
$59$	−1.65985	−0.216093	−0.108047	−	0.994146i	$-0.534460\pi$
$59$	−1.65985	−0.216093	−0.108047	+	0.994146i	$0.534460\pi$
$60$	0	0
$61$	−1.15575	−0.147979	−0.0739893	−	0.997259i	$-0.523573\pi$
$61$	−1.15575	−0.147979	−0.0739893	+	0.997259i	$0.523573\pi$
$62$	0	0
$63$	−17.9501	−2.26151
$64$	0	0
$65$	−1.63103	−0.202305
$66$	0	0
$67$	−1.99239	−0.243409	−0.121705	−	0.992566i	$-0.538836\pi$
$67$	−1.99239	−0.243409	−0.121705	+	0.992566i	$0.538836\pi$
$68$	0	0
$69$	−21.6426	−2.60546
$70$	0	0
$71$	−5.53232	−0.656566	−0.328283	−	0.944579i	$-0.606470\pi$
$71$	−5.53232	−0.656566	−0.328283	+	0.944579i	$0.606470\pi$
$72$	0	0
$73$	1.87248	0.219157	0.109579	−	0.993978i	$-0.465050\pi$
$73$	1.87248	0.219157	0.109579	+	0.993978i	$0.465050\pi$
$74$	0	0
$75$	2.78678	0.321790
$76$	0	0
$77$	4.35274	0.496041
$78$	0	0
$79$	−0.0412422	−0.00464011	−0.00232005	−	0.999997i	$-0.500738\pi$
$79$	−0.0412422	−0.00464011	−0.00232005	+	0.999997i	$0.500738\pi$
$80$	0	0
$81$	−0.582183	−0.0646870
$82$	0	0
$83$	−5.95014	−0.653113	−0.326556	−	0.945178i	$-0.605888\pi$
$83$	−5.95014	−0.653113	−0.326556	+	0.945178i	$0.605888\pi$
$84$	0	0
$85$	−1.38959	−0.150722
$86$	0	0
$87$	−1.81602	−0.194698
$88$	0	0
$89$	−0.794392	−0.0842054	−0.0421027	−	0.999113i	$-0.513406\pi$
$89$	−0.794392	−0.0842054	−0.0421027	+	0.999113i	$0.513406\pi$
$90$	0	0
$91$	6.14274	0.643934
$92$	0	0
$93$	12.0152	1.24592
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−18.9209	−1.92113	−0.960563	−	0.278061i	$-0.910308\pi$
$97$	−18.9209	−1.92113	−0.960563	+	0.278061i	$0.910308\pi$
$98$	0	0
$99$	−5.50849	−0.553624
$100$	0	0
$101$	7.95014	0.791069	0.395534	−	0.918451i	$-0.370559\pi$
$101$	7.95014	0.791069	0.395534	+	0.918451i	$0.370559\pi$
$102$	0	0
$103$	−15.7087	−1.54782	−0.773912	−	0.633293i	$-0.781703\pi$
$103$	−15.7087	−1.54782	−0.773912	+	0.633293i	$0.781703\pi$
$104$	0	0
$105$	−10.4955	−1.02425
$106$	0	0
$107$	1.99239	0.192612	0.0963059	−	0.995352i	$-0.469297\pi$
$107$	1.99239	0.192612	0.0963059	+	0.995352i	$0.469297\pi$
$108$	0	0
$109$	−0.963155	−0.0922535	−0.0461267	−	0.998936i	$-0.514688\pi$
$109$	−0.963155	−0.0922535	−0.0461267	+	0.998936i	$0.514688\pi$
$110$	0	0
$111$	−8.41782	−0.798984
$112$	0	0
$113$	−0.938137	−0.0882525	−0.0441262	−	0.999026i	$-0.514050\pi$
$113$	−0.938137	−0.0882525	−0.0441262	+	0.999026i	$0.514050\pi$
$114$	0	0
$115$	−7.76616	−0.724198
$116$	0	0
$117$	−7.77377	−0.718685
$118$	0	0
$119$	5.23341	0.479746
$120$	0	0
$121$	−9.66424	−0.878567
$122$	0	0
$123$	27.4326	2.47352
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.8146	0.959639	0.479820	−	0.877367i	$-0.340702\pi$
$127$	10.8146	0.959639	0.479820	+	0.877367i	$0.340702\pi$
$128$	0	0
$129$	−27.7289	−2.44139
$130$	0	0
$131$	19.4174	1.69651	0.848253	−	0.529592i	$-0.177655\pi$
$131$	19.4174	1.69651	0.848253	+	0.529592i	$0.177655\pi$
$132$	0	0
$133$	3.76616	0.326568
$134$	0	0
$135$	4.92191	0.423611
$136$	0	0
$137$	−2.22522	−0.190114	−0.0950568	−	0.995472i	$-0.530303\pi$
$137$	−2.22522	−0.190114	−0.0950568	+	0.995472i	$0.530303\pi$
$138$	0	0
$139$	−1.90890	−0.161911	−0.0809554	−	0.996718i	$-0.525797\pi$
$139$	−1.90890	−0.161911	−0.0809554	+	0.996718i	$0.525797\pi$
$140$	0	0
$141$	−10.0977	−0.850380
$142$	0	0
$143$	1.88507	0.157637
$144$	0	0
$145$	−0.651655	−0.0541170
$146$	0	0
$147$	20.0202	1.65124
$148$	0	0
$149$	−19.3675	−1.58665	−0.793325	−	0.608798i	$-0.791652\pi$
$149$	−19.3675	−1.58665	−0.793325	+	0.608798i	$0.791652\pi$
$150$	0	0
$151$	14.3267	1.16589	0.582946	−	0.812511i	$-0.301900\pi$
$151$	14.3267	1.16589	0.582946	+	0.812511i	$0.301900\pi$
$152$	0	0
$153$	−6.62300	−0.535438
$154$	0	0
$155$	4.31150	0.346308
$156$	0	0
$157$	20.2943	1.61966	0.809829	−	0.586665i	$-0.199560\pi$
$157$	20.2943	1.61966	0.809829	+	0.586665i	$0.199560\pi$
$158$	0	0
$159$	−14.5041	−1.15025
$160$	0	0
$161$	29.2486	2.30511
$162$	0	0
$163$	−16.2616	−1.27371	−0.636855	−	0.770984i	$-0.719765\pi$
$163$	−16.2616	−1.27371	−0.636855	+	0.770984i	$0.719765\pi$
$164$	0	0
$165$	−3.22082	−0.250741
$166$	0	0
$167$	−11.1269	−0.861028	−0.430514	−	0.902584i	$-0.641668\pi$
$167$	−11.1269	−0.861028	−0.430514	+	0.902584i	$0.641668\pi$
$168$	0	0
$169$	−10.3397	−0.795364
$170$	0	0
$171$	−4.76616	−0.364478
$172$	0	0
$173$	2.32393	0.176685	0.0883426	−	0.996090i	$-0.471843\pi$
$173$	2.32393	0.176685	0.0883426	+	0.996090i	$0.471843\pi$
$174$	0	0
$175$	−3.76616	−0.284695
$176$	0	0
$177$	−4.62563	−0.347684
$178$	0	0
$179$	17.8851	1.33679	0.668396	−	0.743805i	$-0.266981\pi$
$179$	17.8851	1.33679	0.668396	+	0.743805i	$0.266981\pi$
$180$	0	0
$181$	24.2118	1.79965	0.899824	−	0.436253i	$-0.143695\pi$
$181$	24.2118	1.79965	0.899824	+	0.436253i	$0.143695\pi$
$182$	0	0
$183$	−3.22082	−0.238090
$184$	0	0
$185$	−3.02062	−0.222080
$186$	0	0
$187$	1.60602	0.117444
$188$	0	0
$189$	−18.5367	−1.34835
$190$	0	0
$191$	−15.7575	−1.14018	−0.570088	−	0.821584i	$-0.693091\pi$
$191$	−15.7575	−1.14018	−0.570088	+	0.821584i	$0.693091\pi$
$192$	0	0
$193$	14.4380	1.03927	0.519635	−	0.854388i	$-0.326068\pi$
$193$	14.4380	1.03927	0.519635	+	0.854388i	$0.326068\pi$
$194$	0	0
$195$	−4.54534	−0.325498
$196$	0	0
$197$	−19.0234	−1.35536	−0.677681	−	0.735356i	$-0.737015\pi$
$197$	−19.0234	−1.35536	−0.677681	+	0.735356i	$0.737015\pi$
$198$	0	0
$199$	−13.9866	−0.991481	−0.495741	−	0.868471i	$-0.665103\pi$
$199$	−13.9866	−0.991481	−0.495741	+	0.868471i	$0.665103\pi$
$200$	0	0
$201$	−5.55236	−0.391634
$202$	0	0
$203$	2.45424	0.172254
$204$	0	0
$205$	9.84382	0.687523
$206$	0	0
$207$	−37.0148	−2.57271
$208$	0	0
$209$	1.15575	0.0799449
$210$	0	0
$211$	17.3311	1.19312	0.596562	−	0.802567i	$-0.296533\pi$
$211$	17.3311	1.19312	0.596562	+	0.802567i	$0.296533\pi$
$212$	0	0
$213$	−15.4174	−1.05638
$214$	0	0
$215$	−9.95014	−0.678594
$216$	0	0
$217$	−16.2378	−1.10229
$218$	0	0
$219$	5.21820	0.352613
$220$	0	0
$221$	2.26646	0.152459
$222$	0	0
$223$	3.60983	0.241732	0.120866	−	0.992669i	$-0.461433\pi$
$223$	3.60983	0.241732	0.120866	+	0.992669i	$0.461433\pi$
$224$	0	0
$225$	4.76616	0.317744
$226$	0	0
$227$	−10.1629	−0.674538	−0.337269	−	0.941408i	$-0.609503\pi$
$227$	−10.1629	−0.674538	−0.337269	+	0.941408i	$0.609503\pi$
$228$	0	0
$229$	−13.7940	−0.911531	−0.455765	−	0.890100i	$-0.650634\pi$
$229$	−13.7940	−0.911531	−0.455765	+	0.890100i	$0.650634\pi$
$230$	0	0
$231$	12.1301	0.798105
$232$	0	0
$233$	−19.1059	−1.25167	−0.625834	−	0.779956i	$-0.715241\pi$
$233$	−19.1059	−1.25167	−0.625834	+	0.779956i	$0.715241\pi$
$234$	0	0
$235$	−3.62343	−0.236366
$236$	0	0
$237$	−0.114933	−0.00746570
$238$	0	0
$239$	8.70811	0.563281	0.281641	−	0.959520i	$-0.409121\pi$
$239$	8.70811	0.563281	0.281641	+	0.959520i	$0.409121\pi$
$240$	0	0
$241$	25.4586	1.63993	0.819967	−	0.572410i	$-0.193992\pi$
$241$	25.4586	1.63993	0.819967	+	0.572410i	$0.193992\pi$
$242$	0	0
$243$	−16.3882	−1.05130
$244$	0	0
$245$	7.18398	0.458968
$246$	0	0
$247$	1.63103	0.103780
$248$	0	0
$249$	−16.5818	−1.05083
$250$	0	0
$251$	−23.9828	−1.51378	−0.756889	−	0.653543i	$-0.773282\pi$
$251$	−23.9828	−1.51378	−0.756889	+	0.653543i	$0.773282\pi$
$252$	0	0
$253$	8.97574	0.564300
$254$	0	0
$255$	−3.87248	−0.242504
$256$	0	0
$257$	−9.06186	−0.565263	−0.282632	−	0.959228i	$-0.591207\pi$
$257$	−9.06186	−0.565263	−0.282632	+	0.959228i	$0.591207\pi$
$258$	0	0
$259$	11.3761	0.706880
$260$	0	0
$261$	−3.10589	−0.192250
$262$	0	0
$263$	−0.459059	−0.0283068	−0.0141534	−	0.999900i	$-0.504505\pi$
$263$	−0.459059	−0.0283068	−0.0141534	+	0.999900i	$0.504505\pi$
$264$	0	0
$265$	−5.20460	−0.319716
$266$	0	0
$267$	−2.21380	−0.135482
$268$	0	0
$269$	8.63822	0.526681	0.263341	−	0.964703i	$-0.415176\pi$
$269$	8.63822	0.526681	0.263341	+	0.964703i	$0.415176\pi$
$270$	0	0
$271$	5.29849	0.321860	0.160930	−	0.986966i	$-0.448551\pi$
$271$	5.29849	0.321860	0.160930	+	0.986966i	$0.448551\pi$
$272$	0	0
$273$	17.1185	1.03606
$274$	0	0
$275$	−1.15575	−0.0696943
$276$	0	0
$277$	26.3680	1.58430	0.792149	−	0.610328i	$-0.208962\pi$
$277$	26.3680	1.58430	0.792149	+	0.610328i	$0.208962\pi$
$278$	0	0
$279$	20.5493	1.23025
$280$	0	0
$281$	−9.22082	−0.550068	−0.275034	−	0.961434i	$-0.588689\pi$
$281$	−9.22082	−0.550068	−0.275034	+	0.961434i	$0.588689\pi$
$282$	0	0
$283$	−1.96737	−0.116948	−0.0584741	−	0.998289i	$-0.518624\pi$
$283$	−1.96737	−0.116948	−0.0584741	+	0.998289i	$0.518624\pi$
$284$	0	0
$285$	−2.78678	−0.165075
$286$	0	0
$287$	−37.0734	−2.18838
$288$	0	0
$289$	−15.0690	−0.886415
$290$	0	0
$291$	−52.7285	−3.09100
$292$	0	0
$293$	23.6866	1.38379	0.691894	−	0.721999i	$-0.256776\pi$
$293$	23.6866	1.38379	0.691894	+	0.721999i	$0.256776\pi$
$294$	0	0
$295$	−1.65985	−0.0966399
$296$	0	0
$297$	−5.68850	−0.330080
$298$	0	0
$299$	12.6669	0.732544
$300$	0	0
$301$	37.4739	2.15996
$302$	0	0
$303$	22.1553	1.27279
$304$	0	0
$305$	−1.15575	−0.0661781
$306$	0	0
$307$	−4.47908	−0.255634	−0.127817	−	0.991798i	$-0.540797\pi$
$307$	−4.47908	−0.255634	−0.127817	+	0.991798i	$0.540797\pi$
$308$	0	0
$309$	−43.7767	−2.49037
$310$	0	0
$311$	28.5281	1.61768	0.808840	−	0.588028i	$-0.200096\pi$
$311$	28.5281	1.61768	0.808840	+	0.588028i	$0.200096\pi$
$312$	0	0
$313$	28.8070	1.62827	0.814133	−	0.580678i	$-0.197212\pi$
$313$	28.8070	1.62827	0.814133	+	0.580678i	$0.197212\pi$
$314$	0	0
$315$	−17.9501	−1.01138
$316$	0	0
$317$	23.6550	1.32860	0.664300	−	0.747466i	$-0.268730\pi$
$317$	23.6550	1.32860	0.664300	+	0.747466i	$0.268730\pi$
$318$	0	0
$319$	0.753150	0.0421683
$320$	0	0
$321$	5.55236	0.309903
$322$	0	0
$323$	1.38959	0.0773187
$324$	0	0
$325$	−1.63103	−0.0904735
$326$	0	0
$327$	−2.68410	−0.148431
$328$	0	0
$329$	13.6464	0.752351
$330$	0	0
$331$	−25.3483	−1.39327	−0.696636	−	0.717425i	$-0.745321\pi$
$331$	−25.3483	−1.39327	−0.696636	+	0.717425i	$0.745321\pi$
$332$	0	0
$333$	−14.3968	−0.788938
$334$	0	0
$335$	−1.99239	−0.108856
$336$	0	0
$337$	−29.0935	−1.58482	−0.792411	−	0.609988i	$-0.791174\pi$
$337$	−29.0935	−1.58482	−0.792411	+	0.609988i	$0.791174\pi$
$338$	0	0
$339$	−2.61438	−0.141994
$340$	0	0
$341$	−4.98302	−0.269845
$342$	0	0
$343$	−0.692897	−0.0374129
$344$	0	0
$345$	−21.6426	−1.16520
$346$	0	0
$347$	−7.23823	−0.388569	−0.194284	−	0.980945i	$-0.562238\pi$
$347$	−7.23823	−0.388569	−0.194284	+	0.980945i	$0.562238\pi$
$348$	0	0
$349$	−19.0234	−1.01830	−0.509150	−	0.860678i	$-0.670040\pi$
$349$	−19.0234	−1.01830	−0.509150	+	0.860678i	$0.670040\pi$
$350$	0	0
$351$	−8.02780	−0.428492
$352$	0	0
$353$	10.4379	0.555551	0.277776	−	0.960646i	$-0.410403\pi$
$353$	10.4379	0.555551	0.277776	+	0.960646i	$0.410403\pi$
$354$	0	0
$355$	−5.53232	−0.293625
$356$	0	0
$357$	14.5844	0.771888
$358$	0	0
$359$	−28.4456	−1.50130	−0.750651	−	0.660699i	$-0.770260\pi$
$359$	−28.4456	−1.50130	−0.750651	+	0.660699i	$0.770260\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−26.9322	−1.41357
$364$	0	0
$365$	1.87248	0.0980101
$366$	0	0
$367$	−29.3523	−1.53218	−0.766089	−	0.642734i	$-0.777800\pi$
$367$	−29.3523	−1.53218	−0.766089	+	0.642734i	$0.777800\pi$
$368$	0	0
$369$	46.9173	2.44242
$370$	0	0
$371$	19.6014	1.01765
$372$	0	0
$373$	9.47486	0.490590	0.245295	−	0.969449i	$-0.421115\pi$
$373$	9.47486	0.490590	0.245295	+	0.969449i	$0.421115\pi$
$374$	0	0
$375$	2.78678	0.143909
$376$	0	0
$377$	1.06287	0.0547406
$378$	0	0
$379$	19.8304	1.01862	0.509309	−	0.860584i	$-0.329901\pi$
$379$	19.8304	1.01862	0.509309	+	0.860584i	$0.329901\pi$
$380$	0	0
$381$	30.1379	1.54401
$382$	0	0
$383$	−17.9626	−0.917845	−0.458922	−	0.888476i	$-0.651764\pi$
$383$	−17.9626	−0.917845	−0.458922	+	0.888476i	$0.651764\pi$
$384$	0	0
$385$	4.35274	0.221836
$386$	0	0
$387$	−47.4240	−2.41070
$388$	0	0
$389$	−9.34455	−0.473788	−0.236894	−	0.971536i	$-0.576129\pi$
$389$	−9.34455	−0.473788	−0.236894	+	0.971536i	$0.576129\pi$
$390$	0	0
$391$	10.7918	0.545763
$392$	0	0
$393$	54.1121	2.72959
$394$	0	0
$395$	−0.0412422	−0.00207512
$396$	0	0
$397$	31.4739	1.57963	0.789814	−	0.613347i	$-0.210177\pi$
$397$	31.4739	1.57963	0.789814	+	0.613347i	$0.210177\pi$
$398$	0	0
$399$	10.4955	0.525431
$400$	0	0
$401$	5.35977	0.267654	0.133827	−	0.991005i	$-0.457273\pi$
$401$	5.35977	0.267654	0.133827	+	0.991005i	$0.457273\pi$
$402$	0	0
$403$	−7.03220	−0.350299
$404$	0	0
$405$	−0.582183	−0.0289289
$406$	0	0
$407$	3.49108	0.173047
$408$	0	0
$409$	11.1319	0.550438	0.275219	−	0.961382i	$-0.411250\pi$
$409$	11.1319	0.550438	0.275219	+	0.961382i	$0.411250\pi$
$410$	0	0
$411$	−6.20121	−0.305883
$412$	0	0
$413$	6.25125	0.307604
$414$	0	0
$415$	−5.95014	−0.292081
$416$	0	0
$417$	−5.31969	−0.260506
$418$	0	0
$419$	35.5727	1.73784	0.868920	−	0.494952i	$-0.164814\pi$
$419$	35.5727	1.73784	0.868920	+	0.494952i	$0.164814\pi$
$420$	0	0
$421$	26.8222	1.30723	0.653617	−	0.756826i	$-0.273251\pi$
$421$	26.8222	1.30723	0.653617	+	0.756826i	$0.273251\pi$
$422$	0	0
$423$	−17.2698	−0.839688
$424$	0	0
$425$	−1.38959	−0.0674049
$426$	0	0
$427$	4.35274	0.210644
$428$	0	0
$429$	5.25327	0.253630
$430$	0	0
$431$	33.0887	1.59382	0.796912	−	0.604095i	$-0.206465\pi$
$431$	33.0887	1.59382	0.796912	+	0.604095i	$0.206465\pi$
$432$	0	0
$433$	32.9613	1.58402	0.792009	−	0.610509i	$-0.209035\pi$
$433$	32.9613	1.58402	0.792009	+	0.610509i	$0.209035\pi$
$434$	0	0
$435$	−1.81602	−0.0870715
$436$	0	0
$437$	7.76616	0.371506
$438$	0	0
$439$	−22.4657	−1.07223	−0.536114	−	0.844146i	$-0.680108\pi$
$439$	−22.4657	−1.07223	−0.536114	+	0.844146i	$0.680108\pi$
$440$	0	0
$441$	34.2400	1.63048
$442$	0	0
$443$	−14.4754	−0.687749	−0.343874	−	0.939016i	$-0.611739\pi$
$443$	−14.4754	−0.687749	−0.343874	+	0.939016i	$0.611739\pi$
$444$	0	0
$445$	−0.794392	−0.0376578
$446$	0	0
$447$	−53.9731	−2.55284
$448$	0	0
$449$	−11.2621	−0.531490	−0.265745	−	0.964043i	$-0.585618\pi$
$449$	−11.2621	−0.531490	−0.265745	+	0.964043i	$0.585618\pi$
$450$	0	0
$451$	−11.3770	−0.535722
$452$	0	0
$453$	39.9255	1.87586
$454$	0	0
$455$	6.14274	0.287976
$456$	0	0
$457$	19.3887	0.906967	0.453483	−	0.891265i	$-0.350181\pi$
$457$	19.3887	0.906967	0.453483	+	0.891265i	$0.350181\pi$
$458$	0	0
$459$	−6.83943	−0.319237
$460$	0	0
$461$	22.3106	1.03911	0.519555	−	0.854437i	$-0.326098\pi$
$461$	22.3106	1.03911	0.519555	+	0.854437i	$0.326098\pi$
$462$	0	0
$463$	23.6062	1.09707	0.548536	−	0.836127i	$-0.315185\pi$
$463$	23.6062	1.09707	0.548536	+	0.836127i	$0.315185\pi$
$464$	0	0
$465$	12.0152	0.557192
$466$	0	0
$467$	31.9341	1.47773	0.738866	−	0.673852i	$-0.235361\pi$
$467$	31.9341	1.47773	0.738866	+	0.673852i	$0.235361\pi$
$468$	0	0
$469$	7.50367	0.346487
$470$	0	0
$471$	56.5557	2.60595
$472$	0	0
$473$	11.4999	0.528765
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−24.8060	−1.13579
$478$	0	0
$479$	−4.53275	−0.207107	−0.103553	−	0.994624i	$-0.533021\pi$
$479$	−4.53275	−0.207107	−0.103553	+	0.994624i	$0.533021\pi$
$480$	0	0
$481$	4.92673	0.224640
$482$	0	0
$483$	81.5096	3.70882
$484$	0	0
$485$	−18.9209	−0.859154
$486$	0	0
$487$	−2.70051	−0.122372	−0.0611858	−	0.998126i	$-0.519488\pi$
$487$	−2.70051	−0.122372	−0.0611858	+	0.998126i	$0.519488\pi$
$488$	0	0
$489$	−45.3177	−2.04934
$490$	0	0
$491$	−37.9091	−1.71081	−0.855406	−	0.517957i	$-0.826693\pi$
$491$	−37.9091	−1.71081	−0.855406	+	0.517957i	$0.826693\pi$
$492$	0	0
$493$	0.905531	0.0407831
$494$	0	0
$495$	−5.50849	−0.247588
$496$	0	0
$497$	20.8356	0.934606
$498$	0	0
$499$	5.54094	0.248047	0.124023	−	0.992279i	$-0.460420\pi$
$499$	5.54094	0.248047	0.124023	+	0.992279i	$0.460420\pi$
$500$	0	0
$501$	−31.0084	−1.38535
$502$	0	0
$503$	1.28211	0.0571663	0.0285831	−	0.999591i	$-0.490900\pi$
$503$	1.28211	0.0571663	0.0285831	+	0.999591i	$0.490900\pi$
$504$	0	0
$505$	7.95014	0.353777
$506$	0	0
$507$	−28.8146	−1.27970
$508$	0	0
$509$	42.7935	1.89679	0.948395	−	0.317091i	$-0.102706\pi$
$509$	42.7935	1.89679	0.948395	+	0.317091i	$0.102706\pi$
$510$	0	0
$511$	−7.05206	−0.311965
$512$	0	0
$513$	−4.92191	−0.217308
$514$	0	0
$515$	−15.7087	−0.692208
$516$	0	0
$517$	4.18777	0.184178
$518$	0	0
$519$	6.47629	0.284278
$520$	0	0
$521$	−38.8088	−1.70024	−0.850121	−	0.526587i	$-0.823471\pi$
$521$	−38.8088	−1.70024	−0.850121	+	0.526587i	$0.823471\pi$
$522$	0	0
$523$	−15.4923	−0.677430	−0.338715	−	0.940889i	$-0.609992\pi$
$523$	−15.4923	−0.677430	−0.338715	+	0.940889i	$0.609992\pi$
$524$	0	0
$525$	−10.4955	−0.458060
$526$	0	0
$527$	−5.99121	−0.260981
$528$	0	0
$529$	37.3133	1.62232
$530$	0	0
$531$	−7.91109	−0.343312
$532$	0	0
$533$	−16.0556	−0.695446
$534$	0	0
$535$	1.99239	0.0861386
$536$	0	0
$537$	49.8418	2.15083
$538$	0	0
$539$	−8.30288	−0.357631
$540$	0	0
$541$	−6.90187	−0.296735	−0.148367	−	0.988932i	$-0.547402\pi$
$541$	−6.90187	−0.296735	−0.148367	+	0.988932i	$0.547402\pi$
$542$	0	0
$543$	67.4730	2.89554
$544$	0	0
$545$	−0.963155	−0.0412570
$546$	0	0
$547$	−27.3647	−1.17003	−0.585016	−	0.811022i	$-0.698912\pi$
$547$	−27.3647	−1.17003	−0.585016	+	0.811022i	$0.698912\pi$
$548$	0	0
$549$	−5.50849	−0.235097
$550$	0	0
$551$	0.651655	0.0277614
$552$	0	0
$553$	0.155325	0.00660508
$554$	0	0
$555$	−8.41782	−0.357316
$556$	0	0
$557$	−31.8438	−1.34927	−0.674633	−	0.738153i	$-0.735698\pi$
$557$	−31.8438	−1.34927	−0.674633	+	0.738153i	$0.735698\pi$
$558$	0	0
$559$	16.2290	0.686414
$560$	0	0
$561$	4.47562	0.188961
$562$	0	0
$563$	11.5121	0.485178	0.242589	−	0.970129i	$-0.422003\pi$
$563$	11.5121	0.485178	0.242589	+	0.970129i	$0.422003\pi$
$564$	0	0
$565$	−0.938137	−0.0394677
$566$	0	0
$567$	2.19260	0.0920804
$568$	0	0
$569$	−21.8286	−0.915103	−0.457551	−	0.889183i	$-0.651273\pi$
$569$	−21.8286	−0.915103	−0.457551	+	0.889183i	$0.651273\pi$
$570$	0	0
$571$	−25.5237	−1.06813	−0.534067	−	0.845442i	$-0.679337\pi$
$571$	−25.5237	−1.06813	−0.534067	+	0.845442i	$0.679337\pi$
$572$	0	0
$573$	−43.9129	−1.83449
$574$	0	0
$575$	−7.76616	−0.323871
$576$	0	0
$577$	1.32232	0.0550489	0.0275245	−	0.999621i	$-0.491238\pi$
$577$	1.32232	0.0550489	0.0275245	+	0.999621i	$0.491238\pi$
$578$	0	0
$579$	40.2356	1.67214
$580$	0	0
$581$	22.4092	0.929690
$582$	0	0
$583$	6.01522	0.249125
$584$	0	0
$585$	−7.77377	−0.321406
$586$	0	0
$587$	44.7849	1.84847	0.924236	−	0.381822i	$-0.124703\pi$
$587$	44.7849	1.84847	0.924236	+	0.381822i	$0.124703\pi$
$588$	0	0
$589$	−4.31150	−0.177652
$590$	0	0
$591$	−53.0141	−2.18071
$592$	0	0
$593$	1.22962	0.0504944	0.0252472	−	0.999681i	$-0.491963\pi$
$593$	1.22962	0.0504944	0.0252472	+	0.999681i	$0.491963\pi$
$594$	0	0
$595$	5.23341	0.214549
$596$	0	0
$597$	−38.9775	−1.59524
$598$	0	0
$599$	−11.8590	−0.484547	−0.242274	−	0.970208i	$-0.577893\pi$
$599$	−11.8590	−0.484547	−0.242274	+	0.970208i	$0.577893\pi$
$600$	0	0
$601$	−2.10851	−0.0860079	−0.0430039	−	0.999075i	$-0.513693\pi$
$601$	−2.10851	−0.0860079	−0.0430039	+	0.999075i	$0.513693\pi$
$602$	0	0
$603$	−9.49606	−0.386710
$604$	0	0
$605$	−9.66424	−0.392907
$606$	0	0
$607$	15.4940	0.628884	0.314442	−	0.949277i	$-0.398183\pi$
$607$	15.4940	0.628884	0.314442	+	0.949277i	$0.398183\pi$
$608$	0	0
$609$	6.83943	0.277148
$610$	0	0
$611$	5.90993	0.239090
$612$	0	0
$613$	4.53935	0.183343	0.0916713	−	0.995789i	$-0.470779\pi$
$613$	4.53935	0.183343	0.0916713	+	0.995789i	$0.470779\pi$
$614$	0	0
$615$	27.4326	1.10619
$616$	0	0
$617$	−0.711908	−0.0286603	−0.0143302	−	0.999897i	$-0.504562\pi$
$617$	−0.711908	−0.0286603	−0.0143302	+	0.999897i	$0.504562\pi$
$618$	0	0
$619$	−44.0382	−1.77004	−0.885022	−	0.465549i	$-0.845857\pi$
$619$	−44.0382	−1.77004	−0.885022	+	0.465549i	$0.845857\pi$
$620$	0	0
$621$	−38.2244	−1.53389
$622$	0	0
$623$	2.99181	0.119864
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	3.22082	0.128627
$628$	0	0
$629$	4.19742	0.167362
$630$	0	0
$631$	−22.6448	−0.901476	−0.450738	−	0.892656i	$-0.648839\pi$
$631$	−22.6448	−0.901476	−0.450738	+	0.892656i	$0.648839\pi$
$632$	0	0
$633$	48.2981	1.91968
$634$	0	0
$635$	10.8146	0.429164
$636$	0	0
$637$	−11.7173	−0.464257
$638$	0	0
$639$	−26.3680	−1.04310
$640$	0	0
$641$	−33.4066	−1.31948	−0.659740	−	0.751494i	$-0.729334\pi$
$641$	−33.4066	−1.31948	−0.659740	+	0.751494i	$0.729334\pi$
$642$	0	0
$643$	6.22040	0.245309	0.122654	−	0.992449i	$-0.460859\pi$
$643$	6.22040	0.245309	0.122654	+	0.992449i	$0.460859\pi$
$644$	0	0
$645$	−27.7289	−1.09182
$646$	0	0
$647$	39.3693	1.54777	0.773884	−	0.633328i	$-0.218311\pi$
$647$	39.3693	1.54777	0.773884	+	0.633328i	$0.218311\pi$
$648$	0	0
$649$	1.91837	0.0753025
$650$	0	0
$651$	−45.2513	−1.77354
$652$	0	0
$653$	−21.0494	−0.823728	−0.411864	−	0.911245i	$-0.635122\pi$
$653$	−21.0494	−0.823728	−0.411864	+	0.911245i	$0.635122\pi$
$654$	0	0
$655$	19.4174	0.758700
$656$	0	0
$657$	8.92454	0.348180
$658$	0	0
$659$	−5.17494	−0.201587	−0.100793	−	0.994907i	$-0.532138\pi$
$659$	−5.17494	−0.201587	−0.100793	+	0.994907i	$0.532138\pi$
$660$	0	0
$661$	−29.8868	−1.16246	−0.581232	−	0.813738i	$-0.697429\pi$
$661$	−29.8868	−1.16246	−0.581232	+	0.813738i	$0.697429\pi$
$662$	0	0
$663$	6.31614	0.245299
$664$	0	0
$665$	3.76616	0.146046
$666$	0	0
$667$	5.06086	0.195957
$668$	0	0
$669$	10.0598	0.388935
$670$	0	0
$671$	1.33576	0.0515664
$672$	0	0
$673$	41.5439	1.60140	0.800700	−	0.599066i	$-0.204461\pi$
$673$	41.5439	1.60140	0.800700	+	0.599066i	$0.204461\pi$
$674$	0	0
$675$	4.92191	0.189444
$676$	0	0
$677$	38.6384	1.48499	0.742497	−	0.669850i	$-0.233642\pi$
$677$	38.6384	1.48499	0.742497	+	0.669850i	$0.233642\pi$
$678$	0	0
$679$	71.2592	2.73468
$680$	0	0
$681$	−28.3219	−1.08530
$682$	0	0
$683$	−10.9295	−0.418206	−0.209103	−	0.977894i	$-0.567054\pi$
$683$	−10.9295	−0.418206	−0.209103	+	0.977894i	$0.567054\pi$
$684$	0	0
$685$	−2.22522	−0.0850214
$686$	0	0
$687$	−38.4408	−1.46661
$688$	0	0
$689$	8.48888	0.323401
$690$	0	0
$691$	−11.9349	−0.454026	−0.227013	−	0.973892i	$-0.572896\pi$
$691$	−11.9349	−0.454026	−0.227013	+	0.973892i	$0.572896\pi$
$692$	0	0
$693$	20.7459	0.788071
$694$	0	0
$695$	−1.90890	−0.0724087
$696$	0	0
$697$	−13.6789	−0.518124
$698$	0	0
$699$	−53.2440	−2.01387
$700$	0	0
$701$	−18.4494	−0.696825	−0.348412	−	0.937341i	$-0.613279\pi$
$701$	−18.4494	−0.696825	−0.348412	+	0.937341i	$0.613279\pi$
$702$	0	0
$703$	3.02062	0.113925
$704$	0	0
$705$	−10.0977	−0.380301
$706$	0	0
$707$	−29.9415	−1.12607
$708$	0	0
$709$	−13.7125	−0.514984	−0.257492	−	0.966280i	$-0.582896\pi$
$709$	−13.7125	−0.514984	−0.257492	+	0.966280i	$0.582896\pi$
$710$	0	0
$711$	−0.196567	−0.00737184
$712$	0	0
$713$	−33.4838	−1.25398
$714$	0	0
$715$	1.88507	0.0704975
$716$	0	0
$717$	24.2676	0.906291
$718$	0	0
$719$	−25.5363	−0.952343	−0.476172	−	0.879352i	$-0.657976\pi$
$719$	−25.5363	−0.952343	−0.476172	+	0.879352i	$0.657976\pi$
$720$	0	0
$721$	59.1615	2.20329
$722$	0	0
$723$	70.9477	2.63857
$724$	0	0
$725$	−0.651655	−0.0242018
$726$	0	0
$727$	5.33973	0.198040	0.0990198	−	0.995085i	$-0.468429\pi$
$727$	5.33973	0.198040	0.0990198	+	0.995085i	$0.468429\pi$
$728$	0	0
$729$	−43.9237	−1.62680
$730$	0	0
$731$	13.8266	0.511395
$732$	0	0
$733$	48.6201	1.79583	0.897913	−	0.440173i	$-0.145083\pi$
$733$	48.6201	1.79583	0.897913	+	0.440173i	$0.145083\pi$
$734$	0	0
$735$	20.0202	0.738456
$736$	0	0
$737$	2.30271	0.0848213
$738$	0	0
$739$	−22.3019	−0.820387	−0.410194	−	0.911999i	$-0.634539\pi$
$739$	−22.3019	−0.820387	−0.410194	+	0.911999i	$0.634539\pi$
$740$	0	0
$741$	4.54534	0.166977
$742$	0	0
$743$	17.1433	0.628927	0.314464	−	0.949269i	$-0.398175\pi$
$743$	17.1433	0.628927	0.314464	+	0.949269i	$0.398175\pi$
$744$	0	0
$745$	−19.3675	−0.709572
$746$	0	0
$747$	−28.3593	−1.03761
$748$	0	0
$749$	−7.50367	−0.274178
$750$	0	0
$751$	−5.65605	−0.206392	−0.103196	−	0.994661i	$-0.532907\pi$
$751$	−5.65605	−0.206392	−0.103196	+	0.994661i	$0.532907\pi$
$752$	0	0
$753$	−66.8348	−2.43559
$754$	0	0
$755$	14.3267	0.521403
$756$	0	0
$757$	−39.8114	−1.44697	−0.723485	−	0.690341i	$-0.757461\pi$
$757$	−39.8114	−1.44697	−0.723485	+	0.690341i	$0.757461\pi$
$758$	0	0
$759$	25.0134	0.907931
$760$	0	0
$761$	16.8712	0.611581	0.305790	−	0.952099i	$-0.401079\pi$
$761$	16.8712	0.611581	0.305790	+	0.952099i	$0.401079\pi$
$762$	0	0
$763$	3.62740	0.131321
$764$	0	0
$765$	−6.62300	−0.239455
$766$	0	0
$767$	2.70726	0.0977536
$768$	0	0
$769$	−14.4140	−0.519783	−0.259891	−	0.965638i	$-0.583687\pi$
$769$	−14.4140	−0.519783	−0.259891	+	0.965638i	$0.583687\pi$
$770$	0	0
$771$	−25.2535	−0.909481
$772$	0	0
$773$	−9.24584	−0.332550	−0.166275	−	0.986079i	$-0.553174\pi$
$773$	−9.24584	−0.332550	−0.166275	+	0.986079i	$0.553174\pi$
$774$	0	0
$775$	4.31150	0.154874
$776$	0	0
$777$	31.7029	1.13733
$778$	0	0
$779$	−9.84382	−0.352692
$780$	0	0
$781$	6.39398	0.228795
$782$	0	0
$783$	−3.20739	−0.114623
$784$	0	0
$785$	20.2943	0.724333
$786$	0	0
$787$	53.7345	1.91543	0.957714	−	0.287723i	$-0.0928981\pi$
$787$	53.7345	1.91543	0.957714	+	0.287723i	$0.0928981\pi$
$788$	0	0
$789$	−1.27930	−0.0455442
$790$	0	0
$791$	3.53318	0.125625
$792$	0	0
$793$	1.88507	0.0669407
$794$	0	0
$795$	−14.5041	−0.514407
$796$	0	0
$797$	46.5960	1.65051	0.825257	−	0.564757i	$-0.191030\pi$
$797$	46.5960	1.65051	0.825257	+	0.564757i	$0.191030\pi$
$798$	0	0
$799$	5.03507	0.178128
$800$	0	0
$801$	−3.78620	−0.133779
$802$	0	0
$803$	−2.16412	−0.0763701
$804$	0	0
$805$	29.2486	1.03088
$806$	0	0
$807$	24.0728	0.847404
$808$	0	0
$809$	8.43902	0.296700	0.148350	−	0.988935i	$-0.452604\pi$
$809$	8.43902	0.296700	0.148350	+	0.988935i	$0.452604\pi$
$810$	0	0
$811$	26.6660	0.936371	0.468185	−	0.883630i	$-0.344908\pi$
$811$	26.6660	0.936371	0.468185	+	0.883630i	$0.344908\pi$
$812$	0	0
$813$	14.7657	0.517857
$814$	0	0
$815$	−16.2616	−0.569620
$816$	0	0
$817$	9.95014	0.348111
$818$	0	0
$819$	29.2773	1.02303
$820$	0	0
$821$	51.3265	1.79131	0.895653	−	0.444753i	$-0.146709\pi$
$821$	51.3265	1.79131	0.895653	+	0.444753i	$0.146709\pi$
$822$	0	0
$823$	−31.6340	−1.10269	−0.551346	−	0.834277i	$-0.685886\pi$
$823$	−31.6340	−1.10269	−0.551346	+	0.834277i	$0.685886\pi$
$824$	0	0
$825$	−3.22082	−0.112135
$826$	0	0
$827$	10.5165	0.365695	0.182848	−	0.983141i	$-0.441468\pi$
$827$	10.5165	0.365695	0.182848	+	0.983141i	$0.441468\pi$
$828$	0	0
$829$	12.3653	0.429466	0.214733	−	0.976673i	$-0.431112\pi$
$829$	12.3653	0.429466	0.214733	+	0.976673i	$0.431112\pi$
$830$	0	0
$831$	73.4818	2.54906
$832$	0	0
$833$	−9.98277	−0.345882
$834$	0	0
$835$	−11.1269	−0.385064
$836$	0	0
$837$	21.2208	0.733499
$838$	0	0
$839$	−31.2612	−1.07926	−0.539629	−	0.841903i	$-0.681435\pi$
$839$	−31.2612	−1.07926	−0.539629	+	0.841903i	$0.681435\pi$
$840$	0	0
$841$	−28.5753	−0.985357
$842$	0	0
$843$	−25.6964	−0.885033
$844$	0	0
$845$	−10.3397	−0.355698
$846$	0	0
$847$	36.3971	1.25062
$848$	0	0
$849$	−5.48264	−0.188164
$850$	0	0
$851$	23.4586	0.804152
$852$	0	0
$853$	−38.0132	−1.30155	−0.650774	−	0.759272i	$-0.725555\pi$
$853$	−38.0132	−1.30155	−0.650774	+	0.759272i	$0.725555\pi$
$854$	0	0
$855$	−4.76616	−0.162999
$856$	0	0
$857$	−19.6761	−0.672122	−0.336061	−	0.941840i	$-0.609095\pi$
$857$	−19.6761	−0.672122	−0.336061	+	0.941840i	$0.609095\pi$
$858$	0	0
$859$	−14.4665	−0.493591	−0.246795	−	0.969068i	$-0.579378\pi$
$859$	−14.4665	−0.493591	−0.246795	+	0.969068i	$0.579378\pi$
$860$	0	0
$861$	−103.316	−3.52099
$862$	0	0
$863$	22.1515	0.754046	0.377023	−	0.926204i	$-0.376948\pi$
$863$	22.1515	0.754046	0.377023	+	0.926204i	$0.376948\pi$
$864$	0	0
$865$	2.32393	0.0790160
$866$	0	0
$867$	−41.9942	−1.42620
$868$	0	0
$869$	0.0476656	0.00161695
$870$	0	0
$871$	3.24966	0.110110
$872$	0	0
$873$	−90.1801	−3.05213
$874$	0	0
$875$	−3.76616	−0.127320
$876$	0	0
$877$	3.99899	0.135036	0.0675182	−	0.997718i	$-0.478492\pi$
$877$	3.99899	0.135036	0.0675182	+	0.997718i	$0.478492\pi$
$878$	0	0
$879$	66.0095	2.22645
$880$	0	0
$881$	−54.5223	−1.83690	−0.918451	−	0.395535i	$-0.870559\pi$
$881$	−54.5223	−1.83690	−0.918451	+	0.395535i	$0.870559\pi$
$882$	0	0
$883$	−7.22100	−0.243006	−0.121503	−	0.992591i	$-0.538771\pi$
$883$	−7.22100	−0.243006	−0.121503	+	0.992591i	$0.538771\pi$
$884$	0	0
$885$	−4.62563	−0.155489
$886$	0	0
$887$	−19.2258	−0.645539	−0.322770	−	0.946478i	$-0.604614\pi$
$887$	−19.2258	−0.645539	−0.322770	+	0.946478i	$0.604614\pi$
$888$	0	0
$889$	−40.7295	−1.36602
$890$	0	0
$891$	0.672858	0.0225416
$892$	0	0
$893$	3.62343	0.121253
$894$	0	0
$895$	17.8851	0.597832
$896$	0	0
$897$	35.2998	1.17863
$898$	0	0
$899$	−2.80961	−0.0937057
$900$	0	0
$901$	7.23225	0.240941
$902$	0	0
$903$	104.432	3.47526
$904$	0	0
$905$	24.2118	0.804827
$906$	0	0
$907$	−12.4188	−0.412360	−0.206180	−	0.978514i	$-0.566103\pi$
$907$	−12.4188	−0.412360	−0.206180	+	0.978514i	$0.566103\pi$
$908$	0	0
$909$	37.8917	1.25679
$910$	0	0
$911$	19.4999	0.646060	0.323030	−	0.946389i	$-0.395299\pi$
$911$	19.4999	0.646060	0.323030	+	0.946389i	$0.395299\pi$
$912$	0	0
$913$	6.87688	0.227591
$914$	0	0
$915$	−3.22082	−0.106477
$916$	0	0
$917$	−73.1291	−2.41493
$918$	0	0
$919$	58.4870	1.92931	0.964655	−	0.263517i	$-0.0848825\pi$
$919$	58.4870	1.92931	0.964655	+	0.263517i	$0.0848825\pi$
$920$	0	0
$921$	−12.4822	−0.411303
$922$	0	0
$923$	9.02341	0.297009
$924$	0	0
$925$	−3.02062	−0.0993174
$926$	0	0
$927$	−74.8702	−2.45906
$928$	0	0
$929$	−48.2071	−1.58162	−0.790812	−	0.612059i	$-0.790342\pi$
$929$	−48.2071	−1.58162	−0.790812	+	0.612059i	$0.790342\pi$
$930$	0	0
$931$	−7.18398	−0.235445
$932$	0	0
$933$	79.5017	2.60277
$934$	0	0
$935$	1.60602	0.0525223
$936$	0	0
$937$	−25.5028	−0.833141	−0.416570	−	0.909103i	$-0.636768\pi$
$937$	−25.5028	−0.833141	−0.416570	+	0.909103i	$0.636768\pi$
$938$	0	0
$939$	80.2788	2.61980
$940$	0	0
$941$	−3.54576	−0.115589	−0.0577943	−	0.998329i	$-0.518407\pi$
$941$	−3.54576	−0.115589	−0.0577943	+	0.998329i	$0.518407\pi$
$942$	0	0
$943$	−76.4487	−2.48951
$944$	0	0
$945$	−18.5367	−0.603000
$946$	0	0
$947$	−19.5065	−0.633875	−0.316938	−	0.948446i	$-0.602655\pi$
$947$	−19.5065	−0.633875	−0.316938	+	0.948446i	$0.602655\pi$
$948$	0	0
$949$	−3.05408	−0.0991395
$950$	0	0
$951$	65.9215	2.13765
$952$	0	0
$953$	−5.13470	−0.166329	−0.0831647	−	0.996536i	$-0.526503\pi$
$953$	−5.13470	−0.166329	−0.0831647	+	0.996536i	$0.526503\pi$
$954$	0	0
$955$	−15.7575	−0.509902
$956$	0	0
$957$	2.09887	0.0678467
$958$	0	0
$959$	8.38055	0.270622
$960$	0	0
$961$	−12.4110	−0.400354
$962$	0	0
$963$	9.49606	0.306006
$964$	0	0
$965$	14.4380	0.464776
$966$	0	0
$967$	−37.9753	−1.22120	−0.610602	−	0.791938i	$-0.709072\pi$
$967$	−37.9753	−1.22120	−0.610602	+	0.791938i	$0.709072\pi$
$968$	0	0
$969$	3.87248	0.124402
$970$	0	0
$971$	17.0571	0.547387	0.273694	−	0.961817i	$-0.411755\pi$
$971$	17.0571	0.547387	0.273694	+	0.961817i	$0.411755\pi$
$972$	0	0
$973$	7.18923	0.230476
$974$	0	0
$975$	−4.54534	−0.145567
$976$	0	0
$977$	28.4705	0.910851	0.455425	−	0.890274i	$-0.349487\pi$
$977$	28.4705	0.910851	0.455425	+	0.890274i	$0.349487\pi$
$978$	0	0
$979$	0.918119	0.0293432
$980$	0	0
$981$	−4.59055	−0.146565
$982$	0	0
$983$	−49.9529	−1.59325	−0.796625	−	0.604473i	$-0.793384\pi$
$983$	−49.9529	−1.59325	−0.796625	+	0.604473i	$0.793384\pi$
$984$	0	0
$985$	−19.0234	−0.606136
$986$	0	0
$987$	38.0296	1.21049
$988$	0	0
$989$	77.2744	2.45718
$990$	0	0
$991$	−24.6210	−0.782111	−0.391056	−	0.920367i	$-0.627890\pi$
$991$	−24.6210	−0.782111	−0.391056	+	0.920367i	$0.627890\pi$
$992$	0	0
$993$	−70.6404	−2.24170
$994$	0	0
$995$	−13.9866	−0.443404
$996$	0	0
$997$	−9.04742	−0.286535	−0.143267	−	0.989684i	$-0.545761\pi$
$997$	−9.04742	−0.286535	−0.143267	+	0.989684i	$0.545761\pi$
$998$	0	0
$999$	−14.8672	−0.470378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.ce.1.4		4
4.3	odd	2		6080.2.a.cg.1.1		4
8.3	odd	2		3040.2.a.q.1.4	✓	4
8.5	even	2		3040.2.a.s.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.q.1.4	✓	4	8.3	odd	2
3040.2.a.s.1.1	yes	4	8.5	even	2
6080.2.a.ce.1.4		4	1.1	even	1	trivial
6080.2.a.cg.1.1		4	4.3	odd	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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q+2.78678 q^{3} +1.00000 q^{5} -3.76616 q^{7} +4.76616 q^{9} +O(q^{10})\)	\(q+2.78678 q^{3} +1.00000 q^{5} -3.76616 q^{7} +4.76616 q^{9} -1.15575 q^{11} -1.63103 q^{13} +2.78678 q^{15} -1.38959 q^{17} -1.00000 q^{19} -10.4955 q^{21} -7.76616 q^{23} +1.00000 q^{25} +4.92191 q^{27} -0.651655 q^{29} +4.31150 q^{31} -3.22082 q^{33} -3.76616 q^{35} -3.02062 q^{37} -4.54534 q^{39} +9.84382 q^{41} -9.95014 q^{43} +4.76616 q^{45} -3.62343 q^{47} +7.18398 q^{49} -3.87248 q^{51} -5.20460 q^{53} -1.15575 q^{55} -2.78678 q^{57} -1.65985 q^{59} -1.15575 q^{61} -17.9501 q^{63} -1.63103 q^{65} -1.99239 q^{67} -21.6426 q^{69} -5.53232 q^{71} +1.87248 q^{73} +2.78678 q^{75} +4.35274 q^{77} -0.0412422 q^{79} -0.582183 q^{81} -5.95014 q^{83} -1.38959 q^{85} -1.81602 q^{87} -0.794392 q^{89} +6.14274 q^{91} +12.0152 q^{93} -1.00000 q^{95} -18.9209 q^{97} -5.50849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + 9 q^{9} + 6 q^{11} - 5 q^{13} - q^{15} - 5 q^{17} - 4 q^{19} + 3 q^{21} - 21 q^{23} + 4 q^{25} - q^{27} + q^{29} - 4 q^{31} - 14 q^{33} - 5 q^{35} - 10 q^{37} - 7 q^{39} - 2 q^{41} - 6 q^{43} + 9 q^{45} - 24 q^{47} + 5 q^{49} - 13 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 11 q^{59} + 6 q^{61} - 38 q^{63} - 5 q^{65} - 19 q^{67} + 7 q^{69} - 2 q^{71} + 5 q^{73} - q^{75} - 8 q^{77} + 4 q^{79} - 16 q^{81} + 10 q^{83} - 5 q^{85} - 31 q^{87} + 20 q^{89} + 5 q^{91} + 26 q^{93} - 4 q^{95} - 6 q^{97} + 14 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + 9 * q^9 + 6 * q^11 - 5 * q^13 - q^15 - 5 * q^17 - 4 * q^19 + 3 * q^21 - 21 * q^23 + 4 * q^25 - q^27 + q^29 - 4 * q^31 - 14 * q^33 - 5 * q^35 - 10 * q^37 - 7 * q^39 - 2 * q^41 - 6 * q^43 + 9 * q^45 - 24 * q^47 + 5 * q^49 - 13 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 11 * q^59 + 6 * q^61 - 38 * q^63 - 5 * q^65 - 19 * q^67 + 7 * q^69 - 2 * q^71 + 5 * q^73 - q^75 - 8 * q^77 + 4 * q^79 - 16 * q^81 + 10 * q^83 - 5 * q^85 - 31 * q^87 + 20 * q^89 + 5 * q^91 + 26 * q^93 - 4 * q^95 - 6 * q^97 + 14 * q^99

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