LMFDB - Embedded newform 6040.2.a.l.1.9

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 9x^{6} + 32x^{5} - 17x^{4} - 27x^{3} + 10x^{2} + 3x - 1 $ x^9 - x^8 - 11x^7 + 9x^6 + 32x^5 - 17x^4 - 27x^3 + 10x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.61675$ 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)

$f(q)$	$=$	$q+2.50073 q^{3} +1.00000 q^{5} -2.23459 q^{7} +3.25367 q^{9} +O(q^{10})$	$q+2.50073 q^{3} +1.00000 q^{5} -2.23459 q^{7} +3.25367 q^{9} -0.595816 q^{11} -3.72317 q^{13} +2.50073 q^{15} -2.87712 q^{17} -4.54184 q^{19} -5.58812 q^{21} +6.52105 q^{23} +1.00000 q^{25} +0.634355 q^{27} -3.40521 q^{29} -4.46118 q^{31} -1.48998 q^{33} -2.23459 q^{35} -1.19972 q^{37} -9.31066 q^{39} +6.99016 q^{41} -6.50220 q^{43} +3.25367 q^{45} +1.73067 q^{47} -2.00660 q^{49} -7.19492 q^{51} +1.40445 q^{53} -0.595816 q^{55} -11.3579 q^{57} +4.59179 q^{59} -0.884262 q^{61} -7.27062 q^{63} -3.72317 q^{65} -7.08341 q^{67} +16.3074 q^{69} -4.77071 q^{71} -12.6755 q^{73} +2.50073 q^{75} +1.33141 q^{77} -5.17110 q^{79} -8.17465 q^{81} -1.25705 q^{83} -2.87712 q^{85} -8.51551 q^{87} -0.496952 q^{89} +8.31977 q^{91} -11.1562 q^{93} -4.54184 q^{95} -13.7580 q^{97} -1.93859 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * 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
$3$	2.50073	1.44380	0.721900	−	0.691998i	$-0.243269\pi$
$3$	2.50073	1.44380	0.721900	+	0.691998i	$0.243269\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.23459	−0.844597	−0.422298	−	0.906457i	$-0.638777\pi$
$7$	−2.23459	−0.844597	−0.422298	+	0.906457i	$0.638777\pi$
$8$	0	0
$9$	3.25367	1.08456
$10$	0	0
$11$	−0.595816	−0.179645	−0.0898227	−	0.995958i	$-0.528630\pi$
$11$	−0.595816	−0.179645	−0.0898227	+	0.995958i	$0.528630\pi$
$12$	0	0
$13$	−3.72317	−1.03262	−0.516311	−	0.856401i	$-0.672695\pi$
$13$	−3.72317	−1.03262	−0.516311	+	0.856401i	$0.672695\pi$
$14$	0	0
$15$	2.50073	0.645687
$16$	0	0
$17$	−2.87712	−0.697805	−0.348903	−	0.937159i	$-0.613446\pi$
$17$	−2.87712	−0.697805	−0.348903	+	0.937159i	$0.613446\pi$
$18$	0	0
$19$	−4.54184	−1.04197	−0.520985	−	0.853566i	$-0.674435\pi$
$19$	−4.54184	−1.04197	−0.520985	+	0.853566i	$0.674435\pi$
$20$	0	0
$21$	−5.58812	−1.21943
$22$	0	0
$23$	6.52105	1.35973	0.679866	−	0.733336i	$-0.262038\pi$
$23$	6.52105	1.35973	0.679866	+	0.733336i	$0.262038\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0.634355	0.122082
$28$	0	0
$29$	−3.40521	−0.632331	−0.316165	−	0.948704i	$-0.602395\pi$
$29$	−3.40521	−0.632331	−0.316165	+	0.948704i	$0.602395\pi$
$30$	0	0
$31$	−4.46118	−0.801251	−0.400625	−	0.916242i	$-0.631207\pi$
$31$	−4.46118	−0.801251	−0.400625	+	0.916242i	$0.631207\pi$
$32$	0	0
$33$	−1.48998	−0.259372
$34$	0	0
$35$	−2.23459	−0.377715
$36$	0	0
$37$	−1.19972	−0.197232	−0.0986162	−	0.995126i	$-0.531442\pi$
$37$	−1.19972	−0.197232	−0.0986162	+	0.995126i	$0.531442\pi$
$38$	0	0
$39$	−9.31066	−1.49090
$40$	0	0
$41$	6.99016	1.09168	0.545840	−	0.837890i	$-0.316211\pi$
$41$	6.99016	1.09168	0.545840	+	0.837890i	$0.316211\pi$
$42$	0	0
$43$	−6.50220	−0.991576	−0.495788	−	0.868444i	$-0.665121\pi$
$43$	−6.50220	−0.991576	−0.495788	+	0.868444i	$0.665121\pi$
$44$	0	0
$45$	3.25367	0.485028
$46$	0	0
$47$	1.73067	0.252444	0.126222	−	0.992002i	$-0.459715\pi$
$47$	1.73067	0.252444	0.126222	+	0.992002i	$0.459715\pi$
$48$	0	0
$49$	−2.00660	−0.286657
$50$	0	0
$51$	−7.19492	−1.00749
$52$	0	0
$53$	1.40445	0.192916	0.0964578	−	0.995337i	$-0.469249\pi$
$53$	1.40445	0.192916	0.0964578	+	0.995337i	$0.469249\pi$
$54$	0	0
$55$	−0.595816	−0.0803398
$56$	0	0
$57$	−11.3579	−1.50439
$58$	0	0
$59$	4.59179	0.597801	0.298900	−	0.954284i	$-0.403380\pi$
$59$	4.59179	0.597801	0.298900	+	0.954284i	$0.403380\pi$
$60$	0	0
$61$	−0.884262	−0.113218	−0.0566090	−	0.998396i	$-0.518029\pi$
$61$	−0.884262	−0.113218	−0.0566090	+	0.998396i	$0.518029\pi$
$62$	0	0
$63$	−7.27062	−0.916012
$64$	0	0
$65$	−3.72317	−0.461803
$66$	0	0
$67$	−7.08341	−0.865377	−0.432688	−	0.901544i	$-0.642435\pi$
$67$	−7.08341	−0.865377	−0.432688	+	0.901544i	$0.642435\pi$
$68$	0	0
$69$	16.3074	1.96318
$70$	0	0
$71$	−4.77071	−0.566179	−0.283089	−	0.959094i	$-0.591359\pi$
$71$	−4.77071	−0.566179	−0.283089	+	0.959094i	$0.591359\pi$
$72$	0	0
$73$	−12.6755	−1.48355	−0.741776	−	0.670648i	$-0.766016\pi$
$73$	−12.6755	−1.48355	−0.741776	+	0.670648i	$0.766016\pi$
$74$	0	0
$75$	2.50073	0.288760
$76$	0	0
$77$	1.33141	0.151728
$78$	0	0
$79$	−5.17110	−0.581794	−0.290897	−	0.956754i	$-0.593954\pi$
$79$	−5.17110	−0.581794	−0.290897	+	0.956754i	$0.593954\pi$
$80$	0	0
$81$	−8.17465	−0.908295
$82$	0	0
$83$	−1.25705	−0.137979	−0.0689897	−	0.997617i	$-0.521978\pi$
$83$	−1.25705	−0.137979	−0.0689897	+	0.997617i	$0.521978\pi$
$84$	0	0
$85$	−2.87712	−0.312068
$86$	0	0
$87$	−8.51551	−0.912959
$88$	0	0
$89$	−0.496952	−0.0526769	−0.0263384	−	0.999653i	$-0.508385\pi$
$89$	−0.496952	−0.0526769	−0.0263384	+	0.999653i	$0.508385\pi$
$90$	0	0
$91$	8.31977	0.872149
$92$	0	0
$93$	−11.1562	−1.15685
$94$	0	0
$95$	−4.54184	−0.465983
$96$	0	0
$97$	−13.7580	−1.39691	−0.698457	−	0.715652i	$-0.746130\pi$
$97$	−13.7580	−1.39691	−0.698457	+	0.715652i	$0.746130\pi$
$98$	0	0
$99$	−1.93859	−0.194835
$100$	0	0
$101$	6.72952	0.669613	0.334806	−	0.942287i	$-0.391329\pi$
$101$	6.72952	0.669613	0.334806	+	0.942287i	$0.391329\pi$
$102$	0	0
$103$	17.8215	1.75600	0.878002	−	0.478657i	$-0.158876\pi$
$103$	17.8215	1.75600	0.878002	+	0.478657i	$0.158876\pi$
$104$	0	0
$105$	−5.58812	−0.545345
$106$	0	0
$107$	4.87081	0.470879	0.235440	−	0.971889i	$-0.424347\pi$
$107$	4.87081	0.470879	0.235440	+	0.971889i	$0.424347\pi$
$108$	0	0
$109$	−8.56483	−0.820362	−0.410181	−	0.912004i	$-0.634534\pi$
$109$	−8.56483	−0.820362	−0.410181	+	0.912004i	$0.634534\pi$
$110$	0	0
$111$	−3.00017	−0.284764
$112$	0	0
$113$	−1.92152	−0.180762	−0.0903809	−	0.995907i	$-0.528808\pi$
$113$	−1.92152	−0.180762	−0.0903809	+	0.995907i	$0.528808\pi$
$114$	0	0
$115$	6.52105	0.608091
$116$	0	0
$117$	−12.1140	−1.11994
$118$	0	0
$119$	6.42920	0.589364
$120$	0	0
$121$	−10.6450	−0.967728
$122$	0	0
$123$	17.4805	1.57617
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.79230	−0.868926	−0.434463	−	0.900690i	$-0.643062\pi$
$127$	−9.79230	−0.868926	−0.434463	+	0.900690i	$0.643062\pi$
$128$	0	0
$129$	−16.2603	−1.43164
$130$	0	0
$131$	−13.0867	−1.14339	−0.571693	−	0.820467i	$-0.693713\pi$
$131$	−13.0867	−1.14339	−0.571693	+	0.820467i	$0.693713\pi$
$132$	0	0
$133$	10.1492	0.880044
$134$	0	0
$135$	0.634355	0.0545966
$136$	0	0
$137$	−2.89866	−0.247650	−0.123825	−	0.992304i	$-0.539516\pi$
$137$	−2.89866	−0.247650	−0.123825	+	0.992304i	$0.539516\pi$
$138$	0	0
$139$	−5.72145	−0.485287	−0.242644	−	0.970116i	$-0.578015\pi$
$139$	−5.72145	−0.485287	−0.242644	+	0.970116i	$0.578015\pi$
$140$	0	0
$141$	4.32794	0.364479
$142$	0	0
$143$	2.21833	0.185506
$144$	0	0
$145$	−3.40521	−0.282787
$146$	0	0
$147$	−5.01796	−0.413875
$148$	0	0
$149$	10.8717	0.890640	0.445320	−	0.895371i	$-0.353090\pi$
$149$	10.8717	0.890640	0.445320	+	0.895371i	$0.353090\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−9.36121	−0.756809
$154$	0	0
$155$	−4.46118	−0.358330
$156$	0	0
$157$	−1.54050	−0.122945	−0.0614727	−	0.998109i	$-0.519580\pi$
$157$	−1.54050	−0.122945	−0.0614727	+	0.998109i	$0.519580\pi$
$158$	0	0
$159$	3.51215	0.278531
$160$	0	0
$161$	−14.5719	−1.14843
$162$	0	0
$163$	18.9269	1.48247	0.741235	−	0.671245i	$-0.234240\pi$
$163$	18.9269	1.48247	0.741235	+	0.671245i	$0.234240\pi$
$164$	0	0
$165$	−1.48998	−0.115995
$166$	0	0
$167$	−15.4690	−1.19702	−0.598512	−	0.801114i	$-0.704241\pi$
$167$	−15.4690	−1.19702	−0.598512	+	0.801114i	$0.704241\pi$
$168$	0	0
$169$	0.862007	0.0663082
$170$	0	0
$171$	−14.7776	−1.13007
$172$	0	0
$173$	0.494021	0.0375597	0.0187799	−	0.999824i	$-0.494022\pi$
$173$	0.494021	0.0375597	0.0187799	+	0.999824i	$0.494022\pi$
$174$	0	0
$175$	−2.23459	−0.168919
$176$	0	0
$177$	11.4829	0.863104
$178$	0	0
$179$	−2.14246	−0.160135	−0.0800675	−	0.996789i	$-0.525514\pi$
$179$	−2.14246	−0.160135	−0.0800675	+	0.996789i	$0.525514\pi$
$180$	0	0
$181$	−5.26919	−0.391656	−0.195828	−	0.980638i	$-0.562739\pi$
$181$	−5.26919	−0.391656	−0.195828	+	0.980638i	$0.562739\pi$
$182$	0	0
$183$	−2.21130	−0.163464
$184$	0	0
$185$	−1.19972	−0.0882050
$186$	0	0
$187$	1.71424	0.125357
$188$	0	0
$189$	−1.41752	−0.103110
$190$	0	0
$191$	18.4623	1.33588	0.667941	−	0.744214i	$-0.267176\pi$
$191$	18.4623	1.33588	0.667941	+	0.744214i	$0.267176\pi$
$192$	0	0
$193$	−3.42365	−0.246440	−0.123220	−	0.992379i	$-0.539322\pi$
$193$	−3.42365	−0.246440	−0.123220	+	0.992379i	$0.539322\pi$
$194$	0	0
$195$	−9.31066	−0.666750
$196$	0	0
$197$	11.3266	0.806987	0.403494	−	0.914983i	$-0.367796\pi$
$197$	11.3266	0.806987	0.403494	+	0.914983i	$0.367796\pi$
$198$	0	0
$199$	−9.44925	−0.669840	−0.334920	−	0.942247i	$-0.608709\pi$
$199$	−9.44925	−0.669840	−0.334920	+	0.942247i	$0.608709\pi$
$200$	0	0
$201$	−17.7137	−1.24943
$202$	0	0
$203$	7.60925	0.534064
$204$	0	0
$205$	6.99016	0.488214
$206$	0	0
$207$	21.2173	1.47471
$208$	0	0
$209$	2.70610	0.187185
$210$	0	0
$211$	1.83176	0.126103	0.0630517	−	0.998010i	$-0.479917\pi$
$211$	1.83176	0.126103	0.0630517	+	0.998010i	$0.479917\pi$
$212$	0	0
$213$	−11.9303	−0.817449
$214$	0	0
$215$	−6.50220	−0.443446
$216$	0	0
$217$	9.96891	0.676734
$218$	0	0
$219$	−31.6980	−2.14195
$220$	0	0
$221$	10.7120	0.720569
$222$	0	0
$223$	14.9018	0.997899	0.498950	−	0.866631i	$-0.333719\pi$
$223$	14.9018	0.997899	0.498950	+	0.866631i	$0.333719\pi$
$224$	0	0
$225$	3.25367	0.216911
$226$	0	0
$227$	1.84861	0.122696	0.0613482	−	0.998116i	$-0.480460\pi$
$227$	1.84861	0.122696	0.0613482	+	0.998116i	$0.480460\pi$
$228$	0	0
$229$	−0.282678	−0.0186799	−0.00933993	−	0.999956i	$-0.502973\pi$
$229$	−0.282678	−0.0186799	−0.00933993	+	0.999956i	$0.502973\pi$
$230$	0	0
$231$	3.32949	0.219064
$232$	0	0
$233$	29.3691	1.92404	0.962018	−	0.272985i	$-0.0880109\pi$
$233$	29.3691	1.92404	0.962018	+	0.272985i	$0.0880109\pi$
$234$	0	0
$235$	1.73067	0.112896
$236$	0	0
$237$	−12.9315	−0.839994
$238$	0	0
$239$	−27.3417	−1.76859	−0.884295	−	0.466930i	$-0.845360\pi$
$239$	−27.3417	−1.76859	−0.884295	+	0.466930i	$0.845360\pi$
$240$	0	0
$241$	−15.1842	−0.978099	−0.489050	−	0.872256i	$-0.662656\pi$
$241$	−15.1842	−0.978099	−0.489050	+	0.872256i	$0.662656\pi$
$242$	0	0
$243$	−22.3457	−1.43348
$244$	0	0
$245$	−2.00660	−0.128197
$246$	0	0
$247$	16.9100	1.07596
$248$	0	0
$249$	−3.14355	−0.199215
$250$	0	0
$251$	16.0107	1.01059	0.505293	−	0.862948i	$-0.331384\pi$
$251$	16.0107	1.01059	0.505293	+	0.862948i	$0.331384\pi$
$252$	0	0
$253$	−3.88534	−0.244269
$254$	0	0
$255$	−7.19492	−0.450563
$256$	0	0
$257$	−3.68137	−0.229638	−0.114819	−	0.993386i	$-0.536629\pi$
$257$	−3.68137	−0.229638	−0.114819	+	0.993386i	$0.536629\pi$
$258$	0	0
$259$	2.68088	0.166582
$260$	0	0
$261$	−11.0794	−0.685798
$262$	0	0
$263$	8.37151	0.516210	0.258105	−	0.966117i	$-0.416902\pi$
$263$	8.37151	0.516210	0.258105	+	0.966117i	$0.416902\pi$
$264$	0	0
$265$	1.40445	0.0862745
$266$	0	0
$267$	−1.24275	−0.0760548
$268$	0	0
$269$	25.5313	1.55667	0.778334	−	0.627850i	$-0.216065\pi$
$269$	25.5313	1.55667	0.778334	+	0.627850i	$0.216065\pi$
$270$	0	0
$271$	13.1676	0.799872	0.399936	−	0.916543i	$-0.369032\pi$
$271$	13.1676	0.799872	0.399936	+	0.916543i	$0.369032\pi$
$272$	0	0
$273$	20.8055	1.25921
$274$	0	0
$275$	−0.595816	−0.0359291
$276$	0	0
$277$	15.6240	0.938758	0.469379	−	0.882997i	$-0.344478\pi$
$277$	15.6240	0.938758	0.469379	+	0.882997i	$0.344478\pi$
$278$	0	0
$279$	−14.5152	−0.869001
$280$	0	0
$281$	−10.3803	−0.619236	−0.309618	−	0.950861i	$-0.600201\pi$
$281$	−10.3803	−0.619236	−0.309618	+	0.950861i	$0.600201\pi$
$282$	0	0
$283$	14.6520	0.870969	0.435484	−	0.900196i	$-0.356577\pi$
$283$	14.6520	0.870969	0.435484	+	0.900196i	$0.356577\pi$
$284$	0	0
$285$	−11.3579	−0.672786
$286$	0	0
$287$	−15.6202	−0.922029
$288$	0	0
$289$	−8.72216	−0.513068
$290$	0	0
$291$	−34.4051	−2.01686
$292$	0	0
$293$	−21.1456	−1.23534	−0.617669	−	0.786438i	$-0.711923\pi$
$293$	−21.1456	−1.23534	−0.617669	+	0.786438i	$0.711923\pi$
$294$	0	0
$295$	4.59179	0.267345
$296$	0	0
$297$	−0.377959	−0.0219314
$298$	0	0
$299$	−24.2790	−1.40409
$300$	0	0
$301$	14.5298	0.837482
$302$	0	0
$303$	16.8287	0.966786
$304$	0	0
$305$	−0.884262	−0.0506327
$306$	0	0
$307$	−29.7630	−1.69866	−0.849331	−	0.527860i	$-0.822994\pi$
$307$	−29.7630	−1.69866	−0.849331	+	0.527860i	$0.822994\pi$
$308$	0	0
$309$	44.5668	2.53532
$310$	0	0
$311$	−4.35955	−0.247207	−0.123604	−	0.992332i	$-0.539445\pi$
$311$	−4.35955	−0.247207	−0.123604	+	0.992332i	$0.539445\pi$
$312$	0	0
$313$	9.57090	0.540979	0.270490	−	0.962723i	$-0.412814\pi$
$313$	9.57090	0.540979	0.270490	+	0.962723i	$0.412814\pi$
$314$	0	0
$315$	−7.27062	−0.409653
$316$	0	0
$317$	2.73886	0.153829	0.0769147	−	0.997038i	$-0.475493\pi$
$317$	2.73886	0.153829	0.0769147	+	0.997038i	$0.475493\pi$
$318$	0	0
$319$	2.02888	0.113595
$320$	0	0
$321$	12.1806	0.679855
$322$	0	0
$323$	13.0674	0.727091
$324$	0	0
$325$	−3.72317	−0.206524
$326$	0	0
$327$	−21.4184	−1.18444
$328$	0	0
$329$	−3.86734	−0.213213
$330$	0	0
$331$	−22.3972	−1.23106	−0.615531	−	0.788112i	$-0.711059\pi$
$331$	−22.3972	−1.23106	−0.615531	+	0.788112i	$0.711059\pi$
$332$	0	0
$333$	−3.90348	−0.213910
$334$	0	0
$335$	−7.08341	−0.387008
$336$	0	0
$337$	−14.5160	−0.790738	−0.395369	−	0.918522i	$-0.629383\pi$
$337$	−14.5160	−0.790738	−0.395369	+	0.918522i	$0.629383\pi$
$338$	0	0
$339$	−4.80522	−0.260984
$340$	0	0
$341$	2.65804	0.143941
$342$	0	0
$343$	20.1261	1.08671
$344$	0	0
$345$	16.3074	0.877961
$346$	0	0
$347$	13.5090	0.725200	0.362600	−	0.931945i	$-0.381889\pi$
$347$	13.5090	0.725200	0.362600	+	0.931945i	$0.381889\pi$
$348$	0	0
$349$	−1.19745	−0.0640981	−0.0320491	−	0.999486i	$-0.510203\pi$
$349$	−1.19745	−0.0640981	−0.0320491	+	0.999486i	$0.510203\pi$
$350$	0	0
$351$	−2.36181	−0.126064
$352$	0	0
$353$	13.3936	0.712869	0.356435	−	0.934320i	$-0.383992\pi$
$353$	13.3936	0.712869	0.356435	+	0.934320i	$0.383992\pi$
$354$	0	0
$355$	−4.77071	−0.253203
$356$	0	0
$357$	16.0777	0.850923
$358$	0	0
$359$	30.1532	1.59142	0.795712	−	0.605675i	$-0.207097\pi$
$359$	30.1532	1.59142	0.795712	+	0.605675i	$0.207097\pi$
$360$	0	0
$361$	1.62830	0.0856998
$362$	0	0
$363$	−26.6203	−1.39720
$364$	0	0
$365$	−12.6755	−0.663465
$366$	0	0
$367$	−3.47684	−0.181490	−0.0907449	−	0.995874i	$-0.528925\pi$
$367$	−3.47684	−0.181490	−0.0907449	+	0.995874i	$0.528925\pi$
$368$	0	0
$369$	22.7437	1.18399
$370$	0	0
$371$	−3.13837	−0.162936
$372$	0	0
$373$	−31.7637	−1.64466	−0.822332	−	0.569009i	$-0.807327\pi$
$373$	−31.7637	−1.64466	−0.822332	+	0.569009i	$0.807327\pi$
$374$	0	0
$375$	2.50073	0.129137
$376$	0	0
$377$	12.6782	0.652959
$378$	0	0
$379$	26.1043	1.34089	0.670444	−	0.741960i	$-0.266104\pi$
$379$	26.1043	1.34089	0.670444	+	0.741960i	$0.266104\pi$
$380$	0	0
$381$	−24.4879	−1.25455
$382$	0	0
$383$	−24.7126	−1.26275	−0.631377	−	0.775476i	$-0.717510\pi$
$383$	−24.7126	−1.26275	−0.631377	+	0.775476i	$0.717510\pi$
$384$	0	0
$385$	1.33141	0.0678547
$386$	0	0
$387$	−21.1560	−1.07542
$388$	0	0
$389$	16.5502	0.839126	0.419563	−	0.907726i	$-0.362183\pi$
$389$	16.5502	0.839126	0.419563	+	0.907726i	$0.362183\pi$
$390$	0	0
$391$	−18.7619	−0.948828
$392$	0	0
$393$	−32.7263	−1.65082
$394$	0	0
$395$	−5.17110	−0.260186
$396$	0	0
$397$	12.9094	0.647904	0.323952	−	0.946074i	$-0.394988\pi$
$397$	12.9094	0.647904	0.323952	+	0.946074i	$0.394988\pi$
$398$	0	0
$399$	25.3803	1.27061
$400$	0	0
$401$	2.96235	0.147933	0.0739663	−	0.997261i	$-0.476434\pi$
$401$	2.96235	0.147933	0.0739663	+	0.997261i	$0.476434\pi$
$402$	0	0
$403$	16.6097	0.827389
$404$	0	0
$405$	−8.17465	−0.406202
$406$	0	0
$407$	0.714811	0.0354319
$408$	0	0
$409$	3.79317	0.187560	0.0937801	−	0.995593i	$-0.470105\pi$
$409$	3.79317	0.187560	0.0937801	+	0.995593i	$0.470105\pi$
$410$	0	0
$411$	−7.24879	−0.357556
$412$	0	0
$413$	−10.2608	−0.504900
$414$	0	0
$415$	−1.25705	−0.0617063
$416$	0	0
$417$	−14.3078	−0.700657
$418$	0	0
$419$	5.82390	0.284516	0.142258	−	0.989830i	$-0.454564\pi$
$419$	5.82390	0.284516	0.142258	+	0.989830i	$0.454564\pi$
$420$	0	0
$421$	−32.2079	−1.56972	−0.784859	−	0.619675i	$-0.787265\pi$
$421$	−32.2079	−1.56972	−0.784859	+	0.619675i	$0.787265\pi$
$422$	0	0
$423$	5.63102	0.273790
$424$	0	0
$425$	−2.87712	−0.139561
$426$	0	0
$427$	1.97596	0.0956236
$428$	0	0
$429$	5.54744	0.267833
$430$	0	0
$431$	31.8411	1.53373	0.766865	−	0.641808i	$-0.221815\pi$
$431$	31.8411	1.53373	0.766865	+	0.641808i	$0.221815\pi$
$432$	0	0
$433$	3.26192	0.156758	0.0783790	−	0.996924i	$-0.475026\pi$
$433$	3.26192	0.156758	0.0783790	+	0.996924i	$0.475026\pi$
$434$	0	0
$435$	−8.51551	−0.408288
$436$	0	0
$437$	−29.6175	−1.41680
$438$	0	0
$439$	11.2835	0.538531	0.269265	−	0.963066i	$-0.413219\pi$
$439$	11.2835	0.538531	0.269265	+	0.963066i	$0.413219\pi$
$440$	0	0
$441$	−6.52880	−0.310895
$442$	0	0
$443$	1.59261	0.0756673	0.0378336	−	0.999284i	$-0.487954\pi$
$443$	1.59261	0.0756673	0.0378336	+	0.999284i	$0.487954\pi$
$444$	0	0
$445$	−0.496952	−0.0235578
$446$	0	0
$447$	27.1871	1.28591
$448$	0	0
$449$	−5.41951	−0.255762	−0.127881	−	0.991789i	$-0.540818\pi$
$449$	−5.41951	−0.255762	−0.127881	+	0.991789i	$0.540818\pi$
$450$	0	0
$451$	−4.16485	−0.196115
$452$	0	0
$453$	−2.50073	−0.117495
$454$	0	0
$455$	8.31977	0.390037
$456$	0	0
$457$	−0.152758	−0.00714572	−0.00357286	−	0.999994i	$-0.501137\pi$
$457$	−0.152758	−0.00714572	−0.00357286	+	0.999994i	$0.501137\pi$
$458$	0	0
$459$	−1.82512	−0.0851892
$460$	0	0
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	0	0
$463$	36.5086	1.69670	0.848349	−	0.529438i	$-0.177597\pi$
$463$	36.5086	1.69670	0.848349	+	0.529438i	$0.177597\pi$
$464$	0	0
$465$	−11.1562	−0.517357
$466$	0	0
$467$	22.8318	1.05653	0.528265	−	0.849079i	$-0.322843\pi$
$467$	22.8318	1.05653	0.528265	+	0.849079i	$0.322843\pi$
$468$	0	0
$469$	15.8285	0.730894
$470$	0	0
$471$	−3.85238	−0.177509
$472$	0	0
$473$	3.87411	0.178132
$474$	0	0
$475$	−4.54184	−0.208394
$476$	0	0
$477$	4.56960	0.209228
$478$	0	0
$479$	22.5769	1.03156	0.515782	−	0.856720i	$-0.327502\pi$
$479$	22.5769	1.03156	0.515782	+	0.856720i	$0.327502\pi$
$480$	0	0
$481$	4.46676	0.203667
$482$	0	0
$483$	−36.4404	−1.65810
$484$	0	0
$485$	−13.7580	−0.624719
$486$	0	0
$487$	−18.6400	−0.844658	−0.422329	−	0.906443i	$-0.638787\pi$
$487$	−18.6400	−0.844658	−0.422329	+	0.906443i	$0.638787\pi$
$488$	0	0
$489$	47.3312	2.14039
$490$	0	0
$491$	−7.74615	−0.349579	−0.174789	−	0.984606i	$-0.555924\pi$
$491$	−7.74615	−0.349579	−0.174789	+	0.984606i	$0.555924\pi$
$492$	0	0
$493$	9.79720	0.441244
$494$	0	0
$495$	−1.93859	−0.0871330
$496$	0	0
$497$	10.6606	0.478193
$498$	0	0
$499$	17.8916	0.800936	0.400468	−	0.916311i	$-0.368847\pi$
$499$	17.8916	0.800936	0.400468	+	0.916311i	$0.368847\pi$
$500$	0	0
$501$	−38.6837	−1.72826
$502$	0	0
$503$	3.65161	0.162817	0.0814087	−	0.996681i	$-0.474058\pi$
$503$	3.65161	0.162817	0.0814087	+	0.996681i	$0.474058\pi$
$504$	0	0
$505$	6.72952	0.299460
$506$	0	0
$507$	2.15565	0.0957357
$508$	0	0
$509$	17.2974	0.766695	0.383348	−	0.923604i	$-0.374771\pi$
$509$	17.2974	0.766695	0.383348	+	0.923604i	$0.374771\pi$
$510$	0	0
$511$	28.3245	1.25300
$512$	0	0
$513$	−2.88114	−0.127205
$514$	0	0
$515$	17.8215	0.785309
$516$	0	0
$517$	−1.03116	−0.0453504
$518$	0	0
$519$	1.23541	0.0542287
$520$	0	0
$521$	−2.71324	−0.118869	−0.0594345	−	0.998232i	$-0.518930\pi$
$521$	−2.71324	−0.118869	−0.0594345	+	0.998232i	$0.518930\pi$
$522$	0	0
$523$	41.9196	1.83301	0.916507	−	0.400018i	$-0.130996\pi$
$523$	41.9196	1.83301	0.916507	+	0.400018i	$0.130996\pi$
$524$	0	0
$525$	−5.58812	−0.243886
$526$	0	0
$527$	12.8354	0.559117
$528$	0	0
$529$	19.5240	0.848871
$530$	0	0
$531$	14.9402	0.648348
$532$	0	0
$533$	−26.0256	−1.12729
$534$	0	0
$535$	4.87081	0.210584
$536$	0	0
$537$	−5.35772	−0.231203
$538$	0	0
$539$	1.19556	0.0514965
$540$	0	0
$541$	−1.78080	−0.0765628	−0.0382814	−	0.999267i	$-0.512188\pi$
$541$	−1.78080	−0.0765628	−0.0382814	+	0.999267i	$0.512188\pi$
$542$	0	0
$543$	−13.1768	−0.565472
$544$	0	0
$545$	−8.56483	−0.366877
$546$	0	0
$547$	−14.6949	−0.628309	−0.314154	−	0.949372i	$-0.601721\pi$
$547$	−14.6949	−0.628309	−0.314154	+	0.949372i	$0.601721\pi$
$548$	0	0
$549$	−2.87709	−0.122791
$550$	0	0
$551$	15.4659	0.658869
$552$	0	0
$553$	11.5553	0.491381
$554$	0	0
$555$	−3.00017	−0.127350
$556$	0	0
$557$	11.4666	0.485856	0.242928	−	0.970044i	$-0.421892\pi$
$557$	11.4666	0.485856	0.242928	+	0.970044i	$0.421892\pi$
$558$	0	0
$559$	24.2088	1.02392
$560$	0	0
$561$	4.28685	0.180991
$562$	0	0
$563$	−2.83079	−0.119303	−0.0596517	−	0.998219i	$-0.518999\pi$
$563$	−2.83079	−0.119303	−0.0596517	+	0.998219i	$0.518999\pi$
$564$	0	0
$565$	−1.92152	−0.0808391
$566$	0	0
$567$	18.2670	0.767142
$568$	0	0
$569$	−37.8446	−1.58653	−0.793264	−	0.608878i	$-0.791620\pi$
$569$	−37.8446	−1.58653	−0.793264	+	0.608878i	$0.791620\pi$
$570$	0	0
$571$	11.0219	0.461251	0.230625	−	0.973043i	$-0.425923\pi$
$571$	11.0219	0.461251	0.230625	+	0.973043i	$0.425923\pi$
$572$	0	0
$573$	46.1692	1.92875
$574$	0	0
$575$	6.52105	0.271946
$576$	0	0
$577$	−16.7739	−0.698305	−0.349153	−	0.937066i	$-0.613531\pi$
$577$	−16.7739	−0.698305	−0.349153	+	0.937066i	$0.613531\pi$
$578$	0	0
$579$	−8.56164	−0.355810
$580$	0	0
$581$	2.80900	0.116537
$582$	0	0
$583$	−0.836792	−0.0346564
$584$	0	0
$585$	−12.1140	−0.500851
$586$	0	0
$587$	−5.42080	−0.223741	−0.111870	−	0.993723i	$-0.535684\pi$
$587$	−5.42080	−0.223741	−0.111870	+	0.993723i	$0.535684\pi$
$588$	0	0
$589$	20.2619	0.834879
$590$	0	0
$591$	28.3248	1.16513
$592$	0	0
$593$	−0.636921	−0.0261552	−0.0130776	−	0.999914i	$-0.504163\pi$
$593$	−0.636921	−0.0261552	−0.0130776	+	0.999914i	$0.504163\pi$
$594$	0	0
$595$	6.42920	0.263572
$596$	0	0
$597$	−23.6301	−0.967114
$598$	0	0
$599$	−29.3506	−1.19923	−0.599616	−	0.800288i	$-0.704680\pi$
$599$	−29.3506	−1.19923	−0.599616	+	0.800288i	$0.704680\pi$
$600$	0	0
$601$	−16.3740	−0.667910	−0.333955	−	0.942589i	$-0.608383\pi$
$601$	−16.3740	−0.667910	−0.333955	+	0.942589i	$0.608383\pi$
$602$	0	0
$603$	−23.0471	−0.938549
$604$	0	0
$605$	−10.6450	−0.432781
$606$	0	0
$607$	12.1263	0.492191	0.246095	−	0.969246i	$-0.420852\pi$
$607$	12.1263	0.492191	0.246095	+	0.969246i	$0.420852\pi$
$608$	0	0
$609$	19.0287	0.771082
$610$	0	0
$611$	−6.44358	−0.260679
$612$	0	0
$613$	11.6390	0.470095	0.235047	−	0.971984i	$-0.424475\pi$
$613$	11.6390	0.470095	0.235047	+	0.971984i	$0.424475\pi$
$614$	0	0
$615$	17.4805	0.704883
$616$	0	0
$617$	−21.0453	−0.847251	−0.423626	−	0.905837i	$-0.639243\pi$
$617$	−21.0453	−0.847251	−0.423626	+	0.905837i	$0.639243\pi$
$618$	0	0
$619$	−48.3482	−1.94328	−0.971638	−	0.236472i	$-0.924009\pi$
$619$	−48.3482	−1.94328	−0.971638	+	0.236472i	$0.924009\pi$
$620$	0	0
$621$	4.13666	0.165998
$622$	0	0
$623$	1.11049	0.0444907
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.76724	0.270257
$628$	0	0
$629$	3.45174	0.137630
$630$	0	0
$631$	−35.7760	−1.42422	−0.712111	−	0.702067i	$-0.752261\pi$
$631$	−35.7760	−1.42422	−0.712111	+	0.702067i	$0.752261\pi$
$632$	0	0
$633$	4.58074	0.182068
$634$	0	0
$635$	−9.79230	−0.388596
$636$	0	0
$637$	7.47090	0.296008
$638$	0	0
$639$	−15.5223	−0.614053
$640$	0	0
$641$	−7.52766	−0.297325	−0.148662	−	0.988888i	$-0.547497\pi$
$641$	−7.52766	−0.297325	−0.148662	+	0.988888i	$0.547497\pi$
$642$	0	0
$643$	−9.10077	−0.358899	−0.179450	−	0.983767i	$-0.557432\pi$
$643$	−9.10077	−0.358899	−0.179450	+	0.983767i	$0.557432\pi$
$644$	0	0
$645$	−16.2603	−0.640247
$646$	0	0
$647$	45.0870	1.77255	0.886276	−	0.463158i	$-0.153284\pi$
$647$	45.0870	1.77255	0.886276	+	0.463158i	$0.153284\pi$
$648$	0	0
$649$	−2.73586	−0.107392
$650$	0	0
$651$	24.9296	0.977068
$652$	0	0
$653$	10.5418	0.412534	0.206267	−	0.978496i	$-0.433868\pi$
$653$	10.5418	0.412534	0.206267	+	0.978496i	$0.433868\pi$
$654$	0	0
$655$	−13.0867	−0.511338
$656$	0	0
$657$	−41.2418	−1.60900
$658$	0	0
$659$	22.5415	0.878093	0.439047	−	0.898464i	$-0.355316\pi$
$659$	22.5415	0.878093	0.439047	+	0.898464i	$0.355316\pi$
$660$	0	0
$661$	21.7894	0.847510	0.423755	−	0.905777i	$-0.360712\pi$
$661$	21.7894	0.847510	0.423755	+	0.905777i	$0.360712\pi$
$662$	0	0
$663$	26.7879	1.04036
$664$	0	0
$665$	10.1492	0.393567
$666$	0	0
$667$	−22.2055	−0.859801
$668$	0	0
$669$	37.2655	1.44077
$670$	0	0
$671$	0.526857	0.0203391
$672$	0	0
$673$	−41.8986	−1.61507	−0.807536	−	0.589818i	$-0.799199\pi$
$673$	−41.8986	−1.61507	−0.807536	+	0.589818i	$0.799199\pi$
$674$	0	0
$675$	0.634355	0.0244163
$676$	0	0
$677$	−28.4061	−1.09174	−0.545868	−	0.837871i	$-0.683800\pi$
$677$	−28.4061	−1.09174	−0.545868	+	0.837871i	$0.683800\pi$
$678$	0	0
$679$	30.7435	1.17983
$680$	0	0
$681$	4.62287	0.177149
$682$	0	0
$683$	−22.2452	−0.851189	−0.425595	−	0.904914i	$-0.639935\pi$
$683$	−22.2452	−0.851189	−0.425595	+	0.904914i	$0.639935\pi$
$684$	0	0
$685$	−2.89866	−0.110752
$686$	0	0
$687$	−0.706902	−0.0269700
$688$	0	0
$689$	−5.22900	−0.199209
$690$	0	0
$691$	−17.1019	−0.650586	−0.325293	−	0.945613i	$-0.605463\pi$
$691$	−17.1019	−0.650586	−0.325293	+	0.945613i	$0.605463\pi$
$692$	0	0
$693$	4.33195	0.164557
$694$	0	0
$695$	−5.72145	−0.217027
$696$	0	0
$697$	−20.1116	−0.761780
$698$	0	0
$699$	73.4444	2.77792
$700$	0	0
$701$	−14.1734	−0.535320	−0.267660	−	0.963513i	$-0.586250\pi$
$701$	−14.1734	−0.535320	−0.267660	+	0.963513i	$0.586250\pi$
$702$	0	0
$703$	5.44893	0.205510
$704$	0	0
$705$	4.32794	0.163000
$706$	0	0
$707$	−15.0377	−0.565553
$708$	0	0
$709$	29.0166	1.08974	0.544872	−	0.838520i	$-0.316578\pi$
$709$	29.0166	1.08974	0.544872	+	0.838520i	$0.316578\pi$
$710$	0	0
$711$	−16.8250	−0.630988
$712$	0	0
$713$	−29.0915	−1.08949
$714$	0	0
$715$	2.21833	0.0829607
$716$	0	0
$717$	−68.3744	−2.55349
$718$	0	0
$719$	32.8260	1.22420	0.612101	−	0.790780i	$-0.290325\pi$
$719$	32.8260	1.22420	0.612101	+	0.790780i	$0.290325\pi$
$720$	0	0
$721$	−39.8238	−1.48311
$722$	0	0
$723$	−37.9716	−1.41218
$724$	0	0
$725$	−3.40521	−0.126466
$726$	0	0
$727$	19.6767	0.729768	0.364884	−	0.931053i	$-0.381109\pi$
$727$	19.6767	0.729768	0.364884	+	0.931053i	$0.381109\pi$
$728$	0	0
$729$	−31.3566	−1.16136
$730$	0	0
$731$	18.7076	0.691927
$732$	0	0
$733$	23.8154	0.879643	0.439822	−	0.898085i	$-0.355042\pi$
$733$	23.8154	0.879643	0.439822	+	0.898085i	$0.355042\pi$
$734$	0	0
$735$	−5.01796	−0.185090
$736$	0	0
$737$	4.22041	0.155461
$738$	0	0
$739$	−6.24066	−0.229566	−0.114783	−	0.993391i	$-0.536617\pi$
$739$	−6.24066	−0.229566	−0.114783	+	0.993391i	$0.536617\pi$
$740$	0	0
$741$	42.2875	1.55347
$742$	0	0
$743$	9.18899	0.337111	0.168556	−	0.985692i	$-0.446090\pi$
$743$	9.18899	0.337111	0.168556	+	0.985692i	$0.446090\pi$
$744$	0	0
$745$	10.8717	0.398306
$746$	0	0
$747$	−4.09003	−0.149646
$748$	0	0
$749$	−10.8843	−0.397703
$750$	0	0
$751$	−13.9511	−0.509084	−0.254542	−	0.967062i	$-0.581925\pi$
$751$	−13.9511	−0.509084	−0.254542	+	0.967062i	$0.581925\pi$
$752$	0	0
$753$	40.0385	1.45908
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	−37.2959	−1.35554	−0.677771	−	0.735273i	$-0.737054\pi$
$757$	−37.2959	−1.35554	−0.677771	+	0.735273i	$0.737054\pi$
$758$	0	0
$759$	−9.71621	−0.352676
$760$	0	0
$761$	−35.5858	−1.28999	−0.644993	−	0.764189i	$-0.723140\pi$
$761$	−35.5858	−1.28999	−0.644993	+	0.764189i	$0.723140\pi$
$762$	0	0
$763$	19.1389	0.692875
$764$	0	0
$765$	−9.36121	−0.338455
$766$	0	0
$767$	−17.0960	−0.617302
$768$	0	0
$769$	−29.5041	−1.06395	−0.531973	−	0.846761i	$-0.678549\pi$
$769$	−29.5041	−1.06395	−0.531973	+	0.846761i	$0.678549\pi$
$770$	0	0
$771$	−9.20612	−0.331550
$772$	0	0
$773$	−22.3314	−0.803205	−0.401602	−	0.915814i	$-0.631547\pi$
$773$	−22.3314	−0.803205	−0.401602	+	0.915814i	$0.631547\pi$
$774$	0	0
$775$	−4.46118	−0.160250
$776$	0	0
$777$	6.70417	0.240511
$778$	0	0
$779$	−31.7482	−1.13750
$780$	0	0
$781$	2.84246	0.101711
$782$	0	0
$783$	−2.16011	−0.0771960
$784$	0	0
$785$	−1.54050	−0.0549829
$786$	0	0
$787$	3.07679	0.109676	0.0548379	−	0.998495i	$-0.482536\pi$
$787$	3.07679	0.109676	0.0548379	+	0.998495i	$0.482536\pi$
$788$	0	0
$789$	20.9349	0.745303
$790$	0	0
$791$	4.29382	0.152671
$792$	0	0
$793$	3.29226	0.116911
$794$	0	0
$795$	3.51215	0.124563
$796$	0	0
$797$	−5.24635	−0.185835	−0.0929177	−	0.995674i	$-0.529619\pi$
$797$	−5.24635	−0.185835	−0.0929177	+	0.995674i	$0.529619\pi$
$798$	0	0
$799$	−4.97935	−0.176157
$800$	0	0
$801$	−1.61692	−0.0571310
$802$	0	0
$803$	7.55225	0.266513
$804$	0	0
$805$	−14.5719	−0.513591
$806$	0	0
$807$	63.8469	2.24752
$808$	0	0
$809$	−7.54397	−0.265232	−0.132616	−	0.991168i	$-0.542338\pi$
$809$	−7.54397	−0.265232	−0.132616	+	0.991168i	$0.542338\pi$
$810$	0	0
$811$	30.3175	1.06459	0.532295	−	0.846559i	$-0.321330\pi$
$811$	30.3175	1.06459	0.532295	+	0.846559i	$0.321330\pi$
$812$	0	0
$813$	32.9286	1.15486
$814$	0	0
$815$	18.9269	0.662981
$816$	0	0
$817$	29.5319	1.03319
$818$	0	0
$819$	27.0698	0.945894
$820$	0	0
$821$	−8.68993	−0.303281	−0.151640	−	0.988436i	$-0.548456\pi$
$821$	−8.68993	−0.303281	−0.151640	+	0.988436i	$0.548456\pi$
$822$	0	0
$823$	−2.16280	−0.0753903	−0.0376952	−	0.999289i	$-0.512002\pi$
$823$	−2.16280	−0.0753903	−0.0376952	+	0.999289i	$0.512002\pi$
$824$	0	0
$825$	−1.48998	−0.0518743
$826$	0	0
$827$	17.0630	0.593337	0.296669	−	0.954980i	$-0.404124\pi$
$827$	17.0630	0.593337	0.296669	+	0.954980i	$0.404124\pi$
$828$	0	0
$829$	−34.8709	−1.21112	−0.605558	−	0.795801i	$-0.707050\pi$
$829$	−34.8709	−1.21112	−0.605558	+	0.795801i	$0.707050\pi$
$830$	0	0
$831$	39.0716	1.35538
$832$	0	0
$833$	5.77323	0.200030
$834$	0	0
$835$	−15.4690	−0.535325
$836$	0	0
$837$	−2.82997	−0.0978180
$838$	0	0
$839$	10.5512	0.364267	0.182133	−	0.983274i	$-0.441700\pi$
$839$	10.5512	0.364267	0.182133	+	0.983274i	$0.441700\pi$
$840$	0	0
$841$	−17.4046	−0.600158
$842$	0	0
$843$	−25.9583	−0.894053
$844$	0	0
$845$	0.862007	0.0296539
$846$	0	0
$847$	23.7872	0.817339
$848$	0	0
$849$	36.6406	1.25750
$850$	0	0
$851$	−7.82342	−0.268183
$852$	0	0
$853$	−36.3930	−1.24607	−0.623036	−	0.782193i	$-0.714101\pi$
$853$	−36.3930	−1.24607	−0.623036	+	0.782193i	$0.714101\pi$
$854$	0	0
$855$	−14.7776	−0.505384
$856$	0	0
$857$	−43.5153	−1.48646	−0.743228	−	0.669039i	$-0.766706\pi$
$857$	−43.5153	−1.48646	−0.743228	+	0.669039i	$0.766706\pi$
$858$	0	0
$859$	25.5083	0.870331	0.435165	−	0.900351i	$-0.356690\pi$
$859$	25.5083	0.870331	0.435165	+	0.900351i	$0.356690\pi$
$860$	0	0
$861$	−39.0619	−1.33122
$862$	0	0
$863$	10.3708	0.353025	0.176513	−	0.984298i	$-0.443518\pi$
$863$	10.3708	0.353025	0.176513	+	0.984298i	$0.443518\pi$
$864$	0	0
$865$	0.494021	0.0167972
$866$	0	0
$867$	−21.8118	−0.740767
$868$	0	0
$869$	3.08103	0.104517
$870$	0	0
$871$	26.3728	0.893607
$872$	0	0
$873$	−44.7640	−1.51503
$874$	0	0
$875$	−2.23459	−0.0755430
$876$	0	0
$877$	−34.0213	−1.14882	−0.574408	−	0.818569i	$-0.694768\pi$
$877$	−34.0213	−1.14882	−0.574408	+	0.818569i	$0.694768\pi$
$878$	0	0
$879$	−52.8795	−1.78358
$880$	0	0
$881$	14.9969	0.505259	0.252630	−	0.967563i	$-0.418705\pi$
$881$	14.9969	0.505259	0.252630	+	0.967563i	$0.418705\pi$
$882$	0	0
$883$	−16.2566	−0.547079	−0.273540	−	0.961861i	$-0.588194\pi$
$883$	−16.2566	−0.547079	−0.273540	+	0.961861i	$0.588194\pi$
$884$	0	0
$885$	11.4829	0.385992
$886$	0	0
$887$	29.6093	0.994182	0.497091	−	0.867699i	$-0.334402\pi$
$887$	29.6093	0.994182	0.497091	+	0.867699i	$0.334402\pi$
$888$	0	0
$889$	21.8818	0.733892
$890$	0	0
$891$	4.87059	0.163171
$892$	0	0
$893$	−7.86042	−0.263039
$894$	0	0
$895$	−2.14246	−0.0716145
$896$	0	0
$897$	−60.7152	−2.02722
$898$	0	0
$899$	15.1912	0.506656
$900$	0	0
$901$	−4.04077	−0.134617
$902$	0	0
$903$	36.3351	1.20916
$904$	0	0
$905$	−5.26919	−0.175154
$906$	0	0
$907$	14.8796	0.494070	0.247035	−	0.969007i	$-0.420544\pi$
$907$	14.8796	0.494070	0.247035	+	0.969007i	$0.420544\pi$
$908$	0	0
$909$	21.8956	0.726232
$910$	0	0
$911$	−31.5430	−1.04507	−0.522534	−	0.852618i	$-0.675013\pi$
$911$	−31.5430	−1.04507	−0.522534	+	0.852618i	$0.675013\pi$
$912$	0	0
$913$	0.748972	0.0247874
$914$	0	0
$915$	−2.21130	−0.0731034
$916$	0	0
$917$	29.2434	0.965701
$918$	0	0
$919$	8.19526	0.270337	0.135168	−	0.990823i	$-0.456842\pi$
$919$	8.19526	0.270337	0.135168	+	0.990823i	$0.456842\pi$
$920$	0	0
$921$	−74.4292	−2.45253
$922$	0	0
$923$	17.7622	0.584649
$924$	0	0
$925$	−1.19972	−0.0394465
$926$	0	0
$927$	57.9852	1.90448
$928$	0	0
$929$	9.76470	0.320369	0.160185	−	0.987087i	$-0.448791\pi$
$929$	9.76470	0.320369	0.160185	+	0.987087i	$0.448791\pi$
$930$	0	0
$931$	9.11364	0.298687
$932$	0	0
$933$	−10.9021	−0.356917
$934$	0	0
$935$	1.71424	0.0560615
$936$	0	0
$937$	35.5725	1.16210	0.581052	−	0.813867i	$-0.302641\pi$
$937$	35.5725	1.16210	0.581052	+	0.813867i	$0.302641\pi$
$938$	0	0
$939$	23.9343	0.781065
$940$	0	0
$941$	−51.8087	−1.68892	−0.844458	−	0.535622i	$-0.820077\pi$
$941$	−51.8087	−1.68892	−0.844458	+	0.535622i	$0.820077\pi$
$942$	0	0
$943$	45.5832	1.48439
$944$	0	0
$945$	−1.41752	−0.0461121
$946$	0	0
$947$	−28.2648	−0.918481	−0.459241	−	0.888312i	$-0.651878\pi$
$947$	−28.2648	−0.918481	−0.459241	+	0.888312i	$0.651878\pi$
$948$	0	0
$949$	47.1930	1.53195
$950$	0	0
$951$	6.84915	0.222099
$952$	0	0
$953$	−12.4268	−0.402545	−0.201272	−	0.979535i	$-0.564508\pi$
$953$	−12.4268	−0.402545	−0.201272	+	0.979535i	$0.564508\pi$
$954$	0	0
$955$	18.4623	0.597425
$956$	0	0
$957$	5.07368	0.164009
$958$	0	0
$959$	6.47733	0.209164
$960$	0	0
$961$	−11.0979	−0.357997
$962$	0	0
$963$	15.8480	0.510695
$964$	0	0
$965$	−3.42365	−0.110211
$966$	0	0
$967$	53.2390	1.71205	0.856026	−	0.516934i	$-0.172927\pi$
$967$	53.2390	1.71205	0.856026	+	0.516934i	$0.172927\pi$
$968$	0	0
$969$	32.6782	1.04977
$970$	0	0
$971$	−21.1474	−0.678652	−0.339326	−	0.940669i	$-0.610199\pi$
$971$	−21.1474	−0.678652	−0.339326	+	0.940669i	$0.610199\pi$
$972$	0	0
$973$	12.7851	0.409872
$974$	0	0
$975$	−9.31066	−0.298180
$976$	0	0
$977$	37.2318	1.19115	0.595576	−	0.803299i	$-0.296924\pi$
$977$	37.2318	1.19115	0.595576	+	0.803299i	$0.296924\pi$
$978$	0	0
$979$	0.296092	0.00946315
$980$	0	0
$981$	−27.8671	−0.889728
$982$	0	0
$983$	6.13633	0.195719	0.0978593	−	0.995200i	$-0.468800\pi$
$983$	6.13633	0.195719	0.0978593	+	0.995200i	$0.468800\pi$
$984$	0	0
$985$	11.3266	0.360896
$986$	0	0
$987$	−9.67119	−0.307837
$988$	0	0
$989$	−42.4011	−1.34828
$990$	0	0
$991$	4.68364	0.148781	0.0743903	−	0.997229i	$-0.476299\pi$
$991$	4.68364	0.148781	0.0743903	+	0.997229i	$0.476299\pi$
$992$	0	0
$993$	−56.0095	−1.77741
$994$	0	0
$995$	−9.44925	−0.299561
$996$	0	0
$997$	−10.5068	−0.332753	−0.166376	−	0.986062i	$-0.553207\pi$
$997$	−10.5068	−0.332753	−0.166376	+	0.986062i	$0.553207\pi$
$998$	0	0
$999$	−0.761047	−0.0240785

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.l.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.l.1.9	✓	9	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)

\(f(q)\)	\(=\)	\(q+2.50073 q^{3} +1.00000 q^{5} -2.23459 q^{7} +3.25367 q^{9} +O(q^{10})\)	\(q+2.50073 q^{3} +1.00000 q^{5} -2.23459 q^{7} +3.25367 q^{9} -0.595816 q^{11} -3.72317 q^{13} +2.50073 q^{15} -2.87712 q^{17} -4.54184 q^{19} -5.58812 q^{21} +6.52105 q^{23} +1.00000 q^{25} +0.634355 q^{27} -3.40521 q^{29} -4.46118 q^{31} -1.48998 q^{33} -2.23459 q^{35} -1.19972 q^{37} -9.31066 q^{39} +6.99016 q^{41} -6.50220 q^{43} +3.25367 q^{45} +1.73067 q^{47} -2.00660 q^{49} -7.19492 q^{51} +1.40445 q^{53} -0.595816 q^{55} -11.3579 q^{57} +4.59179 q^{59} -0.884262 q^{61} -7.27062 q^{63} -3.72317 q^{65} -7.08341 q^{67} +16.3074 q^{69} -4.77071 q^{71} -12.6755 q^{73} +2.50073 q^{75} +1.33141 q^{77} -5.17110 q^{79} -8.17465 q^{81} -1.25705 q^{83} -2.87712 q^{85} -8.51551 q^{87} -0.496952 q^{89} +8.31977 q^{91} -11.1562 q^{93} -4.54184 q^{95} -13.7580 q^{97} -1.93859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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