LMFDB - Embedded newform 8015.2.a.j.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.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:	$64.0000972201$
Analytic rank:	$1$
Dimension:	$45$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.556743 q^{2} -0.855264 q^{3} -1.69004 q^{4} -1.00000 q^{5} +0.476163 q^{6} +1.00000 q^{7} +2.05440 q^{8} -2.26852 q^{9} +O(q^{10})$	\(q-0.556743 q^{2} -0.855264 q^{3} -1.69004 q^{4} -1.00000 q^{5} +0.476163 q^{6} +1.00000 q^{7} +2.05440 q^{8} -2.26852 q^{9} +0.556743 q^{10} +3.86363 q^{11} +1.44543 q^{12} +6.16076 q^{13} -0.556743 q^{14} +0.855264 q^{15} +2.23630 q^{16} -0.562314 q^{17} +1.26299 q^{18} -7.14969 q^{19} +1.69004 q^{20} -0.855264 q^{21} -2.15105 q^{22} -7.00506 q^{23} -1.75706 q^{24} +1.00000 q^{25} -3.42996 q^{26} +4.50598 q^{27} -1.69004 q^{28} -0.249350 q^{29} -0.476163 q^{30} +2.99275 q^{31} -5.35385 q^{32} -3.30442 q^{33} +0.313065 q^{34} -1.00000 q^{35} +3.83389 q^{36} -4.66576 q^{37} +3.98054 q^{38} -5.26908 q^{39} -2.05440 q^{40} -5.67158 q^{41} +0.476163 q^{42} +8.89471 q^{43} -6.52967 q^{44} +2.26852 q^{45} +3.90002 q^{46} +3.83207 q^{47} -1.91263 q^{48} +1.00000 q^{49} -0.556743 q^{50} +0.480927 q^{51} -10.4119 q^{52} +4.47201 q^{53} -2.50867 q^{54} -3.86363 q^{55} +2.05440 q^{56} +6.11488 q^{57} +0.138824 q^{58} -4.21343 q^{59} -1.44543 q^{60} -7.41414 q^{61} -1.66619 q^{62} -2.26852 q^{63} -1.49188 q^{64} -6.16076 q^{65} +1.83971 q^{66} -1.37628 q^{67} +0.950332 q^{68} +5.99117 q^{69} +0.556743 q^{70} +1.22054 q^{71} -4.66046 q^{72} -2.45646 q^{73} +2.59763 q^{74} -0.855264 q^{75} +12.0832 q^{76} +3.86363 q^{77} +2.93352 q^{78} -10.7361 q^{79} -2.23630 q^{80} +2.95177 q^{81} +3.15761 q^{82} -4.50624 q^{83} +1.44543 q^{84} +0.562314 q^{85} -4.95207 q^{86} +0.213260 q^{87} +7.93744 q^{88} +9.13167 q^{89} -1.26299 q^{90} +6.16076 q^{91} +11.8388 q^{92} -2.55959 q^{93} -2.13348 q^{94} +7.14969 q^{95} +4.57896 q^{96} +13.0692 q^{97} -0.556743 q^{98} -8.76472 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9}+O(q^{10}) $ 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9	\( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9} + 6 q^{10} - q^{11} - 3 q^{12} - 21 q^{13} - 6 q^{14} + 8 q^{16} - 7 q^{17} - 36 q^{18} - 20 q^{19} - 34 q^{20} - 34 q^{22} - 22 q^{23} - 11 q^{24} + 45 q^{25} - q^{26} + 12 q^{27} + 34 q^{28} + 10 q^{29} - q^{30} - 27 q^{31} - 26 q^{32} - 39 q^{33} - 13 q^{34} - 45 q^{35} - 3 q^{36} - 72 q^{37} + 2 q^{38} - 37 q^{39} + 15 q^{40} - 4 q^{41} + q^{42} - 49 q^{43} + 5 q^{44} - 29 q^{45} - 67 q^{46} + 2 q^{47} + 8 q^{48} + 45 q^{49} - 6 q^{50} - 49 q^{51} - 47 q^{52} - 35 q^{53} - 12 q^{54} + q^{55} - 15 q^{56} - 77 q^{57} - 50 q^{58} + 4 q^{59} + 3 q^{60} - 36 q^{61} + 17 q^{62} + 29 q^{63} + 5 q^{64} + 21 q^{65} - 8 q^{66} - 80 q^{67} + 27 q^{68} + 9 q^{69} + 6 q^{70} - 12 q^{71} - 97 q^{72} - 55 q^{73} + 32 q^{74} - 37 q^{76} - q^{77} + 20 q^{78} - 94 q^{79} - 8 q^{80} - 19 q^{81} - 36 q^{82} + 24 q^{83} - 3 q^{84} + 7 q^{85} - 3 q^{86} - 4 q^{87} - 95 q^{88} + q^{89} + 36 q^{90} - 21 q^{91} - 65 q^{92} - 71 q^{93} - 53 q^{94} + 20 q^{95} - 13 q^{96} - 110 q^{97} - 6 q^{98} - 27 q^{99}+O(q^{100}) \) 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9 + 6 * q^10 - q^11 - 3 * q^12 - 21 * q^13 - 6 * q^14 + 8 * q^16 - 7 * q^17 - 36 * q^18 - 20 * q^19 - 34 * q^20 - 34 * q^22 - 22 * q^23 - 11 * q^24 + 45 * q^25 - q^26 + 12 * q^27 + 34 * q^28 + 10 * q^29 - q^30 - 27 * q^31 - 26 * q^32 - 39 * q^33 - 13 * q^34 - 45 * q^35 - 3 * q^36 - 72 * q^37 + 2 * q^38 - 37 * q^39 + 15 * q^40 - 4 * q^41 + q^42 - 49 * q^43 + 5 * q^44 - 29 * q^45 - 67 * q^46 + 2 * q^47 + 8 * q^48 + 45 * q^49 - 6 * q^50 - 49 * q^51 - 47 * q^52 - 35 * q^53 - 12 * q^54 + q^55 - 15 * q^56 - 77 * q^57 - 50 * q^58 + 4 * q^59 + 3 * q^60 - 36 * q^61 + 17 * q^62 + 29 * q^63 + 5 * q^64 + 21 * q^65 - 8 * q^66 - 80 * q^67 + 27 * q^68 + 9 * q^69 + 6 * q^70 - 12 * q^71 - 97 * q^72 - 55 * q^73 + 32 * q^74 - 37 * q^76 - q^77 + 20 * q^78 - 94 * q^79 - 8 * q^80 - 19 * q^81 - 36 * q^82 + 24 * q^83 - 3 * q^84 + 7 * q^85 - 3 * q^86 - 4 * q^87 - 95 * q^88 + q^89 + 36 * q^90 - 21 * q^91 - 65 * q^92 - 71 * q^93 - 53 * q^94 + 20 * q^95 - 13 * q^96 - 110 * q^97 - 6 * q^98 - 27 * 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 $

$2$	−0.556743	−0.393677	−0.196838	−	0.980436i	$-0.563067\pi$
$2$	−0.556743	−0.393677	−0.196838	+	0.980436i	$0.563067\pi$
$3$	−0.855264	−0.493787	−0.246894	−	0.969043i	$-0.579410\pi$
$3$	−0.855264	−0.493787	−0.246894	+	0.969043i	$0.579410\pi$
$4$	−1.69004	−0.845018
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.476163	0.194393
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.05440	0.726341
$9$	−2.26852	−0.756174
$10$	0.556743	0.176058
$11$	3.86363	1.16493	0.582463	−	0.812857i	$-0.302089\pi$
$11$	3.86363	1.16493	0.582463	+	0.812857i	$0.302089\pi$
$12$	1.44543	0.417259
$13$	6.16076	1.70869	0.854344	−	0.519708i	$-0.173959\pi$
$13$	6.16076	1.70869	0.854344	+	0.519708i	$0.173959\pi$
$14$	−0.556743	−0.148796
$15$	0.855264	0.220828
$16$	2.23630	0.559075
$17$	−0.562314	−0.136381	−0.0681906	−	0.997672i	$-0.521723\pi$
$17$	−0.562314	−0.136381	−0.0681906	+	0.997672i	$0.521723\pi$
$18$	1.26299	0.297688
$19$	−7.14969	−1.64025	−0.820126	−	0.572183i	$-0.806097\pi$
$19$	−7.14969	−1.64025	−0.820126	+	0.572183i	$0.806097\pi$
$20$	1.69004	0.377904
$21$	−0.855264	−0.186634
$22$	−2.15105	−0.458605
$23$	−7.00506	−1.46066	−0.730328	−	0.683097i	$-0.760633\pi$
$23$	−7.00506	−1.46066	−0.730328	+	0.683097i	$0.760633\pi$
$24$	−1.75706	−0.358658
$25$	1.00000	0.200000
$26$	−3.42996	−0.672671
$27$	4.50598	0.867176
$28$	−1.69004	−0.319387
$29$	−0.249350	−0.0463031	−0.0231516	−	0.999732i	$-0.507370\pi$
$29$	−0.249350	−0.0463031	−0.0231516	+	0.999732i	$0.507370\pi$
$30$	−0.476163	−0.0869350
$31$	2.99275	0.537513	0.268756	−	0.963208i	$-0.413387\pi$
$31$	2.99275	0.537513	0.268756	+	0.963208i	$0.413387\pi$
$32$	−5.35385	−0.946436
$33$	−3.30442	−0.575226
$34$	0.313065	0.0536902
$35$	−1.00000	−0.169031
$36$	3.83389	0.638981
$37$	−4.66576	−0.767046	−0.383523	−	0.923531i	$-0.625289\pi$
$37$	−4.66576	−0.767046	−0.383523	+	0.923531i	$0.625289\pi$
$38$	3.98054	0.645730
$39$	−5.26908	−0.843728
$40$	−2.05440	−0.324830
$41$	−5.67158	−0.885752	−0.442876	−	0.896583i	$-0.646042\pi$
$41$	−5.67158	−0.885752	−0.442876	+	0.896583i	$0.646042\pi$
$42$	0.476163	0.0734735
$43$	8.89471	1.35643	0.678215	−	0.734864i	$-0.262754\pi$
$43$	8.89471	1.35643	0.678215	+	0.734864i	$0.262754\pi$
$44$	−6.52967	−0.984385
$45$	2.26852	0.338171
$46$	3.90002	0.575026
$47$	3.83207	0.558964	0.279482	−	0.960151i	$-0.409837\pi$
$47$	3.83207	0.558964	0.279482	+	0.960151i	$0.409837\pi$
$48$	−1.91263	−0.276064
$49$	1.00000	0.142857
$50$	−0.556743	−0.0787354
$51$	0.480927	0.0673433
$52$	−10.4119	−1.44387
$53$	4.47201	0.614277	0.307139	−	0.951665i	$-0.400628\pi$
$53$	4.47201	0.614277	0.307139	+	0.951665i	$0.400628\pi$
$54$	−2.50867	−0.341387
$55$	−3.86363	−0.520971
$56$	2.05440	0.274531
$57$	6.11488	0.809935
$58$	0.138824	0.0182285
$59$	−4.21343	−0.548542	−0.274271	−	0.961652i	$-0.588436\pi$
$59$	−4.21343	−0.548542	−0.274271	+	0.961652i	$0.588436\pi$
$60$	−1.44543	−0.186604
$61$	−7.41414	−0.949284	−0.474642	−	0.880179i	$-0.657422\pi$
$61$	−7.41414	−0.949284	−0.474642	+	0.880179i	$0.657422\pi$
$62$	−1.66619	−0.211606
$63$	−2.26852	−0.285807
$64$	−1.49188	−0.186485
$65$	−6.16076	−0.764148
$66$	1.83971	0.226453
$67$	−1.37628	−0.168140	−0.0840699	−	0.996460i	$-0.526792\pi$
$67$	−1.37628	−0.168140	−0.0840699	+	0.996460i	$0.526792\pi$
$68$	0.950332	0.115245
$69$	5.99117	0.721252
$70$	0.556743	0.0665436
$71$	1.22054	0.144852	0.0724259	−	0.997374i	$-0.476926\pi$
$71$	1.22054	0.144852	0.0724259	+	0.997374i	$0.476926\pi$
$72$	−4.66046	−0.549241
$73$	−2.45646	−0.287507	−0.143753	−	0.989614i	$-0.545917\pi$
$73$	−2.45646	−0.287507	−0.143753	+	0.989614i	$0.545917\pi$
$74$	2.59763	0.301969
$75$	−0.855264	−0.0987574
$76$	12.0832	1.38604
$77$	3.86363	0.440301
$78$	2.93352	0.332156
$79$	−10.7361	−1.20790	−0.603952	−	0.797021i	$-0.706408\pi$
$79$	−10.7361	−1.20790	−0.603952	+	0.797021i	$0.706408\pi$
$80$	−2.23630	−0.250026
$81$	2.95177	0.327974
$82$	3.15761	0.348700
$83$	−4.50624	−0.494624	−0.247312	−	0.968936i	$-0.579547\pi$
$83$	−4.50624	−0.494624	−0.247312	+	0.968936i	$0.579547\pi$
$84$	1.44543	0.157709
$85$	0.562314	0.0609916
$86$	−4.95207	−0.533995
$87$	0.213260	0.0228639
$88$	7.93744	0.846134
$89$	9.13167	0.967955	0.483977	−	0.875081i	$-0.339192\pi$
$89$	9.13167	0.967955	0.483977	+	0.875081i	$0.339192\pi$
$90$	−1.26299	−0.133130
$91$	6.16076	0.645823
$92$	11.8388	1.23428
$93$	−2.55959	−0.265417
$94$	−2.13348	−0.220051
$95$	7.14969	0.733543
$96$	4.57896	0.467338
$97$	13.0692	1.32698	0.663488	−	0.748187i	$-0.269075\pi$
$97$	13.0692	1.32698	0.663488	+	0.748187i	$0.269075\pi$
$98$	−0.556743	−0.0562396
$99$	−8.76472	−0.880888
$100$	−1.69004	−0.169004
$101$	8.64622	0.860331	0.430166	−	0.902750i	$-0.358455\pi$
$101$	8.64622	0.860331	0.430166	+	0.902750i	$0.358455\pi$
$102$	−0.267753	−0.0265115
$103$	−9.46044	−0.932165	−0.466083	−	0.884741i	$-0.654335\pi$
$103$	−9.46044	−0.932165	−0.466083	+	0.884741i	$0.654335\pi$
$104$	12.6567	1.24109
$105$	0.855264	0.0834652
$106$	−2.48976	−0.241827
$107$	7.12539	0.688837	0.344419	−	0.938816i	$-0.388076\pi$
$107$	7.12539	0.688837	0.344419	+	0.938816i	$0.388076\pi$
$108$	−7.61527	−0.732780
$109$	−16.1501	−1.54690	−0.773449	−	0.633859i	$-0.781470\pi$
$109$	−16.1501	−1.54690	−0.773449	+	0.633859i	$0.781470\pi$
$110$	2.15105	0.205094
$111$	3.99046	0.378758
$112$	2.23630	0.211310
$113$	−1.39118	−0.130871	−0.0654356	−	0.997857i	$-0.520844\pi$
$113$	−1.39118	−0.130871	−0.0654356	+	0.997857i	$0.520844\pi$
$114$	−3.40442	−0.318853
$115$	7.00506	0.653225
$116$	0.421411	0.0391270
$117$	−13.9758	−1.29207
$118$	2.34580	0.215948
$119$	−0.562314	−0.0515473
$120$	1.75706	0.160397
$121$	3.92760	0.357054
$122$	4.12777	0.373711
$123$	4.85070	0.437373
$124$	−5.05785	−0.454208
$125$	−1.00000	−0.0894427
$126$	1.26299	0.112516
$127$	9.42289	0.836147	0.418073	−	0.908413i	$-0.362706\pi$
$127$	9.42289	0.836147	0.418073	+	0.908413i	$0.362706\pi$
$128$	11.5383	1.01985
$129$	−7.60732	−0.669787
$130$	3.42996	0.300828
$131$	8.94147	0.781219	0.390610	−	0.920556i	$-0.372264\pi$
$131$	8.94147	0.781219	0.390610	+	0.920556i	$0.372264\pi$
$132$	5.58459	0.486076
$133$	−7.14969	−0.619957
$134$	0.766236	0.0661927
$135$	−4.50598	−0.387813
$136$	−1.15522	−0.0990594
$137$	−2.37581	−0.202979	−0.101490	−	0.994837i	$-0.532361\pi$
$137$	−2.37581	−0.202979	−0.101490	+	0.994837i	$0.532361\pi$
$138$	−3.33555	−0.283940
$139$	−20.1440	−1.70859	−0.854297	−	0.519786i	$-0.826012\pi$
$139$	−20.1440	−1.70859	−0.854297	+	0.519786i	$0.826012\pi$
$140$	1.69004	0.142834
$141$	−3.27743	−0.276009
$142$	−0.679529	−0.0570248
$143$	23.8029	1.99050
$144$	−5.07309	−0.422758
$145$	0.249350	0.0207074
$146$	1.36762	0.113185
$147$	−0.855264	−0.0705410
$148$	7.88531	0.648168
$149$	0.860986	0.0705347	0.0352673	−	0.999378i	$-0.488772\pi$
$149$	0.860986	0.0705347	0.0352673	+	0.999378i	$0.488772\pi$
$150$	0.476163	0.0388785
$151$	−4.24664	−0.345587	−0.172793	−	0.984958i	$-0.555279\pi$
$151$	−4.24664	−0.345587	−0.172793	+	0.984958i	$0.555279\pi$
$152$	−14.6884	−1.19138
$153$	1.27562	0.103128
$154$	−2.15105	−0.173336
$155$	−2.99275	−0.240383
$156$	8.90494	0.712966
$157$	−3.13846	−0.250476	−0.125238	−	0.992127i	$-0.539969\pi$
$157$	−3.13846	−0.250476	−0.125238	+	0.992127i	$0.539969\pi$
$158$	5.97724	0.475524
$159$	−3.82475	−0.303322
$160$	5.35385	0.423259
$161$	−7.00506	−0.552076
$162$	−1.64338	−0.129116
$163$	21.5098	1.68478	0.842389	−	0.538870i	$-0.181148\pi$
$163$	21.5098	1.68478	0.842389	+	0.538870i	$0.181148\pi$
$164$	9.58518	0.748477
$165$	3.30442	0.257249
$166$	2.50882	0.194722
$167$	18.2373	1.41124	0.705622	−	0.708588i	$-0.250668\pi$
$167$	18.2373	1.41124	0.705622	+	0.708588i	$0.250668\pi$
$168$	−1.75706	−0.135560
$169$	24.9550	1.91961
$170$	−0.313065	−0.0240110
$171$	16.2192	1.24032
$172$	−15.0324	−1.14621
$173$	14.1675	1.07713	0.538567	−	0.842583i	$-0.318966\pi$
$173$	14.1675	1.07713	0.538567	+	0.842583i	$0.318966\pi$
$174$	−0.118731	−0.00900098
$175$	1.00000	0.0755929
$176$	8.64022	0.651281
$177$	3.60359	0.270863
$178$	−5.08399	−0.381061
$179$	11.6024	0.867201	0.433600	−	0.901105i	$-0.357243\pi$
$179$	11.6024	0.867201	0.433600	+	0.901105i	$0.357243\pi$
$180$	−3.83389	−0.285761
$181$	2.81785	0.209449	0.104725	−	0.994501i	$-0.466604\pi$
$181$	2.81785	0.209449	0.104725	+	0.994501i	$0.466604\pi$
$182$	−3.42996	−0.254246
$183$	6.34105	0.468744
$184$	−14.3912	−1.06093
$185$	4.66576	0.343034
$186$	1.42503	0.104489
$187$	−2.17257	−0.158874
$188$	−6.47633	−0.472335
$189$	4.50598	0.327762
$190$	−3.98054	−0.288779
$191$	−2.83072	−0.204824	−0.102412	−	0.994742i	$-0.532656\pi$
$191$	−2.83072	−0.204824	−0.102412	+	0.994742i	$0.532656\pi$
$192$	1.27595	0.0920836
$193$	−6.62286	−0.476724	−0.238362	−	0.971176i	$-0.576610\pi$
$193$	−6.62286	−0.476724	−0.238362	+	0.971176i	$0.576610\pi$
$194$	−7.27619	−0.522400
$195$	5.26908	0.377327
$196$	−1.69004	−0.120717
$197$	4.50503	0.320970	0.160485	−	0.987038i	$-0.448694\pi$
$197$	4.50503	0.320970	0.160485	+	0.987038i	$0.448694\pi$
$198$	4.87970	0.346785
$199$	9.81586	0.695828	0.347914	−	0.937526i	$-0.386890\pi$
$199$	9.81586	0.695828	0.347914	+	0.937526i	$0.386890\pi$
$200$	2.05440	0.145268
$201$	1.17709	0.0830252
$202$	−4.81373	−0.338693
$203$	−0.249350	−0.0175009
$204$	−0.812785	−0.0569063
$205$	5.67158	0.396120
$206$	5.26704	0.366972
$207$	15.8911	1.10451
$208$	13.7773	0.955284
$209$	−27.6237	−1.91077
$210$	−0.476163	−0.0328583
$211$	2.57384	0.177191	0.0885953	−	0.996068i	$-0.471762\pi$
$211$	2.57384	0.177191	0.0885953	+	0.996068i	$0.471762\pi$
$212$	−7.55786	−0.519076
$213$	−1.04389	−0.0715259
$214$	−3.96701	−0.271179
$215$	−8.89471	−0.606614
$216$	9.25710	0.629866
$217$	2.99275	0.203161
$218$	8.99145	0.608978
$219$	2.10092	0.141967
$220$	6.52967	0.440230
$221$	−3.46429	−0.233033
$222$	−2.22166	−0.149108
$223$	8.53047	0.571242	0.285621	−	0.958343i	$-0.407800\pi$
$223$	8.53047	0.571242	0.285621	+	0.958343i	$0.407800\pi$
$224$	−5.35385	−0.357719
$225$	−2.26852	−0.151235
$226$	0.774531	0.0515210
$227$	−23.4620	−1.55723	−0.778614	−	0.627503i	$-0.784077\pi$
$227$	−23.4620	−1.55723	−0.778614	+	0.627503i	$0.784077\pi$
$228$	−10.3344	−0.684410
$229$	1.00000	0.0660819
$229$	1.00000	0.0660819
$230$	−3.90002	−0.257160
$231$	−3.30442	−0.217415
$232$	−0.512265	−0.0336319
$233$	3.71880	0.243627	0.121813	−	0.992553i	$-0.461129\pi$
$233$	3.71880	0.243627	0.121813	+	0.992553i	$0.461129\pi$
$234$	7.78095	0.508657
$235$	−3.83207	−0.249976
$236$	7.12085	0.463528
$237$	9.18219	0.596447
$238$	0.313065	0.0202930
$239$	−8.21602	−0.531450	−0.265725	−	0.964049i	$-0.585611\pi$
$239$	−8.21602	−0.531450	−0.265725	+	0.964049i	$0.585611\pi$
$240$	1.91263	0.123459
$241$	−3.15165	−0.203016	−0.101508	−	0.994835i	$-0.532367\pi$
$241$	−3.15165	−0.203016	−0.101508	+	0.994835i	$0.532367\pi$
$242$	−2.18666	−0.140564
$243$	−16.0425	−1.02913
$244$	12.5302	0.802162
$245$	−1.00000	−0.0638877
$246$	−2.70059	−0.172184
$247$	−44.0476	−2.80268
$248$	6.14831	0.390418
$249$	3.85403	0.244239
$250$	0.556743	0.0352115
$251$	−23.3760	−1.47548	−0.737740	−	0.675085i	$-0.764107\pi$
$251$	−23.3760	−1.47548	−0.737740	+	0.675085i	$0.764107\pi$
$252$	3.83389	0.241512
$253$	−27.0649	−1.70156
$254$	−5.24613	−0.329172
$255$	−0.480927	−0.0301168
$256$	−3.44012	−0.215007
$257$	−15.2514	−0.951355	−0.475678	−	0.879620i	$-0.657797\pi$
$257$	−15.2514	−0.951355	−0.475678	+	0.879620i	$0.657797\pi$
$258$	4.23533	0.263680
$259$	−4.66576	−0.289916
$260$	10.4119	0.645719
$261$	0.565656	0.0350132
$262$	−4.97810	−0.307548
$263$	7.92347	0.488582	0.244291	−	0.969702i	$-0.421445\pi$
$263$	7.92347	0.488582	0.244291	+	0.969702i	$0.421445\pi$
$264$	−6.78861	−0.417810
$265$	−4.47201	−0.274713
$266$	3.98054	0.244063
$267$	−7.80999	−0.477963
$268$	2.32597	0.142081
$269$	11.6078	0.707743	0.353872	−	0.935294i	$-0.384865\pi$
$269$	11.6078	0.707743	0.353872	+	0.935294i	$0.384865\pi$
$270$	2.50867	0.152673
$271$	16.3637	0.994022	0.497011	−	0.867744i	$-0.334431\pi$
$271$	16.3637	0.994022	0.497011	+	0.867744i	$0.334431\pi$
$272$	−1.25750	−0.0762473
$273$	−5.26908	−0.318899
$274$	1.32272	0.0799082
$275$	3.86363	0.232985
$276$	−10.1253	−0.609472
$277$	0.393204	0.0236253	0.0118127	−	0.999930i	$-0.496240\pi$
$277$	0.393204	0.0236253	0.0118127	+	0.999930i	$0.496240\pi$
$278$	11.2150	0.672634
$279$	−6.78911	−0.406454
$280$	−2.05440	−0.122774
$281$	−9.14919	−0.545795	−0.272897	−	0.962043i	$-0.587982\pi$
$281$	−9.14919	−0.545795	−0.272897	+	0.962043i	$0.587982\pi$
$282$	1.82469	0.108659
$283$	−9.17599	−0.545456	−0.272728	−	0.962091i	$-0.587926\pi$
$283$	−9.17599	−0.545456	−0.272728	+	0.962091i	$0.587926\pi$
$284$	−2.06276	−0.122402
$285$	−6.11488	−0.362214
$286$	−13.2521	−0.783613
$287$	−5.67158	−0.334783
$288$	12.1453	0.715671
$289$	−16.6838	−0.981400
$290$	−0.138824	−0.00815202
$291$	−11.1776	−0.655243
$292$	4.15151	0.242949
$293$	−29.3668	−1.71563	−0.857813	−	0.513961i	$-0.828177\pi$
$293$	−29.3668	−1.71563	−0.857813	+	0.513961i	$0.828177\pi$
$294$	0.476163	0.0277704
$295$	4.21343	0.245315
$296$	−9.58536	−0.557137
$297$	17.4094	1.01020
$298$	−0.479348	−0.0277679
$299$	−43.1565	−2.49580
$300$	1.44543	0.0834518
$301$	8.89471	0.512682
$302$	2.36429	0.136050
$303$	−7.39481	−0.424820
$304$	−15.9888	−0.917023
$305$	7.41414	0.424533
$306$	−0.710195	−0.0405991
$307$	23.7841	1.35743	0.678715	−	0.734402i	$-0.262537\pi$
$307$	23.7841	1.35743	0.678715	+	0.734402i	$0.262537\pi$
$308$	−6.52967	−0.372062
$309$	8.09118	0.460291
$310$	1.66619	0.0946333
$311$	−19.0205	−1.07855	−0.539276	−	0.842129i	$-0.681302\pi$
$311$	−19.0205	−1.07855	−0.539276	+	0.842129i	$0.681302\pi$
$312$	−10.8248	−0.612834
$313$	−28.7364	−1.62428	−0.812139	−	0.583464i	$-0.801697\pi$
$313$	−28.7364	−1.62428	−0.812139	+	0.583464i	$0.801697\pi$
$314$	1.74731	0.0986067
$315$	2.26852	0.127817
$316$	18.1444	1.02070
$317$	−12.0084	−0.674461	−0.337230	−	0.941422i	$-0.609490\pi$
$317$	−12.0084	−0.674461	−0.337230	+	0.941422i	$0.609490\pi$
$318$	2.12940	0.119411
$319$	−0.963395	−0.0539397
$320$	1.49188	0.0833984
$321$	−6.09409	−0.340139
$322$	3.90002	0.217339
$323$	4.02038	0.223700
$324$	−4.98859	−0.277144
$325$	6.16076	0.341738
$326$	−11.9754	−0.663258
$327$	13.8126	0.763838
$328$	−11.6517	−0.643358
$329$	3.83207	0.211269
$330$	−1.83971	−0.101273
$331$	3.07993	0.169288	0.0846442	−	0.996411i	$-0.473025\pi$
$331$	3.07993	0.169288	0.0846442	+	0.996411i	$0.473025\pi$
$332$	7.61571	0.417967
$333$	10.5844	0.580021
$334$	−10.1535	−0.555575
$335$	1.37628	0.0751944
$336$	−1.91263	−0.104342
$337$	−26.7671	−1.45809	−0.729047	−	0.684463i	$-0.760037\pi$
$337$	−26.7671	−1.45809	−0.729047	+	0.684463i	$0.760037\pi$
$338$	−13.8935	−0.755708
$339$	1.18983	0.0646225
$340$	−0.950332	−0.0515390
$341$	11.5628	0.626163
$342$	−9.02996	−0.488284
$343$	1.00000	0.0539949
$344$	18.2733	0.985231
$345$	−5.99117	−0.322554
$346$	−7.88765	−0.424043
$347$	−22.3549	−1.20007	−0.600036	−	0.799973i	$-0.704847\pi$
$347$	−22.3549	−1.20007	−0.600036	+	0.799973i	$0.704847\pi$
$348$	−0.360417	−0.0193204
$349$	−22.5245	−1.20571	−0.602854	−	0.797852i	$-0.705970\pi$
$349$	−22.5245	−1.20571	−0.602854	+	0.797852i	$0.705970\pi$
$350$	−0.556743	−0.0297592
$351$	27.7603	1.48173
$352$	−20.6853	−1.10253
$353$	−29.0631	−1.54687	−0.773436	−	0.633875i	$-0.781463\pi$
$353$	−29.0631	−1.54687	−0.773436	+	0.633875i	$0.781463\pi$
$354$	−2.00628	−0.106632
$355$	−1.22054	−0.0647797
$356$	−15.4329	−0.817939
$357$	0.480927	0.0254534
$358$	−6.45953	−0.341397
$359$	3.83810	0.202567	0.101284	−	0.994858i	$-0.467705\pi$
$359$	3.83810	0.202567	0.101284	+	0.994858i	$0.467705\pi$
$360$	4.66046	0.245628
$361$	32.1181	1.69043
$362$	−1.56882	−0.0824554
$363$	−3.35913	−0.176309
$364$	−10.4119	−0.545733
$365$	2.45646	0.128577
$366$	−3.53034	−0.184534
$367$	−0.575375	−0.0300343	−0.0150172	−	0.999887i	$-0.504780\pi$
$367$	−0.575375	−0.0300343	−0.0150172	+	0.999887i	$0.504780\pi$
$368$	−15.6654	−0.816615
$369$	12.8661	0.669783
$370$	−2.59763	−0.135044
$371$	4.47201	0.232175
$372$	4.32580	0.224282
$373$	−2.07715	−0.107551	−0.0537754	−	0.998553i	$-0.517126\pi$
$373$	−2.07715	−0.107551	−0.0537754	+	0.998553i	$0.517126\pi$
$374$	1.20957	0.0625451
$375$	0.855264	0.0441657
$376$	7.87261	0.405999
$377$	−1.53619	−0.0791176
$378$	−2.50867	−0.129032
$379$	−20.5495	−1.05556	−0.527778	−	0.849382i	$-0.676975\pi$
$379$	−20.5495	−1.05556	−0.527778	+	0.849382i	$0.676975\pi$
$380$	−12.0832	−0.619857
$381$	−8.05906	−0.412878
$382$	1.57599	0.0806345
$383$	1.94646	0.0994593	0.0497297	−	0.998763i	$-0.484164\pi$
$383$	1.94646	0.0994593	0.0497297	+	0.998763i	$0.484164\pi$
$384$	−9.86829	−0.503589
$385$	−3.86363	−0.196909
$386$	3.68723	0.187675
$387$	−20.1778	−1.02570
$388$	−22.0874	−1.12132
$389$	16.5189	0.837544	0.418772	−	0.908091i	$-0.362461\pi$
$389$	16.5189	0.837544	0.418772	+	0.908091i	$0.362461\pi$
$390$	−2.93352	−0.148545
$391$	3.93904	0.199206
$392$	2.05440	0.103763
$393$	−7.64732	−0.385756
$394$	−2.50814	−0.126359
$395$	10.7361	0.540191
$396$	14.8127	0.744366
$397$	24.3880	1.22400	0.611999	−	0.790859i	$-0.290366\pi$
$397$	24.3880	1.22400	0.611999	+	0.790859i	$0.290366\pi$
$398$	−5.46491	−0.273931
$399$	6.11488	0.306127
$400$	2.23630	0.111815
$401$	−17.0682	−0.852347	−0.426174	−	0.904641i	$-0.640139\pi$
$401$	−17.0682	−0.852347	−0.426174	+	0.904641i	$0.640139\pi$
$402$	−0.655334	−0.0326851
$403$	18.4376	0.918442
$404$	−14.6124	−0.726996
$405$	−2.95177	−0.146674
$406$	0.138824	0.00688971
$407$	−18.0268	−0.893553
$408$	0.988019	0.0489142
$409$	−11.1741	−0.552523	−0.276262	−	0.961082i	$-0.589096\pi$
$409$	−11.1741	−0.552523	−0.276262	+	0.961082i	$0.589096\pi$
$410$	−3.15761	−0.155943
$411$	2.03195	0.100228
$412$	15.9885	0.787697
$413$	−4.21343	−0.207329
$414$	−8.84728	−0.434820
$415$	4.50624	0.221203
$416$	−32.9838	−1.61716
$417$	17.2285	0.843681
$418$	15.3793	0.752228
$419$	−10.2822	−0.502321	−0.251160	−	0.967946i	$-0.580812\pi$
$419$	−10.2822	−0.502321	−0.251160	+	0.967946i	$0.580812\pi$
$420$	−1.44543	−0.0705297
$421$	−28.9006	−1.40853	−0.704264	−	0.709938i	$-0.748723\pi$
$421$	−28.9006	−1.40853	−0.704264	+	0.709938i	$0.748723\pi$
$422$	−1.43297	−0.0697558
$423$	−8.69313	−0.422675
$424$	9.18731	0.446175
$425$	−0.562314	−0.0272763
$426$	0.581177	0.0281581
$427$	−7.41414	−0.358796
$428$	−12.0422	−0.582080
$429$	−20.3577	−0.982881
$430$	4.95207	0.238810
$431$	−17.0915	−0.823268	−0.411634	−	0.911349i	$-0.635042\pi$
$431$	−17.0915	−0.823268	−0.411634	+	0.911349i	$0.635042\pi$
$432$	10.0767	0.484816
$433$	27.2965	1.31179	0.655893	−	0.754854i	$-0.272292\pi$
$433$	27.2965	1.31179	0.655893	+	0.754854i	$0.272292\pi$
$434$	−1.66619	−0.0799797
$435$	−0.213260	−0.0102250
$436$	27.2942	1.30716
$437$	50.0840	2.39584
$438$	−1.16967	−0.0558892
$439$	−12.2084	−0.582674	−0.291337	−	0.956620i	$-0.594100\pi$
$439$	−12.2084	−0.582674	−0.291337	+	0.956620i	$0.594100\pi$
$440$	−7.93744	−0.378403
$441$	−2.26852	−0.108025
$442$	1.92872	0.0917397
$443$	−13.9379	−0.662210	−0.331105	−	0.943594i	$-0.607421\pi$
$443$	−13.9379	−0.662210	−0.331105	+	0.943594i	$0.607421\pi$
$444$	−6.74402	−0.320057
$445$	−9.13167	−0.432882
$446$	−4.74928	−0.224885
$447$	−0.736370	−0.0348291
$448$	−1.49188	−0.0704845
$449$	−17.0168	−0.803070	−0.401535	−	0.915844i	$-0.631523\pi$
$449$	−17.0168	−0.803070	−0.401535	+	0.915844i	$0.631523\pi$
$450$	1.26299	0.0595377
$451$	−21.9129	−1.03184
$452$	2.35115	0.110589
$453$	3.63200	0.170646
$454$	13.0623	0.613045
$455$	−6.16076	−0.288821
$456$	12.5624	0.588289
$457$	−7.26381	−0.339787	−0.169893	−	0.985462i	$-0.554342\pi$
$457$	−7.26381	−0.339787	−0.169893	+	0.985462i	$0.554342\pi$
$458$	−0.556743	−0.0260149
$459$	−2.53378	−0.118267
$460$	−11.8388	−0.551987
$461$	−7.00041	−0.326042	−0.163021	−	0.986623i	$-0.552124\pi$
$461$	−7.00041	−0.326042	−0.163021	+	0.986623i	$0.552124\pi$
$462$	1.83971	0.0855912
$463$	−34.5436	−1.60538	−0.802689	−	0.596398i	$-0.796598\pi$
$463$	−34.5436	−1.60538	−0.802689	+	0.596398i	$0.796598\pi$
$464$	−0.557621	−0.0258869
$465$	2.55959	0.118698
$466$	−2.07042	−0.0959103
$467$	−0.528426	−0.0244526	−0.0122263	−	0.999925i	$-0.503892\pi$
$467$	−0.528426	−0.0244526	−0.0122263	+	0.999925i	$0.503892\pi$
$468$	23.6197	1.09182
$469$	−1.37628	−0.0635508
$470$	2.13348	0.0984100
$471$	2.68421	0.123682
$472$	−8.65608	−0.398429
$473$	34.3658	1.58014
$474$	−5.11212	−0.234808
$475$	−7.14969	−0.328050
$476$	0.950332	0.0435584
$477$	−10.1449	−0.464501
$478$	4.57422	0.209220
$479$	−6.59183	−0.301188	−0.150594	−	0.988596i	$-0.548119\pi$
$479$	−6.59183	−0.301188	−0.150594	+	0.988596i	$0.548119\pi$
$480$	−4.57896	−0.209000
$481$	−28.7446	−1.31064
$482$	1.75466	0.0799227
$483$	5.99117	0.272608
$484$	−6.63779	−0.301718
$485$	−13.0692	−0.593441
$486$	8.93154	0.405143
$487$	−13.2354	−0.599752	−0.299876	−	0.953978i	$-0.596945\pi$
$487$	−13.2354	−0.599752	−0.299876	+	0.953978i	$0.596945\pi$
$488$	−15.2316	−0.689504
$489$	−18.3966	−0.831922
$490$	0.556743	0.0251511
$491$	18.7763	0.847363	0.423682	−	0.905811i	$-0.360738\pi$
$491$	18.7763	0.847363	0.423682	+	0.905811i	$0.360738\pi$
$492$	−8.19786	−0.369588
$493$	0.140213	0.00631488
$494$	24.5232	1.10335
$495$	8.76472	0.393945
$496$	6.69267	0.300510
$497$	1.22054	0.0547488
$498$	−2.14570	−0.0961513
$499$	−28.7909	−1.28886	−0.644429	−	0.764664i	$-0.722905\pi$
$499$	−28.7909	−1.28886	−0.644429	+	0.764664i	$0.722905\pi$
$500$	1.69004	0.0755807
$501$	−15.5977	−0.696854
$502$	13.0144	0.580862
$503$	41.3185	1.84230	0.921150	−	0.389209i	$-0.127252\pi$
$503$	41.3185	1.84230	0.921150	+	0.389209i	$0.127252\pi$
$504$	−4.66046	−0.207593
$505$	−8.64622	−0.384752
$506$	15.0682	0.669863
$507$	−21.3431	−0.947880
$508$	−15.9250	−0.706559
$509$	18.1254	0.803394	0.401697	−	0.915773i	$-0.368421\pi$
$509$	18.1254	0.803394	0.401697	+	0.915773i	$0.368421\pi$
$510$	0.267753	0.0118563
$511$	−2.45646	−0.108667
$512$	−21.1613	−0.935207
$513$	−32.2164	−1.42239
$514$	8.49111	0.374527
$515$	9.46044	0.416877
$516$	12.8567	0.565983
$517$	14.8057	0.651153
$518$	2.59763	0.114133
$519$	−12.1169	−0.531874
$520$	−12.6567	−0.555033
$521$	22.4471	0.983427	0.491714	−	0.870757i	$-0.336371\pi$
$521$	22.4471	0.983427	0.491714	+	0.870757i	$0.336371\pi$
$522$	−0.314925	−0.0137839
$523$	−31.3878	−1.37249	−0.686247	−	0.727369i	$-0.740743\pi$
$523$	−31.3878	−1.37249	−0.686247	+	0.727369i	$0.740743\pi$
$524$	−15.1114	−0.660145
$525$	−0.855264	−0.0373268
$526$	−4.41134	−0.192344
$527$	−1.68286	−0.0733067
$528$	−7.38967	−0.321594
$529$	26.0708	1.13351
$530$	2.48976	0.108148
$531$	9.55826	0.414793
$532$	12.0832	0.523875
$533$	−34.9412	−1.51347
$534$	4.34816	0.188163
$535$	−7.12539	−0.308057
$536$	−2.82744	−0.122127
$537$	−9.92308	−0.428212
$538$	−6.46259	−0.278622
$539$	3.86363	0.166418
$540$	7.61527	0.327709
$541$	40.5482	1.74331	0.871653	−	0.490124i	$-0.163049\pi$
$541$	40.5482	1.74331	0.871653	+	0.490124i	$0.163049\pi$
$542$	−9.11036	−0.391323
$543$	−2.41001	−0.103423
$544$	3.01055	0.129076
$545$	16.1501	0.691794
$546$	2.93352	0.125543
$547$	−30.6615	−1.31099	−0.655495	−	0.755200i	$-0.727540\pi$
$547$	−30.6615	−1.31099	−0.655495	+	0.755200i	$0.727540\pi$
$548$	4.01521	0.171521
$549$	16.8192	0.717824
$550$	−2.15105	−0.0917210
$551$	1.78278	0.0759488
$552$	12.3083	0.523875
$553$	−10.7361	−0.456545
$554$	−0.218914	−0.00930075
$555$	−3.99046	−0.169386
$556$	34.0441	1.44379
$557$	−30.6986	−1.30074	−0.650371	−	0.759617i	$-0.725386\pi$
$557$	−30.6986	−1.30074	−0.650371	+	0.759617i	$0.725386\pi$
$558$	3.77979	0.160011
$559$	54.7982	2.31772
$560$	−2.23630	−0.0945009
$561$	1.85812	0.0784500
$562$	5.09375	0.214867
$563$	8.98508	0.378676	0.189338	−	0.981912i	$-0.439366\pi$
$563$	8.98508	0.378676	0.189338	+	0.981912i	$0.439366\pi$
$564$	5.53898	0.233233
$565$	1.39118	0.0585274
$566$	5.10867	0.214734
$567$	2.95177	0.123963
$568$	2.50749	0.105212
$569$	33.3983	1.40013	0.700064	−	0.714080i	$-0.253155\pi$
$569$	33.3983	1.40013	0.700064	+	0.714080i	$0.253155\pi$
$570$	3.40442	0.142595
$571$	−2.48325	−0.103921	−0.0519604	−	0.998649i	$-0.516547\pi$
$571$	−2.48325	−0.103921	−0.0519604	+	0.998649i	$0.516547\pi$
$572$	−40.2277	−1.68201
$573$	2.42102	0.101139
$574$	3.15761	0.131796
$575$	−7.00506	−0.292131
$576$	3.38436	0.141015
$577$	−9.82018	−0.408819	−0.204410	−	0.978885i	$-0.565527\pi$
$577$	−9.82018	−0.408819	−0.204410	+	0.978885i	$0.565527\pi$
$578$	9.28860	0.386355
$579$	5.66429	0.235400
$580$	−0.421411	−0.0174981
$581$	−4.50624	−0.186950
$582$	6.22306	0.257954
$583$	17.2782	0.715588
$584$	−5.04656	−0.208828
$585$	13.9758	0.577829
$586$	16.3498	0.675403
$587$	−11.1971	−0.462155	−0.231078	−	0.972935i	$-0.574225\pi$
$587$	−11.1971	−0.462155	−0.231078	+	0.972935i	$0.574225\pi$
$588$	1.44543	0.0596084
$589$	−21.3972	−0.881657
$590$	−2.34580	−0.0965750
$591$	−3.85299	−0.158491
$592$	−10.4340	−0.428836
$593$	46.8041	1.92201	0.961006	−	0.276527i	$-0.0891834\pi$
$593$	46.8041	1.92201	0.961006	+	0.276527i	$0.0891834\pi$
$594$	−9.69257	−0.397691
$595$	0.562314	0.0230526
$596$	−1.45510	−0.0596031
$597$	−8.39515	−0.343591
$598$	24.0271	0.982540
$599$	19.6593	0.803255	0.401628	−	0.915803i	$-0.368445\pi$
$599$	19.6593	0.803255	0.401628	+	0.915803i	$0.368445\pi$
$600$	−1.75706	−0.0717316
$601$	−26.8751	−1.09626	−0.548129	−	0.836394i	$-0.684660\pi$
$601$	−26.8751	−1.09626	−0.548129	+	0.836394i	$0.684660\pi$
$602$	−4.95207	−0.201831
$603$	3.12213	0.127143
$604$	7.17698	0.292027
$605$	−3.92760	−0.159680
$606$	4.11701	0.167242
$607$	−25.4318	−1.03225	−0.516123	−	0.856514i	$-0.672625\pi$
$607$	−25.4318	−1.03225	−0.516123	+	0.856514i	$0.672625\pi$
$608$	38.2784	1.55239
$609$	0.213260	0.00864173
$610$	−4.12777	−0.167129
$611$	23.6084	0.955096
$612$	−2.15585	−0.0871451
$613$	5.69494	0.230016	0.115008	−	0.993365i	$-0.463311\pi$
$613$	5.69494	0.230016	0.115008	+	0.993365i	$0.463311\pi$
$614$	−13.2416	−0.534389
$615$	−4.85070	−0.195599
$616$	7.93744	0.319809
$617$	8.96831	0.361051	0.180525	−	0.983570i	$-0.442220\pi$
$617$	8.96831	0.361051	0.180525	+	0.983570i	$0.442220\pi$
$618$	−4.50471	−0.181206
$619$	17.4314	0.700626	0.350313	−	0.936633i	$-0.386075\pi$
$619$	17.4314	0.700626	0.350313	+	0.936633i	$0.386075\pi$
$620$	5.05785	0.203128
$621$	−31.5646	−1.26665
$622$	10.5895	0.424601
$623$	9.13167	0.365852
$624$	−11.7832	−0.471707
$625$	1.00000	0.0400000
$626$	15.9988	0.639441
$627$	23.6256	0.943515
$628$	5.30411	0.211657
$629$	2.62363	0.104611
$630$	−1.26299	−0.0503185
$631$	−22.9663	−0.914272	−0.457136	−	0.889397i	$-0.651125\pi$
$631$	−22.9663	−0.914272	−0.457136	+	0.889397i	$0.651125\pi$
$632$	−22.0562	−0.877351
$633$	−2.20131	−0.0874944
$634$	6.68561	0.265520
$635$	−9.42289	−0.373936
$636$	6.46397	0.256313
$637$	6.16076	0.244098
$638$	0.536364	0.0212348
$639$	−2.76883	−0.109533
$640$	−11.5383	−0.456091
$641$	2.44187	0.0964480	0.0482240	−	0.998837i	$-0.484644\pi$
$641$	2.44187	0.0964480	0.0482240	+	0.998837i	$0.484644\pi$
$642$	3.39284	0.133905
$643$	12.3136	0.485600	0.242800	−	0.970076i	$-0.421934\pi$
$643$	12.3136	0.485600	0.242800	+	0.970076i	$0.421934\pi$
$644$	11.8388	0.466514
$645$	7.60732	0.299538
$646$	−2.23832	−0.0880654
$647$	−34.8544	−1.37027	−0.685133	−	0.728418i	$-0.740256\pi$
$647$	−34.8544	−1.37027	−0.685133	+	0.728418i	$0.740256\pi$
$648$	6.06412	0.238221
$649$	−16.2791	−0.639011
$650$	−3.42996	−0.134534
$651$	−2.55959	−0.100318
$652$	−36.3524	−1.42367
$653$	−35.5654	−1.39178	−0.695890	−	0.718148i	$-0.744990\pi$
$653$	−35.5654	−1.39178	−0.695890	+	0.718148i	$0.744990\pi$
$654$	−7.69007	−0.300705
$655$	−8.94147	−0.349372
$656$	−12.6833	−0.495201
$657$	5.57253	0.217405
$658$	−2.13348	−0.0831716
$659$	14.9792	0.583506	0.291753	−	0.956494i	$-0.405762\pi$
$659$	14.9792	0.583506	0.291753	+	0.956494i	$0.405762\pi$
$660$	−5.58459	−0.217380
$661$	−2.98439	−0.116079	−0.0580397	−	0.998314i	$-0.518485\pi$
$661$	−2.98439	−0.116079	−0.0580397	+	0.998314i	$0.518485\pi$
$662$	−1.71473	−0.0666449
$663$	2.96288	0.115069
$664$	−9.25764	−0.359266
$665$	7.14969	0.277253
$666$	−5.89279	−0.228341
$667$	1.74671	0.0676329
$668$	−30.8217	−1.19253
$669$	−7.29580	−0.282072
$670$	−0.766236	−0.0296023
$671$	−28.6455	−1.10585
$672$	4.57896	0.176637
$673$	0.815263	0.0314261	0.0157130	−	0.999877i	$-0.494998\pi$
$673$	0.815263	0.0314261	0.0157130	+	0.999877i	$0.494998\pi$
$674$	14.9024	0.574018
$675$	4.50598	0.173435
$676$	−42.1748	−1.62211
$677$	41.2696	1.58612	0.793060	−	0.609144i	$-0.208487\pi$
$677$	41.2696	1.58612	0.793060	+	0.609144i	$0.208487\pi$
$678$	−0.662428	−0.0254404
$679$	13.0692	0.501550
$680$	1.15522	0.0443007
$681$	20.0662	0.768939
$682$	−6.43754	−0.246506
$683$	−15.8896	−0.607999	−0.304000	−	0.952672i	$-0.598322\pi$
$683$	−15.8896	−0.607999	−0.304000	+	0.952672i	$0.598322\pi$
$684$	−27.4111	−1.04809
$685$	2.37581	0.0907751
$686$	−0.556743	−0.0212566
$687$	−0.855264	−0.0326304
$688$	19.8912	0.758346
$689$	27.5510	1.04961
$690$	3.33555	0.126982
$691$	19.6018	0.745687	0.372843	−	0.927894i	$-0.378383\pi$
$691$	19.6018	0.745687	0.372843	+	0.927894i	$0.378383\pi$
$692$	−23.9436	−0.910198
$693$	−8.76472	−0.332944
$694$	12.4459	0.472441
$695$	20.1440	0.764106
$696$	0.438122	0.0166070
$697$	3.18921	0.120800
$698$	12.5404	0.474659
$699$	−3.18056	−0.120300
$700$	−1.69004	−0.0638774
$701$	−46.0224	−1.73824	−0.869121	−	0.494600i	$-0.835314\pi$
$701$	−46.0224	−1.73824	−0.869121	+	0.494600i	$0.835314\pi$
$702$	−15.4553	−0.583324
$703$	33.3588	1.25815
$704$	−5.76405	−0.217241
$705$	3.27743	0.123435
$706$	16.1807	0.608968
$707$	8.64622	0.325175
$708$	−6.09021	−0.228884
$709$	11.5798	0.434889	0.217445	−	0.976073i	$-0.430228\pi$
$709$	11.5798	0.434889	0.217445	+	0.976073i	$0.430228\pi$
$710$	0.679529	0.0255023
$711$	24.3551	0.913386
$712$	18.7601	0.703065
$713$	−20.9643	−0.785121
$714$	−0.267753	−0.0100204
$715$	−23.8029	−0.890177
$716$	−19.6084	−0.732801
$717$	7.02687	0.262423
$718$	−2.13684	−0.0797460
$719$	25.5933	0.954468	0.477234	−	0.878776i	$-0.341639\pi$
$719$	25.5933	0.954468	0.477234	+	0.878776i	$0.341639\pi$
$720$	5.07309	0.189063
$721$	−9.46044	−0.352325
$722$	−17.8815	−0.665482
$723$	2.69550	0.100247
$724$	−4.76227	−0.176989
$725$	−0.249350	−0.00926062
$726$	1.87018	0.0694087
$727$	8.38222	0.310879	0.155440	−	0.987845i	$-0.450321\pi$
$727$	8.38222	0.310879	0.155440	+	0.987845i	$0.450321\pi$
$728$	12.6567	0.469088
$729$	4.86526	0.180195
$730$	−1.36762	−0.0506178
$731$	−5.00162	−0.184992
$732$	−10.7166	−0.396097
$733$	−53.1894	−1.96460	−0.982298	−	0.187322i	$-0.940019\pi$
$733$	−53.1894	−1.96460	−0.982298	+	0.187322i	$0.940019\pi$
$734$	0.320336	0.0118238
$735$	0.855264	0.0315469
$736$	37.5040	1.38242
$737$	−5.31744	−0.195870
$738$	−7.16312	−0.263678
$739$	37.5694	1.38201	0.691007	−	0.722848i	$-0.257167\pi$
$739$	37.5694	1.38201	0.691007	+	0.722848i	$0.257167\pi$
$740$	−7.88531	−0.289870
$741$	37.6723	1.38393
$742$	−2.48976	−0.0914020
$743$	9.00351	0.330307	0.165153	−	0.986268i	$-0.447188\pi$
$743$	9.00351	0.330307	0.165153	+	0.986268i	$0.447188\pi$
$744$	−5.25843	−0.192783
$745$	−0.860986	−0.0315441
$746$	1.15644	0.0423403
$747$	10.2225	0.374022
$748$	3.67173	0.134252
$749$	7.12539	0.260356
$750$	−0.476163	−0.0173870
$751$	33.1481	1.20959	0.604795	−	0.796381i	$-0.293255\pi$
$751$	33.1481	1.20959	0.604795	+	0.796381i	$0.293255\pi$
$752$	8.56964	0.312503
$753$	19.9927	0.728573
$754$	0.855261	0.0311468
$755$	4.24664	0.154551
$756$	−7.61527	−0.276965
$757$	−16.0133	−0.582013	−0.291007	−	0.956721i	$-0.593990\pi$
$757$	−16.0133	−0.582013	−0.291007	+	0.956721i	$0.593990\pi$
$758$	11.4408	0.415548
$759$	23.1476	0.840206
$760$	14.6884	0.532803
$761$	−12.0469	−0.436701	−0.218351	−	0.975870i	$-0.570068\pi$
$761$	−12.0469	−0.436701	−0.218351	+	0.975870i	$0.570068\pi$
$762$	4.48683	0.162541
$763$	−16.1501	−0.584672
$764$	4.78403	0.173080
$765$	−1.27562	−0.0461203
$766$	−1.08368	−0.0391548
$767$	−25.9579	−0.937287
$768$	2.94221	0.106168
$769$	7.88376	0.284296	0.142148	−	0.989845i	$-0.454599\pi$
$769$	7.88376	0.284296	0.142148	+	0.989845i	$0.454599\pi$
$770$	2.15105	0.0775184
$771$	13.0440	0.469767
$772$	11.1929	0.402840
$773$	−46.4566	−1.67093	−0.835464	−	0.549545i	$-0.814801\pi$
$773$	−46.4566	−1.67093	−0.835464	+	0.549545i	$0.814801\pi$
$774$	11.2339	0.403794
$775$	2.99275	0.107503
$776$	26.8494	0.963837
$777$	3.99046	0.143157
$778$	−9.19681	−0.329722
$779$	40.5501	1.45286
$780$	−8.90494	−0.318848
$781$	4.71572	0.168742
$782$	−2.19304	−0.0784228
$783$	−1.12357	−0.0401530
$784$	2.23630	0.0798678
$785$	3.13846	0.112016
$786$	4.25759	0.151863
$787$	−27.5677	−0.982683	−0.491342	−	0.870967i	$-0.663493\pi$
$787$	−27.5677	−0.982683	−0.491342	+	0.870967i	$0.663493\pi$
$788$	−7.61367	−0.271226
$789$	−6.77666	−0.241256
$790$	−5.97724	−0.212661
$791$	−1.39118	−0.0494647
$792$	−18.0063	−0.639825
$793$	−45.6768	−1.62203
$794$	−13.5778	−0.481860
$795$	3.82475	0.135650
$796$	−16.5892	−0.587987
$797$	−33.4695	−1.18555	−0.592775	−	0.805368i	$-0.701968\pi$
$797$	−33.4695	−1.18555	−0.592775	+	0.805368i	$0.701968\pi$
$798$	−3.40442	−0.120515
$799$	−2.15483	−0.0762323
$800$	−5.35385	−0.189287
$801$	−20.7154	−0.731942
$802$	9.50263	0.335549
$803$	−9.49084	−0.334924
$804$	−1.98932	−0.0701578
$805$	7.00506	0.246896
$806$	−10.2650	−0.361569
$807$	−9.92778	−0.349474
$808$	17.7628	0.624894
$809$	41.9034	1.47324	0.736622	−	0.676305i	$-0.236420\pi$
$809$	41.9034	1.47324	0.736622	+	0.676305i	$0.236420\pi$
$810$	1.64338	0.0577424
$811$	2.75733	0.0968230	0.0484115	−	0.998827i	$-0.484584\pi$
$811$	2.75733	0.0968230	0.0484115	+	0.998827i	$0.484584\pi$
$812$	0.421411	0.0147886
$813$	−13.9953	−0.490835
$814$	10.0363	0.351771
$815$	−21.5098	−0.753456
$816$	1.07550	0.0376499
$817$	−63.5944	−2.22489
$818$	6.22110	0.217516
$819$	−13.9758	−0.488355
$820$	−9.58518	−0.334729
$821$	−33.1720	−1.15771	−0.578855	−	0.815430i	$-0.696500\pi$
$821$	−33.1720	−1.15771	−0.578855	+	0.815430i	$0.696500\pi$
$822$	−1.13127	−0.0394576
$823$	−1.55583	−0.0542328	−0.0271164	−	0.999632i	$-0.508632\pi$
$823$	−1.55583	−0.0542328	−0.0271164	+	0.999632i	$0.508632\pi$
$824$	−19.4356	−0.677070
$825$	−3.30442	−0.115045
$826$	2.34580	0.0816208
$827$	−5.95155	−0.206956	−0.103478	−	0.994632i	$-0.532997\pi$
$827$	−5.95155	−0.206956	−0.103478	+	0.994632i	$0.532997\pi$
$828$	−26.8566	−0.933331
$829$	28.4169	0.986960	0.493480	−	0.869757i	$-0.335725\pi$
$829$	28.4169	0.986960	0.493480	+	0.869757i	$0.335725\pi$
$830$	−2.50882	−0.0870824
$831$	−0.336293	−0.0116659
$832$	−9.19109	−0.318644
$833$	−0.562314	−0.0194830
$834$	−9.59183	−0.332138
$835$	−18.2373	−0.631128
$836$	46.6851	1.61464
$837$	13.4852	0.466118
$838$	5.72457	0.197752
$839$	−26.0272	−0.898560	−0.449280	−	0.893391i	$-0.648319\pi$
$839$	−26.0272	−0.898560	−0.449280	+	0.893391i	$0.648319\pi$
$840$	1.75706	0.0606242
$841$	−28.9378	−0.997856
$842$	16.0902	0.554505
$843$	7.82497	0.269506
$844$	−4.34989	−0.149729
$845$	−24.9550	−0.858477
$846$	4.83984	0.166397
$847$	3.92760	0.134954
$848$	10.0007	0.343427
$849$	7.84790	0.269339
$850$	0.313065	0.0107380
$851$	32.6839	1.12039
$852$	1.76421	0.0604407
$853$	17.2326	0.590031	0.295016	−	0.955492i	$-0.404675\pi$
$853$	17.2326	0.590031	0.295016	+	0.955492i	$0.404675\pi$
$854$	4.12777	0.141250
$855$	−16.2192	−0.554687
$856$	14.6384	0.500331
$857$	−11.7045	−0.399819	−0.199910	−	0.979814i	$-0.564065\pi$
$857$	−11.7045	−0.399819	−0.199910	+	0.979814i	$0.564065\pi$
$858$	11.3340	0.386938
$859$	−36.9170	−1.25959	−0.629795	−	0.776761i	$-0.716861\pi$
$859$	−36.9170	−1.25959	−0.629795	+	0.776761i	$0.716861\pi$
$860$	15.0324	0.512600
$861$	4.85070	0.165311
$862$	9.51557	0.324102
$863$	−47.8946	−1.63035	−0.815176	−	0.579214i	$-0.803360\pi$
$863$	−47.8946	−1.63035	−0.815176	+	0.579214i	$0.803360\pi$
$864$	−24.1243	−0.820727
$865$	−14.1675	−0.481709
$866$	−15.1971	−0.516420
$867$	14.2691	0.484603
$868$	−5.05785	−0.171675
$869$	−41.4802	−1.40712
$870$	0.118731	0.00402536
$871$	−8.47895	−0.287298
$872$	−33.1788	−1.12358
$873$	−29.6478	−1.00342
$874$	−27.8839	−0.943188
$875$	−1.00000	−0.0338062
$876$	−3.55063	−0.119965
$877$	−28.6718	−0.968179	−0.484089	−	0.875019i	$-0.660849\pi$
$877$	−28.6718	−0.968179	−0.484089	+	0.875019i	$0.660849\pi$
$878$	6.79693	0.229385
$879$	25.1164	0.847154
$880$	−8.64022	−0.291262
$881$	46.1962	1.55639	0.778196	−	0.628022i	$-0.216135\pi$
$881$	46.1962	1.55639	0.778196	+	0.628022i	$0.216135\pi$
$882$	1.26299	0.0425269
$883$	−19.7098	−0.663288	−0.331644	−	0.943405i	$-0.607603\pi$
$883$	−19.7098	−0.663288	−0.331644	+	0.943405i	$0.607603\pi$
$884$	5.85477	0.196917
$885$	−3.60359	−0.121134
$886$	7.75983	0.260697
$887$	−18.2139	−0.611565	−0.305782	−	0.952101i	$-0.598918\pi$
$887$	−18.2139	−0.611565	−0.305782	+	0.952101i	$0.598918\pi$
$888$	8.19801	0.275107
$889$	9.42289	0.316034
$890$	5.08399	0.170416
$891$	11.4045	0.382066
$892$	−14.4168	−0.482710
$893$	−27.3981	−0.916843
$894$	0.409969	0.0137114
$895$	−11.6024	−0.387824
$896$	11.5383	0.385467
$897$	36.9102	1.23240
$898$	9.47396	0.316150
$899$	−0.746241	−0.0248885
$900$	3.83389	0.127796
$901$	−2.51467	−0.0837760
$902$	12.1998	0.406210
$903$	−7.60732	−0.253156
$904$	−2.85805	−0.0950572
$905$	−2.81785	−0.0936686
$906$	−2.02209	−0.0671795
$907$	12.2259	0.405954	0.202977	−	0.979184i	$-0.434938\pi$
$907$	12.2259	0.405954	0.202977	+	0.979184i	$0.434938\pi$
$908$	39.6517	1.31589
$909$	−19.6142	−0.650561
$910$	3.42996	0.113702
$911$	0.636656	0.0210933	0.0105467	−	0.999944i	$-0.496643\pi$
$911$	0.636656	0.0210933	0.0105467	+	0.999944i	$0.496643\pi$
$912$	13.6747	0.452814
$913$	−17.4104	−0.576201
$914$	4.04408	0.133766
$915$	−6.34105	−0.209629
$916$	−1.69004	−0.0558404
$917$	8.94147	0.295273
$918$	1.41066	0.0465588
$919$	28.7809	0.949394	0.474697	−	0.880149i	$-0.342558\pi$
$919$	28.7809	0.949394	0.474697	+	0.880149i	$0.342558\pi$
$920$	14.3912	0.474464
$921$	−20.3417	−0.670281
$922$	3.89743	0.128355
$923$	7.51947	0.247506
$924$	5.58459	0.183720
$925$	−4.66576	−0.153409
$926$	19.2319	0.632000
$927$	21.4612	0.704879
$928$	1.33498	0.0438229
$929$	−11.9577	−0.392321	−0.196160	−	0.980572i	$-0.562847\pi$
$929$	−11.9577	−0.392321	−0.196160	+	0.980572i	$0.562847\pi$
$930$	−1.42503	−0.0467287
$931$	−7.14969	−0.234322
$932$	−6.28492	−0.205869
$933$	16.2675	0.532575
$934$	0.294197	0.00962644
$935$	2.17257	0.0710507
$936$	−28.7120	−0.938481
$937$	10.4353	0.340907	0.170453	−	0.985366i	$-0.445477\pi$
$937$	10.4353	0.340907	0.170453	+	0.985366i	$0.445477\pi$
$938$	0.766236	0.0250185
$939$	24.5772	0.802047
$940$	6.47633	0.211235
$941$	−37.2428	−1.21408	−0.607040	−	0.794672i	$-0.707643\pi$
$941$	−37.2428	−1.21408	−0.607040	+	0.794672i	$0.707643\pi$
$942$	−1.49442	−0.0486907
$943$	39.7297	1.29378
$944$	−9.42248	−0.306676
$945$	−4.50598	−0.146580
$946$	−19.1329	−0.622065
$947$	−2.72490	−0.0885472	−0.0442736	−	0.999019i	$-0.514097\pi$
$947$	−2.72490	−0.0885472	−0.0442736	+	0.999019i	$0.514097\pi$
$948$	−15.5182	−0.504009
$949$	−15.1337	−0.491259
$950$	3.98054	0.129146
$951$	10.2704	0.333040
$952$	−1.15522	−0.0374409
$953$	44.9605	1.45641	0.728206	−	0.685358i	$-0.240354\pi$
$953$	44.9605	1.45641	0.728206	+	0.685358i	$0.240354\pi$
$954$	5.64808	0.182863
$955$	2.83072	0.0916000
$956$	13.8854	0.449085
$957$	0.823957	0.0266347
$958$	3.66995	0.118571
$959$	−2.37581	−0.0767189
$960$	−1.27595	−0.0411811
$961$	−22.0435	−0.711080
$962$	16.0034	0.515970
$963$	−16.1641	−0.520881
$964$	5.32641	0.171552
$965$	6.62286	0.213197
$966$	−3.33555	−0.107319
$967$	−34.4213	−1.10691	−0.553457	−	0.832878i	$-0.686692\pi$
$967$	−34.4213	−1.10691	−0.553457	+	0.832878i	$0.686692\pi$
$968$	8.06887	0.259343
$969$	−3.43848	−0.110460
$970$	7.27619	0.233624
$971$	14.5309	0.466319	0.233159	−	0.972439i	$-0.425094\pi$
$971$	14.5309	0.466319	0.233159	+	0.972439i	$0.425094\pi$
$972$	27.1124	0.869630
$973$	−20.1440	−0.645788
$974$	7.36871	0.236109
$975$	−5.26908	−0.168746
$976$	−16.5802	−0.530720
$977$	−27.4713	−0.878885	−0.439442	−	0.898271i	$-0.644824\pi$
$977$	−27.4713	−0.878885	−0.439442	+	0.898271i	$0.644824\pi$
$978$	10.2422	0.327508
$979$	35.2813	1.12760
$980$	1.69004	0.0539862
$981$	36.6368	1.16972
$982$	−10.4536	−0.333587
$983$	30.8705	0.984617	0.492308	−	0.870421i	$-0.336153\pi$
$983$	30.8705	0.984617	0.492308	+	0.870421i	$0.336153\pi$
$984$	9.96529	0.317682
$985$	−4.50503	−0.143542
$986$	−0.0780627	−0.00248602
$987$	−3.27743	−0.104322
$988$	74.4420	2.36832
$989$	−62.3079	−1.98128
$990$	−4.87970	−0.155087
$991$	−1.62094	−0.0514910	−0.0257455	−	0.999669i	$-0.508196\pi$
$991$	−1.62094	−0.0514910	−0.0257455	+	0.999669i	$0.508196\pi$
$992$	−16.0227	−0.508722
$993$	−2.63416	−0.0835924
$994$	−0.679529	−0.0215533
$995$	−9.81586	−0.311184
$996$	−6.51345	−0.206386
$997$	−0.460283	−0.0145773	−0.00728865	−	0.999973i	$-0.502320\pi$
$997$	−0.460283	−0.0145773	−0.00728865	+	0.999973i	$0.502320\pi$
$998$	16.0291	0.507394
$999$	−21.0238	−0.665164

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.j.1.20	✓	45

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.j.1.20	✓	45	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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

\(f(q)\)	\(=\)	\(q-0.556743 q^{2} -0.855264 q^{3} -1.69004 q^{4} -1.00000 q^{5} +0.476163 q^{6} +1.00000 q^{7} +2.05440 q^{8} -2.26852 q^{9} +O(q^{10})\)	\(q-0.556743 q^{2} -0.855264 q^{3} -1.69004 q^{4} -1.00000 q^{5} +0.476163 q^{6} +1.00000 q^{7} +2.05440 q^{8} -2.26852 q^{9} +0.556743 q^{10} +3.86363 q^{11} +1.44543 q^{12} +6.16076 q^{13} -0.556743 q^{14} +0.855264 q^{15} +2.23630 q^{16} -0.562314 q^{17} +1.26299 q^{18} -7.14969 q^{19} +1.69004 q^{20} -0.855264 q^{21} -2.15105 q^{22} -7.00506 q^{23} -1.75706 q^{24} +1.00000 q^{25} -3.42996 q^{26} +4.50598 q^{27} -1.69004 q^{28} -0.249350 q^{29} -0.476163 q^{30} +2.99275 q^{31} -5.35385 q^{32} -3.30442 q^{33} +0.313065 q^{34} -1.00000 q^{35} +3.83389 q^{36} -4.66576 q^{37} +3.98054 q^{38} -5.26908 q^{39} -2.05440 q^{40} -5.67158 q^{41} +0.476163 q^{42} +8.89471 q^{43} -6.52967 q^{44} +2.26852 q^{45} +3.90002 q^{46} +3.83207 q^{47} -1.91263 q^{48} +1.00000 q^{49} -0.556743 q^{50} +0.480927 q^{51} -10.4119 q^{52} +4.47201 q^{53} -2.50867 q^{54} -3.86363 q^{55} +2.05440 q^{56} +6.11488 q^{57} +0.138824 q^{58} -4.21343 q^{59} -1.44543 q^{60} -7.41414 q^{61} -1.66619 q^{62} -2.26852 q^{63} -1.49188 q^{64} -6.16076 q^{65} +1.83971 q^{66} -1.37628 q^{67} +0.950332 q^{68} +5.99117 q^{69} +0.556743 q^{70} +1.22054 q^{71} -4.66046 q^{72} -2.45646 q^{73} +2.59763 q^{74} -0.855264 q^{75} +12.0832 q^{76} +3.86363 q^{77} +2.93352 q^{78} -10.7361 q^{79} -2.23630 q^{80} +2.95177 q^{81} +3.15761 q^{82} -4.50624 q^{83} +1.44543 q^{84} +0.562314 q^{85} -4.95207 q^{86} +0.213260 q^{87} +7.93744 q^{88} +9.13167 q^{89} -1.26299 q^{90} +6.16076 q^{91} +11.8388 q^{92} -2.55959 q^{93} -2.13348 q^{94} +7.14969 q^{95} +4.57896 q^{96} +13.0692 q^{97} -0.556743 q^{98} -8.76472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9}+O(q^{10}) \) 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9	\( 45 q - 6 q^{2} + 34 q^{4} - 45 q^{5} + q^{6} + 45 q^{7} - 15 q^{8} + 29 q^{9} + 6 q^{10} - q^{11} - 3 q^{12} - 21 q^{13} - 6 q^{14} + 8 q^{16} - 7 q^{17} - 36 q^{18} - 20 q^{19} - 34 q^{20} - 34 q^{22} - 22 q^{23} - 11 q^{24} + 45 q^{25} - q^{26} + 12 q^{27} + 34 q^{28} + 10 q^{29} - q^{30} - 27 q^{31} - 26 q^{32} - 39 q^{33} - 13 q^{34} - 45 q^{35} - 3 q^{36} - 72 q^{37} + 2 q^{38} - 37 q^{39} + 15 q^{40} - 4 q^{41} + q^{42} - 49 q^{43} + 5 q^{44} - 29 q^{45} - 67 q^{46} + 2 q^{47} + 8 q^{48} + 45 q^{49} - 6 q^{50} - 49 q^{51} - 47 q^{52} - 35 q^{53} - 12 q^{54} + q^{55} - 15 q^{56} - 77 q^{57} - 50 q^{58} + 4 q^{59} + 3 q^{60} - 36 q^{61} + 17 q^{62} + 29 q^{63} + 5 q^{64} + 21 q^{65} - 8 q^{66} - 80 q^{67} + 27 q^{68} + 9 q^{69} + 6 q^{70} - 12 q^{71} - 97 q^{72} - 55 q^{73} + 32 q^{74} - 37 q^{76} - q^{77} + 20 q^{78} - 94 q^{79} - 8 q^{80} - 19 q^{81} - 36 q^{82} + 24 q^{83} - 3 q^{84} + 7 q^{85} - 3 q^{86} - 4 q^{87} - 95 q^{88} + q^{89} + 36 q^{90} - 21 q^{91} - 65 q^{92} - 71 q^{93} - 53 q^{94} + 20 q^{95} - 13 q^{96} - 110 q^{97} - 6 q^{98} - 27 q^{99}+O(q^{100}) \) 45 * q - 6 * q^2 + 34 * q^4 - 45 * q^5 + q^6 + 45 * q^7 - 15 * q^8 + 29 * q^9 + 6 * q^10 - q^11 - 3 * q^12 - 21 * q^13 - 6 * q^14 + 8 * q^16 - 7 * q^17 - 36 * q^18 - 20 * q^19 - 34 * q^20 - 34 * q^22 - 22 * q^23 - 11 * q^24 + 45 * q^25 - q^26 + 12 * q^27 + 34 * q^28 + 10 * q^29 - q^30 - 27 * q^31 - 26 * q^32 - 39 * q^33 - 13 * q^34 - 45 * q^35 - 3 * q^36 - 72 * q^37 + 2 * q^38 - 37 * q^39 + 15 * q^40 - 4 * q^41 + q^42 - 49 * q^43 + 5 * q^44 - 29 * q^45 - 67 * q^46 + 2 * q^47 + 8 * q^48 + 45 * q^49 - 6 * q^50 - 49 * q^51 - 47 * q^52 - 35 * q^53 - 12 * q^54 + q^55 - 15 * q^56 - 77 * q^57 - 50 * q^58 + 4 * q^59 + 3 * q^60 - 36 * q^61 + 17 * q^62 + 29 * q^63 + 5 * q^64 + 21 * q^65 - 8 * q^66 - 80 * q^67 + 27 * q^68 + 9 * q^69 + 6 * q^70 - 12 * q^71 - 97 * q^72 - 55 * q^73 + 32 * q^74 - 37 * q^76 - q^77 + 20 * q^78 - 94 * q^79 - 8 * q^80 - 19 * q^81 - 36 * q^82 + 24 * q^83 - 3 * q^84 + 7 * q^85 - 3 * q^86 - 4 * q^87 - 95 * q^88 + q^89 + 36 * q^90 - 21 * q^91 - 65 * q^92 - 71 * q^93 - 53 * q^94 + 20 * q^95 - 13 * q^96 - 110 * q^97 - 6 * q^98 - 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more