LMFDB - Embedded newform 1815.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,2,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, 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("1815.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1815.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:	$14.4928479669$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.09529$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.39026 q^{2} +1.00000 q^{3} -0.0671858 q^{4} -1.00000 q^{5} -1.39026 q^{6} +1.27759 q^{7} +2.87392 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.39026 q^{2} +1.00000 q^{3} -0.0671858 q^{4} -1.00000 q^{5} -1.39026 q^{6} +1.27759 q^{7} +2.87392 q^{8} +1.00000 q^{9} +1.39026 q^{10} -0.0671858 q^{12} -1.41837 q^{13} -1.77618 q^{14} -1.00000 q^{15} -3.86111 q^{16} +5.00000 q^{17} -1.39026 q^{18} +0.158146 q^{19} +0.0671858 q^{20} +1.27759 q^{21} -5.00829 q^{23} +2.87392 q^{24} +1.00000 q^{25} +1.97189 q^{26} +1.00000 q^{27} -0.0858360 q^{28} +6.27515 q^{29} +1.39026 q^{30} +3.04981 q^{31} -0.379898 q^{32} -6.95128 q^{34} -1.27759 q^{35} -0.0671858 q^{36} -4.69991 q^{37} -0.219863 q^{38} -1.41837 q^{39} -2.87392 q^{40} +7.58597 q^{41} -1.77618 q^{42} +5.41324 q^{43} -1.00000 q^{45} +6.96281 q^{46} -8.23211 q^{47} -3.86111 q^{48} -5.36776 q^{49} -1.39026 q^{50} +5.00000 q^{51} +0.0952940 q^{52} +9.36648 q^{53} -1.39026 q^{54} +3.67169 q^{56} +0.158146 q^{57} -8.72406 q^{58} +8.09846 q^{59} +0.0671858 q^{60} -14.2917 q^{61} -4.24002 q^{62} +1.27759 q^{63} +8.25038 q^{64} +1.41837 q^{65} -7.38362 q^{67} -0.335929 q^{68} -5.00829 q^{69} +1.77618 q^{70} -6.77618 q^{71} +2.87392 q^{72} +8.66107 q^{73} +6.53409 q^{74} +1.00000 q^{75} -0.0106252 q^{76} +1.97189 q^{78} -2.54572 q^{79} +3.86111 q^{80} +1.00000 q^{81} -10.5464 q^{82} +10.1221 q^{83} -0.0858360 q^{84} -5.00000 q^{85} -7.52580 q^{86} +6.27515 q^{87} +11.0447 q^{89} +1.39026 q^{90} -1.81209 q^{91} +0.336486 q^{92} +3.04981 q^{93} +11.4447 q^{94} -0.158146 q^{95} -0.379898 q^{96} +6.41196 q^{97} +7.46257 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 5 * q^2 + 4 * q^3 + 9 * q^4 - 4 * q^5 + 5 * q^6 - 2 * q^7 + 15 * q^8 + 4 * q^9	$ 4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9} - 5 q^{10} + 9 q^{12} + 3 q^{13} - 5 q^{14} - 4 q^{15} + 15 q^{16} + 20 q^{17} + 5 q^{18} + 3 q^{19} - 9 q^{20} - 2 q^{21} - 5 q^{23} + 15 q^{24} + 4 q^{25} + 6 q^{26} + 4 q^{27} + 3 q^{28} + 5 q^{29} - 5 q^{30} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} + 9 q^{36} - 7 q^{37} + q^{38} + 3 q^{39} - 15 q^{40} + 20 q^{41} - 5 q^{42} - 2 q^{43} - 4 q^{45} + 7 q^{46} - 20 q^{47} + 15 q^{48} + 8 q^{49} + 5 q^{50} + 20 q^{51} - 7 q^{52} + 6 q^{53} + 5 q^{54} + 10 q^{56} + 3 q^{57} - 21 q^{58} - 5 q^{59} - 9 q^{60} - 7 q^{61} - 12 q^{62} - 2 q^{63} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} - 5 q^{69} + 5 q^{70} - 25 q^{71} + 15 q^{72} + 23 q^{73} - 7 q^{74} + 4 q^{75} - 7 q^{76} + 6 q^{78} - 15 q^{80} + 4 q^{81} + 11 q^{82} + 33 q^{83} + 3 q^{84} - 20 q^{85} - 12 q^{86} + 5 q^{87} + 16 q^{89} - 5 q^{90} - 24 q^{91} - q^{93} - 17 q^{94} - 3 q^{95} + 30 q^{96} + 25 q^{98}+O(q^{100}) $ 4 * q + 5 * q^2 + 4 * q^3 + 9 * q^4 - 4 * q^5 + 5 * q^6 - 2 * q^7 + 15 * q^8 + 4 * q^9 - 5 * q^10 + 9 * q^12 + 3 * q^13 - 5 * q^14 - 4 * q^15 + 15 * q^16 + 20 * q^17 + 5 * q^18 + 3 * q^19 - 9 * q^20 - 2 * q^21 - 5 * q^23 + 15 * q^24 + 4 * q^25 + 6 * q^26 + 4 * q^27 + 3 * q^28 + 5 * q^29 - 5 * q^30 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 + 9 * q^36 - 7 * q^37 + q^38 + 3 * q^39 - 15 * q^40 + 20 * q^41 - 5 * q^42 - 2 * q^43 - 4 * q^45 + 7 * q^46 - 20 * q^47 + 15 * q^48 + 8 * q^49 + 5 * q^50 + 20 * q^51 - 7 * q^52 + 6 * q^53 + 5 * q^54 + 10 * q^56 + 3 * q^57 - 21 * q^58 - 5 * q^59 - 9 * q^60 - 7 * q^61 - 12 * q^62 - 2 * q^63 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 - 5 * q^69 + 5 * q^70 - 25 * q^71 + 15 * q^72 + 23 * q^73 - 7 * q^74 + 4 * q^75 - 7 * q^76 + 6 * q^78 - 15 * q^80 + 4 * q^81 + 11 * q^82 + 33 * q^83 + 3 * q^84 - 20 * q^85 - 12 * q^86 + 5 * q^87 + 16 * q^89 - 5 * q^90 - 24 * q^91 - q^93 - 17 * q^94 - 3 * q^95 + 30 * q^96 + 25 * q^98

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 $

$2$	−1.39026	−0.983060	−0.491530	−	0.870861i	$-0.663562\pi$
$2$	−1.39026	−0.983060	−0.491530	+	0.870861i	$0.663562\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.0671858	−0.0335929
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.39026	−0.567570
$7$	1.27759	0.482884	0.241442	−	0.970415i	$-0.422380\pi$
$7$	1.27759	0.482884	0.241442	+	0.970415i	$0.422380\pi$
$8$	2.87392	1.01608
$9$	1.00000	0.333333
$10$	1.39026	0.439638
$11$	0	0
$11$	0	0
$12$	−0.0671858	−0.0193949
$13$	−1.41837	−0.393384	−0.196692	−	0.980465i	$-0.563020\pi$
$13$	−1.41837	−0.393384	−0.196692	+	0.980465i	$0.563020\pi$
$14$	−1.77618	−0.474704
$15$	−1.00000	−0.258199
$16$	−3.86111	−0.965279
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−1.39026	−0.327687
$19$	0.158146	0.0362811	0.0181406	−	0.999835i	$-0.494225\pi$
$19$	0.158146	0.0362811	0.0181406	+	0.999835i	$0.494225\pi$
$20$	0.0671858	0.0150232
$21$	1.27759	0.278793
$22$	0	0
$23$	−5.00829	−1.04430	−0.522150	−	0.852853i	$-0.674870\pi$
$23$	−5.00829	−1.04430	−0.522150	+	0.852853i	$0.674870\pi$
$24$	2.87392	0.586636
$25$	1.00000	0.200000
$26$	1.97189	0.386720
$27$	1.00000	0.192450
$28$	−0.0858360	−0.0162215
$29$	6.27515	1.16527	0.582633	−	0.812736i	$-0.302023\pi$
$29$	6.27515	1.16527	0.582633	+	0.812736i	$0.302023\pi$
$30$	1.39026	0.253825
$31$	3.04981	0.547763	0.273881	−	0.961763i	$-0.411692\pi$
$31$	3.04981	0.547763	0.273881	+	0.961763i	$0.411692\pi$
$32$	−0.379898	−0.0671570
$33$	0	0
$34$	−6.95128	−1.19214
$35$	−1.27759	−0.215952
$36$	−0.0671858	−0.0111976
$37$	−4.69991	−0.772661	−0.386330	−	0.922360i	$-0.626258\pi$
$37$	−4.69991	−0.772661	−0.386330	+	0.922360i	$0.626258\pi$
$38$	−0.219863	−0.0356665
$39$	−1.41837	−0.227120
$40$	−2.87392	−0.454407
$41$	7.58597	1.18473	0.592365	−	0.805670i	$-0.298194\pi$
$41$	7.58597	1.18473	0.592365	+	0.805670i	$0.298194\pi$
$42$	−1.77618	−0.274070
$43$	5.41324	0.825512	0.412756	−	0.910842i	$-0.364566\pi$
$43$	5.41324	0.825512	0.412756	+	0.910842i	$0.364566\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	6.96281	1.02661
$47$	−8.23211	−1.20078	−0.600388	−	0.799709i	$-0.704987\pi$
$47$	−8.23211	−1.20078	−0.600388	+	0.799709i	$0.704987\pi$
$48$	−3.86111	−0.557304
$49$	−5.36776	−0.766823
$50$	−1.39026	−0.196612
$51$	5.00000	0.700140
$52$	0.0952940	0.0132149
$53$	9.36648	1.28659	0.643293	−	0.765620i	$-0.277568\pi$
$53$	9.36648	1.28659	0.643293	+	0.765620i	$0.277568\pi$
$54$	−1.39026	−0.189190
$55$	0	0
$56$	3.67169	0.490651
$57$	0.158146	0.0209469
$58$	−8.72406	−1.14553
$59$	8.09846	1.05433	0.527165	−	0.849763i	$-0.323255\pi$
$59$	8.09846	1.05433	0.527165	+	0.849763i	$0.323255\pi$
$60$	0.0671858	0.00867365
$61$	−14.2917	−1.82987	−0.914934	−	0.403603i	$-0.867758\pi$
$61$	−14.2917	−1.82987	−0.914934	+	0.403603i	$0.867758\pi$
$62$	−4.24002	−0.538484
$63$	1.27759	0.160961
$64$	8.25038	1.03130
$65$	1.41837	0.175927
$66$	0	0
$67$	−7.38362	−0.902053	−0.451026	−	0.892511i	$-0.648942\pi$
$67$	−7.38362	−0.902053	−0.451026	+	0.892511i	$0.648942\pi$
$68$	−0.335929	−0.0407374
$69$	−5.00829	−0.602927
$70$	1.77618	0.212294
$71$	−6.77618	−0.804185	−0.402092	−	0.915599i	$-0.631717\pi$
$71$	−6.77618	−0.804185	−0.402092	+	0.915599i	$0.631717\pi$
$72$	2.87392	0.338695
$73$	8.66107	1.01370	0.506851	−	0.862034i	$-0.330810\pi$
$73$	8.66107	1.01370	0.506851	+	0.862034i	$0.330810\pi$
$74$	6.53409	0.759572
$75$	1.00000	0.115470
$76$	−0.0106252	−0.00121879
$77$	0	0
$78$	1.97189	0.223273
$79$	−2.54572	−0.286416	−0.143208	−	0.989693i	$-0.545742\pi$
$79$	−2.54572	−0.286416	−0.143208	+	0.989693i	$0.545742\pi$
$80$	3.86111	0.431686
$81$	1.00000	0.111111
$82$	−10.5464	−1.16466
$83$	10.1221	1.11105	0.555524	−	0.831501i	$-0.312518\pi$
$83$	10.1221	1.11105	0.555524	+	0.831501i	$0.312518\pi$
$84$	−0.0858360	−0.00936547
$85$	−5.00000	−0.542326
$86$	−7.52580	−0.811527
$87$	6.27515	0.672766
$88$	0	0
$89$	11.0447	1.17073	0.585367	−	0.810768i	$-0.300950\pi$
$89$	11.0447	1.17073	0.585367	+	0.810768i	$0.300950\pi$
$90$	1.39026	0.146546
$91$	−1.81209	−0.189959
$92$	0.336486	0.0350811
$93$	3.04981	0.316251
$94$	11.4447	1.18044
$95$	−0.158146	−0.0162254
$96$	−0.379898	−0.0387731
$97$	6.41196	0.651036	0.325518	−	0.945536i	$-0.394461\pi$
$97$	6.41196	0.651036	0.325518	+	0.945536i	$0.394461\pi$
$98$	7.46257	0.753833
$99$	0	0
$100$	−0.0671858	−0.00671858
$101$	8.77143	0.872790	0.436395	−	0.899755i	$-0.356255\pi$
$101$	8.77143	0.872790	0.436395	+	0.899755i	$0.356255\pi$
$102$	−6.95128	−0.688280
$103$	11.5666	1.13969	0.569847	−	0.821751i	$-0.307002\pi$
$103$	11.5666	1.13969	0.569847	+	0.821751i	$0.307002\pi$
$104$	−4.07627	−0.399711
$105$	−1.27759	−0.124680
$106$	−13.0218	−1.26479
$107$	15.3467	1.48362	0.741809	−	0.670611i	$-0.233968\pi$
$107$	15.3467	1.48362	0.741809	+	0.670611i	$0.233968\pi$
$108$	−0.0671858	−0.00646496
$109$	14.4004	1.37931	0.689656	−	0.724137i	$-0.257762\pi$
$109$	14.4004	1.37931	0.689656	+	0.724137i	$0.257762\pi$
$110$	0	0
$111$	−4.69991	−0.446096
$112$	−4.93293	−0.466118
$113$	19.3645	1.82166	0.910831	−	0.412780i	$-0.135442\pi$
$113$	19.3645	1.82166	0.910831	+	0.412780i	$0.135442\pi$
$114$	−0.219863	−0.0205921
$115$	5.00829	0.467026
$116$	−0.421601	−0.0391446
$117$	−1.41837	−0.131128
$118$	−11.2589	−1.03647
$119$	6.38796	0.585583
$120$	−2.87392	−0.262352
$121$	0	0
$122$	19.8692	1.79887
$123$	7.58597	0.684004
$124$	−0.204904	−0.0184009
$125$	−1.00000	−0.0894427
$126$	−1.77618	−0.158235
$127$	−6.02415	−0.534557	−0.267278	−	0.963619i	$-0.586124\pi$
$127$	−6.02415	−0.534557	−0.267278	+	0.963619i	$0.586124\pi$
$128$	−10.7104	−0.946671
$129$	5.41324	0.476609
$130$	−1.97189	−0.172946
$131$	−18.7278	−1.63626	−0.818130	−	0.575034i	$-0.804989\pi$
$131$	−18.7278	−1.63626	−0.818130	+	0.575034i	$0.804989\pi$
$132$	0	0
$133$	0.202046	0.0175196
$134$	10.2651	0.886772
$135$	−1.00000	−0.0860663
$136$	14.3696	1.23218
$137$	3.03719	0.259485	0.129742	−	0.991548i	$-0.458585\pi$
$137$	3.03719	0.259485	0.129742	+	0.991548i	$0.458585\pi$
$138$	6.96281	0.592714
$139$	−2.21192	−0.187612	−0.0938062	−	0.995590i	$-0.529903\pi$
$139$	−2.21192	−0.187612	−0.0938062	+	0.995590i	$0.529903\pi$
$140$	0.0858360	0.00725446
$141$	−8.23211	−0.693269
$142$	9.42063	0.790562
$143$	0	0
$144$	−3.86111	−0.321760
$145$	−6.27515	−0.521122
$146$	−12.0411	−0.996529
$147$	−5.36776	−0.442725
$148$	0.315767	0.0259559
$149$	1.48122	0.121346	0.0606730	−	0.998158i	$-0.480675\pi$
$149$	1.48122	0.121346	0.0606730	+	0.998158i	$0.480675\pi$
$150$	−1.39026	−0.113514
$151$	−9.03395	−0.735173	−0.367586	−	0.929989i	$-0.619816\pi$
$151$	−9.03395	−0.735173	−0.367586	+	0.929989i	$0.619816\pi$
$152$	0.454498	0.0368647
$153$	5.00000	0.404226
$154$	0	0
$155$	−3.04981	−0.244967
$156$	0.0952940	0.00762962
$157$	−17.6377	−1.40764	−0.703820	−	0.710379i	$-0.748524\pi$
$157$	−17.6377	−1.40764	−0.703820	+	0.710379i	$0.748524\pi$
$158$	3.53921	0.281564
$159$	9.36648	0.742810
$160$	0.379898	0.0300335
$161$	−6.39855	−0.504276
$162$	−1.39026	−0.109229
$163$	23.5040	1.84098	0.920489	−	0.390770i	$-0.127791\pi$
$163$	23.5040	1.84098	0.920489	+	0.390770i	$0.127791\pi$
$164$	−0.509669	−0.0397985
$165$	0	0
$166$	−14.0724	−1.09223
$167$	0.602731	0.0466407	0.0233203	−	0.999728i	$-0.492576\pi$
$167$	0.602731	0.0466407	0.0233203	+	0.999728i	$0.492576\pi$
$168$	3.67169	0.283277
$169$	−10.9882	−0.845249
$170$	6.95128	0.533139
$171$	0.158146	0.0120937
$172$	−0.363693	−0.0277313
$173$	9.66389	0.734732	0.367366	−	0.930076i	$-0.380260\pi$
$173$	9.66389	0.734732	0.367366	+	0.930076i	$0.380260\pi$
$174$	−8.72406	−0.661370
$175$	1.27759	0.0965768
$176$	0	0
$177$	8.09846	0.608718
$178$	−15.3550	−1.15090
$179$	−1.19571	−0.0893717	−0.0446859	−	0.999001i	$-0.514229\pi$
$179$	−1.19571	−0.0893717	−0.0446859	+	0.999001i	$0.514229\pi$
$180$	0.0671858	0.00500773
$181$	15.5741	1.15761	0.578806	−	0.815466i	$-0.303519\pi$
$181$	15.5741	1.15761	0.578806	+	0.815466i	$0.303519\pi$
$182$	2.51927	0.186741
$183$	−14.2917	−1.05647
$184$	−14.3934	−1.06110
$185$	4.69991	0.345544
$186$	−4.24002	−0.310894
$187$	0	0
$188$	0.553081	0.0403376
$189$	1.27759	0.0929311
$190$	0.219863	0.0159506
$191$	18.4016	1.33149	0.665747	−	0.746178i	$-0.268113\pi$
$191$	18.4016	1.33149	0.665747	+	0.746178i	$0.268113\pi$
$192$	8.25038	0.595420
$193$	1.56799	0.112866	0.0564331	−	0.998406i	$-0.482027\pi$
$193$	1.56799	0.112866	0.0564331	+	0.998406i	$0.482027\pi$
$194$	−8.91428	−0.640008
$195$	1.41837	0.101571
$196$	0.360637	0.0257598
$197$	7.97000	0.567839	0.283920	−	0.958848i	$-0.408365\pi$
$197$	7.97000	0.567839	0.283920	+	0.958848i	$0.408365\pi$
$198$	0	0
$199$	3.53141	0.250335	0.125167	−	0.992136i	$-0.460053\pi$
$199$	3.53141	0.250335	0.125167	+	0.992136i	$0.460053\pi$
$200$	2.87392	0.203217
$201$	−7.38362	−0.520801
$202$	−12.1945	−0.858005
$203$	8.01707	0.562688
$204$	−0.335929	−0.0235197
$205$	−7.58597	−0.529827
$206$	−16.0806	−1.12039
$207$	−5.00829	−0.348100
$208$	5.47647	0.379725
$209$	0	0
$210$	1.77618	0.122568
$211$	20.2550	1.39441	0.697207	−	0.716870i	$-0.254426\pi$
$211$	20.2550	1.39441	0.697207	+	0.716870i	$0.254426\pi$
$212$	−0.629295	−0.0432201
$213$	−6.77618	−0.464296
$214$	−21.3358	−1.45849
$215$	−5.41324	−0.369180
$216$	2.87392	0.195545
$217$	3.89642	0.264506
$218$	−20.0203	−1.35595
$219$	8.66107	0.585261
$220$	0	0
$221$	−7.09183	−0.477048
$222$	6.53409	0.438539
$223$	22.5500	1.51006	0.755029	−	0.655691i	$-0.227623\pi$
$223$	22.5500	1.51006	0.755029	+	0.655691i	$0.227623\pi$
$224$	−0.485354	−0.0324291
$225$	1.00000	0.0666667
$226$	−26.9217	−1.79080
$227$	18.8062	1.24821	0.624105	−	0.781341i	$-0.285464\pi$
$227$	18.8062	1.24821	0.624105	+	0.781341i	$0.285464\pi$
$228$	−0.0106252	−0.000703668 0
$229$	−23.8320	−1.57486	−0.787431	−	0.616403i	$-0.788589\pi$
$229$	−23.8320	−1.57486	−0.787431	+	0.616403i	$0.788589\pi$
$230$	−6.96281	−0.459114
$231$	0	0
$232$	18.0343	1.18401
$233$	−10.0918	−0.661137	−0.330569	−	0.943782i	$-0.607241\pi$
$233$	−10.0918	−0.661137	−0.330569	+	0.943782i	$0.607241\pi$
$234$	1.97189	0.128907
$235$	8.23211	0.537004
$236$	−0.544102	−0.0354180
$237$	−2.54572	−0.165363
$238$	−8.88090	−0.575663
$239$	0.167227	0.0108170	0.00540850	−	0.999985i	$-0.498278\pi$
$239$	0.167227	0.0108170	0.00540850	+	0.999985i	$0.498278\pi$
$240$	3.86111	0.249234
$241$	0.965256	0.0621776	0.0310888	−	0.999517i	$-0.490103\pi$
$241$	0.965256	0.0621776	0.0310888	+	0.999517i	$0.490103\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0.960201	0.0614706
$245$	5.36776	0.342934
$246$	−10.5464	−0.672417
$247$	−0.224308	−0.0142724
$248$	8.76492	0.556573
$249$	10.1221	0.641464
$250$	1.39026	0.0879276
$251$	−23.5102	−1.48395	−0.741975	−	0.670428i	$-0.766111\pi$
$251$	−23.5102	−1.48395	−0.741975	+	0.670428i	$0.766111\pi$
$252$	−0.0858360	−0.00540716
$253$	0	0
$254$	8.37512	0.525502
$255$	−5.00000	−0.313112
$256$	−1.61062	−0.100664
$257$	−6.58273	−0.410620	−0.205310	−	0.978697i	$-0.565820\pi$
$257$	−6.58273	−0.410620	−0.205310	+	0.978697i	$0.565820\pi$
$258$	−7.52580	−0.468536
$259$	−6.00457	−0.373106
$260$	−0.0952940	−0.00590988
$261$	6.27515	0.388422
$262$	26.0365	1.60854
$263$	12.4538	0.767936	0.383968	−	0.923346i	$-0.374557\pi$
$263$	12.4538	0.767936	0.383968	+	0.923346i	$0.374557\pi$
$264$	0	0
$265$	−9.36648	−0.575378
$266$	−0.280895	−0.0172228
$267$	11.0447	0.675924
$268$	0.496074	0.0303026
$269$	20.7729	1.26655	0.633274	−	0.773927i	$-0.281710\pi$
$269$	20.7729	1.26655	0.633274	+	0.773927i	$0.281710\pi$
$270$	1.39026	0.0846083
$271$	14.3903	0.874146	0.437073	−	0.899426i	$-0.356015\pi$
$271$	14.3903	0.874146	0.437073	+	0.899426i	$0.356015\pi$
$272$	−19.3056	−1.17057
$273$	−1.81209	−0.109673
$274$	−4.22247	−0.255089
$275$	0	0
$276$	0.336486	0.0202541
$277$	−17.4872	−1.05070	−0.525352	−	0.850885i	$-0.676066\pi$
$277$	−17.4872	−1.05070	−0.525352	+	0.850885i	$0.676066\pi$
$278$	3.07513	0.184434
$279$	3.04981	0.182588
$280$	−3.67169	−0.219426
$281$	−18.8860	−1.12664	−0.563322	−	0.826238i	$-0.690477\pi$
$281$	−18.8860	−1.12664	−0.563322	+	0.826238i	$0.690477\pi$
$282$	11.4447	0.681525
$283$	−13.5813	−0.807326	−0.403663	−	0.914908i	$-0.632263\pi$
$283$	−13.5813	−0.807326	−0.403663	+	0.914908i	$0.632263\pi$
$284$	0.455263	0.0270149
$285$	−0.158146	−0.00936775
$286$	0	0
$287$	9.69177	0.572087
$288$	−0.379898	−0.0223857
$289$	8.00000	0.470588
$290$	8.72406	0.512295
$291$	6.41196	0.375876
$292$	−0.581901	−0.0340532
$293$	19.9023	1.16271	0.581353	−	0.813651i	$-0.302523\pi$
$293$	19.9023	1.16271	0.581353	+	0.813651i	$0.302523\pi$
$294$	7.46257	0.435226
$295$	−8.09846	−0.471511
$296$	−13.5072	−0.785088
$297$	0	0
$298$	−2.05927	−0.119290
$299$	7.10358	0.410811
$300$	−0.0671858	−0.00387897
$301$	6.91591	0.398626
$302$	12.5595	0.722719
$303$	8.77143	0.503906
$304$	−0.610619	−0.0350214
$305$	14.2917	0.818342
$306$	−6.95128	−0.397378
$307$	−14.9354	−0.852410	−0.426205	−	0.904627i	$-0.640150\pi$
$307$	−14.9354	−0.852410	−0.426205	+	0.904627i	$0.640150\pi$
$308$	0	0
$309$	11.5666	0.658003
$310$	4.24002	0.240817
$311$	−4.99700	−0.283354	−0.141677	−	0.989913i	$-0.545249\pi$
$311$	−4.99700	−0.283354	−0.141677	+	0.989913i	$0.545249\pi$
$312$	−4.07627	−0.230773
$313$	9.39648	0.531120	0.265560	−	0.964094i	$-0.414443\pi$
$313$	9.39648	0.531120	0.265560	+	0.964094i	$0.414443\pi$
$314$	24.5209	1.38379
$315$	−1.27759	−0.0719841
$316$	0.171037	0.00962155
$317$	2.15253	0.120898	0.0604492	−	0.998171i	$-0.480747\pi$
$317$	2.15253	0.120898	0.0604492	+	0.998171i	$0.480747\pi$
$318$	−13.0218	−0.730227
$319$	0	0
$320$	−8.25038	−0.461210
$321$	15.3467	0.856567
$322$	8.89563	0.495734
$323$	0.790729	0.0439973
$324$	−0.0671858	−0.00373254
$325$	−1.41837	−0.0786767
$326$	−32.6766	−1.80979
$327$	14.4004	0.796346
$328$	21.8015	1.20378
$329$	−10.5173	−0.579836
$330$	0	0
$331$	−19.5116	−1.07245	−0.536227	−	0.844074i	$-0.680151\pi$
$331$	−19.5116	−1.07245	−0.536227	+	0.844074i	$0.680151\pi$
$332$	−0.680063	−0.0373233
$333$	−4.69991	−0.257554
$334$	−0.837950	−0.0458506
$335$	7.38362	0.403410
$336$	−4.93293	−0.269113
$337$	31.6047	1.72162	0.860809	−	0.508929i	$-0.169958\pi$
$337$	31.6047	1.72162	0.860809	+	0.508929i	$0.169958\pi$
$338$	15.2765	0.830931
$339$	19.3645	1.05174
$340$	0.335929	0.0182183
$341$	0	0
$342$	−0.219863	−0.0118888
$343$	−15.8009	−0.853171
$344$	15.5572	0.838789
$345$	5.00829	0.269637
$346$	−13.4353	−0.722286
$347$	9.38313	0.503713	0.251856	−	0.967765i	$-0.418959\pi$
$347$	9.38313	0.503713	0.251856	+	0.967765i	$0.418959\pi$
$348$	−0.421601	−0.0226002
$349$	−24.8131	−1.32822	−0.664108	−	0.747637i	$-0.731188\pi$
$349$	−24.8131	−1.32822	−0.664108	+	0.747637i	$0.731188\pi$
$350$	−1.77618	−0.0949408
$351$	−1.41837	−0.0757067
$352$	0	0
$353$	−19.4788	−1.03675	−0.518375	−	0.855153i	$-0.673463\pi$
$353$	−19.4788	−1.03675	−0.518375	+	0.855153i	$0.673463\pi$
$354$	−11.2589	−0.598406
$355$	6.77618	0.359642
$356$	−0.742046	−0.0393284
$357$	6.38796	0.338086
$358$	1.66235	0.0878578
$359$	29.7701	1.57121	0.785603	−	0.618732i	$-0.212353\pi$
$359$	29.7701	1.57121	0.785603	+	0.618732i	$0.212353\pi$
$360$	−2.87392	−0.151469
$361$	−18.9750	−0.998684
$362$	−21.6520	−1.13800
$363$	0	0
$364$	0.121747	0.00638126
$365$	−8.66107	−0.453341
$366$	19.8692	1.03858
$367$	−6.43597	−0.335955	−0.167977	−	0.985791i	$-0.553724\pi$
$367$	−6.43597	−0.335955	−0.167977	+	0.985791i	$0.553724\pi$
$368$	19.3376	1.00804
$369$	7.58597	0.394910
$370$	−6.53409	−0.339691
$371$	11.9665	0.621272
$372$	−0.204904	−0.0106238
$373$	−20.8924	−1.08177	−0.540883	−	0.841098i	$-0.681910\pi$
$373$	−20.8924	−1.08177	−0.540883	+	0.841098i	$0.681910\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−23.6584	−1.22009
$377$	−8.90045	−0.458396
$378$	−1.77618	−0.0913568
$379$	−14.1991	−0.729359	−0.364680	−	0.931133i	$-0.618822\pi$
$379$	−14.1991	−0.729359	−0.364680	+	0.931133i	$0.618822\pi$
$380$	0.0106252	0.000545059 0
$381$	−6.02415	−0.308627
$382$	−25.5830	−1.30894
$383$	−26.4783	−1.35298	−0.676489	−	0.736453i	$-0.736499\pi$
$383$	−26.4783	−1.35298	−0.676489	+	0.736453i	$0.736499\pi$
$384$	−10.7104	−0.546561
$385$	0	0
$386$	−2.17990	−0.110954
$387$	5.41324	0.275171
$388$	−0.430793	−0.0218702
$389$	−36.4965	−1.85044	−0.925222	−	0.379427i	$-0.876121\pi$
$389$	−36.4965	−1.85044	−0.925222	+	0.379427i	$0.876121\pi$
$390$	−1.97189	−0.0998506
$391$	−25.0415	−1.26640
$392$	−15.4265	−0.779157
$393$	−18.7278	−0.944695
$394$	−11.0804	−0.558220
$395$	2.54572	0.128089
$396$	0	0
$397$	−8.03969	−0.403500	−0.201750	−	0.979437i	$-0.564663\pi$
$397$	−8.03969	−0.403500	−0.201750	+	0.979437i	$0.564663\pi$
$398$	−4.90956	−0.246094
$399$	0.202046	0.0101149
$400$	−3.86111	−0.193056
$401$	−28.0902	−1.40276	−0.701379	−	0.712788i	$-0.747432\pi$
$401$	−28.0902	−1.40276	−0.701379	+	0.712788i	$0.747432\pi$
$402$	10.2651	0.511978
$403$	−4.32575	−0.215481
$404$	−0.589316	−0.0293196
$405$	−1.00000	−0.0496904
$406$	−11.1458	−0.553156
$407$	0	0
$408$	14.3696	0.711401
$409$	−5.97950	−0.295667	−0.147834	−	0.989012i	$-0.547230\pi$
$409$	−5.97950	−0.295667	−0.147834	+	0.989012i	$0.547230\pi$
$410$	10.5464	0.520852
$411$	3.03719	0.149813
$412$	−0.777114	−0.0382857
$413$	10.3465	0.509119
$414$	6.96281	0.342203
$415$	−10.1221	−0.496876
$416$	0.538833	0.0264185
$417$	−2.21192	−0.108318
$418$	0	0
$419$	−15.4707	−0.755795	−0.377897	−	0.925847i	$-0.623353\pi$
$419$	−15.4707	−0.755795	−0.377897	+	0.925847i	$0.623353\pi$
$420$	0.0858360	0.00418837
$421$	34.5746	1.68506	0.842532	−	0.538647i	$-0.181064\pi$
$421$	34.5746	1.68506	0.842532	+	0.538647i	$0.181064\pi$
$422$	−28.1597	−1.37079
$423$	−8.23211	−0.400259
$424$	26.9185	1.30728
$425$	5.00000	0.242536
$426$	9.42063	0.456431
$427$	−18.2590	−0.883614
$428$	−1.03108	−0.0498390
$429$	0	0
$430$	7.52580	0.362926
$431$	3.36052	0.161870	0.0809352	−	0.996719i	$-0.474209\pi$
$431$	3.36052	0.161870	0.0809352	+	0.996719i	$0.474209\pi$
$432$	−3.86111	−0.185768
$433$	3.67779	0.176744	0.0883718	−	0.996088i	$-0.471834\pi$
$433$	3.67779	0.176744	0.0883718	+	0.996088i	$0.471834\pi$
$434$	−5.41702	−0.260025
$435$	−6.27515	−0.300870
$436$	−0.967505	−0.0463351
$437$	−0.792040	−0.0378884
$438$	−12.0411	−0.575346
$439$	−1.55432	−0.0741835	−0.0370917	−	0.999312i	$-0.511809\pi$
$439$	−1.55432	−0.0741835	−0.0370917	+	0.999312i	$0.511809\pi$
$440$	0	0
$441$	−5.36776	−0.255608
$442$	9.85946	0.468967
$443$	21.2059	1.00752	0.503761	−	0.863843i	$-0.331949\pi$
$443$	21.2059	1.00752	0.503761	+	0.863843i	$0.331949\pi$
$444$	0.315767	0.0149857
$445$	−11.0447	−0.523569
$446$	−31.3503	−1.48448
$447$	1.48122	0.0700592
$448$	10.5406	0.497997
$449$	15.1106	0.713113	0.356557	−	0.934274i	$-0.383951\pi$
$449$	15.1106	0.713113	0.356557	+	0.934274i	$0.383951\pi$
$450$	−1.39026	−0.0655373
$451$	0	0
$452$	−1.30102	−0.0611949
$453$	−9.03395	−0.424452
$454$	−26.1454	−1.22707
$455$	1.81209	0.0849521
$456$	0.454498	0.0212838
$457$	41.3042	1.93213	0.966064	−	0.258303i	$-0.0831634\pi$
$457$	41.3042	1.93213	0.966064	+	0.258303i	$0.0831634\pi$
$458$	33.1326	1.54818
$459$	5.00000	0.233380
$460$	−0.336486	−0.0156887
$461$	−16.3158	−0.759901	−0.379951	−	0.925007i	$-0.624059\pi$
$461$	−16.3158	−0.759901	−0.379951	+	0.925007i	$0.624059\pi$
$462$	0	0
$463$	8.47904	0.394054	0.197027	−	0.980398i	$-0.436871\pi$
$463$	8.47904	0.394054	0.197027	+	0.980398i	$0.436871\pi$
$464$	−24.2291	−1.12481
$465$	−3.04981	−0.141432
$466$	14.0302	0.649938
$467$	−36.5669	−1.69212	−0.846058	−	0.533090i	$-0.821031\pi$
$467$	−36.5669	−1.69212	−0.846058	+	0.533090i	$0.821031\pi$
$468$	0.0952940	0.00440497
$469$	−9.43325	−0.435587
$470$	−11.4447	−0.527907
$471$	−17.6377	−0.812701
$472$	23.2743	1.07129
$473$	0	0
$474$	3.53921	0.162561
$475$	0.158146	0.00725623
$476$	−0.429180	−0.0196714
$477$	9.36648	0.428862
$478$	−0.232488	−0.0106338
$479$	−38.9513	−1.77973	−0.889866	−	0.456222i	$-0.849202\pi$
$479$	−38.9513	−1.77973	−0.889866	+	0.456222i	$0.849202\pi$
$480$	0.379898	0.0173399
$481$	6.66619	0.303952
$482$	−1.34195	−0.0611243
$483$	−6.39855	−0.291144
$484$	0	0
$485$	−6.41196	−0.291152
$486$	−1.39026	−0.0630633
$487$	8.27209	0.374844	0.187422	−	0.982279i	$-0.439987\pi$
$487$	8.27209	0.374844	0.187422	+	0.982279i	$0.439987\pi$
$488$	−41.0733	−1.85930
$489$	23.5040	1.06289
$490$	−7.46257	−0.337124
$491$	−12.1322	−0.547518	−0.273759	−	0.961798i	$-0.588267\pi$
$491$	−12.1322	−0.547518	−0.273759	+	0.961798i	$0.588267\pi$
$492$	−0.509669	−0.0229777
$493$	31.3757	1.41309
$494$	0.311846	0.0140306
$495$	0	0
$496$	−11.7757	−0.528744
$497$	−8.65719	−0.388328
$498$	−14.0724	−0.630597
$499$	−28.3581	−1.26948	−0.634740	−	0.772725i	$-0.718893\pi$
$499$	−28.3581	−1.26948	−0.634740	+	0.772725i	$0.718893\pi$
$500$	0.0671858	0.00300464
$501$	0.602731	0.0269280
$502$	32.6852	1.45881
$503$	−15.2800	−0.681300	−0.340650	−	0.940190i	$-0.610647\pi$
$503$	−15.2800	−0.681300	−0.340650	+	0.940190i	$0.610647\pi$
$504$	3.67169	0.163550
$505$	−8.77143	−0.390324
$506$	0	0
$507$	−10.9882	−0.488005
$508$	0.404737	0.0179573
$509$	−22.0494	−0.977321	−0.488661	−	0.872474i	$-0.662514\pi$
$509$	−22.0494	−0.977321	−0.488661	+	0.872474i	$0.662514\pi$
$510$	6.95128	0.307808
$511$	11.0653	0.489500
$512$	23.6599	1.04563
$513$	0.158146	0.00698231
$514$	9.15169	0.403664
$515$	−11.5666	−0.509687
$516$	−0.363693	−0.0160107
$517$	0	0
$518$	8.34789	0.366785
$519$	9.66389	0.424198
$520$	4.07627	0.178756
$521$	−23.9380	−1.04874	−0.524371	−	0.851490i	$-0.675700\pi$
$521$	−23.9380	−1.04874	−0.524371	+	0.851490i	$0.675700\pi$
$522$	−8.72406	−0.381842
$523$	−25.3421	−1.10813	−0.554066	−	0.832473i	$-0.686925\pi$
$523$	−25.3421	−1.10813	−0.554066	+	0.832473i	$0.686925\pi$
$524$	1.25824	0.0549667
$525$	1.27759	0.0557586
$526$	−17.3140	−0.754927
$527$	15.2491	0.664260
$528$	0	0
$529$	2.08298	0.0905642
$530$	13.0218	0.565632
$531$	8.09846	0.351443
$532$	−0.0135746	−0.000588534 0
$533$	−10.7597	−0.466053
$534$	−15.3550	−0.664474
$535$	−15.3467	−0.663494
$536$	−21.2199	−0.916561
$537$	−1.19571	−0.0515988
$538$	−28.8797	−1.24509
$539$	0	0
$540$	0.0671858	0.00289122
$541$	−35.0634	−1.50749	−0.753747	−	0.657165i	$-0.771755\pi$
$541$	−35.0634	−1.50749	−0.753747	+	0.657165i	$0.771755\pi$
$542$	−20.0062	−0.859338
$543$	15.5741	0.668347
$544$	−1.89949	−0.0814399
$545$	−14.4004	−0.616847
$546$	2.51927	0.107815
$547$	−1.10787	−0.0473689	−0.0236845	−	0.999719i	$-0.507540\pi$
$547$	−1.10787	−0.0473689	−0.0236845	+	0.999719i	$0.507540\pi$
$548$	−0.204056	−0.00871684
$549$	−14.2917	−0.609956
$550$	0	0
$551$	0.992388	0.0422771
$552$	−14.3934	−0.612625
$553$	−3.25239	−0.138306
$554$	24.3117	1.03291
$555$	4.69991	0.199500
$556$	0.148609	0.00630244
$557$	−5.41988	−0.229648	−0.114824	−	0.993386i	$-0.536630\pi$
$557$	−5.41988	−0.229648	−0.114824	+	0.993386i	$0.536630\pi$
$558$	−4.24002	−0.179495
$559$	−7.67795	−0.324743
$560$	4.93293	0.208454
$561$	0	0
$562$	26.2564	1.10756
$563$	32.7448	1.38003	0.690015	−	0.723795i	$-0.257604\pi$
$563$	32.7448	1.38003	0.690015	+	0.723795i	$0.257604\pi$
$564$	0.553081	0.0232889
$565$	−19.3645	−0.814672
$566$	18.8815	0.793650
$567$	1.27759	0.0536538
$568$	−19.4742	−0.817119
$569$	−3.51736	−0.147455	−0.0737277	−	0.997278i	$-0.523490\pi$
$569$	−3.51736	−0.147455	−0.0737277	+	0.997278i	$0.523490\pi$
$570$	0.219863	0.00920906
$571$	−36.0252	−1.50761	−0.753804	−	0.657099i	$-0.771783\pi$
$571$	−36.0252	−1.50761	−0.753804	+	0.657099i	$0.771783\pi$
$572$	0	0
$573$	18.4016	0.768738
$574$	−13.4740	−0.562396
$575$	−5.00829	−0.208860
$576$	8.25038	0.343766
$577$	15.5501	0.647357	0.323679	−	0.946167i	$-0.395080\pi$
$577$	15.5501	0.647357	0.323679	+	0.946167i	$0.395080\pi$
$578$	−11.1221	−0.462617
$579$	1.56799	0.0651633
$580$	0.421601	0.0175060
$581$	12.9319	0.536507
$582$	−8.91428	−0.369509
$583$	0	0
$584$	24.8912	1.03001
$585$	1.41837	0.0586422
$586$	−27.6694	−1.14301
$587$	−24.2604	−1.00133	−0.500667	−	0.865640i	$-0.666912\pi$
$587$	−24.2604	−1.00133	−0.500667	+	0.865640i	$0.666912\pi$
$588$	0.360637	0.0148724
$589$	0.482315	0.0198735
$590$	11.2589	0.463523
$591$	7.97000	0.327842
$592$	18.1469	0.745833
$593$	27.4019	1.12526	0.562630	−	0.826709i	$-0.309790\pi$
$593$	27.4019	1.12526	0.562630	+	0.826709i	$0.309790\pi$
$594$	0	0
$595$	−6.38796	−0.261881
$596$	−0.0995167	−0.00407636
$597$	3.53141	0.144531
$598$	−9.87581	−0.403852
$599$	10.6773	0.436262	0.218131	−	0.975920i	$-0.430004\pi$
$599$	10.6773	0.436262	0.218131	+	0.975920i	$0.430004\pi$
$600$	2.87392	0.117327
$601$	11.0663	0.451403	0.225701	−	0.974197i	$-0.427533\pi$
$601$	11.0663	0.451403	0.225701	+	0.974197i	$0.427533\pi$
$602$	−9.61489	−0.391874
$603$	−7.38362	−0.300684
$604$	0.606953	0.0246966
$605$	0	0
$606$	−12.1945	−0.495370
$607$	−0.514518	−0.0208836	−0.0104418	−	0.999945i	$-0.503324\pi$
$607$	−0.514518	−0.0208836	−0.0104418	+	0.999945i	$0.503324\pi$
$608$	−0.0600792	−0.00243653
$609$	8.01707	0.324868
$610$	−19.8692	−0.804479
$611$	11.6761	0.472366
$612$	−0.335929	−0.0135791
$613$	19.0977	0.771348	0.385674	−	0.922635i	$-0.373969\pi$
$613$	19.0977	0.771348	0.385674	+	0.922635i	$0.373969\pi$
$614$	20.7641	0.837970
$615$	−7.58597	−0.305896
$616$	0	0
$617$	−4.70745	−0.189515	−0.0947573	−	0.995500i	$-0.530208\pi$
$617$	−4.70745	−0.189515	−0.0947573	+	0.995500i	$0.530208\pi$
$618$	−16.0806	−0.646857
$619$	37.3691	1.50199	0.750996	−	0.660307i	$-0.229574\pi$
$619$	37.3691	1.50199	0.750996	+	0.660307i	$0.229574\pi$
$620$	0.204904	0.00822915
$621$	−5.00829	−0.200976
$622$	6.94711	0.278554
$623$	14.1106	0.565329
$624$	5.47647	0.219234
$625$	1.00000	0.0400000
$626$	−13.0635	−0.522123
$627$	0	0
$628$	1.18500	0.0472867
$629$	−23.4996	−0.936989
$630$	1.77618	0.0707647
$631$	35.5669	1.41590	0.707949	−	0.706264i	$-0.249621\pi$
$631$	35.5669	1.41590	0.707949	+	0.706264i	$0.249621\pi$
$632$	−7.31621	−0.291023
$633$	20.2550	0.805065
$634$	−2.99257	−0.118850
$635$	6.02415	0.239061
$636$	−0.629295	−0.0249532
$637$	7.61344	0.301656
$638$	0	0
$639$	−6.77618	−0.268062
$640$	10.7104	0.423364
$641$	12.2098	0.482260	0.241130	−	0.970493i	$-0.422482\pi$
$641$	12.2098	0.482260	0.241130	+	0.970493i	$0.422482\pi$
$642$	−21.3358	−0.842057
$643$	−20.9596	−0.826565	−0.413282	−	0.910603i	$-0.635618\pi$
$643$	−20.9596	−0.826565	−0.413282	+	0.910603i	$0.635618\pi$
$644$	0.429892	0.0169401
$645$	−5.41324	−0.213146
$646$	−1.09932	−0.0432520
$647$	14.8766	0.584859	0.292430	−	0.956287i	$-0.405536\pi$
$647$	14.8766	0.584859	0.292430	+	0.956287i	$0.405536\pi$
$648$	2.87392	0.112898
$649$	0	0
$650$	1.97189	0.0773440
$651$	3.89642	0.152713
$652$	−1.57914	−0.0618438
$653$	−12.1308	−0.474713	−0.237357	−	0.971423i	$-0.576281\pi$
$653$	−12.1308	−0.474713	−0.237357	+	0.971423i	$0.576281\pi$
$654$	−20.0203	−0.782856
$655$	18.7278	0.731757
$656$	−29.2903	−1.14359
$657$	8.66107	0.337900
$658$	14.6217	0.570014
$659$	7.49994	0.292156	0.146078	−	0.989273i	$-0.453335\pi$
$659$	7.49994	0.292156	0.146078	+	0.989273i	$0.453335\pi$
$660$	0	0
$661$	9.65248	0.375438	0.187719	−	0.982223i	$-0.439891\pi$
$661$	9.65248	0.375438	0.187719	+	0.982223i	$0.439891\pi$
$662$	27.1261	1.05429
$663$	−7.09183	−0.275424
$664$	29.0902	1.12892
$665$	−0.202046	−0.00783500
$666$	6.53409	0.253191
$667$	−31.4278	−1.21689
$668$	−0.0404949	−0.00156680
$669$	22.5500	0.871833
$670$	−10.2651	−0.396577
$671$	0	0
$672$	−0.485354	−0.0187229
$673$	24.4496	0.942463	0.471231	−	0.882010i	$-0.343810\pi$
$673$	24.4496	0.942463	0.471231	+	0.882010i	$0.343810\pi$
$674$	−43.9386	−1.69245
$675$	1.00000	0.0384900
$676$	0.738254	0.0283944
$677$	46.7394	1.79634	0.898171	−	0.439646i	$-0.144896\pi$
$677$	46.7394	1.79634	0.898171	+	0.439646i	$0.144896\pi$
$678$	−26.9217	−1.03392
$679$	8.19187	0.314375
$680$	−14.3696	−0.551049
$681$	18.8062	0.720654
$682$	0	0
$683$	−28.1941	−1.07882	−0.539408	−	0.842045i	$-0.681352\pi$
$683$	−28.1941	−1.07882	−0.539408	+	0.842045i	$0.681352\pi$
$684$	−0.0106252	−0.000406263 0
$685$	−3.03719	−0.116045
$686$	21.9674	0.838718
$687$	−23.8320	−0.909247
$688$	−20.9011	−0.796849
$689$	−13.2851	−0.506122
$690$	−6.96281	−0.265070
$691$	11.5971	0.441175	0.220587	−	0.975367i	$-0.429203\pi$
$691$	11.5971	0.441175	0.220587	+	0.975367i	$0.429203\pi$
$692$	−0.649276	−0.0246818
$693$	0	0
$694$	−13.0450	−0.495180
$695$	2.21192	0.0839028
$696$	18.0343	0.683587
$697$	37.9298	1.43670
$698$	34.4966	1.30572
$699$	−10.0918	−0.381708
$700$	−0.0858360	−0.00324429
$701$	−36.7019	−1.38621	−0.693107	−	0.720835i	$-0.743759\pi$
$701$	−36.7019	−1.38621	−0.693107	+	0.720835i	$0.743759\pi$
$702$	1.97189	0.0744243
$703$	−0.743272	−0.0280330
$704$	0	0
$705$	8.23211	0.310039
$706$	27.0805	1.01919
$707$	11.2063	0.421456
$708$	−0.544102	−0.0204486
$709$	36.4905	1.37043	0.685214	−	0.728341i	$-0.259708\pi$
$709$	36.4905	1.37043	0.685214	+	0.728341i	$0.259708\pi$
$710$	−9.42063	−0.353550
$711$	−2.54572	−0.0954721
$712$	31.7415	1.18956
$713$	−15.2744	−0.572029
$714$	−8.88090	−0.332359
$715$	0	0
$716$	0.0803349	0.00300225
$717$	0.167227	0.00624520
$718$	−41.3881	−1.54459
$719$	−9.32888	−0.347909	−0.173954	−	0.984754i	$-0.555654\pi$
$719$	−9.32888	−0.347909	−0.173954	+	0.984754i	$0.555654\pi$
$720$	3.86111	0.143895
$721$	14.7774	0.550340
$722$	26.3801	0.981766
$723$	0.965256	0.0358983
$724$	−1.04636	−0.0388875
$725$	6.27515	0.233053
$726$	0	0
$727$	−8.46883	−0.314091	−0.157046	−	0.987591i	$-0.550197\pi$
$727$	−8.46883	−0.314091	−0.157046	+	0.987591i	$0.550197\pi$
$728$	−5.20780	−0.193014
$729$	1.00000	0.0370370
$730$	12.0411	0.445661
$731$	27.0662	1.00108
$732$	0.960201	0.0354901
$733$	29.0470	1.07288	0.536438	−	0.843940i	$-0.319770\pi$
$733$	29.0470	1.07288	0.536438	+	0.843940i	$0.319770\pi$
$734$	8.94765	0.330264
$735$	5.36776	0.197993
$736$	1.90264	0.0701322
$737$	0	0
$738$	−10.5464	−0.388220
$739$	−13.7551	−0.505990	−0.252995	−	0.967468i	$-0.581416\pi$
$739$	−13.7551	−0.505990	−0.252995	+	0.967468i	$0.581416\pi$
$740$	−0.315767	−0.0116078
$741$	−0.224308	−0.00824018
$742$	−16.6366	−0.610747
$743$	−34.9135	−1.28085	−0.640427	−	0.768019i	$-0.721242\pi$
$743$	−34.9135	−1.28085	−0.640427	+	0.768019i	$0.721242\pi$
$744$	8.76492	0.321338
$745$	−1.48122	−0.0542676
$746$	29.0458	1.06344
$747$	10.1221	0.370349
$748$	0	0
$749$	19.6068	0.716416
$750$	1.39026	0.0507650
$751$	22.0992	0.806412	0.403206	−	0.915109i	$-0.367896\pi$
$751$	22.0992	0.806412	0.403206	+	0.915109i	$0.367896\pi$
$752$	31.7851	1.15908
$753$	−23.5102	−0.856759
$754$	12.3739	0.450631
$755$	9.03395	0.328779
$756$	−0.0858360	−0.00312182
$757$	−45.0300	−1.63664	−0.818321	−	0.574761i	$-0.805095\pi$
$757$	−45.0300	−1.63664	−0.818321	+	0.574761i	$0.805095\pi$
$758$	19.7404	0.717004
$759$	0	0
$760$	−0.454498	−0.0164864
$761$	33.6001	1.21800	0.609001	−	0.793169i	$-0.291570\pi$
$761$	33.6001	1.21800	0.609001	+	0.793169i	$0.291570\pi$
$762$	8.37512	0.303399
$763$	18.3979	0.666048
$764$	−1.23633	−0.0447287
$765$	−5.00000	−0.180775
$766$	36.8116	1.33006
$767$	−11.4866	−0.414756
$768$	−1.61062	−0.0581182
$769$	24.6086	0.887408	0.443704	−	0.896173i	$-0.353664\pi$
$769$	24.6086	0.887408	0.443704	+	0.896173i	$0.353664\pi$
$770$	0	0
$771$	−6.58273	−0.237071
$772$	−0.105346	−0.00379150
$773$	−23.6816	−0.851768	−0.425884	−	0.904778i	$-0.640037\pi$
$773$	−23.6816	−0.851768	−0.425884	+	0.904778i	$0.640037\pi$
$774$	−7.52580	−0.270509
$775$	3.04981	0.109553
$776$	18.4275	0.661507
$777$	−6.00457	−0.215413
$778$	50.7394	1.81910
$779$	1.19969	0.0429833
$780$	−0.0952940	−0.00341207
$781$	0	0
$782$	34.8141	1.24495
$783$	6.27515	0.224255
$784$	20.7255	0.740198
$785$	17.6377	0.629515
$786$	26.0365	0.928692
$787$	−18.4210	−0.656639	−0.328320	−	0.944567i	$-0.606482\pi$
$787$	−18.4210	−0.656639	−0.328320	+	0.944567i	$0.606482\pi$
$788$	−0.535471	−0.0190754
$789$	12.4538	0.443368
$790$	−3.53921	−0.125919
$791$	24.7399	0.879651
$792$	0	0
$793$	20.2709	0.719840
$794$	11.1772	0.396665
$795$	−9.36648	−0.332195
$796$	−0.237260	−0.00840947
$797$	−6.50665	−0.230477	−0.115239	−	0.993338i	$-0.536763\pi$
$797$	−6.50665	−0.230477	−0.115239	+	0.993338i	$0.536763\pi$
$798$	−0.280895	−0.00994359
$799$	−41.1606	−1.45616
$800$	−0.379898	−0.0134314
$801$	11.0447	0.390245
$802$	39.0526	1.37900
$803$	0	0
$804$	0.496074	0.0174952
$805$	6.39855	0.225519
$806$	6.01390	0.211831
$807$	20.7729	0.731242
$808$	25.2084	0.886828
$809$	11.8505	0.416642	0.208321	−	0.978060i	$-0.433200\pi$
$809$	11.8505	0.416642	0.208321	+	0.978060i	$0.433200\pi$
$810$	1.39026	0.0488486
$811$	7.51391	0.263849	0.131925	−	0.991260i	$-0.457884\pi$
$811$	7.51391	0.263849	0.131925	+	0.991260i	$0.457884\pi$
$812$	−0.538633	−0.0189023
$813$	14.3903	0.504688
$814$	0	0
$815$	−23.5040	−0.823310
$816$	−19.3056	−0.675830
$817$	0.856081	0.0299505
$818$	8.31305	0.290659
$819$	−1.81209	−0.0633196
$820$	0.509669	0.0177984
$821$	−37.3365	−1.30305	−0.651527	−	0.758625i	$-0.725871\pi$
$821$	−37.3365	−1.30305	−0.651527	+	0.758625i	$0.725871\pi$
$822$	−4.22247	−0.147276
$823$	−43.0498	−1.50062	−0.750311	−	0.661085i	$-0.770096\pi$
$823$	−43.0498	−1.50062	−0.750311	+	0.661085i	$0.770096\pi$
$824$	33.2416	1.15803
$825$	0	0
$826$	−14.3843	−0.500495
$827$	−20.0304	−0.696524	−0.348262	−	0.937397i	$-0.613228\pi$
$827$	−20.0304	−0.696524	−0.348262	+	0.937397i	$0.613228\pi$
$828$	0.336486	0.0116937
$829$	−47.9207	−1.66435	−0.832177	−	0.554510i	$-0.812906\pi$
$829$	−47.9207	−1.66435	−0.832177	+	0.554510i	$0.812906\pi$
$830$	14.0724	0.488458
$831$	−17.4872	−0.606624
$832$	−11.7021	−0.405696
$833$	−26.8388	−0.929909
$834$	3.07513	0.106483
$835$	−0.602731	−0.0208584
$836$	0	0
$837$	3.04981	0.105417
$838$	21.5083	0.742992
$839$	2.22093	0.0766750	0.0383375	−	0.999265i	$-0.487794\pi$
$839$	2.22093	0.0766750	0.0383375	+	0.999265i	$0.487794\pi$
$840$	−3.67169	−0.126685
$841$	10.3775	0.357843
$842$	−48.0676	−1.65652
$843$	−18.8860	−0.650468
$844$	−1.36085	−0.0468424
$845$	10.9882	0.378007
$846$	11.4447	0.393479
$847$	0	0
$848$	−36.1651	−1.24191
$849$	−13.5813	−0.466110
$850$	−6.95128	−0.238427
$851$	23.5385	0.806890
$852$	0.455263	0.0155971
$853$	−6.37161	−0.218160	−0.109080	−	0.994033i	$-0.534790\pi$
$853$	−6.37161	−0.218160	−0.109080	+	0.994033i	$0.534790\pi$
$854$	25.3847	0.868646
$855$	−0.158146	−0.00540847
$856$	44.1051	1.50748
$857$	35.9060	1.22653	0.613263	−	0.789878i	$-0.289856\pi$
$857$	35.9060	1.22653	0.613263	+	0.789878i	$0.289856\pi$
$858$	0	0
$859$	56.4697	1.92672	0.963360	−	0.268212i	$-0.0864328\pi$
$859$	56.4697	1.92672	0.963360	+	0.268212i	$0.0864328\pi$
$860$	0.363693	0.0124018
$861$	9.69177	0.330295
$862$	−4.67198	−0.159128
$863$	−31.0742	−1.05778	−0.528888	−	0.848691i	$-0.677391\pi$
$863$	−31.0742	−1.05778	−0.528888	+	0.848691i	$0.677391\pi$
$864$	−0.379898	−0.0129244
$865$	−9.66389	−0.328582
$866$	−5.11308	−0.173749
$867$	8.00000	0.271694
$868$	−0.261784	−0.00888552
$869$	0	0
$870$	8.72406	0.295773
$871$	10.4727	0.354853
$872$	41.3857	1.40150
$873$	6.41196	0.217012
$874$	1.10114	0.0372466
$875$	−1.27759	−0.0431905
$876$	−0.581901	−0.0196606
$877$	16.8224	0.568053	0.284026	−	0.958816i	$-0.408330\pi$
$877$	16.8224	0.568053	0.284026	+	0.958816i	$0.408330\pi$
$878$	2.16090	0.0729268
$879$	19.9023	0.671289
$880$	0	0
$881$	26.5633	0.894940	0.447470	−	0.894299i	$-0.352325\pi$
$881$	26.5633	0.894940	0.447470	+	0.894299i	$0.352325\pi$
$882$	7.46257	0.251278
$883$	−54.3742	−1.82984	−0.914919	−	0.403638i	$-0.867746\pi$
$883$	−54.3742	−1.82984	−0.914919	+	0.403638i	$0.867746\pi$
$884$	0.476470	0.0160254
$885$	−8.09846	−0.272227
$886$	−29.4816	−0.990455
$887$	−41.2262	−1.38424	−0.692120	−	0.721783i	$-0.743323\pi$
$887$	−41.2262	−1.38424	−0.692120	+	0.721783i	$0.743323\pi$
$888$	−13.5072	−0.453271
$889$	−7.69640	−0.258129
$890$	15.3550	0.514699
$891$	0	0
$892$	−1.51504	−0.0507273
$893$	−1.30187	−0.0435655
$894$	−2.05927	−0.0688724
$895$	1.19571	0.0399682
$896$	−13.6835	−0.457132
$897$	7.10358	0.237182
$898$	−21.0076	−0.701033
$899$	19.1380	0.638289
$900$	−0.0671858	−0.00223953
$901$	46.8324	1.56021
$902$	0	0
$903$	6.91591	0.230147
$904$	55.6521	1.85096
$905$	−15.5741	−0.517699
$906$	12.5595	0.417262
$907$	42.4046	1.40802	0.704010	−	0.710190i	$-0.251391\pi$
$907$	42.4046	1.40802	0.704010	+	0.710190i	$0.251391\pi$
$908$	−1.26351	−0.0419310
$909$	8.77143	0.290930
$910$	−2.51927	−0.0835130
$911$	−8.01199	−0.265449	−0.132725	−	0.991153i	$-0.542373\pi$
$911$	−8.01199	−0.265449	−0.132725	+	0.991153i	$0.542373\pi$
$912$	−0.610619	−0.0202196
$913$	0	0
$914$	−57.4234	−1.89940
$915$	14.2917	0.472470
$916$	1.60117	0.0529042
$917$	−23.9265	−0.790123
$918$	−6.95128	−0.229427
$919$	41.2336	1.36017	0.680086	−	0.733132i	$-0.261942\pi$
$919$	41.2336	1.36017	0.680086	+	0.733132i	$0.261942\pi$
$920$	14.3934	0.474537
$921$	−14.9354	−0.492139
$922$	22.6831	0.747028
$923$	9.61110	0.316353
$924$	0	0
$925$	−4.69991	−0.154532
$926$	−11.7880	−0.387379
$927$	11.5666	0.379898
$928$	−2.38391	−0.0782558
$929$	−15.1939	−0.498495	−0.249247	−	0.968440i	$-0.580183\pi$
$929$	−15.1939	−0.498495	−0.249247	+	0.968440i	$0.580183\pi$
$930$	4.24002	0.139036
$931$	−0.848889	−0.0278212
$932$	0.678027	0.0222095
$933$	−4.99700	−0.163594
$934$	50.8375	1.66345
$935$	0	0
$936$	−4.07627	−0.133237
$937$	25.5213	0.833746	0.416873	−	0.908965i	$-0.363126\pi$
$937$	25.5213	0.833746	0.416873	+	0.908965i	$0.363126\pi$
$938$	13.1146	0.428208
$939$	9.39648	0.306643
$940$	−0.553081	−0.0180395
$941$	32.3673	1.05514	0.527572	−	0.849511i	$-0.323103\pi$
$941$	32.3673	1.05514	0.527572	+	0.849511i	$0.323103\pi$
$942$	24.5209	0.798934
$943$	−37.9927	−1.23721
$944$	−31.2691	−1.01772
$945$	−1.27759	−0.0415600
$946$	0	0
$947$	−40.1516	−1.30475	−0.652375	−	0.757896i	$-0.726227\pi$
$947$	−40.1516	−1.30475	−0.652375	+	0.757896i	$0.726227\pi$
$948$	0.171037	0.00555501
$949$	−12.2846	−0.398774
$950$	−0.219863	−0.00713331
$951$	2.15253	0.0698007
$952$	18.3585	0.595001
$953$	35.6513	1.15486	0.577429	−	0.816441i	$-0.304056\pi$
$953$	35.6513	1.15486	0.577429	+	0.816441i	$0.304056\pi$
$954$	−13.0218	−0.421597
$955$	−18.4016	−0.595462
$956$	−0.0112353	−0.000363374 0
$957$	0	0
$958$	54.1524	1.74958
$959$	3.88029	0.125301
$960$	−8.25038	−0.266280
$961$	−21.6986	−0.699956
$962$	−9.26772	−0.298803
$963$	15.3467	0.494539
$964$	−0.0648515	−0.00208873
$965$	−1.56799	−0.0504753
$966$	8.89563	0.286212
$967$	−20.0622	−0.645156	−0.322578	−	0.946543i	$-0.604549\pi$
$967$	−20.0622	−0.645156	−0.322578	+	0.946543i	$0.604549\pi$
$968$	0	0
$969$	0.790729	0.0254019
$970$	8.91428	0.286220
$971$	−41.6307	−1.33599	−0.667996	−	0.744165i	$-0.732848\pi$
$971$	−41.6307	−1.33599	−0.667996	+	0.744165i	$0.732848\pi$
$972$	−0.0671858	−0.00215499
$973$	−2.82593	−0.0905950
$974$	−11.5003	−0.368494
$975$	−1.41837	−0.0454240
$976$	55.1820	1.76633
$977$	19.0722	0.610173	0.305087	−	0.952325i	$-0.401315\pi$
$977$	19.0722	0.610173	0.305087	+	0.952325i	$0.401315\pi$
$978$	−32.6766	−1.04488
$979$	0	0
$980$	−0.360637	−0.0115201
$981$	14.4004	0.459771
$982$	16.8668	0.538243
$983$	26.5563	0.847015	0.423507	−	0.905893i	$-0.360799\pi$
$983$	26.5563	0.847015	0.423507	+	0.905893i	$0.360799\pi$
$984$	21.8015	0.695005
$985$	−7.97000	−0.253945
$986$	−43.6203	−1.38915
$987$	−10.5173	−0.334768
$988$	0.0150703	0.000479452 0
$989$	−27.1111	−0.862082
$990$	0	0
$991$	−24.2494	−0.770307	−0.385153	−	0.922853i	$-0.625851\pi$
$991$	−24.2494	−0.770307	−0.385153	+	0.922853i	$0.625851\pi$
$992$	−1.15862	−0.0367861
$993$	−19.5116	−0.619182
$994$	12.0357	0.381750
$995$	−3.53141	−0.111953
$996$	−0.680063	−0.0215486
$997$	16.0616	0.508675	0.254338	−	0.967116i	$-0.418143\pi$
$997$	16.0616	0.508675	0.254338	+	0.967116i	$0.418143\pi$
$998$	39.4250	1.24798
$999$	−4.69991	−0.148699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.2.a.x.1.1		4
3.2	odd	2		5445.2.a.be.1.4		4
5.4	even	2		9075.2.a.cl.1.4		4
11.5	even	5		165.2.m.a.91.2	✓	8
11.9	even	5		165.2.m.a.136.2	yes	8
11.10	odd	2		1815.2.a.o.1.4		4
33.5	odd	10		495.2.n.d.91.1		8
33.20	odd	10		495.2.n.d.136.1		8
33.32	even	2		5445.2.a.bv.1.1		4
55.9	even	10		825.2.n.k.301.1		8
55.27	odd	20		825.2.bx.h.124.3		16
55.38	odd	20		825.2.bx.h.124.2		16
55.42	odd	20		825.2.bx.h.499.2		16
55.49	even	10		825.2.n.k.751.1		8
55.53	odd	20		825.2.bx.h.499.3		16
55.54	odd	2		9075.2.a.dj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.m.a.91.2	✓	8	11.5	even	5
165.2.m.a.136.2	yes	8	11.9	even	5
495.2.n.d.91.1		8	33.5	odd	10
495.2.n.d.136.1		8	33.20	odd	10
825.2.n.k.301.1		8	55.9	even	10
825.2.n.k.751.1		8	55.49	even	10
825.2.bx.h.124.2		16	55.38	odd	20
825.2.bx.h.124.3		16	55.27	odd	20
825.2.bx.h.499.2		16	55.42	odd	20
825.2.bx.h.499.3		16	55.53	odd	20
1815.2.a.o.1.4		4	11.10	odd	2
1815.2.a.x.1.1		4	1.1	even	1	trivial
5445.2.a.be.1.4		4	3.2	odd	2
5445.2.a.bv.1.1		4	33.32	even	2
9075.2.a.cl.1.4		4	5.4	even	2
9075.2.a.dj.1.1		4	55.54	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39026 q^{2} +1.00000 q^{3} -0.0671858 q^{4} -1.00000 q^{5} -1.39026 q^{6} +1.27759 q^{7} +2.87392 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.39026 q^{2} +1.00000 q^{3} -0.0671858 q^{4} -1.00000 q^{5} -1.39026 q^{6} +1.27759 q^{7} +2.87392 q^{8} +1.00000 q^{9} +1.39026 q^{10} -0.0671858 q^{12} -1.41837 q^{13} -1.77618 q^{14} -1.00000 q^{15} -3.86111 q^{16} +5.00000 q^{17} -1.39026 q^{18} +0.158146 q^{19} +0.0671858 q^{20} +1.27759 q^{21} -5.00829 q^{23} +2.87392 q^{24} +1.00000 q^{25} +1.97189 q^{26} +1.00000 q^{27} -0.0858360 q^{28} +6.27515 q^{29} +1.39026 q^{30} +3.04981 q^{31} -0.379898 q^{32} -6.95128 q^{34} -1.27759 q^{35} -0.0671858 q^{36} -4.69991 q^{37} -0.219863 q^{38} -1.41837 q^{39} -2.87392 q^{40} +7.58597 q^{41} -1.77618 q^{42} +5.41324 q^{43} -1.00000 q^{45} +6.96281 q^{46} -8.23211 q^{47} -3.86111 q^{48} -5.36776 q^{49} -1.39026 q^{50} +5.00000 q^{51} +0.0952940 q^{52} +9.36648 q^{53} -1.39026 q^{54} +3.67169 q^{56} +0.158146 q^{57} -8.72406 q^{58} +8.09846 q^{59} +0.0671858 q^{60} -14.2917 q^{61} -4.24002 q^{62} +1.27759 q^{63} +8.25038 q^{64} +1.41837 q^{65} -7.38362 q^{67} -0.335929 q^{68} -5.00829 q^{69} +1.77618 q^{70} -6.77618 q^{71} +2.87392 q^{72} +8.66107 q^{73} +6.53409 q^{74} +1.00000 q^{75} -0.0106252 q^{76} +1.97189 q^{78} -2.54572 q^{79} +3.86111 q^{80} +1.00000 q^{81} -10.5464 q^{82} +10.1221 q^{83} -0.0858360 q^{84} -5.00000 q^{85} -7.52580 q^{86} +6.27515 q^{87} +11.0447 q^{89} +1.39026 q^{90} -1.81209 q^{91} +0.336486 q^{92} +3.04981 q^{93} +11.4447 q^{94} -0.158146 q^{95} -0.379898 q^{96} +6.41196 q^{97} +7.46257 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 5 * q^2 + 4 * q^3 + 9 * q^4 - 4 * q^5 + 5 * q^6 - 2 * q^7 + 15 * q^8 + 4 * q^9	\( 4 q + 5 q^{2} + 4 q^{3} + 9 q^{4} - 4 q^{5} + 5 q^{6} - 2 q^{7} + 15 q^{8} + 4 q^{9} - 5 q^{10} + 9 q^{12} + 3 q^{13} - 5 q^{14} - 4 q^{15} + 15 q^{16} + 20 q^{17} + 5 q^{18} + 3 q^{19} - 9 q^{20} - 2 q^{21} - 5 q^{23} + 15 q^{24} + 4 q^{25} + 6 q^{26} + 4 q^{27} + 3 q^{28} + 5 q^{29} - 5 q^{30} - q^{31} + 30 q^{32} + 25 q^{34} + 2 q^{35} + 9 q^{36} - 7 q^{37} + q^{38} + 3 q^{39} - 15 q^{40} + 20 q^{41} - 5 q^{42} - 2 q^{43} - 4 q^{45} + 7 q^{46} - 20 q^{47} + 15 q^{48} + 8 q^{49} + 5 q^{50} + 20 q^{51} - 7 q^{52} + 6 q^{53} + 5 q^{54} + 10 q^{56} + 3 q^{57} - 21 q^{58} - 5 q^{59} - 9 q^{60} - 7 q^{61} - 12 q^{62} - 2 q^{63} + 49 q^{64} - 3 q^{65} - 13 q^{67} + 45 q^{68} - 5 q^{69} + 5 q^{70} - 25 q^{71} + 15 q^{72} + 23 q^{73} - 7 q^{74} + 4 q^{75} - 7 q^{76} + 6 q^{78} - 15 q^{80} + 4 q^{81} + 11 q^{82} + 33 q^{83} + 3 q^{84} - 20 q^{85} - 12 q^{86} + 5 q^{87} + 16 q^{89} - 5 q^{90} - 24 q^{91} - q^{93} - 17 q^{94} - 3 q^{95} + 30 q^{96} + 25 q^{98}+O(q^{100}) \) 4 * q + 5 * q^2 + 4 * q^3 + 9 * q^4 - 4 * q^5 + 5 * q^6 - 2 * q^7 + 15 * q^8 + 4 * q^9 - 5 * q^10 + 9 * q^12 + 3 * q^13 - 5 * q^14 - 4 * q^15 + 15 * q^16 + 20 * q^17 + 5 * q^18 + 3 * q^19 - 9 * q^20 - 2 * q^21 - 5 * q^23 + 15 * q^24 + 4 * q^25 + 6 * q^26 + 4 * q^27 + 3 * q^28 + 5 * q^29 - 5 * q^30 - q^31 + 30 * q^32 + 25 * q^34 + 2 * q^35 + 9 * q^36 - 7 * q^37 + q^38 + 3 * q^39 - 15 * q^40 + 20 * q^41 - 5 * q^42 - 2 * q^43 - 4 * q^45 + 7 * q^46 - 20 * q^47 + 15 * q^48 + 8 * q^49 + 5 * q^50 + 20 * q^51 - 7 * q^52 + 6 * q^53 + 5 * q^54 + 10 * q^56 + 3 * q^57 - 21 * q^58 - 5 * q^59 - 9 * q^60 - 7 * q^61 - 12 * q^62 - 2 * q^63 + 49 * q^64 - 3 * q^65 - 13 * q^67 + 45 * q^68 - 5 * q^69 + 5 * q^70 - 25 * q^71 + 15 * q^72 + 23 * q^73 - 7 * q^74 + 4 * q^75 - 7 * q^76 + 6 * q^78 - 15 * q^80 + 4 * q^81 + 11 * q^82 + 33 * q^83 + 3 * q^84 - 20 * q^85 - 12 * q^86 + 5 * q^87 + 16 * q^89 - 5 * q^90 - 24 * q^91 - q^93 - 17 * q^94 - 3 * q^95 + 30 * q^96 + 25 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more