LMFDB - Embedded newform 4334.2.a.b.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 6 x^{14} - 8 x^{13} + 94 x^{12} - 13 x^{11} - 582 x^{10} + 295 x^{9} + 1814 x^{8} - 1056 x^{7} + \cdots - 45 $ x^15 - 6x^14 - 8x^13 + 94x^12 - 13x^11 - 582x^10 + 295x^9 + 1814x^8 - 1056x^7 - 2966x^6 + 1621x^5 + 2328x^4 - 1160x^3 - 589x^2 + 364x - 45
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.49640$ of defining polynomial
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.496403 q^{3} +1.00000 q^{4} -1.51850 q^{5} +0.496403 q^{6} -2.22955 q^{7} +1.00000 q^{8} -2.75358 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.496403 q^{3} +1.00000 q^{4} -1.51850 q^{5} +0.496403 q^{6} -2.22955 q^{7} +1.00000 q^{8} -2.75358 q^{9} -1.51850 q^{10} +1.00000 q^{11} +0.496403 q^{12} +5.83120 q^{13} -2.22955 q^{14} -0.753789 q^{15} +1.00000 q^{16} -1.07159 q^{17} -2.75358 q^{18} -6.01383 q^{19} -1.51850 q^{20} -1.10675 q^{21} +1.00000 q^{22} +6.48472 q^{23} +0.496403 q^{24} -2.69415 q^{25} +5.83120 q^{26} -2.85610 q^{27} -2.22955 q^{28} +3.30200 q^{29} -0.753789 q^{30} +1.42282 q^{31} +1.00000 q^{32} +0.496403 q^{33} -1.07159 q^{34} +3.38557 q^{35} -2.75358 q^{36} +1.31446 q^{37} -6.01383 q^{38} +2.89463 q^{39} -1.51850 q^{40} -4.42379 q^{41} -1.10675 q^{42} -4.90620 q^{43} +1.00000 q^{44} +4.18132 q^{45} +6.48472 q^{46} -13.0209 q^{47} +0.496403 q^{48} -2.02912 q^{49} -2.69415 q^{50} -0.531942 q^{51} +5.83120 q^{52} +6.25507 q^{53} -2.85610 q^{54} -1.51850 q^{55} -2.22955 q^{56} -2.98529 q^{57} +3.30200 q^{58} -11.8690 q^{59} -0.753789 q^{60} -5.34067 q^{61} +1.42282 q^{62} +6.13925 q^{63} +1.00000 q^{64} -8.85468 q^{65} +0.496403 q^{66} +1.83551 q^{67} -1.07159 q^{68} +3.21904 q^{69} +3.38557 q^{70} -9.98592 q^{71} -2.75358 q^{72} -14.2776 q^{73} +1.31446 q^{74} -1.33739 q^{75} -6.01383 q^{76} -2.22955 q^{77} +2.89463 q^{78} +6.92598 q^{79} -1.51850 q^{80} +6.84297 q^{81} -4.42379 q^{82} -7.83273 q^{83} -1.10675 q^{84} +1.62722 q^{85} -4.90620 q^{86} +1.63912 q^{87} +1.00000 q^{88} -4.82911 q^{89} +4.18132 q^{90} -13.0009 q^{91} +6.48472 q^{92} +0.706293 q^{93} -13.0209 q^{94} +9.13202 q^{95} +0.496403 q^{96} -3.47660 q^{97} -2.02912 q^{98} -2.75358 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9	\( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9} - 11 q^{10} + 15 q^{11} - 9 q^{12} - 21 q^{13} - 11 q^{14} - 2 q^{15} + 15 q^{16} - 4 q^{17} + 10 q^{18} - 22 q^{19} - 11 q^{20} - 13 q^{21} + 15 q^{22} - 16 q^{23} - 9 q^{24} + 6 q^{25} - 21 q^{26} - 21 q^{27} - 11 q^{28} - 8 q^{29} - 2 q^{30} - 33 q^{31} + 15 q^{32} - 9 q^{33} - 4 q^{34} - 2 q^{35} + 10 q^{36} - q^{37} - 22 q^{38} + q^{39} - 11 q^{40} - 10 q^{41} - 13 q^{42} - 8 q^{43} + 15 q^{44} - 10 q^{45} - 16 q^{46} - 31 q^{47} - 9 q^{48} + 2 q^{49} + 6 q^{50} + 2 q^{51} - 21 q^{52} - 18 q^{53} - 21 q^{54} - 11 q^{55} - 11 q^{56} + 16 q^{57} - 8 q^{58} - 37 q^{59} - 2 q^{60} - 31 q^{61} - 33 q^{62} - 20 q^{63} + 15 q^{64} - 13 q^{65} - 9 q^{66} + q^{67} - 4 q^{68} - 25 q^{69} - 2 q^{70} - 28 q^{71} + 10 q^{72} - 20 q^{73} - q^{74} - 9 q^{75} - 22 q^{76} - 11 q^{77} + q^{78} - 6 q^{79} - 11 q^{80} + 3 q^{81} - 10 q^{82} - 15 q^{83} - 13 q^{84} - 31 q^{85} - 8 q^{86} - 16 q^{87} + 15 q^{88} - 17 q^{89} - 10 q^{90} - 21 q^{91} - 16 q^{92} + 10 q^{93} - 31 q^{94} - 3 q^{95} - 9 q^{96} - 9 q^{97} + 2 q^{98} + 10 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9 - 11 * q^10 + 15 * q^11 - 9 * q^12 - 21 * q^13 - 11 * q^14 - 2 * q^15 + 15 * q^16 - 4 * q^17 + 10 * q^18 - 22 * q^19 - 11 * q^20 - 13 * q^21 + 15 * q^22 - 16 * q^23 - 9 * q^24 + 6 * q^25 - 21 * q^26 - 21 * q^27 - 11 * q^28 - 8 * q^29 - 2 * q^30 - 33 * q^31 + 15 * q^32 - 9 * q^33 - 4 * q^34 - 2 * q^35 + 10 * q^36 - q^37 - 22 * q^38 + q^39 - 11 * q^40 - 10 * q^41 - 13 * q^42 - 8 * q^43 + 15 * q^44 - 10 * q^45 - 16 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 + 6 * q^50 + 2 * q^51 - 21 * q^52 - 18 * q^53 - 21 * q^54 - 11 * q^55 - 11 * q^56 + 16 * q^57 - 8 * q^58 - 37 * q^59 - 2 * q^60 - 31 * q^61 - 33 * q^62 - 20 * q^63 + 15 * q^64 - 13 * q^65 - 9 * q^66 + q^67 - 4 * q^68 - 25 * q^69 - 2 * q^70 - 28 * q^71 + 10 * q^72 - 20 * q^73 - q^74 - 9 * q^75 - 22 * q^76 - 11 * q^77 + q^78 - 6 * q^79 - 11 * q^80 + 3 * q^81 - 10 * q^82 - 15 * q^83 - 13 * q^84 - 31 * q^85 - 8 * q^86 - 16 * q^87 + 15 * q^88 - 17 * q^89 - 10 * q^90 - 21 * q^91 - 16 * q^92 + 10 * q^93 - 31 * q^94 - 3 * q^95 - 9 * q^96 - 9 * q^97 + 2 * q^98 + 10 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.496403	0.286599	0.143299	−	0.989679i	$-0.454229\pi$
$3$	0.496403	0.286599	0.143299	+	0.989679i	$0.454229\pi$
$4$	1.00000	0.500000
$5$	−1.51850	−0.679094	−0.339547	−	0.940589i	$-0.610274\pi$
$5$	−1.51850	−0.679094	−0.339547	+	0.940589i	$0.610274\pi$
$6$	0.496403	0.202656
$7$	−2.22955	−0.842690	−0.421345	−	0.906900i	$-0.638442\pi$
$7$	−2.22955	−0.842690	−0.421345	+	0.906900i	$0.638442\pi$
$8$	1.00000	0.353553
$9$	−2.75358	−0.917861
$10$	−1.51850	−0.480192
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0.496403	0.143299
$13$	5.83120	1.61728	0.808642	−	0.588301i	$-0.200203\pi$
$13$	5.83120	1.61728	0.808642	+	0.588301i	$0.200203\pi$
$14$	−2.22955	−0.595872
$15$	−0.753789	−0.194628
$16$	1.00000	0.250000
$17$	−1.07159	−0.259899	−0.129950	−	0.991521i	$-0.541482\pi$
$17$	−1.07159	−0.259899	−0.129950	+	0.991521i	$0.541482\pi$
$18$	−2.75358	−0.649026
$19$	−6.01383	−1.37967	−0.689834	−	0.723967i	$-0.742317\pi$
$19$	−6.01383	−1.37967	−0.689834	+	0.723967i	$0.742317\pi$
$20$	−1.51850	−0.339547
$21$	−1.10675	−0.241514
$22$	1.00000	0.213201
$23$	6.48472	1.35216	0.676079	−	0.736829i	$-0.263678\pi$
$23$	6.48472	1.35216	0.676079	+	0.736829i	$0.263678\pi$
$24$	0.496403	0.101328
$25$	−2.69415	−0.538831
$26$	5.83120	1.14359
$27$	−2.85610	−0.549656
$28$	−2.22955	−0.421345
$29$	3.30200	0.613166	0.306583	−	0.951844i	$-0.400814\pi$
$29$	3.30200	0.613166	0.306583	+	0.951844i	$0.400814\pi$
$30$	−0.753789	−0.137622
$31$	1.42282	0.255546	0.127773	−	0.991803i	$-0.459217\pi$
$31$	1.42282	0.255546	0.127773	+	0.991803i	$0.459217\pi$
$32$	1.00000	0.176777
$33$	0.496403	0.0864127
$34$	−1.07159	−0.183777
$35$	3.38557	0.572266
$36$	−2.75358	−0.458931
$37$	1.31446	0.216096	0.108048	−	0.994146i	$-0.465540\pi$
$37$	1.31446	0.216096	0.108048	+	0.994146i	$0.465540\pi$
$38$	−6.01383	−0.975573
$39$	2.89463	0.463511
$40$	−1.51850	−0.240096
$41$	−4.42379	−0.690880	−0.345440	−	0.938441i	$-0.612270\pi$
$41$	−4.42379	−0.690880	−0.345440	+	0.938441i	$0.612270\pi$
$42$	−1.10675	−0.170776
$43$	−4.90620	−0.748188	−0.374094	−	0.927391i	$-0.622046\pi$
$43$	−4.90620	−0.748188	−0.374094	+	0.927391i	$0.622046\pi$
$44$	1.00000	0.150756
$45$	4.18132	0.623315
$46$	6.48472	0.956120
$47$	−13.0209	−1.89929	−0.949647	−	0.313323i	$-0.898558\pi$
$47$	−13.0209	−1.89929	−0.949647	+	0.313323i	$0.898558\pi$
$48$	0.496403	0.0716497
$49$	−2.02912	−0.289874
$50$	−2.69415	−0.381011
$51$	−0.531942	−0.0744868
$52$	5.83120	0.808642
$53$	6.25507	0.859199	0.429600	−	0.903019i	$-0.358655\pi$
$53$	6.25507	0.859199	0.429600	+	0.903019i	$0.358655\pi$
$54$	−2.85610	−0.388666
$55$	−1.51850	−0.204755
$56$	−2.22955	−0.297936
$57$	−2.98529	−0.395411
$58$	3.30200	0.433574
$59$	−11.8690	−1.54522	−0.772608	−	0.634884i	$-0.781048\pi$
$59$	−11.8690	−1.54522	−0.772608	+	0.634884i	$0.781048\pi$
$60$	−0.753789	−0.0973138
$61$	−5.34067	−0.683803	−0.341901	−	0.939736i	$-0.611071\pi$
$61$	−5.34067	−0.683803	−0.341901	+	0.939736i	$0.611071\pi$
$62$	1.42282	0.180698
$63$	6.13925	0.773472
$64$	1.00000	0.125000
$65$	−8.85468	−1.09829
$66$	0.496403	0.0611030
$67$	1.83551	0.224244	0.112122	−	0.993694i	$-0.464235\pi$
$67$	1.83551	0.224244	0.112122	+	0.993694i	$0.464235\pi$
$68$	−1.07159	−0.129950
$69$	3.21904	0.387527
$70$	3.38557	0.404653
$71$	−9.98592	−1.18511	−0.592555	−	0.805530i	$-0.701881\pi$
$71$	−9.98592	−1.18511	−0.592555	+	0.805530i	$0.701881\pi$
$72$	−2.75358	−0.324513
$73$	−14.2776	−1.67107	−0.835534	−	0.549439i	$-0.814841\pi$
$73$	−14.2776	−1.67107	−0.835534	+	0.549439i	$0.814841\pi$
$74$	1.31446	0.152803
$75$	−1.33739	−0.154428
$76$	−6.01383	−0.689834
$77$	−2.22955	−0.254081
$78$	2.89463	0.327752
$79$	6.92598	0.779234	0.389617	−	0.920977i	$-0.372607\pi$
$79$	6.92598	0.779234	0.389617	+	0.920977i	$0.372607\pi$
$80$	−1.51850	−0.169774
$81$	6.84297	0.760330
$82$	−4.42379	−0.488526
$83$	−7.83273	−0.859754	−0.429877	−	0.902887i	$-0.641443\pi$
$83$	−7.83273	−0.859754	−0.429877	+	0.902887i	$0.641443\pi$
$84$	−1.10675	−0.120757
$85$	1.62722	0.176496
$86$	−4.90620	−0.529049
$87$	1.63912	0.175732
$88$	1.00000	0.106600
$89$	−4.82911	−0.511884	−0.255942	−	0.966692i	$-0.582386\pi$
$89$	−4.82911	−0.511884	−0.255942	+	0.966692i	$0.582386\pi$
$90$	4.18132	0.440750
$91$	−13.0009	−1.36287
$92$	6.48472	0.676079
$93$	0.706293	0.0732391
$94$	−13.0209	−1.34300
$95$	9.13202	0.936925
$96$	0.496403	0.0506640
$97$	−3.47660	−0.352995	−0.176497	−	0.984301i	$-0.556477\pi$
$97$	−3.47660	−0.352995	−0.176497	+	0.984301i	$0.556477\pi$
$98$	−2.02912	−0.204972
$99$	−2.75358	−0.276746
$100$	−2.69415	−0.269415
$101$	5.64311	0.561511	0.280755	−	0.959779i	$-0.409415\pi$
$101$	5.64311	0.561511	0.280755	+	0.959779i	$0.409415\pi$
$102$	−0.531942	−0.0526701
$103$	4.54850	0.448177	0.224088	−	0.974569i	$-0.428060\pi$
$103$	4.54850	0.448177	0.224088	+	0.974569i	$0.428060\pi$
$104$	5.83120	0.571796
$105$	1.68061	0.164011
$106$	6.25507	0.607546
$107$	−14.1123	−1.36429	−0.682144	−	0.731218i	$-0.738952\pi$
$107$	−14.1123	−1.36429	−0.682144	+	0.731218i	$0.738952\pi$
$108$	−2.85610	−0.274828
$109$	−3.78441	−0.362480	−0.181240	−	0.983439i	$-0.558011\pi$
$109$	−3.78441	−0.362480	−0.181240	+	0.983439i	$0.558011\pi$
$110$	−1.51850	−0.144783
$111$	0.652501	0.0619327
$112$	−2.22955	−0.210672
$113$	−0.334347	−0.0314527	−0.0157264	−	0.999876i	$-0.505006\pi$
$113$	−0.334347	−0.0314527	−0.0157264	+	0.999876i	$0.505006\pi$
$114$	−2.98529	−0.279598
$115$	−9.84706	−0.918243
$116$	3.30200	0.306583
$117$	−16.0567	−1.48444
$118$	−11.8690	−1.09263
$119$	2.38917	0.219015
$120$	−0.753789	−0.0688112
$121$	1.00000	0.0909091
$122$	−5.34067	−0.483521
$123$	−2.19598	−0.198005
$124$	1.42282	0.127773
$125$	11.6836	1.04501
$126$	6.13925	0.546927
$127$	10.2961	0.913628	0.456814	−	0.889562i	$-0.348990\pi$
$127$	10.2961	0.913628	0.456814	+	0.889562i	$0.348990\pi$
$128$	1.00000	0.0883883
$129$	−2.43545	−0.214430
$130$	−8.85468	−0.776607
$131$	6.39532	0.558762	0.279381	−	0.960180i	$-0.409871\pi$
$131$	6.39532	0.558762	0.279381	+	0.960180i	$0.409871\pi$
$132$	0.496403	0.0432064
$133$	13.4081	1.16263
$134$	1.83551	0.158564
$135$	4.33699	0.373269
$136$	−1.07159	−0.0918883
$137$	−15.7198	−1.34304	−0.671518	−	0.740988i	$-0.734357\pi$
$137$	−15.7198	−1.34304	−0.671518	+	0.740988i	$0.734357\pi$
$138$	3.21904	0.274023
$139$	−6.53072	−0.553928	−0.276964	−	0.960880i	$-0.589328\pi$
$139$	−6.53072	−0.553928	−0.276964	+	0.960880i	$0.589328\pi$
$140$	3.38557	0.286133
$141$	−6.46362	−0.544335
$142$	−9.98592	−0.838000
$143$	5.83120	0.487629
$144$	−2.75358	−0.229465
$145$	−5.01409	−0.416398
$146$	−14.2776	−1.18162
$147$	−1.00726	−0.0830775
$148$	1.31446	0.108048
$149$	−5.26195	−0.431075	−0.215538	−	0.976496i	$-0.569150\pi$
$149$	−5.26195	−0.431075	−0.215538	+	0.976496i	$0.569150\pi$
$150$	−1.33739	−0.109197
$151$	12.7158	1.03480	0.517398	−	0.855745i	$-0.326901\pi$
$151$	12.7158	1.03480	0.517398	+	0.855745i	$0.326901\pi$
$152$	−6.01383	−0.487786
$153$	2.95072	0.238552
$154$	−2.22955	−0.179662
$155$	−2.16055	−0.173540
$156$	2.89463	0.231756
$157$	3.61551	0.288549	0.144275	−	0.989538i	$-0.453915\pi$
$157$	3.61551	0.288549	0.144275	+	0.989538i	$0.453915\pi$
$158$	6.92598	0.551002
$159$	3.10504	0.246245
$160$	−1.51850	−0.120048
$161$	−14.4580	−1.13945
$162$	6.84297	0.537635
$163$	−10.9728	−0.859459	−0.429730	−	0.902958i	$-0.641391\pi$
$163$	−10.9728	−0.859459	−0.429730	+	0.902958i	$0.641391\pi$
$164$	−4.42379	−0.345440
$165$	−0.753789	−0.0586824
$166$	−7.83273	−0.607938
$167$	−19.5002	−1.50897	−0.754485	−	0.656318i	$-0.772113\pi$
$167$	−19.5002	−1.50897	−0.754485	+	0.656318i	$0.772113\pi$
$168$	−1.10675	−0.0853880
$169$	21.0029	1.61561
$170$	1.62722	0.124802
$171$	16.5596	1.26634
$172$	−4.90620	−0.374094
$173$	−3.79375	−0.288433	−0.144217	−	0.989546i	$-0.546066\pi$
$173$	−3.79375	−0.288433	−0.144217	+	0.989546i	$0.546066\pi$
$174$	1.63912	0.124262
$175$	6.00674	0.454067
$176$	1.00000	0.0753778
$177$	−5.89182	−0.442857
$178$	−4.82911	−0.361957
$179$	6.93043	0.518004	0.259002	−	0.965877i	$-0.416606\pi$
$179$	6.93043	0.518004	0.259002	+	0.965877i	$0.416606\pi$
$180$	4.18132	0.311657
$181$	−6.38275	−0.474426	−0.237213	−	0.971458i	$-0.576234\pi$
$181$	−6.38275	−0.474426	−0.237213	+	0.971458i	$0.576234\pi$
$182$	−13.0009	−0.963693
$183$	−2.65113	−0.195977
$184$	6.48472	0.478060
$185$	−1.99601	−0.146749
$186$	0.706293	0.0517879
$187$	−1.07159	−0.0783626
$188$	−13.0209	−0.949647
$189$	6.36781	0.463190
$190$	9.13202	0.662506
$191$	−13.1177	−0.949163	−0.474582	−	0.880211i	$-0.657401\pi$
$191$	−13.1177	−0.949163	−0.474582	+	0.880211i	$0.657401\pi$
$192$	0.496403	0.0358248
$193$	3.04007	0.218829	0.109414	−	0.993996i	$-0.465102\pi$
$193$	3.04007	0.218829	0.109414	+	0.993996i	$0.465102\pi$
$194$	−3.47660	−0.249605
$195$	−4.39549	−0.314768
$196$	−2.02912	−0.144937
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	−2.75358	−0.195689
$199$	−3.06627	−0.217362	−0.108681	−	0.994077i	$-0.534663\pi$
$199$	−3.06627	−0.217362	−0.108681	+	0.994077i	$0.534663\pi$
$200$	−2.69415	−0.190505
$201$	0.911156	0.0642680
$202$	5.64311	0.397048
$203$	−7.36196	−0.516709
$204$	−0.531942	−0.0372434
$205$	6.71753	0.469173
$206$	4.54850	0.316909
$207$	−17.8562	−1.24109
$208$	5.83120	0.404321
$209$	−6.01383	−0.415986
$210$	1.68061	0.115973
$211$	14.2027	0.977757	0.488879	−	0.872352i	$-0.337406\pi$
$211$	14.2027	0.977757	0.488879	+	0.872352i	$0.337406\pi$
$212$	6.25507	0.429600
$213$	−4.95704	−0.339651
$214$	−14.1123	−0.964697
$215$	7.45007	0.508091
$216$	−2.85610	−0.194333
$217$	−3.17224	−0.215346
$218$	−3.78441	−0.256312
$219$	−7.08745	−0.478926
$220$	−1.51850	−0.102377
$221$	−6.24867	−0.420331
$222$	0.652501	0.0437930
$223$	−13.1473	−0.880411	−0.440206	−	0.897897i	$-0.645094\pi$
$223$	−13.1473	−0.880411	−0.440206	+	0.897897i	$0.645094\pi$
$224$	−2.22955	−0.148968
$225$	7.41858	0.494572
$226$	−0.334347	−0.0222404
$227$	4.02185	0.266940	0.133470	−	0.991053i	$-0.457388\pi$
$227$	4.02185	0.266940	0.133470	+	0.991053i	$0.457388\pi$
$228$	−2.98529	−0.197706
$229$	15.2597	1.00839	0.504196	−	0.863589i	$-0.331789\pi$
$229$	15.2597	1.00839	0.504196	+	0.863589i	$0.331789\pi$
$230$	−9.84706	−0.649296
$231$	−1.10675	−0.0728191
$232$	3.30200	0.216787
$233$	0.531295	0.0348063	0.0174032	−	0.999849i	$-0.494460\pi$
$233$	0.531295	0.0348063	0.0174032	+	0.999849i	$0.494460\pi$
$234$	−16.0567	−1.04966
$235$	19.7723	1.28980
$236$	−11.8690	−0.772608
$237$	3.43808	0.223327
$238$	2.38917	0.154867
$239$	−14.5924	−0.943906	−0.471953	−	0.881624i	$-0.656451\pi$
$239$	−14.5924	−0.943906	−0.471953	+	0.881624i	$0.656451\pi$
$240$	−0.753789	−0.0486569
$241$	−29.6308	−1.90869	−0.954344	−	0.298711i	$-0.903443\pi$
$241$	−29.6308	−1.90869	−0.954344	+	0.298711i	$0.903443\pi$
$242$	1.00000	0.0642824
$243$	11.9652	0.767566
$244$	−5.34067	−0.341901
$245$	3.08122	0.196852
$246$	−2.19598	−0.140011
$247$	−35.0679	−2.23131
$248$	1.42282	0.0903492
$249$	−3.88819	−0.246404
$250$	11.6836	0.738935
$251$	6.83584	0.431475	0.215737	−	0.976451i	$-0.430785\pi$
$251$	6.83584	0.431475	0.215737	+	0.976451i	$0.430785\pi$
$252$	6.13925	0.386736
$253$	6.48472	0.407691
$254$	10.2961	0.646033
$255$	0.807755	0.0505836
$256$	1.00000	0.0625000
$257$	13.3608	0.833425	0.416713	−	0.909038i	$-0.363182\pi$
$257$	13.3608	0.833425	0.416713	+	0.909038i	$0.363182\pi$
$258$	−2.43545	−0.151625
$259$	−2.93065	−0.182102
$260$	−8.85468	−0.549144
$261$	−9.09233	−0.562801
$262$	6.39532	0.395104
$263$	16.2037	0.999163	0.499581	−	0.866267i	$-0.333487\pi$
$263$	16.2037	0.999163	0.499581	+	0.866267i	$0.333487\pi$
$264$	0.496403	0.0305515
$265$	−9.49833	−0.583478
$266$	13.4081	0.822105
$267$	−2.39718	−0.146705
$268$	1.83551	0.112122
$269$	27.7964	1.69478	0.847388	−	0.530973i	$-0.178174\pi$
$269$	27.7964	1.69478	0.847388	+	0.530973i	$0.178174\pi$
$270$	4.33699	0.263941
$271$	−18.5674	−1.12789	−0.563946	−	0.825812i	$-0.690717\pi$
$271$	−18.5674	−1.12789	−0.563946	+	0.825812i	$0.690717\pi$
$272$	−1.07159	−0.0649749
$273$	−6.45371	−0.390596
$274$	−15.7198	−0.949670
$275$	−2.69415	−0.162464
$276$	3.21904	0.193763
$277$	2.51646	0.151199	0.0755997	−	0.997138i	$-0.475913\pi$
$277$	2.51646	0.151199	0.0755997	+	0.997138i	$0.475913\pi$
$278$	−6.53072	−0.391687
$279$	−3.91785	−0.234556
$280$	3.38557	0.202327
$281$	29.3540	1.75112	0.875558	−	0.483114i	$-0.160494\pi$
$281$	29.3540	1.75112	0.875558	+	0.483114i	$0.160494\pi$
$282$	−6.46362	−0.384903
$283$	11.9928	0.712897	0.356448	−	0.934315i	$-0.383988\pi$
$283$	11.9928	0.712897	0.356448	+	0.934315i	$0.383988\pi$
$284$	−9.98592	−0.592555
$285$	4.53316	0.268521
$286$	5.83120	0.344806
$287$	9.86305	0.582197
$288$	−2.75358	−0.162256
$289$	−15.8517	−0.932452
$290$	−5.01409	−0.294438
$291$	−1.72579	−0.101168
$292$	−14.2776	−0.835534
$293$	28.1122	1.64233	0.821167	−	0.570688i	$-0.193324\pi$
$293$	28.1122	1.64233	0.821167	+	0.570688i	$0.193324\pi$
$294$	−1.00726	−0.0587447
$295$	18.0231	1.04935
$296$	1.31446	0.0764013
$297$	−2.85610	−0.165728
$298$	−5.26195	−0.304816
$299$	37.8137	2.18682
$300$	−1.33739	−0.0772141
$301$	10.9386	0.630491
$302$	12.7158	0.731711
$303$	2.80126	0.160928
$304$	−6.01383	−0.344917
$305$	8.10981	0.464367
$306$	2.95072	0.168681
$307$	−10.8297	−0.618082	−0.309041	−	0.951049i	$-0.600008\pi$
$307$	−10.8297	−0.618082	−0.309041	+	0.951049i	$0.600008\pi$
$308$	−2.22955	−0.127040
$309$	2.25789	0.128447
$310$	−2.16055	−0.122711
$311$	29.6731	1.68261	0.841304	−	0.540562i	$-0.181788\pi$
$311$	29.6731	1.68261	0.841304	+	0.540562i	$0.181788\pi$
$312$	2.89463	0.163876
$313$	11.7046	0.661584	0.330792	−	0.943704i	$-0.392684\pi$
$313$	11.7046	0.661584	0.330792	+	0.943704i	$0.392684\pi$
$314$	3.61551	0.204035
$315$	−9.32245	−0.525261
$316$	6.92598	0.389617
$317$	−2.05561	−0.115455	−0.0577273	−	0.998332i	$-0.518385\pi$
$317$	−2.05561	−0.115455	−0.0577273	+	0.998332i	$0.518385\pi$
$318$	3.10504	0.174122
$319$	3.30200	0.184876
$320$	−1.51850	−0.0848868
$321$	−7.00539	−0.391003
$322$	−14.4580	−0.805713
$323$	6.44438	0.358575
$324$	6.84297	0.380165
$325$	−15.7101	−0.871442
$326$	−10.9728	−0.607730
$327$	−1.87859	−0.103886
$328$	−4.42379	−0.244263
$329$	29.0307	1.60052
$330$	−0.753789	−0.0414947
$331$	25.0928	1.37923	0.689614	−	0.724177i	$-0.257780\pi$
$331$	25.0928	1.37923	0.689614	+	0.724177i	$0.257780\pi$
$332$	−7.83273	−0.429877
$333$	−3.61947	−0.198346
$334$	−19.5002	−1.06700
$335$	−2.78723	−0.152283
$336$	−1.10675	−0.0603784
$337$	−13.6170	−0.741765	−0.370883	−	0.928680i	$-0.620945\pi$
$337$	−13.6170	−0.741765	−0.370883	+	0.928680i	$0.620945\pi$
$338$	21.0029	1.14241
$339$	−0.165971	−0.00901431
$340$	1.62722	0.0882481
$341$	1.42282	0.0770500
$342$	16.5596	0.895440
$343$	20.1308	1.08696
$344$	−4.90620	−0.264525
$345$	−4.88811	−0.263167
$346$	−3.79375	−0.203953
$347$	30.7835	1.65254	0.826272	−	0.563271i	$-0.190457\pi$
$347$	30.7835	1.65254	0.826272	+	0.563271i	$0.190457\pi$
$348$	1.63912	0.0878662
$349$	3.52795	0.188847	0.0944235	−	0.995532i	$-0.469899\pi$
$349$	3.52795	0.188847	0.0944235	+	0.995532i	$0.469899\pi$
$350$	6.00674	0.321074
$351$	−16.6545	−0.888950
$352$	1.00000	0.0533002
$353$	−15.3527	−0.817140	−0.408570	−	0.912727i	$-0.633972\pi$
$353$	−15.3527	−0.817140	−0.408570	+	0.912727i	$0.633972\pi$
$354$	−5.89182	−0.313147
$355$	15.1636	0.804802
$356$	−4.82911	−0.255942
$357$	1.18599	0.0627693
$358$	6.93043	0.366284
$359$	−8.31396	−0.438794	−0.219397	−	0.975636i	$-0.570409\pi$
$359$	−8.31396	−0.438794	−0.219397	+	0.975636i	$0.570409\pi$
$360$	4.18132	0.220375
$361$	17.1662	0.903485
$362$	−6.38275	−0.335470
$363$	0.496403	0.0260544
$364$	−13.0009	−0.681434
$365$	21.6806	1.13481
$366$	−2.65113	−0.138577
$367$	−30.5996	−1.59729	−0.798644	−	0.601804i	$-0.794449\pi$
$367$	−30.5996	−1.59729	−0.798644	+	0.601804i	$0.794449\pi$
$368$	6.48472	0.338040
$369$	12.1813	0.634132
$370$	−1.99601	−0.103767
$371$	−13.9460	−0.724038
$372$	0.706293	0.0366196
$373$	−1.30456	−0.0675475	−0.0337737	−	0.999430i	$-0.510753\pi$
$373$	−1.30456	−0.0675475	−0.0337737	+	0.999430i	$0.510753\pi$
$374$	−1.07159	−0.0554107
$375$	5.79977	0.299499
$376$	−13.0209	−0.671502
$377$	19.2546	0.991663
$378$	6.36781	0.327525
$379$	19.4767	1.00045	0.500227	−	0.865895i	$-0.333250\pi$
$379$	19.4767	1.00045	0.500227	+	0.865895i	$0.333250\pi$
$380$	9.13202	0.468463
$381$	5.11100	0.261845
$382$	−13.1177	−0.671160
$383$	21.1789	1.08219	0.541095	−	0.840962i	$-0.318010\pi$
$383$	21.1789	1.08219	0.541095	+	0.840962i	$0.318010\pi$
$384$	0.496403	0.0253320
$385$	3.38557	0.172545
$386$	3.04007	0.154735
$387$	13.5096	0.686733
$388$	−3.47660	−0.176497
$389$	−7.33640	−0.371970	−0.185985	−	0.982553i	$-0.559548\pi$
$389$	−7.33640	−0.371970	−0.185985	+	0.982553i	$0.559548\pi$
$390$	−4.39549	−0.222575
$391$	−6.94898	−0.351425
$392$	−2.02912	−0.102486
$393$	3.17466	0.160140
$394$	−1.00000	−0.0503793
$395$	−10.5171	−0.529173
$396$	−2.75358	−0.138373
$397$	−19.7596	−0.991705	−0.495853	−	0.868407i	$-0.665144\pi$
$397$	−19.7596	−0.991705	−0.495853	+	0.868407i	$0.665144\pi$
$398$	−3.06627	−0.153698
$399$	6.65584	0.333209
$400$	−2.69415	−0.134708
$401$	18.8294	0.940296	0.470148	−	0.882588i	$-0.344200\pi$
$401$	18.8294	0.940296	0.470148	+	0.882588i	$0.344200\pi$
$402$	0.911156	0.0454443
$403$	8.29675	0.413290
$404$	5.64311	0.280755
$405$	−10.3911	−0.516336
$406$	−7.36196	−0.365368
$407$	1.31446	0.0651553
$408$	−0.531942	−0.0263351
$409$	14.1843	0.701367	0.350683	−	0.936494i	$-0.385949\pi$
$409$	14.1843	0.701367	0.350683	+	0.936494i	$0.385949\pi$
$410$	6.71753	0.331755
$411$	−7.80338	−0.384912
$412$	4.54850	0.224088
$413$	26.4625	1.30214
$414$	−17.8562	−0.877586
$415$	11.8940	0.583854
$416$	5.83120	0.285898
$417$	−3.24187	−0.158755
$418$	−6.01383	−0.294146
$419$	−34.8319	−1.70165	−0.850825	−	0.525450i	$-0.823897\pi$
$419$	−34.8319	−1.70165	−0.850825	+	0.525450i	$0.823897\pi$
$420$	1.68061	0.0820053
$421$	35.5091	1.73061	0.865304	−	0.501247i	$-0.167125\pi$
$421$	35.5091	1.73061	0.865304	+	0.501247i	$0.167125\pi$
$422$	14.2027	0.691379
$423$	35.8541	1.74329
$424$	6.25507	0.303773
$425$	2.88704	0.140042
$426$	−4.95704	−0.240170
$427$	11.9073	0.576233
$428$	−14.1123	−0.682144
$429$	2.89463	0.139754
$430$	7.45007	0.359274
$431$	−27.3834	−1.31901	−0.659505	−	0.751700i	$-0.729234\pi$
$431$	−27.3834	−1.31901	−0.659505	+	0.751700i	$0.729234\pi$
$432$	−2.85610	−0.137414
$433$	−6.38329	−0.306761	−0.153381	−	0.988167i	$-0.549016\pi$
$433$	−6.38329	−0.306761	−0.153381	+	0.988167i	$0.549016\pi$
$434$	−3.17224	−0.152273
$435$	−2.48901	−0.119339
$436$	−3.78441	−0.181240
$437$	−38.9981	−1.86553
$438$	−7.08745	−0.338651
$439$	−30.1611	−1.43951	−0.719756	−	0.694227i	$-0.755746\pi$
$439$	−30.1611	−1.43951	−0.719756	+	0.694227i	$0.755746\pi$
$440$	−1.51850	−0.0723917
$441$	5.58735	0.266064
$442$	−6.24867	−0.297219
$443$	−16.4321	−0.780712	−0.390356	−	0.920664i	$-0.627648\pi$
$443$	−16.4321	−0.780712	−0.390356	+	0.920664i	$0.627648\pi$
$444$	0.652501	0.0309663
$445$	7.33300	0.347618
$446$	−13.1473	−0.622545
$447$	−2.61205	−0.123546
$448$	−2.22955	−0.105336
$449$	−22.6459	−1.06873	−0.534363	−	0.845255i	$-0.679448\pi$
$449$	−22.6459	−1.06873	−0.534363	+	0.845255i	$0.679448\pi$
$450$	7.41858	0.349715
$451$	−4.42379	−0.208308
$452$	−0.334347	−0.0157264
$453$	6.31216	0.296571
$454$	4.02185	0.188755
$455$	19.7419	0.925516
$456$	−2.98529	−0.139799
$457$	−12.1708	−0.569327	−0.284663	−	0.958628i	$-0.591882\pi$
$457$	−12.1708	−0.569327	−0.284663	+	0.958628i	$0.591882\pi$
$458$	15.2597	0.713040
$459$	3.06057	0.142855
$460$	−9.84706	−0.459122
$461$	−2.73188	−0.127237	−0.0636183	−	0.997974i	$-0.520264\pi$
$461$	−2.73188	−0.127237	−0.0636183	+	0.997974i	$0.520264\pi$
$462$	−1.10675	−0.0514909
$463$	−8.82659	−0.410207	−0.205103	−	0.978740i	$-0.565753\pi$
$463$	−8.82659	−0.410207	−0.205103	+	0.978740i	$0.565753\pi$
$464$	3.30200	0.153291
$465$	−1.07251	−0.0497363
$466$	0.531295	0.0246118
$467$	9.46888	0.438167	0.219084	−	0.975706i	$-0.429693\pi$
$467$	9.46888	0.438167	0.219084	+	0.975706i	$0.429693\pi$
$468$	−16.0567	−0.742221
$469$	−4.09237	−0.188968
$470$	19.7723	0.912026
$471$	1.79475	0.0826979
$472$	−11.8690	−0.546316
$473$	−4.90620	−0.225587
$474$	3.43808	0.157916
$475$	16.2022	0.743408
$476$	2.38917	0.109507
$477$	−17.2238	−0.788626
$478$	−14.5924	−0.667442
$479$	15.0802	0.689034	0.344517	−	0.938780i	$-0.388043\pi$
$479$	15.0802	0.689034	0.344517	+	0.938780i	$0.388043\pi$
$480$	−0.753789	−0.0344056
$481$	7.66487	0.349488
$482$	−29.6308	−1.34965
$483$	−7.17700	−0.326565
$484$	1.00000	0.0454545
$485$	5.27922	0.239717
$486$	11.9652	0.542751
$487$	−31.8317	−1.44243	−0.721215	−	0.692711i	$-0.756416\pi$
$487$	−31.8317	−1.44243	−0.721215	+	0.692711i	$0.756416\pi$
$488$	−5.34067	−0.241761
$489$	−5.44696	−0.246320
$490$	3.08122	0.139195
$491$	−27.3864	−1.23593	−0.617965	−	0.786205i	$-0.712043\pi$
$491$	−27.3864	−1.23593	−0.617965	+	0.786205i	$0.712043\pi$
$492$	−2.19598	−0.0990026
$493$	−3.53840	−0.159361
$494$	−35.0679	−1.57778
$495$	4.18132	0.187936
$496$	1.42282	0.0638865
$497$	22.2641	0.998680
$498$	−3.88819	−0.174234
$499$	−28.3988	−1.27131	−0.635653	−	0.771975i	$-0.719269\pi$
$499$	−28.3988	−1.27131	−0.635653	+	0.771975i	$0.719269\pi$
$500$	11.6836	0.522506
$501$	−9.67995	−0.432468
$502$	6.83584	0.305099
$503$	20.3481	0.907277	0.453639	−	0.891186i	$-0.350126\pi$
$503$	20.3481	0.907277	0.453639	+	0.891186i	$0.350126\pi$
$504$	6.13925	0.273464
$505$	−8.56908	−0.381319
$506$	6.48472	0.288281
$507$	10.4259	0.463030
$508$	10.2961	0.456814
$509$	22.3718	0.991614	0.495807	−	0.868433i	$-0.334872\pi$
$509$	22.3718	0.991614	0.495807	+	0.868433i	$0.334872\pi$
$510$	0.807755	0.0357680
$511$	31.8326	1.40819
$512$	1.00000	0.0441942
$513$	17.1761	0.758343
$514$	13.3608	0.589321
$515$	−6.90690	−0.304354
$516$	−2.43545	−0.107215
$517$	−13.0209	−0.572659
$518$	−2.93065	−0.128765
$519$	−1.88323	−0.0826646
$520$	−8.85468	−0.388304
$521$	7.95074	0.348328	0.174164	−	0.984717i	$-0.444278\pi$
$521$	7.95074	0.348328	0.174164	+	0.984717i	$0.444278\pi$
$522$	−9.09233	−0.397961
$523$	−9.66346	−0.422553	−0.211277	−	0.977426i	$-0.567762\pi$
$523$	−9.66346	−0.422553	−0.211277	+	0.977426i	$0.567762\pi$
$524$	6.39532	0.279381
$525$	2.98177	0.130135
$526$	16.2037	0.706515
$527$	−1.52468	−0.0664163
$528$	0.496403	0.0216032
$529$	19.0516	0.828332
$530$	−9.49833	−0.412581
$531$	32.6823	1.41829
$532$	13.4081	0.581316
$533$	−25.7960	−1.11735
$534$	−2.39718	−0.103736
$535$	21.4295	0.926480
$536$	1.83551	0.0792822
$537$	3.44029	0.148459
$538$	27.7964	1.19839
$539$	−2.02912	−0.0874003
$540$	4.33699	0.186634
$541$	−27.4254	−1.17911	−0.589556	−	0.807728i	$-0.700697\pi$
$541$	−27.4254	−1.17911	−0.589556	+	0.807728i	$0.700697\pi$
$542$	−18.5674	−0.797540
$543$	−3.16842	−0.135970
$544$	−1.07159	−0.0459442
$545$	5.74663	0.246158
$546$	−6.45371	−0.276193
$547$	−41.2540	−1.76389	−0.881947	−	0.471349i	$-0.843767\pi$
$547$	−41.2540	−1.76389	−0.881947	+	0.471349i	$0.843767\pi$
$548$	−15.7198	−0.671518
$549$	14.7060	0.627636
$550$	−2.69415	−0.114879
$551$	−19.8577	−0.845965
$552$	3.21904	0.137011
$553$	−15.4418	−0.656652
$554$	2.51646	0.106914
$555$	−0.990824	−0.0420582
$556$	−6.53072	−0.276964
$557$	−31.9367	−1.35320	−0.676601	−	0.736349i	$-0.736548\pi$
$557$	−31.9367	−1.35320	−0.676601	+	0.736349i	$0.736548\pi$
$558$	−3.91785	−0.165856
$559$	−28.6090	−1.21003
$560$	3.38557	0.143066
$561$	−0.531942	−0.0224586
$562$	29.3540	1.23823
$563$	8.68076	0.365850	0.182925	−	0.983127i	$-0.441443\pi$
$563$	8.68076	0.365850	0.182925	+	0.983127i	$0.441443\pi$
$564$	−6.46362	−0.272167
$565$	0.507707	0.0213594
$566$	11.9928	0.504094
$567$	−15.2567	−0.640723
$568$	−9.98592	−0.419000
$569$	−3.96001	−0.166012	−0.0830061	−	0.996549i	$-0.526452\pi$
$569$	−3.96001	−0.166012	−0.0830061	+	0.996549i	$0.526452\pi$
$570$	4.53316	0.189873
$571$	18.8206	0.787616	0.393808	−	0.919193i	$-0.371158\pi$
$571$	18.8206	0.787616	0.393808	+	0.919193i	$0.371158\pi$
$572$	5.83120	0.243815
$573$	−6.51167	−0.272029
$574$	9.86305	0.411676
$575$	−17.4708	−0.728584
$576$	−2.75358	−0.114733
$577$	−30.2485	−1.25926	−0.629631	−	0.776895i	$-0.716794\pi$
$577$	−30.2485	−1.25926	−0.629631	+	0.776895i	$0.716794\pi$
$578$	−15.8517	−0.659343
$579$	1.50910	0.0627161
$580$	−5.01409	−0.208199
$581$	17.4634	0.724506
$582$	−1.72579	−0.0715364
$583$	6.25507	0.259058
$584$	−14.2776	−0.590811
$585$	24.3821	1.00808
$586$	28.1122	1.16131
$587$	2.29850	0.0948694	0.0474347	−	0.998874i	$-0.484895\pi$
$587$	2.29850	0.0948694	0.0474347	+	0.998874i	$0.484895\pi$
$588$	−1.00726	−0.0415387
$589$	−8.55660	−0.352569
$590$	18.0231	0.742000
$591$	−0.496403	−0.0204193
$592$	1.31446	0.0540239
$593$	11.7360	0.481941	0.240970	−	0.970532i	$-0.422534\pi$
$593$	11.7360	0.481941	0.240970	+	0.970532i	$0.422534\pi$
$594$	−2.85610	−0.117187
$595$	−3.62795	−0.148732
$596$	−5.26195	−0.215538
$597$	−1.52211	−0.0622957
$598$	37.8137	1.54632
$599$	44.5834	1.82163	0.910813	−	0.412819i	$-0.135456\pi$
$599$	44.5834	1.82163	0.910813	+	0.412819i	$0.135456\pi$
$600$	−1.33739	−0.0545986
$601$	−9.93149	−0.405114	−0.202557	−	0.979270i	$-0.564925\pi$
$601$	−9.93149	−0.405114	−0.202557	+	0.979270i	$0.564925\pi$
$602$	10.9386	0.445824
$603$	−5.05424	−0.205825
$604$	12.7158	0.517398
$605$	−1.51850	−0.0617359
$606$	2.80126	0.113793
$607$	−18.3177	−0.743492	−0.371746	−	0.928334i	$-0.621241\pi$
$607$	−18.3177	−0.743492	−0.371746	+	0.928334i	$0.621241\pi$
$608$	−6.01383	−0.243893
$609$	−3.65450	−0.148088
$610$	8.10981	0.328357
$611$	−75.9275	−3.07170
$612$	2.95072	0.119276
$613$	1.20625	0.0487199	0.0243600	−	0.999703i	$-0.492245\pi$
$613$	1.20625	0.0487199	0.0243600	+	0.999703i	$0.492245\pi$
$614$	−10.8297	−0.437050
$615$	3.33460	0.134464
$616$	−2.22955	−0.0898310
$617$	−29.8587	−1.20207	−0.601033	−	0.799224i	$-0.705244\pi$
$617$	−29.8587	−1.20207	−0.601033	+	0.799224i	$0.705244\pi$
$618$	2.25789	0.0908256
$619$	7.21916	0.290163	0.145081	−	0.989420i	$-0.453656\pi$
$619$	7.21916	0.290163	0.145081	+	0.989420i	$0.453656\pi$
$620$	−2.16055	−0.0867699
$621$	−18.5210	−0.743222
$622$	29.6731	1.18978
$623$	10.7667	0.431360
$624$	2.89463	0.115878
$625$	−4.27077	−0.170831
$626$	11.7046	0.467810
$627$	−2.98529	−0.119221
$628$	3.61551	0.144275
$629$	−1.40856	−0.0561631
$630$	−9.32245	−0.371415
$631$	20.1070	0.800448	0.400224	−	0.916417i	$-0.368932\pi$
$631$	20.1070	0.800448	0.400224	+	0.916417i	$0.368932\pi$
$632$	6.92598	0.275501
$633$	7.05029	0.280224
$634$	−2.05561	−0.0816388
$635$	−15.6346	−0.620440
$636$	3.10504	0.123123
$637$	−11.8322	−0.468808
$638$	3.30200	0.130727
$639$	27.4971	1.08777
$640$	−1.51850	−0.0600240
$641$	−31.9038	−1.26012	−0.630061	−	0.776545i	$-0.716970\pi$
$641$	−31.9038	−1.26012	−0.630061	+	0.776545i	$0.716970\pi$
$642$	−7.00539	−0.276481
$643$	13.2304	0.521756	0.260878	−	0.965372i	$-0.415988\pi$
$643$	13.2304	0.521756	0.260878	+	0.965372i	$0.415988\pi$
$644$	−14.4580	−0.569725
$645$	3.69824	0.145618
$646$	6.44438	0.253551
$647$	30.8484	1.21278	0.606388	−	0.795169i	$-0.292618\pi$
$647$	30.8484	1.21278	0.606388	+	0.795169i	$0.292618\pi$
$648$	6.84297	0.268817
$649$	−11.8690	−0.465900
$650$	−15.7101	−0.616203
$651$	−1.57471	−0.0617179
$652$	−10.9728	−0.429730
$653$	45.9356	1.79760	0.898799	−	0.438361i	$-0.144441\pi$
$653$	45.9356	1.79760	0.898799	+	0.438361i	$0.144441\pi$
$654$	−1.87859	−0.0734588
$655$	−9.71130	−0.379452
$656$	−4.42379	−0.172720
$657$	39.3146	1.53381
$658$	29.0307	1.13174
$659$	36.5931	1.42546	0.712732	−	0.701436i	$-0.247458\pi$
$659$	36.5931	1.42546	0.712732	+	0.701436i	$0.247458\pi$
$660$	−0.753789	−0.0293412
$661$	1.84238	0.0716602	0.0358301	−	0.999358i	$-0.488592\pi$
$661$	1.84238	0.0716602	0.0358301	+	0.999358i	$0.488592\pi$
$662$	25.0928	0.975261
$663$	−3.10186	−0.120466
$664$	−7.83273	−0.303969
$665$	−20.3603	−0.789537
$666$	−3.61947	−0.140252
$667$	21.4126	0.829097
$668$	−19.5002	−0.754485
$669$	−6.52639	−0.252325
$670$	−2.78723	−0.107680
$671$	−5.34067	−0.206174
$672$	−1.10675	−0.0426940
$673$	6.40603	0.246934	0.123467	−	0.992349i	$-0.460599\pi$
$673$	6.40603	0.246934	0.123467	+	0.992349i	$0.460599\pi$
$674$	−13.6170	−0.524507
$675$	7.69477	0.296172
$676$	21.0029	0.807803
$677$	12.3288	0.473834	0.236917	−	0.971530i	$-0.423863\pi$
$677$	12.3288	0.473834	0.236917	+	0.971530i	$0.423863\pi$
$678$	−0.165971	−0.00637408
$679$	7.75123	0.297465
$680$	1.62722	0.0624009
$681$	1.99646	0.0765045
$682$	1.42282	0.0544826
$683$	−27.8856	−1.06701	−0.533506	−	0.845796i	$-0.679126\pi$
$683$	−27.8856	−1.06701	−0.533506	+	0.845796i	$0.679126\pi$
$684$	16.5596	0.633172
$685$	23.8706	0.912048
$686$	20.1308	0.768599
$687$	7.57498	0.289004
$688$	−4.90620	−0.187047
$689$	36.4745	1.38957
$690$	−4.88811	−0.186087
$691$	19.3597	0.736478	0.368239	−	0.929731i	$-0.379961\pi$
$691$	19.3597	0.736478	0.368239	+	0.929731i	$0.379961\pi$
$692$	−3.79375	−0.144217
$693$	6.13925	0.233211
$694$	30.7835	1.16853
$695$	9.91691	0.376170
$696$	1.63912	0.0621308
$697$	4.74050	0.179559
$698$	3.52795	0.133535
$699$	0.263737	0.00997544
$700$	6.00674	0.227034
$701$	28.7483	1.08581	0.542905	−	0.839794i	$-0.317324\pi$
$701$	28.7483	1.08581	0.542905	+	0.839794i	$0.317324\pi$
$702$	−16.6545	−0.628583
$703$	−7.90493	−0.298140
$704$	1.00000	0.0376889
$705$	9.81501	0.369655
$706$	−15.3527	−0.577805
$707$	−12.5816	−0.473179
$708$	−5.89182	−0.221428
$709$	−9.13180	−0.342952	−0.171476	−	0.985188i	$-0.554854\pi$
$709$	−9.13180	−0.342952	−0.171476	+	0.985188i	$0.554854\pi$
$710$	15.1636	0.569081
$711$	−19.0713	−0.715228
$712$	−4.82911	−0.180978
$713$	9.22659	0.345539
$714$	1.18599	0.0443846
$715$	−8.85468	−0.331146
$716$	6.93043	0.259002
$717$	−7.24373	−0.270522
$718$	−8.31396	−0.310274
$719$	18.4169	0.686833	0.343417	−	0.939183i	$-0.388416\pi$
$719$	18.4169	0.686833	0.343417	+	0.939183i	$0.388416\pi$
$720$	4.18132	0.155829
$721$	−10.1411	−0.377674
$722$	17.1662	0.638860
$723$	−14.7088	−0.547027
$724$	−6.38275	−0.237213
$725$	−8.89609	−0.330393
$726$	0.496403	0.0184233
$727$	20.3259	0.753845	0.376923	−	0.926245i	$-0.376982\pi$
$727$	20.3259	0.753845	0.376923	+	0.926245i	$0.376982\pi$
$728$	−13.0009	−0.481847
$729$	−14.5894	−0.540347
$730$	21.6806	0.802434
$731$	5.25745	0.194454
$732$	−2.65113	−0.0979884
$733$	−36.7138	−1.35606	−0.678028	−	0.735036i	$-0.737165\pi$
$733$	−36.7138	−1.35606	−0.678028	+	0.735036i	$0.737165\pi$
$734$	−30.5996	−1.12945
$735$	1.52953	0.0564175
$736$	6.48472	0.239030
$737$	1.83551	0.0676121
$738$	12.1813	0.448399
$739$	32.4101	1.19222	0.596112	−	0.802901i	$-0.296711\pi$
$739$	32.4101	1.19222	0.596112	+	0.802901i	$0.296711\pi$
$740$	−1.99601	−0.0733747
$741$	−17.4078	−0.639492
$742$	−13.9460	−0.511973
$743$	24.3080	0.891774	0.445887	−	0.895089i	$-0.352888\pi$
$743$	24.3080	0.891774	0.445887	+	0.895089i	$0.352888\pi$
$744$	0.706293	0.0258939
$745$	7.99027	0.292741
$746$	−1.30456	−0.0477633
$747$	21.5681	0.789135
$748$	−1.07159	−0.0391813
$749$	31.4640	1.14967
$750$	5.79977	0.211778
$751$	38.0965	1.39016	0.695081	−	0.718932i	$-0.255369\pi$
$751$	38.0965	1.39016	0.695081	+	0.718932i	$0.255369\pi$
$752$	−13.0209	−0.474823
$753$	3.39334	0.123660
$754$	19.2546	0.701212
$755$	−19.3089	−0.702724
$756$	6.36781	0.231595
$757$	−3.63943	−0.132277	−0.0661387	−	0.997810i	$-0.521068\pi$
$757$	−3.63943	−0.132277	−0.0661387	+	0.997810i	$0.521068\pi$
$758$	19.4767	0.707427
$759$	3.21904	0.116844
$760$	9.13202	0.331253
$761$	−12.3517	−0.447749	−0.223875	−	0.974618i	$-0.571871\pi$
$761$	−12.3517	−0.447749	−0.223875	+	0.974618i	$0.571871\pi$
$762$	5.11100	0.185152
$763$	8.43752	0.305459
$764$	−13.1177	−0.474582
$765$	−4.48067	−0.161999
$766$	21.1789	0.765224
$767$	−69.2106	−2.49905
$768$	0.496403	0.0179124
$769$	−21.7965	−0.786002	−0.393001	−	0.919538i	$-0.628563\pi$
$769$	−21.7965	−0.786002	−0.393001	+	0.919538i	$0.628563\pi$
$770$	3.38557	0.122008
$771$	6.63236	0.238858
$772$	3.04007	0.109414
$773$	−36.4772	−1.31199	−0.655997	−	0.754763i	$-0.727752\pi$
$773$	−36.4772	−1.31199	−0.655997	+	0.754763i	$0.727752\pi$
$774$	13.5096	0.485594
$775$	−3.83330	−0.137696
$776$	−3.47660	−0.124803
$777$	−1.45478	−0.0521900
$778$	−7.33640	−0.263023
$779$	26.6039	0.953185
$780$	−4.39549	−0.157384
$781$	−9.98592	−0.357324
$782$	−6.94898	−0.248495
$783$	−9.43083	−0.337031
$784$	−2.02912	−0.0724685
$785$	−5.49016	−0.195952
$786$	3.17466	0.113236
$787$	47.0974	1.67884	0.839421	−	0.543481i	$-0.182894\pi$
$787$	47.0974	1.67884	0.839421	+	0.543481i	$0.182894\pi$
$788$	−1.00000	−0.0356235
$789$	8.04357	0.286359
$790$	−10.5171	−0.374182
$791$	0.745443	0.0265049
$792$	−2.75358	−0.0978443
$793$	−31.1425	−1.10590
$794$	−19.7596	−0.701241
$795$	−4.71500	−0.167224
$796$	−3.06627	−0.108681
$797$	−8.56751	−0.303477	−0.151738	−	0.988421i	$-0.548487\pi$
$797$	−8.56751	−0.303477	−0.151738	+	0.988421i	$0.548487\pi$
$798$	6.65584	0.235614
$799$	13.9531	0.493625
$800$	−2.69415	−0.0952527
$801$	13.2973	0.469839
$802$	18.8294	0.664890
$803$	−14.2776	−0.503846
$804$	0.911156	0.0321340
$805$	21.9545	0.773794
$806$	8.29675	0.292240
$807$	13.7982	0.485721
$808$	5.64311	0.198524
$809$	45.9624	1.61595	0.807975	−	0.589217i	$-0.200563\pi$
$809$	45.9624	1.61595	0.807975	+	0.589217i	$0.200563\pi$
$810$	−10.3911	−0.365105
$811$	−12.8067	−0.449704	−0.224852	−	0.974393i	$-0.572190\pi$
$811$	−12.8067	−0.449704	−0.224852	+	0.974393i	$0.572190\pi$
$812$	−7.36196	−0.258354
$813$	−9.21694	−0.323252
$814$	1.31446	0.0460717
$815$	16.6623	0.583654
$816$	−0.531942	−0.0186217
$817$	29.5051	1.03225
$818$	14.1843	0.495941
$819$	35.7992	1.25092
$820$	6.71753	0.234586
$821$	56.5123	1.97229	0.986147	−	0.165875i	$-0.0530448\pi$
$821$	56.5123	1.97229	0.986147	+	0.165875i	$0.0530448\pi$
$822$	−7.80338	−0.272174
$823$	21.1939	0.738773	0.369387	−	0.929276i	$-0.379568\pi$
$823$	21.1939	0.738773	0.369387	+	0.929276i	$0.379568\pi$
$824$	4.54850	0.158454
$825$	−1.33739	−0.0465618
$826$	26.4625	0.920750
$827$	−3.33480	−0.115962	−0.0579812	−	0.998318i	$-0.518466\pi$
$827$	−3.33480	−0.115962	−0.0579812	+	0.998318i	$0.518466\pi$
$828$	−17.8562	−0.620547
$829$	3.31930	0.115284	0.0576420	−	0.998337i	$-0.481642\pi$
$829$	3.31930	0.115284	0.0576420	+	0.998337i	$0.481642\pi$
$830$	11.8940	0.412847
$831$	1.24918	0.0433336
$832$	5.83120	0.202160
$833$	2.17439	0.0753381
$834$	−3.24187	−0.112257
$835$	29.6111	1.02473
$836$	−6.01383	−0.207993
$837$	−4.06371	−0.140462
$838$	−34.8319	−1.20325
$839$	−30.7564	−1.06183	−0.530915	−	0.847425i	$-0.678152\pi$
$839$	−30.7564	−1.06183	−0.530915	+	0.847425i	$0.678152\pi$
$840$	1.68061	0.0579865
$841$	−18.0968	−0.624028
$842$	35.5091	1.22373
$843$	14.5714	0.501867
$844$	14.2027	0.488879
$845$	−31.8929	−1.09715
$846$	35.8541	1.23269
$847$	−2.22955	−0.0766082
$848$	6.25507	0.214800
$849$	5.95326	0.204315
$850$	2.88704	0.0990245
$851$	8.52390	0.292195
$852$	−4.95704	−0.169825
$853$	−25.3568	−0.868200	−0.434100	−	0.900865i	$-0.642934\pi$
$853$	−25.3568	−0.868200	−0.434100	+	0.900865i	$0.642934\pi$
$854$	11.9073	0.407459
$855$	−25.1458	−0.859967
$856$	−14.1123	−0.482348
$857$	18.3283	0.626083	0.313042	−	0.949739i	$-0.398652\pi$
$857$	18.3283	0.626083	0.313042	+	0.949739i	$0.398652\pi$
$858$	2.89463	0.0988209
$859$	−49.8598	−1.70119	−0.850597	−	0.525818i	$-0.823759\pi$
$859$	−49.8598	−1.70119	−0.850597	+	0.525818i	$0.823759\pi$
$860$	7.45007	0.254045
$861$	4.89605	0.166857
$862$	−27.3834	−0.932681
$863$	−50.1994	−1.70881	−0.854403	−	0.519611i	$-0.826077\pi$
$863$	−50.1994	−1.70881	−0.854403	+	0.519611i	$0.826077\pi$
$864$	−2.85610	−0.0971664
$865$	5.76081	0.195874
$866$	−6.38329	−0.216913
$867$	−7.86883	−0.267240
$868$	−3.17224	−0.107673
$869$	6.92598	0.234948
$870$	−2.48901	−0.0843854
$871$	10.7033	0.362666
$872$	−3.78441	−0.128156
$873$	9.57310	0.324000
$874$	−38.9981	−1.31913
$875$	−26.0491	−0.880620
$876$	−7.08745	−0.239463
$877$	28.2789	0.954909	0.477455	−	0.878656i	$-0.341559\pi$
$877$	28.2789	0.954909	0.477455	+	0.878656i	$0.341559\pi$
$878$	−30.1611	−1.01789
$879$	13.9550	0.470690
$880$	−1.51850	−0.0511887
$881$	−7.13214	−0.240288	−0.120144	−	0.992756i	$-0.538336\pi$
$881$	−7.13214	−0.240288	−0.120144	+	0.992756i	$0.538336\pi$
$882$	5.58735	0.188136
$883$	−20.5759	−0.692435	−0.346218	−	0.938154i	$-0.612534\pi$
$883$	−20.5759	−0.692435	−0.346218	+	0.938154i	$0.612534\pi$
$884$	−6.24867	−0.210166
$885$	8.94674	0.300741
$886$	−16.4321	−0.552047
$887$	−3.95094	−0.132659	−0.0663297	−	0.997798i	$-0.521129\pi$
$887$	−3.95094	−0.132659	−0.0663297	+	0.997798i	$0.521129\pi$
$888$	0.652501	0.0218965
$889$	−22.9556	−0.769905
$890$	7.33300	0.245803
$891$	6.84297	0.229248
$892$	−13.1473	−0.440206
$893$	78.3055	2.62039
$894$	−2.61205	−0.0873599
$895$	−10.5239	−0.351774
$896$	−2.22955	−0.0744840
$897$	18.7709	0.626741
$898$	−22.6459	−0.755703
$899$	4.69815	0.156692
$900$	7.41858	0.247286
$901$	−6.70288	−0.223305
$902$	−4.42379	−0.147296
$903$	5.42996	0.180698
$904$	−0.334347	−0.0111202
$905$	9.69221	0.322180
$906$	6.31216	0.209707
$907$	2.27082	0.0754013	0.0377007	−	0.999289i	$-0.487997\pi$
$907$	2.27082	0.0754013	0.0377007	+	0.999289i	$0.487997\pi$
$908$	4.02185	0.133470
$909$	−15.5388	−0.515389
$910$	19.7419	0.654439
$911$	−56.7910	−1.88157	−0.940784	−	0.339006i	$-0.889909\pi$
$911$	−56.7910	−1.88157	−0.940784	+	0.339006i	$0.889909\pi$
$912$	−2.98529	−0.0988528
$913$	−7.83273	−0.259226
$914$	−12.1708	−0.402575
$915$	4.02574	0.133087
$916$	15.2597	0.504196
$917$	−14.2587	−0.470863
$918$	3.06057	0.101014
$919$	−41.3849	−1.36516	−0.682581	−	0.730810i	$-0.739143\pi$
$919$	−41.3849	−1.36516	−0.682581	+	0.730810i	$0.739143\pi$
$920$	−9.84706	−0.324648
$921$	−5.37588	−0.177141
$922$	−2.73188	−0.0899698
$923$	−58.2299	−1.91666
$924$	−1.10675	−0.0364096
$925$	−3.54135	−0.116439
$926$	−8.82659	−0.290060
$927$	−12.5247	−0.411364
$928$	3.30200	0.108393
$929$	39.7912	1.30551	0.652754	−	0.757570i	$-0.273614\pi$
$929$	39.7912	1.30551	0.652754	+	0.757570i	$0.273614\pi$
$930$	−1.07251	−0.0351689
$931$	12.2028	0.399930
$932$	0.531295	0.0174032
$933$	14.7298	0.482233
$934$	9.46888	0.309831
$935$	1.62722	0.0532156
$936$	−16.0567	−0.524829
$937$	−26.6778	−0.871527	−0.435764	−	0.900061i	$-0.643522\pi$
$937$	−26.6778	−0.871527	−0.435764	+	0.900061i	$0.643522\pi$
$938$	−4.09237	−0.133621
$939$	5.81021	0.189609
$940$	19.7723	0.644900
$941$	19.4591	0.634349	0.317175	−	0.948367i	$-0.397266\pi$
$941$	19.4591	0.634349	0.317175	+	0.948367i	$0.397266\pi$
$942$	1.79475	0.0584762
$943$	−28.6870	−0.934179
$944$	−11.8690	−0.386304
$945$	−9.66952	−0.314550
$946$	−4.90620	−0.159514
$947$	−31.0055	−1.00754	−0.503771	−	0.863837i	$-0.668055\pi$
$947$	−31.0055	−1.00754	−0.503771	+	0.863837i	$0.668055\pi$
$948$	3.43808	0.111664
$949$	−83.2555	−2.70259
$950$	16.2022	0.525669
$951$	−1.02041	−0.0330891
$952$	2.38917	0.0774334
$953$	21.0841	0.682981	0.341491	−	0.939885i	$-0.389068\pi$
$953$	21.0841	0.682981	0.341491	+	0.939885i	$0.389068\pi$
$954$	−17.2238	−0.557643
$955$	19.9192	0.644572
$956$	−14.5924	−0.471953
$957$	1.63912	0.0529853
$958$	15.0802	0.487221
$959$	35.0481	1.13176
$960$	−0.753789	−0.0243284
$961$	−28.9756	−0.934696
$962$	7.66487	0.247125
$963$	38.8594	1.25223
$964$	−29.6308	−0.954344
$965$	−4.61635	−0.148606
$966$	−7.17700	−0.230916
$967$	−47.0491	−1.51300	−0.756498	−	0.653996i	$-0.773091\pi$
$967$	−47.0491	−1.51300	−0.756498	+	0.653996i	$0.773091\pi$
$968$	1.00000	0.0321412
$969$	3.19901	0.102767
$970$	5.27922	0.169505
$971$	−45.3067	−1.45396	−0.726981	−	0.686657i	$-0.759077\pi$
$971$	−45.3067	−1.45396	−0.726981	+	0.686657i	$0.759077\pi$
$972$	11.9652	0.383783
$973$	14.5605	0.466790
$974$	−31.8317	−1.01995
$975$	−7.79857	−0.249754
$976$	−5.34067	−0.170951
$977$	59.1166	1.89131	0.945653	−	0.325177i	$-0.105424\pi$
$977$	59.1166	1.89131	0.945653	+	0.325177i	$0.105424\pi$
$978$	−5.44696	−0.174174
$979$	−4.82911	−0.154339
$980$	3.08122	0.0984259
$981$	10.4207	0.332707
$982$	−27.3864	−0.873935
$983$	−41.1405	−1.31218	−0.656089	−	0.754684i	$-0.727790\pi$
$983$	−41.1405	−1.31218	−0.656089	+	0.754684i	$0.727790\pi$
$984$	−2.19598	−0.0700054
$985$	1.51850	0.0483835
$986$	−3.53840	−0.112686
$987$	14.4109	0.458705
$988$	−35.0679	−1.11566
$989$	−31.8153	−1.01167
$990$	4.18132	0.132891
$991$	22.0219	0.699550	0.349775	−	0.936834i	$-0.386258\pi$
$991$	22.0219	0.699550	0.349775	+	0.936834i	$0.386258\pi$
$992$	1.42282	0.0451746
$993$	12.4562	0.395285
$994$	22.2641	0.706174
$995$	4.65614	0.147610
$996$	−3.88819	−0.123202
$997$	51.6742	1.63654	0.818269	−	0.574835i	$-0.194934\pi$
$997$	51.6742	1.63654	0.818269	+	0.574835i	$0.194934\pi$
$998$	−28.3988	−0.898949
$999$	−3.75422	−0.118778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.b.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.b.1.10	✓	15	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:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.496403 q^{3} +1.00000 q^{4} -1.51850 q^{5} +0.496403 q^{6} -2.22955 q^{7} +1.00000 q^{8} -2.75358 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.496403 q^{3} +1.00000 q^{4} -1.51850 q^{5} +0.496403 q^{6} -2.22955 q^{7} +1.00000 q^{8} -2.75358 q^{9} -1.51850 q^{10} +1.00000 q^{11} +0.496403 q^{12} +5.83120 q^{13} -2.22955 q^{14} -0.753789 q^{15} +1.00000 q^{16} -1.07159 q^{17} -2.75358 q^{18} -6.01383 q^{19} -1.51850 q^{20} -1.10675 q^{21} +1.00000 q^{22} +6.48472 q^{23} +0.496403 q^{24} -2.69415 q^{25} +5.83120 q^{26} -2.85610 q^{27} -2.22955 q^{28} +3.30200 q^{29} -0.753789 q^{30} +1.42282 q^{31} +1.00000 q^{32} +0.496403 q^{33} -1.07159 q^{34} +3.38557 q^{35} -2.75358 q^{36} +1.31446 q^{37} -6.01383 q^{38} +2.89463 q^{39} -1.51850 q^{40} -4.42379 q^{41} -1.10675 q^{42} -4.90620 q^{43} +1.00000 q^{44} +4.18132 q^{45} +6.48472 q^{46} -13.0209 q^{47} +0.496403 q^{48} -2.02912 q^{49} -2.69415 q^{50} -0.531942 q^{51} +5.83120 q^{52} +6.25507 q^{53} -2.85610 q^{54} -1.51850 q^{55} -2.22955 q^{56} -2.98529 q^{57} +3.30200 q^{58} -11.8690 q^{59} -0.753789 q^{60} -5.34067 q^{61} +1.42282 q^{62} +6.13925 q^{63} +1.00000 q^{64} -8.85468 q^{65} +0.496403 q^{66} +1.83551 q^{67} -1.07159 q^{68} +3.21904 q^{69} +3.38557 q^{70} -9.98592 q^{71} -2.75358 q^{72} -14.2776 q^{73} +1.31446 q^{74} -1.33739 q^{75} -6.01383 q^{76} -2.22955 q^{77} +2.89463 q^{78} +6.92598 q^{79} -1.51850 q^{80} +6.84297 q^{81} -4.42379 q^{82} -7.83273 q^{83} -1.10675 q^{84} +1.62722 q^{85} -4.90620 q^{86} +1.63912 q^{87} +1.00000 q^{88} -4.82911 q^{89} +4.18132 q^{90} -13.0009 q^{91} +6.48472 q^{92} +0.706293 q^{93} -13.0209 q^{94} +9.13202 q^{95} +0.496403 q^{96} -3.47660 q^{97} -2.02912 q^{98} -2.75358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9	\( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9} - 11 q^{10} + 15 q^{11} - 9 q^{12} - 21 q^{13} - 11 q^{14} - 2 q^{15} + 15 q^{16} - 4 q^{17} + 10 q^{18} - 22 q^{19} - 11 q^{20} - 13 q^{21} + 15 q^{22} - 16 q^{23} - 9 q^{24} + 6 q^{25} - 21 q^{26} - 21 q^{27} - 11 q^{28} - 8 q^{29} - 2 q^{30} - 33 q^{31} + 15 q^{32} - 9 q^{33} - 4 q^{34} - 2 q^{35} + 10 q^{36} - q^{37} - 22 q^{38} + q^{39} - 11 q^{40} - 10 q^{41} - 13 q^{42} - 8 q^{43} + 15 q^{44} - 10 q^{45} - 16 q^{46} - 31 q^{47} - 9 q^{48} + 2 q^{49} + 6 q^{50} + 2 q^{51} - 21 q^{52} - 18 q^{53} - 21 q^{54} - 11 q^{55} - 11 q^{56} + 16 q^{57} - 8 q^{58} - 37 q^{59} - 2 q^{60} - 31 q^{61} - 33 q^{62} - 20 q^{63} + 15 q^{64} - 13 q^{65} - 9 q^{66} + q^{67} - 4 q^{68} - 25 q^{69} - 2 q^{70} - 28 q^{71} + 10 q^{72} - 20 q^{73} - q^{74} - 9 q^{75} - 22 q^{76} - 11 q^{77} + q^{78} - 6 q^{79} - 11 q^{80} + 3 q^{81} - 10 q^{82} - 15 q^{83} - 13 q^{84} - 31 q^{85} - 8 q^{86} - 16 q^{87} + 15 q^{88} - 17 q^{89} - 10 q^{90} - 21 q^{91} - 16 q^{92} + 10 q^{93} - 31 q^{94} - 3 q^{95} - 9 q^{96} - 9 q^{97} + 2 q^{98} + 10 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9 - 11 * q^10 + 15 * q^11 - 9 * q^12 - 21 * q^13 - 11 * q^14 - 2 * q^15 + 15 * q^16 - 4 * q^17 + 10 * q^18 - 22 * q^19 - 11 * q^20 - 13 * q^21 + 15 * q^22 - 16 * q^23 - 9 * q^24 + 6 * q^25 - 21 * q^26 - 21 * q^27 - 11 * q^28 - 8 * q^29 - 2 * q^30 - 33 * q^31 + 15 * q^32 - 9 * q^33 - 4 * q^34 - 2 * q^35 + 10 * q^36 - q^37 - 22 * q^38 + q^39 - 11 * q^40 - 10 * q^41 - 13 * q^42 - 8 * q^43 + 15 * q^44 - 10 * q^45 - 16 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 + 6 * q^50 + 2 * q^51 - 21 * q^52 - 18 * q^53 - 21 * q^54 - 11 * q^55 - 11 * q^56 + 16 * q^57 - 8 * q^58 - 37 * q^59 - 2 * q^60 - 31 * q^61 - 33 * q^62 - 20 * q^63 + 15 * q^64 - 13 * q^65 - 9 * q^66 + q^67 - 4 * q^68 - 25 * q^69 - 2 * q^70 - 28 * q^71 + 10 * q^72 - 20 * q^73 - q^74 - 9 * q^75 - 22 * q^76 - 11 * q^77 + q^78 - 6 * q^79 - 11 * q^80 + 3 * q^81 - 10 * q^82 - 15 * q^83 - 13 * q^84 - 31 * q^85 - 8 * q^86 - 16 * q^87 + 15 * q^88 - 17 * q^89 - 10 * q^90 - 21 * q^91 - 16 * q^92 + 10 * q^93 - 31 * q^94 - 3 * q^95 - 9 * q^96 - 9 * q^97 + 2 * q^98 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more