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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [15,10,Mod(2,15)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("15.2"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.72553754246\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 8.11
Character \(\chi\) \(=\) 15.8
Dual form 15.10.e.a.2.11

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(10.0225 + 10.0225i) q^{2} +(-84.4162 + 112.058i) q^{3} -311.097i q^{4} +(-1254.86 - 615.193i) q^{5} +(-1969.17 + 277.038i) q^{6} +(2707.78 - 2707.78i) q^{7} +(8249.53 - 8249.53i) q^{8} +(-5430.82 - 18919.0i) q^{9} +(-6411.06 - 18742.6i) q^{10} +25188.1i q^{11} +(34860.8 + 26261.6i) q^{12} +(-131855. - 131855. i) q^{13} +54277.7 q^{14} +(174867. - 88683.9i) q^{15} +6080.90 q^{16} +(-274335. - 274335. i) q^{17} +(135185. - 244047. i) q^{18} +504083. i q^{19} +(-191385. + 390382. i) q^{20} +(74846.9 + 532007. i) q^{21} +(-252449. + 252449. i) q^{22} +(-357367. + 357367. i) q^{23} +(228029. + 1.62082e6i) q^{24} +(1.19620e6 + 1.54396e6i) q^{25} -2.64304e6i q^{26} +(2.57846e6 + 988501. i) q^{27} +(-842382. - 842382. i) q^{28} -1.44978e6 q^{29} +(2.64145e6 + 863775. i) q^{30} -1.57256e6 q^{31} +(-4.16281e6 - 4.16281e6i) q^{32} +(-2.82252e6 - 2.12629e6i) q^{33} -5.49908e6i q^{34} +(-5.06368e6 + 1.73207e6i) q^{35} +(-5.88563e6 + 1.68951e6i) q^{36} +(1.41364e7 - 1.41364e7i) q^{37} +(-5.05220e6 + 5.05220e6i) q^{38} +(2.59060e7 - 3.64466e6i) q^{39} +(-1.54270e7 + 5.27692e6i) q^{40} -1.52870e7i q^{41} +(-4.58191e6 + 6.08223e6i) q^{42} +(1.45503e7 + 1.45503e7i) q^{43} +7.83596e6 q^{44} +(-4.82391e6 + 2.70816e7i) q^{45} -7.16345e6 q^{46} +(-2.49426e7 - 2.49426e7i) q^{47} +(-513326. + 681411. i) q^{48} +2.56895e7i q^{49} +(-3.48539e6 + 2.74634e7i) q^{50} +(5.38997e7 - 7.58302e6i) q^{51} +(-4.10196e7 + 4.10196e7i) q^{52} +(-2.38697e7 + 2.38697e7i) q^{53} +(1.59354e7 + 3.57501e7i) q^{54} +(1.54956e7 - 3.16075e7i) q^{55} -4.46758e7i q^{56} +(-5.64864e7 - 4.25528e7i) q^{57} +(-1.45305e7 - 1.45305e7i) q^{58} +1.41279e8 q^{59} +(-2.75893e7 - 5.44007e7i) q^{60} +2.77276e7 q^{61} +(-1.57611e7 - 1.57611e7i) q^{62} +(-6.59338e7 - 3.65229e7i) q^{63} -8.65574e7i q^{64} +(8.43427e7 + 2.46575e8i) q^{65} +(-6.97806e6 - 4.95997e7i) q^{66} +(5.16735e7 - 5.16735e7i) q^{67} +(-8.53449e7 + 8.53449e7i) q^{68} +(-9.87812e6 - 7.02132e7i) q^{69} +(-6.81107e7 - 3.33912e7i) q^{70} -2.50202e8i q^{71} +(-2.00874e8 - 1.11271e8i) q^{72} +(-8.81264e7 - 8.81264e7i) q^{73} +2.83365e8 q^{74} +(-2.73991e8 + 3.70861e6i) q^{75} +1.56819e8 q^{76} +(6.82039e7 + 6.82039e7i) q^{77} +(2.96173e8 + 2.23116e8i) q^{78} +3.89035e8i q^{79} +(-7.63065e6 - 3.74092e6i) q^{80} +(-3.28433e8 + 2.05491e8i) q^{81} +(1.53214e8 - 1.53214e8i) q^{82} +(-9.47356e6 + 9.47356e6i) q^{83} +(1.65506e8 - 2.32846e7i) q^{84} +(1.75482e8 + 5.13020e8i) q^{85} +2.91662e8i q^{86} +(1.22385e8 - 1.62459e8i) q^{87} +(2.07790e8 + 2.07790e8i) q^{88} -4.88281e8 q^{89} +(-3.19774e8 + 2.23078e8i) q^{90} -7.14067e8 q^{91} +(1.11176e8 + 1.11176e8i) q^{92} +(1.32750e8 - 1.76218e8i) q^{93} -4.99976e8i q^{94} +(3.10108e8 - 6.32552e8i) q^{95} +(8.17884e8 - 1.15066e8i) q^{96} +(7.00398e7 - 7.00398e7i) q^{97} +(-2.57474e8 + 2.57474e8i) q^{98} +(4.76533e8 - 1.36792e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 150 q^{3} - 4548 q^{6} - 9760 q^{7} - 39820 q^{10} + 191940 q^{12} + 114260 q^{13} - 40230 q^{15} - 1494724 q^{16} + 994800 q^{18} + 592212 q^{21} + 308260 q^{22} + 5808020 q^{25} + 4786830 q^{27}+ \cdots + 3987642200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0225 + 10.0225i 0.442938 + 0.442938i 0.892998 − 0.450060i \(-0.148597\pi\)
−0.450060 + 0.892998i \(0.648597\pi\)
\(3\) −84.4162 + 112.058i −0.601700 + 0.798722i
\(4\) − 311.097i − 0.607611i
\(5\) −1254.86 − 615.193i −0.897902 − 0.440196i
\(6\) −1969.17 + 277.038i −0.620300 + 0.0872686i
\(7\) 2707.78 − 2707.78i 0.426257 − 0.426257i −0.461094 − 0.887351i \(-0.652543\pi\)
0.887351 + 0.461094i \(0.152543\pi\)
\(8\) 8249.53 − 8249.53i 0.712073 − 0.712073i
\(9\) −5430.82 − 18919.0i −0.275914 − 0.961182i
\(10\) −6411.06 − 18742.6i −0.202735 − 0.592695i
\(11\) 25188.1i 0.518715i 0.965781 + 0.259358i \(0.0835109\pi\)
−0.965781 + 0.259358i \(0.916489\pi\)
\(12\) 34860.8 + 26261.6i 0.485313 + 0.365600i
\(13\) −131855. − 131855.i −1.28042 − 1.28042i −0.940431 − 0.339984i \(-0.889578\pi\)
−0.339984 − 0.940431i \(-0.610422\pi\)
\(14\) 54277.7 0.377611
\(15\) 174867. − 88683.9i 0.891862 − 0.452308i
\(16\) 6080.90 0.0231968
\(17\) −274335. − 274335.i −0.796639 − 0.796639i 0.185925 − 0.982564i \(-0.440472\pi\)
−0.982564 + 0.185925i \(0.940472\pi\)
\(18\) 135185. − 244047.i 0.303531 − 0.547957i
\(19\) 504083.i 0.887383i 0.896180 + 0.443691i \(0.146331\pi\)
−0.896180 + 0.443691i \(0.853669\pi\)
\(20\) −191385. + 390382.i −0.267468 + 0.545575i
\(21\) 74846.9 + 532007.i 0.0839821 + 0.596940i
\(22\) −252449. + 252449.i −0.229759 + 0.229759i
\(23\) −357367. + 357367.i −0.266280 + 0.266280i −0.827599 − 0.561319i \(-0.810294\pi\)
0.561319 + 0.827599i \(0.310294\pi\)
\(24\) 228029. + 1.62082e6i 0.140294 + 0.997202i
\(25\) 1.19620e6 + 1.54396e6i 0.612455 + 0.790505i
\(26\) − 2.64304e6i − 1.13429i
\(27\) 2.57846e6 + 988501.i 0.933735 + 0.357965i
\(28\) −842382. − 842382.i −0.258999 − 0.258999i
\(29\) −1.44978e6 −0.380636 −0.190318 − 0.981722i \(-0.560952\pi\)
−0.190318 + 0.981722i \(0.560952\pi\)
\(30\) 2.64145e6 + 863775.i 0.595384 + 0.194695i
\(31\) −1.57256e6 −0.305831 −0.152915 − 0.988239i \(-0.548866\pi\)
−0.152915 + 0.988239i \(0.548866\pi\)
\(32\) −4.16281e6 − 4.16281e6i −0.701798 − 0.701798i
\(33\) −2.82252e6 − 2.12629e6i −0.414309 − 0.312111i
\(34\) − 5.49908e6i − 0.705723i
\(35\) −5.06368e6 + 1.73207e6i −0.570374 + 0.195100i
\(36\) −5.88563e6 + 1.68951e6i −0.584025 + 0.167649i
\(37\) 1.41364e7 − 1.41364e7i 1.24002 − 1.24002i 0.280034 − 0.959990i \(-0.409654\pi\)
0.959990 − 0.280034i \(-0.0903458\pi\)
\(38\) −5.05220e6 + 5.05220e6i −0.393056 + 0.393056i
\(39\) 2.59060e7 − 3.64466e6i 1.79312 − 0.252270i
\(40\) −1.54270e7 + 5.27692e6i −0.952823 + 0.325920i
\(41\) − 1.52870e7i − 0.844878i −0.906391 − 0.422439i \(-0.861174\pi\)
0.906391 − 0.422439i \(-0.138826\pi\)
\(42\) −4.58191e6 + 6.08223e6i −0.227209 + 0.301607i
\(43\) 1.45503e7 + 1.45503e7i 0.649027 + 0.649027i 0.952758 − 0.303731i \(-0.0982323\pi\)
−0.303731 + 0.952758i \(0.598232\pi\)
\(44\) 7.83596e6 0.315177
\(45\) −4.82391e6 + 2.70816e7i −0.175365 + 0.984504i
\(46\) −7.16345e6 −0.235891
\(47\) −2.49426e7 − 2.49426e7i −0.745592 − 0.745592i 0.228056 − 0.973648i \(-0.426763\pi\)
−0.973648 + 0.228056i \(0.926763\pi\)
\(48\) −513326. + 681411.i −0.0139575 + 0.0185278i
\(49\) 2.56895e7i 0.636609i
\(50\) −3.48539e6 + 2.74634e7i −0.0788653 + 0.621425i
\(51\) 5.38997e7 − 7.58302e6i 1.11563 − 0.156955i
\(52\) −4.10196e7 + 4.10196e7i −0.777995 + 0.777995i
\(53\) −2.38697e7 + 2.38697e7i −0.415534 + 0.415534i −0.883661 − 0.468127i \(-0.844929\pi\)
0.468127 + 0.883661i \(0.344929\pi\)
\(54\) 1.59354e7 + 3.57501e7i 0.255031 + 0.572143i
\(55\) 1.54956e7 − 3.16075e7i 0.228336 − 0.465755i
\(56\) − 4.46758e7i − 0.607052i
\(57\) −5.64864e7 − 4.25528e7i −0.708772 − 0.533938i
\(58\) −1.45305e7 − 1.45305e7i −0.168598 − 0.168598i
\(59\) 1.41279e8 1.51790 0.758951 − 0.651148i \(-0.225712\pi\)
0.758951 + 0.651148i \(0.225712\pi\)
\(60\) −2.75893e7 − 5.44007e7i −0.274828 − 0.541905i
\(61\) 2.77276e7 0.256406 0.128203 − 0.991748i \(-0.459079\pi\)
0.128203 + 0.991748i \(0.459079\pi\)
\(62\) −1.57611e7 − 1.57611e7i −0.135464 − 0.135464i
\(63\) −6.59338e7 − 3.65229e7i −0.527321 − 0.292101i
\(64\) − 8.65574e7i − 0.644903i
\(65\) 8.43427e7 + 2.46575e8i 0.586054 + 1.71332i
\(66\) −6.97806e6 − 4.95997e7i −0.0452676 − 0.321759i
\(67\) 5.16735e7 − 5.16735e7i 0.313279 − 0.313279i −0.532900 − 0.846179i \(-0.678898\pi\)
0.846179 + 0.532900i \(0.178898\pi\)
\(68\) −8.53449e7 + 8.53449e7i −0.484047 + 0.484047i
\(69\) −9.87812e6 − 7.02132e7i −0.0524630 − 0.372905i
\(70\) −6.81107e7 − 3.33912e7i −0.339058 − 0.166223i
\(71\) − 2.50202e8i − 1.16850i −0.811575 − 0.584249i \(-0.801389\pi\)
0.811575 − 0.584249i \(-0.198611\pi\)
\(72\) −2.00874e8 − 1.11271e8i −0.880902 − 0.487961i
\(73\) −8.81264e7 − 8.81264e7i −0.363206 − 0.363206i 0.501786 − 0.864992i \(-0.332677\pi\)
−0.864992 + 0.501786i \(0.832677\pi\)
\(74\) 2.83365e8 1.09851
\(75\) −2.73991e8 + 3.70861e6i −0.999908 + 0.0135343i
\(76\) 1.56819e8 0.539184
\(77\) 6.82039e7 + 6.82039e7i 0.221106 + 0.221106i
\(78\) 2.96173e8 + 2.23116e8i 0.905982 + 0.682502i
\(79\) 3.89035e8i 1.12374i 0.827225 + 0.561871i \(0.189918\pi\)
−0.827225 + 0.561871i \(0.810082\pi\)
\(80\) −7.63065e6 − 3.74092e6i −0.0208284 − 0.0102111i
\(81\) −3.28433e8 + 2.05491e8i −0.847743 + 0.530407i
\(82\) 1.53214e8 − 1.53214e8i 0.374229 − 0.374229i
\(83\) −9.47356e6 + 9.47356e6i −0.0219110 + 0.0219110i −0.717977 − 0.696066i \(-0.754932\pi\)
0.696066 + 0.717977i \(0.254932\pi\)
\(84\) 1.65506e8 − 2.32846e7i 0.362708 − 0.0510285i
\(85\) 1.75482e8 + 5.13020e8i 0.364626 + 1.06598i
\(86\) 2.91662e8i 0.574958i
\(87\) 1.22385e8 − 1.62459e8i 0.229029 − 0.304023i
\(88\) 2.07790e8 + 2.07790e8i 0.369363 + 0.369363i
\(89\) −4.88281e8 −0.824925 −0.412462 − 0.910975i \(-0.635331\pi\)
−0.412462 + 0.910975i \(0.635331\pi\)
\(90\) −3.19774e8 + 2.23078e8i −0.513750 + 0.358398i
\(91\) −7.14067e8 −1.09157
\(92\) 1.11176e8 + 1.11176e8i 0.161795 + 0.161795i
\(93\) 1.32750e8 − 1.76218e8i 0.184018 − 0.244274i
\(94\) − 4.99976e8i − 0.660502i
\(95\) 3.10108e8 − 6.32552e8i 0.390622 − 0.796783i
\(96\) 8.17884e8 − 1.15066e8i 0.982813 − 0.138270i
\(97\) 7.00398e7 − 7.00398e7i 0.0803289 − 0.0803289i −0.665801 − 0.746130i \(-0.731910\pi\)
0.746130 + 0.665801i \(0.231910\pi\)
\(98\) −2.57474e8 + 2.57474e8i −0.281979 + 0.281979i
\(99\) 4.76533e8 − 1.36792e8i 0.498580 − 0.143121i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 15.10.e.a.8.11 yes 32
3.2 odd 2 inner 15.10.e.a.8.6 yes 32
5.2 odd 4 inner 15.10.e.a.2.6 ✓ 32
5.3 odd 4 75.10.e.f.32.11 32
5.4 even 2 75.10.e.f.68.6 32
15.2 even 4 inner 15.10.e.a.2.11 yes 32
15.8 even 4 75.10.e.f.32.6 32
15.14 odd 2 75.10.e.f.68.11 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.e.a.2.6 ✓ 32 5.2 odd 4 inner
15.10.e.a.2.11 yes 32 15.2 even 4 inner
15.10.e.a.8.6 yes 32 3.2 odd 2 inner
15.10.e.a.8.11 yes 32 1.1 even 1 trivial
75.10.e.f.32.6 32 15.8 even 4
75.10.e.f.32.11 32 5.3 odd 4
75.10.e.f.68.6 32 5.4 even 2
75.10.e.f.68.11 32 15.14 odd 2