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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,18,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,338] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 580318x^{4} + 45393344x^{3} + 72695152416x^{2} - 6623241804288x - 149217035286528 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{5}\cdot 5^{7} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-686.218\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q-630.218 q^{2} -6561.00 q^{3} +266103. q^{4} +4.13486e6 q^{6} -2.83896e6 q^{7} -8.50989e7 q^{8} +4.30467e7 q^{9} -1.10053e8 q^{11} -1.74590e9 q^{12} -3.59052e9 q^{13} +1.78917e9 q^{14} +1.87523e10 q^{16} -3.08091e10 q^{17} -2.71288e10 q^{18} -7.55210e10 q^{19} +1.86264e10 q^{21} +6.93576e10 q^{22} -4.68750e11 q^{23} +5.58334e11 q^{24} +2.26281e12 q^{26} -2.82430e11 q^{27} -7.55457e11 q^{28} -4.60936e11 q^{29} -1.07463e12 q^{31} -6.63924e11 q^{32} +7.22060e11 q^{33} +1.94164e13 q^{34} +1.14549e13 q^{36} -1.38949e13 q^{37} +4.75947e13 q^{38} +2.35574e13 q^{39} +7.28512e13 q^{41} -1.17387e13 q^{42} +7.83377e13 q^{43} -2.92855e13 q^{44} +2.95415e14 q^{46} -3.11197e14 q^{47} -1.23034e14 q^{48} -2.24571e14 q^{49} +2.02138e14 q^{51} -9.55449e14 q^{52} -2.62293e14 q^{53} +1.77992e14 q^{54} +2.41593e14 q^{56} +4.95494e14 q^{57} +2.90490e14 q^{58} +1.62771e15 q^{59} -1.90819e15 q^{61} +6.77249e14 q^{62} -1.22208e14 q^{63} -2.03948e15 q^{64} -4.55055e14 q^{66} -2.21897e15 q^{67} -8.19839e15 q^{68} +3.07547e15 q^{69} -6.87948e15 q^{71} -3.66323e15 q^{72} -8.52377e15 q^{73} +8.75682e15 q^{74} -2.00964e16 q^{76} +3.12437e14 q^{77} -1.48463e16 q^{78} -1.57221e16 q^{79} +1.85302e15 q^{81} -4.59121e16 q^{82} +2.13945e15 q^{83} +4.95655e15 q^{84} -4.93698e16 q^{86} +3.02420e15 q^{87} +9.36542e15 q^{88} +3.67059e16 q^{89} +1.01934e16 q^{91} -1.24736e17 q^{92} +7.05062e15 q^{93} +1.96122e17 q^{94} +4.35601e15 q^{96} -4.83530e16 q^{97} +1.41529e17 q^{98} -4.73743e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 338 q^{2} - 39366 q^{3} + 393248 q^{4} - 2217618 q^{6} - 20999794 q^{7} - 25242756 q^{8} + 258280326 q^{9} + 85907324 q^{11} - 2580100128 q^{12} - 344649098 q^{13} + 4962466602 q^{14} + 13788291320 q^{16}+ \cdots + 36\!\cdots\!04 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −630.218 −1.74075 −0.870374 − 0.492391i \(-0.836123\pi\)
−0.870374 + 0.492391i \(0.836123\pi\)
\(3\) −6561.00 −0.577350
\(4\) 266103. 2.03020
\(5\) 0 0
\(6\) 4.13486e6 1.00502
\(7\) −2.83896e6 −0.186134 −0.0930672 − 0.995660i \(-0.529667\pi\)
−0.0930672 + 0.995660i \(0.529667\pi\)
\(8\) −8.50989e7 −1.79333
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) −1.10053e8 −0.154798 −0.0773990 − 0.997000i \(-0.524662\pi\)
−0.0773990 + 0.997000i \(0.524662\pi\)
\(12\) −1.74590e9 −1.17214
\(13\) −3.59052e9 −1.22079 −0.610393 − 0.792099i \(-0.708988\pi\)
−0.610393 + 0.792099i \(0.708988\pi\)
\(14\) 1.78917e9 0.324013
\(15\) 0 0
\(16\) 1.87523e10 1.09153
\(17\) −3.08091e10 −1.07118 −0.535591 − 0.844478i \(-0.679911\pi\)
−0.535591 + 0.844478i \(0.679911\pi\)
\(18\) −2.71288e10 −0.580249
\(19\) −7.55210e10 −1.02015 −0.510073 − 0.860131i \(-0.670382\pi\)
−0.510073 + 0.860131i \(0.670382\pi\)
\(20\) 0 0
\(21\) 1.86264e10 0.107465
\(22\) 6.93576e10 0.269464
\(23\) −4.68750e11 −1.24811 −0.624057 − 0.781379i \(-0.714517\pi\)
−0.624057 + 0.781379i \(0.714517\pi\)
\(24\) 5.58334e11 1.03538
\(25\) 0 0
\(26\) 2.26281e12 2.12508
\(27\) −2.82430e11 −0.192450
\(28\) −7.55457e11 −0.377891
\(29\) −4.60936e11 −0.171103 −0.0855515 − 0.996334i \(-0.527265\pi\)
−0.0855515 + 0.996334i \(0.527265\pi\)
\(30\) 0 0
\(31\) −1.07463e12 −0.226299 −0.113150 − 0.993578i \(-0.536094\pi\)
−0.113150 + 0.993578i \(0.536094\pi\)
\(32\) −6.63924e11 −0.106744
\(33\) 7.22060e11 0.0893727
\(34\) 1.94164e13 1.86466
\(35\) 0 0
\(36\) 1.14549e13 0.676735
\(37\) −1.38949e13 −0.650341 −0.325170 − 0.945655i \(-0.605422\pi\)
−0.325170 + 0.945655i \(0.605422\pi\)
\(38\) 4.75947e13 1.77582
\(39\) 2.35574e13 0.704821
\(40\) 0 0
\(41\) 7.28512e13 1.42487 0.712433 − 0.701741i \(-0.247593\pi\)
0.712433 + 0.701741i \(0.247593\pi\)
\(42\) −1.17387e13 −0.187069
\(43\) 7.83377e13 1.02209 0.511045 − 0.859554i \(-0.329259\pi\)
0.511045 + 0.859554i \(0.329259\pi\)
\(44\) −2.92855e13 −0.314272
\(45\) 0 0
\(46\) 2.95415e14 2.17265
\(47\) −3.11197e14 −1.90635 −0.953177 − 0.302412i \(-0.902208\pi\)
−0.953177 + 0.302412i \(0.902208\pi\)
\(48\) −1.23034e14 −0.630192
\(49\) −2.24571e14 −0.965354
\(50\) 0 0
\(51\) 2.02138e14 0.618447
\(52\) −9.55449e14 −2.47844
\(53\) −2.62293e14 −0.578685 −0.289342 − 0.957226i \(-0.593437\pi\)
−0.289342 + 0.957226i \(0.593437\pi\)
\(54\) 1.77992e14 0.335007
\(55\) 0 0
\(56\) 2.41593e14 0.333800
\(57\) 4.95494e14 0.588982
\(58\) 2.90490e14 0.297847
\(59\) 1.62771e15 1.44322 0.721612 − 0.692298i \(-0.243402\pi\)
0.721612 + 0.692298i \(0.243402\pi\)
\(60\) 0 0
\(61\) −1.90819e15 −1.27443 −0.637217 − 0.770685i \(-0.719914\pi\)
−0.637217 + 0.770685i \(0.719914\pi\)
\(62\) 6.77249e14 0.393930
\(63\) −1.22208e14 −0.0620448
\(64\) −2.03948e15 −0.905711
\(65\) 0 0
\(66\) −4.55055e14 −0.155575
\(67\) −2.21897e15 −0.667599 −0.333799 − 0.942644i \(-0.608331\pi\)
−0.333799 + 0.942644i \(0.608331\pi\)
\(68\) −8.19839e15 −2.17472
\(69\) 3.07547e15 0.720599
\(70\) 0 0
\(71\) −6.87948e15 −1.26433 −0.632164 − 0.774835i \(-0.717833\pi\)
−0.632164 + 0.774835i \(0.717833\pi\)
\(72\) −3.66323e15 −0.597775
\(73\) −8.52377e15 −1.23705 −0.618525 − 0.785765i \(-0.712269\pi\)
−0.618525 + 0.785765i \(0.712269\pi\)
\(74\) 8.75682e15 1.13208
\(75\) 0 0
\(76\) −2.00964e16 −2.07110
\(77\) 3.12437e14 0.0288132
\(78\) −1.48463e16 −1.22692
\(79\) −1.57221e16 −1.16595 −0.582974 − 0.812491i \(-0.698111\pi\)
−0.582974 + 0.812491i \(0.698111\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −4.59121e16 −2.48033
\(83\) 2.13945e15 0.104265 0.0521325 − 0.998640i \(-0.483398\pi\)
0.0521325 + 0.998640i \(0.483398\pi\)
\(84\) 4.95655e15 0.218175
\(85\) 0 0
\(86\) −4.93698e16 −1.77920
\(87\) 3.02420e15 0.0987864
\(88\) 9.36542e15 0.277603
\(89\) 3.67059e16 0.988374 0.494187 − 0.869356i \(-0.335466\pi\)
0.494187 + 0.869356i \(0.335466\pi\)
\(90\) 0 0
\(91\) 1.01934e16 0.227230
\(92\) −1.24736e17 −2.53393
\(93\) 7.05062e15 0.130654
\(94\) 1.96122e17 3.31848
\(95\) 0 0
\(96\) 4.35601e15 0.0616287
\(97\) −4.83530e16 −0.626417 −0.313208 − 0.949684i \(-0.601404\pi\)
−0.313208 + 0.949684i \(0.601404\pi\)
\(98\) 1.41529e17 1.68044
\(99\) −4.73743e15 −0.0515993
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 75.18.a.j.1.1 yes 6
5.2 odd 4 75.18.b.h.49.1 12
5.3 odd 4 75.18.b.h.49.12 12
5.4 even 2 75.18.a.i.1.6 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.6 ✓ 6 5.4 even 2
75.18.a.j.1.1 yes 6 1.1 even 1 trivial
75.18.b.h.49.1 12 5.2 odd 4
75.18.b.h.49.12 12 5.3 odd 4