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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,18,Mod(49,75)] 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("75.49"); 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, 1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-786496] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1179680 x^{10} + 508533387652 x^{8} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{10}\cdot 5^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.12
Root \(630.218i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.h.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+630.218i q^{2} -6561.00i q^{3} -266103. q^{4} +4.13486e6 q^{6} +2.83896e6i q^{7} -8.50989e7i q^{8} -4.30467e7 q^{9} -1.10053e8 q^{11} +1.74590e9i q^{12} -3.59052e9i q^{13} -1.78917e9 q^{14} +1.87523e10 q^{16} +3.08091e10i q^{17} -2.71288e10i q^{18} +7.55210e10 q^{19} +1.86264e10 q^{21} -6.93576e10i q^{22} -4.68750e11i q^{23} -5.58334e11 q^{24} +2.26281e12 q^{26} +2.82430e11i q^{27} -7.55457e11i q^{28} +4.60936e11 q^{29} -1.07463e12 q^{31} +6.63924e11i q^{32} +7.22060e11i q^{33} -1.94164e13 q^{34} +1.14549e13 q^{36} +1.38949e13i q^{37} +4.75947e13i q^{38} -2.35574e13 q^{39} +7.28512e13 q^{41} +1.17387e13i q^{42} +7.83377e13i q^{43} +2.92855e13 q^{44} +2.95415e14 q^{46} +3.11197e14i q^{47} -1.23034e14i q^{48} +2.24571e14 q^{49} +2.02138e14 q^{51} +9.55449e14i q^{52} -2.62293e14i q^{53} -1.77992e14 q^{54} +2.41593e14 q^{56} -4.95494e14i q^{57} +2.90490e14i q^{58} -1.62771e15 q^{59} -1.90819e15 q^{61} -6.77249e14i q^{62} -1.22208e14i q^{63} +2.03948e15 q^{64} -4.55055e14 q^{66} +2.21897e15i q^{67} -8.19839e15i q^{68} -3.07547e15 q^{69} -6.87948e15 q^{71} +3.66323e15i q^{72} -8.52377e15i q^{73} -8.75682e15 q^{74} -2.00964e16 q^{76} -3.12437e14i q^{77} -1.48463e16i q^{78} +1.57221e16 q^{79} +1.85302e15 q^{81} +4.59121e16i q^{82} +2.13945e15i q^{83} -4.95655e15 q^{84} -4.93698e16 q^{86} -3.02420e15i q^{87} +9.36542e15i q^{88} -3.67059e16 q^{89} +1.01934e16 q^{91} +1.24736e17i q^{92} +7.05062e15i q^{93} -1.96122e17 q^{94} +4.35601e15 q^{96} +4.83530e16i q^{97} +1.41529e17i q^{98} +4.73743e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 786496 q^{4} - 4435236 q^{6} - 516560652 q^{9} + 171814648 q^{11} - 9924933204 q^{14} + 27576582640 q^{16} - 54387440804 q^{19} + 275559296868 q^{21} - 331235444232 q^{24} + 1222667429284 q^{26}+ \cdots - 73\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 630.218i 1.74075i 0.492391 + 0.870374i \(0.336123\pi\)
−0.492391 + 0.870374i \(0.663877\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) −266103. −2.03020
\(5\) 0 0
\(6\) 4.13486e6 1.00502
\(7\) 2.83896e6i 0.186134i 0.995660 + 0.0930672i \(0.0296671\pi\)
−0.995660 + 0.0930672i \(0.970333\pi\)
\(8\) − 8.50989e7i − 1.79333i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −1.10053e8 −0.154798 −0.0773990 − 0.997000i \(-0.524662\pi\)
−0.0773990 + 0.997000i \(0.524662\pi\)
\(12\) 1.74590e9i 1.17214i
\(13\) − 3.59052e9i − 1.22079i −0.792099 − 0.610393i \(-0.791012\pi\)
0.792099 − 0.610393i \(-0.208988\pi\)
\(14\) −1.78917e9 −0.324013
\(15\) 0 0
\(16\) 1.87523e10 1.09153
\(17\) 3.08091e10i 1.07118i 0.844478 + 0.535591i \(0.179911\pi\)
−0.844478 + 0.535591i \(0.820089\pi\)
\(18\) − 2.71288e10i − 0.580249i
\(19\) 7.55210e10 1.02015 0.510073 − 0.860131i \(-0.329618\pi\)
0.510073 + 0.860131i \(0.329618\pi\)
\(20\) 0 0
\(21\) 1.86264e10 0.107465
\(22\) − 6.93576e10i − 0.269464i
\(23\) − 4.68750e11i − 1.24811i −0.781379 − 0.624057i \(-0.785483\pi\)
0.781379 − 0.624057i \(-0.214517\pi\)
\(24\) −5.58334e11 −1.03538
\(25\) 0 0
\(26\) 2.26281e12 2.12508
\(27\) 2.82430e11i 0.192450i
\(28\) − 7.55457e11i − 0.377891i
\(29\) 4.60936e11 0.171103 0.0855515 − 0.996334i \(-0.472735\pi\)
0.0855515 + 0.996334i \(0.472735\pi\)
\(30\) 0 0
\(31\) −1.07463e12 −0.226299 −0.113150 − 0.993578i \(-0.536094\pi\)
−0.113150 + 0.993578i \(0.536094\pi\)
\(32\) 6.63924e11i 0.106744i
\(33\) 7.22060e11i 0.0893727i
\(34\) −1.94164e13 −1.86466
\(35\) 0 0
\(36\) 1.14549e13 0.676735
\(37\) 1.38949e13i 0.650341i 0.945655 + 0.325170i \(0.105422\pi\)
−0.945655 + 0.325170i \(0.894578\pi\)
\(38\) 4.75947e13i 1.77582i
\(39\) −2.35574e13 −0.704821
\(40\) 0 0
\(41\) 7.28512e13 1.42487 0.712433 − 0.701741i \(-0.247593\pi\)
0.712433 + 0.701741i \(0.247593\pi\)
\(42\) 1.17387e13i 0.187069i
\(43\) 7.83377e13i 1.02209i 0.859554 + 0.511045i \(0.170741\pi\)
−0.859554 + 0.511045i \(0.829259\pi\)
\(44\) 2.92855e13 0.314272
\(45\) 0 0
\(46\) 2.95415e14 2.17265
\(47\) 3.11197e14i 1.90635i 0.302412 + 0.953177i \(0.402208\pi\)
−0.302412 + 0.953177i \(0.597792\pi\)
\(48\) − 1.23034e14i − 0.630192i
\(49\) 2.24571e14 0.965354
\(50\) 0 0
\(51\) 2.02138e14 0.618447
\(52\) 9.55449e14i 2.47844i
\(53\) − 2.62293e14i − 0.578685i −0.957226 − 0.289342i \(-0.906563\pi\)
0.957226 − 0.289342i \(-0.0934366\pi\)
\(54\) −1.77992e14 −0.335007
\(55\) 0 0
\(56\) 2.41593e14 0.333800
\(57\) − 4.95494e14i − 0.588982i
\(58\) 2.90490e14i 0.297847i
\(59\) −1.62771e15 −1.44322 −0.721612 − 0.692298i \(-0.756598\pi\)
−0.721612 + 0.692298i \(0.756598\pi\)
\(60\) 0 0
\(61\) −1.90819e15 −1.27443 −0.637217 − 0.770685i \(-0.719914\pi\)
−0.637217 + 0.770685i \(0.719914\pi\)
\(62\) − 6.77249e14i − 0.393930i
\(63\) − 1.22208e14i − 0.0620448i
\(64\) 2.03948e15 0.905711
\(65\) 0 0
\(66\) −4.55055e14 −0.155575
\(67\) 2.21897e15i 0.667599i 0.942644 + 0.333799i \(0.108331\pi\)
−0.942644 + 0.333799i \(0.891669\pi\)
\(68\) − 8.19839e15i − 2.17472i
\(69\) −3.07547e15 −0.720599
\(70\) 0 0
\(71\) −6.87948e15 −1.26433 −0.632164 − 0.774835i \(-0.717833\pi\)
−0.632164 + 0.774835i \(0.717833\pi\)
\(72\) 3.66323e15i 0.597775i
\(73\) − 8.52377e15i − 1.23705i −0.785765 − 0.618525i \(-0.787731\pi\)
0.785765 − 0.618525i \(-0.212269\pi\)
\(74\) −8.75682e15 −1.13208
\(75\) 0 0
\(76\) −2.00964e16 −2.07110
\(77\) − 3.12437e14i − 0.0288132i
\(78\) − 1.48463e16i − 1.22692i
\(79\) 1.57221e16 1.16595 0.582974 − 0.812491i \(-0.301889\pi\)
0.582974 + 0.812491i \(0.301889\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 4.59121e16i 2.48033i
\(83\) 2.13945e15i 0.104265i 0.998640 + 0.0521325i \(0.0166018\pi\)
−0.998640 + 0.0521325i \(0.983398\pi\)
\(84\) −4.95655e15 −0.218175
\(85\) 0 0
\(86\) −4.93698e16 −1.77920
\(87\) − 3.02420e15i − 0.0987864i
\(88\) 9.36542e15i 0.277603i
\(89\) −3.67059e16 −0.988374 −0.494187 − 0.869356i \(-0.664534\pi\)
−0.494187 + 0.869356i \(0.664534\pi\)
\(90\) 0 0
\(91\) 1.01934e16 0.227230
\(92\) 1.24736e17i 2.53393i
\(93\) 7.05062e15i 0.130654i
\(94\) −1.96122e17 −3.31848
\(95\) 0 0
\(96\) 4.35601e15 0.0616287
\(97\) 4.83530e16i 0.626417i 0.949684 + 0.313208i \(0.101404\pi\)
−0.949684 + 0.313208i \(0.898596\pi\)
\(98\) 1.41529e17i 1.68044i
\(99\) 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.b.h.49.12 12
5.2 odd 4 75.18.a.j.1.1 yes 6
5.3 odd 4 75.18.a.i.1.6 ✓ 6
5.4 even 2 inner 75.18.b.h.49.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.6 ✓ 6 5.3 odd 4
75.18.a.j.1.1 yes 6 5.2 odd 4
75.18.b.h.49.1 12 5.4 even 2 inner
75.18.b.h.49.12 12 1.1 even 1 trivial