Show commands: Magma / Pari/GP / SageMath

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 1.6
Root \(541.028\) 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)
 
\(f(q)\) \(=\) \(q+597.028 q^{2} -6561.00 q^{3} +225370. q^{4} -3.91710e6 q^{6} +1.98246e7 q^{7} +5.62985e7 q^{8} +4.30467e7 q^{9} +6.04621e8 q^{11} -1.47865e9 q^{12} +2.50897e9 q^{13} +1.18358e10 q^{14} +4.07206e9 q^{16} +8.38830e9 q^{17} +2.57001e10 q^{18} -4.21327e10 q^{19} -1.30069e11 q^{21} +3.60975e11 q^{22} +3.84367e11 q^{23} -3.69374e11 q^{24} +1.49792e12 q^{26} -2.82430e11 q^{27} +4.46787e12 q^{28} -3.28787e12 q^{29} -5.09337e10 q^{31} -4.94802e12 q^{32} -3.96692e12 q^{33} +5.00804e12 q^{34} +9.70144e12 q^{36} -1.62069e13 q^{37} -2.51544e13 q^{38} -1.64613e13 q^{39} +9.05833e13 q^{41} -7.76549e13 q^{42} +1.32629e14 q^{43} +1.36263e14 q^{44} +2.29478e14 q^{46} -1.95095e14 q^{47} -2.67168e13 q^{48} +1.60384e14 q^{49} -5.50356e13 q^{51} +5.65446e14 q^{52} +5.41182e14 q^{53} -1.68618e14 q^{54} +1.11610e15 q^{56} +2.76433e14 q^{57} -1.96295e15 q^{58} -1.37048e15 q^{59} +1.95170e15 q^{61} -3.04088e13 q^{62} +8.53384e14 q^{63} -3.48784e15 q^{64} -2.36836e15 q^{66} +3.10261e15 q^{67} +1.89047e15 q^{68} -2.52183e15 q^{69} -2.23233e15 q^{71} +2.42347e15 q^{72} -8.15042e15 q^{73} -9.67597e15 q^{74} -9.49545e15 q^{76} +1.19864e16 q^{77} -9.82787e15 q^{78} +1.37011e16 q^{79} +1.85302e15 q^{81} +5.40807e16 q^{82} +3.51898e16 q^{83} -2.93137e16 q^{84} +7.91834e16 q^{86} +2.15717e16 q^{87} +3.40393e16 q^{88} +1.60937e16 q^{89} +4.97393e16 q^{91} +8.66247e16 q^{92} +3.34176e14 q^{93} -1.16477e17 q^{94} +3.24640e16 q^{96} -8.81534e15 q^{97} +9.57539e16 q^{98} +2.60270e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 338 q^{2} - 39366 q^{3} + 393248 q^{4} - 2217618 q^{6} - 20999794 q^{7} - 25242756 q^{8} + 258280326 q^{9} + 85907324 q^{11} - 2580100128 q^{12} - 344649098 q^{13} + 4962466602 q^{14} + 13788291320 q^{16}+ \cdots + 36\!\cdots\!04 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 597.028 1.64907 0.824536 − 0.565810i \(-0.191436\pi\)
0.824536 + 0.565810i \(0.191436\pi\)
\(3\) −6561.00 −0.577350
\(4\) 225370. 1.71944
\(5\) 0 0
\(6\) −3.91710e6 −0.952092
\(7\) 1.98246e7 1.29978 0.649892 − 0.760027i \(-0.274814\pi\)
0.649892 + 0.760027i \(0.274814\pi\)
\(8\) 5.62985e7 1.18640
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 6.04621e8 0.850444 0.425222 − 0.905089i \(-0.360196\pi\)
0.425222 + 0.905089i \(0.360196\pi\)
\(12\) −1.47865e9 −0.992717
\(13\) 2.50897e9 0.853054 0.426527 − 0.904475i \(-0.359737\pi\)
0.426527 + 0.904475i \(0.359737\pi\)
\(14\) 1.18358e10 2.14344
\(15\) 0 0
\(16\) 4.07206e9 0.237025
\(17\) 8.38830e9 0.291647 0.145824 − 0.989311i \(-0.453417\pi\)
0.145824 + 0.989311i \(0.453417\pi\)
\(18\) 2.57001e10 0.549690
\(19\) −4.21327e10 −0.569133 −0.284567 − 0.958656i \(-0.591850\pi\)
−0.284567 + 0.958656i \(0.591850\pi\)
\(20\) 0 0
\(21\) −1.30069e11 −0.750431
\(22\) 3.60975e11 1.40244
\(23\) 3.84367e11 1.02343 0.511716 − 0.859154i \(-0.329010\pi\)
0.511716 + 0.859154i \(0.329010\pi\)
\(24\) −3.69374e11 −0.684970
\(25\) 0 0
\(26\) 1.49792e12 1.40675
\(27\) −2.82430e11 −0.192450
\(28\) 4.46787e12 2.23490
\(29\) −3.28787e12 −1.22048 −0.610241 − 0.792216i \(-0.708927\pi\)
−0.610241 + 0.792216i \(0.708927\pi\)
\(30\) 0 0
\(31\) −5.09337e10 −0.0107258 −0.00536292 − 0.999986i \(-0.501707\pi\)
−0.00536292 + 0.999986i \(0.501707\pi\)
\(32\) −4.94802e12 −0.795531
\(33\) −3.96692e12 −0.491004
\(34\) 5.00804e12 0.480947
\(35\) 0 0
\(36\) 9.70144e12 0.573145
\(37\) −1.62069e13 −0.758552 −0.379276 − 0.925284i \(-0.623827\pi\)
−0.379276 + 0.925284i \(0.623827\pi\)
\(38\) −2.51544e13 −0.938541
\(39\) −1.64613e13 −0.492511
\(40\) 0 0
\(41\) 9.05833e13 1.77168 0.885840 − 0.463990i \(-0.153583\pi\)
0.885840 + 0.463990i \(0.153583\pi\)
\(42\) −7.76549e13 −1.23751
\(43\) 1.32629e14 1.73044 0.865222 − 0.501389i \(-0.167177\pi\)
0.865222 + 0.501389i \(0.167177\pi\)
\(44\) 1.36263e14 1.46228
\(45\) 0 0
\(46\) 2.29478e14 1.68771
\(47\) −1.95095e14 −1.19513 −0.597565 − 0.801821i \(-0.703865\pi\)
−0.597565 + 0.801821i \(0.703865\pi\)
\(48\) −2.67168e13 −0.136847
\(49\) 1.60384e14 0.689438
\(50\) 0 0
\(51\) −5.50356e13 −0.168383
\(52\) 5.65446e14 1.46677
\(53\) 5.41182e14 1.19398 0.596992 − 0.802247i \(-0.296362\pi\)
0.596992 + 0.802247i \(0.296362\pi\)
\(54\) −1.68618e14 −0.317364
\(55\) 0 0
\(56\) 1.11610e15 1.54207
\(57\) 2.76433e14 0.328589
\(58\) −1.96295e15 −2.01266
\(59\) −1.37048e15 −1.21515 −0.607575 − 0.794262i \(-0.707858\pi\)
−0.607575 + 0.794262i \(0.707858\pi\)
\(60\) 0 0
\(61\) 1.95170e15 1.30350 0.651748 − 0.758436i \(-0.274036\pi\)
0.651748 + 0.758436i \(0.274036\pi\)
\(62\) −3.04088e13 −0.0176877
\(63\) 8.53384e14 0.433261
\(64\) −3.48784e15 −1.54891
\(65\) 0 0
\(66\) −2.36836e15 −0.809700
\(67\) 3.10261e15 0.933451 0.466726 − 0.884402i \(-0.345434\pi\)
0.466726 + 0.884402i \(0.345434\pi\)
\(68\) 1.89047e15 0.501469
\(69\) −2.52183e15 −0.590879
\(70\) 0 0
\(71\) −2.23233e15 −0.410263 −0.205132 − 0.978734i \(-0.565762\pi\)
−0.205132 + 0.978734i \(0.565762\pi\)
\(72\) 2.42347e15 0.395467
\(73\) −8.15042e15 −1.18287 −0.591433 − 0.806354i \(-0.701438\pi\)
−0.591433 + 0.806354i \(0.701438\pi\)
\(74\) −9.67597e15 −1.25091
\(75\) 0 0
\(76\) −9.49545e15 −0.978588
\(77\) 1.19864e16 1.10539
\(78\) −9.82787e15 −0.812186
\(79\) 1.37011e16 1.01607 0.508035 − 0.861336i \(-0.330372\pi\)
0.508035 + 0.861336i \(0.330372\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 5.40807e16 2.92163
\(83\) 3.51898e16 1.71496 0.857479 − 0.514519i \(-0.172029\pi\)
0.857479 + 0.514519i \(0.172029\pi\)
\(84\) −2.93137e16 −1.29032
\(85\) 0 0
\(86\) 7.91834e16 2.85363
\(87\) 2.15717e16 0.704645
\(88\) 3.40393e16 1.00897
\(89\) 1.60937e16 0.433352 0.216676 − 0.976244i \(-0.430479\pi\)
0.216676 + 0.976244i \(0.430479\pi\)
\(90\) 0 0
\(91\) 4.97393e16 1.10879
\(92\) 8.66247e16 1.75973
\(93\) 3.34176e14 0.00619256
\(94\) −1.16477e17 −1.97085
\(95\) 0 0
\(96\) 3.24640e16 0.459300
\(97\) −8.81534e15 −0.114204 −0.0571018 − 0.998368i \(-0.518186\pi\)
−0.0571018 + 0.998368i \(0.518186\pi\)
\(98\) 9.57539e16 1.13693
\(99\) 2.60270e16 0.283481
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.1.6 yes 6
5.2 odd 4 75.18.b.h.49.11 12
5.3 odd 4 75.18.b.h.49.2 12
5.4 even 2 75.18.a.i.1.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.1 ✓ 6 5.4 even 2
75.18.a.j.1.6 yes 6 1.1 even 1 trivial
75.18.b.h.49.2 12 5.3 odd 4
75.18.b.h.49.11 12 5.2 odd 4