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 = [3,253] 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: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 182396x + 3921120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-436.958\) 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)
 
\(f(q)\) \(=\) \(q-352.958 q^{2} -6561.00 q^{3} -6492.31 q^{4} +2.31576e6 q^{6} -1.07009e7 q^{7} +4.85545e7 q^{8} +4.30467e7 q^{9} +1.00429e9 q^{11} +4.25961e7 q^{12} -3.43757e9 q^{13} +3.77697e9 q^{14} -1.62868e10 q^{16} -4.81590e10 q^{17} -1.51937e10 q^{18} +8.08158e9 q^{19} +7.02085e10 q^{21} -3.54473e11 q^{22} +3.08039e11 q^{23} -3.18566e11 q^{24} +1.21332e12 q^{26} -2.82430e11 q^{27} +6.94735e10 q^{28} +7.97487e11 q^{29} -3.17371e12 q^{31} -6.15585e11 q^{32} -6.58916e12 q^{33} +1.69981e13 q^{34} -2.79473e11 q^{36} -1.61437e13 q^{37} -2.85246e12 q^{38} +2.25539e13 q^{39} +7.63188e13 q^{41} -2.47807e13 q^{42} -1.29020e14 q^{43} -6.52018e12 q^{44} -1.08725e14 q^{46} +2.44935e14 q^{47} +1.06857e14 q^{48} -1.18121e14 q^{49} +3.15971e14 q^{51} +2.23178e13 q^{52} +8.60208e14 q^{53} +9.96859e13 q^{54} -5.19576e14 q^{56} -5.30232e13 q^{57} -2.81480e14 q^{58} +1.21582e14 q^{59} -4.27987e14 q^{61} +1.12019e15 q^{62} -4.60638e14 q^{63} +2.35201e15 q^{64} +2.32570e15 q^{66} +1.16734e15 q^{67} +3.12664e14 q^{68} -2.02104e15 q^{69} +3.01633e15 q^{71} +2.09011e15 q^{72} +7.32396e15 q^{73} +5.69806e15 q^{74} -5.24682e13 q^{76} -1.07468e16 q^{77} -7.96059e15 q^{78} -7.07645e15 q^{79} +1.85302e15 q^{81} -2.69374e16 q^{82} +1.18761e16 q^{83} -4.55816e14 q^{84} +4.55388e16 q^{86} -5.23231e15 q^{87} +4.87629e16 q^{88} +5.69376e16 q^{89} +3.67851e16 q^{91} -1.99989e15 q^{92} +2.08227e16 q^{93} -8.64519e16 q^{94} +4.03885e15 q^{96} +8.11777e16 q^{97} +4.16920e16 q^{98} +4.32315e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 253 q^{2} - 19683 q^{3} - 7087 q^{4} - 1659933 q^{6} + 4332484 q^{7} + 16188513 q^{8} + 129140163 q^{9} + 943563680 q^{11} + 46497807 q^{12} - 4257013150 q^{13} + 1847483988 q^{14} - 21943871359 q^{16}+ \cdots + 40\!\cdots\!80 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −352.958 −0.974919 −0.487460 − 0.873146i \(-0.662076\pi\)
−0.487460 + 0.873146i \(0.662076\pi\)
\(3\) −6561.00 −0.577350
\(4\) −6492.31 −0.0495324
\(5\) 0 0
\(6\) 2.31576e6 0.562870
\(7\) −1.07009e7 −0.701595 −0.350798 − 0.936451i \(-0.614089\pi\)
−0.350798 + 0.936451i \(0.614089\pi\)
\(8\) 4.85545e7 1.02321
\(9\) 4.30467e7 0.333333
\(10\) 0 0
\(11\) 1.00429e9 1.41261 0.706305 − 0.707908i \(-0.250361\pi\)
0.706305 + 0.707908i \(0.250361\pi\)
\(12\) 4.25961e7 0.0285976
\(13\) −3.43757e9 −1.16878 −0.584391 − 0.811473i \(-0.698666\pi\)
−0.584391 + 0.811473i \(0.698666\pi\)
\(14\) 3.77697e9 0.683999
\(15\) 0 0
\(16\) −1.62868e10 −0.948014
\(17\) −4.81590e10 −1.67441 −0.837205 − 0.546889i \(-0.815812\pi\)
−0.837205 + 0.546889i \(0.815812\pi\)
\(18\) −1.51937e10 −0.324973
\(19\) 8.08158e9 0.109167 0.0545834 − 0.998509i \(-0.482617\pi\)
0.0545834 + 0.998509i \(0.482617\pi\)
\(20\) 0 0
\(21\) 7.02085e10 0.405066
\(22\) −3.54473e11 −1.37718
\(23\) 3.08039e11 0.820199 0.410099 − 0.912041i \(-0.365494\pi\)
0.410099 + 0.912041i \(0.365494\pi\)
\(24\) −3.18566e11 −0.590750
\(25\) 0 0
\(26\) 1.21332e12 1.13947
\(27\) −2.82430e11 −0.192450
\(28\) 6.94735e10 0.0347517
\(29\) 7.97487e11 0.296033 0.148017 − 0.988985i \(-0.452711\pi\)
0.148017 + 0.988985i \(0.452711\pi\)
\(30\) 0 0
\(31\) −3.17371e12 −0.668334 −0.334167 − 0.942514i \(-0.608455\pi\)
−0.334167 + 0.942514i \(0.608455\pi\)
\(32\) −6.15585e11 −0.0989722
\(33\) −6.58916e12 −0.815570
\(34\) 1.69981e13 1.63241
\(35\) 0 0
\(36\) −2.79473e11 −0.0165108
\(37\) −1.61437e13 −0.755595 −0.377797 − 0.925888i \(-0.623318\pi\)
−0.377797 + 0.925888i \(0.623318\pi\)
\(38\) −2.85246e12 −0.106429
\(39\) 2.25539e13 0.674796
\(40\) 0 0
\(41\) 7.63188e13 1.49269 0.746344 − 0.665560i \(-0.231807\pi\)
0.746344 + 0.665560i \(0.231807\pi\)
\(42\) −2.47807e13 −0.394907
\(43\) −1.29020e14 −1.68336 −0.841679 − 0.539978i \(-0.818432\pi\)
−0.841679 + 0.539978i \(0.818432\pi\)
\(44\) −6.52018e12 −0.0699700
\(45\) 0 0
\(46\) −1.08725e14 −0.799628
\(47\) 2.44935e14 1.50044 0.750221 − 0.661187i \(-0.229947\pi\)
0.750221 + 0.661187i \(0.229947\pi\)
\(48\) 1.06857e14 0.547336
\(49\) −1.18121e14 −0.507764
\(50\) 0 0
\(51\) 3.15971e14 0.966721
\(52\) 2.23178e13 0.0578926
\(53\) 8.60208e14 1.89784 0.948918 − 0.315523i \(-0.102180\pi\)
0.948918 + 0.315523i \(0.102180\pi\)
\(54\) 9.96859e13 0.187623
\(55\) 0 0
\(56\) −5.19576e14 −0.717879
\(57\) −5.30232e13 −0.0630275
\(58\) −2.81480e14 −0.288609
\(59\) 1.21582e14 0.107802 0.0539012 − 0.998546i \(-0.482834\pi\)
0.0539012 + 0.998546i \(0.482834\pi\)
\(60\) 0 0
\(61\) −4.27987e14 −0.285842 −0.142921 − 0.989734i \(-0.545650\pi\)
−0.142921 + 0.989734i \(0.545650\pi\)
\(62\) 1.12019e15 0.651572
\(63\) −4.60638e14 −0.233865
\(64\) 2.35201e15 1.04450
\(65\) 0 0
\(66\) 2.32570e15 0.795115
\(67\) 1.16734e15 0.351206 0.175603 − 0.984461i \(-0.443813\pi\)
0.175603 + 0.984461i \(0.443813\pi\)
\(68\) 3.12664e14 0.0829376
\(69\) −2.02104e15 −0.473542
\(70\) 0 0
\(71\) 3.01633e15 0.554348 0.277174 − 0.960820i \(-0.410602\pi\)
0.277174 + 0.960820i \(0.410602\pi\)
\(72\) 2.09011e15 0.341070
\(73\) 7.32396e15 1.06292 0.531461 − 0.847082i \(-0.321643\pi\)
0.531461 + 0.847082i \(0.321643\pi\)
\(74\) 5.69806e15 0.736644
\(75\) 0 0
\(76\) −5.24682e13 −0.00540730
\(77\) −1.07468e16 −0.991080
\(78\) −7.96059e15 −0.657872
\(79\) −7.07645e15 −0.524789 −0.262395 − 0.964961i \(-0.584512\pi\)
−0.262395 + 0.964961i \(0.584512\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) −2.69374e16 −1.45525
\(83\) 1.18761e16 0.578773 0.289387 − 0.957212i \(-0.406549\pi\)
0.289387 + 0.957212i \(0.406549\pi\)
\(84\) −4.55816e14 −0.0200639
\(85\) 0 0
\(86\) 4.55388e16 1.64114
\(87\) −5.23231e15 −0.170915
\(88\) 4.87629e16 1.44540
\(89\) 5.69376e16 1.53315 0.766574 − 0.642156i \(-0.221960\pi\)
0.766574 + 0.642156i \(0.221960\pi\)
\(90\) 0 0
\(91\) 3.67851e16 0.820011
\(92\) −1.99989e15 −0.0406264
\(93\) 2.08227e16 0.385863
\(94\) −8.64519e16 −1.46281
\(95\) 0 0
\(96\) 4.03885e15 0.0571416
\(97\) 8.11777e16 1.05166 0.525832 − 0.850588i \(-0.323754\pi\)
0.525832 + 0.850588i \(0.323754\pi\)
\(98\) 4.16920e16 0.495029
\(99\) 4.32315e16 0.470870
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.d.1.1 3
5.2 odd 4 75.18.b.e.49.2 6
5.3 odd 4 75.18.b.e.49.5 6
5.4 even 2 15.18.a.c.1.3 ✓ 3
15.14 odd 2 45.18.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.3 ✓ 3 5.4 even 2
45.18.a.d.1.1 3 15.14 odd 2
75.18.a.d.1.1 3 1.1 even 1 trivial
75.18.b.e.49.2 6 5.2 odd 4
75.18.b.e.49.5 6 5.3 odd 4