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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 = [6,0,0,-596296] 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: \(6\)
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: \( x^{6} - 175350x^{3} + 346592689x^{2} - 3264490950x + 15373811250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(94.0336 + 94.0336i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.d.49.6

$q$-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)
 
\(f(q)\) \(=\) \(q-670.613i q^{2} +6561.00i q^{3} -318650. q^{4} +4.39989e6 q^{6} -2.70804e7i q^{7} +1.25792e8i q^{8} -4.30467e7 q^{9} +1.01644e9 q^{11} -2.09066e9i q^{12} +1.04834e9i q^{13} -1.81605e10 q^{14} +4.25918e10 q^{16} +2.00099e10i q^{17} +2.88677e10i q^{18} -1.64417e10 q^{19} +1.77675e11 q^{21} -6.81636e11i q^{22} -5.92721e11i q^{23} -8.25322e11 q^{24} +7.03032e11 q^{26} -2.82430e11i q^{27} +8.62917e12i q^{28} -4.83212e11 q^{29} -5.20802e12 q^{31} -1.20748e13i q^{32} +6.66885e12i q^{33} +1.34189e13 q^{34} +1.37168e13 q^{36} +3.82367e13i q^{37} +1.10260e13i q^{38} -6.87817e12 q^{39} -6.56402e13 q^{41} -1.19151e14i q^{42} +5.83804e13i q^{43} -3.23888e14 q^{44} -3.97486e14 q^{46} +7.43658e12i q^{47} +2.79445e14i q^{48} -5.00719e14 q^{49} -1.31285e14 q^{51} -3.34054e14i q^{52} +3.01466e14i q^{53} -1.89401e14 q^{54} +3.40650e15 q^{56} -1.07874e14i q^{57} +3.24048e14i q^{58} +3.44389e14 q^{59} -2.43237e15 q^{61} +3.49257e15i q^{62} +1.16572e15i q^{63} -2.51491e15 q^{64} +4.47222e15 q^{66} +1.19632e15i q^{67} -6.37614e15i q^{68} +3.88884e15 q^{69} +5.42254e15 q^{71} -5.41494e15i q^{72} +5.84732e15i q^{73} +2.56420e16 q^{74} +5.23914e15 q^{76} -2.75256e16i q^{77} +4.61259e15i q^{78} -2.03455e15 q^{79} +1.85302e15 q^{81} +4.40192e16i q^{82} +1.18343e16i q^{83} -5.66160e16 q^{84} +3.91507e16 q^{86} -3.17035e15i q^{87} +1.27860e17i q^{88} +2.43170e16 q^{89} +2.83895e16 q^{91} +1.88870e17i q^{92} -3.41698e16i q^{93} +4.98707e15 q^{94} +7.92225e16 q^{96} -1.06105e17i q^{97} +3.35788e17i q^{98} -4.37543e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 596296 q^{4} + 5799924 q^{6} - 258280326 q^{9} + 2099699440 q^{11} - 55172056656 q^{14} + 44785594912 q^{16} + 223792235992 q^{19} - 65119814568 q^{21} - 1428092349264 q^{24} - 2185023682088 q^{26}+ \cdots - 90\!\cdots\!40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 670.613i − 1.85232i −0.377126 − 0.926162i \(-0.623088\pi\)
0.377126 − 0.926162i \(-0.376912\pi\)
\(3\) 6561.00i 0.577350i
\(4\) −318650. −2.43110
\(5\) 0 0
\(6\) 4.39989e6 1.06944
\(7\) − 2.70804e7i − 1.77551i −0.460320 − 0.887753i \(-0.652265\pi\)
0.460320 − 0.887753i \(-0.347735\pi\)
\(8\) 1.25792e8i 2.65087i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 1.01644e9 1.42969 0.714847 − 0.699281i \(-0.246496\pi\)
0.714847 + 0.699281i \(0.246496\pi\)
\(12\) − 2.09066e9i − 1.40360i
\(13\) 1.04834e9i 0.356438i 0.983991 + 0.178219i \(0.0570336\pi\)
−0.983991 + 0.178219i \(0.942966\pi\)
\(14\) −1.81605e10 −3.28881
\(15\) 0 0
\(16\) 4.25918e10 2.47917
\(17\) 2.00099e10i 0.695711i 0.937548 + 0.347855i \(0.113090\pi\)
−0.937548 + 0.347855i \(0.886910\pi\)
\(18\) 2.88677e10i 0.617441i
\(19\) −1.64417e10 −0.222096 −0.111048 − 0.993815i \(-0.535421\pi\)
−0.111048 + 0.993815i \(0.535421\pi\)
\(20\) 0 0
\(21\) 1.77675e11 1.02509
\(22\) − 6.81636e11i − 2.64826i
\(23\) − 5.92721e11i − 1.57821i −0.614261 − 0.789103i \(-0.710546\pi\)
0.614261 − 0.789103i \(-0.289454\pi\)
\(24\) −8.25322e11 −1.53048
\(25\) 0 0
\(26\) 7.03032e11 0.660239
\(27\) − 2.82430e11i − 0.192450i
\(28\) 8.62917e12i 4.31644i
\(29\) −4.83212e11 −0.179372 −0.0896860 − 0.995970i \(-0.528586\pi\)
−0.0896860 + 0.995970i \(0.528586\pi\)
\(30\) 0 0
\(31\) −5.20802e12 −1.09673 −0.548363 − 0.836240i \(-0.684749\pi\)
−0.548363 + 0.836240i \(0.684749\pi\)
\(32\) − 1.20748e13i − 1.94135i
\(33\) 6.66885e12i 0.825434i
\(34\) 1.34189e13 1.28868
\(35\) 0 0
\(36\) 1.37168e13 0.810368
\(37\) 3.82367e13i 1.78964i 0.446427 + 0.894820i \(0.352696\pi\)
−0.446427 + 0.894820i \(0.647304\pi\)
\(38\) 1.10260e13i 0.411393i
\(39\) −6.87817e12 −0.205790
\(40\) 0 0
\(41\) −6.56402e13 −1.28383 −0.641915 − 0.766776i \(-0.721860\pi\)
−0.641915 + 0.766776i \(0.721860\pi\)
\(42\) − 1.19151e14i − 1.89880i
\(43\) 5.83804e13i 0.761703i 0.924636 + 0.380851i \(0.124369\pi\)
−0.924636 + 0.380851i \(0.875631\pi\)
\(44\) −3.23888e14 −3.47573
\(45\) 0 0
\(46\) −3.97486e14 −2.92335
\(47\) 7.43658e12i 0.0455556i 0.999741 + 0.0227778i \(0.00725102\pi\)
−0.999741 + 0.0227778i \(0.992749\pi\)
\(48\) 2.79445e14i 1.43135i
\(49\) −5.00719e14 −2.15242
\(50\) 0 0
\(51\) −1.31285e14 −0.401669
\(52\) − 3.34054e14i − 0.866539i
\(53\) 3.01466e14i 0.665109i 0.943084 + 0.332555i \(0.107911\pi\)
−0.943084 + 0.332555i \(0.892089\pi\)
\(54\) −1.89401e14 −0.356480
\(55\) 0 0
\(56\) 3.40650e15 4.70664
\(57\) − 1.07874e14i − 0.128227i
\(58\) 3.24048e14i 0.332255i
\(59\) 3.44389e14 0.305357 0.152678 − 0.988276i \(-0.451210\pi\)
0.152678 + 0.988276i \(0.451210\pi\)
\(60\) 0 0
\(61\) −2.43237e15 −1.62452 −0.812261 − 0.583294i \(-0.801763\pi\)
−0.812261 + 0.583294i \(0.801763\pi\)
\(62\) 3.49257e15i 2.03149i
\(63\) 1.16572e15i 0.591835i
\(64\) −2.51491e15 −1.11684
\(65\) 0 0
\(66\) 4.47222e15 1.52897
\(67\) 1.19632e15i 0.359924i 0.983674 + 0.179962i \(0.0575975\pi\)
−0.983674 + 0.179962i \(0.942403\pi\)
\(68\) − 6.37614e15i − 1.69135i
\(69\) 3.88884e15 0.911177
\(70\) 0 0
\(71\) 5.42254e15 0.996567 0.498283 − 0.867014i \(-0.333964\pi\)
0.498283 + 0.867014i \(0.333964\pi\)
\(72\) − 5.41494e15i − 0.883623i
\(73\) 5.84732e15i 0.848619i 0.905517 + 0.424309i \(0.139483\pi\)
−0.905517 + 0.424309i \(0.860517\pi\)
\(74\) 2.56420e16 3.31499
\(75\) 0 0
\(76\) 5.23914e15 0.539938
\(77\) − 2.75256e16i − 2.53843i
\(78\) 4.61259e15i 0.381189i
\(79\) −2.03455e15 −0.150882 −0.0754411 − 0.997150i \(-0.524036\pi\)
−0.0754411 + 0.997150i \(0.524036\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 4.40192e16i 2.37807i
\(83\) 1.18343e16i 0.576740i 0.957519 + 0.288370i \(0.0931133\pi\)
−0.957519 + 0.288370i \(0.906887\pi\)
\(84\) −5.66160e16 −2.49210
\(85\) 0 0
\(86\) 3.91507e16 1.41092
\(87\) − 3.17035e15i − 0.103560i
\(88\) 1.27860e17i 3.78993i
\(89\) 2.43170e16 0.654780 0.327390 − 0.944889i \(-0.393831\pi\)
0.327390 + 0.944889i \(0.393831\pi\)
\(90\) 0 0
\(91\) 2.83895e16 0.632858
\(92\) 1.88870e17i 3.83678i
\(93\) − 3.41698e16i − 0.633195i
\(94\) 4.98707e15 0.0843837
\(95\) 0 0
\(96\) 7.92225e16 1.12084
\(97\) − 1.06105e17i − 1.37459i −0.726377 − 0.687296i \(-0.758797\pi\)
0.726377 − 0.687296i \(-0.241203\pi\)
\(98\) 3.35788e17i 3.98698i
\(99\) −4.37543e16 −0.476564
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.d.49.1 6
5.2 odd 4 75.18.a.e.1.3 3
5.3 odd 4 15.18.a.b.1.1 ✓ 3
5.4 even 2 inner 75.18.b.d.49.6 6
15.8 even 4 45.18.a.e.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.b.1.1 ✓ 3 5.3 odd 4
45.18.a.e.1.3 3 15.8 even 4
75.18.a.e.1.3 3 5.2 odd 4
75.18.b.d.49.1 6 1.1 even 1 trivial
75.18.b.d.49.6 6 5.4 even 2 inner