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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 49.5
Root \(-168.575i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.h.49.8

$q$-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)
 
\(f(q)\) \(=\) \(q-168.575i q^{2} -6561.00i q^{3} +102655. q^{4} -1.10602e6 q^{6} +2.46799e6i q^{7} -3.94004e7i q^{8} -4.30467e7 q^{9} -9.18646e8 q^{11} -6.73517e8i q^{12} +5.59047e9i q^{13} +4.16040e8 q^{14} +6.81325e9 q^{16} +1.89701e10i q^{17} +7.25658e9i q^{18} +1.07413e11 q^{19} +1.61925e10 q^{21} +1.54860e11i q^{22} -6.17937e11i q^{23} -2.58506e11 q^{24} +9.42411e11 q^{26} +2.82430e11i q^{27} +2.53350e11i q^{28} +3.24945e11 q^{29} -7.66951e12 q^{31} -6.31282e12i q^{32} +6.02724e12i q^{33} +3.19787e12 q^{34} -4.41894e12 q^{36} +1.16830e13i q^{37} -1.81071e13i q^{38} +3.66791e13 q^{39} +1.26785e13 q^{41} -2.72964e12i q^{42} -6.34524e13i q^{43} -9.43033e13 q^{44} -1.04168e14 q^{46} -2.95498e14i q^{47} -4.47017e13i q^{48} +2.26540e14 q^{49} +1.24463e14 q^{51} +5.73888e14i q^{52} -5.71911e14i q^{53} +4.76104e13 q^{54} +9.72397e13 q^{56} -7.04738e14i q^{57} -5.47774e13i q^{58} +1.31940e15 q^{59} +1.00730e14 q^{61} +1.29288e15i q^{62} -1.06239e14i q^{63} -1.71155e14 q^{64} +1.01604e15 q^{66} -1.83567e15i q^{67} +1.94737e15i q^{68} -4.05428e15 q^{69} +5.00202e15 q^{71} +1.69606e15i q^{72} +2.35185e15i q^{73} +1.96946e15 q^{74} +1.10265e16 q^{76} -2.26721e15i q^{77} -6.18316e15i q^{78} +2.06977e16 q^{79} +1.85302e15 q^{81} -2.13727e15i q^{82} -1.09097e16i q^{83} +1.66223e15 q^{84} -1.06965e16 q^{86} -2.13196e15i q^{87} +3.61950e16i q^{88} -4.23635e16 q^{89} -1.37972e16 q^{91} -6.34341e16i q^{92} +5.03197e16i q^{93} -4.98135e16 q^{94} -4.14184e16 q^{96} -9.31195e16i q^{97} -3.81888e16i q^{98} +3.95447e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 786496 q^{4} - 4435236 q^{6} - 516560652 q^{9} + 171814648 q^{11} - 9924933204 q^{14} + 27576582640 q^{16} - 54387440804 q^{19} + 275559296868 q^{21} - 331235444232 q^{24} + 1222667429284 q^{26}+ \cdots - 73\!\cdots\!08 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 168.575i − 0.465626i −0.972522 − 0.232813i \(-0.925207\pi\)
0.972522 − 0.232813i \(-0.0747930\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) 102655. 0.783193
\(5\) 0 0
\(6\) −1.10602e6 −0.268829
\(7\) 2.46799e6i 0.161812i 0.996722 + 0.0809058i \(0.0257813\pi\)
−0.996722 + 0.0809058i \(0.974219\pi\)
\(8\) − 3.94004e7i − 0.830300i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −9.18646e8 −1.29214 −0.646071 − 0.763277i \(-0.723589\pi\)
−0.646071 + 0.763277i \(0.723589\pi\)
\(12\) − 6.73517e8i − 0.452176i
\(13\) 5.59047e9i 1.90077i 0.311074 + 0.950386i \(0.399311\pi\)
−0.311074 + 0.950386i \(0.600689\pi\)
\(14\) 4.16040e8 0.0753437
\(15\) 0 0
\(16\) 6.81325e9 0.396583
\(17\) 1.89701e10i 0.659559i 0.944058 + 0.329779i \(0.106974\pi\)
−0.944058 + 0.329779i \(0.893026\pi\)
\(18\) 7.25658e9i 0.155209i
\(19\) 1.07413e11 1.45095 0.725474 − 0.688250i \(-0.241621\pi\)
0.725474 + 0.688250i \(0.241621\pi\)
\(20\) 0 0
\(21\) 1.61925e10 0.0934220
\(22\) 1.54860e11i 0.601655i
\(23\) − 6.17937e11i − 1.64535i −0.568514 − 0.822673i \(-0.692482\pi\)
0.568514 − 0.822673i \(-0.307518\pi\)
\(24\) −2.58506e11 −0.479374
\(25\) 0 0
\(26\) 9.42411e11 0.885048
\(27\) 2.82430e11i 0.192450i
\(28\) 2.53350e11i 0.126730i
\(29\) 3.24945e11 0.120622 0.0603110 − 0.998180i \(-0.480791\pi\)
0.0603110 + 0.998180i \(0.480791\pi\)
\(30\) 0 0
\(31\) −7.66951e12 −1.61508 −0.807539 − 0.589815i \(-0.799201\pi\)
−0.807539 + 0.589815i \(0.799201\pi\)
\(32\) − 6.31282e12i − 1.01496i
\(33\) 6.02724e12i 0.746019i
\(34\) 3.19787e12 0.307107
\(35\) 0 0
\(36\) −4.41894e12 −0.261064
\(37\) 1.16830e13i 0.546816i 0.961898 + 0.273408i \(0.0881508\pi\)
−0.961898 + 0.273408i \(0.911849\pi\)
\(38\) − 1.81071e13i − 0.675599i
\(39\) 3.66791e13 1.09741
\(40\) 0 0
\(41\) 1.26785e13 0.247974 0.123987 − 0.992284i \(-0.460432\pi\)
0.123987 + 0.992284i \(0.460432\pi\)
\(42\) − 2.72964e12i − 0.0434997i
\(43\) − 6.34524e13i − 0.827878i −0.910305 − 0.413939i \(-0.864153\pi\)
0.910305 − 0.413939i \(-0.135847\pi\)
\(44\) −9.43033e13 −1.01200
\(45\) 0 0
\(46\) −1.04168e14 −0.766116
\(47\) − 2.95498e14i − 1.81019i −0.425214 − 0.905093i \(-0.639801\pi\)
0.425214 − 0.905093i \(-0.360199\pi\)
\(48\) − 4.47017e13i − 0.228967i
\(49\) 2.26540e14 0.973817
\(50\) 0 0
\(51\) 1.24463e14 0.380796
\(52\) 5.73888e14i 1.48867i
\(53\) − 5.71911e14i − 1.26178i −0.775872 − 0.630890i \(-0.782690\pi\)
0.775872 − 0.630890i \(-0.217310\pi\)
\(54\) 4.76104e13 0.0896097
\(55\) 0 0
\(56\) 9.72397e13 0.134352
\(57\) − 7.04738e14i − 0.837705i
\(58\) − 5.47774e13i − 0.0561647i
\(59\) 1.31940e15 1.16986 0.584932 − 0.811082i \(-0.301121\pi\)
0.584932 + 0.811082i \(0.301121\pi\)
\(60\) 0 0
\(61\) 1.00730e14 0.0672755 0.0336378 − 0.999434i \(-0.489291\pi\)
0.0336378 + 0.999434i \(0.489291\pi\)
\(62\) 1.29288e15i 0.752022i
\(63\) − 1.06239e14i − 0.0539372i
\(64\) −1.71155e14 −0.0760082
\(65\) 0 0
\(66\) 1.01604e15 0.347366
\(67\) − 1.83567e15i − 0.552278i −0.961118 − 0.276139i \(-0.910945\pi\)
0.961118 − 0.276139i \(-0.0890551\pi\)
\(68\) 1.94737e15i 0.516561i
\(69\) −4.05428e15 −0.949942
\(70\) 0 0
\(71\) 5.00202e15 0.919284 0.459642 − 0.888104i \(-0.347978\pi\)
0.459642 + 0.888104i \(0.347978\pi\)
\(72\) 1.69606e15i 0.276767i
\(73\) 2.35185e15i 0.341322i 0.985330 + 0.170661i \(0.0545903\pi\)
−0.985330 + 0.170661i \(0.945410\pi\)
\(74\) 1.96946e15 0.254611
\(75\) 0 0
\(76\) 1.10265e16 1.13637
\(77\) − 2.26721e15i − 0.209084i
\(78\) − 6.18316e15i − 0.510983i
\(79\) 2.06977e16 1.53494 0.767469 − 0.641086i \(-0.221516\pi\)
0.767469 + 0.641086i \(0.221516\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 2.13727e15i − 0.115463i
\(83\) − 1.09097e16i − 0.531677i −0.964018 − 0.265839i \(-0.914351\pi\)
0.964018 − 0.265839i \(-0.0856488\pi\)
\(84\) 1.66223e15 0.0731674
\(85\) 0 0
\(86\) −1.06965e16 −0.385481
\(87\) − 2.13196e15i − 0.0696411i
\(88\) 3.61950e16i 1.07287i
\(89\) −4.23635e16 −1.14071 −0.570356 − 0.821397i \(-0.693195\pi\)
−0.570356 + 0.821397i \(0.693195\pi\)
\(90\) 0 0
\(91\) −1.37972e16 −0.307567
\(92\) − 6.34341e16i − 1.28862i
\(93\) 5.03197e16i 0.932465i
\(94\) −4.98135e16 −0.842869
\(95\) 0 0
\(96\) −4.14184e16 −0.585987
\(97\) − 9.31195e16i − 1.20637i −0.797601 − 0.603186i \(-0.793898\pi\)
0.797601 − 0.603186i \(-0.206102\pi\)
\(98\) − 3.81888e16i − 0.453434i
\(99\) 3.95447e16 0.430714
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.49.5 12
5.2 odd 4 75.18.a.j.1.4 yes 6
5.3 odd 4 75.18.a.i.1.3 ✓ 6
5.4 even 2 inner 75.18.b.h.49.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.18.a.i.1.3 ✓ 6 5.3 odd 4
75.18.a.j.1.4 yes 6 5.2 odd 4
75.18.b.h.49.5 12 1.1 even 1 trivial
75.18.b.h.49.8 12 5.4 even 2 inner