Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,8,Mod(1,432)] 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("432.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-188,0,180] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(134.950331009\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 271x^{2} - 1425x - 1692 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 216)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(19.2088\) of defining polynomial
Character \(\chi\) \(=\) 432.1

$q$-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)
 
\(f(q)\) \(=\) \(q+322.106 q^{5} -235.705 q^{7} -7068.49 q^{11} +5316.66 q^{13} +7661.81 q^{17} +18944.3 q^{19} -12539.1 q^{23} +25627.1 q^{25} +118076. q^{29} -124703. q^{31} -75921.9 q^{35} -350405. q^{37} -58460.6 q^{41} -631759. q^{43} +918884. q^{47} -767986. q^{49} -1.59457e6 q^{53} -2.27680e6 q^{55} -448469. q^{59} +2.01947e6 q^{61} +1.71253e6 q^{65} +3.00192e6 q^{67} -4.21336e6 q^{71} +4.50326e6 q^{73} +1.66608e6 q^{77} -4.55318e6 q^{79} -264682. q^{83} +2.46791e6 q^{85} -2.70653e6 q^{89} -1.25316e6 q^{91} +6.10207e6 q^{95} +1.50515e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 188 q^{5} + 180 q^{7} + 1460 q^{11} - 440 q^{13} - 9928 q^{17} + 2032 q^{19} + 24224 q^{23} + 24296 q^{25} - 58080 q^{29} - 63932 q^{31} - 64260 q^{35} + 351600 q^{37} + 9840 q^{41} - 143800 q^{43}+ \cdots + 25436060 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 322.106 1.15240 0.576200 − 0.817309i \(-0.304535\pi\)
0.576200 + 0.817309i \(0.304535\pi\)
\(6\) 0 0
\(7\) −235.705 −0.259732 −0.129866 − 0.991532i \(-0.541455\pi\)
−0.129866 + 0.991532i \(0.541455\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −7068.49 −1.60122 −0.800612 − 0.599183i \(-0.795492\pi\)
−0.800612 + 0.599183i \(0.795492\pi\)
\(12\) 0 0
\(13\) 5316.66 0.671177 0.335589 − 0.942009i \(-0.391065\pi\)
0.335589 + 0.942009i \(0.391065\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7661.81 0.378234 0.189117 − 0.981955i \(-0.439437\pi\)
0.189117 + 0.981955i \(0.439437\pi\)
\(18\) 0 0
\(19\) 18944.3 0.633638 0.316819 − 0.948486i \(-0.397385\pi\)
0.316819 + 0.948486i \(0.397385\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12539.1 −0.214891 −0.107445 − 0.994211i \(-0.534267\pi\)
−0.107445 + 0.994211i \(0.534267\pi\)
\(24\) 0 0
\(25\) 25627.1 0.328027
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 118076. 0.899016 0.449508 − 0.893276i \(-0.351599\pi\)
0.449508 + 0.893276i \(0.351599\pi\)
\(30\) 0 0
\(31\) −124703. −0.751813 −0.375906 − 0.926658i \(-0.622669\pi\)
−0.375906 + 0.926658i \(0.622669\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −75921.9 −0.299315
\(36\) 0 0
\(37\) −350405. −1.13727 −0.568635 − 0.822590i \(-0.692528\pi\)
−0.568635 + 0.822590i \(0.692528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −58460.6 −0.132471 −0.0662353 − 0.997804i \(-0.521099\pi\)
−0.0662353 + 0.997804i \(0.521099\pi\)
\(42\) 0 0
\(43\) −631759. −1.21175 −0.605873 − 0.795561i \(-0.707176\pi\)
−0.605873 + 0.795561i \(0.707176\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 918884. 1.29098 0.645488 − 0.763770i \(-0.276654\pi\)
0.645488 + 0.763770i \(0.276654\pi\)
\(48\) 0 0
\(49\) −767986. −0.932539
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.59457e6 −1.47122 −0.735611 − 0.677404i \(-0.763105\pi\)
−0.735611 + 0.677404i \(0.763105\pi\)
\(54\) 0 0
\(55\) −2.27680e6 −1.84525
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −448469. −0.284283 −0.142141 − 0.989846i \(-0.545399\pi\)
−0.142141 + 0.989846i \(0.545399\pi\)
\(60\) 0 0
\(61\) 2.01947e6 1.13916 0.569578 − 0.821938i \(-0.307107\pi\)
0.569578 + 0.821938i \(0.307107\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.71253e6 0.773465
\(66\) 0 0
\(67\) 3.00192e6 1.21937 0.609687 − 0.792642i \(-0.291295\pi\)
0.609687 + 0.792642i \(0.291295\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.21336e6 −1.39709 −0.698546 − 0.715565i \(-0.746169\pi\)
−0.698546 + 0.715565i \(0.746169\pi\)
\(72\) 0 0
\(73\) 4.50326e6 1.35487 0.677434 − 0.735584i \(-0.263092\pi\)
0.677434 + 0.735584i \(0.263092\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.66608e6 0.415889
\(78\) 0 0
\(79\) −4.55318e6 −1.03901 −0.519506 − 0.854467i \(-0.673884\pi\)
−0.519506 + 0.854467i \(0.673884\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −264682. −0.0508102 −0.0254051 − 0.999677i \(-0.508088\pi\)
−0.0254051 + 0.999677i \(0.508088\pi\)
\(84\) 0 0
\(85\) 2.46791e6 0.435877
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.70653e6 −0.406957 −0.203478 − 0.979079i \(-0.565225\pi\)
−0.203478 + 0.979079i \(0.565225\pi\)
\(90\) 0 0
\(91\) −1.25316e6 −0.174326
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.10207e6 0.730205
\(96\) 0 0
\(97\) 1.50515e7 1.67448 0.837240 − 0.546836i \(-0.184168\pi\)
0.837240 + 0.546836i \(0.184168\pi\)
\(98\) 0 0
\(99\) 0 0
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 432.8.a.w.1.4 4
3.2 odd 2 432.8.a.z.1.1 4
4.3 odd 2 216.8.a.e.1.4 ✓ 4
12.11 even 2 216.8.a.h.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.8.a.e.1.4 ✓ 4 4.3 odd 2
216.8.a.h.1.1 yes 4 12.11 even 2
432.8.a.w.1.4 4 1.1 even 1 trivial
432.8.a.z.1.1 4 3.2 odd 2