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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [32,7,Mod(31,32)] 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("32.31"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 32.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.36173067584\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.1
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 32.31
Dual form 32.7.c.b.31.4

$q$-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)
 
\(f(q)\) \(=\) \(q-51.1918i q^{3} +142.767 q^{5} -217.616i q^{7} -1891.60 q^{9} -25.7388i q^{11} -1390.16 q^{13} -7308.52i q^{15} +4004.09 q^{17} +129.021i q^{19} -11140.2 q^{21} +3697.19i q^{23} +4757.51 q^{25} +59515.8i q^{27} +40871.1 q^{29} -46174.6i q^{31} -1317.62 q^{33} -31068.5i q^{35} +65959.6 q^{37} +71165.0i q^{39} -41801.8 q^{41} +30093.2i q^{43} -270059. q^{45} -79785.8i q^{47} +70292.1 q^{49} -204977. i q^{51} +13717.2 q^{53} -3674.66i q^{55} +6604.80 q^{57} -300272. i q^{59} +243442. q^{61} +411644. i q^{63} -198470. q^{65} +406008. i q^{67} +189266. q^{69} +249585. i q^{71} -366195. q^{73} -243546. i q^{75} -5601.18 q^{77} +452154. i q^{79} +1.66775e6 q^{81} +512869. i q^{83} +571653. q^{85} -2.09227e6i q^{87} -797498. q^{89} +302522. i q^{91} -2.36376e6 q^{93} +18419.9i q^{95} -8788.19 q^{97} +48687.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 56 q^{5} - 3804 q^{9} - 8696 q^{13} - 5304 q^{17} - 1920 q^{21} + 36588 q^{25} + 77576 q^{29} + 71232 q^{33} + 125256 q^{37} - 209848 q^{41} - 536568 q^{45} + 95556 q^{49} - 68664 q^{53} - 526656 q^{57}+ \cdots + 2918344 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 51.1918i − 1.89599i −0.318279 − 0.947997i \(-0.603105\pi\)
0.318279 − 0.947997i \(-0.396895\pi\)
\(4\) 0 0
\(5\) 142.767 1.14214 0.571069 − 0.820902i \(-0.306529\pi\)
0.571069 + 0.820902i \(0.306529\pi\)
\(6\) 0 0
\(7\) − 217.616i − 0.634450i −0.948350 − 0.317225i \(-0.897249\pi\)
0.948350 − 0.317225i \(-0.102751\pi\)
\(8\) 0 0
\(9\) −1891.60 −2.59479
\(10\) 0 0
\(11\) − 25.7388i − 0.0193379i −0.999953 − 0.00966897i \(-0.996922\pi\)
0.999953 − 0.00966897i \(-0.00307778\pi\)
\(12\) 0 0
\(13\) −1390.16 −0.632755 −0.316378 − 0.948633i \(-0.602467\pi\)
−0.316378 + 0.948633i \(0.602467\pi\)
\(14\) 0 0
\(15\) − 7308.52i − 2.16549i
\(16\) 0 0
\(17\) 4004.09 0.814999 0.407499 − 0.913205i \(-0.366401\pi\)
0.407499 + 0.913205i \(0.366401\pi\)
\(18\) 0 0
\(19\) 129.021i 0.0188104i 0.999956 + 0.00940520i \(0.00299381\pi\)
−0.999956 + 0.00940520i \(0.997006\pi\)
\(20\) 0 0
\(21\) −11140.2 −1.20291
\(22\) 0 0
\(23\) 3697.19i 0.303870i 0.988390 + 0.151935i \(0.0485505\pi\)
−0.988390 + 0.151935i \(0.951449\pi\)
\(24\) 0 0
\(25\) 4757.51 0.304481
\(26\) 0 0
\(27\) 59515.8i 3.02372i
\(28\) 0 0
\(29\) 40871.1 1.67580 0.837901 − 0.545823i \(-0.183783\pi\)
0.837901 + 0.545823i \(0.183783\pi\)
\(30\) 0 0
\(31\) − 46174.6i − 1.54995i −0.631991 − 0.774976i \(-0.717762\pi\)
0.631991 − 0.774976i \(-0.282238\pi\)
\(32\) 0 0
\(33\) −1317.62 −0.0366646
\(34\) 0 0
\(35\) − 31068.5i − 0.724630i
\(36\) 0 0
\(37\) 65959.6 1.30219 0.651093 − 0.758998i \(-0.274311\pi\)
0.651093 + 0.758998i \(0.274311\pi\)
\(38\) 0 0
\(39\) 71165.0i 1.19970i
\(40\) 0 0
\(41\) −41801.8 −0.606518 −0.303259 − 0.952908i \(-0.598075\pi\)
−0.303259 + 0.952908i \(0.598075\pi\)
\(42\) 0 0
\(43\) 30093.2i 0.378498i 0.981929 + 0.189249i \(0.0606053\pi\)
−0.981929 + 0.189249i \(0.939395\pi\)
\(44\) 0 0
\(45\) −270059. −2.96361
\(46\) 0 0
\(47\) − 79785.8i − 0.768479i −0.923234 − 0.384239i \(-0.874464\pi\)
0.923234 − 0.384239i \(-0.125536\pi\)
\(48\) 0 0
\(49\) 70292.1 0.597473
\(50\) 0 0
\(51\) − 204977.i − 1.54523i
\(52\) 0 0
\(53\) 13717.2 0.0921376 0.0460688 − 0.998938i \(-0.485331\pi\)
0.0460688 + 0.998938i \(0.485331\pi\)
\(54\) 0 0
\(55\) − 3674.66i − 0.0220866i
\(56\) 0 0
\(57\) 6604.80 0.0356644
\(58\) 0 0
\(59\) − 300272.i − 1.46204i −0.682357 − 0.731019i \(-0.739045\pi\)
0.682357 − 0.731019i \(-0.260955\pi\)
\(60\) 0 0
\(61\) 243442. 1.07252 0.536261 − 0.844052i \(-0.319836\pi\)
0.536261 + 0.844052i \(0.319836\pi\)
\(62\) 0 0
\(63\) 411644.i 1.64627i
\(64\) 0 0
\(65\) −198470. −0.722694
\(66\) 0 0
\(67\) 406008.i 1.34993i 0.737851 + 0.674964i \(0.235841\pi\)
−0.737851 + 0.674964i \(0.764159\pi\)
\(68\) 0 0
\(69\) 189266. 0.576137
\(70\) 0 0
\(71\) 249585.i 0.697337i 0.937246 + 0.348669i \(0.113366\pi\)
−0.937246 + 0.348669i \(0.886634\pi\)
\(72\) 0 0
\(73\) −366195. −0.941334 −0.470667 − 0.882311i \(-0.655987\pi\)
−0.470667 + 0.882311i \(0.655987\pi\)
\(74\) 0 0
\(75\) − 243546.i − 0.577294i
\(76\) 0 0
\(77\) −5601.18 −0.0122690
\(78\) 0 0
\(79\) 452154.i 0.917076i 0.888675 + 0.458538i \(0.151627\pi\)
−0.888675 + 0.458538i \(0.848373\pi\)
\(80\) 0 0
\(81\) 1.66775e6 3.13816
\(82\) 0 0
\(83\) 512869.i 0.896958i 0.893793 + 0.448479i \(0.148034\pi\)
−0.893793 + 0.448479i \(0.851966\pi\)
\(84\) 0 0
\(85\) 571653. 0.930842
\(86\) 0 0
\(87\) − 2.09227e6i − 3.17731i
\(88\) 0 0
\(89\) −797498. −1.13125 −0.565626 − 0.824662i \(-0.691365\pi\)
−0.565626 + 0.824662i \(0.691365\pi\)
\(90\) 0 0
\(91\) 302522.i 0.401452i
\(92\) 0 0
\(93\) −2.36376e6 −2.93870
\(94\) 0 0
\(95\) 18419.9i 0.0214841i
\(96\) 0 0
\(97\) −8788.19 −0.00962907 −0.00481453 − 0.999988i \(-0.501533\pi\)
−0.00481453 + 0.999988i \(0.501533\pi\)
\(98\) 0 0
\(99\) 48687.6i 0.0501779i
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 32.7.c.b.31.1 ✓ 4
3.2 odd 2 288.7.g.b.127.1 4
4.3 odd 2 inner 32.7.c.b.31.4 yes 4
8.3 odd 2 64.7.c.e.63.1 4
8.5 even 2 64.7.c.e.63.4 4
12.11 even 2 288.7.g.b.127.2 4
16.3 odd 4 256.7.d.d.127.1 4
16.5 even 4 256.7.d.d.127.2 4
16.11 odd 4 256.7.d.g.127.4 4
16.13 even 4 256.7.d.g.127.3 4
24.5 odd 2 576.7.g.l.127.3 4
24.11 even 2 576.7.g.l.127.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.7.c.b.31.1 ✓ 4 1.1 even 1 trivial
32.7.c.b.31.4 yes 4 4.3 odd 2 inner
64.7.c.e.63.1 4 8.3 odd 2
64.7.c.e.63.4 4 8.5 even 2
256.7.d.d.127.1 4 16.3 odd 4
256.7.d.d.127.2 4 16.5 even 4
256.7.d.g.127.3 4 16.13 even 4
256.7.d.g.127.4 4 16.11 odd 4
288.7.g.b.127.1 4 3.2 odd 2
288.7.g.b.127.2 4 12.11 even 2
576.7.g.l.127.3 4 24.5 odd 2
576.7.g.l.127.4 4 24.11 even 2