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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,5,Mod(449,1008)] 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("1008.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,688] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.196922789\)
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: \( x^{12} + 152x^{10} + 8601x^{8} + 233552x^{6} + 3184240x^{4} + 20126976x^{2} + 43243776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{6}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(3.42011i\) of defining polynomial
Character \(\chi\) \(=\) 1008.449
Dual form 1008.5.d.f.449.9

$q$-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)
 
\(f(q)\) \(=\) \(q-28.6324i q^{5} +18.5203 q^{7} +228.980i q^{11} -137.086 q^{13} -71.9404i q^{17} -9.73322 q^{19} -159.211i q^{23} -194.816 q^{25} +237.240i q^{29} +485.848 q^{31} -530.280i q^{35} -182.208 q^{37} -1873.93i q^{41} +475.754 q^{43} -3185.41i q^{47} +343.000 q^{49} -1222.35i q^{53} +6556.27 q^{55} -2467.13i q^{59} +3685.66 q^{61} +3925.12i q^{65} -1255.00 q^{67} -9274.35i q^{71} -6282.76 q^{73} +4240.78i q^{77} -1170.75 q^{79} +5662.77i q^{83} -2059.83 q^{85} +3335.45i q^{89} -2538.87 q^{91} +278.686i q^{95} -13827.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 688 q^{13} - 96 q^{19} - 1500 q^{25} + 608 q^{31} - 2728 q^{37} + 3088 q^{43} + 4116 q^{49} - 8608 q^{55} + 904 q^{61} + 15376 q^{67} + 12608 q^{73} - 11456 q^{79} + 18792 q^{85} - 3920 q^{91} - 23584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 28.6324i − 1.14530i −0.819801 − 0.572649i \(-0.805916\pi\)
0.819801 − 0.572649i \(-0.194084\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 228.980i 1.89240i 0.323583 + 0.946200i \(0.395113\pi\)
−0.323583 + 0.946200i \(0.604887\pi\)
\(12\) 0 0
\(13\) −137.086 −0.811162 −0.405581 − 0.914059i \(-0.632931\pi\)
−0.405581 + 0.914059i \(0.632931\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 71.9404i − 0.248929i −0.992224 − 0.124464i \(-0.960279\pi\)
0.992224 − 0.124464i \(-0.0397213\pi\)
\(18\) 0 0
\(19\) −9.73322 −0.0269618 −0.0134809 − 0.999909i \(-0.504291\pi\)
−0.0134809 + 0.999909i \(0.504291\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 159.211i − 0.300967i −0.988613 − 0.150483i \(-0.951917\pi\)
0.988613 − 0.150483i \(-0.0480830\pi\)
\(24\) 0 0
\(25\) −194.816 −0.311706
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 237.240i 0.282093i 0.990003 + 0.141046i \(0.0450466\pi\)
−0.990003 + 0.141046i \(0.954953\pi\)
\(30\) 0 0
\(31\) 485.848 0.505565 0.252783 − 0.967523i \(-0.418654\pi\)
0.252783 + 0.967523i \(0.418654\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 530.280i − 0.432882i
\(36\) 0 0
\(37\) −182.208 −0.133096 −0.0665480 − 0.997783i \(-0.521199\pi\)
−0.0665480 + 0.997783i \(0.521199\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1873.93i − 1.11477i −0.830254 − 0.557385i \(-0.811805\pi\)
0.830254 − 0.557385i \(-0.188195\pi\)
\(42\) 0 0
\(43\) 475.754 0.257303 0.128652 − 0.991690i \(-0.458935\pi\)
0.128652 + 0.991690i \(0.458935\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 3185.41i − 1.44201i −0.692928 − 0.721007i \(-0.743680\pi\)
0.692928 − 0.721007i \(-0.256320\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1222.35i − 0.435154i −0.976043 − 0.217577i \(-0.930185\pi\)
0.976043 − 0.217577i \(-0.0698152\pi\)
\(54\) 0 0
\(55\) 6556.27 2.16736
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2467.13i − 0.708741i −0.935105 − 0.354371i \(-0.884695\pi\)
0.935105 − 0.354371i \(-0.115305\pi\)
\(60\) 0 0
\(61\) 3685.66 0.990503 0.495251 − 0.868750i \(-0.335076\pi\)
0.495251 + 0.868750i \(0.335076\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3925.12i 0.929022i
\(66\) 0 0
\(67\) −1255.00 −0.279573 −0.139786 − 0.990182i \(-0.544642\pi\)
−0.139786 + 0.990182i \(0.544642\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 9274.35i − 1.83978i −0.392173 − 0.919891i \(-0.628277\pi\)
0.392173 − 0.919891i \(-0.371723\pi\)
\(72\) 0 0
\(73\) −6282.76 −1.17898 −0.589488 − 0.807777i \(-0.700670\pi\)
−0.589488 + 0.807777i \(0.700670\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4240.78i 0.715260i
\(78\) 0 0
\(79\) −1170.75 −0.187590 −0.0937951 − 0.995592i \(-0.529900\pi\)
−0.0937951 + 0.995592i \(0.529900\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5662.77i 0.822002i 0.911635 + 0.411001i \(0.134821\pi\)
−0.911635 + 0.411001i \(0.865179\pi\)
\(84\) 0 0
\(85\) −2059.83 −0.285097
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3335.45i 0.421089i 0.977584 + 0.210545i \(0.0675237\pi\)
−0.977584 + 0.210545i \(0.932476\pi\)
\(90\) 0 0
\(91\) −2538.87 −0.306590
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 278.686i 0.0308793i
\(96\) 0 0
\(97\) −13827.1 −1.46957 −0.734783 − 0.678303i \(-0.762716\pi\)
−0.734783 + 0.678303i \(0.762716\pi\)
\(98\) 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 1008.5.d.f.449.4 12
3.2 odd 2 inner 1008.5.d.f.449.9 12
4.3 odd 2 504.5.d.b.449.4 ✓ 12
12.11 even 2 504.5.d.b.449.9 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.5.d.b.449.4 ✓ 12 4.3 odd 2
504.5.d.b.449.9 yes 12 12.11 even 2
1008.5.d.f.449.4 12 1.1 even 1 trivial
1008.5.d.f.449.9 12 3.2 odd 2 inner