Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,7,Mod(161,288)] 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("288.161"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 288.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,2016] 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: \(66.2555760825\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.1
Root \(0.517638i\) of defining polynomial
Character \(\chi\) \(=\) 288.161
Dual form 288.7.e.g.161.3

$q$-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)
 
\(f(q)\) \(=\) \(q-77.7817i q^{5} -249.415 q^{7} +2469.09i q^{11} +504.000 q^{13} -2773.27i q^{17} +6983.63 q^{19} -12345.4i q^{23} +9575.00 q^{25} -21305.1i q^{29} -5237.72 q^{31} +19400.0i q^{35} -37802.0 q^{37} +26294.5i q^{41} -108246. q^{43} +17283.6i q^{47} -55441.0 q^{49} -44826.3i q^{53} +192050. q^{55} -153083. i q^{59} -293830. q^{61} -39202.0i q^{65} +373624. q^{67} +160491. i q^{71} -389088. q^{73} -615828. i q^{77} -958503. q^{79} +491348. i q^{83} -215710. q^{85} -922394. i q^{89} -125705. q^{91} -543199. i q^{95} -1.52712e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2016 q^{13} + 38300 q^{25} - 151208 q^{37} - 221764 q^{49} - 1175320 q^{61} - 1556352 q^{73} - 862840 q^{85} - 6108480 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(5\) − 77.7817i − 0.622254i −0.950368 − 0.311127i \(-0.899294\pi\)
0.950368 − 0.311127i \(-0.100706\pi\)
\(6\) 0 0
\(7\) −249.415 −0.727158 −0.363579 − 0.931563i \(-0.618445\pi\)
−0.363579 + 0.931563i \(0.618445\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2469.09i 1.85506i 0.373748 + 0.927530i \(0.378072\pi\)
−0.373748 + 0.927530i \(0.621928\pi\)
\(12\) 0 0
\(13\) 504.000 0.229404 0.114702 − 0.993400i \(-0.463409\pi\)
0.114702 + 0.993400i \(0.463409\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2773.27i − 0.564476i −0.959344 − 0.282238i \(-0.908923\pi\)
0.959344 − 0.282238i \(-0.0910769\pi\)
\(18\) 0 0
\(19\) 6983.63 1.01817 0.509085 − 0.860716i \(-0.329984\pi\)
0.509085 + 0.860716i \(0.329984\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 12345.4i − 1.01466i −0.861750 − 0.507332i \(-0.830632\pi\)
0.861750 − 0.507332i \(-0.169368\pi\)
\(24\) 0 0
\(25\) 9575.00 0.612800
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 21305.1i − 0.873555i −0.899570 − 0.436777i \(-0.856120\pi\)
0.899570 − 0.436777i \(-0.143880\pi\)
\(30\) 0 0
\(31\) −5237.72 −0.175816 −0.0879078 − 0.996129i \(-0.528018\pi\)
−0.0879078 + 0.996129i \(0.528018\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 19400.0i 0.452477i
\(36\) 0 0
\(37\) −37802.0 −0.746293 −0.373147 − 0.927772i \(-0.621721\pi\)
−0.373147 + 0.927772i \(0.621721\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 26294.5i 0.381516i 0.981637 + 0.190758i \(0.0610946\pi\)
−0.981637 + 0.190758i \(0.938905\pi\)
\(42\) 0 0
\(43\) −108246. −1.36147 −0.680734 − 0.732531i \(-0.738339\pi\)
−0.680734 + 0.732531i \(0.738339\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 17283.6i 0.166472i 0.996530 + 0.0832359i \(0.0265255\pi\)
−0.996530 + 0.0832359i \(0.973475\pi\)
\(48\) 0 0
\(49\) −55441.0 −0.471241
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 44826.3i − 0.301096i −0.988603 − 0.150548i \(-0.951896\pi\)
0.988603 − 0.150548i \(-0.0481039\pi\)
\(54\) 0 0
\(55\) 192050. 1.15432
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 153083.i − 0.745370i −0.927958 − 0.372685i \(-0.878437\pi\)
0.927958 − 0.372685i \(-0.121563\pi\)
\(60\) 0 0
\(61\) −293830. −1.29451 −0.647257 − 0.762272i \(-0.724084\pi\)
−0.647257 + 0.762272i \(0.724084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 39202.0i − 0.142747i
\(66\) 0 0
\(67\) 373624. 1.24225 0.621127 − 0.783710i \(-0.286675\pi\)
0.621127 + 0.783710i \(0.286675\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 160491.i 0.448409i 0.974542 + 0.224205i \(0.0719784\pi\)
−0.974542 + 0.224205i \(0.928022\pi\)
\(72\) 0 0
\(73\) −389088. −1.00018 −0.500091 − 0.865973i \(-0.666700\pi\)
−0.500091 + 0.865973i \(0.666700\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 615828.i − 1.34892i
\(78\) 0 0
\(79\) −958503. −1.94407 −0.972036 − 0.234833i \(-0.924546\pi\)
−0.972036 + 0.234833i \(0.924546\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 491348.i 0.859320i 0.902991 + 0.429660i \(0.141367\pi\)
−0.902991 + 0.429660i \(0.858633\pi\)
\(84\) 0 0
\(85\) −215710. −0.351248
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 922394.i − 1.30842i −0.756314 − 0.654209i \(-0.773002\pi\)
0.756314 − 0.654209i \(-0.226998\pi\)
\(90\) 0 0
\(91\) −125705. −0.166813
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 543199.i − 0.633560i
\(96\) 0 0
\(97\) −1.52712e6 −1.67324 −0.836619 − 0.547785i \(-0.815471\pi\)
−0.836619 + 0.547785i \(0.815471\pi\)
\(98\) 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 288.7.e.g.161.1 ✓ 4
3.2 odd 2 inner 288.7.e.g.161.3 yes 4
4.3 odd 2 inner 288.7.e.g.161.2 yes 4
8.3 odd 2 576.7.e.o.449.4 4
8.5 even 2 576.7.e.o.449.3 4
12.11 even 2 inner 288.7.e.g.161.4 yes 4
24.5 odd 2 576.7.e.o.449.1 4
24.11 even 2 576.7.e.o.449.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.7.e.g.161.1 ✓ 4 1.1 even 1 trivial
288.7.e.g.161.2 yes 4 4.3 odd 2 inner
288.7.e.g.161.3 yes 4 3.2 odd 2 inner
288.7.e.g.161.4 yes 4 12.11 even 2 inner
576.7.e.o.449.1 4 24.5 odd 2
576.7.e.o.449.2 4 24.11 even 2
576.7.e.o.449.3 4 8.5 even 2
576.7.e.o.449.4 4 8.3 odd 2