Properties

Label 648.2.l.f
Level $648$
Weight $2$
Character orbit 648.l
Analytic conductor $5.174$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,2,Mod(107,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.534694406811304329216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{12} + \beta_{6}) q^{2} + ( - \beta_{8} - \beta_{3} + 1) q^{4} + ( - \beta_{15} + \beta_{14}) q^{5} + ( - \beta_{13} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{15} - \beta_{14} + \beta_{12} + \beta_{7} - \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} + \beta_{6}) q^{2} + ( - \beta_{8} - \beta_{3} + 1) q^{4} + ( - \beta_{15} + \beta_{14}) q^{5} + ( - \beta_{13} + \beta_{11} - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{15} - \beta_{14} + \beta_{12} + \beta_{7} - \beta_{5}) q^{8} + ( - 2 \beta_{13} + \beta_{11} - \beta_{8} - \beta_{4} - \beta_{2}) q^{10} + ( - 2 \beta_{15} + \beta_{10} - \beta_{7} + \beta_{5} + \beta_1) q^{11} + (\beta_{13} + \beta_{9} + 2 \beta_{8} + \beta_{4} + \beta_{3} - 1) q^{13} + ( - 2 \beta_{15} + \beta_{14}) q^{14} + (\beta_{13} - \beta_{11} + \beta_{9} + \beta_{8} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2}) q^{16} + ( - \beta_{15} + \beta_{14} - 2 \beta_{12} + 2 \beta_{10} - \beta_{7}) q^{17} + (\beta_{13} + 2) q^{19} + ( - 2 \beta_{15} + 2 \beta_{10}) q^{20} + ( - 3 \beta_{13} - \beta_{9} - 2 \beta_{8} - 3 \beta_{4} - 2 \beta_{3} + 2) q^{22} + (\beta_{15} - \beta_{14} + 2 \beta_{6} - \beta_1) q^{23} + (2 \beta_{4} - \beta_{3}) q^{25} + ( - \beta_{12} + 2 \beta_{5}) q^{26} + ( - 4 \beta_{13} + \beta_{11} - 2 \beta_{8} - 2 \beta_{4} - 2 \beta_{2} - 1) q^{28} + (4 \beta_{12} + 2 \beta_{10} + 4 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{29} + (2 \beta_{13} + 4 \beta_{8} + 2 \beta_{4} + 2 \beta_{3} - 2) q^{31} + (2 \beta_{15} - 2 \beta_{14} + 2 \beta_{6}) q^{32} + ( - \beta_{13} + \beta_{11} - \beta_{9} - \beta_{8} - 4 \beta_{4} + 4 \beta_{3} - \beta_{2}) q^{34} + (\beta_{15} - \beta_{14} + 4 \beta_{12} - 3 \beta_{10} + \beta_{7} + \beta_{5}) q^{35} + (2 \beta_{13} - \beta_{11} + 3 \beta_{8} + 3 \beta_{4} + 3 \beta_{2} - 1) q^{37} + (\beta_{15} + 2 \beta_{12} - \beta_{10} + \beta_{7} + 2 \beta_{6}) q^{38} + ( - 4 \beta_{13} - 2 \beta_{8} - 4 \beta_{4} + 2 \beta_{3} - 2) q^{40} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_1) q^{41} - 2 \beta_{3} q^{43} + ( - 2 \beta_{15} + 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{10} - 2 \beta_{7} - 2 \beta_{5}) q^{44} + (2 \beta_{13} - \beta_{11} - \beta_{8} - \beta_{4} - \beta_{2} - 2) q^{46} + ( - 2 \beta_{12} - \beta_{10} + 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_1) q^{47} + (4 \beta_{13} + 4 \beta_{4} - 2 \beta_{3} + 2) q^{49} + ( - 2 \beta_{14} - \beta_{6}) q^{50} + ( - 3 \beta_{4} - 5 \beta_{3} + \beta_{2}) q^{52} + (2 \beta_{15} - 2 \beta_{14} + 4 \beta_{12} + 2 \beta_{10} + 2 \beta_{7} - 2 \beta_{5}) q^{53} + (2 \beta_{13} + 4 \beta_{8} + 4 \beta_{4} + 4 \beta_{2} - 2) q^{55} + ( - 3 \beta_{15} - \beta_{12} + 4 \beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_1) q^{56} + ( - 4 \beta_{8} + 4 \beta_{3} - 4) q^{58} + (\beta_{15} + \beta_{14} - \beta_1) q^{59} + (\beta_{13} - \beta_{11} + \beta_{9} + \beta_{8} + \beta_{3} - \beta_{2}) q^{61} + ( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{12} - 2 \beta_{10} - 2 \beta_{7} + 4 \beta_{5}) q^{62} + (4 \beta_{13} - 2 \beta_{11} + 2) q^{64} + (2 \beta_{15} - 2 \beta_{12} + \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_1) q^{65} + (3 \beta_{13} + 3 \beta_{4} - 4 \beta_{3} + 4) q^{67} + ( - 2 \beta_{15} + 4 \beta_{14} + 4 \beta_{6}) q^{68} + (\beta_{13} - \beta_{11} + \beta_{9} + \beta_{8} + 6 \beta_{4} - 10 \beta_{3} + 3 \beta_{2}) q^{70} + ( - 2 \beta_{15} + 2 \beta_{14} - 4 \beta_{12} - 2 \beta_{10} - 2 \beta_{7} + 2 \beta_{5}) q^{71} + (4 \beta_{13} - 1) q^{73} + ( - \beta_{12} - 2 \beta_{10} - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{74} + (\beta_{13} + \beta_{9} - \beta_{8} + \beta_{4} - 3 \beta_{3} + 3) q^{76} + (\beta_{15} - \beta_{14} - 4 \beta_{6} + 2 \beta_1) q^{77} + (3 \beta_{13} - 3 \beta_{11} + 3 \beta_{9} + 3 \beta_{8} + \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{79} + ( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{12} + 4 \beta_{10} - 2 \beta_{7} - 2 \beta_{5}) q^{80} + ( - 4 \beta_{13} - 2 \beta_{11} - 2 \beta_{8} - 2 \beta_{4} - 2 \beta_{2} + 4) q^{82} + (4 \beta_{12} - 2 \beta_{10} + 4 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{83} + (2 \beta_{13} - 4 \beta_{9} + 4 \beta_{8} + 2 \beta_{4} + 2 \beta_{3} - 2) q^{85} - 2 \beta_{6} q^{86} + ( - 2 \beta_{13} + 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{4} + 6 \beta_{3}) q^{88} + (\beta_{15} - \beta_{14} - 6 \beta_{12} + 2 \beta_{10} + \beta_{7} - 4 \beta_{5}) q^{89} + (\beta_{13} - 6) q^{91} + (4 \beta_{15} - 2 \beta_{12} - 2 \beta_{10} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{92}+ \cdots + (4 \beta_{15} - 4 \beta_{14} + 2 \beta_{12} - 4 \beta_{10} + 4 \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 16 q^{16} + 32 q^{19} + 8 q^{22} - 8 q^{25} - 24 q^{28} + 32 q^{34} - 24 q^{40} - 16 q^{43} - 48 q^{46} + 16 q^{49} - 36 q^{52} - 48 q^{58} + 16 q^{64} + 32 q^{67} - 72 q^{70} - 16 q^{73} + 20 q^{76} + 32 q^{82} + 56 q^{88} - 96 q^{91} + 24 q^{94} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + 2\nu^{10} - 4\nu^{6} - 8\nu^{4} + 32\nu^{2} + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} + 6\nu^{10} - 16\nu^{8} - 12\nu^{6} - 24\nu^{4} - 64\nu^{2} + 192 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{15} - 6\nu^{11} + 8\nu^{9} + 12\nu^{7} + 24\nu^{5} + 80\nu^{3} - 192\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{15} + 6\nu^{13} + 2\nu^{11} + 4\nu^{9} - 4\nu^{7} - 32\nu^{5} + 256\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} + 2\nu^{11} - 4\nu^{7} + 88\nu^{5} + 32\nu^{3} - 128\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{14} - 3\nu^{12} - 4\nu^{10} + 6\nu^{8} + 8\nu^{6} + 28\nu^{4} - 64\nu^{2} - 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{14} + \nu^{12} + 8\nu^{10} - 2\nu^{8} + 28\nu^{4} + 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 2\nu^{11} + 8\nu^{9} + 4\nu^{7} + 8\nu^{5} + 144\nu^{3} - 64\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{14} - 2\nu^{12} - 6\nu^{10} - 12\nu^{8} + 76\nu^{6} + 96\nu^{4} - 320 ) / 192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} + 2\nu^{11} - 8\nu^{9} - 4\nu^{7} - 8\nu^{5} + 48\nu^{3} + 64\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -3\nu^{14} + 2\nu^{12} + 6\nu^{10} + 12\nu^{8} + 20\nu^{6} - 96\nu^{4} + 320 ) / 192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3\nu^{15} - 2\nu^{13} - 6\nu^{11} - 12\nu^{9} + 76\nu^{7} + 96\nu^{5} - 128\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -5\nu^{15} - 2\nu^{13} + 10\nu^{11} + 20\nu^{9} + 44\nu^{7} - 160\nu^{5} + 768\nu ) / 384 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{9} - \beta_{4} - 2\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} + 2\beta_{7} + \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{13} + 2\beta_{11} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{15} + 4\beta_{14} + 2\beta_{6} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{13} - 2\beta_{11} + 2\beta_{9} + 2\beta_{8} - 8\beta_{4} + 4\beta_{3} - 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4\beta_{15} - 4\beta_{14} - 12\beta_{12} + 2\beta_{10} + 4\beta_{7} + 2\beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4\beta_{13} + 8\beta_{9} + 4\beta_{8} + 4\beta_{4} + 24\beta_{3} - 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12\beta_{15} + 4\beta_{12} + 4\beta_{10} + 16\beta_{7} + 4\beta_{6} - 16\beta_{5} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -8\beta_{13} + 4\beta_{11} - 36\beta_{8} - 36\beta_{4} - 36\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -8\beta_{15} + 8\beta_{14} + 64\beta_{6} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -8\beta_{13} + 8\beta_{11} - 8\beta_{9} - 8\beta_{8} - 16\beta_{4} + 128\beta_{3} - 32\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -16\beta_{15} + 16\beta_{14} - 40\beta_{12} + 48\beta_{10} - 16\beta_{7} - 56\beta_{5} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(-1\) \(1 - \beta_{3}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.40501 + 0.161069i
−1.32661 0.490008i
−0.841995 + 1.13624i
−0.238945 + 1.39388i
0.238945 1.39388i
0.841995 1.13624i
1.32661 + 0.490008i
1.40501 0.161069i
−1.40501 0.161069i
−1.32661 + 0.490008i
−0.841995 1.13624i
−0.238945 1.39388i
0.238945 + 1.39388i
0.841995 + 1.13624i
1.32661 0.490008i
1.40501 + 0.161069i
−1.40501 0.161069i 0 1.94811 + 0.452606i 1.53819 + 2.66422i 0 −3.45632 1.99551i −2.66422 0.949697i 0 −1.73205 3.99102i
107.2 −1.32661 + 0.490008i 0 1.51978 1.30010i −0.796225 1.37910i 0 1.24653 + 0.719687i −1.37910 + 2.46943i 0 1.73205 + 1.43937i
107.3 −0.841995 1.13624i 0 −0.582088 + 1.91342i −1.53819 2.66422i 0 3.45632 + 1.99551i 2.66422 0.949697i 0 −1.73205 + 3.99102i
107.4 −0.238945 1.39388i 0 −1.88581 + 0.666123i 0.796225 + 1.37910i 0 −1.24653 0.719687i 1.37910 + 2.46943i 0 1.73205 1.43937i
107.5 0.238945 + 1.39388i 0 −1.88581 + 0.666123i −0.796225 1.37910i 0 −1.24653 0.719687i −1.37910 2.46943i 0 1.73205 1.43937i
107.6 0.841995 + 1.13624i 0 −0.582088 + 1.91342i 1.53819 + 2.66422i 0 3.45632 + 1.99551i −2.66422 + 0.949697i 0 −1.73205 + 3.99102i
107.7 1.32661 0.490008i 0 1.51978 1.30010i 0.796225 + 1.37910i 0 1.24653 + 0.719687i 1.37910 2.46943i 0 1.73205 + 1.43937i
107.8 1.40501 + 0.161069i 0 1.94811 + 0.452606i −1.53819 2.66422i 0 −3.45632 1.99551i 2.66422 + 0.949697i 0 −1.73205 3.99102i
539.1 −1.40501 + 0.161069i 0 1.94811 0.452606i 1.53819 2.66422i 0 −3.45632 + 1.99551i −2.66422 + 0.949697i 0 −1.73205 + 3.99102i
539.2 −1.32661 0.490008i 0 1.51978 + 1.30010i −0.796225 + 1.37910i 0 1.24653 0.719687i −1.37910 2.46943i 0 1.73205 1.43937i
539.3 −0.841995 + 1.13624i 0 −0.582088 1.91342i −1.53819 + 2.66422i 0 3.45632 1.99551i 2.66422 + 0.949697i 0 −1.73205 3.99102i
539.4 −0.238945 + 1.39388i 0 −1.88581 0.666123i 0.796225 1.37910i 0 −1.24653 + 0.719687i 1.37910 2.46943i 0 1.73205 + 1.43937i
539.5 0.238945 1.39388i 0 −1.88581 0.666123i −0.796225 + 1.37910i 0 −1.24653 + 0.719687i −1.37910 + 2.46943i 0 1.73205 + 1.43937i
539.6 0.841995 1.13624i 0 −0.582088 1.91342i 1.53819 2.66422i 0 3.45632 1.99551i −2.66422 0.949697i 0 −1.73205 3.99102i
539.7 1.32661 + 0.490008i 0 1.51978 + 1.30010i 0.796225 1.37910i 0 1.24653 0.719687i 1.37910 + 2.46943i 0 1.73205 1.43937i
539.8 1.40501 0.161069i 0 1.94811 0.452606i −1.53819 + 2.66422i 0 −3.45632 + 1.99551i 2.66422 0.949697i 0 −1.73205 + 3.99102i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
24.f even 2 1 inner
72.l even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.l.f 16
3.b odd 2 1 inner 648.2.l.f 16
4.b odd 2 1 2592.2.p.f 16
8.b even 2 1 2592.2.p.f 16
8.d odd 2 1 inner 648.2.l.f 16
9.c even 3 1 216.2.f.a 8
9.c even 3 1 inner 648.2.l.f 16
9.d odd 6 1 216.2.f.a 8
9.d odd 6 1 inner 648.2.l.f 16
12.b even 2 1 2592.2.p.f 16
24.f even 2 1 inner 648.2.l.f 16
24.h odd 2 1 2592.2.p.f 16
36.f odd 6 1 864.2.f.a 8
36.f odd 6 1 2592.2.p.f 16
36.h even 6 1 864.2.f.a 8
36.h even 6 1 2592.2.p.f 16
72.j odd 6 1 864.2.f.a 8
72.j odd 6 1 2592.2.p.f 16
72.l even 6 1 216.2.f.a 8
72.l even 6 1 inner 648.2.l.f 16
72.n even 6 1 864.2.f.a 8
72.n even 6 1 2592.2.p.f 16
72.p odd 6 1 216.2.f.a 8
72.p odd 6 1 inner 648.2.l.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.f.a 8 9.c even 3 1
216.2.f.a 8 9.d odd 6 1
216.2.f.a 8 72.l even 6 1
216.2.f.a 8 72.p odd 6 1
648.2.l.f 16 1.a even 1 1 trivial
648.2.l.f 16 3.b odd 2 1 inner
648.2.l.f 16 8.d odd 2 1 inner
648.2.l.f 16 9.c even 3 1 inner
648.2.l.f 16 9.d odd 6 1 inner
648.2.l.f 16 24.f even 2 1 inner
648.2.l.f 16 72.l even 6 1 inner
648.2.l.f 16 72.p odd 6 1 inner
864.2.f.a 8 36.f odd 6 1
864.2.f.a 8 36.h even 6 1
864.2.f.a 8 72.j odd 6 1
864.2.f.a 8 72.n even 6 1
2592.2.p.f 16 4.b odd 2 1
2592.2.p.f 16 8.b even 2 1
2592.2.p.f 16 12.b even 2 1
2592.2.p.f 16 24.h odd 2 1
2592.2.p.f 16 36.f odd 6 1
2592.2.p.f 16 36.h even 6 1
2592.2.p.f 16 72.j odd 6 1
2592.2.p.f 16 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\):

\( T_{5}^{8} + 12T_{5}^{6} + 120T_{5}^{4} + 288T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{7}^{8} - 18T_{7}^{6} + 291T_{7}^{4} - 594T_{7}^{2} + 1089 \) Copy content Toggle raw display
\( T_{41}^{8} - 112T_{41}^{6} + 11136T_{41}^{4} - 157696T_{41}^{2} + 1982464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{14} - 2 T^{12} + 4 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 12 T^{6} + 120 T^{4} + 288 T^{2} + \cdots + 576)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 18 T^{6} + 291 T^{4} - 594 T^{2} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 28 T^{6} + 696 T^{4} - 2464 T^{2} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 30 T^{6} + 867 T^{4} - 990 T^{2} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 52 T^{2} + 88)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T + 1)^{8} \) Copy content Toggle raw display
$23$ \( (T^{8} + 36 T^{6} + 1272 T^{4} + 864 T^{2} + \cdots + 576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 96 T^{6} + 7680 T^{4} + \cdots + 2359296)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 96 T^{6} + 7104 T^{4} + \cdots + 4460544)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 54 T^{2} + 297)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 112 T^{6} + 11136 T^{4} + \cdots + 1982464)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 4)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 132 T^{6} + 13368 T^{4} + \cdots + 16451136)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 144 T^{2} + 3456)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 28 T^{6} + 696 T^{4} - 2464 T^{2} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 30 T^{6} + 867 T^{4} - 990 T^{2} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 144 T^{2} + 3456)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 47)^{8} \) Copy content Toggle raw display
$79$ \( (T^{8} - 186 T^{6} + 29019 T^{4} + \cdots + 31102929)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 160 T^{6} + 19968 T^{4} + \cdots + 31719424)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 436 T^{2} + 46552)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 111 T^{2} + 214 T + 11449)^{4} \) Copy content Toggle raw display
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