Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,2,Mod(109,648)]
 
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([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.n (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: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} + 2x^{12} + 12x^{10} - 28x^{8} + 48x^{6} + 32x^{4} - 128x^{2} + 256 \) 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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{3} - \beta_{2}) q^{4} + (\beta_{9} + \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{6}) q^{7} + ( - \beta_{15} + \beta_{13} + \beta_{12} + \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{3} - \beta_{2}) q^{4} + (\beta_{9} + \beta_{5} - \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{6}) q^{7} + ( - \beta_{15} + \beta_{13} + \beta_{12} + \beta_{5}) q^{8} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \beta_{2} - \beta_1) q^{10} + ( - \beta_{14} + \beta_{9} - \beta_{5}) q^{11} + (\beta_{8} - \beta_{7} + \beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{15} + \beta_{12} + \beta_{11}) q^{14} + (\beta_{10} - \beta_{7} + \beta_{6}) q^{16} + ( - 2 \beta_{15} + 2 \beta_{13} + 2 \beta_{12} + 2 \beta_{5} - \beta_{4}) q^{17} + (2 \beta_{10} - \beta_{8} - 2 \beta_{6} - 2 \beta_{2} + 2 \beta_1 - 2) q^{19} - 2 \beta_{9} q^{20} + (\beta_{8} - \beta_{7} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{22} + ( - 2 \beta_{15} - \beta_{12} - \beta_{11} + \beta_{9} + \beta_{5} - \beta_{4}) q^{23} + ( - 2 \beta_{10} - 2 \beta_{6} - \beta_{3} + 1) q^{25} + (\beta_{12} + \beta_{5} - 2 \beta_{4}) q^{26} + ( - \beta_{8} + \beta_{6} - \beta_1 - 3) q^{28} + (2 \beta_{14} - 2 \beta_{9} + 2 \beta_{5}) q^{29} - 2 \beta_{3} q^{31} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{5} - 2 \beta_{4}) q^{32} + (\beta_{10} - \beta_{7} + 3 \beta_{6}) q^{34} + (\beta_{14} + \beta_{12} - \beta_{11} + 2 \beta_{5} - 3 \beta_{4}) q^{35} + (\beta_{10} + \beta_{8} - \beta_{6} - \beta_{2} + \beta_1 - 1) q^{37} + (\beta_{14} + \beta_{13} + 2 \beta_{9} + 2 \beta_{5}) q^{38} + ( - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{3} + 2 \beta_1) q^{40} + (6 \beta_{12} + 2 \beta_{11}) q^{41} + (2 \beta_{10} - 2 \beta_{6} + 2 \beta_{3} - 2) q^{43} + ( - 2 \beta_{15} + 2 \beta_{13} - 2 \beta_{4}) q^{44} + ( - \beta_{10} - \beta_{8} + 3 \beta_{6} + \beta_{2} - 3 \beta_1 + 6) q^{46} + ( - \beta_{14} + 2 \beta_{13} - \beta_{9} + 3 \beta_{5}) q^{47} + ( - 2 \beta_{15} + 3 \beta_{12} + 2 \beta_{11}) q^{50} + ( - \beta_{10} + 2 \beta_{7} + 2 \beta_{6} + \beta_{3} - 1) q^{52} + ( - 2 \beta_{14} - 2 \beta_{12} + 2 \beta_{11} - 4 \beta_{5} + 2 \beta_{4}) q^{53} + ( - 2 \beta_{10} - 2 \beta_{6} + 2 \beta_{2} + 2 \beta_1 + 2) q^{55} + ( - 2 \beta_{14} - \beta_{13} + 2 \beta_{9} + 3 \beta_{5}) q^{56} + ( - 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{58} + (\beta_{12} - \beta_{11} - 3 \beta_{9} - 3 \beta_{5} + 3 \beta_{4}) q^{59} + (3 \beta_{10} + \beta_{7} - 3 \beta_{6} + 3 \beta_{3} - 3) q^{61} + (2 \beta_{12} + 2 \beta_{5}) q^{62} + (2 \beta_{8} + 2 \beta_{6} - 2 \beta_1 + 6) q^{64} + (2 \beta_{14} - 2 \beta_{13} + \beta_{9} - 5 \beta_{5}) q^{65} + ( - \beta_{8} + \beta_{7}) q^{67} + ( - 2 \beta_{12} - 4 \beta_{11} + 2 \beta_{9} + 2 \beta_{5} - 2 \beta_{4}) q^{68} + ( - \beta_{10} + 3 \beta_{7} + 3 \beta_{6} - 2 \beta_{3} + 2) q^{70} + ( - 4 \beta_{10} - 4 \beta_{6} + 4 \beta_{2} + 4 \beta_1 + 1) q^{73} + (2 \beta_{14} + 2 \beta_{13} - 2 \beta_{9} + \beta_{5}) q^{74} + (3 \beta_{8} - 3 \beta_{7} - 5 \beta_{3} + 2 \beta_{2} - \beta_1) q^{76} + ( - 2 \beta_{12} + 2 \beta_{11} + \beta_{9} + \beta_{5} - \beta_{4}) q^{77} + ( - \beta_{10} - \beta_{6} - 6 \beta_{3} + 6) q^{79} + (2 \beta_{15} - 2 \beta_{13} - 6 \beta_{12} - 6 \beta_{5} + 4 \beta_{4}) q^{80} + (4 \beta_{10} - 4 \beta_{2} - 12) q^{82} + ( - 2 \beta_{14} - 4 \beta_{5}) q^{83} + ( - 4 \beta_{8} + 4 \beta_{7} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{85} + (2 \beta_{15} + 2 \beta_{11}) q^{86} + ( - 2 \beta_{10} + 4 \beta_{6} - 2 \beta_{3} + 2) q^{88} + (2 \beta_{15} - 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{5} + \beta_{4}) q^{89} + ( - 2 \beta_{10} + 3 \beta_{8} + 2 \beta_{6} + 2 \beta_{2} - 2 \beta_1 + 2) q^{91} + ( - 4 \beta_{14} - 2 \beta_{13} + 2 \beta_{9} - 6 \beta_{5}) q^{92} + (\beta_{8} - \beta_{7} + 6 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{94} + (2 \beta_{15} - 11 \beta_{12} - 3 \beta_{11} - \beta_{9} - \beta_{5} + \beta_{4}) q^{95} + (2 \beta_{10} + 2 \beta_{6} - 3 \beta_{3} + 3) q^{97}+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 \( 16 q - 4 q^{4} - 16 q^{10} + 16 q^{22} + 8 q^{25} - 56 q^{28} - 16 q^{31} - 8 q^{34} + 24 q^{40} + 64 q^{46} - 20 q^{52} + 32 q^{55} - 32 q^{58} + 80 q^{64} + 16 q^{73} - 52 q^{76} + 48 q^{79} - 160 q^{82} - 8 q^{88} + 48 q^{94} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} + 2x^{12} + 12x^{10} - 28x^{8} + 48x^{6} + 32x^{4} - 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 14\nu^{10} + 32\nu^{8} - 28\nu^{6} - 56\nu^{4} + 192\nu^{2} - 448 ) / 192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} + 10\nu^{10} - 16\nu^{8} + 20\nu^{6} + 40\nu^{4} - 96\nu^{2} + 320 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} + 10\nu^{11} + 32\nu^{9} + 20\nu^{7} + 40\nu^{5} + 480\nu^{3} + 320\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 30\nu^{13} + 2\nu^{11} + 4\nu^{9} - 284\nu^{7} + 32\nu^{5} - 1664\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{14} + 6\nu^{12} - 2\nu^{10} - 4\nu^{8} + 44\nu^{6} - 32\nu^{4} + 96\nu^{2} + 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} - 3\nu^{12} + 4\nu^{10} - 22\nu^{8} + 8\nu^{6} + 76\nu^{4} - 192\nu^{2} + 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{14} - 6\nu^{12} + 10\nu^{10} + 20\nu^{8} + 20\nu^{6} + 160\nu^{4} + 320 ) / 192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{15} + 6\nu^{13} - 10\nu^{11} - 20\nu^{9} + 268\nu^{7} - 160\nu^{5} + 2560\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{14} - 3\nu^{12} - 2\nu^{10} + 26\nu^{8} - 52\nu^{6} + 52\nu^{4} + 96\nu^{2} - 256 ) / 96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} - 18\nu^{13} + 22\nu^{11} - 52\nu^{9} - 52\nu^{7} + 256\nu^{5} - 672\nu^{3} + 128\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -5\nu^{15} + 30\nu^{13} - 34\nu^{11} - 20\nu^{9} + 220\nu^{7} - 160\nu^{5} + 480\nu^{3} + 640\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5\nu^{15} - 6\nu^{13} + 10\nu^{11} + 20\nu^{9} - 76\nu^{7} + 160\nu^{5} + 512\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 5\nu^{15} - 6\nu^{13} + 10\nu^{11} + 20\nu^{9} + 20\nu^{7} + 160\nu^{5} + 320\nu ) / 288 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 7\nu^{15} - 6\nu^{13} - 34\nu^{11} + 100\nu^{9} - 164\nu^{7} - 16\nu^{5} + 480\nu^{3} - 896\nu ) / 384 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} + 2\beta_{13} + 2\beta_{9} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} + \beta_{14} + \beta_{13} + 4\beta_{12} - \beta_{11} + 5\beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{7} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{15} + 7\beta_{12} + 5\beta_{11} - 4\beta_{9} - 4\beta_{5} + 4\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{10} + 2\beta_{8} + 2\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{14} - 10\beta_{13} + 2\beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{8} - 2\beta_{7} + 8\beta_{3} + 6\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 16\beta_{15} - 10\beta_{14} - 16\beta_{13} - 46\beta_{12} + 10\beta_{11} - 56\beta_{5} + 20\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -8\beta_{7} - 4\beta_{6} + 24\beta_{3} - 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -20\beta_{15} - 64\beta_{12} - 20\beta_{11} - 20\beta_{9} - 20\beta_{5} + 20\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 20\beta_{10} - 12\beta_{8} + 20\beta_{6} - 20\beta_{2} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -44\beta_{14} + 40\beta_{13} - 80\beta_{9} - 120\beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8\beta_{8} - 8\beta_{7} - 80\beta_{3} - 56\beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -88\beta_{15} + 160\beta_{14} + 88\beta_{13} + 88\beta_{12} - 160\beta_{11} + 248\beta_{5} - 248\beta_{4} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.25842 0.645283i
1.34612 0.433550i
−0.0703759 + 1.41246i
−1.04852 + 0.948998i
1.04852 0.948998i
0.0703759 1.41246i
−1.34612 + 0.433550i
1.25842 + 0.645283i
−1.25842 + 0.645283i
1.34612 + 0.433550i
−0.0703759 1.41246i
−1.04852 0.948998i
1.04852 + 0.948998i
0.0703759 + 1.41246i
−1.34612 0.433550i
1.25842 0.645283i
−1.41246 0.0703759i 0 1.99009 + 0.198807i 0.728953 + 0.420861i 0 −1.32288 2.29129i −2.79694 0.420861i 0 −1.00000 0.645751i
109.2 −0.948998 1.04852i 0 −0.198807 + 1.99009i 2.91009 + 1.68014i 0 1.32288 + 2.29129i 2.27533 1.68014i 0 −1.00000 4.64575i
109.3 −0.645283 + 1.25842i 0 −1.16722 1.62407i 0.728953 + 0.420861i 0 −1.32288 2.29129i 2.79694 0.420861i 0 −1.00000 + 0.645751i
109.4 −0.433550 1.34612i 0 −1.62407 + 1.16722i −2.91009 1.68014i 0 1.32288 + 2.29129i 2.27533 + 1.68014i 0 −1.00000 + 4.64575i
109.5 0.433550 + 1.34612i 0 −1.62407 + 1.16722i 2.91009 + 1.68014i 0 1.32288 + 2.29129i −2.27533 1.68014i 0 −1.00000 + 4.64575i
109.6 0.645283 1.25842i 0 −1.16722 1.62407i −0.728953 0.420861i 0 −1.32288 2.29129i −2.79694 + 0.420861i 0 −1.00000 + 0.645751i
109.7 0.948998 + 1.04852i 0 −0.198807 + 1.99009i −2.91009 1.68014i 0 1.32288 + 2.29129i −2.27533 + 1.68014i 0 −1.00000 4.64575i
109.8 1.41246 + 0.0703759i 0 1.99009 + 0.198807i −0.728953 0.420861i 0 −1.32288 2.29129i 2.79694 + 0.420861i 0 −1.00000 0.645751i
541.1 −1.41246 + 0.0703759i 0 1.99009 0.198807i 0.728953 0.420861i 0 −1.32288 + 2.29129i −2.79694 + 0.420861i 0 −1.00000 + 0.645751i
541.2 −0.948998 + 1.04852i 0 −0.198807 1.99009i 2.91009 1.68014i 0 1.32288 2.29129i 2.27533 + 1.68014i 0 −1.00000 + 4.64575i
541.3 −0.645283 1.25842i 0 −1.16722 + 1.62407i 0.728953 0.420861i 0 −1.32288 + 2.29129i 2.79694 + 0.420861i 0 −1.00000 0.645751i
541.4 −0.433550 + 1.34612i 0 −1.62407 1.16722i −2.91009 + 1.68014i 0 1.32288 2.29129i 2.27533 1.68014i 0 −1.00000 4.64575i
541.5 0.433550 1.34612i 0 −1.62407 1.16722i 2.91009 1.68014i 0 1.32288 2.29129i −2.27533 + 1.68014i 0 −1.00000 4.64575i
541.6 0.645283 + 1.25842i 0 −1.16722 + 1.62407i −0.728953 + 0.420861i 0 −1.32288 + 2.29129i −2.79694 0.420861i 0 −1.00000 0.645751i
541.7 0.948998 1.04852i 0 −0.198807 1.99009i −2.91009 + 1.68014i 0 1.32288 2.29129i −2.27533 1.68014i 0 −1.00000 + 4.64575i
541.8 1.41246 0.0703759i 0 1.99009 0.198807i −0.728953 + 0.420861i 0 −1.32288 + 2.29129i 2.79694 0.420861i 0 −1.00000 + 0.645751i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
24.h odd 2 1 inner
72.j odd 6 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.n.q 16
3.b odd 2 1 inner 648.2.n.q 16
4.b odd 2 1 2592.2.r.q 16
8.b even 2 1 inner 648.2.n.q 16
8.d odd 2 1 2592.2.r.q 16
9.c even 3 1 216.2.d.c 8
9.c even 3 1 inner 648.2.n.q 16
9.d odd 6 1 216.2.d.c 8
9.d odd 6 1 inner 648.2.n.q 16
12.b even 2 1 2592.2.r.q 16
24.f even 2 1 2592.2.r.q 16
24.h odd 2 1 inner 648.2.n.q 16
36.f odd 6 1 864.2.d.c 8
36.f odd 6 1 2592.2.r.q 16
36.h even 6 1 864.2.d.c 8
36.h even 6 1 2592.2.r.q 16
72.j odd 6 1 216.2.d.c 8
72.j odd 6 1 inner 648.2.n.q 16
72.l even 6 1 864.2.d.c 8
72.l even 6 1 2592.2.r.q 16
72.n even 6 1 216.2.d.c 8
72.n even 6 1 inner 648.2.n.q 16
72.p odd 6 1 864.2.d.c 8
72.p odd 6 1 2592.2.r.q 16
144.u even 12 1 6912.2.a.cd 4
144.u even 12 1 6912.2.a.ci 4
144.v odd 12 1 6912.2.a.cd 4
144.v odd 12 1 6912.2.a.ci 4
144.w odd 12 1 6912.2.a.cc 4
144.w odd 12 1 6912.2.a.cj 4
144.x even 12 1 6912.2.a.cc 4
144.x even 12 1 6912.2.a.cj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.c 8 9.c even 3 1
216.2.d.c 8 9.d odd 6 1
216.2.d.c 8 72.j odd 6 1
216.2.d.c 8 72.n even 6 1
648.2.n.q 16 1.a even 1 1 trivial
648.2.n.q 16 3.b odd 2 1 inner
648.2.n.q 16 8.b even 2 1 inner
648.2.n.q 16 9.c even 3 1 inner
648.2.n.q 16 9.d odd 6 1 inner
648.2.n.q 16 24.h odd 2 1 inner
648.2.n.q 16 72.j odd 6 1 inner
648.2.n.q 16 72.n even 6 1 inner
864.2.d.c 8 36.f odd 6 1
864.2.d.c 8 36.h even 6 1
864.2.d.c 8 72.l even 6 1
864.2.d.c 8 72.p odd 6 1
2592.2.r.q 16 4.b odd 2 1
2592.2.r.q 16 8.d odd 2 1
2592.2.r.q 16 12.b even 2 1
2592.2.r.q 16 24.f even 2 1
2592.2.r.q 16 36.f odd 6 1
2592.2.r.q 16 36.h even 6 1
2592.2.r.q 16 72.l even 6 1
2592.2.r.q 16 72.p odd 6 1
6912.2.a.cc 4 144.w odd 12 1
6912.2.a.cc 4 144.x even 12 1
6912.2.a.cd 4 144.u even 12 1
6912.2.a.cd 4 144.v odd 12 1
6912.2.a.ci 4 144.u even 12 1
6912.2.a.ci 4 144.v odd 12 1
6912.2.a.cj 4 144.w odd 12 1
6912.2.a.cj 4 144.x even 12 1

Hecke kernels

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

\( T_{5}^{8} - 12T_{5}^{6} + 136T_{5}^{4} - 96T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{8} - 22T_{13}^{6} + 475T_{13}^{4} - 198T_{13}^{2} + 81 \) Copy content Toggle raw display
\( T_{17}^{4} - 52T_{17}^{2} + 648 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{14} + 2 T^{12} - 12 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 12 T^{6} + 136 T^{4} - 96 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 7 T^{2} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 20 T^{6} + 328 T^{4} - 1440 T^{2} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 22 T^{6} + 475 T^{4} - 198 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 52 T^{2} + 648)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 58 T^{2} + 729)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 76 T^{6} + 5704 T^{4} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 80 T^{6} + 5248 T^{4} + \cdots + 1327104)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 4)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 46 T^{2} + 81)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 160 T^{6} + 20992 T^{4} + \cdots + 21233664)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 64 T^{6} + 3520 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 108 T^{6} + 11016 T^{4} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 80 T^{2} + 1152)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 180 T^{6} + 25672 T^{4} + \cdots + 45265984)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 198 T^{6} + 38475 T^{4} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 9 T^{2} + 81)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T - 111)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 12 T^{3} + 115 T^{2} - 348 T + 841)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 96 T^{6} + 8704 T^{4} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 52 T^{2} + 648)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} + 55 T^{2} + 114 T + 361)^{4} \) Copy content Toggle raw display
show more
show less