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Properties

Label 9216.2.a.bq
Level $9216$
Weight $2$
Character orbit 9216.a
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

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: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.10070523904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 8x^{6} + 24x^{5} + 30x^{4} - 16x^{3} - 20x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 144)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{5} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{5} - 1) q^{7} + (\beta_{7} - \beta_{2}) q^{11} + \beta_1 q^{13} + \beta_{3} q^{17} + (\beta_{4} - \beta_1) q^{19} + (\beta_{6} - \beta_{3}) q^{23} + (2 \beta_{5} + 1) q^{25} + ( - 2 \beta_{7} + \beta_{2}) q^{29} + (\beta_{5} - 3) q^{31} + ( - \beta_{7} - 3 \beta_{2}) q^{35} + ( - 2 \beta_{4} - \beta_1) q^{37} + ( - 2 \beta_{6} - \beta_{3}) q^{41} + ( - 3 \beta_{4} - \beta_1) q^{43} + ( - 3 \beta_{6} - \beta_{3}) q^{47} + (2 \beta_{5} + 1) q^{49} + ( - 2 \beta_{7} - \beta_{2}) q^{53} - 4 q^{55} + ( - 2 \beta_{7} - 2 \beta_{2}) q^{59} + (2 \beta_{4} + \beta_1) q^{61} + (2 \beta_{6} - \beta_{3}) q^{65} + 4 \beta_{4} q^{67} + (2 \beta_{6} + 2 \beta_{3}) q^{71} + (2 \beta_{5} + 2) q^{73} + 2 \beta_{7} q^{77} + (\beta_{5} - 7) q^{79} + (\beta_{7} - \beta_{2}) q^{83} + ( - 2 \beta_{4} - 2 \beta_1) q^{85} + 2 \beta_{6} q^{89} + ( - 7 \beta_{4} - \beta_1) q^{91} + ( - \beta_{6} + \beta_{3}) q^{95} + 4 \beta_{5} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 8 q^{25} - 24 q^{31} + 8 q^{49} - 32 q^{55} + 16 q^{73} - 56 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 8x^{6} + 24x^{5} + 30x^{4} - 16x^{3} - 20x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} - 9\nu^{6} - 10\nu^{5} + 47\nu^{4} + 26\nu^{3} - 12\nu^{2} - 4\nu - 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -18\nu^{7} + 83\nu^{6} + 94\nu^{5} - 492\nu^{4} - 246\nu^{3} + 452\nu^{2} + 106\nu - 66 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20\nu^{7} - 90\nu^{6} - 114\nu^{5} + 532\nu^{4} + 332\nu^{3} - 462\nu^{2} - 168\nu + 64 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\nu^{7} - 91\nu^{6} - 110\nu^{5} + 541\nu^{4} + 302\nu^{3} - 490\nu^{2} - 128\nu + 76 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -38\nu^{7} + 174\nu^{6} + 204\nu^{5} - 1033\nu^{4} - 548\nu^{3} + 942\nu^{2} + 240\nu - 145 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44\nu^{7} - 200\nu^{6} - 242\nu^{5} + 1184\nu^{4} + 668\nu^{3} - 1046\nu^{2} - 284\nu + 152 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -44\nu^{7} + 199\nu^{6} + 248\nu^{5} - 1186\nu^{4} - 698\nu^{3} + 1066\nu^{2} + 314\nu - 154 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} - 9\beta_{6} + 20\beta_{5} + 22\beta_{4} + 3\beta_{3} - 26\beta_{2} + 6\beta _1 + 44 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{7} - 12\beta_{6} + 17\beta_{5} + 21\beta_{4} + 2\beta_{3} - 24\beta_{2} + 9\beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -39\beta_{7} - 105\beta_{6} + 144\beta_{5} + 209\beta_{4} + 25\beta_{3} - 197\beta_{2} + 75\beta _1 + 366 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -176\beta_{7} - 473\beta_{6} + 566\beta_{5} + 890\beta_{4} + 97\beta_{3} - 786\beta_{2} + 344\beta _1 + 1526 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 685 \beta_{7} - 2037 \beta_{6} + 2352 \beta_{5} + 3928 \beta_{4} + 448 \beta_{3} - 3245 \beta_{2} + \cdots + 6168 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.79591
4.21012
0.305093
−1.10912
−0.536630
−1.95084
−0.564373
0.849841
0 0 0 −3.36028 0 −3.64575 0 0 0
1.2 0 0 0 −3.36028 0 −3.64575 0 0 0
1.3 0 0 0 −0.841723 0 1.64575 0 0 0
1.4 0 0 0 −0.841723 0 1.64575 0 0 0
1.5 0 0 0 0.841723 0 1.64575 0 0 0
1.6 0 0 0 0.841723 0 1.64575 0 0 0
1.7 0 0 0 3.36028 0 −3.64575 0 0 0
1.8 0 0 0 3.36028 0 −3.64575 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)

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
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9216.2.a.bq 8
3.b odd 2 1 inner 9216.2.a.bq 8
4.b odd 2 1 9216.2.a.bt 8
8.b even 2 1 inner 9216.2.a.bq 8
8.d odd 2 1 9216.2.a.bt 8
12.b even 2 1 9216.2.a.bt 8
24.f even 2 1 9216.2.a.bt 8
24.h odd 2 1 inner 9216.2.a.bq 8
32.g even 8 2 144.2.k.c ✓ 8
32.g even 8 2 1152.2.k.e 8
32.h odd 8 2 576.2.k.c 8
32.h odd 8 2 1152.2.k.d 8
96.o even 8 2 576.2.k.c 8
96.o even 8 2 1152.2.k.d 8
96.p odd 8 2 144.2.k.c ✓ 8
96.p odd 8 2 1152.2.k.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.k.c ✓ 8 32.g even 8 2
144.2.k.c ✓ 8 96.p odd 8 2
576.2.k.c 8 32.h odd 8 2
576.2.k.c 8 96.o even 8 2
1152.2.k.d 8 32.h odd 8 2
1152.2.k.d 8 96.o even 8 2
1152.2.k.e 8 32.g even 8 2
1152.2.k.e 8 96.p odd 8 2
9216.2.a.bq 8 1.a even 1 1 trivial
9216.2.a.bq 8 3.b odd 2 1 inner
9216.2.a.bq 8 8.b even 2 1 inner
9216.2.a.bq 8 24.h odd 2 1 inner
9216.2.a.bt 8 4.b odd 2 1
9216.2.a.bt 8 8.d odd 2 1
9216.2.a.bt 8 12.b even 2 1
9216.2.a.bt 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\):

\( T_{5}^{4} - 12T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} - 6 \) Copy content Toggle raw display
\( T_{11}^{4} - 24T_{11}^{2} + 32 \) Copy content Toggle raw display
\( T_{13}^{2} - 14 \) Copy content Toggle raw display
\( T_{17}^{4} - 40T_{17}^{2} + 288 \) Copy content Toggle raw display
\( T_{19}^{4} - 32T_{19}^{2} + 144 \) Copy content Toggle raw display
\( T_{67}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 12 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 6)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 24 T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 14)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{2} + 288)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 32 T^{2} + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 80 T^{2} + 1152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 76 T^{2} + 72)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 44 T^{2} + 36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 104 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 64 T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 208 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 108 T^{2} + 2888)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 160 T^{2} + 4608)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 44 T^{2} + 36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 192 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 4 T - 24)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 42)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 24 T^{2} + 32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 96 T^{2} + 512)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 112)^{4} \) Copy content Toggle raw display
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