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Properties

Label 9248.2.a.bv
Level $9248$
Weight $2$
Character orbit 9248.a
Self dual yes
Analytic conductor $73.846$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9248,2,Mod(1,9248)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9248, 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("9248.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9248 = 2^{5} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9248.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: \(73.8456517893\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.12.10455582754471936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 18x^{10} + 122x^{8} - 384x^{6} + 553x^{4} - 294x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 544)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} - \beta_{4} q^{5} + (\beta_{11} - \beta_{6}) q^{7} + ( - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} - \beta_{4} q^{5} + (\beta_{11} - \beta_{6}) q^{7} + ( - \beta_{3} + 2) q^{9} - \beta_{7} q^{11} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{2} + \beta_1) q^{15} + (\beta_{10} - \beta_1) q^{19} + ( - \beta_{5} - \beta_{3} + 3) q^{21} + (\beta_{11} - \beta_{7}) q^{23} + ( - \beta_{5} - 2 \beta_{3} + 5) q^{25} + ( - \beta_{7} - \beta_{6}) q^{27} + ( - \beta_{9} + \beta_{8}) q^{29} + (\beta_{11} + \beta_{7}) q^{31} + (2 \beta_{5} - \beta_{3} + 1) q^{33} + (2 \beta_{10} - \beta_{2} + \beta_1) q^{35} + ( - 2 \beta_{8} - \beta_{4}) q^{37} + ( - \beta_{7} - 3 \beta_{6}) q^{39} + \beta_{8} q^{41} + ( - \beta_{10} - 2 \beta_1) q^{43} + (\beta_{9} + 3 \beta_{8} - 4 \beta_{4}) q^{45} + (2 \beta_{10} + 2 \beta_1) q^{47} + ( - \beta_{5} + \beta_{3} + 2) q^{49} + ( - \beta_{5} + 2) q^{53} + ( - 5 \beta_{10} - \beta_1) q^{55} + ( - \beta_{9} - 7 \beta_{8} + 3 \beta_{4}) q^{57} + \beta_{2} q^{59} + ( - 2 \beta_{8} + \beta_{4}) q^{61} + ( - \beta_{11} + \beta_{7} - 2 \beta_{6}) q^{63} + (\beta_{9} + 3 \beta_{8} - 3 \beta_{4}) q^{65} + ( - \beta_{10} + \beta_{2} + \beta_1) q^{67} + (\beta_{5} - \beta_{3} - 1) q^{69} + (3 \beta_{11} - 5 \beta_{6}) q^{71} + (2 \beta_{9} + 5 \beta_{8}) q^{73} + (2 \beta_{11} - 9 \beta_{6}) q^{75} + (3 \beta_{5} + 3 \beta_{3} - 1) q^{77} + (\beta_{11} + \beta_{7} + 2 \beta_{6}) q^{79} + (2 \beta_{5} + \beta_{3}) q^{81} + ( - 2 \beta_{10} + \beta_{2}) q^{83} + (\beta_{10} + \beta_1) q^{87} + ( - \beta_{5} + \beta_{3} + 9) q^{89} + ( - 2 \beta_{11} + \beta_{7} - \beta_{6}) q^{91} + ( - 3 \beta_{5} + \beta_{3} - 3) q^{93} + (2 \beta_{11} + 2 \beta_{7} + 4 \beta_{6}) q^{95} + ( - \beta_{9} - \beta_{4}) q^{97} + ( - 4 \beta_{11} - 2 \beta_{7} - 3 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 20 q^{9} + 8 q^{13} + 32 q^{21} + 52 q^{25} + 8 q^{33} + 28 q^{49} + 24 q^{53} - 16 q^{69} + 4 q^{81} + 112 q^{89} - 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 18x^{10} + 122x^{8} - 384x^{6} + 553x^{4} - 294x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 9\nu^{4} + 20\nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 12\nu^{6} + 45\nu^{4} - 54\nu^{2} + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 13\nu^{9} + 53\nu^{7} - 63\nu^{5} - 22\nu^{3} + 28\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{8} + 12\nu^{6} - 43\nu^{4} + 42\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 15\nu^{9} + 81\nu^{7} - 193\nu^{5} + 194\nu^{3} - 44\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 15\nu^{9} + 73\nu^{7} - 97\nu^{5} - 150\nu^{3} + 276\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 17\nu^{9} + 105\nu^{7} - 283\nu^{5} + 310\nu^{3} - 100\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} - 15\nu^{7} + 79\nu^{5} - 169\nu^{3} + 120\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} - 15\nu^{8} + 79\nu^{6} - 171\nu^{4} + 126\nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 17\nu^{9} + 107\nu^{7} - 303\nu^{5} + 368\nu^{3} - 148\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + 5\beta_{9} + 3\beta_{8} - 5\beta_{7} + \beta_{6} - 3\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + 6\beta _1 + 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 18\beta_{11} + 27\beta_{9} + 5\beta_{8} - 25\beta_{7} - 3\beta_{6} - 13\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9\beta_{5} + 18\beta_{3} + 2\beta_{2} + 34\beta _1 + 126 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 130\beta_{11} + 149\beta_{9} - 29\beta_{8} - 129\beta_{7} - 35\beta_{6} - 67\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 63\beta_{5} + 130\beta_{3} + 24\beta_{2} + 192\beta _1 + 646 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 866\beta_{11} + 835\beta_{9} - 443\beta_{8} - 685\beta_{7} - 239\beta_{6} - 365\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4\beta_{10} + 405\beta_{5} + 870\beta_{3} + 202\beta_{2} + 1094\beta _1 + 3426 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5546\beta_{11} + 4741\beta_{9} - 3869\beta_{8} - 3725\beta_{7} - 1447\beta_{6} - 2019\beta_{4} ) / 4 \) 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
−2.16202
1.15137
2.03260
1.36695
2.44664
−0.118190
0.118190
−2.44664
−1.36695
−2.03260
−1.15137
2.16202
0 −2.94718 0 −3.52039 0 −0.579290 0 5.68585 0
1.2 0 −2.94718 0 3.52039 0 −0.579290 0 5.68585 0
1.3 0 −2.22292 0 −3.92933 0 −3.82303 0 1.94137 0
1.4 0 −2.22292 0 3.92933 0 −3.82303 0 1.94137 0
1.5 0 −0.610563 0 −0.408946 0 3.61232 0 −2.62721 0
1.6 0 −0.610563 0 0.408946 0 3.61232 0 −2.62721 0
1.7 0 0.610563 0 −0.408946 0 −3.61232 0 −2.62721 0
1.8 0 0.610563 0 0.408946 0 −3.61232 0 −2.62721 0
1.9 0 2.22292 0 −3.92933 0 3.82303 0 1.94137 0
1.10 0 2.22292 0 3.92933 0 3.82303 0 1.94137 0
1.11 0 2.94718 0 −3.52039 0 0.579290 0 5.68585 0
1.12 0 2.94718 0 3.52039 0 0.579290 0 5.68585 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(17\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.b even 2 1 inner
68.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9248.2.a.bv 12
4.b odd 2 1 inner 9248.2.a.bv 12
17.b even 2 1 inner 9248.2.a.bv 12
17.d even 8 2 544.2.o.i ✓ 12
68.d odd 2 1 inner 9248.2.a.bv 12
68.g odd 8 2 544.2.o.i ✓ 12
136.o even 8 2 1088.2.o.w 12
136.p odd 8 2 1088.2.o.w 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.o.i ✓ 12 17.d even 8 2
544.2.o.i ✓ 12 68.g odd 8 2
1088.2.o.w 12 136.o even 8 2
1088.2.o.w 12 136.p odd 8 2
9248.2.a.bv 12 1.a even 1 1 trivial
9248.2.a.bv 12 4.b odd 2 1 inner
9248.2.a.bv 12 17.b even 2 1 inner
9248.2.a.bv 12 68.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9248))\):

\( T_{3}^{6} - 14T_{3}^{4} + 48T_{3}^{2} - 16 \) Copy content Toggle raw display
\( T_{5}^{6} - 28T_{5}^{4} + 196T_{5}^{2} - 32 \) Copy content Toggle raw display
\( T_{7}^{6} - 28T_{7}^{4} + 200T_{7}^{2} - 64 \) Copy content Toggle raw display
\( T_{19}^{6} - 104T_{19}^{4} + 2720T_{19}^{2} - 2048 \) Copy content Toggle raw display
\( T_{43}^{6} - 188T_{43}^{4} + 8384T_{43}^{2} - 46208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 14 T^{4} + \cdots - 16)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 28 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 28 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 62 T^{4} + \cdots - 8464)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 2 T^{2} - 16 T + 16)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} - 104 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 84 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 76 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 92 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 52 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2)^{6} \) Copy content Toggle raw display
$43$ \( (T^{6} - 188 T^{4} + \cdots - 46208)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 224 T^{4} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} - 16 T + 32)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} - 124 T^{4} + \cdots - 67712)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 52 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 212 T^{4} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 404 T^{4} + \cdots - 1364224)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 374 T^{4} + \cdots - 1113032)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 180 T^{4} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 220 T^{4} + \cdots - 15488)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 28 T^{2} + \cdots - 64)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - 118 T^{4} + \cdots - 10952)^{2} \) Copy content Toggle raw display
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