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Properties

Label 7232.2.a.bf
Level $7232$
Weight $2$
Character orbit 7232.a
Self dual yes
Analytic conductor $57.748$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7232,2,Mod(1,7232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7232, 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("7232.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7232 = 2^{6} \cdot 113 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7232.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: \(57.7478107418\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 13x^{7} - 3x^{6} + 42x^{5} + 16x^{4} - 34x^{3} - 4x^{2} + 10x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 904)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_{6} q^{7} + (\beta_{8} + \beta_{7} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_{6} q^{7} + (\beta_{8} + \beta_{7} + 1) q^{9} - \beta_{5} q^{11} + (\beta_{7} - \beta_{3} - 1) q^{13} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - 1) q^{15} + ( - \beta_{8} + \beta_{5} + 2) q^{17} + ( - \beta_{8} + \beta_{6} - \beta_1) q^{19} + ( - \beta_{6} + \beta_{5} + \beta_{3} - 1) q^{21} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{23} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{25} + ( - 2 \beta_{7} - \beta_{4} + \beta_{3} + \cdots - 1) q^{27}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{3} - 9 q^{5} - 2 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{3} - 9 q^{5} - 2 q^{7} + 14 q^{9} + 4 q^{11} - 8 q^{13} + q^{15} + 10 q^{17} - 8 q^{19} - 11 q^{21} - 2 q^{23} + 22 q^{25} - 12 q^{27} - 17 q^{29} - 9 q^{31} + 12 q^{33} + 2 q^{35} - 20 q^{37} - 8 q^{39} + 25 q^{41} - 18 q^{43} - 32 q^{45} - 19 q^{47} + 19 q^{49} - 3 q^{51} - 27 q^{53} - 16 q^{55} + 7 q^{57} + 13 q^{59} - 23 q^{61} + 16 q^{63} + 20 q^{65} - 5 q^{67} - 3 q^{69} - 10 q^{71} + 23 q^{73} - 16 q^{75} - 13 q^{77} + 3 q^{79} + 13 q^{81} + 10 q^{85} + q^{87} + 26 q^{89} - 9 q^{91} + 28 q^{93} + 16 q^{97} - 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 13x^{7} - 3x^{6} + 42x^{5} + 16x^{4} - 34x^{3} - 4x^{2} + 10x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -5\nu^{8} - 2\nu^{7} + 65\nu^{6} + 40\nu^{5} - 203\nu^{4} - 153\nu^{3} + 130\nu^{2} + 64\nu - 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -6\nu^{8} - 3\nu^{7} + 78\nu^{6} + 56\nu^{5} - 242\nu^{4} - 210\nu^{3} + 148\nu^{2} + 92\nu - 39 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -6\nu^{8} - 3\nu^{7} + 78\nu^{6} + 56\nu^{5} - 242\nu^{4} - 210\nu^{3} + 148\nu^{2} + 94\nu - 39 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 12\nu^{8} + 4\nu^{7} - 154\nu^{6} - 88\nu^{5} + 467\nu^{4} + 353\nu^{3} - 272\nu^{2} - 144\nu + 66 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 12\nu^{8} + 5\nu^{7} - 155\nu^{6} - 100\nu^{5} + 476\nu^{4} + 386\nu^{3} - 288\nu^{2} - 162\nu + 74 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -17\nu^{8} - 7\nu^{7} + 219\nu^{6} + 140\nu^{5} - 666\nu^{4} - 536\nu^{3} + 378\nu^{2} + 210\nu - 90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 20\nu^{8} + 6\nu^{7} - 257\nu^{6} - 138\nu^{5} + 784\nu^{4} + 562\nu^{3} - 470\nu^{2} - 228\nu + 108 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 21\nu^{8} + 8\nu^{7} - 270\nu^{6} - 166\nu^{5} + 820\nu^{4} + 650\nu^{3} - 474\nu^{2} - 268\nu + 116 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} - 3\beta_{4} + \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{8} + \beta_{6} - 2\beta_{5} - \beta_{4} + 7\beta_{3} - 8\beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10\beta_{8} + 8\beta_{7} + \beta_{6} + 2\beta_{5} - 27\beta_{4} + 3\beta_{3} + 7\beta _1 + 41 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23\beta_{8} + 10\beta_{6} - 19\beta_{5} - 15\beta_{4} + 59\beta_{3} - 64\beta_{2} - 13\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 48\beta_{8} + 32\beta_{7} + 7\beta_{6} + 9\beta_{5} - 117\beta_{4} + 22\beta_{3} - 4\beta_{2} + 25\beta _1 + 164 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 116\beta_{8} + 4\beta_{7} + 46\beta_{6} - 80\beta_{5} - 94\beta_{4} + 256\beta_{3} - 265\beta_{2} - 60\beta _1 + 105 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 449 \beta_{8} + 265 \beta_{7} + 77 \beta_{6} + 63 \beta_{5} - 1019 \beta_{4} + 258 \beta_{3} + \cdots + 1382 \) 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
0.352794
−1.21304
−0.848935
0.582081
0.377865
1.97698
−1.48860
−2.78876
3.04962
0 −3.30594 0 −0.360640 0 −1.54040 0 7.92925 0
1.2 0 −2.78710 0 −3.92764 0 3.56411 0 4.76794 0
1.3 0 −1.92644 0 2.81163 0 4.04739 0 0.711180 0
1.4 0 −1.62934 0 −3.12824 0 −4.36409 0 −0.345237 0
1.5 0 −0.199732 0 −2.28438 0 2.52842 0 −2.96011 0
1.6 0 0.135658 0 3.11650 0 −2.65853 0 −2.98160 0
1.7 0 1.54482 0 −0.721614 0 −4.12055 0 −0.613532 0
1.8 0 2.32177 0 −4.27152 0 −0.402809 0 2.39062 0
1.9 0 2.84631 0 −0.234095 0 0.946456 0 5.10150 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(113\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7232.2.a.bf 9
4.b odd 2 1 7232.2.a.bg 9
8.b even 2 1 1808.2.a.o 9
8.d odd 2 1 904.2.a.e ✓ 9
24.f even 2 1 8136.2.a.w 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
904.2.a.e ✓ 9 8.d odd 2 1
1808.2.a.o 9 8.b even 2 1
7232.2.a.bf 9 1.a even 1 1 trivial
7232.2.a.bg 9 4.b odd 2 1
8136.2.a.w 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7232))\):

\( T_{3}^{9} + 3T_{3}^{8} - 16T_{3}^{7} - 47T_{3}^{6} + 76T_{3}^{5} + 222T_{3}^{4} - 96T_{3}^{3} - 308T_{3}^{2} - 16T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{9} + 9T_{5}^{8} + 7T_{5}^{7} - 137T_{5}^{6} - 348T_{5}^{5} + 308T_{5}^{4} + 1744T_{5}^{3} + 1680T_{5}^{2} + 576T_{5} + 64 \) Copy content Toggle raw display
\( T_{7}^{9} + 2T_{7}^{8} - 39T_{7}^{7} - 65T_{7}^{6} + 496T_{7}^{5} + 656T_{7}^{4} - 2176T_{7}^{3} - 2176T_{7}^{2} + 2048T_{7} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{9} + 9 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{9} + 2 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( T^{9} - 4 T^{8} + \cdots + 512 \) Copy content Toggle raw display
$13$ \( T^{9} + 8 T^{8} + \cdots - 256 \) Copy content Toggle raw display
$17$ \( T^{9} - 10 T^{8} + \cdots - 31808 \) Copy content Toggle raw display
$19$ \( T^{9} + 8 T^{8} + \cdots + 184 \) Copy content Toggle raw display
$23$ \( T^{9} + 2 T^{8} + \cdots + 536 \) Copy content Toggle raw display
$29$ \( T^{9} + 17 T^{8} + \cdots - 12608 \) Copy content Toggle raw display
$31$ \( T^{9} + 9 T^{8} + \cdots - 512 \) Copy content Toggle raw display
$37$ \( T^{9} + 20 T^{8} + \cdots - 114368 \) Copy content Toggle raw display
$41$ \( T^{9} - 25 T^{8} + \cdots - 80128 \) Copy content Toggle raw display
$43$ \( T^{9} + 18 T^{8} + \cdots - 3759352 \) Copy content Toggle raw display
$47$ \( T^{9} + 19 T^{8} + \cdots + 18472 \) Copy content Toggle raw display
$53$ \( T^{9} + 27 T^{8} + \cdots + 6563584 \) Copy content Toggle raw display
$59$ \( T^{9} - 13 T^{8} + \cdots - 43768 \) Copy content Toggle raw display
$61$ \( T^{9} + 23 T^{8} + \cdots - 3750464 \) Copy content Toggle raw display
$67$ \( T^{9} + 5 T^{8} + \cdots + 2294536 \) Copy content Toggle raw display
$71$ \( T^{9} + 10 T^{8} + \cdots + 71139136 \) Copy content Toggle raw display
$73$ \( T^{9} - 23 T^{8} + \cdots + 18112 \) Copy content Toggle raw display
$79$ \( T^{9} - 3 T^{8} + \cdots + 464276072 \) Copy content Toggle raw display
$83$ \( T^{9} - 359 T^{7} + \cdots + 100496384 \) Copy content Toggle raw display
$89$ \( T^{9} - 26 T^{8} + \cdots - 129728 \) Copy content Toggle raw display
$97$ \( T^{9} - 16 T^{8} + \cdots - 3850048 \) Copy content Toggle raw display
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