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Properties

Label 7225.2.a.br
Level $7225$
Weight $2$
Character orbit 7225.a
Self dual yes
Analytic conductor $57.692$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7225,2,Mod(1,7225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7225 = 5^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7225.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: \(57.6919154604\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 20x^{10} + 135x^{8} - 400x^{6} + 515x^{4} - 222x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 425)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_1 q^{3} + (\beta_{4} + 1) q^{4} + \beta_{11} q^{6} + (\beta_{11} - \beta_{6} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{4} + 1) q^{8} + (\beta_{5} - \beta_{3}) q^{9} + (\beta_{10} - \beta_{9}) q^{11}+ \cdots + (\beta_{11} - \beta_{10} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 12 q^{4} + 12 q^{8} + 4 q^{9} + 12 q^{13} + 4 q^{16} + 4 q^{18} + 12 q^{19} - 8 q^{21} + 12 q^{26} + 48 q^{32} - 4 q^{33} - 20 q^{36} - 12 q^{38} + 56 q^{42} + 16 q^{43} + 36 q^{47} + 16 q^{49}+ \cdots + 76 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 20x^{10} + 135x^{8} - 400x^{6} + 515x^{4} - 222x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 17\nu^{7} + 83\nu^{5} - 135\nu^{3} + 38\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 17\nu^{8} + 83\nu^{6} - 135\nu^{4} + 38\nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{8} + 32\nu^{6} - 139\nu^{4} + 196\nu^{2} - 45 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 17\nu^{8} + 83\nu^{6} - 135\nu^{4} + 43\nu^{2} - 15 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{9} - 32\nu^{7} + 139\nu^{5} - 196\nu^{3} + 40\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{10} - 33\nu^{8} + 155\nu^{6} - 268\nu^{4} + 163\nu^{2} - 45 ) / 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{10} - 51\nu^{8} + 254\nu^{6} - 470\nu^{4} + 274\nu^{2} - 35 ) / 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - 18\nu^{9} + 102\nu^{7} - 245\nu^{5} + 247\nu^{3} - 64\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{11} - 36\nu^{9} + 202\nu^{7} - 463\nu^{5} + 425\nu^{3} - 127\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{11} - 51\nu^{9} + 254\nu^{7} - 470\nu^{5} + 274\nu^{3} - 35\nu ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - 2\beta_{9} + \beta_{6} - 2\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} - 2\beta_{7} + 10\beta_{5} - \beta_{4} - 12\beta_{3} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} + 10\beta_{10} - 26\beta_{9} + 11\beta_{6} - 28\beta_{2} + 60\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{8} - 26\beta_{7} + 98\beta_{5} - 13\beta_{4} - 127\beta_{3} + 132 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{11} + 98\beta_{10} - 277\beta_{9} + 114\beta_{6} - 309\beta_{2} + 569\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 293\beta_{8} - 277\beta_{7} + 971\beta_{5} - 141\beta_{4} - 1296\beta_{3} + 1202 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 293\beta_{11} + 971\beta_{10} - 2821\beta_{9} + 1160\beta_{6} - 3194\beta_{2} + 5600\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3010\beta_{8} - 2821\beta_{7} + 9685\beta_{5} - 1453\beta_{4} - 13068\beta_{3} + 11659 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3010\beta_{11} + 9685\beta_{10} - 28395\beta_{9} + 11700\beta_{6} - 32340\beta_{2} + 55797\beta_1 \) 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
−1.85347
1.85347
−1.62170
1.62170
−0.416780
0.416780
−3.16588
3.16588
−1.80252
1.80252
−0.699410
0.699410
−2.08389 −1.85347 2.34261 0 3.86244 1.37395 −0.713960 0.435362 0
1.2 −2.08389 1.85347 2.34261 0 −3.86244 −1.37395 −0.713960 0.435362 0
1.3 −1.44718 −1.62170 0.0943296 0 2.34689 3.81562 2.75785 −0.370087 0
1.4 −1.44718 1.62170 0.0943296 0 −2.34689 −3.81562 2.75785 −0.370087 0
1.5 −0.0601793 −0.416780 −1.99638 0 0.0250815 1.27391 0.240499 −2.82629 0
1.6 −0.0601793 0.416780 −1.99638 0 −0.0250815 −1.27391 0.240499 −2.82629 0
1.7 1.21647 −3.16588 −0.520205 0 −3.85119 0.961594 −3.06575 7.02277 0
1.8 1.21647 3.16588 −0.520205 0 3.85119 −0.961594 −3.06575 7.02277 0
1.9 1.68228 −1.80252 0.830060 0 −3.03234 −2.72602 −1.96816 0.249077 0
1.10 1.68228 1.80252 0.830060 0 3.03234 2.72602 −1.96816 0.249077 0
1.11 2.69251 −0.699410 5.24959 0 −1.88317 −4.85537 8.74953 −2.51083 0
1.12 2.69251 0.699410 5.24959 0 1.88317 4.85537 8.74953 −2.51083 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
\(5\) \( -1 \)
\(17\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7225.2.a.br 12
5.b even 2 1 7225.2.a.bm 12
17.b even 2 1 inner 7225.2.a.br 12
17.d even 8 2 425.2.e.e yes 12
85.c even 2 1 7225.2.a.bm 12
85.k odd 8 2 425.2.j.a 12
85.m even 8 2 425.2.e.c ✓ 12
85.n odd 8 2 425.2.j.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.2.e.c ✓ 12 85.m even 8 2
425.2.e.e yes 12 17.d even 8 2
425.2.j.a 12 85.k odd 8 2
425.2.j.d 12 85.n odd 8 2
7225.2.a.bm 12 5.b even 2 1
7225.2.a.bm 12 85.c even 2 1
7225.2.a.br 12 1.a even 1 1 trivial
7225.2.a.br 12 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7225))\):

\( T_{2}^{6} - 2T_{2}^{5} - 7T_{2}^{4} + 12T_{2}^{3} + 11T_{2}^{2} - 16T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{12} - 20T_{3}^{10} + 135T_{3}^{8} - 400T_{3}^{6} + 515T_{3}^{4} - 222T_{3}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{12} - 50T_{7}^{10} + 835T_{7}^{8} - 5620T_{7}^{6} + 15395T_{7}^{4} - 17868T_{7}^{2} + 7225 \) Copy content Toggle raw display
\( T_{11}^{12} - 64T_{11}^{10} + 1216T_{11}^{8} - 9188T_{11}^{6} + 24944T_{11}^{4} - 6624T_{11}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{5} - 7 T^{4} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 20 T^{10} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 50 T^{10} + \cdots + 7225 \) Copy content Toggle raw display
$11$ \( T^{12} - 64 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + \cdots + 755)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} - 20 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 156 T^{10} + \cdots + 8584900 \) Copy content Toggle raw display
$29$ \( T^{12} - 58 T^{10} + \cdots + 35344 \) Copy content Toggle raw display
$31$ \( T^{12} - 112 T^{10} + \cdots + 10609 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 5396371600 \) Copy content Toggle raw display
$41$ \( T^{12} - 154 T^{10} + \cdots + 35344 \) Copy content Toggle raw display
$43$ \( (T^{6} - 8 T^{5} + \cdots + 1076)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 18 T^{5} + \cdots + 234104)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 24 T^{5} + \cdots - 24049)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 16 T^{5} + \cdots + 15524)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 80947078144 \) Copy content Toggle raw display
$67$ \( (T^{6} + 10 T^{5} + \cdots - 9824)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 234 T^{10} + \cdots + 72982849 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 762163920400 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1809056089 \) Copy content Toggle raw display
$83$ \( (T^{6} - 26 T^{5} + \cdots + 344)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 12 T^{5} + \cdots - 500)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 41697640000 \) Copy content Toggle raw display
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