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Properties

Label 7581.2.a.bl
Level $7581$
Weight $2$
Character orbit 7581.a
Self dual yes
Analytic conductor $60.535$
Analytic rank $1$
Dimension $12$
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] = [7581,2,Mod(1,7581)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7581, 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("7581.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7581 = 3 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7581.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: \(60.5345897723\)
Analytic rank: \(1\)
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: \( x^{12} - 15x^{10} - x^{9} + 84x^{8} + 6x^{7} - 218x^{6} - 3x^{5} + 261x^{4} - 27x^{3} - 114x^{2} + 33x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 399)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} + \beta_{8} + \beta_{5} + \cdots - 1) q^{5}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} + \beta_{8} + \beta_{5} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} + 12 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 6 q^{4} - 12 q^{5} + 12 q^{7} - 3 q^{8} + 12 q^{9} + 18 q^{10} - 18 q^{11} - 6 q^{12} + 9 q^{13} + 12 q^{15} - 18 q^{16} - 18 q^{17} - 6 q^{20} - 12 q^{21} + 3 q^{22} + 6 q^{23} + 3 q^{24} + 6 q^{25} + 15 q^{26} - 12 q^{27} + 6 q^{28} - 9 q^{29} - 18 q^{30} - 21 q^{32} + 18 q^{33} + 9 q^{34} - 12 q^{35} + 6 q^{36} + 24 q^{37} - 9 q^{39} - 6 q^{40} - 3 q^{41} - 21 q^{43} - 12 q^{45} - 12 q^{46} - 3 q^{47} + 18 q^{48} + 12 q^{49} - 12 q^{50} + 18 q^{51} + 21 q^{52} - 9 q^{53} - 9 q^{55} - 3 q^{56} - 9 q^{58} - 3 q^{59} + 6 q^{60} - 33 q^{61} - 39 q^{62} + 12 q^{63} - 15 q^{64} - 24 q^{65} - 3 q^{66} + 36 q^{67} - 48 q^{68} - 6 q^{69} + 18 q^{70} - 3 q^{72} - 27 q^{73} + 3 q^{74} - 6 q^{75} - 18 q^{77} - 15 q^{78} + 42 q^{79} + 6 q^{80} + 12 q^{81} + 21 q^{82} - 12 q^{83} - 6 q^{84} + 39 q^{85} + 30 q^{86} + 9 q^{87} + 12 q^{88} - 21 q^{89} + 18 q^{90} + 9 q^{91} - 66 q^{92} - 27 q^{94} + 21 q^{96} + 75 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} - x^{9} + 84x^{8} + 6x^{7} - 218x^{6} - 3x^{5} + 261x^{4} - 27x^{3} - 114x^{2} + 33x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + \nu^{10} + 12 \nu^{9} - 11 \nu^{8} - 47 \nu^{7} + 45 \nu^{6} + 59 \nu^{5} - 84 \nu^{4} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 13\nu^{8} - \nu^{7} + 58\nu^{6} + 5\nu^{5} - 103\nu^{4} - 2\nu^{3} + 62\nu^{2} - 7\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} + \nu^{10} + 10 \nu^{9} - 11 \nu^{8} - 25 \nu^{7} + 51 \nu^{6} - 21 \nu^{5} - 118 \nu^{4} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - \nu^{10} - 12 \nu^{9} + 11 \nu^{8} + 49 \nu^{7} - 47 \nu^{6} - 77 \nu^{5} + 98 \nu^{4} + \cdots + 13 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - \nu^{10} - 14 \nu^{9} + 11 \nu^{8} + 73 \nu^{7} - 43 \nu^{6} - 171 \nu^{5} + 70 \nu^{4} + \cdots + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{10} + \nu^{9} + 13\nu^{8} - 11\nu^{7} - 60\nu^{6} + 44\nu^{5} + 115\nu^{4} - 77\nu^{3} - 78\nu^{2} + 48\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} + 12 \nu^{9} - 35 \nu^{8} - 53 \nu^{7} + 141 \nu^{6} + 101 \nu^{5} - 224 \nu^{4} + \cdots - 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 13\nu^{9} - \nu^{8} + 58\nu^{7} + 5\nu^{6} - 103\nu^{5} - 2\nu^{4} + 62\nu^{3} - 7\nu^{2} - 4\nu + 2 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 6\beta_{2} + \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} + \beta_{8} + \beta_{6} + 2\beta_{5} + 8\beta_{3} + 9\beta_{2} + 19\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} - \beta_{10} + 8 \beta_{9} - \beta_{8} + 7 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} + \beta_{4} + \cdots + 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} - \beta_{10} + 19 \beta_{9} + 8 \beta_{8} + 2 \beta_{7} + 10 \beta_{6} + 19 \beta_{5} + \cdots + 62 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 12 \beta_{11} - 10 \beta_{10} + 53 \beta_{9} - 10 \beta_{8} + 41 \beta_{7} + 51 \beta_{6} + 51 \beta_{5} + \cdots + 264 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14 \beta_{11} - 14 \beta_{10} + 136 \beta_{9} + 45 \beta_{8} + 26 \beta_{7} + 75 \beta_{6} + 136 \beta_{5} + \cdots + 394 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 99 \beta_{11} - 73 \beta_{10} + 337 \beta_{9} - 69 \beta_{8} + 232 \beta_{7} + 307 \beta_{6} + \cdots + 1373 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 133 \beta_{11} - 129 \beta_{10} + 887 \beta_{9} + 219 \beta_{8} + 230 \beta_{7} + 511 \beta_{6} + \cdots + 2428 \) 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
2.46692
2.04687
1.50309
1.39087
0.682105
0.403155
−0.0276785
−1.13149
−1.53067
−1.76109
−1.95620
−2.08588
−2.46692 −1.00000 4.08569 −2.87546 2.46692 1.00000 −5.14522 1.00000 7.09352
1.2 −2.04687 −1.00000 2.18968 0.166923 2.04687 1.00000 −0.388245 1.00000 −0.341669
1.3 −1.50309 −1.00000 0.259290 −3.00302 1.50309 1.00000 2.61645 1.00000 4.51382
1.4 −1.39087 −1.00000 −0.0654912 −3.98207 1.39087 1.00000 2.87282 1.00000 5.53853
1.5 −0.682105 −1.00000 −1.53473 0.533658 0.682105 1.00000 2.41106 1.00000 −0.364011
1.6 −0.403155 −1.00000 −1.83747 −2.42975 0.403155 1.00000 1.54710 1.00000 0.979569
1.7 0.0276785 −1.00000 −1.99923 −0.643808 −0.0276785 1.00000 −0.110693 1.00000 −0.0178196
1.8 1.13149 −1.00000 −0.719726 −0.324136 −1.13149 1.00000 −3.07735 1.00000 −0.366757
1.9 1.53067 −1.00000 0.342959 1.99105 −1.53067 1.00000 −2.53639 1.00000 3.04764
1.10 1.76109 −1.00000 1.10144 −3.86228 −1.76109 1.00000 −1.58244 1.00000 −6.80183
1.11 1.95620 −1.00000 1.82670 2.67895 −1.95620 1.00000 −0.338999 1.00000 5.24054
1.12 2.08588 −1.00000 2.35089 −0.250036 −2.08588 1.00000 0.731912 1.00000 −0.521544
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(19\) \(-1\)

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 7581.2.a.bl 12
19.b odd 2 1 7581.2.a.bm 12
19.f odd 18 2 399.2.bo.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.bo.c ✓ 24 19.f odd 18 2
7581.2.a.bl 12 1.a even 1 1 trivial
7581.2.a.bm 12 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):

\( T_{2}^{12} - 15 T_{2}^{10} + T_{2}^{9} + 84 T_{2}^{8} - 6 T_{2}^{7} - 218 T_{2}^{6} + 3 T_{2}^{5} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{12} + 12 T_{5}^{11} + 39 T_{5}^{10} - 55 T_{5}^{9} - 513 T_{5}^{8} - 564 T_{5}^{7} + 1232 T_{5}^{6} + \cdots + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 15 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 12 T^{11} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 18 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} - 9 T^{11} + \cdots - 109432 \) Copy content Toggle raw display
$17$ \( T^{12} + 18 T^{11} + \cdots - 1691 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 140650217 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 240348736 \) Copy content Toggle raw display
$31$ \( T^{12} - 246 T^{10} + \cdots + 38487511 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 160955461 \) Copy content Toggle raw display
$41$ \( T^{12} + 3 T^{11} + \cdots - 4577113 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 243775936 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 561247048 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 138282328 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 19572763832 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 50697185984 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 6956086976 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 1183197833 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 10034139032 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 1162883584 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10123267816 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 100267991 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2961285632 \) Copy content Toggle raw display
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