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Properties

Label 7406.2.a.br
Level $7406$
Weight $2$
Character orbit 7406.a
Self dual yes
Analytic conductor $59.137$
Analytic rank $0$
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] = [7406,2,Mod(1,7406)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7406, 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("7406.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.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: \(59.1372077370\)
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: \( x^{12} - 26 x^{10} - 4 x^{9} + 246 x^{8} + 72 x^{7} - 1016 x^{6} - 396 x^{5} + 1714 x^{4} + 688 x^{3} + \cdots - 23 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{8} q^{5} - \beta_1 q^{6} + q^{7} - q^{8} + ( - \beta_{7} + \beta_{6} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_{8} q^{5} - \beta_1 q^{6} + q^{7} - q^{8} + ( - \beta_{7} + \beta_{6} - \beta_{2} + 1) q^{9} + \beta_{8} q^{10} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{11} + \beta_1 q^{12} + ( - \beta_{10} - \beta_{8} - \beta_{7} + \cdots + 1) q^{13}+ \cdots + (\beta_{11} - 2 \beta_{9} + 3 \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} + 12 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} + 12 q^{7} - 12 q^{8} + 16 q^{9} + 4 q^{13} - 12 q^{14} - 16 q^{15} + 12 q^{16} + 8 q^{17} - 16 q^{18} + 24 q^{19} + 12 q^{25} - 4 q^{26} + 12 q^{27} + 12 q^{28} - 4 q^{29} + 16 q^{30} + 4 q^{31} - 12 q^{32} + 60 q^{33} - 8 q^{34} + 16 q^{36} + 16 q^{37} - 24 q^{38} - 4 q^{39} - 20 q^{41} - 16 q^{43} + 8 q^{45} + 4 q^{47} + 12 q^{49} - 12 q^{50} + 24 q^{51} + 4 q^{52} - 8 q^{53} - 12 q^{54} - 20 q^{55} - 12 q^{56} + 4 q^{58} - 20 q^{59} - 16 q^{60} + 32 q^{61} - 4 q^{62} + 16 q^{63} + 12 q^{64} + 16 q^{65} - 60 q^{66} + 8 q^{68} + 24 q^{71} - 16 q^{72} + 4 q^{73} - 16 q^{74} - 36 q^{75} + 24 q^{76} + 4 q^{78} + 80 q^{79} + 8 q^{81} + 20 q^{82} - 36 q^{83} - 12 q^{85} + 16 q^{86} + 4 q^{87} - 12 q^{89} - 8 q^{90} + 4 q^{91} + 20 q^{93} - 4 q^{94} + 20 q^{95} + 16 q^{97} - 12 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 26 x^{10} - 4 x^{9} + 246 x^{8} + 72 x^{7} - 1016 x^{6} - 396 x^{5} + 1714 x^{4} + 688 x^{3} + \cdots - 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6773 \nu^{11} - 1665 \nu^{10} + 172587 \nu^{9} + 27811 \nu^{8} - 1560265 \nu^{7} + \cdots + 3745583 ) / 2681844 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 92987 \nu^{11} + 570440 \nu^{10} - 2595289 \nu^{9} - 13589924 \nu^{8} + 23436959 \nu^{7} + \cdots + 8906796 ) / 18772908 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 104642 \nu^{11} - 595017 \nu^{10} + 2600322 \nu^{9} + 14331175 \nu^{8} - 21672322 \nu^{7} + \cdots - 28770157 ) / 18772908 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 859 \nu^{11} + 62 \nu^{10} - 18943 \nu^{9} - 9138 \nu^{8} + 129423 \nu^{7} + 141900 \nu^{6} + \cdots + 205540 ) / 77574 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 387252 \nu^{11} + 92987 \nu^{10} - 9498112 \nu^{9} - 4144297 \nu^{8} + 81674068 \nu^{7} + \cdots - 91930273 ) / 18772908 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 434663 \nu^{11} + 104642 \nu^{10} - 10706221 \nu^{9} - 4338974 \nu^{8} + 92595923 \nu^{7} + \cdots - 43057722 ) / 18772908 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 239135 \nu^{11} + 51923 \nu^{10} - 5786413 \nu^{9} - 2046599 \nu^{8} + 48506507 \nu^{7} + \cdots + 27252831 ) / 9386454 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 619433 \nu^{11} - 339278 \nu^{10} - 16802039 \nu^{9} + 5908134 \nu^{8} + 169140069 \nu^{7} + \cdots - 160333798 ) / 18772908 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 401321 \nu^{11} - 254599 \nu^{10} + 10525919 \nu^{9} + 6970461 \nu^{8} - 99036567 \nu^{7} + \cdots + 20766961 ) / 9386454 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 424287 \nu^{11} + 521414 \nu^{10} - 10794625 \nu^{9} - 14521072 \nu^{8} + 95938969 \nu^{7} + \cdots - 55171720 ) / 9386454 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + 2\beta_{8} - \beta_{7} - 2\beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{10} - \beta_{9} - \beta_{8} - 10\beta_{7} + 9\beta_{6} + \beta_{5} - 2\beta_{3} - 7\beta_{2} + \beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{11} + 12 \beta_{10} + \beta_{9} + 24 \beta_{8} - 16 \beta_{7} - 21 \beta_{5} + 10 \beta_{4} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} - 26 \beta_{10} - 14 \beta_{9} - 14 \beta_{8} - 90 \beta_{7} + 76 \beta_{6} + 12 \beta_{5} + \cdots + 222 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 120 \beta_{11} + 117 \beta_{10} + 15 \beta_{9} + 238 \beta_{8} - 186 \beta_{7} + 2 \beta_{6} + \cdots + 116 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 36 \beta_{11} - 272 \beta_{10} - 157 \beta_{9} - 139 \beta_{8} - 804 \beta_{7} + 644 \beta_{6} + \cdots + 1852 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1132 \beta_{11} + 1078 \beta_{10} + 162 \beta_{9} + 2239 \beta_{8} - 1929 \beta_{7} + 56 \beta_{6} + \cdots + 1035 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 462 \beta_{11} - 2660 \beta_{10} - 1638 \beta_{9} - 1228 \beta_{8} - 7197 \beta_{7} + 5530 \beta_{6} + \cdots + 15857 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10397 \beta_{11} + 9730 \beta_{10} + 1540 \beta_{9} + 20677 \beta_{8} - 18926 \beta_{7} + 958 \beta_{6} + \cdots + 8986 \) 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
−2.99022
−2.67278
−2.05584
−1.84157
−0.922058
−0.393237
−0.0633327
1.16789
1.24821
2.52971
2.93591
3.05731
−1.00000 −2.99022 1.00000 3.93256 2.99022 1.00000 −1.00000 5.94142 −3.93256
1.2 −1.00000 −2.67278 1.00000 1.18883 2.67278 1.00000 −1.00000 4.14374 −1.18883
1.3 −1.00000 −2.05584 1.00000 −1.60500 2.05584 1.00000 −1.00000 1.22646 1.60500
1.4 −1.00000 −1.84157 1.00000 −2.61966 1.84157 1.00000 −1.00000 0.391388 2.61966
1.5 −1.00000 −0.922058 1.00000 3.96535 0.922058 1.00000 −1.00000 −2.14981 −3.96535
1.6 −1.00000 −0.393237 1.00000 −1.28814 0.393237 1.00000 −1.00000 −2.84536 1.28814
1.7 −1.00000 −0.0633327 1.00000 −2.68442 0.0633327 1.00000 −1.00000 −2.99599 2.68442
1.8 −1.00000 1.16789 1.00000 −0.795261 −1.16789 1.00000 −1.00000 −1.63603 0.795261
1.9 −1.00000 1.24821 1.00000 2.91031 −1.24821 1.00000 −1.00000 −1.44196 −2.91031
1.10 −1.00000 2.52971 1.00000 −0.745358 −2.52971 1.00000 −1.00000 3.39942 0.745358
1.11 −1.00000 2.93591 1.00000 0.977950 −2.93591 1.00000 −1.00000 5.61957 −0.977950
1.12 −1.00000 3.05731 1.00000 −3.23717 −3.05731 1.00000 −1.00000 6.34715 3.23717
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 7406.2.a.br yes 12
23.b odd 2 1 7406.2.a.bq ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7406.2.a.bq ✓ 12 23.b odd 2 1
7406.2.a.br yes 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):

\( T_{3}^{12} - 26 T_{3}^{10} - 4 T_{3}^{9} + 246 T_{3}^{8} + 72 T_{3}^{7} - 1016 T_{3}^{6} - 396 T_{3}^{5} + \cdots - 23 \) Copy content Toggle raw display
\( T_{5}^{12} - 36 T_{5}^{10} - 24 T_{5}^{9} + 453 T_{5}^{8} + 564 T_{5}^{7} - 2146 T_{5}^{6} - 3804 T_{5}^{5} + \cdots - 1472 \) Copy content Toggle raw display
\( T_{11}^{12} - 64 T_{11}^{10} - 24 T_{11}^{9} + 1219 T_{11}^{8} + 824 T_{11}^{7} - 8024 T_{11}^{6} + \cdots - 92 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 26 T^{10} + \cdots - 23 \) Copy content Toggle raw display
$5$ \( T^{12} - 36 T^{10} + \cdots - 1472 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 64 T^{10} + \cdots - 92 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + \cdots - 28400 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots - 7775 \) Copy content Toggle raw display
$19$ \( T^{12} - 24 T^{11} + \cdots - 3119 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + \cdots + 2679808 \) Copy content Toggle raw display
$31$ \( T^{12} - 4 T^{11} + \cdots + 337936 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1369762048 \) Copy content Toggle raw display
$41$ \( T^{12} + 20 T^{11} + \cdots + 4651264 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1614643729 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 4250660864 \) Copy content Toggle raw display
$53$ \( T^{12} + 8 T^{11} + \cdots + 16122112 \) Copy content Toggle raw display
$59$ \( T^{12} + 20 T^{11} + \cdots + 46720000 \) Copy content Toggle raw display
$61$ \( T^{12} - 32 T^{11} + \cdots - 15494912 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 49354935536 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 241201552 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 6086277503 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 17566304192 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 4751921113 \) Copy content Toggle raw display
$89$ \( T^{12} + 12 T^{11} + \cdots - 6640028 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 3420114044 \) Copy content Toggle raw display
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