Properties

Label 7406.2.a.bn
Level $7406$
Weight $2$
Character orbit 7406.a
Self dual yes
Analytic conductor $59.137$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.8.6120603648.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} - 4x^{5} + 37x^{4} + 24x^{3} - 14x^{2} - 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{7} + \beta_{4} - \beta_{2} + 1) q^{5} - \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + (\beta_{7} + \beta_{4} - \beta_{2} + 1) q^{5} - \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{9} + (\beta_{7} + \beta_{4} - \beta_{2} + 1) q^{10} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{11} - \beta_1 q^{12} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1) q^{13} - q^{14} + (\beta_{5} - \beta_{2} - 2 \beta_1) q^{15} + q^{16} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{18} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{7} + \beta_{4} - \beta_{2} + 1) q^{20} + \beta_1 q^{21} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{22} - \beta_1 q^{24} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{25} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1) q^{26} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + \beta_1 - 1) q^{27} - q^{28} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{29} + (\beta_{5} - \beta_{2} - 2 \beta_1) q^{30} + (3 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} + \beta_1 + 1) q^{31} + q^{32} + ( - \beta_{7} + \beta_{5} - 2 \beta_{2} - 3 \beta_1) q^{33} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{34} + ( - \beta_{7} - \beta_{4} + \beta_{2} - 1) q^{35} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{36} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 + 4) q^{37} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{38} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_1) q^{39} + (\beta_{7} + \beta_{4} - \beta_{2} + 1) q^{40} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{41} + \beta_1 q^{42} + (2 \beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 2) q^{43} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{44} + ( - 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{3} + 4) q^{45} + (4 \beta_{6} + 2 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 2) q^{47} - \beta_1 q^{48} + q^{49} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}) q^{50} + (\beta_{7} - 4 \beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{51} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1) q^{52} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 4) q^{53} + ( - \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + \beta_1 - 1) q^{54} + (3 \beta_{7} - 4 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 1) q^{55} - q^{56} + ( - 3 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + \cdots + 2) q^{57}+ \cdots + (6 \beta_{7} - 4 \beta_{6} + \beta_{5} + 4 \beta_{4} - 6 \beta_{2} + 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{5} - 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{10} + 16 q^{11} + 4 q^{13} - 8 q^{14} + 8 q^{16} + 8 q^{17} + 16 q^{19} + 8 q^{20} + 16 q^{22} + 4 q^{26} - 12 q^{27} - 8 q^{28} - 20 q^{29} + 12 q^{31} + 8 q^{32} - 4 q^{33} + 8 q^{34} - 8 q^{35} + 32 q^{37} + 16 q^{38} - 4 q^{39} + 8 q^{40} + 12 q^{41} + 16 q^{43} + 16 q^{44} + 32 q^{45} + 12 q^{47} + 8 q^{49} + 24 q^{51} + 4 q^{52} + 24 q^{53} - 12 q^{54} + 12 q^{55} - 8 q^{56} - 20 q^{58} - 8 q^{59} + 24 q^{61} + 12 q^{62} + 8 q^{64} - 16 q^{65} - 4 q^{66} + 16 q^{67} + 8 q^{68} - 8 q^{70} - 8 q^{71} + 8 q^{73} + 32 q^{74} - 4 q^{75} + 16 q^{76} - 16 q^{77} - 4 q^{78} + 32 q^{79} + 8 q^{80} - 4 q^{81} + 12 q^{82} + 44 q^{83} - 12 q^{85} + 16 q^{86} + 20 q^{87} + 16 q^{88} - 28 q^{89} + 32 q^{90} - 4 q^{91} - 12 q^{93} + 12 q^{94} + 4 q^{95} + 40 q^{97} + 8 q^{98} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} - 4x^{5} + 37x^{4} + 24x^{3} - 14x^{2} - 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -18\nu^{7} + 2\nu^{6} + 203\nu^{5} + 75\nu^{4} - 521\nu^{3} - 553\nu^{2} - 121\nu + 183 ) / 115 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\nu^{7} + 49\nu^{6} - 259\nu^{5} - 635\nu^{4} + 863\nu^{3} + 2034\nu^{2} - 147\nu - 519 ) / 115 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{7} + 3\nu^{6} + 40\nu^{5} - 14\nu^{4} - 80\nu^{3} - 13\nu^{2} - 55\nu - 36 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31\nu^{7} - 29\nu^{6} - 356\nu^{5} + 235\nu^{4} + 1057\nu^{3} - 434\nu^{2} - 488\nu + 279 ) / 115 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} - 22\nu^{5} + 25\nu^{4} + 64\nu^{3} - 43\nu^{2} - 36\nu + 13 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 98\nu^{7} - 62\nu^{6} - 1118\nu^{5} + 320\nu^{4} + 3271\nu^{3} + 238\nu^{2} - 1309\nu + 77 ) / 115 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{5} + 2\beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} - 9\beta_{6} - 6\beta_{5} + 7\beta_{4} - 8\beta_{3} - 6\beta_{2} + \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{7} - 4\beta_{6} - 21\beta_{5} - 2\beta_{4} - 3\beta_{3} + 18\beta_{2} + 29\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 63\beta_{7} - 73\beta_{6} - 43\beta_{5} + 46\beta_{4} - 59\beta_{3} - 31\beta_{2} + 11\beta _1 + 107 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 116\beta_{7} - 60\beta_{6} - 178\beta_{5} - 19\beta_{4} - 43\beta_{3} + 141\beta_{2} + 181\beta _1 + 62 \) Copy content Toggle raw display

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
2.82352
2.25853
0.588222
0.108789
−0.626427
−0.891667
−1.74089
−2.52007
1.00000 −2.82352 1.00000 2.68420 −2.82352 −1.00000 1.00000 4.97226 2.68420
1.2 1.00000 −2.25853 1.00000 2.22919 −2.25853 −1.00000 1.00000 2.10095 2.22919
1.3 1.00000 −0.588222 1.00000 −1.28422 −0.588222 −1.00000 1.00000 −2.65400 −1.28422
1.4 1.00000 −0.108789 1.00000 −2.75819 −0.108789 −1.00000 1.00000 −2.98816 −2.75819
1.5 1.00000 0.626427 1.00000 2.82633 0.626427 −1.00000 1.00000 −2.60759 2.82633
1.6 1.00000 0.891667 1.00000 −0.166562 0.891667 −1.00000 1.00000 −2.20493 −0.166562
1.7 1.00000 1.74089 1.00000 1.70266 1.74089 −1.00000 1.00000 0.0307003 1.70266
1.8 1.00000 2.52007 1.00000 2.76659 2.52007 −1.00000 1.00000 3.35077 2.76659
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bn yes 8
23.b odd 2 1 7406.2.a.bm 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7406.2.a.bm 8 23.b odd 2 1
7406.2.a.bn yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7406))\):

\( T_{3}^{8} - 12T_{3}^{6} + 4T_{3}^{5} + 37T_{3}^{4} - 24T_{3}^{3} - 14T_{3}^{2} + 8T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} - 8T_{5}^{7} + 12T_{5}^{6} + 56T_{5}^{5} - 187T_{5}^{4} + 68T_{5}^{3} + 294T_{5}^{2} - 236T_{5} - 47 \) Copy content Toggle raw display
\( T_{11}^{8} - 16T_{11}^{7} + 68T_{11}^{6} + 104T_{11}^{5} - 1217T_{11}^{4} + 1336T_{11}^{3} + 4196T_{11}^{2} - 8360T_{11} + 3025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 12 T^{6} + 4 T^{5} + 37 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{7} + 12 T^{6} + 56 T^{5} + \cdots - 47 \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 16 T^{7} + 68 T^{6} + \cdots + 3025 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} - 44 T^{6} + 120 T^{5} + \cdots + 361 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} - 18 T^{6} + 204 T^{5} + \cdots - 479 \) Copy content Toggle raw display
$19$ \( T^{8} - 16 T^{7} + 46 T^{6} + \cdots - 2711 \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 20 T^{7} + 120 T^{6} + \cdots - 2672 \) Copy content Toggle raw display
$31$ \( T^{8} - 12 T^{7} - 46 T^{6} + \cdots + 36697 \) Copy content Toggle raw display
$37$ \( T^{8} - 32 T^{7} + 408 T^{6} + \cdots + 337 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} - 108 T^{6} + \cdots - 9887 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} - 48 T^{6} + 928 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} - 148 T^{6} + \cdots - 5975 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + 136 T^{6} + \cdots - 36848 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} - 216 T^{6} + \cdots - 1350656 \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + 28 T^{6} + \cdots + 144016 \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} - 108 T^{6} + \cdots - 2489456 \) Copy content Toggle raw display
$71$ \( T^{8} + 8 T^{7} - 190 T^{6} + \cdots - 973751 \) Copy content Toggle raw display
$73$ \( T^{8} - 8 T^{7} - 322 T^{6} + \cdots - 4875599 \) Copy content Toggle raw display
$79$ \( T^{8} - 32 T^{7} + 288 T^{6} + \cdots + 484153 \) Copy content Toggle raw display
$83$ \( T^{8} - 44 T^{7} + 604 T^{6} + \cdots + 957889 \) Copy content Toggle raw display
$89$ \( T^{8} + 28 T^{7} + 130 T^{6} + \cdots - 21767 \) Copy content Toggle raw display
$97$ \( T^{8} - 40 T^{7} + 318 T^{6} + \cdots + 757249 \) Copy content Toggle raw display
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