Properties

Label 7623.2.a.cu
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.6988960000.1
Defining polynomial: \( x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{7} + \beta_1) q^{5} - q^{7} + (\beta_{6} + \beta_{5} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{7} + \beta_1) q^{5} - q^{7} + (\beta_{6} + \beta_{5} + \beta_1) q^{8} + ( - \beta_{3} + \beta_{2} + 1) q^{10} + (\beta_{4} + \beta_{3}) q^{13} - \beta_1 q^{14} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{16} + (\beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{17} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{19} + ( - \beta_{7} + \beta_{6} + 2 \beta_1) q^{20} + ( - \beta_{7} - \beta_{6} - 3 \beta_1) q^{23} + ( - \beta_{3} - 2) q^{25} + ( - 2 \beta_{7} + 3 \beta_{5} + \beta_1) q^{26} - \beta_{2} q^{28} + ( - 3 \beta_{6} - 3 \beta_1) q^{29} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{31} + ( - 2 \beta_{7} - 3 \beta_{6} - 4 \beta_1) q^{32} + ( - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 3) q^{34} + ( - \beta_{7} - \beta_1) q^{35} + (\beta_{4} - \beta_{2} - 5) q^{37} + (3 \beta_{7} - \beta_{6} - 6 \beta_{5} - 5 \beta_1) q^{38} + (3 \beta_{3} + 2) q^{40} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 4 \beta_1) q^{41} + (3 \beta_{4} + 3 \beta_{3} + 2) q^{43} + (\beta_{3} - 3 \beta_{2} - 4) q^{46} + (\beta_{6} - 5 \beta_{5}) q^{47} + q^{49} + (\beta_{7} - \beta_{5} - 2 \beta_1) q^{50} + (\beta_{4} + 3 \beta_{3} + \beta_{2} + 1) q^{52} + ( - 4 \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_1) q^{53} + ( - \beta_{6} - \beta_{5} - \beta_1) q^{56} + ( - 3 \beta_{2} - 3) q^{58} + ( - 4 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - \beta_1) q^{59} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4) q^{61} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_1) q^{62} + ( - 2 \beta_{4} - 2 \beta_{2} - 3) q^{64} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{5}) q^{65} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 6) q^{67} + (3 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 7 \beta_1) q^{68} + (\beta_{3} - \beta_{2} - 1) q^{70} + ( - \beta_{6} + 4 \beta_{5} + 3 \beta_1) q^{71} + ( - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3) q^{73} + ( - \beta_{7} - \beta_{6} + \beta_{5} - 7 \beta_1) q^{74} + ( - 2 \beta_{4} - 7 \beta_{3} - 3 \beta_{2} - 6) q^{76} + ( - \beta_{4} - 7 \beta_{3} + 3 \beta_{2} - 1) q^{79} + ( - \beta_{7} - 2 \beta_{6} + 3 \beta_{5} - 2 \beta_1) q^{80} + (2 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 4) q^{82} + ( - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{83} + ( - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{85} + ( - 6 \beta_{7} + 9 \beta_{5} + 5 \beta_1) q^{86} + (3 \beta_{7} + 6 \beta_{6} + \beta_{5} - \beta_1) q^{89} + ( - \beta_{4} - \beta_{3}) q^{91} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - 7 \beta_1) q^{92} + ( - 5 \beta_{4} - 5 \beta_{3} + 4) q^{94} + (4 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 2 \beta_1) q^{95} + ( - \beta_{4} + 5 \beta_{3} - 4 \beta_{2} - 2) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 8 q^{7} + 14 q^{10} - 2 q^{16} - 6 q^{19} - 12 q^{25} - 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} + 4 q^{40} + 16 q^{43} - 42 q^{46} + 8 q^{49} + 2 q^{52} - 30 q^{58} + 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} + 14 q^{73} - 34 q^{76} + 22 q^{79} + 36 q^{82} - 18 q^{85} + 32 q^{94} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 10\nu^{4} - 26\nu^{2} + 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} + 9\nu^{5} - 21\nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} - 9\nu^{5} + 22\nu^{3} - 11\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 26\nu^{5} - 58\nu^{3} + 17\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 5\beta_{6} + 8\beta_{5} + 24\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{4} + 10\beta_{3} + 24\beta_{2} + 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -18\beta_{7} + 24\beta_{6} + 50\beta_{5} + 117\beta_1 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.25947
−1.80692
−0.716111
−0.342036
0.342036
0.716111
1.80692
2.25947
−2.25947 0 3.10522 −1.54336 0 −1.00000 −2.49721 0 3.48718
1.2 −1.80692 0 1.26498 −2.14896 0 −1.00000 1.32813 0 3.88301
1.3 −0.716111 0 −1.48718 1.54336 0 −1.00000 2.49721 0 −1.10522
1.4 −0.342036 0 −1.88301 −2.14896 0 −1.00000 1.32813 0 0.735023
1.5 0.342036 0 −1.88301 2.14896 0 −1.00000 −1.32813 0 0.735023
1.6 0.716111 0 −1.48718 −1.54336 0 −1.00000 −2.49721 0 −1.10522
1.7 1.80692 0 1.26498 2.14896 0 −1.00000 −1.32813 0 3.88301
1.8 2.25947 0 3.10522 1.54336 0 −1.00000 2.49721 0 3.48718
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(11\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.cu 8
3.b odd 2 1 inner 7623.2.a.cu 8
11.b odd 2 1 7623.2.a.cv 8
11.c even 5 2 693.2.m.h 16
33.d even 2 1 7623.2.a.cv 8
33.h odd 10 2 693.2.m.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.h 16 11.c even 5 2
693.2.m.h 16 33.h odd 10 2
7623.2.a.cu 8 1.a even 1 1 trivial
7623.2.a.cu 8 3.b odd 2 1 inner
7623.2.a.cv 8 11.b odd 2 1
7623.2.a.cv 8 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{8} - 9T_{2}^{6} + 22T_{2}^{4} - 11T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} - 7T_{5}^{2} + 11 \) Copy content Toggle raw display
\( T_{13}^{4} - 11T_{13}^{2} + 10T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 9 T^{6} + 22 T^{4} - 11 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 7 T^{2} + 11)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 11 T^{2} + 10 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 49 T^{6} + 246 T^{4} - 299 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{4} + 3 T^{3} - 41 T^{2} - 18 T + 236)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 61 T^{6} + 991 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} - 108 T^{6} + 3483 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$31$ \( (T^{4} + 3 T^{3} - 36 T^{2} - 73 T + 281)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 19 T^{3} + 117 T^{2} + 236 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 148 T^{6} + 5329 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} - 75 T^{2} + 634 T - 596)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 311 T^{6} + \cdots + 11262736 \) Copy content Toggle raw display
$53$ \( T^{8} - 115 T^{6} + 3975 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$59$ \( T^{8} - 241 T^{6} + 14227 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( (T^{4} - 14 T^{3} + 27 T^{2} + 314 T - 1124)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 18 T^{3} + 28 T^{2} - 912 T - 3904)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 212 T^{6} + 14083 T^{4} + \cdots + 2085136 \) Copy content Toggle raw display
$73$ \( (T^{4} - 7 T^{3} - 125 T^{2} + 1106 T - 1436)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 11 T^{3} - 113 T^{2} + 506 T - 484)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 176 T^{6} + 8976 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{8} - 524 T^{6} + \cdots + 71554681 \) Copy content Toggle raw display
$97$ \( (T^{4} + 24 T^{3} + 67 T^{2} - 1584 T - 7744)^{2} \) Copy content Toggle raw display
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