Properties

Label 7623.2.a.co
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 + 1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} + 3) q^{8} + (\beta_{2} + 2 \beta_1 + 3) q^{10} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + \beta_1 q^{14} + (2 \beta_{3} + \beta_1 - 1) q^{16} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{19} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{20} + (2 \beta_{3} + \beta_{2} + 3) q^{23} + (\beta_{2} + 3 \beta_1 - 1) q^{25} + ( - 3 \beta_{2} + \beta_1 + 1) q^{26} + (\beta_{2} + \beta_1 + 1) q^{28} - 3 \beta_{3} q^{29} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{31} + (3 \beta_{2} - 1) q^{32} + (2 \beta_{3} - 3 \beta_{2} - 4) q^{34} + (\beta_1 + 1) q^{35} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{37} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 7) q^{38} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 4) q^{40} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{41} + ( - 6 \beta_{2} - 5) q^{43} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 2) q^{46} + ( - 4 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{47} + q^{49} + (\beta_{3} + 3 \beta_{2} + 2 \beta_1 + 9) q^{50} + (\beta_{3} - 3 \beta_{2} + 3) q^{52} + ( - 3 \beta_{3} + \beta_{2} - 4 \beta_1 + 4) q^{53} + (\beta_{3} + \beta_{2} + 3) q^{56} + ( - 3 \beta_{3} - 6 \beta_{2} - 3) q^{58} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{59} + ( - \beta_{3} - 5 \beta_1 + 7) q^{61} + (2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{62} + ( - \beta_{3} - 3 \beta_1 + 2) q^{64} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{65} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{67} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{68} + (\beta_{2} + 2 \beta_1 + 3) q^{70} + (3 \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 1) q^{71} + ( - 5 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 4) q^{73} + ( - \beta_{2} - \beta_1 + 2) q^{74} + (4 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 7) q^{76} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{79} + (4 \beta_{3} + 5 \beta_{2} + \beta_1 + 4) q^{80} + ( - 5 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{82} + (5 \beta_{3} - 7 \beta_{2} + 3 \beta_1 - 4) q^{83} + (\beta_{3} - \beta_1 - 3) q^{85} + ( - 6 \beta_{3} - 5 \beta_1) q^{86} + ( - 5 \beta_{3} + 3 \beta_{2} + \beta_1 + 4) q^{89} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{91} + (3 \beta_{3} + 7 \beta_{2} + 5 \beta_1 + 6) q^{92} + ( - 2 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 1) q^{94} + (3 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 6) q^{95} + ( - 5 \beta_{3} - 2 \beta_{2} - 6) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8} + 14 q^{10} + 2 q^{14} - 4 q^{16} - 3 q^{17} - 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} + 4 q^{28} + 3 q^{29} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - 7 q^{37} + 20 q^{38} + 13 q^{40} + 4 q^{41} - 8 q^{43} + 3 q^{46} + 14 q^{47} + 4 q^{49} + 33 q^{50} + 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} + 19 q^{61} + 10 q^{62} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{70} + 7 q^{71} + 11 q^{73} + 8 q^{74} + 26 q^{76} - 8 q^{79} + 4 q^{80} + 3 q^{82} - q^{83} - 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} + 20 q^{94} + 17 q^{95} - 15 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 3 \) 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
−1.46673
−0.777484
1.77748
2.46673
−1.46673 0 0.151302 −0.466732 0 1.00000 2.71154 0 0.684570
1.2 −0.777484 0 −1.39552 0.222516 0 1.00000 2.63996 0 −0.173002
1.3 1.77748 0 1.15945 2.77748 0 1.00000 −1.49406 0 4.93693
1.4 2.46673 0 4.08477 3.46673 0 1.00000 5.14256 0 8.55150
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 7623.2.a.co 4
3.b odd 2 1 847.2.a.k 4
11.b odd 2 1 7623.2.a.ch 4
11.d odd 10 2 693.2.m.g 8
21.c even 2 1 5929.2.a.bb 4
33.d even 2 1 847.2.a.l 4
33.f even 10 2 77.2.f.a 8
33.f even 10 2 847.2.f.p 8
33.h odd 10 2 847.2.f.q 8
33.h odd 10 2 847.2.f.s 8
231.h odd 2 1 5929.2.a.bi 4
231.r odd 10 2 539.2.f.d 8
231.be even 30 4 539.2.q.c 16
231.bf odd 30 4 539.2.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.a 8 33.f even 10 2
539.2.f.d 8 231.r odd 10 2
539.2.q.b 16 231.bf odd 30 4
539.2.q.c 16 231.be even 30 4
693.2.m.g 8 11.d odd 10 2
847.2.a.k 4 3.b odd 2 1
847.2.a.l 4 33.d even 2 1
847.2.f.p 8 33.f even 10 2
847.2.f.q 8 33.h odd 10 2
847.2.f.s 8 33.h odd 10 2
5929.2.a.bb 4 21.c even 2 1
5929.2.a.bi 4 231.h odd 2 1
7623.2.a.ch 4 11.b odd 2 1
7623.2.a.co 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 5T_{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{4} - 6T_{5}^{3} + 8T_{5}^{2} + 3T_{5} - 1 \) Copy content Toggle raw display
\( T_{13}^{4} - 32T_{13}^{2} + 65T_{13} - 29 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 4 T^{2} + 5 T + 5 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} + 8 T^{2} + 3 T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 32 T^{2} + 65 T - 29 \) Copy content Toggle raw display
$17$ \( T^{4} + 3 T^{3} - 23 T^{2} - 91 T - 71 \) Copy content Toggle raw display
$19$ \( T^{4} + 3 T^{3} - 29 T^{2} - 135 T - 145 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} - 9 T^{2} + 150 T - 205 \) Copy content Toggle raw display
$29$ \( T^{4} - 3 T^{3} - 54 T^{2} + 405 \) Copy content Toggle raw display
$31$ \( T^{4} + 3 T^{3} - 24 T^{2} + 20 T + 5 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + 6 T^{2} - 10 T - 5 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} - 57 T^{2} + 22 T + 79 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 41)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 14 T^{3} - 26 T^{2} + \cdots - 305 \) Copy content Toggle raw display
$53$ \( T^{4} - 9 T^{3} - 104 T^{2} + \cdots - 869 \) Copy content Toggle raw display
$59$ \( T^{4} - 25 T^{3} + 192 T^{2} + \cdots - 1189 \) Copy content Toggle raw display
$61$ \( T^{4} - 19 T^{3} - 6 T^{2} + 1030 T - 995 \) Copy content Toggle raw display
$67$ \( T^{4} + 15 T^{3} + 67 T^{2} + \cdots - 199 \) Copy content Toggle raw display
$71$ \( T^{4} - 7 T^{3} - 83 T^{2} + 679 T - 991 \) Copy content Toggle raw display
$73$ \( T^{4} - 11 T^{3} - 116 T^{2} + \cdots + 4975 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 8 T^{2} - 161 T - 271 \) Copy content Toggle raw display
$83$ \( T^{4} + T^{3} - 236 T^{2} + 900 T + 2245 \) Copy content Toggle raw display
$89$ \( T^{4} - 17 T^{3} - 44 T^{2} + \cdots - 755 \) Copy content Toggle raw display
$97$ \( T^{4} + 15 T^{3} - 110 T^{2} + \cdots - 4225 \) Copy content Toggle raw display
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