Properties

Label 7623.2.a.cr
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 6
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.3829849.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + q^{7} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + q^{7} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{8} + ( 2 + \beta_{2} ) q^{10} + ( -\beta_{2} - \beta_{4} ) q^{13} + \beta_{1} q^{14} + ( 1 - \beta_{4} ) q^{16} + 2 \beta_{3} q^{17} + ( \beta_{2} - \beta_{4} ) q^{19} + ( 2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{20} + ( 2 \beta_{3} - 2 \beta_{5} ) q^{23} + ( -1 - \beta_{2} - \beta_{4} ) q^{25} + ( -\beta_{1} + \beta_{5} ) q^{26} + ( 1 + \beta_{2} ) q^{28} + ( \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{29} -2 \beta_{4} q^{31} + \beta_{3} q^{32} + 2 q^{34} + ( \beta_{1} - \beta_{3} ) q^{35} + ( 2 + \beta_{2} + \beta_{4} ) q^{37} + ( 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{5} ) q^{38} + ( 4 + \beta_{2} - \beta_{4} ) q^{40} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{41} + ( 4 + 2 \beta_{2} ) q^{43} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{46} + ( \beta_{1} + \beta_{3} + 2 \beta_{5} ) q^{47} + q^{49} + ( -2 \beta_{1} + \beta_{5} ) q^{50} + ( -2 + 2 \beta_{2} + \beta_{4} ) q^{52} + ( 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{53} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{56} + ( -\beta_{2} + 2 \beta_{4} ) q^{58} + ( -3 \beta_{1} - 3 \beta_{3} + 2 \beta_{5} ) q^{59} + ( 4 + 2 \beta_{2} + 2 \beta_{4} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} ) q^{62} + ( -1 + 2 \beta_{4} ) q^{64} + ( \beta_{1} - 5 \beta_{3} ) q^{65} + ( 4 + \beta_{2} - \beta_{4} ) q^{67} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{68} + ( 2 + \beta_{2} ) q^{70} + ( 6 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 8 + \beta_{2} + \beta_{4} ) q^{73} + ( 3 \beta_{1} - \beta_{5} ) q^{74} + ( 10 + 4 \beta_{2} - \beta_{4} ) q^{76} -4 \beta_{2} q^{79} + ( 3 \beta_{1} - 4 \beta_{3} + \beta_{5} ) q^{80} + ( 2 + 2 \beta_{4} ) q^{82} + ( -4 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{83} + ( -4 + 4 \beta_{2} + 2 \beta_{4} ) q^{85} + ( 8 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{86} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -\beta_{2} - \beta_{4} ) q^{91} + ( -6 \beta_{1} - 2 \beta_{5} ) q^{92} + ( 6 + 3 \beta_{2} - 2 \beta_{4} ) q^{94} + ( 3 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{95} + ( -2 + 2 \beta_{4} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{4} + 6q^{7} + O(q^{10}) \) \( 6q + 4q^{4} + 6q^{7} + 10q^{10} + 4q^{13} + 8q^{16} - 2q^{25} + 4q^{28} + 4q^{31} + 12q^{34} + 8q^{37} + 24q^{40} + 20q^{43} + 6q^{49} - 18q^{52} - 2q^{58} + 16q^{61} - 10q^{64} + 24q^{67} + 10q^{70} + 44q^{73} + 54q^{76} + 8q^{79} + 8q^{82} - 36q^{85} + 4q^{91} + 34q^{94} - 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 6 x^{4} + 15 x^{3} - 5 x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{4} - 6 \nu^{2} + 3 \nu + 1 \)
\(\beta_{2}\)\(=\)\( -\nu^{5} + \nu^{4} + 7 \nu^{3} - 8 \nu^{2} - 2 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 8 \nu^{2} + 4 \nu - 1 \)
\(\beta_{4}\)\(=\)\( -2 \nu^{5} + 3 \nu^{4} + 14 \nu^{3} - 24 \nu^{2} - 5 \nu + 8 \)
\(\beta_{5}\)\(=\)\( 3 \nu^{5} - 4 \nu^{4} - 19 \nu^{3} + 32 \nu^{2} - 3 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{4} - 2 \beta_{3} + \beta_{1} + 5\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + \beta_{4} + 5 \beta_{3} + 6 \beta_{2} + 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{4} - 15 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} + 25\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(7 \beta_{5} + 9 \beta_{4} + 34 \beta_{3} + 35 \beta_{2} - 3\)\()/2\)

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
1.81705
−0.430314
−2.63651
0.725011
0.300853
2.22391
−2.45784 0 4.04096 −2.05098 0 1.00000 −5.01636 0 5.04096
1.2 −1.36768 0 −0.129461 −0.636509 0 1.00000 2.91241 0 0.870539
1.3 −0.297484 0 −1.91150 3.06404 0 1.00000 1.16361 0 −0.911503
1.4 0.297484 0 −1.91150 −3.06404 0 1.00000 −1.16361 0 −0.911503
1.5 1.36768 0 −0.129461 0.636509 0 1.00000 −2.91241 0 0.870539
1.6 2.45784 0 4.04096 2.05098 0 1.00000 5.01636 0 5.04096
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

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.cr yes 6
3.b odd 2 1 inner 7623.2.a.cr yes 6
11.b odd 2 1 7623.2.a.cq 6
33.d even 2 1 7623.2.a.cq 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cq 6 11.b odd 2 1
7623.2.a.cq 6 33.d even 2 1
7623.2.a.cr yes 6 1.a even 1 1 trivial
7623.2.a.cr yes 6 3.b odd 2 1 inner

Atkin-Lehner signs

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

Hecke kernels

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

\( T_{2}^{6} - 8 T_{2}^{4} + 12 T_{2}^{2} - 1 \)
\( T_{5}^{6} - 14 T_{5}^{4} + 45 T_{5}^{2} - 16 \)
\( T_{13}^{3} - 2 T_{13}^{2} - 19 T_{13} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} + 8 T^{4} + 15 T^{6} + 32 T^{8} + 64 T^{10} + 64 T^{12} \)
$3$ \( \)
$5$ \( 1 + 16 T^{2} + 140 T^{4} + 834 T^{6} + 3500 T^{8} + 10000 T^{10} + 15625 T^{12} \)
$7$ \( ( 1 - T )^{6} \)
$11$ \( \)
$13$ \( ( 1 - 2 T + 20 T^{2} - 48 T^{3} + 260 T^{4} - 338 T^{5} + 2197 T^{6} )^{2} \)
$17$ \( 1 + 54 T^{2} + 1199 T^{4} + 19316 T^{6} + 346511 T^{8} + 4510134 T^{10} + 24137569 T^{12} \)
$19$ \( ( 1 + 20 T^{2} - 16 T^{3} + 380 T^{4} + 6859 T^{6} )^{2} \)
$23$ \( 1 + 30 T^{2} + 767 T^{4} + 11492 T^{6} + 405743 T^{8} + 8395230 T^{10} + 148035889 T^{12} \)
$29$ \( 1 + 52 T^{2} + 2892 T^{4} + 86614 T^{6} + 2432172 T^{8} + 36778612 T^{10} + 594823321 T^{12} \)
$31$ \( ( 1 - 2 T + 17 T^{2} + 180 T^{3} + 527 T^{4} - 1922 T^{5} + 29791 T^{6} )^{2} \)
$37$ \( ( 1 - 4 T + 96 T^{2} - 262 T^{3} + 3552 T^{4} - 5476 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 + 74 T^{2} + 4479 T^{4} + 233228 T^{6} + 7529199 T^{8} + 209106314 T^{10} + 4750104241 T^{12} \)
$43$ \( ( 1 - 10 T + 125 T^{2} - 828 T^{3} + 5375 T^{4} - 18490 T^{5} + 79507 T^{6} )^{2} \)
$47$ \( 1 + 124 T^{2} + 11060 T^{4} + 591870 T^{6} + 24431540 T^{8} + 605080444 T^{10} + 10779215329 T^{12} \)
$53$ \( 1 + 98 T^{2} + 8247 T^{4} + 518972 T^{6} + 23165823 T^{8} + 773267138 T^{10} + 22164361129 T^{12} \)
$59$ \( 1 + 116 T^{2} + 6924 T^{4} + 354662 T^{6} + 24102444 T^{8} + 1405613876 T^{10} + 42180533641 T^{12} \)
$61$ \( ( 1 - 8 T + 123 T^{2} - 704 T^{3} + 7503 T^{4} - 29768 T^{5} + 226981 T^{6} )^{2} \)
$67$ \( ( 1 - 12 T + 212 T^{2} - 1540 T^{3} + 14204 T^{4} - 53868 T^{5} + 300763 T^{6} )^{2} \)
$71$ \( 1 + 18 T^{2} + 6671 T^{4} - 156772 T^{6} + 33628511 T^{8} + 457410258 T^{10} + 128100283921 T^{12} \)
$73$ \( ( 1 - 22 T + 360 T^{2} - 3448 T^{3} + 26280 T^{4} - 117238 T^{5} + 389017 T^{6} )^{2} \)
$79$ \( ( 1 - 4 T + 93 T^{2} + 8 T^{3} + 7347 T^{4} - 24964 T^{5} + 493039 T^{6} )^{2} \)
$83$ \( 1 + 262 T^{2} + 27815 T^{4} + 2144628 T^{6} + 191617535 T^{8} + 12434080102 T^{10} + 326940373369 T^{12} \)
$89$ \( 1 + 430 T^{2} + 85071 T^{4} + 9709540 T^{6} + 673847391 T^{8} + 26979163630 T^{10} + 496981290961 T^{12} \)
$97$ \( ( 1 + 8 T + 235 T^{2} + 1112 T^{3} + 22795 T^{4} + 75272 T^{5} + 912673 T^{6} )^{2} \)
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