Properties

Label 7623.2.a.da
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 12
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} -\beta_{8} q^{5} + q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{4} -\beta_{8} q^{5} + q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( 2 - \beta_{4} - \beta_{6} ) q^{10} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{13} + \beta_{1} q^{14} + ( 3 + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{10} ) q^{16} + ( -\beta_{1} - \beta_{7} - \beta_{8} ) q^{17} + ( 1 + \beta_{4} ) q^{19} + ( 2 \beta_{1} + \beta_{3} - \beta_{8} - \beta_{9} ) q^{20} + ( \beta_{1} + \beta_{3} ) q^{23} + ( 3 - \beta_{2} - \beta_{6} - \beta_{10} ) q^{25} + ( 4 \beta_{1} + \beta_{3} + \beta_{8} + \beta_{9} ) q^{26} + ( 2 + \beta_{2} ) q^{28} + ( -\beta_{8} - \beta_{11} ) q^{29} + ( -2 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{31} + ( 3 \beta_{1} + \beta_{3} + 2 \beta_{7} - \beta_{8} + \beta_{11} ) q^{32} + ( -2 - 2 \beta_{2} - \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{10} ) q^{34} -\beta_{8} q^{35} + ( -\beta_{2} + 2 \beta_{4} ) q^{37} + ( \beta_{1} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{38} + ( 5 + 3 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{40} + ( -2 \beta_{1} - \beta_{3} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{41} + ( 1 - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{43} + ( 3 + 3 \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{10} ) q^{46} + ( -\beta_{3} - \beta_{9} ) q^{47} + q^{49} + ( \beta_{1} - 2 \beta_{7} - 3 \beta_{8} - \beta_{11} ) q^{50} + ( 9 + 5 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{10} ) q^{52} + ( \beta_{1} + \beta_{3} - \beta_{7} + 2 \beta_{8} ) q^{53} + ( 2 \beta_{1} + \beta_{3} ) q^{56} + ( 2 - \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{10} ) q^{58} + ( \beta_{1} - \beta_{3} - \beta_{7} - \beta_{8} ) q^{59} + ( 3 - 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{61} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{7} + 2 \beta_{8} ) q^{62} + ( 7 + 4 \beta_{2} - 8 \beta_{5} - \beta_{6} + 3 \beta_{10} ) q^{64} + ( -\beta_{1} + 2 \beta_{7} - 2 \beta_{8} ) q^{65} + ( 2 \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{10} ) q^{67} + ( -4 \beta_{1} - \beta_{3} - 3 \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{68} + ( 2 - \beta_{4} - \beta_{6} ) q^{70} + ( -\beta_{3} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{71} + ( 5 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{73} + ( -2 \beta_{1} - \beta_{3} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{74} + ( \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{76} + ( -1 + \beta_{2} + 2 \beta_{5} + 3 \beta_{6} - \beta_{10} ) q^{79} + ( 7 \beta_{1} + \beta_{3} + \beta_{7} ) q^{80} + ( -7 - 3 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{10} ) q^{82} + ( -\beta_{1} - \beta_{3} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{83} + ( 6 - 2 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 2 \beta_{10} ) q^{85} + ( \beta_{1} - \beta_{3} + 2 \beta_{7} - 2 \beta_{9} ) q^{86} + ( -2 \beta_{1} - \beta_{8} - 2 \beta_{9} ) q^{89} + ( 2 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{91} + ( 7 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} - \beta_{8} + \beta_{11} ) q^{92} + ( 1 - 3 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} + \beta_{6} - 2 \beta_{10} ) q^{94} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{7} - 2 \beta_{8} ) q^{95} + ( -2 + 3 \beta_{2} + \beta_{4} - 3 \beta_{5} + \beta_{10} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 20q^{4} + 12q^{7} + O(q^{10}) \) \( 12q + 20q^{4} + 12q^{7} + 20q^{10} + 20q^{13} + 28q^{16} + 12q^{19} + 32q^{25} + 20q^{28} - 16q^{31} - 24q^{34} + 4q^{37} + 48q^{40} + 16q^{43} + 24q^{46} + 12q^{49} + 96q^{52} + 20q^{58} + 44q^{61} + 76q^{64} + 20q^{70} + 52q^{73} - 8q^{79} - 68q^{82} + 72q^{85} + 20q^{91} + 20q^{94} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 22 x^{10} + 181 x^{8} - 692 x^{6} + 1240 x^{4} - 936 x^{2} + 244\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 6 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{8} - 17 \nu^{6} + 94 \nu^{4} - 186 \nu^{2} + 92 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{10} - 21 \nu^{8} + 160 \nu^{6} - 536 \nu^{4} + 748 \nu^{2} - 308 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{10} - 21 \nu^{8} + 164 \nu^{6} - 588 \nu^{4} + 924 \nu^{2} - 412 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} + 21 \nu^{9} - 160 \nu^{7} + 536 \nu^{5} - 748 \nu^{3} + 308 \nu \)\()/8\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{11} - 21 \nu^{9} + 162 \nu^{7} - 562 \nu^{5} + 840 \nu^{3} - 384 \nu \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{11} + 22 \nu^{9} - 178 \nu^{7} + 643 \nu^{5} - 980 \nu^{3} + 438 \nu \)\()/4\)
\(\beta_{10}\)\(=\)\((\)\( 3 \nu^{10} - 63 \nu^{8} + 484 \nu^{6} - 1652 \nu^{4} + 2356 \nu^{2} - 956 \)\()/8\)
\(\beta_{11}\)\(=\)\((\)\( 3 \nu^{11} - 63 \nu^{9} + 482 \nu^{7} - 1626 \nu^{5} + 2264 \nu^{3} - 880 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10} - \beta_{6} - 2 \beta_{5} + 8 \beta_{2} + 23\)
\(\nu^{5}\)\(=\)\(\beta_{11} - \beta_{8} + 2 \beta_{7} + 9 \beta_{3} + 39 \beta_{1}\)
\(\nu^{6}\)\(=\)\(13 \beta_{10} - 11 \beta_{6} - 28 \beta_{5} + 60 \beta_{2} + 149\)
\(\nu^{7}\)\(=\)\(13 \beta_{11} - 9 \beta_{8} + 30 \beta_{7} + 71 \beta_{3} + 269 \beta_{1}\)
\(\nu^{8}\)\(=\)\(127 \beta_{10} - 93 \beta_{6} - 288 \beta_{5} + 4 \beta_{4} + 454 \beta_{2} + 1023\)
\(\nu^{9}\)\(=\)\(127 \beta_{11} + 4 \beta_{9} - 55 \beta_{8} + 318 \beta_{7} + 547 \beta_{3} + 1931 \beta_{1}\)
\(\nu^{10}\)\(=\)\(1123 \beta_{10} - 729 \beta_{6} - 2632 \beta_{5} + 84 \beta_{4} + 3474 \beta_{2} + 7287\)
\(\nu^{11}\)\(=\)\(1123 \beta_{11} + 84 \beta_{9} - 251 \beta_{8} + 2942 \beta_{7} + 4203 \beta_{3} + 14235 \beta_{1}\)

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
−2.79401
−2.41714
−2.09771
−1.60257
−0.870105
−0.790737
0.790737
0.870105
1.60257
2.09771
2.41714
2.79401
−2.79401 0 5.80650 −1.62657 0 1.00000 −10.6354 0 4.54465
1.2 −2.41714 0 3.84255 −4.01054 0 1.00000 −4.45369 0 9.69401
1.3 −2.09771 0 2.40037 2.53213 0 1.00000 −0.839859 0 −5.31167
1.4 −1.60257 0 0.568238 1.76461 0 1.00000 2.29450 0 −2.82792
1.5 −0.870105 0 −1.24292 −0.709862 0 1.00000 2.82168 0 0.617654
1.6 −0.790737 0 −1.37474 −4.15216 0 1.00000 2.66853 0 3.28327
1.7 0.790737 0 −1.37474 4.15216 0 1.00000 −2.66853 0 3.28327
1.8 0.870105 0 −1.24292 0.709862 0 1.00000 −2.82168 0 0.617654
1.9 1.60257 0 0.568238 −1.76461 0 1.00000 −2.29450 0 −2.82792
1.10 2.09771 0 2.40037 −2.53213 0 1.00000 0.839859 0 −5.31167
1.11 2.41714 0 3.84255 4.01054 0 1.00000 4.45369 0 9.69401
1.12 2.79401 0 5.80650 1.62657 0 1.00000 10.6354 0 4.54465
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.da yes 12
3.b odd 2 1 inner 7623.2.a.da yes 12
11.b odd 2 1 7623.2.a.cz 12
33.d even 2 1 7623.2.a.cz 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7623.2.a.cz 12 11.b odd 2 1
7623.2.a.cz 12 33.d even 2 1
7623.2.a.da yes 12 1.a even 1 1 trivial
7623.2.a.da yes 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{12} - 22 T_{2}^{10} + 181 T_{2}^{8} - 692 T_{2}^{6} + 1240 T_{2}^{4} - 936 T_{2}^{2} + 244 \)
\( T_{5}^{12} - 46 T_{5}^{10} + 751 T_{5}^{8} - 5300 T_{5}^{6} + 16771 T_{5}^{4} - 21846 T_{5}^{2} + 7381 \)
\( T_{13}^{6} - 10 T_{13}^{5} + 5 T_{13}^{4} + 150 T_{13}^{3} - 174 T_{13}^{2} - 260 T_{13} - 50 \)