Properties

Label 7623.2.a.cy
Level $7623$
Weight $2$
Character orbit 7623.a
Self dual yes
Analytic conductor $60.870$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{6} q^{5} + q^{7} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{6} q^{5} + q^{7} + (\beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{8} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2}) q^{10} + ( - \beta_{9} + \beta_{5} - \beta_{3}) q^{13} + \beta_1 q^{14} + (\beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{16} + ( - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{7} + \beta_{4} + \beta_{3} + 1) q^{19} + (\beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{3} + 2) q^{20} + (\beta_{6} - \beta_{4} - \beta_{3} + \beta_{2}) q^{23} + (\beta_{9} + \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{25} + ( - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - \beta_1 - 3) q^{26} + (\beta_{2} + 2) q^{28} + ( - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{29} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{31} + (2 \beta_{9} + \beta_{8} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_1 + 7) q^{32} + (\beta_{9} + \beta_{6} - \beta_{4} + 3 \beta_{3} - 3 \beta_1 + 4) q^{34} + \beta_{6} q^{35} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{37} + (\beta_{9} + 2 \beta_{8} - 2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{38} + (2 \beta_{9} + \beta_{8} - 5 \beta_{5} + 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{40} + ( - \beta_{9} - \beta_{6} + \beta_{5} + \beta_{3} - 2) q^{41} + ( - \beta_{9} - \beta_{8} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{43} + (\beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{46} + ( - 2 \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{47} + q^{49} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - 6 \beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{50} + ( - 2 \beta_{9} - 2 \beta_{8} - \beta_{7} - 3 \beta_{6} + 5 \beta_{5} - 7 \beta_{3} + \cdots - 7) q^{52}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25 \nu^{9} + 31 \nu^{8} + 558 \nu^{7} - 505 \nu^{6} - 4093 \nu^{5} + 2537 \nu^{4} + 11069 \nu^{3} - 4152 \nu^{2} - 8301 \nu + 385 ) / 2024 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25 \nu^{9} - 31 \nu^{8} - 558 \nu^{7} + 505 \nu^{6} + 4093 \nu^{5} - 2537 \nu^{4} - 9045 \nu^{3} + 4152 \nu^{2} - 3843 \nu - 2409 ) / 2024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31 \nu^{9} - 83 \nu^{8} + 530 \nu^{7} + 993 \nu^{6} - 2687 \nu^{5} - 3169 \nu^{4} + 3727 \nu^{3} + 1976 \nu^{2} + 1365 \nu + 275 ) / 2024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 97 \nu^{9} - 325 \nu^{8} + 2246 \nu^{7} + 5327 \nu^{6} - 16569 \nu^{5} - 27479 \nu^{4} + 43393 \nu^{3} + 43072 \nu^{2} - 29941 \nu - 5995 ) / 2024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 229 \nu^{9} + 203 \nu^{8} + 3654 \nu^{7} - 2197 \nu^{6} - 19033 \nu^{5} + 4861 \nu^{4} + 34681 \nu^{3} + 5848 \nu^{2} - 13617 \nu - 7403 ) / 2024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 285 \nu^{9} + 151 \nu^{8} + 4742 \nu^{7} - 1709 \nu^{6} - 25813 \nu^{5} + 6253 \nu^{4} + 49477 \nu^{3} - 12520 \nu^{2} - 24601 \nu + 9449 ) / 2024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 70 \nu^{9} + 65 \nu^{8} - 1360 \nu^{7} - 1116 \nu^{6} + 8981 \nu^{5} + 5850 \nu^{4} - 22543 \nu^{3} - 8412 \nu^{2} + 16260 \nu - 1331 ) / 506 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{3} + 8\beta_{2} + 2\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{5} + 9\beta_{4} + 11\beta_{3} + 39\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + 11 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 14 \beta_{5} + \beta_{4} - 12 \beta_{3} + 58 \beta_{2} + 26 \beta _1 + 158 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26 \beta_{9} + 13 \beta_{8} - \beta_{7} + 27 \beta_{6} + 10 \beta_{5} + 69 \beta_{4} + 104 \beta_{3} + 3 \beta_{2} + 266 \beta _1 + 157 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 18 \beta_{9} + 97 \beta_{8} - 80 \beta_{7} + 30 \beta_{6} - 145 \beta_{5} + 16 \beta_{4} - 92 \beta_{3} + 412 \beta_{2} + 257 \beta _1 + 1095 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 255 \beta_{9} + 126 \beta_{8} - 21 \beta_{7} + 272 \beta_{6} + 61 \beta_{5} + 509 \beta_{4} + 909 \beta_{3} + 52 \beta_{2} + 1873 \beta _1 + 1444 \) 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
−2.65195
−2.39396
−2.09767
−1.33330
−0.112481
0.473713
0.871604
1.80545
2.63994
2.79866
−2.65195 0 5.03286 −1.71311 0 1.00000 −8.04301 0 4.54308
1.2 −2.39396 0 3.73106 3.93829 0 1.00000 −4.14409 0 −9.42812
1.3 −2.09767 0 2.40021 −3.15947 0 1.00000 −0.839503 0 6.62751
1.4 −1.33330 0 −0.222305 0.873210 0 1.00000 2.96300 0 −1.16425
1.5 −0.112481 0 −1.98735 −1.06131 0 1.00000 0.448501 0 0.119378
1.6 0.473713 0 −1.77560 −3.75881 0 1.00000 −1.78855 0 −1.78060
1.7 0.871604 0 −1.24031 4.06436 0 1.00000 −2.82426 0 3.54252
1.8 1.80545 0 1.25966 −2.77715 0 1.00000 −1.33666 0 −5.01400
1.9 2.63994 0 4.96928 −3.08369 0 1.00000 7.83871 0 −8.14075
1.10 2.79866 0 5.83249 1.67767 0 1.00000 10.7258 0 4.69523
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.cy 10
3.b odd 2 1 2541.2.a.br 10
11.b odd 2 1 7623.2.a.cx 10
11.d odd 10 2 693.2.m.j 20
33.d even 2 1 2541.2.a.bq 10
33.f even 10 2 231.2.j.g 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.g 20 33.f even 10 2
693.2.m.j 20 11.d odd 10 2
2541.2.a.bq 10 33.d even 2 1
2541.2.a.br 10 3.b odd 2 1
7623.2.a.cx 10 11.b odd 2 1
7623.2.a.cy 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{10} - 19T_{2}^{8} - T_{2}^{7} + 124T_{2}^{6} + 6T_{2}^{5} - 316T_{2}^{4} + 17T_{2}^{3} + 253T_{2}^{2} - 70T_{2} - 11 \) Copy content Toggle raw display
\( T_{5}^{10} + 5 T_{5}^{9} - 28 T_{5}^{8} - 179 T_{5}^{7} + 108 T_{5}^{6} + 1873 T_{5}^{5} + 1751 T_{5}^{4} - 4812 T_{5}^{3} - 6768 T_{5}^{2} + 2392 T_{5} + 4336 \) Copy content Toggle raw display
\( T_{13}^{10} - 6 T_{13}^{9} - 65 T_{13}^{8} + 334 T_{13}^{7} + 1540 T_{13}^{6} - 4808 T_{13}^{5} - 19424 T_{13}^{4} + 12448 T_{13}^{3} + 90048 T_{13}^{2} + 74752 T_{13} + 2816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 19 T^{8} - T^{7} + 124 T^{6} + \cdots - 11 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 5 T^{9} - 28 T^{8} + \cdots + 4336 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} - 65 T^{8} + \cdots + 2816 \) Copy content Toggle raw display
$17$ \( T^{10} + 8 T^{9} - 34 T^{8} + \cdots - 8336 \) Copy content Toggle raw display
$19$ \( T^{10} - 121 T^{8} + 136 T^{7} + \cdots - 50944 \) Copy content Toggle raw display
$23$ \( T^{10} - 106 T^{8} - 38 T^{7} + \cdots - 600380 \) Copy content Toggle raw display
$29$ \( T^{10} - 14 T^{9} - 90 T^{8} + \cdots - 3572144 \) Copy content Toggle raw display
$31$ \( T^{10} - 26 T^{9} + 172 T^{8} + \cdots + 532400 \) Copy content Toggle raw display
$37$ \( T^{10} - 24 T^{9} + 70 T^{8} + \cdots - 1675684 \) Copy content Toggle raw display
$41$ \( T^{10} + 19 T^{9} + 74 T^{8} + \cdots - 10096 \) Copy content Toggle raw display
$43$ \( T^{10} + 6 T^{9} - 118 T^{8} + \cdots - 21296 \) Copy content Toggle raw display
$47$ \( T^{10} + 15 T^{9} - 137 T^{8} + \cdots - 2694400 \) Copy content Toggle raw display
$53$ \( T^{10} - T^{9} - 276 T^{8} + \cdots - 240496 \) Copy content Toggle raw display
$59$ \( T^{10} + 23 T^{9} + \cdots - 159106816 \) Copy content Toggle raw display
$61$ \( T^{10} - 395 T^{8} + \cdots - 87904256 \) Copy content Toggle raw display
$67$ \( T^{10} - 38 T^{9} + 269 T^{8} + \cdots + 52960256 \) Copy content Toggle raw display
$71$ \( T^{10} + 26 T^{9} - 78 T^{8} + \cdots + 17073920 \) Copy content Toggle raw display
$73$ \( T^{10} + T^{9} - 337 T^{8} + \cdots - 7615744 \) Copy content Toggle raw display
$79$ \( T^{10} - 5 T^{9} - 238 T^{8} + \cdots + 1785296 \) Copy content Toggle raw display
$83$ \( T^{10} + 6 T^{9} - 340 T^{8} + \cdots - 779264 \) Copy content Toggle raw display
$89$ \( T^{10} - 9 T^{9} - 250 T^{8} + \cdots + 106384 \) Copy content Toggle raw display
$97$ \( T^{10} - 24 T^{9} + 99 T^{8} + \cdots + 481024 \) Copy content Toggle raw display
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