Properties

Label 7623.2.a.cl
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: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} - \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{4} + 2 \beta_{2} q^{5} + q^{7} + ( -2 - \beta_{1} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} - \beta_{2} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{4} + 2 \beta_{2} q^{5} + q^{7} + ( -2 - \beta_{1} ) q^{8} + ( -2 - 2 \beta_{1} ) q^{10} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} ) q^{14} + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{16} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{17} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{19} + ( -2 + 4 \beta_{1} + 2 \beta_{3} ) q^{20} + ( 3 + 4 \beta_{1} - 3 \beta_{3} ) q^{23} + ( 3 + 4 \beta_{2} - 4 \beta_{3} ) q^{25} + ( 2 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{26} + ( \beta_{1} + \beta_{3} ) q^{28} + ( 2 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{29} + ( -4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{32} + ( 2 - 2 \beta_{1} + 6 \beta_{3} ) q^{34} + 2 \beta_{2} q^{35} + ( -3 + 4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{37} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{38} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{40} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{41} + ( -6 \beta_{1} + 5 \beta_{3} ) q^{43} + ( 3 - \beta_{1} - 4 \beta_{2} - 7 \beta_{3} ) q^{46} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{47} + q^{49} + ( -1 + \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{50} + ( -8 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{52} + ( 2 + 2 \beta_{1} - 5 \beta_{3} ) q^{53} + ( -2 - \beta_{1} ) q^{56} + ( 6 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{58} + ( 6 - 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -6 + 8 \beta_{1} + 8 \beta_{2} ) q^{62} + ( 1 - \beta_{1} - 5 \beta_{2} - 2 \beta_{3} ) q^{64} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{65} + ( -8 - 2 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -2 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{68} + ( -2 - 2 \beta_{1} ) q^{70} + ( 6 - 4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{71} + ( 2 - 2 \beta_{2} ) q^{73} + ( -7 - 3 \beta_{1} - 5 \beta_{3} ) q^{74} + ( -2 \beta_{2} + 4 \beta_{3} ) q^{76} + ( 4 + 2 \beta_{1} + \beta_{3} ) q^{79} + ( 10 + 2 \beta_{2} - 4 \beta_{3} ) q^{80} + ( -6 - 12 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{82} + ( 2 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 8 - 4 \beta_{1} + 4 \beta_{2} - 8 \beta_{3} ) q^{85} + ( -4 \beta_{1} + \beta_{2} + 11 \beta_{3} ) q^{86} + ( 2 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 1 + 8 \beta_{1} + 5 \beta_{2} ) q^{92} + ( -4 \beta_{1} + 10 \beta_{3} ) q^{94} + ( 4 + 4 \beta_{2} - 4 \beta_{3} ) q^{95} + ( 8 + 4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{97} + ( 1 - \beta_{1} - \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + 3q^{4} + 4q^{5} + 4q^{7} - 9q^{8} + O(q^{10}) \) \( 4q + q^{2} + 3q^{4} + 4q^{5} + 4q^{7} - 9q^{8} - 10q^{10} - 6q^{13} + q^{14} - 3q^{16} + 8q^{17} + 10q^{19} + 10q^{23} + 12q^{25} + 20q^{26} + 3q^{28} - 18q^{31} - 2q^{32} + 18q^{34} + 4q^{35} - 2q^{37} + 8q^{38} - 6q^{40} + 10q^{41} + 4q^{43} - 11q^{46} - 4q^{47} + 4q^{49} - 9q^{50} - 20q^{52} - 9q^{56} + 14q^{58} + 16q^{59} - 14q^{61} - 11q^{64} - 28q^{65} - 28q^{67} - 16q^{68} - 10q^{70} + 18q^{71} + 4q^{73} - 41q^{74} + 4q^{76} + 20q^{79} + 36q^{80} - 24q^{82} + 6q^{83} + 20q^{85} + 20q^{86} + 16q^{89} - 6q^{91} + 22q^{92} + 16q^{94} + 16q^{95} + 32q^{97} + q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)

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
2.09529
−1.35567
0.737640
−0.477260
−2.39026 0 3.71333 2.58993 0 1.00000 −4.09529 0 −6.19059
1.2 0.162147 0 −1.97371 4.38705 0 1.00000 −0.644326 0 0.711349
1.3 1.45589 0 0.119606 −2.38705 0 1.00000 −2.73764 0 −3.47528
1.4 1.77222 0 1.14077 −0.589926 0 1.00000 −1.52274 0 −1.04548
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.cl 4
3.b odd 2 1 2541.2.a.bm 4
11.b odd 2 1 7623.2.a.ci 4
11.d odd 10 2 693.2.m.f 8
33.d even 2 1 2541.2.a.bn 4
33.f even 10 2 231.2.j.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.f 8 33.f even 10 2
693.2.m.f 8 11.d odd 10 2
2541.2.a.bm 4 3.b odd 2 1
2541.2.a.bn 4 33.d even 2 1
7623.2.a.ci 4 11.b odd 2 1
7623.2.a.cl 4 1.a even 1 1 trivial

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\( T_{2}^{4} - T_{2}^{3} - 5 T_{2}^{2} + 7 T_{2} - 1 \)
\( T_{5}^{4} - 4 T_{5}^{3} - 8 T_{5}^{2} + 24 T_{5} + 16 \)
\( T_{13}^{4} + 6 T_{13}^{3} - 8 T_{13}^{2} - 16 T_{13} + 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 3 T^{2} + T^{3} + 3 T^{4} + 2 T^{5} + 12 T^{6} - 8 T^{7} + 16 T^{8} \)
$3$ \( \)
$5$ \( 1 - 4 T + 12 T^{2} - 36 T^{3} + 86 T^{4} - 180 T^{5} + 300 T^{6} - 500 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( \)
$13$ \( 1 + 6 T + 44 T^{2} + 218 T^{3} + 822 T^{4} + 2834 T^{5} + 7436 T^{6} + 13182 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 8 T + 48 T^{2} - 264 T^{3} + 1358 T^{4} - 4488 T^{5} + 13872 T^{6} - 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 10 T + 100 T^{2} - 570 T^{3} + 3062 T^{4} - 10830 T^{5} + 36100 T^{6} - 68590 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 10 T + 83 T^{2} - 500 T^{3} + 2869 T^{4} - 11500 T^{5} + 43907 T^{6} - 121670 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 37 T^{2} + 493 T^{4} + 31117 T^{6} + 707281 T^{8} \)
$31$ \( 1 + 18 T + 192 T^{2} + 1362 T^{3} + 8238 T^{4} + 42222 T^{5} + 184512 T^{6} + 536238 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 2 T + 71 T^{2} + 4 T^{3} + 2797 T^{4} + 148 T^{5} + 97199 T^{6} + 101306 T^{7} + 1874161 T^{8} \)
$41$ \( 1 - 10 T + 60 T^{2} + 290 T^{3} - 2938 T^{4} + 11890 T^{5} + 100860 T^{6} - 689210 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 4 T + 69 T^{2} - 392 T^{3} + 4097 T^{4} - 16856 T^{5} + 127581 T^{6} - 318028 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 4 T + 108 T^{2} - 4 T^{3} + 4758 T^{4} - 188 T^{5} + 238572 T^{6} + 415292 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 161 T^{2} + 20 T^{3} + 11657 T^{4} + 1060 T^{5} + 452249 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 16 T + 308 T^{2} - 2936 T^{3} + 29398 T^{4} - 173224 T^{5} + 1072148 T^{6} - 3286064 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 14 T + 260 T^{2} + 2306 T^{3} + 24294 T^{4} + 140666 T^{5} + 967460 T^{6} + 3177734 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 448632 T^{5} + 2428549 T^{6} + 8421364 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 18 T + 327 T^{2} - 3632 T^{3} + 36173 T^{4} - 257872 T^{5} + 1648407 T^{6} - 6442398 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 4 T + 284 T^{2} - 852 T^{3} + 30822 T^{4} - 62196 T^{5} + 1513436 T^{6} - 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 20 T + 445 T^{2} - 5040 T^{3} + 57977 T^{4} - 398160 T^{5} + 2777245 T^{6} - 9860780 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 6 T + 212 T^{2} - 982 T^{3} + 24278 T^{4} - 81506 T^{5} + 1460468 T^{6} - 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 16 T + 308 T^{2} - 3096 T^{3} + 38678 T^{4} - 275544 T^{5} + 2439668 T^{6} - 11279504 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 32 T + 656 T^{2} - 8944 T^{3} + 99982 T^{4} - 867568 T^{5} + 6172304 T^{6} - 29205536 T^{7} + 88529281 T^{8} \)
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