Properties

Label 7524.2.l.b
Level 7524
Weight 2
Character orbit 7524.l
Analytic conductor 60.079
Analytic rank 0
Dimension 8
CM discriminant -627
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 7524 = 2^{2} \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7524.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(60.0794424808\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.488455618816.6
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q -\beta_{6} q^{11} -\beta_{1} q^{13} + \beta_{3} q^{17} -\beta_{4} q^{19} -5 q^{25} + 7 q^{49} -\beta_{2} q^{53} + ( -\beta_{2} - 2 \beta_{5} ) q^{59} + ( -\beta_{2} + 2 \beta_{5} ) q^{71} + ( 3 \beta_{1} - 2 \beta_{4} ) q^{79} + ( -\beta_{3} + 2 \beta_{6} ) q^{83} + ( -\beta_{2} - 4 \beta_{5} ) q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 40q^{25} + 56q^{49} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 14 x^{5} + 105 x^{4} - 238 x^{3} - 426 x^{2} + 548 x + 3140\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 107 \nu^{7} + 1001 \nu^{6} - 4543 \nu^{5} - 13153 \nu^{4} + 51884 \nu^{3} + 72002 \nu^{2} - 258088 \nu - 983740 \)\()/137550\)
\(\beta_{2}\)\(=\)\((\)\( 39 \nu^{7} - 595 \nu^{6} + 1995 \nu^{5} - 749 \nu^{4} - 406 \nu^{3} - 76426 \nu^{2} + 98312 \nu + 158560 \)\()/45850\)
\(\beta_{3}\)\(=\)\((\)\( 73 \nu^{7} + 1120 \nu^{6} - 4025 \nu^{5} + 5887 \nu^{4} + 4648 \nu^{3} + 61978 \nu^{2} - 65216 \nu + 73420 \)\()/68775\)
\(\beta_{4}\)\(=\)\((\)\( 214 \nu^{7} - 749 \nu^{6} - 833 \nu^{5} + 3955 \nu^{4} + 29491 \nu^{3} - 48566 \nu^{2} - 285092 \nu + 150790 \)\()/137550\)
\(\beta_{5}\)\(=\)\((\)\( -78 \nu^{7} + 273 \nu^{6} - 1239 \nu^{5} + 2415 \nu^{4} - 5607 \nu^{3} + 6132 \nu^{2} - 46236 \nu + 22170 \)\()/45850\)
\(\beta_{6}\)\(=\)\((\)\( -146 \nu^{7} + 511 \nu^{6} - 203 \nu^{5} - 770 \nu^{4} - 17549 \nu^{3} + 27349 \nu^{2} - 18122 \nu + 4465 \)\()/68775\)
\(\beta_{7}\)\(=\)\((\)\( -24 \nu^{7} + 84 \nu^{6} + 42 \nu^{5} - 315 \nu^{4} - 1302 \nu^{3} + 2310 \nu^{2} + 8628 \nu - 4253 \)\()/917\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{6} + \beta_{5} - 3 \beta_{4} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_{1} + 6\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} - 36 \beta_{6} + 2 \beta_{5} + 12 \beta_{4} - 3 \beta_{3} - 9 \beta_{2} - 3 \beta_{1} + 15\)\()/6\)
\(\nu^{4}\)\(=\)\(\beta_{7} - 10 \beta_{6} + 8 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 8 \beta_{1} - 49\)
\(\nu^{5}\)\(=\)\((\)\(25 \beta_{7} + 52 \beta_{6} - 254 \beta_{5} + 272 \beta_{4} + 35 \beta_{3} - 45 \beta_{2} - 115 \beta_{1} - 767\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(30 \beta_{7} + 248 \beta_{6} - 296 \beta_{5} + 304 \beta_{4} + 229 \beta_{3} + 132 \beta_{2} - 26 \beta_{1} - 954\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-217 \beta_{7} + 3104 \beta_{6} - 2458 \beta_{5} + 52 \beta_{4} + 1477 \beta_{3} + 1071 \beta_{2} + 217 \beta_{1} - 3805\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7524\mathbb{Z}\right)^\times\).

\(n\) \(2377\) \(3763\) \(4105\) \(6689\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2089.1
−1.67945 2.15831i
2.67945 2.15831i
−1.67945 1.15831i
2.67945 1.15831i
−1.67945 + 2.15831i
2.67945 + 2.15831i
−1.67945 + 1.15831i
2.67945 + 1.15831i
0 0 0 0 0 0 0 0 0
2089.2 0 0 0 0 0 0 0 0 0
2089.3 0 0 0 0 0 0 0 0 0
2089.4 0 0 0 0 0 0 0 0 0
2089.5 0 0 0 0 0 0 0 0 0
2089.6 0 0 0 0 0 0 0 0 0
2089.7 0 0 0 0 0 0 0 0 0
2089.8 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2089.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
627.b odd 2 1 CM by \(\Q(\sqrt{-627}) \)
3.b odd 2 1 inner
11.b odd 2 1 inner
19.b odd 2 1 inner
33.d even 2 1 inner
57.d even 2 1 inner
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7524.2.l.b 8
3.b odd 2 1 inner 7524.2.l.b 8
11.b odd 2 1 inner 7524.2.l.b 8
19.b odd 2 1 inner 7524.2.l.b 8
33.d even 2 1 inner 7524.2.l.b 8
57.d even 2 1 inner 7524.2.l.b 8
209.d even 2 1 inner 7524.2.l.b 8
627.b odd 2 1 CM 7524.2.l.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7524.2.l.b 8 1.a even 1 1 trivial
7524.2.l.b 8 3.b odd 2 1 inner
7524.2.l.b 8 11.b odd 2 1 inner
7524.2.l.b 8 19.b odd 2 1 inner
7524.2.l.b 8 33.d even 2 1 inner
7524.2.l.b 8 57.d even 2 1 inner
7524.2.l.b 8 209.d even 2 1 inner
7524.2.l.b 8 627.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(7524, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{8} \)
$11$ \( ( 1 + 11 T^{2} )^{4} \)
$13$ \( ( 1 - 7 T^{2} - 120 T^{4} - 1183 T^{6} + 28561 T^{8} )^{2} \)
$17$ \( ( 1 + 23 T^{2} + 240 T^{4} + 6647 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{8} \)
$29$ \( ( 1 + 29 T^{2} )^{8} \)
$31$ \( ( 1 - 31 T^{2} )^{8} \)
$37$ \( ( 1 - 37 T^{2} )^{8} \)
$41$ \( ( 1 + 41 T^{2} )^{8} \)
$43$ \( ( 1 - 43 T^{2} )^{8} \)
$47$ \( ( 1 + 47 T^{2} )^{8} \)
$53$ \( ( 1 - 103 T^{2} + 7800 T^{4} - 289327 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{8} \)
$67$ \( ( 1 - 67 T^{2} )^{8} \)
$71$ \( ( 1 - 67 T^{2} - 552 T^{4} - 337747 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{8} \)
$79$ \( ( 1 - 139 T^{2} + 13080 T^{4} - 867499 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 109 T^{2} + 4992 T^{4} - 750901 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 31 T^{2} - 6960 T^{4} - 245551 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 97 T^{2} )^{8} \)
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