Properties

Label 7524.2.l.b
Level 7524
Weight 2
Character orbit 7524.l
Analytic conductor 60.079
Analytic rank 0
Dimension 8
CM disc. -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 \(2\) and degree \(1\))

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: \( 2^{4}\cdot 3^{4} \)
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.

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
627.b Odd 1 CM by \(\Q(\sqrt{-627}) \) yes
3.b Odd 1 yes
11.b Odd 1 no
19.b Odd 1 no
33.d Even 1 no
57.d Even 1 no
209.d Even 1 no

Hecke kernels

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