Properties

Label 1575.2.b.d
Level 1575
Weight 2
Character orbit 1575.b
Analytic conductor 12.576
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{2} q^{7} -2 \beta_{1} q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{2} q^{7} -2 \beta_{1} q^{8} -\beta_{1} q^{11} + ( 1 - 2 \beta_{2} ) q^{13} + ( \beta_{1} - \beta_{3} ) q^{14} -4 q^{16} + ( \beta_{1} - 2 \beta_{3} ) q^{17} + ( -1 + 2 \beta_{2} ) q^{19} -2 q^{22} + 5 \beta_{1} q^{23} + ( \beta_{1} - 2 \beta_{3} ) q^{26} -\beta_{1} q^{29} + ( 1 - 2 \beta_{2} ) q^{31} + ( -2 + 4 \beta_{2} ) q^{34} + 8 q^{37} + ( -\beta_{1} + 2 \beta_{3} ) q^{38} + 5 q^{43} + 10 q^{46} + ( \beta_{1} - 2 \beta_{3} ) q^{47} + ( -7 + \beta_{2} ) q^{49} -4 \beta_{1} q^{53} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{56} -2 q^{58} + ( \beta_{1} - 2 \beta_{3} ) q^{59} + ( -1 + 2 \beta_{2} ) q^{61} + ( \beta_{1} - 2 \beta_{3} ) q^{62} -8 q^{64} -7 q^{67} + 2 \beta_{1} q^{71} + ( 2 - 4 \beta_{2} ) q^{73} -8 \beta_{1} q^{74} + ( \beta_{1} - \beta_{3} ) q^{77} + 14 q^{79} + ( -\beta_{1} + 2 \beta_{3} ) q^{83} -5 \beta_{1} q^{86} -4 q^{88} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{89} + ( -14 + \beta_{2} ) q^{91} + ( -2 + 4 \beta_{2} ) q^{94} + ( -1 + 2 \beta_{2} ) q^{97} + ( 6 \beta_{1} + \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{7} + O(q^{10}) \) \( 4q - 2q^{7} - 16q^{16} - 8q^{22} + 32q^{37} + 20q^{43} + 40q^{46} - 26q^{49} - 8q^{58} - 32q^{64} - 28q^{67} + 56q^{79} - 16q^{88} - 54q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{2} - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 6 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{2} + 2\)\()/3\)
\(\nu^{3}\)\(=\)\(2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.22474 + 0.707107i
−1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.2 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 0 0
251.3 1.41421i 0 0 0 0 −0.500000 2.59808i 2.82843i 0 0
251.4 1.41421i 0 0 0 0 −0.500000 + 2.59808i 2.82843i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.b.d 4
3.b odd 2 1 inner 1575.2.b.d 4
5.b even 2 1 1575.2.b.e yes 4
5.c odd 4 2 1575.2.g.b 8
7.b odd 2 1 inner 1575.2.b.d 4
15.d odd 2 1 1575.2.b.e yes 4
15.e even 4 2 1575.2.g.b 8
21.c even 2 1 inner 1575.2.b.d 4
35.c odd 2 1 1575.2.b.e yes 4
35.f even 4 2 1575.2.g.b 8
105.g even 2 1 1575.2.b.e yes 4
105.k odd 4 2 1575.2.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.2.b.d 4 1.a even 1 1 trivial
1575.2.b.d 4 3.b odd 2 1 inner
1575.2.b.d 4 7.b odd 2 1 inner
1575.2.b.d 4 21.c even 2 1 inner
1575.2.b.e yes 4 5.b even 2 1
1575.2.b.e yes 4 15.d odd 2 1
1575.2.b.e yes 4 35.c odd 2 1
1575.2.b.e yes 4 105.g even 2 1
1575.2.g.b 8 5.c odd 4 2
1575.2.g.b 8 15.e even 4 2
1575.2.g.b 8 35.f even 4 2
1575.2.g.b 8 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{2} + 2 \)
\( T_{37} - 8 \)
\( T_{47}^{2} - 54 \)
\( T_{67} + 7 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \)
$3$ 1
$5$ 1
$7$ \( ( 1 + T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 4 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 35 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 8 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 40 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 74 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 64 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 95 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 7 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 134 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 112 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 38 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \)
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