Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{3} q^{7} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( -1 + 2 \beta_{2} ) q^{11} -2 \beta_{1} q^{13} -\beta_{2} q^{14} + 3 \beta_{2} q^{16} -4 \beta_{1} q^{17} + ( 2 + 4 \beta_{2} ) q^{19} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( 2 - 2 \beta_{2} ) q^{26} + ( \beta_{1} + \beta_{3} ) q^{28} + 5 q^{29} + 6 \beta_{2} q^{31} + ( \beta_{1} + 5 \beta_{3} ) q^{32} + ( 4 - 4 \beta_{2} ) q^{34} + 3 \beta_{3} q^{37} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{38} + ( -6 + 2 \beta_{2} ) q^{41} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{43} + ( 1 - \beta_{2} ) q^{44} + ( -2 - 3 \beta_{2} ) q^{46} + 2 \beta_{3} q^{47} - q^{49} -2 \beta_{3} q^{52} + ( 4 \beta_{1} + 6 \beta_{3} ) q^{53} + ( -1 - 2 \beta_{2} ) q^{56} + 5 \beta_{1} q^{58} + ( 8 + 6 \beta_{2} ) q^{59} + ( -6 - 6 \beta_{2} ) q^{61} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{62} + ( -1 + 2 \beta_{2} ) q^{64} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{67} -4 \beta_{3} q^{68} + ( -5 - 6 \beta_{2} ) q^{71} + ( -2 \beta_{1} + 10 \beta_{3} ) q^{73} -3 \beta_{2} q^{74} + ( 6 + 2 \beta_{2} ) q^{76} + ( 2 \beta_{1} - \beta_{3} ) q^{77} + ( 5 + 10 \beta_{2} ) q^{79} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{82} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -2 + 5 \beta_{2} ) q^{86} + ( -4 \beta_{1} + 3 \beta_{3} ) q^{88} + ( 16 + 2 \beta_{2} ) q^{89} + 2 \beta_{2} q^{91} + ( 5 \beta_{1} + 7 \beta_{3} ) q^{92} -2 \beta_{2} q^{94} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{97} -\beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + O(q^{10}) \) \( 4q + 2q^{4} - 8q^{11} + 2q^{14} - 6q^{16} + 12q^{26} + 20q^{29} - 12q^{31} + 24q^{34} - 28q^{41} + 6q^{44} - 2q^{46} - 4q^{49} + 20q^{59} - 12q^{61} - 8q^{64} - 8q^{71} + 6q^{74} + 20q^{76} - 18q^{86} + 60q^{89} - 4q^{91} + 4q^{94} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1324.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 0 −0.618034 0 0 1.00000i 2.23607i 0 0
1324.2 0.618034i 0 1.61803 0 0 1.00000i 2.23607i 0 0
1324.3 0.618034i 0 1.61803 0 0 1.00000i 2.23607i 0 0
1324.4 1.61803i 0 −0.618034 0 0 1.00000i 2.23607i 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
5.b 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.d.k 4
3.b odd 2 1 175.2.b.c 4
5.b even 2 1 inner 1575.2.d.k 4
5.c odd 4 1 1575.2.a.n 2
5.c odd 4 1 1575.2.a.s 2
12.b even 2 1 2800.2.g.s 4
15.d odd 2 1 175.2.b.c 4
15.e even 4 1 175.2.a.d 2
15.e even 4 1 175.2.a.e yes 2
21.c even 2 1 1225.2.b.k 4
60.h even 2 1 2800.2.g.s 4
60.l odd 4 1 2800.2.a.bh 2
60.l odd 4 1 2800.2.a.bp 2
105.g even 2 1 1225.2.b.k 4
105.k odd 4 1 1225.2.a.n 2
105.k odd 4 1 1225.2.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 15.e even 4 1
175.2.a.e yes 2 15.e even 4 1
175.2.b.c 4 3.b odd 2 1
175.2.b.c 4 15.d odd 2 1
1225.2.a.n 2 105.k odd 4 1
1225.2.a.u 2 105.k odd 4 1
1225.2.b.k 4 21.c even 2 1
1225.2.b.k 4 105.g even 2 1
1575.2.a.n 2 5.c odd 4 1
1575.2.a.s 2 5.c odd 4 1
1575.2.d.k 4 1.a even 1 1 trivial
1575.2.d.k 4 5.b even 2 1 inner
2800.2.a.bh 2 60.l odd 4 1
2800.2.a.bp 2 60.l odd 4 1
2800.2.g.s 4 12.b even 2 1
2800.2.g.s 4 60.h even 2 1

Hecke kernels

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

\( T_{2}^{4} + 3 T_{2}^{2} + 1 \)
\( T_{11}^{2} + 4 T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -1 + 4 T + T^{2} )^{2} \)
$13$ \( 16 + 12 T^{2} + T^{4} \)
$17$ \( 256 + 48 T^{2} + T^{4} \)
$19$ \( ( -20 + T^{2} )^{2} \)
$23$ \( 121 + 42 T^{2} + T^{4} \)
$29$ \( ( -5 + T )^{4} \)
$31$ \( ( -36 + 6 T + T^{2} )^{2} \)
$37$ \( ( 9 + T^{2} )^{2} \)
$41$ \( ( 44 + 14 T + T^{2} )^{2} \)
$43$ \( 121 + 42 T^{2} + T^{4} \)
$47$ \( ( 4 + T^{2} )^{2} \)
$53$ \( 16 + 72 T^{2} + T^{4} \)
$59$ \( ( -20 - 10 T + T^{2} )^{2} \)
$61$ \( ( -36 + 6 T + T^{2} )^{2} \)
$67$ \( 1 + 18 T^{2} + T^{4} \)
$71$ \( ( -41 + 4 T + T^{2} )^{2} \)
$73$ \( 13456 + 252 T^{2} + T^{4} \)
$79$ \( ( -125 + T^{2} )^{2} \)
$83$ \( 1936 + 92 T^{2} + T^{4} \)
$89$ \( ( 220 - 30 T + T^{2} )^{2} \)
$97$ \( 16 + 28 T^{2} + T^{4} \)
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