Properties

Label 1575.2.d.d
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: \(\Q(i, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -3 q^{4} + \beta_{1} q^{7} + \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} -3 q^{4} + \beta_{1} q^{7} + \beta_{2} q^{8} + ( -2 + 2 \beta_{3} ) q^{11} + 2 \beta_{2} q^{13} + \beta_{3} q^{14} - q^{16} + 2 \beta_{1} q^{17} + ( -2 - 2 \beta_{3} ) q^{19} + ( -10 \beta_{1} + 2 \beta_{2} ) q^{22} + 4 \beta_{1} q^{23} + 10 q^{26} -3 \beta_{1} q^{28} -2 q^{29} + ( 6 + 2 \beta_{3} ) q^{31} + 3 \beta_{2} q^{32} + 2 \beta_{3} q^{34} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{37} + ( 10 \beta_{1} + 2 \beta_{2} ) q^{38} + 2 q^{41} + 4 \beta_{2} q^{43} + ( 6 - 6 \beta_{3} ) q^{44} + 4 \beta_{3} q^{46} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{47} - q^{49} -6 \beta_{2} q^{52} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{53} -\beta_{3} q^{56} + 2 \beta_{2} q^{58} + 4 \beta_{3} q^{59} -2 q^{61} + ( -10 \beta_{1} - 6 \beta_{2} ) q^{62} + 13 q^{64} -4 \beta_{1} q^{67} -6 \beta_{1} q^{68} + ( -10 - 2 \beta_{3} ) q^{71} + ( 8 \beta_{1} - 2 \beta_{2} ) q^{73} + ( 20 + 2 \beta_{3} ) q^{74} + ( 6 + 6 \beta_{3} ) q^{76} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -4 + 4 \beta_{3} ) q^{79} -2 \beta_{2} q^{82} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{83} + 20 q^{86} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{88} -2 q^{89} -2 \beta_{3} q^{91} -12 \beta_{1} q^{92} + ( 20 - 4 \beta_{3} ) q^{94} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{97} + \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{4} + O(q^{10}) \) \( 4q - 12q^{4} - 8q^{11} - 4q^{16} - 8q^{19} + 40q^{26} - 8q^{29} + 24q^{31} + 8q^{41} + 24q^{44} - 4q^{49} - 8q^{61} + 52q^{64} - 40q^{71} + 80q^{74} + 24q^{76} - 16q^{79} + 80q^{86} - 8q^{89} + 80q^{94} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

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
2.23607i 0 −3.00000 0 0 1.00000i 2.23607i 0 0
1324.2 2.23607i 0 −3.00000 0 0 1.00000i 2.23607i 0 0
1324.3 2.23607i 0 −3.00000 0 0 1.00000i 2.23607i 0 0
1324.4 2.23607i 0 −3.00000 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.d 4
3.b odd 2 1 525.2.d.c 4
5.b even 2 1 inner 1575.2.d.d 4
5.c odd 4 1 315.2.a.d 2
5.c odd 4 1 1575.2.a.r 2
15.d odd 2 1 525.2.d.c 4
15.e even 4 1 105.2.a.b 2
15.e even 4 1 525.2.a.g 2
20.e even 4 1 5040.2.a.bw 2
35.f even 4 1 2205.2.a.w 2
60.l odd 4 1 1680.2.a.v 2
60.l odd 4 1 8400.2.a.cx 2
105.k odd 4 1 735.2.a.k 2
105.k odd 4 1 3675.2.a.y 2
105.w odd 12 2 735.2.i.i 4
105.x even 12 2 735.2.i.k 4
120.q odd 4 1 6720.2.a.cs 2
120.w even 4 1 6720.2.a.cx 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.a.b 2 15.e even 4 1
315.2.a.d 2 5.c odd 4 1
525.2.a.g 2 15.e even 4 1
525.2.d.c 4 3.b odd 2 1
525.2.d.c 4 15.d odd 2 1
735.2.a.k 2 105.k odd 4 1
735.2.i.i 4 105.w odd 12 2
735.2.i.k 4 105.x even 12 2
1575.2.a.r 2 5.c odd 4 1
1575.2.d.d 4 1.a even 1 1 trivial
1575.2.d.d 4 5.b even 2 1 inner
1680.2.a.v 2 60.l odd 4 1
2205.2.a.w 2 35.f even 4 1
3675.2.a.y 2 105.k odd 4 1
5040.2.a.bw 2 20.e even 4 1
6720.2.a.cs 2 120.q odd 4 1
6720.2.a.cx 2 120.w even 4 1
8400.2.a.cx 2 60.l odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 5 \)
\( T_{11}^{2} + 4 T_{11} - 16 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} + 4 T^{4} )^{2} \)
$3$ 1
$5$ 1
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( 1 + 4 T + 6 T^{2} + 44 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 6 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 4 T + 22 T^{2} + 76 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 30 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 2 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 20 T^{2} + 1558 T^{4} + 27380 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 6 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 4 T^{2} - 698 T^{4} + 8836 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 44 T^{2} + 982 T^{4} - 123596 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 38 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 118 T^{2} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 20 T + 222 T^{2} + 1420 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 124 T^{2} + 9382 T^{4} - 660796 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 8 T + 94 T^{2} + 632 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 44 T^{2} - 6218 T^{4} - 303116 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 2 T + 89 T^{2} )^{4} \)
$97$ \( 1 - 316 T^{2} + 42502 T^{4} - 2973244 T^{6} + 88529281 T^{8} \)
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