Properties

Label 1575.2.a.q
Level 1575
Weight 2
Character orbit 1575.a
Self dual yes
Analytic conductor 12.576
Analytic rank 1
Dimension 2
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.5764383184\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + q^{4} - q^{7} -\beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + q^{4} - q^{7} -\beta q^{8} -2 \beta q^{11} -2 q^{13} -\beta q^{14} -5 q^{16} + 2 \beta q^{17} -4 q^{19} -6 q^{22} -2 \beta q^{23} -2 \beta q^{26} - q^{28} -4 q^{31} -3 \beta q^{32} + 6 q^{34} -2 q^{37} -4 \beta q^{38} -6 \beta q^{41} + 4 q^{43} -2 \beta q^{44} -6 q^{46} + 4 \beta q^{47} + q^{49} -2 q^{52} -4 \beta q^{53} + \beta q^{56} + 4 \beta q^{59} -10 q^{61} -4 \beta q^{62} + q^{64} + 4 q^{67} + 2 \beta q^{68} + 6 \beta q^{71} -14 q^{73} -2 \beta q^{74} -4 q^{76} + 2 \beta q^{77} + 8 q^{79} -18 q^{82} + 4 \beta q^{86} + 6 q^{88} + 2 \beta q^{89} + 2 q^{91} -2 \beta q^{92} + 12 q^{94} -14 q^{97} + \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} - 2q^{7} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{7} - 4q^{13} - 10q^{16} - 8q^{19} - 12q^{22} - 2q^{28} - 8q^{31} + 12q^{34} - 4q^{37} + 8q^{43} - 12q^{46} + 2q^{49} - 4q^{52} - 20q^{61} + 2q^{64} + 8q^{67} - 28q^{73} - 8q^{76} + 16q^{79} - 36q^{82} + 12q^{88} + 4q^{91} + 24q^{94} - 28q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
−1.73205 0 1.00000 0 0 −1.00000 1.73205 0 0
1.2 1.73205 0 1.00000 0 0 −1.00000 −1.73205 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.a.q 2
3.b odd 2 1 inner 1575.2.a.q 2
5.b even 2 1 63.2.a.b 2
5.c odd 4 2 1575.2.d.i 4
15.d odd 2 1 63.2.a.b 2
15.e even 4 2 1575.2.d.i 4
20.d odd 2 1 1008.2.a.n 2
35.c odd 2 1 441.2.a.g 2
35.i odd 6 2 441.2.e.i 4
35.j even 6 2 441.2.e.j 4
40.e odd 2 1 4032.2.a.bq 2
40.f even 2 1 4032.2.a.bt 2
45.h odd 6 2 567.2.f.j 4
45.j even 6 2 567.2.f.j 4
55.d odd 2 1 7623.2.a.bi 2
60.h even 2 1 1008.2.a.n 2
105.g even 2 1 441.2.a.g 2
105.o odd 6 2 441.2.e.j 4
105.p even 6 2 441.2.e.i 4
120.i odd 2 1 4032.2.a.bt 2
120.m even 2 1 4032.2.a.bq 2
140.c even 2 1 7056.2.a.cm 2
165.d even 2 1 7623.2.a.bi 2
420.o odd 2 1 7056.2.a.cm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 5.b even 2 1
63.2.a.b 2 15.d odd 2 1
441.2.a.g 2 35.c odd 2 1
441.2.a.g 2 105.g even 2 1
441.2.e.i 4 35.i odd 6 2
441.2.e.i 4 105.p even 6 2
441.2.e.j 4 35.j even 6 2
441.2.e.j 4 105.o odd 6 2
567.2.f.j 4 45.h odd 6 2
567.2.f.j 4 45.j even 6 2
1008.2.a.n 2 20.d odd 2 1
1008.2.a.n 2 60.h even 2 1
1575.2.a.q 2 1.a even 1 1 trivial
1575.2.a.q 2 3.b odd 2 1 inner
1575.2.d.i 4 5.c odd 4 2
1575.2.d.i 4 15.e even 4 2
4032.2.a.bq 2 40.e odd 2 1
4032.2.a.bq 2 120.m even 2 1
4032.2.a.bt 2 40.f even 2 1
4032.2.a.bt 2 120.i odd 2 1
7056.2.a.cm 2 140.c even 2 1
7056.2.a.cm 2 420.o odd 2 1
7623.2.a.bi 2 55.d odd 2 1
7623.2.a.bi 2 165.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} - 3 \)
\( T_{11}^{2} - 12 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 4 T^{4} \)
$3$ 1
$5$ 1
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 + 10 T^{2} + 121 T^{4} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 22 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 4 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 34 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 - 26 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 46 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 58 T^{2} + 2809 T^{4} \)
$59$ \( 1 + 70 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 34 T^{2} + 5041 T^{4} \)
$73$ \( ( 1 + 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 + 83 T^{2} )^{2} \)
$89$ \( 1 + 166 T^{2} + 7921 T^{4} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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