Properties

Label 1575.2.b.a
Level 1575
Weight 2
Character orbit 1575.b
Analytic conductor 12.576
Analytic rank 0
Dimension 4
CM discriminant -7
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: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -2 + \beta_{2} ) q^{4} + \beta_{2} q^{7} + ( -2 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -2 + \beta_{2} ) q^{4} + \beta_{2} q^{7} + ( -2 \beta_{1} + \beta_{3} ) q^{8} + ( \beta_{1} + \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{3} ) q^{14} + ( 3 - 2 \beta_{2} ) q^{16} + ( -5 - \beta_{2} ) q^{22} + ( -3 \beta_{1} + \beta_{3} ) q^{23} + ( 7 - 2 \beta_{2} ) q^{28} + ( -3 \beta_{1} - \beta_{3} ) q^{29} + 3 \beta_{1} q^{32} -4 \beta_{2} q^{37} + 2 \beta_{2} q^{43} + ( -\beta_{1} + \beta_{3} ) q^{44} + ( 11 - 5 \beta_{2} ) q^{46} + 7 q^{49} + ( 5 \beta_{1} - \beta_{3} ) q^{53} + 7 \beta_{1} q^{56} + ( 13 - \beta_{2} ) q^{58} + ( -6 - \beta_{2} ) q^{64} + 4 q^{67} + ( -\beta_{1} + 3 \beta_{3} ) q^{71} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{74} + ( \beta_{1} + 3 \beta_{3} ) q^{77} + 8 q^{79} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{86} + ( -7 - 5 \beta_{2} ) q^{88} + ( 15 \beta_{1} - 3 \beta_{3} ) q^{92} + 7 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 12q^{16} - 20q^{22} + 28q^{28} + 44q^{46} + 28q^{49} + 52q^{58} - 24q^{64} + 16q^{67} + 32q^{79} - 28q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 6 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 6 \beta_{1}\)

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
2.57794i
1.16372i
1.16372i
2.57794i
2.57794i 0 −4.64575 0 0 −2.64575 6.82058i 0 0
251.2 1.16372i 0 0.645751 0 0 2.64575 3.07892i 0 0
251.3 1.16372i 0 0.645751 0 0 2.64575 3.07892i 0 0
251.4 2.57794i 0 −4.64575 0 0 −2.64575 6.82058i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.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.a 4
3.b odd 2 1 inner 1575.2.b.a 4
5.b even 2 1 63.2.c.a 4
5.c odd 4 2 1575.2.g.d 8
7.b odd 2 1 CM 1575.2.b.a 4
15.d odd 2 1 63.2.c.a 4
15.e even 4 2 1575.2.g.d 8
20.d odd 2 1 1008.2.k.a 4
21.c even 2 1 inner 1575.2.b.a 4
35.c odd 2 1 63.2.c.a 4
35.f even 4 2 1575.2.g.d 8
35.i odd 6 2 441.2.p.b 8
35.j even 6 2 441.2.p.b 8
40.e odd 2 1 4032.2.k.b 4
40.f even 2 1 4032.2.k.c 4
45.h odd 6 2 567.2.o.f 8
45.j even 6 2 567.2.o.f 8
60.h even 2 1 1008.2.k.a 4
105.g even 2 1 63.2.c.a 4
105.k odd 4 2 1575.2.g.d 8
105.o odd 6 2 441.2.p.b 8
105.p even 6 2 441.2.p.b 8
120.i odd 2 1 4032.2.k.c 4
120.m even 2 1 4032.2.k.b 4
140.c even 2 1 1008.2.k.a 4
280.c odd 2 1 4032.2.k.c 4
280.n even 2 1 4032.2.k.b 4
315.z even 6 2 567.2.o.f 8
315.bg odd 6 2 567.2.o.f 8
420.o odd 2 1 1008.2.k.a 4
840.b odd 2 1 4032.2.k.b 4
840.u even 2 1 4032.2.k.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.c.a 4 5.b even 2 1
63.2.c.a 4 15.d odd 2 1
63.2.c.a 4 35.c odd 2 1
63.2.c.a 4 105.g even 2 1
441.2.p.b 8 35.i odd 6 2
441.2.p.b 8 35.j even 6 2
441.2.p.b 8 105.o odd 6 2
441.2.p.b 8 105.p even 6 2
567.2.o.f 8 45.h odd 6 2
567.2.o.f 8 45.j even 6 2
567.2.o.f 8 315.z even 6 2
567.2.o.f 8 315.bg odd 6 2
1008.2.k.a 4 20.d odd 2 1
1008.2.k.a 4 60.h even 2 1
1008.2.k.a 4 140.c even 2 1
1008.2.k.a 4 420.o odd 2 1
1575.2.b.a 4 1.a even 1 1 trivial
1575.2.b.a 4 3.b odd 2 1 inner
1575.2.b.a 4 7.b odd 2 1 CM
1575.2.b.a 4 21.c even 2 1 inner
1575.2.g.d 8 5.c odd 4 2
1575.2.g.d 8 15.e even 4 2
1575.2.g.d 8 35.f even 4 2
1575.2.g.d 8 105.k odd 4 2
4032.2.k.b 4 40.e odd 2 1
4032.2.k.b 4 120.m even 2 1
4032.2.k.b 4 280.n even 2 1
4032.2.k.b 4 840.b odd 2 1
4032.2.k.c 4 40.f even 2 1
4032.2.k.c 4 120.i odd 2 1
4032.2.k.c 4 280.c odd 2 1
4032.2.k.c 4 840.u even 2 1

Hecke kernels

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

\( T_{2}^{4} + 8 T_{2}^{2} + 9 \)
\( T_{37}^{2} - 112 \)
\( T_{47} \)
\( T_{67} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} + 16 T^{8} \)
$3$ 1
$5$ 1
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 - 206 T^{4} + 14641 T^{8} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( 1 - 734 T^{4} + 279841 T^{8} \)
$29$ \( 1 + 1234 T^{4} + 707281 T^{8} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 - 38 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 + 58 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( 1 - 5582 T^{4} + 7890481 T^{8} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{4} \)
$71$ \( 1 + 2914 T^{4} + 25411681 T^{8} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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