Properties

Label 1575.1.h.b
Level 1575
Weight 1
Character orbit 1575.h
Self dual yes
Analytic conductor 0.786
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -3, -7, 21
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\)
Artin image $D_4$
Artin field Galois closure of 4.0.4725.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{4} + q^{7} + O(q^{10}) \) \( q - q^{4} + q^{7} + q^{16} - q^{28} + 2q^{37} + 2q^{43} + q^{49} - q^{64} - 2q^{67} + 2q^{79} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1126.1
0
0 0 −1.00000 0 0 1.00000 0 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 CM by \(\Q(\sqrt{-3}) \)
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
21.c even 2 1 RM by \(\Q(\sqrt{21}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.h.b 1
3.b odd 2 1 CM 1575.1.h.b 1
5.b even 2 1 63.1.d.a 1
5.c odd 4 2 1575.1.e.b 2
7.b odd 2 1 CM 1575.1.h.b 1
15.d odd 2 1 63.1.d.a 1
15.e even 4 2 1575.1.e.b 2
20.d odd 2 1 1008.1.f.a 1
21.c even 2 1 RM 1575.1.h.b 1
35.c odd 2 1 63.1.d.a 1
35.f even 4 2 1575.1.e.b 2
35.i odd 6 2 441.1.m.a 2
35.j even 6 2 441.1.m.a 2
45.h odd 6 2 567.1.l.b 2
45.j even 6 2 567.1.l.b 2
60.h even 2 1 1008.1.f.a 1
105.g even 2 1 63.1.d.a 1
105.k odd 4 2 1575.1.e.b 2
105.o odd 6 2 441.1.m.a 2
105.p even 6 2 441.1.m.a 2
140.c even 2 1 1008.1.f.a 1
315.q odd 6 2 3969.1.t.c 2
315.r even 6 2 3969.1.t.c 2
315.u even 6 2 3969.1.k.b 2
315.v odd 6 2 3969.1.k.b 2
315.z even 6 2 567.1.l.b 2
315.bg odd 6 2 567.1.l.b 2
315.bn odd 6 2 3969.1.k.b 2
315.bo even 6 2 3969.1.k.b 2
315.bq even 6 2 3969.1.t.c 2
315.br odd 6 2 3969.1.t.c 2
420.o odd 2 1 1008.1.f.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.1.d.a 1 5.b even 2 1
63.1.d.a 1 15.d odd 2 1
63.1.d.a 1 35.c odd 2 1
63.1.d.a 1 105.g even 2 1
441.1.m.a 2 35.i odd 6 2
441.1.m.a 2 35.j even 6 2
441.1.m.a 2 105.o odd 6 2
441.1.m.a 2 105.p even 6 2
567.1.l.b 2 45.h odd 6 2
567.1.l.b 2 45.j even 6 2
567.1.l.b 2 315.z even 6 2
567.1.l.b 2 315.bg odd 6 2
1008.1.f.a 1 20.d odd 2 1
1008.1.f.a 1 60.h even 2 1
1008.1.f.a 1 140.c even 2 1
1008.1.f.a 1 420.o odd 2 1
1575.1.e.b 2 5.c odd 4 2
1575.1.e.b 2 15.e even 4 2
1575.1.e.b 2 35.f even 4 2
1575.1.e.b 2 105.k odd 4 2
1575.1.h.b 1 1.a even 1 1 trivial
1575.1.h.b 1 3.b odd 2 1 CM
1575.1.h.b 1 7.b odd 2 1 CM
1575.1.h.b 1 21.c even 2 1 RM
3969.1.k.b 2 315.u even 6 2
3969.1.k.b 2 315.v odd 6 2
3969.1.k.b 2 315.bn odd 6 2
3969.1.k.b 2 315.bo even 6 2
3969.1.t.c 2 315.q odd 6 2
3969.1.t.c 2 315.r even 6 2
3969.1.t.c 2 315.bq even 6 2
3969.1.t.c 2 315.br odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2} \)
\( T_{37} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 - T \)
$11$ \( 1 + T^{2} \)
$13$ \( ( 1 - T )( 1 + T ) \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( ( 1 - T )( 1 + T ) \)
$23$ \( 1 + T^{2} \)
$29$ \( 1 + T^{2} \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( ( 1 - T )^{2} \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 - T )^{2} \)
$47$ \( ( 1 - T )( 1 + T ) \)
$53$ \( 1 + T^{2} \)
$59$ \( ( 1 - T )( 1 + T ) \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( ( 1 + T )^{2} \)
$71$ \( 1 + T^{2} \)
$73$ \( ( 1 - T )( 1 + T ) \)
$79$ \( ( 1 - T )^{2} \)
$83$ \( ( 1 - T )( 1 + T ) \)
$89$ \( ( 1 - T )( 1 + T ) \)
$97$ \( ( 1 - T )( 1 + T ) \)
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