Properties

Label 1575.1.e.b
Level 1575
Weight 1
Character orbit 1575.e
Analytic conductor 0.786
Analytic rank 0
Dimension 2
Projective image \(D_{2}\)
CM/RM discs -3, -7, 21
Inner twists 8

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 $D_4:C_2$
Artin field Galois closure of 8.0.558140625.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
874.1
1.00000i
1.00000i
0 0 1.00000 0 0 1.00000i 0 0 0
874.2 0 0 1.00000 0 0 1.00000i 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 \(\Q(\sqrt{21}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.e.b 2
3.b odd 2 1 CM 1575.1.e.b 2
5.b even 2 1 inner 1575.1.e.b 2
5.c odd 4 1 63.1.d.a 1
5.c odd 4 1 1575.1.h.b 1
7.b odd 2 1 CM 1575.1.e.b 2
15.d odd 2 1 inner 1575.1.e.b 2
15.e even 4 1 63.1.d.a 1
15.e even 4 1 1575.1.h.b 1
20.e even 4 1 1008.1.f.a 1
21.c even 2 1 RM 1575.1.e.b 2
35.c odd 2 1 inner 1575.1.e.b 2
35.f even 4 1 63.1.d.a 1
35.f even 4 1 1575.1.h.b 1
35.k even 12 2 441.1.m.a 2
35.l odd 12 2 441.1.m.a 2
45.k odd 12 2 567.1.l.b 2
45.l even 12 2 567.1.l.b 2
60.l odd 4 1 1008.1.f.a 1
105.g even 2 1 inner 1575.1.e.b 2
105.k odd 4 1 63.1.d.a 1
105.k odd 4 1 1575.1.h.b 1
105.w odd 12 2 441.1.m.a 2
105.x even 12 2 441.1.m.a 2
140.j odd 4 1 1008.1.f.a 1
315.bs even 12 2 3969.1.t.c 2
315.bt odd 12 2 3969.1.t.c 2
315.bu odd 12 2 3969.1.t.c 2
315.bv even 12 2 3969.1.t.c 2
315.bw odd 12 2 3969.1.k.b 2
315.bx even 12 2 3969.1.k.b 2
315.cb even 12 2 567.1.l.b 2
315.cf odd 12 2 567.1.l.b 2
315.cg even 12 2 3969.1.k.b 2
315.ch odd 12 2 3969.1.k.b 2
420.w even 4 1 1008.1.f.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.1.d.a 1 5.c odd 4 1
63.1.d.a 1 15.e even 4 1
63.1.d.a 1 35.f even 4 1
63.1.d.a 1 105.k odd 4 1
441.1.m.a 2 35.k even 12 2
441.1.m.a 2 35.l odd 12 2
441.1.m.a 2 105.w odd 12 2
441.1.m.a 2 105.x even 12 2
567.1.l.b 2 45.k odd 12 2
567.1.l.b 2 45.l even 12 2
567.1.l.b 2 315.cb even 12 2
567.1.l.b 2 315.cf odd 12 2
1008.1.f.a 1 20.e even 4 1
1008.1.f.a 1 60.l odd 4 1
1008.1.f.a 1 140.j odd 4 1
1008.1.f.a 1 420.w even 4 1
1575.1.e.b 2 1.a even 1 1 trivial
1575.1.e.b 2 3.b odd 2 1 CM
1575.1.e.b 2 5.b even 2 1 inner
1575.1.e.b 2 7.b odd 2 1 CM
1575.1.e.b 2 15.d odd 2 1 inner
1575.1.e.b 2 21.c even 2 1 RM
1575.1.e.b 2 35.c odd 2 1 inner
1575.1.e.b 2 105.g even 2 1 inner
1575.1.h.b 1 5.c odd 4 1
1575.1.h.b 1 15.e even 4 1
1575.1.h.b 1 35.f even 4 1
1575.1.h.b 1 105.k odd 4 1
3969.1.k.b 2 315.bw odd 12 2
3969.1.k.b 2 315.bx even 12 2
3969.1.k.b 2 315.cg even 12 2
3969.1.k.b 2 315.ch odd 12 2
3969.1.t.c 2 315.bs even 12 2
3969.1.t.c 2 315.bt odd 12 2
3969.1.t.c 2 315.bu odd 12 2
3969.1.t.c 2 315.bv even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$3$ 1
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$23$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$43$ \( ( 1 + T^{2} )^{2} \)
$47$ \( ( 1 + T^{2} )^{2} \)
$53$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$59$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$61$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$67$ \( ( 1 + T^{2} )^{2} \)
$71$ \( ( 1 + T^{2} )^{2} \)
$73$ \( ( 1 + T^{2} )^{2} \)
$79$ \( ( 1 + T )^{4} \)
$83$ \( ( 1 + T^{2} )^{2} \)
$89$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$97$ \( ( 1 + T^{2} )^{2} \)
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