Properties

Label 1475.1.d.a
Level $1475$
Weight $1$
Character orbit 1475.d
Analytic conductor $0.736$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -59
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1475 = 5^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1475.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.736120893634\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 59)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.59.1
Artin image $C_4\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{3} - q^{4} + i q^{7} +O(q^{10})\) \( q -i q^{3} - q^{4} + i q^{7} + i q^{12} + q^{16} -2 i q^{17} + q^{19} + q^{21} -i q^{27} -i q^{28} + q^{29} - q^{41} -i q^{48} -2 q^{51} -i q^{53} -i q^{57} - q^{59} - q^{64} + 2 i q^{68} + 2 q^{71} - q^{76} + q^{79} - q^{81} - q^{84} -i q^{87} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{16} + 2q^{19} + 2q^{21} + 2q^{29} - 2q^{41} - 4q^{51} - 2q^{59} - 2q^{64} + 4q^{71} - 2q^{76} + 2q^{79} - 2q^{81} - 2q^{84} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1475\mathbb{Z}\right)^\times\).

\(n\) \(651\) \(827\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1474.1
1.00000i
1.00000i
0 1.00000i −1.00000 0 0 1.00000i 0 0 0
1474.2 0 1.00000i −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
59.b odd 2 1 CM by \(\Q(\sqrt{-59}) \)
5.b even 2 1 inner
295.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1475.1.d.a 2
5.b even 2 1 inner 1475.1.d.a 2
5.c odd 4 1 59.1.b.a 1
5.c odd 4 1 1475.1.c.b 1
15.e even 4 1 531.1.c.a 1
20.e even 4 1 944.1.h.a 1
35.f even 4 1 2891.1.c.e 1
35.k even 12 2 2891.1.g.b 2
35.l odd 12 2 2891.1.g.d 2
40.i odd 4 1 3776.1.h.b 1
40.k even 4 1 3776.1.h.a 1
59.b odd 2 1 CM 1475.1.d.a 2
295.d odd 2 1 inner 1475.1.d.a 2
295.e even 4 1 59.1.b.a 1
295.e even 4 1 1475.1.c.b 1
295.k odd 116 28 3481.1.d.a 28
295.l even 116 28 3481.1.d.a 28
885.k odd 4 1 531.1.c.a 1
1180.l odd 4 1 944.1.h.a 1
2065.l odd 4 1 2891.1.c.e 1
2065.v even 12 2 2891.1.g.d 2
2065.x odd 12 2 2891.1.g.b 2
2360.q odd 4 1 3776.1.h.a 1
2360.w even 4 1 3776.1.h.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 5.c odd 4 1
59.1.b.a 1 295.e even 4 1
531.1.c.a 1 15.e even 4 1
531.1.c.a 1 885.k odd 4 1
944.1.h.a 1 20.e even 4 1
944.1.h.a 1 1180.l odd 4 1
1475.1.c.b 1 5.c odd 4 1
1475.1.c.b 1 295.e even 4 1
1475.1.d.a 2 1.a even 1 1 trivial
1475.1.d.a 2 5.b even 2 1 inner
1475.1.d.a 2 59.b odd 2 1 CM
1475.1.d.a 2 295.d odd 2 1 inner
2891.1.c.e 1 35.f even 4 1
2891.1.c.e 1 2065.l odd 4 1
2891.1.g.b 2 35.k even 12 2
2891.1.g.b 2 2065.x odd 12 2
2891.1.g.d 2 35.l odd 12 2
2891.1.g.d 2 2065.v even 12 2
3481.1.d.a 28 295.k odd 116 28
3481.1.d.a 28 295.l even 116 28
3776.1.h.a 1 40.k even 4 1
3776.1.h.a 1 2360.q odd 4 1
3776.1.h.b 1 40.i odd 4 1
3776.1.h.b 1 2360.w even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1475, [\chi])\).

Hecke characteristic polynomials

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