Properties

Label 1575.1.h.d
Level 1575
Weight 1
Character orbit 1575.h
Self dual yes
Analytic conductor 0.786
Analytic rank 0
Dimension 2
Projective image \(D_{6}\)
CM discriminant -7
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.5788125.1

$q$-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.

\(f(q)\) \(=\) \( q -\beta q^{2} + 2 q^{4} - q^{7} -\beta q^{8} +O(q^{10})\) \( q -\beta q^{2} + 2 q^{4} - q^{7} -\beta q^{8} -\beta q^{11} + \beta q^{14} + q^{16} + 3 q^{22} + \beta q^{23} -2 q^{28} + \beta q^{29} + q^{37} + q^{43} -2 \beta q^{44} -3 q^{46} + q^{49} + \beta q^{56} -3 q^{58} - q^{64} - q^{67} + \beta q^{71} -\beta q^{74} + \beta q^{77} - q^{79} -\beta q^{86} + 3 q^{88} + 2 \beta q^{92} -\beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} - 2q^{7} + O(q^{10}) \) \( 2q + 4q^{4} - 2q^{7} + 2q^{16} + 6q^{22} - 4q^{28} + 2q^{37} + 2q^{43} - 6q^{46} + 2q^{49} - 6q^{58} - 2q^{64} - 2q^{67} - 2q^{79} + 6q^{88} + O(q^{100}) \)

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
1.73205
−1.73205
−1.73205 0 2.00000 0 0 −1.00000 −1.73205 0 0
1126.2 1.73205 0 2.00000 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
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 1575.1.h.d 2
3.b odd 2 1 inner 1575.1.h.d 2
5.b even 2 1 1575.1.h.e yes 2
5.c odd 4 2 1575.1.e.c 4
7.b odd 2 1 CM 1575.1.h.d 2
15.d odd 2 1 1575.1.h.e yes 2
15.e even 4 2 1575.1.e.c 4
21.c even 2 1 inner 1575.1.h.d 2
35.c odd 2 1 1575.1.h.e yes 2
35.f even 4 2 1575.1.e.c 4
105.g even 2 1 1575.1.h.e yes 2
105.k odd 4 2 1575.1.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.e.c 4 5.c odd 4 2
1575.1.e.c 4 15.e even 4 2
1575.1.e.c 4 35.f even 4 2
1575.1.e.c 4 105.k odd 4 2
1575.1.h.d 2 1.a even 1 1 trivial
1575.1.h.d 2 3.b odd 2 1 inner
1575.1.h.d 2 7.b odd 2 1 CM
1575.1.h.d 2 21.c even 2 1 inner
1575.1.h.e yes 2 5.b even 2 1
1575.1.h.e yes 2 15.d odd 2 1
1575.1.h.e yes 2 35.c odd 2 1
1575.1.h.e yes 2 105.g even 2 1

Hecke kernels

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

\( T_{2}^{2} - 3 \)
\( T_{37} - 1 \)

Hecke characteristic polynomials

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