Properties

Label 1575.1.e.c
Level 1575
Weight 1
Character orbit 1575.e
Analytic conductor 0.786
Analytic rank 0
Dimension 4
Projective image \(D_{6}\)
CM discriminant -7
Inner twists 8

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.5788125.1

$q$-expansion

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

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

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
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 0 0
874.2 1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 0 0
874.3 1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 0 0
874.4 1.73205i 0 −2.00000 0 0 1.00000i 1.73205i 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
5.b even 2 1 inner
15.d odd 2 1 inner
21.c even 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.c 4
3.b odd 2 1 inner 1575.1.e.c 4
5.b even 2 1 inner 1575.1.e.c 4
5.c odd 4 1 1575.1.h.d 2
5.c odd 4 1 1575.1.h.e yes 2
7.b odd 2 1 CM 1575.1.e.c 4
15.d odd 2 1 inner 1575.1.e.c 4
15.e even 4 1 1575.1.h.d 2
15.e even 4 1 1575.1.h.e yes 2
21.c even 2 1 inner 1575.1.e.c 4
35.c odd 2 1 inner 1575.1.e.c 4
35.f even 4 1 1575.1.h.d 2
35.f even 4 1 1575.1.h.e yes 2
105.g even 2 1 inner 1575.1.e.c 4
105.k odd 4 1 1575.1.h.d 2
105.k odd 4 1 1575.1.h.e yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.e.c 4 1.a even 1 1 trivial
1575.1.e.c 4 3.b odd 2 1 inner
1575.1.e.c 4 5.b even 2 1 inner
1575.1.e.c 4 7.b odd 2 1 CM
1575.1.e.c 4 15.d odd 2 1 inner
1575.1.e.c 4 21.c even 2 1 inner
1575.1.e.c 4 35.c odd 2 1 inner
1575.1.e.c 4 105.g even 2 1 inner
1575.1.h.d 2 5.c odd 4 1
1575.1.h.d 2 15.e even 4 1
1575.1.h.d 2 35.f even 4 1
1575.1.h.d 2 105.k odd 4 1
1575.1.h.e yes 2 5.c odd 4 1
1575.1.h.e yes 2 15.e even 4 1
1575.1.h.e yes 2 35.f even 4 1
1575.1.h.e yes 2 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1575, [\chi])\).

Hecke characteristic polynomials

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