Properties

Label 1575.1.n.b
Level 1575
Weight 1
Character orbit 1575.n
Analytic conductor 0.786
Analytic rank 0
Dimension 4
Projective image \(D_{4}\)
CM discriminant -7
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.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.786027394897\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.33075.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{8}^{2} ) q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} +O(q^{10})\) \( q + ( 1 + \zeta_{8}^{2} ) q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{7} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{14} + q^{16} -2 \zeta_{8}^{3} q^{22} + ( -1 + \zeta_{8}^{2} ) q^{23} + \zeta_{8} q^{28} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + ( 1 + \zeta_{8}^{2} ) q^{32} + 2 \zeta_{8}^{3} q^{37} + 2 \zeta_{8} q^{43} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{44} -2 q^{46} -\zeta_{8}^{2} q^{49} + ( -1 + \zeta_{8}^{2} ) q^{53} -2 \zeta_{8} q^{58} + \zeta_{8}^{2} q^{64} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{71} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{74} + ( -1 - \zeta_{8}^{2} ) q^{77} -2 \zeta_{8}^{2} q^{79} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{86} + ( -1 - \zeta_{8}^{2} ) q^{92} + ( 1 - \zeta_{8}^{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{16} - 4q^{23} + 4q^{32} - 8q^{46} - 4q^{53} - 4q^{77} - 4q^{92} + 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
818.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
1.00000 + 1.00000i 0 1.00000i 0 0 −0.707107 + 0.707107i 0 0 0
818.2 1.00000 + 1.00000i 0 1.00000i 0 0 0.707107 0.707107i 0 0 0
1007.1 1.00000 1.00000i 0 1.00000i 0 0 −0.707107 0.707107i 0 0 0
1007.2 1.00000 1.00000i 0 1.00000i 0 0 0.707107 + 0.707107i 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.c odd 4 1 inner
15.d odd 2 1 inner
15.e even 4 1 inner
35.f even 4 1 inner
105.g even 2 1 inner
105.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.1.n.b yes 4
3.b odd 2 1 1575.1.n.a 4
5.b even 2 1 1575.1.n.a 4
5.c odd 4 1 1575.1.n.a 4
5.c odd 4 1 inner 1575.1.n.b yes 4
7.b odd 2 1 CM 1575.1.n.b yes 4
15.d odd 2 1 inner 1575.1.n.b yes 4
15.e even 4 1 1575.1.n.a 4
15.e even 4 1 inner 1575.1.n.b yes 4
21.c even 2 1 1575.1.n.a 4
35.c odd 2 1 1575.1.n.a 4
35.f even 4 1 1575.1.n.a 4
35.f even 4 1 inner 1575.1.n.b yes 4
105.g even 2 1 inner 1575.1.n.b yes 4
105.k odd 4 1 1575.1.n.a 4
105.k odd 4 1 inner 1575.1.n.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1575.1.n.a 4 3.b odd 2 1
1575.1.n.a 4 5.b even 2 1
1575.1.n.a 4 5.c odd 4 1
1575.1.n.a 4 15.e even 4 1
1575.1.n.a 4 21.c even 2 1
1575.1.n.a 4 35.c odd 2 1
1575.1.n.a 4 35.f even 4 1
1575.1.n.a 4 105.k odd 4 1
1575.1.n.b yes 4 1.a even 1 1 trivial
1575.1.n.b yes 4 5.c odd 4 1 inner
1575.1.n.b yes 4 7.b odd 2 1 CM
1575.1.n.b yes 4 15.d odd 2 1 inner
1575.1.n.b yes 4 15.e even 4 1 inner
1575.1.n.b yes 4 35.f even 4 1 inner
1575.1.n.b yes 4 105.g even 2 1 inner
1575.1.n.b yes 4 105.k odd 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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