Properties

Label 3525.1.l.b
Level $3525$
Weight $1$
Character orbit 3525.l
Analytic conductor $1.759$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -47
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3525.l (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.75920416953\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.186384375.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{24}^{9} q^{2} -\zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{3} q^{8} -\zeta_{24}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{24}^{9} q^{2} -\zeta_{24}^{7} q^{3} + \zeta_{24}^{4} q^{6} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{7} -\zeta_{24}^{3} q^{8} -\zeta_{24}^{2} q^{9} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{14} + q^{16} + \zeta_{24}^{9} q^{17} -\zeta_{24}^{11} q^{18} + ( -1 + \zeta_{24}^{8} ) q^{21} + \zeta_{24}^{10} q^{24} + \zeta_{24}^{9} q^{27} -\zeta_{24}^{6} q^{34} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{42} + \zeta_{24}^{9} q^{47} -\zeta_{24}^{7} q^{48} + ( \zeta_{24}^{2} + \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{49} + \zeta_{24}^{4} q^{51} + 2 \zeta_{24}^{3} q^{53} -\zeta_{24}^{6} q^{54} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{56} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{59} -2 q^{61} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{63} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{71} + \zeta_{24}^{5} q^{72} + 2 \zeta_{24}^{6} q^{79} + \zeta_{24}^{4} q^{81} -\zeta_{24}^{3} q^{83} -\zeta_{24}^{6} q^{94} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{6} + O(q^{10}) \) \( 8 q + 4 q^{6} + 8 q^{16} - 12 q^{21} + 4 q^{51} - 16 q^{61} + 4 q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3525\mathbb{Z}\right)^\times\).

\(n\) \(1552\) \(2026\) \(2351\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1268.1
−0.258819 + 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.707107 0.707107i −0.965926 0.258819i 0 0 0.500000 + 0.866025i 1.22474 1.22474i −0.707107 + 0.707107i 0.866025 + 0.500000i 0
1268.2 −0.707107 0.707107i 0.258819 + 0.965926i 0 0 0.500000 0.866025i −1.22474 + 1.22474i −0.707107 + 0.707107i −0.866025 + 0.500000i 0
1268.3 0.707107 + 0.707107i −0.258819 0.965926i 0 0 0.500000 0.866025i 1.22474 1.22474i 0.707107 0.707107i −0.866025 + 0.500000i 0
1268.4 0.707107 + 0.707107i 0.965926 + 0.258819i 0 0 0.500000 + 0.866025i −1.22474 + 1.22474i 0.707107 0.707107i 0.866025 + 0.500000i 0
1832.1 −0.707107 + 0.707107i −0.965926 + 0.258819i 0 0 0.500000 0.866025i 1.22474 + 1.22474i −0.707107 0.707107i 0.866025 0.500000i 0
1832.2 −0.707107 + 0.707107i 0.258819 0.965926i 0 0 0.500000 + 0.866025i −1.22474 1.22474i −0.707107 0.707107i −0.866025 0.500000i 0
1832.3 0.707107 0.707107i −0.258819 + 0.965926i 0 0 0.500000 + 0.866025i 1.22474 + 1.22474i 0.707107 + 0.707107i −0.866025 0.500000i 0
1832.4 0.707107 0.707107i 0.965926 0.258819i 0 0 0.500000 0.866025i −1.22474 1.22474i 0.707107 + 0.707107i 0.866025 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1832.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner
141.c even 2 1 inner
235.b odd 2 1 inner
235.e even 4 2 inner
705.g even 2 1 inner
705.l odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3525.1.l.b 8
3.b odd 2 1 inner 3525.1.l.b 8
5.b even 2 1 inner 3525.1.l.b 8
5.c odd 4 2 inner 3525.1.l.b 8
15.d odd 2 1 inner 3525.1.l.b 8
15.e even 4 2 inner 3525.1.l.b 8
47.b odd 2 1 CM 3525.1.l.b 8
141.c even 2 1 inner 3525.1.l.b 8
235.b odd 2 1 inner 3525.1.l.b 8
235.e even 4 2 inner 3525.1.l.b 8
705.g even 2 1 inner 3525.1.l.b 8
705.l odd 4 2 inner 3525.1.l.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.l.b 8 1.a even 1 1 trivial
3525.1.l.b 8 3.b odd 2 1 inner
3525.1.l.b 8 5.b even 2 1 inner
3525.1.l.b 8 5.c odd 4 2 inner
3525.1.l.b 8 15.d odd 2 1 inner
3525.1.l.b 8 15.e even 4 2 inner
3525.1.l.b 8 47.b odd 2 1 CM
3525.1.l.b 8 141.c even 2 1 inner
3525.1.l.b 8 235.b odd 2 1 inner
3525.1.l.b 8 235.e even 4 2 inner
3525.1.l.b 8 705.g even 2 1 inner
3525.1.l.b 8 705.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3525, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( 1 - T^{4} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 9 + T^{4} )^{2} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( ( 1 + T^{4} )^{2} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( ( 1 + T^{4} )^{2} \)
$53$ \( ( 16 + T^{4} )^{2} \)
$59$ \( ( -3 + T^{2} )^{4} \)
$61$ \( ( 2 + T )^{8} \)
$67$ \( T^{8} \)
$71$ \( ( 3 + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( ( 4 + T^{2} )^{4} \)
$83$ \( ( 1 + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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