Properties

Label 3525.1.l.a
Level $3525$
Weight $1$
Character orbit 3525.l
Analytic conductor $1.759$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -15, -47, 705
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-15}, \sqrt{-47})\)
Artin image: $OD_{16}:C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -2 \zeta_{8} q^{2} + \zeta_{8}^{3} q^{3} + 3 \zeta_{8}^{2} q^{4} + 2 q^{6} -4 \zeta_{8}^{3} q^{8} -\zeta_{8}^{2} q^{9} +O(q^{10})\) \( q -2 \zeta_{8} q^{2} + \zeta_{8}^{3} q^{3} + 3 \zeta_{8}^{2} q^{4} + 2 q^{6} -4 \zeta_{8}^{3} q^{8} -\zeta_{8}^{2} q^{9} -3 \zeta_{8} q^{12} -5 q^{16} -2 \zeta_{8} q^{17} + 2 \zeta_{8}^{3} q^{18} + 4 \zeta_{8}^{2} q^{24} + \zeta_{8} q^{27} + 6 \zeta_{8} q^{32} + 4 \zeta_{8}^{2} q^{34} + 3 q^{36} + \zeta_{8} q^{47} -5 \zeta_{8}^{3} q^{48} + \zeta_{8}^{2} q^{49} + 2 q^{51} + 2 \zeta_{8}^{3} q^{53} -2 \zeta_{8}^{2} q^{54} -2 q^{61} -7 \zeta_{8}^{2} q^{64} -6 \zeta_{8}^{3} q^{68} -4 \zeta_{8} q^{72} -2 \zeta_{8}^{2} q^{79} - q^{81} + 2 \zeta_{8}^{3} q^{83} -2 \zeta_{8}^{2} q^{94} -6 q^{96} -2 \zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{6} + O(q^{10}) \) \( 4q + 8q^{6} - 20q^{16} + 12q^{36} + 8q^{51} - 8q^{61} - 4q^{81} - 24q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(1552\) \(2026\) \(2351\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1268.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−1.41421 1.41421i −0.707107 + 0.707107i 3.00000i 0 2.00000 0 2.82843 2.82843i 1.00000i 0
1268.2 1.41421 + 1.41421i 0.707107 0.707107i 3.00000i 0 2.00000 0 −2.82843 + 2.82843i 1.00000i 0
1832.1 −1.41421 + 1.41421i −0.707107 0.707107i 3.00000i 0 2.00000 0 2.82843 + 2.82843i 1.00000i 0
1832.2 1.41421 1.41421i 0.707107 + 0.707107i 3.00000i 0 2.00000 0 −2.82843 2.82843i 1.00000i 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
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)
705.g even 2 1 RM by \(\Q(\sqrt{705}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 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.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.a 4
3.b odd 2 1 inner 3525.1.l.a 4
5.b even 2 1 inner 3525.1.l.a 4
5.c odd 4 2 inner 3525.1.l.a 4
15.d odd 2 1 CM 3525.1.l.a 4
15.e even 4 2 inner 3525.1.l.a 4
47.b odd 2 1 CM 3525.1.l.a 4
141.c even 2 1 inner 3525.1.l.a 4
235.b odd 2 1 inner 3525.1.l.a 4
235.e even 4 2 inner 3525.1.l.a 4
705.g even 2 1 RM 3525.1.l.a 4
705.l odd 4 2 inner 3525.1.l.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.l.a 4 1.a even 1 1 trivial
3525.1.l.a 4 3.b odd 2 1 inner
3525.1.l.a 4 5.b even 2 1 inner
3525.1.l.a 4 5.c odd 4 2 inner
3525.1.l.a 4 15.d odd 2 1 CM
3525.1.l.a 4 15.e even 4 2 inner
3525.1.l.a 4 47.b odd 2 1 CM
3525.1.l.a 4 141.c even 2 1 inner
3525.1.l.a 4 235.b odd 2 1 inner
3525.1.l.a 4 235.e even 4 2 inner
3525.1.l.a 4 705.g even 2 1 RM
3525.1.l.a 4 705.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 16 \) acting on \(S_{1}^{\mathrm{new}}(3525, [\chi])\).

Hecke characteristic polynomials

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