Properties

Label 3525.1.l.c
Level $3525$
Weight $1$
Character orbit 3525.l
Analytic conductor $1.759$
Analytic rank $0$
Dimension $16$
Projective image $D_{10}$
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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.0.3705507759375.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{40}^{13} + \zeta_{40}^{17} ) q^{2} -\zeta_{40}^{11} q^{3} + ( -\zeta_{40}^{6} + \zeta_{40}^{10} - \zeta_{40}^{14} ) q^{4} + ( -\zeta_{40}^{4} + \zeta_{40}^{8} ) q^{6} + ( \zeta_{40}^{13} + \zeta_{40}^{17} ) q^{7} + ( \zeta_{40}^{3} - \zeta_{40}^{7} + \zeta_{40}^{11} + \zeta_{40}^{19} ) q^{8} -\zeta_{40}^{2} q^{9} +O(q^{10})\) \( q + ( -\zeta_{40}^{13} + \zeta_{40}^{17} ) q^{2} -\zeta_{40}^{11} q^{3} + ( -\zeta_{40}^{6} + \zeta_{40}^{10} - \zeta_{40}^{14} ) q^{4} + ( -\zeta_{40}^{4} + \zeta_{40}^{8} ) q^{6} + ( \zeta_{40}^{13} + \zeta_{40}^{17} ) q^{7} + ( \zeta_{40}^{3} - \zeta_{40}^{7} + \zeta_{40}^{11} + \zeta_{40}^{19} ) q^{8} -\zeta_{40}^{2} q^{9} + ( \zeta_{40} - \zeta_{40}^{5} + \zeta_{40}^{17} ) q^{12} + ( \zeta_{40}^{6} - \zeta_{40}^{14} ) q^{14} + ( -1 + \zeta_{40}^{4} - \zeta_{40}^{8} + \zeta_{40}^{12} - \zeta_{40}^{16} ) q^{16} + ( \zeta_{40} + \zeta_{40}^{9} ) q^{17} + ( \zeta_{40}^{15} - \zeta_{40}^{19} ) q^{18} + ( \zeta_{40}^{4} + \zeta_{40}^{8} ) q^{21} + ( \zeta_{40}^{2} + \zeta_{40}^{10} - \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{24} + \zeta_{40}^{13} q^{27} + ( \zeta_{40}^{11} - \zeta_{40}^{19} ) q^{28} + ( -\zeta_{40} + 2 \zeta_{40}^{5} - \zeta_{40}^{9} + \zeta_{40}^{13} - \zeta_{40}^{17} ) q^{32} + ( \zeta_{40}^{2} - \zeta_{40}^{6} - \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{34} + ( \zeta_{40}^{8} - \zeta_{40}^{12} + \zeta_{40}^{16} ) q^{36} + ( \zeta_{40} - \zeta_{40}^{9} ) q^{37} + ( -\zeta_{40}^{5} - \zeta_{40}^{17} ) q^{42} + \zeta_{40}^{5} q^{47} + ( \zeta_{40}^{3} - \zeta_{40}^{7} + \zeta_{40}^{11} - \zeta_{40}^{15} + \zeta_{40}^{19} ) q^{48} + ( -\zeta_{40}^{6} - \zeta_{40}^{10} - \zeta_{40}^{14} ) q^{49} + ( 1 - \zeta_{40}^{12} ) q^{51} + ( -\zeta_{40}^{3} + \zeta_{40}^{7} ) q^{53} + ( \zeta_{40}^{6} - \zeta_{40}^{10} ) q^{54} + ( -\zeta_{40}^{8} - \zeta_{40}^{12} ) q^{56} + ( \zeta_{40}^{2} - \zeta_{40}^{18} ) q^{59} + ( -\zeta_{40}^{8} + \zeta_{40}^{12} ) q^{61} + ( -\zeta_{40}^{15} - \zeta_{40}^{19} ) q^{63} + ( -2 \zeta_{40}^{2} + \zeta_{40}^{6} - \zeta_{40}^{10} + \zeta_{40}^{14} - 2 \zeta_{40}^{18} ) q^{64} + ( \zeta_{40}^{3} - \zeta_{40}^{7} + \zeta_{40}^{11} - 2 \zeta_{40}^{15} + \zeta_{40}^{19} ) q^{68} + ( \zeta_{40}^{4} + \zeta_{40}^{16} ) q^{71} + ( \zeta_{40} - \zeta_{40}^{5} + \zeta_{40}^{9} - \zeta_{40}^{13} ) q^{72} + ( -\zeta_{40}^{2} + \zeta_{40}^{6} - \zeta_{40}^{14} + \zeta_{40}^{18} ) q^{74} + ( \zeta_{40}^{6} + \zeta_{40}^{14} ) q^{79} + \zeta_{40}^{4} q^{81} + 2 \zeta_{40}^{15} q^{83} + ( \zeta_{40}^{2} - \zeta_{40}^{10} ) q^{84} + ( \zeta_{40}^{2} - \zeta_{40}^{18} ) q^{89} + ( -\zeta_{40}^{2} - \zeta_{40}^{18} ) q^{94} + ( -1 + \zeta_{40}^{4} - \zeta_{40}^{8} + \zeta_{40}^{12} - 2 \zeta_{40}^{16} ) q^{96} + ( -\zeta_{40}^{13} - \zeta_{40}^{17} ) q^{97} + ( \zeta_{40}^{11} + \zeta_{40}^{19} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{6} + O(q^{10}) \) \( 16q - 8q^{6} - 12q^{36} + 12q^{51} + 8q^{61} + 4q^{81} + 4q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(1552\) \(2026\) \(2351\)
\(\chi(n)\) \(-\zeta_{40}^{10}\) \(-1\) \(-1\)

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
0.891007 0.453990i
−0.453990 + 0.891007i
0.987688 + 0.156434i
0.156434 + 0.987688i
−0.156434 0.987688i
−0.987688 0.156434i
0.453990 0.891007i
−0.891007 + 0.453990i
0.891007 + 0.453990i
−0.453990 0.891007i
0.987688 0.156434i
0.156434 0.987688i
−0.156434 + 0.987688i
−0.987688 + 0.156434i
0.453990 + 0.891007i
−0.891007 0.453990i
−1.14412 1.14412i −0.453990 0.891007i 1.61803i 0 −0.500000 + 1.53884i 0.831254 0.831254i 0.707107 0.707107i −0.587785 + 0.809017i 0
1268.2 −1.14412 1.14412i 0.891007 + 0.453990i 1.61803i 0 −0.500000 1.53884i −0.831254 + 0.831254i 0.707107 0.707107i 0.587785 + 0.809017i 0
1268.3 −0.437016 0.437016i 0.156434 0.987688i 0.618034i 0 −0.500000 + 0.363271i −1.34500 + 1.34500i −0.707107 + 0.707107i −0.951057 0.309017i 0
1268.4 −0.437016 0.437016i 0.987688 0.156434i 0.618034i 0 −0.500000 0.363271i 1.34500 1.34500i −0.707107 + 0.707107i 0.951057 0.309017i 0
1268.5 0.437016 + 0.437016i −0.987688 + 0.156434i 0.618034i 0 −0.500000 0.363271i −1.34500 + 1.34500i 0.707107 0.707107i 0.951057 0.309017i 0
1268.6 0.437016 + 0.437016i −0.156434 + 0.987688i 0.618034i 0 −0.500000 + 0.363271i 1.34500 1.34500i 0.707107 0.707107i −0.951057 0.309017i 0
1268.7 1.14412 + 1.14412i −0.891007 0.453990i 1.61803i 0 −0.500000 1.53884i 0.831254 0.831254i −0.707107 + 0.707107i 0.587785 + 0.809017i 0
1268.8 1.14412 + 1.14412i 0.453990 + 0.891007i 1.61803i 0 −0.500000 + 1.53884i −0.831254 + 0.831254i −0.707107 + 0.707107i −0.587785 + 0.809017i 0
1832.1 −1.14412 + 1.14412i −0.453990 + 0.891007i 1.61803i 0 −0.500000 1.53884i 0.831254 + 0.831254i 0.707107 + 0.707107i −0.587785 0.809017i 0
1832.2 −1.14412 + 1.14412i 0.891007 0.453990i 1.61803i 0 −0.500000 + 1.53884i −0.831254 0.831254i 0.707107 + 0.707107i 0.587785 0.809017i 0
1832.3 −0.437016 + 0.437016i 0.156434 + 0.987688i 0.618034i 0 −0.500000 0.363271i −1.34500 1.34500i −0.707107 0.707107i −0.951057 + 0.309017i 0
1832.4 −0.437016 + 0.437016i 0.987688 + 0.156434i 0.618034i 0 −0.500000 + 0.363271i 1.34500 + 1.34500i −0.707107 0.707107i 0.951057 + 0.309017i 0
1832.5 0.437016 0.437016i −0.987688 0.156434i 0.618034i 0 −0.500000 + 0.363271i −1.34500 1.34500i 0.707107 + 0.707107i 0.951057 + 0.309017i 0
1832.6 0.437016 0.437016i −0.156434 0.987688i 0.618034i 0 −0.500000 0.363271i 1.34500 + 1.34500i 0.707107 + 0.707107i −0.951057 + 0.309017i 0
1832.7 1.14412 1.14412i −0.891007 + 0.453990i 1.61803i 0 −0.500000 + 1.53884i 0.831254 + 0.831254i −0.707107 0.707107i 0.587785 0.809017i 0
1832.8 1.14412 1.14412i 0.453990 0.891007i 1.61803i 0 −0.500000 1.53884i −0.831254 0.831254i −0.707107 0.707107i −0.587785 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1832.8
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.c 16
3.b odd 2 1 inner 3525.1.l.c 16
5.b even 2 1 inner 3525.1.l.c 16
5.c odd 4 2 inner 3525.1.l.c 16
15.d odd 2 1 inner 3525.1.l.c 16
15.e even 4 2 inner 3525.1.l.c 16
47.b odd 2 1 CM 3525.1.l.c 16
141.c even 2 1 inner 3525.1.l.c 16
235.b odd 2 1 inner 3525.1.l.c 16
235.e even 4 2 inner 3525.1.l.c 16
705.g even 2 1 inner 3525.1.l.c 16
705.l odd 4 2 inner 3525.1.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3525.1.l.c 16 1.a even 1 1 trivial
3525.1.l.c 16 3.b odd 2 1 inner
3525.1.l.c 16 5.b even 2 1 inner
3525.1.l.c 16 5.c odd 4 2 inner
3525.1.l.c 16 15.d odd 2 1 inner
3525.1.l.c 16 15.e even 4 2 inner
3525.1.l.c 16 47.b odd 2 1 CM
3525.1.l.c 16 141.c even 2 1 inner
3525.1.l.c 16 235.b odd 2 1 inner
3525.1.l.c 16 235.e even 4 2 inner
3525.1.l.c 16 705.g even 2 1 inner
3525.1.l.c 16 705.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 7 T_{2}^{4} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3525, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 7 T^{4} + T^{8} )^{2} \)
$3$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$5$ \( T^{16} \)
$7$ \( ( 25 + 15 T^{4} + T^{8} )^{2} \)
$11$ \( T^{16} \)
$13$ \( T^{16} \)
$17$ \( ( 1 + 7 T^{4} + T^{8} )^{2} \)
$19$ \( T^{16} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( ( 25 + 15 T^{4} + T^{8} )^{2} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( ( 1 + T^{4} )^{4} \)
$53$ \( ( 1 + 7 T^{4} + T^{8} )^{2} \)
$59$ \( ( 5 - 5 T^{2} + T^{4} )^{4} \)
$61$ \( ( -1 - T + T^{2} )^{8} \)
$67$ \( T^{16} \)
$71$ \( ( 5 + 5 T^{2} + T^{4} )^{4} \)
$73$ \( T^{16} \)
$79$ \( ( 1 + 3 T^{2} + T^{4} )^{4} \)
$83$ \( ( 16 + T^{4} )^{4} \)
$89$ \( ( 5 - 5 T^{2} + T^{4} )^{4} \)
$97$ \( ( 25 + 15 T^{4} + T^{8} )^{2} \)
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