Properties

Label 1225.1.q.b
Level $1225$
Weight $1$
Character orbit 1225.q
Analytic conductor $0.611$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -7
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1225.q (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.611354640475\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.2.153125.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{2} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{4} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{8} -\zeta_{24}^{10} q^{9} +O(q^{10})\) \( q + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{2} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{4} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{8} -\zeta_{24}^{10} q^{9} -\zeta_{24}^{8} q^{11} + \zeta_{24}^{4} q^{16} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{18} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{22} + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{23} + \zeta_{24}^{6} q^{29} + ( -1 - \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{36} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{37} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{43} + ( -\zeta_{24}^{2} + \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{44} + ( -1 + \zeta_{24}^{4} + 2 \zeta_{24}^{8} ) q^{46} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{58} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24} + \zeta_{24}^{9} ) q^{67} + q^{71} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{72} + ( 2 \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{10} ) q^{74} + \zeta_{24}^{10} q^{79} -\zeta_{24}^{8} q^{81} + ( -1 - 2 \zeta_{24}^{4} - \zeta_{24}^{8} ) q^{86} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{88} + ( -2 \zeta_{24} - 2 \zeta_{24}^{5} ) q^{92} -\zeta_{24}^{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{11} + 4q^{16} - 16q^{36} - 12q^{46} + 8q^{71} + 4q^{81} - 12q^{86} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
−0.258819 + 0.965926i
0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−1.67303 0.448288i 0 1.73205 + 1.00000i 0 0 0 −1.22474 1.22474i −0.866025 + 0.500000i 0
18.2 1.67303 + 0.448288i 0 1.73205 + 1.00000i 0 0 0 1.22474 + 1.22474i −0.866025 + 0.500000i 0
557.1 −0.448288 + 1.67303i 0 −1.73205 1.00000i 0 0 0 1.22474 1.22474i 0.866025 0.500000i 0
557.2 0.448288 1.67303i 0 −1.73205 1.00000i 0 0 0 −1.22474 + 1.22474i 0.866025 0.500000i 0
618.1 −0.448288 1.67303i 0 −1.73205 + 1.00000i 0 0 0 1.22474 + 1.22474i 0.866025 + 0.500000i 0
618.2 0.448288 + 1.67303i 0 −1.73205 + 1.00000i 0 0 0 −1.22474 1.22474i 0.866025 + 0.500000i 0
1157.1 −1.67303 + 0.448288i 0 1.73205 1.00000i 0 0 0 −1.22474 + 1.22474i −0.866025 0.500000i 0
1157.2 1.67303 0.448288i 0 1.73205 1.00000i 0 0 0 1.22474 1.22474i −0.866025 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1157.2
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.b even 2 1 inner
5.c odd 4 2 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner
35.i odd 6 1 inner
35.j even 6 1 inner
35.k even 12 2 inner
35.l odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.1.q.b 8
5.b even 2 1 inner 1225.1.q.b 8
5.c odd 4 2 inner 1225.1.q.b 8
7.b odd 2 1 CM 1225.1.q.b 8
7.c even 3 1 1225.1.g.b 4
7.c even 3 1 inner 1225.1.q.b 8
7.d odd 6 1 1225.1.g.b 4
7.d odd 6 1 inner 1225.1.q.b 8
35.c odd 2 1 inner 1225.1.q.b 8
35.f even 4 2 inner 1225.1.q.b 8
35.i odd 6 1 1225.1.g.b 4
35.i odd 6 1 inner 1225.1.q.b 8
35.j even 6 1 1225.1.g.b 4
35.j even 6 1 inner 1225.1.q.b 8
35.k even 12 2 1225.1.g.b 4
35.k even 12 2 inner 1225.1.q.b 8
35.l odd 12 2 1225.1.g.b 4
35.l odd 12 2 inner 1225.1.q.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.1.g.b 4 7.c even 3 1
1225.1.g.b 4 7.d odd 6 1
1225.1.g.b 4 35.i odd 6 1
1225.1.g.b 4 35.j even 6 1
1225.1.g.b 4 35.k even 12 2
1225.1.g.b 4 35.l odd 12 2
1225.1.q.b 8 1.a even 1 1 trivial
1225.1.q.b 8 5.b even 2 1 inner
1225.1.q.b 8 5.c odd 4 2 inner
1225.1.q.b 8 7.b odd 2 1 CM
1225.1.q.b 8 7.c even 3 1 inner
1225.1.q.b 8 7.d odd 6 1 inner
1225.1.q.b 8 35.c odd 2 1 inner
1225.1.q.b 8 35.f even 4 2 inner
1225.1.q.b 8 35.i odd 6 1 inner
1225.1.q.b 8 35.j even 6 1 inner
1225.1.q.b 8 35.k even 12 2 inner
1225.1.q.b 8 35.l odd 12 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 9 T_{2}^{4} + 81 \) acting on \(S_{1}^{\mathrm{new}}(1225, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 81 - 9 T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 1 - T + T^{2} )^{4} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( 81 - 9 T^{4} + T^{8} \)
$29$ \( ( 1 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( 81 - 9 T^{4} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( ( 9 + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( 81 - 9 T^{4} + T^{8} \)
$71$ \( ( -1 + T )^{8} \)
$73$ \( T^{8} \)
$79$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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