Properties

Label 1225.1.j.a
Level $1225$
Weight $1$
Character orbit 1225.j
Analytic conductor $0.611$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -7
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1225.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.611354640475\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.175.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} -\zeta_{12}^{4} q^{11} + \zeta_{12}^{2} q^{16} -\zeta_{12} q^{18} + \zeta_{12}^{3} q^{22} -\zeta_{12}^{5} q^{23} + q^{29} + \zeta_{12}^{5} q^{37} + \zeta_{12}^{3} q^{43} + \zeta_{12}^{4} q^{46} + 2 \zeta_{12} q^{53} + \zeta_{12}^{5} q^{58} - q^{64} + \zeta_{12} q^{67} - q^{71} -\zeta_{12}^{5} q^{72} -\zeta_{12}^{4} q^{74} -\zeta_{12}^{2} q^{79} + \zeta_{12}^{4} q^{81} -\zeta_{12}^{2} q^{86} -\zeta_{12} q^{88} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{9} + 2q^{11} + 2q^{16} + 4q^{29} - 2q^{46} - 4q^{64} - 4q^{71} + 2q^{74} - 2q^{79} - 2q^{81} - 2q^{86} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
374.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 + 0.500000i 0 0 0 0 0 1.00000i 0.500000 + 0.866025i 0
374.2 0.866025 0.500000i 0 0 0 0 0 1.00000i 0.500000 + 0.866025i 0
999.1 −0.866025 0.500000i 0 0 0 0 0 1.00000i 0.500000 0.866025i 0
999.2 0.866025 + 0.500000i 0 0 0 0 0 1.00000i 0.500000 0.866025i 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.b even 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.1.j.a 4
5.b even 2 1 inner 1225.1.j.a 4
5.c odd 4 1 1225.1.i.a 2
5.c odd 4 1 1225.1.i.b 2
7.b odd 2 1 CM 1225.1.j.a 4
7.c even 3 1 175.1.c.a 2
7.c even 3 1 inner 1225.1.j.a 4
7.d odd 6 1 175.1.c.a 2
7.d odd 6 1 inner 1225.1.j.a 4
21.g even 6 1 1575.1.e.a 2
21.h odd 6 1 1575.1.e.a 2
28.f even 6 1 2800.1.p.a 2
28.g odd 6 1 2800.1.p.a 2
35.c odd 2 1 inner 1225.1.j.a 4
35.f even 4 1 1225.1.i.a 2
35.f even 4 1 1225.1.i.b 2
35.i odd 6 1 175.1.c.a 2
35.i odd 6 1 inner 1225.1.j.a 4
35.j even 6 1 175.1.c.a 2
35.j even 6 1 inner 1225.1.j.a 4
35.k even 12 1 175.1.d.a 1
35.k even 12 1 175.1.d.b yes 1
35.k even 12 1 1225.1.i.a 2
35.k even 12 1 1225.1.i.b 2
35.l odd 12 1 175.1.d.a 1
35.l odd 12 1 175.1.d.b yes 1
35.l odd 12 1 1225.1.i.a 2
35.l odd 12 1 1225.1.i.b 2
105.o odd 6 1 1575.1.e.a 2
105.p even 6 1 1575.1.e.a 2
105.w odd 12 1 1575.1.h.a 1
105.w odd 12 1 1575.1.h.c 1
105.x even 12 1 1575.1.h.a 1
105.x even 12 1 1575.1.h.c 1
140.p odd 6 1 2800.1.p.a 2
140.s even 6 1 2800.1.p.a 2
140.w even 12 1 2800.1.f.a 1
140.w even 12 1 2800.1.f.b 1
140.x odd 12 1 2800.1.f.a 1
140.x odd 12 1 2800.1.f.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.1.c.a 2 7.c even 3 1
175.1.c.a 2 7.d odd 6 1
175.1.c.a 2 35.i odd 6 1
175.1.c.a 2 35.j even 6 1
175.1.d.a 1 35.k even 12 1
175.1.d.a 1 35.l odd 12 1
175.1.d.b yes 1 35.k even 12 1
175.1.d.b yes 1 35.l odd 12 1
1225.1.i.a 2 5.c odd 4 1
1225.1.i.a 2 35.f even 4 1
1225.1.i.a 2 35.k even 12 1
1225.1.i.a 2 35.l odd 12 1
1225.1.i.b 2 5.c odd 4 1
1225.1.i.b 2 35.f even 4 1
1225.1.i.b 2 35.k even 12 1
1225.1.i.b 2 35.l odd 12 1
1225.1.j.a 4 1.a even 1 1 trivial
1225.1.j.a 4 5.b even 2 1 inner
1225.1.j.a 4 7.b odd 2 1 CM
1225.1.j.a 4 7.c even 3 1 inner
1225.1.j.a 4 7.d odd 6 1 inner
1225.1.j.a 4 35.c odd 2 1 inner
1225.1.j.a 4 35.i odd 6 1 inner
1225.1.j.a 4 35.j even 6 1 inner
1575.1.e.a 2 21.g even 6 1
1575.1.e.a 2 21.h odd 6 1
1575.1.e.a 2 105.o odd 6 1
1575.1.e.a 2 105.p even 6 1
1575.1.h.a 1 105.w odd 12 1
1575.1.h.a 1 105.x even 12 1
1575.1.h.c 1 105.w odd 12 1
1575.1.h.c 1 105.x even 12 1
2800.1.f.a 1 140.w even 12 1
2800.1.f.a 1 140.x odd 12 1
2800.1.f.b 1 140.w even 12 1
2800.1.f.b 1 140.x odd 12 1
2800.1.p.a 2 28.f even 6 1
2800.1.p.a 2 28.g odd 6 1
2800.1.p.a 2 140.p odd 6 1
2800.1.p.a 2 140.s even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1225, [\chi])\).

Hecke characteristic polynomials

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