Properties

Label 1225.2.b.c
Level $1225$
Weight $2$
Character orbit 1225.b
Analytic conductor $9.782$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9} +O(q^{10})\) \( q + i q^{2} + q^{4} + 3 i q^{8} + 3 q^{9} + 4 q^{11} - q^{16} + 3 i q^{18} + 4 i q^{22} -8 i q^{23} -2 q^{29} + 5 i q^{32} + 3 q^{36} -6 i q^{37} + 12 i q^{43} + 4 q^{44} + 8 q^{46} + 10 i q^{53} -2 i q^{58} -7 q^{64} + 4 i q^{67} + 16 q^{71} + 9 i q^{72} + 6 q^{74} -8 q^{79} + 9 q^{81} -12 q^{86} + 12 i q^{88} -8 i q^{92} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{4} + 6 q^{9} + 8 q^{11} - 2 q^{16} - 4 q^{29} + 6 q^{36} + 8 q^{44} + 16 q^{46} - 14 q^{64} + 32 q^{71} + 12 q^{74} - 16 q^{79} + 18 q^{81} - 24 q^{86} + 24 q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
1.00000i 0 1.00000 0 0 0 3.00000i 3.00000 0
99.2 1.00000i 0 1.00000 0 0 0 3.00000i 3.00000 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
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.2.b.c 2
5.b even 2 1 inner 1225.2.b.c 2
5.c odd 4 1 49.2.a.a 1
5.c odd 4 1 1225.2.a.c 1
7.b odd 2 1 CM 1225.2.b.c 2
15.e even 4 1 441.2.a.c 1
20.e even 4 1 784.2.a.f 1
35.c odd 2 1 inner 1225.2.b.c 2
35.f even 4 1 49.2.a.a 1
35.f even 4 1 1225.2.a.c 1
35.k even 12 2 49.2.c.a 2
35.l odd 12 2 49.2.c.a 2
40.i odd 4 1 3136.2.a.n 1
40.k even 4 1 3136.2.a.o 1
55.e even 4 1 5929.2.a.c 1
60.l odd 4 1 7056.2.a.bg 1
65.h odd 4 1 8281.2.a.d 1
105.k odd 4 1 441.2.a.c 1
105.w odd 12 2 441.2.e.d 2
105.x even 12 2 441.2.e.d 2
140.j odd 4 1 784.2.a.f 1
140.w even 12 2 784.2.i.f 2
140.x odd 12 2 784.2.i.f 2
280.s even 4 1 3136.2.a.n 1
280.y odd 4 1 3136.2.a.o 1
385.l odd 4 1 5929.2.a.c 1
420.w even 4 1 7056.2.a.bg 1
455.s even 4 1 8281.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.2.a.a 1 5.c odd 4 1
49.2.a.a 1 35.f even 4 1
49.2.c.a 2 35.k even 12 2
49.2.c.a 2 35.l odd 12 2
441.2.a.c 1 15.e even 4 1
441.2.a.c 1 105.k odd 4 1
441.2.e.d 2 105.w odd 12 2
441.2.e.d 2 105.x even 12 2
784.2.a.f 1 20.e even 4 1
784.2.a.f 1 140.j odd 4 1
784.2.i.f 2 140.w even 12 2
784.2.i.f 2 140.x odd 12 2
1225.2.a.c 1 5.c odd 4 1
1225.2.a.c 1 35.f even 4 1
1225.2.b.c 2 1.a even 1 1 trivial
1225.2.b.c 2 5.b even 2 1 inner
1225.2.b.c 2 7.b odd 2 1 CM
1225.2.b.c 2 35.c odd 2 1 inner
3136.2.a.n 1 40.i odd 4 1
3136.2.a.n 1 280.s even 4 1
3136.2.a.o 1 40.k even 4 1
3136.2.a.o 1 280.y odd 4 1
5929.2.a.c 1 55.e even 4 1
5929.2.a.c 1 385.l odd 4 1
7056.2.a.bg 1 60.l odd 4 1
7056.2.a.bg 1 420.w even 4 1
8281.2.a.d 1 65.h odd 4 1
8281.2.a.d 1 455.s even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1225, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{19} \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -4 + T )^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( ( 2 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 36 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -16 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( ( 8 + T )^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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