Properties

Label 1225.2.b.a
Level $1225$
Weight $2$
Character orbit 1225.b
Analytic conductor $9.782$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 3 i q^{3} -2 q^{4} -6 q^{6} -6 q^{9} +O(q^{10})\) \( q + 2 i q^{2} + 3 i q^{3} -2 q^{4} -6 q^{6} -6 q^{9} + q^{11} -6 i q^{12} + 3 i q^{13} -4 q^{16} + 3 i q^{17} -12 i q^{18} -6 q^{19} + 2 i q^{22} -4 i q^{23} -6 q^{26} -9 i q^{27} + q^{29} + 6 q^{31} -8 i q^{32} + 3 i q^{33} -6 q^{34} + 12 q^{36} -12 i q^{38} -9 q^{39} + 6 q^{41} -6 i q^{43} -2 q^{44} + 8 q^{46} + 9 i q^{47} -12 i q^{48} -9 q^{51} -6 i q^{52} -10 i q^{53} + 18 q^{54} -18 i q^{57} + 2 i q^{58} + 6 q^{59} + 12 i q^{62} + 8 q^{64} -6 q^{66} + 14 i q^{67} -6 i q^{68} + 12 q^{69} -8 q^{71} + 6 i q^{73} + 12 q^{76} -18 i q^{78} + q^{79} + 9 q^{81} + 12 i q^{82} + 12 i q^{83} + 12 q^{86} + 3 i q^{87} -12 q^{89} + 8 i q^{92} + 18 i q^{93} -18 q^{94} + 24 q^{96} + 15 i q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 12q^{6} - 12q^{9} + O(q^{10}) \) \( 2q - 4q^{4} - 12q^{6} - 12q^{9} + 2q^{11} - 8q^{16} - 12q^{19} - 12q^{26} + 2q^{29} + 12q^{31} - 12q^{34} + 24q^{36} - 18q^{39} + 12q^{41} - 4q^{44} + 16q^{46} - 18q^{51} + 36q^{54} + 12q^{59} + 16q^{64} - 12q^{66} + 24q^{69} - 16q^{71} + 24q^{76} + 2q^{79} + 18q^{81} + 24q^{86} - 24q^{89} - 36q^{94} + 48q^{96} - 12q^{99} + O(q^{100}) \)

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
2.00000i 3.00000i −2.00000 0 −6.00000 0 0 −6.00000 0
99.2 2.00000i 3.00000i −2.00000 0 −6.00000 0 0 −6.00000 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
5.b even 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.a 2
5.b even 2 1 inner 1225.2.b.a 2
5.c odd 4 1 245.2.a.b yes 1
5.c odd 4 1 1225.2.a.h 1
7.b odd 2 1 1225.2.b.b 2
15.e even 4 1 2205.2.a.l 1
20.e even 4 1 3920.2.a.a 1
35.c odd 2 1 1225.2.b.b 2
35.f even 4 1 245.2.a.a 1
35.f even 4 1 1225.2.a.j 1
35.k even 12 2 245.2.e.d 2
35.l odd 12 2 245.2.e.c 2
105.k odd 4 1 2205.2.a.j 1
140.j odd 4 1 3920.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 35.f even 4 1
245.2.a.b yes 1 5.c odd 4 1
245.2.e.c 2 35.l odd 12 2
245.2.e.d 2 35.k even 12 2
1225.2.a.h 1 5.c odd 4 1
1225.2.a.j 1 35.f even 4 1
1225.2.b.a 2 1.a even 1 1 trivial
1225.2.b.a 2 5.b even 2 1 inner
1225.2.b.b 2 7.b odd 2 1
1225.2.b.b 2 35.c odd 2 1
2205.2.a.j 1 105.k odd 4 1
2205.2.a.l 1 15.e even 4 1
3920.2.a.a 1 20.e even 4 1
3920.2.a.bj 1 140.j odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \)
\( T_{19} + 6 \)
\( T_{31} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -1 + T )^{2} \)
$13$ \( 9 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -1 + T )^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( 81 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( 196 + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( ( -1 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( 12 + T )^{2} \)
$97$ \( 225 + T^{2} \)
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