Properties

Label 2450.2.c.q
Level 2450
Weight 2
Character orbit 2450.c
Analytic conductor 19.563
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q -i q^{2} + 2 i q^{3} - q^{4} + 2 q^{6} + i q^{8} - q^{9} +O(q^{10})\) \( q -i q^{2} + 2 i q^{3} - q^{4} + 2 q^{6} + i q^{8} - q^{9} + 3 q^{11} -2 i q^{12} + i q^{13} + q^{16} -6 i q^{17} + i q^{18} + q^{19} -3 i q^{22} -9 i q^{23} -2 q^{24} + q^{26} + 4 i q^{27} -6 q^{29} + 8 q^{31} -i q^{32} + 6 i q^{33} -6 q^{34} + q^{36} -7 i q^{37} -i q^{38} -2 q^{39} + 3 q^{41} -2 i q^{43} -3 q^{44} -9 q^{46} + 9 i q^{47} + 2 i q^{48} + 12 q^{51} -i q^{52} -9 i q^{53} + 4 q^{54} + 2 i q^{57} + 6 i q^{58} + 8 q^{61} -8 i q^{62} - q^{64} + 6 q^{66} + 8 i q^{67} + 6 i q^{68} + 18 q^{69} -i q^{72} + 4 i q^{73} -7 q^{74} - q^{76} + 2 i q^{78} + 10 q^{79} -11 q^{81} -3 i q^{82} -2 q^{86} -12 i q^{87} + 3 i q^{88} -6 q^{89} + 9 i q^{92} + 16 i q^{93} + 9 q^{94} + 2 q^{96} -10 i q^{97} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + 6q^{11} + 2q^{16} + 2q^{19} - 4q^{24} + 2q^{26} - 12q^{29} + 16q^{31} - 12q^{34} + 2q^{36} - 4q^{39} + 6q^{41} - 6q^{44} - 18q^{46} + 24q^{51} + 8q^{54} + 16q^{61} - 2q^{64} + 12q^{66} + 36q^{69} - 14q^{74} - 2q^{76} + 20q^{79} - 22q^{81} - 4q^{86} - 12q^{89} + 18q^{94} + 4q^{96} - 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

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

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 2.00000i −1.00000 0 2.00000 0 1.00000i −1.00000 0
99.2 1.00000i 2.00000i −1.00000 0 2.00000 0 1.00000i −1.00000 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 2450.2.c.q 2
5.b even 2 1 inner 2450.2.c.q 2
5.c odd 4 1 490.2.a.a 1
5.c odd 4 1 2450.2.a.bf 1
7.b odd 2 1 2450.2.c.e 2
7.c even 3 2 350.2.j.d 4
15.e even 4 1 4410.2.a.x 1
20.e even 4 1 3920.2.a.bh 1
35.c odd 2 1 2450.2.c.e 2
35.f even 4 1 490.2.a.d 1
35.f even 4 1 2450.2.a.v 1
35.j even 6 2 350.2.j.d 4
35.k even 12 2 490.2.e.g 2
35.l odd 12 2 70.2.e.d 2
35.l odd 12 2 350.2.e.b 2
105.k odd 4 1 4410.2.a.bg 1
105.x even 12 2 630.2.k.d 2
140.j odd 4 1 3920.2.a.e 1
140.w even 12 2 560.2.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.d 2 35.l odd 12 2
350.2.e.b 2 35.l odd 12 2
350.2.j.d 4 7.c even 3 2
350.2.j.d 4 35.j even 6 2
490.2.a.a 1 5.c odd 4 1
490.2.a.d 1 35.f even 4 1
490.2.e.g 2 35.k even 12 2
560.2.q.b 2 140.w even 12 2
630.2.k.d 2 105.x even 12 2
2450.2.a.v 1 35.f even 4 1
2450.2.a.bf 1 5.c odd 4 1
2450.2.c.e 2 7.b odd 2 1
2450.2.c.e 2 35.c odd 2 1
2450.2.c.q 2 1.a even 1 1 trivial
2450.2.c.q 2 5.b even 2 1 inner
3920.2.a.e 1 140.j odd 4 1
3920.2.a.bh 1 20.e even 4 1
4410.2.a.x 1 15.e even 4 1
4410.2.a.bg 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2450, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{11} - 3 \)
\( T_{13}^{2} + 1 \)
\( T_{19} - 1 \)
\( T_{31} - 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ 1
$7$ 1
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 - 25 T^{2} + 169 T^{4} \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2} \)
$23$ \( 1 + 35 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 25 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 82 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 13 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 25 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 70 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 130 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{2} \)
$89$ \( ( 1 + 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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