Properties

Label 2450.2.c.g
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{8} + 2 q^{9} - 6 q^{11} - i q^{12} - 4 i q^{13} + q^{16} + 2 i q^{18} - 2 q^{19} - 6 i q^{22} - 3 i q^{23} + q^{24} + 4 q^{26} + 5 i q^{27} + 3 q^{29} + 8 q^{31} + i q^{32} - 6 i q^{33} - 2 q^{36} + 4 i q^{37} - 2 i q^{38} + 4 q^{39} + 9 q^{41} - 7 i q^{43} + 6 q^{44} + 3 q^{46} + i q^{48} + 4 i q^{52} - 6 i q^{53} - 5 q^{54} - 2 i q^{57} + 3 i q^{58} + 6 q^{59} + 5 q^{61} + 8 i q^{62} - q^{64} + 6 q^{66} - 5 i q^{67} + 3 q^{69} - 6 q^{71} - 2 i q^{72} - 16 i q^{73} - 4 q^{74} + 2 q^{76} + 4 i q^{78} - 2 q^{79} + q^{81} + 9 i q^{82} + 3 i q^{83} + 7 q^{86} + 3 i q^{87} + 6 i q^{88} + 15 q^{89} + 3 i q^{92} + 8 i q^{93} - q^{96} - 14 i q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{6} + 4 q^{9} - 12 q^{11} + 2 q^{16} - 4 q^{19} + 2 q^{24} + 8 q^{26} + 6 q^{29} + 16 q^{31} - 4 q^{36} + 8 q^{39} + 18 q^{41} + 12 q^{44} + 6 q^{46} - 10 q^{54} + 12 q^{59} + 10 q^{61} - 2 q^{64} + 12 q^{66} + 6 q^{69} - 12 q^{71} - 8 q^{74} + 4 q^{76} - 4 q^{79} + 2 q^{81} + 14 q^{86} + 30 q^{89} - 2 q^{96} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{19} + 2 \) Copy content Toggle raw display
\( T_{31} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T + 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 49 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( (T - 5)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 25 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T + 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9 \) Copy content Toggle raw display
$89$ \( (T - 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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