Properties

Label 2450.2.c.b
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: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
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.

\(f(q)\) \(=\) \( q + i q^{2} + 2 i q^{3} - q^{4} -2 q^{6} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} + 2 i q^{3} - q^{4} -2 q^{6} -i q^{8} - q^{9} -4 q^{11} -2 i q^{12} + 2 i q^{13} + q^{16} -8 i q^{17} -i q^{18} -6 q^{19} -4 i q^{22} + 4 i q^{23} + 2 q^{24} -2 q^{26} + 4 i q^{27} + 6 q^{29} -4 q^{31} + i q^{32} -8 i q^{33} + 8 q^{34} + q^{36} -10 i q^{37} -6 i q^{38} -4 q^{39} -4 q^{41} -4 i q^{43} + 4 q^{44} -4 q^{46} -4 i q^{47} + 2 i q^{48} + 16 q^{51} -2 i q^{52} -10 i q^{53} -4 q^{54} -12 i q^{57} + 6 i q^{58} -14 q^{59} + 10 q^{61} -4 i q^{62} - q^{64} + 8 q^{66} -4 i q^{67} + 8 i q^{68} -8 q^{69} + 12 q^{71} + i q^{72} + 4 i q^{73} + 10 q^{74} + 6 q^{76} -4 i q^{78} -4 q^{79} -11 q^{81} -4 i q^{82} -2 i q^{83} + 4 q^{86} + 12 i q^{87} + 4 i q^{88} + 8 q^{89} -4 i q^{92} -8 i q^{93} + 4 q^{94} -2 q^{96} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 4q^{6} - 2q^{9} - 8q^{11} + 2q^{16} - 12q^{19} + 4q^{24} - 4q^{26} + 12q^{29} - 8q^{31} + 16q^{34} + 2q^{36} - 8q^{39} - 8q^{41} + 8q^{44} - 8q^{46} + 32q^{51} - 8q^{54} - 28q^{59} + 20q^{61} - 2q^{64} + 16q^{66} - 16q^{69} + 24q^{71} + 20q^{74} + 12q^{76} - 8q^{79} - 22q^{81} + 8q^{86} + 16q^{89} + 8q^{94} - 4q^{96} + 8q^{99} + O(q^{100}) \)

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.b 2
5.b even 2 1 inner 2450.2.c.b 2
5.c odd 4 1 490.2.a.f 1
5.c odd 4 1 2450.2.a.n 1
7.b odd 2 1 2450.2.c.n 2
15.e even 4 1 4410.2.a.s 1
20.e even 4 1 3920.2.a.bg 1
35.c odd 2 1 2450.2.c.n 2
35.f even 4 1 490.2.a.i yes 1
35.f even 4 1 2450.2.a.d 1
35.k even 12 2 490.2.e.b 2
35.l odd 12 2 490.2.e.e 2
105.k odd 4 1 4410.2.a.i 1
140.j odd 4 1 3920.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.f 1 5.c odd 4 1
490.2.a.i yes 1 35.f even 4 1
490.2.e.b 2 35.k even 12 2
490.2.e.e 2 35.l odd 12 2
2450.2.a.d 1 35.f even 4 1
2450.2.a.n 1 5.c odd 4 1
2450.2.c.b 2 1.a even 1 1 trivial
2450.2.c.b 2 5.b even 2 1 inner
2450.2.c.n 2 7.b odd 2 1
2450.2.c.n 2 35.c odd 2 1
3920.2.a.j 1 140.j odd 4 1
3920.2.a.bg 1 20.e even 4 1
4410.2.a.i 1 105.k odd 4 1
4410.2.a.s 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 \(S_{2}^{\mathrm{new}}(2450, [\chi])\):

\( T_{3}^{2} + 4 \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 4 \)
\( T_{19} + 6 \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 64 + T^{2} \)
$19$ \( ( 6 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -6 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 100 + T^{2} \)
$41$ \( ( 4 + T )^{2} \)
$43$ \( 16 + T^{2} \)
$47$ \( 16 + T^{2} \)
$53$ \( 100 + T^{2} \)
$59$ \( ( 14 + T )^{2} \)
$61$ \( ( -10 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( 4 + T )^{2} \)
$83$ \( 4 + T^{2} \)
$89$ \( ( -8 + T )^{2} \)
$97$ \( T^{2} \)
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