Properties

Label 2550.2.d.m
Level $2550$
Weight $2$
Character orbit 2550.d
Analytic conductor $20.362$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2550.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.3618525154\)
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 102)
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.

\(f(q)\) \(=\) \( q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{8} - q^{9} +O(q^{10})\) \( q -i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{8} - q^{9} -4 q^{11} -i q^{12} -2 i q^{13} + q^{16} -i q^{17} + i q^{18} -4 q^{19} + 4 i q^{22} - q^{24} -2 q^{26} -i q^{27} + 10 q^{29} + 8 q^{31} -i q^{32} -4 i q^{33} - q^{34} + q^{36} + 2 i q^{37} + 4 i q^{38} + 2 q^{39} + 10 q^{41} + 12 i q^{43} + 4 q^{44} + i q^{48} + 7 q^{49} + q^{51} + 2 i q^{52} + 6 i q^{53} - q^{54} -4 i q^{57} -10 i q^{58} -12 q^{59} -10 q^{61} -8 i q^{62} - q^{64} -4 q^{66} + 12 i q^{67} + i q^{68} -i q^{72} + 10 i q^{73} + 2 q^{74} + 4 q^{76} -2 i q^{78} + 8 q^{79} + q^{81} -10 i q^{82} + 4 i q^{83} + 12 q^{86} + 10 i q^{87} -4 i q^{88} + 6 q^{89} + 8 i q^{93} + q^{96} + 14 i q^{97} -7 i q^{98} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 8q^{11} + 2q^{16} - 8q^{19} - 2q^{24} - 4q^{26} + 20q^{29} + 16q^{31} - 2q^{34} + 2q^{36} + 4q^{39} + 20q^{41} + 8q^{44} + 14q^{49} + 2q^{51} - 2q^{54} - 24q^{59} - 20q^{61} - 2q^{64} - 8q^{66} + 4q^{74} + 8q^{76} + 16q^{79} + 2q^{81} + 24q^{86} + 12q^{89} + 2q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(851\) \(1327\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 0 1.00000i −1.00000 0
2449.2 1.00000i 1.00000i −1.00000 0 1.00000 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 2550.2.d.m 2
5.b even 2 1 inner 2550.2.d.m 2
5.c odd 4 1 102.2.a.c 1
5.c odd 4 1 2550.2.a.c 1
15.e even 4 1 306.2.a.b 1
15.e even 4 1 7650.2.a.ca 1
20.e even 4 1 816.2.a.b 1
35.f even 4 1 4998.2.a.be 1
40.i odd 4 1 3264.2.a.m 1
40.k even 4 1 3264.2.a.bc 1
60.l odd 4 1 2448.2.a.p 1
85.f odd 4 1 1734.2.b.b 2
85.g odd 4 1 1734.2.a.j 1
85.i odd 4 1 1734.2.b.b 2
85.k odd 8 2 1734.2.f.e 4
85.n odd 8 2 1734.2.f.e 4
120.q odd 4 1 9792.2.a.l 1
120.w even 4 1 9792.2.a.k 1
255.o even 4 1 5202.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 5.c odd 4 1
306.2.a.b 1 15.e even 4 1
816.2.a.b 1 20.e even 4 1
1734.2.a.j 1 85.g odd 4 1
1734.2.b.b 2 85.f odd 4 1
1734.2.b.b 2 85.i odd 4 1
1734.2.f.e 4 85.k odd 8 2
1734.2.f.e 4 85.n odd 8 2
2448.2.a.p 1 60.l odd 4 1
2550.2.a.c 1 5.c odd 4 1
2550.2.d.m 2 1.a even 1 1 trivial
2550.2.d.m 2 5.b even 2 1 inner
3264.2.a.m 1 40.i odd 4 1
3264.2.a.bc 1 40.k even 4 1
4998.2.a.be 1 35.f even 4 1
5202.2.a.c 1 255.o even 4 1
7650.2.a.ca 1 15.e even 4 1
9792.2.a.k 1 120.w even 4 1
9792.2.a.l 1 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2550, [\chi])\):

\( T_{7} \)
\( T_{11} + 4 \)
\( T_{13}^{2} + 4 \)
\( T_{19} + 4 \)