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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(2550, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} + 4 \) 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
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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