Properties

Label 550.2.b.d
Level $550$
Weight $2$
Character orbit 550.b
Analytic conductor $4.392$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.39177211117\)
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: yes
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.

\(f(q)\) \(=\) \( q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} + 4 i q^{7} - i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - 2 i q^{3} - q^{4} + 2 q^{6} + 4 i q^{7} - i q^{8} - q^{9} - q^{11} + 2 i q^{12} + 5 i q^{13} - 4 q^{14} + q^{16} - i q^{18} + 7 q^{19} + 8 q^{21} - i q^{22} + 3 i q^{23} - 2 q^{24} - 5 q^{26} - 4 i q^{27} - 4 i q^{28} - 3 q^{29} + 5 q^{31} + i q^{32} + 2 i q^{33} + q^{36} + 4 i q^{37} + 7 i q^{38} + 10 q^{39} + 12 q^{41} + 8 i q^{42} + 5 i q^{43} + q^{44} - 3 q^{46} - 2 i q^{48} - 9 q^{49} - 5 i q^{52} + 6 i q^{53} + 4 q^{54} + 4 q^{56} - 14 i q^{57} - 3 i q^{58} - 12 q^{59} - 10 q^{61} + 5 i q^{62} - 4 i q^{63} - q^{64} - 2 q^{66} - 14 i q^{67} + 6 q^{69} + 3 q^{71} + i q^{72} + 8 i q^{73} - 4 q^{74} - 7 q^{76} - 4 i q^{77} + 10 i q^{78} + 4 q^{79} - 11 q^{81} + 12 i q^{82} - 15 i q^{83} - 8 q^{84} - 5 q^{86} + 6 i q^{87} + i q^{88} - 3 q^{89} - 20 q^{91} - 3 i q^{92} - 10 i q^{93} + 2 q^{96} + 13 i q^{97} - 9 i q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 2 q^{11} - 8 q^{14} + 2 q^{16} + 14 q^{19} + 16 q^{21} - 4 q^{24} - 10 q^{26} - 6 q^{29} + 10 q^{31} + 2 q^{36} + 20 q^{39} + 24 q^{41} + 2 q^{44} - 6 q^{46} - 18 q^{49} + 8 q^{54} + 8 q^{56} - 24 q^{59} - 20 q^{61} - 2 q^{64} - 4 q^{66} + 12 q^{69} + 6 q^{71} - 8 q^{74} - 14 q^{76} + 8 q^{79} - 22 q^{81} - 16 q^{84} - 10 q^{86} - 6 q^{89} - 40 q^{91} + 4 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
1.00000i 2.00000i −1.00000 0 2.00000 4.00000i 1.00000i −1.00000 0
199.2 1.00000i 2.00000i −1.00000 0 2.00000 4.00000i 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 550.2.b.d 2
3.b odd 2 1 4950.2.c.ba 2
4.b odd 2 1 4400.2.b.e 2
5.b even 2 1 inner 550.2.b.d 2
5.c odd 4 1 550.2.a.a 1
5.c odd 4 1 550.2.a.m yes 1
15.d odd 2 1 4950.2.c.ba 2
15.e even 4 1 4950.2.a.u 1
15.e even 4 1 4950.2.a.y 1
20.d odd 2 1 4400.2.b.e 2
20.e even 4 1 4400.2.a.d 1
20.e even 4 1 4400.2.a.bc 1
55.e even 4 1 6050.2.a.n 1
55.e even 4 1 6050.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.a.a 1 5.c odd 4 1
550.2.a.m yes 1 5.c odd 4 1
550.2.b.d 2 1.a even 1 1 trivial
550.2.b.d 2 5.b even 2 1 inner
4400.2.a.d 1 20.e even 4 1
4400.2.a.bc 1 20.e even 4 1
4400.2.b.e 2 4.b odd 2 1
4400.2.b.e 2 20.d odd 2 1
4950.2.a.u 1 15.e even 4 1
4950.2.a.y 1 15.e even 4 1
4950.2.c.ba 2 3.b odd 2 1
4950.2.c.ba 2 15.d odd 2 1
6050.2.a.n 1 55.e even 4 1
6050.2.a.bb 1 55.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 \(S_{2}^{\mathrm{new}}(550, [\chi])\):

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T - 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 25 \) 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
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 225 \) Copy content Toggle raw display
$89$ \( (T + 3)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 169 \) Copy content Toggle raw display
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