Properties

Label 2200.2.b.b
Level $2200$
Weight $2$
Character orbit 2200.b
Analytic conductor $17.567$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.5670884447\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} -i q^{7} -6 q^{9} +O(q^{10})\) \( q + 3 i q^{3} -i q^{7} -6 q^{9} - q^{11} -6 i q^{13} -3 i q^{17} + 5 q^{19} + 3 q^{21} -2 i q^{23} -9 i q^{27} + 5 q^{29} + 5 q^{31} -3 i q^{33} + i q^{37} + 18 q^{39} -2 q^{41} + 12 i q^{43} + 2 i q^{47} + 6 q^{49} + 9 q^{51} -13 i q^{53} + 15 i q^{57} -2 q^{59} + q^{61} + 6 i q^{63} -16 i q^{67} + 6 q^{69} + 15 q^{71} + 10 i q^{73} + i q^{77} -2 q^{79} + 9 q^{81} -14 i q^{83} + 15 i q^{87} -9 q^{89} -6 q^{91} + 15 i q^{93} + 16 i q^{97} + 6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9} + O(q^{10}) \) \( 2 q - 12 q^{9} - 2 q^{11} + 10 q^{19} + 6 q^{21} + 10 q^{29} + 10 q^{31} + 36 q^{39} - 4 q^{41} + 12 q^{49} + 18 q^{51} - 4 q^{59} + 2 q^{61} + 12 q^{69} + 30 q^{71} - 4 q^{79} + 18 q^{81} - 18 q^{89} - 12 q^{91} + 12 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
1.00000i
1.00000i
0 3.00000i 0 0 0 1.00000i 0 −6.00000 0
1849.2 0 3.00000i 0 0 0 1.00000i 0 −6.00000 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 2200.2.b.b 2
4.b odd 2 1 4400.2.b.a 2
5.b even 2 1 inner 2200.2.b.b 2
5.c odd 4 1 440.2.a.d 1
5.c odd 4 1 2200.2.a.a 1
15.e even 4 1 3960.2.a.f 1
20.d odd 2 1 4400.2.b.a 2
20.e even 4 1 880.2.a.a 1
20.e even 4 1 4400.2.a.be 1
40.i odd 4 1 3520.2.a.a 1
40.k even 4 1 3520.2.a.bh 1
55.e even 4 1 4840.2.a.i 1
60.l odd 4 1 7920.2.a.e 1
220.i odd 4 1 9680.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.d 1 5.c odd 4 1
880.2.a.a 1 20.e even 4 1
2200.2.a.a 1 5.c odd 4 1
2200.2.b.b 2 1.a even 1 1 trivial
2200.2.b.b 2 5.b even 2 1 inner
3520.2.a.a 1 40.i odd 4 1
3520.2.a.bh 1 40.k even 4 1
3960.2.a.f 1 15.e even 4 1
4400.2.a.be 1 20.e even 4 1
4400.2.b.a 2 4.b odd 2 1
4400.2.b.a 2 20.d odd 2 1
4840.2.a.i 1 55.e even 4 1
7920.2.a.e 1 60.l odd 4 1
9680.2.a.a 1 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2200, [\chi])\):

\( T_{3}^{2} + 9 \)
\( T_{7}^{2} + 1 \)
\( T_{13}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 9 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 4 + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( 4 + T^{2} \)
$53$ \( 169 + T^{2} \)
$59$ \( ( 2 + T )^{2} \)
$61$ \( ( -1 + T )^{2} \)
$67$ \( 256 + T^{2} \)
$71$ \( ( -15 + T )^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( 2 + T )^{2} \)
$83$ \( 196 + T^{2} \)
$89$ \( ( 9 + T )^{2} \)
$97$ \( 256 + T^{2} \)
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