Properties

Label 4400.2.b.b
Level $4400$
Weight $2$
Character orbit 4400.b
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
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.

\(f(q)\) \(=\) \( q + 3 i q^{3} - 2 i q^{7} - 6 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} - 2 i q^{7} - 6 q^{9} + q^{11} + 6 i q^{17} + 4 q^{19} + 6 q^{21} - i q^{23} - 9 i q^{27} + 8 q^{29} + 7 q^{31} + 3 i q^{33} + i q^{37} + 4 q^{41} - 6 i q^{43} - 8 i q^{47} + 3 q^{49} - 18 q^{51} + 2 i q^{53} + 12 i q^{57} - q^{59} + 4 q^{61} + 12 i q^{63} - 5 i q^{67} + 3 q^{69} - 3 q^{71} + 16 i q^{73} - 2 i q^{77} + 2 q^{79} + 9 q^{81} + 2 i q^{83} + 24 i q^{87} - 15 q^{89} + 21 i q^{93} + 7 i q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} + 2 q^{11} + 8 q^{19} + 12 q^{21} + 16 q^{29} + 14 q^{31} + 8 q^{41} + 6 q^{49} - 36 q^{51} - 2 q^{59} + 8 q^{61} + 6 q^{69} - 6 q^{71} + 4 q^{79} + 18 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
1.00000i
1.00000i
0 3.00000i 0 0 0 2.00000i 0 −6.00000 0
4049.2 0 3.00000i 0 0 0 2.00000i 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 4400.2.b.b 2
4.b odd 2 1 2200.2.b.a 2
5.b even 2 1 inner 4400.2.b.b 2
5.c odd 4 1 176.2.a.c 1
5.c odd 4 1 4400.2.a.a 1
15.e even 4 1 1584.2.a.q 1
20.d odd 2 1 2200.2.b.a 2
20.e even 4 1 88.2.a.a 1
20.e even 4 1 2200.2.a.k 1
35.f even 4 1 8624.2.a.c 1
40.i odd 4 1 704.2.a.b 1
40.k even 4 1 704.2.a.l 1
55.e even 4 1 1936.2.a.l 1
60.l odd 4 1 792.2.a.g 1
80.i odd 4 1 2816.2.c.d 2
80.j even 4 1 2816.2.c.i 2
80.s even 4 1 2816.2.c.i 2
80.t odd 4 1 2816.2.c.d 2
120.q odd 4 1 6336.2.a.h 1
120.w even 4 1 6336.2.a.k 1
140.j odd 4 1 4312.2.a.l 1
220.i odd 4 1 968.2.a.a 1
220.v even 20 4 968.2.i.j 4
220.w odd 20 4 968.2.i.i 4
440.t even 4 1 7744.2.a.b 1
440.w odd 4 1 7744.2.a.bk 1
660.q even 4 1 8712.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 20.e even 4 1
176.2.a.c 1 5.c odd 4 1
704.2.a.b 1 40.i odd 4 1
704.2.a.l 1 40.k even 4 1
792.2.a.g 1 60.l odd 4 1
968.2.a.a 1 220.i odd 4 1
968.2.i.i 4 220.w odd 20 4
968.2.i.j 4 220.v even 20 4
1584.2.a.q 1 15.e even 4 1
1936.2.a.l 1 55.e even 4 1
2200.2.a.k 1 20.e even 4 1
2200.2.b.a 2 4.b odd 2 1
2200.2.b.a 2 20.d odd 2 1
2816.2.c.d 2 80.i odd 4 1
2816.2.c.d 2 80.t odd 4 1
2816.2.c.i 2 80.j even 4 1
2816.2.c.i 2 80.s even 4 1
4312.2.a.l 1 140.j odd 4 1
4400.2.a.a 1 5.c odd 4 1
4400.2.b.b 2 1.a even 1 1 trivial
4400.2.b.b 2 5.b even 2 1 inner
6336.2.a.h 1 120.q odd 4 1
6336.2.a.k 1 120.w even 4 1
7744.2.a.b 1 440.t even 4 1
7744.2.a.bk 1 440.w odd 4 1
8624.2.a.c 1 35.f even 4 1
8712.2.a.x 1 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4400, [\chi])\):

\( T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T - 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T - 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T + 1)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 25 \) Copy content Toggle raw display
$71$ \( (T + 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T + 15)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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