Properties

Label 4400.2.b.h
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
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.

\(f(q)\) \(=\) \( q + i q^{3} - 2 i q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - 2 i q^{7} + 2 q^{9} - q^{11} + 4 i q^{13} + 2 i q^{17} + 2 q^{21} + i q^{23} + 5 i q^{27} - 7 q^{31} - i q^{33} - 3 i q^{37} - 4 q^{39} - 8 q^{41} + 6 i q^{43} + 8 i q^{47} + 3 q^{49} - 2 q^{51} - 6 i q^{53} + 5 q^{59} + 12 q^{61} - 4 i q^{63} - 7 i q^{67} - q^{69} + 3 q^{71} + 4 i q^{73} + 2 i q^{77} - 10 q^{79} + q^{81} + 6 i q^{83} - 15 q^{89} + 8 q^{91} - 7 i q^{93} + 7 i q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} - 2 q^{11} + 4 q^{21} - 14 q^{31} - 8 q^{39} - 16 q^{41} + 6 q^{49} - 4 q^{51} + 10 q^{59} + 24 q^{61} - 2 q^{69} + 6 q^{71} - 20 q^{79} + 2 q^{81} - 30 q^{89} + 16 q^{91} - 4 q^{99}+O(q^{100}) \) 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 1.00000i 0 0 0 2.00000i 0 2.00000 0
4049.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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.h 2
4.b odd 2 1 275.2.b.a 2
5.b even 2 1 inner 4400.2.b.h 2
5.c odd 4 1 176.2.a.b 1
5.c odd 4 1 4400.2.a.i 1
12.b even 2 1 2475.2.c.a 2
15.e even 4 1 1584.2.a.g 1
20.d odd 2 1 275.2.b.a 2
20.e even 4 1 11.2.a.a 1
20.e even 4 1 275.2.a.b 1
35.f even 4 1 8624.2.a.j 1
40.i odd 4 1 704.2.a.c 1
40.k even 4 1 704.2.a.h 1
55.e even 4 1 1936.2.a.i 1
60.h even 2 1 2475.2.c.a 2
60.l odd 4 1 99.2.a.d 1
60.l odd 4 1 2475.2.a.a 1
80.i odd 4 1 2816.2.c.f 2
80.j even 4 1 2816.2.c.j 2
80.s even 4 1 2816.2.c.j 2
80.t odd 4 1 2816.2.c.f 2
120.q odd 4 1 6336.2.a.br 1
120.w even 4 1 6336.2.a.bu 1
140.j odd 4 1 539.2.a.a 1
140.w even 12 2 539.2.e.h 2
140.x odd 12 2 539.2.e.g 2
180.v odd 12 2 891.2.e.b 2
180.x even 12 2 891.2.e.k 2
220.i odd 4 1 121.2.a.d 1
220.i odd 4 1 3025.2.a.a 1
220.v even 20 4 121.2.c.e 4
220.w odd 20 4 121.2.c.a 4
260.p even 4 1 1859.2.a.b 1
340.r even 4 1 3179.2.a.a 1
380.j odd 4 1 3971.2.a.b 1
420.w even 4 1 4851.2.a.t 1
440.t even 4 1 7744.2.a.k 1
440.w odd 4 1 7744.2.a.x 1
460.k odd 4 1 5819.2.a.a 1
580.o even 4 1 9251.2.a.d 1
660.q even 4 1 1089.2.a.b 1
1540.x even 4 1 5929.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 20.e even 4 1
99.2.a.d 1 60.l odd 4 1
121.2.a.d 1 220.i odd 4 1
121.2.c.a 4 220.w odd 20 4
121.2.c.e 4 220.v even 20 4
176.2.a.b 1 5.c odd 4 1
275.2.a.b 1 20.e even 4 1
275.2.b.a 2 4.b odd 2 1
275.2.b.a 2 20.d odd 2 1
539.2.a.a 1 140.j odd 4 1
539.2.e.g 2 140.x odd 12 2
539.2.e.h 2 140.w even 12 2
704.2.a.c 1 40.i odd 4 1
704.2.a.h 1 40.k even 4 1
891.2.e.b 2 180.v odd 12 2
891.2.e.k 2 180.x even 12 2
1089.2.a.b 1 660.q even 4 1
1584.2.a.g 1 15.e even 4 1
1859.2.a.b 1 260.p even 4 1
1936.2.a.i 1 55.e even 4 1
2475.2.a.a 1 60.l odd 4 1
2475.2.c.a 2 12.b even 2 1
2475.2.c.a 2 60.h even 2 1
2816.2.c.f 2 80.i odd 4 1
2816.2.c.f 2 80.t odd 4 1
2816.2.c.j 2 80.j even 4 1
2816.2.c.j 2 80.s even 4 1
3025.2.a.a 1 220.i odd 4 1
3179.2.a.a 1 340.r even 4 1
3971.2.a.b 1 380.j odd 4 1
4400.2.a.i 1 5.c odd 4 1
4400.2.b.h 2 1.a even 1 1 trivial
4400.2.b.h 2 5.b even 2 1 inner
4851.2.a.t 1 420.w even 4 1
5819.2.a.a 1 460.k odd 4 1
5929.2.a.h 1 1540.x even 4 1
6336.2.a.br 1 120.q odd 4 1
6336.2.a.bu 1 120.w even 4 1
7744.2.a.k 1 440.t even 4 1
7744.2.a.x 1 440.w odd 4 1
8624.2.a.j 1 35.f even 4 1
9251.2.a.d 1 580.o even 4 1

Hecke kernels

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

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 5)^{2} \) Copy content Toggle raw display
$61$ \( (T - 12)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 49 \) Copy content Toggle raw display
$71$ \( (T - 3)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T + 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) 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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