Properties

Label 4400.2.b.s
Level $4400$
Weight $2$
Character orbit 4400.b
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \(x^{4} + 11 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1100)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{1} + 3 \beta_{2} ) q^{7} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{1} + 3 \beta_{2} ) q^{7} + ( -3 + \beta_{3} ) q^{9} + q^{11} + \beta_{2} q^{13} + ( -\beta_{1} - 2 \beta_{2} ) q^{17} + ( 3 - 2 \beta_{3} ) q^{19} + ( -3 - 2 \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{2} ) q^{23} + 5 \beta_{2} q^{27} + ( -5 + \beta_{3} ) q^{29} + ( 5 - 2 \beta_{3} ) q^{31} + \beta_{1} q^{33} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{37} + ( 1 - \beta_{3} ) q^{39} + ( -5 - 2 \beta_{3} ) q^{41} -10 \beta_{2} q^{43} + ( 2 \beta_{1} + 7 \beta_{2} ) q^{47} + ( -2 - 5 \beta_{3} ) q^{49} + ( 4 + \beta_{3} ) q^{51} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{53} + ( 3 \beta_{1} - 10 \beta_{2} ) q^{57} + ( -7 + 2 \beta_{3} ) q^{59} + ( 6 + \beta_{3} ) q^{61} -\beta_{2} q^{63} + 4 \beta_{2} q^{67} + ( -7 + 2 \beta_{3} ) q^{69} -6 \beta_{3} q^{71} + ( -\beta_{1} + 5 \beta_{2} ) q^{73} + ( \beta_{1} + 3 \beta_{2} ) q^{77} + ( -3 + 7 \beta_{3} ) q^{79} + ( -4 - 2 \beta_{3} ) q^{81} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{87} + ( -1 - \beta_{3} ) q^{89} + ( -2 - \beta_{3} ) q^{91} + ( 5 \beta_{1} - 10 \beta_{2} ) q^{93} + ( -\beta_{1} - 9 \beta_{2} ) q^{97} + ( -3 + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9} + O(q^{10}) \) \( 4 q - 10 q^{9} + 4 q^{11} + 8 q^{19} - 16 q^{21} - 18 q^{29} + 16 q^{31} + 2 q^{39} - 24 q^{41} - 18 q^{49} + 18 q^{51} - 24 q^{59} + 26 q^{61} - 24 q^{69} - 12 q^{71} + 2 q^{79} - 20 q^{81} - 6 q^{89} - 10 q^{91} - 10 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 11 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 6 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 6\)
\(\nu^{3}\)\(=\)\(5 \beta_{2} - 6 \beta_{1}\)

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
2.79129i
1.79129i
1.79129i
2.79129i
0 2.79129i 0 0 0 0.208712i 0 −4.79129 0
4049.2 0 1.79129i 0 0 0 4.79129i 0 −0.208712 0
4049.3 0 1.79129i 0 0 0 4.79129i 0 −0.208712 0
4049.4 0 2.79129i 0 0 0 0.208712i 0 −4.79129 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.s 4
4.b odd 2 1 1100.2.b.d 4
5.b even 2 1 inner 4400.2.b.s 4
5.c odd 4 1 4400.2.a.bi 2
5.c odd 4 1 4400.2.a.bu 2
12.b even 2 1 9900.2.c.x 4
20.d odd 2 1 1100.2.b.d 4
20.e even 4 1 1100.2.a.g 2
20.e even 4 1 1100.2.a.h yes 2
60.h even 2 1 9900.2.c.x 4
60.l odd 4 1 9900.2.a.bh 2
60.l odd 4 1 9900.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.2.a.g 2 20.e even 4 1
1100.2.a.h yes 2 20.e even 4 1
1100.2.b.d 4 4.b odd 2 1
1100.2.b.d 4 20.d odd 2 1
4400.2.a.bi 2 5.c odd 4 1
4400.2.a.bu 2 5.c odd 4 1
4400.2.b.s 4 1.a even 1 1 trivial
4400.2.b.s 4 5.b even 2 1 inner
9900.2.a.bh 2 60.l odd 4 1
9900.2.a.bz 2 60.l odd 4 1
9900.2.c.x 4 12.b even 2 1
9900.2.c.x 4 60.h even 2 1

Hecke kernels

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

\( T_{3}^{4} + 11 T_{3}^{2} + 25 \)
\( T_{7}^{4} + 23 T_{7}^{2} + 1 \)
\( T_{13}^{2} + 1 \)
\( T_{17}^{4} + 15 T_{17}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 25 + 11 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + 23 T^{2} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( 9 + 15 T^{2} + T^{4} \)
$19$ \( ( -17 - 4 T + T^{2} )^{2} \)
$23$ \( 9 + 15 T^{2} + T^{4} \)
$29$ \( ( 15 + 9 T + T^{2} )^{2} \)
$31$ \( ( -5 - 8 T + T^{2} )^{2} \)
$37$ \( 289 + 50 T^{2} + T^{4} \)
$41$ \( ( 15 + 12 T + T^{2} )^{2} \)
$43$ \( ( 100 + T^{2} )^{2} \)
$47$ \( 225 + 114 T^{2} + T^{4} \)
$53$ \( 729 + 135 T^{2} + T^{4} \)
$59$ \( ( 15 + 12 T + T^{2} )^{2} \)
$61$ \( ( 37 - 13 T + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( -180 + 6 T + T^{2} )^{2} \)
$73$ \( 625 + 71 T^{2} + T^{4} \)
$79$ \( ( -257 - T + T^{2} )^{2} \)
$83$ \( 16641 + 267 T^{2} + T^{4} \)
$89$ \( ( -3 + 3 T + T^{2} )^{2} \)
$97$ \( 4489 + 155 T^{2} + T^{4} \)
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