Properties

Label 4400.2.b.ba
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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2200)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} + q^{11} + ( 2 \beta_{1} + \beta_{3} ) q^{13} -3 \beta_{1} q^{17} + ( 1 + 4 \beta_{2} ) q^{19} + q^{21} + ( -7 \beta_{1} - 5 \beta_{3} ) q^{23} + ( 4 \beta_{1} + \beta_{3} ) q^{27} + ( 6 + \beta_{2} ) q^{29} + ( 1 - 2 \beta_{2} ) q^{31} + \beta_{1} q^{33} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{37} + ( -2 + \beta_{2} ) q^{39} + ( -5 - 8 \beta_{2} ) q^{41} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{47} + ( 5 - \beta_{2} ) q^{49} + ( 3 - 3 \beta_{2} ) q^{51} + ( -3 \beta_{1} - \beta_{3} ) q^{53} + ( -3 \beta_{1} + 4 \beta_{3} ) q^{57} + ( -7 - 2 \beta_{2} ) q^{59} + ( -1 - 5 \beta_{2} ) q^{61} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{63} + ( 7 - 2 \beta_{2} ) q^{69} + ( 10 + 6 \beta_{2} ) q^{71} + ( -5 \beta_{1} - \beta_{3} ) q^{73} + ( -\beta_{1} - \beta_{3} ) q^{77} + ( -2 - 3 \beta_{2} ) q^{79} + ( 2 + 6 \beta_{2} ) q^{81} + ( -3 \beta_{1} - 6 \beta_{3} ) q^{83} + ( 5 \beta_{1} + \beta_{3} ) q^{87} + ( -2 - 9 \beta_{2} ) q^{89} + ( 3 + \beta_{2} ) q^{91} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{93} + ( 5 \beta_{1} + 13 \beta_{3} ) q^{97} + ( 2 + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9} + O(q^{10}) \) \( 4 q + 6 q^{9} + 4 q^{11} - 4 q^{19} + 4 q^{21} + 22 q^{29} + 8 q^{31} - 10 q^{39} - 4 q^{41} + 22 q^{49} + 18 q^{51} - 24 q^{59} + 6 q^{61} + 32 q^{69} + 28 q^{71} - 2 q^{79} - 4 q^{81} + 10 q^{89} + 10 q^{91} + 6 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

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
1.61803i
0.618034i
0.618034i
1.61803i
0 1.61803i 0 0 0 0.618034i 0 0.381966 0
4049.2 0 0.618034i 0 0 0 1.61803i 0 2.61803 0
4049.3 0 0.618034i 0 0 0 1.61803i 0 2.61803 0
4049.4 0 1.61803i 0 0 0 0.618034i 0 0.381966 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.ba 4
4.b odd 2 1 2200.2.b.j 4
5.b even 2 1 inner 4400.2.b.ba 4
5.c odd 4 1 4400.2.a.bk 2
5.c odd 4 1 4400.2.a.bq 2
20.d odd 2 1 2200.2.b.j 4
20.e even 4 1 2200.2.a.n 2
20.e even 4 1 2200.2.a.r yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.n 2 20.e even 4 1
2200.2.a.r yes 2 20.e even 4 1
2200.2.b.j 4 4.b odd 2 1
2200.2.b.j 4 20.d odd 2 1
4400.2.a.bk 2 5.c odd 4 1
4400.2.a.bq 2 5.c odd 4 1
4400.2.b.ba 4 1.a even 1 1 trivial
4400.2.b.ba 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{3}^{4} + 3 T_{3}^{2} + 1 \)
\( T_{7}^{4} + 3 T_{7}^{2} + 1 \)
\( T_{13}^{2} + 5 \)
\( T_{17}^{4} + 27 T_{17}^{2} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + 3 T^{2} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( 5 + T^{2} )^{2} \)
$17$ \( 81 + 27 T^{2} + T^{4} \)
$19$ \( ( -19 + 2 T + T^{2} )^{2} \)
$23$ \( 3481 + 127 T^{2} + T^{4} \)
$29$ \( ( 29 - 11 T + T^{2} )^{2} \)
$31$ \( ( -1 - 4 T + T^{2} )^{2} \)
$37$ \( 121 + 42 T^{2} + T^{4} \)
$41$ \( ( -79 + 2 T + T^{2} )^{2} \)
$43$ \( 16 + 72 T^{2} + T^{4} \)
$47$ \( 1681 + 98 T^{2} + T^{4} \)
$53$ \( 121 + 23 T^{2} + T^{4} \)
$59$ \( ( 31 + 12 T + T^{2} )^{2} \)
$61$ \( ( -29 - 3 T + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( ( 4 - 14 T + T^{2} )^{2} \)
$73$ \( 841 + 67 T^{2} + T^{4} \)
$79$ \( ( -11 + T + T^{2} )^{2} \)
$83$ \( 81 + 63 T^{2} + T^{4} \)
$89$ \( ( -95 - 5 T + T^{2} )^{2} \)
$97$ \( 6241 + 283 T^{2} + T^{4} \)
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