Properties

Label 4400.2.b.t
Level $4400$
Weight $2$
Character orbit 4400.b
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{1} q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{1} q^{7} + ( -2 + \beta_{3} ) q^{9} - q^{11} -\beta_{2} q^{13} + ( \beta_{1} + \beta_{2} ) q^{17} + ( 1 - \beta_{3} ) q^{19} + ( 5 - \beta_{3} ) q^{21} -2 \beta_{1} q^{23} + ( \beta_{1} + 2 \beta_{2} ) q^{27} + ( -5 + 3 \beta_{3} ) q^{29} + ( -5 + \beta_{3} ) q^{31} -\beta_{1} q^{33} + ( 3 \beta_{1} + \beta_{2} ) q^{37} + ( -2 + 2 \beta_{3} ) q^{39} + 2 q^{41} -4 \beta_{1} q^{43} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{47} + ( 2 + \beta_{3} ) q^{49} + ( -3 - \beta_{3} ) q^{51} + ( \beta_{1} - \beta_{2} ) q^{53} + ( \beta_{1} - 2 \beta_{2} ) q^{57} + ( -2 - 2 \beta_{3} ) q^{59} + ( 7 + 3 \beta_{3} ) q^{61} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{63} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{67} + ( 10 - 2 \beta_{3} ) q^{69} + ( 1 + 3 \beta_{3} ) q^{71} + \beta_{2} q^{73} + \beta_{1} q^{77} + ( -6 + 6 \beta_{3} ) q^{79} -7 q^{81} -2 \beta_{1} q^{83} + ( -5 \beta_{1} + 6 \beta_{2} ) q^{87} + ( 11 - \beta_{3} ) q^{89} + ( 2 - 2 \beta_{3} ) q^{91} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -2 \beta_{1} + \beta_{2} ) q^{97} + ( 2 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9} + O(q^{10}) \) \( 4 q - 6 q^{9} - 4 q^{11} + 2 q^{19} + 18 q^{21} - 14 q^{29} - 18 q^{31} - 4 q^{39} + 8 q^{41} + 10 q^{49} - 14 q^{51} - 12 q^{59} + 34 q^{61} + 36 q^{69} + 10 q^{71} - 12 q^{79} - 28 q^{81} + 42 q^{89} + 4 q^{91} + 6 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9 x^{2} + 16\):

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

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
2.56155i
1.56155i
1.56155i
2.56155i
0 2.56155i 0 0 0 2.56155i 0 −3.56155 0
4049.2 0 1.56155i 0 0 0 1.56155i 0 0.561553 0
4049.3 0 1.56155i 0 0 0 1.56155i 0 0.561553 0
4049.4 0 2.56155i 0 0 0 2.56155i 0 −3.56155 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.t 4
4.b odd 2 1 2200.2.b.i 4
5.b even 2 1 inner 4400.2.b.t 4
5.c odd 4 1 880.2.a.o 2
5.c odd 4 1 4400.2.a.bj 2
15.e even 4 1 7920.2.a.bu 2
20.d odd 2 1 2200.2.b.i 4
20.e even 4 1 440.2.a.e 2
20.e even 4 1 2200.2.a.s 2
40.i odd 4 1 3520.2.a.bk 2
40.k even 4 1 3520.2.a.bp 2
55.e even 4 1 9680.2.a.bs 2
60.l odd 4 1 3960.2.a.w 2
220.i odd 4 1 4840.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.e 2 20.e even 4 1
880.2.a.o 2 5.c odd 4 1
2200.2.a.s 2 20.e even 4 1
2200.2.b.i 4 4.b odd 2 1
2200.2.b.i 4 20.d odd 2 1
3520.2.a.bk 2 40.i odd 4 1
3520.2.a.bp 2 40.k even 4 1
3960.2.a.w 2 60.l odd 4 1
4400.2.a.bj 2 5.c odd 4 1
4400.2.b.t 4 1.a even 1 1 trivial
4400.2.b.t 4 5.b even 2 1 inner
4840.2.a.j 2 220.i odd 4 1
7920.2.a.bu 2 15.e even 4 1
9680.2.a.bs 2 55.e even 4 1

Hecke kernels

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

\( T_{3}^{4} + 9 T_{3}^{2} + 16 \)
\( T_{7}^{4} + 9 T_{7}^{2} + 16 \)
\( T_{13}^{2} + 4 \)
\( T_{17}^{4} + 13 T_{17}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 16 + 9 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 + 9 T^{2} + T^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( ( 4 + T^{2} )^{2} \)
$17$ \( 4 + 13 T^{2} + T^{4} \)
$19$ \( ( -4 - T + T^{2} )^{2} \)
$23$ \( 256 + 36 T^{2} + T^{4} \)
$29$ \( ( -26 + 7 T + T^{2} )^{2} \)
$31$ \( ( 16 + 9 T + T^{2} )^{2} \)
$37$ \( 1444 + 77 T^{2} + T^{4} \)
$41$ \( ( -2 + T )^{4} \)
$43$ \( 4096 + 144 T^{2} + T^{4} \)
$47$ \( 4096 + 196 T^{2} + T^{4} \)
$53$ \( 4 + 21 T^{2} + T^{4} \)
$59$ \( ( -8 + 6 T + T^{2} )^{2} \)
$61$ \( ( 34 - 17 T + T^{2} )^{2} \)
$67$ \( 4096 + 144 T^{2} + T^{4} \)
$71$ \( ( -32 - 5 T + T^{2} )^{2} \)
$73$ \( ( 4 + T^{2} )^{2} \)
$79$ \( ( -144 + 6 T + T^{2} )^{2} \)
$83$ \( 256 + 36 T^{2} + T^{4} \)
$89$ \( ( 106 - 21 T + T^{2} )^{2} \)
$97$ \( 64 + 52 T^{2} + T^{4} \)
show more
show less