Properties

Label 4400.2.b.bb
Level $4400$
Weight $2$
Character orbit 4400.b
Analytic conductor $35.134$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.96668224.1
Defining polynomial: \(x^{6} + 15 x^{4} + 61 x^{2} + 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{2} - \beta_{4} ) q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{2} - \beta_{4} ) q^{7} + ( -2 + \beta_{3} ) q^{9} - q^{11} -\beta_{2} q^{13} + ( \beta_{1} - 2 \beta_{2} ) q^{17} + ( 3 - 2 \beta_{5} ) q^{19} + ( -1 - \beta_{3} + 3 \beta_{5} ) q^{21} + ( \beta_{1} + 3 \beta_{2} ) q^{23} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{27} + ( -4 - \beta_{3} + 2 \beta_{5} ) q^{29} + ( 1 - 2 \beta_{3} ) q^{31} -\beta_{1} q^{33} + ( \beta_{1} - 3 \beta_{2} - \beta_{4} ) q^{37} + \beta_{5} q^{39} + ( 1 + \beta_{3} + \beta_{5} ) q^{41} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{43} + ( 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{47} + ( -3 - \beta_{5} ) q^{49} + ( -5 + \beta_{3} + 2 \beta_{5} ) q^{51} + ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} ) q^{53} + ( 3 \beta_{1} - 10 \beta_{2} - 2 \beta_{4} ) q^{57} + ( 1 + \beta_{3} - 3 \beta_{5} ) q^{59} + ( -1 - \beta_{3} + 2 \beta_{5} ) q^{61} + ( \beta_{1} + 11 \beta_{2} + \beta_{4} ) q^{63} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{67} + ( -5 + \beta_{3} - 3 \beta_{5} ) q^{69} + ( -6 + \beta_{3} + \beta_{5} ) q^{71} + ( 7 \beta_{2} + 3 \beta_{4} ) q^{73} + ( \beta_{2} + \beta_{4} ) q^{77} + ( 4 + 2 \beta_{3} - \beta_{5} ) q^{79} + ( -2 + \beta_{3} + \beta_{5} ) q^{81} + ( -4 \beta_{1} + 8 \beta_{2} - \beta_{4} ) q^{83} + ( -2 \beta_{1} + 9 \beta_{2} + 3 \beta_{4} ) q^{87} + ( -10 - \beta_{3} ) q^{89} + ( -1 + \beta_{3} ) q^{91} + ( 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{93} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{4} ) q^{97} + ( 2 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{9} + O(q^{10}) \) \( 6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 15 x^{4} + 61 x^{2} + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 9 \nu^{3} + 13 \nu \)\()/6\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 5 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 3 \nu^{3} - 29 \nu \)\()/6\)
\(\beta_{5}\)\(=\)\( \nu^{4} + 8 \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 5\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{2} - 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{5} - 8 \beta_{3} + 34\)
\(\nu^{5}\)\(=\)\(9 \beta_{4} - 3 \beta_{2} + 50 \beta_{1}\)

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
2.75153i
2.59261i
0.841083i
0.841083i
2.59261i
2.75153i
0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
4049.2 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.3 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.4 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.5 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.6 0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4049.6
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.bb 6
4.b odd 2 1 2200.2.b.m 6
5.b even 2 1 inner 4400.2.b.bb 6
5.c odd 4 1 4400.2.a.by 3
5.c odd 4 1 4400.2.a.bz 3
20.d odd 2 1 2200.2.b.m 6
20.e even 4 1 2200.2.a.u 3
20.e even 4 1 2200.2.a.v yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.u 3 20.e even 4 1
2200.2.a.v yes 3 20.e even 4 1
2200.2.b.m 6 4.b odd 2 1
2200.2.b.m 6 20.d odd 2 1
4400.2.a.by 3 5.c odd 4 1
4400.2.a.bz 3 5.c odd 4 1
4400.2.b.bb 6 1.a even 1 1 trivial
4400.2.b.bb 6 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 \(S_{2}^{\mathrm{new}}(4400, [\chi])\):

\( T_{3}^{6} + 15 T_{3}^{4} + 61 T_{3}^{2} + 36 \)
\( T_{7}^{6} + 31 T_{7}^{4} + 313 T_{7}^{2} + 1024 \)
\( T_{13}^{2} + 1 \)
\( T_{17}^{6} + 23 T_{17}^{4} + 41 T_{17}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 36 + 61 T^{2} + 15 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1024 + 313 T^{2} + 31 T^{4} + T^{6} \)
$11$ \( ( 1 + T )^{6} \)
$13$ \( ( 1 + T^{2} )^{3} \)
$17$ \( 16 + 41 T^{2} + 23 T^{4} + T^{6} \)
$19$ \( ( 27 - 13 T - 7 T^{2} + T^{3} )^{2} \)
$23$ \( 81 + 496 T^{2} + 48 T^{4} + T^{6} \)
$29$ \( ( -201 - 8 T + 10 T^{2} + T^{3} )^{2} \)
$31$ \( ( 207 - 53 T - 3 T^{2} + T^{3} )^{2} \)
$37$ \( 1936 + 705 T^{2} + 66 T^{4} + T^{6} \)
$41$ \( ( 24 - 17 T - 4 T^{2} + T^{3} )^{2} \)
$43$ \( 258064 + 12280 T^{2} + 193 T^{4} + T^{6} \)
$47$ \( 36864 + 5929 T^{2} + 154 T^{4} + T^{6} \)
$53$ \( 20164 + 21301 T^{2} + 307 T^{4} + T^{6} \)
$59$ \( ( 192 - 77 T + T^{3} )^{2} \)
$61$ \( ( -114 - 41 T + T^{2} + T^{3} )^{2} \)
$67$ \( 9216 + 2944 T^{2} + 228 T^{4} + T^{6} \)
$71$ \( ( 52 + 74 T + 17 T^{2} + T^{3} )^{2} \)
$73$ \( 1106704 + 44625 T^{2} + 399 T^{4} + T^{6} \)
$79$ \( ( 144 - 21 T - 11 T^{2} + T^{3} )^{2} \)
$83$ \( 687241 + 33196 T^{2} + 388 T^{4} + T^{6} \)
$89$ \( ( 879 + 286 T + 30 T^{2} + T^{3} )^{2} \)
$97$ \( 529 + 1484 T^{2} + 164 T^{4} + T^{6} \)
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