Properties

Label 880.2.bd.d
Level 880
Weight 2
Character orbit 880.bd
Analytic conductor 7.027
Analytic rank 0
Dimension 4
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 880.bd (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} + ( -1 + 3 \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + ( -\beta_{1} + 2 \beta_{2} ) q^{5} + ( -1 + 3 \beta_{2} - 2 \beta_{3} ) q^{9} + ( 2 \beta_{1} - \beta_{2} ) q^{11} + ( 1 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{15} + ( -5 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{23} + ( -1 - 3 \beta_{3} ) q^{25} + ( -6 + \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{27} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{31} + ( -5 + \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{33} + ( -2 - 3 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{37} + ( -6 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} ) q^{45} + ( 7 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{47} -7 \beta_{2} q^{49} + ( -5 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{53} + ( -4 + 3 \beta_{3} ) q^{55} + ( 1 + 2 \beta_{3} ) q^{59} + ( -8 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 4 + \beta_{2} + 8 \beta_{3} ) q^{69} + 3 q^{71} + ( -9 + \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{75} + ( -2 + 6 \beta_{1} - 3 \beta_{2} ) q^{81} -9 \beta_{2} q^{89} + ( 15 - 3 \beta_{1} + 18 \beta_{2} - 3 \beta_{3} ) q^{93} + ( -7 - 3 \beta_{1} + 10 \beta_{2} + 3 \beta_{3} ) q^{97} + ( 3 - 11 \beta_{2} + 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + O(q^{10}) \) \( 4q + 2q^{3} + 8q^{15} - 18q^{23} + 2q^{25} - 22q^{27} - 22q^{33} - 14q^{37} - 18q^{45} + 24q^{47} - 12q^{53} - 22q^{55} - 26q^{67} + 12q^{71} - 32q^{75} - 8q^{81} + 66q^{93} - 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
1.65831 0.500000i
−1.65831 0.500000i
1.65831 + 0.500000i
−1.65831 + 0.500000i
0 −1.15831 + 1.15831i 0 −1.65831 1.50000i 0 0 0 0.316625i 0
417.2 0 2.15831 2.15831i 0 1.65831 1.50000i 0 0 0 6.31662i 0
593.1 0 −1.15831 1.15831i 0 −1.65831 + 1.50000i 0 0 0 0.316625i 0
593.2 0 2.15831 + 2.15831i 0 1.65831 + 1.50000i 0 0 0 6.31662i 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.c odd 4 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bd.d 4
4.b odd 2 1 55.2.e.b 4
5.c odd 4 1 inner 880.2.bd.d 4
11.b odd 2 1 CM 880.2.bd.d 4
12.b even 2 1 495.2.k.a 4
20.d odd 2 1 275.2.e.a 4
20.e even 4 1 55.2.e.b 4
20.e even 4 1 275.2.e.a 4
44.c even 2 1 55.2.e.b 4
44.g even 10 4 605.2.m.a 16
44.h odd 10 4 605.2.m.a 16
55.e even 4 1 inner 880.2.bd.d 4
60.l odd 4 1 495.2.k.a 4
132.d odd 2 1 495.2.k.a 4
220.g even 2 1 275.2.e.a 4
220.i odd 4 1 55.2.e.b 4
220.i odd 4 1 275.2.e.a 4
220.v even 20 4 605.2.m.a 16
220.w odd 20 4 605.2.m.a 16
660.q even 4 1 495.2.k.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.b 4 4.b odd 2 1
55.2.e.b 4 20.e even 4 1
55.2.e.b 4 44.c even 2 1
55.2.e.b 4 220.i odd 4 1
275.2.e.a 4 20.d odd 2 1
275.2.e.a 4 20.e even 4 1
275.2.e.a 4 220.g even 2 1
275.2.e.a 4 220.i odd 4 1
495.2.k.a 4 12.b even 2 1
495.2.k.a 4 60.l odd 4 1
495.2.k.a 4 132.d odd 2 1
495.2.k.a 4 660.q even 4 1
605.2.m.a 16 44.g even 10 4
605.2.m.a 16 44.h odd 10 4
605.2.m.a 16 220.v even 20 4
605.2.m.a 16 220.w odd 20 4
880.2.bd.d 4 1.a even 1 1 trivial
880.2.bd.d 4 5.c odd 4 1 inner
880.2.bd.d 4 11.b odd 2 1 CM
880.2.bd.d 4 55.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\):

\( T_{3}^{4} - 2 T_{3}^{3} + 2 T_{3}^{2} + 10 T_{3} + 25 \)
\( T_{7} \)
\( T_{13} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - T + 3 T^{2} )^{2}( 1 - 5 T^{2} + 9 T^{4} ) \)
$5$ \( 1 - T^{2} + 25 T^{4} \)
$7$ \( ( 1 + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 9 T + 23 T^{2} )^{2}( 1 + 35 T^{2} + 529 T^{4} ) \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 37 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{2}( 1 - 25 T^{2} + 1369 T^{4} ) \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )^{2}( 1 + 50 T^{2} + 2209 T^{4} ) \)
$53$ \( ( 1 + 6 T + 53 T^{2} )^{2}( 1 - 70 T^{2} + 2809 T^{4} ) \)
$59$ \( ( 1 - 15 T + 59 T^{2} )^{2}( 1 + 15 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 + 13 T + 67 T^{2} )^{2}( 1 + 35 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 - 3 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 97 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 17 T + 97 T^{2} )^{2}( 1 + 95 T^{2} + 9409 T^{4} ) \)
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