Properties

Label 855.2.be.a
Level $855$
Weight $2$
Character orbit 855.be
Analytic conductor $6.827$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.be (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{4} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} -4 \zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + 2 \zeta_{12} q^{2} + 2 \zeta_{12}^{2} q^{4} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{5} -4 \zeta_{12}^{3} q^{7} + ( -2 \zeta_{12} - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + q^{11} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{13} + ( 8 - 8 \zeta_{12}^{2} ) q^{14} + ( 4 - 4 \zeta_{12}^{2} ) q^{16} -2 \zeta_{12} q^{17} + ( 2 + 3 \zeta_{12}^{2} ) q^{19} + ( -2 - 4 \zeta_{12}^{3} ) q^{20} + 2 \zeta_{12} q^{22} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{23} + ( 4 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{25} + 4 q^{26} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{28} -9 \zeta_{12}^{2} q^{29} -7 q^{31} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{32} -4 \zeta_{12}^{2} q^{34} + ( -8 + 4 \zeta_{12} + 8 \zeta_{12}^{2} ) q^{35} + 2 \zeta_{12}^{3} q^{37} + ( 4 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{38} + ( 2 - 2 \zeta_{12}^{2} ) q^{41} + 2 \zeta_{12} q^{43} + 2 \zeta_{12}^{2} q^{44} + 12 q^{46} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{47} -9 q^{49} + ( 8 + 6 \zeta_{12}^{3} ) q^{50} + 4 \zeta_{12} q^{52} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{53} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{55} -18 \zeta_{12}^{3} q^{58} + ( -9 + 9 \zeta_{12}^{2} ) q^{59} + 7 \zeta_{12}^{2} q^{61} -14 \zeta_{12} q^{62} + 8 q^{64} + ( -4 + 2 \zeta_{12}^{3} ) q^{65} + ( -10 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{67} -4 \zeta_{12}^{3} q^{68} + ( -16 \zeta_{12} + 8 \zeta_{12}^{2} + 16 \zeta_{12}^{3} ) q^{70} + ( 1 - \zeta_{12}^{2} ) q^{71} + 10 \zeta_{12} q^{73} + ( -4 + 4 \zeta_{12}^{2} ) q^{74} + ( -6 + 10 \zeta_{12}^{2} ) q^{76} -4 \zeta_{12}^{3} q^{77} + ( 1 - \zeta_{12}^{2} ) q^{79} + ( -8 \zeta_{12} + 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{80} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{82} -6 \zeta_{12}^{3} q^{83} + ( 2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + 4 \zeta_{12}^{2} q^{86} + 11 \zeta_{12}^{2} q^{89} -8 \zeta_{12}^{2} q^{91} + 12 \zeta_{12} q^{92} -12 q^{94} + ( -5 - 4 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{95} + 6 \zeta_{12} q^{97} -18 \zeta_{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 2 q^{5} + O(q^{10}) \) \( 4 q + 4 q^{4} - 2 q^{5} - 8 q^{10} + 4 q^{11} + 16 q^{14} + 8 q^{16} + 14 q^{19} - 8 q^{20} + 6 q^{25} + 16 q^{26} - 18 q^{29} - 28 q^{31} - 8 q^{34} - 16 q^{35} + 4 q^{41} + 4 q^{44} + 48 q^{46} - 36 q^{49} + 32 q^{50} - 2 q^{55} - 18 q^{59} + 14 q^{61} + 32 q^{64} - 16 q^{65} + 16 q^{70} + 2 q^{71} - 8 q^{74} - 4 q^{76} + 2 q^{79} + 8 q^{80} + 8 q^{85} + 8 q^{86} + 22 q^{89} - 16 q^{91} - 48 q^{94} - 16 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.73205 + 1.00000i 0 1.00000 1.73205i 1.23205 1.86603i 0 4.00000i 0 0 −0.267949 + 4.46410i
64.2 1.73205 1.00000i 0 1.00000 1.73205i −2.23205 + 0.133975i 0 4.00000i 0 0 −3.73205 + 2.46410i
334.1 −1.73205 1.00000i 0 1.00000 + 1.73205i 1.23205 + 1.86603i 0 4.00000i 0 0 −0.267949 4.46410i
334.2 1.73205 + 1.00000i 0 1.00000 + 1.73205i −2.23205 0.133975i 0 4.00000i 0 0 −3.73205 2.46410i
\(n\): 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
19.c even 3 1 inner
95.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.be.a 4
3.b odd 2 1 95.2.i.a 4
5.b even 2 1 inner 855.2.be.a 4
15.d odd 2 1 95.2.i.a 4
15.e even 4 1 475.2.e.a 2
15.e even 4 1 475.2.e.c 2
19.c even 3 1 inner 855.2.be.a 4
57.f even 6 1 1805.2.b.a 2
57.h odd 6 1 95.2.i.a 4
57.h odd 6 1 1805.2.b.b 2
95.i even 6 1 inner 855.2.be.a 4
285.n odd 6 1 95.2.i.a 4
285.n odd 6 1 1805.2.b.b 2
285.q even 6 1 1805.2.b.a 2
285.v even 12 1 475.2.e.a 2
285.v even 12 1 475.2.e.c 2
285.v even 12 1 9025.2.a.b 1
285.v even 12 1 9025.2.a.i 1
285.w odd 12 1 9025.2.a.a 1
285.w odd 12 1 9025.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 3.b odd 2 1
95.2.i.a 4 15.d odd 2 1
95.2.i.a 4 57.h odd 6 1
95.2.i.a 4 285.n odd 6 1
475.2.e.a 2 15.e even 4 1
475.2.e.a 2 285.v even 12 1
475.2.e.c 2 15.e even 4 1
475.2.e.c 2 285.v even 12 1
855.2.be.a 4 1.a even 1 1 trivial
855.2.be.a 4 5.b even 2 1 inner
855.2.be.a 4 19.c even 3 1 inner
855.2.be.a 4 95.i even 6 1 inner
1805.2.b.a 2 57.f even 6 1
1805.2.b.a 2 285.q even 6 1
1805.2.b.b 2 57.h odd 6 1
1805.2.b.b 2 285.n odd 6 1
9025.2.a.a 1 285.w odd 12 1
9025.2.a.b 1 285.v even 12 1
9025.2.a.i 1 285.v even 12 1
9025.2.a.j 1 285.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4 T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 25 + 10 T - T^{2} + 2 T^{3} + T^{4} \)
$7$ \( ( 16 + T^{2} )^{2} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( 16 - 4 T^{2} + T^{4} \)
$19$ \( ( 19 - 7 T + T^{2} )^{2} \)
$23$ \( 1296 - 36 T^{2} + T^{4} \)
$29$ \( ( 81 + 9 T + T^{2} )^{2} \)
$31$ \( ( 7 + T )^{4} \)
$37$ \( ( 4 + T^{2} )^{2} \)
$41$ \( ( 4 - 2 T + T^{2} )^{2} \)
$43$ \( 16 - 4 T^{2} + T^{4} \)
$47$ \( 1296 - 36 T^{2} + T^{4} \)
$53$ \( 256 - 16 T^{2} + T^{4} \)
$59$ \( ( 81 + 9 T + T^{2} )^{2} \)
$61$ \( ( 49 - 7 T + T^{2} )^{2} \)
$67$ \( 10000 - 100 T^{2} + T^{4} \)
$71$ \( ( 1 - T + T^{2} )^{2} \)
$73$ \( 10000 - 100 T^{2} + T^{4} \)
$79$ \( ( 1 - T + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 121 - 11 T + T^{2} )^{2} \)
$97$ \( 1296 - 36 T^{2} + T^{4} \)
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