Properties

Label 855.2.c.d
Level $855$
Weight $2$
Character orbit 855.c
Analytic conductor $6.827$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.16516096.1
Defining polynomial: \(x^{6} + 9 x^{4} + 13 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} +O(q^{10})\) \( q -\beta_{5} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( \beta_{3} + \beta_{4} ) q^{7} + ( \beta_{1} + \beta_{3} + \beta_{4} ) q^{8} + ( 2 - 2 \beta_{1} - \beta_{5} ) q^{10} + ( \beta_{3} - \beta_{4} ) q^{11} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} + ( 2 - 2 \beta_{2} ) q^{14} + ( -1 - \beta_{3} + \beta_{4} ) q^{16} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{17} + q^{19} + ( -1 + \beta_{2} - 2 \beta_{4} ) q^{20} + ( 4 \beta_{1} + 2 \beta_{5} ) q^{22} + 2 \beta_{1} q^{23} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{25} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} ) q^{26} + ( -2 \beta_{1} - 4 \beta_{5} ) q^{28} + 6 q^{29} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{32} + ( -2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{34} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{37} -\beta_{5} q^{38} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{40} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{41} + ( \beta_{3} + \beta_{4} ) q^{43} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{44} + ( -2 - 2 \beta_{3} + 2 \beta_{4} ) q^{46} + ( 4 \beta_{1} - \beta_{3} - \beta_{4} + 4 \beta_{5} ) q^{47} + ( 1 + \beta_{3} - \beta_{4} ) q^{49} + ( -4 \beta_{3} - \beta_{5} ) q^{50} + ( -3 \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{52} + ( -\beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{53} + ( -5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{55} + ( -6 + 2 \beta_{3} - 2 \beta_{4} ) q^{56} -6 \beta_{5} q^{58} + ( 4 + 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} ) q^{61} + ( -6 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{62} + ( 1 - 3 \beta_{2} ) q^{64} + ( 4 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{65} + ( 3 \beta_{1} - 5 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{67} + ( -6 \beta_{1} - 6 \beta_{5} ) q^{68} + ( 2 - 2 \beta_{2} + 4 \beta_{4} ) q^{70} + ( -8 + 2 \beta_{2} ) q^{71} + ( -2 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{73} + ( -6 + 3 \beta_{2} - \beta_{3} + \beta_{4} ) q^{74} + ( -1 + \beta_{2} ) q^{76} + ( -4 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{77} + ( 4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{79} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{80} + ( -6 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} ) q^{82} + ( -2 \beta_{1} + 4 \beta_{5} ) q^{83} + ( 3 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{85} + ( 2 - 2 \beta_{2} ) q^{86} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{88} + ( 2 - 8 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{89} + ( -4 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{91} + ( -4 \beta_{1} - 2 \beta_{5} ) q^{92} + ( 6 - 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} ) q^{94} + ( -\beta_{3} + \beta_{5} ) q^{95} + ( -\beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{97} + ( 4 \beta_{1} + \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 8q^{4} + q^{5} + O(q^{10}) \) \( 6q - 8q^{4} + q^{5} + 12q^{10} - 2q^{11} + 16q^{14} - 4q^{16} + 6q^{19} - 10q^{20} + 3q^{25} - 8q^{26} + 36q^{29} + 8q^{34} + 3q^{35} + 8q^{40} - 12q^{41} + 20q^{44} - 8q^{46} + 4q^{49} + 4q^{50} - 33q^{55} - 40q^{56} + 20q^{59} - 14q^{61} + 12q^{64} + 20q^{65} + 20q^{70} - 52q^{71} - 40q^{74} - 8q^{76} + 24q^{79} + 32q^{80} + 13q^{85} + 16q^{86} + 24q^{89} - 24q^{91} + 48q^{94} + q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 9 x^{4} + 13 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} + 8 \nu^{2} + 5 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + \nu^{4} + 10 \nu^{3} + 6 \nu^{2} + 19 \nu - 1 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 10 \nu^{3} - 6 \nu^{2} + 19 \nu + 1 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} - 8 \nu^{3} - 7 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} - \beta_{3} + \beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} - 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-8 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 19\)
\(\nu^{5}\)\(=\)\(-10 \beta_{5} - 8 \beta_{4} - 8 \beta_{3} + 41 \beta_{1}\)

Character values

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

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
514.1
1.30397i
2.68667i
0.285442i
0.285442i
2.68667i
1.30397i
2.41987i 0 −3.85577 2.07772 + 0.826491i 0 3.18676i 4.49073i 0 2.00000 5.02781i
514.2 1.82254i 0 −1.32164 −1.94827 + 1.09737i 0 1.45033i 1.23634i 0 2.00000 + 3.55080i
514.3 0.906968i 0 1.17741 0.370556 + 2.20515i 0 2.59637i 2.88181i 0 2.00000 0.336083i
514.4 0.906968i 0 1.17741 0.370556 2.20515i 0 2.59637i 2.88181i 0 2.00000 + 0.336083i
514.5 1.82254i 0 −1.32164 −1.94827 1.09737i 0 1.45033i 1.23634i 0 2.00000 3.55080i
514.6 2.41987i 0 −3.85577 2.07772 0.826491i 0 3.18676i 4.49073i 0 2.00000 + 5.02781i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 514.6
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 855.2.c.d 6
3.b odd 2 1 95.2.b.b 6
5.b even 2 1 inner 855.2.c.d 6
5.c odd 4 2 4275.2.a.br 6
12.b even 2 1 1520.2.d.h 6
15.d odd 2 1 95.2.b.b 6
15.e even 4 2 475.2.a.j 6
57.d even 2 1 1805.2.b.e 6
60.h even 2 1 1520.2.d.h 6
60.l odd 4 2 7600.2.a.ck 6
285.b even 2 1 1805.2.b.e 6
285.j odd 4 2 9025.2.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 3.b odd 2 1
95.2.b.b 6 15.d odd 2 1
475.2.a.j 6 15.e even 4 2
855.2.c.d 6 1.a even 1 1 trivial
855.2.c.d 6 5.b even 2 1 inner
1520.2.d.h 6 12.b even 2 1
1520.2.d.h 6 60.h even 2 1
1805.2.b.e 6 57.d even 2 1
1805.2.b.e 6 285.b even 2 1
4275.2.a.br 6 5.c odd 4 2
7600.2.a.ck 6 60.l odd 4 2
9025.2.a.bx 6 285.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\):

\( T_{2}^{6} + 10 T_{2}^{4} + 27 T_{2}^{2} + 16 \)
\( T_{11}^{3} + T_{11}^{2} - 16 T_{11} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 27 T^{2} + 10 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 125 - 25 T - 5 T^{2} + 2 T^{3} - T^{4} - T^{5} + T^{6} \)
$7$ \( 144 + 104 T^{2} + 19 T^{4} + T^{6} \)
$11$ \( ( -12 - 16 T + T^{2} + T^{3} )^{2} \)
$13$ \( 576 + 236 T^{2} + 28 T^{4} + T^{6} \)
$17$ \( 5184 + 1008 T^{2} + 59 T^{4} + T^{6} \)
$19$ \( ( -1 + T )^{6} \)
$23$ \( 64 + 208 T^{2} + 36 T^{4} + T^{6} \)
$29$ \( ( -6 + T )^{6} \)
$31$ \( ( 128 - 56 T + T^{3} )^{2} \)
$37$ \( 1296 + 764 T^{2} + 56 T^{4} + T^{6} \)
$41$ \( ( 24 - 44 T + 6 T^{2} + T^{3} )^{2} \)
$43$ \( 144 + 104 T^{2} + 19 T^{4} + T^{6} \)
$47$ \( 85264 + 7464 T^{2} + 187 T^{4} + T^{6} \)
$53$ \( 64 + 2476 T^{2} + 156 T^{4} + T^{6} \)
$59$ \( ( 48 + 8 T - 10 T^{2} + T^{3} )^{2} \)
$61$ \( ( -776 - 104 T + 7 T^{2} + T^{3} )^{2} \)
$67$ \( 484416 + 28556 T^{2} + 340 T^{4} + T^{6} \)
$71$ \( ( 432 + 200 T + 26 T^{2} + T^{3} )^{2} \)
$73$ \( 5184 + 1616 T^{2} + 131 T^{4} + T^{6} \)
$79$ \( ( 32 - 8 T - 12 T^{2} + T^{3} )^{2} \)
$83$ \( 141376 + 11728 T^{2} + 228 T^{4} + T^{6} \)
$89$ \( ( 3456 - 284 T - 12 T^{2} + T^{3} )^{2} \)
$97$ \( 576 + 236 T^{2} + 28 T^{4} + T^{6} \)
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