Properties

Label 855.2.k.h
Level $855$
Weight $2$
Character orbit 855.k
Analytic conductor $6.827$
Analytic rank $0$
Dimension $8$
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.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.4601315889.1
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 1) q^{4} + \beta_{5} q^{5} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + ( - 2 \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 3) q^{8} + ( - \beta_{7} + \beta_{4}) q^{10} + (\beta_{6} - 2 \beta_{3} - \beta_{2}) q^{11} + (\beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{13} + (2 \beta_{7} + \beta_{2} + \beta_1) q^{14} + (4 \beta_{7} - \beta_{5} + 2 \beta_{2} + \beta_1) q^{16} + ( - \beta_{7} - 2 \beta_1) q^{17} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 1) q^{19} + ( - \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{20} + ( - \beta_{7} + \beta_{2} - 3 \beta_1) q^{22} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_1) q^{23} + (\beta_{5} - 1) q^{25} + (\beta_{4} + 2 \beta_{3}) q^{26} + (3 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 4) q^{28} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_1) q^{29} + ( - \beta_{6} - 3 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{31} + (5 \beta_{7} + 3 \beta_{6} - 6 \beta_{5} - 5 \beta_{4} - \beta_{3} - \beta_1 + 6) q^{32} + ( - \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 3) q^{34} + ( - \beta_{5} + \beta_{2}) q^{35} + (\beta_{6} - 2 \beta_{4} - \beta_{2} - 1) q^{37} + (2 \beta_{7} + \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{38} + (\beta_{7} - 3 \beta_{5} + 2 \beta_{2} + \beta_1) q^{40} + ( - 2 \beta_{7} - 3 \beta_{5} + 3 \beta_{2} + 2 \beta_1) q^{41} + (\beta_{7} + \beta_{5} - \beta_{2} - 4 \beta_1) q^{43} + ( - \beta_{6} + 3 \beta_{5} - 3) q^{44} + ( - \beta_{6} + \beta_{2} + 3) q^{46} + ( - 2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 3) q^{47} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{49} + \beta_{4} q^{50} + (\beta_{7} + \beta_{5} + \beta_{2}) q^{52} + ( - 3 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{53} + ( - \beta_{2} + 2 \beta_1) q^{55} + (3 \beta_{6} - 5 \beta_{4} - 3 \beta_{2} + 9) q^{56} + ( - 5 \beta_{6} + 4 \beta_{4} - \beta_{3} + 5 \beta_{2} - 6) q^{58} - 5 \beta_1 q^{59} + ( - 4 \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{4} + 1) q^{61} + ( - \beta_{7} + 9 \beta_{5} - \beta_{2}) q^{62} + (3 \beta_{6} - 6 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 13) q^{64} + (\beta_{6} - \beta_{3} - \beta_{2} - 2) q^{65} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 2) q^{67} + ( - 2 \beta_{6} + 5 \beta_{4} - \beta_{3} + 2 \beta_{2} - 3) q^{68} + (2 \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1) q^{70} + (3 \beta_{7} + 6 \beta_{5} - \beta_1) q^{71} + ( - \beta_{7} + 4 \beta_{5} - \beta_{2} + 3 \beta_1) q^{73} + ( - 2 \beta_{7} + 6 \beta_{5} - 3 \beta_{2} - \beta_1) q^{74} + (2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 5 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 8) q^{76} + ( - 2 \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 3) q^{77} + ( - 2 \beta_{7} - 5 \beta_{5} + \beta_{2} + \beta_1) q^{79} + (4 \beta_{7} + 2 \beta_{6} - \beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{80} + (4 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_1 - 6) q^{82} + (2 \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{83} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{85} + ( - \beta_{7} - 4 \beta_{6} - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_1 + 3) q^{86} + ( - 3 \beta_{6} + 2 \beta_{4} + 5 \beta_{3} + 3 \beta_{2}) q^{88} + (2 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_1 + 3) q^{89} + (\beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_1 - 1) q^{91} + ( - \beta_{2} - \beta_1) q^{92} + ( - \beta_{6} + 2 \beta_{4} + 5 \beta_{3} + \beta_{2} - 6) q^{94} + (\beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{95} + (\beta_{5} - 4 \beta_{2} - 5 \beta_1) q^{97} + (\beta_{7} + 3 \beta_{5} - \beta_{2} - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8} - q^{10} + 4 q^{11} - 7 q^{13} - q^{14} - 7 q^{16} - q^{17} + 5 q^{19} - 10 q^{20} - 2 q^{22} + 2 q^{23} - 4 q^{25} - 6 q^{26} + 19 q^{28} - q^{29} + 30 q^{32} - 15 q^{34} - 4 q^{35} - 4 q^{37} - 13 q^{38} - 12 q^{40} - 8 q^{41} - q^{43} - 12 q^{44} + 24 q^{46} - 12 q^{47} - 20 q^{49} - 2 q^{50} + 3 q^{52} - 5 q^{53} + 2 q^{55} + 82 q^{56} - 54 q^{58} - 5 q^{59} + 37 q^{62} + 112 q^{64} - 14 q^{65} - 4 q^{67} - 32 q^{68} + q^{70} + 20 q^{71} + 20 q^{73} + 25 q^{74} + 63 q^{76} - 28 q^{77} - 17 q^{79} + 7 q^{80} - 21 q^{82} - 2 q^{83} + q^{85} + 8 q^{86} - 14 q^{88} + 11 q^{89} - 6 q^{91} - q^{92} - 62 q^{94} + 4 q^{95} - q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{5} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} - 12\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
1.07988 + 1.87040i
−1.02359 1.77290i
0.689667 + 1.19454i
−0.245959 0.426014i
1.07988 1.87040i
−1.02359 + 1.77290i
0.689667 1.19454i
−0.245959 + 0.426014i
−0.832272 1.44154i 0 −0.385355 + 0.667454i 0.500000 + 0.866025i 0 −2.43525 −2.04621 0 0.832272 1.44154i
406.2 −0.595455 1.03136i 0 0.290867 0.503797i 0.500000 + 0.866025i 0 −0.609175 −3.07461 0 0.595455 1.03136i
406.3 0.548719 + 0.950409i 0 0.397815 0.689035i 0.500000 + 0.866025i 0 1.89307 3.06803 0 −0.548719 + 0.950409i
406.4 1.37901 + 2.38851i 0 −2.80333 + 4.85550i 0.500000 + 0.866025i 0 −2.84864 −9.94721 0 −1.37901 + 2.38851i
676.1 −0.832272 + 1.44154i 0 −0.385355 0.667454i 0.500000 0.866025i 0 −2.43525 −2.04621 0 0.832272 + 1.44154i
676.2 −0.595455 + 1.03136i 0 0.290867 + 0.503797i 0.500000 0.866025i 0 −0.609175 −3.07461 0 0.595455 + 1.03136i
676.3 0.548719 0.950409i 0 0.397815 + 0.689035i 0.500000 0.866025i 0 1.89307 3.06803 0 −0.548719 0.950409i
676.4 1.37901 2.38851i 0 −2.80333 4.85550i 0.500000 0.866025i 0 −2.84864 −9.94721 0 −1.37901 2.38851i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 676.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.k.h 8
3.b odd 2 1 95.2.e.c 8
12.b even 2 1 1520.2.q.o 8
15.d odd 2 1 475.2.e.e 8
15.e even 4 2 475.2.j.c 16
19.c even 3 1 inner 855.2.k.h 8
57.f even 6 1 1805.2.a.i 4
57.h odd 6 1 95.2.e.c 8
57.h odd 6 1 1805.2.a.o 4
228.m even 6 1 1520.2.q.o 8
285.n odd 6 1 475.2.e.e 8
285.n odd 6 1 9025.2.a.bg 4
285.q even 6 1 9025.2.a.bp 4
285.v even 12 2 475.2.j.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 3.b odd 2 1
95.2.e.c 8 57.h odd 6 1
475.2.e.e 8 15.d odd 2 1
475.2.e.e 8 285.n odd 6 1
475.2.j.c 16 15.e even 4 2
475.2.j.c 16 285.v even 12 2
855.2.k.h 8 1.a even 1 1 trivial
855.2.k.h 8 19.c even 3 1 inner
1520.2.q.o 8 12.b even 2 1
1520.2.q.o 8 228.m even 6 1
1805.2.a.i 4 57.f even 6 1
1805.2.a.o 4 57.h odd 6 1
9025.2.a.bg 4 285.n odd 6 1
9025.2.a.bp 4 285.q even 6 1

Hecke kernels

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

\( T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} + 4T_{2}^{5} + 31T_{2}^{4} + 6T_{2}^{3} + 37T_{2}^{2} + 6T_{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{4} + 4T_{7}^{3} - T_{7}^{2} - 15T_{7} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 7 T^{6} + 4 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - T^{2} - 15 T - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 25 T^{2} + 19 T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} + 42 T^{6} + 95 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} + 31 T^{6} - 64 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + 31 T^{6} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + 21 T^{6} + 48 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + 64 T^{6} - 451 T^{5} + \cdots + 19881 \) Copy content Toggle raw display
$31$ \( (T^{4} - 67 T^{2} - 5 T + 1063)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 31 T^{2} - 123 T - 118)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + 151 T^{6} + \cdots + 5008644 \) Copy content Toggle raw display
$43$ \( T^{8} + T^{7} + 99 T^{6} + \cdots + 630436 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + 199 T^{6} + \cdots + 5363856 \) Copy content Toggle raw display
$53$ \( T^{8} + 5 T^{7} + 111 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{7} + 150 T^{6} + \cdots + 3515625 \) Copy content Toggle raw display
$61$ \( T^{8} + 130 T^{6} + 176 T^{5} + \cdots + 9296401 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + 52 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + 309 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + 305 T^{6} + \cdots + 2979076 \) Copy content Toggle raw display
$79$ \( T^{8} + 17 T^{7} + 217 T^{6} + \cdots + 33856 \) Copy content Toggle raw display
$83$ \( (T^{4} + T^{3} - 62 T^{2} - 55 T + 366)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 11 T^{7} + 211 T^{6} + \cdots + 14561856 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + 267 T^{6} + \cdots + 55383364 \) Copy content Toggle raw display
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