Properties

Label 855.2.be.d
Level $855$
Weight $2$
Character orbit 855.be
Analytic conductor $6.827$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6x^{10} + 29x^{8} - 40x^{6} + 43x^{4} - 7x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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 basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{9} - \beta_{8} + \beta_{3} - \beta_{2}) q^{4} + (\beta_{9} + \beta_1) q^{5} + ( - 2 \beta_{11} - \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{7} + ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{9} - \beta_{8} + \beta_{3} - \beta_{2}) q^{4} + (\beta_{9} + \beta_1) q^{5} + ( - 2 \beta_{11} - \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{7} + ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{8} + (2 \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_1) q^{10} - \beta_{3} q^{11} + (\beta_{11} + \beta_{10} + 2 \beta_{7} + 2 \beta_1) q^{13} + ( - 2 \beta_{9} - 2 \beta_{8} + 3 \beta_{6} - 3) q^{14} + ( - \beta_{9} + 2 \beta_{6} - 2) q^{16} + (\beta_{5} - 2 \beta_{4}) q^{17} + (\beta_{9} - 2 \beta_{8} + \beta_{6} - 2 \beta_{3} - \beta_{2} - 1) q^{19} + ( - \beta_{11} + \beta_{7} + \beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{20} + ( - 2 \beta_{5} - \beta_{4} + \beta_1) q^{22} + ( - \beta_{10} + \beta_{7} + \beta_1) q^{23} + (2 \beta_{10} + 2 \beta_{8} - \beta_{6} + 2 \beta_{2}) q^{25} + (2 \beta_{3} - \beta_{2} + 3) q^{26} + ( - 7 \beta_{11} - 2 \beta_{10} - 5 \beta_{7} - 5 \beta_1) q^{28} + (3 \beta_{9} + 3 \beta_{6} - 3 \beta_{3}) q^{29} + ( - \beta_{2} + 5) q^{31} + (\beta_{10} - 3 \beta_{7} - 3 \beta_1) q^{32} + (\beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{3} - \beta_{2}) q^{34} + ( - \beta_{8} + 4 \beta_{5} + \beta_{4} - 2 \beta_1) q^{35} + (2 \beta_{10} + 3 \beta_{7} - 2 \beta_{4}) q^{37} + ( - 3 \beta_{11} + \beta_{10} - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} - \beta_1) q^{38} + ( - \beta_{9} - \beta_{8} - 2 \beta_{6} + 5 \beta_{5} - \beta_{4} - \beta_1 + 2) q^{40} + ( - 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{6} + 3) q^{41} + (\beta_{5} - 5 \beta_1) q^{43} + (\beta_{9} + 2 \beta_{8} - 3 \beta_{6} - \beta_{3} + 2 \beta_{2}) q^{44} + ( - \beta_{3} + 1) q^{46} + ( - 4 \beta_{11} + 4 \beta_{10} - \beta_{7} - \beta_1) q^{47} + ( - 4 \beta_{3} + 3 \beta_{2} + 1) q^{49} + (5 \beta_{11} + 3 \beta_{7} - 5 \beta_{5} + 2 \beta_{3} + 5 \beta_1) q^{50} + (7 \beta_{5} - 5 \beta_1) q^{52} + (\beta_{11} + 2 \beta_{10} - 3 \beta_{7} - 3 \beta_1) q^{53} + ( - \beta_{8} + 3 \beta_{6} - \beta_{4} - 3) q^{55} + ( - 5 \beta_{3} + 3 \beta_{2} - 6) q^{56} + (3 \beta_{11} + 3 \beta_{10} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_1) q^{58} + ( - \beta_{9} - 4 \beta_{8} + 3 \beta_{6} - 3) q^{59} + (2 \beta_{9} + 2 \beta_{8} + \beta_{6} - 2 \beta_{3} + 2 \beta_{2}) q^{61} + (7 \beta_{5} - \beta_1) q^{62} + (3 \beta_{3} + 1) q^{64} + (3 \beta_{11} + \beta_{10} + 3 \beta_{7} - 3 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 3) q^{65} + (2 \beta_{11} + 4 \beta_{7} + 4 \beta_1) q^{67} + ( - 3 \beta_{10} + 3 \beta_{4}) q^{68} + ( - 2 \beta_{11} - 5 \beta_{9} - 4 \beta_{8} - \beta_{7} + 6 \beta_{6} + 5 \beta_{3} + \cdots - \beta_1) q^{70}+ \cdots + ( - 13 \beta_{5} - 4 \beta_{4} + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} + 2 q^{5} + 6 q^{10} - 4 q^{11} - 22 q^{14} - 14 q^{16} - 12 q^{19} + 40 q^{20} - 6 q^{25} + 44 q^{26} + 12 q^{29} + 60 q^{31} + 10 q^{34} + 10 q^{40} + 12 q^{41} - 20 q^{44} + 8 q^{46} - 4 q^{49} + 8 q^{50} - 18 q^{55} - 92 q^{56} - 20 q^{59} + 2 q^{61} + 24 q^{64} + 40 q^{65} + 46 q^{70} - 2 q^{71} + 22 q^{74} - 70 q^{76} + 24 q^{79} + 22 q^{80} + 2 q^{85} - 16 q^{86} - 36 q^{89} + 24 q^{91} - 60 q^{94} - 46 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6x^{10} + 29x^{8} - 40x^{6} + 43x^{4} - 7x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -36\nu^{10} + 174\nu^{8} - 841\nu^{6} + 258\nu^{4} - 42\nu^{2} - 2207 ) / 1205 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 138\nu^{10} - 667\nu^{8} + 3023\nu^{6} - 989\nu^{4} + 161\nu^{2} + 2636 ) / 1205 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 138\nu^{11} - 667\nu^{9} + 3023\nu^{7} - 989\nu^{5} + 161\nu^{3} + 2636\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 174\nu^{11} - 841\nu^{9} + 3864\nu^{7} - 1247\nu^{5} + 203\nu^{3} + 7253\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 203\nu^{10} - 1182\nu^{8} + 5713\nu^{6} - 7279\nu^{4} + 8471\nu^{2} - 174 ) / 1205 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -203\nu^{11} + 1182\nu^{9} - 5713\nu^{7} + 7279\nu^{5} - 8471\nu^{3} + 174\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -74\nu^{10} + 438\nu^{8} - 2117\nu^{6} + 2860\nu^{4} - 3139\nu^{2} + 511 ) / 241 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -429\nu^{10} + 2676\nu^{8} - 12934\nu^{6} + 19342\nu^{4} - 19178\nu^{2} + 3122 ) / 1205 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -429\nu^{11} + 2676\nu^{9} - 12934\nu^{7} + 19342\nu^{5} - 19178\nu^{3} + 3122\nu ) / 1205 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -1002\nu^{11} + 6048\nu^{9} - 29232\nu^{7} + 40921\nu^{5} - 43344\nu^{3} + 7056\nu ) / 1205 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 2\beta_{6} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - 3\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 5\beta_{8} + 7\beta_{6} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{11} - 6\beta_{10} - 12\beta_{7} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{3} - 23\beta_{2} - 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{5} - 29\beta_{4} - 75\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 29\beta_{9} - 104\beta_{8} - 127\beta_{6} - 29\beta_{3} - 104\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -104\beta_{11} + 133\beta_{10} + 231\beta_{7} + 104\beta_{5} - 133\beta_{4} - 104\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 133\beta_{9} - 468\beta_{8} - 566\beta_{6} + 566 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -468\beta_{11} + 601\beta_{10} + 1034\beta_{7} + 1034\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{6}\)

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
−0.352587 + 0.203566i
−1.00376 + 0.579521i
−1.83525 + 1.05958i
1.83525 1.05958i
1.00376 0.579521i
0.352587 0.203566i
−0.352587 0.203566i
−1.00376 0.579521i
−1.83525 1.05958i
1.83525 + 1.05958i
1.00376 + 0.579521i
0.352587 + 0.203566i
−2.12713 + 1.22810i 0 2.01647 3.49262i 0.746759 + 2.10769i 0 4.50527i 4.99330i 0 −4.17691 3.56624i
64.2 −0.747190 + 0.431391i 0 −0.627804 + 1.08739i −1.95987 1.07652i 0 0.566520i 2.80888i 0 1.92880 0.0411078i
64.3 −0.408663 + 0.235942i 0 −0.888663 + 1.53921i −1.47848 + 1.67752i 0 1.17540i 1.78246i 0 0.208403 1.03438i
64.4 0.408663 0.235942i 0 −0.888663 + 1.53921i 2.19202 0.441641i 0 1.17540i 1.78246i 0 0.791597 0.697672i
64.5 0.747190 0.431391i 0 −0.627804 + 1.08739i 0.0476457 2.23556i 0 0.566520i 2.80888i 0 −0.928799 1.69094i
64.6 2.12713 1.22810i 0 2.01647 3.49262i 1.45193 + 1.70056i 0 4.50527i 4.99330i 0 5.17691 + 1.83419i
334.1 −2.12713 1.22810i 0 2.01647 + 3.49262i 0.746759 2.10769i 0 4.50527i 4.99330i 0 −4.17691 + 3.56624i
334.2 −0.747190 0.431391i 0 −0.627804 1.08739i −1.95987 + 1.07652i 0 0.566520i 2.80888i 0 1.92880 + 0.0411078i
334.3 −0.408663 0.235942i 0 −0.888663 1.53921i −1.47848 1.67752i 0 1.17540i 1.78246i 0 0.208403 + 1.03438i
334.4 0.408663 + 0.235942i 0 −0.888663 1.53921i 2.19202 + 0.441641i 0 1.17540i 1.78246i 0 0.791597 + 0.697672i
334.5 0.747190 + 0.431391i 0 −0.627804 1.08739i 0.0476457 + 2.23556i 0 0.566520i 2.80888i 0 −0.928799 + 1.69094i
334.6 2.12713 + 1.22810i 0 2.01647 + 3.49262i 1.45193 1.70056i 0 4.50527i 4.99330i 0 5.17691 1.83419i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 334.6
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.d 12
3.b odd 2 1 95.2.i.b 12
5.b even 2 1 inner 855.2.be.d 12
15.d odd 2 1 95.2.i.b 12
15.e even 4 2 475.2.e.g 12
19.c even 3 1 inner 855.2.be.d 12
57.f even 6 1 1805.2.b.g 6
57.h odd 6 1 95.2.i.b 12
57.h odd 6 1 1805.2.b.f 6
95.i even 6 1 inner 855.2.be.d 12
285.n odd 6 1 95.2.i.b 12
285.n odd 6 1 1805.2.b.f 6
285.q even 6 1 1805.2.b.g 6
285.v even 12 2 475.2.e.g 12
285.v even 12 2 9025.2.a.bu 6
285.w odd 12 2 9025.2.a.bt 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.b 12 3.b odd 2 1
95.2.i.b 12 15.d odd 2 1
95.2.i.b 12 57.h odd 6 1
95.2.i.b 12 285.n odd 6 1
475.2.e.g 12 15.e even 4 2
475.2.e.g 12 285.v even 12 2
855.2.be.d 12 1.a even 1 1 trivial
855.2.be.d 12 5.b even 2 1 inner
855.2.be.d 12 19.c even 3 1 inner
855.2.be.d 12 95.i even 6 1 inner
1805.2.b.f 6 57.h odd 6 1
1805.2.b.f 6 285.n odd 6 1
1805.2.b.g 6 57.f even 6 1
1805.2.b.g 6 285.q even 6 1
9025.2.a.bt 6 285.w odd 12 2
9025.2.a.bu 6 285.v even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 7T_{2}^{10} + 43T_{2}^{8} - 40T_{2}^{6} + 29T_{2}^{4} - 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 7 T^{10} + 43 T^{8} - 40 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{11} + 5 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 22 T^{4} + 35 T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + T^{2} - 4 T - 3)^{4} \) Copy content Toggle raw display
$13$ \( T^{12} - 31 T^{10} + 722 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{12} - 35 T^{10} + 1027 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + 30 T^{4} + 115 T^{3} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{10} + 137 T^{8} - 82 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + 63 T^{4} + 108 T^{3} + \cdots + 729)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 15 T^{2} + 70 T - 97)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 98 T^{4} + 887 T^{2} + 729)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 6 T^{5} + 77 T^{4} + 240 T^{3} + \cdots + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 127 T^{10} + 13613 T^{8} + \cdots + 194481 \) Copy content Toggle raw display
$47$ \( T^{12} - 214 T^{10} + \cdots + 47562811921 \) Copy content Toggle raw display
$53$ \( T^{12} - 99 T^{10} + \cdots + 131079601 \) Copy content Toggle raw display
$59$ \( (T^{6} + 10 T^{5} + 155 T^{4} + \cdots + 84681)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - T^{5} + 42 T^{4} - 185 T^{3} + \cdots + 12769)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 76 T^{10} + \cdots + 207360000 \) Copy content Toggle raw display
$71$ \( (T^{6} + T^{5} + 125 T^{4} - 1078 T^{3} + \cdots + 227529)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 236 T^{10} + \cdots + 9845600625 \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + 176 T^{4} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 459 T^{4} + 56302 T^{2} + \cdots + 966289)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 18 T^{5} + 296 T^{4} + 648 T^{3} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 529 T^{10} + \cdots + 1534548635361 \) Copy content Toggle raw display
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