↑

Properties

Label 855.2.k.j
Level $855$
Weight $2$
Character orbit 855.k
Analytic conductor $6.827$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(406,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("855.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 10 x^{9} + 44 x^{8} - 20 x^{7} + 119 x^{6} + 13 x^{5} + 83 x^{4} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{9} + \beta_{4} + \beta_{2} - \beta_1) q^{4} - \beta_{4} q^{5} + (\beta_{8} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{2} + (\beta_{9} + \beta_{4} + \beta_{2} - \beta_1) q^{4} - \beta_{4} q^{5} + (\beta_{8} - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{5} + \cdots - \beta_{2}) q^{8}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8} - 3 q^{10} - 8 q^{13} - 10 q^{14} - 3 q^{16} - 4 q^{17} - 6 q^{19} + 10 q^{20} + 2 q^{23} - 6 q^{25} + 40 q^{26} - 26 q^{28} - 4 q^{29} + 24 q^{31} - 15 q^{32} + 7 q^{34} - 2 q^{35} + 29 q^{38} - 6 q^{40} - 12 q^{41} - 4 q^{43} - 6 q^{44} + 48 q^{46} - 6 q^{47} + 32 q^{49} + 6 q^{50} - 20 q^{52} - 26 q^{53} + 44 q^{56} - 20 q^{58} - 16 q^{59} + 20 q^{61} + 25 q^{62} + 28 q^{64} + 16 q^{65} - 12 q^{67} - 54 q^{68} - 10 q^{70} + 8 q^{71} - 4 q^{73} + 16 q^{74} - 66 q^{76} - 48 q^{77} - 12 q^{79} - 3 q^{80} + 26 q^{82} + 44 q^{83} - 4 q^{85} + 44 q^{86} - 32 q^{88} + 8 q^{89} + 2 q^{91} - 36 q^{92} - 14 q^{94} + 6 q^{95} + 30 q^{97} - 49 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 13 x^{10} - 10 x^{9} + 44 x^{8} - 20 x^{7} + 119 x^{6} + 13 x^{5} + 83 x^{4} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9467 \nu^{11} - 219232 \nu^{10} + 319009 \nu^{9} - 2066475 \nu^{8} - 2131461 \nu^{7} + \cdots - 31535764 ) / 24043688 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 119083 \nu^{11} - 330656 \nu^{10} + 1240077 \nu^{9} - 175781 \nu^{8} + 2267093 \nu^{7} + \cdots + 20278716 ) / 12021844 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1873019 \nu^{11} + 5628524 \nu^{10} - 24130015 \nu^{9} + 18411181 \nu^{8} - 80346361 \nu^{7} + \cdots + 4981404 ) / 24043688 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 489392 \nu^{11} - 929709 \nu^{10} + 4518956 \nu^{9} + 2914321 \nu^{8} + 12727847 \nu^{7} + \cdots + 12751576 ) / 6010922 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2233513 \nu^{11} - 9651444 \nu^{10} + 38772881 \nu^{9} - 62266695 \nu^{8} + 135956727 \nu^{7} + \cdots - 11971084 ) / 24043688 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2481235 \nu^{11} - 3945300 \nu^{10} + 21441087 \nu^{9} + 21286535 \nu^{8} + 70637065 \nu^{7} + \cdots + 121494172 ) / 24043688 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2992771 \nu^{11} - 6744800 \nu^{10} + 29254579 \nu^{9} + 8845171 \nu^{8} + 69415229 \nu^{7} + \cdots + 108328060 ) / 24043688 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33067 \nu^{11} - 101560 \nu^{10} + 429903 \nu^{9} - 344149 \nu^{8} + 1403197 \nu^{7} + \cdots + 271084 ) / 212776 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 723138 \nu^{11} + 2635302 \nu^{10} - 10862733 \nu^{9} + 13343636 \nu^{8} - 36937486 \nu^{7} + \cdots + 2856086 ) / 3005461 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3942893 \nu^{11} + 13919486 \nu^{10} - 58052365 \nu^{9} + 68012787 \nu^{8} - 199825235 \nu^{7} + \cdots + 14863148 ) / 12021844 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{4} - \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - 2\beta_{10} - 9\beta_{9} - 3\beta_{6} - 10\beta_{4} + 9\beta_{3} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{11} - 9 \beta_{10} - 24 \beta_{9} + 3 \beta_{8} + 9 \beta_{7} - 13 \beta_{6} - 13 \beta_{5} + \cdots + 55 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{8} + 24\beta_{7} - 40\beta_{5} - 81\beta_{3} + 91\beta_{2} + 161 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40\beta_{11} + 81\beta_{10} + 236\beta_{9} + 134\beta_{6} + 182\beta_{4} - 236\beta_{3} + 286\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 134 \beta_{11} + 236 \beta_{10} + 737 \beta_{9} - 134 \beta_{8} - 236 \beta_{7} + 410 \beta_{6} + \cdots - 1391 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -410\beta_{8} - 737\beta_{7} + 1281\beta_{5} + 2202\beta_{3} - 2495\beta_{2} - 4133 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1281\beta_{11} - 2202\beta_{10} - 6715\beta_{9} - 3893\beta_{6} - 4990\beta_{4} + 6715\beta_{3} - 7390\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3893 \beta_{11} - 6715 \beta_{10} - 20200 \beta_{9} + 3893 \beta_{8} + 6715 \beta_{7} - 11889 \beta_{6} + \cdots + 37107 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
−0.832197 − 1.44141i
−0.398236 − 0.689765i
−0.156312 − 0.270740i
0.414953 + 0.718719i
0.964458 + 1.67049i
1.50733 + 2.61078i
−0.832197 + 1.44141i
−0.398236 + 0.689765i
−0.156312 + 0.270740i
0.414953 − 0.718719i
0.964458 − 1.67049i
1.50733 − 2.61078i
−1.33220 − 2.30743i 0 −2.54950 + 4.41586i −0.500000 − 0.866025i 0 4.51669 8.25696 0 −1.33220 + 2.30743i
406.2 −0.898236 − 1.55579i 0 −0.613656 + 1.06288i −0.500000 − 0.866025i 0 −3.41478 −1.38811 0 −0.898236 + 1.55579i
406.3 −0.656312 − 1.13677i 0 0.138510 − 0.239907i −0.500000 − 0.866025i 0 3.11486 −2.98887 0 −0.656312 + 1.13677i
406.4 −0.0850473 − 0.147306i 0 0.985534 − 1.70699i −0.500000 − 0.866025i 0 −0.218469 −0.675457 0 −0.0850473 + 0.147306i
406.5 0.464458 + 0.804466i 0 0.568557 − 0.984769i −0.500000 − 0.866025i 0 −3.66299 2.91412 0 0.464458 − 0.804466i
406.6 1.00733 + 1.74475i 0 −1.02944 + 1.78305i −0.500000 − 0.866025i 0 1.66469 −0.118641 0 1.00733 − 1.74475i
676.1 −1.33220 + 2.30743i 0 −2.54950 − 4.41586i −0.500000 + 0.866025i 0 4.51669 8.25696 0 −1.33220 − 2.30743i
676.2 −0.898236 + 1.55579i 0 −0.613656 − 1.06288i −0.500000 + 0.866025i 0 −3.41478 −1.38811 0 −0.898236 − 1.55579i
676.3 −0.656312 + 1.13677i 0 0.138510 + 0.239907i −0.500000 + 0.866025i 0 3.11486 −2.98887 0 −0.656312 − 1.13677i
676.4 −0.0850473 + 0.147306i 0 0.985534 + 1.70699i −0.500000 + 0.866025i 0 −0.218469 −0.675457 0 −0.0850473 − 0.147306i
676.5 0.464458 − 0.804466i 0 0.568557 + 0.984769i −0.500000 + 0.866025i 0 −3.66299 2.91412 0 0.464458 + 0.804466i
676.6 1.00733 − 1.74475i 0 −1.02944 − 1.78305i −0.500000 + 0.866025i 0 1.66469 −0.118641 0 1.00733 + 1.74475i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.6
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.j ✓ 12
3.b odd 2 1 855.2.k.k yes 12
19.c even 3 1 inner 855.2.k.j ✓ 12
57.h odd 6 1 855.2.k.k yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
855.2.k.j ✓ 12 1.a even 1 1 trivial
855.2.k.j ✓ 12 19.c even 3 1 inner
855.2.k.k yes 12 3.b odd 2 1
855.2.k.k yes 12 57.h odd 6 1

Hecke kernels

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

\( T_{2}^{12} + 3 T_{2}^{11} + 13 T_{2}^{10} + 18 T_{2}^{9} + 62 T_{2}^{8} + 78 T_{2}^{7} + 189 T_{2}^{6} + \cdots + 4 \) Copy content Toggle raw display
\( T_{7}^{6} - 2T_{7}^{5} - 27T_{7}^{4} + 44T_{7}^{3} + 180T_{7}^{2} - 256T_{7} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} - 27 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 27 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} + \cdots + 678976 \) Copy content Toggle raw display
$17$ \( T^{12} + 4 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + \cdots + 4194304 \) Copy content Toggle raw display
$31$ \( (T^{6} - 12 T^{5} + \cdots + 47)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 125 T^{4} + \cdots - 14104)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 12 T^{11} + \cdots + 1183744 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 130690624 \) Copy content Toggle raw display
$47$ \( T^{12} + 6 T^{11} + \cdots + 66455104 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 254466304 \) Copy content Toggle raw display
$59$ \( T^{12} + 16 T^{11} + \cdots + 14622976 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 253064464 \) Copy content Toggle raw display
$67$ \( T^{12} + 12 T^{11} + \cdots + 27541504 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 19394461696 \) Copy content Toggle raw display
$73$ \( T^{12} + 4 T^{11} + \cdots + 984064 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 3586572544 \) Copy content Toggle raw display
$83$ \( (T^{6} - 22 T^{5} + \cdots - 15872)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1185562624 \) Copy content Toggle raw display
$97$ \( T^{12} - 30 T^{11} + \cdots + 16384 \) Copy content Toggle raw display
show more
show less