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Properties

Label 855.2.k.i
Level $855$
Weight $2$
Character orbit 855.k
Analytic conductor $6.827$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 285)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{8} - \beta_{4} - \beta_{3} - 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{7} - \beta_{6} + 1) q^{7} + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + ( - \beta_{8} - \beta_{4} - \beta_{3} - 1) q^{4} - \beta_{4} q^{5} + ( - \beta_{7} - \beta_{6} + 1) q^{7} + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{8}+ \cdots + (5 \beta_{9} - 4 \beta_{8} + \cdots + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - 7 q^{4} + 5 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - 7 q^{4} + 5 q^{5} + 4 q^{7} + 12 q^{8} + q^{10} - 10 q^{11} + 8 q^{13} - 4 q^{14} - 7 q^{16} + 10 q^{17} + 5 q^{19} - 14 q^{20} - 2 q^{22} - 2 q^{23} - 5 q^{25} + 4 q^{26} - 10 q^{28} - 7 q^{29} - 18 q^{31} - 23 q^{32} - 25 q^{34} + 2 q^{35} + 12 q^{37} + 37 q^{38} + 6 q^{40} + 12 q^{41} + 8 q^{43} + 16 q^{44} - 40 q^{46} + 6 q^{47} + 30 q^{49} + 2 q^{50} + 4 q^{52} + 8 q^{53} - 5 q^{55} - 72 q^{56} + 76 q^{58} - q^{59} - 7 q^{61} + 15 q^{62} + 28 q^{64} + 16 q^{65} + 14 q^{67} + 2 q^{68} + 4 q^{70} - 27 q^{71} - 26 q^{73} - 18 q^{74} - 56 q^{76} - 28 q^{77} - 23 q^{79} + 7 q^{80} + 36 q^{82} + 24 q^{83} - 10 q^{85} - 2 q^{86} - 9 q^{89} - 46 q^{91} - 52 q^{92} + 90 q^{94} - 2 q^{95} - 10 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 9x^{8} - 2x^{7} + 56x^{6} - 18x^{5} + 125x^{4} + x^{3} + 189x^{2} - 52x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} + 3\nu^{7} - 9\nu^{6} + 15\nu^{5} - 45\nu^{4} + 135\nu^{3} - 86\nu^{2} + 24\nu - 72 ) / 234 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 3\nu^{8} + 9\nu^{7} - 15\nu^{6} + 45\nu^{5} - 135\nu^{4} + 86\nu^{3} - 258\nu^{2} + 72\nu - 702 ) / 234 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{9} - 9 \nu^{8} + 79 \nu^{7} - 12 \nu^{6} + 486 \nu^{5} - 132 \nu^{4} + 1035 \nu^{3} + \cdots - 420 ) / 468 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28 \nu^{9} - 44 \nu^{8} + 275 \nu^{7} - 229 \nu^{6} + 1713 \nu^{5} - 1551 \nu^{4} + 4223 \nu^{3} + \cdots - 3932 ) / 702 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31 \nu^{9} - 5 \nu^{8} + 275 \nu^{7} + 158 \nu^{6} + 1830 \nu^{5} + 906 \nu^{4} + 4319 \nu^{3} + \cdots + 2332 ) / 702 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34 \nu^{9} - \nu^{8} - 257 \nu^{7} - 248 \nu^{6} - 1740 \nu^{5} - 1176 \nu^{4} - 3254 \nu^{3} + \cdots - 1306 ) / 702 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9 \nu^{9} + 9 \nu^{8} - 79 \nu^{7} + 12 \nu^{6} - 486 \nu^{5} + 132 \nu^{4} - 1035 \nu^{3} + \cdots + 420 ) / 156 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49 \nu^{9} + 56 \nu^{8} - 428 \nu^{7} + 163 \nu^{6} - 2595 \nu^{5} + 1389 \nu^{4} - 5228 \nu^{3} + \cdots + 4160 ) / 702 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 8\beta_{8} - 2\beta_{6} - 14\beta_{4} - 8\beta_{3} + \beta_{2} - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{9} - 11\beta_{8} - \beta_{6} + 8\beta_{5} - 11\beta_{4} - 28\beta_{2} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} + 3\beta_{7} + 12\beta_{6} + 57\beta_{3} - 24\beta_{2} + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 66 \beta_{9} + 96 \beta_{8} + 54 \beta_{7} + 78 \beta_{6} - 54 \beta_{5} + 96 \beta_{4} + 96 \beta_{3} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -174\beta_{9} + 397\beta_{8} + 66\beta_{6} - 42\beta_{5} + 495\beta_{4} + 156\beta_{2} + 48\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -108\beta_{9} - 355\beta_{7} - 463\beta_{6} - 769\beta_{3} + 1181\beta_{2} - 426 \) 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
1.34580 − 2.33099i
0.823305 − 1.42601i
0.145349 − 0.251751i
−0.690702 + 1.19633i
−1.12375 + 1.94639i
1.34580 + 2.33099i
0.823305 + 1.42601i
0.145349 + 0.251751i
−0.690702 − 1.19633i
−1.12375 − 1.94639i
−1.34580 − 2.33099i 0 −2.62233 + 4.54201i 0.500000 + 0.866025i 0 −0.797044 8.73329 0 1.34580 − 2.33099i
406.2 −0.823305 − 1.42601i 0 −0.355663 + 0.616027i 0.500000 + 0.866025i 0 4.47988 −2.12194 0 0.823305 − 1.42601i
406.3 −0.145349 − 0.251751i 0 0.957748 − 1.65887i 0.500000 + 0.866025i 0 −0.486575 −1.13822 0 0.145349 − 0.251751i
406.4 0.690702 + 1.19633i 0 0.0458624 − 0.0794360i 0.500000 + 0.866025i 0 −4.36264 2.88952 0 −0.690702 + 1.19633i
406.5 1.12375 + 1.94639i 0 −1.52562 + 2.64245i 0.500000 + 0.866025i 0 3.16638 −2.36264 0 −1.12375 + 1.94639i
676.1 −1.34580 + 2.33099i 0 −2.62233 − 4.54201i 0.500000 − 0.866025i 0 −0.797044 8.73329 0 1.34580 + 2.33099i
676.2 −0.823305 + 1.42601i 0 −0.355663 − 0.616027i 0.500000 − 0.866025i 0 4.47988 −2.12194 0 0.823305 + 1.42601i
676.3 −0.145349 + 0.251751i 0 0.957748 + 1.65887i 0.500000 − 0.866025i 0 −0.486575 −1.13822 0 0.145349 + 0.251751i
676.4 0.690702 − 1.19633i 0 0.0458624 + 0.0794360i 0.500000 − 0.866025i 0 −4.36264 2.88952 0 −0.690702 − 1.19633i
676.5 1.12375 − 1.94639i 0 −1.52562 − 2.64245i 0.500000 − 0.866025i 0 3.16638 −2.36264 0 −1.12375 − 1.94639i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.5
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.i 10
3.b odd 2 1 285.2.i.f ✓ 10
19.c even 3 1 inner 855.2.k.i 10
57.f even 6 1 5415.2.a.z 5
57.h odd 6 1 285.2.i.f ✓ 10
57.h odd 6 1 5415.2.a.y 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.i.f ✓ 10 3.b odd 2 1
285.2.i.f ✓ 10 57.h odd 6 1
855.2.k.i 10 1.a even 1 1 trivial
855.2.k.i 10 19.c even 3 1 inner
5415.2.a.y 5 57.h odd 6 1
5415.2.a.z 5 57.f even 6 1

Hecke kernels

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

\( T_{2}^{10} + T_{2}^{9} + 9T_{2}^{8} + 2T_{2}^{7} + 56T_{2}^{6} + 18T_{2}^{5} + 125T_{2}^{4} - T_{2}^{3} + 189T_{2}^{2} + 52T_{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{5} - 2T_{7}^{4} - 23T_{7}^{3} + 36T_{7}^{2} + 72T_{7} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$7$ \( (T^{5} - 2 T^{4} - 23 T^{3} + \cdots + 24)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 5 T^{4} + \cdots + 992)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 8 T^{9} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{10} - 10 T^{9} + \cdots + 25240576 \) Copy content Toggle raw display
$19$ \( T^{10} - 5 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} + 2 T^{9} + \cdots + 147456 \) Copy content Toggle raw display
$29$ \( T^{10} + 7 T^{9} + \cdots + 9048064 \) Copy content Toggle raw display
$31$ \( (T^{5} + 9 T^{4} + \cdots + 117)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 6 T^{4} + \cdots - 9024)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 12 T^{9} + \cdots + 36864 \) Copy content Toggle raw display
$43$ \( T^{10} - 8 T^{9} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( T^{10} - 6 T^{9} + \cdots + 4562496 \) Copy content Toggle raw display
$53$ \( T^{10} - 8 T^{9} + \cdots + 18731584 \) Copy content Toggle raw display
$59$ \( T^{10} + T^{9} + \cdots + 20647936 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 591267856 \) Copy content Toggle raw display
$67$ \( T^{10} - 14 T^{9} + \cdots + 70157376 \) Copy content Toggle raw display
$71$ \( T^{10} + 27 T^{9} + \cdots + 37161216 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 2135179264 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 11935125504 \) Copy content Toggle raw display
$83$ \( (T^{5} - 12 T^{4} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 126157824 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 4)^{5} \) Copy content Toggle raw display
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