Properties

Label 1638.2.p.k
Level $1638$
Weight $2$
Character orbit 1638.p
Analytic conductor $13.079$
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] = [1638,2,Mod(919,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.919");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.p (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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.23207289578928.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 182)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{9} + \beta_{8} + \beta_{6} + 1) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{5} + \beta_{3} + \beta_1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{9} + \beta_{8} + \beta_{6} + 1) q^{5} + ( - \beta_{9} + \beta_{8} - \beta_{5} + \beta_{3} + \beta_1) q^{7} - q^{8} + ( - \beta_{9} + \beta_{8} + 1) q^{10} + ( - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{9} + 2 \beta_{8} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{13} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{3}) q^{14} + \beta_{2} q^{16} + (\beta_{9} + 2 \beta_{8} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{17} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{3} + 2 \beta_1 + 1) q^{19} - \beta_{6} q^{20} + (\beta_{9} - \beta_{6} + \beta_{4} + \beta_{2} - \beta_1) q^{22} + (\beta_{5} + \beta_{2}) q^{23} + (\beta_{9} + 2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_1) q^{25} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{26} + (\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_1) q^{28} + (\beta_{8} + \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2}) q^{29} + ( - \beta_{9} - 2 \beta_{7} + \beta_{5} - \beta_{4} + 5 \beta_{2} + \beta_1) q^{31} + (\beta_{2} + 1) q^{32} + (2 \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{34} + (\beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{3} + 3 \beta_{2} + 3) q^{35} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{2}) q^{37} + ( - 2 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{38} + (\beta_{9} - \beta_{8} - \beta_{6} - 1) q^{40} + (2 \beta_{8} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{41} + ( - \beta_{7} + 3 \beta_{5} - \beta_{2}) q^{43} + (\beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_1) q^{44} + (\beta_{4} + \beta_{2} + 1) q^{46} + (2 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1 + 2) q^{47} + (\beta_{9} + \beta_{8} + 2 \beta_{6} - 3 \beta_{5} - \beta_{2} + 3 \beta_1 - 2) q^{49} + (\beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{50} + (\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2}) q^{52} + ( - \beta_{9} - \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_1) q^{53} + (2 \beta_{9} - 2 \beta_{8} - \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{55}+ \cdots + ( - 2 \beta_{9} + \beta_{8} + 3 \beta_{6} - 3 \beta_{4} + 2 \beta_1 + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{2} - 5 q^{4} + 2 q^{5} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{2} - 5 q^{4} + 2 q^{5} - 10 q^{8} + 4 q^{10} - 8 q^{11} - 6 q^{13} + 3 q^{14} - 5 q^{16} + 3 q^{17} + 10 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} - 3 q^{25} - 6 q^{26} + 3 q^{28} - q^{29} - 21 q^{31} + 5 q^{32} + 6 q^{34} + 22 q^{35} + 10 q^{37} + 5 q^{38} - 2 q^{40} + 9 q^{41} + 9 q^{43} + 4 q^{44} + 4 q^{46} + 13 q^{47} - 8 q^{49} + 3 q^{50} + 15 q^{53} + 7 q^{55} - 2 q^{58} - q^{59} + 30 q^{61} + 21 q^{62} + 10 q^{64} + 25 q^{65} + 3 q^{68} + 26 q^{70} + q^{71} - 4 q^{73} - 10 q^{74} - 5 q^{76} - 16 q^{77} - q^{79} - 4 q^{80} + 18 q^{82} - 96 q^{83} + 12 q^{85} - 9 q^{86} + 8 q^{88} + 9 q^{89} + 10 q^{91} + 8 q^{92} + 26 q^{94} - 6 q^{95} - 19 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{9} - \nu^{8} - 21\nu^{7} + 35\nu^{6} + 68\nu^{5} - 111\nu^{4} - 201\nu^{3} + 351\nu^{2} + 405\nu - 891 ) / 81 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{9} + 2\nu^{8} + 45\nu^{7} - 28\nu^{6} - 136\nu^{5} + 99\nu^{4} + 393\nu^{3} - 342\nu^{2} - 675\nu + 567 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 2\nu^{7} - 8\nu^{6} - 5\nu^{5} + 26\nu^{4} + 13\nu^{3} - 69\nu^{2} - 24\nu + 162 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} - 2\nu^{7} + 8\nu^{6} + 5\nu^{5} - 26\nu^{4} - 13\nu^{3} + 78\nu^{2} + 33\nu - 171 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{9} + 20\nu^{8} + 15\nu^{7} - 88\nu^{6} - 37\nu^{5} + 276\nu^{4} + 87\nu^{3} - 783\nu^{2} + 108\nu + 972 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{9} + 5\nu^{8} - 30\nu^{7} - 31\nu^{6} + 92\nu^{5} + 96\nu^{4} - 273\nu^{3} - 243\nu^{2} + 459\nu + 729 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{9} - 4\nu^{8} - 57\nu^{7} + 59\nu^{6} + 164\nu^{5} - 174\nu^{4} - 480\nu^{3} + 567\nu^{2} + 891\nu - 1215 ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6\nu^{9} + 2\nu^{8} + 35\nu^{7} - 24\nu^{6} - 109\nu^{5} + 77\nu^{4} + 306\nu^{3} - 273\nu^{2} - 531\nu + 432 ) / 27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23 \nu^{9} + 37 \nu^{8} + 141 \nu^{7} - 245 \nu^{6} - 410 \nu^{5} + 780 \nu^{4} + 1119 \nu^{3} - 2421 \nu^{2} - 1620 \nu + 3969 ) / 81 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 10 \beta_{2} - 6 \beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{9} - 2\beta_{8} + 9\beta_{7} - 10\beta_{6} + \beta_{5} - 2\beta_{4} + 9\beta_{3} + 5\beta_{2} - 6\beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{9} - 19 \beta_{8} - 12 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 7 \beta_{4} - 9 \beta_{3} + 22 \beta_{2} - 9 \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{9} - 8\beta_{8} + 24\beta_{7} - 13\beta_{6} + \beta_{5} - 17\beta_{4} + 35\beta_{2} - 3\beta _1 + 41 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19 \beta_{9} - 28 \beta_{8} - 21 \beta_{7} + \beta_{6} - 31 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 19 \beta_{2} + 51 \beta _1 - 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 49 \beta_{9} + 46 \beta_{8} - 21 \beta_{7} + 41 \beta_{6} + 64 \beta_{5} - 32 \beta_{4} - 12 \beta_{3} + 47 \beta_{2} + 63 \beta _1 + 185 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23 \beta_{9} + 29 \beta_{8} - 45 \beta_{7} + 151 \beta_{6} + 2 \beta_{5} + 50 \beta_{4} + 3 \beta_{3} + 118 \beta_{2} + 111 \beta _1 - 104 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(\beta_{2}\) \(-1 - \beta_{2}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
919.1
1.16412 + 1.28251i
1.67396 0.444815i
1.13823 1.30554i
−1.73100 0.0603688i
−1.24530 1.20384i
1.16412 1.28251i
1.67396 + 0.444815i
1.13823 + 1.30554i
−1.73100 + 0.0603688i
−1.24530 + 1.20384i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.69274 + 2.93192i 0 −2.60522 + 0.461340i −1.00000 0 −3.38549
919.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.451759 + 0.782469i 0 −0.686411 2.55516i −1.00000 0 −0.903518
919.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.561520 0.972581i 0 2.64005 + 0.173642i −1.00000 0 1.12304
919.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.917780 1.58964i 0 1.05365 2.42690i −1.00000 0 1.83556
919.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.66520 2.88422i 0 −0.402064 + 2.61502i −1.00000 0 3.33041
991.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.69274 2.93192i 0 −2.60522 0.461340i −1.00000 0 −3.38549
991.2 0.500000 0.866025i 0 −0.500000 0.866025i −0.451759 0.782469i 0 −0.686411 + 2.55516i −1.00000 0 −0.903518
991.3 0.500000 0.866025i 0 −0.500000 0.866025i 0.561520 + 0.972581i 0 2.64005 0.173642i −1.00000 0 1.12304
991.4 0.500000 0.866025i 0 −0.500000 0.866025i 0.917780 + 1.58964i 0 1.05365 + 2.42690i −1.00000 0 1.83556
991.5 0.500000 0.866025i 0 −0.500000 0.866025i 1.66520 + 2.88422i 0 −0.402064 2.61502i −1.00000 0 3.33041
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 919.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.p.k 10
3.b odd 2 1 182.2.h.d yes 10
7.c even 3 1 1638.2.m.j 10
13.c even 3 1 1638.2.m.j 10
21.c even 2 1 1274.2.h.s 10
21.g even 6 1 1274.2.e.s 10
21.g even 6 1 1274.2.g.q 10
21.h odd 6 1 182.2.e.d 10
21.h odd 6 1 1274.2.g.p 10
39.i odd 6 1 182.2.e.d 10
91.g even 3 1 inner 1638.2.p.k 10
273.r even 6 1 1274.2.g.q 10
273.s odd 6 1 1274.2.g.p 10
273.bf even 6 1 1274.2.h.s 10
273.bm odd 6 1 182.2.h.d yes 10
273.bn even 6 1 1274.2.e.s 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.e.d 10 21.h odd 6 1
182.2.e.d 10 39.i odd 6 1
182.2.h.d yes 10 3.b odd 2 1
182.2.h.d yes 10 273.bm odd 6 1
1274.2.e.s 10 21.g even 6 1
1274.2.e.s 10 273.bn even 6 1
1274.2.g.p 10 21.h odd 6 1
1274.2.g.p 10 273.s odd 6 1
1274.2.g.q 10 21.g even 6 1
1274.2.g.q 10 273.r even 6 1
1274.2.h.s 10 21.c even 2 1
1274.2.h.s 10 273.bf even 6 1
1638.2.m.j 10 7.c even 3 1
1638.2.m.j 10 13.c even 3 1
1638.2.p.k 10 1.a even 1 1 trivial
1638.2.p.k 10 91.g even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 2 T_{5}^{9} + 16 T_{5}^{8} - 26 T_{5}^{7} + 187 T_{5}^{6} - 293 T_{5}^{5} + 667 T_{5}^{4} - 329 T_{5}^{3} + 574 T_{5}^{2} - 147 T_{5} + 441 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + 16 T^{8} - 26 T^{7} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{10} + 4 T^{8} - 4 T^{7} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( (T^{5} + 4 T^{4} - 24 T^{3} - 65 T^{2} + \cdots - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + 10 T^{8} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} - 3 T^{9} + 75 T^{8} + \cdots + 6718464 \) Copy content Toggle raw display
$19$ \( (T^{5} - 5 T^{4} - 78 T^{3} + 390 T^{2} + \cdots - 7089)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 4 T^{9} + 19 T^{8} + 22 T^{7} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{10} + T^{9} + 49 T^{8} + \cdots + 103041 \) Copy content Toggle raw display
$31$ \( T^{10} + 21 T^{9} + 316 T^{8} + \cdots + 32228329 \) Copy content Toggle raw display
$37$ \( T^{10} - 10 T^{9} + 157 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{10} - 9 T^{9} + 132 T^{8} + \cdots + 7733961 \) Copy content Toggle raw display
$43$ \( T^{10} - 9 T^{9} + 130 T^{8} + \cdots + 6538249 \) Copy content Toggle raw display
$47$ \( T^{10} - 13 T^{9} + 166 T^{8} + \cdots + 9529569 \) Copy content Toggle raw display
$53$ \( T^{10} - 15 T^{9} + 294 T^{8} + \cdots + 5948721 \) Copy content Toggle raw display
$59$ \( T^{10} + T^{9} + 130 T^{8} + \cdots + 52012944 \) Copy content Toggle raw display
$61$ \( (T^{5} - 15 T^{4} - 79 T^{3} + 2378 T^{2} + \cdots + 22213)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} - 220 T^{3} - 139 T^{2} + \cdots - 4703)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - T^{9} + 214 T^{8} + \cdots + 1069878681 \) Copy content Toggle raw display
$73$ \( T^{10} + 4 T^{9} + 132 T^{8} + \cdots + 41744521 \) Copy content Toggle raw display
$79$ \( T^{10} + T^{9} + 186 T^{8} + \cdots + 66049 \) Copy content Toggle raw display
$83$ \( (T^{5} + 48 T^{4} + 837 T^{3} + \cdots + 20412)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 9 T^{9} + 171 T^{8} + \cdots + 44836416 \) Copy content Toggle raw display
$97$ \( T^{10} + 19 T^{9} + 318 T^{8} + \cdots + 42523441 \) Copy content Toggle raw display
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