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Properties

Label 1638.2.m.k
Level $1638$
Weight $2$
Character orbit 1638.m
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(289,1638)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.289");
 
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.m (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: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 15x^{8} + 14x^{7} + 110x^{6} + 36x^{5} + 233x^{4} + 164x^{3} + 345x^{2} + 76x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 546)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{4} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + ( - \beta_{2} - \beta_1) q^{5} + \beta_{4} q^{7} + q^{8} + ( - \beta_{2} - \beta_1) q^{10} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{7} - 2 \beta_{6} + \cdots + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 10 q^{4} + 2 q^{5} - 2 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} + 10 q^{4} + 2 q^{5} - 2 q^{7} + 10 q^{8} + 2 q^{10} - 6 q^{11} - 4 q^{13} - 2 q^{14} + 10 q^{16} + 8 q^{17} + 3 q^{19} + 2 q^{20} - 6 q^{22} + 12 q^{23} - q^{25} - 4 q^{26} - 2 q^{28} - 10 q^{31} + 10 q^{32} + 8 q^{34} - 16 q^{35} - 2 q^{37} + 3 q^{38} + 2 q^{40} + 4 q^{41} + 3 q^{43} - 6 q^{44} + 12 q^{46} + 15 q^{47} + 4 q^{49} - q^{50} - 4 q^{52} + 17 q^{53} + 3 q^{55} - 2 q^{56} + 4 q^{59} + 11 q^{61} - 10 q^{62} + 10 q^{64} + 4 q^{65} - q^{67} + 8 q^{68} - 16 q^{70} - 18 q^{71} + 12 q^{73} - 2 q^{74} + 3 q^{76} - 18 q^{77} - 4 q^{79} + 2 q^{80} + 4 q^{82} + q^{85} + 3 q^{86} - 6 q^{88} + 14 q^{89} + 26 q^{91} + 12 q^{92} + 15 q^{94} + 48 q^{95} - 6 q^{97} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 15x^{8} + 14x^{7} + 110x^{6} + 36x^{5} + 233x^{4} + 164x^{3} + 345x^{2} + 76x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 503 \nu^{9} - 2241 \nu^{8} + 8466 \nu^{7} - 67528 \nu^{6} + 19422 \nu^{5} - 156870 \nu^{4} + \cdots + 438544 ) / 2044008 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3064 \nu^{9} + 9207 \nu^{8} - 34782 \nu^{7} - 28346 \nu^{6} - 79794 \nu^{5} + 644490 \nu^{4} + \cdots + 3620228 ) / 1022004 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3977 \nu^{9} - 10833 \nu^{8} - 12699 \nu^{7} - 334006 \nu^{6} - 625302 \nu^{5} - 1723536 \nu^{4} + \cdots - 3118196 ) / 1022004 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27409 \nu^{9} + 54315 \nu^{8} - 413376 \nu^{7} - 375260 \nu^{6} - 3082518 \nu^{5} + \cdots - 2149816 ) / 2044008 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14545 \nu^{9} + 29941 \nu^{8} - 216152 \nu^{7} - 200758 \nu^{6} - 1521222 \nu^{5} + \cdots - 1074620 ) / 340668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45928 \nu^{9} - 96426 \nu^{8} + 695481 \nu^{7} + 561428 \nu^{6} + 5037264 \nu^{5} + 744876 \nu^{4} + \cdots + 165664 ) / 1022004 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 97465 \nu^{9} - 178473 \nu^{8} + 1399728 \nu^{7} + 1662752 \nu^{6} + 10555542 \nu^{5} + \cdots + 5377120 ) / 2044008 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 51344 \nu^{9} - 117222 \nu^{8} + 805587 \nu^{7} + 484042 \nu^{6} + 5520312 \nu^{5} + 367938 \nu^{4} + \cdots + 742892 ) / 1022004 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 4\beta_{5} - \beta_{4} + 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{8} - 3\beta_{6} - 3\beta_{4} - 2\beta_{3} + 11\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -16\beta_{9} - 16\beta_{8} + 6\beta_{7} - 18\beta_{6} - 40\beta_{5} + 2\beta_{4} - 37\beta _1 - 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 65 \beta_{9} + 4 \beta_{8} + 34 \beta_{7} - 4 \beta_{6} - 126 \beta_{5} + 65 \beta_{4} + \cdots - 168 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -30\beta_{9} + 297\beta_{8} + 267\beta_{6} + 267\beta_{4} + 126\beta_{3} - 657\beta_{2} + 592 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1080\beta_{9} + 1080\beta_{8} - 564\beta_{7} + 1176\beta_{6} + 2280\beta_{5} - 96\beta_{4} + 2797\beta _1 + 2280 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5005 \beta_{9} - 468 \beta_{8} - 2256 \beta_{7} + 468 \beta_{6} + 9784 \beta_{5} - 5005 \beta_{4} + \cdots + 11402 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1788\beta_{9} - 20451\beta_{8} - 18663\beta_{6} - 18663\beta_{4} - 9542\beta_{3} + 47603\beta_{2} - 39726 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(\beta_{5}\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.10337 − 1.91109i
−0.623307 − 1.07960i
−0.114009 − 0.197470i
0.769836 + 1.33339i
2.07085 + 3.58682i
−1.10337 + 1.91109i
−0.623307 + 1.07960i
−0.114009 + 0.197470i
0.769836 − 1.33339i
2.07085 − 3.58682i
1.00000 0 1.00000 −1.10337 + 1.91109i 0 1.19230 − 2.36187i 1.00000 0 −1.10337 + 1.91109i
289.2 1.00000 0 1.00000 −0.623307 + 1.07960i 0 2.30301 + 1.30235i 1.00000 0 −0.623307 + 1.07960i
289.3 1.00000 0 1.00000 −0.114009 + 0.197470i 0 −2.59452 + 0.518144i 1.00000 0 −0.114009 + 0.197470i
289.4 1.00000 0 1.00000 0.769836 − 1.33339i 0 −2.22250 + 1.43544i 1.00000 0 0.769836 − 1.33339i
289.5 1.00000 0 1.00000 2.07085 − 3.58682i 0 0.321703 − 2.62612i 1.00000 0 2.07085 − 3.58682i
1621.1 1.00000 0 1.00000 −1.10337 − 1.91109i 0 1.19230 + 2.36187i 1.00000 0 −1.10337 − 1.91109i
1621.2 1.00000 0 1.00000 −0.623307 − 1.07960i 0 2.30301 − 1.30235i 1.00000 0 −0.623307 − 1.07960i
1621.3 1.00000 0 1.00000 −0.114009 − 0.197470i 0 −2.59452 − 0.518144i 1.00000 0 −0.114009 − 0.197470i
1621.4 1.00000 0 1.00000 0.769836 + 1.33339i 0 −2.22250 − 1.43544i 1.00000 0 0.769836 + 1.33339i
1621.5 1.00000 0 1.00000 2.07085 + 3.58682i 0 0.321703 + 2.62612i 1.00000 0 2.07085 + 3.58682i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h 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.m.k 10
3.b odd 2 1 546.2.j.e ✓ 10
7.c even 3 1 1638.2.p.j 10
13.c even 3 1 1638.2.p.j 10
21.h odd 6 1 546.2.k.e yes 10
39.i odd 6 1 546.2.k.e yes 10
91.h even 3 1 inner 1638.2.m.k 10
273.s odd 6 1 546.2.j.e ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.e ✓ 10 3.b odd 2 1
546.2.j.e ✓ 10 273.s odd 6 1
546.2.k.e yes 10 21.h odd 6 1
546.2.k.e yes 10 39.i odd 6 1
1638.2.m.k 10 1.a even 1 1 trivial
1638.2.m.k 10 91.h even 3 1 inner
1638.2.p.j 10 7.c even 3 1
1638.2.p.j 10 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - 2 T_{5}^{9} + 15 T_{5}^{8} + 14 T_{5}^{7} + 110 T_{5}^{6} + 36 T_{5}^{5} + 233 T_{5}^{4} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{10} + 2 T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 6 T^{9} + \cdots + 900 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T^{5} - 4 T^{4} - 20 T^{3} + \cdots - 28)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} + \cdots + 12321 \) Copy content Toggle raw display
$23$ \( (T^{5} - 6 T^{4} + \cdots - 360)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 60 T^{8} + \cdots + 236196 \) Copy content Toggle raw display
$31$ \( T^{10} + 10 T^{9} + \cdots + 1254400 \) Copy content Toggle raw display
$37$ \( (T^{5} + T^{4} - 38 T^{3} + \cdots - 565)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 4 T^{9} + \cdots + 770884 \) Copy content Toggle raw display
$43$ \( T^{10} - 3 T^{9} + \cdots + 308025 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 271854144 \) Copy content Toggle raw display
$53$ \( T^{10} - 17 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{5} - 2 T^{4} + \cdots - 1268)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} - 11 T^{9} + \cdots + 6400 \) Copy content Toggle raw display
$67$ \( T^{10} + T^{9} + \cdots + 400 \) Copy content Toggle raw display
$71$ \( T^{10} + 18 T^{9} + \cdots + 202500 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 37941975369 \) Copy content Toggle raw display
$79$ \( T^{10} + 4 T^{9} + \cdots + 817216 \) Copy content Toggle raw display
$83$ \( (T^{5} - 60 T^{3} + \cdots - 486)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} - 7 T^{4} + \cdots - 453250)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 6 T^{9} + \cdots + 1734489 \) Copy content Toggle raw display
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