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Properties

Label 338.2.c.h
Level $338$
Weight $2$
Character orbit 338.c
Analytic conductor $2.699$
Analytic rank $0$
Dimension $6$
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] = [338,2,Mod(191,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.c (of order \(3\), degree \(2\), not 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: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{3}+ \cdots + (3 \beta_{5} + \beta_{4} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{5} - \beta_{4} + \cdots + \beta_1) q^{3}+ \cdots + ( - 6 \beta_{3} + 4 \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} + 4 q^{5} - 3 q^{6} + 4 q^{7} + 6 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{2} - 3 q^{3} - 3 q^{4} + 4 q^{5} - 3 q^{6} + 4 q^{7} + 6 q^{8} - 8 q^{9} - 2 q^{10} - 3 q^{11} + 6 q^{12} - 8 q^{14} + 12 q^{15} - 3 q^{16} - 5 q^{17} + 16 q^{18} - q^{19} - 2 q^{20} - 8 q^{21} - 3 q^{22} - 3 q^{24} + 10 q^{25} + 12 q^{27} + 4 q^{28} + 10 q^{29} + 12 q^{30} - 32 q^{31} - 3 q^{32} + 4 q^{33} + 10 q^{34} + 12 q^{35} - 8 q^{36} + 14 q^{37} + 2 q^{38} + 4 q^{40} - 7 q^{41} + 4 q^{42} - 11 q^{43} + 6 q^{44} + 4 q^{45} - 4 q^{47} - 3 q^{48} - 3 q^{49} - 5 q^{50} - 32 q^{51} - 6 q^{54} - 16 q^{55} + 4 q^{56} + 16 q^{57} + 10 q^{58} + 3 q^{59} - 24 q^{60} + 4 q^{61} + 16 q^{62} + 6 q^{63} + 6 q^{64} - 8 q^{66} - 21 q^{67} - 5 q^{68} + 14 q^{69} - 24 q^{70} + 12 q^{71} - 8 q^{72} + 2 q^{73} + 14 q^{74} + 23 q^{75} - q^{76} - 36 q^{77} - 36 q^{79} - 2 q^{80} + 25 q^{81} - 7 q^{82} - 2 q^{83} + 4 q^{84} - 36 q^{85} + 22 q^{86} + 10 q^{87} - 3 q^{88} - 25 q^{89} - 8 q^{90} + 2 q^{93} + 2 q^{94} + 18 q^{95} + 6 q^{96} - 23 q^{97} - 3 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(-1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 − 1.07992i
0.222521 − 0.385418i
0.900969 − 1.56052i
−0.623490 + 1.07992i
−0.500000 + 0.866025i −1.34601 + 2.33136i −0.500000 − 0.866025i −2.49396 −1.34601 − 2.33136i −0.801938 − 1.38900i 1.00000 −2.12349 − 3.67799i 1.24698 − 2.15983i
191.2 −0.500000 + 0.866025i −1.17845 + 2.04113i −0.500000 − 0.866025i 0.890084 −1.17845 − 2.04113i 2.24698 + 3.89188i 1.00000 −1.27748 − 2.21266i −0.445042 + 0.770835i
191.3 −0.500000 + 0.866025i 1.02446 − 1.77441i −0.500000 − 0.866025i 3.60388 1.02446 + 1.77441i 0.554958 + 0.961216i 1.00000 −0.599031 − 1.03755i −1.80194 + 3.12105i
315.1 −0.500000 − 0.866025i −1.34601 − 2.33136i −0.500000 + 0.866025i −2.49396 −1.34601 + 2.33136i −0.801938 + 1.38900i 1.00000 −2.12349 + 3.67799i 1.24698 + 2.15983i
315.2 −0.500000 − 0.866025i −1.17845 − 2.04113i −0.500000 + 0.866025i 0.890084 −1.17845 + 2.04113i 2.24698 − 3.89188i 1.00000 −1.27748 + 2.21266i −0.445042 − 0.770835i
315.3 −0.500000 − 0.866025i 1.02446 + 1.77441i −0.500000 + 0.866025i 3.60388 1.02446 − 1.77441i 0.554958 − 0.961216i 1.00000 −0.599031 + 1.03755i −1.80194 − 3.12105i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.2.c.h 6
13.b even 2 1 338.2.c.i 6
13.c even 3 1 338.2.a.h yes 3
13.c even 3 1 inner 338.2.c.h 6
13.d odd 4 2 338.2.e.e 12
13.e even 6 1 338.2.a.g ✓ 3
13.e even 6 1 338.2.c.i 6
13.f odd 12 2 338.2.b.d 6
13.f odd 12 2 338.2.e.e 12
39.h odd 6 1 3042.2.a.bi 3
39.i odd 6 1 3042.2.a.z 3
39.k even 12 2 3042.2.b.n 6
52.i odd 6 1 2704.2.a.v 3
52.j odd 6 1 2704.2.a.w 3
52.l even 12 2 2704.2.f.m 6
65.l even 6 1 8450.2.a.bx 3
65.n even 6 1 8450.2.a.bn 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.2.a.g ✓ 3 13.e even 6 1
338.2.a.h yes 3 13.c even 3 1
338.2.b.d 6 13.f odd 12 2
338.2.c.h 6 1.a even 1 1 trivial
338.2.c.h 6 13.c even 3 1 inner
338.2.c.i 6 13.b even 2 1
338.2.c.i 6 13.e even 6 1
338.2.e.e 12 13.d odd 4 2
338.2.e.e 12 13.f odd 12 2
2704.2.a.v 3 52.i odd 6 1
2704.2.a.w 3 52.j odd 6 1
2704.2.f.m 6 52.l even 12 2
3042.2.a.z 3 39.i odd 6 1
3042.2.a.bi 3 39.h odd 6 1
3042.2.b.n 6 39.k even 12 2
8450.2.a.bn 3 65.n even 6 1
8450.2.a.bx 3 65.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{6} + 3T_{3}^{5} + 13T_{3}^{4} + 14T_{3}^{3} + 55T_{3}^{2} + 52T_{3} + 169 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 8T_{5} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$5$ \( (T^{3} - 2 T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 4 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 5 T^{5} + \cdots + 9409 \) Copy content Toggle raw display
$19$ \( T^{6} + T^{5} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{6} + 28 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$29$ \( T^{6} - 10 T^{5} + \cdots + 10816 \) Copy content Toggle raw display
$31$ \( (T^{3} + 16 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 14 T^{5} + \cdots + 3136 \) Copy content Toggle raw display
$41$ \( T^{6} + 7 T^{5} + \cdots + 8281 \) Copy content Toggle raw display
$43$ \( T^{6} + 11 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( (T^{3} + 2 T^{2} - 36 T - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 28 T - 56)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$61$ \( T^{6} - 4 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( T^{6} + 21 T^{5} + \cdots + 82369 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( (T^{3} - T^{2} - 86 T - 251)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 18 T^{2} + \cdots - 232)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + T^{2} - 16 T + 13)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 25 T^{5} + \cdots + 573049 \) Copy content Toggle raw display
$97$ \( T^{6} + 23 T^{5} + \cdots + 779689 \) Copy content Toggle raw display
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