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Properties

Label 338.2.c.d
Level $338$
Weight $2$
Character orbit 338.c
Analytic conductor $2.699$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} - 3 q^{5} + \zeta_{6} q^{6} + \zeta_{6} q^{7} - q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} - 3 q^{5} + \zeta_{6} q^{6} + \zeta_{6} q^{7} - q^{8} + 2 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{10} + (6 \zeta_{6} - 6) q^{11} + q^{12} + q^{14} + ( - 3 \zeta_{6} + 3) q^{15} + (\zeta_{6} - 1) q^{16} + 3 \zeta_{6} q^{17} + 2 q^{18} - 2 \zeta_{6} q^{19} + 3 \zeta_{6} q^{20} - q^{21} + 6 \zeta_{6} q^{22} + ( - \zeta_{6} + 1) q^{24} + 4 q^{25} - 5 q^{27} + ( - \zeta_{6} + 1) q^{28} + (6 \zeta_{6} - 6) q^{29} - 3 \zeta_{6} q^{30} - 4 q^{31} + \zeta_{6} q^{32} - 6 \zeta_{6} q^{33} + 3 q^{34} - 3 \zeta_{6} q^{35} + ( - 2 \zeta_{6} + 2) q^{36} + ( - 7 \zeta_{6} + 7) q^{37} - 2 q^{38} + 3 q^{40} + (\zeta_{6} - 1) q^{42} + \zeta_{6} q^{43} + 6 q^{44} - 6 \zeta_{6} q^{45} + 3 q^{47} - \zeta_{6} q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + ( - 4 \zeta_{6} + 4) q^{50} - 3 q^{51} + (5 \zeta_{6} - 5) q^{54} + ( - 18 \zeta_{6} + 18) q^{55} - \zeta_{6} q^{56} + 2 q^{57} + 6 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} - 3 q^{60} - 8 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{62} + (2 \zeta_{6} - 2) q^{63} + q^{64} - 6 q^{66} + (14 \zeta_{6} - 14) q^{67} + ( - 3 \zeta_{6} + 3) q^{68} - 3 q^{70} + 3 \zeta_{6} q^{71} - 2 \zeta_{6} q^{72} + 2 q^{73} - 7 \zeta_{6} q^{74} + (4 \zeta_{6} - 4) q^{75} + (2 \zeta_{6} - 2) q^{76} - 6 q^{77} + 8 q^{79} + ( - 3 \zeta_{6} + 3) q^{80} + (\zeta_{6} - 1) q^{81} + 12 q^{83} + \zeta_{6} q^{84} - 9 \zeta_{6} q^{85} + q^{86} - 6 \zeta_{6} q^{87} + ( - 6 \zeta_{6} + 6) q^{88} + ( - 6 \zeta_{6} + 6) q^{89} - 6 q^{90} + ( - 4 \zeta_{6} + 4) q^{93} + ( - 3 \zeta_{6} + 3) q^{94} + 6 \zeta_{6} q^{95} - q^{96} + 10 \zeta_{6} q^{97} - 6 \zeta_{6} q^{98} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{3} - q^{4} - 6 q^{5} + q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{3} - q^{4} - 6 q^{5} + q^{6} + q^{7} - 2 q^{8} + 2 q^{9} - 3 q^{10} - 6 q^{11} + 2 q^{12} + 2 q^{14} + 3 q^{15} - q^{16} + 3 q^{17} + 4 q^{18} - 2 q^{19} + 3 q^{20} - 2 q^{21} + 6 q^{22} + q^{24} + 8 q^{25} - 10 q^{27} + q^{28} - 6 q^{29} - 3 q^{30} - 8 q^{31} + q^{32} - 6 q^{33} + 6 q^{34} - 3 q^{35} + 2 q^{36} + 7 q^{37} - 4 q^{38} + 6 q^{40} - q^{42} + q^{43} + 12 q^{44} - 6 q^{45} + 6 q^{47} - q^{48} + 6 q^{49} + 4 q^{50} - 6 q^{51} - 5 q^{54} + 18 q^{55} - q^{56} + 4 q^{57} + 6 q^{58} + 6 q^{59} - 6 q^{60} - 8 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} - 12 q^{66} - 14 q^{67} + 3 q^{68} - 6 q^{70} + 3 q^{71} - 2 q^{72} + 4 q^{73} - 7 q^{74} - 4 q^{75} - 2 q^{76} - 12 q^{77} + 16 q^{79} + 3 q^{80} - q^{81} + 24 q^{83} + q^{84} - 9 q^{85} + 2 q^{86} - 6 q^{87} + 6 q^{88} + 6 q^{89} - 12 q^{90} + 4 q^{93} + 3 q^{94} + 6 q^{95} - 2 q^{96} + 10 q^{97} - 6 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 − 0.866025i
0.500000 − 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i −3.00000 0.500000 + 0.866025i 0.500000 + 0.866025i −1.00000 1.00000 + 1.73205i −1.50000 + 2.59808i
315.1 0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i −3.00000 0.500000 − 0.866025i 0.500000 − 0.866025i −1.00000 1.00000 − 1.73205i −1.50000 − 2.59808i
\(n\): e.g. 2-40 or 990-1000
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.d 2
13.b even 2 1 338.2.c.a 2
13.c even 3 1 26.2.a.a ✓ 1
13.c even 3 1 inner 338.2.c.d 2
13.d odd 4 2 338.2.e.a 4
13.e even 6 1 338.2.a.f 1
13.e even 6 1 338.2.c.a 2
13.f odd 12 2 338.2.b.c 2
13.f odd 12 2 338.2.e.a 4
39.h odd 6 1 3042.2.a.a 1
39.i odd 6 1 234.2.a.e 1
39.k even 12 2 3042.2.b.a 2
52.i odd 6 1 2704.2.a.f 1
52.j odd 6 1 208.2.a.a 1
52.l even 12 2 2704.2.f.d 2
65.l even 6 1 8450.2.a.c 1
65.n even 6 1 650.2.a.j 1
65.q odd 12 2 650.2.b.d 2
91.g even 3 1 1274.2.f.p 2
91.h even 3 1 1274.2.f.p 2
91.m odd 6 1 1274.2.f.r 2
91.n odd 6 1 1274.2.a.d 1
91.v odd 6 1 1274.2.f.r 2
104.n odd 6 1 832.2.a.i 1
104.r even 6 1 832.2.a.d 1
117.f even 3 1 2106.2.e.ba 2
117.h even 3 1 2106.2.e.ba 2
117.k odd 6 1 2106.2.e.b 2
117.u odd 6 1 2106.2.e.b 2
143.k odd 6 1 3146.2.a.n 1
156.p even 6 1 1872.2.a.q 1
195.x odd 6 1 5850.2.a.p 1
195.bl even 12 2 5850.2.e.a 2
208.bg odd 12 2 3328.2.b.j 2
208.bj even 12 2 3328.2.b.m 2
221.l even 6 1 7514.2.a.c 1
247.u odd 6 1 9386.2.a.j 1
260.v odd 6 1 5200.2.a.x 1
312.bh odd 6 1 7488.2.a.g 1
312.bn even 6 1 7488.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a ✓ 1 13.c even 3 1
208.2.a.a 1 52.j odd 6 1
234.2.a.e 1 39.i odd 6 1
338.2.a.f 1 13.e even 6 1
338.2.b.c 2 13.f odd 12 2
338.2.c.a 2 13.b even 2 1
338.2.c.a 2 13.e even 6 1
338.2.c.d 2 1.a even 1 1 trivial
338.2.c.d 2 13.c even 3 1 inner
338.2.e.a 4 13.d odd 4 2
338.2.e.a 4 13.f odd 12 2
650.2.a.j 1 65.n even 6 1
650.2.b.d 2 65.q odd 12 2
832.2.a.d 1 104.r even 6 1
832.2.a.i 1 104.n odd 6 1
1274.2.a.d 1 91.n odd 6 1
1274.2.f.p 2 91.g even 3 1
1274.2.f.p 2 91.h even 3 1
1274.2.f.r 2 91.m odd 6 1
1274.2.f.r 2 91.v odd 6 1
1872.2.a.q 1 156.p even 6 1
2106.2.e.b 2 117.k odd 6 1
2106.2.e.b 2 117.u odd 6 1
2106.2.e.ba 2 117.f even 3 1
2106.2.e.ba 2 117.h even 3 1
2704.2.a.f 1 52.i odd 6 1
2704.2.f.d 2 52.l even 12 2
3042.2.a.a 1 39.h odd 6 1
3042.2.b.a 2 39.k even 12 2
3146.2.a.n 1 143.k odd 6 1
3328.2.b.j 2 208.bg odd 12 2
3328.2.b.m 2 208.bj even 12 2
5200.2.a.x 1 260.v odd 6 1
5850.2.a.p 1 195.x odd 6 1
5850.2.e.a 2 195.bl even 12 2
7488.2.a.g 1 312.bh odd 6 1
7488.2.a.h 1 312.bn even 6 1
7514.2.a.c 1 221.l even 6 1
8450.2.a.c 1 65.l even 6 1
9386.2.a.j 1 247.u odd 6 1

Hecke kernels

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

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( (T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( (T - 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
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