Properties

Label 338.2.e.c
Level $338$
Weight $2$
Character orbit 338.e
Analytic conductor $2.699$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.e (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} + \zeta_{12}^{2} q^{4} + 3 \zeta_{12}^{3} q^{5} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{6} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + 2 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( 1 - \zeta_{12}^{2} ) q^{3} + \zeta_{12}^{2} q^{4} + 3 \zeta_{12}^{3} q^{5} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{6} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + 2 \zeta_{12}^{2} q^{9} + ( -3 + 3 \zeta_{12}^{2} ) q^{10} + q^{12} -3 q^{14} + 3 \zeta_{12} q^{15} + ( -1 + \zeta_{12}^{2} ) q^{16} -3 \zeta_{12}^{2} q^{17} + 2 \zeta_{12}^{3} q^{18} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{19} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{20} + 3 \zeta_{12}^{3} q^{21} + ( 6 - 6 \zeta_{12}^{2} ) q^{23} + \zeta_{12} q^{24} -4 q^{25} + 5 q^{27} -3 \zeta_{12} q^{28} + 3 \zeta_{12}^{2} q^{30} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} -3 \zeta_{12}^{3} q^{34} -9 \zeta_{12}^{2} q^{35} + ( -2 + 2 \zeta_{12}^{2} ) q^{36} + 3 \zeta_{12} q^{37} + 6 q^{38} -3 q^{40} + ( -3 + 3 \zeta_{12}^{2} ) q^{42} + \zeta_{12}^{2} q^{43} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{45} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{46} -3 \zeta_{12}^{3} q^{47} + \zeta_{12}^{2} q^{48} + ( 2 - 2 \zeta_{12}^{2} ) q^{49} -4 \zeta_{12} q^{50} -3 q^{51} -6 q^{53} + 5 \zeta_{12} q^{54} -3 \zeta_{12}^{2} q^{56} -6 \zeta_{12}^{3} q^{57} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{59} + 3 \zeta_{12}^{3} q^{60} + 8 \zeta_{12}^{2} q^{61} -6 \zeta_{12} q^{63} - q^{64} -12 \zeta_{12} q^{67} + ( 3 - 3 \zeta_{12}^{2} ) q^{68} -6 \zeta_{12}^{2} q^{69} -9 \zeta_{12}^{3} q^{70} + ( -15 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{71} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{72} -6 \zeta_{12}^{3} q^{73} + 3 \zeta_{12}^{2} q^{74} + ( -4 + 4 \zeta_{12}^{2} ) q^{75} + 6 \zeta_{12} q^{76} + 10 q^{79} -3 \zeta_{12} q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} -6 \zeta_{12}^{3} q^{83} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{84} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{85} + \zeta_{12}^{3} q^{86} -6 \zeta_{12} q^{89} -6 q^{90} + 6 q^{92} + ( 3 - 3 \zeta_{12}^{2} ) q^{94} + 18 \zeta_{12}^{2} q^{95} + \zeta_{12}^{3} q^{96} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{97} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{4} + 4 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{3} + 2 q^{4} + 4 q^{9} - 6 q^{10} + 4 q^{12} - 12 q^{14} - 2 q^{16} - 6 q^{17} + 12 q^{23} - 16 q^{25} + 20 q^{27} + 6 q^{30} - 18 q^{35} - 4 q^{36} + 24 q^{38} - 12 q^{40} - 6 q^{42} + 2 q^{43} + 2 q^{48} + 4 q^{49} - 12 q^{51} - 24 q^{53} - 6 q^{56} + 16 q^{61} - 4 q^{64} + 6 q^{68} - 12 q^{69} + 6 q^{74} - 8 q^{75} + 40 q^{79} - 2 q^{81} - 24 q^{90} + 24 q^{92} + 6 q^{94} + 36 q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(171\)
\(\chi(n)\) \(\zeta_{12}^{2}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 3.00000i −0.866025 0.500000i 2.59808 + 1.50000i 1.00000i 1.00000 1.73205i −1.50000 2.59808i
23.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 3.00000i 0.866025 + 0.500000i −2.59808 1.50000i 1.00000i 1.00000 1.73205i −1.50000 2.59808i
147.1 −0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 3.00000i −0.866025 + 0.500000i 2.59808 1.50000i 1.00000i 1.00000 + 1.73205i −1.50000 + 2.59808i
147.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 3.00000i 0.866025 0.500000i −2.59808 + 1.50000i 1.00000i 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.2.e.c 4
13.b even 2 1 inner 338.2.e.c 4
13.c even 3 1 26.2.b.a 2
13.c even 3 1 inner 338.2.e.c 4
13.d odd 4 1 338.2.c.b 2
13.d odd 4 1 338.2.c.f 2
13.e even 6 1 26.2.b.a 2
13.e even 6 1 inner 338.2.e.c 4
13.f odd 12 1 338.2.a.b 1
13.f odd 12 1 338.2.a.d 1
13.f odd 12 1 338.2.c.b 2
13.f odd 12 1 338.2.c.f 2
39.h odd 6 1 234.2.b.b 2
39.i odd 6 1 234.2.b.b 2
39.k even 12 1 3042.2.a.g 1
39.k even 12 1 3042.2.a.j 1
52.i odd 6 1 208.2.f.a 2
52.j odd 6 1 208.2.f.a 2
52.l even 12 1 2704.2.a.j 1
52.l even 12 1 2704.2.a.k 1
65.l even 6 1 650.2.d.b 2
65.n even 6 1 650.2.d.b 2
65.q odd 12 1 650.2.c.a 2
65.q odd 12 1 650.2.c.d 2
65.r odd 12 1 650.2.c.a 2
65.r odd 12 1 650.2.c.d 2
65.s odd 12 1 8450.2.a.h 1
65.s odd 12 1 8450.2.a.u 1
91.g even 3 1 1274.2.n.d 4
91.h even 3 1 1274.2.n.d 4
91.k even 6 1 1274.2.n.d 4
91.l odd 6 1 1274.2.n.c 4
91.m odd 6 1 1274.2.n.c 4
91.n odd 6 1 1274.2.d.c 2
91.p odd 6 1 1274.2.n.c 4
91.t odd 6 1 1274.2.d.c 2
91.u even 6 1 1274.2.n.d 4
91.v odd 6 1 1274.2.n.c 4
104.n odd 6 1 832.2.f.b 2
104.p odd 6 1 832.2.f.b 2
104.r even 6 1 832.2.f.d 2
104.s even 6 1 832.2.f.d 2
156.p even 6 1 1872.2.c.f 2
156.r even 6 1 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 13.c even 3 1
26.2.b.a 2 13.e even 6 1
208.2.f.a 2 52.i odd 6 1
208.2.f.a 2 52.j odd 6 1
234.2.b.b 2 39.h odd 6 1
234.2.b.b 2 39.i odd 6 1
338.2.a.b 1 13.f odd 12 1
338.2.a.d 1 13.f odd 12 1
338.2.c.b 2 13.d odd 4 1
338.2.c.b 2 13.f odd 12 1
338.2.c.f 2 13.d odd 4 1
338.2.c.f 2 13.f odd 12 1
338.2.e.c 4 1.a even 1 1 trivial
338.2.e.c 4 13.b even 2 1 inner
338.2.e.c 4 13.c even 3 1 inner
338.2.e.c 4 13.e even 6 1 inner
650.2.c.a 2 65.q odd 12 1
650.2.c.a 2 65.r odd 12 1
650.2.c.d 2 65.q odd 12 1
650.2.c.d 2 65.r odd 12 1
650.2.d.b 2 65.l even 6 1
650.2.d.b 2 65.n even 6 1
832.2.f.b 2 104.n odd 6 1
832.2.f.b 2 104.p odd 6 1
832.2.f.d 2 104.r even 6 1
832.2.f.d 2 104.s even 6 1
1274.2.d.c 2 91.n odd 6 1
1274.2.d.c 2 91.t odd 6 1
1274.2.n.c 4 91.l odd 6 1
1274.2.n.c 4 91.m odd 6 1
1274.2.n.c 4 91.p odd 6 1
1274.2.n.c 4 91.v odd 6 1
1274.2.n.d 4 91.g even 3 1
1274.2.n.d 4 91.h even 3 1
1274.2.n.d 4 91.k even 6 1
1274.2.n.d 4 91.u even 6 1
1872.2.c.f 2 156.p even 6 1
1872.2.c.f 2 156.r even 6 1
2704.2.a.j 1 52.l even 12 1
2704.2.a.k 1 52.l even 12 1
3042.2.a.g 1 39.k even 12 1
3042.2.a.j 1 39.k even 12 1
8450.2.a.h 1 65.s odd 12 1
8450.2.a.u 1 65.s odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(338, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 9 + T^{2} )^{2} \)
$7$ \( 81 - 9 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 9 + 3 T + T^{2} )^{2} \)
$19$ \( 1296 - 36 T^{2} + T^{4} \)
$23$ \( ( 36 - 6 T + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( 81 - 9 T^{2} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( 1 - T + T^{2} )^{2} \)
$47$ \( ( 9 + T^{2} )^{2} \)
$53$ \( ( 6 + T )^{4} \)
$59$ \( 1296 - 36 T^{2} + T^{4} \)
$61$ \( ( 64 - 8 T + T^{2} )^{2} \)
$67$ \( 20736 - 144 T^{2} + T^{4} \)
$71$ \( 50625 - 225 T^{2} + T^{4} \)
$73$ \( ( 36 + T^{2} )^{2} \)
$79$ \( ( -10 + T )^{4} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( 1296 - 36 T^{2} + T^{4} \)
$97$ \( 20736 - 144 T^{2} + T^{4} \)
show more
show less