Properties

Label 338.2.c.b
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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Newspace parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 338.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.69894358832\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

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

Character values

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

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

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.

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 1.50000 + 2.59808i 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 1.50000 2.59808i 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.b 2
13.b even 2 1 338.2.c.f 2
13.c even 3 1 338.2.a.d 1
13.c even 3 1 inner 338.2.c.b 2
13.d odd 4 2 338.2.e.c 4
13.e even 6 1 338.2.a.b 1
13.e even 6 1 338.2.c.f 2
13.f odd 12 2 26.2.b.a 2
13.f odd 12 2 338.2.e.c 4
39.h odd 6 1 3042.2.a.j 1
39.i odd 6 1 3042.2.a.g 1
39.k even 12 2 234.2.b.b 2
52.i odd 6 1 2704.2.a.k 1
52.j odd 6 1 2704.2.a.j 1
52.l even 12 2 208.2.f.a 2
65.l even 6 1 8450.2.a.u 1
65.n even 6 1 8450.2.a.h 1
65.o even 12 2 650.2.c.d 2
65.s odd 12 2 650.2.d.b 2
65.t even 12 2 650.2.c.a 2
91.w even 12 2 1274.2.n.c 4
91.x odd 12 2 1274.2.n.d 4
91.ba even 12 2 1274.2.n.c 4
91.bc even 12 2 1274.2.d.c 2
91.bd odd 12 2 1274.2.n.d 4
104.u even 12 2 832.2.f.b 2
104.x odd 12 2 832.2.f.d 2
156.v odd 12 2 1872.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.b.a 2 13.f odd 12 2
208.2.f.a 2 52.l even 12 2
234.2.b.b 2 39.k even 12 2
338.2.a.b 1 13.e even 6 1
338.2.a.d 1 13.c even 3 1
338.2.c.b 2 1.a even 1 1 trivial
338.2.c.b 2 13.c even 3 1 inner
338.2.c.f 2 13.b even 2 1
338.2.c.f 2 13.e even 6 1
338.2.e.c 4 13.d odd 4 2
338.2.e.c 4 13.f odd 12 2
650.2.c.a 2 65.t even 12 2
650.2.c.d 2 65.o even 12 2
650.2.d.b 2 65.s odd 12 2
832.2.f.b 2 104.u even 12 2
832.2.f.d 2 104.x odd 12 2
1274.2.d.c 2 91.bc even 12 2
1274.2.n.c 4 91.w even 12 2
1274.2.n.c 4 91.ba even 12 2
1274.2.n.d 4 91.x odd 12 2
1274.2.n.d 4 91.bd odd 12 2
1872.2.c.f 2 156.v odd 12 2
2704.2.a.j 1 52.j odd 6 1
2704.2.a.k 1 52.i odd 6 1
3042.2.a.g 1 39.i odd 6 1
3042.2.a.j 1 39.h odd 6 1
8450.2.a.h 1 65.n even 6 1
8450.2.a.u 1 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 \(S_{2}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{5} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( ( 3 + T )^{2} \)
$7$ \( 9 - 3 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( 9 - 3 T + T^{2} \)
$19$ \( 36 - 6 T + T^{2} \)
$23$ \( 36 + 6 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( 9 - 3 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1 + T + T^{2} \)
$47$ \( ( 3 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( 36 + 6 T + T^{2} \)
$61$ \( 64 - 8 T + T^{2} \)
$67$ \( 144 - 12 T + T^{2} \)
$71$ \( 225 + 15 T + T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( ( -10 + T )^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 36 + 6 T + T^{2} \)
$97$ \( 144 - 12 T + T^{2} \)
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