Properties

Label 338.2.c.a
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: \( x^{2} - x + 1 \) 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

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

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 −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.a 2
13.b even 2 1 338.2.c.d 2
13.c even 3 1 338.2.a.f 1
13.c even 3 1 inner 338.2.c.a 2
13.d odd 4 2 338.2.e.a 4
13.e even 6 1 26.2.a.a 1
13.e even 6 1 338.2.c.d 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 1 234.2.a.e 1
39.i odd 6 1 3042.2.a.a 1
39.k even 12 2 3042.2.b.a 2
52.i odd 6 1 208.2.a.a 1
52.j odd 6 1 2704.2.a.f 1
52.l even 12 2 2704.2.f.d 2
65.l even 6 1 650.2.a.j 1
65.n even 6 1 8450.2.a.c 1
65.r odd 12 2 650.2.b.d 2
91.k even 6 1 1274.2.f.p 2
91.l odd 6 1 1274.2.f.r 2
91.p odd 6 1 1274.2.f.r 2
91.t odd 6 1 1274.2.a.d 1
91.u even 6 1 1274.2.f.p 2
104.p odd 6 1 832.2.a.i 1
104.s even 6 1 832.2.a.d 1
117.l even 6 1 2106.2.e.ba 2
117.m odd 6 1 2106.2.e.b 2
117.r even 6 1 2106.2.e.ba 2
117.v odd 6 1 2106.2.e.b 2
143.i odd 6 1 3146.2.a.n 1
156.r even 6 1 1872.2.a.q 1
195.y odd 6 1 5850.2.a.p 1
195.bf even 12 2 5850.2.e.a 2
208.bh even 12 2 3328.2.b.m 2
208.bi odd 12 2 3328.2.b.j 2
221.n even 6 1 7514.2.a.c 1
247.m odd 6 1 9386.2.a.j 1
260.w odd 6 1 5200.2.a.x 1
312.ba even 6 1 7488.2.a.h 1
312.bg odd 6 1 7488.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 13.e even 6 1
208.2.a.a 1 52.i odd 6 1
234.2.a.e 1 39.h odd 6 1
338.2.a.f 1 13.c even 3 1
338.2.b.c 2 13.f odd 12 2
338.2.c.a 2 1.a even 1 1 trivial
338.2.c.a 2 13.c even 3 1 inner
338.2.c.d 2 13.b even 2 1
338.2.c.d 2 13.e even 6 1
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.l even 6 1
650.2.b.d 2 65.r odd 12 2
832.2.a.d 1 104.s even 6 1
832.2.a.i 1 104.p odd 6 1
1274.2.a.d 1 91.t odd 6 1
1274.2.f.p 2 91.k even 6 1
1274.2.f.p 2 91.u even 6 1
1274.2.f.r 2 91.l odd 6 1
1274.2.f.r 2 91.p odd 6 1
1872.2.a.q 1 156.r even 6 1
2106.2.e.b 2 117.m odd 6 1
2106.2.e.b 2 117.v odd 6 1
2106.2.e.ba 2 117.l even 6 1
2106.2.e.ba 2 117.r even 6 1
2704.2.a.f 1 52.j odd 6 1
2704.2.f.d 2 52.l even 12 2
3042.2.a.a 1 39.i odd 6 1
3042.2.b.a 2 39.k even 12 2
3146.2.a.n 1 143.i odd 6 1
3328.2.b.j 2 208.bi odd 12 2
3328.2.b.m 2 208.bh even 12 2
5200.2.a.x 1 260.w odd 6 1
5850.2.a.p 1 195.y odd 6 1
5850.2.e.a 2 195.bf even 12 2
7488.2.a.g 1 312.bg odd 6 1
7488.2.a.h 1 312.ba even 6 1
7514.2.a.c 1 221.n even 6 1
8450.2.a.c 1 65.n even 6 1
9386.2.a.j 1 247.m 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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