Properties

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

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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: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

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

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

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 −0.866025 0.500000i 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 0.866025 + 0.500000i 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 −0.866025 + 0.500000i 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 0.866025 0.500000i 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.a 4
13.b even 2 1 inner 338.2.e.a 4
13.c even 3 1 338.2.b.c 2
13.c even 3 1 inner 338.2.e.a 4
13.d odd 4 1 338.2.c.a 2
13.d odd 4 1 338.2.c.d 2
13.e even 6 1 338.2.b.c 2
13.e even 6 1 inner 338.2.e.a 4
13.f odd 12 1 26.2.a.a 1
13.f odd 12 1 338.2.a.f 1
13.f odd 12 1 338.2.c.a 2
13.f odd 12 1 338.2.c.d 2
39.h odd 6 1 3042.2.b.a 2
39.i odd 6 1 3042.2.b.a 2
39.k even 12 1 234.2.a.e 1
39.k even 12 1 3042.2.a.a 1
52.i odd 6 1 2704.2.f.d 2
52.j odd 6 1 2704.2.f.d 2
52.l even 12 1 208.2.a.a 1
52.l even 12 1 2704.2.a.f 1
65.o even 12 1 650.2.b.d 2
65.s odd 12 1 650.2.a.j 1
65.s odd 12 1 8450.2.a.c 1
65.t even 12 1 650.2.b.d 2
91.w even 12 1 1274.2.f.r 2
91.x odd 12 1 1274.2.f.p 2
91.ba even 12 1 1274.2.f.r 2
91.bc even 12 1 1274.2.a.d 1
91.bd odd 12 1 1274.2.f.p 2
104.u even 12 1 832.2.a.i 1
104.x odd 12 1 832.2.a.d 1
117.w odd 12 1 2106.2.e.ba 2
117.x even 12 1 2106.2.e.b 2
117.bb odd 12 1 2106.2.e.ba 2
117.bc even 12 1 2106.2.e.b 2
143.o even 12 1 3146.2.a.n 1
156.v odd 12 1 1872.2.a.q 1
195.bc odd 12 1 5850.2.e.a 2
195.bh even 12 1 5850.2.a.p 1
195.bn odd 12 1 5850.2.e.a 2
208.be odd 12 1 3328.2.b.m 2
208.bf even 12 1 3328.2.b.j 2
208.bk even 12 1 3328.2.b.j 2
208.bl odd 12 1 3328.2.b.m 2
221.w odd 12 1 7514.2.a.c 1
247.bd even 12 1 9386.2.a.j 1
260.bc even 12 1 5200.2.a.x 1
312.bo even 12 1 7488.2.a.g 1
312.bq odd 12 1 7488.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 13.f odd 12 1
208.2.a.a 1 52.l even 12 1
234.2.a.e 1 39.k even 12 1
338.2.a.f 1 13.f odd 12 1
338.2.b.c 2 13.c even 3 1
338.2.b.c 2 13.e even 6 1
338.2.c.a 2 13.d odd 4 1
338.2.c.a 2 13.f odd 12 1
338.2.c.d 2 13.d odd 4 1
338.2.c.d 2 13.f odd 12 1
338.2.e.a 4 1.a even 1 1 trivial
338.2.e.a 4 13.b even 2 1 inner
338.2.e.a 4 13.c even 3 1 inner
338.2.e.a 4 13.e even 6 1 inner
650.2.a.j 1 65.s odd 12 1
650.2.b.d 2 65.o even 12 1
650.2.b.d 2 65.t even 12 1
832.2.a.d 1 104.x odd 12 1
832.2.a.i 1 104.u even 12 1
1274.2.a.d 1 91.bc even 12 1
1274.2.f.p 2 91.x odd 12 1
1274.2.f.p 2 91.bd odd 12 1
1274.2.f.r 2 91.w even 12 1
1274.2.f.r 2 91.ba even 12 1
1872.2.a.q 1 156.v odd 12 1
2106.2.e.b 2 117.x even 12 1
2106.2.e.b 2 117.bc even 12 1
2106.2.e.ba 2 117.w odd 12 1
2106.2.e.ba 2 117.bb odd 12 1
2704.2.a.f 1 52.l even 12 1
2704.2.f.d 2 52.i odd 6 1
2704.2.f.d 2 52.j odd 6 1
3042.2.a.a 1 39.k even 12 1
3042.2.b.a 2 39.h odd 6 1
3042.2.b.a 2 39.i odd 6 1
3146.2.a.n 1 143.o even 12 1
3328.2.b.j 2 208.bf even 12 1
3328.2.b.j 2 208.bk even 12 1
3328.2.b.m 2 208.be odd 12 1
3328.2.b.m 2 208.bl odd 12 1
5200.2.a.x 1 260.bc even 12 1
5850.2.a.p 1 195.bh even 12 1
5850.2.e.a 2 195.bc odd 12 1
5850.2.e.a 2 195.bn odd 12 1
7488.2.a.g 1 312.bo even 12 1
7488.2.a.h 1 312.bq odd 12 1
7514.2.a.c 1 221.w odd 12 1
8450.2.a.c 1 65.s odd 12 1
9386.2.a.j 1 247.bd even 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 \(S_{2}^{\mathrm{new}}(338, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 196 T^{2} + 38416 \) Copy content Toggle raw display
$71$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$73$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$97$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
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