Properties

Label 342.2.u.d
Level $342$
Weight $2$
Character orbit 342.u
Analytic conductor $2.731$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.u (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + ( 1 + 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( 1 - 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{18} q^{2} + \zeta_{18}^{2} q^{4} + ( 1 + 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( 1 - 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{7} + \zeta_{18}^{3} q^{8} + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{10} + ( \zeta_{18}^{2} - 4 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{11} + ( -3 + 4 \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{13} + ( -1 + \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{14} + \zeta_{18}^{4} q^{16} + ( -4 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{17} + ( \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{19} + ( 2 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{20} + ( \zeta_{18}^{3} - 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{22} + ( 2 + \zeta_{18} + 4 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{23} + ( 4 + \zeta_{18} - 5 \zeta_{18}^{3} - 5 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{25} + ( -3 \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{26} + ( -1 - \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{28} + ( -4 - 4 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{29} + ( 1 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{31} + \zeta_{18}^{5} q^{32} + ( -1 - 4 \zeta_{18}^{2} - \zeta_{18}^{4} ) q^{34} + ( 2 \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{35} + ( 1 + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{37} + ( -2 + \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{38} + ( 1 + 2 \zeta_{18} + \zeta_{18}^{2} ) q^{40} + ( 3 - 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{41} + ( -3 - \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{43} + ( -1 + \zeta_{18}^{3} + \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{44} + ( 2 \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} + \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{46} + ( 4 + 4 \zeta_{18} - 5 \zeta_{18}^{2} - \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{47} + ( -5 \zeta_{18} + 7 \zeta_{18}^{2} + 7 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{49} + ( 1 + 4 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{50} + ( 1 - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{52} + ( -4 - 2 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{53} + ( -2 - 6 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{55} + ( 1 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{56} + ( -2 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{58} + ( -2 - 7 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{5} ) q^{59} + ( -3 + 3 \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{61} + ( -2 + \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{62} + ( -1 + \zeta_{18}^{3} ) q^{64} + ( -5 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{65} + ( -4 - 4 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{67} + ( -\zeta_{18} - 4 \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{68} + ( -2 + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{70} + ( -3 + 3 \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{71} + ( 7 - 7 \zeta_{18}^{2} - 10 \zeta_{18}^{3} + \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{73} + ( 2 + \zeta_{18} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{74} + ( 2 - 2 \zeta_{18} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{76} + ( -7 + 4 \zeta_{18} + 4 \zeta_{18}^{2} + 6 \zeta_{18}^{4} - 10 \zeta_{18}^{5} ) q^{77} + ( 5 - 5 \zeta_{18}^{2} - 5 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{79} + ( \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{80} + ( 3 \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{82} + ( 1 + 11 \zeta_{18} - 6 \zeta_{18}^{2} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{83} + ( -7 - 7 \zeta_{18} - 9 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 9 \zeta_{18}^{5} ) q^{85} + ( -3 - 3 \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{86} + ( 4 - \zeta_{18} - 4 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{88} + ( 5 - 11 \zeta_{18} + 6 \zeta_{18}^{3} + 6 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{89} + ( -6 + 10 \zeta_{18} - 9 \zeta_{18}^{2} + 10 \zeta_{18}^{3} - 6 \zeta_{18}^{4} ) q^{91} + ( -2 + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{92} + ( -5 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{94} + ( 7 + 5 \zeta_{18} + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{95} + ( -1 + 4 \zeta_{18} + 7 \zeta_{18}^{2} + 8 \zeta_{18}^{3} - 8 \zeta_{18}^{5} ) q^{97} + ( 5 - 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 7 \zeta_{18}^{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{5} + 3q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 3q^{5} + 3q^{7} + 3q^{8} + 6q^{10} - 12q^{11} - 21q^{13} - 3q^{17} + 6q^{19} + 12q^{20} + 3q^{22} + 15q^{23} + 9q^{25} - 9q^{28} - 15q^{29} + 3q^{31} - 6q^{34} + 6q^{35} + 6q^{37} + 6q^{40} + 9q^{41} - 9q^{43} - 3q^{44} + 12q^{46} + 21q^{47} + 3q^{50} + 15q^{52} - 30q^{53} - 9q^{55} + 6q^{56} - 12q^{58} - 27q^{59} - 9q^{61} - 9q^{62} - 3q^{64} + 6q^{65} - 15q^{67} - 12q^{68} - 6q^{70} - 9q^{71} + 12q^{73} + 6q^{74} + 9q^{76} - 42q^{77} + 15q^{79} + 3q^{80} - 9q^{82} + 3q^{83} - 36q^{85} - 18q^{86} + 12q^{88} + 48q^{89} - 6q^{91} - 3q^{92} - 30q^{94} + 48q^{95} + 18q^{97} + 36q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(\zeta_{18}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.173648 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
0.939693 + 0.342020i
−0.173648 + 0.984808i 0 −0.939693 0.342020i −1.55303 + 0.565258i 0 −0.0923963 0.160035i 0.500000 0.866025i 0 −0.286989 1.62760i
73.1 −0.766044 0.642788i 0 0.173648 + 0.984808i 0.0812519 0.460802i 0 2.20574 + 3.82045i 0.500000 0.866025i 0 −0.358441 + 0.300767i
199.1 −0.173648 0.984808i 0 −0.939693 + 0.342020i −1.55303 0.565258i 0 −0.0923963 + 0.160035i 0.500000 + 0.866025i 0 −0.286989 + 1.62760i
253.1 −0.766044 + 0.642788i 0 0.173648 0.984808i 0.0812519 + 0.460802i 0 2.20574 3.82045i 0.500000 + 0.866025i 0 −0.358441 0.300767i
271.1 0.939693 0.342020i 0 0.766044 0.642788i 2.97178 + 2.49362i 0 −0.613341 1.06234i 0.500000 0.866025i 0 3.64543 + 1.32683i
289.1 0.939693 + 0.342020i 0 0.766044 + 0.642788i 2.97178 2.49362i 0 −0.613341 + 1.06234i 0.500000 + 0.866025i 0 3.64543 1.32683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.u.d 6
3.b odd 2 1 114.2.i.b 6
12.b even 2 1 912.2.bo.c 6
19.e even 9 1 inner 342.2.u.d 6
19.e even 9 1 6498.2.a.bo 3
19.f odd 18 1 6498.2.a.bt 3
57.j even 18 1 2166.2.a.n 3
57.l odd 18 1 114.2.i.b 6
57.l odd 18 1 2166.2.a.t 3
228.v even 18 1 912.2.bo.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.i.b 6 3.b odd 2 1
114.2.i.b 6 57.l odd 18 1
342.2.u.d 6 1.a even 1 1 trivial
342.2.u.d 6 19.e even 9 1 inner
912.2.bo.c 6 12.b even 2 1
912.2.bo.c 6 228.v even 18 1
2166.2.a.n 3 57.j even 18 1
2166.2.a.t 3 57.l odd 18 1
6498.2.a.bo 3 19.e even 9 1
6498.2.a.bt 3 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 3 T_{5}^{5} + 30 T_{5}^{3} + 36 T_{5}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{3} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 9 + 36 T^{2} + 30 T^{3} - 3 T^{5} + T^{6} \)
$7$ \( 1 + 6 T + 33 T^{2} + 20 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \)
$11$ \( 2601 + 2295 T + 1413 T^{2} + 438 T^{3} + 99 T^{4} + 12 T^{5} + T^{6} \)
$13$ \( 361 - 57 T + 1407 T^{2} + 800 T^{3} + 186 T^{4} + 21 T^{5} + T^{6} \)
$17$ \( 2601 - 459 T + 495 T^{2} + 24 T^{3} - 18 T^{4} + 3 T^{5} + T^{6} \)
$19$ \( 6859 - 2166 T - 228 T^{2} + 169 T^{3} - 12 T^{4} - 6 T^{5} + T^{6} \)
$23$ \( 9 - 162 T + 846 T^{2} - 246 T^{3} + 108 T^{4} - 15 T^{5} + T^{6} \)
$29$ \( 2601 - 1836 T + 576 T^{2} + 138 T^{3} + 72 T^{4} + 15 T^{5} + T^{6} \)
$31$ \( 289 - 102 T + 87 T^{2} - 16 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \)
$37$ \( ( 19 - 9 T - 3 T^{2} + T^{3} )^{2} \)
$41$ \( 729 + 729 T + 486 T^{2} + 216 T^{3} + 27 T^{4} - 9 T^{5} + T^{6} \)
$43$ \( 1 + 45 T + 576 T^{2} + 80 T^{3} + 45 T^{4} + 9 T^{5} + T^{6} \)
$47$ \( 25281 - 27189 T + 10836 T^{2} - 1968 T^{3} + 261 T^{4} - 21 T^{5} + T^{6} \)
$53$ \( 47961 + 11826 T + 7218 T^{2} + 2373 T^{3} + 378 T^{4} + 30 T^{5} + T^{6} \)
$59$ \( 23409 + 31671 T + 19197 T^{2} + 3312 T^{3} + 360 T^{4} + 27 T^{5} + T^{6} \)
$61$ \( 1 - 36 T + 549 T^{2} + 323 T^{3} + 72 T^{4} + 9 T^{5} + T^{6} \)
$67$ \( 7921 + 8544 T + 4200 T^{2} + 1070 T^{3} + 156 T^{4} + 15 T^{5} + T^{6} \)
$71$ \( 6561 + 6561 T + 2916 T^{2} + 648 T^{3} + 81 T^{4} + 9 T^{5} + T^{6} \)
$73$ \( 546121 + 281559 T + 32586 T^{2} - 2404 T^{3} + 246 T^{4} - 12 T^{5} + T^{6} \)
$79$ \( 5329 + 7665 T + 3180 T^{2} + 152 T^{3} + 105 T^{4} - 15 T^{5} + T^{6} \)
$83$ \( 3583449 - 511110 T + 78579 T^{2} - 2976 T^{3} + 279 T^{4} - 3 T^{5} + T^{6} \)
$89$ \( 3583449 - 698517 T + 95616 T^{2} - 12174 T^{3} + 1044 T^{4} - 48 T^{5} + T^{6} \)
$97$ \( 1630729 + 34479 T + 3636 T^{2} - 613 T^{3} + 99 T^{4} - 18 T^{5} + T^{6} \)
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