Properties

Label 342.4.g.h
Level $342$
Weight $4$
Character orbit 342.g
Analytic conductor $20.179$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 342.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.1786532220\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.627014547.1
Defining polynomial: \(x^{6} + 26 x^{4} + 169 x^{2} + 147\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{1} q^{2} + ( -4 + 4 \beta_{1} ) q^{4} + ( 3 \beta_{1} - \beta_{2} - \beta_{5} ) q^{5} + ( 2 - \beta_{2} ) q^{7} -8 q^{8} +O(q^{10})\) \( q + 2 \beta_{1} q^{2} + ( -4 + 4 \beta_{1} ) q^{4} + ( 3 \beta_{1} - \beta_{2} - \beta_{5} ) q^{5} + ( 2 - \beta_{2} ) q^{7} -8 q^{8} + ( -6 + 6 \beta_{1} - 2 \beta_{5} ) q^{10} + ( 15 - \beta_{3} ) q^{11} + ( 1 - \beta_{1} + 6 \beta_{5} ) q^{13} + ( 4 \beta_{1} - 2 \beta_{2} - 2 \beta_{5} ) q^{14} -16 \beta_{1} q^{16} + ( -30 \beta_{1} - 7 \beta_{2} + \beta_{4} - 7 \beta_{5} ) q^{17} + ( 10 - 6 \beta_{1} + 4 \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{19} + ( -12 + 4 \beta_{2} ) q^{20} + ( 30 \beta_{1} + 2 \beta_{4} ) q^{22} + ( 3 - 3 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 9 \beta_{5} ) q^{23} + ( 29 - 29 \beta_{1} + \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{25} + ( 2 - 12 \beta_{2} ) q^{26} + ( -8 + 8 \beta_{1} - 4 \beta_{5} ) q^{28} + ( 36 - 36 \beta_{1} - \beta_{3} - \beta_{4} - 15 \beta_{5} ) q^{29} + ( -28 + 24 \beta_{2} - \beta_{3} ) q^{31} + ( 32 - 32 \beta_{1} ) q^{32} + ( 60 - 60 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 14 \beta_{5} ) q^{34} + ( 93 \beta_{1} - 4 \beta_{2} + \beta_{4} - 4 \beta_{5} ) q^{35} + ( 77 - 17 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 12 + 8 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 8 \beta_{5} ) q^{38} + ( -24 \beta_{1} + 8 \beta_{2} + 8 \beta_{5} ) q^{40} + ( -222 \beta_{1} + 20 \beta_{2} + 2 \beta_{4} + 20 \beta_{5} ) q^{41} + ( 28 \beta_{1} - 21 \beta_{2} + 2 \beta_{4} - 21 \beta_{5} ) q^{43} + ( -60 + 60 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} ) q^{44} + ( 6 + 18 \beta_{2} - 4 \beta_{3} ) q^{46} + ( 102 - 102 \beta_{1} + 6 \beta_{3} + 6 \beta_{4} + 10 \beta_{5} ) q^{47} + ( -252 - 3 \beta_{2} - \beta_{3} ) q^{49} + ( 58 + 10 \beta_{2} + 2 \beta_{3} ) q^{50} + ( 4 \beta_{1} - 24 \beta_{2} - 24 \beta_{5} ) q^{52} + ( -435 + 435 \beta_{1} + \beta_{3} + \beta_{4} - 18 \beta_{5} ) q^{53} + ( 66 \beta_{1} - 55 \beta_{2} + 5 \beta_{4} - 55 \beta_{5} ) q^{55} + ( -16 + 8 \beta_{2} ) q^{56} + ( 72 + 30 \beta_{2} - 2 \beta_{3} ) q^{58} + ( 81 \beta_{1} - 9 \beta_{2} - 9 \beta_{5} ) q^{59} + ( 157 - 157 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 32 \beta_{5} ) q^{61} + ( -56 \beta_{1} + 48 \beta_{2} + 2 \beta_{4} + 48 \beta_{5} ) q^{62} + 64 q^{64} + ( 525 - 13 \beta_{2} - 6 \beta_{3} ) q^{65} + ( 244 - 244 \beta_{1} - \beta_{3} - \beta_{4} - 12 \beta_{5} ) q^{67} + ( 120 + 28 \beta_{2} + 4 \beta_{3} ) q^{68} + ( -186 + 186 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} ) q^{70} + ( -18 \beta_{1} + 19 \beta_{2} - 11 \beta_{4} + 19 \beta_{5} ) q^{71} + ( 217 \beta_{1} + 66 \beta_{2} + 4 \beta_{4} + 66 \beta_{5} ) q^{73} + ( 154 \beta_{1} - 34 \beta_{2} + 6 \beta_{4} - 34 \beta_{5} ) q^{74} + ( -16 + 40 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} + 28 \beta_{5} ) q^{76} + ( 51 - 55 \beta_{2} - 4 \beta_{3} ) q^{77} + ( 172 \beta_{1} + 30 \beta_{2} - 3 \beta_{4} + 30 \beta_{5} ) q^{79} + ( 48 - 48 \beta_{1} + 16 \beta_{5} ) q^{80} + ( 444 - 444 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} + 40 \beta_{5} ) q^{82} + ( -846 - 37 \beta_{2} + 5 \beta_{3} ) q^{83} + ( -540 + 540 \beta_{1} + 12 \beta_{3} + 12 \beta_{4} - 24 \beta_{5} ) q^{85} + ( -56 + 56 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} - 42 \beta_{5} ) q^{86} + ( -120 + 8 \beta_{3} ) q^{88} + ( 543 - 543 \beta_{1} - 11 \beta_{3} - 11 \beta_{4} - 34 \beta_{5} ) q^{89} + ( 524 - 524 \beta_{1} - 6 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} ) q^{91} + ( 12 \beta_{1} + 36 \beta_{2} + 8 \beta_{4} + 36 \beta_{5} ) q^{92} + ( 204 - 20 \beta_{2} + 12 \beta_{3} ) q^{94} + ( -222 - 357 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} + 44 \beta_{5} ) q^{95} + ( -320 \beta_{1} + 13 \beta_{2} + \beta_{4} + 13 \beta_{5} ) q^{97} + ( -504 \beta_{1} - 6 \beta_{2} + 2 \beta_{4} - 6 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 12 q^{4} + 10 q^{5} + 14 q^{7} - 48 q^{8} + O(q^{10}) \) \( 6 q + 6 q^{2} - 12 q^{4} + 10 q^{5} + 14 q^{7} - 48 q^{8} - 20 q^{10} + 88 q^{11} + 9 q^{13} + 14 q^{14} - 48 q^{16} - 84 q^{17} + 32 q^{19} - 80 q^{20} + 88 q^{22} - 2 q^{23} + 83 q^{25} + 36 q^{26} - 28 q^{28} + 92 q^{29} - 218 q^{31} + 96 q^{32} + 168 q^{34} + 282 q^{35} + 490 q^{37} + 74 q^{38} - 80 q^{40} - 688 q^{41} + 103 q^{43} - 176 q^{44} - 8 q^{46} + 322 q^{47} - 1508 q^{49} + 332 q^{50} + 36 q^{52} - 1322 q^{53} + 248 q^{55} - 112 q^{56} + 368 q^{58} + 252 q^{59} + 435 q^{61} - 218 q^{62} + 384 q^{64} + 3164 q^{65} + 719 q^{67} + 672 q^{68} - 564 q^{70} - 62 q^{71} + 581 q^{73} + 490 q^{74} + 20 q^{76} + 408 q^{77} + 489 q^{79} + 160 q^{80} + 1376 q^{82} - 4992 q^{83} - 1632 q^{85} - 206 q^{86} - 704 q^{88} + 1584 q^{89} + 1573 q^{91} - 8 q^{92} + 1288 q^{94} - 2362 q^{95} - 974 q^{97} - 1508 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 26 x^{4} + 169 x^{2} + 147\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 13 \nu + 7 \)\()/14\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 40 \nu^{2} + 119 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( 10 \nu^{4} + 116 \nu^{2} - 119 \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{5} - 10 \nu^{4} + 43 \nu^{3} - 116 \nu^{2} + 431 \nu + 119 \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 2 \nu^{4} + 43 \nu^{3} - 40 \nu^{2} + 179 \nu - 119 \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2}\)\()/36\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 5 \beta_{2} - 102\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(26 \beta_{5} - 26 \beta_{4} - 13 \beta_{3} + 13 \beta_{2} + 504 \beta_{1} - 252\)\()/36\)
\(\nu^{4}\)\(=\)\((\)\(10 \beta_{3} - 29 \beta_{2} + 663\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(-64 \beta_{5} + 190 \beta_{4} + 95 \beta_{3} - 32 \beta_{2} - 5418 \beta_{1} + 2709\)\()/18\)

Character values

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

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
1.01248i
4.00355i
2.99107i
1.01248i
4.00355i
2.99107i
1.00000 1.73205i 0 −2.00000 3.46410i −4.22121 + 7.31135i 0 −9.44242 −8.00000 0 8.44242 + 14.6227i
163.2 1.00000 1.73205i 0 −2.00000 3.46410i 2.09405 3.62701i 0 3.18810 −8.00000 0 −4.18810 7.25401i
163.3 1.00000 1.73205i 0 −2.00000 3.46410i 7.12716 12.3446i 0 13.2543 −8.00000 0 −14.2543 24.6892i
235.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.22121 7.31135i 0 −9.44242 −8.00000 0 8.44242 14.6227i
235.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.09405 + 3.62701i 0 3.18810 −8.00000 0 −4.18810 + 7.25401i
235.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 7.12716 + 12.3446i 0 13.2543 −8.00000 0 −14.2543 + 24.6892i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.4.g.h 6
3.b odd 2 1 114.4.e.d 6
19.c even 3 1 inner 342.4.g.h 6
57.f even 6 1 2166.4.a.t 3
57.h odd 6 1 114.4.e.d 6
57.h odd 6 1 2166.4.a.u 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.d 6 3.b odd 2 1
114.4.e.d 6 57.h odd 6 1
342.4.g.h 6 1.a even 1 1 trivial
342.4.g.h 6 19.c even 3 1 inner
2166.4.a.t 3 57.f even 6 1
2166.4.a.u 3 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 10 T_{5}^{5} + 196 T_{5}^{4} - 48 T_{5}^{3} + 14256 T_{5}^{2} - 48384 T_{5} + 254016 \) acting on \(S_{4}^{\mathrm{new}}(342, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{3} \)
$3$ \( T^{6} \)
$5$ \( 254016 - 48384 T + 14256 T^{2} - 48 T^{3} + 196 T^{4} - 10 T^{5} + T^{6} \)
$7$ \( ( 399 - 113 T - 7 T^{2} + T^{3} )^{2} \)
$11$ \( ( 217224 - 4740 T - 44 T^{2} + T^{3} )^{2} \)
$13$ \( 1420159225 - 174443865 T + 21766806 T^{2} - 33709 T^{3} + 4710 T^{4} - 9 T^{5} + T^{6} \)
$17$ \( 673712640000 + 8273664000 T + 170553600 T^{2} + 794880 T^{3} + 17136 T^{4} + 84 T^{5} + T^{6} \)
$19$ \( 322687697779 - 1505468192 T - 8443429 T^{2} + 1033600 T^{3} - 1231 T^{4} - 32 T^{5} + T^{6} \)
$23$ \( 11401114176 - 3223780992 T + 911343312 T^{2} - 273936 T^{3} + 30196 T^{4} + 2 T^{5} + T^{6} \)
$29$ \( 1696079056896 + 39257616384 T + 788845824 T^{2} + 5377920 T^{3} + 38608 T^{4} - 92 T^{5} + T^{6} \)
$31$ \( ( -9721957 - 73489 T + 109 T^{2} + T^{3} )^{2} \)
$37$ \( ( -1317799 - 71005 T - 245 T^{2} + T^{3} )^{2} \)
$41$ \( 105675768336384 - 910385464320 T + 14915425536 T^{2} + 81489024 T^{3} + 384784 T^{4} + 688 T^{5} + T^{6} \)
$43$ \( 39054112950241 + 495553041713 T + 5644333322 T^{2} + 20666249 T^{3} + 89906 T^{4} - 103 T^{5} + T^{6} \)
$47$ \( 136793983242816 + 1943436862944 T + 23844396384 T^{2} + 76896600 T^{3} + 269848 T^{4} - 322 T^{5} + T^{6} \)
$53$ \( 4486100708289600 + 35729472185280 T + 196021376784 T^{2} + 571261536 T^{3} + 1214236 T^{4} + 1322 T^{5} + T^{6} \)
$59$ \( 134994517056 + 3928411872 T + 206907696 T^{2} - 3429216 T^{3} + 52812 T^{4} - 252 T^{5} + T^{6} \)
$61$ \( 5456924641 - 10530828147 T + 20354632134 T^{2} + 61864553 T^{3} + 331782 T^{4} - 435 T^{5} + T^{6} \)
$67$ \( 92437321911489 - 1437617323191 T + 15445546402 T^{2} - 88281047 T^{3} + 367434 T^{4} - 719 T^{5} + T^{6} \)
$71$ \( 12330917402872896 + 79753610220768 T + 522713246112 T^{2} + 177560184 T^{3} + 722056 T^{4} + 62 T^{5} + T^{6} \)
$73$ \( 4992458989302225 + 36054247673115 T + 219322540726 T^{2} + 437780959 T^{3} + 847830 T^{4} - 581 T^{5} + T^{6} \)
$79$ \( 2685185649775729 - 4885526114337 T + 34228288914 T^{2} - 57534145 T^{3} + 333402 T^{4} - 489 T^{5} + T^{6} \)
$83$ \( ( 372278592 + 1783728 T + 2496 T^{2} + T^{3} )^{2} \)
$89$ \( 156435026228942400 + 28918743806880 T + 631847538576 T^{2} - 906853104 T^{3} + 2435940 T^{4} - 1584 T^{5} + T^{6} \)
$97$ \( 585649680040000 + 7025318060000 T + 60703095200 T^{2} + 234351800 T^{3} + 658376 T^{4} + 974 T^{5} + T^{6} \)
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