# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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