# Properties

 Label 342.6.a.i Level $342$ Weight $6$ Character orbit 342.a Self dual yes Analytic conductor $54.851$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 342.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$54.8512663760$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1441})$$ Defining polynomial: $$x^{2} - x - 360$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{1441})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 16 q^{4} + ( 24 - 3 \beta ) q^{5} + ( 59 - 4 \beta ) q^{7} + 64 q^{8} +O(q^{10})$$ $$q + 4 q^{2} + 16 q^{4} + ( 24 - 3 \beta ) q^{5} + ( 59 - 4 \beta ) q^{7} + 64 q^{8} + ( 96 - 12 \beta ) q^{10} + ( -330 - \beta ) q^{11} + ( 809 - 5 \beta ) q^{13} + ( 236 - 16 \beta ) q^{14} + 256 q^{16} + ( -27 - 10 \beta ) q^{17} + 361 q^{19} + ( 384 - 48 \beta ) q^{20} + ( -1320 - 4 \beta ) q^{22} + ( 1617 - 49 \beta ) q^{23} + ( 691 - 135 \beta ) q^{25} + ( 3236 - 20 \beta ) q^{26} + ( 944 - 64 \beta ) q^{28} + ( 1083 + 315 \beta ) q^{29} + ( -748 + 316 \beta ) q^{31} + 1024 q^{32} + ( -108 - 40 \beta ) q^{34} + ( 5736 - 261 \beta ) q^{35} + ( 5330 - 172 \beta ) q^{37} + 1444 q^{38} + ( 1536 - 192 \beta ) q^{40} + ( -8616 + 602 \beta ) q^{41} + ( 5792 - 281 \beta ) q^{43} + ( -5280 - 16 \beta ) q^{44} + ( 6468 - 196 \beta ) q^{46} + ( 5520 + 1115 \beta ) q^{47} + ( -7566 - 456 \beta ) q^{49} + ( 2764 - 540 \beta ) q^{50} + ( 12944 - 80 \beta ) q^{52} + ( -10593 + 601 \beta ) q^{53} + ( -6840 + 969 \beta ) q^{55} + ( 3776 - 256 \beta ) q^{56} + ( 4332 + 1260 \beta ) q^{58} + ( 39327 - 73 \beta ) q^{59} + ( 21398 + 825 \beta ) q^{61} + ( -2992 + 1264 \beta ) q^{62} + 4096 q^{64} + ( 24816 - 2532 \beta ) q^{65} + ( 5453 - 3101 \beta ) q^{67} + ( -432 - 160 \beta ) q^{68} + ( 22944 - 1044 \beta ) q^{70} + ( 31878 - 1268 \beta ) q^{71} + ( 6617 + 2984 \beta ) q^{73} + ( 21320 - 688 \beta ) q^{74} + 5776 q^{76} + ( -18030 + 1265 \beta ) q^{77} + ( 33494 + 134 \beta ) q^{79} + ( 6144 - 768 \beta ) q^{80} + ( -34464 + 2408 \beta ) q^{82} + ( 4134 + 2446 \beta ) q^{83} + ( 10152 - 129 \beta ) q^{85} + ( 23168 - 1124 \beta ) q^{86} + ( -21120 - 64 \beta ) q^{88} + ( -61956 - 4276 \beta ) q^{89} + ( 54931 - 3511 \beta ) q^{91} + ( 25872 - 784 \beta ) q^{92} + ( 22080 + 4460 \beta ) q^{94} + ( 8664 - 1083 \beta ) q^{95} + ( 90590 - 2622 \beta ) q^{97} + ( -30264 - 1824 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{2} + 32q^{4} + 45q^{5} + 114q^{7} + 128q^{8} + O(q^{10})$$ $$2q + 8q^{2} + 32q^{4} + 45q^{5} + 114q^{7} + 128q^{8} + 180q^{10} - 661q^{11} + 1613q^{13} + 456q^{14} + 512q^{16} - 64q^{17} + 722q^{19} + 720q^{20} - 2644q^{22} + 3185q^{23} + 1247q^{25} + 6452q^{26} + 1824q^{28} + 2481q^{29} - 1180q^{31} + 2048q^{32} - 256q^{34} + 11211q^{35} + 10488q^{37} + 2888q^{38} + 2880q^{40} - 16630q^{41} + 11303q^{43} - 10576q^{44} + 12740q^{46} + 12155q^{47} - 15588q^{49} + 4988q^{50} + 25808q^{52} - 20585q^{53} - 12711q^{55} + 7296q^{56} + 9924q^{58} + 78581q^{59} + 43621q^{61} - 4720q^{62} + 8192q^{64} + 47100q^{65} + 7805q^{67} - 1024q^{68} + 44844q^{70} + 62488q^{71} + 16218q^{73} + 41952q^{74} + 11552q^{76} - 34795q^{77} + 67122q^{79} + 11520q^{80} - 66520q^{82} + 10714q^{83} + 20175q^{85} + 45212q^{86} - 42304q^{88} - 128188q^{89} + 106351q^{91} + 50960q^{92} + 48620q^{94} + 16245q^{95} + 178558q^{97} - 62352q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 19.4803 −18.4803
4.00000 0 16.0000 −34.4408 0 −18.9210 64.0000 0 −137.763
1.2 4.00000 0 16.0000 79.4408 0 132.921 64.0000 0 317.763
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.6.a.i 2
3.b odd 2 1 38.6.a.c 2
12.b even 2 1 304.6.a.f 2
15.d odd 2 1 950.6.a.d 2
57.d even 2 1 722.6.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.a.c 2 3.b odd 2 1
304.6.a.f 2 12.b even 2 1
342.6.a.i 2 1.a even 1 1 trivial
722.6.a.c 2 57.d even 2 1
950.6.a.d 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 45 T_{5} - 2736$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(342))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -4 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-2736 - 45 T + T^{2}$$
$7$ $$-2515 - 114 T + T^{2}$$
$11$ $$108870 + 661 T + T^{2}$$
$13$ $$641436 - 1613 T + T^{2}$$
$17$ $$-35001 + 64 T + T^{2}$$
$19$ $$( -361 + T )^{2}$$
$23$ $$1671096 - 3185 T + T^{2}$$
$29$ $$-34206966 - 2481 T + T^{2}$$
$31$ $$-35625024 + 1180 T + T^{2}$$
$37$ $$16841900 - 10488 T + T^{2}$$
$41$ $$-61416816 + 16630 T + T^{2}$$
$43$ $$3493752 - 11303 T + T^{2}$$
$47$ $$-410935800 - 12155 T + T^{2}$$
$53$ $$-24187104 + 20585 T + T^{2}$$
$59$ $$1541823618 - 78581 T + T^{2}$$
$61$ $$230502754 - 43621 T + T^{2}$$
$67$ $$-3449006904 - 7805 T + T^{2}$$
$71$ $$396968940 - 62488 T + T^{2}$$
$73$ $$-3142002343 - 16218 T + T^{2}$$
$79$ $$1119872072 - 67122 T + T^{2}$$
$83$ $$-2126648040 - 10714 T + T^{2}$$
$89$ $$-2478833568 + 128188 T + T^{2}$$
$97$ $$5494062880 - 178558 T + T^{2}$$