# Properties

 Label 342.8.a.g Level $342$ Weight $8$ Character orbit 342.a Self dual yes Analytic conductor $106.836$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 342.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$106.835678716$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{633})$$ Defining polynomial: $$x^{2} - x - 158$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{633})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -8 q^{2} + 64 q^{4} + ( -62 - 31 \beta ) q^{5} + ( -1105 - 28 \beta ) q^{7} -512 q^{8} +O(q^{10})$$ $$q -8 q^{2} + 64 q^{4} + ( -62 - 31 \beta ) q^{5} + ( -1105 - 28 \beta ) q^{7} -512 q^{8} + ( 496 + 248 \beta ) q^{10} + ( 1572 + 151 \beta ) q^{11} + ( -6489 - 449 \beta ) q^{13} + ( 8840 + 224 \beta ) q^{14} + 4096 q^{16} + ( 16233 - 210 \beta ) q^{17} -6859 q^{19} + ( -3968 - 1984 \beta ) q^{20} + ( -12576 - 1208 \beta ) q^{22} + ( 39163 + 4199 \beta ) q^{23} + ( 77557 + 4805 \beta ) q^{25} + ( 51912 + 3592 \beta ) q^{26} + ( -70720 - 1792 \beta ) q^{28} + ( 7961 - 3173 \beta ) q^{29} + ( 126188 + 6568 \beta ) q^{31} -32768 q^{32} + ( -129864 + 1680 \beta ) q^{34} + ( 205654 + 36859 \beta ) q^{35} + ( -78934 + 8608 \beta ) q^{37} + 54872 q^{38} + ( 31744 + 15872 \beta ) q^{40} + ( -195404 + 51678 \beta ) q^{41} + ( -21290 - 41289 \beta ) q^{43} + ( 100608 + 9664 \beta ) q^{44} + ( -313304 - 33592 \beta ) q^{46} + ( -745730 + 20435 \beta ) q^{47} + ( 521354 + 62664 \beta ) q^{49} + ( -620456 - 38440 \beta ) q^{50} + ( -415296 - 28736 \beta ) q^{52} + ( 487913 - 30183 \beta ) q^{53} + ( -837062 - 62775 \beta ) q^{55} + ( 565760 + 14336 \beta ) q^{56} + ( -63688 + 25384 \beta ) q^{58} + ( 541721 - 114433 \beta ) q^{59} + ( -715104 - 76547 \beta ) q^{61} + ( -1009504 - 52544 \beta ) q^{62} + 262144 q^{64} + ( 2601520 + 242916 \beta ) q^{65} + ( -979769 + 111319 \beta ) q^{67} + ( 1038912 - 13440 \beta ) q^{68} + ( -1645232 - 294872 \beta ) q^{70} + ( 1854214 - 291244 \beta ) q^{71} + ( -1347935 + 196048 \beta ) q^{73} + ( 631472 - 68864 \beta ) q^{74} -438976 q^{76} + ( -2405084 - 215099 \beta ) q^{77} + ( 1214050 + 208826 \beta ) q^{79} + ( -253952 - 126976 \beta ) q^{80} + ( 1563232 - 413424 \beta ) q^{82} + ( 5088786 - 118218 \beta ) q^{83} + ( 22134 - 483693 \beta ) q^{85} + ( 170320 + 330312 \beta ) q^{86} + ( -804864 - 77312 \beta ) q^{88} + ( 1524580 + 457000 \beta ) q^{89} + ( 9156721 + 690409 \beta ) q^{91} + ( 2506432 + 268736 \beta ) q^{92} + ( 5965840 - 163480 \beta ) q^{94} + ( 425258 + 212629 \beta ) q^{95} + ( 2555234 + 783058 \beta ) q^{97} + ( -4170832 - 501312 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 16q^{2} + 128q^{4} - 155q^{5} - 2238q^{7} - 1024q^{8} + O(q^{10})$$ $$2q - 16q^{2} + 128q^{4} - 155q^{5} - 2238q^{7} - 1024q^{8} + 1240q^{10} + 3295q^{11} - 13427q^{13} + 17904q^{14} + 8192q^{16} + 32256q^{17} - 13718q^{19} - 9920q^{20} - 26360q^{22} + 82525q^{23} + 159919q^{25} + 107416q^{26} - 143232q^{28} + 12749q^{29} + 258944q^{31} - 65536q^{32} - 258048q^{34} + 448167q^{35} - 149260q^{37} + 109744q^{38} + 79360q^{40} - 339130q^{41} - 83869q^{43} + 210880q^{44} - 660200q^{46} - 1471025q^{47} + 1105372q^{49} - 1279352q^{50} - 859328q^{52} + 945643q^{53} - 1736899q^{55} + 1145856q^{56} - 101992q^{58} + 969009q^{59} - 1506755q^{61} - 2071552q^{62} + 524288q^{64} + 5445956q^{65} - 1848219q^{67} + 2064384q^{68} - 3585336q^{70} + 3417184q^{71} - 2499822q^{73} + 1194080q^{74} - 877952q^{76} - 5025267q^{77} + 2636926q^{79} - 634880q^{80} + 2713040q^{82} + 10059354q^{83} - 439425q^{85} + 670952q^{86} - 1687040q^{88} + 3506160q^{89} + 19003851q^{91} + 5281600q^{92} + 11768200q^{94} + 1063145q^{95} + 5893526q^{97} - 8842976q^{98} + O(q^{100})$$

## 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
 13.0797 −12.0797
−8.00000 0 64.0000 −467.472 0 −1471.23 −512.000 0 3739.78
1.2 −8.00000 0 64.0000 312.472 0 −766.767 −512.000 0 −2499.78
 $$n$$: 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 342.8.a.g 2
3.b odd 2 1 38.8.a.d 2
12.b even 2 1 304.8.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.d 2 3.b odd 2 1
304.8.a.d 2 12.b even 2 1
342.8.a.g 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 155 T_{5} - 146072$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(342))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$-146072 + 155 T + T^{2}$$
$7$ $$1128093 + 2238 T + T^{2}$$
$11$ $$-894002 - 3295 T + T^{2}$$
$13$ $$13167724 + 13427 T + T^{2}$$
$17$ $$253133559 - 32256 T + T^{2}$$
$19$ $$( 6859 + T )^{2}$$
$23$ $$-1087606952 - 82525 T + T^{2}$$
$29$ $$-1552615514 - 12749 T + T^{2}$$
$31$ $$9936311536 - 258944 T + T^{2}$$
$37$ $$-6156318428 + 149260 T + T^{2}$$
$41$ $$-393872642768 + 339130 T + T^{2}$$
$43$ $$-268023173408 + 83869 T + T^{2}$$
$47$ $$474895142800 + 1471025 T + T^{2}$$
$53$ $$79392286228 - 945643 T + T^{2}$$
$59$ $$-1837525132614 - 969009 T + T^{2}$$
$61$ $$-359679230318 + 1506755 T + T^{2}$$
$67$ $$-1107042934188 + 1848219 T + T^{2}$$
$71$ $$-10503963815108 - 3417184 T + T^{2}$$
$73$ $$-4520032488687 + 2499822 T + T^{2}$$
$79$ $$-5162668519808 - 2636926 T + T^{2}$$
$83$ $$23086028557656 - 10059354 T + T^{2}$$
$89$ $$-29977064763600 - 3506160 T + T^{2}$$
$97$ $$-88352296135184 - 5893526 T + T^{2}$$