# Properties

 Label 304.8.a.d Level $304$ Weight $8$ Character orbit 304.a Self dual yes Analytic conductor $94.965$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 304.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$94.9650477472$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{633})$$ Defining polynomial: $$x^{2} - x - 158$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 36 - 3 \beta ) q^{3} + ( 93 - 31 \beta ) q^{5} + ( 1133 - 28 \beta ) q^{7} + ( 531 - 207 \beta ) q^{9} +O(q^{10})$$ $$q + ( 36 - 3 \beta ) q^{3} + ( 93 - 31 \beta ) q^{5} + ( 1133 - 28 \beta ) q^{7} + ( 531 - 207 \beta ) q^{9} + ( 1723 - 151 \beta ) q^{11} + ( -6938 + 449 \beta ) q^{13} + ( 18042 - 1302 \beta ) q^{15} + ( -16023 - 210 \beta ) q^{17} + 6859 q^{19} + ( 54060 - 4323 \beta ) q^{21} + ( 43362 - 4199 \beta ) q^{23} + ( 82362 - 4805 \beta ) q^{25} + ( 38502 - 1863 \beta ) q^{27} + ( -4788 - 3173 \beta ) q^{29} + ( -132756 + 6568 \beta ) q^{31} + ( 133602 - 10152 \beta ) q^{33} + ( 242513 - 36859 \beta ) q^{35} + ( -70326 - 8608 \beta ) q^{37} + ( -462594 + 35631 \beta ) q^{39} + ( 143726 + 51678 \beta ) q^{41} + ( 62579 - 41289 \beta ) q^{43} + ( 1063269 - 29295 \beta ) q^{45} + ( -725295 - 20435 \beta ) q^{47} + ( 584018 - 62664 \beta ) q^{49} + ( -477288 + 41139 \beta ) q^{51} + ( -457730 - 30183 \beta ) q^{53} + ( 899837 - 62775 \beta ) q^{55} + ( 246924 - 20577 \beta ) q^{57} + ( 427288 + 114433 \beta ) q^{59} + ( -791651 + 76547 \beta ) q^{61} + ( 1517391 - 243603 \beta ) q^{63} + ( -2844436 + 242916 \beta ) q^{65} + ( 868450 + 111319 \beta ) q^{67} + ( 3551358 - 268653 \beta ) q^{69} + ( 1562970 + 291244 \beta ) q^{71} + ( -1151887 - 196048 \beta ) q^{73} + ( 5242602 - 405651 \beta ) q^{75} + ( 2620183 - 215099 \beta ) q^{77} + ( -1422876 + 208826 \beta ) q^{79} + ( 1107837 + 275724 \beta ) q^{81} + ( 4970568 + 118218 \beta ) q^{83} + ( -461559 + 483693 \beta ) q^{85} + ( 1331634 - 90345 \beta ) q^{87} + ( -1981580 + 457000 \beta ) q^{89} + ( -9847130 + 690409 \beta ) q^{91} + ( -7892448 + 615012 \beta ) q^{93} + ( 637887 - 212629 \beta ) q^{95} + ( 3338292 - 783058 \beta ) q^{97} + ( 5853519 - 405585 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + O(q^{10})$$ $$2q + 69q^{3} + 155q^{5} + 2238q^{7} + 855q^{9} + 3295q^{11} - 13427q^{13} + 34782q^{15} - 32256q^{17} + 13718q^{19} + 103797q^{21} + 82525q^{23} + 159919q^{25} + 75141q^{27} - 12749q^{29} - 258944q^{31} + 257052q^{33} + 448167q^{35} - 149260q^{37} - 889557q^{39} + 339130q^{41} + 83869q^{43} + 2097243q^{45} - 1471025q^{47} + 1105372q^{49} - 913437q^{51} - 945643q^{53} + 1736899q^{55} + 473271q^{57} + 969009q^{59} - 1506755q^{61} + 2791179q^{63} - 5445956q^{65} + 1848219q^{67} + 6834063q^{69} + 3417184q^{71} - 2499822q^{73} + 10079553q^{75} + 5025267q^{77} - 2636926q^{79} + 2491398q^{81} + 10059354q^{83} - 439425q^{85} + 2572923q^{87} - 3506160q^{89} - 19003851q^{91} - 15169884q^{93} + 1063145q^{95} + 5893526q^{97} + 11301453q^{99} + O(q^{100})$$

## 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
0 −3.23924 0 −312.472 0 766.767 0 −2176.51 0
1.2 0 72.2392 0 467.472 0 1471.23 0 3031.51 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 304.8.a.d 2
4.b odd 2 1 38.8.a.d 2
12.b even 2 1 342.8.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.d 2 4.b odd 2 1
304.8.a.d 2 1.a even 1 1 trivial
342.8.a.g 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 69 T_{3} - 234$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(304))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-234 - 69 T + T^{2}$$
$5$ $$-146072 - 155 T + T^{2}$$
$7$ $$1128093 - 2238 T + T^{2}$$
$11$ $$-894002 - 3295 T + T^{2}$$
$13$ $$13167724 + 13427 T + T^{2}$$
$17$ $$253133559 + 32256 T + T^{2}$$
$19$ $$( -6859 + T )^{2}$$
$23$ $$-1087606952 - 82525 T + T^{2}$$
$29$ $$-1552615514 + 12749 T + T^{2}$$
$31$ $$9936311536 + 258944 T + T^{2}$$
$37$ $$-6156318428 + 149260 T + T^{2}$$
$41$ $$-393872642768 - 339130 T + T^{2}$$
$43$ $$-268023173408 - 83869 T + T^{2}$$
$47$ $$474895142800 + 1471025 T + T^{2}$$
$53$ $$79392286228 + 945643 T + T^{2}$$
$59$ $$-1837525132614 - 969009 T + T^{2}$$
$61$ $$-359679230318 + 1506755 T + T^{2}$$
$67$ $$-1107042934188 - 1848219 T + T^{2}$$
$71$ $$-10503963815108 - 3417184 T + T^{2}$$
$73$ $$-4520032488687 + 2499822 T + T^{2}$$
$79$ $$-5162668519808 + 2636926 T + T^{2}$$
$83$ $$23086028557656 - 10059354 T + T^{2}$$
$89$ $$-29977064763600 + 3506160 T + T^{2}$$
$97$ $$-88352296135184 - 5893526 T + T^{2}$$