# Properties

 Label 304.6.a.f Level $304$ Weight $6$ Character orbit 304.a Self dual yes Analytic conductor $48.757$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 304.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.7566812231$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1441})$$ Defining polynomial: $$x^{2} - x - 360$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + ( -24 + 3 \beta ) q^{5} + ( -59 + 4 \beta ) q^{7} + ( 118 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + ( -24 + 3 \beta ) q^{5} + ( -59 + 4 \beta ) q^{7} + ( 118 + 3 \beta ) q^{9} + ( -330 - \beta ) q^{11} + ( 809 - 5 \beta ) q^{13} + ( -1056 + 18 \beta ) q^{15} + ( 27 + 10 \beta ) q^{17} -361 q^{19} + ( -1381 + 51 \beta ) q^{21} + ( 1617 - 49 \beta ) q^{23} + ( 691 - 135 \beta ) q^{25} + ( -955 + 119 \beta ) q^{27} + ( -1083 - 315 \beta ) q^{29} + ( 748 - 316 \beta ) q^{31} + ( 690 + 332 \beta ) q^{33} + ( 5736 - 261 \beta ) q^{35} + ( 5330 - 172 \beta ) q^{37} + ( 991 - 799 \beta ) q^{39} + ( 8616 - 602 \beta ) q^{41} + ( -5792 + 281 \beta ) q^{43} + ( 408 + 291 \beta ) q^{45} + ( 5520 + 1115 \beta ) q^{47} + ( -7566 - 456 \beta ) q^{49} + ( -3627 - 47 \beta ) q^{51} + ( 10593 - 601 \beta ) q^{53} + ( 6840 - 969 \beta ) q^{55} + ( 361 + 361 \beta ) q^{57} + ( 39327 - 73 \beta ) q^{59} + ( 21398 + 825 \beta ) q^{61} + ( -2642 + 307 \beta ) q^{63} + ( -24816 + 2532 \beta ) q^{65} + ( -5453 + 3101 \beta ) q^{67} + ( 16023 - 1519 \beta ) q^{69} + ( 31878 - 1268 \beta ) q^{71} + ( 6617 + 2984 \beta ) q^{73} + ( 47909 - 421 \beta ) q^{75} + ( 18030 - 1265 \beta ) q^{77} + ( -33494 - 134 \beta ) q^{79} + ( -70559 - 12 \beta ) q^{81} + ( 4134 + 2446 \beta ) q^{83} + ( 10152 - 129 \beta ) q^{85} + ( 114483 + 1713 \beta ) q^{87} + ( 61956 + 4276 \beta ) q^{89} + ( -54931 + 3511 \beta ) q^{91} + ( 113012 - 116 \beta ) q^{93} + ( 8664 - 1083 \beta ) q^{95} + ( 90590 - 2622 \beta ) q^{97} + ( -40020 - 1111 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 45q^{5} - 114q^{7} + 239q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 45q^{5} - 114q^{7} + 239q^{9} - 661q^{11} + 1613q^{13} - 2094q^{15} + 64q^{17} - 722q^{19} - 2711q^{21} + 3185q^{23} + 1247q^{25} - 1791q^{27} - 2481q^{29} + 1180q^{31} + 1712q^{33} + 11211q^{35} + 10488q^{37} + 1183q^{39} + 16630q^{41} - 11303q^{43} + 1107q^{45} + 12155q^{47} - 15588q^{49} - 7301q^{51} + 20585q^{53} + 12711q^{55} + 1083q^{57} + 78581q^{59} + 43621q^{61} - 4977q^{63} - 47100q^{65} - 7805q^{67} + 30527q^{69} + 62488q^{71} + 16218q^{73} + 95397q^{75} + 34795q^{77} - 67122q^{79} - 141130q^{81} + 10714q^{83} + 20175q^{85} + 230679q^{87} + 128188q^{89} - 106351q^{91} + 225908q^{93} + 16245q^{95} + 178558q^{97} - 81151q^{99} + O(q^{100})$$

## 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
 19.4803 −18.4803
0 −20.4803 0 34.4408 0 18.9210 0 176.441 0
1.2 0 17.4803 0 −79.4408 0 −132.921 0 62.5592 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 304.6.a.f 2
4.b odd 2 1 38.6.a.c 2
12.b even 2 1 342.6.a.i 2
20.d odd 2 1 950.6.a.d 2
76.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 4.b odd 2 1
304.6.a.f 2 1.a even 1 1 trivial
342.6.a.i 2 12.b even 2 1
722.6.a.c 2 76.d even 2 1
950.6.a.d 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} - 358$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(304))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-358 + 3 T + T^{2}$$
$5$ $$-2736 + 45 T + T^{2}$$
$7$ $$-2515 + 114 T + T^{2}$$
$11$ $$108870 + 661 T + T^{2}$$
$13$ $$641436 - 1613 T + T^{2}$$
$17$ $$-35001 - 64 T + T^{2}$$
$19$ $$( 361 + T )^{2}$$
$23$ $$1671096 - 3185 T + T^{2}$$
$29$ $$-34206966 + 2481 T + T^{2}$$
$31$ $$-35625024 - 1180 T + T^{2}$$
$37$ $$16841900 - 10488 T + T^{2}$$
$41$ $$-61416816 - 16630 T + T^{2}$$
$43$ $$3493752 + 11303 T + T^{2}$$
$47$ $$-410935800 - 12155 T + T^{2}$$
$53$ $$-24187104 - 20585 T + T^{2}$$
$59$ $$1541823618 - 78581 T + T^{2}$$
$61$ $$230502754 - 43621 T + T^{2}$$
$67$ $$-3449006904 + 7805 T + T^{2}$$
$71$ $$396968940 - 62488 T + T^{2}$$
$73$ $$-3142002343 - 16218 T + T^{2}$$
$79$ $$1119872072 + 67122 T + T^{2}$$
$83$ $$-2126648040 - 10714 T + T^{2}$$
$89$ $$-2478833568 - 128188 T + T^{2}$$
$97$ $$5494062880 - 178558 T + T^{2}$$