# Properties

 Label 304.5.e.b Level $304$ Weight $5$ Character orbit 304.e Self dual yes Analytic conductor $31.424$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$31.4244687775$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(-1 + 3\sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -17 - 3 \beta ) q^{5} + ( -39 - 5 \beta ) q^{7} + 81 q^{9} +O(q^{10})$$ $$q + ( -17 - 3 \beta ) q^{5} + ( -39 - 5 \beta ) q^{7} + 81 q^{9} + ( -119 - 5 \beta ) q^{11} + ( 159 - 35 \beta ) q^{17} -361 q^{19} + 158 q^{23} + ( 816 + 93 \beta ) q^{25} + ( 2583 + 187 \beta ) q^{35} + ( 1721 - 85 \beta ) q^{43} + ( -1377 - 243 \beta ) q^{45} + ( 441 - 325 \beta ) q^{47} + ( 2320 + 365 \beta ) q^{49} + ( 3943 + 427 \beta ) q^{55} + ( -1841 - 515 \beta ) q^{61} + ( -3159 - 405 \beta ) q^{63} + ( 4879 - 275 \beta ) q^{73} + ( 7841 + 765 \beta ) q^{77} + 6561 q^{81} + 5678 q^{83} + ( 10737 + 13 \beta ) q^{85} + ( 6137 + 1083 \beta ) q^{95} + ( -9639 - 405 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 31q^{5} - 73q^{7} + 162q^{9} + O(q^{10})$$ $$2q - 31q^{5} - 73q^{7} + 162q^{9} - 233q^{11} + 353q^{17} - 722q^{19} + 316q^{23} + 1539q^{25} + 4979q^{35} + 3527q^{43} - 2511q^{45} + 1207q^{47} + 4275q^{49} + 7459q^{55} - 3167q^{61} - 5913q^{63} + 10033q^{73} + 14917q^{77} + 13122q^{81} + 11356q^{83} + 21461q^{85} + 11191q^{95} - 18873q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 4.27492 −3.27492
0 0 0 −49.4743 0 −93.1238 0 81.0000 0
113.2 0 0 0 18.4743 0 20.1238 0 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 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.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.5.e.b 2
4.b odd 2 1 76.5.c.a 2
12.b even 2 1 684.5.h.b 2
19.b odd 2 1 CM 304.5.e.b 2
76.d even 2 1 76.5.c.a 2
228.b odd 2 1 684.5.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.c.a 2 4.b odd 2 1
76.5.c.a 2 76.d even 2 1
304.5.e.b 2 1.a even 1 1 trivial
304.5.e.b 2 19.b odd 2 1 CM
684.5.h.b 2 12.b even 2 1
684.5.h.b 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{5}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}$$ $$T_{5}^{2} + 31 T_{5} - 914$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-914 + 31 T + T^{2}$$
$7$ $$-1874 + 73 T + T^{2}$$
$11$ $$10366 + 233 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$-125954 - 353 T + T^{2}$$
$19$ $$( 361 + T )^{2}$$
$23$ $$( -158 + T )^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$2183326 - 3527 T + T^{2}$$
$47$ $$-13182194 - 1207 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$-31507634 + 3167 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$15466366 - 10033 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -5678 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$