# Properties

 Label 304.3.e.c Level $304$ Weight $3$ Character orbit 304.e Analytic conductor $8.283$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.28340003655$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - q^{5} -5 q^{7} + q^{9} +O(q^{10})$$ $$q + \beta q^{3} - q^{5} -5 q^{7} + q^{9} -5 q^{11} + 6 \beta q^{13} -\beta q^{15} -25 q^{17} -19 q^{19} -5 \beta q^{21} + 10 q^{23} -24 q^{25} + 10 \beta q^{27} -15 \beta q^{29} -15 \beta q^{31} -5 \beta q^{33} + 5 q^{35} + 9 \beta q^{37} -48 q^{39} + 15 \beta q^{41} -5 q^{43} - q^{45} -5 q^{47} -24 q^{49} -25 \beta q^{51} + 9 \beta q^{53} + 5 q^{55} -19 \beta q^{57} + 30 \beta q^{59} + 95 q^{61} -5 q^{63} -6 \beta q^{65} + 39 \beta q^{67} + 10 \beta q^{69} -25 q^{73} -24 \beta q^{75} + 25 q^{77} + 15 \beta q^{79} -71 q^{81} + 130 q^{83} + 25 q^{85} + 120 q^{87} + 45 \beta q^{89} -30 \beta q^{91} + 120 q^{93} + 19 q^{95} -6 \beta q^{97} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 10q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{5} - 10q^{7} + 2q^{9} - 10q^{11} - 50q^{17} - 38q^{19} + 20q^{23} - 48q^{25} + 10q^{35} - 96q^{39} - 10q^{43} - 2q^{45} - 10q^{47} - 48q^{49} + 10q^{55} + 190q^{61} - 10q^{63} - 50q^{73} + 50q^{77} - 142q^{81} + 260q^{83} + 50q^{85} + 240q^{87} + 240q^{93} + 38q^{95} - 10q^{99} + O(q^{100})$$

## 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
 − 1.41421i 1.41421i
0 2.82843i 0 −1.00000 0 −5.00000 0 1.00000 0
113.2 0 2.82843i 0 −1.00000 0 −5.00000 0 1.00000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.e.c 2
3.b odd 2 1 2736.3.o.h 2
4.b odd 2 1 38.3.b.a 2
8.b even 2 1 1216.3.e.i 2
8.d odd 2 1 1216.3.e.j 2
12.b even 2 1 342.3.d.a 2
19.b odd 2 1 inner 304.3.e.c 2
20.d odd 2 1 950.3.c.a 2
20.e even 4 2 950.3.d.a 4
57.d even 2 1 2736.3.o.h 2
76.d even 2 1 38.3.b.a 2
152.b even 2 1 1216.3.e.j 2
152.g odd 2 1 1216.3.e.i 2
228.b odd 2 1 342.3.d.a 2
380.d even 2 1 950.3.c.a 2
380.j odd 4 2 950.3.d.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 4.b odd 2 1
38.3.b.a 2 76.d even 2 1
304.3.e.c 2 1.a even 1 1 trivial
304.3.e.c 2 19.b odd 2 1 inner
342.3.d.a 2 12.b even 2 1
342.3.d.a 2 228.b odd 2 1
950.3.c.a 2 20.d odd 2 1
950.3.c.a 2 380.d even 2 1
950.3.d.a 4 20.e even 4 2
950.3.d.a 4 380.j odd 4 2
1216.3.e.i 2 8.b even 2 1
1216.3.e.i 2 152.g odd 2 1
1216.3.e.j 2 8.d odd 2 1
1216.3.e.j 2 152.b even 2 1
2736.3.o.h 2 3.b odd 2 1
2736.3.o.h 2 57.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{2} + 8$$ $$T_{5} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$8 + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$288 + T^{2}$$
$17$ $$( 25 + T )^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$( -10 + T )^{2}$$
$29$ $$1800 + T^{2}$$
$31$ $$1800 + T^{2}$$
$37$ $$648 + T^{2}$$
$41$ $$1800 + T^{2}$$
$43$ $$( 5 + T )^{2}$$
$47$ $$( 5 + T )^{2}$$
$53$ $$648 + T^{2}$$
$59$ $$7200 + T^{2}$$
$61$ $$( -95 + T )^{2}$$
$67$ $$12168 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 25 + T )^{2}$$
$79$ $$1800 + T^{2}$$
$83$ $$( -130 + T )^{2}$$
$89$ $$16200 + T^{2}$$
$97$ $$288 + T^{2}$$