# Properties

 Label 304.3.e.a Level $304$ Weight $3$ Character orbit 304.e Self dual yes Analytic conductor $8.283$ Analytic rank $0$ Dimension $1$ CM discriminant -19 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: yes Analytic conductor: $$8.28340003655$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 9 q^{5} + 5 q^{7} + 9 q^{9} + O(q^{10})$$ $$q - 9 q^{5} + 5 q^{7} + 9 q^{9} - 3 q^{11} + 15 q^{17} + 19 q^{19} + 30 q^{23} + 56 q^{25} - 45 q^{35} + 85 q^{43} - 81 q^{45} - 75 q^{47} - 24 q^{49} + 27 q^{55} + 103 q^{61} + 45 q^{63} - 25 q^{73} - 15 q^{77} + 81 q^{81} - 90 q^{83} - 135 q^{85} - 171 q^{95} - 27 q^{99} + O(q^{100})$$

## 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
 0
0 0 0 −9.00000 0 5.00000 0 9.00000 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 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.e.a 1
3.b odd 2 1 2736.3.o.a 1
4.b odd 2 1 19.3.b.a 1
8.b even 2 1 1216.3.e.b 1
8.d odd 2 1 1216.3.e.a 1
12.b even 2 1 171.3.c.a 1
19.b odd 2 1 CM 304.3.e.a 1
20.d odd 2 1 475.3.c.a 1
20.e even 4 2 475.3.d.a 2
57.d even 2 1 2736.3.o.a 1
76.d even 2 1 19.3.b.a 1
76.f even 6 2 361.3.d.a 2
76.g odd 6 2 361.3.d.a 2
76.k even 18 6 361.3.f.a 6
76.l odd 18 6 361.3.f.a 6
152.b even 2 1 1216.3.e.a 1
152.g odd 2 1 1216.3.e.b 1
228.b odd 2 1 171.3.c.a 1
380.d even 2 1 475.3.c.a 1
380.j odd 4 2 475.3.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 4.b odd 2 1
19.3.b.a 1 76.d even 2 1
171.3.c.a 1 12.b even 2 1
171.3.c.a 1 228.b odd 2 1
304.3.e.a 1 1.a even 1 1 trivial
304.3.e.a 1 19.b odd 2 1 CM
361.3.d.a 2 76.f even 6 2
361.3.d.a 2 76.g odd 6 2
361.3.f.a 6 76.k even 18 6
361.3.f.a 6 76.l odd 18 6
475.3.c.a 1 20.d odd 2 1
475.3.c.a 1 380.d even 2 1
475.3.d.a 2 20.e even 4 2
475.3.d.a 2 380.j odd 4 2
1216.3.e.a 1 8.d odd 2 1
1216.3.e.a 1 152.b even 2 1
1216.3.e.b 1 8.b even 2 1
1216.3.e.b 1 152.g odd 2 1
2736.3.o.a 1 3.b odd 2 1
2736.3.o.a 1 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 $$S_{3}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}$$ $$T_{5} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$9 + T$$
$7$ $$-5 + T$$
$11$ $$3 + T$$
$13$ $$T$$
$17$ $$-15 + T$$
$19$ $$-19 + T$$
$23$ $$-30 + T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$-85 + T$$
$47$ $$75 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$-103 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$25 + T$$
$79$ $$T$$
$83$ $$90 + T$$
$89$ $$T$$
$97$ $$T$$