# Properties

 Label 304.7.e.a Level $304$ Weight $7$ Character orbit 304.e Self dual yes Analytic conductor $69.936$ Analytic rank $0$ Dimension $1$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.9364414204$$ 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

 $$f(q)$$ $$=$$ $$q - 54q^{5} - 610q^{7} + 729q^{9} + O(q^{10})$$ $$q - 54q^{5} - 610q^{7} + 729q^{9} + 1062q^{11} - 9630q^{17} + 6859q^{19} - 20610q^{23} - 12709q^{25} + 32940q^{35} + 142630q^{43} - 39366q^{45} + 75150q^{47} + 254451q^{49} - 57348q^{55} - 57062q^{61} - 444690q^{63} + 384050q^{73} - 647820q^{77} + 531441q^{81} + 1131030q^{83} + 520020q^{85} - 370386q^{95} + 774198q^{99} + O(q^{100})$$

## 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 −54.0000 0 −610.000 0 729.000 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 304.7.e.a 1
4.b odd 2 1 19.7.b.a 1
12.b even 2 1 171.7.c.a 1
19.b odd 2 1 CM 304.7.e.a 1
76.d even 2 1 19.7.b.a 1
228.b odd 2 1 171.7.c.a 1

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}$$ $$T_{5} + 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$54 + T$$
$7$ $$610 + T$$
$11$ $$-1062 + T$$
$13$ $$T$$
$17$ $$9630 + T$$
$19$ $$-6859 + T$$
$23$ $$20610 + T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$-142630 + T$$
$47$ $$-75150 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$57062 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-384050 + T$$
$79$ $$T$$
$83$ $$-1131030 + T$$
$89$ $$T$$
$97$ $$T$$