# Properties

 Label 304.2.h.a Level $304$ Weight $2$ Character orbit 304.h Analytic conductor $2.427$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,2,Mod(303,304)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("304.303");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$2.42745222145$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 5$$ x^2 - x + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} - \beta q^{7} - 3 q^{9} +O(q^{10})$$ q - q^5 - b * q^7 - 3 * q^9 $$q - q^{5} - \beta q^{7} - 3 q^{9} - \beta q^{11} + 7 q^{17} + \beta q^{19} - 2 \beta q^{23} - 4 q^{25} + \beta q^{35} + 3 \beta q^{43} + 3 q^{45} - \beta q^{47} - 12 q^{49} + \beta q^{55} + 15 q^{61} + 3 \beta q^{63} + 11 q^{73} - 19 q^{77} + 9 q^{81} + 2 \beta q^{83} - 7 q^{85} - \beta q^{95} + 3 \beta q^{99} +O(q^{100})$$ q - q^5 - b * q^7 - 3 * q^9 - b * q^11 + 7 * q^17 + b * q^19 - 2*b * q^23 - 4 * q^25 + b * q^35 + 3*b * q^43 + 3 * q^45 - b * q^47 - 12 * q^49 + b * q^55 + 15 * q^61 + 3*b * q^63 + 11 * q^73 - 19 * q^77 + 9 * q^81 + 2*b * q^83 - 7 * q^85 - b * q^95 + 3*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 6 * q^9 $$2 q - 2 q^{5} - 6 q^{9} + 14 q^{17} - 8 q^{25} + 6 q^{45} - 24 q^{49} + 30 q^{61} + 22 q^{73} - 38 q^{77} + 18 q^{81} - 14 q^{85}+O(q^{100})$$ 2 * q - 2 * q^5 - 6 * q^9 + 14 * q^17 - 8 * q^25 + 6 * q^45 - 24 * q^49 + 30 * q^61 + 22 * q^73 - 38 * q^77 + 18 * q^81 - 14 * q^85

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
303.1
 0.5 + 2.17945i 0.5 − 2.17945i
0 0 0 −1.00000 0 4.35890i 0 −3.00000 0
303.2 0 0 0 −1.00000 0 4.35890i 0 −3.00000 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 $$\Q(\sqrt{-19})$$
4.b odd 2 1 inner
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.h.a 2
3.b odd 2 1 2736.2.k.g 2
4.b odd 2 1 inner 304.2.h.a 2
8.b even 2 1 1216.2.h.a 2
8.d odd 2 1 1216.2.h.a 2
12.b even 2 1 2736.2.k.g 2
19.b odd 2 1 CM 304.2.h.a 2
57.d even 2 1 2736.2.k.g 2
76.d even 2 1 inner 304.2.h.a 2
152.b even 2 1 1216.2.h.a 2
152.g odd 2 1 1216.2.h.a 2
228.b odd 2 1 2736.2.k.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.h.a 2 1.a even 1 1 trivial
304.2.h.a 2 4.b odd 2 1 inner
304.2.h.a 2 19.b odd 2 1 CM
304.2.h.a 2 76.d even 2 1 inner
1216.2.h.a 2 8.b even 2 1
1216.2.h.a 2 8.d odd 2 1
1216.2.h.a 2 152.b even 2 1
1216.2.h.a 2 152.g odd 2 1
2736.2.k.g 2 3.b odd 2 1
2736.2.k.g 2 12.b even 2 1
2736.2.k.g 2 57.d even 2 1
2736.2.k.g 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 $$S_{2}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}$$ T3 $$T_{5} + 1$$ T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 19$$
$11$ $$T^{2} + 19$$
$13$ $$T^{2}$$
$17$ $$(T - 7)^{2}$$
$19$ $$T^{2} + 19$$
$23$ $$T^{2} + 76$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 171$$
$47$ $$T^{2} + 19$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 15)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 76$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$