Properties

 Label 304.1.e.a Level $304$ Weight $1$ Character orbit 304.e Self dual yes Analytic conductor $0.152$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -19 Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,1,Mod(113,304)]

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([0, 0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not 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: yes Analytic conductor: $$0.151715763840$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.76.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.92416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{5} + q^{7} + q^{9}+O(q^{10})$$ q - q^5 + q^7 + q^9 $$q - q^{5} + q^{7} + q^{9} + q^{11} - q^{17} - q^{19} - 2 q^{23} - q^{35} + q^{43} - q^{45} + q^{47} - q^{55} - q^{61} + q^{63} - q^{73} + q^{77} + q^{81} - 2 q^{83} + q^{85} + q^{95} + q^{99}+O(q^{100})$$ q - q^5 + q^7 + q^9 + q^11 - q^17 - q^19 - 2 * q^23 - q^35 + q^43 - q^45 + q^47 - q^55 - q^61 + q^63 - q^73 + q^77 + q^81 - 2 * q^83 + q^85 + q^95 + q^99

Character values

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

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$1$$ $$0$$ $$0$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 0
0 0 0 −1.00000 0 1.00000 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 CM by $$\Q(\sqrt{-19})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.1.e.a 1
3.b odd 2 1 2736.1.o.b 1
4.b odd 2 1 76.1.c.a 1
8.b even 2 1 1216.1.e.b 1
8.d odd 2 1 1216.1.e.a 1
12.b even 2 1 684.1.h.a 1
19.b odd 2 1 CM 304.1.e.a 1
20.d odd 2 1 1900.1.e.a 1
20.e even 4 2 1900.1.g.a 2
28.d even 2 1 3724.1.e.c 1
28.f even 6 2 3724.1.bc.b 2
28.g odd 6 2 3724.1.bc.c 2
57.d even 2 1 2736.1.o.b 1
76.d even 2 1 76.1.c.a 1
76.f even 6 2 1444.1.h.a 2
76.g odd 6 2 1444.1.h.a 2
76.k even 18 6 1444.1.j.a 6
76.l odd 18 6 1444.1.j.a 6
152.b even 2 1 1216.1.e.a 1
152.g odd 2 1 1216.1.e.b 1
228.b odd 2 1 684.1.h.a 1
380.d even 2 1 1900.1.e.a 1
380.j odd 4 2 1900.1.g.a 2
532.b odd 2 1 3724.1.e.c 1
532.t even 6 2 3724.1.bc.c 2
532.bh odd 6 2 3724.1.bc.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 4.b odd 2 1
76.1.c.a 1 76.d even 2 1
304.1.e.a 1 1.a even 1 1 trivial
304.1.e.a 1 19.b odd 2 1 CM
684.1.h.a 1 12.b even 2 1
684.1.h.a 1 228.b odd 2 1
1216.1.e.a 1 8.d odd 2 1
1216.1.e.a 1 152.b even 2 1
1216.1.e.b 1 8.b even 2 1
1216.1.e.b 1 152.g odd 2 1
1444.1.h.a 2 76.f even 6 2
1444.1.h.a 2 76.g odd 6 2
1444.1.j.a 6 76.k even 18 6
1444.1.j.a 6 76.l odd 18 6
1900.1.e.a 1 20.d odd 2 1
1900.1.e.a 1 380.d even 2 1
1900.1.g.a 2 20.e even 4 2
1900.1.g.a 2 380.j odd 4 2
2736.1.o.b 1 3.b odd 2 1
2736.1.o.b 1 57.d even 2 1
3724.1.e.c 1 28.d even 2 1
3724.1.e.c 1 532.b odd 2 1
3724.1.bc.b 2 28.f even 6 2
3724.1.bc.b 2 532.bh odd 6 2
3724.1.bc.c 2 28.g odd 6 2
3724.1.bc.c 2 532.t even 6 2

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(304, [\chi])$$.

Hecke characteristic polynomials

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