# Properties

 Label 304.2.a.f Level $304$ Weight $2$ Character orbit 304.a Self dual yes Analytic conductor $2.427$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,2,Mod(1,304)]

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, 0]))

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

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

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.a (trivial)

## 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: $$2.42745222145$$ 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) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + 3 q^{5} + q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 + 3 * q^5 + q^7 + q^9 $$q + 2 q^{3} + 3 q^{5} + q^{7} + q^{9} - 3 q^{11} - 4 q^{13} + 6 q^{15} - 3 q^{17} - q^{19} + 2 q^{21} + 4 q^{25} - 4 q^{27} + 6 q^{29} + 4 q^{31} - 6 q^{33} + 3 q^{35} + 2 q^{37} - 8 q^{39} - 6 q^{41} + q^{43} + 3 q^{45} + 3 q^{47} - 6 q^{49} - 6 q^{51} + 12 q^{53} - 9 q^{55} - 2 q^{57} + 6 q^{59} - q^{61} + q^{63} - 12 q^{65} + 4 q^{67} - 6 q^{71} - 7 q^{73} + 8 q^{75} - 3 q^{77} - 8 q^{79} - 11 q^{81} - 12 q^{83} - 9 q^{85} + 12 q^{87} + 12 q^{89} - 4 q^{91} + 8 q^{93} - 3 q^{95} + 8 q^{97} - 3 q^{99}+O(q^{100})$$ q + 2 * q^3 + 3 * q^5 + q^7 + q^9 - 3 * q^11 - 4 * q^13 + 6 * q^15 - 3 * q^17 - q^19 + 2 * q^21 + 4 * q^25 - 4 * q^27 + 6 * q^29 + 4 * q^31 - 6 * q^33 + 3 * q^35 + 2 * q^37 - 8 * q^39 - 6 * q^41 + q^43 + 3 * q^45 + 3 * q^47 - 6 * q^49 - 6 * q^51 + 12 * q^53 - 9 * q^55 - 2 * q^57 + 6 * q^59 - q^61 + q^63 - 12 * q^65 + 4 * q^67 - 6 * q^71 - 7 * q^73 + 8 * q^75 - 3 * q^77 - 8 * q^79 - 11 * q^81 - 12 * q^83 - 9 * q^85 + 12 * q^87 + 12 * q^89 - 4 * q^91 + 8 * q^93 - 3 * q^95 + 8 * q^97 - 3 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 2.00000 0 3.00000 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.a.f 1
3.b odd 2 1 2736.2.a.c 1
4.b odd 2 1 19.2.a.a 1
5.b even 2 1 7600.2.a.c 1
8.b even 2 1 1216.2.a.b 1
8.d odd 2 1 1216.2.a.o 1
12.b even 2 1 171.2.a.b 1
19.b odd 2 1 5776.2.a.c 1
20.d odd 2 1 475.2.a.b 1
20.e even 4 2 475.2.b.a 2
28.d even 2 1 931.2.a.a 1
28.f even 6 2 931.2.f.b 2
28.g odd 6 2 931.2.f.c 2
44.c even 2 1 2299.2.a.b 1
52.b odd 2 1 3211.2.a.a 1
60.h even 2 1 4275.2.a.i 1
68.d odd 2 1 5491.2.a.b 1
76.d even 2 1 361.2.a.b 1
76.f even 6 2 361.2.c.a 2
76.g odd 6 2 361.2.c.c 2
76.k even 18 6 361.2.e.e 6
76.l odd 18 6 361.2.e.d 6
84.h odd 2 1 8379.2.a.j 1
228.b odd 2 1 3249.2.a.d 1
380.d even 2 1 9025.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 4.b odd 2 1
171.2.a.b 1 12.b even 2 1
304.2.a.f 1 1.a even 1 1 trivial
361.2.a.b 1 76.d even 2 1
361.2.c.a 2 76.f even 6 2
361.2.c.c 2 76.g odd 6 2
361.2.e.d 6 76.l odd 18 6
361.2.e.e 6 76.k even 18 6
475.2.a.b 1 20.d odd 2 1
475.2.b.a 2 20.e even 4 2
931.2.a.a 1 28.d even 2 1
931.2.f.b 2 28.f even 6 2
931.2.f.c 2 28.g odd 6 2
1216.2.a.b 1 8.b even 2 1
1216.2.a.o 1 8.d odd 2 1
2299.2.a.b 1 44.c even 2 1
2736.2.a.c 1 3.b odd 2 1
3211.2.a.a 1 52.b odd 2 1
3249.2.a.d 1 228.b odd 2 1
4275.2.a.i 1 60.h even 2 1
5491.2.a.b 1 68.d odd 2 1
5776.2.a.c 1 19.b odd 2 1
7600.2.a.c 1 5.b even 2 1
8379.2.a.j 1 84.h odd 2 1
9025.2.a.d 1 380.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(304))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{5} - 3$$ T5 - 3 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T - 3$$
$7$ $$T - 1$$
$11$ $$T + 3$$
$13$ $$T + 4$$
$17$ $$T + 3$$
$19$ $$T + 1$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T - 4$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T - 1$$
$47$ $$T - 3$$
$53$ $$T - 12$$
$59$ $$T - 6$$
$61$ $$T + 1$$
$67$ $$T - 4$$
$71$ $$T + 6$$
$73$ $$T + 7$$
$79$ $$T + 8$$
$83$ $$T + 12$$
$89$ $$T - 12$$
$97$ $$T - 8$$