# Properties

 Label 304.2.i.b Level $304$ Weight $2$ Character orbit 304.i Analytic conductor $2.427$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,2,Mod(49,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("304.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.i (of order $$3$$, degree $$2$$, 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: no Analytic conductor: $$2.42745222145$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + (3 \zeta_{6} - 3) q^{5} + 2 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z + 1) * q^3 + (3*z - 3) * q^5 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + (3 \zeta_{6} - 3) q^{5} + 2 \zeta_{6} q^{9} + 4 q^{11} + 5 \zeta_{6} q^{13} + 3 \zeta_{6} q^{15} + ( - 5 \zeta_{6} + 5) q^{17} + (2 \zeta_{6} - 5) q^{19} - \zeta_{6} q^{23} - 4 \zeta_{6} q^{25} + 5 q^{27} - 3 \zeta_{6} q^{29} - 4 q^{31} + ( - 4 \zeta_{6} + 4) q^{33} + 2 q^{37} + 5 q^{39} + ( - 5 \zeta_{6} + 5) q^{41} + (11 \zeta_{6} - 11) q^{43} - 6 q^{45} - 5 \zeta_{6} q^{47} - 7 q^{49} - 5 \zeta_{6} q^{51} + 9 \zeta_{6} q^{53} + (12 \zeta_{6} - 12) q^{55} + (5 \zeta_{6} - 3) q^{57} + ( - 13 \zeta_{6} + 13) q^{59} + \zeta_{6} q^{61} - 15 q^{65} - 5 \zeta_{6} q^{67} - q^{69} + ( - \zeta_{6} + 1) q^{71} + ( - 9 \zeta_{6} + 9) q^{73} - 4 q^{75} + ( - 17 \zeta_{6} + 17) q^{79} + (\zeta_{6} - 1) q^{81} - 16 q^{83} + 15 \zeta_{6} q^{85} - 3 q^{87} - 3 \zeta_{6} q^{89} + (4 \zeta_{6} - 4) q^{93} + ( - 15 \zeta_{6} + 9) q^{95} + ( - 13 \zeta_{6} + 13) q^{97} + 8 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (3*z - 3) * q^5 + 2*z * q^9 + 4 * q^11 + 5*z * q^13 + 3*z * q^15 + (-5*z + 5) * q^17 + (2*z - 5) * q^19 - z * q^23 - 4*z * q^25 + 5 * q^27 - 3*z * q^29 - 4 * q^31 + (-4*z + 4) * q^33 + 2 * q^37 + 5 * q^39 + (-5*z + 5) * q^41 + (11*z - 11) * q^43 - 6 * q^45 - 5*z * q^47 - 7 * q^49 - 5*z * q^51 + 9*z * q^53 + (12*z - 12) * q^55 + (5*z - 3) * q^57 + (-13*z + 13) * q^59 + z * q^61 - 15 * q^65 - 5*z * q^67 - q^69 + (-z + 1) * q^71 + (-9*z + 9) * q^73 - 4 * q^75 + (-17*z + 17) * q^79 + (z - 1) * q^81 - 16 * q^83 + 15*z * q^85 - 3 * q^87 - 3*z * q^89 + (4*z - 4) * q^93 + (-15*z + 9) * q^95 + (-13*z + 13) * q^97 + 8*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^9 $$2 q + q^{3} - 3 q^{5} + 2 q^{9} + 8 q^{11} + 5 q^{13} + 3 q^{15} + 5 q^{17} - 8 q^{19} - q^{23} - 4 q^{25} + 10 q^{27} - 3 q^{29} - 8 q^{31} + 4 q^{33} + 4 q^{37} + 10 q^{39} + 5 q^{41} - 11 q^{43} - 12 q^{45} - 5 q^{47} - 14 q^{49} - 5 q^{51} + 9 q^{53} - 12 q^{55} - q^{57} + 13 q^{59} + q^{61} - 30 q^{65} - 5 q^{67} - 2 q^{69} + q^{71} + 9 q^{73} - 8 q^{75} + 17 q^{79} - q^{81} - 32 q^{83} + 15 q^{85} - 6 q^{87} - 3 q^{89} - 4 q^{93} + 3 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^9 + 8 * q^11 + 5 * q^13 + 3 * q^15 + 5 * q^17 - 8 * q^19 - q^23 - 4 * q^25 + 10 * q^27 - 3 * q^29 - 8 * q^31 + 4 * q^33 + 4 * q^37 + 10 * q^39 + 5 * q^41 - 11 * q^43 - 12 * q^45 - 5 * q^47 - 14 * q^49 - 5 * q^51 + 9 * q^53 - 12 * q^55 - q^57 + 13 * q^59 + q^61 - 30 * q^65 - 5 * q^67 - 2 * q^69 + q^71 + 9 * q^73 - 8 * q^75 + 17 * q^79 - q^81 - 32 * q^83 + 15 * q^85 - 6 * q^87 - 3 * q^89 - 4 * q^93 + 3 * q^95 + 13 * q^97 + 8 * q^99

## Character values

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

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0.500000 0.866025i 0 −1.50000 + 2.59808i 0 0 0 1.00000 + 1.73205i 0
273.1 0 0.500000 + 0.866025i 0 −1.50000 2.59808i 0 0 0 1.00000 1.73205i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.i.b 2
3.b odd 2 1 2736.2.s.q 2
4.b odd 2 1 152.2.i.a 2
8.b even 2 1 1216.2.i.e 2
8.d odd 2 1 1216.2.i.i 2
12.b even 2 1 1368.2.s.g 2
19.c even 3 1 inner 304.2.i.b 2
19.c even 3 1 5776.2.a.h 1
19.d odd 6 1 5776.2.a.o 1
57.h odd 6 1 2736.2.s.q 2
76.f even 6 1 2888.2.a.c 1
76.g odd 6 1 152.2.i.a 2
76.g odd 6 1 2888.2.a.e 1
152.k odd 6 1 1216.2.i.i 2
152.p even 6 1 1216.2.i.e 2
228.m even 6 1 1368.2.s.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.a 2 4.b odd 2 1
152.2.i.a 2 76.g odd 6 1
304.2.i.b 2 1.a even 1 1 trivial
304.2.i.b 2 19.c even 3 1 inner
1216.2.i.e 2 8.b even 2 1
1216.2.i.e 2 152.p even 6 1
1216.2.i.i 2 8.d odd 2 1
1216.2.i.i 2 152.k odd 6 1
1368.2.s.g 2 12.b even 2 1
1368.2.s.g 2 228.m even 6 1
2736.2.s.q 2 3.b odd 2 1
2736.2.s.q 2 57.h odd 6 1
2888.2.a.c 1 76.f even 6 1
2888.2.a.e 1 76.g odd 6 1
5776.2.a.h 1 19.c even 3 1
5776.2.a.o 1 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2}$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$T^{2} - 5T + 25$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T + 4)^{2}$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 5T + 25$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$T^{2} + 5T + 25$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2} - 13T + 169$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$T^{2} - T + 1$$
$73$ $$T^{2} - 9T + 81$$
$79$ $$T^{2} - 17T + 289$$
$83$ $$(T + 16)^{2}$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} - 13T + 169$$