# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.42745222145$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} + ( 4 - 4 \zeta_{6} ) q^{5} + 2 \zeta_{6} q^{9} -3 q^{11} -2 \zeta_{6} q^{13} -4 \zeta_{6} q^{15} + ( -2 + 2 \zeta_{6} ) q^{17} + ( 2 - 5 \zeta_{6} ) q^{19} + 6 \zeta_{6} q^{23} -11 \zeta_{6} q^{25} + 5 q^{27} + 4 \zeta_{6} q^{29} + 10 q^{31} + ( -3 + 3 \zeta_{6} ) q^{33} + 2 q^{37} -2 q^{39} + ( -9 + 9 \zeta_{6} ) q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} + 8 q^{45} -12 \zeta_{6} q^{47} -7 q^{49} + 2 \zeta_{6} q^{51} + 2 \zeta_{6} q^{53} + ( -12 + 12 \zeta_{6} ) q^{55} + ( -3 - 2 \zeta_{6} ) q^{57} + ( -1 + \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} -8 q^{65} + 9 \zeta_{6} q^{67} + 6 q^{69} + ( -6 + 6 \zeta_{6} ) q^{71} + ( 9 - 9 \zeta_{6} ) q^{73} -11 q^{75} + ( -4 + 4 \zeta_{6} ) q^{79} + ( -1 + \zeta_{6} ) q^{81} + 5 q^{83} + 8 \zeta_{6} q^{85} + 4 q^{87} + 18 \zeta_{6} q^{89} + ( 10 - 10 \zeta_{6} ) q^{93} + ( -12 - 8 \zeta_{6} ) q^{95} + ( -1 + \zeta_{6} ) q^{97} -6 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} + 4q^{5} + 2q^{9} - 6q^{11} - 2q^{13} - 4q^{15} - 2q^{17} - q^{19} + 6q^{23} - 11q^{25} + 10q^{27} + 4q^{29} + 20q^{31} - 3q^{33} + 4q^{37} - 4q^{39} - 9q^{41} - 4q^{43} + 16q^{45} - 12q^{47} - 14q^{49} + 2q^{51} + 2q^{53} - 12q^{55} - 8q^{57} - q^{59} + 8q^{61} - 16q^{65} + 9q^{67} + 12q^{69} - 6q^{71} + 9q^{73} - 22q^{75} - 4q^{79} - q^{81} + 10q^{83} + 8q^{85} + 8q^{87} + 18q^{89} + 10q^{93} - 32q^{95} - q^{97} - 6q^{99} + O(q^{100})$$

## Character values

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

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

## 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.

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 2.00000 3.46410i 0 0 0 1.00000 + 1.73205i 0
273.1 0 0.500000 + 0.866025i 0 2.00000 + 3.46410i 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.d 2
3.b odd 2 1 2736.2.s.a 2
4.b odd 2 1 152.2.i.b 2
8.b even 2 1 1216.2.i.b 2
8.d odd 2 1 1216.2.i.f 2
12.b even 2 1 1368.2.s.a 2
19.c even 3 1 inner 304.2.i.d 2
19.c even 3 1 5776.2.a.e 1
19.d odd 6 1 5776.2.a.j 1
57.h odd 6 1 2736.2.s.a 2
76.f even 6 1 2888.2.a.a 1
76.g odd 6 1 152.2.i.b 2
76.g odd 6 1 2888.2.a.d 1
152.k odd 6 1 1216.2.i.f 2
152.p even 6 1 1216.2.i.b 2
228.m even 6 1 1368.2.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.i.b 2 4.b odd 2 1
152.2.i.b 2 76.g odd 6 1
304.2.i.d 2 1.a even 1 1 trivial
304.2.i.d 2 19.c even 3 1 inner
1216.2.i.b 2 8.b even 2 1
1216.2.i.b 2 152.p even 6 1
1216.2.i.f 2 8.d odd 2 1
1216.2.i.f 2 152.k odd 6 1
1368.2.s.a 2 12.b even 2 1
1368.2.s.a 2 228.m even 6 1
2736.2.s.a 2 3.b odd 2 1
2736.2.s.a 2 57.h odd 6 1
2888.2.a.a 1 76.f even 6 1
2888.2.a.d 1 76.g odd 6 1
5776.2.a.e 1 19.c even 3 1
5776.2.a.j 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 $$S_{2}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{5}^{2} - 4 T_{5} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$16 - 4 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$4 + 2 T + T^{2}$$
$17$ $$4 + 2 T + T^{2}$$
$19$ $$19 + T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$16 - 4 T + T^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$81 + 9 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$144 + 12 T + T^{2}$$
$53$ $$4 - 2 T + T^{2}$$
$59$ $$1 + T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$81 - 9 T + T^{2}$$
$71$ $$36 + 6 T + T^{2}$$
$73$ $$81 - 9 T + T^{2}$$
$79$ $$16 + 4 T + T^{2}$$
$83$ $$( -5 + T )^{2}$$
$89$ $$324 - 18 T + T^{2}$$
$97$ $$1 + T + T^{2}$$