# Properties

 Label 304.3.e.d Level $304$ Weight $3$ Character orbit 304.e Analytic conductor $8.283$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.28340003655$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-13})$$ Defining polynomial: $$x^{2} + 13$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 4 q^{5} + 5 q^{7} -4 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 4 q^{5} + 5 q^{7} -4 q^{9} + 10 q^{11} + \beta q^{13} + 4 \beta q^{15} + 15 q^{17} + ( 6 - 5 \beta ) q^{19} + 5 \beta q^{21} -35 q^{23} -9 q^{25} + 5 \beta q^{27} + 5 \beta q^{29} + 10 \beta q^{31} + 10 \beta q^{33} + 20 q^{35} -6 \beta q^{37} -13 q^{39} + 10 \beta q^{41} + 20 q^{43} -16 q^{45} -10 q^{47} -24 q^{49} + 15 \beta q^{51} -21 \beta q^{53} + 40 q^{55} + ( 65 + 6 \beta ) q^{57} -5 \beta q^{59} -40 q^{61} -20 q^{63} + 4 \beta q^{65} -11 \beta q^{67} -35 \beta q^{69} -30 \beta q^{71} + 105 q^{73} -9 \beta q^{75} + 50 q^{77} + 10 \beta q^{79} -101 q^{81} + 40 q^{83} + 60 q^{85} -65 q^{87} + 5 \beta q^{91} -130 q^{93} + ( 24 - 20 \beta ) q^{95} + 34 \beta q^{97} -40 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 8q^{5} + 10q^{7} - 8q^{9} + O(q^{10})$$ $$2q + 8q^{5} + 10q^{7} - 8q^{9} + 20q^{11} + 30q^{17} + 12q^{19} - 70q^{23} - 18q^{25} + 40q^{35} - 26q^{39} + 40q^{43} - 32q^{45} - 20q^{47} - 48q^{49} + 80q^{55} + 130q^{57} - 80q^{61} - 40q^{63} + 210q^{73} + 100q^{77} - 202q^{81} + 80q^{83} + 120q^{85} - 130q^{87} - 260q^{93} + 48q^{95} - 80q^{99} + O(q^{100})$$

## Character values

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

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

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1
 − 3.60555i 3.60555i
0 3.60555i 0 4.00000 0 5.00000 0 −4.00000 0
113.2 0 3.60555i 0 4.00000 0 5.00000 0 −4.00000 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.e.d 2
3.b odd 2 1 2736.3.o.d 2
4.b odd 2 1 19.3.b.b 2
8.b even 2 1 1216.3.e.h 2
8.d odd 2 1 1216.3.e.g 2
12.b even 2 1 171.3.c.b 2
19.b odd 2 1 inner 304.3.e.d 2
20.d odd 2 1 475.3.c.b 2
20.e even 4 2 475.3.d.b 4
57.d even 2 1 2736.3.o.d 2
76.d even 2 1 19.3.b.b 2
76.f even 6 2 361.3.d.b 4
76.g odd 6 2 361.3.d.b 4
76.k even 18 6 361.3.f.d 12
76.l odd 18 6 361.3.f.d 12
152.b even 2 1 1216.3.e.g 2
152.g odd 2 1 1216.3.e.h 2
228.b odd 2 1 171.3.c.b 2
380.d even 2 1 475.3.c.b 2
380.j odd 4 2 475.3.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 4.b odd 2 1
19.3.b.b 2 76.d even 2 1
171.3.c.b 2 12.b even 2 1
171.3.c.b 2 228.b odd 2 1
304.3.e.d 2 1.a even 1 1 trivial
304.3.e.d 2 19.b odd 2 1 inner
361.3.d.b 4 76.f even 6 2
361.3.d.b 4 76.g odd 6 2
361.3.f.d 12 76.k even 18 6
361.3.f.d 12 76.l odd 18 6
475.3.c.b 2 20.d odd 2 1
475.3.c.b 2 380.d even 2 1
475.3.d.b 4 20.e even 4 2
475.3.d.b 4 380.j odd 4 2
1216.3.e.g 2 8.d odd 2 1
1216.3.e.g 2 152.b even 2 1
1216.3.e.h 2 8.b even 2 1
1216.3.e.h 2 152.g odd 2 1
2736.3.o.d 2 3.b odd 2 1
2736.3.o.d 2 57.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 $$S_{3}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{2} + 13$$ $$T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$13 + T^{2}$$
$5$ $$( -4 + T )^{2}$$
$7$ $$( -5 + T )^{2}$$
$11$ $$( -10 + T )^{2}$$
$13$ $$13 + T^{2}$$
$17$ $$( -15 + T )^{2}$$
$19$ $$361 - 12 T + T^{2}$$
$23$ $$( 35 + T )^{2}$$
$29$ $$325 + T^{2}$$
$31$ $$1300 + T^{2}$$
$37$ $$468 + T^{2}$$
$41$ $$1300 + T^{2}$$
$43$ $$( -20 + T )^{2}$$
$47$ $$( 10 + T )^{2}$$
$53$ $$5733 + T^{2}$$
$59$ $$325 + T^{2}$$
$61$ $$( 40 + T )^{2}$$
$67$ $$1573 + T^{2}$$
$71$ $$11700 + T^{2}$$
$73$ $$( -105 + T )^{2}$$
$79$ $$1300 + T^{2}$$
$83$ $$( -40 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$15028 + T^{2}$$