# Properties

 Label 304.3.e.f 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: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 7 q^{5} -11 q^{7} -23 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 7 q^{5} -11 q^{7} -23 q^{9} -3 q^{11} + 2 \beta q^{13} + 7 \beta q^{15} -17 q^{17} + 19 q^{19} -11 \beta q^{21} -2 q^{23} + 24 q^{25} -14 \beta q^{27} + 7 \beta q^{29} + \beta q^{31} -3 \beta q^{33} -77 q^{35} + 7 \beta q^{37} -64 q^{39} -7 \beta q^{41} + 21 q^{43} -161 q^{45} + 5 q^{47} + 72 q^{49} -17 \beta q^{51} -\beta q^{53} -21 q^{55} + 19 \beta q^{57} + 6 \beta q^{59} + 23 q^{61} + 253 q^{63} + 14 \beta q^{65} + 7 \beta q^{67} -2 \beta q^{69} + 16 \beta q^{71} + 39 q^{73} + 24 \beta q^{75} + 33 q^{77} -17 \beta q^{79} + 241 q^{81} + 6 q^{83} -119 q^{85} -224 q^{87} -21 \beta q^{89} -22 \beta q^{91} -32 q^{93} + 133 q^{95} + 30 \beta q^{97} + 69 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 14q^{5} - 22q^{7} - 46q^{9} + O(q^{10})$$ $$2q + 14q^{5} - 22q^{7} - 46q^{9} - 6q^{11} - 34q^{17} + 38q^{19} - 4q^{23} + 48q^{25} - 154q^{35} - 128q^{39} + 42q^{43} - 322q^{45} + 10q^{47} + 144q^{49} - 42q^{55} + 46q^{61} + 506q^{63} + 78q^{73} + 66q^{77} + 482q^{81} + 12q^{83} - 238q^{85} - 448q^{87} - 64q^{93} + 266q^{95} + 138q^{99} + O(q^{100})$$

## 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
 − 1.41421i 1.41421i
0 5.65685i 0 7.00000 0 −11.0000 0 −23.0000 0
113.2 0 5.65685i 0 7.00000 0 −11.0000 0 −23.0000 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.f 2
3.b odd 2 1 2736.3.o.b 2
4.b odd 2 1 152.3.e.a 2
8.b even 2 1 1216.3.e.c 2
8.d odd 2 1 1216.3.e.d 2
12.b even 2 1 1368.3.o.a 2
19.b odd 2 1 inner 304.3.e.f 2
57.d even 2 1 2736.3.o.b 2
76.d even 2 1 152.3.e.a 2
152.b even 2 1 1216.3.e.d 2
152.g odd 2 1 1216.3.e.c 2
228.b odd 2 1 1368.3.o.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.e.a 2 4.b odd 2 1
152.3.e.a 2 76.d even 2 1
304.3.e.f 2 1.a even 1 1 trivial
304.3.e.f 2 19.b odd 2 1 inner
1216.3.e.c 2 8.b even 2 1
1216.3.e.c 2 152.g odd 2 1
1216.3.e.d 2 8.d odd 2 1
1216.3.e.d 2 152.b even 2 1
1368.3.o.a 2 12.b even 2 1
1368.3.o.a 2 228.b odd 2 1
2736.3.o.b 2 3.b odd 2 1
2736.3.o.b 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 $$S_{3}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{2} + 32$$ $$T_{5} - 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$32 + T^{2}$$
$5$ $$( -7 + T )^{2}$$
$7$ $$( 11 + T )^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$128 + T^{2}$$
$17$ $$( 17 + T )^{2}$$
$19$ $$( -19 + T )^{2}$$
$23$ $$( 2 + T )^{2}$$
$29$ $$1568 + T^{2}$$
$31$ $$32 + T^{2}$$
$37$ $$1568 + T^{2}$$
$41$ $$1568 + T^{2}$$
$43$ $$( -21 + T )^{2}$$
$47$ $$( -5 + T )^{2}$$
$53$ $$32 + T^{2}$$
$59$ $$1152 + T^{2}$$
$61$ $$( -23 + T )^{2}$$
$67$ $$1568 + T^{2}$$
$71$ $$8192 + T^{2}$$
$73$ $$( -39 + T )^{2}$$
$79$ $$9248 + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$14112 + T^{2}$$
$97$ $$28800 + T^{2}$$