# Properties

 Label 304.3.e.b 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: $$\Q(\sqrt{-29})$$ Defining polynomial: $$x^{2} + 29$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-29}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -4 q^{5} + q^{7} -20 q^{9} +O(q^{10})$$ $$q + \beta q^{3} -4 q^{5} + q^{7} -20 q^{9} -14 q^{11} -3 \beta q^{13} -4 \beta q^{15} + 23 q^{17} + ( -10 + 3 \beta ) q^{19} + \beta q^{21} + q^{23} -9 q^{25} -11 \beta q^{27} + 9 \beta q^{29} -6 \beta q^{31} -14 \beta q^{33} -4 q^{35} -6 \beta q^{37} + 87 q^{39} -6 \beta q^{41} -68 q^{43} + 80 q^{45} -26 q^{47} -48 q^{49} + 23 \beta q^{51} + 15 \beta q^{53} + 56 q^{55} + ( -87 - 10 \beta ) q^{57} + 3 \beta q^{59} -40 q^{61} -20 q^{63} + 12 \beta q^{65} -3 \beta q^{67} + \beta q^{69} -6 \beta q^{71} -7 q^{73} -9 \beta q^{75} -14 q^{77} + 18 \beta q^{79} + 139 q^{81} -32 q^{83} -92 q^{85} -261 q^{87} + 24 \beta q^{89} -3 \beta q^{91} + 174 q^{93} + ( 40 - 12 \beta ) q^{95} + 18 \beta q^{97} + 280 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} + 2q^{7} - 40q^{9} + O(q^{10})$$ $$2q - 8q^{5} + 2q^{7} - 40q^{9} - 28q^{11} + 46q^{17} - 20q^{19} + 2q^{23} - 18q^{25} - 8q^{35} + 174q^{39} - 136q^{43} + 160q^{45} - 52q^{47} - 96q^{49} + 112q^{55} - 174q^{57} - 80q^{61} - 40q^{63} - 14q^{73} - 28q^{77} + 278q^{81} - 64q^{83} - 184q^{85} - 522q^{87} + 348q^{93} + 80q^{95} + 560q^{99} + O(q^{100})$$

## 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
 − 5.38516i 5.38516i
0 5.38516i 0 −4.00000 0 1.00000 0 −20.0000 0
113.2 0 5.38516i 0 −4.00000 0 1.00000 0 −20.0000 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.b 2
3.b odd 2 1 2736.3.o.i 2
4.b odd 2 1 76.3.c.a 2
8.b even 2 1 1216.3.e.l 2
8.d odd 2 1 1216.3.e.k 2
12.b even 2 1 684.3.h.c 2
19.b odd 2 1 inner 304.3.e.b 2
20.d odd 2 1 1900.3.e.b 2
20.e even 4 2 1900.3.g.b 4
57.d even 2 1 2736.3.o.i 2
76.d even 2 1 76.3.c.a 2
152.b even 2 1 1216.3.e.k 2
152.g odd 2 1 1216.3.e.l 2
228.b odd 2 1 684.3.h.c 2
380.d even 2 1 1900.3.e.b 2
380.j odd 4 2 1900.3.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.a 2 4.b odd 2 1
76.3.c.a 2 76.d even 2 1
304.3.e.b 2 1.a even 1 1 trivial
304.3.e.b 2 19.b odd 2 1 inner
684.3.h.c 2 12.b even 2 1
684.3.h.c 2 228.b odd 2 1
1216.3.e.k 2 8.d odd 2 1
1216.3.e.k 2 152.b even 2 1
1216.3.e.l 2 8.b even 2 1
1216.3.e.l 2 152.g odd 2 1
1900.3.e.b 2 20.d odd 2 1
1900.3.e.b 2 380.d even 2 1
1900.3.g.b 4 20.e even 4 2
1900.3.g.b 4 380.j odd 4 2
2736.3.o.i 2 3.b odd 2 1
2736.3.o.i 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 $$S_{3}^{\mathrm{new}}(304, [\chi])$$:

 $$T_{3}^{2} + 29$$ $$T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$29 + T^{2}$$
$5$ $$( 4 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( 14 + T )^{2}$$
$13$ $$261 + T^{2}$$
$17$ $$( -23 + T )^{2}$$
$19$ $$361 + 20 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$2349 + T^{2}$$
$31$ $$1044 + T^{2}$$
$37$ $$1044 + T^{2}$$
$41$ $$1044 + T^{2}$$
$43$ $$( 68 + T )^{2}$$
$47$ $$( 26 + T )^{2}$$
$53$ $$6525 + T^{2}$$
$59$ $$261 + T^{2}$$
$61$ $$( 40 + T )^{2}$$
$67$ $$261 + T^{2}$$
$71$ $$1044 + T^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$9396 + T^{2}$$
$83$ $$( 32 + T )^{2}$$
$89$ $$16704 + T^{2}$$
$97$ $$9396 + T^{2}$$