# Properties

 Label 304.3.r.a Level $304$ Weight $3$ Character orbit 304.r Analytic conductor $8.283$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 304.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.28340003655$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{7} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( -2 + 4 \beta_{1} - 2 \beta_{3} ) q^{7} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{9} + ( 10 - 4 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -5 + 6 \beta_{1} - 5 \beta_{2} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} ) q^{15} + ( 2 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{17} + ( -5 + 6 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{19} + ( 12 - 8 \beta_{1} - 6 \beta_{2} + 8 \beta_{3} ) q^{21} + ( 5 - 9 \beta_{1} - 5 \beta_{2} + 18 \beta_{3} ) q^{23} + ( 24 - 24 \beta_{2} ) q^{25} + ( 9 - 18 \beta_{2} + 7 \beta_{3} ) q^{27} + ( 11 + 18 \beta_{1} + 11 \beta_{2} ) q^{29} + ( -18 + 36 \beta_{2} ) q^{31} + ( -28 + 16 \beta_{1} + 14 \beta_{2} - 16 \beta_{3} ) q^{33} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{35} + ( 2 - 4 \beta_{2} + 18 \beta_{3} ) q^{37} + ( 27 - 22 \beta_{1} + 11 \beta_{3} ) q^{39} + ( 6 - 42 \beta_{1} - 3 \beta_{2} + 42 \beta_{3} ) q^{41} + ( 23 \beta_{1} - 19 \beta_{2} + 23 \beta_{3} ) q^{43} + ( -4 - 4 \beta_{1} + 2 \beta_{3} ) q^{45} + ( 35 + 19 \beta_{1} - 35 \beta_{2} - 38 \beta_{3} ) q^{47} + ( -21 - 16 \beta_{1} + 8 \beta_{3} ) q^{49} + ( -3 + \beta_{1} - 3 \beta_{2} ) q^{51} + ( -7 + 48 \beta_{1} - 7 \beta_{2} ) q^{53} + ( -2 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 20 - 24 \beta_{1} + 11 \beta_{2} + 2 \beta_{3} ) q^{57} + ( -14 + 33 \beta_{1} + 7 \beta_{2} - 33 \beta_{3} ) q^{59} + ( 37 + 16 \beta_{1} - 37 \beta_{2} - 32 \beta_{3} ) q^{61} + ( -16 - 4 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} ) q^{63} + ( 5 - 10 \beta_{2} + 6 \beta_{3} ) q^{65} + ( -17 - 63 \beta_{1} - 17 \beta_{2} ) q^{67} + ( -23 + 46 \beta_{2} - 32 \beta_{3} ) q^{69} + ( -102 + 9 \beta_{1} + 51 \beta_{2} - 9 \beta_{3} ) q^{71} + ( -32 \beta_{1} + 49 \beta_{2} - 32 \beta_{3} ) q^{73} + ( -24 + 48 \beta_{2} - 24 \beta_{3} ) q^{75} + ( -44 + 48 \beta_{1} - 24 \beta_{3} ) q^{77} + ( 42 + 21 \beta_{1} - 21 \beta_{2} - 21 \beta_{3} ) q^{79} + ( -34 \beta_{1} + 5 \beta_{2} - 34 \beta_{3} ) q^{81} + ( 16 + 12 \beta_{1} - 6 \beta_{3} ) q^{83} + ( -7 - 2 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{85} + ( 3 - 14 \beta_{1} + 7 \beta_{3} ) q^{87} + ( -1 - 6 \beta_{1} - \beta_{2} ) q^{89} + ( 34 - 42 \beta_{1} + 34 \beta_{2} ) q^{91} + ( 18 \beta_{1} - 54 \beta_{2} + 18 \beta_{3} ) q^{93} + ( -2 - 9 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} ) q^{95} + ( -162 - 6 \beta_{1} + 81 \beta_{2} + 6 \beta_{3} ) q^{97} + ( -16 - 12 \beta_{1} + 16 \beta_{2} + 24 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} + 2 q^{5} - 8 q^{7} - 8 q^{9} + O(q^{10})$$ $$4 q - 6 q^{3} + 2 q^{5} - 8 q^{7} - 8 q^{9} + 40 q^{11} - 30 q^{13} - 6 q^{15} + 14 q^{17} - 16 q^{19} + 36 q^{21} + 10 q^{23} + 48 q^{25} + 66 q^{29} - 84 q^{33} - 4 q^{35} + 108 q^{39} + 18 q^{41} - 38 q^{43} - 16 q^{45} + 70 q^{47} - 84 q^{49} - 18 q^{51} - 42 q^{53} + 20 q^{55} + 102 q^{57} - 42 q^{59} + 74 q^{61} - 32 q^{63} - 102 q^{67} - 306 q^{71} + 98 q^{73} - 176 q^{77} + 126 q^{79} + 10 q^{81} + 64 q^{83} - 14 q^{85} + 12 q^{87} - 6 q^{89} + 204 q^{91} - 108 q^{93} - 14 q^{95} - 486 q^{97} - 32 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$97$$ $$191$$ $$229$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
0 −2.72474 1.57313i 0 0.500000 0.866025i 0 −6.89898 0 0.449490 + 0.778539i 0
65.2 0 −0.275255 0.158919i 0 0.500000 0.866025i 0 2.89898 0 −4.44949 7.70674i 0
145.1 0 −2.72474 + 1.57313i 0 0.500000 + 0.866025i 0 −6.89898 0 0.449490 0.778539i 0
145.2 0 −0.275255 + 0.158919i 0 0.500000 + 0.866025i 0 2.89898 0 −4.44949 + 7.70674i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.r.a 4
4.b odd 2 1 38.3.d.a 4
12.b even 2 1 342.3.m.a 4
19.d odd 6 1 inner 304.3.r.a 4
76.f even 6 1 38.3.d.a 4
76.f even 6 1 722.3.b.b 4
76.g odd 6 1 722.3.b.b 4
228.n odd 6 1 342.3.m.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.d.a 4 4.b odd 2 1
38.3.d.a 4 76.f even 6 1
304.3.r.a 4 1.a even 1 1 trivial
304.3.r.a 4 19.d odd 6 1 inner
342.3.m.a 4 12.b even 2 1
342.3.m.a 4 228.n odd 6 1
722.3.b.b 4 76.f even 6 1
722.3.b.b 4 76.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 6 T_{3}^{3} + 13 T_{3}^{2} + 6 T_{3} + 1$$ acting on $$S_{3}^{\mathrm{new}}(304, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + 6 T + 13 T^{2} + 6 T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$( -20 + 4 T + T^{2} )^{2}$$
$11$ $$( 76 - 20 T + T^{2} )^{2}$$
$13$ $$9 + 90 T + 303 T^{2} + 30 T^{3} + T^{4}$$
$17$ $$625 - 350 T + 171 T^{2} - 14 T^{3} + T^{4}$$
$19$ $$130321 + 5776 T + 570 T^{2} + 16 T^{3} + T^{4}$$
$23$ $$212521 + 4610 T + 561 T^{2} - 10 T^{3} + T^{4}$$
$29$ $$81225 + 18810 T + 1167 T^{2} - 66 T^{3} + T^{4}$$
$31$ $$( 972 + T^{2} )^{2}$$
$37$ $$404496 + 1320 T^{2} + T^{4}$$
$41$ $$12257001 + 63018 T - 3393 T^{2} - 18 T^{3} + T^{4}$$
$43$ $$7912969 - 106894 T + 4257 T^{2} + 38 T^{3} + T^{4}$$
$47$ $$885481 + 65870 T + 5841 T^{2} - 70 T^{3} + T^{4}$$
$53$ $$19900521 - 187362 T - 3873 T^{2} + 42 T^{3} + T^{4}$$
$59$ $$4124961 - 85302 T - 1443 T^{2} + 42 T^{3} + T^{4}$$
$61$ $$27889 + 12358 T + 5643 T^{2} - 74 T^{3} + T^{4}$$
$67$ $$49999041 - 721242 T - 3603 T^{2} + 102 T^{3} + T^{4}$$
$71$ $$58384881 + 2338146 T + 38853 T^{2} + 306 T^{3} + T^{4}$$
$73$ $$14010049 + 366814 T + 13347 T^{2} - 98 T^{3} + T^{4}$$
$79$ $$194481 - 55566 T + 5733 T^{2} - 126 T^{3} + T^{4}$$
$83$ $$( 40 - 32 T + T^{2} )^{2}$$
$89$ $$4761 - 414 T - 57 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$384591321 + 9530946 T + 98343 T^{2} + 486 T^{3} + T^{4}$$