# Properties

 Label 1216.3.g.c Level $1216$ Weight $3$ Character orbit 1216.g Analytic conductor $33.134$ Analytic rank $0$ Dimension $8$ CM discriminant -19 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1216.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1336001462$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2702336256.1 Defining polynomial: $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{3} - \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} -9 q^{9} +O(q^{10})$$ $$q + ( -\beta_{3} - \beta_{6} ) q^{5} + ( \beta_{1} - \beta_{7} ) q^{7} -9 q^{9} + ( 5 \beta_{2} - 4 \beta_{5} ) q^{11} + ( -11 - 7 \beta_{4} ) q^{17} -19 \beta_{5} q^{19} + ( -2 \beta_{1} + 7 \beta_{7} ) q^{23} + ( -36 + 9 \beta_{4} ) q^{25} + ( -11 \beta_{2} - 14 \beta_{5} ) q^{35} + ( 3 \beta_{2} - 44 \beta_{5} ) q^{43} + ( 9 \beta_{3} + 9 \beta_{6} ) q^{45} + ( -11 \beta_{1} + \beta_{7} ) q^{47} + ( 20 + 15 \beta_{4} ) q^{49} + ( \beta_{1} - 34 \beta_{7} ) q^{55} + ( 11 \beta_{3} + 13 \beta_{6} ) q^{61} + ( -9 \beta_{1} + 9 \beta_{7} ) q^{63} + ( -29 - 33 \beta_{4} ) q^{73} + ( 21 \beta_{3} - 6 \beta_{6} ) q^{77} + 81 q^{81} + 90 \beta_{5} q^{83} + ( 11 \beta_{3} - 38 \beta_{6} ) q^{85} + ( -19 \beta_{1} - 19 \beta_{7} ) q^{95} + ( -45 \beta_{2} + 36 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 72q^{9} + O(q^{10})$$ $$8q - 72q^{9} - 60q^{17} - 324q^{25} + 100q^{49} - 100q^{73} + 648q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} - 84 \nu^{5} - 356 \nu^{3} - 925 \nu$$$$)/500$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 14 \nu^{5} - 126 \nu^{3} - 855 \nu$$$$)/350$$ $$\beta_{3}$$ $$=$$ $$($$$$-19 \nu^{6} - 196 \nu^{4} - 1764 \nu^{2} - 4975$$$$)/700$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{6} - 9 \nu^{4} - 31 \nu^{2} - 125$$$$)/25$$ $$\beta_{5}$$ $$=$$ $$($$$$9 \nu^{7} + 56 \nu^{5} + 154 \nu^{3} + 625 \nu$$$$)/1750$$ $$\beta_{6}$$ $$=$$ $$($$$$-17 \nu^{6} - 28 \nu^{4} - 252 \nu^{2} - 325$$$$)/350$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} + 4 \nu^{5} + 36 \nu^{3} + 45 \nu$$$$)/50$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{7} + 8 \beta_{5} - 8 \beta_{2} + 2 \beta_{1}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{6} + 8 \beta_{4} - 10 \beta_{3} - 32$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{7} - 28 \beta_{5} - 2 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$47 \beta_{6} - 72 \beta_{4} + 22 \beta_{3} - 160$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-89 \beta_{7} + 360 \beta_{5} + 88 \beta_{2} - 90 \beta_{1}$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$($$$$-49 \beta_{6} + 14 \beta_{3} + 54$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$283 \beta_{7} + 2232 \beta_{5} + 8 \beta_{2} + 558 \beta_{1}$$$$)/16$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
417.1
 −0.656712 + 2.13746i 0.656712 − 2.13746i −1.52274 + 1.63746i 1.52274 − 1.63746i −1.52274 − 1.63746i 1.52274 + 1.63746i −0.656712 − 2.13746i 0.656712 + 2.13746i
0 0 0 9.97368i 0 −2.20822 0 −9.00000 0
417.2 0 0 0 9.97368i 0 2.20822 0 −9.00000 0
417.3 0 0 0 5.61478i 0 −10.8685 0 −9.00000 0
417.4 0 0 0 5.61478i 0 10.8685 0 −9.00000 0
417.5 0 0 0 5.61478i 0 −10.8685 0 −9.00000 0
417.6 0 0 0 5.61478i 0 10.8685 0 −9.00000 0
417.7 0 0 0 9.97368i 0 −2.20822 0 −9.00000 0
417.8 0 0 0 9.97368i 0 2.20822 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 417.8 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 CM by $$\Q(\sqrt{-19})$$
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
76.d even 2 1 inner
152.b even 2 1 inner
152.g odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.g.c 8
4.b odd 2 1 inner 1216.3.g.c 8
8.b even 2 1 inner 1216.3.g.c 8
8.d odd 2 1 inner 1216.3.g.c 8
19.b odd 2 1 CM 1216.3.g.c 8
76.d even 2 1 inner 1216.3.g.c 8
152.b even 2 1 inner 1216.3.g.c 8
152.g odd 2 1 inner 1216.3.g.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.3.g.c 8 1.a even 1 1 trivial
1216.3.g.c 8 4.b odd 2 1 inner
1216.3.g.c 8 8.b even 2 1 inner
1216.3.g.c 8 8.d odd 2 1 inner
1216.3.g.c 8 19.b odd 2 1 CM
1216.3.g.c 8 76.d even 2 1 inner
1216.3.g.c 8 152.b even 2 1 inner
1216.3.g.c 8 152.g odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1216, [\chi])$$:

 $$T_{3}$$ $$T_{5}^{4} + 131 T_{5}^{2} + 3136$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 3136 + 131 T^{2} + T^{4} )^{2}$$
$7$ $$( 576 - 123 T^{2} + T^{4} )^{2}$$
$11$ $$( 125316 + 717 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$( -642 + 15 T + T^{2} )^{4}$$
$19$ $$( 361 + T^{2} )^{4}$$
$23$ $$( -1216 + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 2815684 + 3869 T^{2} + T^{4} )^{2}$$
$47$ $$( 11669056 - 10043 T^{2} + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 47444544 + 18051 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( -15362 + 25 T + T^{2} )^{4}$$
$79$ $$T^{8}$$
$83$ $$( 8100 + T^{2} )^{4}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$