# Properties

 Label 912.2.ch.a Level $912$ Weight $2$ Character orbit 912.ch Analytic conductor $7.282$ Analytic rank $0$ Dimension $6$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.ch (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{18} - \zeta_{18}^{4} ) q^{3} + ( 3 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} + ( 3 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{18} - \zeta_{18}^{4} ) q^{3} + ( 3 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{7} + ( 3 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{9} + ( -4 + \zeta_{18} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{13} + ( -3 + 5 \zeta_{18}^{3} ) q^{19} + ( -1 + 4 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 5 \zeta_{18}^{5} ) q^{21} -5 \zeta_{18}^{5} q^{25} + ( 6 - 3 \zeta_{18}^{3} ) q^{27} + ( -5 \zeta_{18} - 6 \zeta_{18}^{2} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{31} + ( 7 \zeta_{18} + 7 \zeta_{18}^{2} - 4 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{37} + ( -5 \zeta_{18} + 5 \zeta_{18}^{2} + 7 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{39} + ( -7 + \zeta_{18}^{2} + \zeta_{18}^{3} - 7 \zeta_{18}^{5} ) q^{43} + ( -8 \zeta_{18} + 5 \zeta_{18}^{2} + 7 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{49} + ( -\zeta_{18} + 8 \zeta_{18}^{4} ) q^{57} + ( 5 + 4 \zeta_{18} + 4 \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{61} + ( 9 + 3 \zeta_{18} - 6 \zeta_{18}^{3} + 6 \zeta_{18}^{4} ) q^{63} + ( -2 + 2 \zeta_{18} - 7 \zeta_{18}^{3} - 9 \zeta_{18}^{4} ) q^{67} + ( -9 + 9 \zeta_{18}^{2} + \zeta_{18}^{3} - 8 \zeta_{18}^{5} ) q^{73} + ( 5 - 10 \zeta_{18}^{3} ) q^{75} + ( -3 - 3 \zeta_{18}^{2} + 10 \zeta_{18}^{3} - 7 \zeta_{18}^{5} ) q^{79} + ( 9 \zeta_{18} - 9 \zeta_{18}^{4} ) q^{81} + ( -11 - 6 \zeta_{18} - \zeta_{18}^{2} + 6 \zeta_{18}^{3} + 11 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{91} + ( -7 - 11 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{93} -19 \zeta_{18} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + O(q^{10})$$ $$6q - 15q^{13} - 3q^{19} + 9q^{21} + 27q^{27} - 39q^{43} + 21q^{49} + 42q^{61} + 36q^{63} - 33q^{67} - 51q^{73} + 12q^{79} - 48q^{91} - 54q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$\zeta_{18}^{2}$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.173648 − 0.984808i 0.939693 + 0.342020i 0.939693 − 0.342020i −0.766044 + 0.642788i −0.766044 − 0.642788i −0.173648 + 0.984808i
0 −1.11334 1.32683i 0 0 0 −4.52481 2.61240i 0 −0.520945 + 2.95442i 0
479.1 0 1.70574 0.300767i 0 0 0 2.89053 + 1.66885i 0 2.81908 1.02606i 0
575.1 0 1.70574 + 0.300767i 0 0 0 2.89053 1.66885i 0 2.81908 + 1.02606i 0
671.1 0 −0.592396 + 1.62760i 0 0 0 1.63429 + 0.943555i 0 −2.29813 1.92836i 0
719.1 0 −0.592396 1.62760i 0 0 0 1.63429 0.943555i 0 −2.29813 + 1.92836i 0
815.1 0 −1.11334 + 1.32683i 0 0 0 −4.52481 + 2.61240i 0 −0.520945 2.95442i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 815.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
76.l odd 18 1 inner
228.v even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ch.a 6
3.b odd 2 1 CM 912.2.ch.a 6
4.b odd 2 1 912.2.ch.b yes 6
12.b even 2 1 912.2.ch.b yes 6
19.e even 9 1 912.2.ch.b yes 6
57.l odd 18 1 912.2.ch.b yes 6
76.l odd 18 1 inner 912.2.ch.a 6
228.v even 18 1 inner 912.2.ch.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ch.a 6 1.a even 1 1 trivial
912.2.ch.a 6 3.b odd 2 1 CM
912.2.ch.a 6 76.l odd 18 1 inner
912.2.ch.a 6 228.v even 18 1 inner
912.2.ch.b yes 6 4.b odd 2 1
912.2.ch.b yes 6 12.b even 2 1
912.2.ch.b yes 6 19.e even 9 1
912.2.ch.b yes 6 57.l odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}$$ $$T_{7}^{6} - 21 T_{7}^{4} + 441 T_{7}^{2} - 1197 T_{7} + 1083$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$27 - 9 T^{3} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$1083 - 1197 T + 441 T^{2} - 21 T^{4} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$361 + 1254 T + 1920 T^{2} + 604 T^{3} + 114 T^{4} + 15 T^{5} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$( 19 + T + T^{2} )^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$118803 - 55521 T + 8649 T^{2} - 93 T^{4} + T^{6}$$
$37$ $$( -323 - 111 T + T^{3} )^{2}$$
$41$ $$T^{6}$$
$43$ $$312987 + 139536 T + 36990 T^{2} + 6000 T^{3} + 636 T^{4} + 39 T^{5} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$516961 - 101379 T + 31332 T^{2} - 6967 T^{3} + 771 T^{4} - 42 T^{5} + T^{6}$$
$67$ $$189003 - 72288 T + 20070 T^{2} + 5880 T^{3} + 564 T^{4} + 33 T^{5} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$73441 + 37398 T + 53652 T^{2} + 11440 T^{3} + 1086 T^{4} + 51 T^{5} + T^{6}$$
$79$ $$48387 - 104013 T + 59598 T^{2} - 2463 T^{3} + 285 T^{4} - 12 T^{5} + T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$47045881 + 6859 T^{3} + T^{6}$$