# Properties

 Label 912.2.bn.m Level $912$ Weight $2$ Character orbit 912.bn Analytic conductor $7.282$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(65,912)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("912.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 912.bn (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 57) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - \beta_{5} q^{5} + (\beta_{3} - 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{6} + \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (-b4 - b3 + b1) * q^3 - b5 * q^5 + (b3 - 2*b1 + 2) * q^7 + (-2*b6 + b2 - 1) * q^9 $$q + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} - \beta_{5} q^{5} + (\beta_{3} - 2 \beta_1 + 2) q^{7} + ( - 2 \beta_{6} + \beta_{2} - 1) q^{9} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4}) q^{11} + (\beta_{2} - 2 \beta_1 + 1) q^{13} + ( - \beta_{7} + \beta_{4} + 2 \beta_1) q^{15} + ( - 4 \beta_{7} + 2 \beta_{4}) q^{17} + ( - 3 \beta_{3} - 2 \beta_1) q^{19} + (2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{21} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4}) q^{23} + (3 \beta_{2} - 3) q^{25} + ( - \beta_{7} + 5 \beta_{3}) q^{27} + (2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{4}) q^{29} + ( - \beta_{3} + 4 \beta_{2} - 2) q^{31} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{33} + (2 \beta_{7} - 2 \beta_{5} - \beta_{4}) q^{35} + (6 \beta_{3} + 2 \beta_{2} - 1) q^{37} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{39} + 4 \beta_{4} q^{41} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{43} + ( - \beta_{6} + \beta_{5} + 4) q^{45} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{4}) q^{47} + (4 \beta_{3} - 8 \beta_1) q^{49} + (2 \beta_{6} - 4 \beta_{5} + 4 \beta_{2} + 4) q^{51} + (6 \beta_{7} - \beta_{6} + 2 \beta_{5} - 6 \beta_{4}) q^{53} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{55} + (2 \beta_{6} - 5 \beta_{5} - 3 \beta_{2} - 2) q^{57} + (8 \beta_{6} - 4 \beta_{5} + \beta_{4}) q^{59} + ( - 4 \beta_{3} - 7 \beta_{2} + 2 \beta_1 + 7) q^{61} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{63} + (2 \beta_{7} - \beta_{6} - \beta_{5}) q^{65} + \beta_1 q^{67} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + 4 \beta_{3} + 4 \beta_{2} - 2) q^{69} + ( - 4 \beta_{6} + 2 \beta_{5} - 6 \beta_{4}) q^{71} + 3 \beta_{2} q^{73} + (3 \beta_{7} + 3 \beta_{3}) q^{75} + (\beta_{6} - \beta_{5}) q^{77} + (7 \beta_{3} + 2 \beta_{2} - 7 \beta_1 - 4) q^{79} + (4 \beta_{5} + 7 \beta_{2}) q^{81} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4}) q^{83} + ( - 8 \beta_{3} + 4 \beta_1) q^{85} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + 8 \beta_1 - 4) q^{87} + (3 \beta_{6} - 6 \beta_{5}) q^{89} + (4 \beta_{2} - 7 \beta_1 + 4) q^{91} + (4 \beta_{7} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1) q^{93} + (2 \beta_{7} + 3 \beta_{4}) q^{95} + (4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{97} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 8) q^{99}+O(q^{100})$$ q + (-b4 - b3 + b1) * q^3 - b5 * q^5 + (b3 - 2*b1 + 2) * q^7 + (-2*b6 + b2 - 1) * q^9 + (-b7 + 2*b6 - 2*b5 + 2*b4) * q^11 + (b2 - 2*b1 + 1) * q^13 + (-b7 + b4 + 2*b1) * q^15 + (-4*b7 + 2*b4) * q^17 + (-3*b3 - 2*b1) * q^19 + (2*b6 - b5 - 2*b4 - 2*b3 + b2 + 2*b1 - 2) * q^21 + (-b7 - 2*b6 - b4) * q^23 + (3*b2 - 3) * q^25 + (-b7 + 5*b3) * q^27 + (2*b7 + 2*b6 - 4*b5 - 2*b4) * q^29 + (-b3 + 4*b2 - 2) * q^31 + (2*b6 - b5 + 2*b4 - 4*b3 - 2*b2 + 4*b1 + 4) * q^33 + (2*b7 - 2*b5 - b4) * q^35 + (6*b3 + 2*b2 - 1) * q^37 + (b7 + 2*b6 - 2*b5 - 2*b4 - b3 + 2*b1 - 2) * q^39 + 4*b4 * q^41 + (b3 + 4*b2 + b1) * q^43 + (-b6 + b5 + 4) * q^45 + (2*b7 - 4*b6 + 2*b4) * q^47 + (4*b3 - 8*b1) * q^49 + (2*b6 - 4*b5 + 4*b2 + 4) * q^51 + (6*b7 - b6 + 2*b5 - 6*b4) * q^53 + (-2*b3 - 4*b2 - 2*b1) * q^55 + (2*b6 - 5*b5 - 3*b2 - 2) * q^57 + (8*b6 - 4*b5 + b4) * q^59 + (-4*b3 - 7*b2 + 2*b1 + 7) * q^61 + (2*b7 - 4*b6 + 2*b4 - 2*b3 + 2*b2 + b1 - 2) * q^63 + (2*b7 - b6 - b5) * q^65 + b1 * q^67 + (-2*b7 - b6 - b5 + 4*b3 + 4*b2 - 2) * q^69 + (-4*b6 + 2*b5 - 6*b4) * q^71 + 3*b2 * q^73 + (3*b7 + 3*b3) * q^75 + (b6 - b5) * q^77 + (7*b3 + 2*b2 - 7*b1 - 4) * q^79 + (4*b5 + 7*b2) * q^81 + (2*b7 + 2*b6 - 2*b5 - 4*b4) * q^83 + (-8*b3 + 4*b1) * q^85 + (-2*b7 - 2*b6 + 2*b5 + 4*b4 - 4*b3 + 8*b1 - 4) * q^87 + (3*b6 - 6*b5) * q^89 + (4*b2 - 7*b1 + 4) * q^91 + (4*b7 - b5 - 2*b4 + 2*b3 - b2 + 2*b1) * q^93 + (2*b7 + 3*b4) * q^95 + (4*b3 - 2*b2 - 4*b1 + 4) * q^97 + (-b7 - 2*b6 - b4 - 8*b3 - 8*b2 + 4*b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 16 q^{7} - 4 q^{9}+O(q^{10})$$ 8 * q + 16 * q^7 - 4 * q^9 $$8 q + 16 q^{7} - 4 q^{9} + 12 q^{13} - 12 q^{21} - 12 q^{25} + 24 q^{33} - 16 q^{39} + 16 q^{43} + 32 q^{45} + 48 q^{51} - 16 q^{55} - 28 q^{57} + 28 q^{61} - 8 q^{63} + 12 q^{73} - 24 q^{79} + 28 q^{81} - 32 q^{87} + 48 q^{91} - 4 q^{93} + 24 q^{97} + 32 q^{99}+O(q^{100})$$ 8 * q + 16 * q^7 - 4 * q^9 + 12 * q^13 - 12 * q^21 - 12 * q^25 + 24 * q^33 - 16 * q^39 + 16 * q^43 + 32 * q^45 + 48 * q^51 - 16 * q^55 - 28 * q^57 + 28 * q^61 - 8 * q^63 + 12 * q^73 - 24 * q^79 + 28 * q^81 - 32 * q^87 + 48 * q^91 - 4 * q^93 + 24 * q^97 + 32 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$\zeta_{24}^{7} + \zeta_{24}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}$$ -v^7 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3}$$ -v^7 + v^5 + v^3 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{6} - \beta_{5} ) / 2$$ (b7 + b6 - b5) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2$$ (-b7 + b6 + b4) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 2

## Character values

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

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1$$ $$-1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i
0 −1.57313 + 0.724745i 0 1.22474 + 0.707107i 0 3.73205 0 1.94949 2.28024i 0
65.2 0 −0.158919 1.72474i 0 −1.22474 0.707107i 0 3.73205 0 −2.94949 + 0.548188i 0
65.3 0 0.158919 + 1.72474i 0 −1.22474 0.707107i 0 0.267949 0 −2.94949 + 0.548188i 0
65.4 0 1.57313 0.724745i 0 1.22474 + 0.707107i 0 0.267949 0 1.94949 2.28024i 0
449.1 0 −1.57313 0.724745i 0 1.22474 0.707107i 0 3.73205 0 1.94949 + 2.28024i 0
449.2 0 −0.158919 + 1.72474i 0 −1.22474 + 0.707107i 0 3.73205 0 −2.94949 0.548188i 0
449.3 0 0.158919 1.72474i 0 −1.22474 + 0.707107i 0 0.267949 0 −2.94949 0.548188i 0
449.4 0 1.57313 + 0.724745i 0 1.22474 0.707107i 0 0.267949 0 1.94949 + 2.28024i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.4 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 inner
19.d odd 6 1 inner
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.m 8
3.b odd 2 1 inner 912.2.bn.m 8
4.b odd 2 1 57.2.f.a 8
12.b even 2 1 57.2.f.a 8
19.d odd 6 1 inner 912.2.bn.m 8
57.f even 6 1 inner 912.2.bn.m 8
76.f even 6 1 57.2.f.a 8
76.f even 6 1 1083.2.d.b 8
76.g odd 6 1 1083.2.d.b 8
228.m even 6 1 1083.2.d.b 8
228.n odd 6 1 57.2.f.a 8
228.n odd 6 1 1083.2.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.f.a 8 4.b odd 2 1
57.2.f.a 8 12.b even 2 1
57.2.f.a 8 76.f even 6 1
57.2.f.a 8 228.n odd 6 1
912.2.bn.m 8 1.a even 1 1 trivial
912.2.bn.m 8 3.b odd 2 1 inner
912.2.bn.m 8 19.d odd 6 1 inner
912.2.bn.m 8 57.f even 6 1 inner
1083.2.d.b 8 76.f even 6 1
1083.2.d.b 8 76.g odd 6 1
1083.2.d.b 8 228.m even 6 1
1083.2.d.b 8 228.n odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 2T_{5}^{2} + 4$$ T5^4 - 2*T5^2 + 4 $$T_{7}^{2} - 4T_{7} + 1$$ T7^2 - 4*T7 + 1 $$T_{17}^{4} - 24T_{17}^{2} + 576$$ T17^4 - 24*T17^2 + 576

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 2 T^{6} - 5 T^{4} + 18 T^{2} + \cdots + 81$$
$5$ $$(T^{4} - 2 T^{2} + 4)^{2}$$
$7$ $$(T^{2} - 4 T + 1)^{4}$$
$11$ $$(T^{4} + 28 T^{2} + 4)^{2}$$
$13$ $$(T^{4} - 6 T^{3} + 11 T^{2} + 6 T + 1)^{2}$$
$17$ $$(T^{4} - 24 T^{2} + 576)^{2}$$
$19$ $$(T^{4} + 26 T^{2} + 361)^{2}$$
$23$ $$T^{8} - 28 T^{6} + 780 T^{4} + \cdots + 16$$
$29$ $$T^{8} + 64 T^{6} + 3840 T^{4} + \cdots + 65536$$
$31$ $$(T^{4} + 26 T^{2} + 121)^{2}$$
$37$ $$(T^{4} + 78 T^{2} + 1089)^{2}$$
$41$ $$(T^{4} + 32 T^{2} + 1024)^{2}$$
$43$ $$(T^{4} - 8 T^{3} + 51 T^{2} - 104 T + 169)^{2}$$
$47$ $$T^{8} - 112 T^{6} + 12480 T^{4} + \cdots + 4096$$
$53$ $$T^{8} + 156 T^{6} + \cdots + 18974736$$
$59$ $$T^{8} + 196 T^{6} + \cdots + 78074896$$
$61$ $$(T^{4} - 14 T^{3} + 159 T^{2} - 518 T + 1369)^{2}$$
$67$ $$(T^{4} - T^{2} + 1)^{2}$$
$71$ $$T^{8} + 192 T^{6} + 34560 T^{4} + \cdots + 5308416$$
$73$ $$(T^{2} - 3 T + 9)^{4}$$
$79$ $$(T^{4} + 12 T^{3} + 11 T^{2} - 444 T + 1369)^{2}$$
$83$ $$(T^{4} + 64 T^{2} + 256)^{2}$$
$89$ $$(T^{4} + 54 T^{2} + 2916)^{2}$$
$97$ $$(T^{4} - 12 T^{3} + 44 T^{2} + 48 T + 16)^{2}$$