# Properties

 Label 416.2.w.c Level $416$ Weight $2$ Character orbit 416.w Analytic conductor $3.322$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(225,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("416.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$416 = 2^{5} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 416.w (of order $$6$$, degree $$2$$, 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: $$3.32177672409$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.56070144.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ x^8 - 4*x^7 + 16*x^6 - 34*x^5 + 63*x^4 - 74*x^3 + 70*x^2 - 38*x + 13 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + (\beta_{5} - \beta_{4}) q^{5} + \beta_{3} q^{7} + (2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 2) q^{9}+O(q^{10})$$ q + b6 * q^3 + (b5 - b4) * q^5 + b3 * q^7 + (2*b5 - b4 - 2*b2 - 2) * q^9 $$q + \beta_{6} q^{3} + (\beta_{5} - \beta_{4}) q^{5} + \beta_{3} q^{7} + (2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 2) q^{9} + ( - \beta_{6} + 2 \beta_1) q^{11} + (4 \beta_{2} + 1) q^{13} + (\beta_{7} + \beta_{6} - 2 \beta_1) q^{15} + (2 \beta_{5} - \beta_{4} - 3 \beta_{2} - 3) q^{17} + \beta_{3} q^{19} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 1) q^{21} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{3}) q^{23} + q^{25} + (\beta_{7} - \beta_{3} - 2 \beta_1) q^{27} + 3 \beta_{2} q^{29} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{31} + ( - 3 \beta_{4} - 5 \beta_{2} + 5) q^{33} + ( - 2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{35}+ \cdots + (3 \beta_{7} + 10 \beta_{6} + \cdots - 5 \beta_1) q^{99}+O(q^{100})$$ q + b6 * q^3 + (b5 - b4) * q^5 + b3 * q^7 + (2*b5 - b4 - 2*b2 - 2) * q^9 + (-b6 + 2*b1) * q^11 + (4*b2 + 1) * q^13 + (b7 + b6 - 2*b1) * q^15 + (2*b5 - b4 - 3*b2 - 3) * q^17 + b3 * q^19 + (2*b5 - 2*b4 + 2*b2 + 1) * q^21 + (-b7 - 2*b6 - 2*b3) * q^23 + q^25 + (b7 - b3 - 2*b1) * q^27 + 3*b2 * q^29 + (-2*b7 - 2*b3) * q^31 + (-3*b4 - 5*b2 + 5) * q^33 + (-2*b7 + b6 - b3 - b1) * q^35 + (3*b5 + b2 + 2) * q^37 + (-3*b6 + 4*b1) * q^39 + (-b5 - 3*b2 - 6) * q^41 + (-3*b6 + 3*b1) * q^43 + (2*b4 + 4*b2 - 4) * q^45 + (3*b7 + 2*b6 + 3*b3 - b1) * q^47 + (-3*b5 + 6*b4 - 4*b2) * q^49 + (b7 - b3 - 6*b1) * q^51 + (-b5 - b4 + 6) * q^53 + (b7 + 3*b6 + 2*b3) * q^55 + (2*b5 - 2*b4 + 2*b2 + 1) * q^57 + (2*b6 + 3*b3 + 2*b1) * q^59 + (-3*b2 - 3) * q^61 + (-b7 + b6 - 2*b1) * q^63 + (-3*b5 - b4) * q^65 + b7 * q^67 + (-8*b5 + 4*b4 + 7*b2 + 7) * q^69 + (3*b6 + 3*b1) * q^71 + (-3*b5 + 3*b4) * q^73 + b6 * q^75 + (-2*b5 - 2*b4 - 3) * q^77 + (-b7 + b3 + 3*b1) * q^79 + (-b5 + 2*b4 + b2) * q^81 + (-4*b6 + 2*b1) * q^83 + (3*b4 + 4*b2 - 4) * q^85 + (-3*b6 + 3*b1) * q^87 + (2*b5 - 3*b2 - 6) * q^89 + (-4*b7 - 3*b3) * q^91 + (-4*b5 - 2*b2 - 4) * q^93 + (-2*b7 + b6 - b3 - b1) * q^95 + (6*b4 - b2 + 1) * q^97 + (3*b7 + 10*b6 + 3*b3 - 5*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^9 $$8 q - 8 q^{9} - 8 q^{13} - 12 q^{17} + 8 q^{25} - 12 q^{29} + 60 q^{33} + 12 q^{37} - 36 q^{41} - 48 q^{45} + 16 q^{49} + 48 q^{53} - 12 q^{61} + 28 q^{69} - 24 q^{77} - 4 q^{81} - 48 q^{85} - 36 q^{89} - 24 q^{93} + 12 q^{97}+O(q^{100})$$ 8 * q - 8 * q^9 - 8 * q^13 - 12 * q^17 + 8 * q^25 - 12 * q^29 + 60 * q^33 + 12 * q^37 - 36 * q^41 - 48 * q^45 + 16 * q^49 + 48 * q^53 - 12 * q^61 + 28 * q^69 - 24 * q^77 - 4 * q^81 - 48 * q^85 - 36 * q^89 - 24 * q^93 + 12 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{4} - 2\nu^{3} + 7\nu^{2} - 6\nu + 6$$ v^4 - 2*v^3 + 7*v^2 - 6*v + 6 $$\beta_{2}$$ $$=$$ $$( 8\nu^{7} - 28\nu^{6} + 114\nu^{5} - 215\nu^{4} + 378\nu^{3} - 366\nu^{2} + 266\nu - 97 ) / 37$$ (8*v^7 - 28*v^6 + 114*v^5 - 215*v^4 + 378*v^3 - 366*v^2 + 266*v - 97) / 37 $$\beta_{3}$$ $$=$$ $$( -24\nu^{7} + 47\nu^{6} - 194\nu^{5} + 127\nu^{4} - 172\nu^{3} - 234\nu^{2} + 238\nu - 153 ) / 37$$ (-24*v^7 + 47*v^6 - 194*v^5 + 127*v^4 - 172*v^3 - 234*v^2 + 238*v - 153) / 37 $$\beta_{4}$$ $$=$$ $$( -6\nu^{7} - 16\nu^{6} + 44\nu^{5} - 292\nu^{4} + 512\nu^{3} - 854\nu^{2} + 670\nu - 362 ) / 37$$ (-6*v^7 - 16*v^6 + 44*v^5 - 292*v^4 + 512*v^3 - 854*v^2 + 670*v - 362) / 37 $$\beta_{5}$$ $$=$$ $$( 6\nu^{7} - 58\nu^{6} + 178\nu^{5} - 522\nu^{4} + 746\nu^{3} - 996\nu^{2} + 588\nu - 304 ) / 37$$ (6*v^7 - 58*v^6 + 178*v^5 - 522*v^4 + 746*v^3 - 996*v^2 + 588*v - 304) / 37 $$\beta_{6}$$ $$=$$ $$( -18\nu^{7} + 63\nu^{6} - 238\nu^{5} + 456\nu^{4} - 758\nu^{3} + 805\nu^{2} - 654\nu + 283 ) / 37$$ (-18*v^7 + 63*v^6 - 238*v^5 + 456*v^4 - 758*v^3 + 805*v^2 - 654*v + 283) / 37 $$\beta_{7}$$ $$=$$ $$( -24\nu^{7} + 121\nu^{6} - 416\nu^{5} + 978\nu^{4} - 1504\nu^{3} + 1727\nu^{2} - 1094\nu + 365 ) / 37$$ (-24*v^7 + 121*v^6 - 416*v^5 + 978*v^4 - 1504*v^3 + 1727*v^2 - 1094*v + 365) / 37
 $$\nu$$ $$=$$ $$( \beta_{7} - 2\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 4$$ (b7 - 2*b6 + b5 - b4 + b3 + b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{6} - \beta_{4} + \beta_{3} + \beta _1 - 4 ) / 2$$ (-b6 - b4 + b3 + b1 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( -5\beta_{7} + 3\beta_{6} - 7\beta_{5} + 4\beta_{4} - 2\beta_{3} - 6\beta_{2} - 16 ) / 4$$ (-5*b7 + 3*b6 - 7*b5 + 4*b4 - 2*b3 - 6*b2 - 16) / 4 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 2\beta_{6} - 2\beta_{5} + 4\beta_{4} - 3\beta_{3} - 3\beta_{2} - \beta _1 + 3$$ -b7 + 2*b6 - 2*b5 + 4*b4 - 3*b3 - 3*b2 - b1 + 3 $$\nu^{5}$$ $$=$$ $$( 22\beta_{7} - 3\beta_{6} + 26\beta_{5} - \beta_{4} - 3\beta_{3} + 30\beta_{2} - \beta _1 + 82 ) / 4$$ (22*b7 - 3*b6 + 26*b5 - b4 - 3*b3 + 30*b2 - b1 + 82) / 4 $$\nu^{6}$$ $$=$$ $$( 21\beta_{7} - 15\beta_{6} + 31\beta_{5} - 40\beta_{4} + 28\beta_{3} + 60\beta_{2} + 4\beta _1 + 20 ) / 2$$ (21*b7 - 15*b6 + 31*b5 - 40*b4 + 28*b3 + 60*b2 + 4*b1 + 20) / 2 $$\nu^{7}$$ $$=$$ $$( -71\beta_{7} - 14\beta_{6} - 71\beta_{5} - 83\beta_{4} + 69\beta_{3} - 28\beta_{2} - 7\beta _1 - 334 ) / 4$$ (-71*b7 - 14*b6 - 71*b5 - 83*b4 + 69*b3 - 28*b2 - 7*b1 - 334) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$287$$ $$353$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 + \beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
225.1
 0.5 − 1.56488i 0.5 − 1.19293i 0.5 + 2.19293i 0.5 + 0.564882i 0.5 + 1.56488i 0.5 + 1.19293i 0.5 − 2.19293i 0.5 − 0.564882i
0 −1.45466 + 2.51954i 0 2.00000i 0 −0.675108 + 0.389774i 0 −2.73205 4.73205i 0
225.2 0 −0.619657 + 1.07328i 0 2.00000i 0 4.00552 2.31259i 0 0.732051 + 1.26795i 0
225.3 0 0.619657 1.07328i 0 2.00000i 0 −4.00552 + 2.31259i 0 0.732051 + 1.26795i 0
225.4 0 1.45466 2.51954i 0 2.00000i 0 0.675108 0.389774i 0 −2.73205 4.73205i 0
257.1 0 −1.45466 2.51954i 0 2.00000i 0 −0.675108 0.389774i 0 −2.73205 + 4.73205i 0
257.2 0 −0.619657 1.07328i 0 2.00000i 0 4.00552 + 2.31259i 0 0.732051 1.26795i 0
257.3 0 0.619657 + 1.07328i 0 2.00000i 0 −4.00552 2.31259i 0 0.732051 1.26795i 0
257.4 0 1.45466 + 2.51954i 0 2.00000i 0 0.675108 + 0.389774i 0 −2.73205 + 4.73205i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 225.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.e even 6 1 inner
52.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.w.c 8
4.b odd 2 1 inner 416.2.w.c 8
8.b even 2 1 832.2.w.h 8
8.d odd 2 1 832.2.w.h 8
13.e even 6 1 inner 416.2.w.c 8
13.f odd 12 1 5408.2.a.bg 4
13.f odd 12 1 5408.2.a.bk 4
52.i odd 6 1 inner 416.2.w.c 8
52.l even 12 1 5408.2.a.bg 4
52.l even 12 1 5408.2.a.bk 4
104.p odd 6 1 832.2.w.h 8
104.s even 6 1 832.2.w.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.w.c 8 1.a even 1 1 trivial
416.2.w.c 8 4.b odd 2 1 inner
416.2.w.c 8 13.e even 6 1 inner
416.2.w.c 8 52.i odd 6 1 inner
832.2.w.h 8 8.b even 2 1
832.2.w.h 8 8.d odd 2 1
832.2.w.h 8 104.p odd 6 1
832.2.w.h 8 104.s even 6 1
5408.2.a.bg 4 13.f odd 12 1
5408.2.a.bg 4 52.l even 12 1
5408.2.a.bk 4 13.f odd 12 1
5408.2.a.bk 4 52.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 10T_{3}^{6} + 87T_{3}^{4} + 130T_{3}^{2} + 169$$ acting on $$S_{2}^{\mathrm{new}}(416, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 10 T^{6} + \cdots + 169$$
$5$ $$(T^{2} + 4)^{4}$$
$7$ $$T^{8} - 22 T^{6} + \cdots + 169$$
$11$ $$T^{8} - 30 T^{6} + \cdots + 13689$$
$13$ $$(T^{2} + 2 T + 13)^{4}$$
$17$ $$(T^{4} + 6 T^{3} + 39 T^{2} + \cdots + 9)^{2}$$
$19$ $$T^{8} - 22 T^{6} + \cdots + 169$$
$23$ $$T^{8} + 82 T^{6} + \cdots + 2474329$$
$29$ $$(T^{2} + 3 T + 9)^{4}$$
$31$ $$(T^{4} + 88 T^{2} + 208)^{2}$$
$37$ $$(T^{4} - 6 T^{3} + \cdots + 1089)^{2}$$
$41$ $$(T^{4} + 18 T^{3} + \cdots + 529)^{2}$$
$43$ $$T^{8} + 90 T^{6} + \cdots + 1108809$$
$47$ $$(T^{4} + 192 T^{2} + 7488)^{2}$$
$53$ $$(T^{2} - 12 T + 24)^{4}$$
$59$ $$T^{8} - 246 T^{6} + \cdots + 200420649$$
$61$ $$(T^{2} + 3 T + 9)^{4}$$
$67$ $$T^{8} - 22 T^{6} + \cdots + 169$$
$71$ $$T^{8} - 270 T^{6} + \cdots + 89813529$$
$73$ $$(T^{2} + 36)^{4}$$
$79$ $$(T^{4} - 120 T^{2} + 1872)^{2}$$
$83$ $$(T^{4} + 120 T^{2} + 1872)^{2}$$
$89$ $$(T^{4} + 18 T^{3} + \cdots + 121)^{2}$$
$97$ $$(T^{4} - 6 T^{3} + \cdots + 19881)^{2}$$