# Properties

 Label 612.2.b.a Level $612$ Weight $2$ Character orbit 612.b Analytic conductor $4.887$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [612,2,Mod(577,612)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(612, base_ring=CyclotomicField(2))

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("612.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$612 = 2^{2} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 612.b (of order $$2$$, degree $$1$$, 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: $$4.88684460370$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} + 3 \beta q^{7}+O(q^{10})$$ q + 2*b * q^5 + 3*b * q^7 $$q + 2 \beta q^{5} + 3 \beta q^{7} - \beta q^{11} - 4 q^{13} + ( - 2 \beta - 3) q^{17} - 4 q^{19} - \beta q^{23} - 3 q^{25} + 2 \beta q^{29} - 3 \beta q^{31} - 12 q^{35} + 6 \beta q^{37} + 8 \beta q^{41} + 8 q^{43} + 12 q^{47} - 11 q^{49} + 6 q^{53} + 4 q^{55} + 6 \beta q^{61} - 8 \beta q^{65} - 4 q^{67} + 5 \beta q^{71} + 6 q^{77} - 3 \beta q^{79} + ( - 6 \beta + 8) q^{85} - 12 q^{89} - 12 \beta q^{91} - 8 \beta q^{95} +O(q^{100})$$ q + 2*b * q^5 + 3*b * q^7 - b * q^11 - 4 * q^13 + (-2*b - 3) * q^17 - 4 * q^19 - b * q^23 - 3 * q^25 + 2*b * q^29 - 3*b * q^31 - 12 * q^35 + 6*b * q^37 + 8*b * q^41 + 8 * q^43 + 12 * q^47 - 11 * q^49 + 6 * q^53 + 4 * q^55 + 6*b * q^61 - 8*b * q^65 - 4 * q^67 + 5*b * q^71 + 6 * q^77 - 3*b * q^79 + (-6*b + 8) * q^85 - 12 * q^89 - 12*b * q^91 - 8*b * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{13} - 6 q^{17} - 8 q^{19} - 6 q^{25} - 24 q^{35} + 16 q^{43} + 24 q^{47} - 22 q^{49} + 12 q^{53} + 8 q^{55} - 8 q^{67} + 12 q^{77} + 16 q^{85} - 24 q^{89}+O(q^{100})$$ 2 * q - 8 * q^13 - 6 * q^17 - 8 * q^19 - 6 * q^25 - 24 * q^35 + 16 * q^43 + 24 * q^47 - 22 * q^49 + 12 * q^53 + 8 * q^55 - 8 * q^67 + 12 * q^77 + 16 * q^85 - 24 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/612\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$137$$ $$307$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 − 1.41421i 1.41421i
0 0 0 2.82843i 0 4.24264i 0 0 0
577.2 0 0 0 2.82843i 0 4.24264i 0 0 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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 612.2.b.a 2
3.b odd 2 1 68.2.b.a 2
4.b odd 2 1 2448.2.c.d 2
12.b even 2 1 272.2.b.c 2
15.d odd 2 1 1700.2.c.a 2
15.e even 4 2 1700.2.g.a 4
17.b even 2 1 inner 612.2.b.a 2
21.c even 2 1 3332.2.b.a 2
24.f even 2 1 1088.2.b.f 2
24.h odd 2 1 1088.2.b.e 2
51.c odd 2 1 68.2.b.a 2
51.f odd 4 2 1156.2.a.c 2
51.g odd 8 2 1156.2.e.a 2
51.g odd 8 2 1156.2.e.b 2
51.i even 16 8 1156.2.h.d 8
68.d odd 2 1 2448.2.c.d 2
204.h even 2 1 272.2.b.c 2
204.l even 4 2 4624.2.a.n 2
255.h odd 2 1 1700.2.c.a 2
255.o even 4 2 1700.2.g.a 4
357.c even 2 1 3332.2.b.a 2
408.b odd 2 1 1088.2.b.e 2
408.h even 2 1 1088.2.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 3.b odd 2 1
68.2.b.a 2 51.c odd 2 1
272.2.b.c 2 12.b even 2 1
272.2.b.c 2 204.h even 2 1
612.2.b.a 2 1.a even 1 1 trivial
612.2.b.a 2 17.b even 2 1 inner
1088.2.b.e 2 24.h odd 2 1
1088.2.b.e 2 408.b odd 2 1
1088.2.b.f 2 24.f even 2 1
1088.2.b.f 2 408.h even 2 1
1156.2.a.c 2 51.f odd 4 2
1156.2.e.a 2 51.g odd 8 2
1156.2.e.b 2 51.g odd 8 2
1156.2.h.d 8 51.i even 16 8
1700.2.c.a 2 15.d odd 2 1
1700.2.c.a 2 255.h odd 2 1
1700.2.g.a 4 15.e even 4 2
1700.2.g.a 4 255.o even 4 2
2448.2.c.d 2 4.b odd 2 1
2448.2.c.d 2 68.d odd 2 1
3332.2.b.a 2 21.c even 2 1
3332.2.b.a 2 357.c even 2 1
4624.2.a.n 2 204.l even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(612, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 8$$
$7$ $$T^{2} + 18$$
$11$ $$T^{2} + 2$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2} + 6T + 17$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 2$$
$29$ $$T^{2} + 8$$
$31$ $$T^{2} + 18$$
$37$ $$T^{2} + 72$$
$41$ $$T^{2} + 128$$
$43$ $$(T - 8)^{2}$$
$47$ $$(T - 12)^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 72$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 50$$
$73$ $$T^{2}$$
$79$ $$T^{2} + 18$$
$83$ $$T^{2}$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2}$$