# Properties

 Label 684.4.a.g Level $684$ Weight $4$ Character orbit 684.a Self dual yes Analytic conductor $40.357$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,4,Mod(1,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("684.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 684.a (trivial)

## 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: yes Analytic conductor: $$40.3573064439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{5} + (4 \beta - 17) q^{7}+O(q^{10})$$ q + 5*b * q^5 + (4*b - 17) * q^7 $$q + 5 \beta q^{5} + (4 \beta - 17) q^{7} + ( - 5 \beta + 38) q^{11} + (11 \beta - 23) q^{13} + (6 \beta - 3) q^{17} - 19 q^{19} + (59 \beta - 27) q^{23} + (25 \beta + 75) q^{25} + ( - 65 \beta - 45) q^{29} + (80 \beta - 84) q^{31} + ( - 65 \beta + 160) q^{35} + (8 \beta + 186) q^{37} + (30 \beta + 56) q^{41} + ( - 117 \beta + 136) q^{43} + (23 \beta + 216) q^{47} + ( - 120 \beta + 74) q^{49} + ( - 123 \beta + 199) q^{53} + (165 \beta - 200) q^{55} + (35 \beta + 419) q^{59} + ( - 175 \beta + 310) q^{61} + ( - 60 \beta + 440) q^{65} + ( - 61 \beta + 353) q^{67} + (20 \beta + 846) q^{71} + ( - 64 \beta - 463) q^{73} + (217 \beta - 806) q^{77} + (10 \beta - 642) q^{79} + ( - 114 \beta + 102) q^{83} + (15 \beta + 240) q^{85} + ( - 80 \beta + 484) q^{89} + ( - 235 \beta + 743) q^{91} - 95 \beta q^{95} + (458 \beta + 126) q^{97} +O(q^{100})$$ q + 5*b * q^5 + (4*b - 17) * q^7 + (-5*b + 38) * q^11 + (11*b - 23) * q^13 + (6*b - 3) * q^17 - 19 * q^19 + (59*b - 27) * q^23 + (25*b + 75) * q^25 + (-65*b - 45) * q^29 + (80*b - 84) * q^31 + (-65*b + 160) * q^35 + (8*b + 186) * q^37 + (30*b + 56) * q^41 + (-117*b + 136) * q^43 + (23*b + 216) * q^47 + (-120*b + 74) * q^49 + (-123*b + 199) * q^53 + (165*b - 200) * q^55 + (35*b + 419) * q^59 + (-175*b + 310) * q^61 + (-60*b + 440) * q^65 + (-61*b + 353) * q^67 + (20*b + 846) * q^71 + (-64*b - 463) * q^73 + (217*b - 806) * q^77 + (10*b - 642) * q^79 + (-114*b + 102) * q^83 + (15*b + 240) * q^85 + (-80*b + 484) * q^89 + (-235*b + 743) * q^91 - 95*b * q^95 + (458*b + 126) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{5} - 30 q^{7}+O(q^{10})$$ 2 * q + 5 * q^5 - 30 * q^7 $$2 q + 5 q^{5} - 30 q^{7} + 71 q^{11} - 35 q^{13} - 38 q^{19} + 5 q^{23} + 175 q^{25} - 155 q^{29} - 88 q^{31} + 255 q^{35} + 380 q^{37} + 142 q^{41} + 155 q^{43} + 455 q^{47} + 28 q^{49} + 275 q^{53} - 235 q^{55} + 873 q^{59} + 445 q^{61} + 820 q^{65} + 645 q^{67} + 1712 q^{71} - 990 q^{73} - 1395 q^{77} - 1274 q^{79} + 90 q^{83} + 495 q^{85} + 888 q^{89} + 1251 q^{91} - 95 q^{95} + 710 q^{97}+O(q^{100})$$ 2 * q + 5 * q^5 - 30 * q^7 + 71 * q^11 - 35 * q^13 - 38 * q^19 + 5 * q^23 + 175 * q^25 - 155 * q^29 - 88 * q^31 + 255 * q^35 + 380 * q^37 + 142 * q^41 + 155 * q^43 + 455 * q^47 + 28 * q^49 + 275 * q^53 - 235 * q^55 + 873 * q^59 + 445 * q^61 + 820 * q^65 + 645 * q^67 + 1712 * q^71 - 990 * q^73 - 1395 * q^77 - 1274 * q^79 + 90 * q^83 + 495 * q^85 + 888 * q^89 + 1251 * q^91 - 95 * q^95 + 710 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.37228 3.37228
0 0 0 −11.8614 0 −26.4891 0 0 0
1.2 0 0 0 16.8614 0 −3.51087 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.4.a.g 2
3.b odd 2 1 76.4.a.a 2
12.b even 2 1 304.4.a.f 2
15.d odd 2 1 1900.4.a.b 2
15.e even 4 2 1900.4.c.b 4
24.f even 2 1 1216.4.a.h 2
24.h odd 2 1 1216.4.a.o 2
57.d even 2 1 1444.4.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.a 2 3.b odd 2 1
304.4.a.f 2 12.b even 2 1
684.4.a.g 2 1.a even 1 1 trivial
1216.4.a.h 2 24.f even 2 1
1216.4.a.o 2 24.h odd 2 1
1444.4.a.d 2 57.d even 2 1
1900.4.a.b 2 15.d odd 2 1
1900.4.c.b 4 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 5T_{5} - 200$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(684))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 5T - 200$$
$7$ $$T^{2} + 30T + 93$$
$11$ $$T^{2} - 71T + 1054$$
$13$ $$T^{2} + 35T - 692$$
$17$ $$T^{2} - 297$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} - 5T - 28712$$
$29$ $$T^{2} + 155T - 28850$$
$31$ $$T^{2} + 88T - 50864$$
$37$ $$T^{2} - 380T + 35572$$
$41$ $$T^{2} - 142T - 2384$$
$43$ $$T^{2} - 155T - 106928$$
$47$ $$T^{2} - 455T + 47392$$
$53$ $$T^{2} - 275T - 105908$$
$59$ $$T^{2} - 873T + 180426$$
$61$ $$T^{2} - 445T - 203150$$
$67$ $$T^{2} - 645T + 73308$$
$71$ $$T^{2} - 1712 T + 729436$$
$73$ $$T^{2} + 990T + 211233$$
$79$ $$T^{2} + 1274 T + 404944$$
$83$ $$T^{2} - 90T - 105192$$
$89$ $$T^{2} - 888T + 144336$$
$97$ $$T^{2} - 710 T - 1604528$$