# Properties

 Label 4368.2.h.f Level $4368$ Weight $2$ Character orbit 4368.h Analytic conductor $34.879$ 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] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4368.h (of order $$2$$, degree $$1$$, 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: $$34.8786556029$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + 2 i q^{5} - i q^{7} + q^{9} +O(q^{10})$$ q - q^3 + 2*i * q^5 - i * q^7 + q^9 $$q - q^{3} + 2 i q^{5} - i q^{7} + q^{9} + (3 i + 2) q^{13} - 2 i q^{15} - 2 q^{17} - 4 i q^{19} + i q^{21} + 6 q^{23} + q^{25} - q^{27} + 2 q^{35} - 2 i q^{37} + ( - 3 i - 2) q^{39} - 4 q^{43} + 2 i q^{45} - 8 i q^{47} - q^{49} + 2 q^{51} + 4 q^{53} + 4 i q^{57} + 6 i q^{59} + 12 q^{61} - i q^{63} + (4 i - 6) q^{65} + 2 i q^{67} - 6 q^{69} - 14 i q^{73} - q^{75} + q^{81} + 14 i q^{83} - 4 i q^{85} + 4 i q^{89} + ( - 2 i + 3) q^{91} + 8 q^{95} - 2 i q^{97} +O(q^{100})$$ q - q^3 + 2*i * q^5 - i * q^7 + q^9 + (3*i + 2) * q^13 - 2*i * q^15 - 2 * q^17 - 4*i * q^19 + i * q^21 + 6 * q^23 + q^25 - q^27 + 2 * q^35 - 2*i * q^37 + (-3*i - 2) * q^39 - 4 * q^43 + 2*i * q^45 - 8*i * q^47 - q^49 + 2 * q^51 + 4 * q^53 + 4*i * q^57 + 6*i * q^59 + 12 * q^61 - i * q^63 + (4*i - 6) * q^65 + 2*i * q^67 - 6 * q^69 - 14*i * q^73 - q^75 + q^81 + 14*i * q^83 - 4*i * q^85 + 4*i * q^89 + (-2*i + 3) * q^91 + 8 * q^95 - 2*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 4 q^{17} + 12 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{35} - 4 q^{39} - 8 q^{43} - 2 q^{49} + 4 q^{51} + 8 q^{53} + 24 q^{61} - 12 q^{65} - 12 q^{69} - 2 q^{75} + 2 q^{81} + 6 q^{91} + 16 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 4 * q^17 + 12 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^35 - 4 * q^39 - 8 * q^43 - 2 * q^49 + 4 * q^51 + 8 * q^53 + 24 * q^61 - 12 * q^65 - 12 * q^69 - 2 * q^75 + 2 * q^81 + 6 * q^91 + 16 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$$.

 $$n$$ $$1093$$ $$1249$$ $$1457$$ $$2017$$ $$3823$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.00000i 1.00000i
0 −1.00000 0 2.00000i 0 1.00000i 0 1.00000 0
337.2 0 −1.00000 0 2.00000i 0 1.00000i 0 1.00000 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.f 2
4.b odd 2 1 546.2.c.b 2
12.b even 2 1 1638.2.c.b 2
13.b even 2 1 inner 4368.2.h.f 2
28.d even 2 1 3822.2.c.c 2
52.b odd 2 1 546.2.c.b 2
52.f even 4 1 7098.2.a.k 1
52.f even 4 1 7098.2.a.bc 1
156.h even 2 1 1638.2.c.b 2
364.h even 2 1 3822.2.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.b 2 4.b odd 2 1
546.2.c.b 2 52.b odd 2 1
1638.2.c.b 2 12.b even 2 1
1638.2.c.b 2 156.h even 2 1
3822.2.c.c 2 28.d even 2 1
3822.2.c.c 2 364.h even 2 1
4368.2.h.f 2 1.a even 1 1 trivial
4368.2.h.f 2 13.b even 2 1 inner
7098.2.a.k 1 52.f even 4 1
7098.2.a.bc 1 52.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4368, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{11}$$ T11 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 4T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T - 4)^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$T^{2} + 16$$
$97$ $$T^{2} + 4$$