# Properties

 Label 7623.2.a.g Level $7623$ Weight $2$ Character orbit 7623.a Self dual yes Analytic conductor $60.870$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7623 = 3^{2} \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7623.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: $$60.8699614608$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + 2 * q^5 + q^7 + 3 * q^8 $$q - q^{2} - q^{4} + 2 q^{5} + q^{7} + 3 q^{8} - 2 q^{10} + 2 q^{13} - q^{14} - q^{16} - 6 q^{17} - 4 q^{19} - 2 q^{20} - q^{25} - 2 q^{26} - q^{28} - 2 q^{29} - 5 q^{32} + 6 q^{34} + 2 q^{35} + 6 q^{37} + 4 q^{38} + 6 q^{40} + 2 q^{41} + 4 q^{43} + q^{49} + q^{50} - 2 q^{52} - 6 q^{53} + 3 q^{56} + 2 q^{58} - 12 q^{59} + 2 q^{61} + 7 q^{64} + 4 q^{65} + 4 q^{67} + 6 q^{68} - 2 q^{70} + 6 q^{73} - 6 q^{74} + 4 q^{76} + 16 q^{79} - 2 q^{80} - 2 q^{82} - 12 q^{83} - 12 q^{85} - 4 q^{86} + 14 q^{89} + 2 q^{91} - 8 q^{95} + 18 q^{97} - q^{98}+O(q^{100})$$ q - q^2 - q^4 + 2 * q^5 + q^7 + 3 * q^8 - 2 * q^10 + 2 * q^13 - q^14 - q^16 - 6 * q^17 - 4 * q^19 - 2 * q^20 - q^25 - 2 * q^26 - q^28 - 2 * q^29 - 5 * q^32 + 6 * q^34 + 2 * q^35 + 6 * q^37 + 4 * q^38 + 6 * q^40 + 2 * q^41 + 4 * q^43 + q^49 + q^50 - 2 * q^52 - 6 * q^53 + 3 * q^56 + 2 * q^58 - 12 * q^59 + 2 * q^61 + 7 * q^64 + 4 * q^65 + 4 * q^67 + 6 * q^68 - 2 * q^70 + 6 * q^73 - 6 * q^74 + 4 * q^76 + 16 * q^79 - 2 * q^80 - 2 * q^82 - 12 * q^83 - 12 * q^85 - 4 * q^86 + 14 * q^89 + 2 * q^91 - 8 * q^95 + 18 * q^97 - q^98

## 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
 0
−1.00000 0 −1.00000 2.00000 0 1.00000 3.00000 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 7623.2.a.g 1
3.b odd 2 1 2541.2.a.j 1
11.b odd 2 1 63.2.a.a 1
33.d even 2 1 21.2.a.a 1
44.c even 2 1 1008.2.a.l 1
55.d odd 2 1 1575.2.a.c 1
55.e even 4 2 1575.2.d.a 2
77.b even 2 1 441.2.a.f 1
77.h odd 6 2 441.2.e.a 2
77.i even 6 2 441.2.e.b 2
88.b odd 2 1 4032.2.a.h 1
88.g even 2 1 4032.2.a.k 1
99.g even 6 2 567.2.f.g 2
99.h odd 6 2 567.2.f.b 2
132.d odd 2 1 336.2.a.a 1
165.d even 2 1 525.2.a.d 1
165.l odd 4 2 525.2.d.a 2
231.h odd 2 1 147.2.a.a 1
231.k odd 6 2 147.2.e.c 2
231.l even 6 2 147.2.e.b 2
264.m even 2 1 1344.2.a.g 1
264.p odd 2 1 1344.2.a.s 1
308.g odd 2 1 7056.2.a.p 1
429.e even 2 1 3549.2.a.c 1
528.s odd 4 2 5376.2.c.l 2
528.x even 4 2 5376.2.c.r 2
561.h even 2 1 6069.2.a.b 1
627.b odd 2 1 7581.2.a.d 1
660.g odd 2 1 8400.2.a.bn 1
924.n even 2 1 2352.2.a.v 1
924.y even 6 2 2352.2.q.e 2
924.z odd 6 2 2352.2.q.x 2
1155.e odd 2 1 3675.2.a.n 1
1848.b odd 2 1 9408.2.a.bv 1
1848.e even 2 1 9408.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 33.d even 2 1
63.2.a.a 1 11.b odd 2 1
147.2.a.a 1 231.h odd 2 1
147.2.e.b 2 231.l even 6 2
147.2.e.c 2 231.k odd 6 2
336.2.a.a 1 132.d odd 2 1
441.2.a.f 1 77.b even 2 1
441.2.e.a 2 77.h odd 6 2
441.2.e.b 2 77.i even 6 2
525.2.a.d 1 165.d even 2 1
525.2.d.a 2 165.l odd 4 2
567.2.f.b 2 99.h odd 6 2
567.2.f.g 2 99.g even 6 2
1008.2.a.l 1 44.c even 2 1
1344.2.a.g 1 264.m even 2 1
1344.2.a.s 1 264.p odd 2 1
1575.2.a.c 1 55.d odd 2 1
1575.2.d.a 2 55.e even 4 2
2352.2.a.v 1 924.n even 2 1
2352.2.q.e 2 924.y even 6 2
2352.2.q.x 2 924.z odd 6 2
2541.2.a.j 1 3.b odd 2 1
3549.2.a.c 1 429.e even 2 1
3675.2.a.n 1 1155.e odd 2 1
4032.2.a.h 1 88.b odd 2 1
4032.2.a.k 1 88.g even 2 1
5376.2.c.l 2 528.s odd 4 2
5376.2.c.r 2 528.x even 4 2
6069.2.a.b 1 561.h even 2 1
7056.2.a.p 1 308.g odd 2 1
7581.2.a.d 1 627.b odd 2 1
7623.2.a.g 1 1.a even 1 1 trivial
8400.2.a.bn 1 660.g odd 2 1
9408.2.a.m 1 1848.e even 2 1
9408.2.a.bv 1 1848.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7623))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 2$$ T5 - 2 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T - 2$$
$43$ $$T - 4$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T - 2$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T - 6$$
$79$ $$T - 16$$
$83$ $$T + 12$$
$89$ $$T - 14$$
$97$ $$T - 18$$
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