# Properties

 Label 7400.2.a.g Level $7400$ Weight $2$ Character orbit 7400.a Self dual yes Analytic conductor $59.089$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7400,2,Mod(1,7400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7400 = 2^{3} \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7400.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: $$59.0892974957$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 296) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + 3 q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + 3 * q^7 - 2 * q^9 $$q + q^{3} + 3 q^{7} - 2 q^{9} - 3 q^{11} - 2 q^{17} - 2 q^{19} + 3 q^{21} + 6 q^{23} - 5 q^{27} - 2 q^{29} - 4 q^{31} - 3 q^{33} - q^{37} + 7 q^{41} - 4 q^{43} - q^{47} + 2 q^{49} - 2 q^{51} - 9 q^{53} - 2 q^{57} + 8 q^{59} - 4 q^{61} - 6 q^{63} - 12 q^{67} + 6 q^{69} - 5 q^{71} + 13 q^{73} - 9 q^{77} - 10 q^{79} + q^{81} + q^{83} - 2 q^{87} - 2 q^{89} - 4 q^{93} + 12 q^{97} + 6 q^{99}+O(q^{100})$$ q + q^3 + 3 * q^7 - 2 * q^9 - 3 * q^11 - 2 * q^17 - 2 * q^19 + 3 * q^21 + 6 * q^23 - 5 * q^27 - 2 * q^29 - 4 * q^31 - 3 * q^33 - q^37 + 7 * q^41 - 4 * q^43 - q^47 + 2 * q^49 - 2 * q^51 - 9 * q^53 - 2 * q^57 + 8 * q^59 - 4 * q^61 - 6 * q^63 - 12 * q^67 + 6 * q^69 - 5 * q^71 + 13 * q^73 - 9 * q^77 - 10 * q^79 + q^81 + q^83 - 2 * q^87 - 2 * q^89 - 4 * q^93 + 12 * q^97 + 6 * q^99

## 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
0 1.00000 0 0 0 3.00000 0 −2.00000 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
$$2$$ $$+1$$
$$5$$ $$+1$$
$$37$$ $$+1$$

## 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 7400.2.a.g 1
5.b even 2 1 296.2.a.b 1
15.d odd 2 1 2664.2.a.c 1
20.d odd 2 1 592.2.a.d 1
40.e odd 2 1 2368.2.a.f 1
40.f even 2 1 2368.2.a.k 1
60.h even 2 1 5328.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.a.b 1 5.b even 2 1
592.2.a.d 1 20.d odd 2 1
2368.2.a.f 1 40.e odd 2 1
2368.2.a.k 1 40.f even 2 1
2664.2.a.c 1 15.d odd 2 1
5328.2.a.m 1 60.h even 2 1
7400.2.a.g 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7400))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 3$$ T7 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 3$$
$11$ $$T + 3$$
$13$ $$T$$
$17$ $$T + 2$$
$19$ $$T + 2$$
$23$ $$T - 6$$
$29$ $$T + 2$$
$31$ $$T + 4$$
$37$ $$T + 1$$
$41$ $$T - 7$$
$43$ $$T + 4$$
$47$ $$T + 1$$
$53$ $$T + 9$$
$59$ $$T - 8$$
$61$ $$T + 4$$
$67$ $$T + 12$$
$71$ $$T + 5$$
$73$ $$T - 13$$
$79$ $$T + 10$$
$83$ $$T - 1$$
$89$ $$T + 2$$
$97$ $$T - 12$$