# Properties

 Label 3136.2.a.e Level $3136$ Weight $2$ Character orbit 3136.a Self dual yes Analytic conductor $25.041$ Analytic rank $0$ 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] = [3136,2,Mod(1,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, 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("3136.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3136 = 2^{6} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3136.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: $$25.0410860739$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{9}+O(q^{10})$$ q - 2 * q^3 + q^9 $$q - 2 q^{3} + q^{9} - 4 q^{13} - 6 q^{17} + 2 q^{19} - 5 q^{25} + 4 q^{27} + 6 q^{29} + 4 q^{31} - 2 q^{37} + 8 q^{39} - 6 q^{41} - 8 q^{43} + 12 q^{47} + 12 q^{51} - 6 q^{53} - 4 q^{57} - 6 q^{59} + 8 q^{61} + 4 q^{67} - 2 q^{73} + 10 q^{75} + 8 q^{79} - 11 q^{81} - 6 q^{83} - 12 q^{87} + 6 q^{89} - 8 q^{93} + 10 q^{97}+O(q^{100})$$ q - 2 * q^3 + q^9 - 4 * q^13 - 6 * q^17 + 2 * q^19 - 5 * q^25 + 4 * q^27 + 6 * q^29 + 4 * q^31 - 2 * q^37 + 8 * q^39 - 6 * q^41 - 8 * q^43 + 12 * q^47 + 12 * q^51 - 6 * q^53 - 4 * q^57 - 6 * q^59 + 8 * q^61 + 4 * q^67 - 2 * q^73 + 10 * q^75 + 8 * q^79 - 11 * q^81 - 6 * q^83 - 12 * q^87 + 6 * q^89 - 8 * q^93 + 10 * 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
 0
0 −2.00000 0 0 0 0 0 1.00000 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
$$2$$ $$1$$
$$7$$ $$-1$$

## 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 3136.2.a.e 1
4.b odd 2 1 3136.2.a.z 1
7.b odd 2 1 448.2.a.g 1
8.b even 2 1 98.2.a.a 1
8.d odd 2 1 784.2.a.b 1
21.c even 2 1 4032.2.a.w 1
24.f even 2 1 7056.2.a.bd 1
24.h odd 2 1 882.2.a.i 1
28.d even 2 1 448.2.a.a 1
40.f even 2 1 2450.2.a.t 1
40.i odd 4 2 2450.2.c.c 2
56.e even 2 1 112.2.a.c 1
56.h odd 2 1 14.2.a.a 1
56.j odd 6 2 98.2.c.b 2
56.k odd 6 2 784.2.i.i 2
56.m even 6 2 784.2.i.c 2
56.p even 6 2 98.2.c.a 2
84.h odd 2 1 4032.2.a.r 1
112.j even 4 2 1792.2.b.g 2
112.l odd 4 2 1792.2.b.c 2
168.e odd 2 1 1008.2.a.h 1
168.i even 2 1 126.2.a.b 1
168.s odd 6 2 882.2.g.d 2
168.ba even 6 2 882.2.g.c 2
280.c odd 2 1 350.2.a.f 1
280.n even 2 1 2800.2.a.g 1
280.s even 4 2 350.2.c.d 2
280.y odd 4 2 2800.2.g.h 2
504.bn odd 6 2 1134.2.f.l 2
504.cc even 6 2 1134.2.f.f 2
616.o even 2 1 1694.2.a.e 1
728.l odd 2 1 2366.2.a.j 1
728.ba even 4 2 2366.2.d.b 2
840.u even 2 1 3150.2.a.i 1
840.bp odd 4 2 3150.2.g.j 2
952.e odd 2 1 4046.2.a.f 1
1064.f even 2 1 5054.2.a.c 1
1288.m even 2 1 7406.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 56.h odd 2 1
98.2.a.a 1 8.b even 2 1
98.2.c.a 2 56.p even 6 2
98.2.c.b 2 56.j odd 6 2
112.2.a.c 1 56.e even 2 1
126.2.a.b 1 168.i even 2 1
350.2.a.f 1 280.c odd 2 1
350.2.c.d 2 280.s even 4 2
448.2.a.a 1 28.d even 2 1
448.2.a.g 1 7.b odd 2 1
784.2.a.b 1 8.d odd 2 1
784.2.i.c 2 56.m even 6 2
784.2.i.i 2 56.k odd 6 2
882.2.a.i 1 24.h odd 2 1
882.2.g.c 2 168.ba even 6 2
882.2.g.d 2 168.s odd 6 2
1008.2.a.h 1 168.e odd 2 1
1134.2.f.f 2 504.cc even 6 2
1134.2.f.l 2 504.bn odd 6 2
1694.2.a.e 1 616.o even 2 1
1792.2.b.c 2 112.l odd 4 2
1792.2.b.g 2 112.j even 4 2
2366.2.a.j 1 728.l odd 2 1
2366.2.d.b 2 728.ba even 4 2
2450.2.a.t 1 40.f even 2 1
2450.2.c.c 2 40.i odd 4 2
2800.2.a.g 1 280.n even 2 1
2800.2.g.h 2 280.y odd 4 2
3136.2.a.e 1 1.a even 1 1 trivial
3136.2.a.z 1 4.b odd 2 1
3150.2.a.i 1 840.u even 2 1
3150.2.g.j 2 840.bp odd 4 2
4032.2.a.r 1 84.h odd 2 1
4032.2.a.w 1 21.c even 2 1
4046.2.a.f 1 952.e odd 2 1
5054.2.a.c 1 1064.f even 2 1
7056.2.a.bd 1 24.f even 2 1
7406.2.a.a 1 1288.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3136))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{5}$$ T5 $$T_{11}$$ T11

## Hecke characteristic polynomials

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