# Properties

 Label 3150.2.a.bt Level $3150$ Weight $2$ Character orbit 3150.a Self dual yes Analytic conductor $25.153$ 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] = [3150,2,Mod(1,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3150.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: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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 $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^7 + q^8 $$q + q^{2} + q^{4} - q^{7} + q^{8} + 2 \beta q^{11} + (\beta + 2) q^{13} - q^{14} + q^{16} - 2 q^{17} + ( - \beta + 4) q^{19} + 2 \beta q^{22} + ( - 2 \beta + 2) q^{23} + (\beta + 2) q^{26} - q^{28} + ( - 2 \beta - 2) q^{29} + (2 \beta + 4) q^{31} + q^{32} - 2 q^{34} + 2 q^{37} + ( - \beta + 4) q^{38} + ( - 2 \beta + 6) q^{41} + ( - 2 \beta + 4) q^{43} + 2 \beta q^{44} + ( - 2 \beta + 2) q^{46} + ( - 2 \beta - 4) q^{47} + q^{49} + (\beta + 2) q^{52} + (2 \beta + 6) q^{53} - q^{56} + ( - 2 \beta - 2) q^{58} + ( - \beta + 4) q^{59} + ( - \beta + 6) q^{61} + (2 \beta + 4) q^{62} + q^{64} - 8 q^{67} - 2 q^{68} + ( - 2 \beta + 6) q^{71} + (2 \beta - 2) q^{73} + 2 q^{74} + ( - \beta + 4) q^{76} - 2 \beta q^{77} + (2 \beta + 2) q^{79} + ( - 2 \beta + 6) q^{82} + \beta q^{83} + ( - 2 \beta + 4) q^{86} + 2 \beta q^{88} + 10 q^{89} + ( - \beta - 2) q^{91} + ( - 2 \beta + 2) q^{92} + ( - 2 \beta - 4) q^{94} + (4 \beta + 6) q^{97} + q^{98} +O(q^{100})$$ q + q^2 + q^4 - q^7 + q^8 + 2*b * q^11 + (b + 2) * q^13 - q^14 + q^16 - 2 * q^17 + (-b + 4) * q^19 + 2*b * q^22 + (-2*b + 2) * q^23 + (b + 2) * q^26 - q^28 + (-2*b - 2) * q^29 + (2*b + 4) * q^31 + q^32 - 2 * q^34 + 2 * q^37 + (-b + 4) * q^38 + (-2*b + 6) * q^41 + (-2*b + 4) * q^43 + 2*b * q^44 + (-2*b + 2) * q^46 + (-2*b - 4) * q^47 + q^49 + (b + 2) * q^52 + (2*b + 6) * q^53 - q^56 + (-2*b - 2) * q^58 + (-b + 4) * q^59 + (-b + 6) * q^61 + (2*b + 4) * q^62 + q^64 - 8 * q^67 - 2 * q^68 + (-2*b + 6) * q^71 + (2*b - 2) * q^73 + 2 * q^74 + (-b + 4) * q^76 - 2*b * q^77 + (2*b + 2) * q^79 + (-2*b + 6) * q^82 + b * q^83 + (-2*b + 4) * q^86 + 2*b * q^88 + 10 * q^89 + (-b - 2) * q^91 + (-2*b + 2) * q^92 + (-2*b - 4) * q^94 + (4*b + 6) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{19} + 4 q^{23} + 4 q^{26} - 2 q^{28} - 4 q^{29} + 8 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{37} + 8 q^{38} + 12 q^{41} + 8 q^{43} + 4 q^{46} - 8 q^{47} + 2 q^{49} + 4 q^{52} + 12 q^{53} - 2 q^{56} - 4 q^{58} + 8 q^{59} + 12 q^{61} + 8 q^{62} + 2 q^{64} - 16 q^{67} - 4 q^{68} + 12 q^{71} - 4 q^{73} + 4 q^{74} + 8 q^{76} + 4 q^{79} + 12 q^{82} + 8 q^{86} + 20 q^{89} - 4 q^{91} + 4 q^{92} - 8 q^{94} + 12 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^19 + 4 * q^23 + 4 * q^26 - 2 * q^28 - 4 * q^29 + 8 * q^31 + 2 * q^32 - 4 * q^34 + 4 * q^37 + 8 * q^38 + 12 * q^41 + 8 * q^43 + 4 * q^46 - 8 * q^47 + 2 * q^49 + 4 * q^52 + 12 * q^53 - 2 * q^56 - 4 * q^58 + 8 * q^59 + 12 * q^61 + 8 * q^62 + 2 * q^64 - 16 * q^67 - 4 * q^68 + 12 * q^71 - 4 * q^73 + 4 * q^74 + 8 * q^76 + 4 * q^79 + 12 * q^82 + 8 * q^86 + 20 * q^89 - 4 * q^91 + 4 * q^92 - 8 * q^94 + 12 * q^97 + 2 * 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
 −2.44949 2.44949
1.00000 0 1.00000 0 0 −1.00000 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 −1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$+1$$

## 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 3150.2.a.bt 2
3.b odd 2 1 350.2.a.g 2
5.b even 2 1 3150.2.a.bs 2
5.c odd 4 2 630.2.g.g 4
12.b even 2 1 2800.2.a.bm 2
15.d odd 2 1 350.2.a.h 2
15.e even 4 2 70.2.c.a 4
20.e even 4 2 5040.2.t.t 4
21.c even 2 1 2450.2.a.bl 2
60.h even 2 1 2800.2.a.bl 2
60.l odd 4 2 560.2.g.e 4
105.g even 2 1 2450.2.a.bq 2
105.k odd 4 2 490.2.c.e 4
105.w odd 12 4 490.2.i.f 8
105.x even 12 4 490.2.i.c 8
120.q odd 4 2 2240.2.g.i 4
120.w even 4 2 2240.2.g.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 15.e even 4 2
350.2.a.g 2 3.b odd 2 1
350.2.a.h 2 15.d odd 2 1
490.2.c.e 4 105.k odd 4 2
490.2.i.c 8 105.x even 12 4
490.2.i.f 8 105.w odd 12 4
560.2.g.e 4 60.l odd 4 2
630.2.g.g 4 5.c odd 4 2
2240.2.g.i 4 120.q odd 4 2
2240.2.g.j 4 120.w even 4 2
2450.2.a.bl 2 21.c even 2 1
2450.2.a.bq 2 105.g even 2 1
2800.2.a.bl 2 60.h even 2 1
2800.2.a.bm 2 12.b even 2 1
3150.2.a.bs 2 5.b even 2 1
3150.2.a.bt 2 1.a even 1 1 trivial
5040.2.t.t 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3150))$$:

 $$T_{11}^{2} - 24$$ T11^2 - 24 $$T_{13}^{2} - 4T_{13} - 2$$ T13^2 - 4*T13 - 2 $$T_{17} + 2$$ T17 + 2 $$T_{19}^{2} - 8T_{19} + 10$$ T19^2 - 8*T19 + 10 $$T_{29}^{2} + 4T_{29} - 20$$ T29^2 + 4*T29 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 24$$
$13$ $$T^{2} - 4T - 2$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} - 8T + 10$$
$23$ $$T^{2} - 4T - 20$$
$29$ $$T^{2} + 4T - 20$$
$31$ $$T^{2} - 8T - 8$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 12T + 12$$
$43$ $$T^{2} - 8T - 8$$
$47$ $$T^{2} + 8T - 8$$
$53$ $$T^{2} - 12T + 12$$
$59$ $$T^{2} - 8T + 10$$
$61$ $$T^{2} - 12T + 30$$
$67$ $$(T + 8)^{2}$$
$71$ $$T^{2} - 12T + 12$$
$73$ $$T^{2} + 4T - 20$$
$79$ $$T^{2} - 4T - 20$$
$83$ $$T^{2} - 6$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} - 12T - 60$$