# Properties

 Label 3150.2.g.c Level $3150$ Weight $2$ Character orbit 3150.g Analytic conductor $25.153$ 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] = [3150,2,Mod(2899,3150)]

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, 1, 0]))

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

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

chi := DirichletCharacter("3150.2899");

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.g (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: $$25.1528766367$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + i q^{7} - i q^{8} +O(q^{10})$$ q + i * q^2 - q^4 + i * q^7 - i * q^8 $$q + i q^{2} - q^{4} + i q^{7} - i q^{8} - 4 q^{11} - 6 i q^{13} - q^{14} + q^{16} + 2 i q^{17} - 4 i q^{22} + 6 q^{26} - i q^{28} + 6 q^{29} + 8 q^{31} + i q^{32} - 2 q^{34} + 10 i q^{37} - 2 q^{41} + 4 i q^{43} + 4 q^{44} + 8 i q^{47} - q^{49} + 6 i q^{52} + 2 i q^{53} + q^{56} + 6 i q^{58} - 8 q^{59} - 14 q^{61} + 8 i q^{62} - q^{64} + 12 i q^{67} - 2 i q^{68} + 16 q^{71} + 2 i q^{73} - 10 q^{74} - 4 i q^{77} + 8 q^{79} - 2 i q^{82} - 8 i q^{83} - 4 q^{86} + 4 i q^{88} + 10 q^{89} + 6 q^{91} - 8 q^{94} - 2 i q^{97} - i q^{98} +O(q^{100})$$ q + i * q^2 - q^4 + i * q^7 - i * q^8 - 4 * q^11 - 6*i * q^13 - q^14 + q^16 + 2*i * q^17 - 4*i * q^22 + 6 * q^26 - i * q^28 + 6 * q^29 + 8 * q^31 + i * q^32 - 2 * q^34 + 10*i * q^37 - 2 * q^41 + 4*i * q^43 + 4 * q^44 + 8*i * q^47 - q^49 + 6*i * q^52 + 2*i * q^53 + q^56 + 6*i * q^58 - 8 * q^59 - 14 * q^61 + 8*i * q^62 - q^64 + 12*i * q^67 - 2*i * q^68 + 16 * q^71 + 2*i * q^73 - 10 * q^74 - 4*i * q^77 + 8 * q^79 - 2*i * q^82 - 8*i * q^83 - 4 * q^86 + 4*i * q^88 + 10 * q^89 + 6 * q^91 - 8 * q^94 - 2*i * q^97 - i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 8 q^{11} - 2 q^{14} + 2 q^{16} + 12 q^{26} + 12 q^{29} + 16 q^{31} - 4 q^{34} - 4 q^{41} + 8 q^{44} - 2 q^{49} + 2 q^{56} - 16 q^{59} - 28 q^{61} - 2 q^{64} + 32 q^{71} - 20 q^{74} + 16 q^{79} - 8 q^{86} + 20 q^{89} + 12 q^{91} - 16 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 - 8 * q^11 - 2 * q^14 + 2 * q^16 + 12 * q^26 + 12 * q^29 + 16 * q^31 - 4 * q^34 - 4 * q^41 + 8 * q^44 - 2 * q^49 + 2 * q^56 - 16 * q^59 - 28 * q^61 - 2 * q^64 + 32 * q^71 - 20 * q^74 + 16 * q^79 - 8 * q^86 + 20 * q^89 + 12 * q^91 - 16 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2899.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 0
2899.2 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.g.c 2
3.b odd 2 1 350.2.c.b 2
5.b even 2 1 inner 3150.2.g.c 2
5.c odd 4 1 630.2.a.d 1
5.c odd 4 1 3150.2.a.bj 1
12.b even 2 1 2800.2.g.n 2
15.d odd 2 1 350.2.c.b 2
15.e even 4 1 70.2.a.a 1
15.e even 4 1 350.2.a.b 1
20.e even 4 1 5040.2.a.bm 1
21.c even 2 1 2450.2.c.k 2
35.f even 4 1 4410.2.a.b 1
60.h even 2 1 2800.2.g.n 2
60.l odd 4 1 560.2.a.d 1
60.l odd 4 1 2800.2.a.m 1
105.g even 2 1 2450.2.c.k 2
105.k odd 4 1 490.2.a.h 1
105.k odd 4 1 2450.2.a.l 1
105.w odd 12 2 490.2.e.c 2
105.x even 12 2 490.2.e.d 2
120.q odd 4 1 2240.2.a.q 1
120.w even 4 1 2240.2.a.n 1
165.l odd 4 1 8470.2.a.j 1
420.w even 4 1 3920.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 15.e even 4 1
350.2.a.b 1 15.e even 4 1
350.2.c.b 2 3.b odd 2 1
350.2.c.b 2 15.d odd 2 1
490.2.a.h 1 105.k odd 4 1
490.2.e.c 2 105.w odd 12 2
490.2.e.d 2 105.x even 12 2
560.2.a.d 1 60.l odd 4 1
630.2.a.d 1 5.c odd 4 1
2240.2.a.n 1 120.w even 4 1
2240.2.a.q 1 120.q odd 4 1
2450.2.a.l 1 105.k odd 4 1
2450.2.c.k 2 21.c even 2 1
2450.2.c.k 2 105.g even 2 1
2800.2.a.m 1 60.l odd 4 1
2800.2.g.n 2 12.b even 2 1
2800.2.g.n 2 60.h even 2 1
3150.2.a.bj 1 5.c odd 4 1
3150.2.g.c 2 1.a even 1 1 trivial
3150.2.g.c 2 5.b even 2 1 inner
3920.2.a.t 1 420.w even 4 1
4410.2.a.b 1 35.f even 4 1
5040.2.a.bm 1 20.e even 4 1
8470.2.a.j 1 165.l odd 4 1

## Hecke kernels

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

 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 4$$ T17^2 + 4 $$T_{19}$$ T19 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 4$$
$59$ $$(T + 8)^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 4$$