# Properties

 Label 3150.2.g.f Level $3150$ Weight $2$ Character orbit 3150.g Analytic conductor $25.153$ Analytic rank $0$ Dimension $2$ 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 350) 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 $$q + i q^{2} - q^{4} - i q^{7} - i q^{8} - 3 q^{11} + 2 i q^{13} + q^{14} + q^{16} + 3 i q^{17} + 7 q^{19} - 3 i q^{22} - 2 q^{26} + i q^{28} - 6 q^{29} - 4 q^{31} + i q^{32} - 3 q^{34} - 8 i q^{37} + 7 i q^{38} + 9 q^{41} + 8 i q^{43} + 3 q^{44} - 6 i q^{47} - q^{49} - 2 i q^{52} + 12 i q^{53} - q^{56} - 6 i q^{58} + 12 q^{59} - 10 q^{61} - 4 i q^{62} - q^{64} + 7 i q^{67} - 3 i q^{68} - 6 q^{71} + 5 i q^{73} + 8 q^{74} - 7 q^{76} + 3 i q^{77} - 14 q^{79} + 9 i q^{82} + 9 i q^{83} - 8 q^{86} + 3 i q^{88} - 15 q^{89} + 2 q^{91} + 6 q^{94} + 10 i q^{97} - i q^{98} +O(q^{100})$$ q + i * q^2 - q^4 - i * q^7 - i * q^8 - 3 * q^11 + 2*i * q^13 + q^14 + q^16 + 3*i * q^17 + 7 * q^19 - 3*i * q^22 - 2 * q^26 + i * q^28 - 6 * q^29 - 4 * q^31 + i * q^32 - 3 * q^34 - 8*i * q^37 + 7*i * q^38 + 9 * q^41 + 8*i * q^43 + 3 * q^44 - 6*i * q^47 - q^49 - 2*i * q^52 + 12*i * q^53 - q^56 - 6*i * q^58 + 12 * q^59 - 10 * q^61 - 4*i * q^62 - q^64 + 7*i * q^67 - 3*i * q^68 - 6 * q^71 + 5*i * q^73 + 8 * q^74 - 7 * q^76 + 3*i * q^77 - 14 * q^79 + 9*i * q^82 + 9*i * q^83 - 8 * q^86 + 3*i * q^88 - 15 * q^89 + 2 * q^91 + 6 * q^94 + 10*i * q^97 - i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 6 q^{11} + 2 q^{14} + 2 q^{16} + 14 q^{19} - 4 q^{26} - 12 q^{29} - 8 q^{31} - 6 q^{34} + 18 q^{41} + 6 q^{44} - 2 q^{49} - 2 q^{56} + 24 q^{59} - 20 q^{61} - 2 q^{64} - 12 q^{71} + 16 q^{74} - 14 q^{76} - 28 q^{79} - 16 q^{86} - 30 q^{89} + 4 q^{91} + 12 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 - 6 * q^11 + 2 * q^14 + 2 * q^16 + 14 * q^19 - 4 * q^26 - 12 * q^29 - 8 * q^31 - 6 * q^34 + 18 * q^41 + 6 * q^44 - 2 * q^49 - 2 * q^56 + 24 * q^59 - 20 * q^61 - 2 * q^64 - 12 * q^71 + 16 * q^74 - 14 * q^76 - 28 * q^79 - 16 * q^86 - 30 * q^89 + 4 * q^91 + 12 * 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.f 2
3.b odd 2 1 350.2.c.c 2
5.b even 2 1 inner 3150.2.g.f 2
5.c odd 4 1 3150.2.a.m 1
5.c odd 4 1 3150.2.a.x 1
12.b even 2 1 2800.2.g.i 2
15.d odd 2 1 350.2.c.c 2
15.e even 4 1 350.2.a.a 1
15.e even 4 1 350.2.a.e yes 1
21.c even 2 1 2450.2.c.h 2
60.h even 2 1 2800.2.g.i 2
60.l odd 4 1 2800.2.a.h 1
60.l odd 4 1 2800.2.a.x 1
105.g even 2 1 2450.2.c.h 2
105.k odd 4 1 2450.2.a.m 1
105.k odd 4 1 2450.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 15.e even 4 1
350.2.a.e yes 1 15.e even 4 1
350.2.c.c 2 3.b odd 2 1
350.2.c.c 2 15.d odd 2 1
2450.2.a.m 1 105.k odd 4 1
2450.2.a.x 1 105.k odd 4 1
2450.2.c.h 2 21.c even 2 1
2450.2.c.h 2 105.g even 2 1
2800.2.a.h 1 60.l odd 4 1
2800.2.a.x 1 60.l odd 4 1
2800.2.g.i 2 12.b even 2 1
2800.2.g.i 2 60.h even 2 1
3150.2.a.m 1 5.c odd 4 1
3150.2.a.x 1 5.c odd 4 1
3150.2.g.f 2 1.a even 1 1 trivial
3150.2.g.f 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{11} + 3$$ T11 + 3 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}^{2} + 9$$ T17^2 + 9 $$T_{19} - 7$$ T19 - 7 $$T_{29} + 6$$ T29 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 9$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T - 9)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 144$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 25$$
$79$ $$(T + 14)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$(T + 15)^{2}$$
$97$ $$T^{2} + 100$$