# Properties

 Label 1950.2.e.g Level $1950$ Weight $2$ Character orbit 1950.e Analytic conductor $15.571$ 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] = [1950,2,Mod(1249,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1950.e (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: $$15.5708283941$$ 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 390) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 + i * q^3 - q^4 - q^6 - i * q^8 - q^9 $$q + i q^{2} + i q^{3} - q^{4} - q^{6} - i q^{8} - q^{9} + 4 q^{11} - i q^{12} - i q^{13} + q^{16} - 6 i q^{17} - i q^{18} - 4 q^{19} + 4 i q^{22} - 8 i q^{23} + q^{24} + q^{26} - i q^{27} - 6 q^{29} - 8 q^{31} + i q^{32} + 4 i q^{33} + 6 q^{34} + q^{36} - 10 i q^{37} - 4 i q^{38} + q^{39} - 6 q^{41} - 4 i q^{43} - 4 q^{44} + 8 q^{46} + i q^{48} + 7 q^{49} + 6 q^{51} + i q^{52} + 10 i q^{53} + q^{54} - 4 i q^{57} - 6 i q^{58} - 4 q^{59} - 2 q^{61} - 8 i q^{62} - q^{64} - 4 q^{66} - 12 i q^{67} + 6 i q^{68} + 8 q^{69} + 16 q^{71} + i q^{72} - 2 i q^{73} + 10 q^{74} + 4 q^{76} + i q^{78} + 16 q^{79} + q^{81} - 6 i q^{82} + 12 i q^{83} + 4 q^{86} - 6 i q^{87} - 4 i q^{88} - 10 q^{89} + 8 i q^{92} - 8 i q^{93} - q^{96} - 6 i q^{97} + 7 i q^{98} - 4 q^{99} +O(q^{100})$$ q + i * q^2 + i * q^3 - q^4 - q^6 - i * q^8 - q^9 + 4 * q^11 - i * q^12 - i * q^13 + q^16 - 6*i * q^17 - i * q^18 - 4 * q^19 + 4*i * q^22 - 8*i * q^23 + q^24 + q^26 - i * q^27 - 6 * q^29 - 8 * q^31 + i * q^32 + 4*i * q^33 + 6 * q^34 + q^36 - 10*i * q^37 - 4*i * q^38 + q^39 - 6 * q^41 - 4*i * q^43 - 4 * q^44 + 8 * q^46 + i * q^48 + 7 * q^49 + 6 * q^51 + i * q^52 + 10*i * q^53 + q^54 - 4*i * q^57 - 6*i * q^58 - 4 * q^59 - 2 * q^61 - 8*i * q^62 - q^64 - 4 * q^66 - 12*i * q^67 + 6*i * q^68 + 8 * q^69 + 16 * q^71 + i * q^72 - 2*i * q^73 + 10 * q^74 + 4 * q^76 + i * q^78 + 16 * q^79 + q^81 - 6*i * q^82 + 12*i * q^83 + 4 * q^86 - 6*i * q^87 - 4*i * q^88 - 10 * q^89 + 8*i * q^92 - 8*i * q^93 - q^96 - 6*i * q^97 + 7*i * q^98 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} - 8 q^{19} + 2 q^{24} + 2 q^{26} - 12 q^{29} - 16 q^{31} + 12 q^{34} + 2 q^{36} + 2 q^{39} - 12 q^{41} - 8 q^{44} + 16 q^{46} + 14 q^{49} + 12 q^{51} + 2 q^{54} - 8 q^{59} - 4 q^{61} - 2 q^{64} - 8 q^{66} + 16 q^{69} + 32 q^{71} + 20 q^{74} + 8 q^{76} + 32 q^{79} + 2 q^{81} + 8 q^{86} - 20 q^{89} - 2 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 - 8 * q^19 + 2 * q^24 + 2 * q^26 - 12 * q^29 - 16 * q^31 + 12 * q^34 + 2 * q^36 + 2 * q^39 - 12 * q^41 - 8 * q^44 + 16 * q^46 + 14 * q^49 + 12 * q^51 + 2 * q^54 - 8 * q^59 - 4 * q^61 - 2 * q^64 - 8 * q^66 + 16 * q^69 + 32 * q^71 + 20 * q^74 + 8 * q^76 + 32 * q^79 + 2 * q^81 + 8 * q^86 - 20 * q^89 - 2 * q^96 - 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$1301$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i −1.00000 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 1950.2.e.g 2
3.b odd 2 1 5850.2.e.e 2
5.b even 2 1 inner 1950.2.e.g 2
5.c odd 4 1 390.2.a.f 1
5.c odd 4 1 1950.2.a.k 1
15.d odd 2 1 5850.2.e.e 2
15.e even 4 1 1170.2.a.a 1
15.e even 4 1 5850.2.a.bo 1
20.e even 4 1 3120.2.a.w 1
60.l odd 4 1 9360.2.a.p 1
65.f even 4 1 5070.2.b.d 2
65.h odd 4 1 5070.2.a.a 1
65.k even 4 1 5070.2.b.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.f 1 5.c odd 4 1
1170.2.a.a 1 15.e even 4 1
1950.2.a.k 1 5.c odd 4 1
1950.2.e.g 2 1.a even 1 1 trivial
1950.2.e.g 2 5.b even 2 1 inner
3120.2.a.w 1 20.e even 4 1
5070.2.a.a 1 65.h odd 4 1
5070.2.b.d 2 65.f even 4 1
5070.2.b.d 2 65.k even 4 1
5850.2.a.bo 1 15.e even 4 1
5850.2.e.e 2 3.b odd 2 1
5850.2.e.e 2 15.d odd 2 1
9360.2.a.p 1 60.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 $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11} - 4$$ T11 - 4 $$T_{17}^{2} + 36$$ T17^2 + 36 $$T_{31} + 8$$ T31 + 8

## Hecke characteristic polynomials

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