# Properties

 Label 1950.2.e.j 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: yes 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - i q^{3} - q^{4} + q^{6} + 3 i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 - i * q^3 - q^4 + q^6 + 3*i * q^7 - i * q^8 - q^9 $$q + i q^{2} - i q^{3} - q^{4} + q^{6} + 3 i q^{7} - i q^{8} - q^{9} - 3 q^{11} + i q^{12} - i q^{13} - 3 q^{14} + q^{16} + 3 i q^{17} - i q^{18} + 3 q^{21} - 3 i q^{22} - 4 i q^{23} - q^{24} + q^{26} + i q^{27} - 3 i q^{28} - 5 q^{29} - 3 q^{31} + i q^{32} + 3 i q^{33} - 3 q^{34} + q^{36} - 12 i q^{37} - q^{39} + 2 q^{41} + 3 i q^{42} - 4 i q^{43} + 3 q^{44} + 4 q^{46} + 3 i q^{47} - i q^{48} - 2 q^{49} + 3 q^{51} + i q^{52} - 9 i q^{53} - q^{54} + 3 q^{56} - 5 i q^{58} - 15 q^{59} - 3 q^{61} - 3 i q^{62} - 3 i q^{63} - q^{64} - 3 q^{66} - 7 i q^{67} - 3 i q^{68} - 4 q^{69} - 8 q^{71} + i q^{72} + 16 i q^{73} + 12 q^{74} - 9 i q^{77} - i q^{78} + q^{81} + 2 i q^{82} + i q^{83} - 3 q^{84} + 4 q^{86} + 5 i q^{87} + 3 i q^{88} + 3 q^{91} + 4 i q^{92} + 3 i q^{93} - 3 q^{94} + q^{96} - 2 i q^{97} - 2 i q^{98} + 3 q^{99} +O(q^{100})$$ q + i * q^2 - i * q^3 - q^4 + q^6 + 3*i * q^7 - i * q^8 - q^9 - 3 * q^11 + i * q^12 - i * q^13 - 3 * q^14 + q^16 + 3*i * q^17 - i * q^18 + 3 * q^21 - 3*i * q^22 - 4*i * q^23 - q^24 + q^26 + i * q^27 - 3*i * q^28 - 5 * q^29 - 3 * q^31 + i * q^32 + 3*i * q^33 - 3 * q^34 + q^36 - 12*i * q^37 - q^39 + 2 * q^41 + 3*i * q^42 - 4*i * q^43 + 3 * q^44 + 4 * q^46 + 3*i * q^47 - i * q^48 - 2 * q^49 + 3 * q^51 + i * q^52 - 9*i * q^53 - q^54 + 3 * q^56 - 5*i * q^58 - 15 * q^59 - 3 * q^61 - 3*i * q^62 - 3*i * q^63 - q^64 - 3 * q^66 - 7*i * q^67 - 3*i * q^68 - 4 * q^69 - 8 * q^71 + i * q^72 + 16*i * q^73 + 12 * q^74 - 9*i * q^77 - i * q^78 + q^81 + 2*i * q^82 + i * q^83 - 3 * q^84 + 4 * q^86 + 5*i * q^87 + 3*i * q^88 + 3 * q^91 + 4*i * q^92 + 3*i * q^93 - 3 * q^94 + q^96 - 2*i * q^97 - 2*i * q^98 + 3 * 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 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 6 q^{11} - 6 q^{14} + 2 q^{16} + 6 q^{21} - 2 q^{24} + 2 q^{26} - 10 q^{29} - 6 q^{31} - 6 q^{34} + 2 q^{36} - 2 q^{39} + 4 q^{41} + 6 q^{44} + 8 q^{46} - 4 q^{49} + 6 q^{51} - 2 q^{54} + 6 q^{56} - 30 q^{59} - 6 q^{61} - 2 q^{64} - 6 q^{66} - 8 q^{69} - 16 q^{71} + 24 q^{74} + 2 q^{81} - 6 q^{84} + 8 q^{86} + 6 q^{91} - 6 q^{94} + 2 q^{96} + 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 6 * q^11 - 6 * q^14 + 2 * q^16 + 6 * q^21 - 2 * q^24 + 2 * q^26 - 10 * q^29 - 6 * q^31 - 6 * q^34 + 2 * q^36 - 2 * q^39 + 4 * q^41 + 6 * q^44 + 8 * q^46 - 4 * q^49 + 6 * q^51 - 2 * q^54 + 6 * q^56 - 30 * q^59 - 6 * q^61 - 2 * q^64 - 6 * q^66 - 8 * q^69 - 16 * q^71 + 24 * q^74 + 2 * q^81 - 6 * q^84 + 8 * q^86 + 6 * q^91 - 6 * q^94 + 2 * q^96 + 6 * 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 3.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 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.j 2
3.b odd 2 1 5850.2.e.z 2
5.b even 2 1 inner 1950.2.e.j 2
5.c odd 4 1 1950.2.a.a 1
5.c odd 4 1 1950.2.a.bb yes 1
15.d odd 2 1 5850.2.e.z 2
15.e even 4 1 5850.2.a.w 1
15.e even 4 1 5850.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.a 1 5.c odd 4 1
1950.2.a.bb yes 1 5.c odd 4 1
1950.2.e.j 2 1.a even 1 1 trivial
1950.2.e.j 2 5.b even 2 1 inner
5850.2.a.w 1 15.e even 4 1
5850.2.a.bf 1 15.e even 4 1
5850.2.e.z 2 3.b odd 2 1
5850.2.e.z 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11} + 3$$ T11 + 3 $$T_{17}^{2} + 9$$ T17^2 + 9 $$T_{31} + 3$$ T31 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 144$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 81$$
$59$ $$(T + 15)^{2}$$
$61$ $$(T + 3)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 256$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 1$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 4$$