Properties

 Label 1950.2.e.n 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + i q^{3} - q^{4} + q^{6} + 4 i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ q - i * q^2 + i * q^3 - q^4 + q^6 + 4*i * q^7 + i * q^8 - q^9 $$q - i q^{2} + i q^{3} - q^{4} + q^{6} + 4 i q^{7} + i q^{8} - q^{9} + 4 q^{11} - i q^{12} + i q^{13} + 4 q^{14} + q^{16} + 4 i q^{17} + i q^{18} - 7 q^{19} - 4 q^{21} - 4 i q^{22} + 4 i q^{23} - q^{24} + q^{26} - i q^{27} - 4 i q^{28} - 5 q^{29} + 4 q^{31} - i q^{32} + 4 i q^{33} + 4 q^{34} + q^{36} - 9 i q^{37} + 7 i q^{38} - q^{39} - 5 q^{41} + 4 i q^{42} - 10 i q^{43} - 4 q^{44} + 4 q^{46} - 3 i q^{47} + i q^{48} - 9 q^{49} - 4 q^{51} - i q^{52} + 9 i q^{53} - q^{54} - 4 q^{56} - 7 i q^{57} + 5 i q^{58} + 6 q^{59} + 4 q^{61} - 4 i q^{62} - 4 i q^{63} - q^{64} + 4 q^{66} + 7 i q^{67} - 4 i q^{68} - 4 q^{69} - 15 q^{71} - i q^{72} + 12 i q^{73} - 9 q^{74} + 7 q^{76} + 16 i q^{77} + i q^{78} - 7 q^{79} + q^{81} + 5 i q^{82} + 6 i q^{83} + 4 q^{84} - 10 q^{86} - 5 i q^{87} + 4 i q^{88} - 14 q^{89} - 4 q^{91} - 4 i q^{92} + 4 i q^{93} - 3 q^{94} + q^{96} + 16 i q^{97} + 9 i q^{98} - 4 q^{99} +O(q^{100})$$ q - i * q^2 + i * q^3 - q^4 + q^6 + 4*i * q^7 + i * q^8 - q^9 + 4 * q^11 - i * q^12 + i * q^13 + 4 * q^14 + q^16 + 4*i * q^17 + i * q^18 - 7 * q^19 - 4 * q^21 - 4*i * q^22 + 4*i * q^23 - q^24 + q^26 - i * q^27 - 4*i * q^28 - 5 * q^29 + 4 * q^31 - i * q^32 + 4*i * q^33 + 4 * q^34 + q^36 - 9*i * q^37 + 7*i * q^38 - q^39 - 5 * q^41 + 4*i * q^42 - 10*i * q^43 - 4 * q^44 + 4 * q^46 - 3*i * q^47 + i * q^48 - 9 * q^49 - 4 * q^51 - i * q^52 + 9*i * q^53 - q^54 - 4 * q^56 - 7*i * q^57 + 5*i * q^58 + 6 * q^59 + 4 * q^61 - 4*i * q^62 - 4*i * q^63 - q^64 + 4 * q^66 + 7*i * q^67 - 4*i * q^68 - 4 * q^69 - 15 * q^71 - i * q^72 + 12*i * q^73 - 9 * q^74 + 7 * q^76 + 16*i * q^77 + i * q^78 - 7 * q^79 + q^81 + 5*i * q^82 + 6*i * q^83 + 4 * q^84 - 10 * q^86 - 5*i * q^87 + 4*i * q^88 - 14 * q^89 - 4 * q^91 - 4*i * q^92 + 4*i * q^93 - 3 * q^94 + q^96 + 16*i * q^97 + 9*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 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 8 q^{11} + 8 q^{14} + 2 q^{16} - 14 q^{19} - 8 q^{21} - 2 q^{24} + 2 q^{26} - 10 q^{29} + 8 q^{31} + 8 q^{34} + 2 q^{36} - 2 q^{39} - 10 q^{41} - 8 q^{44} + 8 q^{46} - 18 q^{49} - 8 q^{51} - 2 q^{54} - 8 q^{56} + 12 q^{59} + 8 q^{61} - 2 q^{64} + 8 q^{66} - 8 q^{69} - 30 q^{71} - 18 q^{74} + 14 q^{76} - 14 q^{79} + 2 q^{81} + 8 q^{84} - 20 q^{86} - 28 q^{89} - 8 q^{91} - 6 q^{94} + 2 q^{96} - 8 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 8 * q^11 + 8 * q^14 + 2 * q^16 - 14 * q^19 - 8 * q^21 - 2 * q^24 + 2 * q^26 - 10 * q^29 + 8 * q^31 + 8 * q^34 + 2 * q^36 - 2 * q^39 - 10 * q^41 - 8 * q^44 + 8 * q^46 - 18 * q^49 - 8 * q^51 - 2 * q^54 - 8 * q^56 + 12 * q^59 + 8 * q^61 - 2 * q^64 + 8 * q^66 - 8 * q^69 - 30 * q^71 - 18 * q^74 + 14 * q^76 - 14 * q^79 + 2 * q^81 + 8 * q^84 - 20 * q^86 - 28 * q^89 - 8 * q^91 - 6 * q^94 + 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 4.00000i 1.00000i −1.00000 0
1249.2 1.00000i 1.00000i −1.00000 0 1.00000 4.00000i 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.n 2
3.b odd 2 1 5850.2.e.c 2
5.b even 2 1 inner 1950.2.e.n 2
5.c odd 4 1 1950.2.a.e 1
5.c odd 4 1 1950.2.a.x yes 1
15.d odd 2 1 5850.2.e.c 2
15.e even 4 1 5850.2.a.a 1
15.e even 4 1 5850.2.a.by 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.a.e 1 5.c odd 4 1
1950.2.a.x yes 1 5.c odd 4 1
1950.2.e.n 2 1.a even 1 1 trivial
1950.2.e.n 2 5.b even 2 1 inner
5850.2.a.a 1 15.e even 4 1
5850.2.a.by 1 15.e even 4 1
5850.2.e.c 2 3.b odd 2 1
5850.2.e.c 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 $$S_{2}^{\mathrm{new}}(1950, [\chi])$$:

 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{11} - 4$$ T11 - 4 $$T_{17}^{2} + 16$$ T17^2 + 16 $$T_{31} - 4$$ T31 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 4)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 16$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 81$$
$41$ $$(T + 5)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 81$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 4)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T + 15)^{2}$$
$73$ $$T^{2} + 144$$
$79$ $$(T + 7)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 256$$