# Properties

 Label 950.2.b.e Level $950$ Weight $2$ Character orbit 950.b Analytic conductor $7.586$ 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] = [950,2,Mod(799,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("950.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$950 = 2 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 950.b (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: $$7.58578819202$$ 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 190) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{7} + i q^{8} + 2 q^{9} +O(q^{10})$$ q - i * q^2 + i * q^3 - q^4 + q^6 + i * q^7 + i * q^8 + 2 * q^9 $$q - i q^{2} + i q^{3} - q^{4} + q^{6} + i q^{7} + i q^{8} + 2 q^{9} - i q^{12} - i q^{13} + q^{14} + q^{16} + 3 i q^{17} - 2 i q^{18} - q^{19} - q^{21} + 3 i q^{23} - q^{24} - q^{26} + 5 i q^{27} - i q^{28} + 3 q^{29} + 2 q^{31} - i q^{32} + 3 q^{34} - 2 q^{36} + 10 i q^{37} + i q^{38} + q^{39} + 6 q^{41} + i q^{42} + 2 i q^{43} + 3 q^{46} + i q^{48} + 6 q^{49} - 3 q^{51} + i q^{52} + 3 i q^{53} + 5 q^{54} - q^{56} - i q^{57} - 3 i q^{58} - 3 q^{59} + 8 q^{61} - 2 i q^{62} + 2 i q^{63} - q^{64} + 7 i q^{67} - 3 i q^{68} - 3 q^{69} + 12 q^{71} + 2 i q^{72} - 13 i q^{73} + 10 q^{74} + q^{76} - i q^{78} - 14 q^{79} + q^{81} - 6 i q^{82} + 6 i q^{83} + q^{84} + 2 q^{86} + 3 i q^{87} - 6 q^{89} + q^{91} - 3 i q^{92} + 2 i q^{93} + q^{96} + 10 i q^{97} - 6 i q^{98} +O(q^{100})$$ q - i * q^2 + i * q^3 - q^4 + q^6 + i * q^7 + i * q^8 + 2 * q^9 - i * q^12 - i * q^13 + q^14 + q^16 + 3*i * q^17 - 2*i * q^18 - q^19 - q^21 + 3*i * q^23 - q^24 - q^26 + 5*i * q^27 - i * q^28 + 3 * q^29 + 2 * q^31 - i * q^32 + 3 * q^34 - 2 * q^36 + 10*i * q^37 + i * q^38 + q^39 + 6 * q^41 + i * q^42 + 2*i * q^43 + 3 * q^46 + i * q^48 + 6 * q^49 - 3 * q^51 + i * q^52 + 3*i * q^53 + 5 * q^54 - q^56 - i * q^57 - 3*i * q^58 - 3 * q^59 + 8 * q^61 - 2*i * q^62 + 2*i * q^63 - q^64 + 7*i * q^67 - 3*i * q^68 - 3 * q^69 + 12 * q^71 + 2*i * q^72 - 13*i * q^73 + 10 * q^74 + q^76 - i * q^78 - 14 * q^79 + q^81 - 6*i * q^82 + 6*i * q^83 + q^84 + 2 * q^86 + 3*i * q^87 - 6 * q^89 + q^91 - 3*i * q^92 + 2*i * q^93 + q^96 + 10*i * q^97 - 6*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 $$2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} + 2 q^{14} + 2 q^{16} - 2 q^{19} - 2 q^{21} - 2 q^{24} - 2 q^{26} + 6 q^{29} + 4 q^{31} + 6 q^{34} - 4 q^{36} + 2 q^{39} + 12 q^{41} + 6 q^{46} + 12 q^{49} - 6 q^{51} + 10 q^{54} - 2 q^{56} - 6 q^{59} + 16 q^{61} - 2 q^{64} - 6 q^{69} + 24 q^{71} + 20 q^{74} + 2 q^{76} - 28 q^{79} + 2 q^{81} + 2 q^{84} + 4 q^{86} - 12 q^{89} + 2 q^{91} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 + 2 * q^14 + 2 * q^16 - 2 * q^19 - 2 * q^21 - 2 * q^24 - 2 * q^26 + 6 * q^29 + 4 * q^31 + 6 * q^34 - 4 * q^36 + 2 * q^39 + 12 * q^41 + 6 * q^46 + 12 * q^49 - 6 * q^51 + 10 * q^54 - 2 * q^56 - 6 * q^59 + 16 * q^61 - 2 * q^64 - 6 * q^69 + 24 * q^71 + 20 * q^74 + 2 * q^76 - 28 * q^79 + 2 * q^81 + 2 * q^84 + 4 * q^86 - 12 * q^89 + 2 * q^91 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
799.1
 1.00000i − 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i 2.00000 0
799.2 1.00000i 1.00000i −1.00000 0 1.00000 1.00000i 1.00000i 2.00000 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 950.2.b.e 2
5.b even 2 1 inner 950.2.b.e 2
5.c odd 4 1 190.2.a.c 1
5.c odd 4 1 950.2.a.a 1
15.e even 4 1 1710.2.a.d 1
15.e even 4 1 8550.2.a.bd 1
20.e even 4 1 1520.2.a.d 1
20.e even 4 1 7600.2.a.m 1
35.f even 4 1 9310.2.a.o 1
40.i odd 4 1 6080.2.a.h 1
40.k even 4 1 6080.2.a.p 1
95.g even 4 1 3610.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 5.c odd 4 1
950.2.a.a 1 5.c odd 4 1
950.2.b.e 2 1.a even 1 1 trivial
950.2.b.e 2 5.b even 2 1 inner
1520.2.a.d 1 20.e even 4 1
1710.2.a.d 1 15.e even 4 1
3610.2.a.b 1 95.g even 4 1
6080.2.a.h 1 40.i odd 4 1
6080.2.a.p 1 40.k even 4 1
7600.2.a.m 1 20.e even 4 1
8550.2.a.bd 1 15.e even 4 1
9310.2.a.o 1 35.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(950, [\chi])$$:

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11}$$ T11 $$T_{13}^{2} + 1$$ T13^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T - 3)^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 9$$
$59$ $$(T + 3)^{2}$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 169$$
$79$ $$(T + 14)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 100$$