# Properties

 Label 350.2.c.a Level $350$ Weight $2$ Character orbit 350.c Analytic conductor $2.795$ 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] = [350,2,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.c (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: $$2.79476407074$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + 3 i q^{3} - q^{4} - 3 q^{6} - i q^{7} - i q^{8} - 6 q^{9} +O(q^{10})$$ q + i * q^2 + 3*i * q^3 - q^4 - 3 * q^6 - i * q^7 - i * q^8 - 6 * q^9 $$q + i q^{2} + 3 i q^{3} - q^{4} - 3 q^{6} - i q^{7} - i q^{8} - 6 q^{9} - 5 q^{11} - 3 i q^{12} + 6 i q^{13} + q^{14} + q^{16} - i q^{17} - 6 i q^{18} + 3 q^{19} + 3 q^{21} - 5 i q^{22} + 3 q^{24} - 6 q^{26} - 9 i q^{27} + i q^{28} + 6 q^{29} - 4 q^{31} + i q^{32} - 15 i q^{33} + q^{34} + 6 q^{36} + 8 i q^{37} + 3 i q^{38} - 18 q^{39} + 11 q^{41} + 3 i q^{42} + 8 i q^{43} + 5 q^{44} + 2 i q^{47} + 3 i q^{48} - q^{49} + 3 q^{51} - 6 i q^{52} - 4 i q^{53} + 9 q^{54} - q^{56} + 9 i q^{57} + 6 i q^{58} - 4 q^{59} - 2 q^{61} - 4 i q^{62} + 6 i q^{63} - q^{64} + 15 q^{66} + 9 i q^{67} + i q^{68} - 10 q^{71} + 6 i q^{72} + 7 i q^{73} - 8 q^{74} - 3 q^{76} + 5 i q^{77} - 18 i q^{78} + 2 q^{79} + 9 q^{81} + 11 i q^{82} - 11 i q^{83} - 3 q^{84} - 8 q^{86} + 18 i q^{87} + 5 i q^{88} + 11 q^{89} + 6 q^{91} - 12 i q^{93} - 2 q^{94} - 3 q^{96} - 10 i q^{97} - i q^{98} + 30 q^{99} +O(q^{100})$$ q + i * q^2 + 3*i * q^3 - q^4 - 3 * q^6 - i * q^7 - i * q^8 - 6 * q^9 - 5 * q^11 - 3*i * q^12 + 6*i * q^13 + q^14 + q^16 - i * q^17 - 6*i * q^18 + 3 * q^19 + 3 * q^21 - 5*i * q^22 + 3 * q^24 - 6 * q^26 - 9*i * q^27 + i * q^28 + 6 * q^29 - 4 * q^31 + i * q^32 - 15*i * q^33 + q^34 + 6 * q^36 + 8*i * q^37 + 3*i * q^38 - 18 * q^39 + 11 * q^41 + 3*i * q^42 + 8*i * q^43 + 5 * q^44 + 2*i * q^47 + 3*i * q^48 - q^49 + 3 * q^51 - 6*i * q^52 - 4*i * q^53 + 9 * q^54 - q^56 + 9*i * q^57 + 6*i * q^58 - 4 * q^59 - 2 * q^61 - 4*i * q^62 + 6*i * q^63 - q^64 + 15 * q^66 + 9*i * q^67 + i * q^68 - 10 * q^71 + 6*i * q^72 + 7*i * q^73 - 8 * q^74 - 3 * q^76 + 5*i * q^77 - 18*i * q^78 + 2 * q^79 + 9 * q^81 + 11*i * q^82 - 11*i * q^83 - 3 * q^84 - 8 * q^86 + 18*i * q^87 + 5*i * q^88 + 11 * q^89 + 6 * q^91 - 12*i * q^93 - 2 * q^94 - 3 * q^96 - 10*i * q^97 - i * q^98 + 30 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 $$2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 10 q^{11} + 2 q^{14} + 2 q^{16} + 6 q^{19} + 6 q^{21} + 6 q^{24} - 12 q^{26} + 12 q^{29} - 8 q^{31} + 2 q^{34} + 12 q^{36} - 36 q^{39} + 22 q^{41} + 10 q^{44} - 2 q^{49} + 6 q^{51} + 18 q^{54} - 2 q^{56} - 8 q^{59} - 4 q^{61} - 2 q^{64} + 30 q^{66} - 20 q^{71} - 16 q^{74} - 6 q^{76} + 4 q^{79} + 18 q^{81} - 6 q^{84} - 16 q^{86} + 22 q^{89} + 12 q^{91} - 4 q^{94} - 6 q^{96} + 60 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 - 10 * q^11 + 2 * q^14 + 2 * q^16 + 6 * q^19 + 6 * q^21 + 6 * q^24 - 12 * q^26 + 12 * q^29 - 8 * q^31 + 2 * q^34 + 12 * q^36 - 36 * q^39 + 22 * q^41 + 10 * q^44 - 2 * q^49 + 6 * q^51 + 18 * q^54 - 2 * q^56 - 8 * q^59 - 4 * q^61 - 2 * q^64 + 30 * q^66 - 20 * q^71 - 16 * q^74 - 6 * q^76 + 4 * q^79 + 18 * q^81 - 6 * q^84 - 16 * q^86 + 22 * q^89 + 12 * q^91 - 4 * q^94 - 6 * q^96 + 60 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 1.00000i 1.00000i
1.00000i 3.00000i −1.00000 0 −3.00000 1.00000i 1.00000i −6.00000 0
99.2 1.00000i 3.00000i −1.00000 0 −3.00000 1.00000i 1.00000i −6.00000 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 350.2.c.a 2
3.b odd 2 1 3150.2.g.v 2
4.b odd 2 1 2800.2.g.a 2
5.b even 2 1 inner 350.2.c.a 2
5.c odd 4 1 350.2.a.c 1
5.c odd 4 1 350.2.a.d yes 1
7.b odd 2 1 2450.2.c.r 2
15.d odd 2 1 3150.2.g.v 2
15.e even 4 1 3150.2.a.j 1
15.e even 4 1 3150.2.a.bq 1
20.d odd 2 1 2800.2.g.a 2
20.e even 4 1 2800.2.a.b 1
20.e even 4 1 2800.2.a.bg 1
35.c odd 2 1 2450.2.c.r 2
35.f even 4 1 2450.2.a.a 1
35.f even 4 1 2450.2.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 5.c odd 4 1
350.2.a.d yes 1 5.c odd 4 1
350.2.c.a 2 1.a even 1 1 trivial
350.2.c.a 2 5.b even 2 1 inner
2450.2.a.a 1 35.f even 4 1
2450.2.a.bg 1 35.f even 4 1
2450.2.c.r 2 7.b odd 2 1
2450.2.c.r 2 35.c odd 2 1
2800.2.a.b 1 20.e even 4 1
2800.2.a.bg 1 20.e even 4 1
2800.2.g.a 2 4.b odd 2 1
2800.2.g.a 2 20.d odd 2 1
3150.2.a.j 1 15.e even 4 1
3150.2.a.bq 1 15.e even 4 1
3150.2.g.v 2 3.b odd 2 1
3150.2.g.v 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 1$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T - 11)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 16$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 81$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} + 49$$
$79$ $$(T - 2)^{2}$$
$83$ $$T^{2} + 121$$
$89$ $$(T - 11)^{2}$$
$97$ $$T^{2} + 100$$