# Properties

 Label 350.2.a.a Level $350$ Weight $2$ Character orbit 350.a Self dual yes Analytic conductor $2.795$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(1,350)]

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([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## 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: yes Analytic conductor: $$2.79476407074$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 −1.00000 1.00000 0 1.00000 −1.00000 −1.00000 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$5$$ $$+1$$
$$7$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.a.a 1
3.b odd 2 1 3150.2.a.x 1
4.b odd 2 1 2800.2.a.x 1
5.b even 2 1 350.2.a.e yes 1
5.c odd 4 2 350.2.c.c 2
7.b odd 2 1 2450.2.a.m 1
15.d odd 2 1 3150.2.a.m 1
15.e even 4 2 3150.2.g.f 2
20.d odd 2 1 2800.2.a.h 1
20.e even 4 2 2800.2.g.i 2
35.c odd 2 1 2450.2.a.x 1
35.f even 4 2 2450.2.c.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 1.a even 1 1 trivial
350.2.a.e yes 1 5.b even 2 1
350.2.c.c 2 5.c odd 4 2
2450.2.a.m 1 7.b odd 2 1
2450.2.a.x 1 35.c odd 2 1
2450.2.c.h 2 35.f even 4 2
2800.2.a.h 1 20.d odd 2 1
2800.2.a.x 1 4.b odd 2 1
2800.2.g.i 2 20.e even 4 2
3150.2.a.m 1 15.d odd 2 1
3150.2.a.x 1 3.b odd 2 1
3150.2.g.f 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(350))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T + 2$$
$17$ $$T + 3$$
$19$ $$T + 7$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T + 8$$
$41$ $$T + 9$$
$43$ $$T + 8$$
$47$ $$T - 6$$
$53$ $$T - 12$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T - 7$$
$71$ $$T - 6$$
$73$ $$T + 5$$
$79$ $$T - 14$$
$83$ $$T - 9$$
$89$ $$T + 15$$
$97$ $$T - 10$$