# Properties

 Label 350.2.a.f Level $350$ Weight $2$ Character orbit 350.a Self dual yes Analytic conductor $2.795$ Analytic rank $0$ 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 # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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)

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

## 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 2.00000 1.00000 0 2.00000 −1.00000 1.00000 1.00000 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.f 1
3.b odd 2 1 3150.2.a.i 1
4.b odd 2 1 2800.2.a.g 1
5.b even 2 1 14.2.a.a 1
5.c odd 4 2 350.2.c.d 2
7.b odd 2 1 2450.2.a.t 1
15.d odd 2 1 126.2.a.b 1
15.e even 4 2 3150.2.g.j 2
20.d odd 2 1 112.2.a.c 1
20.e even 4 2 2800.2.g.h 2
35.c odd 2 1 98.2.a.a 1
35.f even 4 2 2450.2.c.c 2
35.i odd 6 2 98.2.c.a 2
35.j even 6 2 98.2.c.b 2
40.e odd 2 1 448.2.a.a 1
40.f even 2 1 448.2.a.g 1
45.h odd 6 2 1134.2.f.f 2
45.j even 6 2 1134.2.f.l 2
55.d odd 2 1 1694.2.a.e 1
60.h even 2 1 1008.2.a.h 1
65.d even 2 1 2366.2.a.j 1
65.g odd 4 2 2366.2.d.b 2
80.k odd 4 2 1792.2.b.g 2
80.q even 4 2 1792.2.b.c 2
85.c even 2 1 4046.2.a.f 1
95.d odd 2 1 5054.2.a.c 1
105.g even 2 1 882.2.a.i 1
105.o odd 6 2 882.2.g.c 2
105.p even 6 2 882.2.g.d 2
115.c odd 2 1 7406.2.a.a 1
120.i odd 2 1 4032.2.a.w 1
120.m even 2 1 4032.2.a.r 1
140.c even 2 1 784.2.a.b 1
140.p odd 6 2 784.2.i.c 2
140.s even 6 2 784.2.i.i 2
280.c odd 2 1 3136.2.a.e 1
280.n even 2 1 3136.2.a.z 1
420.o odd 2 1 7056.2.a.bd 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 5.b even 2 1
98.2.a.a 1 35.c odd 2 1
98.2.c.a 2 35.i odd 6 2
98.2.c.b 2 35.j even 6 2
112.2.a.c 1 20.d odd 2 1
126.2.a.b 1 15.d odd 2 1
350.2.a.f 1 1.a even 1 1 trivial
350.2.c.d 2 5.c odd 4 2
448.2.a.a 1 40.e odd 2 1
448.2.a.g 1 40.f even 2 1
784.2.a.b 1 140.c even 2 1
784.2.i.c 2 140.p odd 6 2
784.2.i.i 2 140.s even 6 2
882.2.a.i 1 105.g even 2 1
882.2.g.c 2 105.o odd 6 2
882.2.g.d 2 105.p even 6 2
1008.2.a.h 1 60.h even 2 1
1134.2.f.f 2 45.h odd 6 2
1134.2.f.l 2 45.j even 6 2
1694.2.a.e 1 55.d odd 2 1
1792.2.b.c 2 80.q even 4 2
1792.2.b.g 2 80.k odd 4 2
2366.2.a.j 1 65.d even 2 1
2366.2.d.b 2 65.g odd 4 2
2450.2.a.t 1 7.b odd 2 1
2450.2.c.c 2 35.f even 4 2
2800.2.a.g 1 4.b odd 2 1
2800.2.g.h 2 20.e even 4 2
3136.2.a.e 1 280.c odd 2 1
3136.2.a.z 1 280.n even 2 1
3150.2.a.i 1 3.b odd 2 1
3150.2.g.j 2 15.e even 4 2
4032.2.a.r 1 120.m even 2 1
4032.2.a.w 1 120.i odd 2 1
4046.2.a.f 1 85.c even 2 1
5054.2.a.c 1 95.d odd 2 1
7056.2.a.bd 1 420.o odd 2 1
7406.2.a.a 1 115.c odd 2 1

## Hecke kernels

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

 $$T_{3} - 2$$ T3 - 2 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

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