# Properties

 Label 350.2.a.g Level $350$ Weight $2$ Character orbit 350.a Self dual yes Analytic conductor $2.795$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} - \beta q^{6} - q^{7} - q^{8} + 3 q^{9} +O(q^{10})$$ q - q^2 + b * q^3 + q^4 - b * q^6 - q^7 - q^8 + 3 * q^9 $$q - q^{2} + \beta q^{3} + q^{4} - \beta q^{6} - q^{7} - q^{8} + 3 q^{9} + 2 \beta q^{11} + \beta q^{12} + ( - \beta + 2) q^{13} + q^{14} + q^{16} + 2 q^{17} - 3 q^{18} + (\beta + 4) q^{19} - \beta q^{21} - 2 \beta q^{22} + ( - 2 \beta - 2) q^{23} - \beta q^{24} + (\beta - 2) q^{26} - q^{28} + ( - 2 \beta + 2) q^{29} + ( - 2 \beta + 4) q^{31} - q^{32} + 12 q^{33} - 2 q^{34} + 3 q^{36} + 2 q^{37} + ( - \beta - 4) q^{38} + (2 \beta - 6) q^{39} + ( - 2 \beta - 6) q^{41} + \beta q^{42} + (2 \beta + 4) q^{43} + 2 \beta q^{44} + (2 \beta + 2) q^{46} + ( - 2 \beta + 4) q^{47} + \beta q^{48} + q^{49} + 2 \beta q^{51} + ( - \beta + 2) q^{52} + (2 \beta - 6) q^{53} + q^{56} + (4 \beta + 6) q^{57} + (2 \beta - 2) q^{58} + ( - \beta - 4) q^{59} + (\beta + 6) q^{61} + (2 \beta - 4) q^{62} - 3 q^{63} + q^{64} - 12 q^{66} - 8 q^{67} + 2 q^{68} + ( - 2 \beta - 12) q^{69} + ( - 2 \beta - 6) q^{71} - 3 q^{72} + ( - 2 \beta - 2) q^{73} - 2 q^{74} + (\beta + 4) q^{76} - 2 \beta q^{77} + ( - 2 \beta + 6) q^{78} + ( - 2 \beta + 2) q^{79} - 9 q^{81} + (2 \beta + 6) q^{82} + \beta q^{83} - \beta q^{84} + ( - 2 \beta - 4) q^{86} + (2 \beta - 12) q^{87} - 2 \beta q^{88} - 10 q^{89} + (\beta - 2) q^{91} + ( - 2 \beta - 2) q^{92} + (4 \beta - 12) q^{93} + (2 \beta - 4) q^{94} - \beta q^{96} + ( - 4 \beta + 6) q^{97} - q^{98} + 6 \beta q^{99} +O(q^{100})$$ q - q^2 + b * q^3 + q^4 - b * q^6 - q^7 - q^8 + 3 * q^9 + 2*b * q^11 + b * q^12 + (-b + 2) * q^13 + q^14 + q^16 + 2 * q^17 - 3 * q^18 + (b + 4) * q^19 - b * q^21 - 2*b * q^22 + (-2*b - 2) * q^23 - b * q^24 + (b - 2) * q^26 - q^28 + (-2*b + 2) * q^29 + (-2*b + 4) * q^31 - q^32 + 12 * q^33 - 2 * q^34 + 3 * q^36 + 2 * q^37 + (-b - 4) * q^38 + (2*b - 6) * q^39 + (-2*b - 6) * q^41 + b * q^42 + (2*b + 4) * q^43 + 2*b * q^44 + (2*b + 2) * q^46 + (-2*b + 4) * q^47 + b * q^48 + q^49 + 2*b * q^51 + (-b + 2) * q^52 + (2*b - 6) * q^53 + q^56 + (4*b + 6) * q^57 + (2*b - 2) * q^58 + (-b - 4) * q^59 + (b + 6) * q^61 + (2*b - 4) * q^62 - 3 * q^63 + q^64 - 12 * q^66 - 8 * q^67 + 2 * q^68 + (-2*b - 12) * q^69 + (-2*b - 6) * q^71 - 3 * q^72 + (-2*b - 2) * q^73 - 2 * q^74 + (b + 4) * q^76 - 2*b * q^77 + (-2*b + 6) * q^78 + (-2*b + 2) * q^79 - 9 * q^81 + (2*b + 6) * q^82 + b * q^83 - b * q^84 + (-2*b - 4) * q^86 + (2*b - 12) * q^87 - 2*b * q^88 - 10 * q^89 + (b - 2) * q^91 + (-2*b - 2) * q^92 + (4*b - 12) * q^93 + (2*b - 4) * q^94 - b * q^96 + (-4*b + 6) * q^97 - q^98 + 6*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 4 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 8 q^{19} - 4 q^{23} - 4 q^{26} - 2 q^{28} + 4 q^{29} + 8 q^{31} - 2 q^{32} + 24 q^{33} - 4 q^{34} + 6 q^{36} + 4 q^{37} - 8 q^{38} - 12 q^{39} - 12 q^{41} + 8 q^{43} + 4 q^{46} + 8 q^{47} + 2 q^{49} + 4 q^{52} - 12 q^{53} + 2 q^{56} + 12 q^{57} - 4 q^{58} - 8 q^{59} + 12 q^{61} - 8 q^{62} - 6 q^{63} + 2 q^{64} - 24 q^{66} - 16 q^{67} + 4 q^{68} - 24 q^{69} - 12 q^{71} - 6 q^{72} - 4 q^{73} - 4 q^{74} + 8 q^{76} + 12 q^{78} + 4 q^{79} - 18 q^{81} + 12 q^{82} - 8 q^{86} - 24 q^{87} - 20 q^{89} - 4 q^{91} - 4 q^{92} - 24 q^{93} - 8 q^{94} + 12 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 + 6 * q^9 + 4 * q^13 + 2 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 8 * q^19 - 4 * q^23 - 4 * q^26 - 2 * q^28 + 4 * q^29 + 8 * q^31 - 2 * q^32 + 24 * q^33 - 4 * q^34 + 6 * q^36 + 4 * q^37 - 8 * q^38 - 12 * q^39 - 12 * q^41 + 8 * q^43 + 4 * q^46 + 8 * q^47 + 2 * q^49 + 4 * q^52 - 12 * q^53 + 2 * q^56 + 12 * q^57 - 4 * q^58 - 8 * q^59 + 12 * q^61 - 8 * q^62 - 6 * q^63 + 2 * q^64 - 24 * q^66 - 16 * q^67 + 4 * q^68 - 24 * q^69 - 12 * q^71 - 6 * q^72 - 4 * q^73 - 4 * q^74 + 8 * q^76 + 12 * q^78 + 4 * q^79 - 18 * q^81 + 12 * q^82 - 8 * q^86 - 24 * q^87 - 20 * q^89 - 4 * q^91 - 4 * q^92 - 24 * q^93 - 8 * q^94 + 12 * q^97 - 2 * 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
 −2.44949 2.44949
−1.00000 −2.44949 1.00000 0 2.44949 −1.00000 −1.00000 3.00000 0
1.2 −1.00000 2.44949 1.00000 0 −2.44949 −1.00000 −1.00000 3.00000 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.g 2
3.b odd 2 1 3150.2.a.bt 2
4.b odd 2 1 2800.2.a.bm 2
5.b even 2 1 350.2.a.h 2
5.c odd 4 2 70.2.c.a 4
7.b odd 2 1 2450.2.a.bl 2
15.d odd 2 1 3150.2.a.bs 2
15.e even 4 2 630.2.g.g 4
20.d odd 2 1 2800.2.a.bl 2
20.e even 4 2 560.2.g.e 4
35.c odd 2 1 2450.2.a.bq 2
35.f even 4 2 490.2.c.e 4
35.k even 12 4 490.2.i.f 8
35.l odd 12 4 490.2.i.c 8
40.i odd 4 2 2240.2.g.j 4
40.k even 4 2 2240.2.g.i 4
60.l odd 4 2 5040.2.t.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 5.c odd 4 2
350.2.a.g 2 1.a even 1 1 trivial
350.2.a.h 2 5.b even 2 1
490.2.c.e 4 35.f even 4 2
490.2.i.c 8 35.l odd 12 4
490.2.i.f 8 35.k even 12 4
560.2.g.e 4 20.e even 4 2
630.2.g.g 4 15.e even 4 2
2240.2.g.i 4 40.k even 4 2
2240.2.g.j 4 40.i odd 4 2
2450.2.a.bl 2 7.b odd 2 1
2450.2.a.bq 2 35.c odd 2 1
2800.2.a.bl 2 20.d odd 2 1
2800.2.a.bm 2 4.b odd 2 1
3150.2.a.bs 2 15.d odd 2 1
3150.2.a.bt 2 3.b odd 2 1
5040.2.t.t 4 60.l odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 6$$ T3^2 - 6 $$T_{13}^{2} - 4T_{13} - 2$$ T13^2 - 4*T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 6$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 24$$
$13$ $$T^{2} - 4T - 2$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} - 8T + 10$$
$23$ $$T^{2} + 4T - 20$$
$29$ $$T^{2} - 4T - 20$$
$31$ $$T^{2} - 8T - 8$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} + 12T + 12$$
$43$ $$T^{2} - 8T - 8$$
$47$ $$T^{2} - 8T - 8$$
$53$ $$T^{2} + 12T + 12$$
$59$ $$T^{2} + 8T + 10$$
$61$ $$T^{2} - 12T + 30$$
$67$ $$(T + 8)^{2}$$
$71$ $$T^{2} + 12T + 12$$
$73$ $$T^{2} + 4T - 20$$
$79$ $$T^{2} - 4T - 20$$
$83$ $$T^{2} - 6$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} - 12T - 60$$