# Properties

 Label 350.2.j.a Level $350$ Weight $2$ Character orbit 350.j Analytic conductor $2.795$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(149,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2]))

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

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

chi := DirichletCharacter("350.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 350.j (of order $$6$$, degree $$2$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} - 2 q^{6} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10})$$ q + z * q^2 + (2*z^3 - 2*z) * q^3 + z^2 * q^4 - 2 * q^6 + (z^3 + 2*z) * q^7 + z^3 * q^8 + (-z^2 + 1) * q^9 $$q + \zeta_{12} q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{3} + \zeta_{12}^{2} q^{4} - 2 q^{6} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} - 3 \zeta_{12}^{2} q^{11} - 2 \zeta_{12} q^{12} + 5 \zeta_{12}^{3} q^{13} + (3 \zeta_{12}^{2} - 1) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{17} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{18} + (\zeta_{12}^{2} - 1) q^{19} + ( - 2 \zeta_{12}^{2} - 4) q^{21} - 3 \zeta_{12}^{3} q^{22} - 3 \zeta_{12} q^{23} - 2 \zeta_{12}^{2} q^{24} + (5 \zeta_{12}^{2} - 5) q^{26} - 4 \zeta_{12}^{3} q^{27} + (3 \zeta_{12}^{3} - \zeta_{12}) q^{28} + 6 q^{29} + 4 \zeta_{12}^{2} q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + 6 \zeta_{12} q^{33} - 6 q^{34} + q^{36} + 11 \zeta_{12} q^{37} + (\zeta_{12}^{3} - \zeta_{12}) q^{38} - 10 \zeta_{12}^{2} q^{39} + 3 q^{41} + ( - 2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{42} - 10 \zeta_{12}^{3} q^{43} + ( - 3 \zeta_{12}^{2} + 3) q^{44} - 3 \zeta_{12}^{2} q^{46} + 3 \zeta_{12} q^{47} - 2 \zeta_{12}^{3} q^{48} + (8 \zeta_{12}^{2} - 5) q^{49} + ( - 12 \zeta_{12}^{2} + 12) q^{51} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{52} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{53} + ( - 4 \zeta_{12}^{2} + 4) q^{54} + (2 \zeta_{12}^{2} - 3) q^{56} - 2 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + ( - 4 \zeta_{12}^{2} + 4) q^{61} + 4 \zeta_{12}^{3} q^{62} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{63} - q^{64} + 6 \zeta_{12}^{2} q^{66} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{67} - 6 \zeta_{12} q^{68} + 6 q^{69} + 12 q^{71} + \zeta_{12} q^{72} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{73} + 11 \zeta_{12}^{2} q^{74} - q^{76} + ( - 9 \zeta_{12}^{3} + 3 \zeta_{12}) q^{77} - 10 \zeta_{12}^{3} q^{78} + (10 \zeta_{12}^{2} - 10) q^{79} + 11 \zeta_{12}^{2} q^{81} + 3 \zeta_{12} q^{82} - 12 \zeta_{12}^{3} q^{83} + ( - 6 \zeta_{12}^{2} + 2) q^{84} + ( - 10 \zeta_{12}^{2} + 10) q^{86} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{87} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{88} + ( - 6 \zeta_{12}^{2} + 6) q^{89} + (10 \zeta_{12}^{2} - 15) q^{91} - 3 \zeta_{12}^{3} q^{92} - 8 \zeta_{12} q^{93} + 3 \zeta_{12}^{2} q^{94} + ( - 2 \zeta_{12}^{2} + 2) q^{96} - 14 \zeta_{12}^{3} q^{97} + (8 \zeta_{12}^{3} - 5 \zeta_{12}) q^{98} - 3 q^{99} +O(q^{100})$$ q + z * q^2 + (2*z^3 - 2*z) * q^3 + z^2 * q^4 - 2 * q^6 + (z^3 + 2*z) * q^7 + z^3 * q^8 + (-z^2 + 1) * q^9 - 3*z^2 * q^11 - 2*z * q^12 + 5*z^3 * q^13 + (3*z^2 - 1) * q^14 + (z^2 - 1) * q^16 + (6*z^3 - 6*z) * q^17 + (-z^3 + z) * q^18 + (z^2 - 1) * q^19 + (-2*z^2 - 4) * q^21 - 3*z^3 * q^22 - 3*z * q^23 - 2*z^2 * q^24 + (5*z^2 - 5) * q^26 - 4*z^3 * q^27 + (3*z^3 - z) * q^28 + 6 * q^29 + 4*z^2 * q^31 + (z^3 - z) * q^32 + 6*z * q^33 - 6 * q^34 + q^36 + 11*z * q^37 + (z^3 - z) * q^38 - 10*z^2 * q^39 + 3 * q^41 + (-2*z^3 - 4*z) * q^42 - 10*z^3 * q^43 + (-3*z^2 + 3) * q^44 - 3*z^2 * q^46 + 3*z * q^47 - 2*z^3 * q^48 + (8*z^2 - 5) * q^49 + (-12*z^2 + 12) * q^51 + (5*z^3 - 5*z) * q^52 + (-3*z^3 + 3*z) * q^53 + (-4*z^2 + 4) * q^54 + (2*z^2 - 3) * q^56 - 2*z^3 * q^57 + 6*z * q^58 + (-4*z^2 + 4) * q^61 + 4*z^3 * q^62 + (-2*z^3 + 3*z) * q^63 - q^64 + 6*z^2 * q^66 + (-4*z^3 + 4*z) * q^67 - 6*z * q^68 + 6 * q^69 + 12 * q^71 + z * q^72 + (4*z^3 - 4*z) * q^73 + 11*z^2 * q^74 - q^76 + (-9*z^3 + 3*z) * q^77 - 10*z^3 * q^78 + (10*z^2 - 10) * q^79 + 11*z^2 * q^81 + 3*z * q^82 - 12*z^3 * q^83 + (-6*z^2 + 2) * q^84 + (-10*z^2 + 10) * q^86 + (12*z^3 - 12*z) * q^87 + (-3*z^3 + 3*z) * q^88 + (-6*z^2 + 6) * q^89 + (10*z^2 - 15) * q^91 - 3*z^3 * q^92 - 8*z * q^93 + 3*z^2 * q^94 + (-2*z^2 + 2) * q^96 - 14*z^3 * q^97 + (8*z^3 - 5*z) * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 8 q^{6} + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 8 * q^6 + 2 * q^9 $$4 q + 2 q^{4} - 8 q^{6} + 2 q^{9} - 6 q^{11} + 2 q^{14} - 2 q^{16} - 2 q^{19} - 20 q^{21} - 4 q^{24} - 10 q^{26} + 24 q^{29} + 8 q^{31} - 24 q^{34} + 4 q^{36} - 20 q^{39} + 12 q^{41} + 6 q^{44} - 6 q^{46} - 4 q^{49} + 24 q^{51} + 8 q^{54} - 8 q^{56} + 8 q^{61} - 4 q^{64} + 12 q^{66} + 24 q^{69} + 48 q^{71} + 22 q^{74} - 4 q^{76} - 20 q^{79} + 22 q^{81} - 4 q^{84} + 20 q^{86} + 12 q^{89} - 40 q^{91} + 6 q^{94} + 4 q^{96} - 12 q^{99}+O(q^{100})$$ 4 * q + 2 * q^4 - 8 * q^6 + 2 * q^9 - 6 * q^11 + 2 * q^14 - 2 * q^16 - 2 * q^19 - 20 * q^21 - 4 * q^24 - 10 * q^26 + 24 * q^29 + 8 * q^31 - 24 * q^34 + 4 * q^36 - 20 * q^39 + 12 * q^41 + 6 * q^44 - 6 * q^46 - 4 * q^49 + 24 * q^51 + 8 * q^54 - 8 * q^56 + 8 * q^61 - 4 * q^64 + 12 * q^66 + 24 * q^69 + 48 * q^71 + 22 * q^74 - 4 * q^76 - 20 * q^79 + 22 * q^81 - 4 * q^84 + 20 * q^86 + 12 * q^89 - 40 * q^91 + 6 * q^94 + 4 * q^96 - 12 * q^99

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 1.73205 1.00000i 0.500000 + 0.866025i 0 −2.00000 −1.73205 2.00000i 1.00000i 0.500000 0.866025i 0
149.2 0.866025 + 0.500000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0 −2.00000 1.73205 + 2.00000i 1.00000i 0.500000 0.866025i 0
249.1 −0.866025 + 0.500000i 1.73205 + 1.00000i 0.500000 0.866025i 0 −2.00000 −1.73205 + 2.00000i 1.00000i 0.500000 + 0.866025i 0
249.2 0.866025 0.500000i −1.73205 1.00000i 0.500000 0.866025i 0 −2.00000 1.73205 2.00000i 1.00000i 0.500000 + 0.866025i 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
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.j.a 4
5.b even 2 1 inner 350.2.j.a 4
5.c odd 4 1 70.2.e.b 2
5.c odd 4 1 350.2.e.h 2
7.c even 3 1 inner 350.2.j.a 4
7.c even 3 1 2450.2.c.f 2
7.d odd 6 1 2450.2.c.p 2
15.e even 4 1 630.2.k.e 2
20.e even 4 1 560.2.q.d 2
35.f even 4 1 490.2.e.a 2
35.i odd 6 1 2450.2.c.p 2
35.j even 6 1 inner 350.2.j.a 4
35.j even 6 1 2450.2.c.f 2
35.k even 12 1 490.2.a.j 1
35.k even 12 1 490.2.e.a 2
35.k even 12 1 2450.2.a.f 1
35.l odd 12 1 70.2.e.b 2
35.l odd 12 1 350.2.e.h 2
35.l odd 12 1 490.2.a.g 1
35.l odd 12 1 2450.2.a.p 1
105.w odd 12 1 4410.2.a.c 1
105.x even 12 1 630.2.k.e 2
105.x even 12 1 4410.2.a.m 1
140.w even 12 1 560.2.q.d 2
140.w even 12 1 3920.2.a.be 1
140.x odd 12 1 3920.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 5.c odd 4 1
70.2.e.b 2 35.l odd 12 1
350.2.e.h 2 5.c odd 4 1
350.2.e.h 2 35.l odd 12 1
350.2.j.a 4 1.a even 1 1 trivial
350.2.j.a 4 5.b even 2 1 inner
350.2.j.a 4 7.c even 3 1 inner
350.2.j.a 4 35.j even 6 1 inner
490.2.a.g 1 35.l odd 12 1
490.2.a.j 1 35.k even 12 1
490.2.e.a 2 35.f even 4 1
490.2.e.a 2 35.k even 12 1
560.2.q.d 2 20.e even 4 1
560.2.q.d 2 140.w even 12 1
630.2.k.e 2 15.e even 4 1
630.2.k.e 2 105.x even 12 1
2450.2.a.f 1 35.k even 12 1
2450.2.a.p 1 35.l odd 12 1
2450.2.c.f 2 7.c even 3 1
2450.2.c.f 2 35.j even 6 1
2450.2.c.p 2 7.d odd 6 1
2450.2.c.p 2 35.i odd 6 1
3920.2.a.g 1 140.x odd 12 1
3920.2.a.be 1 140.w even 12 1
4410.2.a.c 1 105.w odd 12 1
4410.2.a.m 1 105.x even 12 1

## Hecke kernels

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

 $$T_{3}^{4} - 4T_{3}^{2} + 16$$ T3^4 - 4*T3^2 + 16 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9 $$T_{13}^{2} + 25$$ T13^2 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$(T^{2} + 3 T + 9)^{2}$$
$13$ $$(T^{2} + 25)^{2}$$
$17$ $$T^{4} - 36T^{2} + 1296$$
$19$ $$(T^{2} + T + 1)^{2}$$
$23$ $$T^{4} - 9T^{2} + 81$$
$29$ $$(T - 6)^{4}$$
$31$ $$(T^{2} - 4 T + 16)^{2}$$
$37$ $$T^{4} - 121 T^{2} + 14641$$
$41$ $$(T - 3)^{4}$$
$43$ $$(T^{2} + 100)^{2}$$
$47$ $$T^{4} - 9T^{2} + 81$$
$53$ $$T^{4} - 9T^{2} + 81$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 4 T + 16)^{2}$$
$67$ $$T^{4} - 16T^{2} + 256$$
$71$ $$(T - 12)^{4}$$
$73$ $$T^{4} - 16T^{2} + 256$$
$79$ $$(T^{2} + 10 T + 100)^{2}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$(T^{2} - 6 T + 36)^{2}$$
$97$ $$(T^{2} + 196)^{2}$$