# Properties

 Label 784.2.x.f Level $784$ Weight $2$ Character orbit 784.x Analytic conductor $6.260$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(165,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 8]))

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

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

chi := DirichletCharacter("784.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 784.x (of order $$12$$, degree $$4$$, 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: $$6.26027151847$$ Analytic rank: $$0$$ Dimension: $$4$$ 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: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{5} + 2 q^{6} + ( - 2 \zeta_{12}^{3} + 2) q^{8} - \zeta_{12} q^{9} +O(q^{10})$$ q + (-z^2 + z + 1) * q^2 + (-z^3 + z^2 + z) * q^3 + (-2*z^3 + 2*z) * q^4 + (-z^2 + z + 1) * q^5 + 2 * q^6 + (-2*z^3 + 2) * q^8 - z * q^9 $$q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{5} + 2 q^{6} + ( - 2 \zeta_{12}^{3} + 2) q^{8} - \zeta_{12} q^{9} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{10} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{11} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{12} + (\zeta_{12}^{3} - 1) q^{13} + 2 q^{15} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + 2 \zeta_{12}^{2} q^{17} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{18} + (3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{19} + ( - 2 \zeta_{12}^{3} + 2) q^{20} - 2 \zeta_{12}^{3} q^{22} - 6 \zeta_{12} q^{23} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{24} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{25} + (2 \zeta_{12}^{2} - 2) q^{26} + ( - 4 \zeta_{12}^{3} - 4) q^{27} + ( - 3 \zeta_{12}^{3} + 3) q^{29} + ( - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{30} + 8 \zeta_{12}^{2} q^{31} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{32} + ( - 2 \zeta_{12}^{2} + 2) q^{33} + (2 \zeta_{12}^{3} + 2) q^{34} - 2 q^{36} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{37} + 6 \zeta_{12}^{2} q^{38} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{39} + ( - 4 \zeta_{12}^{2} + 4) q^{40} + (5 \zeta_{12}^{3} + 5) q^{43} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{44} + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{45} + (6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12}) q^{46} + (8 \zeta_{12}^{2} - 8) q^{47} + ( - 4 \zeta_{12}^{3} + 4) q^{48} + (3 \zeta_{12}^{3} - 3) q^{50} + (2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{51} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{52} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12}) q^{53} - 8 \zeta_{12} q^{54} - 2 \zeta_{12}^{3} q^{55} + 6 \zeta_{12}^{3} q^{57} + ( - 6 \zeta_{12}^{2} + 6) q^{58} + (3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{59} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{60} + ( - 9 \zeta_{12}^{2} - 9 \zeta_{12} + 9) q^{61} + (8 \zeta_{12}^{3} + 8) q^{62} - 8 \zeta_{12}^{3} q^{64} + (2 \zeta_{12}^{2} - 2) q^{65} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{66} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12}) q^{67} + 4 \zeta_{12} q^{68} + ( - 6 \zeta_{12}^{3} - 6) q^{69} - 10 \zeta_{12}^{3} q^{71} + (2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{72} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{73} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{74} + (3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{75} + (6 \zeta_{12}^{3} + 6) q^{76} + (2 \zeta_{12}^{3} - 2) q^{78} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12}) q^{80} - 5 \zeta_{12}^{2} q^{81} + (\zeta_{12}^{3} - 1) q^{83} + (2 \zeta_{12}^{3} + 2) q^{85} + 10 \zeta_{12} q^{86} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{87} - 4 \zeta_{12}^{2} q^{88} - 4 \zeta_{12} q^{89} - 2 q^{90} - 12 q^{92} + (8 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{93} + (8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12}) q^{94} + 6 \zeta_{12}^{2} q^{95} + ( - 8 \zeta_{12}^{2} + 8) q^{96} - 2 q^{97} + (\zeta_{12}^{3} - 1) q^{99} +O(q^{100})$$ q + (-z^2 + z + 1) * q^2 + (-z^3 + z^2 + z) * q^3 + (-2*z^3 + 2*z) * q^4 + (-z^2 + z + 1) * q^5 + 2 * q^6 + (-2*z^3 + 2) * q^8 - z * q^9 + (-2*z^3 + 2*z) * q^10 + (-z^3 - z^2 + z) * q^11 + (-2*z^2 + 2*z + 2) * q^12 + (z^3 - 1) * q^13 + 2 * q^15 + (-4*z^2 + 4) * q^16 + 2*z^2 * q^17 + (z^3 - z^2 - z) * q^18 + (3*z^2 + 3*z - 3) * q^19 + (-2*z^3 + 2) * q^20 - 2*z^3 * q^22 - 6*z * q^23 + (-4*z^3 + 4*z) * q^24 + (3*z^3 - 3*z) * q^25 + (2*z^2 - 2) * q^26 + (-4*z^3 - 4) * q^27 + (-3*z^3 + 3) * q^29 + (-2*z^2 + 2*z + 2) * q^30 + 8*z^2 * q^31 + (-4*z^3 - 4*z^2 + 4*z) * q^32 + (-2*z^2 + 2) * q^33 + (2*z^3 + 2) * q^34 - 2 * q^36 + (3*z^2 - 3*z - 3) * q^37 + 6*z^2 * q^38 + (2*z^3 - 2*z) * q^39 + (-4*z^2 + 4) * q^40 + (5*z^3 + 5) * q^43 + (-2*z^2 - 2*z + 2) * q^44 + (z^3 - z^2 - z) * q^45 + (6*z^3 - 6*z^2 - 6*z) * q^46 + (8*z^2 - 8) * q^47 + (-4*z^3 + 4) * q^48 + (3*z^3 - 3) * q^50 + (2*z^2 + 2*z - 2) * q^51 + (2*z^3 + 2*z^2 - 2*z) * q^52 + (5*z^3 + 5*z^2 - 5*z) * q^53 - 8*z * q^54 - 2*z^3 * q^55 + 6*z^3 * q^57 + (-6*z^2 + 6) * q^58 + (3*z^3 + 3*z^2 - 3*z) * q^59 + (-4*z^3 + 4*z) * q^60 + (-9*z^2 - 9*z + 9) * q^61 + (8*z^3 + 8) * q^62 - 8*z^3 * q^64 + (2*z^2 - 2) * q^65 + (-2*z^3 - 2*z^2 + 2*z) * q^66 + (-5*z^3 + 5*z^2 + 5*z) * q^67 + 4*z * q^68 + (-6*z^3 - 6) * q^69 - 10*z^3 * q^71 + (2*z^2 - 2*z - 2) * q^72 + (4*z^3 - 4*z) * q^73 + (6*z^3 - 6*z) * q^74 + (3*z^2 - 3*z - 3) * q^75 + (6*z^3 + 6) * q^76 + (2*z^3 - 2) * q^78 + (-4*z^3 - 4*z^2 + 4*z) * q^80 - 5*z^2 * q^81 + (z^3 - 1) * q^83 + (2*z^3 + 2) * q^85 + 10*z * q^86 + (-6*z^3 + 6*z) * q^87 - 4*z^2 * q^88 - 4*z * q^89 - 2 * q^90 - 12 * q^92 + (8*z^2 + 8*z - 8) * q^93 + (8*z^3 + 8*z^2 - 8*z) * q^94 + 6*z^2 * q^95 + (-8*z^2 + 8) * q^96 - 2 * q^97 + (z^3 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{3} + 2 q^{5} + 8 q^{6} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^3 + 2 * q^5 + 8 * q^6 + 8 * q^8 $$4 q + 2 q^{2} + 2 q^{3} + 2 q^{5} + 8 q^{6} + 8 q^{8} - 2 q^{11} + 4 q^{12} - 4 q^{13} + 8 q^{15} + 8 q^{16} + 4 q^{17} - 2 q^{18} - 6 q^{19} + 8 q^{20} - 4 q^{26} - 16 q^{27} + 12 q^{29} + 4 q^{30} + 16 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{34} - 8 q^{36} - 6 q^{37} + 12 q^{38} + 8 q^{40} + 20 q^{43} + 4 q^{44} - 2 q^{45} - 12 q^{46} - 16 q^{47} + 16 q^{48} - 12 q^{50} - 4 q^{51} + 4 q^{52} + 10 q^{53} + 12 q^{58} + 6 q^{59} + 18 q^{61} + 32 q^{62} - 4 q^{65} - 4 q^{66} + 10 q^{67} - 24 q^{69} - 4 q^{72} - 6 q^{75} + 24 q^{76} - 8 q^{78} - 8 q^{80} - 10 q^{81} - 4 q^{83} + 8 q^{85} - 8 q^{88} - 8 q^{90} - 48 q^{92} - 16 q^{93} + 16 q^{94} + 12 q^{95} + 16 q^{96} - 8 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^3 + 2 * q^5 + 8 * q^6 + 8 * q^8 - 2 * q^11 + 4 * q^12 - 4 * q^13 + 8 * q^15 + 8 * q^16 + 4 * q^17 - 2 * q^18 - 6 * q^19 + 8 * q^20 - 4 * q^26 - 16 * q^27 + 12 * q^29 + 4 * q^30 + 16 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^34 - 8 * q^36 - 6 * q^37 + 12 * q^38 + 8 * q^40 + 20 * q^43 + 4 * q^44 - 2 * q^45 - 12 * q^46 - 16 * q^47 + 16 * q^48 - 12 * q^50 - 4 * q^51 + 4 * q^52 + 10 * q^53 + 12 * q^58 + 6 * q^59 + 18 * q^61 + 32 * q^62 - 4 * q^65 - 4 * q^66 + 10 * q^67 - 24 * q^69 - 4 * q^72 - 6 * q^75 + 24 * q^76 - 8 * q^78 - 8 * q^80 - 10 * q^81 - 4 * q^83 + 8 * q^85 - 8 * q^88 - 8 * q^90 - 48 * q^92 - 16 * q^93 + 16 * q^94 + 12 * q^95 + 16 * q^96 - 8 * q^97 - 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$687$$ $$689$$ $$\chi(n)$$ $$\zeta_{12}^{3}$$ $$1$$ $$-1 + \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1
 −0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
−0.366025 + 1.36603i −0.366025 1.36603i −1.73205 1.00000i −0.366025 + 1.36603i 2.00000 0 2.00000 2.00000i 0.866025 0.500000i −1.73205 1.00000i
373.1 1.36603 0.366025i 1.36603 + 0.366025i 1.73205 1.00000i 1.36603 0.366025i 2.00000 0 2.00000 2.00000i −0.866025 0.500000i 1.73205 1.00000i
557.1 1.36603 + 0.366025i 1.36603 0.366025i 1.73205 + 1.00000i 1.36603 + 0.366025i 2.00000 0 2.00000 + 2.00000i −0.866025 + 0.500000i 1.73205 + 1.00000i
765.1 −0.366025 1.36603i −0.366025 + 1.36603i −1.73205 + 1.00000i −0.366025 1.36603i 2.00000 0 2.00000 + 2.00000i 0.866025 + 0.500000i −1.73205 + 1.00000i
 $$n$$: 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
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.f 4
7.b odd 2 1 784.2.x.c 4
7.c even 3 1 16.2.e.a 2
7.c even 3 1 inner 784.2.x.f 4
7.d odd 6 1 784.2.m.b 2
7.d odd 6 1 784.2.x.c 4
16.e even 4 1 inner 784.2.x.f 4
21.h odd 6 1 144.2.k.a 2
28.g odd 6 1 64.2.e.a 2
35.j even 6 1 400.2.l.c 2
35.l odd 12 1 400.2.q.a 2
35.l odd 12 1 400.2.q.b 2
56.k odd 6 1 128.2.e.a 2
56.p even 6 1 128.2.e.b 2
84.n even 6 1 576.2.k.a 2
112.l odd 4 1 784.2.x.c 4
112.u odd 12 1 64.2.e.a 2
112.u odd 12 1 128.2.e.a 2
112.w even 12 1 16.2.e.a 2
112.w even 12 1 128.2.e.b 2
112.w even 12 1 inner 784.2.x.f 4
112.x odd 12 1 784.2.m.b 2
112.x odd 12 1 784.2.x.c 4
140.p odd 6 1 1600.2.l.a 2
140.w even 12 1 1600.2.q.a 2
140.w even 12 1 1600.2.q.b 2
168.s odd 6 1 1152.2.k.b 2
168.v even 6 1 1152.2.k.a 2
224.bd even 24 2 1024.2.a.b 2
224.bd even 24 2 1024.2.b.e 2
224.bf odd 24 2 1024.2.a.e 2
224.bf odd 24 2 1024.2.b.b 2
336.bt odd 12 1 144.2.k.a 2
336.bt odd 12 1 1152.2.k.b 2
336.bu even 12 1 576.2.k.a 2
336.bu even 12 1 1152.2.k.a 2
560.cf even 12 1 1600.2.q.a 2
560.cg odd 12 1 400.2.q.a 2
560.cr even 12 1 400.2.l.c 2
560.cs odd 12 1 1600.2.l.a 2
560.cy odd 12 1 400.2.q.b 2
560.db even 12 1 1600.2.q.b 2
672.ce odd 24 2 9216.2.a.d 2
672.ch even 24 2 9216.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 7.c even 3 1
16.2.e.a 2 112.w even 12 1
64.2.e.a 2 28.g odd 6 1
64.2.e.a 2 112.u odd 12 1
128.2.e.a 2 56.k odd 6 1
128.2.e.a 2 112.u odd 12 1
128.2.e.b 2 56.p even 6 1
128.2.e.b 2 112.w even 12 1
144.2.k.a 2 21.h odd 6 1
144.2.k.a 2 336.bt odd 12 1
400.2.l.c 2 35.j even 6 1
400.2.l.c 2 560.cr even 12 1
400.2.q.a 2 35.l odd 12 1
400.2.q.a 2 560.cg odd 12 1
400.2.q.b 2 35.l odd 12 1
400.2.q.b 2 560.cy odd 12 1
576.2.k.a 2 84.n even 6 1
576.2.k.a 2 336.bu even 12 1
784.2.m.b 2 7.d odd 6 1
784.2.m.b 2 112.x odd 12 1
784.2.x.c 4 7.b odd 2 1
784.2.x.c 4 7.d odd 6 1
784.2.x.c 4 112.l odd 4 1
784.2.x.c 4 112.x odd 12 1
784.2.x.f 4 1.a even 1 1 trivial
784.2.x.f 4 7.c even 3 1 inner
784.2.x.f 4 16.e even 4 1 inner
784.2.x.f 4 112.w even 12 1 inner
1024.2.a.b 2 224.bd even 24 2
1024.2.a.e 2 224.bf odd 24 2
1024.2.b.b 2 224.bf odd 24 2
1024.2.b.e 2 224.bd even 24 2
1152.2.k.a 2 168.v even 6 1
1152.2.k.a 2 336.bu even 12 1
1152.2.k.b 2 168.s odd 6 1
1152.2.k.b 2 336.bt odd 12 1
1600.2.l.a 2 140.p odd 6 1
1600.2.l.a 2 560.cs odd 12 1
1600.2.q.a 2 140.w even 12 1
1600.2.q.a 2 560.cf even 12 1
1600.2.q.b 2 140.w even 12 1
1600.2.q.b 2 560.db even 12 1
9216.2.a.d 2 672.ce odd 24 2
9216.2.a.s 2 672.ch even 24 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(784, [\chi])$$:

 $$T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4$$ T3^4 - 2*T3^3 + 2*T3^2 - 4*T3 + 4 $$T_{5}^{4} - 2T_{5}^{3} + 2T_{5}^{2} - 4T_{5} + 4$$ T5^4 - 2*T5^3 + 2*T5^2 - 4*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$5$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$13$ $$(T^{2} + 2 T + 2)^{2}$$
$17$ $$(T^{2} - 2 T + 4)^{2}$$
$19$ $$T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} - 6 T + 18)^{2}$$
$31$ $$(T^{2} - 8 T + 64)^{2}$$
$37$ $$T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 10 T + 50)^{2}$$
$47$ $$(T^{2} + 8 T + 64)^{2}$$
$53$ $$T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500$$
$59$ $$T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324$$
$61$ $$T^{4} - 18 T^{3} + 162 T^{2} + \cdots + 26244$$
$67$ $$T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500$$
$71$ $$(T^{2} + 100)^{2}$$
$73$ $$T^{4} - 16T^{2} + 256$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 2 T + 2)^{2}$$
$89$ $$T^{4} - 16T^{2} + 256$$
$97$ $$(T + 2)^{4}$$