# Properties

 Label 966.2.i.h Level $966$ Weight $2$ Character orbit 966.i Analytic conductor $7.714$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(277,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("966.277");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.i (of order $$3$$, degree $$2$$, 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: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} - 1) q^{4} - \beta_{2} q^{5} + q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b2 - 1) * q^3 + (-b2 - 1) * q^4 - b2 * q^5 + q^6 + (2*b3 - b2 + b1 - 1) * q^7 + q^8 + b2 * q^9 $$q + \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{3} + ( - \beta_{2} - 1) q^{4} - \beta_{2} q^{5} + q^{6} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + q^{8} + \beta_{2} q^{9} + (\beta_{2} + 1) q^{10} + (3 \beta_{2} + 3) q^{11} + \beta_{2} q^{12} + \beta_{3} q^{13} + ( - \beta_{3} - 2 \beta_1 + 1) q^{14} - q^{15} + \beta_{2} q^{16} - \beta_1 q^{17} + ( - \beta_{2} - 1) q^{18} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{19} - q^{20} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{21} - 3 q^{22} - \beta_{2} q^{23} + ( - \beta_{2} - 1) q^{24} + (4 \beta_{2} + 4) q^{25} + ( - \beta_{3} - \beta_1) q^{26} + q^{27} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{28} + (3 \beta_{3} - 5) q^{29} - \beta_{2} q^{30} + (\beta_{2} + 4 \beta_1 + 1) q^{31} + ( - \beta_{2} - 1) q^{32} - 3 \beta_{2} q^{33} - \beta_{3} q^{34} + (\beta_{3} + 2 \beta_1 - 1) q^{35} + q^{36} + (5 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{37} + (2 \beta_{2} - \beta_1 + 2) q^{38} + \beta_1 q^{39} - \beta_{2} q^{40} + ( - 3 \beta_{3} + 6) q^{41} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{42} + ( - 4 \beta_{3} - 2) q^{43} - 3 \beta_{2} q^{44} + (\beta_{2} + 1) q^{45} + (\beta_{2} + 1) q^{46} + (8 \beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{47} + q^{48} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{49} - 4 q^{50} + (\beta_{3} + \beta_1) q^{51} + \beta_1 q^{52} + (5 \beta_{2} + 2 \beta_1 + 5) q^{53} + \beta_{2} q^{54} + 3 q^{55} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{56} + ( - \beta_{3} - 2) q^{57} + ( - 3 \beta_{3} - 5 \beta_{2} - 3 \beta_1) q^{58} + (5 \beta_{2} + 3 \beta_1 + 5) q^{59} + (\beta_{2} + 1) q^{60} - 6 \beta_{2} q^{61} + (4 \beta_{3} - 1) q^{62} + ( - \beta_{3} - 2 \beta_1 + 1) q^{63} + q^{64} + (\beta_{3} + \beta_1) q^{65} + (3 \beta_{2} + 3) q^{66} + (10 \beta_{2} + 4 \beta_1 + 10) q^{67} + (\beta_{3} + \beta_1) q^{68} - q^{69} + (\beta_{3} - \beta_{2} - \beta_1) q^{70} + (7 \beta_{3} - 4) q^{71} + \beta_{2} q^{72} + 4 \beta_1 q^{73} + (2 \beta_{2} - 5 \beta_1 + 2) q^{74} - 4 \beta_{2} q^{75} + ( - \beta_{3} - 2) q^{76} + (3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{77} + \beta_{3} q^{78} + (\beta_{3} + \beta_{2} + \beta_1) q^{79} + (\beta_{2} + 1) q^{80} + ( - \beta_{2} - 1) q^{81} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{82} + (8 \beta_{3} + 1) q^{83} + ( - \beta_{3} - 2 \beta_1 + 1) q^{84} + \beta_{3} q^{85} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{86} + (5 \beta_{2} + 3 \beta_1 + 5) q^{87} + (3 \beta_{2} + 3) q^{88} + ( - 9 \beta_{3} + 2 \beta_{2} - 9 \beta_1) q^{89} - q^{90} + ( - 2 \beta_{2} + \beta_1 + 2) q^{91} - q^{92} + ( - 4 \beta_{3} - \beta_{2} - 4 \beta_1) q^{93} + ( - 2 \beta_{2} - 8 \beta_1 - 2) q^{94} + ( - 2 \beta_{2} + \beta_1 - 2) q^{95} + \beta_{2} q^{96} + (\beta_{3} - 5) q^{97} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{98} - 3 q^{99}+O(q^{100})$$ q + b2 * q^2 + (-b2 - 1) * q^3 + (-b2 - 1) * q^4 - b2 * q^5 + q^6 + (2*b3 - b2 + b1 - 1) * q^7 + q^8 + b2 * q^9 + (b2 + 1) * q^10 + (3*b2 + 3) * q^11 + b2 * q^12 + b3 * q^13 + (-b3 - 2*b1 + 1) * q^14 - q^15 + b2 * q^16 - b1 * q^17 + (-b2 - 1) * q^18 + (b3 - 2*b2 + b1) * q^19 - q^20 + (-b3 + b2 + b1) * q^21 - 3 * q^22 - b2 * q^23 + (-b2 - 1) * q^24 + (4*b2 + 4) * q^25 + (-b3 - b1) * q^26 + q^27 + (-b3 + b2 + b1) * q^28 + (3*b3 - 5) * q^29 - b2 * q^30 + (b2 + 4*b1 + 1) * q^31 + (-b2 - 1) * q^32 - 3*b2 * q^33 - b3 * q^34 + (b3 + 2*b1 - 1) * q^35 + q^36 + (5*b3 - 2*b2 + 5*b1) * q^37 + (2*b2 - b1 + 2) * q^38 + b1 * q^39 - b2 * q^40 + (-3*b3 + 6) * q^41 + (2*b3 - b2 + b1 - 1) * q^42 + (-4*b3 - 2) * q^43 - 3*b2 * q^44 + (b2 + 1) * q^45 + (b2 + 1) * q^46 + (8*b3 + 2*b2 + 8*b1) * q^47 + q^48 + (-2*b3 - 5*b2 + 2*b1) * q^49 - 4 * q^50 + (b3 + b1) * q^51 + b1 * q^52 + (5*b2 + 2*b1 + 5) * q^53 + b2 * q^54 + 3 * q^55 + (2*b3 - b2 + b1 - 1) * q^56 + (-b3 - 2) * q^57 + (-3*b3 - 5*b2 - 3*b1) * q^58 + (5*b2 + 3*b1 + 5) * q^59 + (b2 + 1) * q^60 - 6*b2 * q^61 + (4*b3 - 1) * q^62 + (-b3 - 2*b1 + 1) * q^63 + q^64 + (b3 + b1) * q^65 + (3*b2 + 3) * q^66 + (10*b2 + 4*b1 + 10) * q^67 + (b3 + b1) * q^68 - q^69 + (b3 - b2 - b1) * q^70 + (7*b3 - 4) * q^71 + b2 * q^72 + 4*b1 * q^73 + (2*b2 - 5*b1 + 2) * q^74 - 4*b2 * q^75 + (-b3 - 2) * q^76 + (3*b3 - 3*b2 - 3*b1) * q^77 + b3 * q^78 + (b3 + b2 + b1) * q^79 + (b2 + 1) * q^80 + (-b2 - 1) * q^81 + (3*b3 + 6*b2 + 3*b1) * q^82 + (8*b3 + 1) * q^83 + (-b3 - 2*b1 + 1) * q^84 + b3 * q^85 + (4*b3 - 2*b2 + 4*b1) * q^86 + (5*b2 + 3*b1 + 5) * q^87 + (3*b2 + 3) * q^88 + (-9*b3 + 2*b2 - 9*b1) * q^89 - q^90 + (-2*b2 + b1 + 2) * q^91 - q^92 + (-4*b3 - b2 - 4*b1) * q^93 + (-2*b2 - 8*b1 - 2) * q^94 + (-2*b2 + b1 - 2) * q^95 + b2 * q^96 + (b3 - 5) * q^97 + (4*b3 + 5*b2 + 2*b1 + 5) * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} + 2 q^{10} + 6 q^{11} - 2 q^{12} + 4 q^{14} - 4 q^{15} - 2 q^{16} - 2 q^{18} + 4 q^{19} - 4 q^{20} - 2 q^{21} - 12 q^{22} + 2 q^{23} - 2 q^{24} + 8 q^{25} + 4 q^{27} - 2 q^{28} - 20 q^{29} + 2 q^{30} + 2 q^{31} - 2 q^{32} + 6 q^{33} - 4 q^{35} + 4 q^{36} + 4 q^{37} + 4 q^{38} + 2 q^{40} + 24 q^{41} - 2 q^{42} - 8 q^{43} + 6 q^{44} + 2 q^{45} + 2 q^{46} - 4 q^{47} + 4 q^{48} + 10 q^{49} - 16 q^{50} + 10 q^{53} - 2 q^{54} + 12 q^{55} - 2 q^{56} - 8 q^{57} + 10 q^{58} + 10 q^{59} + 2 q^{60} + 12 q^{61} - 4 q^{62} + 4 q^{63} + 4 q^{64} + 6 q^{66} + 20 q^{67} - 4 q^{69} + 2 q^{70} - 16 q^{71} - 2 q^{72} + 4 q^{74} + 8 q^{75} - 8 q^{76} + 6 q^{77} - 2 q^{79} + 2 q^{80} - 2 q^{81} - 12 q^{82} + 4 q^{83} + 4 q^{84} + 4 q^{86} + 10 q^{87} + 6 q^{88} - 4 q^{89} - 4 q^{90} + 12 q^{91} - 4 q^{92} + 2 q^{93} - 4 q^{94} - 4 q^{95} - 2 q^{96} - 20 q^{97} + 10 q^{98} - 12 q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^9 + 2 * q^10 + 6 * q^11 - 2 * q^12 + 4 * q^14 - 4 * q^15 - 2 * q^16 - 2 * q^18 + 4 * q^19 - 4 * q^20 - 2 * q^21 - 12 * q^22 + 2 * q^23 - 2 * q^24 + 8 * q^25 + 4 * q^27 - 2 * q^28 - 20 * q^29 + 2 * q^30 + 2 * q^31 - 2 * q^32 + 6 * q^33 - 4 * q^35 + 4 * q^36 + 4 * q^37 + 4 * q^38 + 2 * q^40 + 24 * q^41 - 2 * q^42 - 8 * q^43 + 6 * q^44 + 2 * q^45 + 2 * q^46 - 4 * q^47 + 4 * q^48 + 10 * q^49 - 16 * q^50 + 10 * q^53 - 2 * q^54 + 12 * q^55 - 2 * q^56 - 8 * q^57 + 10 * q^58 + 10 * q^59 + 2 * q^60 + 12 * q^61 - 4 * q^62 + 4 * q^63 + 4 * q^64 + 6 * q^66 + 20 * q^67 - 4 * q^69 + 2 * q^70 - 16 * q^71 - 2 * q^72 + 4 * q^74 + 8 * q^75 - 8 * q^76 + 6 * q^77 - 2 * q^79 + 2 * q^80 - 2 * q^81 - 12 * q^82 + 4 * q^83 + 4 * q^84 + 4 * q^86 + 10 * q^87 + 6 * q^88 - 4 * q^89 - 4 * q^90 + 12 * q^91 - 4 * q^92 + 2 * q^93 - 4 * q^94 - 4 * q^95 - 2 * q^96 - 20 * q^97 + 10 * q^98 - 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i −0.707107 − 1.22474i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 −2.62132 0.358719i 1.00000 −0.500000 0.866025i 0.500000 0.866025i
277.2 −0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 1.62132 + 2.09077i 1.00000 −0.500000 0.866025i 0.500000 0.866025i
415.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 −2.62132 + 0.358719i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
415.2 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 1.62132 2.09077i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.i.h 4
7.c even 3 1 inner 966.2.i.h 4
7.c even 3 1 6762.2.a.cc 2
7.d odd 6 1 6762.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.h 4 1.a even 1 1 trivial
966.2.i.h 4 7.c even 3 1 inner
6762.2.a.ca 2 7.d odd 6 1
6762.2.a.cc 2 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$:

 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} + 2 T^{3} + \cdots + 49$$
$11$ $$(T^{2} - 3 T + 9)^{2}$$
$13$ $$(T^{2} - 2)^{2}$$
$17$ $$T^{4} + 2T^{2} + 4$$
$19$ $$T^{4} - 4 T^{3} + \cdots + 4$$
$23$ $$(T^{2} - T + 1)^{2}$$
$29$ $$(T^{2} + 10 T + 7)^{2}$$
$31$ $$T^{4} - 2 T^{3} + \cdots + 961$$
$37$ $$T^{4} - 4 T^{3} + \cdots + 2116$$
$41$ $$(T^{2} - 12 T + 18)^{2}$$
$43$ $$(T^{2} + 4 T - 28)^{2}$$
$47$ $$T^{4} + 4 T^{3} + \cdots + 15376$$
$53$ $$T^{4} - 10 T^{3} + \cdots + 289$$
$59$ $$T^{4} - 10 T^{3} + \cdots + 49$$
$61$ $$(T^{2} - 6 T + 36)^{2}$$
$67$ $$T^{4} - 20 T^{3} + \cdots + 4624$$
$71$ $$(T^{2} + 8 T - 82)^{2}$$
$73$ $$T^{4} + 32T^{2} + 1024$$
$79$ $$T^{4} + 2 T^{3} + \cdots + 1$$
$83$ $$(T^{2} - 2 T - 127)^{2}$$
$89$ $$T^{4} + 4 T^{3} + \cdots + 24964$$
$97$ $$(T^{2} + 10 T + 23)^{2}$$