# Properties

 Label 966.2.a.n Level $966$ Weight $2$ Character orbit 966.a Self dual yes Analytic conductor $7.714$ 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] = [966,2,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("966.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.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: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −2.70156 3.70156
−1.00000 1.00000 1.00000 −2.70156 −1.00000 1.00000 −1.00000 1.00000 2.70156
1.2 −1.00000 1.00000 1.00000 3.70156 −1.00000 1.00000 −1.00000 1.00000 −3.70156
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## 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 966.2.a.n 2
3.b odd 2 1 2898.2.a.bb 2
4.b odd 2 1 7728.2.a.bc 2
7.b odd 2 1 6762.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.n 2 1.a even 1 1 trivial
2898.2.a.bb 2 3.b odd 2 1
6762.2.a.bo 2 7.b odd 2 1
7728.2.a.bc 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(966))$$:

 $$T_{5}^{2} - T_{5} - 10$$ T5^2 - T5 - 10 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} - 5T_{13} - 4$$ T13^2 - 5*T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - T - 10$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} - 5T - 4$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} - 2T - 40$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} - 7T + 2$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} - 15T + 46$$
$41$ $$T^{2} + 7T + 2$$
$43$ $$T^{2} - 3T - 8$$
$47$ $$T^{2} + 3T - 90$$
$53$ $$T^{2} + 6T - 32$$
$59$ $$T^{2} + 2T - 40$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} - 4T - 160$$
$71$ $$T^{2} + 2T - 40$$
$73$ $$T^{2} + 10T - 16$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 14T + 8$$
$89$ $$T^{2} + 10T - 16$$
$97$ $$T^{2} - 19T + 80$$