# Properties

 Label 7942.2.a.bi Level $7942$ Weight $2$ Character orbit 7942.a Self dual yes Analytic conductor $63.417$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7942 = 2 \cdot 11 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7942.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: $$63.4171892853$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 418) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4

## 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
 2.66908 −0.523976 −2.14510
1.00000 −2.66908 1.00000 −4.12398 −2.66908 −4.21417 1.00000 4.12398 −4.12398
1.2 1.00000 0.523976 1.00000 2.72545 0.523976 −4.67750 1.00000 −2.72545 2.72545
1.3 1.00000 2.14510 1.00000 −1.60147 2.14510 2.89167 1.00000 1.60147 −1.60147
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$+1$$
$$19$$ $$-1$$

## 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 7942.2.a.bi 3
19.b odd 2 1 418.2.a.g 3
57.d even 2 1 3762.2.a.bg 3
76.d even 2 1 3344.2.a.q 3
209.d even 2 1 4598.2.a.bo 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.a.g 3 19.b odd 2 1
3344.2.a.q 3 76.d even 2 1
3762.2.a.bg 3 57.d even 2 1
4598.2.a.bo 3 209.d even 2 1
7942.2.a.bi 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7942))$$:

 $$T_{3}^{3} - 6T_{3} + 3$$ T3^3 - 6*T3 + 3 $$T_{5}^{3} + 3T_{5}^{2} - 9T_{5} - 18$$ T5^3 + 3*T5^2 - 9*T5 - 18 $$T_{13}^{3} - 18T_{13} + 29$$ T13^3 - 18*T13 + 29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} - 6T + 3$$
$5$ $$T^{3} + 3 T^{2} + \cdots - 18$$
$7$ $$T^{3} + 6 T^{2} + \cdots - 57$$
$11$ $$(T + 1)^{3}$$
$13$ $$T^{3} - 18T + 29$$
$17$ $$T^{3} + 9 T^{2} + \cdots - 4$$
$19$ $$T^{3}$$
$23$ $$T^{3} + 15 T^{2} + \cdots + 76$$
$29$ $$T^{3} + 6 T^{2} + \cdots - 147$$
$31$ $$T^{3} - 3 T^{2} + \cdots + 134$$
$37$ $$T^{3} - 6 T^{2} + \cdots + 96$$
$41$ $$T^{3} - 9 T^{2} + \cdots + 122$$
$43$ $$T^{3} + 21 T^{2} + \cdots - 184$$
$47$ $$T^{3} + 12 T^{2} + \cdots - 672$$
$53$ $$T^{3} - 9 T^{2} + \cdots + 452$$
$59$ $$T^{3} - 3 T^{2} + \cdots + 64$$
$61$ $$T^{3} - 24 T^{2} + \cdots - 192$$
$67$ $$T^{3} - 96T - 123$$
$71$ $$T^{3} + 21 T^{2} + \cdots - 1306$$
$73$ $$T^{3} + 3 T^{2} + \cdots - 12$$
$79$ $$T^{3} - 6 T^{2} + \cdots + 944$$
$83$ $$T^{3} + 33 T^{2} + \cdots + 1104$$
$89$ $$T^{3} + 6 T^{2} + \cdots - 144$$
$97$ $$T^{3} + 12 T^{2} + \cdots - 256$$