# Properties

 Label 322.2.c.b Level $322$ Weight $2$ Character orbit 322.c Analytic conductor $2.571$ 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] = [322,2,Mod(321,322)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(322, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("322.321");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$322 = 2 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 322.c (of order $$2$$, degree $$1$$, 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: $$2.57118294509$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-7})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} + 9x^{2} - 8x + 2$$ x^4 - 2*x^3 + 9*x^2 - 8*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 9x^{2} - 8x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 4$$ v^2 - v + 4 $$\beta_{2}$$ $$=$$ $$2\nu^{3} - 3\nu^{2} + 17\nu - 8$$ 2*v^3 - 3*v^2 + 17*v - 8 $$\beta_{3}$$ $$=$$ $$-4\nu^{3} + 6\nu^{2} - 32\nu + 15$$ -4*v^3 + 6*v^2 - 32*v + 15
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 1 ) / 2$$ (b3 + 2*b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2\beta_{2} + 2\beta _1 - 7 ) / 2$$ (b3 + 2*b2 + 2*b1 - 7) / 2 $$\nu^{3}$$ $$=$$ $$( -7\beta_{3} - 13\beta_{2} + 3\beta _1 - 11 ) / 2$$ (-7*b3 - 13*b2 + 3*b1 - 11) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/322\mathbb{Z}\right)^\times$$.

 $$n$$ $$185$$ $$281$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 0.5 − 2.73709i 0.5 − 0.0913379i 0.5 + 2.73709i 0.5 + 0.0913379i
−1.00000 1.41421i 1.00000 −3.74166 1.41421i 2.64575i −1.00000 1.00000 3.74166
321.2 −1.00000 1.41421i 1.00000 3.74166 1.41421i 2.64575i −1.00000 1.00000 −3.74166
321.3 −1.00000 1.41421i 1.00000 −3.74166 1.41421i 2.64575i −1.00000 1.00000 3.74166
321.4 −1.00000 1.41421i 1.00000 3.74166 1.41421i 2.64575i −1.00000 1.00000 −3.74166
 $$n$$: 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.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.c.b 4
3.b odd 2 1 2898.2.g.e 4
4.b odd 2 1 2576.2.f.c 4
7.b odd 2 1 inner 322.2.c.b 4
21.c even 2 1 2898.2.g.e 4
23.b odd 2 1 inner 322.2.c.b 4
28.d even 2 1 2576.2.f.c 4
69.c even 2 1 2898.2.g.e 4
92.b even 2 1 2576.2.f.c 4
161.c even 2 1 inner 322.2.c.b 4
483.c odd 2 1 2898.2.g.e 4
644.h odd 2 1 2576.2.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.c.b 4 1.a even 1 1 trivial
322.2.c.b 4 7.b odd 2 1 inner
322.2.c.b 4 23.b odd 2 1 inner
322.2.c.b 4 161.c even 2 1 inner
2576.2.f.c 4 4.b odd 2 1
2576.2.f.c 4 28.d even 2 1
2576.2.f.c 4 92.b even 2 1
2576.2.f.c 4 644.h odd 2 1
2898.2.g.e 4 3.b odd 2 1
2898.2.g.e 4 21.c even 2 1
2898.2.g.e 4 69.c even 2 1
2898.2.g.e 4 483.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(322, [\chi])$$:

 $$T_{3}^{2} + 2$$ T3^2 + 2 $$T_{5}^{2} - 14$$ T5^2 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T^{2} + 2)^{2}$$
$5$ $$(T^{2} - 14)^{2}$$
$7$ $$(T^{2} + 7)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 32)^{2}$$
$17$ $$(T^{2} - 14)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 8 T + 23)^{2}$$
$29$ $$(T + 8)^{4}$$
$31$ $$(T^{2} + 2)^{2}$$
$37$ $$(T^{2} + 112)^{2}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$(T^{2} + 28)^{2}$$
$47$ $$(T^{2} + 50)^{2}$$
$53$ $$(T^{2} + 112)^{2}$$
$59$ $$(T^{2} + 98)^{2}$$
$61$ $$(T^{2} - 14)^{2}$$
$67$ $$T^{4}$$
$71$ $$(T + 8)^{4}$$
$73$ $$(T^{2} + 128)^{2}$$
$79$ $$(T^{2} + 112)^{2}$$
$83$ $$(T^{2} - 224)^{2}$$
$89$ $$(T^{2} - 126)^{2}$$
$97$ $$(T^{2} - 14)^{2}$$