# Properties

 Label 930.2.h.a Level $930$ Weight $2$ Character orbit 930.h Analytic conductor $7.426$ 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] = [930,2,Mod(371,930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("930.371");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.h (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: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

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

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
371.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
1.00000i −1.22474 1.22474i −1.00000 1.00000i −1.22474 + 1.22474i −4.00000 1.00000i 3.00000i 1.00000
371.2 1.00000i 1.22474 + 1.22474i −1.00000 1.00000i 1.22474 1.22474i −4.00000 1.00000i 3.00000i 1.00000
371.3 1.00000i −1.22474 + 1.22474i −1.00000 1.00000i −1.22474 1.22474i −4.00000 1.00000i 3.00000i 1.00000
371.4 1.00000i 1.22474 1.22474i −1.00000 1.00000i 1.22474 + 1.22474i −4.00000 1.00000i 3.00000i 1.00000
 $$n$$: 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
3.b odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.h.a 4
3.b odd 2 1 inner 930.2.h.a 4
31.b odd 2 1 inner 930.2.h.a 4
93.c even 2 1 inner 930.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.a 4 1.a even 1 1 trivial
930.2.h.a 4 3.b odd 2 1 inner
930.2.h.a 4 31.b odd 2 1 inner
930.2.h.a 4 93.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 9$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T + 4)^{4}$$
$11$ $$(T^{2} - 24)^{2}$$
$13$ $$(T^{2} + 6)^{2}$$
$17$ $$(T^{2} - 54)^{2}$$
$19$ $$(T + 2)^{4}$$
$23$ $$(T^{2} - 54)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 10 T + 31)^{2}$$
$37$ $$(T^{2} + 6)^{2}$$
$41$ $$(T^{2} + 144)^{2}$$
$43$ $$(T^{2} + 6)^{2}$$
$47$ $$(T^{2} + 144)^{2}$$
$53$ $$(T^{2} - 6)^{2}$$
$59$ $$(T^{2} + 36)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T - 4)^{4}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 6)^{2}$$
$79$ $$(T^{2} + 24)^{2}$$
$83$ $$(T^{2} - 150)^{2}$$
$89$ $$(T^{2} - 24)^{2}$$
$97$ $$(T - 14)^{4}$$