# Properties

 Label 930.2.d.g Level $930$ Weight $2$ Character orbit 930.d Analytic conductor $7.426$ 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] = [930,2,Mod(559,930)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("930.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.d (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: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 4$$ (b3 + b2) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 4$$ (-b3 + b2) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 4.82843i 1.00000i −1.00000 −1.00000 2.00000i
559.2 1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 0.828427i 1.00000i −1.00000 −1.00000 2.00000i
559.3 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i −1.00000 0.828427i 1.00000i −1.00000 −1.00000 + 2.00000i
559.4 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i −1.00000 4.82843i 1.00000i −1.00000 −1.00000 + 2.00000i
 $$n$$: 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
5.b 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.d.g 4
3.b odd 2 1 2790.2.d.i 4
5.b even 2 1 inner 930.2.d.g 4
5.c odd 4 1 4650.2.a.cc 2
5.c odd 4 1 4650.2.a.cf 2
15.d odd 2 1 2790.2.d.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.g 4 1.a even 1 1 trivial
930.2.d.g 4 5.b even 2 1 inner
2790.2.d.i 4 3.b odd 2 1
2790.2.d.i 4 15.d odd 2 1
4650.2.a.cc 2 5.c odd 4 1
4650.2.a.cf 2 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{4} + 24T_{7}^{2} + 16$$ T7^4 + 24*T7^2 + 16 $$T_{11}^{2} - 4T_{11} - 4$$ T11^2 - 4*T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} - 4 T + 5)^{2}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T^{2} - 4 T - 4)^{2}$$
$13$ $$T^{4} + 24T^{2} + 16$$
$17$ $$T^{4} + 24T^{2} + 16$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 72)^{2}$$
$29$ $$(T^{2} + 8 T - 16)^{2}$$
$31$ $$(T - 1)^{4}$$
$37$ $$T^{4} + 152T^{2} + 4624$$
$41$ $$(T^{2} - 4 T - 28)^{2}$$
$43$ $$T^{4} + 96T^{2} + 256$$
$47$ $$(T^{2} + 32)^{2}$$
$53$ $$T^{4} + 136T^{2} + 16$$
$59$ $$(T^{2} + 12 T + 28)^{2}$$
$61$ $$(T^{2} + 4 T - 4)^{2}$$
$67$ $$T^{4} + 304 T^{2} + 18496$$
$71$ $$(T^{2} - 8)^{2}$$
$73$ $$T^{4} + 192T^{2} + 1024$$
$79$ $$(T^{2} - 128)^{2}$$
$83$ $$T^{4} + 96T^{2} + 256$$
$89$ $$(T^{2} + 4 T - 4)^{2}$$
$97$ $$(T^{2} + 128)^{2}$$