# Properties

 Label 930.2.d.i Level $930$ Weight $2$ Character orbit 930.d Analytic conductor $7.426$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.11669056.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 7x^{4} + 8x^{3} - x^{2} + 54x + 58$$ x^6 - 2*x^5 + 7*x^4 + 8*x^3 - x^2 + 54*x + 58 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_1 - 1) q^{5} + q^{6} + 2 \beta_{3} q^{7} - \beta_{3} q^{8} - q^{9}+O(q^{10})$$ q + b3 * q^2 - b3 * q^3 - q^4 + (b1 - 1) * q^5 + q^6 + 2*b3 * q^7 - b3 * q^8 - q^9 $$q + \beta_{3} q^{2} - \beta_{3} q^{3} - q^{4} + (\beta_1 - 1) q^{5} + q^{6} + 2 \beta_{3} q^{7} - \beta_{3} q^{8} - q^{9} + (\beta_{5} - \beta_{3}) q^{10} + ( - \beta_{2} + \beta_1 + 2) q^{11} + \beta_{3} q^{12} + (\beta_{5} + \beta_{4} - 4 \beta_{3}) q^{13} - 2 q^{14} + ( - \beta_{5} + \beta_{3}) q^{15} + q^{16} + ( - 2 \beta_{5} - 2 \beta_{4} + \cdots + \beta_1) q^{17}+ \cdots + (\beta_{2} - \beta_1 - 2) q^{99}+O(q^{100})$$ q + b3 * q^2 - b3 * q^3 - q^4 + (b1 - 1) * q^5 + q^6 + 2*b3 * q^7 - b3 * q^8 - q^9 + (b5 - b3) * q^10 + (-b2 + b1 + 2) * q^11 + b3 * q^12 + (b5 + b4 - 4*b3) * q^13 - 2 * q^14 + (-b5 + b3) * q^15 + q^16 + (-2*b5 - 2*b4 + 2*b3 + b2 + b1) * q^17 - b3 * q^18 + (-b5 + b4 + 2*b2 - 2*b1 + 2) * q^19 + (-b1 + 1) * q^20 + 2 * q^21 + (b5 + b4 + 2*b3) * q^22 + (2*b5 + 2*b4 - 2*b3) * q^23 - q^24 + (b4 + b3 + 2*b2 - 2*b1) * q^25 + (b2 - b1 + 4) * q^26 + b3 * q^27 - 2*b3 * q^28 + (2*b2 - 2*b1 + 2) * q^29 + (b1 - 1) * q^30 - q^31 + b3 * q^32 + (-b5 - b4 - 2*b3) * q^33 + (b5 - b4 - 2*b2 + 2*b1 - 2) * q^34 + (2*b5 - 2*b3) * q^35 + q^36 + (-2*b3 + 2*b2 + 2*b1) * q^37 + (-2*b5 - 2*b4 + 2*b3 + b2 + b1) * q^38 + (-b2 + b1 - 4) * q^39 + (-b5 + b3) * q^40 + 2 * q^41 + 2*b3 * q^42 + (2*b2 + 2*b1) * q^43 + (b2 - b1 - 2) * q^44 + (-b1 + 1) * q^45 + (2*b2 - 2*b1 + 2) * q^46 + (2*b5 + 2*b4 - 2*b3 - b2 - b1) * q^47 - b3 * q^48 + 3 * q^49 + (-2*b5 - 2*b4 + b2 - 1) * q^50 + (-b5 + b4 + 2*b2 - 2*b1 + 2) * q^51 + (-b5 - b4 + 4*b3) * q^52 + (-2*b5 - 2*b4 - 2*b3 + 2*b2 + 2*b1) * q^53 - q^54 + (b4 + b3 + 2*b2 + 2*b1 + 1) * q^55 + 2 * q^56 + (2*b5 + 2*b4 - 2*b3 - b2 - b1) * q^57 + (-2*b5 - 2*b4 + 2*b3) * q^58 + (-2*b5 + 2*b4 + 2*b2 - 2*b1 + 2) * q^59 + (b5 - b3) * q^60 + (-3*b5 + 3*b4 + 2*b2 - 2*b1 + 6) * q^61 - b3 * q^62 - 2*b3 * q^63 - q^64 + (-4*b5 - 2*b4 + 7*b3 + b2 - 1) * q^65 + (-b2 + b1 + 2) * q^66 + (b5 + b4 - 6*b3 + 2*b2 + 2*b1) * q^67 + (2*b5 + 2*b4 - 2*b3 - b2 - b1) * q^68 + (-2*b2 + 2*b1 - 2) * q^69 + (-2*b1 + 2) * q^70 + (2*b5 - 2*b4 + b2 - b1 - 2) * q^71 + b3 * q^72 + (4*b5 + 4*b4 - 2*b3) * q^73 + (2*b5 - 2*b4 + 2) * q^74 + (2*b5 + 2*b4 - b2 + 1) * q^75 + (b5 - b4 - 2*b2 + 2*b1 - 2) * q^76 + (2*b5 + 2*b4 + 4*b3) * q^77 + (b5 + b4 - 4*b3) * q^78 + (b5 - b4 + 2) * q^79 + (b1 - 1) * q^80 + q^81 + 2*b3 * q^82 + (-4*b5 - 4*b4 + 2*b3 - b2 - b1) * q^83 - 2 * q^84 + (2*b5 + 5*b4 - 7*b3 - 2*b1 - 3) * q^85 + (2*b5 - 2*b4) * q^86 + (2*b5 + 2*b4 - 2*b3) * q^87 + (-b5 - b4 - 2*b3) * q^88 + (2*b5 - 2*b4 - 4*b2 + 4*b1 - 2) * q^89 + (-b5 + b3) * q^90 + (2*b2 - 2*b1 + 8) * q^91 + (-2*b5 - 2*b4 + 2*b3) * q^92 + b3 * q^93 + (-b5 + b4 + 2*b2 - 2*b1 + 2) * q^94 + (2*b5 + 3*b3 - 5*b2 + 2*b1 - 7) * q^95 + q^96 + (-b5 - b4 - 6*b3) * q^97 + 3*b3 * q^98 + (b2 - b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9 $$6 q - 6 q^{4} - 4 q^{5} + 6 q^{6} - 6 q^{9} + 16 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} + 4 q^{20} + 12 q^{21} - 6 q^{24} - 8 q^{25} + 20 q^{26} + 4 q^{29} - 4 q^{30} - 6 q^{31} - 4 q^{34} + 6 q^{36} - 20 q^{39} + 12 q^{41} - 16 q^{44} + 4 q^{45} + 4 q^{46} + 18 q^{49} - 8 q^{50} + 4 q^{51} - 6 q^{54} + 6 q^{55} + 12 q^{56} + 4 q^{59} + 28 q^{61} - 6 q^{64} - 8 q^{65} + 16 q^{66} - 4 q^{69} + 8 q^{70} - 16 q^{71} + 12 q^{74} + 8 q^{75} - 4 q^{76} + 12 q^{79} - 4 q^{80} + 6 q^{81} - 12 q^{84} - 22 q^{85} + 4 q^{89} + 40 q^{91} + 4 q^{94} - 28 q^{95} + 6 q^{96} - 16 q^{99}+O(q^{100})$$ 6 * q - 6 * q^4 - 4 * q^5 + 6 * q^6 - 6 * q^9 + 16 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 + 4 * q^20 + 12 * q^21 - 6 * q^24 - 8 * q^25 + 20 * q^26 + 4 * q^29 - 4 * q^30 - 6 * q^31 - 4 * q^34 + 6 * q^36 - 20 * q^39 + 12 * q^41 - 16 * q^44 + 4 * q^45 + 4 * q^46 + 18 * q^49 - 8 * q^50 + 4 * q^51 - 6 * q^54 + 6 * q^55 + 12 * q^56 + 4 * q^59 + 28 * q^61 - 6 * q^64 - 8 * q^65 + 16 * q^66 - 4 * q^69 + 8 * q^70 - 16 * q^71 + 12 * q^74 + 8 * q^75 - 4 * q^76 + 12 * q^79 - 4 * q^80 + 6 * q^81 - 12 * q^84 - 22 * q^85 + 4 * q^89 + 40 * q^91 + 4 * q^94 - 28 * q^95 + 6 * q^96 - 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 7x^{4} + 8x^{3} - x^{2} + 54x + 58$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - \nu^{4} + 6\nu^{3} + 14\nu^{2} + 13\nu + 42 ) / 25$$ (v^5 - v^4 + 6*v^3 + 14*v^2 + 13*v + 42) / 25 $$\beta_{3}$$ $$=$$ $$( 4\nu^{5} - 14\nu^{4} + 49\nu^{3} - 29\nu^{2} + 27\nu + 213 ) / 25$$ (4*v^5 - 14*v^4 + 49*v^3 - 29*v^2 + 27*v + 213) / 25 $$\beta_{4}$$ $$=$$ $$( -6\nu^{5} + 16\nu^{4} - 61\nu^{3} + 26\nu^{2} - 53\nu - 272 ) / 25$$ (-6*v^5 + 16*v^4 - 61*v^3 + 26*v^2 - 53*v - 272) / 25 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 21\nu^{4} - 61\nu^{3} + 31\nu^{2} - 3\nu - 232 ) / 25$$ (-6*v^5 + 21*v^4 - 61*v^3 + 31*v^2 - 3*v - 232) / 25
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2\beta_{2} - 1$$ b4 + b3 + 2*b2 - 1 $$\nu^{3}$$ $$=$$ $$2\beta_{5} + \beta_{4} + 4\beta_{3} + 2\beta_{2} - 3\beta _1 - 8$$ 2*b5 + b4 + 4*b3 + 2*b2 - 3*b1 - 8 $$\nu^{4}$$ $$=$$ $$5\beta_{5} - 6\beta_{4} - \beta_{3} - 2\beta_{2} - 10\beta _1 - 7$$ 5*b5 - 6*b4 - b3 - 2*b2 - 10*b1 - 7 $$\nu^{5}$$ $$=$$ $$-7\beta_{5} - 26\beta_{4} - 39\beta_{3} - 17\beta_{2} - 5\beta _1 + 13$$ -7*b5 - 26*b4 - 39*b3 - 17*b2 - 5*b1 + 13

## 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
 −1.23545 − 0.0526623i 0.630356 + 2.20530i 1.60509 − 2.15264i −1.23545 + 0.0526623i 0.630356 − 2.20530i 1.60509 + 2.15264i
1.00000i 1.00000i −1.00000 −2.23545 0.0526623i 1.00000 2.00000i 1.00000i −1.00000 −0.0526623 + 2.23545i
559.2 1.00000i 1.00000i −1.00000 −0.369644 + 2.20530i 1.00000 2.00000i 1.00000i −1.00000 2.20530 + 0.369644i
559.3 1.00000i 1.00000i −1.00000 0.605092 2.15264i 1.00000 2.00000i 1.00000i −1.00000 −2.15264 0.605092i
559.4 1.00000i 1.00000i −1.00000 −2.23545 + 0.0526623i 1.00000 2.00000i 1.00000i −1.00000 −0.0526623 2.23545i
559.5 1.00000i 1.00000i −1.00000 −0.369644 2.20530i 1.00000 2.00000i 1.00000i −1.00000 2.20530 0.369644i
559.6 1.00000i 1.00000i −1.00000 0.605092 + 2.15264i 1.00000 2.00000i 1.00000i −1.00000 −2.15264 + 0.605092i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.6 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.i 6
3.b odd 2 1 2790.2.d.j 6
5.b even 2 1 inner 930.2.d.i 6
5.c odd 4 1 4650.2.a.ci 3
5.c odd 4 1 4650.2.a.cp 3
15.d odd 2 1 2790.2.d.j 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.i 6 1.a even 1 1 trivial
930.2.d.i 6 5.b even 2 1 inner
2790.2.d.j 6 3.b odd 2 1
2790.2.d.j 6 15.d odd 2 1
4650.2.a.ci 3 5.c odd 4 1
4650.2.a.cp 3 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{3} - 8T_{11}^{2} + 13T_{11} + 8$$ T11^3 - 8*T11^2 + 13*T11 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6} + 4 T^{5} + \cdots + 125$$
$7$ $$(T^{2} + 4)^{3}$$
$11$ $$(T^{3} - 8 T^{2} + 13 T + 8)^{2}$$
$13$ $$T^{6} + 50 T^{4} + \cdots + 196$$
$17$ $$T^{6} + 74 T^{4} + \cdots + 16$$
$19$ $$(T^{3} - 2 T^{2} - 35 T - 4)^{2}$$
$23$ $$T^{6} + 68 T^{4} + \cdots + 256$$
$29$ $$(T^{3} - 2 T^{2} - 32 T - 16)^{2}$$
$31$ $$(T + 1)^{6}$$
$37$ $$T^{6} + 164 T^{4} + \cdots + 16384$$
$41$ $$(T - 2)^{6}$$
$43$ $$T^{6} + 152 T^{4} + \cdots + 256$$
$47$ $$T^{6} + 74 T^{4} + \cdots + 16$$
$53$ $$T^{6} + 188 T^{4} + \cdots + 64$$
$59$ $$(T^{3} - 2 T^{2} + \cdots + 280)^{2}$$
$61$ $$(T^{3} - 14 T^{2} + \cdots + 1372)^{2}$$
$67$ $$T^{6} + 286 T^{4} + \cdots + 425104$$
$71$ $$(T^{3} + 8 T^{2} - 79 T + 20)^{2}$$
$73$ $$T^{6} + 268 T^{4} + \cdots + 153664$$
$79$ $$(T^{3} - 6 T^{2} - 7 T + 28)^{2}$$
$83$ $$T^{6} + 370 T^{4} + \cdots + 59536$$
$89$ $$(T^{3} - 2 T^{2} + \cdots + 320)^{2}$$
$97$ $$T^{6} + 150 T^{4} + \cdots + 55696$$