# Properties

 Label 930.2.h.b Level $930$ Weight $2$ Character orbit 930.h Analytic conductor $7.426$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
371.1
 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.00000i −1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −1.00000
371.2 1.00000i 1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −1.00000
371.3 1.00000i −1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −1.00000
371.4 1.00000i 1.73205 −1.00000 1.00000i 1.73205i 2.00000 1.00000i 3.00000 −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.b 4
3.b odd 2 1 inner 930.2.h.b 4
31.b odd 2 1 inner 930.2.h.b 4
93.c even 2 1 inner 930.2.h.b 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T - 2)^{4}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 48)^{2}$$
$19$ $$(T - 4)^{4}$$
$23$ $$(T^{2} - 12)^{2}$$
$29$ $$(T^{2} - 12)^{2}$$
$31$ $$(T^{2} - 4 T + 31)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 144)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + 36)^{2}$$
$61$ $$T^{4}$$
$67$ $$(T + 8)^{4}$$
$71$ $$(T^{2} + 144)^{2}$$
$73$ $$(T^{2} + 108)^{2}$$
$79$ $$(T^{2} + 108)^{2}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$(T^{2} - 192)^{2}$$
$97$ $$(T + 10)^{4}$$