# Properties

 Label 930.2.n.b Level $930$ Weight $2$ Character orbit 930.n Analytic conductor $7.426$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.1816890625.5 Defining polynomial: $$x^{8} - x^{7} + 8 x^{6} - 15 x^{5} + 41 x^{4} + 15 x^{3} + 62 x^{2} + 133 x + 361$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 8 x^{6} - 15 x^{5} + 41 x^{4} + 15 x^{3} + 62 x^{2} + 133 x + 361$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$128795 \nu^{7} + 470503 \nu^{6} - 1652991 \nu^{5} + 5286232 \nu^{4} - 16599235 \nu^{3} + 104742046 \nu^{2} - 43752185 \nu + 22451179$$$$)/ 303029309$$ $$\beta_{3}$$ $$=$$ $$($$$$214762 \nu^{7} + 1363340 \nu^{6} + 2203394 \nu^{5} - 2198831 \nu^{4} + 4048801 \nu^{3} - 23952807 \nu^{2} + 137220400 \nu - 89195880$$$$)/ 303029309$$ $$\beta_{4}$$ $$=$$ $$($$$$-17568 \nu^{7} - 45727 \nu^{6} - 524081 \nu^{5} - 229321 \nu^{4} - 419871 \nu^{3} - 455350 \nu^{2} - 298585 \nu - 20195518$$$$)/15948911$$ $$\beta_{5}$$ $$=$$ $$($$$$-31542 \nu^{7} + 141229 \nu^{6} - 379903 \nu^{5} + 1151570 \nu^{4} - 5411059 \nu^{3} + 2723025 \nu^{2} - 280076 \nu + 2447105$$$$)/15948911$$ $$\beta_{6}$$ $$=$$ $$($$$$-719365 \nu^{7} + 3230557 \nu^{6} - 7084160 \nu^{5} + 28988770 \nu^{4} - 51773137 \nu^{3} + 72822958 \nu^{2} + 36218701 \nu + 100589097$$$$)/ 303029309$$ $$\beta_{7}$$ $$=$$ $$($$$$132168 \nu^{7} - 69960 \nu^{6} + 957805 \nu^{5} - 1172588 \nu^{4} + 4400707 \nu^{3} + 4253649 \nu^{2} + 10329718 \nu + 13667935$$$$)/15948911$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$11 \beta_{7} + 21 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 31 \beta_{3} - 31 \beta_{2} - 21$$ $$\nu^{5}$$ $$=$$ $$-12 \beta_{7} - 19 \beta_{6} - 42 \beta_{4} + 19 \beta_{2} + 12 \beta_{1} - 38$$ $$\nu^{6}$$ $$=$$ $$-31 \beta_{7} - 31 \beta_{5} + 222 \beta_{3} + 90 \beta_{2} - 68 \beta_{1} + 90$$ $$\nu^{7}$$ $$=$$ $$321 \beta_{7} + 223 \beta_{6} + 200 \beta_{5} + 321 \beta_{4} - 223 \beta_{3} - 526 \beta_{2} - 200 \beta_{1}$$

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
481.1
 0.594055 + 1.82831i −0.903072 − 2.77937i 1.94631 − 1.41408i −1.13729 + 0.826292i 1.94631 + 1.41408i −1.13729 − 0.826292i 0.594055 − 1.82831i −0.903072 + 2.77937i
0.809017 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i −1.00000 1.00000 −0.785038 + 2.41610i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
481.2 0.809017 0.587785i 0.809017 + 0.587785i 0.309017 0.951057i −1.00000 1.00000 0.712089 2.19158i −0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i
721.1 −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 0.587785i −1.00000 1.00000 −3.25533 2.36513i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
721.2 −0.309017 + 0.951057i −0.309017 0.951057i −0.809017 0.587785i −1.00000 1.00000 −0.171724 0.124765i 0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i
841.1 −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 −3.25533 + 2.36513i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
841.2 −0.309017 0.951057i −0.309017 + 0.951057i −0.809017 + 0.587785i −1.00000 1.00000 −0.171724 + 0.124765i 0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i
901.1 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 −0.785038 2.41610i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
901.2 0.809017 + 0.587785i 0.809017 0.587785i 0.309017 + 0.951057i −1.00000 1.00000 0.712089 + 2.19158i −0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 901.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.n.b 8
31.d even 5 1 inner 930.2.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.n.b 8 1.a even 1 1 trivial
930.2.n.b 8 31.d even 5 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$3$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$5$ $$( 1 + T )^{8}$$
$7$ $$25 + 200 T + 635 T^{2} + 275 T^{3} + 206 T^{4} + 73 T^{5} + 29 T^{6} + 7 T^{7} + T^{8}$$
$11$ $$361 + 133 T + 62 T^{2} + 15 T^{3} + 41 T^{4} - 15 T^{5} + 8 T^{6} - T^{7} + T^{8}$$
$13$ $$1 - 2 T + 27 T^{2} - 155 T^{3} + 426 T^{4} - 195 T^{5} + 63 T^{6} - 11 T^{7} + T^{8}$$
$17$ $$121 - 121 T + 583 T^{2} + 275 T^{3} + 36 T^{4} - 35 T^{5} + 27 T^{6} - 7 T^{7} + T^{8}$$
$19$ $$28561 + 26364 T + 13013 T^{2} + 2340 T^{3} + 841 T^{4} - 180 T^{5} + 77 T^{6} - 12 T^{7} + T^{8}$$
$23$ $$25 - 150 T + 335 T^{2} + 120 T^{3} + 121 T^{4} + 96 T^{5} + 51 T^{6} + 6 T^{7} + T^{8}$$
$29$ $$14641 - 22143 T + 15392 T^{2} - 5505 T^{3} + 1561 T^{4} - 375 T^{5} + 78 T^{6} - 9 T^{7} + T^{8}$$
$31$ $$923521 + 446865 T + 178746 T^{2} + 46035 T^{3} + 9651 T^{4} + 1485 T^{5} + 186 T^{6} + 15 T^{7} + T^{8}$$
$37$ $$( -155 + 120 T + 49 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$41$ $$361 + 2318 T + 41443 T^{2} - 10429 T^{3} + 180 T^{4} + 431 T^{5} + 123 T^{6} + 13 T^{7} + T^{8}$$
$43$ $$3568321 - 3375643 T + 1419943 T^{2} - 276721 T^{3} + 35430 T^{4} - 3191 T^{5} + 273 T^{6} - 13 T^{7} + T^{8}$$
$47$ $$1890625 + 378125 T + 281875 T^{2} - 12375 T^{3} + 1400 T^{4} + 75 T^{5} + 115 T^{6} + 15 T^{7} + T^{8}$$
$53$ $$130321 - 138985 T + 93709 T^{2} - 40015 T^{3} + 11676 T^{4} - 2255 T^{5} + 309 T^{6} - 25 T^{7} + T^{8}$$
$59$ $$185761 + 11637 T + 19658 T^{2} + 4929 T^{3} + 1105 T^{4} - 81 T^{5} + 18 T^{6} - 3 T^{7} + T^{8}$$
$61$ $$( -761 + 312 T + 13 T^{2} - 14 T^{3} + T^{4} )^{2}$$
$67$ $$( 919 - 77 T - 102 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$71$ $$18931201 - 4503285 T + 2167239 T^{2} - 61745 T^{3} - 1494 T^{4} + 55 T^{5} + 189 T^{6} + 15 T^{7} + T^{8}$$
$73$ $$3025 + 24200 T + 470965 T^{2} + 166980 T^{3} + 28121 T^{4} + 2604 T^{5} + 301 T^{6} + 24 T^{7} + T^{8}$$
$79$ $$3568321 - 1375192 T + 834353 T^{2} - 99354 T^{3} + 13655 T^{4} + 66 T^{5} - 37 T^{6} - 2 T^{7} + T^{8}$$
$83$ $$502681 - 262330 T + 40614 T^{2} + 17960 T^{3} + 11796 T^{4} + 2460 T^{5} + 309 T^{6} + 20 T^{7} + T^{8}$$
$89$ $$31798321 - 727431 T + 1729708 T^{2} - 285825 T^{3} + 38321 T^{4} - 5175 T^{5} + 622 T^{6} - 37 T^{7} + T^{8}$$
$97$ $$3025 + 6325 T + 10785 T^{2} + 9275 T^{3} + 4476 T^{4} + 1171 T^{5} + 181 T^{6} + 16 T^{7} + T^{8}$$