# Properties

 Label 930.2.z.a Level $930$ Weight $2$ Character orbit 930.z Analytic conductor $7.426$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.z (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{20}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{20}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.587785 − 0.809017i −0.587785 + 0.809017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.951057 + 0.309017i −0.951057 − 0.309017i 0.587785 + 0.809017i −0.587785 − 0.809017i
−0.587785 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 2.23607 −1.00000 −3.07768 1.00000i 0.951057 0.309017i −0.309017 0.951057i −1.31433 1.80902i
109.2 0.587785 + 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 2.23607 −1.00000 3.07768 + 1.00000i −0.951057 + 0.309017i −0.309017 0.951057i 1.31433 + 1.80902i
349.1 −0.951057 0.309017i 0.951057 0.309017i 0.809017 + 0.587785i −2.23607 −1.00000 −0.726543 + 1.00000i −0.587785 0.809017i 0.809017 0.587785i 2.12663 + 0.690983i
349.2 0.951057 + 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.23607 −1.00000 0.726543 1.00000i 0.587785 + 0.809017i 0.809017 0.587785i −2.12663 0.690983i
469.1 −0.951057 + 0.309017i 0.951057 + 0.309017i 0.809017 0.587785i −2.23607 −1.00000 −0.726543 1.00000i −0.587785 + 0.809017i 0.809017 + 0.587785i 2.12663 0.690983i
469.2 0.951057 0.309017i −0.951057 0.309017i 0.809017 0.587785i −2.23607 −1.00000 0.726543 + 1.00000i 0.587785 0.809017i 0.809017 + 0.587785i −2.12663 + 0.690983i
529.1 −0.587785 + 0.809017i 0.587785 + 0.809017i −0.309017 0.951057i 2.23607 −1.00000 −3.07768 + 1.00000i 0.951057 + 0.309017i −0.309017 + 0.951057i −1.31433 + 1.80902i
529.2 0.587785 0.809017i −0.587785 0.809017i −0.309017 0.951057i 2.23607 −1.00000 3.07768 1.00000i −0.951057 0.309017i −0.309017 + 0.951057i 1.31433 1.80902i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.d even 5 1 inner
155.n even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.z.a 8
5.b even 2 1 inner 930.2.z.a 8
31.d even 5 1 inner 930.2.z.a 8
155.n even 10 1 inner 930.2.z.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.z.a 8 1.a even 1 1 trivial
930.2.z.a 8 5.b even 2 1 inner
930.2.z.a 8 31.d even 5 1 inner
930.2.z.a 8 155.n even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 16 T_{7}^{6} + 96 T_{7}^{4} + 64 T_{7}^{2} + 256$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$3$ $$1 - T^{2} + T^{4} - T^{6} + T^{8}$$
$5$ $$( -5 + T^{2} )^{4}$$
$7$ $$256 + 64 T^{2} + 96 T^{4} - 16 T^{6} + T^{8}$$
$11$ $$( 121 - 66 T + 16 T^{2} - T^{3} + T^{4} )^{2}$$
$13$ $$1 - 76 T^{2} + 2206 T^{4} + 29 T^{6} + T^{8}$$
$17$ $$6561 + 729 T^{2} + 486 T^{4} - 36 T^{6} + T^{8}$$
$19$ $$( 400 + 200 T + 60 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$23$ $$625 - 1375 T^{2} + 1150 T^{4} + 20 T^{6} + T^{8}$$
$29$ $$( 1 - 11 T + 46 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$31$ $$( 961 + 341 T + 91 T^{2} + 11 T^{3} + T^{4} )^{2}$$
$37$ $$( 2025 + 135 T^{2} + T^{4} )^{2}$$
$41$ $$( 1936 + 704 T + 136 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$43$ $$65536 - 4096 T^{2} + 256 T^{4} - 16 T^{6} + T^{8}$$
$47$ $$390625 - 62500 T^{2} + 3750 T^{4} + 25 T^{6} + T^{8}$$
$53$ $$3748096 - 147136 T^{2} + 2656 T^{4} - 16 T^{6} + T^{8}$$
$59$ $$( 961 - 31 T + 76 T^{2} + 14 T^{3} + T^{4} )^{2}$$
$61$ $$( 20 - 10 T + T^{2} )^{4}$$
$67$ $$( 64 + T^{2} )^{4}$$
$71$ $$( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$73$ $$3748096 + 240064 T^{2} + 48016 T^{4} - 356 T^{6} + T^{8}$$
$79$ $$( 961 - 372 T + 94 T^{2} - 13 T^{3} + T^{4} )^{2}$$
$83$ $$160000 - 32000 T^{2} + 2400 T^{4} + 20 T^{6} + T^{8}$$
$89$ $$( 1296 - 864 T + 216 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$97$ $$65536 + 45056 T^{2} + 82176 T^{4} - 464 T^{6} + T^{8}$$
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