# Properties

 Label 930.2.v.b Level $930$ Weight $2$ Character orbit 930.v Analytic conductor $7.426$ Analytic rank $0$ Dimension $80$ 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.v (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$20$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 20q^{4} + 12q^{7} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 20q^{4} + 12q^{7} + 8q^{9} - 20q^{10} + 10q^{13} - 20q^{16} - 4q^{18} - 12q^{19} - 30q^{21} - 10q^{22} - 80q^{25} + 60q^{27} + 18q^{28} + 8q^{31} - 20q^{33} + 10q^{34} - 8q^{36} - 46q^{39} + 20q^{40} + 50q^{43} - 16q^{45} - 10q^{46} - 8q^{49} - 20q^{51} + 10q^{52} - 10q^{55} + 10q^{58} + 44q^{63} + 20q^{64} - 10q^{66} - 132q^{67} - 16q^{69} + 12q^{70} + 4q^{72} - 20q^{73} - 18q^{76} + 48q^{78} - 120q^{79} + 48q^{81} - 20q^{82} + 20q^{84} - 136q^{87} - 12q^{90} - 110q^{91} - 20q^{93} - 28q^{94} + 22q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1 −0.587785 + 0.809017i −1.71066 + 0.271374i −0.309017 0.951057i 1.00000i 0.785954 1.54346i 0.226704 + 0.697724i 0.951057 + 0.309017i 2.85271 0.928458i −0.809017 0.587785i
401.2 −0.587785 + 0.809017i −1.32297 1.11792i −0.309017 0.951057i 1.00000i 1.68204 0.413212i −1.31298 4.04094i 0.951057 + 0.309017i 0.500516 + 2.95795i −0.809017 0.587785i
401.3 −0.587785 + 0.809017i −1.23647 + 1.21291i −0.309017 0.951057i 1.00000i −0.254481 1.71325i −0.489949 1.50791i 0.951057 + 0.309017i 0.0577223 2.99944i −0.809017 0.587785i
401.4 −0.587785 + 0.809017i −0.829675 + 1.52041i −0.309017 0.951057i 1.00000i −0.742365 1.56489i 1.36101 + 4.18875i 0.951057 + 0.309017i −1.62328 2.52289i −0.809017 0.587785i
401.5 −0.587785 + 0.809017i −0.335845 1.69918i −0.309017 0.951057i 1.00000i 1.57207 + 0.727048i 0.678803 + 2.08914i 0.951057 + 0.309017i −2.77442 + 1.14132i −0.809017 0.587785i
401.6 −0.587785 + 0.809017i 0.240855 + 1.71522i −0.309017 0.951057i 1.00000i −1.52922 0.813327i −0.762356 2.34629i 0.951057 + 0.309017i −2.88398 + 0.826239i −0.809017 0.587785i
401.7 −0.587785 + 0.809017i 0.305332 1.70493i −0.309017 0.951057i 1.00000i 1.19984 + 1.24915i 0.0989466 + 0.304526i 0.951057 + 0.309017i −2.81354 1.04114i −0.809017 0.587785i
401.8 −0.587785 + 0.809017i 1.57188 + 0.727458i −0.309017 0.951057i 1.00000i −1.51245 + 0.844088i 1.04544 + 3.21754i 0.951057 + 0.309017i 1.94161 + 2.28695i −0.809017 0.587785i
401.9 −0.587785 + 0.809017i 1.60896 0.641293i −0.309017 0.951057i 1.00000i −0.426905 + 1.67862i 1.11324 + 3.42620i 0.951057 + 0.309017i 2.17749 2.06362i −0.809017 0.587785i
401.10 −0.587785 + 0.809017i 1.70860 0.284052i −0.309017 0.951057i 1.00000i −0.774487 + 1.54925i −0.458862 1.41223i 0.951057 + 0.309017i 2.83863 0.970663i −0.809017 0.587785i
401.11 0.587785 0.809017i −1.71583 0.236463i −0.309017 0.951057i 1.00000i −1.19984 + 1.24915i 0.0989466 + 0.304526i −0.951057 0.309017i 2.88817 + 0.811463i −0.809017 0.587785i
401.12 0.587785 0.809017i −1.51223 0.844483i −0.309017 0.951057i 1.00000i −1.57207 + 0.727048i 0.678803 + 2.08914i −0.951057 0.309017i 1.57370 + 2.55411i −0.809017 0.587785i
401.13 0.587785 0.809017i −1.10710 + 1.33204i −0.309017 0.951057i 1.00000i 0.426905 + 1.67862i 1.11324 + 3.42620i −0.951057 0.309017i −0.548656 2.94940i −0.809017 0.587785i
401.14 0.587785 0.809017i −0.798136 + 1.53720i −0.309017 0.951057i 1.00000i 0.774487 + 1.54925i −0.458862 1.41223i −0.951057 0.309017i −1.72596 2.45379i −0.809017 0.587785i
401.15 0.587785 0.809017i −0.654383 1.60368i −0.309017 0.951057i 1.00000i −1.68204 0.413212i −1.31298 4.04094i −0.951057 0.309017i −2.14357 + 2.09884i −0.809017 0.587785i
401.16 0.587785 0.809017i 0.206116 + 1.71974i −0.309017 0.951057i 1.00000i 1.51245 + 0.844088i 1.04544 + 3.21754i −0.951057 0.309017i −2.91503 + 0.708934i −0.809017 0.587785i
401.17 0.587785 0.809017i 0.786715 1.54307i −0.309017 0.951057i 1.00000i −0.785954 1.54346i 0.226704 + 0.697724i −0.951057 0.309017i −1.76216 2.42792i −0.809017 0.587785i
401.18 0.587785 0.809017i 1.53563 0.801146i −0.309017 0.951057i 1.00000i 0.254481 1.71325i −0.489949 1.50791i −0.951057 0.309017i 1.71633 2.46053i −0.809017 0.587785i
401.19 0.587785 0.809017i 1.55685 + 0.759099i −0.309017 0.951057i 1.00000i 1.52922 0.813327i −0.762356 2.34629i −0.951057 0.309017i 1.84754 + 2.36360i −0.809017 0.587785i
401.20 0.587785 0.809017i 1.70238 0.319236i −0.309017 0.951057i 1.00000i 0.742365 1.56489i 1.36101 + 4.18875i −0.951057 0.309017i 2.79618 1.08692i −0.809017 0.587785i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 821.20 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.f odd 10 1 inner
93.k 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.v.b 80
3.b odd 2 1 inner 930.2.v.b 80
31.f odd 10 1 inner 930.2.v.b 80
93.k even 10 1 inner 930.2.v.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.v.b 80 1.a even 1 1 trivial
930.2.v.b 80 3.b odd 2 1 inner
930.2.v.b 80 31.f odd 10 1 inner
930.2.v.b 80 93.k even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$11\!\cdots\!31$$$$T_{7}^{19} +$$$$40\!\cdots\!35$$$$T_{7}^{18} -$$$$55\!\cdots\!67$$$$T_{7}^{17} +$$$$17\!\cdots\!83$$$$T_{7}^{16} -$$$$15\!\cdots\!06$$$$T_{7}^{15} +$$$$57\!\cdots\!73$$$$T_{7}^{14} -$$$$24\!\cdots\!94$$$$T_{7}^{13} +$$$$16\!\cdots\!66$$$$T_{7}^{12} -$$$$12\!\cdots\!97$$$$T_{7}^{11} +$$$$32\!\cdots\!57$$$$T_{7}^{10} -$$$$13\!\cdots\!64$$$$T_{7}^{9} +$$$$36\!\cdots\!57$$$$T_{7}^{8} -$$$$22\!\cdots\!72$$$$T_{7}^{7} +$$$$40\!\cdots\!82$$$$T_{7}^{6} -$$$$44\!\cdots\!99$$$$T_{7}^{5} +$$$$11\!\cdots\!75$$$$T_{7}^{4} +$$$$54\!\cdots\!54$$$$T_{7}^{3} +$$$$25\!\cdots\!52$$$$T_{7}^{2} + 959152476760 T_{7} +$$$$31\!\cdots\!16$$">$$T_{7}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.