# Properties

 Label 930.2.v.a 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 20q^{4} + 12q^{7} - 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 20q^{4} + 12q^{7} - 8q^{9} + 20q^{10} - 10q^{13} - 20q^{16} - 12q^{18} - 12q^{19} + 30q^{21} + 10q^{22} - 80q^{25} + 18q^{28} + 16q^{31} + 72q^{33} + 10q^{34} + 8q^{36} + 54q^{39} - 20q^{40} + 50q^{43} + 8q^{45} - 10q^{46} + 8q^{49} - 44q^{51} - 10q^{52} - 10q^{55} + 30q^{58} + 28q^{63} + 20q^{64} + 66q^{66} + 60q^{67} + 20q^{69} - 12q^{70} + 12q^{72} - 20q^{73} - 18q^{76} + 8q^{78} + 40q^{79} + 16q^{81} - 4q^{82} - 60q^{84} - 72q^{87} - 12q^{90} + 10q^{91} + 104q^{93} - 12q^{94} - 178q^{97} + O(q^{100})$$

## 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 −1.73130 0.0510207i −0.309017 0.951057i 1.00000i 1.05891 1.37066i −0.425822 1.31054i 0.951057 + 0.309017i 2.99479 + 0.176664i 0.809017 + 0.587785i
401.2 −0.587785 + 0.809017i −1.71414 0.248453i −0.309017 0.951057i 1.00000i 1.20855 1.24073i 1.53704 + 4.73053i 0.951057 + 0.309017i 2.87654 + 0.851766i 0.809017 + 0.587785i
401.3 −0.587785 + 0.809017i −0.950729 + 1.44780i −0.309017 0.951057i 1.00000i −0.612467 1.62015i −0.531003 1.63426i 0.951057 + 0.309017i −1.19223 2.75292i 0.809017 + 0.587785i
401.4 −0.587785 + 0.809017i −0.922996 1.46563i −0.309017 0.951057i 1.00000i 1.72824 + 0.114757i 0.0841834 + 0.259090i 0.951057 + 0.309017i −1.29616 + 2.70555i 0.809017 + 0.587785i
401.5 −0.587785 + 0.809017i −0.438530 1.67562i −0.309017 0.951057i 1.00000i 1.61336 + 0.630124i 1.20094 + 3.69613i 0.951057 + 0.309017i −2.61538 + 1.46962i 0.809017 + 0.587785i
401.6 −0.587785 + 0.809017i 0.106510 + 1.72877i −0.309017 0.951057i 1.00000i −1.46121 0.929979i 0.417905 + 1.28618i 0.951057 + 0.309017i −2.97731 + 0.368264i 0.809017 + 0.587785i
401.7 −0.587785 + 0.809017i 1.13234 1.31065i −0.309017 0.951057i 1.00000i 0.394770 + 1.68646i 0.208734 + 0.642416i 0.951057 + 0.309017i −0.435626 2.96820i 0.809017 + 0.587785i
401.8 −0.587785 + 0.809017i 1.35636 + 1.07717i −0.309017 0.951057i 1.00000i −1.66869 + 0.464177i 0.584024 + 1.79744i 0.951057 + 0.309017i 0.679429 + 2.92205i 0.809017 + 0.587785i
401.9 −0.587785 + 0.809017i 1.47839 + 0.902413i −0.309017 0.951057i 1.00000i −1.59905 + 0.665622i −1.34568 4.14157i 0.951057 + 0.309017i 1.37130 + 2.66824i 0.809017 + 0.587785i
401.10 −0.587785 + 0.809017i 1.68409 0.404771i −0.309017 0.951057i 1.00000i −0.662417 + 1.60038i −0.230328 0.708876i 0.951057 + 0.309017i 2.67232 1.36334i 0.809017 + 0.587785i
401.11 0.587785 0.809017i −1.59642 + 0.671902i −0.309017 0.951057i 1.00000i −0.394770 + 1.68646i 0.208734 + 0.642416i −0.951057 0.309017i 2.09709 2.14527i 0.809017 + 0.587785i
401.12 0.587785 0.809017i −1.45809 0.934861i −0.309017 0.951057i 1.00000i −1.61336 + 0.630124i 1.20094 + 3.69613i −0.951057 0.309017i 1.25207 + 2.72623i 0.809017 + 0.587785i
401.13 0.587785 0.809017i −1.10868 1.33073i −0.309017 0.951057i 1.00000i −1.72824 + 0.114757i 0.0841834 + 0.259090i −0.951057 0.309017i −0.541668 + 2.95069i 0.809017 + 0.587785i
401.14 0.587785 0.809017i −0.905373 + 1.47658i −0.309017 0.951057i 1.00000i 0.662417 + 1.60038i −0.230328 0.708876i −0.951057 0.309017i −1.36060 2.67372i 0.809017 + 0.587785i
401.15 0.587785 0.809017i 0.293405 1.70702i −0.309017 0.951057i 1.00000i −1.20855 1.24073i 1.53704 + 4.73053i −0.951057 0.309017i −2.82783 1.00170i 0.809017 + 0.587785i
401.16 0.587785 0.809017i 0.401396 + 1.68490i −0.309017 0.951057i 1.00000i 1.59905 + 0.665622i −1.34568 4.14157i −0.951057 0.309017i −2.67776 + 1.35262i 0.809017 + 0.587785i
401.17 0.587785 0.809017i 0.486477 1.66233i −0.309017 0.951057i 1.00000i −1.05891 1.37066i −0.425822 1.31054i −0.951057 0.309017i −2.52668 1.61737i 0.809017 + 0.587785i
401.18 0.587785 0.809017i 0.605307 + 1.62284i −0.309017 0.951057i 1.00000i 1.66869 + 0.464177i 0.584024 + 1.79744i −0.951057 0.309017i −2.26721 + 1.96463i 0.809017 + 0.587785i
401.19 0.587785 0.809017i 1.61125 + 0.635517i −0.309017 0.951057i 1.00000i 1.46121 0.929979i 0.417905 + 1.28618i −0.951057 0.309017i 2.19224 + 2.04795i 0.809017 + 0.587785i
401.20 0.587785 0.809017i 1.67073 0.456803i −0.309017 0.951057i 1.00000i 0.612467 1.62015i −0.531003 1.63426i −0.951057 0.309017i 2.58266 1.52639i 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.a 80
3.b odd 2 1 inner 930.2.v.a 80
31.f odd 10 1 inner 930.2.v.a 80
93.k even 10 1 inner 930.2.v.a 80

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$25\!\cdots\!65$$$$T_{7}^{14} + 217726535038 T_{7}^{13} +$$$$53\!\cdots\!30$$$$T_{7}^{12} + 65440814015 T_{7}^{11} +$$$$84\!\cdots\!93$$$$T_{7}^{10} - 646762248628 T_{7}^{9} +$$$$11\!\cdots\!81$$$$T_{7}^{8} -$$$$14\!\cdots\!44$$$$T_{7}^{7} +$$$$15\!\cdots\!42$$$$T_{7}^{6} -$$$$21\!\cdots\!95$$$$T_{7}^{5} +$$$$94\!\cdots\!39$$$$T_{7}^{4} -$$$$15\!\cdots\!70$$$$T_{7}^{3} +$$$$27\!\cdots\!08$$$$T_{7}^{2} - 368852411048 T_{7} + 156850849936$$">$$T_{7}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.