# Properties

 Label 930.2.z.d Level $930$ Weight $2$ Character orbit 930.z Analytic conductor $7.426$ Analytic rank $0$ Dimension $72$ 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.z (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q + 18q^{4} + 4q^{5} + 72q^{6} + 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q + 18q^{4} + 4q^{5} + 72q^{6} + 18q^{9} - 6q^{11} - 4q^{14} - 18q^{16} + 12q^{19} - 4q^{20} - 16q^{21} + 18q^{24} - 4q^{25} + 12q^{26} + 24q^{29} + 4q^{30} + 22q^{31} + 14q^{34} - 22q^{35} + 72q^{36} + 8q^{39} - 28q^{41} - 4q^{44} - 4q^{45} - 22q^{46} + 42q^{49} - 32q^{50} - 4q^{51} + 18q^{54} + 14q^{55} + 24q^{56} - 28q^{59} + 56q^{61} + 18q^{64} + 46q^{65} - 6q^{66} - 8q^{69} + 46q^{70} - 68q^{71} + 8q^{74} + 32q^{75} - 12q^{76} + 26q^{79} - 6q^{80} - 18q^{81} - 4q^{84} - 32q^{85} + 80q^{86} + 32q^{89} - 44q^{91} + 12q^{94} + 74q^{95} - 18q^{96} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i −2.20868 0.348916i 1.00000 0.804232 + 0.261311i 0.951057 0.309017i −0.309017 0.951057i 1.01595 + 1.99195i
109.2 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i −1.90693 + 1.16774i 1.00000 0.507889 + 0.165023i 0.951057 0.309017i −0.309017 0.951057i 2.06559 + 0.856357i
109.3 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i −0.635868 + 2.14375i 1.00000 2.45458 + 0.797540i 0.951057 0.309017i −0.309017 0.951057i 2.10809 0.745638i
109.4 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i −0.244369 2.22267i 1.00000 4.86212 + 1.57980i 0.951057 0.309017i −0.309017 0.951057i −1.65455 + 1.50415i
109.5 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 0.218072 + 2.22541i 1.00000 −0.288858 0.0938558i 0.951057 0.309017i −0.309017 0.951057i 1.67221 1.48449i
109.6 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 0.836694 2.07363i 1.00000 −0.595482 0.193484i 0.951057 0.309017i −0.309017 0.951057i −2.16940 + 0.541950i
109.7 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.25767 1.84886i 1.00000 −2.43606 0.791524i 0.951057 0.309017i −0.309017 0.951057i −2.23500 + 0.0692568i
109.8 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 2.06791 + 0.850729i 1.00000 −4.57919 1.48787i 0.951057 0.309017i −0.309017 0.951057i −0.527234 2.17302i
109.9 −0.587785 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 2.23353 + 0.106448i 1.00000 4.25057 + 1.38109i 0.951057 0.309017i −0.309017 0.951057i −1.22672 1.86953i
109.10 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i −2.20868 + 0.348916i 1.00000 −0.804232 0.261311i −0.951057 + 0.309017i −0.309017 0.951057i −1.58051 1.58177i
109.11 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i −1.90693 1.16774i 1.00000 −0.507889 0.165023i −0.951057 + 0.309017i −0.309017 0.951057i −0.176142 2.22912i
109.12 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i −0.635868 2.14375i 1.00000 −2.45458 0.797540i −0.951057 + 0.309017i −0.309017 0.951057i 1.36058 1.77449i
109.13 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i −0.244369 + 2.22267i 1.00000 −4.86212 1.57980i −0.951057 + 0.309017i −0.309017 0.951057i −1.94182 + 1.10876i
109.14 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 0.218072 2.22541i 1.00000 0.288858 + 0.0938558i −0.951057 + 0.309017i −0.309017 0.951057i 1.92857 1.13164i
109.15 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 0.836694 + 2.07363i 1.00000 0.595482 + 0.193484i −0.951057 + 0.309017i −0.309017 0.951057i −1.18581 + 1.89575i
109.16 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 1.25767 + 1.84886i 1.00000 2.43606 + 0.791524i −0.951057 + 0.309017i −0.309017 0.951057i −0.756519 + 2.10421i
109.17 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 2.06791 0.850729i 1.00000 4.57919 + 1.48787i −0.951057 + 0.309017i −0.309017 0.951057i 1.90374 + 1.17293i
109.18 0.587785 + 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 2.23353 0.106448i 1.00000 −4.25057 1.38109i −0.951057 + 0.309017i −0.309017 0.951057i 1.39896 + 1.74440i
349.1 −0.951057 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.22153 0.254605i 1.00000 −2.23337 + 3.07398i −0.587785 0.809017i 0.809017 0.587785i 2.03412 + 0.928633i
349.2 −0.951057 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i −2.16020 + 0.577517i 1.00000 2.84298 3.91303i −0.587785 0.809017i 0.809017 0.587785i 2.23294 + 0.118288i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.18 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.d 72
5.b even 2 1 inner 930.2.z.d 72
31.d even 5 1 inner 930.2.z.d 72
155.n even 10 1 inner 930.2.z.d 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.z.d 72 1.a even 1 1 trivial
930.2.z.d 72 5.b even 2 1 inner
930.2.z.d 72 31.d even 5 1 inner
930.2.z.d 72 155.n even 10 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$16\!\cdots\!55$$$$T_{7}^{56} -$$$$29\!\cdots\!80$$$$T_{7}^{54} +$$$$45\!\cdots\!30$$$$T_{7}^{52} -$$$$63\!\cdots\!00$$$$T_{7}^{50} +$$$$80\!\cdots\!85$$$$T_{7}^{48} -$$$$84\!\cdots\!20$$$$T_{7}^{46} +$$$$82\!\cdots\!90$$$$T_{7}^{44} -$$$$68\!\cdots\!70$$$$T_{7}^{42} +$$$$45\!\cdots\!65$$$$T_{7}^{40} -$$$$24\!\cdots\!20$$$$T_{7}^{38} +$$$$12\!\cdots\!65$$$$T_{7}^{36} -$$$$55\!\cdots\!20$$$$T_{7}^{34} +$$$$20\!\cdots\!10$$$$T_{7}^{32} -$$$$61\!\cdots\!00$$$$T_{7}^{30} +$$$$20\!\cdots\!55$$$$T_{7}^{28} -$$$$34\!\cdots\!80$$$$T_{7}^{26} +$$$$78\!\cdots\!50$$$$T_{7}^{24} -$$$$12\!\cdots\!76$$$$T_{7}^{22} +$$$$11\!\cdots\!39$$$$T_{7}^{20} -$$$$66\!\cdots\!92$$$$T_{7}^{18} +$$$$23\!\cdots\!83$$$$T_{7}^{16} -$$$$52\!\cdots\!28$$$$T_{7}^{14} +$$$$80\!\cdots\!47$$$$T_{7}^{12} -$$$$10\!\cdots\!34$$$$T_{7}^{10} +$$$$12\!\cdots\!41$$$$T_{7}^{8} -$$$$10\!\cdots\!80$$$$T_{7}^{6} +$$$$61\!\cdots\!00$$$$T_{7}^{4} -$$$$33\!\cdots\!00$$$$T_{7}^{2} +$$$$20\!\cdots\!00$$">$$T_{7}^{72} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.