# Properties

 Label 930.2.y.a Level $930$ Weight $2$ Character orbit 930.y Analytic conductor $7.426$ Analytic rank $0$ Dimension $128$ 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.y (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$128q - 32q^{2} - 32q^{4} + 2q^{5} - 32q^{8} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128q - 32q^{2} - 32q^{4} + 2q^{5} - 32q^{8} - 4q^{9} + 2q^{10} + 25q^{15} - 32q^{16} + 6q^{18} - 8q^{19} - 3q^{20} - 20q^{23} - 10q^{25} - 48q^{31} + 128q^{32} - 8q^{33} + 10q^{34} + 16q^{35} - 4q^{36} + 12q^{38} + 4q^{39} - 3q^{40} + 37q^{45} + 10q^{46} + 6q^{47} + 46q^{49} - 5q^{50} + 34q^{51} - 20q^{53} - 25q^{60} - 8q^{62} + 36q^{63} - 32q^{64} - 8q^{66} + 8q^{69} + 16q^{70} + 6q^{72} + 5q^{75} + 12q^{76} + 50q^{77} + 4q^{78} - 10q^{79} + 2q^{80} - 24q^{81} - 40q^{83} - 30q^{85} - 4q^{87} - 53q^{90} + 20q^{91} - 26q^{93} - 4q^{94} - 26q^{95} - 124q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −0.809017 0.587785i −1.68775 + 0.389232i 0.309017 + 0.951057i −1.12975 + 1.92968i 1.59420 + 0.677139i 1.54573 0.502238i 0.309017 0.951057i 2.69700 1.31385i 2.04823 0.897090i
29.2 −0.809017 0.587785i −1.67374 0.445633i 0.309017 + 0.951057i −1.13834 1.92463i 1.09215 + 1.34433i −3.43231 + 1.11523i 0.309017 0.951057i 2.60282 + 1.49175i −0.210331 + 2.22615i
29.3 −0.809017 0.587785i −1.66935 0.461806i 0.309017 + 0.951057i 1.02596 + 1.98681i 1.07909 + 1.35483i −2.19882 + 0.714440i 0.309017 0.951057i 2.57347 + 1.54183i 0.337800 2.21041i
29.4 −0.809017 0.587785i −1.61743 0.619623i 0.309017 + 0.951057i −0.184552 2.22844i 0.944320 + 1.45198i 2.03784 0.662134i 0.309017 0.951057i 2.23213 + 2.00439i −1.16054 + 1.91132i
29.5 −0.809017 0.587785i −1.60626 + 0.648031i 0.309017 + 0.951057i 2.20040 0.397802i 1.68039 + 0.419865i 4.23373 1.37562i 0.309017 0.951057i 2.16011 2.08181i −2.01398 0.971534i
29.6 −0.809017 0.587785i −1.51893 + 0.832372i 0.309017 + 0.951057i 1.72517 1.42260i 1.71810 + 0.219404i 0.914445 0.297121i 0.309017 0.951057i 1.61431 2.52863i −2.23187 + 0.136884i
29.7 −0.809017 0.587785i −1.32727 + 1.11282i 0.309017 + 0.951057i −2.11788 0.717333i 1.72788 0.120141i 0.344014 0.111777i 0.309017 0.951057i 0.523274 2.95401i 1.29177 + 1.82520i
29.8 −0.809017 0.587785i −1.26585 1.18221i 0.309017 + 0.951057i 1.86663 + 1.23113i 0.329206 + 1.70048i 3.84247 1.24850i 0.309017 0.951057i 0.204746 + 2.99301i −0.786495 2.09319i
29.9 −0.809017 0.587785i −1.22753 + 1.22195i 0.309017 + 0.951057i 0.560987 2.16455i 1.71134 0.267056i −4.05181 + 1.31651i 0.309017 0.951057i 0.0136636 2.99997i −1.72614 + 1.42142i
29.10 −0.809017 0.587785i −1.02011 1.39978i 0.309017 + 0.951057i −2.08406 + 0.810354i 0.00251367 + 1.73205i −2.70101 + 0.877611i 0.309017 0.951057i −0.918766 + 2.85585i 2.16236 + 0.569392i
29.11 −0.809017 0.587785i −1.01604 1.40273i 0.309017 + 0.951057i −2.08406 0.810354i −0.00251367 + 1.73205i 2.70101 0.877611i 0.309017 0.951057i −0.935328 + 2.85047i 1.20973 + 1.88057i
29.12 −0.809017 0.587785i −0.733182 1.56922i 0.309017 + 0.951057i 1.86663 1.23113i −0.329206 + 1.70048i −3.84247 + 1.24850i 0.309017 0.951057i −1.92489 + 2.30105i −2.23378 0.101171i
29.13 −0.809017 0.587785i −0.515138 + 1.65367i 0.309017 + 0.951057i −1.43197 + 1.71740i 1.38876 1.03506i −4.13806 + 1.34454i 0.309017 0.951057i −2.46927 1.70374i 2.16795 0.547714i
29.14 −0.809017 0.587785i −0.292097 + 1.70724i 0.309017 + 0.951057i 1.90023 + 1.17861i 1.23980 1.20950i −0.119229 + 0.0387399i 0.309017 0.951057i −2.82936 0.997360i −0.844545 2.07045i
29.15 −0.809017 0.587785i −0.0894846 1.72974i 0.309017 + 0.951057i −0.184552 + 2.22844i −0.944320 + 1.45198i −2.03784 + 0.662134i 0.309017 0.951057i −2.98398 + 0.309570i 1.45915 1.69437i
29.16 −0.809017 0.587785i 0.0766541 1.73035i 0.309017 + 0.951057i 1.02596 1.98681i −1.07909 + 1.35483i 2.19882 0.714440i 0.309017 0.951057i −2.98825 0.265277i −1.99783 + 1.00432i
29.17 −0.809017 0.587785i 0.0933927 1.72953i 0.309017 + 0.951057i −1.13834 + 1.92463i −1.09215 + 1.34433i 3.43231 1.11523i 0.309017 0.951057i −2.98256 0.323051i 2.05220 0.887956i
29.18 −0.809017 0.587785i 0.116082 + 1.72816i 0.309017 + 0.951057i −1.22659 + 1.86962i 0.921873 1.46634i 3.97153 1.29043i 0.309017 0.951057i −2.97305 + 0.401214i 2.09127 0.791586i
29.19 −0.809017 0.587785i 0.195625 + 1.72097i 0.309017 + 0.951057i −0.323295 2.21257i 0.853296 1.50728i 1.33289 0.433081i 0.309017 0.951057i −2.92346 + 0.673329i −1.03897 + 1.98004i
29.20 −0.809017 0.587785i 0.317374 + 1.70273i 0.309017 + 0.951057i 2.12610 0.692603i 0.744076 1.56408i −0.896862 + 0.291408i 0.309017 0.951057i −2.79855 + 1.08080i −2.12715 0.689362i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
31.f odd 10 1 inner
465.w 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.y.a 128
3.b odd 2 1 930.2.y.b yes 128
5.b even 2 1 930.2.y.b yes 128
15.d odd 2 1 inner 930.2.y.a 128
31.f odd 10 1 inner 930.2.y.a 128
93.k even 10 1 930.2.y.b yes 128
155.m odd 10 1 930.2.y.b yes 128
465.w even 10 1 inner 930.2.y.a 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.y.a 128 1.a even 1 1 trivial
930.2.y.a 128 15.d odd 2 1 inner
930.2.y.a 128 31.f odd 10 1 inner
930.2.y.a 128 465.w even 10 1 inner
930.2.y.b yes 128 3.b odd 2 1
930.2.y.b yes 128 5.b even 2 1
930.2.y.b yes 128 93.k even 10 1
930.2.y.b yes 128 155.m odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$34\!\cdots\!89$$$$T_{17}^{50} +$$$$32\!\cdots\!90$$$$T_{17}^{49} +$$$$11\!\cdots\!41$$$$T_{17}^{48} -$$$$13\!\cdots\!05$$$$T_{17}^{47} -$$$$33\!\cdots\!67$$$$T_{17}^{46} +$$$$45\!\cdots\!95$$$$T_{17}^{45} +$$$$89\!\cdots\!11$$$$T_{17}^{44} -$$$$14\!\cdots\!50$$$$T_{17}^{43} -$$$$21\!\cdots\!91$$$$T_{17}^{42} +$$$$39\!\cdots\!40$$$$T_{17}^{41} +$$$$43\!\cdots\!63$$$$T_{17}^{40} -$$$$98\!\cdots\!25$$$$T_{17}^{39} -$$$$78\!\cdots\!58$$$$T_{17}^{38} +$$$$21\!\cdots\!55$$$$T_{17}^{37} +$$$$12\!\cdots\!60$$$$T_{17}^{36} -$$$$40\!\cdots\!95$$$$T_{17}^{35} -$$$$16\!\cdots\!98$$$$T_{17}^{34} +$$$$67\!\cdots\!85$$$$T_{17}^{33} +$$$$18\!\cdots\!15$$$$T_{17}^{32} -$$$$94\!\cdots\!15$$$$T_{17}^{31} -$$$$14\!\cdots\!89$$$$T_{17}^{30} +$$$$10\!\cdots\!30$$$$T_{17}^{29} +$$$$97\!\cdots\!33$$$$T_{17}^{28} -$$$$10\!\cdots\!30$$$$T_{17}^{27} -$$$$34\!\cdots\!86$$$$T_{17}^{26} +$$$$95\!\cdots\!65$$$$T_{17}^{25} -$$$$44\!\cdots\!47$$$$T_{17}^{24} -$$$$66\!\cdots\!80$$$$T_{17}^{23} +$$$$97\!\cdots\!27$$$$T_{17}^{22} +$$$$22\!\cdots\!90$$$$T_{17}^{21} -$$$$30\!\cdots\!60$$$$T_{17}^{20} -$$$$21\!\cdots\!05$$$$T_{17}^{19} +$$$$59\!\cdots\!18$$$$T_{17}^{18} +$$$$40\!\cdots\!90$$$$T_{17}^{17} -$$$$21\!\cdots\!05$$$$T_{17}^{16} +$$$$24\!\cdots\!95$$$$T_{17}^{15} +$$$$28\!\cdots\!83$$$$T_{17}^{14} -$$$$79\!\cdots\!55$$$$T_{17}^{13} +$$$$19\!\cdots\!33$$$$T_{17}^{12} +$$$$91\!\cdots\!80$$$$T_{17}^{11} -$$$$52\!\cdots\!88$$$$T_{17}^{10} -$$$$94\!\cdots\!20$$$$T_{17}^{9} +$$$$66\!\cdots\!84$$$$T_{17}^{8} +$$$$12\!\cdots\!60$$$$T_{17}^{7} -$$$$11\!\cdots\!88$$$$T_{17}^{6} -$$$$10\!\cdots\!00$$$$T_{17}^{5} +$$$$20\!\cdots\!16$$$$T_{17}^{4} -$$$$10\!\cdots\!80$$$$T_{17}^{3} +$$$$21\!\cdots\!68$$$$T_{17}^{2} +$$$$49\!\cdots\!20$$$$T_{17} +$$$$22\!\cdots\!16$$">$$T_{17}^{64} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.