# Properties

 Label 930.2.bs.a Level $930$ Weight $2$ Character orbit 930.bs Analytic conductor $7.426$ Analytic rank $0$ Dimension $1024$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.bs (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$1024$$ Relative dimension: $$64$$ over $$\Q(\zeta_{60})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$1024q - 8q^{3} + 24q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$1024q - 8q^{3} + 24q^{7} + 8q^{10} + 8q^{12} - 8q^{15} + 256q^{16} + 44q^{22} + 16q^{25} - 20q^{27} + 4q^{28} + 48q^{30} - 32q^{31} - 52q^{33} + 112q^{37} + 12q^{42} + 40q^{45} + 24q^{46} - 12q^{48} + 16q^{51} - 20q^{55} - 40q^{57} - 56q^{58} - 12q^{60} + 32q^{61} + 224q^{63} - 32q^{66} + 24q^{67} - 24q^{70} - 88q^{73} + 224q^{75} - 96q^{76} - 32q^{78} + 48q^{81} - 24q^{85} + 20q^{87} - 36q^{88} + 8q^{90} - 80q^{91} - 252q^{93} - 32q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −0.453990 0.891007i −1.71996 0.204302i −0.587785 + 0.809017i −1.77453 1.36053i 0.598811 + 1.62525i −3.08239 + 2.49607i 0.987688 + 0.156434i 2.91652 + 0.702783i −0.406621 + 2.19879i
107.2 −0.453990 0.891007i −1.70228 0.319750i −0.587785 + 0.809017i 1.94583 + 1.10171i 0.487920 + 1.66191i 0.0931562 0.0754364i 0.987688 + 0.156434i 2.79552 + 1.08861i 0.0982390 2.23391i
107.3 −0.453990 0.891007i −1.69507 + 0.355982i −0.587785 + 0.809017i 2.01411 0.971260i 1.08673 + 1.34871i 2.26525 1.83436i 0.987688 + 0.156434i 2.74655 1.20683i −1.77979 1.35365i
107.4 −0.453990 0.891007i −1.67186 + 0.452624i −0.587785 + 0.809017i 2.07413 + 0.835445i 1.16230 + 1.28416i −1.71190 + 1.38627i 0.987688 + 0.156434i 2.59026 1.51345i −0.197250 2.22735i
107.5 −0.453990 0.891007i −1.61622 0.622763i −0.587785 + 0.809017i −0.949618 + 2.02441i 0.178863 + 1.72279i 0.463591 0.375409i 0.987688 + 0.156434i 2.22433 + 2.01304i 2.23488 0.0729461i
107.6 −0.453990 0.891007i −1.52373 0.823562i −0.587785 + 0.809017i −1.29986 1.81944i −0.0420417 + 1.73154i 2.38916 1.93471i 0.987688 + 0.156434i 1.64349 + 2.50977i −1.03101 + 1.98419i
107.7 −0.453990 0.891007i −1.49231 + 0.879203i −0.587785 + 0.809017i −2.08977 + 0.795522i 1.46087 + 0.930512i 3.09036 2.50253i 0.987688 + 0.156434i 1.45400 2.62409i 1.65755 + 1.50084i
107.8 −0.453990 0.891007i −1.19166 1.25696i −0.587785 + 0.809017i 1.48838 1.66875i −0.578959 + 1.63242i −2.67101 + 2.16294i 0.987688 + 0.156434i −0.159900 + 2.99574i −2.16258 0.568562i
107.9 −0.453990 0.891007i −1.02426 + 1.39674i −0.587785 + 0.809017i −2.05905 0.871953i 1.70951 + 0.278513i −0.725093 + 0.587168i 0.987688 + 0.156434i −0.901788 2.86125i 0.157874 + 2.23049i
107.10 −0.453990 0.891007i −1.00357 + 1.41168i −0.587785 + 0.809017i 0.618932 2.14870i 1.71343 + 0.253300i −0.574024 + 0.464836i 0.987688 + 0.156434i −0.985684 2.83345i −2.19550 + 0.424018i
107.11 −0.453990 0.891007i −0.827104 + 1.52181i −0.587785 + 0.809017i 0.791841 + 2.09117i 1.73144 + 0.0460690i 2.74072 2.21939i 0.987688 + 0.156434i −1.63180 2.51739i 1.50376 1.65491i
107.12 −0.453990 0.891007i −0.695665 1.58621i −0.587785 + 0.809017i 1.24127 + 1.85990i −1.09749 + 1.33996i 3.78072 3.06157i 0.987688 + 0.156434i −2.03210 + 2.20694i 1.09366 1.95036i
107.13 −0.453990 0.891007i −0.602074 1.62404i −0.587785 + 0.809017i −1.44079 1.71001i −1.17369 + 1.27375i 0.478316 0.387333i 0.987688 + 0.156434i −2.27501 + 1.95559i −0.869520 + 2.06008i
107.14 −0.453990 0.891007i −0.510421 1.65513i −0.587785 + 0.809017i 1.93109 + 1.12734i −1.24301 + 1.20620i −2.49173 + 2.01776i 0.987688 + 0.156434i −2.47894 + 1.68963i 0.127771 2.23241i
107.15 −0.453990 0.891007i −0.205138 1.71986i −0.587785 + 0.809017i 1.37425 1.76393i −1.43928 + 0.963580i 1.73979 1.40885i 0.987688 + 0.156434i −2.91584 + 0.705618i −2.19557 0.423656i
107.16 −0.453990 0.891007i −0.148049 1.72571i −0.587785 + 0.809017i −1.13774 + 1.92498i −1.47041 + 0.915369i −1.33909 + 1.08437i 0.987688 + 0.156434i −2.95616 + 0.510979i 2.23169 + 0.139806i
107.17 −0.453990 0.891007i 0.0494968 + 1.73134i −0.587785 + 0.809017i 0.307868 + 2.21477i 1.52017 0.830115i −0.186966 + 0.151402i 0.987688 + 0.156434i −2.99510 + 0.171392i 1.83361 1.27980i
107.18 −0.453990 0.891007i 0.0504550 + 1.73132i −0.587785 + 0.809017i −2.07141 + 0.842184i 1.51971 0.830957i −3.23873 + 2.62267i 0.987688 + 0.156434i −2.99491 + 0.174707i 1.69079 + 1.46329i
107.19 −0.453990 0.891007i 0.546386 + 1.64361i −0.587785 + 0.809017i 0.869573 2.06006i 1.21642 1.23302i −1.46148 + 1.18348i 0.987688 + 0.156434i −2.40292 + 1.79610i −2.23030 + 0.160452i
107.20 −0.453990 0.891007i 0.559329 1.63925i −0.587785 + 0.809017i −2.23601 + 0.0166336i −1.71452 + 0.245839i −0.828800 + 0.671149i 0.987688 + 0.156434i −2.37430 1.83376i 1.02995 + 1.98474i
See next 80 embeddings (of 1024 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 917.64 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
5.c odd 4 1 inner
15.e even 4 1 inner
31.g even 15 1 inner
93.o odd 30 1 inner
155.w odd 60 1 inner
465.bt even 60 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.bs.a 1024
3.b odd 2 1 inner 930.2.bs.a 1024
5.c odd 4 1 inner 930.2.bs.a 1024
15.e even 4 1 inner 930.2.bs.a 1024
31.g even 15 1 inner 930.2.bs.a 1024
93.o odd 30 1 inner 930.2.bs.a 1024
155.w odd 60 1 inner 930.2.bs.a 1024
465.bt even 60 1 inner 930.2.bs.a 1024

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.bs.a 1024 1.a even 1 1 trivial
930.2.bs.a 1024 3.b odd 2 1 inner
930.2.bs.a 1024 5.c odd 4 1 inner
930.2.bs.a 1024 15.e even 4 1 inner
930.2.bs.a 1024 31.g even 15 1 inner
930.2.bs.a 1024 93.o odd 30 1 inner
930.2.bs.a 1024 155.w odd 60 1 inner
930.2.bs.a 1024 465.bt even 60 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(930, [\chi])$$.