# Properties

 Label 930.2.j.h Level $930$ Weight $2$ Character orbit 930.j Analytic conductor $7.426$ Analytic rank $0$ Dimension $40$ 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.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{3} + 8q^{6} - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{3} + 8q^{6} - 8q^{7} + 4q^{12} + 24q^{13} - 12q^{15} - 40q^{16} - 56q^{21} + 16q^{22} - 48q^{25} + 8q^{27} - 8q^{28} + 4q^{30} - 40q^{31} - 20q^{33} - 8q^{37} - 8q^{40} + 12q^{42} - 16q^{43} + 76q^{45} - 8q^{46} + 4q^{48} + 16q^{51} - 24q^{52} - 48q^{55} + 56q^{57} - 56q^{58} + 12q^{60} + 128q^{61} - 24q^{63} + 16q^{66} + 8q^{67} + 32q^{70} + 248q^{73} - 8q^{75} + 56q^{76} + 8q^{78} + 40q^{81} - 8q^{82} - 48q^{85} + 12q^{87} + 16q^{88} + 16q^{90} - 192q^{91} + 4q^{93} - 8q^{96} + 24q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
497.1 −0.707107 + 0.707107i −1.73016 + 0.0809240i 1.00000i −0.662050 2.13581i 1.16619 1.28063i 2.89525 + 2.89525i 0.707107 + 0.707107i 2.98690 0.280023i 1.97839 + 1.04211i
497.2 −0.707107 + 0.707107i −1.56098 + 0.750564i 1.00000i 0.256652 + 2.22129i 0.573050 1.63451i 0.463267 + 0.463267i 0.707107 + 0.707107i 1.87331 2.34323i −1.75217 1.38921i
497.3 −0.707107 + 0.707107i −1.50338 + 0.860137i 1.00000i 2.19538 0.424609i 0.454844 1.67126i −0.358790 0.358790i 0.707107 + 0.707107i 1.52033 2.58623i −1.25213 + 1.85261i
497.4 −0.707107 + 0.707107i −1.30199 1.14229i 1.00000i 1.36562 1.77061i 1.72837 0.112923i −0.0800893 0.0800893i 0.707107 + 0.707107i 0.390345 + 2.97450i 0.286372 + 2.21765i
497.5 −0.707107 + 0.707107i −0.404727 1.68410i 1.00000i −2.01952 + 0.959968i 1.47702 + 0.904654i −2.95632 2.95632i 0.707107 + 0.707107i −2.67239 + 1.36320i 0.749217 2.10682i
497.6 −0.707107 + 0.707107i 0.0792035 1.73024i 1.00000i −1.29721 + 1.82133i 1.16746 + 1.27947i 2.17556 + 2.17556i 0.707107 + 0.707107i −2.98745 0.274082i −0.370615 2.20514i
497.7 −0.707107 + 0.707107i 0.0980849 + 1.72927i 1.00000i −0.278761 + 2.21862i −1.29214 1.15342i −1.59354 1.59354i 0.707107 + 0.707107i −2.98076 + 0.339231i −1.37169 1.76592i
497.8 −0.707107 + 0.707107i 0.877144 + 1.49353i 1.00000i −1.98408 1.03122i −1.67632 0.435847i 2.39520 + 2.39520i 0.707107 + 0.707107i −1.46124 + 2.62007i 2.13214 0.673776i
497.9 −0.707107 + 0.707107i 1.33303 1.10591i 1.00000i −0.368595 2.20548i −0.160600 + 1.72459i −2.86656 2.86656i 0.707107 + 0.707107i 0.553937 2.94842i 1.82015 + 1.29887i
497.10 −0.707107 + 0.707107i 1.69956 + 0.333903i 1.00000i 1.37834 + 1.76073i −1.43788 + 0.965667i −2.07397 2.07397i 0.707107 + 0.707107i 2.77702 + 1.13498i −2.21966 0.270390i
497.11 0.707107 0.707107i −1.72927 0.0980849i 1.00000i 0.278761 2.21862i −1.29214 + 1.15342i −1.59354 1.59354i −0.707107 0.707107i 2.98076 + 0.339231i −1.37169 1.76592i
497.12 0.707107 0.707107i −1.49353 0.877144i 1.00000i 1.98408 + 1.03122i −1.67632 + 0.435847i 2.39520 + 2.39520i −0.707107 0.707107i 1.46124 + 2.62007i 2.13214 0.673776i
497.13 0.707107 0.707107i −0.860137 + 1.50338i 1.00000i −2.19538 + 0.424609i 0.454844 + 1.67126i −0.358790 0.358790i −0.707107 0.707107i −1.52033 2.58623i −1.25213 + 1.85261i
497.14 0.707107 0.707107i −0.750564 + 1.56098i 1.00000i −0.256652 2.22129i 0.573050 + 1.63451i 0.463267 + 0.463267i −0.707107 0.707107i −1.87331 2.34323i −1.75217 1.38921i
497.15 0.707107 0.707107i −0.333903 1.69956i 1.00000i −1.37834 1.76073i −1.43788 0.965667i −2.07397 2.07397i −0.707107 0.707107i −2.77702 + 1.13498i −2.21966 0.270390i
497.16 0.707107 0.707107i −0.0809240 + 1.73016i 1.00000i 0.662050 + 2.13581i 1.16619 + 1.28063i 2.89525 + 2.89525i −0.707107 0.707107i −2.98690 0.280023i 1.97839 + 1.04211i
497.17 0.707107 0.707107i 1.10591 1.33303i 1.00000i 0.368595 + 2.20548i −0.160600 1.72459i −2.86656 2.86656i −0.707107 0.707107i −0.553937 2.94842i 1.82015 + 1.29887i
497.18 0.707107 0.707107i 1.14229 + 1.30199i 1.00000i −1.36562 + 1.77061i 1.72837 + 0.112923i −0.0800893 0.0800893i −0.707107 0.707107i −0.390345 + 2.97450i 0.286372 + 2.21765i
497.19 0.707107 0.707107i 1.68410 + 0.404727i 1.00000i 2.01952 0.959968i 1.47702 0.904654i −2.95632 2.95632i −0.707107 0.707107i 2.67239 + 1.36320i 0.749217 2.10682i
497.20 0.707107 0.707107i 1.73024 0.0792035i 1.00000i 1.29721 1.82133i 1.16746 1.27947i 2.17556 + 2.17556i −0.707107 0.707107i 2.98745 0.274082i −0.370615 2.20514i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 683.20 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
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.h 40
3.b odd 2 1 inner 930.2.j.h 40
5.c odd 4 1 inner 930.2.j.h 40
15.e even 4 1 inner 930.2.j.h 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.h 40 1.a even 1 1 trivial
930.2.j.h 40 3.b odd 2 1 inner
930.2.j.h 40 5.c odd 4 1 inner
930.2.j.h 40 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{20} + \cdots$$ $$T_{11}^{20} + \cdots$$ $$30\!\cdots\!96$$$$T_{17}^{24} +$$$$61\!\cdots\!60$$$$T_{17}^{20} +$$$$36\!\cdots\!36$$$$T_{17}^{16} +$$$$59\!\cdots\!48$$$$T_{17}^{12} +$$$$23\!\cdots\!52$$$$T_{17}^{8} +$$$$28\!\cdots\!52$$$$T_{17}^{4} +$$$$10\!\cdots\!96$$">$$T_{17}^{40} + \cdots$$