# Properties

 Label 930.2.j.g 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{3} - 8q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 4q^{3} - 8q^{7} + 8q^{10} + 4q^{12} - 20q^{13} - 44q^{15} - 40q^{16} + 16q^{18} - 32q^{21} - 4q^{22} + 8q^{25} + 8q^{27} - 8q^{28} - 4q^{30} + 40q^{31} + 48q^{33} + 64q^{37} - 4q^{40} - 20q^{42} - 48q^{43} - 4q^{45} - 80q^{46} + 4q^{48} + 48q^{51} + 20q^{52} + 164q^{55} + 40q^{57} + 56q^{58} + 24q^{60} - 128q^{61} + 56q^{63} + 8q^{66} - 4q^{67} - 56q^{70} - 16q^{72} - 48q^{73} - 48q^{75} - 40q^{76} - 36q^{78} - 24q^{81} - 16q^{82} - 56q^{85} - 12q^{87} - 4q^{88} - 28q^{90} - 32q^{91} - 4q^{93} - 52q^{97} + O(q^{100})$$

## 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 −1.65310 + 0.516967i 1.00000i −0.633448 2.14447i 0.803369 1.53447i −2.44874 2.44874i 0.707107 + 0.707107i 2.46549 1.70920i 1.96428 + 1.06845i
497.2 −0.707107 + 0.707107i −1.47409 0.909430i 1.00000i 2.03224 0.932730i 1.68540 0.399274i −2.61297 2.61297i 0.707107 + 0.707107i 1.34588 + 2.68116i −0.777474 + 2.09655i
497.3 −0.707107 + 0.707107i −0.919406 + 1.46789i 1.00000i 2.22068 + 0.261841i −0.387835 1.68807i 3.32500 + 3.32500i 0.707107 + 0.707107i −1.30939 2.69917i −1.75541 + 1.38511i
497.4 −0.707107 + 0.707107i −0.731811 + 1.56986i 1.00000i −1.45516 + 1.69779i −0.592589 1.62753i 1.76311 + 1.76311i 0.707107 + 0.707107i −1.92891 2.29768i −0.171568 2.22948i
497.5 −0.707107 + 0.707107i −0.673165 1.59589i 1.00000i −1.55932 1.60266i 1.60446 + 0.652462i −0.700251 0.700251i 0.707107 + 0.707107i −2.09370 + 2.14859i 2.23586 + 0.0306496i
497.6 −0.707107 + 0.707107i 0.0444250 1.73148i 1.00000i −0.142194 2.23154i 1.19293 + 1.25576i 2.34383 + 2.34383i 0.707107 + 0.707107i −2.99605 0.153842i 1.67849 + 1.47739i
497.7 −0.707107 + 0.707107i 0.765006 + 1.55395i 1.00000i 1.73301 + 1.41304i −1.63975 0.557869i 1.07773 + 1.07773i 0.707107 + 0.707107i −1.82953 + 2.37756i −2.22459 + 0.226250i
497.8 −0.707107 + 0.707107i 0.853874 + 1.50695i 1.00000i −2.20708 0.358875i −1.66935 0.461795i −1.72132 1.72132i 0.707107 + 0.707107i −1.54180 + 2.57349i 1.81441 1.30688i
497.9 −0.707107 + 0.707107i 1.05623 1.37273i 1.00000i −0.0887404 + 2.23431i 0.223801 + 1.71753i −1.91575 1.91575i 0.707107 + 0.707107i −0.768771 2.89983i −1.51714 1.64264i
497.10 −0.707107 + 0.707107i 1.73204 0.00608872i 1.00000i −2.02131 + 0.956187i −1.22043 + 1.22904i −1.11064 1.11064i 0.707107 + 0.707107i 2.99993 0.0210918i 0.753158 2.10541i
497.11 0.707107 0.707107i −1.56986 + 0.731811i 1.00000i 1.45516 1.69779i −0.592589 + 1.62753i 1.76311 + 1.76311i −0.707107 0.707107i 1.92891 2.29768i −0.171568 2.22948i
497.12 0.707107 0.707107i −1.55395 0.765006i 1.00000i −1.73301 1.41304i −1.63975 + 0.557869i 1.07773 + 1.07773i −0.707107 0.707107i 1.82953 + 2.37756i −2.22459 + 0.226250i
497.13 0.707107 0.707107i −1.50695 0.853874i 1.00000i 2.20708 + 0.358875i −1.66935 + 0.461795i −1.72132 1.72132i −0.707107 0.707107i 1.54180 + 2.57349i 1.81441 1.30688i
497.14 0.707107 0.707107i −1.46789 + 0.919406i 1.00000i −2.22068 0.261841i −0.387835 + 1.68807i 3.32500 + 3.32500i −0.707107 0.707107i 1.30939 2.69917i −1.75541 + 1.38511i
497.15 0.707107 0.707107i −0.516967 + 1.65310i 1.00000i 0.633448 + 2.14447i 0.803369 + 1.53447i −2.44874 2.44874i −0.707107 0.707107i −2.46549 1.70920i 1.96428 + 1.06845i
497.16 0.707107 0.707107i 0.00608872 1.73204i 1.00000i 2.02131 0.956187i −1.22043 1.22904i −1.11064 1.11064i −0.707107 0.707107i −2.99993 0.0210918i 0.753158 2.10541i
497.17 0.707107 0.707107i 0.909430 + 1.47409i 1.00000i −2.03224 + 0.932730i 1.68540 + 0.399274i −2.61297 2.61297i −0.707107 0.707107i −1.34588 + 2.68116i −0.777474 + 2.09655i
497.18 0.707107 0.707107i 1.37273 1.05623i 1.00000i 0.0887404 2.23431i 0.223801 1.71753i −1.91575 1.91575i −0.707107 0.707107i 0.768771 2.89983i −1.51714 1.64264i
497.19 0.707107 0.707107i 1.59589 + 0.673165i 1.00000i 1.55932 + 1.60266i 1.60446 0.652462i −0.700251 0.700251i −0.707107 0.707107i 2.09370 + 2.14859i 2.23586 + 0.0306496i
497.20 0.707107 0.707107i 1.73148 0.0444250i 1.00000i 0.142194 + 2.23154i 1.19293 1.25576i 2.34383 + 2.34383i −0.707107 0.707107i 2.99605 0.153842i 1.67849 + 1.47739i
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.g 40
3.b odd 2 1 inner 930.2.j.g 40
5.c odd 4 1 inner 930.2.j.g 40
15.e even 4 1 inner 930.2.j.g 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.g 40 1.a even 1 1 trivial
930.2.j.g 40 3.b odd 2 1 inner
930.2.j.g 40 5.c odd 4 1 inner
930.2.j.g 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{20} + \cdots$$ $$T_{11}^{20} + \cdots$$ $$84\!\cdots\!96$$$$T_{17}^{24} +$$$$24\!\cdots\!16$$$$T_{17}^{20} +$$$$26\!\cdots\!36$$$$T_{17}^{16} +$$$$10\!\cdots\!40$$$$T_{17}^{12} +$$$$99\!\cdots\!44$$$$T_{17}^{8} +$$$$16\!\cdots\!80$$$$T_{17}^{4} +$$$$44\!\cdots\!76$$">$$T_{17}^{40} + \cdots$$