# Properties

 Label 930.2.o.d Level $930$ Weight $2$ Character orbit 930.o Analytic conductor $7.426$ Analytic rank $0$ Dimension $36$ 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.o (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 36q^{4} - 8q^{7} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 36q^{4} - 8q^{7} + 8q^{9} - 18q^{10} + 36q^{16} - 4q^{18} - 4q^{19} + 72q^{21} - 18q^{22} + 18q^{25} + 8q^{28} + 52q^{31} - 36q^{33} + 18q^{34} - 8q^{36} - 54q^{37} - 8q^{39} + 18q^{40} + 12q^{42} - 18q^{43} + 4q^{45} - 62q^{49} - 10q^{51} - 18q^{55} + 12q^{57} + 88q^{63} - 36q^{64} - 4q^{66} + 10q^{67} + 46q^{69} + 16q^{70} + 4q^{72} + 24q^{73} + 4q^{76} + 92q^{78} + 42q^{79} + 4q^{81} - 12q^{82} - 72q^{84} - 16q^{87} + 18q^{88} + 8q^{90} + 48q^{93} - 92q^{94} - 80q^{97} - 18q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1 1.00000i −1.71477 + 0.244025i −1.00000 −0.866025 0.500000i 0.244025 + 1.71477i 0.0693993 + 0.120203i 1.00000i 2.88090 0.836897i −0.500000 + 0.866025i
161.2 1.00000i −1.57795 + 0.714201i −1.00000 −0.866025 0.500000i 0.714201 + 1.57795i −2.21698 3.83992i 1.00000i 1.97984 2.25394i −0.500000 + 0.866025i
161.3 1.00000i −1.06943 1.36247i −1.00000 −0.866025 0.500000i −1.36247 + 1.06943i −2.35643 4.08145i 1.00000i −0.712629 + 2.91413i −0.500000 + 0.866025i
161.4 1.00000i −0.763610 + 1.55464i −1.00000 −0.866025 0.500000i 1.55464 + 0.763610i 0.262744 + 0.455085i 1.00000i −1.83380 2.37427i −0.500000 + 0.866025i
161.5 1.00000i 0.475358 + 1.66554i −1.00000 −0.866025 0.500000i 1.66554 0.475358i −1.64767 2.85386i 1.00000i −2.54807 + 1.58346i −0.500000 + 0.866025i
161.6 1.00000i 0.652600 1.60440i −1.00000 −0.866025 0.500000i −1.60440 0.652600i 2.20176 + 3.81356i 1.00000i −2.14823 2.09407i −0.500000 + 0.866025i
161.7 1.00000i 0.761277 1.55578i −1.00000 −0.866025 0.500000i −1.55578 0.761277i −0.777017 1.34583i 1.00000i −1.84092 2.36876i −0.500000 + 0.866025i
161.8 1.00000i 1.55555 + 0.761750i −1.00000 −0.866025 0.500000i 0.761750 1.55555i 0.295093 + 0.511116i 1.00000i 1.83947 + 2.36988i −0.500000 + 0.866025i
161.9 1.00000i 1.68098 0.417505i −1.00000 −0.866025 0.500000i −0.417505 1.68098i 2.16910 + 3.75699i 1.00000i 2.65138 1.40363i −0.500000 + 0.866025i
161.10 1.00000i −1.72816 0.116014i −1.00000 0.866025 + 0.500000i 0.116014 1.72816i 0.262744 + 0.455085i 1.00000i 2.97308 + 0.400981i −0.500000 + 0.866025i
161.11 1.00000i −1.40749 + 1.00944i −1.00000 0.866025 + 0.500000i −1.00944 1.40749i −2.21698 3.83992i 1.00000i 0.962053 2.84156i −0.500000 + 0.866025i
161.12 1.00000i −1.20472 1.24444i −1.00000 0.866025 + 0.500000i 1.24444 1.20472i −1.64767 2.85386i 1.00000i −0.0972822 + 2.99842i −0.500000 + 0.866025i
161.13 1.00000i −1.06872 + 1.36303i −1.00000 0.866025 + 0.500000i −1.36303 1.06872i 0.0693993 + 0.120203i 1.00000i −0.715677 2.91338i −0.500000 + 0.866025i
161.14 1.00000i 0.118080 1.72802i −1.00000 0.866025 + 0.500000i 1.72802 + 0.118080i 0.295093 + 0.511116i 1.00000i −2.97211 0.408090i −0.500000 + 0.866025i
161.15 1.00000i 0.645214 + 1.60739i −1.00000 0.866025 + 0.500000i −1.60739 + 0.645214i −2.35643 4.08145i 1.00000i −2.16740 + 2.07422i −0.500000 + 0.866025i
161.16 1.00000i 1.20206 1.24702i −1.00000 0.866025 + 0.500000i 1.24702 + 1.20206i 2.16910 + 3.75699i 1.00000i −0.110108 2.99798i −0.500000 + 0.866025i
161.17 1.00000i 1.71575 + 0.237034i −1.00000 0.866025 + 0.500000i −0.237034 + 1.71575i 2.20176 + 3.81356i 1.00000i 2.88763 + 0.813386i −0.500000 + 0.866025i
161.18 1.00000i 1.72799 + 0.118606i −1.00000 0.866025 + 0.500000i −0.118606 + 1.72799i −0.777017 1.34583i 1.00000i 2.97187 + 0.409900i −0.500000 + 0.866025i
491.1 1.00000i −1.72816 + 0.116014i −1.00000 0.866025 0.500000i 0.116014 + 1.72816i 0.262744 0.455085i 1.00000i 2.97308 0.400981i −0.500000 0.866025i
491.2 1.00000i −1.40749 1.00944i −1.00000 0.866025 0.500000i −1.00944 + 1.40749i −2.21698 + 3.83992i 1.00000i 0.962053 + 2.84156i −0.500000 0.866025i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 491.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.e odd 6 1 inner
93.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.o.d 36
3.b odd 2 1 inner 930.2.o.d 36
31.e odd 6 1 inner 930.2.o.d 36
93.g even 6 1 inner 930.2.o.d 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.o.d 36 1.a even 1 1 trivial
930.2.o.d 36 3.b odd 2 1 inner
930.2.o.d 36 31.e odd 6 1 inner
930.2.o.d 36 93.g even 6 1 inner

## Hecke kernels

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

 $$T_{7}^{18} + \cdots$$ $$48\!\cdots\!50$$$$T_{11}^{20} +$$$$44\!\cdots\!92$$$$T_{11}^{18} +$$$$29\!\cdots\!87$$$$T_{11}^{16} +$$$$12\!\cdots\!75$$$$T_{11}^{14} +$$$$32\!\cdots\!13$$$$T_{11}^{12} +$$$$14\!\cdots\!52$$$$T_{11}^{10} +$$$$46\!\cdots\!21$$$$T_{11}^{8} +$$$$69\!\cdots\!47$$$$T_{11}^{6} +$$$$75\!\cdots\!44$$$$T_{11}^{4} + 169253154993 T_{11}^{2} + 3170478249$$">$$T_{11}^{36} + \cdots$$