# Properties

 Label 930.2.k.a Level $930$ Weight $2$ Character orbit 930.k Analytic conductor $7.426$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 4q^{7} + 4q^{10} - 4q^{15} - 32q^{16} - 8q^{17} + 4q^{22} + 32q^{24} + 8q^{25} + 4q^{28} - 8q^{29} - 20q^{31} - 4q^{33} - 24q^{35} - 32q^{36} - 4q^{37} + 16q^{38} - 16q^{41} - 4q^{42} + 16q^{43} + 8q^{44} - 8q^{47} - 16q^{50} + 24q^{53} + 32q^{54} + 28q^{55} - 16q^{57} - 20q^{58} + 16q^{62} + 4q^{63} - 56q^{65} - 8q^{66} + 32q^{67} - 8q^{68} - 28q^{70} + 16q^{71} + 20q^{73} - 24q^{74} + 16q^{75} - 16q^{76} + 40q^{77} + 56q^{79} - 32q^{81} + 16q^{82} - 72q^{83} + 32q^{85} + 20q^{87} + 4q^{88} + 64q^{89} - 16q^{93} + 32q^{95} - 4q^{97} + 16q^{98} + 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
247.1 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −2.23600 + 0.0168647i 1.00000i −2.83917 2.83917i 0.707107 0.707107i 1.00000i 1.59302 + 1.56917i
247.2 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −2.14949 + 0.616200i 1.00000i 0.0344550 + 0.0344550i 0.707107 0.707107i 1.00000i 1.95564 + 1.08420i
247.3 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −1.64980 1.50935i 1.00000i 2.19620 + 2.19620i 0.707107 0.707107i 1.00000i 0.0993121 + 2.23386i
247.4 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i −0.899839 + 2.04702i 1.00000i 2.95191 + 2.95191i 0.707107 0.707107i 1.00000i 2.08374 0.811179i
247.5 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.06571 + 1.96577i 1.00000i −2.96361 2.96361i 0.707107 0.707107i 1.00000i 0.636441 2.14358i
247.6 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.06824 1.96440i 1.00000i −1.05833 1.05833i 0.707107 0.707107i 1.00000i −2.14440 + 0.633681i
247.7 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 1.94894 1.09618i 1.00000i −1.54434 1.54434i 0.707107 0.707107i 1.00000i −2.15323 0.602992i
247.8 −0.707107 0.707107i 0.707107 + 0.707107i 1.00000i 2.14513 + 0.631187i 1.00000i 2.22289 + 2.22289i 0.707107 0.707107i 1.00000i −1.07052 1.96316i
247.9 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.96413 1.06873i 1.00000i −1.95397 1.95397i −0.707107 + 0.707107i 1.00000i −0.633142 2.14456i
247.10 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.87799 + 1.21373i 1.00000i −0.853491 0.853491i −0.707107 + 0.707107i 1.00000i −2.18618 0.469704i
247.11 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i −1.03553 1.98184i 1.00000i 0.181215 + 0.181215i −0.707107 + 0.707107i 1.00000i 0.669145 2.13360i
247.12 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0.0490162 + 2.23553i 1.00000i 3.32380 + 3.32380i −0.707107 + 0.707107i 1.00000i −1.54610 + 1.61542i
247.13 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0.433540 + 2.19364i 1.00000i 0.130349 + 0.130349i −0.707107 + 0.707107i 1.00000i −1.24458 + 1.85769i
247.14 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 0.975733 2.01195i 1.00000i −3.17444 3.17444i −0.707107 + 0.707107i 1.00000i 2.11261 0.732717i
247.15 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 1.89248 1.19101i 1.00000i 2.48213 + 2.48213i −0.707107 + 0.707107i 1.00000i 2.18036 + 0.496012i
247.16 0.707107 + 0.707107i −0.707107 0.707107i 1.00000i 2.23399 0.0964676i 1.00000i −1.13560 1.13560i −0.707107 + 0.707107i 1.00000i 1.64788 + 1.51145i
433.1 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −2.23600 0.0168647i 1.00000i −2.83917 + 2.83917i 0.707107 + 0.707107i 1.00000i 1.59302 1.56917i
433.2 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −2.14949 0.616200i 1.00000i 0.0344550 0.0344550i 0.707107 + 0.707107i 1.00000i 1.95564 1.08420i
433.3 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −1.64980 + 1.50935i 1.00000i 2.19620 2.19620i 0.707107 + 0.707107i 1.00000i 0.0993121 2.23386i
433.4 −0.707107 + 0.707107i 0.707107 0.707107i 1.00000i −0.899839 2.04702i 1.00000i 2.95191 2.95191i 0.707107 + 0.707107i 1.00000i 2.08374 + 0.811179i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 433.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
155.f 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.k.a 32
5.c odd 4 1 930.2.k.b yes 32
31.b odd 2 1 930.2.k.b yes 32
155.f even 4 1 inner 930.2.k.a 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.k.a 32 1.a even 1 1 trivial
930.2.k.a 32 155.f even 4 1 inner
930.2.k.b yes 32 5.c odd 4 1
930.2.k.b yes 32 31.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$10\!\cdots\!64$$$$T_{13}^{7} + 603015118848 T_{13}^{6} + 269881638912 T_{13}^{5} + 284265676800 T_{13}^{4} + 265442033664 T_{13}^{3} + 152882380800 T_{13}^{2} + 45864714240 T_{13} + 6879707136$$">$$T_{13}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.