# Properties

 Label 1380.2.r.c Level $1380$ Weight $2$ Character orbit 1380.r Analytic conductor $11.019$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.r (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0193554789$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 8q^{3} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 8q^{3} + 32q^{13} + 24q^{21} - 32q^{25} - 28q^{27} - 32q^{31} - 44q^{33} + 24q^{37} - 32q^{43} + 88q^{45} + 16q^{51} + 8q^{55} + 16q^{57} - 32q^{61} - 12q^{63} - 16q^{67} - 32q^{73} + 4q^{75} - 64q^{81} - 32q^{85} + 64q^{91} + 8q^{93} - 96q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
737.1 0 −1.72892 0.104128i 0 1.51193 + 1.64744i 0 3.27368 + 3.27368i 0 2.97831 + 0.360057i 0
737.2 0 −1.71368 0.251606i 0 0.257544 2.22119i 0 −2.77975 2.77975i 0 2.87339 + 0.862345i 0
737.3 0 −1.67967 0.422758i 0 2.05360 0.884710i 0 0.924264 + 0.924264i 0 2.64255 + 1.42018i 0
737.4 0 −1.57061 0.730184i 0 −0.115647 + 2.23308i 0 −0.487335 0.487335i 0 1.93366 + 2.29368i 0
737.5 0 −1.53876 + 0.795128i 0 1.56652 1.59562i 0 0.769646 + 0.769646i 0 1.73554 2.44702i 0
737.6 0 −1.50371 0.859570i 0 −0.917834 2.03901i 0 0.160200 + 0.160200i 0 1.52228 + 2.58509i 0
737.7 0 −1.43700 + 0.966965i 0 1.81165 + 1.31070i 0 −2.64771 2.64771i 0 1.12996 2.77906i 0
737.8 0 −1.43142 + 0.975211i 0 0.832554 + 2.07530i 0 0.0455472 + 0.0455472i 0 1.09793 2.79187i 0
737.9 0 −1.31796 1.12382i 0 2.16179 + 0.571544i 0 −3.03605 3.03605i 0 0.474038 + 2.96231i 0
737.10 0 −1.29281 1.15267i 0 −0.532114 + 2.17183i 0 −0.268444 0.268444i 0 0.342726 + 2.98036i 0
737.11 0 −1.21045 1.23887i 0 −2.23574 + 0.0384947i 0 1.22712 + 1.22712i 0 −0.0696199 + 2.99919i 0
737.12 0 −0.975211 + 1.43142i 0 −0.832554 2.07530i 0 0.0455472 + 0.0455472i 0 −1.09793 2.79187i 0
737.13 0 −0.966965 + 1.43700i 0 −1.81165 1.31070i 0 −2.64771 2.64771i 0 −1.12996 2.77906i 0
737.14 0 −0.795128 + 1.53876i 0 −1.56652 + 1.59562i 0 0.769646 + 0.769646i 0 −1.73554 2.44702i 0
737.15 0 −0.727046 1.57207i 0 1.08694 1.95412i 0 2.39152 + 2.39152i 0 −1.94281 + 2.28593i 0
737.16 0 −0.136960 1.72663i 0 −2.01176 0.976134i 0 −0.818985 0.818985i 0 −2.96248 + 0.472958i 0
737.17 0 −0.118851 1.72797i 0 0.238514 + 2.22331i 0 2.86460 + 2.86460i 0 −2.97175 + 0.410741i 0
737.18 0 −0.0819537 1.73011i 0 −1.50202 + 1.65649i 0 −2.00070 2.00070i 0 −2.98657 + 0.283578i 0
737.19 0 0.104128 + 1.72892i 0 −1.51193 1.64744i 0 3.27368 + 3.27368i 0 −2.97831 + 0.360057i 0
737.20 0 0.251606 + 1.71368i 0 −0.257544 + 2.22119i 0 −2.77975 2.77975i 0 −2.87339 + 0.862345i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1013.40 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 1380.2.r.c 80
3.b odd 2 1 inner 1380.2.r.c 80
5.c odd 4 1 inner 1380.2.r.c 80
15.e even 4 1 inner 1380.2.r.c 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.r.c 80 1.a even 1 1 trivial
1380.2.r.c 80 3.b odd 2 1 inner
1380.2.r.c 80 5.c odd 4 1 inner
1380.2.r.c 80 15.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$39\!\cdots\!12$$$$T_{7}^{16} -$$$$75\!\cdots\!60$$$$T_{7}^{15} +$$$$76\!\cdots\!16$$$$T_{7}^{14} -$$$$30\!\cdots\!48$$$$T_{7}^{13} +$$$$62\!\cdots\!20$$$$T_{7}^{12} -$$$$13\!\cdots\!20$$$$T_{7}^{11} +$$$$14\!\cdots\!64$$$$T_{7}^{10} -$$$$26\!\cdots\!20$$$$T_{7}^{9} + 293682355920 T_{7}^{8} -$$$$22\!\cdots\!56$$$$T_{7}^{7} +$$$$44\!\cdots\!60$$$$T_{7}^{6} + 480708836352 T_{7}^{5} + 78849339840 T_{7}^{4} - 105410460672 T_{7}^{3} + 42138206208 T_{7}^{2} - 3386105856 T_{7} + 136048896$$">$$T_{7}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1380, [\chi])$$.