# Properties

 Label 1148.2.i.e Level $1148$ Weight $2$ Character orbit 1148.i Analytic conductor $9.167$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$1148 = 2^{2} \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1148.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.16682615204$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$15$$ over $$\Q(\zeta_{3})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q + q^{3} - 3q^{5} + 3q^{7} - 30q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + q^{3} - 3q^{5} + 3q^{7} - 30q^{9} - 9q^{11} + 14q^{13} + 4q^{15} - 3q^{17} - 7q^{19} - 3q^{21} + q^{23} - 32q^{25} + 22q^{27} + 36q^{29} - 30q^{31} + 16q^{33} - 47q^{35} - 23q^{37} - 5q^{39} - 30q^{41} + 24q^{43} + 13q^{45} + 16q^{47} - 31q^{49} - 29q^{51} - 33q^{53} + 74q^{55} + 32q^{57} + 10q^{59} - q^{61} - 75q^{63} - 16q^{65} - 20q^{67} + 42q^{69} + 10q^{71} + 3q^{73} + 51q^{75} - 15q^{77} - 25q^{79} - 43q^{81} + 36q^{83} + 72q^{85} + 53q^{87} + 11q^{89} - 41q^{91} - 65q^{93} + 30q^{95} + 32q^{97} - 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
165.1 0 −1.62107 + 2.80777i 0 −0.748394 1.29626i 0 2.53169 + 0.768468i 0 −3.75571 6.50509i 0
165.2 0 −1.45317 + 2.51697i 0 −0.253051 0.438298i 0 −2.53278 0.764863i 0 −2.72341 4.71709i 0
165.3 0 −1.36871 + 2.37067i 0 0.431527 + 0.747427i 0 −2.57022 + 0.627666i 0 −2.24672 3.89144i 0
165.4 0 −1.23867 + 2.14545i 0 1.74922 + 3.02973i 0 1.19623 2.35988i 0 −1.56862 2.71694i 0
165.5 0 −1.05884 + 1.83397i 0 −1.96828 3.40916i 0 1.39753 2.24653i 0 −0.742303 1.28571i 0
165.6 0 −0.370691 + 0.642056i 0 1.24331 + 2.15348i 0 0.670716 + 2.55932i 0 1.22518 + 2.12207i 0
165.7 0 −0.224795 + 0.389356i 0 0.393901 + 0.682257i 0 1.75718 + 1.97796i 0 1.39893 + 2.42303i 0
165.8 0 0.257541 0.446074i 0 −1.09982 1.90494i 0 −2.48493 + 0.908374i 0 1.36735 + 2.36831i 0
165.9 0 0.637220 1.10370i 0 −2.18714 3.78824i 0 1.89439 + 1.84696i 0 0.687901 + 1.19148i 0
165.10 0 0.702089 1.21605i 0 −1.80784 3.13127i 0 0.606375 2.57533i 0 0.514141 + 0.890519i 0
165.11 0 0.864978 1.49819i 0 0.0691753 + 0.119815i 0 0.329472 2.62516i 0 0.00362599 + 0.00628040i 0
165.12 0 0.966125 1.67338i 0 0.237733 + 0.411765i 0 −1.29492 + 2.30720i 0 −0.366793 0.635305i 0
165.13 0 1.14590 1.98477i 0 1.96151 + 3.39744i 0 −0.412096 + 2.61346i 0 −1.12620 1.95063i 0
165.14 0 1.55023 2.68508i 0 1.65684 + 2.86974i 0 −1.55882 2.13778i 0 −3.30645 5.72694i 0
165.15 0 1.71186 2.96502i 0 −1.17870 2.04157i 0 1.97017 1.76591i 0 −4.36091 7.55332i 0
821.1 0 −1.62107 2.80777i 0 −0.748394 + 1.29626i 0 2.53169 0.768468i 0 −3.75571 + 6.50509i 0
821.2 0 −1.45317 2.51697i 0 −0.253051 + 0.438298i 0 −2.53278 + 0.764863i 0 −2.72341 + 4.71709i 0
821.3 0 −1.36871 2.37067i 0 0.431527 0.747427i 0 −2.57022 0.627666i 0 −2.24672 + 3.89144i 0
821.4 0 −1.23867 2.14545i 0 1.74922 3.02973i 0 1.19623 + 2.35988i 0 −1.56862 + 2.71694i 0
821.5 0 −1.05884 1.83397i 0 −1.96828 + 3.40916i 0 1.39753 + 2.24653i 0 −0.742303 + 1.28571i 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 821.15 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.i.e 30
7.c even 3 1 inner 1148.2.i.e 30
7.c even 3 1 8036.2.a.q 15
7.d odd 6 1 8036.2.a.r 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.i.e 30 1.a even 1 1 trivial
1148.2.i.e 30 7.c even 3 1 inner
8036.2.a.q 15 7.c even 3 1
8036.2.a.r 15 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1148, [\chi])$$:

 $$T_{3}^{30} - \cdots$$ $$16\!\cdots\!66$$$$T_{11}^{14} +$$$$35\!\cdots\!24$$$$T_{11}^{13} +$$$$93\!\cdots\!08$$$$T_{11}^{12} +$$$$14\!\cdots\!70$$$$T_{11}^{11} +$$$$34\!\cdots\!42$$$$T_{11}^{10} +$$$$28\!\cdots\!75$$$$T_{11}^{9} +$$$$59\!\cdots\!84$$$$T_{11}^{8} -$$$$70\!\cdots\!83$$$$T_{11}^{7} +$$$$85\!\cdots\!27$$$$T_{11}^{6} -$$$$15\!\cdots\!25$$$$T_{11}^{5} +$$$$36\!\cdots\!63$$$$T_{11}^{4} -$$$$36\!\cdots\!28$$$$T_{11}^{3} +$$$$11\!\cdots\!07$$$$T_{11}^{2} -$$$$15\!\cdots\!02$$$$T_{11} + 220991189409$$">$$T_{11}^{30} + \cdots$$