# Properties

 Label 441.2.u.d Level $441$ Weight $2$ Character orbit 441.u Analytic conductor $3.521$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.u (of order $$7$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$6$$ over $$\Q(\zeta_{7})$$ Twist minimal: no (minimal twist has level 147) Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q + q^{2} - 9q^{4} + 4q^{5} - 6q^{7} + 15q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + q^{2} - 9q^{4} + 4q^{5} - 6q^{7} + 15q^{8} + 10q^{10} + 7q^{11} - 12q^{13} + q^{14} - 3q^{16} + 3q^{17} + 6q^{19} - 25q^{20} - 21q^{22} + 20q^{23} - 2q^{25} - 6q^{26} - q^{28} + 22q^{29} + 16q^{31} - 26q^{32} + 6q^{34} + 9q^{35} + 32q^{37} - 17q^{38} - 21q^{40} + 5q^{41} - 34q^{43} - 2q^{44} - 32q^{46} + 7q^{47} + 20q^{49} - 236q^{50} + 20q^{52} + 32q^{53} - 17q^{55} + 39q^{56} - 53q^{58} + q^{59} + 14q^{61} + 60q^{62} - 21q^{64} + 39q^{65} - 22q^{67} + 110q^{68} - 40q^{70} - 36q^{71} - 11q^{73} + 46q^{74} - 101q^{76} + 17q^{77} - 14q^{79} + 112q^{80} + 2q^{82} - 12q^{83} - 44q^{85} - 184q^{86} + 204q^{88} - 12q^{89} - 16q^{91} + 105q^{92} - 5q^{94} - 18q^{95} + 172q^{97} - q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1 −0.603790 + 2.64538i 0 −4.83152 2.32674i 0.940078 1.17882i 0 2.57696 0.599410i 5.68875 7.13347i 0 2.55081 + 3.19862i
64.2 −0.368659 + 1.61520i 0 −0.671021 0.323147i −0.768029 + 0.963077i 0 −1.85883 + 1.88275i −1.29659 + 1.62588i 0 −1.27242 1.59557i
64.3 0.0515349 0.225789i 0 1.75361 + 0.844495i 2.14953 2.69543i 0 1.10205 + 2.40530i 0.569846 0.714564i 0 −0.497824 0.624251i
64.4 0.132418 0.580160i 0 1.48289 + 0.714121i −0.595901 + 0.747236i 0 −0.416022 2.61284i 1.35272 1.69625i 0 0.354609 + 0.444665i
64.5 0.441918 1.93617i 0 −1.75153 0.843490i −2.19876 + 2.75716i 0 1.51600 + 2.16835i 0.0692834 0.0868787i 0 4.36666 + 5.47561i
64.6 0.569099 2.49338i 0 −4.09115 1.97020i 1.87405 2.34998i 0 −2.56325 + 0.655556i −4.05157 + 5.08051i 0 −4.79289 6.01009i
127.1 −2.24272 1.08004i 0 2.61635 + 3.28080i 0.222387 + 0.974342i 0 −2.64454 + 0.0800852i −1.21655 5.33004i 0 0.553574 2.42537i
127.2 −1.54657 0.744788i 0 0.590184 + 0.740067i −0.580458 2.54315i 0 2.64218 + 0.137475i 0.402375 + 1.76292i 0 −0.996391 + 4.36547i
127.3 −0.402966 0.194058i 0 −1.12226 1.40727i 0.162101 + 0.710213i 0 −1.42812 2.22721i 0.378189 + 1.65695i 0 0.0725012 0.317649i
127.4 1.11145 + 0.535248i 0 −0.298141 0.373858i 0.661767 + 2.89939i 0 −0.553581 + 2.58719i −0.680276 2.98049i 0 −0.816369 + 3.57675i
127.5 1.60360 + 0.772253i 0 0.728177 + 0.913105i −0.987738 4.32756i 0 −2.63181 + 0.271221i −0.329556 1.44388i 0 1.75804 7.70246i
127.6 2.37817 + 1.14527i 0 3.09710 + 3.88364i 0.398449 + 1.74572i 0 0.566955 2.58429i 1.74291 + 7.63618i 0 −1.05174 + 4.60796i
190.1 −1.64011 + 2.05663i 0 −1.09473 4.79635i −3.33882 1.60789i 0 −2.39773 + 1.11843i 6.91975 + 3.33238i 0 8.78287 4.22961i
190.2 −1.34250 + 1.68344i 0 −0.586625 2.57017i 2.99194 + 1.44084i 0 2.53827 0.746462i 1.23434 + 0.594424i 0 −6.44225 + 3.10242i
190.3 −0.494936 + 0.620630i 0 0.304822 + 1.33551i −1.66830 0.803409i 0 0.814382 2.51730i −2.41013 1.16066i 0 1.32432 0.637759i
190.4 0.385632 0.483568i 0 0.359916 + 1.57690i 3.63892 + 1.75241i 0 −2.64500 0.0631583i 2.01584 + 0.970778i 0 2.25069 1.08388i
190.5 0.894323 1.12145i 0 −0.0127847 0.0560133i −1.03743 0.499602i 0 2.02504 + 1.70270i 2.51042 + 1.20895i 0 −1.48808 + 0.716620i
190.6 1.57410 1.97386i 0 −0.973283 4.26423i 0.136210 + 0.0655954i 0 0.357057 2.62155i −5.39974 2.60038i 0 0.343884 0.165606i
253.1 −1.64011 2.05663i 0 −1.09473 + 4.79635i −3.33882 + 1.60789i 0 −2.39773 1.11843i 6.91975 3.33238i 0 8.78287 + 4.22961i
253.2 −1.34250 1.68344i 0 −0.586625 + 2.57017i 2.99194 1.44084i 0 2.53827 + 0.746462i 1.23434 0.594424i 0 −6.44225 3.10242i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.u.d 36
3.b odd 2 1 147.2.i.b 36
49.e even 7 1 inner 441.2.u.d 36
147.k even 14 1 7203.2.a.g 18
147.l odd 14 1 147.2.i.b 36
147.l odd 14 1 7203.2.a.h 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.i.b 36 3.b odd 2 1
147.2.i.b 36 147.l odd 14 1
441.2.u.d 36 1.a even 1 1 trivial
441.2.u.d 36 49.e even 7 1 inner
7203.2.a.g 18 147.k even 14 1
7203.2.a.h 18 147.l odd 14 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$.