# Properties

 Label 731.2.f.d Level 731 Weight 2 Character orbit 731.f Analytic conductor 5.837 Analytic rank 0 Dimension 68 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$731 = 17 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 731.f (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$5.83706438776$$ Analytic rank: $$0$$ Dimension: $$68$$ Relative dimension: $$34$$ over $$\Q(i)$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$68q - 76q^{4} - 4q^{5} + 4q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$68q - 76q^{4} - 4q^{5} + 4q^{6} + 2q^{10} - 6q^{11} - 10q^{12} - 24q^{13} - 22q^{14} + 84q^{16} - 2q^{17} + 28q^{18} + 10q^{20} - 36q^{21} + 8q^{22} + 14q^{23} - 62q^{24} - 12q^{27} - 58q^{28} + 2q^{29} + 160q^{30} - 26q^{31} + 44q^{33} + 16q^{34} + 56q^{35} - 6q^{37} - 56q^{38} - 24q^{39} + 70q^{40} + 6q^{41} + 14q^{44} + 10q^{45} + 2q^{46} - 68q^{47} - 58q^{48} + 40q^{50} + 16q^{51} + 4q^{52} + 26q^{54} - 16q^{55} + 50q^{56} + 18q^{57} - 94q^{58} + 22q^{61} - 48q^{62} + 16q^{63} + 60q^{64} - 22q^{65} + 24q^{67} + 20q^{68} + 8q^{69} - 14q^{71} - 84q^{72} + 34q^{73} + 26q^{74} - 102q^{75} + 40q^{78} + 4q^{79} - 30q^{80} - 92q^{81} - 76q^{82} + 108q^{84} + 8q^{85} + 8q^{86} + 16q^{88} - 72q^{89} + 132q^{90} + 12q^{91} - 174q^{92} + 50q^{95} + 10q^{96} - 16q^{97} - 28q^{98} - 86q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
259.1 2.72425i −1.14824 + 1.14824i −5.42152 2.76466 2.76466i 3.12810 + 3.12810i 1.65149 + 1.65149i 9.32106i 0.363073i −7.53160 7.53160i
259.2 2.38680i −1.32556 + 1.32556i −3.69681 −0.620851 + 0.620851i 3.16384 + 3.16384i −1.89151 1.89151i 4.04996i 0.514207i 1.48185 + 1.48185i
259.3 2.33650i −0.692341 + 0.692341i −3.45924 0.878052 0.878052i 1.61766 + 1.61766i −2.34961 2.34961i 3.40953i 2.04133i −2.05157 2.05157i
259.4 2.33024i 2.15102 2.15102i −3.43001 −2.49284 + 2.49284i −5.01238 5.01238i 0.968194 + 0.968194i 3.33226i 6.25374i 5.80892 + 5.80892i
259.5 2.24230i 0.751257 0.751257i −3.02793 −1.86200 + 1.86200i −1.68455 1.68455i 2.49628 + 2.49628i 2.30493i 1.87123i 4.17518 + 4.17518i
259.6 2.17355i −2.09748 + 2.09748i −2.72432 0.426175 0.426175i 4.55898 + 4.55898i 2.68508 + 2.68508i 1.57435i 5.79886i −0.926312 0.926312i
259.7 2.06339i 0.856325 0.856325i −2.25758 −2.54838 + 2.54838i −1.76693 1.76693i −3.18040 3.18040i 0.531478i 1.53342i 5.25831 + 5.25831i
259.8 1.84807i 1.37079 1.37079i −1.41535 2.26460 2.26460i −2.53331 2.53331i −1.84433 1.84433i 1.08047i 0.758116i −4.18514 4.18514i
259.9 1.48494i 1.38481 1.38481i −0.205050 −0.478281 + 0.478281i −2.05636 2.05636i −1.44950 1.44950i 2.66539i 0.835396i 0.710219 + 0.710219i
259.10 1.19080i −1.82510 + 1.82510i 0.581989 0.840782 0.840782i 2.17334 + 2.17334i 1.07906 + 1.07906i 3.07464i 3.66201i −1.00121 1.00121i
259.11 1.12754i 1.76939 1.76939i 0.728657 −0.189570 + 0.189570i −1.99505 1.99505i 3.45806 + 3.45806i 3.07667i 3.26148i 0.213748 + 0.213748i
259.12 0.810360i −0.0502309 + 0.0502309i 1.34332 0.925022 0.925022i 0.0407051 + 0.0407051i −0.994972 0.994972i 2.70929i 2.99495i −0.749601 0.749601i
259.13 0.660947i −1.41031 + 1.41031i 1.56315 −0.660156 + 0.660156i 0.932141 + 0.932141i −2.22740 2.22740i 2.35505i 0.977951i 0.436328 + 0.436328i
259.14 0.436020i 0.705146 0.705146i 1.80989 2.58895 2.58895i −0.307458 0.307458i 0.364609 + 0.364609i 1.66119i 2.00554i −1.12884 1.12884i
259.15 0.236577i 2.31349 2.31349i 1.94403 −2.11402 + 2.11402i −0.547317 0.547317i −3.33868 3.33868i 0.933066i 7.70443i 0.500128 + 0.500128i
259.16 0.166310i −1.09373 + 1.09373i 1.97234 −0.776668 + 0.776668i 0.181899 + 0.181899i −0.631548 0.631548i 0.660639i 0.607490i 0.129167 + 0.129167i
259.17 0.0920143i −0.256358 + 0.256358i 1.99153 −2.26856 + 2.26856i 0.0235886 + 0.0235886i 3.14718 + 3.14718i 0.367278i 2.86856i 0.208740 + 0.208740i
259.18 0.323065i 1.32352 1.32352i 1.89563 −1.60497 + 1.60497i 0.427583 + 0.427583i 1.08526 + 1.08526i 1.25854i 0.503415i −0.518509 0.518509i
259.19 0.374564i −0.409333 + 0.409333i 1.85970 2.01640 2.01640i −0.153322 0.153322i 1.28867 + 1.28867i 1.44571i 2.66489i 0.755271 + 0.755271i
259.20 0.488618i −2.30085 + 2.30085i 1.76125 −0.0210259 + 0.0210259i −1.12424 1.12424i −2.78195 2.78195i 1.83782i 7.58778i −0.0102736 0.0102736i
See all 68 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 302.34 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(731, [\chi])$$:

 $$T_{2}^{68} + \cdots$$ $$T_{3}^{68} + \cdots$$