# Properties

 Label 115.2.g.c Level $115$ Weight $2$ Character orbit 115.g Analytic conductor $0.918$ Analytic rank $0$ Dimension $50$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 115.g (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.918279623245$$ Analytic rank: $$0$$ Dimension: $$50$$ Relative dimension: $$5$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$50q - 5q^{2} - 2q^{3} - 11q^{4} - 5q^{5} - 11q^{6} - 5q^{7} - 2q^{8} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$50q - 5q^{2} - 2q^{3} - 11q^{4} - 5q^{5} - 11q^{6} - 5q^{7} - 2q^{8} + 3q^{9} - 5q^{10} - 16q^{11} - 9q^{12} - 14q^{13} - 12q^{14} - 2q^{15} + 27q^{16} + 38q^{17} - 42q^{18} - 5q^{19} - 11q^{20} - 9q^{21} + 6q^{22} - 8q^{23} + 102q^{24} - 5q^{25} - 19q^{26} + 7q^{27} - 34q^{28} - 38q^{29} - 11q^{30} + 2q^{31} + 49q^{32} - 2q^{33} - 31q^{34} + 6q^{35} - 59q^{36} - 35q^{37} + 30q^{38} + 32q^{39} + 42q^{40} - 11q^{41} - 102q^{42} + 6q^{43} - 55q^{44} + 58q^{45} + 153q^{46} - 10q^{47} + 84q^{48} + 6q^{50} - 20q^{51} - 97q^{52} - 29q^{53} + 19q^{54} + 17q^{55} + 77q^{56} - 49q^{57} - 12q^{58} - 50q^{59} + 2q^{60} + 4q^{61} + 126q^{62} + 74q^{63} - 44q^{64} - 14q^{65} - 144q^{66} - 43q^{67} + 54q^{68} - 50q^{69} - 12q^{70} - 25q^{71} - 14q^{72} - 20q^{73} - 47q^{74} - 2q^{75} - 26q^{76} + 150q^{77} + 174q^{78} + 72q^{79} - 28q^{80} - 71q^{81} - 11q^{82} + 36q^{83} + 100q^{84} - 6q^{85} - 20q^{86} + 85q^{87} - 45q^{88} - 24q^{89} - 42q^{90} + 38q^{91} + 74q^{92} + 100q^{93} + 150q^{94} - 5q^{95} - 169q^{96} - 14q^{97} - 44q^{98} + 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −1.02998 + 2.25535i 2.16804 + 0.636595i −2.71600 3.13443i 0.841254 0.540641i −3.66879 + 4.23400i −0.617798 + 4.29688i 5.10871 1.50005i 1.77140 + 1.13841i 0.352856 + 2.45417i
6.2 −0.923360 + 2.02188i −2.90696 0.853560i −1.92567 2.22234i 0.841254 0.540641i 4.40996 5.08937i 0.283764 1.97362i 2.00598 0.589008i 5.19808 + 3.34060i 0.316329 + 2.20012i
6.3 −0.366526 + 0.802579i 0.817387 + 0.240007i 0.799929 + 0.923167i 0.841254 0.540641i −0.492218 + 0.568049i 0.192580 1.33943i −2.72725 + 0.800794i −1.91324 1.22957i 0.125566 + 0.873331i
6.4 0.418214 0.915761i 0.166345 + 0.0488434i 0.646006 + 0.745531i 0.841254 0.540641i 0.114297 0.131906i −0.333337 + 2.31841i 2.88481 0.847058i −2.49848 1.60567i −0.143274 0.996491i
6.5 0.975815 2.13674i −2.16380 0.635350i −2.30371 2.65862i 0.841254 0.540641i −3.46905 + 4.00349i −0.0703937 + 0.489599i −3.42104 + 1.00451i 1.75461 + 1.12762i −0.334299 2.32510i
16.1 −1.81966 + 2.10000i 1.42355 0.914863i −0.814204 5.66292i 0.415415 + 0.909632i −0.669173 + 4.65420i 3.21768 0.944795i 8.69851 + 5.59019i −0.0567125 + 0.124183i −2.66614 0.782849i
16.2 −1.08398 + 1.25098i −1.71985 + 1.10528i −0.105306 0.732422i 0.415415 + 0.909632i 0.481599 3.34959i −1.10450 + 0.324311i −1.75463 1.12763i 0.489993 1.07293i −1.58823 0.466346i
16.3 −0.553310 + 0.638554i 2.80987 1.80579i 0.183031 + 1.27301i 0.415415 + 0.909632i −0.401632 + 2.79341i −3.04231 + 0.893304i −2.33575 1.50110i 3.38822 7.41917i −0.810702 0.238044i
16.4 0.588656 0.679345i −1.58059 + 1.01578i 0.169636 + 1.17984i 0.415415 + 0.909632i −0.240356 + 1.67171i 2.75076 0.807695i 2.41379 + 1.55125i 0.220205 0.482181i 0.862490 + 0.253250i
16.5 1.29801 1.49799i 0.749526 0.481691i −0.274499 1.90918i 0.415415 + 0.909632i 0.251328 1.74802i −3.44343 + 1.01108i 0.118700 + 0.0762838i −0.916482 + 2.00682i 1.90183 + 0.558429i
26.1 −0.362593 2.52189i 0.804293 1.76116i −4.30948 + 1.26538i −0.654861 + 0.755750i −4.73308 1.38976i 0.564436 0.362741i 2.63693 + 5.77406i −0.490203 0.565724i 2.14337 + 1.37746i
26.2 −0.166938 1.16108i −0.526900 + 1.15375i 0.598752 0.175809i −0.654861 + 0.755750i 1.42755 + 0.419168i 4.23310 2.72045i −1.27866 2.79988i 0.911065 + 1.05142i 0.986805 + 0.634181i
26.3 −0.0393306 0.273550i 0.955665 2.09261i 1.84570 0.541947i −0.654861 + 0.755750i −0.610022 0.179119i −2.63800 + 1.69534i −0.450453 0.986355i −1.50116 1.73243i 0.232492 + 0.149413i
26.4 0.185934 + 1.29320i −0.426013 + 0.932839i 0.281191 0.0825651i −0.654861 + 0.755750i −1.28556 0.377474i −1.03230 + 0.663422i 1.24453 + 2.72515i 1.27588 + 1.47244i −1.09910 0.706346i
26.5 0.395474 + 2.75058i 0.0237855 0.0520830i −5.49030 + 1.61210i −0.654861 + 0.755750i 0.152665 + 0.0448264i 1.15709 0.743614i −4.29671 9.40848i 1.96244 + 2.26477i −2.33773 1.50237i
31.1 −0.362593 + 2.52189i 0.804293 + 1.76116i −4.30948 1.26538i −0.654861 0.755750i −4.73308 + 1.38976i 0.564436 + 0.362741i 2.63693 5.77406i −0.490203 + 0.565724i 2.14337 1.37746i
31.2 −0.166938 + 1.16108i −0.526900 1.15375i 0.598752 + 0.175809i −0.654861 0.755750i 1.42755 0.419168i 4.23310 + 2.72045i −1.27866 + 2.79988i 0.911065 1.05142i 0.986805 0.634181i
31.3 −0.0393306 + 0.273550i 0.955665 + 2.09261i 1.84570 + 0.541947i −0.654861 0.755750i −0.610022 + 0.179119i −2.63800 1.69534i −0.450453 + 0.986355i −1.50116 + 1.73243i 0.232492 0.149413i
31.4 0.185934 1.29320i −0.426013 0.932839i 0.281191 + 0.0825651i −0.654861 0.755750i −1.28556 + 0.377474i −1.03230 0.663422i 1.24453 2.72515i 1.27588 1.47244i −1.09910 + 0.706346i
31.5 0.395474 2.75058i 0.0237855 + 0.0520830i −5.49030 1.61210i −0.654861 0.755750i 0.152665 0.0448264i 1.15709 + 0.743614i −4.29671 + 9.40848i 1.96244 2.26477i −2.33773 + 1.50237i
See all 50 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.g.c 50
5.b even 2 1 575.2.k.d 50
5.c odd 4 2 575.2.p.d 100
23.c even 11 1 inner 115.2.g.c 50
23.c even 11 1 2645.2.a.y 25
23.d odd 22 1 2645.2.a.x 25
115.j even 22 1 575.2.k.d 50
115.k odd 44 2 575.2.p.d 100

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.c 50 1.a even 1 1 trivial
115.2.g.c 50 23.c even 11 1 inner
575.2.k.d 50 5.b even 2 1
575.2.k.d 50 115.j even 22 1
575.2.p.d 100 5.c odd 4 2
575.2.p.d 100 115.k odd 44 2
2645.2.a.x 25 23.d odd 22 1
2645.2.a.y 25 23.c even 11 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{50} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(115, [\chi])$$.