# Properties

 Label 115.2.l.a Level $115$ Weight $2$ Character orbit 115.l Analytic conductor $0.918$ Analytic rank $0$ Dimension $200$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$115 = 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 115.l (of order $$44$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.918279623245$$ Analytic rank: $$0$$ Dimension: $$200$$ Relative dimension: $$10$$ over $$\Q(\zeta_{44})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$200q - 18q^{2} - 14q^{3} - 22q^{5} - 36q^{6} - 22q^{7} - 26q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$200q - 18q^{2} - 14q^{3} - 22q^{5} - 36q^{6} - 22q^{7} - 26q^{8} - 22q^{10} - 44q^{11} - 6q^{12} - 26q^{13} - 22q^{15} - 52q^{16} - 22q^{17} + 58q^{18} - 22q^{20} - 44q^{21} + 22q^{23} - 10q^{25} - 28q^{26} - 26q^{27} + 66q^{28} - 22q^{30} - 40q^{31} - 46q^{32} - 14q^{35} - 12q^{36} + 66q^{37} - 22q^{38} - 22q^{40} - 8q^{41} + 198q^{42} - 22q^{43} - 76q^{46} + 52q^{47} + 18q^{48} - 82q^{50} - 44q^{51} + 158q^{52} - 22q^{53} - 10q^{55} + 88q^{56} + 66q^{57} - 58q^{58} - 22q^{60} + 44q^{61} + 38q^{62} - 22q^{63} - 22q^{65} + 132q^{66} - 22q^{67} + 32q^{70} + 132q^{71} - 28q^{72} + 34q^{73} + 38q^{75} + 132q^{76} - 10q^{77} + 22q^{78} + 176q^{80} - 48q^{81} - 50q^{82} - 22q^{83} + 202q^{85} - 46q^{87} - 110q^{88} + 396q^{90} + 50q^{92} - 36q^{93} + 68q^{95} + 148q^{96} - 88q^{97} - 50q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −2.48666 + 0.177849i 0.639470 2.93959i 4.17220 0.599872i 0.983237 + 2.00829i −1.06734 + 7.42350i 1.40495 2.57297i −5.39608 + 1.17384i −5.50340 2.51332i −2.80215 4.81907i
7.2 −2.43652 + 0.174263i −0.128656 + 0.591424i 3.92662 0.564562i −1.44483 1.70660i 0.210410 1.46344i −0.0458364 + 0.0839431i −4.69506 + 1.02135i 2.39567 + 1.09406i 3.81775 + 3.90638i
7.3 −1.30561 + 0.0933793i 0.136266 0.626406i −0.283736 + 0.0407951i −1.37401 + 1.76411i −0.119418 + 0.830568i −2.05467 + 3.76286i 2.92471 0.636232i 2.35508 + 1.07553i 1.62920 2.43155i
7.4 −1.05653 + 0.0755644i 0.0308132 0.141646i −0.869101 + 0.124958i 2.18100 0.493172i −0.0218516 + 0.151981i 0.717546 1.31409i 2.97883 0.648004i 2.70978 + 1.23752i −2.26703 + 0.685856i
7.5 −0.643856 + 0.0460495i −0.683154 + 3.14041i −1.56721 + 0.225331i −0.267157 2.22005i 0.295239 2.05343i −1.03991 + 1.90445i 2.26018 0.491672i −6.66656 3.04452i 0.274243 + 1.41709i
7.6 −0.107969 + 0.00772212i 0.390334 1.79434i −1.96805 + 0.282962i −2.08676 0.803398i −0.0282880 + 0.196748i 2.36052 4.32296i 0.421846 0.0917670i −0.338390 0.154538i 0.231510 + 0.0706281i
7.7 0.632652 0.0452482i −0.364069 + 1.67360i −1.58144 + 0.227377i 1.17051 + 1.90523i −0.154602 + 1.07528i 0.0937908 0.171765i −2.22976 + 0.485055i 0.0605071 + 0.0276327i 0.826736 + 1.15238i
7.8 1.60196 0.114574i 0.0509080 0.234020i 0.573502 0.0824571i 0.888172 2.05211i 0.0547398 0.380723i −0.562120 + 1.02945i −2.22942 + 0.484980i 2.67672 + 1.22242i 1.18770 3.38916i
7.9 1.82025 0.130187i 0.475590 2.18625i 1.31673 0.189318i −1.13658 + 1.92566i 0.581073 4.04145i −0.696418 + 1.27539i −1.19425 + 0.259794i −1.82462 0.833276i −1.81818 + 3.65316i
7.10 2.13477 0.152682i −0.476413 + 2.19004i 2.55431 0.367254i −2.17992 0.497948i −0.682656 + 4.74797i 1.43707 2.63180i 1.21416 0.264125i −1.84040 0.840481i −4.72966 0.730172i
17.1 −2.39977 + 0.895067i −0.939581 + 0.513050i 3.44624 2.98618i 1.67567 + 1.48058i 1.79556 2.07219i 0.380432 + 0.508198i −3.14238 + 5.75484i −1.00233 + 1.55966i −5.34644 2.05322i
17.2 −1.75302 + 0.653844i 0.975483 0.532654i 1.13408 0.982690i 0.872739 2.05872i −1.36177 + 1.57157i −1.17575 1.57062i 0.447790 0.820066i −0.954075 + 1.48457i −0.183850 + 4.17962i
17.3 −1.54777 + 0.577290i −2.80985 + 1.53430i 0.550844 0.477309i −1.14879 1.91841i 3.46329 3.99685i 1.53995 + 2.05714i 1.00633 1.84296i 3.91929 6.09853i 2.88555 + 2.30607i
17.4 −1.07057 + 0.399303i −0.256216 + 0.139905i −0.524814 + 0.454754i −1.94143 + 1.10943i 0.218434 0.252086i −1.96263 2.62176i 1.47546 2.70211i −1.57585 + 2.45207i 1.63545 1.96295i
17.5 −0.117766 + 0.0439246i −0.632714 + 0.345488i −1.49956 + 1.29938i 2.23369 + 0.103094i 0.0593371 0.0684786i 1.63018 + 2.17766i 0.239998 0.439523i −1.34096 + 2.08657i −0.267582 + 0.0859729i
17.6 0.0418710 0.0156171i 2.62848 1.43526i −1.50999 + 1.30841i −0.593814 2.15578i 0.0876426 0.101145i 1.31081 + 1.75104i −0.0856252 + 0.156811i 3.22702 5.02134i −0.0585306 0.0809910i
17.7 0.864755 0.322537i 1.68339 0.919199i −0.867728 + 0.751890i 1.28668 + 1.82879i 1.15924 1.33784i −2.26318 3.02326i −1.39250 + 2.55018i 0.366945 0.570978i 1.70251 + 1.16645i
17.8 1.03990 0.387863i −2.64829 + 1.44608i −0.580545 + 0.503045i −0.406014 + 2.19890i −2.19308 + 2.53095i −0.573224 0.765738i −1.47241 + 2.69652i 3.30039 5.13550i 0.430656 + 2.44411i
17.9 1.79955 0.671199i 0.756071 0.412846i 1.27639 1.10600i −2.21107 0.333414i 1.08349 1.25041i −0.0833334 0.111320i −0.286357 + 0.524424i −1.22072 + 1.89948i −4.20273 + 0.884071i
17.10 2.04194 0.761604i −1.49195 + 0.814667i 2.07797 1.80057i 1.48286 1.67366i −2.42602 + 2.79977i 0.435377 + 0.581596i 0.782870 1.43372i −0.0596878 + 0.0928760i 1.75324 4.54686i
See next 80 embeddings (of 200 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.d odd 22 1 inner
115.l even 44 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.l.a 200
5.b even 2 1 575.2.r.b 200
5.c odd 4 1 inner 115.2.l.a 200
5.c odd 4 1 575.2.r.b 200
23.d odd 22 1 inner 115.2.l.a 200
115.i odd 22 1 575.2.r.b 200
115.l even 44 1 inner 115.2.l.a 200
115.l even 44 1 575.2.r.b 200

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.l.a 200 1.a even 1 1 trivial
115.2.l.a 200 5.c odd 4 1 inner
115.2.l.a 200 23.d odd 22 1 inner
115.2.l.a 200 115.l even 44 1 inner
575.2.r.b 200 5.b even 2 1
575.2.r.b 200 5.c odd 4 1
575.2.r.b 200 115.i odd 22 1
575.2.r.b 200 115.l even 44 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(115, [\chi])$$.