# Properties

 Label 99.2.m.b Level $99$ Weight $2$ Character orbit 99.m Analytic conductor $0.791$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 99.m (of order $$15$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.790518980011$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$9$$ over $$\Q(\zeta_{15})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72 q - q^{2} - 12 q^{3} + 11 q^{4} - 8 q^{5} - 7 q^{6} - 2 q^{7} + 6 q^{8} - 22 q^{9}+O(q^{10})$$ 72 * q - q^2 - 12 * q^3 + 11 * q^4 - 8 * q^5 - 7 * q^6 - 2 * q^7 + 6 * q^8 - 22 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$72 q - q^{2} - 12 q^{3} + 11 q^{4} - 8 q^{5} - 7 q^{6} - 2 q^{7} + 6 q^{8} - 22 q^{9} - 8 q^{10} - 2 q^{11} - 16 q^{12} - 11 q^{13} - 10 q^{14} - 20 q^{15} - 9 q^{16} - 20 q^{17} - 21 q^{18} + 8 q^{19} - 45 q^{20} - 16 q^{21} - 16 q^{22} + 20 q^{23} + 36 q^{24} + 11 q^{25} - 12 q^{26} + 27 q^{27} - 54 q^{28} - 23 q^{29} - 6 q^{30} + 3 q^{31} + 18 q^{32} + 28 q^{33} + 8 q^{34} + 18 q^{35} + 75 q^{36} - 42 q^{37} - q^{38} - 11 q^{39} - 25 q^{40} + 10 q^{41} + 61 q^{42} - 8 q^{43} + 38 q^{44} + 40 q^{45} - 18 q^{46} - 34 q^{47} - 50 q^{48} + q^{49} + 42 q^{51} - 27 q^{52} + 4 q^{53} + 34 q^{54} + 18 q^{55} + 114 q^{56} - 21 q^{57} + q^{58} - 16 q^{59} - 63 q^{60} - 3 q^{61} + 184 q^{62} - 10 q^{63} + 26 q^{64} + 84 q^{65} + 93 q^{66} + 10 q^{67} - 23 q^{68} + 42 q^{69} - 46 q^{70} - 48 q^{71} + 2 q^{72} - 40 q^{73} + 68 q^{74} + 28 q^{75} + 16 q^{76} - 26 q^{77} - 64 q^{78} + 19 q^{79} - 56 q^{80} + 14 q^{81} + 94 q^{82} + 7 q^{83} - 15 q^{84} + 25 q^{85} - 77 q^{86} - 120 q^{87} + 18 q^{88} - 56 q^{89} - 64 q^{90} + 20 q^{91} + 50 q^{92} - 53 q^{93} - 63 q^{94} - 77 q^{95} - 97 q^{96} - 33 q^{97} - 328 q^{98} - 182 q^{99}+O(q^{100})$$ 72 * q - q^2 - 12 * q^3 + 11 * q^4 - 8 * q^5 - 7 * q^6 - 2 * q^7 + 6 * q^8 - 22 * q^9 - 8 * q^10 - 2 * q^11 - 16 * q^12 - 11 * q^13 - 10 * q^14 - 20 * q^15 - 9 * q^16 - 20 * q^17 - 21 * q^18 + 8 * q^19 - 45 * q^20 - 16 * q^21 - 16 * q^22 + 20 * q^23 + 36 * q^24 + 11 * q^25 - 12 * q^26 + 27 * q^27 - 54 * q^28 - 23 * q^29 - 6 * q^30 + 3 * q^31 + 18 * q^32 + 28 * q^33 + 8 * q^34 + 18 * q^35 + 75 * q^36 - 42 * q^37 - q^38 - 11 * q^39 - 25 * q^40 + 10 * q^41 + 61 * q^42 - 8 * q^43 + 38 * q^44 + 40 * q^45 - 18 * q^46 - 34 * q^47 - 50 * q^48 + q^49 + 42 * q^51 - 27 * q^52 + 4 * q^53 + 34 * q^54 + 18 * q^55 + 114 * q^56 - 21 * q^57 + q^58 - 16 * q^59 - 63 * q^60 - 3 * q^61 + 184 * q^62 - 10 * q^63 + 26 * q^64 + 84 * q^65 + 93 * q^66 + 10 * q^67 - 23 * q^68 + 42 * q^69 - 46 * q^70 - 48 * q^71 + 2 * q^72 - 40 * q^73 + 68 * q^74 + 28 * q^75 + 16 * q^76 - 26 * q^77 - 64 * q^78 + 19 * q^79 - 56 * q^80 + 14 * q^81 + 94 * q^82 + 7 * q^83 - 15 * q^84 + 25 * q^85 - 77 * q^86 - 120 * q^87 + 18 * q^88 - 56 * q^89 - 64 * q^90 + 20 * q^91 + 50 * q^92 - 53 * q^93 - 63 * q^94 - 77 * q^95 - 97 * q^96 - 33 * q^97 - 328 * q^98 - 182 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.231548 + 2.20304i −0.679127 + 1.59336i −2.84345 0.604395i 0.0650661 + 0.619063i −3.35297 1.86508i 1.72584 1.91674i 0.620850 1.91078i −2.07757 2.16418i −1.37888
4.2 −0.161615 + 1.53767i 1.18421 1.26398i −0.382004 0.0811974i −0.0618842 0.588789i 1.75219 + 2.02520i 0.483079 0.536514i −0.768973 + 2.36665i −0.195290 2.99364i 0.915362
4.3 −0.150377 + 1.43074i −1.39014 1.03320i −0.0681222 0.0144798i 0.364652 + 3.46943i 1.68730 1.83357i −0.226061 + 0.251066i −0.858159 + 2.64114i 0.864983 + 2.87260i −5.01870
4.4 −0.0603978 + 0.574647i 0.641057 + 1.60905i 1.62972 + 0.346409i −0.314048 2.98797i −0.963355 + 0.271198i −0.779082 + 0.865259i −0.654602 + 2.01466i −2.17809 + 2.06299i 1.73599
4.5 0.0512706 0.487807i −1.72269 0.179809i 1.72097 + 0.365803i −0.178553 1.69882i −0.176036 + 0.831122i 2.28209 2.53451i 0.569818 1.75372i 2.93534 + 0.619512i −0.837851
4.6 0.0694404 0.660681i −0.106925 1.72875i 1.52462 + 0.324067i −0.116719 1.11051i −1.14958 0.0494014i −2.61265 + 2.90164i 0.730548 2.24840i −2.97713 + 0.369694i −0.741796
4.7 0.122179 1.16246i −0.613763 + 1.61966i 0.619915 + 0.131767i 0.331034 + 3.14958i 1.80780 + 0.911362i 0.166529 0.184949i 0.951310 2.92783i −2.24659 1.98817i 3.70170
4.8 0.209960 1.99763i 1.72741 + 0.126717i −1.99016 0.423021i −0.000569867 0.00542192i 0.615820 3.42412i −2.56130 + 2.84461i −0.0214877 + 0.0661323i 2.96789 + 0.437784i −0.0109507
4.9 0.255617 2.43204i −0.195654 1.72096i −3.89317 0.827519i 0.313886 + 2.98642i −4.23546 + 0.0359288i 2.60423 2.89229i −1.49636 + 4.60531i −2.92344 + 0.673427i 7.34333
16.1 −2.27811 0.484227i 1.65119 0.523037i 3.12821 + 1.39277i 0.310092 0.0659121i −4.01486 + 0.391985i 0.148273 + 1.41072i −2.68358 1.94973i 2.45286 1.72727i −0.738340
16.2 −2.09212 0.444695i −1.47961 0.900424i 2.35214 + 1.04724i −1.63936 + 0.348457i 2.69511 + 2.54177i 0.135431 + 1.28854i −0.994512 0.722555i 1.37847 + 2.66455i 3.58471
16.3 −0.984579 0.209279i −1.59971 + 0.664019i −0.901493 0.401371i 3.90391 0.829803i 1.71401 0.318993i −0.288110 2.74118i 2.43227 + 1.76714i 2.11816 2.12448i −4.01737
16.4 −0.662860 0.140895i 0.607840 1.62189i −1.40756 0.626686i −0.284440 + 0.0604596i −0.631430 + 0.989445i −0.350744 3.33711i 1.94121 + 1.41037i −2.26106 1.97170i 0.197063
16.5 −0.278917 0.0592857i −1.04736 + 1.37950i −1.75281 0.780402i −2.64954 + 0.563177i 0.373912 0.322674i 0.425779 + 4.05102i 0.904002 + 0.656796i −0.806061 2.88968i 0.772391
16.6 1.42554 + 0.303007i −1.06518 1.36579i 0.113254 + 0.0504238i 2.94034 0.624989i −1.10462 2.26974i 0.172491 + 1.64114i −2.21193 1.60706i −0.730763 + 2.90964i 4.38094
16.7 1.45109 + 0.308439i 0.266086 + 1.71149i 0.183441 + 0.0816732i 0.0707467 0.0150377i −0.141775 + 2.56560i −0.203853 1.93954i −2.15937 1.56887i −2.85840 + 0.910807i 0.107298
16.8 1.85327 + 0.393925i 0.886115 1.48822i 1.45235 + 0.646630i −3.43625 + 0.730397i 2.22846 2.40902i 0.358742 + 3.41320i −0.628765 0.456825i −1.42960 2.63747i −6.65603
16.9 2.54483 + 0.540921i −1.71102 + 0.269100i 4.35649 + 1.93963i −2.00724 + 0.426651i −4.49982 0.240711i −0.333407 3.17215i 5.82773 + 4.23409i 2.85517 0.920871i −5.33887
25.1 −0.231548 2.20304i −0.679127 1.59336i −2.84345 + 0.604395i 0.0650661 0.619063i −3.35297 + 1.86508i 1.72584 + 1.91674i 0.620850 + 1.91078i −2.07757 + 2.16418i −1.37888
25.2 −0.161615 1.53767i 1.18421 + 1.26398i −0.382004 + 0.0811974i −0.0618842 + 0.588789i 1.75219 2.02520i 0.483079 + 0.536514i −0.768973 2.36665i −0.195290 + 2.99364i 0.915362
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.m.b 72
3.b odd 2 1 297.2.n.b 72
9.c even 3 1 inner 99.2.m.b 72
9.c even 3 1 891.2.f.f 36
9.d odd 6 1 297.2.n.b 72
9.d odd 6 1 891.2.f.e 36
11.c even 5 1 inner 99.2.m.b 72
11.c even 5 1 1089.2.e.p 36
11.d odd 10 1 1089.2.e.o 36
33.h odd 10 1 297.2.n.b 72
99.m even 15 1 inner 99.2.m.b 72
99.m even 15 1 891.2.f.f 36
99.m even 15 1 1089.2.e.p 36
99.m even 15 1 9801.2.a.cm 18
99.n odd 30 1 297.2.n.b 72
99.n odd 30 1 891.2.f.e 36
99.n odd 30 1 9801.2.a.cp 18
99.o odd 30 1 1089.2.e.o 36
99.o odd 30 1 9801.2.a.co 18
99.p even 30 1 9801.2.a.cn 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.m.b 72 1.a even 1 1 trivial
99.2.m.b 72 9.c even 3 1 inner
99.2.m.b 72 11.c even 5 1 inner
99.2.m.b 72 99.m even 15 1 inner
297.2.n.b 72 3.b odd 2 1
297.2.n.b 72 9.d odd 6 1
297.2.n.b 72 33.h odd 10 1
297.2.n.b 72 99.n odd 30 1
891.2.f.e 36 9.d odd 6 1
891.2.f.e 36 99.n odd 30 1
891.2.f.f 36 9.c even 3 1
891.2.f.f 36 99.m even 15 1
1089.2.e.o 36 11.d odd 10 1
1089.2.e.o 36 99.o odd 30 1
1089.2.e.p 36 11.c even 5 1
1089.2.e.p 36 99.m even 15 1
9801.2.a.cm 18 99.m even 15 1
9801.2.a.cn 18 99.p even 30 1
9801.2.a.co 18 99.o odd 30 1
9801.2.a.cp 18 99.n odd 30 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{72} + T_{2}^{71} - 14 T_{2}^{70} - 19 T_{2}^{69} + 76 T_{2}^{68} + 112 T_{2}^{67} - 78 T_{2}^{66} + 80 T_{2}^{65} - 1644 T_{2}^{64} - 5509 T_{2}^{63} + 18853 T_{2}^{62} + 47090 T_{2}^{61} - 97690 T_{2}^{60} - 201363 T_{2}^{59} + \cdots + 9150625$$ acting on $$S_{2}^{\mathrm{new}}(99, [\chi])$$.