# Properties

 Label 124.2.p.a Level $124$ Weight $2$ Character orbit 124.p Analytic conductor $0.990$ Analytic rank $0$ Dimension $112$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 124.p (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.990144985064$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$14$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 33 q^{6} - 9 q^{8} - 8 q^{9}+O(q^{10})$$ 112 * q - 6 * q^2 - 10 * q^4 - 8 * q^5 - 33 * q^6 - 9 * q^8 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$112 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 33 q^{6} - 9 q^{8} - 8 q^{9} + 4 q^{10} - 31 q^{12} - 2 q^{13} - 16 q^{14} - 18 q^{16} - 14 q^{17} - q^{18} + 29 q^{20} + 6 q^{21} - 23 q^{22} - 16 q^{24} - 24 q^{25} + 9 q^{26} - 16 q^{28} - 20 q^{29} - 26 q^{32} - 32 q^{33} - 30 q^{34} - 5 q^{36} - 12 q^{37} - 6 q^{38} + 25 q^{40} - 18 q^{41} + 37 q^{42} + 59 q^{44} - 54 q^{45} + 30 q^{46} - 28 q^{48} - 68 q^{49} + 47 q^{50} - 5 q^{52} - 38 q^{53} + 110 q^{54} - 14 q^{56} - 60 q^{57} + 15 q^{58} + 155 q^{60} + 19 q^{62} + 95 q^{64} + 36 q^{65} + 74 q^{66} + 174 q^{68} + 64 q^{70} + 21 q^{72} - 50 q^{73} + 55 q^{74} + 46 q^{76} - 20 q^{77} + 41 q^{78} - 26 q^{80} - 14 q^{81} - 102 q^{82} - 8 q^{84} + 30 q^{85} - 30 q^{86} - 87 q^{88} - 40 q^{89} + 21 q^{90} - 102 q^{93} + 72 q^{94} + 30 q^{96} + 20 q^{98}+O(q^{100})$$ 112 * q - 6 * q^2 - 10 * q^4 - 8 * q^5 - 33 * q^6 - 9 * q^8 - 8 * q^9 + 4 * q^10 - 31 * q^12 - 2 * q^13 - 16 * q^14 - 18 * q^16 - 14 * q^17 - q^18 + 29 * q^20 + 6 * q^21 - 23 * q^22 - 16 * q^24 - 24 * q^25 + 9 * q^26 - 16 * q^28 - 20 * q^29 - 26 * q^32 - 32 * q^33 - 30 * q^34 - 5 * q^36 - 12 * q^37 - 6 * q^38 + 25 * q^40 - 18 * q^41 + 37 * q^42 + 59 * q^44 - 54 * q^45 + 30 * q^46 - 28 * q^48 - 68 * q^49 + 47 * q^50 - 5 * q^52 - 38 * q^53 + 110 * q^54 - 14 * q^56 - 60 * q^57 + 15 * q^58 + 155 * q^60 + 19 * q^62 + 95 * q^64 + 36 * q^65 + 74 * q^66 + 174 * q^68 + 64 * q^70 + 21 * q^72 - 50 * q^73 + 55 * q^74 + 46 * q^76 - 20 * q^77 + 41 * q^78 - 26 * q^80 - 14 * q^81 - 102 * q^82 - 8 * q^84 + 30 * q^85 - 30 * q^86 - 87 * q^88 - 40 * q^89 + 21 * q^90 - 102 * q^93 + 72 * q^94 + 30 * q^96 + 20 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −1.40876 + 0.124087i −0.130272 + 1.23946i 1.96920 0.349618i 1.03203 1.78753i 0.0297212 1.76226i 0.311298 + 1.46454i −2.73075 + 0.736882i 1.41516 + 0.300802i −1.23207 + 2.64625i
3.2 −1.37846 + 0.315988i 0.298401 2.83910i 1.80030 0.871153i −0.269294 + 0.466430i 0.485787 + 4.00788i −0.376548 1.77152i −2.20637 + 1.76972i −5.03700 1.07065i 0.223824 0.728049i
3.3 −1.20776 0.735750i 0.0575319 0.547380i 0.917345 + 1.77721i −1.88683 + 3.26808i −0.472219 + 0.618772i 0.727319 + 3.42177i 0.199654 2.82137i 2.63813 + 0.560751i 4.68331 2.55881i
3.4 −0.952505 1.04534i −0.0194475 + 0.185031i −0.185469 + 1.99138i 0.691301 1.19737i 0.211944 0.155913i −0.900538 4.23670i 2.25833 1.70292i 2.90058 + 0.616538i −1.91012 + 0.417855i
3.5 −0.726490 + 1.21335i −0.298401 + 2.83910i −0.944425 1.76297i −0.269294 + 0.466430i −3.22803 2.42464i 0.376548 + 1.77152i 2.82521 + 0.134864i −5.03700 1.07065i −0.370303 0.665604i
3.6 −0.553345 + 1.30146i 0.130272 1.23946i −1.38762 1.44032i 1.03203 1.78753i 1.54102 + 0.855391i −0.311298 1.46454i 2.64235 1.00895i 1.41516 + 0.300802i 1.75533 + 2.33227i
3.7 −0.506641 1.32035i 0.308762 2.93767i −1.48663 + 1.33788i 1.56132 2.70429i −4.03518 + 1.08067i 0.883280 + 4.15550i 2.51966 + 1.28504i −5.60015 1.19035i −4.36163 0.691384i
3.8 −0.182213 1.40243i −0.257581 + 2.45072i −1.93360 + 0.511081i −0.786115 + 1.36159i 3.48389 0.0853156i 0.234244 + 1.10203i 1.06908 + 2.61860i −3.00525 0.638785i 2.05277 + 0.854368i
3.9 0.326523 + 1.37600i −0.0575319 + 0.547380i −1.78677 + 0.898592i −1.88683 + 3.26808i −0.771981 + 0.0995678i −0.727319 3.42177i −1.81988 2.16518i 2.63813 + 0.560751i −5.11298 1.52918i
3.10 0.699837 + 1.22891i 0.0194475 0.185031i −1.02046 + 1.72008i 0.691301 1.19737i 0.240997 0.105592i 0.900538 + 4.23670i −2.82798 0.0502801i 2.90058 + 0.616538i 1.95526 + 0.0115868i
3.11 0.878788 1.10803i 0.184585 1.75621i −0.455463 1.94745i −0.842418 + 1.45911i −1.78372 1.74786i 0.0707307 + 0.332762i −2.55809 1.20673i −0.115759 0.0246052i 0.876432 + 2.21567i
3.12 1.09916 + 0.889853i −0.308762 + 2.93767i 0.416322 + 1.95619i 1.56132 2.70429i −2.95348 + 2.95423i −0.883280 4.15550i −1.28312 + 2.52064i −5.60015 1.19035i 4.12257 1.58311i
3.13 1.27748 + 0.606669i 0.257581 2.45072i 1.26391 + 1.55001i −0.786115 + 1.36159i 1.81583 2.97448i −0.234244 1.10203i 0.674270 + 2.74688i −3.00525 0.638785i −1.83028 + 1.26249i
3.14 1.32536 0.493377i −0.184585 + 1.75621i 1.51316 1.30780i −0.842418 + 1.45911i 0.621832 + 2.41868i −0.0707307 0.332762i 1.36024 2.47987i −0.115759 0.0246052i −0.396616 + 2.34948i
11.1 −1.39926 + 0.205086i 1.36017 + 1.51062i 1.91588 0.573940i 0.360804 + 0.624930i −2.21305 1.83481i 0.886730 0.0931991i −2.56311 + 1.19601i −0.118331 + 1.12584i −0.633024 0.800447i
11.2 −1.28829 0.583353i −0.963066 1.06959i 1.31940 + 1.50306i 1.81486 + 3.14343i 0.616761 + 1.93976i 0.437436 0.0459763i −0.822957 2.70606i 0.0970520 0.923388i −0.504342 5.10836i
11.3 −1.12427 0.857914i −0.304683 0.338385i 0.527967 + 1.92905i −1.49536 2.59004i 0.0522411 + 0.641828i −1.34213 + 0.141064i 1.06138 2.62173i 0.291913 2.77737i −0.540843 + 4.19480i
11.4 −0.875050 + 1.11099i −0.584722 0.649399i −0.468577 1.94433i −0.567618 0.983143i 1.23313 0.0813606i 3.71586 0.390553i 2.57015 + 1.18081i 0.233766 2.22413i 1.58895 + 0.229684i
11.5 −0.683091 1.23830i 2.11915 + 2.35355i −1.06677 + 1.69174i −0.103778 0.179749i 1.46683 4.23183i −1.63343 + 0.171681i 2.82359 + 0.165369i −0.734833 + 6.99146i −0.151693 + 0.251293i
11.6 −0.266023 + 1.38897i 0.974705 + 1.08252i −1.85846 0.738995i 1.14850 + 1.98927i −1.76288 + 1.06586i −2.24092 + 0.235530i 1.52083 2.38476i 0.0917868 0.873293i −3.06856 + 1.06604i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 115.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.h odd 30 1 inner
124.p even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.p.a 112
4.b odd 2 1 inner 124.2.p.a 112
31.h odd 30 1 inner 124.2.p.a 112
124.p even 30 1 inner 124.2.p.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.p.a 112 1.a even 1 1 trivial
124.2.p.a 112 4.b odd 2 1 inner
124.2.p.a 112 31.h odd 30 1 inner
124.2.p.a 112 124.p even 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(124, [\chi])$$.