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 $4$

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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.

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

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:

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 \(S_{2}^{\mathrm{new}}(124, [\chi])\).