Properties

 Label 124.3.n.a Level $124$ Weight $3$ Character orbit 124.n Analytic conductor $3.379$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 124.n (of order $$30$$, degree $$8$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$3.37875527807$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9}+O(q^{10})$$ 240 * q - 6 * q^2 - 10 * q^4 - 8 * q^5 - 5 * q^6 - 27 * q^8 - 96 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$240 q - 6 q^{2} - 10 q^{4} - 8 q^{5} - 5 q^{6} - 27 q^{8} - 96 q^{9} - 4 q^{10} + 27 q^{12} - 26 q^{13} + 10 q^{14} + 46 q^{16} - 18 q^{17} - 11 q^{18} + 143 q^{20} + 90 q^{21} + 77 q^{22} - 54 q^{24} - 464 q^{25} - 27 q^{26} - 52 q^{28} - 12 q^{29} + 206 q^{30} + 154 q^{32} + 72 q^{33} - 168 q^{34} + 23 q^{36} - 48 q^{37} - 78 q^{38} + 85 q^{40} - 18 q^{41} - 91 q^{42} - 493 q^{44} - 30 q^{45} + 198 q^{46} - 314 q^{48} + 48 q^{49} - 563 q^{50} - 551 q^{52} + 46 q^{53} - 600 q^{54} - 90 q^{56} - 44 q^{57} - 125 q^{58} - 77 q^{60} + 208 q^{61} - 17 q^{62} - 529 q^{64} + 132 q^{65} + 788 q^{66} + 364 q^{68} + 36 q^{69} + 586 q^{70} + 1113 q^{72} + 214 q^{73} + 351 q^{74} + 824 q^{76} + 456 q^{77} + 123 q^{78} + 410 q^{80} + 90 q^{81} - 718 q^{82} - 412 q^{84} + 394 q^{85} + 680 q^{86} - 141 q^{88} + 12 q^{89} + 193 q^{90} - 520 q^{92} + 82 q^{93} - 876 q^{94} + 888 q^{96} - 548 q^{97} + 32 q^{98}+O(q^{100})$$ 240 * q - 6 * q^2 - 10 * q^4 - 8 * q^5 - 5 * q^6 - 27 * q^8 - 96 * q^9 - 4 * q^10 + 27 * q^12 - 26 * q^13 + 10 * q^14 + 46 * q^16 - 18 * q^17 - 11 * q^18 + 143 * q^20 + 90 * q^21 + 77 * q^22 - 54 * q^24 - 464 * q^25 - 27 * q^26 - 52 * q^28 - 12 * q^29 + 206 * q^30 + 154 * q^32 + 72 * q^33 - 168 * q^34 + 23 * q^36 - 48 * q^37 - 78 * q^38 + 85 * q^40 - 18 * q^41 - 91 * q^42 - 493 * q^44 - 30 * q^45 + 198 * q^46 - 314 * q^48 + 48 * q^49 - 563 * q^50 - 551 * q^52 + 46 * q^53 - 600 * q^54 - 90 * q^56 - 44 * q^57 - 125 * q^58 - 77 * q^60 + 208 * q^61 - 17 * q^62 - 529 * q^64 + 132 * q^65 + 788 * q^66 + 364 * q^68 + 36 * q^69 + 586 * q^70 + 1113 * q^72 + 214 * q^73 + 351 * q^74 + 824 * q^76 + 456 * q^77 + 123 * q^78 + 410 * q^80 + 90 * q^81 - 718 * q^82 - 412 * q^84 + 394 * q^85 + 680 * q^86 - 141 * q^88 + 12 * q^89 + 193 * q^90 - 520 * q^92 + 82 * q^93 - 876 * q^94 + 888 * q^96 - 548 * q^97 + 32 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.99926 + 0.0544568i −0.0879632 0.413834i 3.99407 0.217747i −0.624847 + 1.08227i 0.198397 + 0.822572i −2.70372 + 6.07265i −7.97332 + 0.652836i 8.05839 3.58783i 1.19029 2.19776i
7.2 −1.99505 + 0.140610i 0.741912 + 3.49042i 3.96046 0.561050i 2.31670 4.01264i −1.97094 6.85925i 5.02348 11.2829i −7.82243 + 1.67620i −3.41071 + 1.51855i −4.05771 + 8.33116i
7.3 −1.86891 + 0.712155i −1.15158 5.41775i 2.98567 2.66191i 4.16039 7.20601i 6.01048 + 9.30520i 0.685204 1.53899i −3.68426 + 7.10114i −19.8040 + 8.81730i −2.64361 + 16.4302i
7.4 −1.82229 + 0.824162i −0.469244 2.20762i 2.64151 3.00373i −2.85641 + 4.94745i 2.67454 + 3.63620i 3.33744 7.49600i −2.33806 + 7.65072i 3.56852 1.58881i 1.12772 11.3698i
7.5 −1.76073 0.948596i −1.13434 5.33667i 2.20033 + 3.34044i −3.56746 + 6.17902i −3.06507 + 10.4725i −0.864598 + 1.94192i −0.705452 7.96884i −18.9714 + 8.44661i 12.1427 7.49550i
7.6 −1.73916 0.987589i −0.272138 1.28031i 2.04934 + 3.43515i 2.35737 4.08308i −0.791127 + 2.49542i −1.03353 + 2.32136i −0.171606 7.99816i 6.65678 2.96379i −8.13224 + 4.77301i
7.7 −1.60673 1.19097i 0.931035 + 4.38017i 1.16319 + 3.82714i −0.973390 + 1.68596i 3.72072 8.14661i −1.13057 + 2.53929i 2.68905 7.53452i −10.0972 + 4.49556i 3.57190 1.54962i
7.8 −1.59315 + 1.20909i 0.932435 + 4.38676i 1.07622 3.85250i −1.13754 + 1.97029i −6.78947 5.86135i −2.59346 + 5.82500i 2.94342 + 7.43884i −10.1523 + 4.52010i −0.569970 4.51434i
7.9 −1.15685 + 1.63147i −0.354741 1.66892i −1.32341 3.77473i −1.19665 + 2.07266i 3.13318 + 1.35194i −1.19159 + 2.67635i 7.68935 + 2.20769i 5.56244 2.47656i −1.99714 4.35005i
7.10 −1.07956 1.68361i 0.0275934 + 0.129817i −1.66909 + 3.63512i −2.86810 + 4.96769i 0.188772 0.186602i 4.37751 9.83205i 7.92202 1.11424i 8.20582 3.65347i 11.4599 0.534167i
7.11 −0.864102 + 1.80370i 0.286400 + 1.34741i −2.50666 3.11716i 3.60353 6.24150i −2.67780 0.647717i 1.82861 4.10713i 7.78842 1.82771i 6.48843 2.88883i 8.14397 + 11.8930i
7.12 −0.673616 1.88315i −0.736057 3.46287i −3.09248 + 2.53704i 3.42633 5.93457i −6.02528 + 3.71875i 2.34572 5.26857i 6.86076 + 4.11461i −3.22781 + 1.43711i −13.4837 2.45465i
7.13 −0.561919 1.91944i 0.736057 + 3.46287i −3.36849 + 2.15714i 3.42633 5.93457i 6.23317 3.35867i −2.34572 + 5.26857i 6.03332 + 5.25348i −3.22781 + 1.43711i −13.3164 3.24188i
7.14 −0.329666 + 1.97264i −0.974786 4.58601i −3.78264 1.30063i −0.229111 + 0.396833i 9.36791 0.411054i −4.30093 + 9.66006i 3.81268 7.03303i −11.8594 + 5.28013i −0.707279 0.582777i
7.15 −0.199060 + 1.99007i 0.547049 + 2.57366i −3.92075 0.792285i −4.34583 + 7.52719i −5.23066 + 0.576352i 2.27507 5.10989i 2.35717 7.64485i 1.89744 0.844793i −14.1146 10.1469i
7.16 −0.116218 1.99662i −0.0275934 0.129817i −3.97299 + 0.464087i −2.86810 + 4.96769i −0.255988 + 0.0701807i −4.37751 + 9.83205i 1.38834 + 7.87861i 8.20582 3.65347i 10.2519 + 5.14916i
7.17 0.595729 + 1.90922i −0.631742 2.97211i −3.29021 + 2.27475i 1.43502 2.48553i 5.29806 2.97670i 3.93413 8.83620i −6.30307 4.92660i −0.212439 + 0.0945840i 5.60029 + 1.25906i
7.18 0.599843 1.90793i −0.931035 4.38017i −3.28038 2.28891i −0.973390 + 1.68596i −8.91553 0.851069i 1.13057 2.53929i −6.33479 + 4.88573i −10.0972 + 4.49556i 2.63281 + 2.86847i
7.19 0.640254 + 1.89475i 0.631742 + 2.97211i −3.18015 + 2.42624i 1.43502 2.48553i −5.22693 + 3.09990i −3.93413 + 8.83620i −6.63323 4.47217i −0.212439 + 0.0945840i 5.62823 + 1.12763i
7.20 0.826518 1.82123i 0.272138 + 1.28031i −2.63374 3.01055i 2.35737 4.08308i 2.55666 + 0.562572i 1.03353 2.32136i −7.65973 + 2.30836i 6.65678 2.96379i −5.48781 7.66804i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 111.30 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.g even 15 1 inner
124.n odd 30 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.3.n.a 240
4.b odd 2 1 inner 124.3.n.a 240
31.g even 15 1 inner 124.3.n.a 240
124.n odd 30 1 inner 124.3.n.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.3.n.a 240 1.a even 1 1 trivial
124.3.n.a 240 4.b odd 2 1 inner
124.3.n.a 240 31.g even 15 1 inner
124.3.n.a 240 124.n odd 30 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(124, [\chi])$$.