# Properties

 Label 124.4.j.a Level $124$ Weight $4$ Character orbit 124.j Analytic conductor $7.316$ Analytic rank $0$ Dimension $184$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 124.j (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.31623684071$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$46$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9}+O(q^{10})$$ 184 * q - 3 * q^2 - 15 * q^4 - 16 * q^5 - 15 * q^8 - 384 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$184 q - 3 q^{2} - 15 q^{4} - 16 q^{5} - 15 q^{8} - 384 q^{9} - 84 q^{10} + 190 q^{12} - 10 q^{13} - 156 q^{14} + 73 q^{16} - 10 q^{17} - 254 q^{18} - 37 q^{20} - 310 q^{21} - 330 q^{22} - 5 q^{24} + 3952 q^{25} + 28 q^{28} - 10 q^{29} - 958 q^{32} + 126 q^{33} + 935 q^{34} + 472 q^{36} - 656 q^{38} - 114 q^{40} - 6 q^{41} - 3340 q^{42} + 380 q^{44} - 2124 q^{45} - 125 q^{46} + 1810 q^{48} + 1192 q^{49} + 2508 q^{50} - 1215 q^{52} - 10 q^{53} - 280 q^{54} - 4376 q^{56} - 3205 q^{58} - 3405 q^{60} - 2292 q^{62} + 807 q^{64} - 1260 q^{65} - 260 q^{66} + 822 q^{69} + 3505 q^{70} - 3582 q^{72} - 10 q^{73} + 5105 q^{74} + 5675 q^{76} - 10 q^{77} + 4581 q^{78} - 3626 q^{80} - 2328 q^{81} - 3854 q^{82} + 7085 q^{84} - 1260 q^{85} + 6945 q^{86} - 1710 q^{89} - 728 q^{90} + 614 q^{93} + 4742 q^{94} + 5990 q^{96} + 7766 q^{97} - 6416 q^{98}+O(q^{100})$$ 184 * q - 3 * q^2 - 15 * q^4 - 16 * q^5 - 15 * q^8 - 384 * q^9 - 84 * q^10 + 190 * q^12 - 10 * q^13 - 156 * q^14 + 73 * q^16 - 10 * q^17 - 254 * q^18 - 37 * q^20 - 310 * q^21 - 330 * q^22 - 5 * q^24 + 3952 * q^25 + 28 * q^28 - 10 * q^29 - 958 * q^32 + 126 * q^33 + 935 * q^34 + 472 * q^36 - 656 * q^38 - 114 * q^40 - 6 * q^41 - 3340 * q^42 + 380 * q^44 - 2124 * q^45 - 125 * q^46 + 1810 * q^48 + 1192 * q^49 + 2508 * q^50 - 1215 * q^52 - 10 * q^53 - 280 * q^54 - 4376 * q^56 - 3205 * q^58 - 3405 * q^60 - 2292 * q^62 + 807 * q^64 - 1260 * q^65 - 260 * q^66 + 822 * q^69 + 3505 * q^70 - 3582 * q^72 - 10 * q^73 + 5105 * q^74 + 5675 * q^76 - 10 * q^77 + 4581 * q^78 - 3626 * q^80 - 2328 * q^81 - 3854 * q^82 + 7085 * q^84 - 1260 * q^85 + 6945 * q^86 - 1710 * q^89 - 728 * q^90 + 614 * q^93 + 4742 * q^94 + 5990 * q^96 + 7766 * q^97 - 6416 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
15.1 −2.82840 0.0132683i 7.87115 + 5.71872i 7.99965 + 0.0750562i −13.1003 −22.1869 16.2793i 25.7893 + 8.37945i −22.6252 0.318431i 20.9077 + 64.3474i 37.0530 + 0.173820i
15.2 −2.81316 0.293474i 0.939320 + 0.682456i 7.82775 + 1.65118i 16.2568 −2.44218 2.19552i 15.2434 + 4.95289i −21.5361 6.94228i −7.92688 24.3964i −45.7331 4.77097i
15.3 −2.75945 + 0.620848i 2.96859 + 2.15681i 7.22910 3.42639i −2.68692 −9.53072 4.10855i −18.6273 6.05236i −17.8210 + 13.9431i −4.18275 12.8732i 7.41440 1.66817i
15.4 −2.72510 + 0.757503i −3.17557 2.30719i 6.85238 4.12855i −16.6602 10.4015 + 3.88182i −3.04000 0.987756i −15.5460 + 16.4414i −3.58232 11.0253i 45.4007 12.6201i
15.5 −2.70634 0.822040i −5.26813 3.82752i 6.64850 + 4.44943i 9.59704 11.1109 + 14.6892i −30.4915 9.90729i −14.3355 17.5070i 4.75980 + 14.6492i −25.9728 7.88915i
15.6 −2.63900 + 1.01768i −7.46913 5.42664i 5.92867 5.37130i 5.42895 25.2336 + 6.71977i 16.7763 + 5.45096i −10.1795 + 20.2083i 17.9960 + 55.3861i −14.3270 + 5.52491i
15.7 −2.61727 1.07234i −2.61926 1.90300i 5.70016 + 5.61321i −5.30736 4.81463 + 7.78942i 21.9191 + 7.12195i −8.89956 20.8038i −5.10436 15.7096i 13.8908 + 5.69130i
15.8 −2.48461 1.35157i 3.71840 + 2.70158i 4.34653 + 6.71623i −7.20285 −5.58740 11.7380i −18.8171 6.11405i −1.72198 22.5618i −1.81547 5.58743i 17.8962 + 9.73514i
15.9 −2.41107 + 1.47877i 3.63646 + 2.64204i 3.62648 7.13083i 11.6709 −12.6747 0.992650i 10.8099 + 3.51236i 1.80118 + 22.5556i −2.10001 6.46315i −28.1393 + 17.2586i
15.10 −2.16768 1.81692i 6.84261 + 4.97145i 1.39764 + 7.87697i 15.5435 −5.79986 23.2089i 2.17443 + 0.706516i 11.2822 19.6141i 13.7626 + 42.3568i −33.6933 28.2412i
15.11 −2.15145 + 1.83609i −3.63646 2.64204i 1.25751 7.90055i 11.6709 12.6747 0.992650i −10.8099 3.51236i 11.8007 + 19.3066i −2.10001 6.46315i −25.1094 + 21.4289i
15.12 −1.78541 2.19370i −5.33512 3.87619i −1.62460 + 7.83331i −18.0276 1.02221 + 18.6242i −11.4547 3.72185i 20.0845 10.4218i 5.09519 + 15.6814i 32.1868 + 39.5472i
15.13 −1.78336 + 2.19536i 7.46913 + 5.42664i −1.63923 7.83026i 5.42895 −25.2336 + 6.71977i −16.7763 5.45096i 20.1136 + 10.3655i 17.9960 + 55.3861i −9.68179 + 11.9185i
15.14 −1.75505 2.21806i 3.06772 + 2.22883i −1.83962 + 7.78562i −5.95936 −0.440306 10.7161i 2.34935 + 0.763351i 20.4976 9.58373i −3.90024 12.0037i 10.4590 + 13.2182i
15.15 −1.60159 2.33129i −7.03574 5.11177i −2.86983 + 7.46754i 11.4470 −0.648641 + 24.5893i 19.7410 + 6.41424i 22.0053 5.26953i 15.0281 + 46.2517i −18.3334 26.6862i
15.16 −1.56253 + 2.35765i 3.17557 + 2.30719i −3.11699 7.36779i −16.6602 −10.4015 + 3.88182i 3.04000 + 0.987756i 22.2410 + 4.16365i −3.58232 11.0253i 26.0321 39.2788i
15.17 −1.44318 + 2.43254i −2.96859 2.15681i −3.83448 7.02117i −2.68692 9.53072 4.10855i 18.6273 + 6.05236i 22.6131 + 0.805281i −4.18275 12.8732i 3.87770 6.53602i
15.18 −1.11434 2.59966i −0.687892 0.499783i −5.51648 + 5.79383i 12.6815 −0.532718 + 2.34522i −14.1641 4.60220i 21.2093 + 7.88465i −8.12005 24.9909i −14.1316 32.9677i
15.19 −0.861403 + 2.69406i −7.87115 5.71872i −6.51597 4.64135i −13.1003 22.1869 16.2793i −25.7893 8.37945i 18.1170 13.5564i 20.9077 + 64.3474i 11.2847 35.2932i
15.20 −0.618335 2.76001i 2.24034 + 1.62770i −7.23532 + 3.41322i −9.14807 3.10720 7.18983i 31.6620 + 10.2876i 13.8944 + 17.8591i −5.97375 18.3853i 5.65657 + 25.2488i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.46 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.f odd 10 1 inner
124.j even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.j.a 184
4.b odd 2 1 inner 124.4.j.a 184
31.f odd 10 1 inner 124.4.j.a 184
124.j even 10 1 inner 124.4.j.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.j.a 184 1.a even 1 1 trivial
124.4.j.a 184 4.b odd 2 1 inner
124.4.j.a 184 31.f odd 10 1 inner
124.4.j.a 184 124.j even 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(124, [\chi])$$.