# Properties

 Label 124.3.l.a Level $124$ Weight $3$ Character orbit 124.l Analytic conductor $3.379$ Analytic rank $0$ Dimension $120$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 124.l (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.37875527807$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$30$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9}+O(q^{10})$$ 120 * q - 3 * q^2 + q^4 - 16 * q^5 - 10 * q^6 + 27 * q^8 + 72 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$120 q - 3 q^{2} + q^{4} - 16 q^{5} - 10 q^{6} + 27 q^{8} + 72 q^{9} - 26 q^{10} - 66 q^{12} - 22 q^{13} - 34 q^{14} - 55 q^{16} - 6 q^{17} + 74 q^{18} - 47 q^{20} - 114 q^{21} - 56 q^{22} + 15 q^{24} + 440 q^{25} - 48 q^{26} - 8 q^{28} - 6 q^{29} - 254 q^{30} - 178 q^{32} - 90 q^{33} + 171 q^{34} - 8 q^{36} - 96 q^{37} - 42 q^{38} + 50 q^{40} - 6 q^{41} + 268 q^{42} + 196 q^{44} - 120 q^{45} - 231 q^{46} - 28 q^{48} + 48 q^{49} - 394 q^{50} - 7 q^{52} + 122 q^{53} - 126 q^{54} - 432 q^{56} - 196 q^{57} - 49 q^{58} - 163 q^{60} + 80 q^{61} + 200 q^{62} + 19 q^{64} - 156 q^{65} + 490 q^{66} + 266 q^{68} - 522 q^{69} + 65 q^{70} + 642 q^{72} + 122 q^{73} + 177 q^{74} + 517 q^{76} - 186 q^{77} + 303 q^{78} - 602 q^{80} - 168 q^{81} + 406 q^{82} + 769 q^{84} - 508 q^{85} - 677 q^{86} - 108 q^{88} - 30 q^{89} + 662 q^{90} + 910 q^{92} - 250 q^{93} + 354 q^{94} - 1230 q^{96} + 530 q^{97} + 76 q^{98}+O(q^{100})$$ 120 * q - 3 * q^2 + q^4 - 16 * q^5 - 10 * q^6 + 27 * q^8 + 72 * q^9 - 26 * q^10 - 66 * q^12 - 22 * q^13 - 34 * q^14 - 55 * q^16 - 6 * q^17 + 74 * q^18 - 47 * q^20 - 114 * q^21 - 56 * q^22 + 15 * q^24 + 440 * q^25 - 48 * q^26 - 8 * q^28 - 6 * q^29 - 254 * q^30 - 178 * q^32 - 90 * q^33 + 171 * q^34 - 8 * q^36 - 96 * q^37 - 42 * q^38 + 50 * q^40 - 6 * q^41 + 268 * q^42 + 196 * q^44 - 120 * q^45 - 231 * q^46 - 28 * q^48 + 48 * q^49 - 394 * q^50 - 7 * q^52 + 122 * q^53 - 126 * q^54 - 432 * q^56 - 196 * q^57 - 49 * q^58 - 163 * q^60 + 80 * q^61 + 200 * q^62 + 19 * q^64 - 156 * q^65 + 490 * q^66 + 266 * q^68 - 522 * q^69 + 65 * q^70 + 642 * q^72 + 122 * q^73 + 177 * q^74 + 517 * q^76 - 186 * q^77 + 303 * q^78 - 602 * q^80 - 168 * q^81 + 406 * q^82 + 769 * q^84 - 508 * q^85 - 677 * q^86 - 108 * q^88 - 30 * q^89 + 662 * q^90 + 910 * q^92 - 250 * q^93 + 354 * q^94 - 1230 * q^96 + 530 * q^97 + 76 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1 −1.99963 0.0384241i 2.41735 + 0.785446i 3.99705 + 0.153668i 7.23273 −4.80363 1.66349i 0.574978 + 0.791389i −7.98671 0.460862i −2.05448 1.49267i −14.4628 0.277911i
35.2 −1.99455 0.147518i −3.91906 1.27338i 3.95648 + 0.588464i −2.43619 7.62893 + 3.11795i −5.14011 7.07475i −7.80459 1.75737i 6.45639 + 4.69084i 4.85910 + 0.359381i
35.3 −1.86773 0.715251i 1.22091 + 0.396697i 2.97683 + 2.67179i −6.33427 −1.99659 1.61418i −0.534611 0.735828i −3.64892 7.11937i −5.94791 4.32141i 11.8307 + 4.53060i
35.4 −1.84215 + 0.778768i −2.10823 0.685006i 2.78704 2.86922i 0.107615 4.41714 0.379938i 2.89081 + 3.97885i −2.89970 + 7.45599i −3.30575 2.40177i −0.198243 + 0.0838070i
35.5 −1.73407 + 0.996495i 3.94056 + 1.28037i 2.01400 3.45598i −5.71773 −8.10909 + 1.70651i 5.97197 + 8.21971i −0.0485371 + 7.99985i 6.60755 + 4.80067i 9.91495 5.69769i
35.6 −1.71695 1.02571i −4.06445 1.32062i 1.89584 + 3.52219i 3.01115 5.62389 + 6.43639i 6.03869 + 8.31155i 0.357682 7.99200i 7.49458 + 5.44513i −5.16999 3.08856i
35.7 −1.48076 + 1.34437i 0.177366 + 0.0576297i 0.385317 3.98140i 0.656973 −0.340113 + 0.153110i −4.86752 6.69957i 4.78192 + 6.41352i −7.25302 5.26962i −0.972821 + 0.883217i
35.8 −1.36046 1.46600i 4.92224 + 1.59933i −0.298302 + 3.98886i 0.357497 −4.35189 9.39182i 0.118689 + 0.163361i 6.25349 4.98937i 14.3894 + 10.4545i −0.486360 0.524090i
35.9 −1.18822 1.60877i −1.95963 0.636724i −1.17627 + 3.82314i 7.70420 1.30413 + 3.90916i −7.26049 9.99321i 7.54822 2.65037i −3.84641 2.79458i −9.15428 12.3943i
35.10 −1.07058 + 1.68934i −5.00352 1.62574i −1.70773 3.61713i −7.77294 8.10307 6.71216i 3.31847 + 4.56749i 7.93882 + 0.987476i 15.1110 + 10.9788i 8.32152 13.1311i
35.11 −0.904691 1.78369i −1.26908 0.412350i −2.36307 + 3.22737i −3.39555 0.412626 + 2.63670i 4.74424 + 6.52988i 7.89446 + 1.29520i −5.84061 4.24345i 3.07193 + 6.05660i
35.12 −0.784035 + 1.83992i 5.17239 + 1.68061i −2.77058 2.88512i 5.84028 −7.14752 + 8.19911i −1.86042 2.56065i 7.48060 2.83560i 16.6481 + 12.0955i −4.57898 + 10.7456i
35.13 −0.646421 + 1.89265i −1.68966 0.549003i −3.16428 2.44690i 6.67872 2.13130 2.84305i 4.01648 + 5.52822i 6.67660 4.40716i −4.72761 3.43481i −4.31726 + 12.6405i
35.14 −0.335855 + 1.97160i 1.85229 + 0.601846i −3.77440 1.32434i −8.62176 −1.80870 + 3.44984i −5.31885 7.32077i 3.87873 6.99682i −4.21239 3.06048i 2.89566 16.9986i
35.15 −0.316513 1.97480i 1.26908 + 0.412350i −3.79964 + 1.25010i −3.39555 0.412626 2.63670i −4.74424 6.52988i 3.67133 + 7.10784i −5.84061 4.24345i 1.07474 + 6.70552i
35.16 0.0156782 1.99994i 1.95963 + 0.636724i −3.99951 0.0627107i 7.70420 1.30413 3.90916i 7.26049 + 9.99321i −0.188122 + 7.99779i −3.84641 2.79458i 0.120788 15.4079i
35.17 0.238942 1.98568i −4.92224 1.59933i −3.88581 0.948924i 0.357497 −4.35189 + 9.39182i −0.118689 0.163361i −2.81274 + 7.48923i 14.3894 + 10.4545i 0.0854211 0.709872i
35.18 0.470632 + 1.94384i −2.99163 0.972039i −3.55701 + 1.82966i 0.689285 0.481530 6.27271i −2.85110 3.92420i −5.23061 6.05315i 0.723827 + 0.525891i 0.324400 + 1.33986i
35.19 0.761810 + 1.84923i 2.99163 + 0.972039i −2.83929 + 2.81752i 0.689285 0.481530 + 6.27271i 2.85110 + 3.92420i −7.37324 3.10408i 0.723827 + 0.525891i 0.525104 + 1.27465i
35.20 0.786145 1.83902i 4.06445 + 1.32062i −2.76395 2.89146i 3.01115 5.62389 6.43639i −6.03869 8.31155i −7.49031 + 2.80984i 7.49458 + 5.44513i 2.36720 5.53755i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 95.30 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.d even 5 1 inner
124.l odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.3.l.a 120
4.b odd 2 1 inner 124.3.l.a 120
31.d even 5 1 inner 124.3.l.a 120
124.l odd 10 1 inner 124.3.l.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.3.l.a 120 1.a even 1 1 trivial
124.3.l.a 120 4.b odd 2 1 inner
124.3.l.a 120 31.d even 5 1 inner
124.3.l.a 120 124.l odd 10 1 inner

## Hecke kernels

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