# Properties

 Label 124.5.b.a Level $124$ Weight $5$ Character orbit 124.b Analytic conductor $12.818$ Analytic rank $0$ Dimension $60$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 124.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.8178754224$$ Analytic rank: $$0$$ Dimension: $$60$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$60 q + 6 q^{2} - 6 q^{4} + 24 q^{5} + 45 q^{8} - 1732 q^{9}+O(q^{10})$$ 60 * q + 6 * q^2 - 6 * q^4 + 24 * q^5 + 45 * q^8 - 1732 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q + 6 q^{2} - 6 q^{4} + 24 q^{5} + 45 q^{8} - 1732 q^{9} + 53 q^{10} + 130 q^{12} + 120 q^{13} - 231 q^{14} - 590 q^{16} - 648 q^{17} + 230 q^{18} + 1113 q^{20} + 608 q^{21} + 1080 q^{22} - 1028 q^{24} + 8340 q^{25} - 1554 q^{26} - 165 q^{28} - 168 q^{29} - 2238 q^{30} - 1674 q^{32} - 1120 q^{33} + 1844 q^{34} + 1966 q^{36} - 2248 q^{37} - 5055 q^{38} - 1716 q^{40} + 6072 q^{41} + 5794 q^{42} - 120 q^{44} - 4040 q^{45} + 8850 q^{46} + 2276 q^{48} - 17604 q^{49} - 4539 q^{50} + 5896 q^{52} + 3480 q^{53} + 5148 q^{54} - 396 q^{56} - 10912 q^{57} - 7484 q^{58} + 22812 q^{60} + 2200 q^{61} - 19299 q^{64} - 9168 q^{65} - 468 q^{66} - 21930 q^{68} + 6496 q^{69} - 9615 q^{70} - 10079 q^{72} + 13752 q^{73} - 2106 q^{74} + 20099 q^{76} + 16608 q^{77} + 39460 q^{78} - 5787 q^{80} + 20732 q^{81} - 30525 q^{82} - 21760 q^{84} - 21200 q^{85} - 13398 q^{86} + 34690 q^{88} + 22296 q^{89} - 25419 q^{90} - 18852 q^{92} + 3196 q^{94} + 20790 q^{96} + 12120 q^{97} + 21921 q^{98}+O(q^{100})$$ 60 * q + 6 * q^2 - 6 * q^4 + 24 * q^5 + 45 * q^8 - 1732 * q^9 + 53 * q^10 + 130 * q^12 + 120 * q^13 - 231 * q^14 - 590 * q^16 - 648 * q^17 + 230 * q^18 + 1113 * q^20 + 608 * q^21 + 1080 * q^22 - 1028 * q^24 + 8340 * q^25 - 1554 * q^26 - 165 * q^28 - 168 * q^29 - 2238 * q^30 - 1674 * q^32 - 1120 * q^33 + 1844 * q^34 + 1966 * q^36 - 2248 * q^37 - 5055 * q^38 - 1716 * q^40 + 6072 * q^41 + 5794 * q^42 - 120 * q^44 - 4040 * q^45 + 8850 * q^46 + 2276 * q^48 - 17604 * q^49 - 4539 * q^50 + 5896 * q^52 + 3480 * q^53 + 5148 * q^54 - 396 * q^56 - 10912 * q^57 - 7484 * q^58 + 22812 * q^60 + 2200 * q^61 - 19299 * q^64 - 9168 * q^65 - 468 * q^66 - 21930 * q^68 + 6496 * q^69 - 9615 * q^70 - 10079 * q^72 + 13752 * q^73 - 2106 * q^74 + 20099 * q^76 + 16608 * q^77 + 39460 * q^78 - 5787 * q^80 + 20732 * q^81 - 30525 * q^82 - 21760 * q^84 - 21200 * q^85 - 13398 * q^86 + 34690 * q^88 + 22296 * q^89 - 25419 * q^90 - 18852 * q^92 + 3196 * q^94 + 20790 * q^96 + 12120 * q^97 + 21921 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1 −3.98801 0.309498i 16.5469i 15.8084 + 2.46856i 5.81989 5.12124 65.9893i 22.5351i −62.2801 14.7373i −192.801 −23.2098 1.80124i
63.2 −3.98801 + 0.309498i 16.5469i 15.8084 2.46856i 5.81989 5.12124 + 65.9893i 22.5351i −62.2801 + 14.7373i −192.801 −23.2098 + 1.80124i
63.3 −3.93534 0.716318i 7.84728i 14.9738 + 5.63791i 18.4905 −5.62115 + 30.8817i 83.1629i −54.8883 32.9131i 19.4202 −72.7665 13.2451i
63.4 −3.93534 + 0.716318i 7.84728i 14.9738 5.63791i 18.4905 −5.62115 30.8817i 83.1629i −54.8883 + 32.9131i 19.4202 −72.7665 + 13.2451i
63.5 −3.71949 1.47153i 6.32028i 11.6692 + 10.9467i −34.2840 9.30050 23.5082i 45.8539i −27.2950 57.8877i 41.0540 127.519 + 50.4501i
63.6 −3.71949 + 1.47153i 6.32028i 11.6692 10.9467i −34.2840 9.30050 + 23.5082i 45.8539i −27.2950 + 57.8877i 41.0540 127.519 50.4501i
63.7 −3.64645 1.64420i 1.82065i 10.5932 + 11.9910i 43.7825 −2.99351 + 6.63891i 53.1613i −18.9122 61.1419i 77.6852 −159.651 71.9870i
63.8 −3.64645 + 1.64420i 1.82065i 10.5932 11.9910i 43.7825 −2.99351 6.63891i 53.1613i −18.9122 + 61.1419i 77.6852 −159.651 + 71.9870i
63.9 −3.40182 2.10420i 9.06141i 7.14471 + 14.3162i 5.08300 −19.0670 + 30.8253i 38.2255i 5.81911 63.7349i −1.10919 −17.2914 10.6956i
63.10 −3.40182 + 2.10420i 9.06141i 7.14471 14.3162i 5.08300 −19.0670 30.8253i 38.2255i 5.81911 + 63.7349i −1.10919 −17.2914 + 10.6956i
63.11 −3.23786 2.34867i 10.1900i 4.96753 + 15.2093i −24.8949 23.9329 32.9938i 72.3138i 19.6374 60.9128i −22.8358 80.6064 + 58.4699i
63.12 −3.23786 + 2.34867i 10.1900i 4.96753 15.2093i −24.8949 23.9329 + 32.9938i 72.3138i 19.6374 + 60.9128i −22.8358 80.6064 58.4699i
63.13 −3.22841 2.36165i 13.9390i 4.84523 + 15.2487i −45.4494 −32.9189 + 45.0007i 19.7116i 20.3698 60.6718i −113.295 146.729 + 107.335i
63.14 −3.22841 + 2.36165i 13.9390i 4.84523 15.2487i −45.4494 −32.9189 45.0007i 19.7116i 20.3698 + 60.6718i −113.295 146.729 107.335i
63.15 −2.98311 2.66478i 10.1030i 1.79785 + 15.8987i 29.6620 26.9222 30.1382i 35.0486i 37.0033 52.2183i −21.0701 −88.4850 79.0429i
63.16 −2.98311 + 2.66478i 10.1030i 1.79785 15.8987i 29.6620 26.9222 + 30.1382i 35.0486i 37.0033 + 52.2183i −21.0701 −88.4850 + 79.0429i
63.17 −2.06596 3.42518i 0.636071i −7.46365 + 14.1525i −5.80492 2.17865 1.31409i 53.1144i 63.8944 3.67417i 80.5954 11.9927 + 19.8829i
63.18 −2.06596 + 3.42518i 0.636071i −7.46365 14.1525i −5.80492 2.17865 + 1.31409i 53.1144i 63.8944 + 3.67417i 80.5954 11.9927 19.8829i
63.19 −1.86128 3.54057i 8.93980i −9.07128 + 13.1800i 0.767532 −31.6520 + 16.6395i 40.1374i 63.5488 + 7.58587i 1.07989 −1.42859 2.71750i
63.20 −1.86128 + 3.54057i 8.93980i −9.07128 13.1800i 0.767532 −31.6520 16.6395i 40.1374i 63.5488 7.58587i 1.07989 −1.42859 + 2.71750i
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 63.60 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.5.b.a 60
4.b odd 2 1 inner 124.5.b.a 60

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.5.b.a 60 1.a even 1 1 trivial
124.5.b.a 60 4.b odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(124, [\chi])$$.