# Properties

 Label 124.6.f.a Level $124$ Weight $6$ Character orbit 124.f Analytic conductor $19.888$ Analytic rank $0$ Dimension $56$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 124.f (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.8875936568$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56 q + 2 q^{3} - 58 q^{5} + 104 q^{7} - 1234 q^{9}+O(q^{10})$$ 56 * q + 2 * q^3 - 58 * q^5 + 104 * q^7 - 1234 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$56 q + 2 q^{3} - 58 q^{5} + 104 q^{7} - 1234 q^{9} - 509 q^{11} - 117 q^{13} + 89 q^{15} - 3504 q^{17} + 262 q^{19} + 352 q^{21} - 2448 q^{23} + 49618 q^{25} + 14324 q^{27} - 9888 q^{29} - 12771 q^{31} + 27699 q^{33} + 13840 q^{35} + 76096 q^{37} + 33520 q^{39} - 4843 q^{41} - 40778 q^{43} + 56692 q^{45} + 38922 q^{47} - 17126 q^{49} - 69292 q^{51} - 41728 q^{53} - 172096 q^{55} + 57066 q^{57} - 58198 q^{59} + 176328 q^{61} - 37444 q^{63} + 143863 q^{65} + 9812 q^{67} - 9250 q^{69} - 67356 q^{71} - 63512 q^{73} - 198012 q^{75} - 74257 q^{77} + 137651 q^{79} + 196077 q^{81} + 156427 q^{83} + 238828 q^{85} - 558144 q^{87} - 99292 q^{89} - 243609 q^{91} - 325925 q^{93} - 75077 q^{95} - 476340 q^{97} + 745812 q^{99}+O(q^{100})$$ 56 * q + 2 * q^3 - 58 * q^5 + 104 * q^7 - 1234 * q^9 - 509 * q^11 - 117 * q^13 + 89 * q^15 - 3504 * q^17 + 262 * q^19 + 352 * q^21 - 2448 * q^23 + 49618 * q^25 + 14324 * q^27 - 9888 * q^29 - 12771 * q^31 + 27699 * q^33 + 13840 * q^35 + 76096 * q^37 + 33520 * q^39 - 4843 * q^41 - 40778 * q^43 + 56692 * q^45 + 38922 * q^47 - 17126 * q^49 - 69292 * q^51 - 41728 * q^53 - 172096 * q^55 + 57066 * q^57 - 58198 * q^59 + 176328 * q^61 - 37444 * q^63 + 143863 * q^65 + 9812 * q^67 - 9250 * q^69 - 67356 * q^71 - 63512 * q^73 - 198012 * q^75 - 74257 * q^77 + 137651 * q^79 + 196077 * q^81 + 156427 * q^83 + 238828 * q^85 - 558144 * q^87 - 99292 * q^89 - 243609 * q^91 - 325925 * q^93 - 75077 * q^95 - 476340 * q^97 + 745812 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1 0 −21.7500 + 15.8023i 0 −26.6888 0 −46.2791 142.432i 0 148.259 456.295i 0
33.2 0 −21.3000 + 15.4753i 0 92.0001 0 26.8893 + 82.7567i 0 139.112 428.142i 0
33.3 0 −18.4148 + 13.3791i 0 −27.4655 0 63.1359 + 194.312i 0 85.0116 261.639i 0
33.4 0 −13.2483 + 9.62542i 0 −92.0437 0 −28.3654 87.2998i 0 7.77640 23.9333i 0
33.5 0 −9.06757 + 6.58797i 0 43.6459 0 −45.9020 141.272i 0 −36.2717 + 111.633i 0
33.6 0 −7.61067 + 5.52948i 0 −0.875792 0 42.6014 + 131.114i 0 −47.7439 + 146.941i 0
33.7 0 0.167144 0.121437i 0 −87.7524 0 19.9011 + 61.2493i 0 −75.0779 + 231.066i 0
33.8 0 1.51931 1.10384i 0 18.9004 0 −11.6309 35.7963i 0 −74.0013 + 227.753i 0
33.9 0 5.76166 4.18609i 0 104.754 0 44.7223 + 137.641i 0 −59.4177 + 182.869i 0
33.10 0 14.1347 10.2695i 0 −78.4837 0 −67.8106 208.700i 0 19.2372 59.2059i 0
33.11 0 14.7756 10.7351i 0 −30.7250 0 −8.20943 25.2660i 0 27.9851 86.1294i 0
33.12 0 14.8483 10.7879i 0 −36.3819 0 65.9182 + 202.875i 0 29.0018 89.2585i 0
33.13 0 16.8446 12.2383i 0 64.7441 0 −38.0058 116.970i 0 58.8724 181.191i 0
33.14 0 23.8399 17.3207i 0 9.44947 0 30.2776 + 93.1850i 0 193.243 594.741i 0
97.1 0 −8.98126 + 27.6415i 0 −66.9725 0 161.853 117.593i 0 −486.797 353.679i 0
97.2 0 −8.90517 + 27.4073i 0 53.9543 0 −68.0214 + 49.4205i 0 −475.267 345.302i 0
97.3 0 −4.51320 + 13.8902i 0 −108.120 0 −183.214 + 133.113i 0 24.0223 + 17.4532i 0
97.4 0 −4.50907 + 13.8775i 0 78.4326 0 −14.7862 + 10.7428i 0 24.3383 + 17.6828i 0
97.5 0 −4.47947 + 13.7864i 0 7.66625 0 −38.1042 + 27.6843i 0 26.5924 + 19.3205i 0
97.6 0 −3.34224 + 10.2864i 0 −12.0574 0 62.6660 45.5295i 0 101.953 + 74.0728i 0
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.6.f.a 56
31.d even 5 1 inner 124.6.f.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.6.f.a 56 1.a even 1 1 trivial
124.6.f.a 56 31.d even 5 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(124, [\chi])$$.