Properties

 Label 125.2.g.a Level $125$ Weight $2$ Character orbit 125.g Analytic conductor $0.998$ Analytic rank $0$ Dimension $220$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$125 = 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 125.g (of order $$25$$, degree $$20$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$0.998130025266$$ Analytic rank: $$0$$ Dimension: $$220$$ Relative dimension: $$11$$ over $$\Q(\zeta_{25})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{25}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$220 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 15 q^{5} - 20 q^{6} - 15 q^{7} - 5 q^{8} - 20 q^{9}+O(q^{10})$$ 220 * q - 20 * q^2 - 20 * q^3 - 20 * q^4 - 15 * q^5 - 20 * q^6 - 15 * q^7 - 5 * q^8 - 20 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$220 q - 20 q^{2} - 20 q^{3} - 20 q^{4} - 15 q^{5} - 20 q^{6} - 15 q^{7} - 5 q^{8} - 20 q^{9} - 20 q^{10} - 15 q^{11} - 100 q^{12} - 20 q^{13} - 10 q^{14} - 15 q^{17} + 10 q^{18} - 10 q^{19} + 5 q^{20} - 5 q^{21} + 25 q^{22} - 90 q^{23} + 15 q^{24} - 45 q^{25} + 15 q^{26} + 10 q^{27} + 30 q^{28} - 10 q^{29} + 40 q^{30} - 10 q^{31} + 20 q^{32} + 15 q^{33} + 5 q^{34} + 10 q^{35} - 210 q^{36} + 20 q^{37} + 30 q^{38} + 45 q^{40} - 5 q^{41} + 70 q^{42} + 25 q^{43} + 15 q^{44} - 100 q^{45} - 100 q^{47} + 105 q^{48} - 20 q^{49} + 60 q^{50} + 25 q^{51} + 85 q^{52} + 20 q^{53} + 60 q^{54} + 15 q^{55} - 105 q^{56} + 40 q^{57} - 225 q^{58} + 5 q^{59} - 35 q^{60} + 65 q^{62} + 85 q^{63} + 45 q^{64} + 45 q^{65} + 55 q^{66} - 105 q^{67} + 80 q^{68} + 40 q^{69} + 95 q^{70} - 85 q^{71} + 160 q^{72} + 40 q^{73} + 35 q^{74} + 55 q^{75} + 35 q^{76} + 75 q^{77} - 10 q^{78} - 330 q^{80} + 55 q^{81} - 95 q^{82} + 50 q^{83} + 115 q^{84} + 85 q^{85} + 40 q^{86} + 95 q^{87} + 165 q^{88} + 55 q^{89} + 170 q^{90} - 105 q^{91} + 155 q^{92} - 185 q^{93} + 60 q^{94} + 30 q^{95} + 135 q^{96} + 65 q^{97} + 135 q^{98} + 45 q^{99}+O(q^{100})$$ 220 * q - 20 * q^2 - 20 * q^3 - 20 * q^4 - 15 * q^5 - 20 * q^6 - 15 * q^7 - 5 * q^8 - 20 * q^9 - 20 * q^10 - 15 * q^11 - 100 * q^12 - 20 * q^13 - 10 * q^14 - 15 * q^17 + 10 * q^18 - 10 * q^19 + 5 * q^20 - 5 * q^21 + 25 * q^22 - 90 * q^23 + 15 * q^24 - 45 * q^25 + 15 * q^26 + 10 * q^27 + 30 * q^28 - 10 * q^29 + 40 * q^30 - 10 * q^31 + 20 * q^32 + 15 * q^33 + 5 * q^34 + 10 * q^35 - 210 * q^36 + 20 * q^37 + 30 * q^38 + 45 * q^40 - 5 * q^41 + 70 * q^42 + 25 * q^43 + 15 * q^44 - 100 * q^45 - 100 * q^47 + 105 * q^48 - 20 * q^49 + 60 * q^50 + 25 * q^51 + 85 * q^52 + 20 * q^53 + 60 * q^54 + 15 * q^55 - 105 * q^56 + 40 * q^57 - 225 * q^58 + 5 * q^59 - 35 * q^60 + 65 * q^62 + 85 * q^63 + 45 * q^64 + 45 * q^65 + 55 * q^66 - 105 * q^67 + 80 * q^68 + 40 * q^69 + 95 * q^70 - 85 * q^71 + 160 * q^72 + 40 * q^73 + 35 * q^74 + 55 * q^75 + 35 * q^76 + 75 * q^77 - 10 * q^78 - 330 * q^80 + 55 * q^81 - 95 * q^82 + 50 * q^83 + 115 * q^84 + 85 * q^85 + 40 * q^86 + 95 * q^87 + 165 * q^88 + 55 * q^89 + 170 * q^90 - 105 * q^91 + 155 * q^92 - 185 * q^93 + 60 * q^94 + 30 * q^95 + 135 * q^96 + 65 * q^97 + 135 * q^98 + 45 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1 −2.40956 0.618669i 0.245395 1.28641i 3.67060 + 2.01793i −1.14792 1.91893i −1.38715 + 2.94785i 1.60052 1.16284i −3.96915 3.72728i 1.19471 + 0.473019i 1.57879 + 5.33394i
6.2 −2.40901 0.618528i −0.622000 + 3.26064i 3.66812 + 2.01657i −1.51593 + 1.64376i 3.51519 7.47017i 0.637759 0.463359i −3.96312 3.72162i −7.45553 2.95185i 4.66861 3.02218i
6.3 −1.61844 0.415545i −0.244888 + 1.28375i 0.694058 + 0.381562i 1.36515 1.77098i 0.929792 1.97591i −1.25281 + 0.910218i 1.47138 + 1.38172i 1.20129 + 0.475623i −2.94533 + 2.29894i
6.4 −1.17253 0.301054i 0.595442 3.12142i −0.468423 0.257518i 2.22254 0.245569i −1.63789 + 3.48069i −0.353637 + 0.256932i 2.23663 + 2.10034i −6.59936 2.61287i −2.67992 0.381170i
6.5 −0.903097 0.231876i −0.0620903 + 0.325489i −0.990795 0.544694i −1.86049 + 1.24039i 0.131547 0.279551i −3.30296 + 2.39974i 2.12785 + 1.99818i 2.68724 + 1.06395i 1.96782 0.688789i
6.6 −0.822898 0.211284i −0.0412084 + 0.216022i −1.12009 0.615777i 0.708288 + 2.12093i 0.0795524 0.169058i 3.77476 2.74253i 2.03027 + 1.90655i 2.74436 + 1.08657i −0.134730 1.89496i
6.7 0.362552 + 0.0930876i 0.327867 1.71874i −1.62983 0.896009i −1.66215 1.49575i 0.278863 0.592613i 0.0276395 0.0200812i −1.05322 0.989036i −0.0572451 0.0226649i −0.463380 0.697013i
6.8 0.784012 + 0.201300i −0.488943 + 2.56313i −1.17846 0.647864i 1.66759 + 1.48968i −0.899294 + 1.91110i −0.523443 + 0.380304i −1.97363 1.85336i −3.54123 1.40207i 1.00754 + 1.50361i
6.9 1.56052 + 0.400675i 0.408202 2.13987i 0.522084 + 0.287018i 0.485910 + 2.18263i 1.49440 3.17576i −0.836629 + 0.607847i −1.64922 1.54872i −1.62309 0.642625i −0.116252 + 3.60075i
6.10 1.85071 + 0.475182i −0.137817 + 0.722462i 1.44673 + 0.795344i 1.17007 1.90550i −0.598361 + 1.27158i −2.81109 + 2.04237i −0.486202 0.456575i 2.28637 + 0.905238i 3.07092 2.97055i
6.11 1.93284 + 0.496268i −0.289066 + 1.51534i 1.73696 + 0.954900i −2.21785 + 0.284821i −1.31073 + 2.78545i 2.58545 1.87844i −0.0259901 0.0244063i 0.576636 + 0.228306i −4.42809 0.550137i
11.1 −1.97173 + 1.85157i 3.22832e−5 0 5.08701e-5i 0.333796 5.30554i 2.21278 0.321903i −0.000157843 0 4.05273e-5i 0.678693 2.08880i 5.71721 + 6.91092i 1.27734 2.71448i −3.76696 + 4.73182i
11.2 −1.60644 + 1.50855i 0.822947 + 1.29676i 0.179354 2.85076i −2.23260 0.124575i −3.27824 0.841710i −1.28662 + 3.95980i 1.20298 + 1.45415i 0.273000 0.580155i 3.77447 3.16786i
11.3 −1.26877 + 1.19146i −1.66333 2.62099i 0.0646335 1.02732i 0.461407 + 2.18795i 5.23318 + 1.34365i −0.283105 + 0.871308i −1.07687 1.30172i −2.82557 + 6.00465i −3.19226 2.22626i
11.4 −1.01409 + 0.952292i 1.42906 + 2.25184i −0.00406678 + 0.0646397i 0.248313 + 2.22224i −3.59359 0.922678i 1.42081 4.37282i −1.83091 2.21319i −1.75122 + 3.72153i −2.36803 2.01708i
11.5 −0.992063 + 0.931610i −0.836274 1.31776i −0.00928759 + 0.147622i −0.216983 2.22552i 2.05727 + 0.528217i 0.230945 0.710774i −1.86327 2.25231i 0.240210 0.510472i 2.28857 + 2.00571i
11.6 0.0548468 0.0515046i −0.203470 0.320618i −0.125226 + 1.99040i −0.553871 + 2.16639i −0.0276730 0.00710521i −1.02685 + 3.16032i 0.191565 + 0.231562i 1.21594 2.58401i 0.0812007 + 0.147346i
11.7 0.436521 0.409921i 1.22515 + 1.93053i −0.103065 + 1.63818i −1.37573 1.76278i 1.32617 + 0.340503i 0.159126 0.489740i 1.38994 + 1.68015i −0.948615 + 2.01591i −1.32313 0.205550i
11.8 0.444774 0.417671i −0.468839 0.738773i −0.102206 + 1.62452i 2.19457 + 0.428772i −0.517091 0.132766i 0.739388 2.27560i 1.41089 + 1.70548i 0.951363 2.02175i 1.15518 0.725902i
11.9 1.03453 0.971489i −1.68964 2.66244i 0.000881745 0.0140149i −2.20603 + 0.365290i −4.33451 1.11291i 1.16348 3.58083i 1.79652 + 2.17162i −2.95638 + 6.28262i −1.92733 + 2.52104i
See next 80 embeddings (of 220 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
125.g even 25 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 125.2.g.a 220
5.b even 2 1 625.2.g.a 220
5.c odd 4 2 625.2.h.b 440
125.g even 25 1 inner 125.2.g.a 220
125.h even 50 1 625.2.g.a 220
125.i odd 100 2 625.2.h.b 440

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.g.a 220 1.a even 1 1 trivial
125.2.g.a 220 125.g even 25 1 inner
625.2.g.a 220 5.b even 2 1
625.2.g.a 220 125.h even 50 1
625.2.h.b 440 5.c odd 4 2
625.2.h.b 440 125.i odd 100 2

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(125, [\chi])$$.