# Properties

 Label 460.2.q.a Level $460$ Weight $2$ Character orbit 460.q Analytic conductor $3.673$ Analytic rank $0$ Dimension $480$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$460 = 2^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 460.q (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.67311849298$$ Analytic rank: $$0$$ Dimension: $$480$$ Relative dimension: $$48$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9}+O(q^{10})$$ 480 * q - 4 * q^2 + 2 * q^6 + 2 * q^8 + 48 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$480 q - 4 q^{2} + 2 q^{6} + 2 q^{8} + 48 q^{9} + 6 q^{12} - 24 q^{16} + 26 q^{18} + 4 q^{24} + 48 q^{25} - 14 q^{26} + 40 q^{29} + 16 q^{32} - 22 q^{34} - 152 q^{36} - 110 q^{38} - 88 q^{40} - 8 q^{41} - 22 q^{44} - 132 q^{46} - 190 q^{48} - 56 q^{49} - 18 q^{50} - 50 q^{52} - 90 q^{54} - 110 q^{56} - 136 q^{58} - 22 q^{60} + 30 q^{62} - 30 q^{64} + 4 q^{69} + 110 q^{72} + 22 q^{74} - 176 q^{77} + 66 q^{78} - 8 q^{81} + 154 q^{82} + 308 q^{84} + 32 q^{85} + 176 q^{88} - 88 q^{89} + 134 q^{92} - 448 q^{93} + 204 q^{94} + 6 q^{96} - 88 q^{97} + 132 q^{98}+O(q^{100})$$ 480 * q - 4 * q^2 + 2 * q^6 + 2 * q^8 + 48 * q^9 + 6 * q^12 - 24 * q^16 + 26 * q^18 + 4 * q^24 + 48 * q^25 - 14 * q^26 + 40 * q^29 + 16 * q^32 - 22 * q^34 - 152 * q^36 - 110 * q^38 - 88 * q^40 - 8 * q^41 - 22 * q^44 - 132 * q^46 - 190 * q^48 - 56 * q^49 - 18 * q^50 - 50 * q^52 - 90 * q^54 - 110 * q^56 - 136 * q^58 - 22 * q^60 + 30 * q^62 - 30 * q^64 + 4 * q^69 + 110 * q^72 + 22 * q^74 - 176 * q^77 + 66 * q^78 - 8 * q^81 + 154 * q^82 + 308 * q^84 + 32 * q^85 + 176 * q^88 - 88 * q^89 + 134 * q^92 - 448 * q^93 + 204 * q^94 + 6 * q^96 - 88 * q^97 + 132 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −1.41401 0.0241273i 0.0682998 + 0.00982003i 1.99884 + 0.0682323i 0.281733 + 0.959493i −0.0963395 0.0155335i 0.126885 + 0.146433i −2.82472 0.144707i −2.87391 0.843856i −0.375222 1.36353i
11.2 −1.41004 + 0.108605i −2.31335 0.332609i 1.97641 0.306274i −0.281733 0.959493i 3.29803 + 0.217751i −1.52622 1.76136i −2.75355 + 0.646505i 2.36248 + 0.693685i 0.501459 + 1.32232i
11.3 −1.39626 0.224613i 1.77069 + 0.254587i 1.89910 + 0.627237i 0.281733 + 0.959493i −2.41517 0.753191i −0.375831 0.433732i −2.51075 1.30235i 0.192056 + 0.0563926i −0.177858 1.40298i
11.4 −1.31872 0.510869i 0.994266 + 0.142954i 1.47803 + 1.34738i −0.281733 0.959493i −1.23812 0.696455i −3.13881 3.62238i −1.26076 2.53189i −1.91035 0.560929i −0.118649 + 1.40923i
11.5 −1.30815 0.537340i −0.501133 0.0720521i 1.42253 + 1.40585i −0.281733 0.959493i 0.616843 + 0.363534i 1.51820 + 1.75210i −1.10547 2.60345i −2.63254 0.772982i −0.147024 + 1.40655i
11.6 −1.27312 + 0.615774i −0.608389 0.0874731i 1.24165 1.56790i −0.281733 0.959493i 0.828413 0.263266i 0.613272 + 0.707753i −0.615286 + 2.76069i −2.51599 0.738762i 0.949508 + 1.04806i
11.7 −1.27090 + 0.620337i −1.04773 0.150641i 1.23036 1.57677i 0.281733 + 0.959493i 1.42501 0.458497i 3.09996 + 3.57755i −0.585539 + 2.76715i −1.80343 0.529535i −0.953262 1.04465i
11.8 −1.19302 + 0.759408i 3.06496 + 0.440674i 0.846600 1.81198i 0.281733 + 0.959493i −3.99121 + 1.80182i 1.05960 + 1.22284i 0.366018 + 2.80464i 6.32130 + 1.85610i −1.06476 0.930746i
11.9 −1.19089 0.762752i −2.15085 0.309246i 0.836418 + 1.81670i 0.281733 + 0.959493i 2.32554 + 2.00885i −2.11824 2.44458i 0.389615 2.80146i 1.65206 + 0.485089i 0.396344 1.35754i
11.10 −1.18638 + 0.769741i −3.06496 0.440674i 0.814997 1.82641i 0.281733 + 0.959493i 3.97541 1.83642i −1.05960 1.22284i 0.438968 + 2.79416i 6.32130 + 1.85610i −1.07280 0.921463i
11.11 −1.18308 0.774803i 3.11182 + 0.447412i 0.799362 + 1.83331i −0.281733 0.959493i −3.33488 2.94037i 1.26361 + 1.45829i 0.474742 2.78830i 6.60476 + 1.93933i −0.410105 + 1.35345i
11.12 −1.09223 + 0.898352i 1.04773 + 0.150641i 0.385926 1.96241i 0.281733 + 0.959493i −1.27969 + 0.776697i −3.09996 3.57755i 1.34142 + 2.49010i −1.80343 0.529535i −1.16968 0.794890i
11.13 −1.08900 + 0.902265i 0.608389 + 0.0874731i 0.371836 1.96513i −0.281733 0.959493i −0.741459 + 0.453670i −0.613272 0.707753i 1.36814 + 2.47552i −2.51599 0.738762i 1.17252 + 0.790689i
11.14 −0.745947 1.20148i 0.934803 + 0.134404i −0.887125 + 1.79249i 0.281733 + 0.959493i −0.535829 1.22341i 2.39561 + 2.76468i 2.81539 0.271235i −2.02269 0.593915i 0.942657 1.05423i
11.15 −0.684541 + 1.23750i 2.31335 + 0.332609i −1.06281 1.69424i −0.281733 0.959493i −1.99519 + 2.63508i 1.52622 + 1.76136i 2.82415 0.155447i 2.36248 + 0.693685i 1.38023 + 0.308169i
11.16 −0.636057 1.26310i −3.26795 0.469861i −1.19086 + 1.60681i −0.281733 0.959493i 1.48512 + 4.42662i 0.616457 + 0.711429i 2.78703 + 0.482157i 7.58027 + 2.22577i −1.03274 + 0.966150i
11.17 −0.565453 + 1.29625i −0.0682998 0.00982003i −1.36053 1.46594i 0.281733 + 0.959493i 0.0513495 0.0829808i −0.126885 0.146433i 2.66953 0.934662i −2.87391 0.843856i −1.40305 0.177353i
11.18 −0.520455 1.31496i −1.91127 0.274799i −1.45825 + 1.36876i 0.281733 + 0.959493i 0.633379 + 2.65627i 1.03497 + 1.19442i 2.55882 + 1.20517i 0.698960 + 0.205233i 1.11507 0.869841i
11.19 −0.497467 1.32383i 3.04838 + 0.438291i −1.50505 + 1.31712i 0.281733 + 0.959493i −0.936246 4.25357i −1.74188 2.01024i 2.49236 + 1.33721i 6.22203 + 1.82695i 1.13005 0.850282i
11.20 −0.375713 + 1.36339i −1.77069 0.254587i −1.71768 1.02449i 0.281733 + 0.959493i 1.01237 2.31850i 0.375831 + 0.433732i 2.04214 1.95696i 0.192056 + 0.0563926i −1.41402 + 0.0236179i
See next 80 embeddings (of 480 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 451.48 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
23.d odd 22 1 inner
92.h even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 460.2.q.a 480
4.b odd 2 1 inner 460.2.q.a 480
23.d odd 22 1 inner 460.2.q.a 480
92.h even 22 1 inner 460.2.q.a 480

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.q.a 480 1.a even 1 1 trivial
460.2.q.a 480 4.b odd 2 1 inner
460.2.q.a 480 23.d odd 22 1 inner
460.2.q.a 480 92.h even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(460, [\chi])$$.