# Properties

 Label 720.2.bm.h Level $720$ Weight $2$ Character orbit 720.bm Analytic conductor $5.749$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.bm (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(i)$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q+O(q^{10})$$ 48 * q $$\operatorname{Tr}(f)(q) =$$ $$48 q + 12 q^{10} + 16 q^{14} - 4 q^{16} + 8 q^{19} + 40 q^{26} - 48 q^{31} - 28 q^{34} - 24 q^{35} - 16 q^{40} + 40 q^{44} - 4 q^{46} + 48 q^{49} + 32 q^{50} - 48 q^{56} + 32 q^{59} + 16 q^{61} + 48 q^{64} - 16 q^{65} - 40 q^{74} + 60 q^{76} - 96 q^{79} - 72 q^{80} - 16 q^{86} - 32 q^{91} + 44 q^{94} + 48 q^{95}+O(q^{100})$$ 48 * q + 12 * q^10 + 16 * q^14 - 4 * q^16 + 8 * q^19 + 40 * q^26 - 48 * q^31 - 28 * q^34 - 24 * q^35 - 16 * q^40 + 40 * q^44 - 4 * q^46 + 48 * q^49 + 32 * q^50 - 48 * q^56 + 32 * q^59 + 16 * q^61 + 48 * q^64 - 16 * q^65 - 40 * q^74 + 60 * q^76 - 96 * q^79 - 72 * q^80 - 16 * q^86 - 32 * q^91 + 44 * q^94 + 48 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.40976 0.112110i 0 1.97486 + 0.316097i 0.466917 + 2.18678i 0 −1.00010 −2.74865 0.667024i 0 −0.413083 3.13518i
109.2 −1.39033 0.258835i 0 1.86601 + 0.719729i −2.19925 + 0.404088i 0 −1.81567 −2.40807 1.48364i 0 3.16227 + 0.00742837i
109.3 −1.36038 + 0.386472i 0 1.70128 1.05150i 1.03097 1.98421i 0 3.91927 −1.90801 + 2.08794i 0 −0.635669 + 3.09773i
109.4 −1.22294 0.710230i 0 0.991146 + 1.73713i −0.607542 2.15195i 0 −2.25286 0.0216568 2.82834i 0 −0.785395 + 3.06319i
109.5 −1.20386 + 0.742113i 0 0.898535 1.78679i −0.860885 2.06370i 0 −0.707398 0.244298 + 2.81786i 0 2.56788 + 1.84553i
109.6 −1.06224 0.933621i 0 0.256702 + 1.98346i 1.24079 1.86022i 0 −1.58988 1.57912 2.34657i 0 −3.05476 + 0.817572i
109.7 −0.903247 + 1.08818i 0 −0.368289 1.96580i 2.09919 + 0.770325i 0 3.05002 2.47181 + 1.37484i 0 −2.73434 + 1.58851i
109.8 −0.750333 1.19875i 0 −0.874002 + 1.79892i −1.95942 + 1.07735i 0 1.22137 2.81225 0.302081i 0 2.76169 + 1.54047i
109.9 −0.550383 + 1.30272i 0 −1.39416 1.43399i 2.23019 + 0.162008i 0 −2.93661 2.63541 1.02695i 0 −1.43851 + 2.81615i
109.10 −0.456856 + 1.33839i 0 −1.58257 1.22290i −1.65754 1.50085i 0 2.58977 2.35972 1.55940i 0 2.76598 1.53277i
109.11 −0.382275 + 1.36157i 0 −1.70773 1.04099i −1.75308 + 1.38805i 0 −4.66030 2.07020 1.92725i 0 −1.21977 2.91756i
109.12 −0.345118 1.37146i 0 −1.76179 + 0.946629i 0.561697 + 2.16437i 0 −4.51614 1.90629 + 2.08952i 0 2.77449 1.51731i
109.13 0.345118 + 1.37146i 0 −1.76179 + 0.946629i −2.16437 0.561697i 0 4.51614 −1.90629 2.08952i 0 0.0233792 3.16219i
109.14 0.382275 1.36157i 0 −1.70773 1.04099i −1.38805 + 1.75308i 0 4.66030 −2.07020 + 1.92725i 0 1.85633 + 2.56009i
109.15 0.456856 1.33839i 0 −1.58257 1.22290i 1.50085 + 1.65754i 0 −2.58977 −2.35972 + 1.55940i 0 2.90411 1.25146i
109.16 0.550383 1.30272i 0 −1.39416 1.43399i −0.162008 2.23019i 0 2.93661 −2.63541 + 1.02695i 0 −2.99448 1.01641i
109.17 0.750333 + 1.19875i 0 −0.874002 + 1.79892i −1.07735 + 1.95942i 0 −1.22137 −2.81225 + 0.302081i 0 −3.15722 + 0.178735i
109.18 0.903247 1.08818i 0 −0.368289 1.96580i −0.770325 2.09919i 0 −3.05002 −2.47181 1.37484i 0 −2.98010 1.05783i
109.19 1.06224 + 0.933621i 0 0.256702 + 1.98346i 1.86022 1.24079i 0 1.58988 −1.57912 + 2.34657i 0 3.13443 + 0.418728i
109.20 1.20386 0.742113i 0 0.898535 1.78679i 2.06370 + 0.860885i 0 0.707398 −0.244298 2.81786i 0 3.12328 0.495122i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 469.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.bm.h 48
3.b odd 2 1 240.2.bl.a 48
5.b even 2 1 inner 720.2.bm.h 48
12.b even 2 1 960.2.bl.a 48
15.d odd 2 1 240.2.bl.a 48
16.e even 4 1 inner 720.2.bm.h 48
24.f even 2 1 1920.2.bl.b 48
24.h odd 2 1 1920.2.bl.a 48
48.i odd 4 1 240.2.bl.a 48
48.i odd 4 1 1920.2.bl.a 48
48.k even 4 1 960.2.bl.a 48
48.k even 4 1 1920.2.bl.b 48
60.h even 2 1 960.2.bl.a 48
80.q even 4 1 inner 720.2.bm.h 48
120.i odd 2 1 1920.2.bl.a 48
120.m even 2 1 1920.2.bl.b 48
240.t even 4 1 960.2.bl.a 48
240.t even 4 1 1920.2.bl.b 48
240.bm odd 4 1 240.2.bl.a 48
240.bm odd 4 1 1920.2.bl.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bl.a 48 3.b odd 2 1
240.2.bl.a 48 15.d odd 2 1
240.2.bl.a 48 48.i odd 4 1
240.2.bl.a 48 240.bm odd 4 1
720.2.bm.h 48 1.a even 1 1 trivial
720.2.bm.h 48 5.b even 2 1 inner
720.2.bm.h 48 16.e even 4 1 inner
720.2.bm.h 48 80.q even 4 1 inner
960.2.bl.a 48 12.b even 2 1
960.2.bl.a 48 48.k even 4 1
960.2.bl.a 48 60.h even 2 1
960.2.bl.a 48 240.t even 4 1
1920.2.bl.a 48 24.h odd 2 1
1920.2.bl.a 48 48.i odd 4 1
1920.2.bl.a 48 120.i odd 2 1
1920.2.bl.a 48 240.bm odd 4 1
1920.2.bl.b 48 24.f even 2 1
1920.2.bl.b 48 48.k even 4 1
1920.2.bl.b 48 120.m even 2 1
1920.2.bl.b 48 240.t even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{24} - 96 T_{7}^{22} + 3920 T_{7}^{20} - 89504 T_{7}^{18} + 1265760 T_{7}^{16} - 11615360 T_{7}^{14} + 70472192 T_{7}^{12} - 282693120 T_{7}^{10} + 737605888 T_{7}^{8} - 1208670208 T_{7}^{6} + \cdots + 115605504$$ T7^24 - 96*T7^22 + 3920*T7^20 - 89504*T7^18 + 1265760*T7^16 - 11615360*T7^14 + 70472192*T7^12 - 282693120*T7^10 + 737605888*T7^8 - 1208670208*T7^6 + 1167380480*T7^4 - 589266944*T7^2 + 115605504 $$T_{11}^{24} - 96 T_{11}^{21} + 1744 T_{11}^{20} - 2944 T_{11}^{19} + 4608 T_{11}^{18} - 79872 T_{11}^{17} + 919488 T_{11}^{16} - 2620416 T_{11}^{15} + 3964928 T_{11}^{14} - 5127168 T_{11}^{13} + 56228864 T_{11}^{12} + \cdots + 1849688064$$ T11^24 - 96*T11^21 + 1744*T11^20 - 2944*T11^19 + 4608*T11^18 - 79872*T11^17 + 919488*T11^16 - 2620416*T11^15 + 3964928*T11^14 - 5127168*T11^13 + 56228864*T11^12 - 164290560*T11^11 + 244645888*T11^10 - 93888512*T11^9 + 1012240384*T11^8 - 2842886144*T11^7 + 4023123968*T11^6 - 706543616*T11^5 + 6182731776*T11^4 - 14541914112*T11^3 + 16798711808*T11^2 - 7883194368*T11 + 1849688064 $$T_{13}^{48} + 4928 T_{13}^{44} + 8830400 T_{13}^{40} + 7352414208 T_{13}^{36} + 3017663342080 T_{13}^{32} + 618814265262080 T_{13}^{28} + \cdots + 4294967296$$ T13^48 + 4928*T13^44 + 8830400*T13^40 + 7352414208*T13^36 + 3017663342080*T13^32 + 618814265262080*T13^28 + 60670318798159872*T13^24 + 2358344545513504768*T13^20 + 15334903448544018432*T13^16 + 16161148863149768704*T13^12 + 1750941450406723584*T13^8 + 1166385342316544*T13^4 + 4294967296