Properties

Label 880.2.cg.a
Level $880$
Weight $2$
Character orbit 880.cg
Analytic conductor $7.027$
Analytic rank $0$
Dimension $1120$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(237,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 15, 5, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cg (of order \(20\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(1120\)
Relative dimension: \(140\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 10 q^{2} - 12 q^{3} - 6 q^{5} - 20 q^{6} - 10 q^{8} - 264 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 10 q^{2} - 12 q^{3} - 6 q^{5} - 20 q^{6} - 10 q^{8} - 264 q^{9} - 16 q^{11} + 12 q^{12} - 20 q^{13} - 12 q^{15} - 12 q^{16} - 20 q^{17} - 10 q^{18} - 6 q^{20} + 6 q^{22} - 44 q^{26} + 12 q^{27} - 10 q^{28} - 20 q^{30} - 24 q^{31} - 16 q^{33} - 32 q^{34} - 10 q^{35} + 20 q^{36} - 12 q^{37} + 30 q^{38} - 80 q^{40} + 42 q^{42} + 80 q^{44} + 16 q^{45} - 20 q^{46} - 12 q^{47} + 6 q^{48} + 60 q^{50} - 20 q^{51} - 50 q^{52} + 16 q^{56} + 54 q^{58} - 16 q^{59} - 40 q^{60} - 20 q^{61} - 160 q^{62} - 20 q^{63} - 24 q^{66} - 10 q^{68} - 36 q^{69} + 46 q^{70} + 140 q^{72} - 140 q^{74} + 54 q^{75} - 132 q^{78} - 58 q^{80} - 240 q^{81} + 6 q^{82} - 220 q^{83} - 10 q^{85} - 20 q^{86} - 98 q^{88} + 100 q^{90} + 4 q^{91} - 116 q^{92} + 12 q^{93} - 20 q^{95} - 20 q^{96} - 12 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
237.1 −1.41407 0.0198985i −1.45345 1.05599i 1.99921 + 0.0562758i −0.196156 + 2.22745i 2.03427 + 1.52217i 1.02883 + 0.162951i −2.82591 0.119359i 0.0703399 + 0.216484i 0.321702 3.14587i
237.2 −1.41096 + 0.0958386i 1.41736 + 1.02977i 1.98163 0.270449i 1.47958 + 1.67656i −2.09853 1.31713i 4.63365 + 0.733898i −2.77009 + 0.571511i 0.0214232 + 0.0659338i −2.24830 2.22376i
237.3 −1.40836 0.128540i −0.487010 0.353834i 1.96695 + 0.362061i 2.23512 0.0651494i 0.640404 + 0.560925i −5.04818 0.799553i −2.72364 0.762745i −0.815070 2.50853i −3.15623 0.195548i
237.4 −1.40170 + 0.187685i −0.902146 0.655448i 1.92955 0.526158i −1.09161 1.95151i 1.38756 + 0.749424i −2.96306 0.469302i −2.60590 + 1.09967i −0.542795 1.67055i 1.89638 + 2.53056i
237.5 −1.40154 0.188892i 1.67325 + 1.21569i 1.92864 + 0.529481i 1.55118 1.61054i −2.11550 2.01990i 0.977944 + 0.154891i −2.60305 1.10640i 0.394824 + 1.21514i −2.47826 + 1.96423i
237.6 −1.39666 0.222097i −0.422825 0.307201i 1.90135 + 0.620391i −0.105986 2.23355i 0.522317 + 0.522965i 4.02525 + 0.637538i −2.51775 1.28876i −0.842642 2.59339i −0.348040 + 3.14307i
237.7 −1.39433 + 0.236295i −2.08493 1.51479i 1.88833 0.658947i −0.842305 + 2.07136i 3.26503 + 1.61947i −0.0369532 0.00585281i −2.47725 + 1.36499i 1.12529 + 3.46330i 0.685003 3.08719i
237.8 −1.38816 + 0.270193i −0.172111 0.125046i 1.85399 0.750144i 1.95249 + 1.08986i 0.272704 + 0.127081i 0.496601 + 0.0786539i −2.37096 + 1.54226i −0.913065 2.81013i −3.00484 0.985354i
237.9 −1.38601 0.281012i 2.10621 + 1.53025i 1.84206 + 0.778972i −1.47245 + 1.68282i −2.48922 2.71282i 2.08045 + 0.329510i −2.33423 1.59731i 1.16740 + 3.59290i 2.51373 1.91864i
237.10 −1.38334 0.293885i 2.51165 + 1.82482i 1.82726 + 0.813087i −1.94695 1.09971i −2.93818 3.26249i −2.63309 0.417041i −2.28877 1.66178i 2.05137 + 6.31348i 2.37011 + 2.09346i
237.11 −1.36973 + 0.351921i −2.62600 1.90790i 1.75230 0.964072i 1.57452 1.58773i 4.26833 + 1.68916i 0.0803937 + 0.0127331i −2.06090 + 1.93719i 2.32873 + 7.16710i −1.59790 + 2.72887i
237.12 −1.36711 + 0.361964i −1.06810 0.776017i 1.73796 0.989688i 1.47475 1.68081i 1.74109 + 0.674286i 2.78066 + 0.440414i −2.01775 + 1.98209i −0.388425 1.19545i −1.40775 + 2.83165i
237.13 −1.36287 + 0.377621i 0.965913 + 0.701777i 1.71481 1.02929i −2.19119 0.445737i −1.58141 0.591679i −0.698960 0.110704i −1.94837 + 2.05033i −0.486554 1.49746i 3.15462 0.219960i
237.14 −1.35870 0.392342i −2.14022 1.55496i 1.69213 + 1.06615i 2.23480 + 0.0754042i 2.29784 + 2.95242i 0.486486 + 0.0770518i −1.88081 2.11248i 1.23559 + 3.80274i −3.00683 0.979257i
237.15 −1.35823 + 0.393971i 0.454023 + 0.329867i 1.68957 1.07021i −2.18674 0.467100i −0.746625 0.269163i 1.07630 + 0.170470i −1.87320 + 2.11923i −0.829726 2.55364i 3.15411 0.227083i
237.16 −1.35284 0.412093i 0.989602 + 0.718988i 1.66036 + 1.11499i 0.266189 2.22017i −1.04249 1.38048i −3.25032 0.514800i −1.78672 2.19263i −0.464682 1.43014i −1.27503 + 2.89384i
237.17 −1.33558 0.465005i −0.657131 0.477434i 1.56754 + 1.24210i −2.22153 + 0.254585i 0.655641 + 0.943219i 1.88906 + 0.299198i −1.51599 2.38784i −0.723173 2.22570i 3.08541 + 0.693003i
237.18 −1.32114 + 0.504581i 1.26975 + 0.922525i 1.49080 1.33324i −0.0549814 + 2.23539i −2.14299 0.578089i −4.67509 0.740462i −1.29681 + 2.51362i −0.165847 0.510425i −1.05530 2.98100i
237.19 −1.31878 + 0.510715i 2.48263 + 1.80374i 1.47834 1.34704i 2.01503 + 0.969365i −4.19523 1.11081i −1.58504 0.251045i −1.26165 + 2.53145i 1.98294 + 6.10288i −3.15244 0.249273i
237.20 −1.31197 0.527959i −0.695819 0.505542i 1.44252 + 1.38533i −1.93533 + 1.12004i 0.645987 + 1.03062i −3.08510 0.488632i −1.16114 2.57910i −0.698459 2.14964i 3.13043 0.447677i
See next 80 embeddings (of 1120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 237.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
80.t odd 4 1 inner
880.cg even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.cg.a 1120
5.c odd 4 1 880.2.cy.a yes 1120
11.d odd 10 1 inner 880.2.cg.a 1120
16.e even 4 1 880.2.cy.a yes 1120
55.l even 20 1 880.2.cy.a yes 1120
80.t odd 4 1 inner 880.2.cg.a 1120
176.u odd 20 1 880.2.cy.a yes 1120
880.cg even 20 1 inner 880.2.cg.a 1120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.cg.a 1120 1.a even 1 1 trivial
880.2.cg.a 1120 11.d odd 10 1 inner
880.2.cg.a 1120 80.t odd 4 1 inner
880.2.cg.a 1120 880.cg even 20 1 inner
880.2.cy.a yes 1120 5.c odd 4 1
880.2.cy.a yes 1120 16.e even 4 1
880.2.cy.a yes 1120 55.l even 20 1
880.2.cy.a yes 1120 176.u odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(880, [\chi])\).