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Properties

Label 547.2.f.a
Level $547$
Weight $2$
Character orbit 547.f
Analytic conductor $4.368$
Analytic rank $0$
Dimension $528$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [547,2,Mod(46,547)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(26))
 
chi = DirichletCharacter(H, H._module([14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("547.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.f (of order \(13\), degree \(12\), 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: \(4.36781699056\)
Analytic rank: \(0\)
Dimension: \(528\)
Relative dimension: \(44\) over \(\Q(\zeta_{13})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{13}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 528 q - 8 q^{2} - 46 q^{3} - 50 q^{4} - 9 q^{5} - 21 q^{6} - 3 q^{7} + 14 q^{8} + 466 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 528 q - 8 q^{2} - 46 q^{3} - 50 q^{4} - 9 q^{5} - 21 q^{6} - 3 q^{7} + 14 q^{8} + 466 q^{9} + 7 q^{10} - 16 q^{11} - 53 q^{12} - 12 q^{13} - 38 q^{15} - 38 q^{16} + 7 q^{17} - 23 q^{18} + 5 q^{19} + 45 q^{20} + 4 q^{21} + 43 q^{22} + 2 q^{23} - 51 q^{24} - 69 q^{25} - 4 q^{26} - 142 q^{27} - 2 q^{28} - 72 q^{29} - 50 q^{30} + 25 q^{31} + 66 q^{32} - 4 q^{34} + 33 q^{35} - 55 q^{36} + 21 q^{37} + 40 q^{38} - 142 q^{39} + 78 q^{40} - 140 q^{41} - 248 q^{42} + 23 q^{43} + 64 q^{44} - 163 q^{45} + 14 q^{46} + 37 q^{47} - 35 q^{48} + q^{49} + 70 q^{50} - 72 q^{51} - 74 q^{52} - 17 q^{53} - 76 q^{54} + 6 q^{55} - 38 q^{56} + 5 q^{57} + 29 q^{58} + 30 q^{59} - 108 q^{60} - 94 q^{61} - 75 q^{62} - 77 q^{63} - 252 q^{64} - 122 q^{65} + 79 q^{66} + 43 q^{67} - 193 q^{68} + 14 q^{69} - 267 q^{70} + 61 q^{71} + 49 q^{72} + 11 q^{73} + 85 q^{74} + 120 q^{75} - 23 q^{76} + 31 q^{77} - 90 q^{78} + 65 q^{79} + 58 q^{80} + 280 q^{81} + 10 q^{82} - 8 q^{83} - 136 q^{84} + 33 q^{85} + 91 q^{86} - 70 q^{87} - 198 q^{88} + 41 q^{89} - 68 q^{90} + 38 q^{91} + 43 q^{92} + 132 q^{93} + 8 q^{94} - 49 q^{95} + 262 q^{96} - 14 q^{97} + 41 q^{98} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −0.335888 + 2.76629i 1.13158 −5.59764 − 1.37970i −0.153172 − 0.403882i −0.380085 + 3.13028i −0.393363 − 3.23964i 3.72053 − 9.81022i −1.71952 1.16870 − 0.288060i
46.2 −0.311246 + 2.56334i −2.48877 −4.53197 − 1.11703i 0.771833 + 2.03516i 0.774620 − 6.37957i −0.224653 − 1.85018i 2.44259 − 6.44059i 3.19397 −5.45703 + 1.34504i
46.3 −0.305966 + 2.51985i 0.544519 −4.31417 − 1.06335i 0.299331 + 0.789270i −0.166604 + 1.37211i 0.466827 + 3.84467i 2.19924 − 5.79891i −2.70350 −2.08043 + 0.512780i
46.4 −0.304286 + 2.50602i −1.95070 −4.24565 − 1.04646i −1.15766 − 3.05250i 0.593571 − 4.88850i 0.109988 + 0.905836i 2.12398 − 5.60049i 0.805247 8.00187 − 1.97228i
46.5 −0.268734 + 2.21322i 2.12681 −2.88424 − 0.710902i −0.947553 − 2.49849i −0.571546 + 4.70710i −0.292661 − 2.41028i 0.767309 − 2.02323i 1.52333 5.78435 − 1.42572i
46.6 −0.255736 + 2.10617i 1.99966 −2.42868 − 0.598615i 0.984637 + 2.59628i −0.511385 + 4.21164i 0.216641 + 1.78420i 0.377195 − 0.994582i 0.998658 −5.72001 + 1.40986i
46.7 −0.245329 + 2.02046i 0.136449 −2.08020 − 0.512724i 1.38996 + 3.66502i −0.0334750 + 0.275691i −0.418952 − 3.45038i 0.102818 − 0.271108i −2.98138 −7.74605 + 1.90923i
46.8 −0.236626 + 1.94879i −2.82258 −1.79991 − 0.443638i 0.468139 + 1.23438i 0.667896 − 5.50062i 0.189324 + 1.55922i −0.101791 + 0.268400i 4.96697 −2.51632 + 0.620218i
46.9 −0.234924 + 1.93478i 0.630029 −1.74629 − 0.430421i −1.10146 − 2.90432i −0.148009 + 1.21897i −0.214681 − 1.76806i −0.139228 + 0.367113i −2.60306 5.87797 − 1.44879i
46.10 −0.192434 + 1.58483i 2.35170 −0.532783 − 0.131319i −0.750458 − 1.97880i −0.452547 + 3.72706i 0.413115 + 3.40231i −0.821590 + 2.16636i 2.53051 3.28047 − 0.808564i
46.11 −0.188009 + 1.54840i −0.908881 −0.420298 − 0.103594i −0.0244243 − 0.0644017i 0.170878 − 1.40731i −0.367947 − 3.03032i −0.866778 + 2.28551i −2.17394 0.104311 − 0.0257104i
46.12 −0.167985 + 1.38348i 2.55202 0.0560731 + 0.0138208i 0.714436 + 1.88381i −0.428702 + 3.53068i −0.0490013 − 0.403562i −1.01693 + 2.68142i 3.51279 −2.72624 + 0.671958i
46.13 −0.166856 + 1.37418i −1.48276 0.0813480 + 0.0200505i −0.394072 − 1.03908i 0.247408 − 2.03759i −0.324019 − 2.66853i −1.02287 + 2.69708i −0.801408 1.49364 − 0.368149i
46.14 −0.149053 + 1.22756i 0.296875 0.457198 + 0.112689i −0.175522 − 0.462814i −0.0442500 + 0.364432i 0.391638 + 3.22543i −1.08347 + 2.85688i −2.91187 0.594294 − 0.146480i
46.15 −0.142855 + 1.17652i −1.73902 0.578096 + 0.142488i 1.43759 + 3.79062i 0.248428 − 2.04599i 0.544189 + 4.48180i −1.09075 + 2.87607i 0.0242007 −4.66510 + 1.14984i
46.16 −0.137944 + 1.13607i −2.73564 0.670252 + 0.165202i −1.11149 − 2.93075i 0.377365 − 3.10788i 0.373963 + 3.07987i −1.09177 + 2.87876i 4.48371 3.48287 − 0.858450i
46.17 −0.0945775 + 0.778916i 3.19456 1.34412 + 0.331295i −0.702506 − 1.85236i −0.302134 + 2.48830i −0.187976 − 1.54812i −0.941647 + 2.48292i 7.20523 1.50927 − 0.372002i
46.18 −0.0547827 + 0.451176i 1.64947 1.74132 + 0.429198i 0.558494 + 1.47263i −0.0903624 + 0.744201i −0.347117 − 2.85877i −0.611367 + 1.61204i −0.279256 −0.695011 + 0.171305i
46.19 −0.0443771 + 0.365478i −0.533473 1.81028 + 0.446194i 0.569048 + 1.50046i 0.0236740 − 0.194973i 0.225842 + 1.85997i −0.504513 + 1.33029i −2.71541 −0.573637 + 0.141389i
46.20 −0.0420575 + 0.346375i −3.30487 1.82368 + 0.449496i 1.10374 + 2.91033i 0.138995 − 1.14473i −0.459333 − 3.78294i −0.479850 + 1.26526i 7.92218 −1.05449 + 0.259908i
See next 80 embeddings (of 528 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
547.f even 13 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.2.f.a ✓ 528
547.f even 13 1 inner 547.2.f.a ✓ 528
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.f.a ✓ 528 1.a even 1 1 trivial
547.2.f.a ✓ 528 547.f even 13 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(547, [\chi])\).