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Properties

Label 639.2.q.a
Level $639$
Weight $2$
Character orbit 639.q
Analytic conductor $5.102$
Analytic rank $0$
Dimension $560$
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] = [639,2,Mod(25,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 24]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.25");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.q (of order \(15\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(70\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 3 q^{2} - 8 q^{3} + 65 q^{4} - q^{5} - q^{6} - 3 q^{7} - 32 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q - 3 q^{2} - 8 q^{3} + 65 q^{4} - q^{5} - q^{6} - 3 q^{7} - 32 q^{8} - 20 q^{9} - 20 q^{10} - 17 q^{11} - 32 q^{12} - 3 q^{13} - 9 q^{14} - 40 q^{15} + 65 q^{16} - 8 q^{17} - 8 q^{18} - 12 q^{19} - 80 q^{20} + 12 q^{21} - 11 q^{22} + 44 q^{23} - 19 q^{24} + 65 q^{25} - 128 q^{26} + 10 q^{27} - 36 q^{28} + 9 q^{29} + 26 q^{30} - 12 q^{31} - 16 q^{32} - 26 q^{33} - 4 q^{34} + 60 q^{35} - 49 q^{36} - 44 q^{37} - 13 q^{38} + 78 q^{39} + 27 q^{40} + 24 q^{41} + 4 q^{42} - 3 q^{43} - 28 q^{44} + 2 q^{45} - 35 q^{47} + 138 q^{48} + 55 q^{49} + 23 q^{50} - 28 q^{51} - 7 q^{52} - 20 q^{53} + 23 q^{54} - 44 q^{55} - 55 q^{56} - 22 q^{57} + 7 q^{58} + 21 q^{59} + 133 q^{60} - 3 q^{61} + 12 q^{62} - 106 q^{63} - 164 q^{64} - 11 q^{65} - 102 q^{66} - 12 q^{67} + 27 q^{68} + 44 q^{69} - 10 q^{70} - 48 q^{71} + 262 q^{72} - 12 q^{73} + 14 q^{74} - 9 q^{75} - 3 q^{76} - 51 q^{77} + 22 q^{78} - 21 q^{79} - 200 q^{80} - 136 q^{81} - 36 q^{82} - 25 q^{83} - 54 q^{84} + 19 q^{85} - 65 q^{86} + 18 q^{87} + 37 q^{88} + 44 q^{89} + 93 q^{90} + 36 q^{91} + 49 q^{92} - 84 q^{93} + 98 q^{94} + 43 q^{95} + 60 q^{96} - 2 q^{97} - 26 q^{98} - 57 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −0.284962 − 2.71123i −1.12502 + 1.31694i −5.31327 + 1.12937i 2.66233 − 2.95681i 3.89111 + 2.67491i 0.344607 + 3.27872i 2.89120 + 8.89820i −0.468662 − 2.96317i −8.77526 − 6.37560i
25.2 −0.281613 − 2.67937i 0.999232 + 1.41476i −5.14341 + 1.09327i −0.258148 + 0.286703i 3.50926 − 3.07572i −0.135890 − 1.29291i 2.71265 + 8.34868i −1.00307 + 2.82734i 0.840881 + 0.610936i
25.3 −0.274998 − 2.61643i −0.725700 − 1.57269i −4.81381 + 1.02321i 0.988067 − 1.09736i −3.91528 + 2.33123i 0.297511 + 2.83063i 2.37499 + 7.30948i −1.94672 + 2.28260i −3.14289 − 2.28344i
25.4 −0.273354 − 2.60079i 1.42146 − 0.989673i −4.73308 + 1.00605i −1.90180 + 2.11216i −2.96249 − 3.42638i 0.412578 + 3.92541i 2.29409 + 7.06048i 1.04110 − 2.81356i 6.01315 + 4.36881i
25.5 −0.262595 − 2.49843i 1.72626 − 0.141478i −4.21688 + 0.896326i 0.483230 − 0.536681i −0.806780 − 4.27579i 0.0193685 + 0.184279i 1.79412 + 5.52173i 2.95997 − 0.488456i −1.46775 − 1.06638i
25.6 −0.258913 − 2.46339i −1.72789 − 0.120037i −4.04497 + 0.859786i −1.60764 + 1.78546i 0.151673 + 4.28754i 0.136363 + 1.29740i 1.63444 + 5.03029i 2.97118 + 0.414822i 4.81454 + 3.49797i
25.7 −0.253974 − 2.41640i 0.640653 − 1.60921i −3.81819 + 0.811581i 1.13296 − 1.25828i −4.05121 − 1.13938i −0.291369 − 2.77219i 1.42918 + 4.39856i −2.17913 − 2.06189i −3.32824 − 2.41811i
25.8 −0.250077 − 2.37932i −1.21247 + 1.23689i −3.64235 + 0.774205i −0.500973 + 0.556387i 3.24618 + 2.57555i −0.341604 − 3.25015i 1.27435 + 3.92204i −0.0598106 − 2.99940i 1.44911 + 1.05284i
25.9 −0.234677 − 2.23280i −1.35820 − 1.07485i −2.97402 + 0.632149i 1.62948 − 1.80972i −2.08119 + 3.28482i −0.325636 − 3.09822i 0.721848 + 2.22162i 0.689397 + 2.91971i −4.42315 − 3.21360i
25.10 −0.211631 − 2.01353i 1.47571 + 0.906803i −2.05324 + 0.436429i 2.72701 − 3.02866i 1.51357 − 3.16329i −0.109006 − 1.03712i 0.0620060 + 0.190835i 1.35541 + 2.67635i −6.67542 − 4.84998i
25.11 −0.211408 − 2.01142i 0.115035 + 1.72823i −2.04481 + 0.434638i 0.453416 − 0.503570i 3.45186 − 0.596745i 0.289376 + 2.75323i 0.0565559 + 0.174061i −2.97353 + 0.397613i −1.10874 − 0.805550i
25.12 −0.211050 − 2.00801i 0.170730 + 1.72362i −2.03125 + 0.431755i −2.67315 + 2.96883i 3.42500 − 0.706596i −0.122036 − 1.16110i 0.0478099 + 0.147144i −2.94170 + 0.588546i 6.52560 + 4.74113i
25.13 −0.187950 − 1.78823i 1.61438 − 0.627520i −1.20613 + 0.256371i 0.522105 − 0.579856i −1.42557 − 2.76893i −0.270361 − 2.57231i −0.426131 − 1.31150i 2.21244 − 2.02611i −1.13504 − 0.824657i
25.14 −0.182793 − 1.73916i 1.63279 + 0.577924i −1.03496 + 0.219987i −2.35482 + 2.61529i 0.706638 − 2.94532i 0.159861 + 1.52097i −0.509004 − 1.56655i 2.33201 + 1.88726i 4.97884 + 3.61734i
25.15 −0.180140 − 1.71391i 1.24094 − 1.20833i −0.948758 + 0.201665i 2.56044 − 2.84366i −2.29451 − 1.90921i 0.515054 + 4.90041i −0.548547 − 1.68825i 0.0798867 − 2.99894i −5.33502 − 3.87612i
25.16 −0.180103 − 1.71356i −1.15017 − 1.29503i −0.947559 + 0.201410i −2.00217 + 2.22364i −2.01197 + 2.20412i −0.378221 − 3.59853i −0.549087 − 1.68991i −0.354227 + 2.97901i 4.17093 + 3.03036i
25.17 −0.176758 − 1.68174i 0.293181 − 1.70706i −0.840714 + 0.178699i −1.05588 + 1.17268i −2.92265 − 0.191319i 0.0838549 + 0.797826i −0.595969 − 1.83421i −2.82809 − 1.00095i 2.15878 + 1.56844i
25.18 −0.171985 − 1.63633i −1.00235 + 1.41255i −0.691691 + 0.147023i −0.143378 + 0.159238i 2.48378 + 1.39723i 0.376844 + 3.58543i −0.657337 − 2.02308i −0.990602 − 2.83173i 0.285224 + 0.207227i
25.19 −0.167099 − 1.58985i −1.70372 − 0.311976i −0.543390 + 0.115501i 0.798931 − 0.887303i −0.211303 + 2.76079i 0.305799 + 2.90948i −0.713562 − 2.19612i 2.80534 + 1.06304i −1.54418 − 1.12191i
25.20 −0.147942 − 1.40758i −1.48500 + 0.891500i −0.00309440 0.000657735i 1.65391 − 1.83686i 1.47455 + 1.95836i −0.326147 − 3.10308i −0.873339 − 2.68786i 1.41046 − 2.64776i −2.83020 − 2.05626i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
71.c even 5 1 inner
639.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.q.a ✓ 560
9.c even 3 1 inner 639.2.q.a ✓ 560
71.c even 5 1 inner 639.2.q.a ✓ 560
639.q even 15 1 inner 639.2.q.a ✓ 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
639.2.q.a ✓ 560 1.a even 1 1 trivial
639.2.q.a ✓ 560 9.c even 3 1 inner
639.2.q.a ✓ 560 71.c even 5 1 inner
639.2.q.a ✓ 560 639.q even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(639, [\chi])\).