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Properties

Label 639.2.r.a
Level $639$
Weight $2$
Character orbit 639.r
Analytic conductor $5.102$
Analytic rank $0$
Dimension $840$
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(103,639)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(639, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([14, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("639.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 639 = 3^{2} \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 639.r (of order \(21\), 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: \(5.10244068916\)
Analytic rank: \(0\)
Dimension: \(840\)
Relative dimension: \(70\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 5 q^{2} - 5 q^{3} + 63 q^{4} - 10 q^{5} - 20 q^{6} - 5 q^{7} - 40 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 840 q - 5 q^{2} - 5 q^{3} + 63 q^{4} - 10 q^{5} - 20 q^{6} - 5 q^{7} - 40 q^{8} + 3 q^{9} - 9 q^{11} + 9 q^{12} - 5 q^{13} + 38 q^{14} + 7 q^{15} + 63 q^{16} - 156 q^{17} + 24 q^{18} - 20 q^{19} + 54 q^{20} - 34 q^{21} - 13 q^{22} - 13 q^{23} + 27 q^{24} - 378 q^{25} - 24 q^{26} - 80 q^{27} - 16 q^{28} - 13 q^{29} + 32 q^{30} - 20 q^{31} - 13 q^{32} - 20 q^{33} - 15 q^{34} + 146 q^{35} + 28 q^{36} - 32 q^{37} - 15 q^{38} + 12 q^{39} - 30 q^{40} - 13 q^{41} + 21 q^{42} - 5 q^{43} - 36 q^{44} + 33 q^{45} - 56 q^{46} - 17 q^{47} + 9 q^{48} + 53 q^{49} + 14 q^{50} - 14 q^{51} + q^{52} - 68 q^{53} - 50 q^{54} + 18 q^{55} + 49 q^{56} - 94 q^{57} + 5 q^{58} - 56 q^{59} + 27 q^{60} - 5 q^{61} - 32 q^{62} + 62 q^{63} - 192 q^{64} + 43 q^{65} - 194 q^{66} + q^{67} - 23 q^{68} + 101 q^{69} + 126 q^{70} - 56 q^{71} - 234 q^{72} - 20 q^{73} + 22 q^{74} - 2 q^{75} - 12 q^{76} - 27 q^{77} - 32 q^{78} + 7 q^{79} + 122 q^{80} + 123 q^{81} - 64 q^{82} + 23 q^{83} - 133 q^{84} - 60 q^{85} - 103 q^{86} + 23 q^{87} - 32 q^{88} + 46 q^{89} - 57 q^{90} - 72 q^{91} - 47 q^{92} + 9 q^{93} - 19 q^{94} - 42 q^{95} + 50 q^{96} + q^{97} - 138 q^{98} - 10 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1 −2.68213 + 0.827327i 0.00247450 + 1.73205i 4.85686 − 3.31135i −1.26735 + 2.19511i −1.43961 − 4.64353i −2.56653 + 2.38139i −6.78710 + 8.51075i −2.99999 + 0.00857192i 1.58311 − 6.93607i
103.2 −2.64268 + 0.815158i −0.938218 − 1.45593i 4.66679 − 3.18176i 0.864947 − 1.49813i 3.66623 + 3.08277i 0.485477 − 0.450457i −6.29061 + 7.88817i −1.23949 + 2.73197i −1.06456 + 4.66415i
103.3 −2.50362 + 0.772265i 1.72530 − 0.152730i 4.01926 − 2.74028i −1.04580 + 1.81137i −4.20156 + 1.71477i −0.376828 + 0.349645i −4.67936 + 5.86773i 2.95335 − 0.527010i 1.21942 − 5.34263i
103.4 −2.39019 + 0.737275i 0.418257 + 1.68079i 3.51694 − 2.39781i 1.34468 − 2.32906i −2.23892 − 3.70904i 2.02738 − 1.88113i −3.51921 + 4.41296i −2.65012 + 1.40601i −1.49688 + 6.55828i
103.5 −2.36278 + 0.728820i 1.72372 − 0.169698i 3.39905 − 2.31743i 0.422014 − 0.730949i −3.94908 + 1.65724i 3.69219 − 3.42585i −3.25889 + 4.08651i 2.94240 − 0.585024i −0.464393 + 2.03464i
103.6 −2.32012 + 0.715661i 0.160018 − 1.72464i 3.21829 − 2.19419i −1.83007 + 3.16977i 0.863000 + 4.11589i 1.93930 − 1.79941i −2.86885 + 3.59743i −2.94879 − 0.551947i 1.97749 − 8.66393i
103.7 −2.31412 + 0.713811i −1.66466 + 0.478457i 3.19315 − 2.17705i 1.26198 − 2.18582i 3.51068 − 2.29546i −1.72873 + 1.60402i −2.81550 + 3.53052i 2.54216 − 1.59293i −1.36012 + 5.95906i
103.8 −2.21500 + 0.683238i −1.66928 − 0.462058i 2.78694 − 1.90010i −1.13095 + 1.95886i 4.01316 − 0.117058i −0.143451 + 0.133103i −1.98438 + 2.48833i 2.57301 + 1.54261i 1.16669 − 5.11159i
103.9 −2.17225 + 0.670050i 1.33450 − 1.10413i 2.61722 − 1.78439i 0.353763 − 0.612736i −2.15904 + 3.29263i −2.37045 + 2.19945i −1.65493 + 2.07522i 0.561786 − 2.94693i −0.357898 + 1.56805i
103.10 −2.07494 + 0.640034i 0.821383 − 1.52490i 2.24325 − 1.52942i 1.63579 − 2.83327i −0.728329 + 3.68979i 0.655093 − 0.607838i −0.968024 + 1.21386i −1.65066 − 2.50506i −1.58078 + 6.92583i
103.11 −2.04720 + 0.631477i 1.37069 + 1.05887i 2.13978 − 1.45888i −0.0940679 + 0.162930i −3.47473 − 1.30215i −1.33665 + 1.24023i −0.787801 + 0.987871i 0.757602 + 2.90276i 0.0896888 − 0.392952i
103.12 −1.90413 + 0.587346i 0.662324 + 1.60041i 1.62825 − 1.11012i −1.68889 + 2.92525i −2.20115 − 2.65838i 1.83537 − 1.70298i 0.0364321 − 0.0456844i −2.12265 + 2.11999i 1.49774 − 6.56201i
103.13 −1.89240 + 0.583729i −0.372336 − 1.69156i 1.58798 − 1.08266i −0.738385 + 1.27892i 1.69202 + 2.98377i −3.66353 + 3.39926i 0.0963914 − 0.120871i −2.72273 + 1.25966i 0.650779 − 2.85125i
103.14 −1.81776 + 0.560703i −1.36264 + 1.06921i 1.33737 − 0.911803i −1.02362 + 1.77297i 1.87744 − 2.70760i 1.90757 − 1.76997i 0.452329 − 0.567203i 0.713585 − 2.91390i 0.866589 − 3.79677i
103.15 −1.76005 + 0.542904i −1.48737 − 0.887543i 1.15056 − 0.784441i 0.550864 − 0.954124i 3.09970 + 0.754623i 2.69506 − 2.50065i 0.697614 − 0.874780i 1.42453 + 2.64021i −0.451551 + 1.97837i
103.16 −1.71905 + 0.530256i −0.975321 + 1.43135i 1.02148 − 0.696431i −1.23167 + 2.13331i 0.917643 − 2.97772i −2.80560 + 2.60322i 0.856600 − 1.07414i −1.09750 − 2.79204i 0.986094 − 4.32036i
103.17 −1.49915 + 0.462426i −0.794478 − 1.53909i 0.381130 − 0.259850i 1.98966 − 3.44620i 1.90276 + 1.93994i −1.05851 + 0.982154i 1.50511 − 1.88735i −1.73761 + 2.44555i −1.38919 + 6.08643i
103.18 −1.48356 + 0.457619i 1.40943 + 1.00673i 0.339068 − 0.231173i 0.454481 − 0.787185i −2.55168 − 0.848571i 0.159915 − 0.148380i 1.53874 − 1.92952i 0.972979 + 2.83784i −0.314021 + 1.37582i
103.19 −1.28643 + 0.396810i −0.377219 + 1.69047i −0.155042 + 0.105706i 0.453166 − 0.784906i −0.185532 − 2.32436i 0.971929 − 0.901819i 1.83623 − 2.30256i −2.71541 − 1.27536i −0.271506 + 1.18954i
103.20 −1.28468 + 0.396270i −1.64784 + 0.533514i −0.159117 + 0.108484i 1.91206 − 3.31178i 1.90552 − 1.33838i 3.21151 − 2.97984i 1.83787 − 2.30461i 2.43072 − 1.75829i −1.14402 + 5.01226i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 103.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
71.d even 7 1 inner
639.r even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 639.2.r.a ✓ 840
9.c even 3 1 inner 639.2.r.a ✓ 840
71.d even 7 1 inner 639.2.r.a ✓ 840
639.r even 21 1 inner 639.2.r.a ✓ 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
639.2.r.a ✓ 840 1.a even 1 1 trivial
639.2.r.a ✓ 840 9.c even 3 1 inner
639.2.r.a ✓ 840 71.d even 7 1 inner
639.2.r.a ✓ 840 639.r even 21 1 inner

Hecke kernels

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