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Properties

Label 637.2.bk.a
Level $637$
Weight $2$
Character orbit 637.bk
Analytic conductor $5.086$
Analytic rank $0$
Dimension $768$
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] = [637,2,Mod(22,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([24, 28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bk (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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(768\)
Relative dimension: \(64\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 5 q^{2} - 7 q^{3} + 59 q^{4} - 20 q^{5} - 13 q^{6} - 7 q^{7} - 28 q^{8} + 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 768 q - 5 q^{2} - 7 q^{3} + 59 q^{4} - 20 q^{5} - 13 q^{6} - 7 q^{7} - 28 q^{8} + 57 q^{9} - 5 q^{10} - q^{11} - 32 q^{12} - 12 q^{13} - 36 q^{14} + q^{15} + 71 q^{16} - 17 q^{17} - 84 q^{18} - 6 q^{19} + 3 q^{20} - 24 q^{21} - 5 q^{22} - 5 q^{23} + 49 q^{24} - 140 q^{25} - 30 q^{26} + 8 q^{27} - 49 q^{28} - 9 q^{29} - 34 q^{30} - 52 q^{31} - 25 q^{32} - 29 q^{33} - 132 q^{34} - 11 q^{35} + 19 q^{36} - 2 q^{37} + 24 q^{38} + 2 q^{39} - 20 q^{40} - 23 q^{41} - 45 q^{42} + 9 q^{43} + 56 q^{44} - 32 q^{45} + 9 q^{46} + 70 q^{47} + 16 q^{48} + 29 q^{49} - 106 q^{50} - 48 q^{51} - 22 q^{52} + 38 q^{53} - 28 q^{54} - 34 q^{55} + 33 q^{56} + 58 q^{57} - 13 q^{58} + 31 q^{59} + 68 q^{60} - 52 q^{61} - 3 q^{62} + 3 q^{63} - 44 q^{64} - 34 q^{65} - 180 q^{66} - 42 q^{67} + 88 q^{68} - 87 q^{69} + 64 q^{70} + 97 q^{71} + 7 q^{72} + 30 q^{73} - 11 q^{74} + 21 q^{75} + 67 q^{76} + 6 q^{77} + 104 q^{78} - 56 q^{79} + 20 q^{80} + 21 q^{81} + 70 q^{82} + 58 q^{83} - 29 q^{84} - 59 q^{85} + 24 q^{86} - 73 q^{87} + 53 q^{88} - 10 q^{89} - 60 q^{90} + 166 q^{91} - 108 q^{92} + 80 q^{93} - 88 q^{94} - 78 q^{95} - 628 q^{96} - 72 q^{97} + 163 q^{98} - 136 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −0.205726 + 2.74522i 2.33271 + 2.16444i −5.51627 − 0.831443i −0.210616 + 0.922768i −6.42178 + 5.95854i −2.64541 + 0.0428035i 2.19217 − 9.60454i 0.532554 + 7.10644i −2.48987 − 0.768025i
22.2 −0.199715 + 2.66502i 1.23016 + 1.14142i −5.08477 − 0.766406i 0.819496 − 3.59045i −3.28759 + 3.05044i 2.63760 + 0.207475i 1.86862 − 8.18697i −0.0137398 − 0.183344i 9.40494 + 2.90104i
22.3 −0.199533 + 2.66259i −1.36737 − 1.26873i −5.07189 − 0.764464i −0.171022 + 0.749297i 3.65095 − 3.38758i 1.16974 + 2.37312i 1.85918 − 8.14559i 0.0358242 + 0.478041i −1.96094 − 0.604871i
22.4 −0.197989 + 2.64198i −1.16973 − 1.08535i −4.96320 − 0.748082i 0.247343 − 1.08368i 3.09908 − 2.87552i 0.323833 − 2.62586i 1.77999 − 7.79865i −0.0339077 − 0.452467i 2.81409 + 0.868033i
22.5 −0.186192 + 2.48456i −0.0859999 − 0.0797962i −4.16070 − 0.627125i −0.670560 + 2.93792i 0.214271 − 0.198814i −2.03661 − 1.68885i 1.22398 − 5.36263i −0.223162 − 2.97789i −7.17457 − 2.21306i
22.6 −0.182742 + 2.43852i 0.612941 + 0.568726i −3.93531 − 0.593152i −0.250501 + 1.09752i −1.49886 + 1.39074i −0.870989 + 2.49827i 1.07727 − 4.71983i −0.171943 − 2.29442i −2.63054 − 0.811413i
22.7 −0.171345 + 2.28644i 0.558412 + 0.518130i −3.22078 − 0.485455i 0.764784 − 3.35074i −1.28035 + 1.18799i −2.31223 + 1.28592i 0.641412 − 2.81021i −0.180826 − 2.41295i 7.53021 + 2.32276i
22.8 −0.170656 + 2.27725i −2.52387 − 2.34181i −3.17906 − 0.479167i −0.369130 + 1.61727i 5.76359 − 5.34783i −2.62806 + 0.305460i 0.617395 − 2.70498i 0.661659 + 8.82922i −3.61992 − 1.11660i
22.9 −0.162796 + 2.17237i 1.76404 + 1.63679i −2.71501 − 0.409222i −0.779088 + 3.41341i −3.84289 + 3.56568i 2.62091 − 0.361720i 0.361469 − 1.58370i 0.208567 + 2.78314i −7.28834 − 2.24815i
22.10 −0.162556 + 2.16916i 0.119920 + 0.111270i −2.70118 − 0.407137i 0.226059 − 0.990429i −0.260856 + 0.242039i −0.692147 − 2.55361i 0.354163 − 1.55169i −0.222190 − 2.96492i 2.11165 + 0.651359i
22.11 −0.156158 + 2.08378i −1.57409 − 1.46054i −2.34011 − 0.352715i −0.779205 + 3.41392i 3.28926 − 3.05198i 2.43753 − 1.02880i 0.170438 − 0.746737i 0.120385 + 1.60642i −6.99220 − 2.15681i
22.12 −0.149443 + 1.99418i 2.14621 + 1.99139i −1.97674 − 0.297946i 0.234428 − 1.02710i −4.29192 + 3.98232i 1.39347 − 2.24905i −0.000412804 0.00180861i 0.416384 + 5.55626i 2.01318 + 0.620982i
22.13 −0.148405 + 1.98033i −2.38794 − 2.21568i −1.92203 − 0.289700i 0.814611 − 3.56904i 4.74217 − 4.40009i 1.74180 − 1.99152i −0.0248601 + 0.108919i 0.568807 + 7.59020i 6.94700 + 2.14287i
22.14 −0.144813 + 1.93240i 1.05401 + 0.977981i −1.73553 − 0.261589i −0.105375 + 0.461679i −2.04248 + 1.89515i 1.20309 + 2.35639i −0.105588 + 0.462610i −0.0696939 − 0.930001i −0.876889 − 0.270484i
22.15 −0.138959 + 1.85428i −1.22102 − 1.13294i −1.44139 − 0.217255i 0.404790 − 1.77350i 2.27047 − 2.10669i −2.19908 + 1.47108i −0.224401 + 0.983166i −0.0168543 − 0.224905i 3.23232 + 0.997039i
22.16 −0.138957 + 1.85425i −0.818792 − 0.759728i −1.44128 − 0.217238i −0.0538882 + 0.236100i 1.52250 − 1.41268i 2.50396 − 0.854519i −0.224443 + 0.983351i −0.130957 − 1.74749i −0.430300 − 0.132730i
22.17 −0.111608 + 1.48930i 0.965347 + 0.895712i −0.227897 − 0.0343499i −0.789778 + 3.46025i −1.44172 + 1.33772i −1.48401 − 2.19037i −0.588067 + 2.57649i −0.0945937 − 1.26227i −5.06520 − 1.56241i
22.18 −0.0985114 + 1.31454i 2.34368 + 2.17462i 0.259343 + 0.0390896i 0.675077 − 2.95770i −3.08950 + 2.86664i 1.10353 + 2.40462i −0.663600 + 2.90742i 0.539684 + 7.20159i 3.82153 + 1.17879i
22.19 −0.0965689 + 1.28862i −1.17706 − 1.09215i 0.326441 + 0.0492031i −0.159670 + 0.699561i 1.52104 − 1.41132i −2.58279 − 0.573758i −0.670027 + 2.93558i −0.0315157 − 0.420548i −0.886050 − 0.273310i
22.20 −0.0955489 + 1.27501i 1.88545 + 1.74944i 0.361138 + 0.0544328i −0.420604 + 1.84279i −2.41071 + 2.23681i −2.62589 + 0.323561i −0.672933 + 2.94831i 0.270182 + 3.60533i −2.30938 − 0.712351i
See next 80 embeddings (of 768 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
49.e even 7 1 inner
637.bk even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bk.a ✓ 768
13.c even 3 1 inner 637.2.bk.a ✓ 768
49.e even 7 1 inner 637.2.bk.a ✓ 768
637.bk even 21 1 inner 637.2.bk.a ✓ 768
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bk.a ✓ 768 1.a even 1 1 trivial
637.2.bk.a ✓ 768 13.c even 3 1 inner
637.2.bk.a ✓ 768 49.e even 7 1 inner
637.2.bk.a ✓ 768 637.bk even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(637, [\chi])\).