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Properties

Label 637.2.bl.a
Level $637$
Weight $2$
Character orbit 637.bl
Analytic conductor $5.086$
Analytic rank $0$
Dimension $324$
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] = [637,2,Mod(53,637)]
 
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([10, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bl (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: \(324\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 324 q - 3 q^{2} + 25 q^{4} + q^{5} - 24 q^{6} - 21 q^{7} + 24 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 324 q - 3 q^{2} + 25 q^{4} + q^{5} - 24 q^{6} - 21 q^{7} + 24 q^{8} + 11 q^{9} - 5 q^{10} - 18 q^{11} - 40 q^{12} + 54 q^{13} - 15 q^{14} + 6 q^{15} + 29 q^{16} - 6 q^{17} + 49 q^{18} - 24 q^{19} + 11 q^{20} - 6 q^{22} - 42 q^{23} + 20 q^{24} + 14 q^{25} + 3 q^{26} - 33 q^{27} + 7 q^{28} - 22 q^{29} + 57 q^{30} - 31 q^{31} - 139 q^{32} + 6 q^{33} + 50 q^{34} + 42 q^{35} - 78 q^{36} - 4 q^{37} - 90 q^{38} + 40 q^{40} + 8 q^{41} + 20 q^{42} + 34 q^{43} - 256 q^{44} + 19 q^{45} + 85 q^{46} + 34 q^{47} - 10 q^{48} + 51 q^{49} - 54 q^{50} + 74 q^{51} - 25 q^{52} + 10 q^{53} + 111 q^{54} - 10 q^{55} - 196 q^{56} - 5 q^{57} - 21 q^{58} + 65 q^{59} + 87 q^{60} + 3 q^{61} - 54 q^{62} - 35 q^{63} - 28 q^{64} - q^{65} - 110 q^{66} + 135 q^{67} - 158 q^{68} + 42 q^{69} + 44 q^{70} + 9 q^{71} - 133 q^{72} + 31 q^{73} - 97 q^{74} - 315 q^{75} - 177 q^{76} + 6 q^{77} - 25 q^{78} + 43 q^{79} - 20 q^{80} - 259 q^{81} + 96 q^{82} + 59 q^{83} + 285 q^{84} + 6 q^{85} + 95 q^{86} - 206 q^{87} + 228 q^{88} + 43 q^{89} - 61 q^{90} - 7 q^{91} + 53 q^{92} - 10 q^{93} - 36 q^{94} - 17 q^{95} + 277 q^{96} + 66 q^{97} + 260 q^{98} - 206 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −2.27310 − 1.54978i 2.32870 + 2.16072i 2.03451 + 5.18386i 1.52353 − 0.469945i −1.94475 − 8.52050i −2.61577 − 0.397162i 2.18478 − 9.57215i 0.529951 + 7.07171i −4.19144 − 1.29289i
53.2 −2.04606 − 1.39498i −2.21248 − 2.05288i 1.50972 + 3.84670i 1.22875 − 0.379019i 1.66314 + 7.28669i 2.38208 − 1.15138i 1.17502 − 5.14808i 0.456553 + 6.09228i −3.04283 − 0.938588i
53.3 −1.99966 − 1.36335i 0.0697391 + 0.0647085i 1.40925 + 3.59072i 4.18596 − 1.29120i −0.0512347 − 0.224474i 0.780361 − 2.52805i 1.00027 − 4.38249i −0.223514 − 2.98259i −10.1309 − 3.12496i
53.4 −1.98190 − 1.35124i 0.272118 + 0.252488i 1.37140 + 3.49428i −2.27380 + 0.701373i −0.198139 − 0.868102i −1.25655 + 2.32832i 0.936090 − 4.10128i −0.213893 − 2.85420i 5.45416 + 1.68238i
53.5 −1.79391 − 1.22307i 1.34252 + 1.24567i 0.991536 + 2.52639i 1.96924 − 0.607429i −0.884812 − 3.87661i 1.60270 + 2.10508i 0.344956 − 1.51135i 0.0264586 + 0.353066i −4.27556 − 1.31883i
53.6 −1.44647 − 0.986187i −2.38493 − 2.21289i 0.389031 + 0.991234i −0.723247 + 0.223092i 1.26741 + 5.55286i −0.372756 + 2.61936i −0.364300 + 1.59610i 0.566810 + 7.56355i 1.26617 + 0.390561i
53.7 −1.40283 − 0.956431i −0.499322 − 0.463303i 0.322480 + 0.821666i −2.80314 + 0.864654i 0.257345 + 1.12750i 2.49640 − 0.876348i −0.422130 + 1.84947i −0.189518 − 2.52894i 4.75930 + 1.46805i
53.8 −1.31734 − 0.898146i −1.50703 − 1.39832i 0.198032 + 0.504577i 0.675664 − 0.208415i 0.729374 + 3.19559i −1.94868 − 1.78960i −0.517257 + 2.26625i 0.0916520 + 1.22301i −1.07726 − 0.332292i
53.9 −1.15236 − 0.785668i 1.86576 + 1.73118i −0.0200159 − 0.0509996i −1.72793 + 0.532996i −0.789908 − 3.46081i −0.964248 − 2.46378i −0.637707 + 2.79398i 0.259914 + 3.46832i 2.40996 + 0.743375i
53.10 −0.971314 − 0.662231i 1.44449 + 1.34029i −0.225781 − 0.575279i −0.154277 + 0.0475881i −0.515472 − 2.25843i 2.49647 + 0.876147i −0.684849 + 3.00052i 0.0659823 + 0.880472i 0.181366 + 0.0559439i
53.11 −0.639784 − 0.436198i 1.84518 + 1.71208i −0.511626 − 1.30360i −1.40912 + 0.434655i −0.433714 − 1.90023i −2.31520 + 1.28057i −0.585909 + 2.56703i 0.249293 + 3.32659i 1.09113 + 0.336568i
53.12 −0.483489 − 0.329637i −1.12540 − 1.04422i −0.605581 − 1.54300i −2.74195 + 0.845781i 0.199905 + 0.875842i 1.46937 + 2.20022i −0.476262 + 2.08664i −0.0480579 − 0.641288i 1.60451 + 0.494925i
53.13 −0.434073 − 0.295946i 0.0114514 + 0.0106254i −0.629847 − 1.60482i 3.15579 − 0.973431i −0.00182622 − 0.00800119i 0.290350 + 2.62977i −0.435349 + 1.90739i −0.224172 − 2.99137i −1.65792 − 0.511402i
53.14 −0.178924 − 0.121988i −0.267794 − 0.248477i −0.713549 − 1.81809i 0.350054 − 0.107977i 0.0176035 + 0.0771261i 1.05341 − 2.42700i −0.190490 + 0.834589i −0.214217 − 2.85853i −0.0758047 − 0.0233827i
53.15 −0.0762337 − 0.0519753i −1.17809 − 1.09311i −0.727572 − 1.85382i 0.443733 − 0.136874i 0.0329956 + 0.144563i −2.48207 + 0.916143i −0.0819496 + 0.359045i −0.0311795 − 0.416062i −0.0409415 − 0.0126288i
53.16 0.395693 + 0.269779i 1.67664 + 1.55569i −0.646890 − 1.64825i 2.78605 − 0.859383i 0.243741 + 1.06790i −1.76663 − 1.96952i 0.401828 − 1.76052i 0.166745 + 2.22506i 1.33427 + 0.411566i
53.17 0.500003 + 0.340896i −1.93585 − 1.79621i −0.596889 − 1.52085i −4.16664 + 1.28524i −0.355611 − 1.55803i −2.59900 − 0.495162i 0.489325 − 2.14387i 0.296968 + 3.96277i −2.52146 − 0.777768i
53.18 0.726317 + 0.495195i −1.98880 − 1.84534i −0.448363 − 1.14241i 1.60817 − 0.496054i −0.530699 − 2.32514i 0.921177 − 2.48021i 0.631282 − 2.76583i 0.325867 + 4.34840i 1.41368 + 0.436064i
53.19 1.01838 + 0.694320i 0.670270 + 0.621920i −0.175663 − 0.447582i 0.228303 − 0.0704221i 0.250779 + 1.09873i 2.28590 + 1.33217i 0.680410 − 2.98107i −0.161712 − 2.15790i 0.281395 + 0.0867987i
53.20 1.14097 + 0.777904i 2.47075 + 2.29252i −0.0339921 − 0.0866106i −0.390328 + 0.120400i 1.03571 + 4.53772i 2.64566 − 0.0219120i 0.643161 − 2.81787i 0.624765 + 8.33691i −0.539015 − 0.166264i
See next 80 embeddings (of 324 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g 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.bl.a ✓ 324
49.g even 21 1 inner 637.2.bl.a ✓ 324
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bl.a ✓ 324 1.a even 1 1 trivial
637.2.bl.a ✓ 324 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{324} + 3 T_{2}^{323} - 35 T_{2}^{322} - 130 T_{2}^{321} + 459 T_{2}^{320} + 2521 T_{2}^{319} + \cdots + 57722629246681 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display