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Properties

Label 637.2.bn.a
Level $637$
Weight $2$
Character orbit 637.bn
Analytic conductor $5.086$
Analytic rank $0$
Dimension $744$
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(34,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([6, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bn (of order \(28\), 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: \(744\)
Relative dimension: \(62\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 744 q - 10 q^{2} - 28 q^{3} - 14 q^{5} - 14 q^{6} - 6 q^{7} - 18 q^{8} + 88 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 744 q - 10 q^{2} - 28 q^{3} - 14 q^{5} - 14 q^{6} - 6 q^{7} - 18 q^{8} + 88 q^{9} - 2 q^{11} - 14 q^{13} - 36 q^{14} - 22 q^{15} + 68 q^{16} - 12 q^{18} - 14 q^{20} - 14 q^{21} - 12 q^{22} - 98 q^{24} - 14 q^{26} - 28 q^{27} - 78 q^{28} - 12 q^{29} + 2 q^{32} - 14 q^{33} - 70 q^{34} + 32 q^{35} + 72 q^{37} + 38 q^{39} - 28 q^{40} - 14 q^{41} - 236 q^{42} - 38 q^{44} - 84 q^{45} - 46 q^{46} - 140 q^{47} - 228 q^{50} + 182 q^{52} - 12 q^{53} - 56 q^{54} - 112 q^{55} + 60 q^{57} - 2 q^{58} - 42 q^{59} - 14 q^{60} - 28 q^{61} + 120 q^{63} + 18 q^{65} - 168 q^{66} - 88 q^{67} - 158 q^{70} - 134 q^{71} + 18 q^{72} + 28 q^{73} - 60 q^{74} - 70 q^{76} - 98 q^{78} - 32 q^{79} - 72 q^{81} - 112 q^{83} - 10 q^{84} + 66 q^{85} - 150 q^{86} + 196 q^{87} + 56 q^{89} + 186 q^{91} + 120 q^{92} - 58 q^{93} + 252 q^{94} + 434 q^{96} - 126 q^{98} - 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.33473 + 1.46701i 1.51398 + 1.20736i 2.43110 − 5.04822i −0.139377 − 1.23701i −5.30595 − 0.597837i 2.38034 − 1.15498i 1.11238 + 9.87263i 0.166857 + 0.731049i 2.14011 + 2.68362i
34.2 −2.21682 + 1.39292i 1.72466 + 1.37537i 2.10631 − 4.37379i 0.283171 + 2.51321i −5.73906 − 0.646637i −2.41648 − 1.07732i 0.836771 + 7.42655i 0.415248 + 1.81932i −4.12845 − 5.17691i
34.3 −2.20425 + 1.38502i −0.369211 − 0.294436i 2.07268 − 4.30396i −0.334380 − 2.96770i 1.22164 + 0.137645i −2.64363 + 0.105979i 0.809433 + 7.18392i −0.617939 − 2.70737i 4.84740 + 6.07844i
34.4 −2.17716 + 1.36800i −0.966508 − 0.770764i 2.00083 − 4.15477i 0.346391 + 3.07431i 3.15865 + 0.355894i 1.00216 − 2.44861i 0.751811 + 6.67251i −0.327503 − 1.43488i −4.95980 − 6.21939i
34.5 −2.07919 + 1.30644i 0.380902 + 0.303759i 1.74847 − 3.63074i −0.146201 − 1.29757i −1.18881 − 0.133947i 1.43462 + 2.22303i 0.558071 + 4.95302i −0.614746 − 2.69338i 1.99917 + 2.50688i
34.6 −2.06845 + 1.29969i −2.27805 − 1.81669i 1.72151 − 3.57476i −0.132855 − 1.17912i 7.07317 + 0.796955i −2.57217 − 0.619635i 0.538194 + 4.77660i 1.22161 + 5.35222i 1.80730 + 2.26628i
34.7 −2.00031 + 1.25688i −1.54358 − 1.23096i 1.55374 − 3.22637i −0.169684 − 1.50599i 4.63482 + 0.522219i 2.31480 − 1.28129i 0.418186 + 3.71150i 0.199804 + 0.875398i 2.23227 + 2.79918i
34.8 −1.93709 + 1.21715i −2.51071 − 2.00223i 1.40307 − 2.91351i 0.128683 + 1.14209i 7.30048 + 0.822567i 1.71198 + 2.01721i 0.316021 + 2.80477i 1.62721 + 7.12927i −1.63937 − 2.05570i
34.9 −1.91628 + 1.20408i 2.31613 + 1.84705i 1.35457 − 2.81278i −0.245020 − 2.17461i −6.66237 − 0.750668i −1.46938 + 2.20021i 0.284298 + 2.52321i 1.28530 + 5.63125i 3.08793 + 3.87215i
34.10 −1.73330 + 1.08911i 0.996127 + 0.794385i 0.950415 − 1.97356i 0.362217 + 3.21476i −2.59176 − 0.292021i 1.96371 + 1.77308i 0.0436582 + 0.387478i −0.306341 − 1.34217i −4.12904 − 5.17766i
34.11 −1.72428 + 1.08344i 0.213720 + 0.170436i 0.931534 − 1.93435i 0.224719 + 1.99444i −0.553169 − 0.0623272i −2.64028 − 0.170031i 0.0335091 + 0.297402i −0.650935 − 2.85193i −2.54832 − 3.19550i
34.12 −1.58739 + 0.997422i 1.86943 + 1.49082i 0.657183 − 1.36465i −0.126715 − 1.12462i −4.45448 − 0.501899i −0.277433 − 2.63117i −0.101877 − 0.904181i 0.604655 + 2.64917i 1.32287 + 1.65883i
34.13 −1.50698 + 0.946899i −1.16768 − 0.931196i 0.506607 − 1.05198i 0.331360 + 2.94090i 2.64142 + 0.297617i −1.77240 + 1.96433i −0.165872 − 1.47215i −0.171205 − 0.750100i −3.28409 − 4.11812i
34.14 −1.50091 + 0.943082i −0.649834 − 0.518226i 0.495548 − 1.02902i −0.298340 − 2.64784i 1.46407 + 0.164961i 0.724402 + 2.54465i −0.170262 − 1.51111i −0.513836 − 2.25126i 2.94491 + 3.69280i
34.15 −1.47628 + 0.927607i 0.750543 + 0.598538i 0.451175 − 0.936874i 0.135425 + 1.20193i −1.66322 − 0.187400i 2.58987 − 0.540908i −0.187432 − 1.66351i −0.462496 − 2.02633i −1.31485 − 1.64876i
34.16 −1.32960 + 0.835441i −1.14707 − 0.914759i 0.202098 − 0.419660i −0.399879 − 3.54902i 2.28937 + 0.257950i 1.72222 − 2.00848i −0.269739 − 2.39400i −0.188574 − 0.826196i 3.49668 + 4.38469i
34.17 −1.17757 + 0.739914i 0.128966 + 0.102847i −0.0285769 + 0.0593406i −0.106521 − 0.945403i −0.227964 − 0.0256854i −0.888002 − 2.49228i −0.321681 − 2.85500i −0.661508 − 2.89826i 0.824954 + 1.03446i
34.18 −1.13810 + 0.715116i −2.25309 − 1.79678i −0.0838860 + 0.174191i 0.459788 + 4.08073i 3.84914 + 0.433694i −0.439146 − 2.60905i −0.330084 − 2.92957i 1.18043 + 5.17181i −3.44148 − 4.31548i
34.19 −0.934726 + 0.587327i 0.556038 + 0.443426i −0.339008 + 0.703957i −0.349731 − 3.10395i −0.780179 − 0.0879051i −2.13288 + 1.56551i −0.343776 − 3.05110i −0.555011 − 2.43166i 2.14994 + 2.69594i
34.20 −0.929398 + 0.583979i 1.22322 + 0.975483i −0.345019 + 0.716440i 0.0456818 + 0.405438i −1.70652 − 0.192278i −1.96755 + 1.76883i −0.343519 − 3.04882i −0.122871 − 0.538331i −0.279224 − 0.350135i
See next 80 embeddings (of 744 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
49.f odd 14 1 inner
637.bn even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bn.a ✓ 744
13.d odd 4 1 inner 637.2.bn.a ✓ 744
49.f odd 14 1 inner 637.2.bn.a ✓ 744
637.bn even 28 1 inner 637.2.bn.a ✓ 744
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bn.a ✓ 744 1.a even 1 1 trivial
637.2.bn.a ✓ 744 13.d odd 4 1 inner
637.2.bn.a ✓ 744 49.f odd 14 1 inner
637.2.bn.a ✓ 744 637.bn even 28 1 inner

Hecke kernels

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