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Properties

Label 637.2.cc.a
Level $637$
Weight $2$
Character orbit 637.cc
Analytic conductor $5.086$
Analytic rank $0$
Dimension $1536$
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(5,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(84))
 
chi = DirichletCharacter(H, H._module([58, 63]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.cc (of order \(84\), degree \(24\), 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: \(1536\)
Relative dimension: \(64\) over \(\Q(\zeta_{84})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1536 q - 26 q^{2} - 44 q^{3} - 22 q^{5} - 28 q^{6} - 22 q^{7} - 12 q^{8} - 176 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1536 q - 26 q^{2} - 44 q^{3} - 22 q^{5} - 28 q^{6} - 22 q^{7} - 12 q^{8} - 176 q^{9} - 22 q^{11} - 28 q^{13} - 84 q^{14} - 8 q^{15} - 180 q^{16} - 18 q^{18} - 54 q^{19} - 28 q^{20} - 2 q^{21} - 48 q^{22} + 68 q^{24} - 52 q^{26} - 56 q^{27} + 62 q^{28} - 48 q^{29} - 66 q^{31} - 32 q^{32} - 76 q^{33} + 28 q^{34} - 140 q^{35} - 62 q^{37} - 16 q^{39} + 76 q^{40} - 28 q^{41} + 128 q^{42} - 226 q^{44} + 66 q^{45} - 32 q^{46} + 68 q^{47} + 156 q^{50} + 48 q^{52} - 48 q^{53} - 64 q^{54} + 28 q^{55} - 62 q^{57} - 22 q^{58} + 54 q^{59} - 88 q^{60} - 8 q^{61} - 150 q^{63} - 12 q^{65} + 72 q^{66} - 30 q^{67} - 36 q^{68} + 62 q^{70} + 104 q^{71} - 54 q^{72} - 136 q^{73} + 12 q^{74} + 28 q^{76} + 68 q^{78} - 16 q^{79} - 180 q^{80} + 64 q^{81} + 70 q^{83} + 30 q^{84} - 96 q^{85} + 30 q^{86} + 416 q^{87} - 38 q^{89} - 196 q^{91} - 180 q^{92} + 38 q^{93} - 264 q^{94} - 350 q^{96} - 390 q^{98} - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.63766 + 2.21895i 1.73964 − 2.55158i −1.65230 − 5.35663i 0.940466 − 0.809336i 2.81290 + 8.03880i −1.64446 − 2.07262i 9.38585 + 3.28425i −2.38821 − 6.08505i 0.255714 + 3.41226i
5.2 −1.60364 + 2.17285i −1.54581 + 2.26729i −1.56013 − 5.05780i 1.55493 − 1.33812i −2.44757 − 6.99474i 2.20094 + 1.46828i 8.39375 + 2.93710i −1.65505 − 4.21700i 0.414004 + 5.52450i
5.3 −1.57654 + 2.13614i 0.252953 − 0.371014i −1.48810 − 4.82430i −1.76353 + 1.51764i 0.393746 + 1.12526i −2.26746 + 1.36331i 7.63958 + 2.67320i 1.02236 + 2.60492i −0.461611 − 6.15977i
5.4 −1.54555 + 2.09414i 0.593096 − 0.869913i −1.40720 − 4.56204i −1.35096 + 1.16260i 0.905062 + 2.58652i 2.32068 − 1.27060i 6.81513 + 2.38472i 0.691038 + 1.76074i −0.346667 − 4.62595i
5.5 −1.50015 + 2.03263i −1.46486 + 2.14855i −1.29163 − 4.18737i −1.42651 + 1.22761i −2.16971 − 6.20067i −2.43746 − 1.02898i 5.68000 + 1.98752i −1.37445 − 3.50203i −0.355302 − 4.74118i
5.6 −1.49274 + 2.02259i 0.837048 − 1.22772i −1.27309 − 4.12726i 2.40257 − 2.06758i 1.23369 + 3.52568i 0.937355 + 2.47414i 5.50271 + 1.92548i 0.289366 + 0.737293i 0.595454 + 7.94578i
5.7 −1.36834 + 1.85404i −0.493898 + 0.724416i −0.975588 − 3.16278i 3.27431 − 2.81777i −0.667273 − 1.90696i −1.25371 − 2.32985i 2.84887 + 0.996861i 0.815181 + 2.07705i 0.743880 + 9.92638i
5.8 −1.35490 + 1.83582i −0.508081 + 0.745218i −0.944978 − 3.06354i 1.28061 − 1.10205i −0.679688 − 1.94244i −1.53884 + 2.15220i 2.59722 + 0.908805i 0.798820 + 2.03536i 0.288078 + 3.84414i
5.9 −1.35325 + 1.83359i 1.40042 − 2.05404i −0.941247 − 3.05145i −2.95043 + 2.53905i 1.87114 + 5.34742i 2.53254 + 0.765672i 2.56683 + 0.898172i −1.16188 − 2.96042i −0.662903 − 8.84583i
5.10 −1.34462 + 1.82190i −1.12975 + 1.65704i −0.921798 − 2.98840i −2.71346 + 2.33512i −1.49987 − 4.28638i 1.54732 − 2.14611i 2.40944 + 0.843101i −0.373414 − 0.951442i −0.605776 − 8.08351i
5.11 −1.22641 + 1.66173i 0.404276 − 0.592964i −0.667749 − 2.16479i −0.181173 + 0.155912i 0.489537 + 1.39902i −1.36433 − 2.26685i 0.517442 + 0.181061i 0.907856 + 2.31318i −0.0368907 − 0.492273i
5.12 −1.18544 + 1.60622i −0.759813 + 1.11444i −0.585148 − 1.89700i 0.466503 − 0.401459i −0.889319 − 2.54153i 2.57227 − 0.619199i −0.0278862 − 0.00975780i 0.431360 + 1.09909i 0.0918169 + 1.22521i
5.13 −1.16139 + 1.57363i 1.48120 − 2.17253i −0.537965 − 1.74404i 0.658683 − 0.566843i 1.69849 + 4.85401i −1.88360 + 1.85797i −0.322828 − 0.112962i −1.42988 − 3.64328i 0.127011 + 1.69485i
5.14 −1.14612 + 1.55294i −1.52877 + 2.24229i −0.508510 − 1.64855i −1.44363 + 1.24234i −1.72999 − 4.94402i 0.162718 + 2.64074i −0.500632 − 0.175179i −1.59472 − 4.06329i −0.274709 − 3.66574i
5.15 −0.965249 + 1.30787i 1.90895 − 2.79991i −0.189298 − 0.613688i 1.01089 − 0.869942i 1.81930 + 5.19926i 2.12237 − 1.57972i −2.08321 − 0.728945i −3.09940 − 7.89713i 0.162006 + 2.16182i
5.16 −0.964100 + 1.30631i −1.85936 + 2.72718i −0.187445 − 0.607682i 2.21985 − 1.91034i −1.76993 − 5.05817i −0.991663 − 2.45288i −2.09036 − 0.731448i −2.88426 − 7.34898i 0.355332 + 4.74157i
5.17 −0.951593 + 1.28936i 0.675753 − 0.991148i −0.167416 − 0.542751i 1.44494 − 1.24347i 0.634907 + 1.81446i 2.61034 + 0.431429i −2.16602 − 0.757923i 0.570291 + 1.45308i 0.228291 + 3.04633i
5.18 −0.935405 + 1.26743i 0.854263 − 1.25297i −0.141882 − 0.459971i −0.846774 + 0.728708i 0.788973 + 2.25475i −1.18137 − 2.36736i −2.25797 − 0.790099i 0.255845 + 0.651882i −0.131509 − 1.75486i
5.19 −0.930356 + 1.26059i −0.305422 + 0.447972i −0.134009 − 0.434446i −1.50993 + 1.29940i −0.280557 − 0.801785i 1.92699 + 1.81293i −2.28529 − 0.799657i 0.988627 + 2.51898i −0.233235 − 3.11230i
5.20 −0.800052 + 1.08403i 0.176501 − 0.258879i 0.0544684 + 0.176582i −2.40607 + 2.07059i 0.139424 + 0.398450i −0.602973 + 2.57613i −2.77838 − 0.972197i 1.06016 + 2.70124i −0.319605 − 4.26483i
See next 80 embeddings (of 1536 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner
49.h odd 42 1 inner
637.cc even 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.cc.a ✓ 1536
13.d odd 4 1 inner 637.2.cc.a ✓ 1536
49.h odd 42 1 inner 637.2.cc.a ✓ 1536
637.cc even 84 1 inner 637.2.cc.a ✓ 1536
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.cc.a ✓ 1536 1.a even 1 1 trivial
637.2.cc.a ✓ 1536 13.d odd 4 1 inner
637.2.cc.a ✓ 1536 49.h odd 42 1 inner
637.2.cc.a ✓ 1536 637.cc even 84 1 inner

Hecke kernels

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