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Properties

Label 637.2.cd.a
Level $637$
Weight $2$
Character orbit 637.cd
Analytic conductor $5.086$
Analytic rank $0$
Dimension $1512$
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(45,637)]
 
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([62, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.cd (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: \(1512\)
Relative dimension: \(63\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1512 q - 26 q^{2} - 8 q^{3} - 42 q^{4} - 22 q^{5} - 16 q^{6} - 22 q^{7} - 24 q^{8} - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1512 q - 26 q^{2} - 8 q^{3} - 42 q^{4} - 22 q^{5} - 16 q^{6} - 22 q^{7} - 24 q^{8} - 132 q^{9} - 36 q^{10} - 34 q^{11} + 8 q^{12} - 28 q^{13} - 36 q^{14} - 26 q^{15} + 206 q^{16} - 30 q^{17} - 56 q^{19} - 64 q^{20} - 22 q^{21} - 6 q^{22} - 42 q^{23} - 178 q^{24} - 52 q^{26} - 56 q^{27} - 198 q^{28} - 6 q^{29} - 36 q^{30} - 38 q^{31} - 38 q^{32} - 16 q^{33} + 40 q^{34} + 34 q^{35} - 48 q^{36} - 116 q^{37} - 26 q^{39} - 104 q^{40} - 10 q^{41} + 164 q^{42} - 90 q^{43} - 10 q^{44} + 48 q^{45} - 44 q^{46} - 22 q^{47} + 12 q^{48} + 8 q^{49} + 186 q^{50} + 36 q^{51} - 198 q^{52} - 72 q^{53} + 44 q^{54} - 62 q^{55} + 156 q^{56} + 20 q^{57} - 22 q^{58} - 126 q^{59} - 106 q^{60} - 44 q^{61} - 78 q^{62} - 54 q^{63} + 336 q^{64} - 48 q^{65} + 18 q^{66} - 4 q^{67} - 220 q^{70} + 134 q^{71} + 90 q^{72} - 110 q^{73} - 30 q^{74} - 2 q^{75} + 80 q^{76} + 50 q^{78} - 16 q^{79} - 72 q^{80} + 92 q^{81} + 12 q^{82} + 4 q^{83} - 132 q^{84} - 102 q^{85} - 210 q^{86} + 98 q^{87} + 36 q^{88} + 112 q^{89} - 198 q^{90} + 282 q^{91} - 120 q^{92} + 24 q^{93} - 276 q^{94} - 42 q^{95} - 410 q^{96} - 190 q^{97} + 168 q^{98} - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −0.912177 + 2.60685i −0.678770 − 2.20052i −4.39994 − 3.50884i 2.55764 + 1.35175i 6.35558 + 0.237809i 0.0907706 − 2.64419i 8.48351 − 5.33054i −1.90283 + 1.29733i −5.85685 + 5.43436i
45.2 −0.906485 + 2.59059i 0.283596 + 0.919396i −4.32576 − 3.44968i −1.90488 − 1.00676i −2.63885 − 0.0987388i 2.43015 − 1.04613i 8.21007 − 5.15873i 1.71385 − 1.16849i 4.33483 − 4.02214i
45.3 −0.860537 + 2.45927i 0.0311116 + 0.100861i −3.74383 − 2.98561i 1.06552 + 0.563143i −0.274818 − 0.0102830i 1.09134 + 2.41018i 6.15187 − 3.86548i 2.46951 − 1.68368i −2.30184 + 2.13579i
45.4 −0.851545 + 2.43358i 0.681244 + 2.20854i −3.63351 − 2.89762i −0.464587 − 0.245541i −5.95476 − 0.222811i −2.59578 − 0.511811i 5.77953 − 3.63152i −1.93483 + 1.31914i 0.993161 − 0.921518i
45.5 −0.820645 + 2.34527i 0.964380 + 3.12644i −3.26317 − 2.60229i 2.05426 + 1.08571i −8.12376 − 0.303970i 2.60171 − 0.480738i 4.57325 − 2.87356i −6.36590 + 4.34020i −4.23209 + 3.92680i
45.6 −0.794090 + 2.26938i −0.567821 − 1.84083i −2.95584 − 2.35721i 0.881582 + 0.465930i 4.62844 + 0.173184i −2.40569 + 1.10121i 3.62505 − 2.27777i −0.587516 + 0.400561i −1.75743 + 1.63065i
45.7 −0.791706 + 2.26257i −0.760720 − 2.46619i −2.92875 − 2.33560i −1.52285 − 0.804851i 6.18220 + 0.231322i 2.62492 + 0.331380i 3.54382 − 2.22673i −3.02471 + 2.06221i 3.02668 − 2.80835i
45.8 −0.771025 + 2.20346i −0.214733 − 0.696146i −2.69711 − 2.15087i −3.11188 − 1.64468i 1.69950 + 0.0635906i −1.80252 − 1.93673i 2.86561 − 1.80058i 2.04021 − 1.39099i 6.02332 − 5.58882i
45.9 −0.700409 + 2.00165i −0.160098 − 0.519025i −1.95238 − 1.55697i −1.11302 − 0.588248i 1.15104 + 0.0430689i −0.688717 + 2.55454i 0.892755 − 0.560955i 2.23496 − 1.52377i 1.95704 − 1.81587i
45.10 −0.680872 + 1.94582i 0.131858 + 0.427472i −1.75897 − 1.40273i 2.82139 + 1.49115i −0.921563 − 0.0344824i 0.653278 − 2.56383i 0.436044 − 0.273985i 2.31337 − 1.57723i −4.82251 + 4.47464i
45.11 −0.656078 + 1.87496i 0.109059 + 0.353561i −1.52139 − 1.21327i 0.264504 + 0.139794i −0.734466 − 0.0274818i −0.602445 − 2.57625i −0.0909495 + 0.0571473i 2.36560 − 1.61284i −0.435645 + 0.404219i
45.12 −0.596636 + 1.70509i −0.670343 − 2.17320i −0.987685 − 0.787652i 3.41069 + 1.80260i 4.10544 + 0.153615i 2.41009 + 1.09155i −1.12684 + 0.708042i −1.79472 + 1.22362i −5.10854 + 4.74003i
45.13 −0.589869 + 1.68575i 0.698369 + 2.26406i −0.930140 − 0.741762i 0.152158 + 0.0804177i −4.22857 − 0.158222i 0.718950 + 2.54620i −1.22537 + 0.769948i −2.15951 + 1.47233i −0.225317 + 0.209064i
45.14 −0.564151 + 1.61225i 0.320996 + 1.04064i −0.717425 − 0.572127i −3.24639 − 1.71577i −1.85887 − 0.0695540i 2.63133 + 0.275858i −1.56544 + 0.983630i 1.49881 − 1.02187i 4.59770 − 4.26605i
45.15 −0.556696 + 1.59094i 0.639923 + 2.07458i −0.657532 − 0.524365i 3.69012 + 1.95028i −3.65678 − 0.136827i −1.31376 + 2.29653i −1.65408 + 1.03933i −1.41566 + 0.965180i −5.15707 + 4.78506i
45.16 −0.513867 + 1.46855i 0.427969 + 1.38744i −0.328909 − 0.262296i 0.134210 + 0.0709320i −2.25744 − 0.0844676i 2.64569 + 0.0179741i −2.08055 + 1.30730i 0.736877 − 0.502395i −0.173133 + 0.160644i
45.17 −0.499991 + 1.42889i 0.671952 + 2.17841i −0.228082 − 0.181889i −0.110409 − 0.0583530i −3.44869 − 0.129041i −2.63253 − 0.264154i −2.18968 + 1.37587i −1.81525 + 1.23762i 0.138584 − 0.128587i
45.18 −0.482252 + 1.37820i 0.997358 + 3.23336i −0.103199 − 0.0822981i −3.01694 − 1.59450i −4.93718 − 0.184736i 0.215766 − 2.63694i −2.30947 + 1.45114i −6.98115 + 4.75967i 3.65246 − 3.38899i
45.19 −0.465355 + 1.32991i −0.910373 − 2.95136i 0.0115601 + 0.00921885i 0.490240 + 0.259099i 4.34869 + 0.162716i −0.439374 + 2.60901i −2.40367 + 1.51032i −5.40302 + 3.68371i −0.572714 + 0.531401i
45.20 −0.400428 + 1.14436i −0.0227722 − 0.0738258i 0.414452 + 0.330514i −2.33481 − 1.23399i 0.0936017 + 0.00350233i −2.56690 + 0.641122i −2.59731 + 1.63200i 2.47378 − 1.68660i 2.34705 − 2.17774i
See next 80 embeddings (of 1512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.63
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
637.cd 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.cd.a ✓ 1512
13.f odd 12 1 637.2.ch.a yes 1512
49.h odd 42 1 637.2.ch.a yes 1512
637.cd even 84 1 inner 637.2.cd.a ✓ 1512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.cd.a ✓ 1512 1.a even 1 1 trivial
637.2.cd.a ✓ 1512 637.cd even 84 1 inner
637.2.ch.a yes 1512 13.f odd 12 1
637.2.ch.a yes 1512 49.h odd 42 1

Hecke kernels

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