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Properties

Label 637.2.ch.a
Level $637$
Weight $2$
Character orbit 637.ch
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(24,637)]
 
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([74, 49]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.ch (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.

\(\operatorname{Tr}(f)(q) = \) \( 1512 q - 26 q^{2} - 14 q^{3} - 36 q^{4} - 22 q^{5} - 40 q^{6} - 30 q^{7} - 24 q^{8} + 222 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1512 q - 26 q^{2} - 14 q^{3} - 36 q^{4} - 22 q^{5} - 40 q^{6} - 30 q^{7} - 24 q^{8} + 222 q^{9} - 30 q^{10} - 22 q^{11} - 8 q^{12} - 28 q^{13} - 36 q^{14} - 26 q^{15} - 124 q^{16} - 36 q^{17} - 18 q^{18} - 34 q^{19} + 8 q^{20} - 26 q^{21} - 6 q^{22} - 36 q^{23} + 44 q^{24} - 52 q^{26} - 56 q^{27} + 74 q^{28} - 6 q^{29} - 4 q^{31} - 8 q^{32} - 46 q^{33} + 16 q^{34} - 68 q^{35} - 48 q^{36} + 30 q^{37} - 100 q^{39} - 104 q^{40} - 46 q^{41} - 94 q^{42} - 90 q^{43} + 158 q^{44} + 30 q^{45} - 62 q^{46} - 112 q^{47} - 12 q^{48} - 50 q^{49} + 186 q^{50} - 36 q^{51} + 28 q^{52} - 72 q^{53} - 82 q^{54} + 76 q^{55} + 234 q^{56} - 106 q^{57} - 34 q^{58} + 6 q^{59} - 4 q^{60} - 14 q^{61} - 6 q^{62} - 144 q^{63} - 336 q^{64} - 18 q^{65} + 18 q^{66} - 10 q^{67} - 72 q^{68} - 84 q^{69} + 44 q^{70} - 202 q^{71} - 78 q^{72} + 42 q^{73} + 36 q^{74} - 22 q^{75} - 192 q^{76} + 50 q^{78} - 16 q^{79} - 54 q^{80} - 226 q^{81} + 66 q^{82} - 158 q^{83} + 28 q^{84} - 102 q^{85} + 102 q^{86} + 56 q^{87} - 42 q^{88} - 68 q^{89} + 198 q^{90} - 154 q^{91} - 120 q^{92} - 186 q^{93} + 126 q^{94} - 12 q^{95} - 494 q^{96} + 190 q^{97} - 56 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −0.103138 − 2.75641i 2.46397 + 1.96495i −5.59274 + 0.419118i −0.512878 + 0.378521i 5.16207 − 6.99436i −2.61447 − 0.405622i 1.11441 + 9.89066i 1.54255 + 6.75836i 1.09626 + 1.37466i
24.2 −0.101438 − 2.71100i −1.03314 − 0.823903i −5.34480 + 0.400537i −1.76647 + 1.30372i −2.12880 + 2.88442i 0.374408 + 2.61913i 1.02053 + 9.05743i −0.278997 − 1.22237i 3.71356 + 4.65665i
24.3 −0.100790 − 2.69368i −0.571812 − 0.456005i −5.25134 + 0.393534i 1.57911 − 1.16544i −1.17070 + 1.58624i 1.26545 − 2.32350i 0.985723 + 8.74854i −0.548534 − 2.40328i −3.29848 − 4.13616i
24.4 −0.0924062 − 2.46961i 0.155526 + 0.124028i −4.09602 + 0.306954i −0.104022 + 0.0767716i 0.291929 − 0.395550i −1.50283 − 2.17750i 0.583151 + 5.17560i −0.658757 − 2.88620i 0.199208 + 0.249799i
24.5 −0.0884620 − 2.36420i −1.98335 − 1.58167i −3.58719 + 0.268823i −1.63675 + 1.20798i −3.56392 + 4.82893i 2.57781 − 0.595740i 0.423097 + 3.75509i 0.764432 + 3.34919i 3.00069 + 3.76275i
24.6 −0.0877980 − 2.34645i −2.51131 − 2.00271i −3.50372 + 0.262568i −0.0550584 + 0.0406350i −4.47877 + 6.06851i −2.43885 + 1.02568i 0.397916 + 3.53160i 1.62830 + 7.13407i 0.100182 + 0.125624i
24.7 −0.0875328 − 2.33936i 1.29379 + 1.03176i −3.47055 + 0.260082i −3.03501 + 2.23994i 2.30042 − 3.11696i 1.75517 − 1.97974i 0.387996 + 3.44356i −0.0582065 − 0.255019i 5.50569 + 6.90392i
24.8 −0.0873425 − 2.33428i 0.631225 + 0.503385i −3.44682 + 0.258303i 1.81645 − 1.34060i 1.11991 − 1.51742i 1.92257 + 1.81761i 0.380927 + 3.38082i −0.522514 − 2.28929i −3.28798 − 4.12300i
24.9 −0.0867721 − 2.31903i 1.20700 + 0.962547i −3.37597 + 0.252994i 2.93028 − 2.16264i 2.12744 − 2.88258i −2.48024 + 0.921084i 0.359980 + 3.19491i −0.137221 − 0.601204i −5.26951 − 6.60776i
24.10 −0.0833672 − 2.22804i 1.35323 + 1.07916i −2.96279 + 0.222030i −2.00261 + 1.47799i 2.29160 − 3.10500i −0.153912 + 2.64127i 0.242420 + 2.15153i −0.000931790 − 0.00408244i 3.45997 + 4.33867i
24.11 −0.0805101 − 2.15168i −2.08962 − 1.66641i −2.62883 + 0.197003i 3.30646 − 2.44028i −3.41735 + 4.63035i 2.56373 + 0.653664i 0.153375 + 1.36124i 0.922001 + 4.03955i −5.51689 − 6.91796i
24.12 −0.0788910 − 2.10841i 2.55145 + 2.03471i −2.44475 + 0.183209i 0.934958 − 0.690030i 4.08871 − 5.54001i 2.42732 − 1.05268i 0.106683 + 0.946841i 1.70227 + 7.45812i −1.52862 − 1.91683i
24.13 −0.0715410 − 1.91197i −0.884372 − 0.705263i −1.65612 + 0.124109i 0.656993 − 0.484883i −1.28518 + 1.74135i −2.28437 + 1.33478i −0.0726730 − 0.644991i −0.382845 − 1.67735i −0.974085 − 1.22146i
24.14 −0.0653250 − 1.74585i −1.54240 − 1.23002i −1.04931 + 0.0786348i −2.24914 + 1.65994i −2.04668 + 2.77315i −1.20114 − 2.35739i −0.185389 − 1.64537i 0.198481 + 0.869602i 3.04493 + 3.81822i
24.15 −0.0561126 − 1.49964i 0.0741105 + 0.0591011i −0.251370 + 0.0188376i 0.913620 − 0.674282i 0.0844720 − 0.114456i 1.94915 − 1.78908i −0.293694 − 2.60660i −0.665563 − 2.91602i −1.06245 − 1.33227i
24.16 −0.0549190 − 1.46774i 2.18772 + 1.74465i −0.156841 + 0.0117536i −0.617888 + 0.456022i 2.44055 − 3.30683i 1.96082 + 1.77629i −0.303035 − 2.68951i 1.07476 + 4.70884i 0.703256 + 0.881855i
24.17 −0.0537978 − 1.43778i 1.23595 + 0.985640i −0.0699018 + 0.00523841i −0.576558 + 0.425520i 1.35064 − 1.83005i −1.73625 − 1.99635i −0.310893 − 2.75925i −0.111468 − 0.488371i 0.642820 + 0.806071i
24.18 −0.0523779 − 1.39983i −0.477286 − 0.380623i 0.0376325 − 0.00282017i −2.47641 + 1.82768i −0.507807 + 0.688054i 1.82721 + 1.91345i −0.319600 − 2.83653i −0.584635 − 2.56145i 2.68814 + 3.37082i
24.19 −0.0500348 − 1.33721i −1.29721 − 1.03449i 0.208783 − 0.0156461i 3.30756 − 2.44109i −1.31843 + 1.78641i −1.69017 − 2.03551i −0.331018 − 2.93787i −0.0549759 − 0.240865i −3.42974 − 4.30076i
24.20 −0.0449070 − 1.20017i −2.02728 − 1.61670i 0.556026 − 0.0416684i −0.867402 + 0.640172i −1.84927 + 2.50567i 1.56731 − 2.13156i −0.343918 − 3.05236i 0.828575 + 3.63022i 0.807265 + 1.01228i
See next 80 embeddings (of 1512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.63
Significant digits:
Format:

Inner twists

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

Hecke kernels

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