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Properties

Label 931.2.cd.a
Level $931$
Weight $2$
Character orbit 931.cd
Analytic conductor $7.434$
Analytic rank $0$
Dimension $3276$
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] = [931,2,Mod(10,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(126))
 
chi = DirichletCharacter(H, H._module([39, 119]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.cd (of order \(126\), degree \(36\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(3276\)
Relative dimension: \(91\) over \(\Q(\zeta_{126})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{126}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3276 q - 39 q^{2} - 33 q^{3} - 39 q^{4} - 33 q^{5} - 42 q^{6} - 24 q^{7} - 45 q^{8} - 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3276 q - 39 q^{2} - 33 q^{3} - 39 q^{4} - 33 q^{5} - 42 q^{6} - 24 q^{7} - 45 q^{8} - 39 q^{9} - 33 q^{10} - 30 q^{11} - 69 q^{12} - 12 q^{13} - 27 q^{14} - 39 q^{15} - 27 q^{16} - 60 q^{17} - 75 q^{19} - 168 q^{20} - 12 q^{21} - 30 q^{22} - 39 q^{23} - 6 q^{24} - 39 q^{25} - 63 q^{26} - 75 q^{27} - 9 q^{28} - 42 q^{29} + 18 q^{30} - 54 q^{31} + 24 q^{32} - 33 q^{33} - 6 q^{34} - 135 q^{35} - 111 q^{36} + 36 q^{37} - 60 q^{38} - 96 q^{39} - 51 q^{40} - 96 q^{41} + 51 q^{42} - 54 q^{43} + 66 q^{44} - 99 q^{45} - 63 q^{46} - 87 q^{47} - 63 q^{48} + 24 q^{49} - 63 q^{50} - 39 q^{51} + 69 q^{52} - 57 q^{53} - 96 q^{54} + 3 q^{55} + 54 q^{56} + 24 q^{57} - 78 q^{58} - 78 q^{59} + 114 q^{60} + 210 q^{61} + 3 q^{62} + 132 q^{63} - 225 q^{64} - 108 q^{65} - 261 q^{66} - 27 q^{67} - 54 q^{68} - 63 q^{69} - 111 q^{70} - 75 q^{71} - 393 q^{72} - 102 q^{73} + 51 q^{74} + 21 q^{75} - 96 q^{76} - 66 q^{77} - 603 q^{78} - 78 q^{79} - 18 q^{80} - 78 q^{81} - 138 q^{82} - 57 q^{83} + 594 q^{84} - 342 q^{85} + 288 q^{86} - 21 q^{87} - 54 q^{88} + 93 q^{89} - 66 q^{90} + 144 q^{91} - 36 q^{92} - 252 q^{93} + 540 q^{94} + 33 q^{95} + 567 q^{96} + 27 q^{97} - 138 q^{98} + 84 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.81392 + 0.0701747i −0.334855 + 1.19015i 5.91568 − 0.295239i −1.75022 − 0.492436i 0.858736 − 3.37247i 0.392693 − 2.61645i −11.0117 + 0.825211i 1.25555 + 0.767252i 4.95953 + 1.26285i
10.2 −2.70425 + 0.0674398i 0.559973 − 1.99026i 5.31092 − 0.265057i −2.75384 − 0.774809i −1.38008 + 5.41993i −0.540377 + 2.58998i −8.94912 + 0.670644i −1.08770 − 0.664679i 7.49932 + 1.90956i
10.3 −2.66486 + 0.0664574i 0.0818661 − 0.290969i 5.09953 − 0.254507i 2.48997 + 0.700568i −0.198824 + 0.780832i 1.81475 + 1.92527i −8.25615 + 0.618713i 2.48191 + 1.51666i −6.68197 − 1.70144i
10.4 −2.63329 + 0.0656703i 0.590052 − 2.09717i 4.93241 − 0.246167i 3.63142 + 1.02172i −1.41606 + 5.56121i −2.64404 + 0.0951003i −7.71883 + 0.578447i −1.49008 − 0.910572i −9.62969 − 2.45202i
10.5 −2.63152 + 0.0656260i −0.550482 + 1.95653i 4.92307 − 0.245700i 1.56890 + 0.441420i 1.32021 − 5.18477i −2.63286 + 0.260819i −7.68908 + 0.576217i −0.965109 − 0.589766i −4.15756 − 1.05865i
10.6 −2.46323 + 0.0614290i 0.575263 − 2.04461i 4.06620 − 0.202935i 1.23834 + 0.348415i −1.29141 + 5.07167i 2.42502 − 1.05797i −5.08929 + 0.381390i −1.28962 − 0.788069i −3.07172 − 0.782156i
10.7 −2.44784 + 0.0610454i 0.518226 − 1.84189i 3.99069 − 0.199167i −0.930533 − 0.261811i −1.15610 + 4.54028i −2.20798 − 1.45768i −4.87291 + 0.365174i −0.564114 − 0.344723i 2.29378 + 0.584068i
10.8 −2.44081 + 0.0608699i −0.875194 + 3.11063i 3.95631 − 0.197451i −2.85455 − 0.803147i 1.94684 − 7.64570i 1.80935 + 1.93035i −4.77509 + 0.357844i −6.35015 − 3.88050i 7.01630 + 1.78657i
10.9 −2.43325 + 0.0606814i −0.868768 + 3.08779i 3.91949 − 0.195614i 2.53450 + 0.713097i 1.92656 − 7.56606i 1.84342 − 1.89784i −4.67082 + 0.350030i −6.21979 − 3.80083i −6.21032 − 1.58134i
10.10 −2.37993 + 0.0593518i −0.333986 + 1.18706i 3.66304 − 0.182815i 1.18985 + 0.334771i 0.724409 − 2.84493i 1.35635 + 2.27163i −3.95890 + 0.296679i 1.26232 + 0.771387i −2.85162 − 0.726112i
10.11 −2.37474 + 0.0592223i 0.00353276 − 0.0125562i 3.63836 − 0.181583i −3.26083 − 0.917455i −0.00764577 + 0.0300268i 2.64374 − 0.103241i −3.89172 + 0.291644i 2.55973 + 1.56422i 7.79794 + 1.98560i
10.12 −2.34795 + 0.0585543i −0.672931 + 2.39174i 3.51194 − 0.175274i −0.325321 − 0.0915310i 1.43996 − 5.65509i −2.26062 + 1.37462i −3.55137 + 0.266139i −2.70771 − 1.65465i 0.769197 + 0.195862i
10.13 −2.21097 + 0.0551382i −0.00372443 + 0.0132374i 2.88785 − 0.144126i −1.67497 − 0.471264i 0.00750472 − 0.0294729i −2.54235 − 0.732442i −1.96605 + 0.147335i 2.55971 + 1.56421i 3.72930 + 0.949597i
10.14 −2.05922 + 0.0513537i 0.102179 − 0.363164i 2.24024 − 0.111806i −0.305267 − 0.0858889i −0.191758 + 0.753083i 1.10090 − 2.40583i −0.499196 + 0.0374096i 2.43842 + 1.49009i 0.633023 + 0.161188i
10.15 −2.00688 + 0.0500485i 0.592979 − 2.10757i 2.02755 − 0.101191i −0.162097 − 0.0456071i −1.08456 + 4.25933i −0.791357 + 2.52463i −0.0602131 + 0.00451235i −1.53037 − 0.935191i 0.327593 + 0.0834154i
10.16 −1.96592 + 0.0490271i −0.302038 + 1.07351i 1.86494 − 0.0930754i 3.76285 + 1.05870i 0.541152 − 2.12524i −1.22449 − 2.34534i 0.260303 − 0.0195070i 1.49869 + 0.915828i −7.44938 − 1.89685i
10.17 −1.95335 + 0.0487136i 0.839959 − 2.98539i 1.81570 − 0.0906177i −2.53356 − 0.712832i −1.49531 + 5.87244i 2.17827 + 1.50171i 0.354709 − 0.0265818i −5.64716 − 3.45091i 4.98365 + 1.26899i
10.18 −1.88800 + 0.0470837i 0.418641 − 1.48794i 1.56481 − 0.0780962i 1.88853 + 0.531350i −0.720336 + 2.82894i 0.836581 − 2.51001i 0.815935 − 0.0611458i 0.521172 + 0.318482i −3.59056 − 0.914269i
10.19 −1.81767 + 0.0453297i −0.0742973 + 0.264068i 1.30434 − 0.0650968i −3.76238 − 1.05857i 0.123078 − 0.483356i −2.29218 + 1.32133i 1.25839 − 0.0943035i 2.49566 + 1.52507i 6.88673 + 1.75358i
10.20 −1.76552 + 0.0440293i 0.710087 − 2.52380i 1.11761 − 0.0557776i 3.96239 + 1.11485i −1.14255 + 4.48708i 2.50379 + 0.854993i 1.55155 − 0.116273i −3.30546 − 2.01992i −7.04478 − 1.79382i
See next 80 embeddings (of 3276 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.91
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
931.cd even 126 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.cd.a ✓ 3276
19.f odd 18 1 931.2.ci.a yes 3276
49.h odd 42 1 931.2.ci.a yes 3276
931.cd even 126 1 inner 931.2.cd.a ✓ 3276
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.cd.a ✓ 3276 1.a even 1 1 trivial
931.2.cd.a ✓ 3276 931.cd even 126 1 inner
931.2.ci.a yes 3276 19.f odd 18 1
931.2.ci.a yes 3276 49.h odd 42 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(931, [\chi])\).