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Properties

Label 931.2.bt.b
Level $931$
Weight $2$
Character orbit 931.bt
Analytic conductor $7.434$
Analytic rank $0$
Dimension $1080$
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] = [931,2,Mod(75,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([17, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.75");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.bt (of order \(42\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(1080\)
Relative dimension: \(90\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1080 q - 122 q^{4} - 22 q^{5} - 28 q^{6} - 18 q^{7} + 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1080 q - 122 q^{4} - 22 q^{5} - 28 q^{6} - 18 q^{7} + 70 q^{9} + 2 q^{11} + 54 q^{16} - 34 q^{17} - 15 q^{19} - 70 q^{20} - 26 q^{23} - 64 q^{24} - 112 q^{25} - 24 q^{26} - 16 q^{28} - 38 q^{30} + 38 q^{35} + 52 q^{36} + 32 q^{38} - 10 q^{39} - 22 q^{42} - 40 q^{43} - 210 q^{44} - 34 q^{45} + 80 q^{47} - 74 q^{49} - 106 q^{54} - 56 q^{55} - 22 q^{57} + 88 q^{58} - 76 q^{61} + 56 q^{62} + 56 q^{63} + 224 q^{64} - 276 q^{66} + 54 q^{68} - 10 q^{73} - 44 q^{74} - 147 q^{76} - 64 q^{77} - 234 q^{80} + 62 q^{81} + 100 q^{82} + 42 q^{83} + 52 q^{85} + 8 q^{87} - 92 q^{92} + 310 q^{93} - 145 q^{95} - 272 q^{96} - 228 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
75.1 −2.77022 − 0.207599i 3.19840 − 0.986577i 5.65334 + 0.852103i 0.763879 − 0.823266i −9.06508 + 2.06905i −0.441329 − 2.60868i −10.0674 − 2.29782i 6.77774 − 4.62099i −2.28702 + 2.12204i
75.2 −2.71541 − 0.203492i −0.996127 + 0.307265i 5.35438 + 0.807044i 1.39683 − 1.50543i 2.76742 − 0.631646i −2.17415 + 1.50767i −9.06562 − 2.06917i −1.58086 + 1.07781i −4.09932 + 3.80361i
75.3 −2.70368 − 0.202613i −1.99061 + 0.614022i 5.29119 + 0.797519i −1.93434 + 2.08473i 5.50639 − 1.25680i 0.314202 − 2.62703i −8.85756 − 2.02168i 1.10679 − 0.754599i 5.65225 − 5.24452i
75.4 −2.66763 − 0.199911i 0.356956 − 0.110106i 5.09863 + 0.768494i 2.10566 − 2.26937i −0.974237 + 0.222363i 2.61318 − 0.413866i −8.23154 − 1.87880i −2.36342 + 1.61135i −6.07081 + 5.63289i
75.5 −2.66328 − 0.199585i 0.822069 − 0.253575i 5.07556 + 0.765017i −1.47857 + 1.59352i −2.24001 + 0.511267i −2.56391 + 0.652964i −8.15737 − 1.86187i −1.86722 + 1.27305i 4.25589 − 3.94889i
75.6 −2.53546 − 0.190006i 2.05226 − 0.633037i 4.41478 + 0.665420i −2.71127 + 2.92205i −5.32369 + 1.21510i 1.62388 + 2.08878i −6.10940 − 1.39443i 1.33230 − 0.908347i 7.42952 − 6.89358i
75.7 −2.45226 − 0.183772i 1.18206 − 0.364617i 4.00216 + 0.603228i −0.384220 + 0.414091i −2.96573 + 0.676908i 1.30643 − 2.30070i −4.90852 − 1.12034i −1.21440 + 0.827962i 1.01831 − 0.944851i
75.8 −2.35040 − 0.176138i 1.46457 − 0.451759i 3.51567 + 0.529902i 1.29186 − 1.39229i −3.52188 + 0.803846i 1.85710 + 1.88446i −3.57410 − 0.815765i −0.537850 + 0.366700i −3.28161 + 3.04489i
75.9 −2.29775 − 0.172192i −2.27835 + 0.702779i 3.27233 + 0.493224i 0.314022 − 0.338435i 5.35609 − 1.22249i −2.12600 − 1.57485i −2.94121 − 0.671312i 2.21828 − 1.51240i −0.779819 + 0.723566i
75.10 −2.28269 − 0.171064i −1.68919 + 0.521045i 3.20373 + 0.482885i −0.499928 + 0.538794i 3.94501 − 0.900423i 2.64362 − 0.106258i −2.76712 − 0.631577i 0.103144 − 0.0703223i 1.23335 − 1.14438i
75.11 −2.24808 − 0.168471i 2.79293 − 0.861505i 3.04783 + 0.459387i 0.104201 − 0.112301i −6.42388 + 1.46621i −0.946337 + 2.47072i −2.37866 − 0.542914i 4.57955 − 3.12228i −0.253171 + 0.234908i
75.12 −2.22740 − 0.166920i −0.945215 + 0.291560i 2.95577 + 0.445511i 2.11105 − 2.27517i 2.15404 − 0.491645i −0.718668 − 2.54628i −2.15403 − 0.491642i −1.67029 + 1.13879i −5.08191 + 4.71533i
75.13 −2.20181 − 0.165002i −2.90179 + 0.895082i 2.84306 + 0.428522i −2.97362 + 3.20480i 6.53686 − 1.49199i −1.63629 + 2.07908i −1.88391 − 0.429991i 5.14047 − 3.50471i 7.07613 − 6.56569i
75.14 −2.18423 − 0.163685i −3.02509 + 0.933118i 2.76640 + 0.416967i 0.273904 − 0.295198i 6.76023 − 1.54298i 0.552537 + 2.58741i −1.70332 − 0.388771i 5.80177 − 3.95558i −0.646588 + 0.599946i
75.15 −2.13744 − 0.160179i −1.48940 + 0.459420i 2.56532 + 0.386660i 2.01715 − 2.17397i 3.25710 − 0.743411i −0.980082 + 2.45753i −1.24190 − 0.283455i −0.471461 + 0.321437i −4.65976 + 4.32363i
75.16 −2.00208 − 0.150035i 1.82442 − 0.562758i 2.00814 + 0.302679i 2.67100 − 2.87866i −3.73706 + 0.852959i −2.42980 − 1.04694i −0.0603327 − 0.0137706i 0.533085 − 0.363451i −5.77946 + 5.36255i
75.17 −1.90876 − 0.143042i 1.48010 − 0.456549i 1.64524 + 0.247980i −0.975644 + 1.05149i −2.89045 + 0.659727i 1.54478 − 2.14794i 0.627347 + 0.143188i −0.496468 + 0.338486i 2.01268 − 1.86749i
75.18 −1.87731 − 0.140685i 0.0264497 − 0.00815864i 1.52683 + 0.230133i −1.35861 + 1.46423i −0.0508020 + 0.0115952i −2.48364 − 0.911885i 0.836789 + 0.190992i −2.47808 + 1.68953i 2.75652 − 2.55768i
75.19 −1.84753 − 0.138453i −0.621008 + 0.191556i 1.41655 + 0.213510i −2.47211 + 2.66430i 1.17385 − 0.267925i 2.64026 + 0.170443i 1.02497 + 0.233942i −2.12976 + 1.45205i 4.93619 − 4.58012i
75.20 −1.76785 − 0.132482i 0.0555758 − 0.0171429i 1.13007 + 0.170330i −0.406115 + 0.437688i −0.100521 + 0.0229432i 0.114957 + 2.64325i 1.48149 + 0.338141i −2.47592 + 1.68805i 0.775934 − 0.719962i
See next 80 embeddings (of 1080 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 75.90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
49.h odd 42 1 inner
931.bt even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.bt.b ✓ 1080
19.b odd 2 1 inner 931.2.bt.b ✓ 1080
49.h odd 42 1 inner 931.2.bt.b ✓ 1080
931.bt even 42 1 inner 931.2.bt.b ✓ 1080
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.bt.b ✓ 1080 1.a even 1 1 trivial
931.2.bt.b ✓ 1080 19.b odd 2 1 inner
931.2.bt.b ✓ 1080 49.h odd 42 1 inner
931.2.bt.b ✓ 1080 931.bt even 42 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{1080} + 151 T_{2}^{1078} + 11066 T_{2}^{1076} + 520677 T_{2}^{1074} + 17499762 T_{2}^{1072} + \cdots + 50\!\cdots\!25 \) acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\). Copy content Toggle raw display