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Properties

Label 931.2.ch.a
Level $931$
Weight $2$
Character orbit 931.ch
Analytic conductor $7.434$
Analytic rank $0$
Dimension $3312$
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(13,931)]
 
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([99, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.ch (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: \(3312\)
Relative dimension: \(92\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 3312 q - 30 q^{2} - 42 q^{3} - 30 q^{4} - 42 q^{5} - 42 q^{6} - 15 q^{7} - 45 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3312 q - 30 q^{2} - 42 q^{3} - 30 q^{4} - 42 q^{5} - 42 q^{6} - 15 q^{7} - 45 q^{8} - 30 q^{9} - 42 q^{10} - 30 q^{11} - 63 q^{12} - 42 q^{13} - 63 q^{14} - 12 q^{15} - 18 q^{16} - 42 q^{17} + 84 q^{20} - 30 q^{21} - 30 q^{22} - 30 q^{23} - 42 q^{24} - 30 q^{25} - 63 q^{26} - 63 q^{27} - 48 q^{28} - 6 q^{29} - 36 q^{30} + 24 q^{32} - 42 q^{33} - 42 q^{34} - 18 q^{35} + 132 q^{36} - 42 q^{38} - 132 q^{39} - 42 q^{40} - 42 q^{41} - 39 q^{42} + 60 q^{43} + 102 q^{44} - 126 q^{45} - 135 q^{46} - 42 q^{47} - 39 q^{49} - 198 q^{50} - 30 q^{51} + 126 q^{52} + 6 q^{53} + 84 q^{54} - 42 q^{55} - 138 q^{57} - 60 q^{58} - 42 q^{59} + 6 q^{60} - 42 q^{61} - 42 q^{62} - 111 q^{63} - 291 q^{64} - 153 q^{65} - 252 q^{66} - 78 q^{67} - 63 q^{69} + 69 q^{70} + 60 q^{71} + 552 q^{72} - 42 q^{73} - 84 q^{74} - 42 q^{76} - 30 q^{77} - 18 q^{78} - 162 q^{79} - 96 q^{81} - 147 q^{82} - 21 q^{83} - 594 q^{84} + 90 q^{85} + 72 q^{86} - 21 q^{87} - 45 q^{88} - 105 q^{89} + 231 q^{90} + 207 q^{91} - 18 q^{92} - 165 q^{93} + 42 q^{95} + 630 q^{96} - 102 q^{98} - 384 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.67410 + 2.21012i −0.106405 − 0.316698i −1.54033 − 5.47466i −0.0679315 − 0.0599499i 0.878072 + 0.295016i −2.12508 + 1.57608i 9.51645 + 3.73493i 2.30242 − 1.74402i 0.246220 − 0.0497745i
13.2 −1.64442 + 2.17094i −0.685090 − 2.03907i −1.46717 − 5.21464i 2.06803 + 1.82504i 5.55327 + 1.86580i −0.961854 − 2.46472i 8.66296 + 3.39996i −1.29705 + 0.982476i −7.36277 + 1.48842i
13.3 −1.60077 + 2.11331i 0.813476 + 2.42119i −1.36193 − 4.84060i 1.33773 + 1.18055i −6.41891 − 2.15664i 0.853736 + 2.50422i 7.47408 + 2.93336i −2.80901 + 2.12774i −4.63627 + 0.937243i
13.4 −1.59937 + 2.11146i 0.799033 + 2.37820i −1.35859 − 4.82873i 0.915587 + 0.808010i −6.29941 − 2.11649i −0.673699 − 2.55854i 7.43711 + 2.91885i −2.62599 + 1.98911i −3.17044 + 0.640917i
13.5 −1.58328 + 2.09022i −0.335617 − 0.998912i −1.32055 − 4.69353i −0.213325 − 0.188260i 2.61932 + 0.880043i 2.55804 − 0.675589i 7.01948 + 2.75494i 1.50621 − 1.14091i 0.731257 − 0.147827i
13.6 −1.49393 + 1.97227i 0.557528 + 1.65940i −1.11632 − 3.96762i −1.73070 − 1.52735i −4.10568 − 1.37944i 2.53945 + 0.742422i 4.88658 + 1.91784i −0.0513648 + 0.0389074i 5.59790 − 1.13164i
13.7 −1.46986 + 1.94048i −0.428937 − 1.27667i −1.06330 − 3.77920i −2.71207 − 2.39342i 3.10782 + 1.04417i 0.660463 + 2.56199i 4.36425 + 1.71284i 0.945508 − 0.716195i 8.63072 − 1.74474i
13.8 −1.46094 + 1.92871i −1.00278 − 2.98462i −1.04389 − 3.71020i 0.0970164 + 0.0856175i 7.22146 + 2.42628i −0.939808 + 2.47321i 4.17633 + 1.63909i −5.51101 + 4.17443i −0.306866 + 0.0620343i
13.9 −1.41437 + 1.86723i 1.07206 + 3.19084i −0.944422 − 3.35668i −3.21682 − 2.83886i −7.47433 − 2.51124i −2.45230 + 0.993081i 3.24243 + 1.27256i −6.64072 + 5.03015i 9.85058 − 1.99134i
13.10 −1.41193 + 1.86400i −0.327101 − 0.973565i −0.939289 − 3.33843i −2.50582 − 2.21140i 2.27657 + 0.764888i −2.43387 − 1.03744i 3.19557 + 1.25417i 1.55056 − 1.17451i 7.66010 − 1.54852i
13.11 −1.41136 + 1.86325i 0.229127 + 0.681963i −0.938094 − 3.33419i 2.92719 + 2.58326i −1.59405 − 0.535572i 2.41755 − 1.07491i 3.18469 + 1.24990i 1.97882 − 1.49890i −8.94457 + 1.80818i
13.12 −1.38723 + 1.83140i −0.802911 − 2.38974i −0.887927 − 3.15588i 1.76134 + 1.55439i 5.49039 + 1.84467i 2.28208 + 1.33869i 2.73410 + 1.07305i −2.67481 + 2.02609i −5.29008 + 1.06941i
13.13 −1.37925 + 1.82087i 0.258398 + 0.769084i −0.871540 − 3.09764i 0.406350 + 0.358606i −1.75680 − 0.590253i −2.03529 − 1.69045i 2.58973 + 1.01639i 1.86668 − 1.41395i −1.21343 + 0.245301i
13.14 −1.23383 + 1.62888i 0.380366 + 1.13210i −0.589236 − 2.09427i 0.00169082 + 0.00149216i −2.31337 − 0.777250i −2.63059 − 0.282848i 0.333991 + 0.131082i 1.25442 − 0.950185i −0.00451673 0.000913077i
13.15 −1.21429 + 1.60309i −0.473788 − 1.41016i −0.553701 − 1.96797i −1.27972 − 1.12936i 2.83592 + 0.952818i 0.870066 − 2.49860i 0.0830836 + 0.0326079i 0.627329 − 0.475183i 3.36442 − 0.680132i
13.16 −1.20917 + 1.59633i 0.484707 + 1.44266i −0.544491 − 1.93524i −0.972153 − 0.857930i −2.88905 − 0.970671i −0.988118 + 2.45431i 0.0193428 + 0.00759149i 0.545078 − 0.412881i 2.54504 − 0.514492i
13.17 −1.14478 + 1.51132i −0.211928 − 0.630771i −0.431884 − 1.53501i 1.34440 + 1.18644i 1.19591 + 0.401804i 0.181276 + 2.63953i −0.715470 − 0.280801i 2.03844 − 1.54406i −3.33213 + 0.673604i
13.18 −1.13557 + 1.49916i 0.329027 + 0.979299i −0.416284 − 1.47956i 2.64271 + 2.33220i −1.84176 − 0.618799i 0.722888 − 2.54508i −0.810556 − 0.318120i 1.54063 − 1.16698i −6.49734 + 1.31346i
13.19 −1.11886 + 1.47710i −0.675252 − 2.00978i −0.388298 − 1.38009i 2.64849 + 2.33731i 3.72417 + 1.25125i −2.51561 − 0.819568i −0.976866 − 0.383392i −1.19187 + 0.902808i −6.41573 + 1.29697i
13.20 −1.08550 + 1.43305i 1.05264 + 3.13301i −0.333660 − 1.18590i 0.305915 + 0.269971i −5.63241 − 1.89239i 2.06305 − 1.65645i −1.28534 − 0.504457i −6.31633 + 4.78443i −0.718952 + 0.145339i
See next 80 embeddings (of 3312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner
49.f odd 14 1 inner
931.ch 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.ch.a ✓ 3312
19.f odd 18 1 inner 931.2.ch.a ✓ 3312
49.f odd 14 1 inner 931.2.ch.a ✓ 3312
931.ch even 126 1 inner 931.2.ch.a ✓ 3312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.ch.a ✓ 3312 1.a even 1 1 trivial
931.2.ch.a ✓ 3312 19.f odd 18 1 inner
931.2.ch.a ✓ 3312 49.f odd 14 1 inner
931.2.ch.a ✓ 3312 931.ch even 126 1 inner

Hecke kernels

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