↑

Properties

Label 731.2.k.b
Level $731$
Weight $2$
Character orbit 731.k
Analytic conductor $5.837$
Analytic rank $0$
Dimension $180$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [731,2,Mod(35,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.35");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.k (of order \(7\), degree \(6\), 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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(30\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 180 q + 2 q^{3} - 34 q^{4} + 2 q^{5} + 6 q^{6} - 54 q^{7} + 14 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 180 q + 2 q^{3} - 34 q^{4} + 2 q^{5} + 6 q^{6} - 54 q^{7} + 14 q^{8} - 18 q^{9} - 8 q^{10} + 4 q^{11} - 4 q^{12} - 5 q^{13} + 24 q^{14} + 11 q^{15} - 58 q^{16} + 30 q^{17} - 84 q^{18} - 4 q^{20} - 28 q^{21} - 49 q^{22} - 23 q^{23} - 6 q^{24} - 14 q^{25} + 14 q^{26} + 20 q^{27} - 14 q^{28} - 60 q^{29} - 5 q^{30} + 36 q^{31} + 39 q^{32} - 39 q^{33} - 14 q^{35} + 224 q^{36} - 184 q^{37} + 32 q^{38} + 58 q^{39} + 22 q^{40} - 48 q^{41} - 58 q^{42} + 2 q^{43} + 134 q^{44} - 54 q^{45} - 5 q^{46} - 35 q^{47} + 44 q^{48} + 174 q^{49} - 70 q^{50} - 2 q^{51} - 22 q^{52} + 8 q^{53} + 70 q^{54} + 51 q^{55} - 61 q^{56} + 72 q^{57} + 29 q^{58} + 35 q^{59} + 96 q^{60} - 40 q^{61} + 20 q^{62} - 35 q^{63} - 18 q^{64} - 17 q^{65} - 218 q^{66} + 16 q^{67} + 27 q^{68} + 50 q^{69} + 72 q^{70} - 20 q^{71} - 143 q^{72} - 4 q^{73} - 35 q^{74} - 45 q^{75} - 148 q^{76} + 40 q^{77} - 220 q^{78} - 12 q^{79} - 222 q^{80} + 12 q^{81} + 50 q^{82} + 16 q^{83} - 170 q^{84} - 2 q^{85} + 108 q^{86} + 78 q^{87} - 14 q^{88} + 55 q^{89} + 105 q^{90} - 53 q^{91} - 28 q^{92} - 142 q^{93} + 108 q^{94} - 49 q^{95} - 148 q^{96} + 73 q^{97} + 6 q^{98} - 73 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −0.609444 + 2.67015i 0.133331 + 0.584162i −4.95634 − 2.38685i −0.259403 + 0.325281i −1.64106 1.45101 5.97861 − 7.49695i 2.37944 − 1.14588i −0.710458 − 0.890886i
35.2 −0.560314 + 2.45489i −0.473877 − 2.07619i −3.91062 − 1.88326i −2.10087 + 2.63441i 5.36236 2.59720 3.67444 − 4.60760i −1.38311 + 0.666071i −5.29005 − 6.63352i
35.3 −0.520342 + 2.27977i 0.631123 + 2.76513i −3.12464 − 1.50475i 0.571583 − 0.716742i −6.63225 0.491347 2.14042 − 2.68400i −4.54473 + 2.18863i 1.33659 + 1.67603i
35.4 −0.487718 + 2.13683i 0.382826 + 1.67727i −2.52625 − 1.21658i 2.58588 − 3.24259i −3.77076 −1.56444 1.09861 − 1.37762i 0.0362221 − 0.0174436i 5.66770 + 7.10707i
35.5 −0.464976 + 2.03719i 0.214156 + 0.938278i −2.13202 − 1.02673i −1.50470 + 1.88683i −2.01103 −4.68221 0.477312 − 0.598530i 1.86840 − 0.899776i −3.14419 − 3.94269i
35.6 −0.452186 + 1.98116i −0.537392 − 2.35447i −1.91857 − 0.923934i 0.431020 − 0.540482i 4.90757 −1.66996 0.164014 − 0.205667i −2.55183 + 1.22889i 0.875878 + 1.09832i
35.7 −0.437378 + 1.91628i −0.377133 − 1.65233i −1.67888 − 0.808507i 1.35653 − 1.70103i 3.33126 −1.53495 −0.167382 + 0.209890i 0.114954 − 0.0553592i 2.66633 + 3.34348i
35.8 −0.335627 + 1.47048i −0.201066 − 0.880929i −0.247722 − 0.119297i −1.49293 + 1.87208i 1.36287 3.95170 −1.62225 + 2.03423i 1.96730 − 0.947401i −2.25178 − 2.82365i
35.9 −0.317607 + 1.39153i 0.469666 + 2.05774i −0.0335331 − 0.0161487i −2.08668 + 2.61661i −3.01257 −1.13533 −1.74671 + 2.19030i −1.31080 + 0.631250i −2.97834 − 3.73472i
35.10 −0.263173 + 1.15304i 0.751127 + 3.29090i 0.541703 + 0.260871i 0.905581 − 1.13556i −3.99221 4.03940 −1.91815 + 2.40528i −7.56295 + 3.64213i 1.07102 + 1.34302i
35.11 −0.224709 + 0.984515i 0.253079 + 1.10881i 0.883161 + 0.425308i −0.00986560 + 0.0123711i −1.14851 1.32938 −1.87642 + 2.35296i 1.53749 − 0.740415i −0.00996262 − 0.0124927i
35.12 −0.212222 + 0.929804i −0.151920 − 0.665606i 0.982440 + 0.473118i 2.43514 − 3.05357i 0.651124 4.10878 −1.83767 + 2.30436i 2.28296 − 1.09941i 2.32243 + 2.91224i
35.13 −0.0957949 + 0.419705i −0.643935 − 2.82126i 1.63496 + 0.787356i 0.574144 − 0.719953i 1.24578 1.75614 −1.02390 + 1.28393i −4.84197 + 2.33177i 0.247168 + 0.309939i
35.14 −0.0946208 + 0.414561i −0.485779 − 2.12833i 1.63903 + 0.789315i −0.816059 + 1.02331i 0.928289 −4.53691 −1.01255 + 1.26970i −1.59092 + 0.766147i −0.347007 − 0.435132i
35.15 −0.0268087 + 0.117457i 0.561962 + 2.46211i 1.78886 + 0.861470i −1.64843 + 2.06707i −0.304257 −2.08745 −0.299375 + 0.375405i −3.04330 + 1.46558i −0.198599 − 0.249035i
35.16 0.0248321 − 0.108796i 0.413289 + 1.81074i 1.79072 + 0.862364i 1.10115 − 1.38080i 0.207265 −1.59919 0.277445 − 0.347905i −0.405063 + 0.195068i −0.122882 − 0.154090i
35.17 0.0787180 − 0.344886i −0.0523067 − 0.229171i 1.68919 + 0.813470i −0.573925 + 0.719679i −0.0831553 0.480778 0.854650 − 1.07170i 2.65312 − 1.27768i 0.203029 + 0.254590i
35.18 0.195877 − 0.858195i 0.00959066 + 0.0420194i 1.10381 + 0.531566i 1.96536 − 2.46448i 0.0379394 −5.12558 1.77007 − 2.21960i 2.70123 − 1.30085i −1.73003 − 2.16939i
35.19 0.205703 − 0.901243i −0.218156 − 0.955803i 1.03201 + 0.496991i 0.408806 − 0.512626i −0.906286 4.36242 1.81293 − 2.27334i 1.83694 − 0.884623i −0.377908 − 0.473882i
35.20 0.231641 − 1.01488i −0.681367 − 2.98526i 0.825605 + 0.397590i 2.03864 − 2.55637i −3.18753 −0.963834 1.89284 − 2.37354i −5.74463 + 2.76647i −2.12219 − 2.66115i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.k.b ✓ 180
43.e even 7 1 inner 731.2.k.b ✓ 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.k.b ✓ 180 1.a even 1 1 trivial
731.2.k.b ✓ 180 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{180} + 47 T_{2}^{178} - 14 T_{2}^{177} + 1230 T_{2}^{176} - 677 T_{2}^{175} + \cdots + 234897341502321 \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\). Copy content Toggle raw display