↑

Properties

Label 31.8.d.a
Level $31$
Weight $8$
Character orbit 31.d
Analytic conductor $9.684$
Analytic rank $0$
Dimension $68$
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] = [31,8,Mod(2,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.2");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 31.d (of order \(5\), degree \(4\), 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: \(9.68393579001\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 3 q^{2} - 31 q^{3} - 847 q^{4} - 150 q^{5} + 1910 q^{6} - 3083 q^{7} - 4145 q^{8} - 11638 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 3 q^{2} - 31 q^{3} - 847 q^{4} - 150 q^{5} + 1910 q^{6} - 3083 q^{7} - 4145 q^{8} - 11638 q^{9} - 4066 q^{10} - 3350 q^{11} + 18302 q^{12} - 4566 q^{13} - 22674 q^{14} - 17419 q^{15} - 23735 q^{16} - 48783 q^{17} - 55866 q^{18} - 130237 q^{19} + 2681 q^{20} - 51409 q^{21} - 65256 q^{22} - 87781 q^{23} - 129545 q^{24} + 626450 q^{25} + 197136 q^{26} - 356005 q^{27} + 420876 q^{28} - 55255 q^{29} - 257790 q^{30} - 401087 q^{31} + 1917102 q^{32} - 302252 q^{33} + 103435 q^{34} - 98916 q^{35} + 1209728 q^{36} - 804288 q^{37} - 949850 q^{38} - 200855 q^{39} + 2072206 q^{40} + 162104 q^{41} + 779504 q^{42} - 1128511 q^{43} + 2219244 q^{44} + 1449628 q^{45} - 1933151 q^{46} + 2374327 q^{47} - 5053016 q^{48} + 1015654 q^{49} + 2382174 q^{50} + 3785939 q^{51} - 3360083 q^{52} + 251589 q^{53} - 14853886 q^{54} - 3785370 q^{55} - 11112192 q^{56} + 4804900 q^{57} - 100225 q^{58} - 2363755 q^{59} + 351853 q^{60} + 6256200 q^{61} - 5290808 q^{62} + 20270344 q^{63} - 6330397 q^{64} - 12056037 q^{65} + 1282378 q^{66} + 6302792 q^{67} - 8621214 q^{68} - 4288703 q^{69} + 6635885 q^{70} - 4167799 q^{71} + 8824730 q^{72} + 12083369 q^{73} + 14129813 q^{74} + 8150887 q^{75} + 11150225 q^{76} - 4669056 q^{77} + 27304243 q^{78} + 4078852 q^{79} - 22381710 q^{80} + 16789559 q^{81} + 19537574 q^{82} - 36261506 q^{83} - 38222583 q^{84} - 16248678 q^{85} + 24931667 q^{86} - 53910770 q^{87} + 46023300 q^{88} + 15742217 q^{89} - 22759926 q^{90} + 70088614 q^{91} - 2843138 q^{92} - 4194616 q^{93} + 83662642 q^{94} + 14177479 q^{95} - 25089026 q^{96} - 1390083 q^{97} - 129391244 q^{98} + 57003176 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −17.2320 − 12.5198i −42.7608 + 31.0676i 100.642 + 309.746i 40.3913 1125.81 −23.6257 − 72.7126i 1301.18 − 4004.61i 187.475 − 576.990i −696.022 − 505.689i
2.2 −15.9006 − 11.5524i 45.6187 − 33.1439i 79.8149 + 245.645i 13.9295 −1108.25 −360.523 − 1109.58i 791.292 − 2435.35i 306.724 − 943.999i −221.487 − 160.920i
2.3 −12.4066 − 9.01390i 28.8666 − 20.9728i 33.1185 + 101.928i 201.404 −547.183 351.308 + 1081.21i −98.6941 + 303.749i −282.398 + 869.131i −2498.73 − 1815.44i
2.4 −11.9130 − 8.65533i −20.6864 + 15.0296i 27.4516 + 84.4874i −409.647 376.525 58.3819 + 179.681i −178.214 + 548.487i −473.779 + 1458.14i 4880.14 + 3545.63i
2.5 −9.16083 − 6.65573i −54.0123 + 39.2422i 0.0678316 + 0.208764i 420.552 755.983 165.897 + 510.578i −447.120 + 1376.09i 701.556 − 2159.17i −3852.60 − 2799.08i
2.6 −6.53547 − 4.74830i −23.3099 + 16.9356i −19.3881 − 59.6704i 72.9858 232.756 −523.451 − 1611.02i −476.153 + 1465.45i −419.285 + 1290.43i −476.997 − 346.559i
2.7 −6.50431 − 4.72565i 64.4742 − 46.8433i −19.5800 − 60.2610i −455.989 −640.725 −46.4660 − 143.008i −475.425 + 1463.21i 1286.81 − 3960.40i 2965.89 + 2154.85i
2.8 −1.86128 − 1.35230i 38.9638 − 28.3089i −37.9185 − 116.701i 324.781 −110.805 −166.467 − 512.332i −178.239 + 548.564i 40.9670 − 126.083i −604.509 − 439.201i
2.9 −0.759382 − 0.551723i −71.2807 + 51.7885i −39.2819 − 120.897i −287.819 82.7022 104.281 + 320.945i −73.9993 + 227.746i 1723.07 − 5303.08i 218.565 + 158.797i
2.10 0.383353 + 0.278522i 1.41506 − 1.02810i −39.4848 − 121.522i −156.006 0.828815 292.863 + 901.339i 37.4526 − 115.267i −674.875 + 2077.05i −59.8055 − 43.4513i
2.11 6.76632 + 4.91602i −39.4447 + 28.6582i −17.9383 − 55.2085i 36.1221 −407.780 −99.6682 − 306.747i 480.846 − 1479.89i 58.7676 − 180.868i 244.414 + 177.577i
2.12 8.84710 + 6.42779i −23.8554 + 17.3319i −2.59956 − 8.00064i 498.304 −322.457 132.869 + 408.928i 460.977 − 1418.74i −407.137 + 1253.04i 4408.54 + 3202.99i
2.13 9.07596 + 6.59407i 60.4266 − 43.9025i −0.662924 − 2.04027i 29.5340 837.926 134.864 + 415.069i 451.176 − 1388.58i 1048.13 − 3225.80i 268.049 + 194.749i
2.14 9.19874 + 6.68328i 11.9988 − 8.71763i 0.396461 + 1.22018i −419.743 168.636 −512.116 − 1576.13i 445.234 − 1370.29i −607.846 + 1870.76i −3861.11 − 2805.26i
2.15 14.9505 + 10.8621i −55.6238 + 40.4130i 65.9758 + 203.052i −82.3490 −1270.57 −12.9897 − 39.9782i −488.265 + 1502.72i 784.970 − 2415.89i −1231.15 − 894.486i
2.16 15.2852 + 11.1053i 11.3840 − 8.27093i 70.7538 + 217.758i −260.275 265.857 436.162 + 1342.37i −589.470 + 1814.20i −614.634 + 1891.65i −3978.35 − 2890.44i
2.17 16.4573 + 11.9570i 29.3303 − 21.3097i 88.3210 + 271.824i 316.946 737.499 −324.733 − 999.424i −992.034 + 3053.17i −269.657 + 829.918i 5216.09 + 3789.71i
4.1 −6.21729 − 19.1348i −6.66015 + 20.4978i −223.933 + 162.697i −211.276 433.631 148.175 − 107.656i 2421.98 + 1759.67i 1393.52 + 1012.45i 1313.56 + 4042.73i
4.2 −5.93558 − 18.2678i 20.1570 − 62.0369i −194.929 + 141.624i 237.413 −1252.92 −1297.74 + 942.862i 1755.12 + 1275.17i −1672.95 − 1215.47i −1409.18 − 4337.02i
4.3 −4.87036 − 14.9894i 4.50379 − 13.8612i −97.4085 + 70.7714i 487.176 −229.707 1239.70 − 900.695i −96.8606 − 70.3733i 1597.47 + 1160.63i −2372.72 − 7302.50i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.8.d.a ✓ 68
31.d even 5 1 inner 31.8.d.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.8.d.a ✓ 68 1.a even 1 1 trivial
31.8.d.a ✓ 68 31.d even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(31, [\chi])\).