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Properties

Label 71.7.f.a
Level $71$
Weight $7$
Character orbit 71.f
Analytic conductor $16.334$
Analytic rank $0$
Dimension $210$
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] = [71,7,Mod(23,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.23");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 71.f (of order \(14\), 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: \(16.3338399370\)
Analytic rank: \(0\)
Dimension: \(210\)
Relative dimension: \(35\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 210 q + 5 q^{2} - 85 q^{3} - 1003 q^{4} - 156 q^{5} - 387 q^{6} - 7 q^{7} + 1829 q^{8} - 9656 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 210 q + 5 q^{2} - 85 q^{3} - 1003 q^{4} - 156 q^{5} - 387 q^{6} - 7 q^{7} + 1829 q^{8} - 9656 q^{9} + 2708 q^{10} - 7 q^{11} + 17648 q^{12} - 7 q^{13} - 6818 q^{15} - 67247 q^{16} + 32795 q^{18} + 6715 q^{19} - 47406 q^{20} - 15967 q^{21} + 35273 q^{22} - 7 q^{23} - 64762 q^{24} + 571062 q^{25} - 67032 q^{26} - 54475 q^{27} + 441 q^{28} + 77251 q^{29} - 216605 q^{30} + 159089 q^{31} - 156595 q^{32} + 108857 q^{33} + 441 q^{34} + 83342 q^{35} + 120113 q^{36} - 117221 q^{37} - 49370 q^{38} - 123431 q^{39} + 150270 q^{40} + 34153 q^{41} + 5096 q^{42} - 438653 q^{43} + 319193 q^{44} + 183196 q^{45} + 466865 q^{47} - 1339763 q^{48} + 516630 q^{49} - 357978 q^{50} + 883036 q^{51} - 279433 q^{52} - 7 q^{53} - 191954 q^{54} + 220402 q^{55} + 863289 q^{56} - 1954170 q^{57} - 422322 q^{58} + 347305 q^{59} - 1465204 q^{60} - 823711 q^{61} + 204638 q^{62} + 3116169 q^{63} - 1002963 q^{64} + 1374618 q^{65} + 2177 q^{67} - 870905 q^{68} - 3364879 q^{69} - 1034902 q^{71} - 1816024 q^{72} + 946123 q^{73} - 3010867 q^{74} - 66639 q^{75} + 1712038 q^{76} + 1246641 q^{77} - 2820615 q^{78} + 211771 q^{79} + 3897450 q^{80} + 296810 q^{81} + 6903092 q^{82} + 3285467 q^{83} + 730758 q^{84} - 4065625 q^{86} - 580259 q^{87} - 1356605 q^{89} - 2408685 q^{90} - 560527 q^{91} + 6785891 q^{92} - 4058103 q^{93} - 4628155 q^{94} + 1738174 q^{95} - 7396410 q^{96} + 5779613 q^{97} + 967549 q^{98} - 5767559 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −9.76772 − 12.2483i −4.50438 + 19.7350i −40.3720 + 176.881i −76.6555 285.718 − 137.594i 82.2456 + 65.5886i 1657.50 − 798.209i 287.627 + 138.514i 748.750 + 938.902i
23.2 −9.31969 − 11.6865i 10.0091 − 43.8526i −35.4769 + 155.434i 218.827 −605.765 + 291.721i 431.382 + 344.016i 1285.21 − 618.924i −1166.06 − 561.544i −2039.40 − 2557.33i
23.3 −8.84883 − 11.0961i 9.88765 − 43.3206i −30.5799 + 133.979i −236.723 −568.183 + 273.623i −279.203 − 222.657i 938.875 − 452.138i −1122.10 − 540.377i 2094.72 + 2626.70i
23.4 −8.09950 − 10.1564i −9.00950 + 39.4732i −23.3102 + 102.129i 193.828 473.880 − 228.208i 82.5048 + 65.7954i 477.004 − 229.713i −820.156 − 394.966i −1569.91 − 1968.60i
23.5 −8.09096 − 10.1457i 4.22534 − 18.5124i −23.2311 + 101.782i 57.7742 −222.009 + 106.914i −348.149 − 277.639i 472.346 − 227.470i 331.950 + 159.859i −467.449 − 586.162i
23.6 −7.51525 − 9.42383i 0.0244697 − 0.107209i −18.0882 + 79.2495i 42.4704 −1.19421 + 0.575102i 9.64465 + 7.69135i 187.740 − 90.4110i 656.795 + 316.296i −319.176 − 400.234i
23.7 −6.86624 − 8.60999i −9.71929 + 42.5830i −12.7454 + 55.8412i −90.0385 433.374 − 208.702i −519.808 − 414.533i −66.7035 + 32.1227i −1062.04 − 511.452i 618.226 + 775.230i
23.8 −6.78986 − 8.51421i 4.37666 − 19.1754i −12.1483 + 53.2253i −102.041 −192.981 + 92.9345i 324.590 + 258.852i −92.2877 + 44.4434i 308.265 + 148.453i 692.845 + 868.800i
23.9 −6.03912 − 7.57282i −7.01303 + 30.7261i −6.63527 + 29.0710i −159.208 275.036 − 132.450i 275.818 + 219.957i −298.293 + 143.651i −238.104 − 114.665i 961.477 + 1205.65i
23.10 −4.94588 − 6.20194i −2.87925 + 12.6148i 0.239040 − 1.04730i 213.429 92.4769 − 44.5345i 139.087 + 110.918i −465.086 + 223.974i 505.962 + 243.659i −1055.59 − 1323.67i
23.11 −3.97956 − 4.99021i 9.62190 − 42.1563i 5.17605 − 22.6778i −54.3606 −248.659 + 119.748i 218.194 + 174.004i −501.805 + 241.657i −1027.77 − 494.946i 216.331 + 271.270i
23.12 −3.96285 − 4.96926i 8.53950 − 37.4140i 5.25199 − 23.0105i 117.615 −219.761 + 105.831i −228.542 − 182.256i −501.653 + 241.583i −670.078 − 322.693i −466.092 − 584.461i
23.13 −3.07316 − 3.85362i −9.17390 + 40.1935i 8.83527 − 38.7099i 10.7787 183.083 − 88.1682i 407.931 + 325.314i −460.539 + 221.784i −874.550 − 421.161i −33.1247 − 41.5371i
23.14 −2.96444 − 3.71729i 0.679010 − 2.97494i 9.21101 − 40.3561i −189.370 −13.0716 + 6.29494i −243.219 − 193.961i −451.480 + 217.421i 648.417 + 312.261i 561.374 + 703.941i
23.15 −2.88266 − 3.61474i −3.82081 + 16.7401i 9.48472 − 41.5553i 96.7128 71.5250 − 34.4446i −275.425 − 219.644i −444.149 + 213.891i 391.175 + 188.380i −278.790 − 349.591i
23.16 −1.16926 − 1.46620i 2.69750 − 11.8185i 13.4588 − 58.9667i 125.878 −20.4824 + 9.86380i 404.217 + 322.352i −210.330 + 101.289i 524.405 + 252.540i −147.184 − 184.563i
23.17 −0.0656390 − 0.0823087i −11.6511 + 51.0468i 14.2389 − 62.3846i 82.1460 4.96636 − 2.39167i −150.847 − 120.297i −12.1399 + 5.84626i −1813.22 − 873.202i −5.39198 − 6.76133i
23.18 0.0708731 + 0.0888721i −5.43457 + 23.8104i 14.2385 − 62.3828i −35.6000 −2.50125 + 1.20454i −20.7898 − 16.5793i 13.1078 − 6.31236i 119.405 + 57.5023i −2.52309 − 3.16385i
23.19 1.17367 + 1.47173i 2.34762 − 10.2856i 13.4528 − 58.9407i −67.9748 17.8930 − 8.61680i 283.876 + 226.383i 211.078 − 101.650i 556.524 + 268.008i −79.7799 − 100.041i
23.20 1.30043 + 1.63069i 10.1967 − 44.6747i 13.2733 − 58.1542i −144.883 86.1106 − 41.4687i −20.8750 − 16.6472i 232.360 − 111.899i −1235.05 − 594.767i −188.410 − 236.258i
See next 80 embeddings (of 210 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.7.f.a ✓ 210
71.f odd 14 1 inner 71.7.f.a ✓ 210
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.7.f.a ✓ 210 1.a even 1 1 trivial
71.7.f.a ✓ 210 71.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(71, [\chi])\).