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Properties

Label 62.8.g.b
Level $62$
Weight $8$
Character orbit 62.g
Analytic conductor $19.368$
Analytic rank $0$
Dimension $72$
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] = [62,8,Mod(7,62)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(62, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([28]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("62.7");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 62 = 2 \cdot 31 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 62.g (of order \(15\), degree \(8\), 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: \(19.3678715800\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 144 q^{2} + 53 q^{3} - 1152 q^{4} + 97 q^{5} + 656 q^{6} - 3516 q^{7} + 9216 q^{8} + 4100 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 144 q^{2} + 53 q^{3} - 1152 q^{4} + 97 q^{5} + 656 q^{6} - 3516 q^{7} + 9216 q^{8} + 4100 q^{9} - 3616 q^{10} - 6385 q^{11} + 3392 q^{12} - 7108 q^{13} - 25872 q^{14} - 37456 q^{15} - 73728 q^{16} - 8376 q^{17} - 70280 q^{18} - 49913 q^{19} + 2048 q^{20} + 323383 q^{21} - 67960 q^{22} - 345829 q^{23} + 6144 q^{24} - 372591 q^{25} + 227584 q^{26} - 71038 q^{27} - 142784 q^{28} - 324657 q^{29} - 926672 q^{30} - 1450215 q^{31} - 2359296 q^{32} + 83140 q^{33} + 542768 q^{34} + 220312 q^{35} - 1649280 q^{36} - 1077276 q^{37} + 242544 q^{38} - 2082538 q^{39} - 16384 q^{40} - 561831 q^{41} + 2046856 q^{42} + 541504 q^{43} + 69440 q^{44} - 3560198 q^{45} - 1210168 q^{46} + 701593 q^{47} - 49152 q^{48} + 2939863 q^{49} - 2047672 q^{50} + 2343644 q^{51} - 454912 q^{52} + 5870237 q^{53} + 568304 q^{54} + 4095901 q^{55} + 358912 q^{56} - 2524936 q^{57} - 4505944 q^{58} + 3564614 q^{59} - 2397184 q^{60} + 6621588 q^{61} - 6588440 q^{62} + 37503466 q^{63} - 4718592 q^{64} + 2623034 q^{65} - 1265000 q^{66} + 6997086 q^{67} + 1117376 q^{68} + 18122809 q^{69} - 1762496 q^{70} + 17340555 q^{71} - 4497920 q^{72} + 893895 q^{73} - 7099272 q^{74} - 4691620 q^{75} + 11121408 q^{76} - 12966527 q^{77} - 31068816 q^{78} + 16057037 q^{79} - 2510848 q^{80} + 17531660 q^{81} + 31965608 q^{82} + 35981510 q^{83} + 13155712 q^{84} + 8693293 q^{85} - 8631272 q^{86} - 7183623 q^{87} + 263680 q^{88} - 23416374 q^{89} - 76756056 q^{90} + 13467668 q^{91} + 24903424 q^{92} - 88528776 q^{93} - 19578704 q^{94} - 9538258 q^{95} - 1736704 q^{96} - 11254011 q^{97} + 30592696 q^{98} + 20676441 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.47214 − 7.60845i −85.5242 + 18.1787i −51.7771 + 37.6183i 85.4938 − 148.080i 349.740 + 605.767i 403.909 + 179.832i 414.217 + 300.946i 4986.00 − 2219.91i −1338.01 − 284.403i
7.2 −2.47214 − 7.60845i −49.8104 + 10.5875i −51.7771 + 37.6183i −198.975 + 344.634i 203.693 + 352.806i 769.064 + 342.409i 414.217 + 300.946i 371.054 − 165.204i 3114.03 + 661.907i
7.3 −2.47214 − 7.60845i −34.2192 + 7.27353i −51.7771 + 37.6183i 141.827 − 245.651i 139.935 + 242.374i 360.576 + 160.539i 414.217 + 300.946i −879.871 + 391.744i −2219.64 − 471.799i
7.4 −2.47214 − 7.60845i −27.9180 + 5.93415i −51.7771 + 37.6183i −188.074 + 325.754i 114.167 + 197.743i −1642.75 − 731.401i 414.217 + 300.946i −1253.72 + 558.194i 2943.43 + 625.645i
7.5 −2.47214 − 7.60845i −20.1699 + 4.28724i −51.7771 + 37.6183i 104.818 − 181.551i 82.4820 + 142.863i −243.185 − 108.273i 414.217 + 300.946i −1609.48 + 716.587i −1640.45 − 348.687i
7.6 −2.47214 − 7.60845i 34.2775 − 7.28590i −51.7771 + 37.6183i −132.279 + 229.114i −140.173 − 242.787i 455.779 + 202.926i 414.217 + 300.946i −876.063 + 390.048i 2070.21 + 440.037i
7.7 −2.47214 − 7.60845i 49.2561 − 10.4697i −51.7771 + 37.6183i 158.162 − 273.945i −201.426 − 348.880i −1044.21 − 464.913i 414.217 + 300.946i 318.621 − 141.859i −2475.30 − 526.141i
7.8 −2.47214 − 7.60845i 61.5772 − 13.0886i −51.7771 + 37.6183i −130.560 + 226.137i −251.812 − 436.150i −320.081 − 142.509i 414.217 + 300.946i 1622.52 − 722.391i 2043.32 + 434.320i
7.9 −2.47214 − 7.60845i 66.6958 − 14.1766i −51.7771 + 37.6183i 217.612 − 376.916i −272.743 − 472.405i 1419.54 + 632.018i 414.217 + 300.946i 2249.43 − 1001.51i −3405.71 − 723.907i
9.1 −2.47214 + 7.60845i −85.5242 − 18.1787i −51.7771 − 37.6183i 85.4938 + 148.080i 349.740 − 605.767i 403.909 − 179.832i 414.217 − 300.946i 4986.00 + 2219.91i −1338.01 + 284.403i
9.2 −2.47214 + 7.60845i −49.8104 − 10.5875i −51.7771 − 37.6183i −198.975 − 344.634i 203.693 − 352.806i 769.064 − 342.409i 414.217 − 300.946i 371.054 + 165.204i 3114.03 − 661.907i
9.3 −2.47214 + 7.60845i −34.2192 − 7.27353i −51.7771 − 37.6183i 141.827 + 245.651i 139.935 − 242.374i 360.576 − 160.539i 414.217 − 300.946i −879.871 − 391.744i −2219.64 + 471.799i
9.4 −2.47214 + 7.60845i −27.9180 − 5.93415i −51.7771 − 37.6183i −188.074 − 325.754i 114.167 − 197.743i −1642.75 + 731.401i 414.217 − 300.946i −1253.72 − 558.194i 2943.43 − 625.645i
9.5 −2.47214 + 7.60845i −20.1699 − 4.28724i −51.7771 − 37.6183i 104.818 + 181.551i 82.4820 − 142.863i −243.185 + 108.273i 414.217 − 300.946i −1609.48 − 716.587i −1640.45 + 348.687i
9.6 −2.47214 + 7.60845i 34.2775 + 7.28590i −51.7771 − 37.6183i −132.279 − 229.114i −140.173 + 242.787i 455.779 − 202.926i 414.217 − 300.946i −876.063 − 390.048i 2070.21 − 440.037i
9.7 −2.47214 + 7.60845i 49.2561 + 10.4697i −51.7771 − 37.6183i 158.162 + 273.945i −201.426 + 348.880i −1044.21 + 464.913i 414.217 − 300.946i 318.621 + 141.859i −2475.30 + 526.141i
9.8 −2.47214 + 7.60845i 61.5772 + 13.0886i −51.7771 − 37.6183i −130.560 − 226.137i −251.812 + 436.150i −320.081 + 142.509i 414.217 − 300.946i 1622.52 + 722.391i 2043.32 − 434.320i
9.9 −2.47214 + 7.60845i 66.6958 + 14.1766i −51.7771 − 37.6183i 217.612 + 376.916i −272.743 + 472.405i 1419.54 − 632.018i 414.217 − 300.946i 2249.43 + 1001.51i −3405.71 + 723.907i
19.1 6.47214 + 4.70228i −64.0587 − 28.5208i 19.7771 + 60.8676i −246.786 + 427.446i −280.484 − 485.813i 122.648 − 136.214i −158.217 + 486.941i 1826.70 + 2028.75i −3607.20 + 1606.03i
19.2 6.47214 + 4.70228i −59.9156 − 26.6762i 19.7771 + 60.8676i 85.4015 − 147.920i −262.343 − 454.392i −1205.22 + 1338.54i −158.217 + 486.941i 1414.88 + 1571.38i 1248.29 − 555.775i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 62.8.g.b ✓ 72
31.g even 15 1 inner 62.8.g.b ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.8.g.b ✓ 72 1.a even 1 1 trivial
62.8.g.b ✓ 72 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - 53 T_{3}^{71} - 10487 T_{3}^{70} + 594124 T_{3}^{69} + 14614744 T_{3}^{68} + \cdots + 51\!\cdots\!25 \) acting on \(S_{8}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display