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Properties

Label 62.8.g.a
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 144 q^{2} - q^{3} - 1152 q^{4} + 45 q^{5} + 1072 q^{6} - 1482 q^{7} - 9216 q^{8} + 3248 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 144 q^{2} - q^{3} - 1152 q^{4} + 45 q^{5} + 1072 q^{6} - 1482 q^{7} - 9216 q^{8} + 3248 q^{9} + 3200 q^{10} + 11631 q^{11} - 64 q^{12} - 28132 q^{13} + 15024 q^{14} + 90398 q^{15} - 73728 q^{16} - 44152 q^{17} + 74824 q^{18} + 9481 q^{19} + 7040 q^{20} - 258569 q^{21} - 184152 q^{22} + 64303 q^{23} - 33792 q^{24} - 436711 q^{25} + 312864 q^{26} + 357302 q^{27} - 157248 q^{28} + 230841 q^{29} - 475216 q^{30} - 333457 q^{31} + 2359296 q^{32} - 1495744 q^{33} + 460944 q^{34} + 615600 q^{35} - 1612928 q^{36} + 1937054 q^{37} - 804192 q^{38} - 530658 q^{39} + 56320 q^{40} - 1873479 q^{41} - 446312 q^{42} + 207918 q^{43} - 155456 q^{44} + 2652772 q^{45} + 85464 q^{46} + 2813457 q^{47} - 270336 q^{48} - 1511049 q^{49} + 432792 q^{50} - 8968912 q^{51} - 1800448 q^{52} - 3285879 q^{53} + 2858416 q^{54} + 6512707 q^{55} - 474624 q^{56} + 10645924 q^{57} + 3637048 q^{58} - 5688600 q^{59} + 5785472 q^{60} - 17087360 q^{61} - 1761816 q^{62} + 1321290 q^{63} - 4718592 q^{64} - 6893278 q^{65} + 7949288 q^{66} + 1131492 q^{67} + 5116352 q^{68} + 5725947 q^{69} + 4924800 q^{70} + 5739133 q^{71} + 4788736 q^{72} + 10538747 q^{73} - 7278248 q^{74} + 13301242 q^{75} + 5015424 q^{76} - 30037767 q^{77} + 353776 q^{78} - 40212351 q^{79} - 2723840 q^{80} + 15047212 q^{81} - 18811272 q^{82} + 39365888 q^{83} + 6416384 q^{84} + 19290231 q^{85} + 24468024 q^{86} - 7650093 q^{87} + 6390272 q^{88} + 21784446 q^{89} - 30629464 q^{90} + 7869428 q^{91} - 9598208 q^{92} + 9495164 q^{93} - 15104944 q^{94} - 40502384 q^{95} - 32768 q^{96} - 42979683 q^{97} - 16303512 q^{98} + 68444739 q^{99}+O(q^{100}) \) 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 −72.0578 + 15.3164i −51.7771 + 37.6183i 96.6908 − 167.473i −294.670 − 510.384i 1531.40 + 681.823i −414.217 − 300.946i 2959.81 − 1317.79i 1513.25 + 321.650i
7.2 2.47214 + 7.60845i −62.1350 + 13.2072i −51.7771 + 37.6183i 16.6583 − 28.8530i −254.093 − 440.101i −836.156 − 372.280i −414.217 − 300.946i 1688.41 − 751.727i 260.708 + 55.4153i
7.3 2.47214 + 7.60845i −29.3157 + 6.23124i −51.7771 + 37.6183i −252.563 + 437.451i −119.882 − 207.642i −130.016 − 57.8870i −414.217 − 300.946i −1177.34 + 524.187i −3952.70 − 840.171i
7.4 2.47214 + 7.60845i −16.7353 + 3.55719i −51.7771 + 37.6183i −25.7246 + 44.5564i −68.4366 − 118.536i 286.559 + 127.584i −414.217 − 300.946i −1730.51 + 770.472i −402.600 − 85.5753i
7.5 2.47214 + 7.60845i 4.85646 − 1.03227i −51.7771 + 37.6183i 192.123 − 332.767i 19.8598 + 34.3983i −652.602 − 290.557i −414.217 − 300.946i −1975.40 + 879.507i 3006.79 + 639.114i
7.6 2.47214 + 7.60845i 32.5995 − 6.92924i −51.7771 + 37.6183i 229.451 − 397.421i 133.311 + 230.902i 689.985 + 307.201i −414.217 − 300.946i −983.209 + 437.753i 3591.00 + 763.290i
7.7 2.47214 + 7.60845i 54.9934 − 11.6892i −51.7771 + 37.6183i −80.6739 + 139.731i 224.888 + 389.517i −1485.55 − 661.409i −414.217 − 300.946i 889.710 − 396.125i −1262.58 − 268.369i
7.8 2.47214 + 7.60845i 58.2152 − 12.3740i −51.7771 + 37.6183i −181.520 + 314.402i 238.063 + 412.337i 1138.16 + 506.742i −414.217 − 300.946i 1237.97 − 551.178i −2840.85 − 603.842i
7.9 2.47214 + 7.60845i 76.5640 − 16.2742i −51.7771 + 37.6183i 42.5489 − 73.6968i 313.098 + 542.301i −5.92267 − 2.63694i −414.217 − 300.946i 3599.27 − 1602.50i 665.905 + 141.542i
9.1 2.47214 − 7.60845i −72.0578 − 15.3164i −51.7771 − 37.6183i 96.6908 + 167.473i −294.670 + 510.384i 1531.40 − 681.823i −414.217 + 300.946i 2959.81 + 1317.79i 1513.25 − 321.650i
9.2 2.47214 − 7.60845i −62.1350 − 13.2072i −51.7771 − 37.6183i 16.6583 + 28.8530i −254.093 + 440.101i −836.156 + 372.280i −414.217 + 300.946i 1688.41 + 751.727i 260.708 − 55.4153i
9.3 2.47214 − 7.60845i −29.3157 − 6.23124i −51.7771 − 37.6183i −252.563 − 437.451i −119.882 + 207.642i −130.016 + 57.8870i −414.217 + 300.946i −1177.34 − 524.187i −3952.70 + 840.171i
9.4 2.47214 − 7.60845i −16.7353 − 3.55719i −51.7771 − 37.6183i −25.7246 − 44.5564i −68.4366 + 118.536i 286.559 − 127.584i −414.217 + 300.946i −1730.51 − 770.472i −402.600 + 85.5753i
9.5 2.47214 − 7.60845i 4.85646 + 1.03227i −51.7771 − 37.6183i 192.123 + 332.767i 19.8598 − 34.3983i −652.602 + 290.557i −414.217 + 300.946i −1975.40 − 879.507i 3006.79 − 639.114i
9.6 2.47214 − 7.60845i 32.5995 + 6.92924i −51.7771 − 37.6183i 229.451 + 397.421i 133.311 − 230.902i 689.985 − 307.201i −414.217 + 300.946i −983.209 − 437.753i 3591.00 − 763.290i
9.7 2.47214 − 7.60845i 54.9934 + 11.6892i −51.7771 − 37.6183i −80.6739 − 139.731i 224.888 − 389.517i −1485.55 + 661.409i −414.217 + 300.946i 889.710 + 396.125i −1262.58 + 268.369i
9.8 2.47214 − 7.60845i 58.2152 + 12.3740i −51.7771 − 37.6183i −181.520 − 314.402i 238.063 − 412.337i 1138.16 − 506.742i −414.217 + 300.946i 1237.97 + 551.178i −2840.85 + 603.842i
9.9 2.47214 − 7.60845i 76.5640 + 16.2742i −51.7771 − 37.6183i 42.5489 + 73.6968i 313.098 − 542.301i −5.92267 + 2.63694i −414.217 + 300.946i 3599.27 + 1602.50i 665.905 − 141.542i
19.1 −6.47214 − 4.70228i −68.5039 − 30.4999i 19.7771 + 60.8676i −145.456 + 251.937i 299.947 + 519.524i 731.830 − 812.780i 158.217 − 486.941i 2299.15 + 2553.47i 2126.09 − 946.595i
19.2 −6.47214 − 4.70228i −67.1786 − 29.9099i 19.7771 + 60.8676i 218.286 − 378.083i 294.145 + 509.474i −325.973 + 362.029i 158.217 − 486.941i 2154.98 + 2393.35i −3190.63 + 1420.56i
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.a ✓ 72
31.g even 15 1 inner 62.8.g.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.8.g.a ✓ 72 1.a even 1 1 trivial
62.8.g.a ✓ 72 31.g even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} + T_{3}^{71} - 11465 T_{3}^{70} - 323022 T_{3}^{69} + 24785134 T_{3}^{68} + \cdots + 28\!\cdots\!25 \) acting on \(S_{8}^{\mathrm{new}}(62, [\chi])\). Copy content Toggle raw display