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Properties

Label 61.8.k.a
Level $61$
Weight $8$
Character orbit 61.k
Analytic conductor $19.055$
Analytic rank $0$
Dimension $280$
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] = [61,8,Mod(4,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 61.k (of order \(30\), 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: \(19.0554865545\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(35\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 7 q^{2} - 60 q^{3} - 2237 q^{4} - 1098 q^{5} + 2039 q^{6} + 5925 q^{7} - 10 q^{8} - 44400 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 7 q^{2} - 60 q^{3} - 2237 q^{4} - 1098 q^{5} + 2039 q^{6} + 5925 q^{7} - 10 q^{8} - 44400 q^{9} - 11733 q^{10} - 50225 q^{12} + 40386 q^{13} - 13077 q^{14} - 654 q^{15} + 133719 q^{16} - 13686 q^{17} + 205552 q^{18} - 168155 q^{19} - 3590 q^{20} + 59103 q^{21} + 347407 q^{22} + 240510 q^{23} - 614565 q^{24} + 950673 q^{25} + 452090 q^{26} - 538713 q^{27} - 1253770 q^{28} + 808719 q^{29} - 1854316 q^{30} - 348875 q^{31} - 1306674 q^{32} - 63835 q^{33} + 165226 q^{34} - 285838 q^{35} - 4574402 q^{36} - 10 q^{37} - 164730 q^{38} - 1136779 q^{39} - 2743827 q^{40} + 1083891 q^{41} + 5146751 q^{42} + 2901896 q^{43} - 357997 q^{44} + 7770065 q^{45} + 6253419 q^{46} + 4027404 q^{47} + 680973 q^{48} + 1658306 q^{49} - 8466427 q^{51} - 8938417 q^{52} - 4663960 q^{53} - 19114987 q^{54} - 8023720 q^{55} + 7124374 q^{56} + 14676958 q^{57} + 20413002 q^{58} + 1799311 q^{59} + 17790088 q^{60} - 2443106 q^{61} - 31684450 q^{62} - 8417990 q^{63} + 3645482 q^{64} - 12073343 q^{65} + 18858044 q^{66} - 9216841 q^{67} + 18527752 q^{68} + 12904330 q^{69} + 28493422 q^{70} + 12854831 q^{71} + 749005 q^{73} + 20054262 q^{74} - 14563615 q^{75} - 15640939 q^{76} - 14019507 q^{77} + 32579143 q^{78} - 1267587 q^{79} - 42514175 q^{80} - 46428706 q^{81} + 9864252 q^{82} - 41019463 q^{83} - 51596810 q^{84} + 75031860 q^{85} - 11991974 q^{86} + 9621163 q^{87} - 52364358 q^{88} + 7174395 q^{89} - 71628123 q^{90} - 3841920 q^{91} + 70313477 q^{92} - 30121491 q^{93} - 102870190 q^{94} + 14858391 q^{95} + 101949444 q^{96} + 46389100 q^{97} + 84253655 q^{98} + 127214585 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −22.3704 − 2.35123i 23.5452 + 17.1066i 369.706 + 78.5834i 13.2121 − 14.6735i −486.495 − 438.042i −299.515 + 672.721i −5347.44 − 1737.49i −414.080 − 1274.41i −330.061 + 297.188i
4.2 −20.1873 − 2.12177i −54.3152 − 39.4623i 277.823 + 59.0530i −258.320 + 286.893i 1012.75 + 911.882i −4.30841 + 9.67685i −3012.15 − 978.708i 717.048 + 2206.85i 5823.50 − 5243.50i
4.3 −18.4887 − 1.94324i −39.3769 − 28.6090i 212.852 + 45.2430i 102.648 − 114.002i 672.431 + 605.460i 586.619 − 1317.57i −1584.31 − 514.772i 56.2439 + 173.101i −2119.36 + 1908.28i
4.4 −18.3230 − 1.92582i −48.9231 − 35.5447i 206.820 + 43.9610i 255.825 − 284.122i 827.964 + 745.502i −376.576 + 845.803i −1462.07 − 475.054i 454.222 + 1397.95i −5234.65 + 4713.30i
4.5 −18.1111 − 1.90355i 28.7250 + 20.8700i 199.186 + 42.3383i 262.089 − 291.079i −480.515 − 432.658i 423.140 − 950.387i −1309.98 − 425.638i −286.248 − 880.981i −5300.81 + 4772.87i
4.6 −17.2245 − 1.81037i 35.7941 + 26.0059i 168.203 + 35.7527i −259.548 + 288.257i −569.455 − 512.739i 334.683 − 751.710i −724.109 − 235.277i −70.9123 − 218.246i 4992.43 − 4495.21i
4.7 −16.3593 − 1.71943i 71.4731 + 51.9282i 139.468 + 29.6449i 73.9058 − 82.0807i −1079.96 − 972.404i −296.425 + 665.782i −228.156 − 74.1325i 1736.04 + 5342.98i −1350.18 + 1215.71i
4.8 −15.4812 − 1.62714i −7.88263 − 5.72707i 111.816 + 23.7672i 23.1417 − 25.7015i 112.714 + 101.488i −251.425 + 564.710i 202.615 + 65.8335i −646.484 − 1989.67i −400.080 + 360.234i
4.9 −13.1192 − 1.37888i −14.6901 − 10.6729i 45.0095 + 9.56707i −238.059 + 264.391i 178.005 + 160.277i −519.837 + 1167.57i 1028.57 + 334.203i −573.934 − 1766.39i 3487.71 − 3140.35i
4.10 −9.55874 − 1.00466i 32.1225 + 23.3383i −34.8428 − 7.40606i 117.153 − 130.112i −283.603 − 255.357i −343.257 + 770.967i 1495.66 + 485.968i −188.646 − 580.592i −1250.56 + 1126.01i
4.11 −8.43552 − 0.886609i −24.8267 − 18.0377i −54.8309 − 11.6547i −86.8519 + 96.4588i 193.434 + 174.169i 489.842 − 1100.20i 1484.75 + 482.425i −384.811 − 1184.33i 818.163 − 736.677i
4.12 −8.18508 − 0.860287i 19.3379 + 14.0498i −58.9474 − 12.5297i 310.083 − 344.383i −146.196 − 131.635i 145.721 − 327.295i 1473.61 + 478.806i −499.263 − 1536.57i −2834.33 + 2552.04i
4.13 −8.04495 − 0.845558i −60.1212 − 43.6806i −61.1967 − 13.0078i 184.705 − 205.136i 446.737 + 402.244i −99.8404 + 224.245i 1466.07 + 476.356i 1030.74 + 3172.30i −1659.40 + 1494.13i
4.14 −7.93518 − 0.834021i −69.1989 − 50.2759i −62.9314 − 13.3765i −215.468 + 239.302i 507.174 + 456.662i 51.7877 − 116.317i 1459.53 + 474.229i 1585.00 + 4878.11i 1909.36 − 1719.20i
4.15 −6.72757 − 0.707096i 58.4048 + 42.4336i −80.4427 − 17.0986i −103.444 + 114.886i −362.918 − 326.773i 488.155 − 1096.41i 1352.59 + 439.482i 934.695 + 2876.70i 777.161 − 699.759i
4.16 −2.81062 − 0.295408i 44.2542 + 32.1526i −117.391 − 24.9521i −225.809 + 250.786i −114.884 − 103.442i −351.558 + 789.612i 666.606 + 216.593i 248.827 + 765.812i 708.748 − 638.160i
4.17 0.641807 + 0.0674566i 2.06335 + 1.49911i −124.796 − 26.5261i −93.9757 + 104.371i 1.22315 + 1.10133i 266.020 − 597.492i −156.866 − 50.9689i −673.810 − 2073.77i −67.3548 + 60.6465i
4.18 0.882056 + 0.0927078i −16.0556 − 11.6650i −124.433 − 26.4491i 221.610 − 246.122i −13.0805 − 11.7777i −163.494 + 367.213i −215.274 − 69.9467i −554.112 − 1705.38i 218.289 − 196.549i
4.19 1.62861 + 0.171174i −39.5814 − 28.7576i −122.580 − 26.0551i 25.9408 − 28.8101i −59.5400 − 53.6101i −674.746 + 1515.50i −394.526 − 128.189i 63.8690 + 196.568i 47.1789 − 42.4800i
4.20 1.96131 + 0.206142i 66.1438 + 48.0563i −121.399 − 25.8041i 222.448 − 247.054i 119.822 + 107.888i 456.781 − 1025.95i −472.856 − 153.640i 1389.78 + 4277.30i 487.217 − 438.692i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.k even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.8.k.a ✓ 280
61.k even 30 1 inner 61.8.k.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.8.k.a ✓ 280 1.a even 1 1 trivial
61.8.k.a ✓ 280 61.k even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(61, [\chi])\).