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Properties

Label 41.10.d.a
Level $41$
Weight $10$
Character orbit 41.d
Analytic conductor $21.116$
Analytic rank $0$
Dimension $120$
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] = [41,10,Mod(10,41)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(41, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("41.10");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 41 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 41.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: \(21.1164692827\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 29 q^{2} + 138 q^{3} - 7171 q^{4} - 1029 q^{5} - 3713 q^{6} - 3 q^{7} + 11266 q^{8} + 674522 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 29 q^{2} + 138 q^{3} - 7171 q^{4} - 1029 q^{5} - 3713 q^{6} - 3 q^{7} + 11266 q^{8} + 674522 q^{9} - 12854 q^{10} + 24738 q^{11} + 39274 q^{12} - 46315 q^{13} + 871034 q^{14} + 360228 q^{15} - 914555 q^{16} - 530149 q^{17} - 25426 q^{18} + 1653832 q^{19} + 4372080 q^{20} - 2130428 q^{21} + 2343001 q^{22} + 2652951 q^{23} + 1106688 q^{24} - 14812735 q^{25} - 6118947 q^{26} + 4180704 q^{27} - 396988 q^{28} - 21623877 q^{29} + 17457778 q^{30} + 6573049 q^{31} - 69342624 q^{32} - 7358251 q^{33} - 8916139 q^{34} + 16136455 q^{35} - 45797760 q^{36} + 33933035 q^{37} + 53400620 q^{38} - 67954622 q^{39} + 110831318 q^{40} - 35740729 q^{41} - 79466024 q^{42} - 76181258 q^{43} + 155633242 q^{44} - 35853333 q^{45} + 140852222 q^{46} + 88286859 q^{47} + 73075535 q^{48} - 281729819 q^{49} - 11977752 q^{50} + 84186546 q^{51} + 255219799 q^{52} - 127910215 q^{53} - 34429074 q^{54} + 130107222 q^{55} + 553640573 q^{56} + 128307432 q^{57} + 300749586 q^{58} - 164829871 q^{59} - 874588408 q^{60} + 578922433 q^{61} + 819367194 q^{62} - 369190165 q^{63} - 338871554 q^{64} - 443441755 q^{65} - 296767968 q^{66} + 13004448 q^{67} - 836712288 q^{68} - 787228706 q^{69} - 2139157547 q^{70} + 599147 q^{71} + 894599137 q^{72} - 176992838 q^{73} - 246331571 q^{74} + 1233144981 q^{75} + 1237713308 q^{76} + 851523751 q^{77} - 983629818 q^{78} + 793058716 q^{79} + 191186357 q^{80} + 1401569296 q^{81} - 2456158137 q^{82} - 3738142678 q^{83} - 593901940 q^{84} - 1539481470 q^{85} + 2544218019 q^{86} - 1184135808 q^{87} + 6713998444 q^{88} + 2672032677 q^{89} + 2669557922 q^{90} - 1251852986 q^{91} - 5662411827 q^{92} + 6666427220 q^{93} + 3586951096 q^{94} - 7531966557 q^{95} + 5903395820 q^{96} + 6521978543 q^{97} + 1686956314 q^{98} + 2271695038 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −34.2137 − 24.8577i −41.9582 394.453 + 1214.00i −674.710 − 2076.54i 1435.54 + 1042.98i −9389.18 + 6821.64i 9990.55 − 30747.7i −17922.5 −28533.7 + 87817.8i
10.2 −33.6267 − 24.4313i 166.346 375.655 + 1156.15i 227.380 + 699.804i −5593.68 − 4064.05i 3881.07 − 2819.77i 9037.81 − 27815.5i 7988.03 9451.04 − 29087.3i
10.3 −31.7851 − 23.0932i −142.014 318.779 + 981.101i 58.5912 + 180.325i 4513.93 + 3279.56i 4453.00 − 3235.30i 6308.27 − 19414.8i 484.977 2301.96 − 7084.72i
10.4 −26.1779 − 19.0193i −205.556 165.329 + 508.831i 643.988 + 1981.99i 5381.01 + 3909.53i −5442.19 + 3953.99i 230.140 − 708.297i 22570.1 20837.9 − 64132.6i
10.5 −25.2980 − 18.3801i 93.6679 143.945 + 443.016i 397.038 + 1221.96i −2369.61 − 1721.62i −5047.66 + 3667.34i −446.284 + 1373.52i −10909.3 12415.4 − 38210.6i
10.6 −24.1200 − 17.5242i 114.608 116.459 + 358.424i −763.544 − 2349.95i −2764.35 − 2008.42i 6932.87 − 5037.03i −1244.96 + 3831.60i −6547.91 −22764.2 + 70061.1i
10.7 −23.0350 − 16.7359i 229.332 92.3033 + 284.080i −455.949 − 1403.27i −5282.66 − 3838.08i −2896.00 + 2104.07i −1876.74 + 5776.01i 32910.2 −12982.1 + 39954.9i
10.8 −19.4722 − 14.1474i −278.982 20.8012 + 64.0196i −689.358 − 2121.63i 5432.38 + 3946.86i −1648.02 + 1197.35i −3307.45 + 10179.3i 58147.8 −16592.1 + 51065.3i
10.9 −18.7982 − 13.6577i −71.5815 8.62315 + 26.5393i −121.496 − 373.927i 1345.60 + 977.638i 3811.19 − 2768.99i −3475.93 + 10697.8i −14559.1 −2823.07 + 8688.52i
10.10 −11.6828 − 8.48804i −62.5806 −93.7761 − 288.613i −316.643 − 974.527i 731.115 + 531.186i −5013.91 + 3642.82i −3638.96 + 11199.6i −15766.7 −4572.55 + 14072.9i
10.11 −11.3768 − 8.26575i 268.173 −97.1071 − 298.865i 340.119 + 1046.78i −3050.95 − 2216.65i 1513.79 − 1099.83i −3590.50 + 11050.4i 52233.5 4782.94 − 14720.4i
10.12 −11.3764 − 8.26544i 42.8468 −97.1116 − 298.879i 806.283 + 2481.48i −487.443 − 354.148i 6472.02 − 4702.20i −3590.43 + 11050.2i −17847.2 11338.0 − 34894.6i
10.13 −5.65927 − 4.11170i 121.246 −143.095 − 440.403i −176.680 − 543.765i −686.163 − 498.526i −5870.47 + 4265.15i −2107.75 + 6486.99i −4982.45 −1235.92 + 3803.77i
10.14 −3.62869 − 2.63640i −215.857 −152.000 − 467.808i 172.941 + 532.258i 783.279 + 569.086i 6732.17 − 4891.21i −1391.42 + 4282.34i 26911.5 775.693 − 2387.34i
10.15 −0.361875 − 0.262918i −167.755 −158.155 − 486.751i 437.652 + 1346.95i 60.7065 + 44.1058i −5225.25 + 3796.36i −141.514 + 435.534i 8458.88 195.763 − 602.496i
10.16 1.34918 + 0.980237i 112.079 −157.357 − 484.296i −405.189 − 1247.04i 151.215 + 109.864i 7699.97 − 5594.35i 526.276 − 1619.71i −7121.19 675.724 − 2079.67i
10.17 3.99041 + 2.89921i 168.752 −150.699 − 463.803i −11.6202 − 35.7634i 673.389 + 489.246i 2028.30 − 1473.64i 1523.70 − 4689.47i 8794.10 57.3159 − 176.400i
10.18 8.53429 + 6.20052i −109.650 −123.829 − 381.107i −672.208 − 2068.84i −935.782 − 679.886i −190.967 + 138.746i 2975.29 − 9157.00i −7659.94 7091.09 − 21824.1i
10.19 11.6428 + 8.45900i 62.0715 −94.2162 − 289.968i 494.520 + 1521.98i 722.687 + 525.063i −6852.98 + 4978.98i 3632.84 − 11180.7i −15830.1 −7116.80 + 21903.2i
10.20 16.4075 + 11.9208i −27.3820 −31.1144 − 95.7601i 379.991 + 1169.49i −449.270 − 326.414i 5079.83 − 3690.71i 3839.79 − 11817.7i −18933.2 −7706.54 + 23718.3i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.30
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 41.10.d.a ✓ 120
41.d even 5 1 inner 41.10.d.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.10.d.a ✓ 120 1.a even 1 1 trivial
41.10.d.a ✓ 120 41.d even 5 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(41, [\chi])\).