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Properties

Label 25.6.e.a
Level $25$
Weight $6$
Character orbit 25.e
Analytic conductor $4.010$
Analytic rank $0$
Dimension $48$
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] = [25,6,Mod(4,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.4");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.e (of order \(10\), 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: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} - 5 q^{3} + 189 q^{4} - 60 q^{5} - 139 q^{6} + 610 q^{8} + 1031 q^{9} + 435 q^{10} - 724 q^{11} + 315 q^{12} - 5 q^{13} - 497 q^{14} - 105 q^{15} - 6967 q^{16} - 960 q^{17} + 4075 q^{19} + 9155 q^{20} + 2016 q^{21} - 5450 q^{22} - 10995 q^{23} - 20720 q^{24} - 16610 q^{25} + 21666 q^{26} - 13805 q^{27} + 14235 q^{28} + 17195 q^{29} + 39595 q^{30} + 8631 q^{31} + 28460 q^{33} - 24067 q^{34} - 4145 q^{35} - 43827 q^{36} + 19980 q^{37} - 77385 q^{38} + 6037 q^{39} + 93210 q^{40} - 26514 q^{41} + 182885 q^{42} + 26748 q^{44} - 94185 q^{45} - 14559 q^{46} - 7700 q^{47} - 127830 q^{48} - 130126 q^{49} - 231115 q^{50} + 121116 q^{51} - 189260 q^{52} + 45840 q^{53} + 19985 q^{54} + 79370 q^{55} - 43280 q^{56} + 337370 q^{58} + 47640 q^{59} + 71480 q^{60} + 7791 q^{61} + 135320 q^{62} - 35790 q^{63} + 68204 q^{64} + 197485 q^{65} - 14798 q^{66} - 66180 q^{67} + 17957 q^{69} - 129930 q^{70} - 46694 q^{71} - 50545 q^{72} - 18985 q^{73} - 183292 q^{74} - 202210 q^{75} + 273360 q^{76} - 132990 q^{77} - 665225 q^{78} - 54065 q^{79} + 362810 q^{80} - 382452 q^{81} + 161155 q^{83} + 474258 q^{84} - 83075 q^{85} + 74691 q^{86} + 897680 q^{87} + 578430 q^{88} + 436475 q^{89} + 775655 q^{90} + 181046 q^{91} - 1412870 q^{92} - 336327 q^{94} - 362785 q^{95} - 766024 q^{96} - 475210 q^{97} - 361660 q^{98} - 451848 q^{99}+O(q^{100}) \) 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 −9.71189 − 3.15559i −6.12321 + 8.42787i 58.4746 + 42.4843i −38.3382 + 40.6839i 86.0628 − 62.5283i − 211.819i −241.764 − 332.759i 41.5558 + 127.895i 500.718 − 274.138i
4.2 −8.74738 − 2.84220i 16.5578 − 22.7899i 42.5500 + 30.9144i 53.9635 − 14.5924i −209.611 + 152.291i − 122.989i −111.339 − 153.244i −170.126 − 523.593i −513.514 − 25.7300i
4.3 −6.91215 − 2.24589i 1.65280 − 2.27488i 16.8453 + 12.2388i −34.0654 − 44.3232i −16.5335 + 12.0123i 197.082i 47.7523 + 65.7253i 72.6478 + 223.587i 135.920 + 382.876i
4.4 −5.43609 − 1.76629i −17.9990 + 24.7735i 0.542750 + 0.394331i 37.4160 − 41.5336i 141.602 − 102.880i − 28.6321i 105.256 + 144.873i −214.672 − 660.693i −276.757 + 159.693i
4.5 −4.17440 − 1.35635i −0.0707334 + 0.0973562i −10.3026 − 7.48525i 26.6799 + 49.1241i 0.427319 − 0.310465i 51.4533i 115.412 + 158.851i 75.0867 + 231.093i −44.7435 − 241.251i
4.6 −0.580095 − 0.188484i 15.1807 − 20.8945i −25.5876 − 18.5905i −52.0512 + 20.3881i −12.7445 + 9.25945i 10.2303i 22.8118 + 31.3978i −131.033 − 403.279i 34.0375 − 2.01618i
4.7 0.835458 + 0.271457i 2.21975 − 3.05523i −25.2642 − 18.3555i 40.9701 − 38.0322i 2.68387 − 1.94995i − 153.044i −32.6474 − 44.9353i 70.6840 + 217.543i 44.5529 − 20.6527i
4.8 2.22114 + 0.721693i −9.09397 + 12.5168i −21.4759 − 15.6032i −55.6985 − 4.76195i −29.2323 + 21.2385i − 43.5518i −80.3681 − 110.617i 1.12167 + 3.45216i −120.278 − 50.7742i
4.9 5.71541 + 1.85705i −10.7874 + 14.8476i 3.32876 + 2.41849i 37.8738 + 41.1166i −89.2272 + 64.8274i 177.090i −98.5002 − 135.574i −28.9918 − 89.2276i 140.109 + 305.332i
4.10 6.57040 + 2.13485i 10.0602 − 13.8467i 12.7241 + 9.24458i 15.9125 − 53.5891i 95.6602 − 69.5012i 188.942i −66.0770 − 90.9472i −15.4317 − 47.4939i 218.956 − 318.131i
4.11 8.05151 + 2.61610i 8.11728 − 11.1725i 32.0944 + 23.3179i −0.713405 + 55.8971i 94.5847 − 68.7198i − 150.001i 38.1710 + 52.5378i 16.1571 + 49.7264i −151.976 + 448.190i
4.12 10.3591 + 3.36586i −9.28713 + 12.7826i 70.0926 + 50.9253i −25.7064 − 49.6405i −139.231 + 101.157i − 71.6578i 349.815 + 481.478i −2.05400 − 6.32156i −99.2113 − 600.754i
9.1 −6.42181 + 8.83886i 13.2720 − 4.31235i −26.9973 − 83.0892i −22.7744 − 51.0522i −47.1143 + 145.003i − 134.700i 575.283 + 186.921i −39.0403 + 28.3645i 597.496 + 126.548i
9.2 −5.77566 + 7.94951i −24.0141 + 7.80267i −19.9480 − 61.3935i 11.5397 + 54.6977i 76.6701 − 235.966i 104.735i 304.214 + 98.8452i 319.207 − 231.917i −501.469 − 224.180i
9.3 −3.62228 + 4.98565i 10.7412 − 3.49003i −1.84718 − 5.68504i 55.8518 + 2.36063i −21.5076 + 66.1937i 133.004i −152.517 − 49.5557i −93.3979 + 67.8576i −214.080 + 269.907i
9.4 −3.32714 + 4.57941i 3.38178 − 1.09881i −0.0126365 − 0.0388913i −43.0053 + 35.7147i −6.21976 + 19.1425i − 56.5747i −172.049 − 55.9023i −186.362 + 135.400i −20.4679 − 315.767i
9.5 −2.41181 + 3.31957i −18.6046 + 6.04499i 4.68583 + 14.4215i 9.93343 − 55.0121i 24.8039 − 76.3384i − 138.174i −184.051 − 59.8017i 112.997 − 82.0971i 158.659 + 165.653i
9.6 0.123616 − 0.170142i 28.0358 − 9.10938i 9.87488 + 30.3917i −30.6717 − 46.7359i 1.91577 − 5.89614i 86.5992i 12.7921 + 4.15640i 506.434 − 367.946i −11.7433 − 0.558733i
9.7 0.958035 − 1.31862i −8.25369 + 2.68179i 9.06761 + 27.9072i −55.8091 + 3.21620i −4.37106 + 13.4527i 127.304i 95.0904 + 30.8967i −135.660 + 98.5625i −49.2261 + 76.6723i
9.8 1.61555 − 2.22361i 12.9949 − 4.22230i 7.55409 + 23.2491i 30.9389 + 46.5595i 11.6052 − 35.7170i − 221.003i 147.549 + 47.9417i −45.5512 + 33.0949i 153.514 + 6.42313i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.6.e.a ✓ 48
25.e even 10 1 inner 25.6.e.a ✓ 48
25.f odd 20 2 625.6.a.g 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.e.a ✓ 48 1.a even 1 1 trivial
25.6.e.a ✓ 48 25.e even 10 1 inner
625.6.a.g 48 25.f odd 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(25, [\chi])\).