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Properties

Label 25.10.e.a
Level $25$
Weight $10$
Character orbit 25.e
Analytic conductor $12.876$
Analytic rank $0$
Dimension $88$
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] = [25,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(12.8758959041\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 5 q^{2} - 5 q^{3} + 5629 q^{4} + 2835 q^{5} - 3619 q^{6} - 1790 q^{8} + 124211 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 5 q^{2} - 5 q^{3} + 5629 q^{4} + 2835 q^{5} - 3619 q^{6} - 1790 q^{8} + 124211 q^{9} + 69155 q^{10} - 47269 q^{11} - 378885 q^{12} - 5 q^{13} + 450943 q^{14} - 340065 q^{15} - 456887 q^{16} + 754715 q^{17} - 153395 q^{19} + 2608515 q^{20} - 1363014 q^{21} - 4340650 q^{22} - 6248245 q^{23} - 6899120 q^{24} + 1884015 q^{25} + 8735826 q^{26} + 17345395 q^{27} + 20042235 q^{28} + 8026945 q^{29} - 39990965 q^{30} - 12756879 q^{31} + 25719035 q^{33} + 10679853 q^{34} + 1252350 q^{35} - 60651027 q^{36} - 22388070 q^{37} + 111106215 q^{38} + 19415527 q^{39} - 89605990 q^{40} - 22944949 q^{41} - 53343715 q^{42} + 125932348 q^{44} + 233340405 q^{45} + 105780001 q^{46} - 75065775 q^{47} - 517872630 q^{48} - 459062716 q^{49} - 89699115 q^{50} + 201283386 q^{51} + 41182740 q^{52} + 294123440 q^{53} + 566108225 q^{54} + 155005685 q^{55} + 417972880 q^{56} + 158032970 q^{58} - 277069545 q^{59} - 687869800 q^{60} - 583888659 q^{61} + 652432920 q^{62} + 705265410 q^{63} - 517602996 q^{64} - 524434150 q^{65} - 1190732558 q^{66} + 489872995 q^{67} + 1281376547 q^{69} + 1024836390 q^{70} - 303972519 q^{71} - 978271345 q^{72} - 1108095485 q^{73} - 2241070572 q^{74} - 2756781835 q^{75} + 2430099440 q^{76} - 486793490 q^{77} + 3200010775 q^{78} + 994068455 q^{79} + 2285739530 q^{80} - 337835307 q^{81} + 2682633855 q^{83} - 358955502 q^{84} + 193954220 q^{85} - 1092217509 q^{86} - 2939033245 q^{87} - 8655162370 q^{88} - 102303250 q^{89} + 8349604895 q^{90} - 1492351214 q^{91} + 13308728730 q^{92} + 2699214873 q^{94} - 5425957335 q^{95} + 1477784936 q^{96} - 3536629235 q^{97} - 6574768060 q^{98} - 5182967688 q^{99}+O(q^{100}) \) 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 −42.2022 − 13.7123i 55.8298 − 76.8432i 1178.78 + 856.432i 831.807 + 1123.04i −3409.84 + 2477.39i − 1560.23i −24649.1 − 33926.6i 3294.48 + 10139.4i −19704.5 − 58800.8i
4.2 −40.8886 − 13.2855i −134.924 + 185.707i 1081.16 + 785.505i −741.082 − 1184.87i 7984.06 − 5800.76i 4412.95i −20832.6 − 28673.6i −10200.2 − 31393.0i 14560.2 + 58293.4i
4.3 −30.3597 − 9.86446i 135.396 − 186.356i 410.186 + 298.018i −1195.97 − 723.035i −5948.88 + 4322.12i − 8910.35i 93.4736 + 128.655i −10314.3 − 31744.1i 29176.9 + 33748.7i
4.4 −30.1028 − 9.78098i 39.2775 − 54.0608i 396.292 + 287.923i 753.740 − 1176.86i −1711.13 + 1243.21i 3312.88i 412.194 + 567.336i 4702.53 + 14472.9i −34200.5 + 28054.4i
4.5 −28.1942 − 9.16084i −26.0220 + 35.8162i 296.774 + 215.619i −1228.02 + 667.153i 1061.77 − 771.424i 1361.49i 2529.54 + 3481.61i 5476.73 + 16855.6i 40734.7 − 7560.15i
4.6 −28.1495 − 9.14633i −105.690 + 145.470i 294.523 + 213.984i 1051.04 + 921.105i 4305.65 − 3128.24i − 9484.78i 2573.93 + 3542.70i −3908.75 − 12029.9i −21161.6 − 35541.8i
4.7 −18.7556 − 6.09407i 149.845 − 206.243i −99.5816 − 72.3503i 242.651 + 1376.32i −4067.29 + 2955.06i 11341.3i 7361.71 + 10132.5i −14000.6 − 43089.3i 3836.29 − 27292.4i
4.8 −15.5560 − 5.05445i −93.5826 + 128.805i −197.775 − 143.692i 1395.34 − 78.5121i 2106.81 − 1530.69i 10735.5i 7272.74 + 10010.1i −1750.75 − 5388.24i −22102.7 − 5831.32i
4.9 −10.0159 − 3.25436i −102.524 + 141.112i −324.490 − 235.755i −474.431 − 1314.55i 1486.09 − 1079.71i − 8099.58i 5652.18 + 7779.56i −3319.02 − 10214.9i 473.830 + 14710.4i
4.10 −6.34561 − 2.06181i 89.8886 − 123.721i −378.201 − 274.779i 1384.84 − 188.011i −825.488 + 599.752i − 8999.66i 3841.34 + 5287.15i −1144.56 − 3522.59i −9175.29 − 1662.23i
4.11 −4.16977 − 1.35484i 16.1156 − 22.1812i −398.665 − 289.647i −168.001 + 1387.41i −97.2502 + 70.6564i − 2755.50i 2589.37 + 3563.96i 5850.09 + 18004.7i 2580.24 − 5557.56i
4.12 −0.601335 − 0.195386i 41.6879 − 57.3784i −413.893 − 300.711i −982.821 − 993.573i −36.2793 + 26.3584i 4837.34i 380.416 + 523.598i 4527.98 + 13935.7i 396.875 + 789.499i
4.13 3.53448 + 1.14842i −151.796 + 208.929i −403.043 − 292.828i −934.066 + 1039.54i −776.456 + 564.128i 6310.76i −2206.68 − 3037.24i −14526.9 − 44709.3i −4495.27 + 2601.53i
4.14 17.4639 + 5.67435i −24.5126 + 33.7387i −141.428 − 102.754i 1082.14 + 884.360i −619.530 + 450.115i 1906.75i −7412.98 − 10203.1i 5544.95 + 17065.6i 13880.2 + 21584.8i
4.15 18.2657 + 5.93488i −83.7000 + 115.203i −115.804 − 84.1365i 749.968 − 1179.27i −2212.56 + 1607.52i − 1702.40i −7395.78 − 10179.4i −183.704 − 565.384i 20697.5 − 17089.2i
4.16 18.3940 + 5.97657i 125.970 − 173.382i −111.598 − 81.0803i −1332.68 + 420.817i 3353.31 − 2436.32i − 4129.61i −7388.61 − 10169.5i −8110.68 − 24962.1i −27028.3 − 224.357i
4.17 21.2451 + 6.90294i 120.972 − 166.504i −10.5142 − 7.63901i 985.407 − 991.009i 3719.43 − 2702.33i 6762.38i −6893.29 − 9487.81i −7006.92 − 21565.1i 27775.9 − 14251.9i
4.18 25.6352 + 8.32940i −68.5538 + 94.3563i 173.570 + 126.106i −1306.59 + 495.925i −2543.33 + 1847.83i − 8082.53i −4712.72 − 6486.50i 1878.90 + 5782.67i −37625.6 + 1830.03i
4.19 35.1664 + 11.4263i −43.4833 + 59.8496i 691.902 + 502.696i −1170.94 − 762.906i −2213.01 + 1607.85i 11580.7i 7459.96 + 10267.8i 4391.20 + 13514.7i −32460.6 − 40208.2i
4.20 35.9786 + 11.6902i 70.7637 − 97.3979i 743.586 + 540.247i 298.253 + 1365.35i 3684.58 − 2677.00i 2617.09i 9052.79 + 12460.1i 1603.53 + 4935.17i −5230.39 + 52609.9i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.22
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 25.10.e.a ✓ 88
25.e even 10 1 inner 25.10.e.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.10.e.a ✓ 88 1.a even 1 1 trivial
25.10.e.a ✓ 88 25.e even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(25, [\chi])\).