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Properties

Label 25.8.e.a
Level $25$
Weight $8$
Character orbit 25.e
Analytic conductor $7.810$
Analytic rank $0$
Dimension $64$
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] = [25,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(7.80962563710\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 5 q^{2} - 5 q^{3} + 893 q^{4} - 510 q^{5} + 173 q^{6} - 2510 q^{8} + 6587 q^{9} - 5805 q^{10} - 472 q^{11} + 16635 q^{12} - 5 q^{13} - 19409 q^{14} + 375 q^{15} - 42071 q^{16} + 31910 q^{17} - 48685 q^{19} - 171405 q^{20} + 2268 q^{21} + 197230 q^{22} + 356525 q^{23} + 544720 q^{24} + 170220 q^{25} - 708942 q^{26} - 319805 q^{27} - 565125 q^{28} - 332885 q^{29} + 66235 q^{30} - 39057 q^{31} - 161110 q^{33} + 1156301 q^{34} + 337515 q^{35} + 1920669 q^{36} - 1480960 q^{37} - 2136945 q^{38} + 1307509 q^{39} - 282470 q^{40} + 1955408 q^{41} - 684355 q^{42} - 3558164 q^{44} - 1351725 q^{45} - 2982527 q^{46} + 1018980 q^{47} + 8967210 q^{48} - 3298162 q^{49} + 8550165 q^{50} - 3922092 q^{51} + 8058740 q^{52} + 316580 q^{53} - 6933535 q^{54} - 2889990 q^{55} + 480880 q^{56} - 12912030 q^{58} - 3625740 q^{59} - 10459240 q^{60} + 656223 q^{61} + 15439920 q^{62} + 16010850 q^{63} + 572908 q^{64} - 9903445 q^{65} + 8105266 q^{66} - 25482280 q^{67} - 9378031 q^{69} + 16449030 q^{70} + 2533578 q^{71} + 8757215 q^{72} + 8477055 q^{73} + 20547876 q^{74} + 28650170 q^{75} - 22436480 q^{76} + 20387770 q^{77} - 23613305 q^{78} - 5343505 q^{79} - 44517070 q^{80} + 7686054 q^{81} - 20224125 q^{83} + 10924626 q^{84} + 2795615 q^{85} + 27540603 q^{86} - 3545560 q^{87} + 36319550 q^{88} - 17844415 q^{89} - 67695745 q^{90} + 9123478 q^{91} - 88054710 q^{92} - 15200999 q^{94} + 42119295 q^{95} - 8739232 q^{96} + 41971560 q^{97} + 60378980 q^{98} + 57867624 q^{99}+O(q^{100}) \) 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 −19.9513 − 6.48258i 31.2993 − 43.0798i 252.478 + 183.436i −278.479 − 23.9682i −903.732 + 656.600i 928.752i −2269.82 − 3124.13i −200.404 − 616.780i 5400.65 + 2283.46i
4.2 −17.4863 − 5.68165i −14.4903 + 19.9442i 169.936 + 123.466i 110.554 − 256.716i 366.698 − 266.422i − 1022.55i −886.753 − 1220.51i 488.018 + 1501.97i −3391.75 + 3860.88i
4.3 −16.8860 − 5.48660i −41.4495 + 57.0503i 151.480 + 110.057i 37.1286 + 277.032i 1012.93 − 735.935i 1074.80i −618.235 − 850.927i −860.857 − 2649.45i 893.006 − 4881.67i
4.4 −14.0786 − 4.57442i 21.9300 − 30.1841i 73.7277 + 53.5663i 222.231 + 169.524i −446.819 + 324.633i 198.096i 320.787 + 441.525i 245.667 + 756.085i −2353.23 − 3403.24i
4.5 −8.19949 − 2.66417i −29.1952 + 40.1838i −43.4204 − 31.5468i −266.824 − 83.2459i 346.442 − 251.705i 135.919i 920.627 + 1267.13i −86.5550 − 266.389i 1966.04 + 1393.44i
4.6 −8.12319 − 2.63939i 22.8316 − 31.4250i −44.5343 − 32.3560i −192.399 + 202.750i −268.408 + 195.010i − 1367.16i 918.973 + 1264.86i 209.571 + 644.993i 2098.03 − 1139.16i
4.7 −5.59051 − 1.81647i 42.7242 − 58.8048i −75.5999 − 54.9266i 12.2594 − 279.240i −345.667 + 251.142i 531.360i 765.126 + 1053.11i −956.829 − 2944.82i −575.766 + 1538.82i
4.8 −2.30461 − 0.748813i −13.2509 + 18.2384i −98.8037 − 71.7851i 254.041 − 116.569i 44.1954 − 32.1098i 591.855i 356.264 + 490.355i 518.770 + 1596.61i −672.753 + 78.4160i
4.9 2.28224 + 0.741543i −38.5064 + 52.9996i −98.8955 − 71.8518i 151.535 + 234.867i −127.182 + 92.4033i − 1410.01i −352.965 − 485.815i −650.387 − 2001.69i 171.674 + 648.390i
4.10 5.26830 + 1.71178i 7.67468 − 10.5633i −78.7293 − 57.2002i −159.549 + 229.497i 58.5145 − 42.5133i 1574.96i −733.623 − 1009.75i 623.138 + 1917.82i −1233.40 + 935.950i
4.11 8.51765 + 2.76755i 43.2379 − 59.5118i −38.6632 − 28.0904i 222.049 + 169.762i 532.987 − 387.238i − 484.106i −925.395 − 1273.70i −996.322 − 3066.36i 1421.51 + 2060.51i
4.12 9.15239 + 2.97379i 10.6184 − 14.6149i −28.6314 − 20.8020i −161.344 − 228.239i 140.645 − 102.185i − 933.124i −924.215 − 1272.07i 574.974 + 1769.59i −797.944 − 2568.74i
4.13 11.2396 + 3.65197i −50.2306 + 69.1364i 9.43754 + 6.85677i −36.2236 − 277.151i −817.055 + 593.625i 865.676i −808.113 − 1112.27i −1580.92 − 4865.56i 605.009 − 3247.36i
4.14 17.1896 + 5.58523i −25.9999 + 35.7858i 160.732 + 116.779i −164.394 + 226.052i −646.798 + 469.926i − 372.101i 750.840 + 1033.44i 71.1926 + 219.108i −4088.41 + 2967.56i
4.15 17.3799 + 5.64708i −0.558679 + 0.768956i 166.618 + 121.055i 271.839 − 65.0263i −14.0522 + 10.2095i 190.361i 837.308 + 1152.46i 675.541 + 2079.10i 5091.76 + 404.946i
4.16 19.7814 + 6.42736i 47.2089 − 64.9775i 246.438 + 179.048i −275.144 − 49.2038i 1351.49 − 981.916i 664.316i 2159.21 + 2971.90i −1317.57 − 4055.07i −5126.47 − 2741.77i
9.1 −12.2463 + 16.8556i 48.7475 − 15.8390i −94.5855 − 291.104i 150.427 + 235.578i −330.002 + 1015.64i 983.801i 3528.76 + 1146.56i 356.126 − 258.741i −5812.99 − 349.422i
9.2 −10.6987 + 14.7255i −47.1398 + 15.3166i −62.8244 − 193.353i 144.406 − 239.316i 278.790 − 858.026i 217.774i 1303.58 + 423.558i 218.237 − 158.559i 1979.10 + 4686.82i
9.3 −10.3019 + 14.1794i −26.5031 + 8.61137i −55.3708 − 170.414i −226.305 + 164.046i 150.928 − 464.510i − 1206.49i 853.170 + 277.212i −1141.06 + 829.031i 5.30280 − 4898.85i
9.4 −7.81540 + 10.7570i 37.7432 − 12.2635i −15.0779 − 46.4050i −244.231 − 135.928i −163.060 + 501.847i 876.288i −1001.62 − 325.445i −495.164 + 359.757i 3370.93 − 1564.85i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
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 25.8.e.a ✓ 64
25.e even 10 1 inner 25.8.e.a ✓ 64
25.f odd 20 2 625.8.a.g 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.8.e.a ✓ 64 1.a even 1 1 trivial
25.8.e.a ✓ 64 25.e even 10 1 inner
625.8.a.g 64 25.f odd 20 2

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(25, [\chi])\).