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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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 25.e (of order \(10\), degree \(4\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 64q - 5q^{2} - 5q^{3} + 893q^{4} - 510q^{5} + 173q^{6} - 2510q^{8} + 6587q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 64q - 5q^{2} - 5q^{3} + 893q^{4} - 510q^{5} + 173q^{6} - 2510q^{8} + 6587q^{9} - 5805q^{10} - 472q^{11} + 16635q^{12} - 5q^{13} - 19409q^{14} + 375q^{15} - 42071q^{16} + 31910q^{17} - 48685q^{19} - 171405q^{20} + 2268q^{21} + 197230q^{22} + 356525q^{23} + 544720q^{24} + 170220q^{25} - 708942q^{26} - 319805q^{27} - 565125q^{28} - 332885q^{29} + 66235q^{30} - 39057q^{31} - 161110q^{33} + 1156301q^{34} + 337515q^{35} + 1920669q^{36} - 1480960q^{37} - 2136945q^{38} + 1307509q^{39} - 282470q^{40} + 1955408q^{41} - 684355q^{42} - 3558164q^{44} - 1351725q^{45} - 2982527q^{46} + 1018980q^{47} + 8967210q^{48} - 3298162q^{49} + 8550165q^{50} - 3922092q^{51} + 8058740q^{52} + 316580q^{53} - 6933535q^{54} - 2889990q^{55} + 480880q^{56} - 12912030q^{58} - 3625740q^{59} - 10459240q^{60} + 656223q^{61} + 15439920q^{62} + 16010850q^{63} + 572908q^{64} - 9903445q^{65} + 8105266q^{66} - 25482280q^{67} - 9378031q^{69} + 16449030q^{70} + 2533578q^{71} + 8757215q^{72} + 8477055q^{73} + 20547876q^{74} + 28650170q^{75} - 22436480q^{76} + 20387770q^{77} - 23613305q^{78} - 5343505q^{79} - 44517070q^{80} + 7686054q^{81} - 20224125q^{83} + 10924626q^{84} + 2795615q^{85} + 27540603q^{86} - 3545560q^{87} + 36319550q^{88} - 17844415q^{89} - 67695745q^{90} + 9123478q^{91} - 88054710q^{92} - 15200999q^{94} + 42119295q^{95} - 8739232q^{96} + 41971560q^{97} + 60378980q^{98} + 57867624q^{99} + O(q^{100}) \)

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.

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 19.16
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 \(S_{8}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database