Properties

Label 25.5.f.a
Level $25$
Weight $5$
Character orbit 25.f
Analytic conductor $2.584$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.58424907710\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72q - 8q^{2} + 2q^{3} - 10q^{4} - 50q^{5} - 6q^{6} + 42q^{7} + 50q^{8} - 10q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q - 8q^{2} + 2q^{3} - 10q^{4} - 50q^{5} - 6q^{6} + 42q^{7} + 50q^{8} - 10q^{9} - 6q^{11} - 178q^{12} - 288q^{13} - 10q^{14} + 410q^{15} + 762q^{16} + 1042q^{17} - 28q^{18} - 1310q^{19} - 3790q^{20} - 6q^{21} - 3226q^{22} - 578q^{23} + 1110q^{25} - 976q^{26} + 5120q^{27} + 10318q^{28} + 3290q^{29} + 5270q^{30} - 6q^{31} - 1298q^{32} - 2806q^{33} - 8010q^{34} - 5360q^{35} - 2758q^{36} + 4692q^{37} + 14480q^{38} + 8790q^{39} + 2540q^{40} + 1434q^{41} - 23886q^{42} - 14958q^{43} - 22060q^{44} - 23420q^{45} - 6q^{46} - 12158q^{47} - 14168q^{48} + 8050q^{50} - 16q^{51} + 29372q^{52} + 22852q^{53} + 53940q^{54} + 18070q^{55} - 1030q^{56} + 34150q^{57} + 23000q^{58} + 23240q^{59} + 40610q^{60} + 2634q^{61} - 13716q^{62} - 42428q^{63} - 66060q^{64} - 41870q^{65} + 5178q^{66} - 25558q^{67} - 71982q^{68} - 39560q^{69} - 38130q^{70} + 7974q^{71} - 37860q^{72} - 4228q^{73} + 21210q^{75} - 1040q^{76} + 34374q^{77} + 51844q^{78} + 26590q^{79} + 110710q^{80} - 14368q^{81} + 145614q^{82} + 99832q^{83} + 149990q^{84} + 48860q^{85} - 6q^{86} - 31320q^{87} - 95490q^{88} - 93760q^{89} - 207250q^{90} - 6q^{91} - 199958q^{92} - 134396q^{93} - 130410q^{94} - 50290q^{95} + 17694q^{96} - 40108q^{97} - 25052q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −7.04610 1.11599i −6.53588 12.8274i 33.1852 + 10.7825i −24.8595 + 2.64690i 31.7372 + 97.6770i 31.4475 + 31.4475i −120.091 61.1893i −74.2134 + 102.146i 178.116 + 9.09265i
2.2 −5.62994 0.891695i 2.94614 + 5.78212i 15.6842 + 5.09611i 0.584324 24.9932i −11.4307 35.1800i −65.0193 65.0193i −2.49547 1.27151i 22.8574 31.4606i −25.5760 + 140.189i
2.3 −4.54749 0.720251i 0.590325 + 1.15858i 4.94395 + 1.60639i 22.6193 + 10.6475i −1.85003 5.69380i 49.8348 + 49.8348i 44.3120 + 22.5781i 46.6168 64.1625i −95.1920 64.7107i
2.4 −1.28906 0.204167i 7.17656 + 14.0848i −13.5969 4.41790i −24.2814 + 5.95099i −6.37537 19.6214i 39.9737 + 39.9737i 35.2313 + 17.9513i −99.2677 + 136.630i 32.5152 2.71373i
2.5 −0.977279 0.154786i −2.72438 5.34689i −14.2858 4.64173i −12.9766 + 21.3684i 1.83485 + 5.64710i −52.4931 52.4931i 27.3486 + 13.9348i 26.4436 36.3965i 15.9893 18.8743i
2.6 1.02878 + 0.162942i −4.17666 8.19716i −14.1851 4.60901i 1.67139 24.9441i −2.96119 9.11361i 26.8728 + 26.8728i −28.6915 14.6190i −2.13832 + 2.94315i 5.78393 25.3896i
2.7 3.78337 + 0.599226i 3.54623 + 6.95987i −1.26212 0.410089i 23.2986 + 9.06514i 9.24615 + 28.4567i −13.0105 13.0105i −59.1377 30.1322i 11.7465 16.1677i 82.7149 + 48.2579i
2.8 6.19314 + 0.980897i 1.94321 + 3.81377i 22.1759 + 7.20540i −22.2324 11.4333i 8.29367 + 25.5253i −2.13863 2.13863i 40.8803 + 20.8296i 36.8419 50.7085i −126.474 92.6155i
2.9 6.53352 + 1.03481i −7.66178 15.0371i 26.3992 + 8.57763i 14.4844 + 20.3765i −34.4979 106.174i 5.89577 + 5.89577i 69.3001 + 35.3102i −119.800 + 164.891i 73.5482 + 148.119i
3.1 −3.10514 6.09417i 2.14164 + 13.5218i −18.0925 + 24.9022i −10.3349 + 22.7638i 75.7541 55.0385i 9.33601 9.33601i 99.8507 + 15.8148i −101.217 + 32.8873i 170.818 7.70230i
3.2 −2.65089 5.20267i −2.30541 14.5558i −10.6360 + 14.6392i 13.2120 + 21.2237i −69.6176 + 50.5802i 28.1685 28.1685i 12.0823 + 1.91365i −129.521 + 42.0839i 75.3962 124.999i
3.3 −2.14039 4.20075i −0.349755 2.20827i −3.66045 + 5.03818i −17.0385 18.2946i −8.52776 + 6.19578i −46.9663 + 46.9663i −45.5061 7.20746i 72.2815 23.4857i −40.3819 + 110.732i
3.4 −1.10795 2.17447i 0.976269 + 6.16392i 5.90379 8.12588i 23.7728 7.73659i 12.3216 8.95217i 23.7798 23.7798i −62.7773 9.94294i 39.9948 12.9951i −43.1620 43.1215i
3.5 0.312556 + 0.613425i −1.24611 7.86762i 9.12596 12.5608i −24.9872 + 0.799781i 4.43671 3.22346i 47.0743 47.0743i 21.4373 + 3.39533i 16.6890 5.42258i −8.30050 15.0778i
3.6 0.958433 + 1.88103i 1.41659 + 8.94397i 6.78488 9.33859i −5.24264 + 24.4441i −15.4662 + 11.2368i −27.3860 + 27.3860i 57.4312 + 9.09622i −0.952289 + 0.309417i −51.0049 + 13.5665i
3.7 1.47174 + 2.88846i −2.00120 12.6351i 3.22739 4.44212i 22.9144 9.99657i 33.5507 24.3760i −45.6970 + 45.6970i 68.8109 + 10.8986i −78.6049 + 25.5403i 62.5988 + 51.4749i
3.8 2.50963 + 4.92542i 1.48300 + 9.36327i −8.55700 + 11.7777i −1.80390 24.9348i −42.3963 + 30.8027i 0.684509 0.684509i 7.87292 + 1.24695i −8.43590 + 2.74099i 118.288 71.4622i
3.9 3.33979 + 6.55470i −0.874157 5.51921i −22.4054 + 30.8383i 2.16102 + 24.9064i 33.2573 24.1628i 37.7565 37.7565i −160.710 25.4540i 47.3380 15.3811i −156.037 + 97.3470i
8.1 −6.66808 3.39756i 11.5382 + 1.82747i 23.5154 + 32.3661i −4.94350 24.5064i −70.7286 51.3874i 32.8567 32.8567i −28.1051 177.449i 52.7545 + 17.1410i −50.2981 + 180.206i
8.2 −5.42141 2.76234i −11.3884 1.80375i 12.3565 + 17.0073i 24.3635 + 5.60550i 56.7588 + 41.2377i −9.79605 + 9.79605i −4.78031 30.1817i 49.4076 + 16.0535i −116.600 97.6900i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.5.f.a 72
5.b even 2 1 125.5.f.c 72
5.c odd 4 1 125.5.f.a 72
5.c odd 4 1 125.5.f.b 72
25.d even 5 1 125.5.f.b 72
25.e even 10 1 125.5.f.a 72
25.f odd 20 1 inner 25.5.f.a 72
25.f odd 20 1 125.5.f.c 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.5.f.a 72 1.a even 1 1 trivial
25.5.f.a 72 25.f odd 20 1 inner
125.5.f.a 72 5.c odd 4 1
125.5.f.a 72 25.e even 10 1
125.5.f.b 72 5.c odd 4 1
125.5.f.b 72 25.d even 5 1
125.5.f.c 72 5.b even 2 1
125.5.f.c 72 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database