Properties

Label 25.9.f.a
Level $25$
Weight $9$
Character orbit 25.f
Analytic conductor $10.184$
Analytic rank $0$
Dimension $152$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1844652515\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 152q - 8q^{2} + 62q^{3} - 10q^{4} - 230q^{5} - 6q^{6} + 2342q^{7} + 8210q^{8} - 10q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 152q - 8q^{2} + 62q^{3} - 10q^{4} - 230q^{5} - 6q^{6} + 2342q^{7} + 8210q^{8} - 10q^{9} - 30880q^{10} - 6q^{11} + 45902q^{12} + 119132q^{13} - 10q^{14} - 241450q^{15} + 524282q^{16} - 105158q^{17} + 454052q^{18} + 718890q^{19} + 340370q^{20} - 6q^{21} - 1986746q^{22} - 899198q^{23} + 2032310q^{25} - 913936q^{26} + 1904300q^{27} - 998002q^{28} - 1956910q^{29} - 4500490q^{30} - 6q^{31} - 3033938q^{32} + 1003994q^{33} + 10303990q^{34} + 4124560q^{35} - 8100358q^{36} + 9534392q^{37} - 14259280q^{38} - 22510410q^{39} - 4123060q^{40} - 4374726q^{41} + 52199634q^{42} + 13583342q^{43} - 2138860q^{44} - 33978380q^{45} - 6q^{46} - 11197058q^{47} - 32869688q^{48} + 13239250q^{50} - 16q^{51} + 93509212q^{52} + 53717512q^{53} + 57025140q^{54} - 14642290q^{55} - 262150q^{56} - 93310490q^{57} - 97437640q^{58} + 24222740q^{59} - 66375070q^{60} - 18369126q^{61} + 6782124q^{62} + 96012652q^{63} + 112817140q^{64} + 89469970q^{65} + 6718458q^{66} + 74549142q^{67} - 62262702q^{68} - 128418860q^{69} - 177099410q^{70} + 60703854q^{71} - 444425220q^{72} - 171988248q^{73} + 257329110q^{75} - 262160q^{76} + 292751454q^{77} + 385000804q^{78} + 138506190q^{79} + 245270230q^{80} + 110809772q^{81} - 73650066q^{82} - 663197528q^{83} - 1517250010q^{84} - 616093620q^{85} - 6q^{86} + 494144820q^{87} + 1287340030q^{88} + 1051406240q^{89} + 1480613390q^{90} - 6q^{91} - 461835158q^{92} - 908140736q^{93} - 1160168810q^{94} - 920404690q^{95} + 172437534q^{96} - 732746208q^{97} - 167741852q^{98} + O(q^{100}) \)

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 −29.9573 4.74476i −5.22252 10.2498i 631.454 + 205.172i −612.487 124.436i 107.820 + 331.835i −2520.14 2520.14i −11024.8 5617.40i 3778.68 5200.90i 17758.0 + 6633.88i
2.2 −27.0051 4.27718i 48.0141 + 94.2330i 467.510 + 151.903i 614.484 + 114.167i −893.573 2750.14i −244.540 244.540i −5738.84 2924.09i −2718.05 + 3741.07i −16105.9 5711.36i
2.3 −23.9462 3.79271i −30.2881 59.4438i 315.567 + 102.534i 109.848 615.271i 499.834 + 1538.33i 2716.43 + 2716.43i −1637.61 834.402i 1240.27 1707.08i −4964.00 + 14316.8i
2.4 −22.6508 3.58754i −57.6629 113.170i 256.720 + 83.4132i 188.239 + 595.979i 900.112 + 2770.26i 118.643 + 118.643i −284.659 145.041i −5625.95 + 7743.46i −2125.68 14174.7i
2.5 −19.1984 3.04073i 29.7932 + 58.4725i 115.862 + 37.6458i −323.288 + 534.893i −394.183 1213.17i 1431.87 + 1431.87i 2323.80 + 1184.04i 1325.06 1823.79i 7833.07 9286.05i
2.6 −13.4028 2.12279i −7.81684 15.3414i −68.3420 22.2057i 533.420 325.711i 72.2008 + 222.211i −1613.32 1613.32i 3964.09 + 2019.80i 3682.20 5068.12i −7840.74 + 3233.10i
2.7 −12.3344 1.95358i 51.8119 + 101.687i −95.1487 30.9157i −327.816 532.129i −440.417 1355.46i −741.865 741.865i 3961.73 + 2018.60i −3799.22 + 5229.18i 3003.87 + 7203.93i
2.8 −8.69439 1.37706i −50.1228 98.3716i −169.774 55.1630i −621.648 64.6469i 300.324 + 924.303i −2315.10 2315.10i 3408.01 + 1736.47i −3308.21 + 4553.36i 5315.82 + 1418.11i
2.9 −2.09064 0.331126i −2.63044 5.16253i −239.209 77.7238i 357.844 + 512.418i 3.78987 + 11.6640i −242.301 242.301i 957.181 + 487.708i 3836.73 5280.80i −578.450 1189.78i
2.10 −0.758404 0.120119i −20.5624 40.3560i −242.910 78.9262i −609.477 138.431i 10.7471 + 33.0761i 3191.92 + 3191.92i 349.890 + 178.278i 2650.67 3648.33i 445.602 + 178.197i
2.11 5.41297 + 0.857331i 59.4837 + 116.743i −214.905 69.8269i 608.428 142.970i 221.896 + 682.926i 2643.50 + 2643.50i −2353.49 1199.16i −6234.25 + 8580.71i 3415.98 252.267i
2.12 6.52417 + 1.03333i −68.4455 134.332i −201.974 65.6252i 526.695 336.477i −307.741 947.129i 561.415 + 561.415i −2756.59 1404.55i −9503.78 + 13080.8i 3783.94 1650.99i
2.13 8.03121 + 1.27202i 50.0921 + 98.3113i −180.588 58.6767i −377.779 + 497.904i 277.246 + 853.277i −1907.75 1907.75i −3230.44 1645.99i −3299.43 + 4541.28i −3667.36 + 3518.23i
2.14 14.1151 + 2.23562i 5.30786 + 10.4173i −49.2316 15.9963i −28.9465 624.329i 51.6321 + 158.907i −772.344 772.344i −3918.91 1996.78i 3776.11 5197.37i 987.178 8877.20i
2.15 18.1127 + 2.86878i −39.2192 76.9719i 76.3708 + 24.8144i −368.107 + 505.096i −489.551 1506.68i −920.836 920.836i −2870.87 1462.78i −530.076 + 729.587i −8116.43 + 8092.66i
2.16 23.0472 + 3.65032i −22.7623 44.6735i 274.378 + 89.1509i 502.411 + 371.763i −361.534 1112.69i 2000.67 + 2000.67i 675.676 + 344.274i 2378.86 3274.22i 10222.1 + 10402.1i
2.17 25.9250 + 4.10612i 40.5801 + 79.6429i 411.775 + 133.794i −624.993 2.91064i 725.016 + 2231.37i 2152.18 + 2152.18i 4138.76 + 2108.80i −839.788 + 1155.87i −16191.0 2641.75i
2.18 26.2450 + 4.15680i 39.3019 + 77.1344i 428.052 + 139.082i 614.814 + 112.380i 710.848 + 2187.76i −2106.70 2106.70i 4595.04 + 2341.29i −548.613 + 755.101i 15668.7 + 5505.08i
2.19 30.6740 + 4.85828i −51.4523 100.981i 673.820 + 218.937i −200.608 591.930i −1087.65 3347.46i −1139.13 1139.13i 12521.2 + 6379.87i −3693.34 + 5083.44i −3277.69 19131.5i
3.1 −13.5902 26.6723i 9.50733 + 60.0269i −376.245 + 517.857i −624.928 + 9.49038i 1471.85 1069.36i −194.417 + 194.417i 11356.7 + 1798.72i 2727.04 886.068i 8746.04 + 16539.3i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.19
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 25.9.f.a 152
25.f odd 20 1 inner 25.9.f.a 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.9.f.a 152 1.a even 1 1 trivial
25.9.f.a 152 25.f odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database