Properties

Label 25.7.f.a
Level 25
Weight 7
Character orbit 25.f
Analytic conductor 5.751
Analytic rank 0
Dimension 112
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 25.f (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.75135209050\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112q - 40q^{3} - 10q^{4} + 60q^{5} - 6q^{6} - 560q^{7} - 1870q^{8} - 10q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 112q - 40q^{3} - 10q^{4} + 60q^{5} - 6q^{6} - 560q^{7} - 1870q^{8} - 10q^{9} + 4360q^{10} - 6q^{11} - 3490q^{12} - 1970q^{13} - 10q^{14} + 80q^{15} + 22522q^{16} - 1210q^{17} - 47740q^{18} - 19210q^{19} + 4290q^{20} - 6q^{21} + 116790q^{22} + 68920q^{23} - 83690q^{25} + 32544q^{26} - 147970q^{27} - 191010q^{28} - 24810q^{29} + 128390q^{30} - 6q^{31} + 335470q^{32} + 133250q^{33} + 32990q^{34} - 126400q^{35} - 156358q^{36} + 64990q^{37} - 26320q^{38} - 414810q^{39} - 609460q^{40} + 108554q^{41} - 345630q^{42} + 228280q^{43} + 711540q^{44} + 1026100q^{45} - 6q^{46} + 470440q^{47} + 948520q^{48} - 453750q^{50} - 16q^{51} - 1534020q^{52} - 922770q^{53} - 2582460q^{54} - 1058910q^{55} - 16390q^{56} + 283510q^{57} + 1902200q^{58} + 2457990q^{59} + 4557170q^{60} - 388446q^{61} - 686820q^{62} - 2340440q^{63} - 5040460q^{64} - 1346350q^{65} + 186618q^{66} + 913000q^{67} + 3094930q^{68} + 2802790q^{69} + 3485630q^{70} - 758886q^{71} + 3144300q^{72} + 74290q^{73} - 890640q^{75} - 16400q^{76} - 3146750q^{77} - 5681900q^{78} - 2289610q^{79} - 6528330q^{80} + 1577702q^{81} + 1401150q^{82} + 26480q^{83} - 1650010q^{84} - 3408980q^{85} - 6q^{86} - 735210q^{87} + 2183870q^{88} + 2906240q^{89} + 3580070q^{90} - 6q^{91} + 1367450q^{92} + 5352610q^{93} + 14186390q^{94} + 5887710q^{95} - 1245186q^{96} + 7532890q^{97} + 5872380q^{98} + O(q^{100}) \)

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 −14.1993 2.24894i −7.91375 15.5316i 135.693 + 44.0895i 119.661 + 36.1404i 77.4397 + 238.335i −114.344 114.344i −1007.79 513.497i 249.892 343.947i −1617.83 782.279i
2.2 −13.7993 2.18560i 17.7803 + 34.8958i 124.777 + 40.5426i −51.5406 113.880i −169.088 520.400i 372.252 + 372.252i −836.528 426.232i −473.084 + 651.144i 462.331 + 1684.11i
2.3 −10.0011 1.58401i 0.769697 + 1.51062i 36.6447 + 11.9066i −99.9207 + 75.1056i −5.30496 16.3270i −35.6779 35.6779i 229.788 + 117.083i 426.806 587.448i 1118.28 592.860i
2.4 −8.11471 1.28524i −17.5114 34.3681i 3.32897 + 1.08165i −25.9891 122.268i 97.9287 + 301.394i −97.8049 97.8049i 442.881 + 225.659i −446.022 + 613.897i 53.7489 + 1025.57i
2.5 −6.66372 1.05543i 21.4438 + 42.0858i −17.5764 5.71093i 78.2365 + 97.4887i −98.4768 303.080i −344.251 344.251i 495.828 + 252.637i −882.884 + 1215.19i −418.454 732.210i
2.6 −3.34875 0.530390i 6.21036 + 12.1885i −49.9348 16.2248i 97.0656 78.7608i −14.3323 44.1103i 180.242 + 180.242i 351.955 + 179.330i 318.504 438.383i −366.823 + 212.268i
2.7 −1.86717 0.295730i −17.0200 33.4037i −57.4688 18.6727i 17.1804 + 123.814i 21.9008 + 67.4036i 280.929 + 280.929i 209.583 + 106.788i −397.628 + 547.288i 4.53684 236.262i
2.8 1.78809 + 0.283205i 6.80972 + 13.3648i −57.7506 18.7643i −112.961 53.5248i 8.39138 + 25.8260i −120.010 120.010i −201.185 102.509i 296.249 407.752i −186.825 127.698i
2.9 5.41905 + 0.858292i −10.2785 20.1728i −32.2382 10.4748i 124.517 + 10.9798i −38.3857 118.139i −443.961 443.961i −478.580 243.849i 127.203 175.080i 665.339 + 166.372i
2.10 7.09592 + 1.12388i 11.5411 + 22.6507i −11.7786 3.82710i 2.31555 + 124.979i 56.4381 + 173.699i 273.850 + 273.850i −488.964 249.139i 48.6381 66.9445i −124.030 + 889.441i
2.11 9.00256 + 1.42587i −17.4856 34.3174i 18.1453 + 5.89578i −121.632 28.8228i −108.483 333.877i 37.0243 + 37.0243i −364.817 185.883i −443.445 + 610.349i −1053.90 432.909i
2.12 11.0471 + 1.74968i 23.3278 + 45.7833i 58.1084 + 18.8806i 16.4012 123.919i 177.597 + 546.587i −146.394 146.394i −28.9129 14.7319i −1123.43 + 1546.27i 398.004 1340.25i
2.13 12.4157 + 1.96646i −5.46221 10.7202i 89.4159 + 29.0530i 84.3839 92.2191i −46.7365 143.840i 324.444 + 324.444i 336.207 + 171.306i 343.409 472.662i 1229.03 979.030i
2.14 14.8618 + 2.35387i 1.72543 + 3.38635i 154.464 + 50.1883i −45.2307 + 116.530i 17.6719 + 54.3886i −314.203 314.203i 1319.42 + 672.277i 420.005 578.088i −946.504 + 1625.37i
3.1 −6.29988 12.3642i 0.183941 + 1.16136i −75.5670 + 104.009i 89.4657 + 87.2977i 13.2005 9.59071i −399.540 + 399.540i 884.879 + 140.151i 692.005 224.846i 515.744 1656.14i
3.2 −6.01647 11.8080i −4.30463 27.1784i −65.6123 + 90.3075i −122.996 22.2949i −295.023 + 214.347i 251.807 251.807i 623.392 + 98.7355i −26.8129 + 8.71203i 476.741 + 1586.47i
3.3 −5.46404 10.7238i 6.12174 + 38.6511i −47.5254 + 65.4131i 32.7052 120.646i 381.037 276.839i 241.406 241.406i 200.362 + 31.7343i −763.115 + 247.951i −1472.48 + 308.489i
3.4 −2.99542 5.87885i −4.81859 30.4234i 12.0299 16.5578i 95.2241 80.9775i −164.421 + 119.459i −61.2445 + 61.2445i −550.448 87.1824i −209.045 + 67.9227i −761.292 317.246i
3.5 −2.81051 5.51594i 4.27590 + 26.9970i 15.0917 20.7719i −123.424 + 19.7866i 136.896 99.4609i −262.093 + 262.093i −548.317 86.8449i −17.2342 + 5.59972i 456.026 + 625.189i
3.6 −2.36346 4.63854i 1.57805 + 9.96344i 21.6881 29.8511i 24.7920 + 122.517i 42.4862 30.8680i 325.037 325.037i −518.804 82.1705i 596.540 193.828i 509.705 404.562i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.14
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 25.7.f.a 112
25.f odd 20 1 inner 25.7.f.a 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.7.f.a 112 1.a even 1 1 trivial
25.7.f.a 112 25.f odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database