Properties

Label 25.6.d.a
Level 25
Weight 6
Character orbit 25.d
Analytic conductor 4.010
Analytic rank 0
Dimension 44
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44q - 7q^{2} - q^{3} - 147q^{4} + 115q^{5} + 133q^{6} - 202q^{7} - 500q^{8} - 508q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 44q - 7q^{2} - q^{3} - 147q^{4} + 115q^{5} + 133q^{6} - 202q^{7} - 500q^{8} - 508q^{9} - 55q^{10} + 718q^{11} - 437q^{12} - 291q^{13} - 689q^{14} - 1125q^{15} + 2609q^{16} + 718q^{17} + 8544q^{18} + 4075q^{19} + 1095q^{20} - 2022q^{21} - 3704q^{22} - 1211q^{23} - 19620q^{24} - 2405q^{25} - 17882q^{26} + 9485q^{27} + 9371q^{28} + 14245q^{29} + 14235q^{30} - 8637q^{31} + 30678q^{32} - 21682q^{33} - 5639q^{34} - 36235q^{35} + 31069q^{36} - 50777q^{37} + 6525q^{38} + 7009q^{39} - 3470q^{40} + 5558q^{41} + 19541q^{42} + 61194q^{43} + 26876q^{44} + 76400q^{45} + 14553q^{46} - 46662q^{47} - 150806q^{48} - 24482q^{49} - 21335q^{50} - 121132q^{51} + 72648q^{52} - 93141q^{53} + 18655q^{54} + 11760q^{55} + 47370q^{56} + 292150q^{57} - 38730q^{58} + 47640q^{59} + 26010q^{60} - 26747q^{61} - 46744q^{62} + 78674q^{63} + 31148q^{64} - 61320q^{65} + 45896q^{66} + 123558q^{67} + 123886q^{68} - 117721q^{69} + 12280q^{70} + 110788q^{71} - 586175q^{72} - 146691q^{73} - 179364q^{74} - 108580q^{75} - 269280q^{76} - 153084q^{77} + 405473q^{78} - 54065q^{79} - 360820q^{80} + 277619q^{81} - 353334q^{82} - 54791q^{83} + 374516q^{84} + 764240q^{85} - 74697q^{86} + 229940q^{87} + 997600q^{88} + 217000q^{89} - 21245q^{90} - 181052q^{91} + 801448q^{92} - 380432q^{93} - 165099q^{94} - 61205q^{95} + 925078q^{96} - 562302q^{97} - 982324q^{98} - 680776q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −3.12228 9.60940i 3.16943 2.30272i −56.7033 + 41.1973i −51.1989 22.4426i −32.0236 23.2665i 54.0512 311.350 + 226.209i −70.3484 + 216.510i −55.8019 + 562.063i
6.2 −2.46926 7.59959i −14.2493 + 10.3527i −25.7680 + 18.7216i 27.5974 + 48.6147i 113.862 + 82.7252i 40.0850 −0.963073 0.699714i 20.7724 63.9307i 301.307 329.771i
6.3 −1.94032 5.97168i 9.69103 7.04095i −6.00755 + 4.36474i 46.5920 30.8899i −60.8499 44.2101i −78.1631 −124.833 90.6962i −30.7500 + 94.6386i −274.868 218.296i
6.4 −1.01935 3.13724i 20.3462 14.7824i 17.0853 12.4132i −31.9140 + 45.8966i −67.1158 48.7625i 70.1855 −141.758 102.993i 120.358 370.424i 176.520 + 53.3371i
6.5 −0.579819 1.78450i −11.3233 + 8.22686i 23.0403 16.7398i −55.7493 + 4.12531i 21.2463 + 15.4363i −222.130 −91.8068 66.7016i −14.5552 + 44.7962i 39.6861 + 97.0926i
6.6 −0.447636 1.37768i −17.2377 + 12.5239i 24.1909 17.5757i 14.3787 54.0209i 24.9702 + 18.1419i 250.563 −72.5441 52.7064i 65.1987 200.661i −80.8600 + 4.37248i
6.7 0.952916 + 2.93277i 0.594983 0.432280i 18.1954 13.2198i 28.1715 + 48.2842i 1.83475 + 1.33302i 57.2936 135.942 + 98.7675i −74.9240 + 230.592i −114.762 + 128.631i
6.8 1.31554 + 4.04881i 13.2842 9.65154i 11.2263 8.15638i −5.21916 55.6575i 56.5532 + 41.0883i −22.2636 158.004 + 114.797i 8.22668 25.3191i 218.481 94.3511i
6.9 2.36682 + 7.28432i −19.7149 + 14.3237i −21.5710 + 15.6723i 54.0142 14.4039i −151.000 109.708i −209.197 33.0689 + 24.0259i 108.418 333.675i 232.764 + 359.365i
6.10 2.71092 + 8.34335i −5.61116 + 4.07675i −36.3739 + 26.4272i −55.5748 + 6.03628i −49.2252 35.7642i 102.277 −91.9847 66.8308i −60.2259 + 185.356i −201.022 447.316i
6.11 3.27755 + 10.0873i 20.2415 14.7063i −65.1220 + 47.3139i 47.0312 + 30.2170i 214.689 + 155.981i −69.7231 −416.125 302.333i 118.352 364.249i −150.659 + 573.454i
11.1 −8.62958 + 6.26976i 1.83172 5.63746i 25.2713 77.7769i 54.8570 + 10.7567i 19.5385 + 60.1334i −120.405 164.084 + 504.997i 168.165 + 122.179i −540.835 + 251.114i
11.2 −6.12278 + 4.44846i −7.99178 + 24.5962i 7.81108 24.0400i −12.4773 + 54.4914i −60.4831 186.148i 130.915 −15.7226 48.3892i −344.512 250.302i −166.007 389.144i
11.3 −6.01899 + 4.37305i 1.80419 5.55274i 7.21609 22.2089i −36.8592 42.0286i 13.4230 + 41.3117i 204.573 −19.8827 61.1927i 169.013 + 122.795i 405.648 + 91.7822i
11.4 −4.49484 + 3.26569i 6.79188 20.9033i −0.349697 + 1.07626i −34.6743 + 43.8485i 37.7352 + 116.137i −153.453 −56.8829 175.068i −194.225 141.113i 12.6596 310.328i
11.5 −3.39495 + 2.46657i −4.95437 + 15.2480i −4.44687 + 13.6860i 23.5016 50.7215i −20.7904 63.9863i −192.901 −60.1569 185.144i −11.3638 8.25627i 45.3216 + 230.165i
11.6 −0.208693 + 0.151625i 1.88200 5.79220i −9.86798 + 30.3705i 51.9690 + 20.5968i 0.485479 + 1.49415i 127.575 −5.09638 15.6850i 166.583 + 121.030i −13.9686 + 3.58137i
11.7 2.06882 1.50308i −3.25658 + 10.0227i −7.86780 + 24.2146i −54.0584 + 14.2370i 8.32772 + 25.6301i −36.6142 45.4065 + 139.747i 106.742 + 77.5524i −90.4375 + 110.708i
11.8 2.28450 1.65978i 7.91379 24.3561i −7.42450 + 22.8503i −0.762635 55.8965i −22.3469 68.7767i −43.3673 48.8885 + 150.463i −334.002 242.667i −94.5184 126.430i
11.9 5.56327 4.04195i −7.98941 + 24.5889i 4.72404 14.5391i 52.9093 + 18.0446i 54.9398 + 169.087i −5.99482 35.5141 + 109.301i −344.191 250.069i 367.284 113.470i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.6.d.a 44
25.d even 5 1 inner 25.6.d.a 44
25.d even 5 1 625.6.a.d 22
25.e even 10 1 625.6.a.c 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.d.a 44 1.a even 1 1 trivial
25.6.d.a 44 25.d even 5 1 inner
625.6.a.c 22 25.e even 10 1
625.6.a.d 22 25.d even 5 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database