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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,6,Mod(6,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.6");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.d (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 6.11
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 \(S_{6}^{\mathrm{new}}(25, [\chi])\).