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Properties

Label 125.4.a.c.1.3
Level $125$
Weight $4$
Character 125.1
Self dual yes
Analytic conductor $7.375$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [125,4,Mod(1,125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("125.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 125.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.37523875072\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.497918125.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 29x^{4} - 6x^{3} + 216x^{2} + 280x + 80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.45688\) of defining polynomial
Character \(\chi\) \(=\) 125.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.288727 q^{2} -4.57894 q^{3} -7.91664 q^{4} -1.32206 q^{6} +18.9048 q^{7} -4.59556 q^{8} -6.03327 q^{9} -7.94853 q^{11} +36.2498 q^{12} +43.9341 q^{13} +5.45833 q^{14} +62.0062 q^{16} +124.854 q^{17} -1.74197 q^{18} +93.4871 q^{19} -86.5642 q^{21} -2.29495 q^{22} -23.6463 q^{23} +21.0428 q^{24} +12.6849 q^{26} +151.258 q^{27} -149.663 q^{28} -171.051 q^{29} -70.6154 q^{31} +54.6673 q^{32} +36.3959 q^{33} +36.0486 q^{34} +47.7632 q^{36} -275.854 q^{37} +26.9922 q^{38} -201.172 q^{39} +259.564 q^{41} -24.9934 q^{42} +433.498 q^{43} +62.9256 q^{44} -6.82730 q^{46} -249.622 q^{47} -283.923 q^{48} +14.3927 q^{49} -571.698 q^{51} -347.810 q^{52} +112.842 q^{53} +43.6721 q^{54} -86.8782 q^{56} -428.072 q^{57} -49.3870 q^{58} +844.191 q^{59} +768.836 q^{61} -20.3885 q^{62} -114.058 q^{63} -480.266 q^{64} +10.5085 q^{66} -385.978 q^{67} -988.422 q^{68} +108.275 q^{69} +350.291 q^{71} +27.7262 q^{72} +773.075 q^{73} -79.6465 q^{74} -740.103 q^{76} -150.266 q^{77} -58.0836 q^{78} -497.416 q^{79} -529.701 q^{81} +74.9431 q^{82} +349.967 q^{83} +685.297 q^{84} +125.162 q^{86} +783.233 q^{87} +36.5279 q^{88} -1377.29 q^{89} +830.566 q^{91} +187.199 q^{92} +323.344 q^{93} -72.0726 q^{94} -250.318 q^{96} +1205.85 q^{97} +4.15555 q^{98} +47.9557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 7 q^{2} + 14 q^{3} + 33 q^{4} - 13 q^{6} + 67 q^{7} + 60 q^{8} + 92 q^{9} + 27 q^{11} + 112 q^{12} + 149 q^{13} - 14 q^{14} + 41 q^{16} + 72 q^{17} + 179 q^{18} - 105 q^{19} - 148 q^{21} + 394 q^{22}+ \cdots - 2511 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.288727 0.102080 0.0510401 − 0.998697i \(-0.483746\pi\)
0.0510401 + 0.998697i \(0.483746\pi\)
\(3\) −4.57894 −0.881218 −0.440609 − 0.897699i \(-0.645237\pi\)
−0.440609 + 0.897699i \(0.645237\pi\)
\(4\) −7.91664 −0.989580
\(5\) 0 0
\(6\) −1.32206 −0.0899550
\(7\) 18.9048 1.02076 0.510382 − 0.859948i \(-0.329504\pi\)
0.510382 + 0.859948i \(0.329504\pi\)
\(8\) −4.59556 −0.203097
\(9\) −6.03327 −0.223455
\(10\) 0 0
\(11\) −7.94853 −0.217870 −0.108935 − 0.994049i \(-0.534744\pi\)
−0.108935 + 0.994049i \(0.534744\pi\)
\(12\) 36.2498 0.872036
\(13\) 43.9341 0.937317 0.468658 − 0.883380i \(-0.344738\pi\)
0.468658 + 0.883380i \(0.344738\pi\)
\(14\) 5.45833 0.104200
\(15\) 0 0
\(16\) 62.0062 0.968847
\(17\) 124.854 1.78126 0.890632 − 0.454725i \(-0.150263\pi\)
0.890632 + 0.454725i \(0.150263\pi\)
\(18\) −1.74197 −0.0228103
\(19\) 93.4871 1.12881 0.564405 − 0.825498i \(-0.309106\pi\)
0.564405 + 0.825498i \(0.309106\pi\)
\(20\) 0 0
\(21\) −86.5642 −0.899517
\(22\) −2.29495 −0.0222402
\(23\) −23.6463 −0.214373 −0.107187 − 0.994239i \(-0.534184\pi\)
−0.107187 + 0.994239i \(0.534184\pi\)
\(24\) 21.0428 0.178973
\(25\) 0 0
\(26\) 12.6849 0.0956815
\(27\) 151.258 1.07813
\(28\) −149.663 −1.01013
\(29\) −171.051 −1.09529 −0.547644 − 0.836711i \(-0.684475\pi\)
−0.547644 + 0.836711i \(0.684475\pi\)
\(30\) 0 0
\(31\) −70.6154 −0.409126 −0.204563 − 0.978853i \(-0.565577\pi\)
−0.204563 + 0.978853i \(0.565577\pi\)
\(32\) 54.6673 0.301997
\(33\) 36.3959 0.191991
\(34\) 36.0486 0.181832
\(35\) 0 0
\(36\) 47.7632 0.221126
\(37\) −275.854 −1.22568 −0.612840 − 0.790207i \(-0.709973\pi\)
−0.612840 + 0.790207i \(0.709973\pi\)
\(38\) 26.9922 0.115229
\(39\) −201.172 −0.825980
\(40\) 0 0
\(41\) 259.564 0.988710 0.494355 − 0.869260i \(-0.335404\pi\)
0.494355 + 0.869260i \(0.335404\pi\)
\(42\) −24.9934 −0.0918229
\(43\) 433.498 1.53739 0.768695 − 0.639616i \(-0.220907\pi\)
0.768695 + 0.639616i \(0.220907\pi\)
\(44\) 62.9256 0.215600
\(45\) 0 0
\(46\) −6.82730 −0.0218833
\(47\) −249.622 −0.774706 −0.387353 − 0.921932i \(-0.626610\pi\)
−0.387353 + 0.921932i \(0.626610\pi\)
\(48\) −283.923 −0.853766
\(49\) 14.3927 0.0419612
\(50\) 0 0
\(51\) −571.698 −1.56968
\(52\) −347.810 −0.927549
\(53\) 112.842 0.292452 0.146226 − 0.989251i \(-0.453287\pi\)
0.146226 + 0.989251i \(0.453287\pi\)
\(54\) 43.6721 0.110056
\(55\) 0 0
\(56\) −86.8782 −0.207314
\(57\) −428.072 −0.994728
\(58\) −49.3870 −0.111807
\(59\) 844.191 1.86279 0.931393 − 0.364016i \(-0.118595\pi\)
0.931393 + 0.364016i \(0.118595\pi\)
\(60\) 0 0
\(61\) 768.836 1.61376 0.806880 − 0.590715i \(-0.201154\pi\)
0.806880 + 0.590715i \(0.201154\pi\)
\(62\) −20.3885 −0.0417637
\(63\) −114.058 −0.228095
\(64\) −480.266 −0.938020
\(65\) 0 0
\(66\) 10.5085 0.0195985
\(67\) −385.978 −0.703802 −0.351901 − 0.936037i \(-0.614465\pi\)
−0.351901 + 0.936037i \(0.614465\pi\)
\(68\) −988.422 −1.76270
\(69\) 108.275 0.188910
\(70\) 0 0
\(71\) 350.291 0.585520 0.292760 − 0.956186i \(-0.405426\pi\)
0.292760 + 0.956186i \(0.405426\pi\)
\(72\) 27.7262 0.0453829
\(73\) 773.075 1.23947 0.619737 − 0.784810i \(-0.287239\pi\)
0.619737 + 0.784810i \(0.287239\pi\)
\(74\) −79.6465 −0.125118
\(75\) 0 0
\(76\) −740.103 −1.11705
\(77\) −150.266 −0.222394
\(78\) −58.0836 −0.0843163
\(79\) −497.416 −0.708401 −0.354201 − 0.935169i \(-0.615247\pi\)
−0.354201 + 0.935169i \(0.615247\pi\)
\(80\) 0 0
\(81\) −529.701 −0.726613
\(82\) 74.9431 0.100928
\(83\) 349.967 0.462818 0.231409 − 0.972857i \(-0.425666\pi\)
0.231409 + 0.972857i \(0.425666\pi\)
\(84\) 685.297 0.890143
\(85\) 0 0
\(86\) 125.162 0.156937
\(87\) 783.233 0.965188
\(88\) 36.5279 0.0442487
\(89\) −1377.29 −1.64036 −0.820181 − 0.572104i \(-0.806127\pi\)
−0.820181 + 0.572104i \(0.806127\pi\)
\(90\) 0 0
\(91\) 830.566 0.956780
\(92\) 187.199 0.212139
\(93\) 323.344 0.360529
\(94\) −72.0726 −0.0790822
\(95\) 0 0
\(96\) −250.318 −0.266125
\(97\) 1205.85 1.26222 0.631111 − 0.775693i \(-0.282599\pi\)
0.631111 + 0.775693i \(0.282599\pi\)
\(98\) 4.15555 0.00428341
\(99\) 47.9557 0.0486841
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 125.4.a.c.1.3 yes 6
3.2 odd 2 1125.4.a.f.1.4 6
4.3 odd 2 2000.4.a.k.1.5 6
5.2 odd 4 125.4.b.c.124.7 12
5.3 odd 4 125.4.b.c.124.6 12
5.4 even 2 125.4.a.b.1.4 ✓ 6
15.14 odd 2 1125.4.a.k.1.3 6
20.19 odd 2 2000.4.a.n.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
125.4.a.b.1.4 ✓ 6 5.4 even 2
125.4.a.c.1.3 yes 6 1.1 even 1 trivial
125.4.b.c.124.6 12 5.3 odd 4
125.4.b.c.124.7 12 5.2 odd 4
1125.4.a.f.1.4 6 3.2 odd 2
1125.4.a.k.1.3 6 15.14 odd 2
2000.4.a.k.1.5 6 4.3 odd 2
2000.4.a.n.1.2 6 20.19 odd 2