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Properties

Label 125.4.b.c.124.7
Level $125$
Weight $4$
Character 125.124
Analytic conductor $7.375$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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(124,125)] 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("125.124"); 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([1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 125 = 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 125.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.37523875072\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 59x^{10} + 1261x^{8} + 11844x^{6} + 45376x^{4} + 43840x^{2} + 6400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 124.7
Root \(-3.45688i\) of defining polynomial
Character \(\chi\) \(=\) 125.124
Dual form 125.4.b.c.124.6

$q$-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)
 
\(f(q)\) \(=\) \(q+0.288727i q^{2} +4.57894i q^{3} +7.91664 q^{4} -1.32206 q^{6} +18.9048i q^{7} +4.59556i q^{8} +6.03327 q^{9} -7.94853 q^{11} +36.2498i q^{12} -43.9341i q^{13} -5.45833 q^{14} +62.0062 q^{16} +124.854i q^{17} +1.74197i q^{18} -93.4871 q^{19} -86.5642 q^{21} -2.29495i q^{22} +23.6463i q^{23} -21.0428 q^{24} +12.6849 q^{26} +151.258i q^{27} +149.663i q^{28} +171.051 q^{29} -70.6154 q^{31} +54.6673i q^{32} -36.3959i q^{33} -36.0486 q^{34} +47.7632 q^{36} -275.854i q^{37} -26.9922i q^{38} +201.172 q^{39} +259.564 q^{41} -24.9934i q^{42} -433.498i q^{43} -62.9256 q^{44} -6.82730 q^{46} -249.622i q^{47} +283.923i q^{48} -14.3927 q^{49} -571.698 q^{51} -347.810i q^{52} -112.842i q^{53} -43.6721 q^{54} -86.8782 q^{56} -428.072i q^{57} +49.3870i q^{58} -844.191 q^{59} +768.836 q^{61} -20.3885i q^{62} +114.058i q^{63} +480.266 q^{64} +10.5085 q^{66} -385.978i q^{67} +988.422i q^{68} -108.275 q^{69} +350.291 q^{71} +27.7262i q^{72} -773.075i q^{73} +79.6465 q^{74} -740.103 q^{76} -150.266i q^{77} +58.0836i q^{78} +497.416 q^{79} -529.701 q^{81} +74.9431i q^{82} -349.967i q^{83} -685.297 q^{84} +125.162 q^{86} +783.233i q^{87} -36.5279i q^{88} +1377.29 q^{89} +830.566 q^{91} +187.199i q^{92} -323.344i q^{93} +72.0726 q^{94} -250.318 q^{96} +1205.85i q^{97} -4.15555i q^{98} -47.9557 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 66 q^{4} - 26 q^{6} - 184 q^{9} + 54 q^{11} + 28 q^{14} + 82 q^{16} + 210 q^{19} - 296 q^{21} + 950 q^{24} - 316 q^{26} + 60 q^{29} - 336 q^{31} - 502 q^{34} + 2392 q^{36} - 68 q^{39} + 774 q^{41}+ \cdots + 5022 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/125\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.288727i 0.102080i 0.998697 + 0.0510401i \(0.0162536\pi\)
−0.998697 + 0.0510401i \(0.983746\pi\)
\(3\) 4.57894i 0.881218i 0.897699 + 0.440609i \(0.145237\pi\)
−0.897699 + 0.440609i \(0.854763\pi\)
\(4\) 7.91664 0.989580
\(5\) 0 0
\(6\) −1.32206 −0.0899550
\(7\) 18.9048i 1.02076i 0.859948 + 0.510382i \(0.170496\pi\)
−0.859948 + 0.510382i \(0.829504\pi\)
\(8\) 4.59556i 0.203097i
\(9\) 6.03327 0.223455
\(10\) 0 0
\(11\) −7.94853 −0.217870 −0.108935 − 0.994049i \(-0.534744\pi\)
−0.108935 + 0.994049i \(0.534744\pi\)
\(12\) 36.2498i 0.872036i
\(13\) − 43.9341i − 0.937317i −0.883380 − 0.468658i \(-0.844738\pi\)
0.883380 − 0.468658i \(-0.155262\pi\)
\(14\) −5.45833 −0.104200
\(15\) 0 0
\(16\) 62.0062 0.968847
\(17\) 124.854i 1.78126i 0.454725 + 0.890632i \(0.349737\pi\)
−0.454725 + 0.890632i \(0.650263\pi\)
\(18\) 1.74197i 0.0228103i
\(19\) −93.4871 −1.12881 −0.564405 − 0.825498i \(-0.690894\pi\)
−0.564405 + 0.825498i \(0.690894\pi\)
\(20\) 0 0
\(21\) −86.5642 −0.899517
\(22\) − 2.29495i − 0.0222402i
\(23\) 23.6463i 0.214373i 0.994239 + 0.107187i \(0.0341842\pi\)
−0.994239 + 0.107187i \(0.965816\pi\)
\(24\) −21.0428 −0.178973
\(25\) 0 0
\(26\) 12.6849 0.0956815
\(27\) 151.258i 1.07813i
\(28\) 149.663i 1.01013i
\(29\) 171.051 1.09529 0.547644 − 0.836711i \(-0.315525\pi\)
0.547644 + 0.836711i \(0.315525\pi\)
\(30\) 0 0
\(31\) −70.6154 −0.409126 −0.204563 − 0.978853i \(-0.565577\pi\)
−0.204563 + 0.978853i \(0.565577\pi\)
\(32\) 54.6673i 0.301997i
\(33\) − 36.3959i − 0.191991i
\(34\) −36.0486 −0.181832
\(35\) 0 0
\(36\) 47.7632 0.221126
\(37\) − 275.854i − 1.22568i −0.790207 − 0.612840i \(-0.790027\pi\)
0.790207 − 0.612840i \(-0.209973\pi\)
\(38\) − 26.9922i − 0.115229i
\(39\) 201.172 0.825980
\(40\) 0 0
\(41\) 259.564 0.988710 0.494355 − 0.869260i \(-0.335404\pi\)
0.494355 + 0.869260i \(0.335404\pi\)
\(42\) − 24.9934i − 0.0918229i
\(43\) − 433.498i − 1.53739i −0.639616 − 0.768695i \(-0.720907\pi\)
0.639616 − 0.768695i \(-0.279093\pi\)
\(44\) −62.9256 −0.215600
\(45\) 0 0
\(46\) −6.82730 −0.0218833
\(47\) − 249.622i − 0.774706i −0.921932 − 0.387353i \(-0.873390\pi\)
0.921932 − 0.387353i \(-0.126610\pi\)
\(48\) 283.923i 0.853766i
\(49\) −14.3927 −0.0419612
\(50\) 0 0
\(51\) −571.698 −1.56968
\(52\) − 347.810i − 0.927549i
\(53\) − 112.842i − 0.292452i −0.989251 − 0.146226i \(-0.953287\pi\)
0.989251 − 0.146226i \(-0.0467127\pi\)
\(54\) −43.6721 −0.110056
\(55\) 0 0
\(56\) −86.8782 −0.207314
\(57\) − 428.072i − 0.994728i
\(58\) 49.3870i 0.111807i
\(59\) −844.191 −1.86279 −0.931393 − 0.364016i \(-0.881405\pi\)
−0.931393 + 0.364016i \(0.881405\pi\)
\(60\) 0 0
\(61\) 768.836 1.61376 0.806880 − 0.590715i \(-0.201154\pi\)
0.806880 + 0.590715i \(0.201154\pi\)
\(62\) − 20.3885i − 0.0417637i
\(63\) 114.058i 0.228095i
\(64\) 480.266 0.938020
\(65\) 0 0
\(66\) 10.5085 0.0195985
\(67\) − 385.978i − 0.703802i −0.936037 − 0.351901i \(-0.885535\pi\)
0.936037 − 0.351901i \(-0.114465\pi\)
\(68\) 988.422i 1.76270i
\(69\) −108.275 −0.188910
\(70\) 0 0
\(71\) 350.291 0.585520 0.292760 − 0.956186i \(-0.405426\pi\)
0.292760 + 0.956186i \(0.405426\pi\)
\(72\) 27.7262i 0.0453829i
\(73\) − 773.075i − 1.23947i −0.784810 − 0.619737i \(-0.787239\pi\)
0.784810 − 0.619737i \(-0.212761\pi\)
\(74\) 79.6465 0.125118
\(75\) 0 0
\(76\) −740.103 −1.11705
\(77\) − 150.266i − 0.222394i
\(78\) 58.0836i 0.0843163i
\(79\) 497.416 0.708401 0.354201 − 0.935169i \(-0.384753\pi\)
0.354201 + 0.935169i \(0.384753\pi\)
\(80\) 0 0
\(81\) −529.701 −0.726613
\(82\) 74.9431i 0.100928i
\(83\) − 349.967i − 0.462818i −0.972857 − 0.231409i \(-0.925666\pi\)
0.972857 − 0.231409i \(-0.0743336\pi\)
\(84\) −685.297 −0.890143
\(85\) 0 0
\(86\) 125.162 0.156937
\(87\) 783.233i 0.965188i
\(88\) − 36.5279i − 0.0442487i
\(89\) 1377.29 1.64036 0.820181 − 0.572104i \(-0.193873\pi\)
0.820181 + 0.572104i \(0.193873\pi\)
\(90\) 0 0
\(91\) 830.566 0.956780
\(92\) 187.199i 0.212139i
\(93\) − 323.344i − 0.360529i
\(94\) 72.0726 0.0790822
\(95\) 0 0
\(96\) −250.318 −0.266125
\(97\) 1205.85i 1.26222i 0.775693 + 0.631111i \(0.217401\pi\)
−0.775693 + 0.631111i \(0.782599\pi\)
\(98\) − 4.15555i − 0.00428341i
\(99\) −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.b.c.124.7 12
5.2 odd 4 125.4.a.b.1.4 ✓ 6
5.3 odd 4 125.4.a.c.1.3 yes 6
5.4 even 2 inner 125.4.b.c.124.6 12
15.2 even 4 1125.4.a.k.1.3 6
15.8 even 4 1125.4.a.f.1.4 6
20.3 even 4 2000.4.a.k.1.5 6
20.7 even 4 2000.4.a.n.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
125.4.a.b.1.4 ✓ 6 5.2 odd 4
125.4.a.c.1.3 yes 6 5.3 odd 4
125.4.b.c.124.6 12 5.4 even 2 inner
125.4.b.c.124.7 12 1.1 even 1 trivial
1125.4.a.f.1.4 6 15.8 even 4
1125.4.a.k.1.3 6 15.2 even 4
2000.4.a.k.1.5 6 20.3 even 4
2000.4.a.n.1.2 6 20.7 even 4