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Properties

Label 1125.4.a.k.1.2
Level $1125$
Weight $4$
Character 1125.1
Self dual yes
Analytic conductor $66.377$
Analytic rank $1$
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] = [1125,4,Mod(1,1125)] 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("1125.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1125 = 3^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1125.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,7,0,33,0,0,-67,60,0,0,-27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.3771487565\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 125)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.420609\) of defining polynomial
Character \(\chi\) \(=\) 1125.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.95323 q^{2} -4.18489 q^{4} -1.57558 q^{7} +23.7999 q^{8} -10.9328 q^{11} -62.1138 q^{13} +3.07746 q^{14} -13.0076 q^{16} +44.4961 q^{17} +22.1536 q^{19} +21.3542 q^{22} +154.321 q^{23} +121.323 q^{26} +6.59362 q^{28} -176.002 q^{29} +186.223 q^{31} -164.992 q^{32} -86.9112 q^{34} -306.999 q^{37} -43.2712 q^{38} +69.8885 q^{41} +368.738 q^{43} +45.7524 q^{44} -301.425 q^{46} +126.036 q^{47} -340.518 q^{49} +259.940 q^{52} -520.791 q^{53} -37.4986 q^{56} +343.772 q^{58} +640.769 q^{59} +561.941 q^{61} -363.736 q^{62} +426.329 q^{64} -738.233 q^{67} -186.211 q^{68} +1097.12 q^{71} +211.022 q^{73} +599.641 q^{74} -92.7106 q^{76} +17.2254 q^{77} +945.464 q^{79} -136.508 q^{82} +597.022 q^{83} -720.230 q^{86} -260.199 q^{88} -732.800 q^{89} +97.8651 q^{91} -645.817 q^{92} -246.178 q^{94} -1348.49 q^{97} +665.109 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 7 q^{2} + 33 q^{4} - 67 q^{7} + 60 q^{8} - 27 q^{11} - 149 q^{13} + 14 q^{14} + 41 q^{16} + 72 q^{17} - 105 q^{19} - 394 q^{22} + 269 q^{23} + 158 q^{26} - 676 q^{28} + 30 q^{29} - 168 q^{31} - 33 q^{32}+ \cdots - 789 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.95323 −0.690571 −0.345286 − 0.938498i \(-0.612218\pi\)
−0.345286 + 0.938498i \(0.612218\pi\)
\(3\) 0 0
\(4\) −4.18489 −0.523111
\(5\) 0 0
\(6\) 0 0
\(7\) −1.57558 −0.0850731 −0.0425366 − 0.999095i \(-0.513544\pi\)
−0.0425366 + 0.999095i \(0.513544\pi\)
\(8\) 23.7999 1.05182
\(9\) 0 0
\(10\) 0 0
\(11\) −10.9328 −0.299669 −0.149834 − 0.988711i \(-0.547874\pi\)
−0.149834 + 0.988711i \(0.547874\pi\)
\(12\) 0 0
\(13\) −62.1138 −1.32518 −0.662588 − 0.748984i \(-0.730542\pi\)
−0.662588 + 0.748984i \(0.730542\pi\)
\(14\) 3.07746 0.0587491
\(15\) 0 0
\(16\) −13.0076 −0.203243
\(17\) 44.4961 0.634818 0.317409 − 0.948289i \(-0.397187\pi\)
0.317409 + 0.948289i \(0.397187\pi\)
\(18\) 0 0
\(19\) 22.1536 0.267494 0.133747 − 0.991015i \(-0.457299\pi\)
0.133747 + 0.991015i \(0.457299\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 21.3542 0.206942
\(23\) 154.321 1.39905 0.699526 − 0.714607i \(-0.253395\pi\)
0.699526 + 0.714607i \(0.253395\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 121.323 0.915128
\(27\) 0 0
\(28\) 6.59362 0.0445027
\(29\) −176.002 −1.12699 −0.563495 − 0.826119i \(-0.690544\pi\)
−0.563495 + 0.826119i \(0.690544\pi\)
\(30\) 0 0
\(31\) 186.223 1.07892 0.539461 − 0.842010i \(-0.318628\pi\)
0.539461 + 0.842010i \(0.318628\pi\)
\(32\) −164.992 −0.911463
\(33\) 0 0
\(34\) −86.9112 −0.438387
\(35\) 0 0
\(36\) 0 0
\(37\) −306.999 −1.36406 −0.682032 − 0.731322i \(-0.738904\pi\)
−0.682032 + 0.731322i \(0.738904\pi\)
\(38\) −43.2712 −0.184724
\(39\) 0 0
\(40\) 0 0
\(41\) 69.8885 0.266214 0.133107 − 0.991102i \(-0.457505\pi\)
0.133107 + 0.991102i \(0.457505\pi\)
\(42\) 0 0
\(43\) 368.738 1.30772 0.653860 − 0.756615i \(-0.273148\pi\)
0.653860 + 0.756615i \(0.273148\pi\)
\(44\) 45.7524 0.156760
\(45\) 0 0
\(46\) −301.425 −0.966145
\(47\) 126.036 0.391154 0.195577 − 0.980688i \(-0.437342\pi\)
0.195577 + 0.980688i \(0.437342\pi\)
\(48\) 0 0
\(49\) −340.518 −0.992763
\(50\) 0 0
\(51\) 0 0
\(52\) 259.940 0.693214
\(53\) −520.791 −1.34974 −0.674869 − 0.737938i \(-0.735800\pi\)
−0.674869 + 0.737938i \(0.735800\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −37.4986 −0.0894814
\(57\) 0 0
\(58\) 343.772 0.778268
\(59\) 640.769 1.41391 0.706957 − 0.707256i \(-0.250067\pi\)
0.706957 + 0.707256i \(0.250067\pi\)
\(60\) 0 0
\(61\) 561.941 1.17950 0.589748 − 0.807588i \(-0.299227\pi\)
0.589748 + 0.807588i \(0.299227\pi\)
\(62\) −363.736 −0.745073
\(63\) 0 0
\(64\) 426.329 0.832673
\(65\) 0 0
\(66\) 0 0
\(67\) −738.233 −1.34611 −0.673056 − 0.739592i \(-0.735019\pi\)
−0.673056 + 0.739592i \(0.735019\pi\)
\(68\) −186.211 −0.332080
\(69\) 0 0
\(70\) 0 0
\(71\) 1097.12 1.83386 0.916930 − 0.399047i \(-0.130659\pi\)
0.916930 + 0.399047i \(0.130659\pi\)
\(72\) 0 0
\(73\) 211.022 0.338333 0.169166 − 0.985588i \(-0.445892\pi\)
0.169166 + 0.985588i \(0.445892\pi\)
\(74\) 599.641 0.941984
\(75\) 0 0
\(76\) −92.7106 −0.139929
\(77\) 17.2254 0.0254937
\(78\) 0 0
\(79\) 945.464 1.34649 0.673247 − 0.739418i \(-0.264899\pi\)
0.673247 + 0.739418i \(0.264899\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −136.508 −0.183839
\(83\) 597.022 0.789539 0.394769 − 0.918780i \(-0.370825\pi\)
0.394769 + 0.918780i \(0.370825\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −720.230 −0.903074
\(87\) 0 0
\(88\) −260.199 −0.315196
\(89\) −732.800 −0.872772 −0.436386 − 0.899760i \(-0.643742\pi\)
−0.436386 + 0.899760i \(0.643742\pi\)
\(90\) 0 0
\(91\) 97.8651 0.112737
\(92\) −645.817 −0.731860
\(93\) 0 0
\(94\) −246.178 −0.270120
\(95\) 0 0
\(96\) 0 0
\(97\) −1348.49 −1.41153 −0.705763 − 0.708448i \(-0.749396\pi\)
−0.705763 + 0.708448i \(0.749396\pi\)
\(98\) 665.109 0.685573
\(99\) 0 0
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 1125.4.a.k.1.2 6
3.2 odd 2 125.4.a.b.1.5 ✓ 6
5.4 even 2 1125.4.a.f.1.5 6
12.11 even 2 2000.4.a.n.1.3 6
15.2 even 4 125.4.b.c.124.8 12
15.8 even 4 125.4.b.c.124.5 12
15.14 odd 2 125.4.a.c.1.2 yes 6
60.59 even 2 2000.4.a.k.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
125.4.a.b.1.5 ✓ 6 3.2 odd 2
125.4.a.c.1.2 yes 6 15.14 odd 2
125.4.b.c.124.5 12 15.8 even 4
125.4.b.c.124.8 12 15.2 even 4
1125.4.a.f.1.5 6 5.4 even 2
1125.4.a.k.1.2 6 1.1 even 1 trivial
2000.4.a.k.1.4 6 60.59 even 2
2000.4.a.n.1.3 6 12.11 even 2