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Properties

Label 1125.4.a.j.1.2
Level $1125$
Weight $4$
Character 1125.1
Self dual yes
Analytic conductor $66.377$
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] = [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 = [6,5,0,25,0,0,-28,96,0,0,77] 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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.920588125.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 31x^{4} + 12x^{3} + 231x^{2} + 20x - 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 375)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.74458\) 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)
 
\(f(q)\) \(=\) \(q-2.77928 q^{2} -0.275604 q^{4} -13.3180 q^{7} +23.0002 q^{8} +15.7384 q^{11} -33.9089 q^{13} +37.0144 q^{14} -61.7192 q^{16} -55.9930 q^{17} -107.375 q^{19} -43.7414 q^{22} +100.280 q^{23} +94.2424 q^{26} +3.67048 q^{28} +235.184 q^{29} +156.601 q^{31} -12.4668 q^{32} +155.620 q^{34} -161.272 q^{37} +298.426 q^{38} -253.195 q^{41} -266.095 q^{43} -4.33756 q^{44} -278.707 q^{46} -374.102 q^{47} -165.631 q^{49} +9.34543 q^{52} +222.885 q^{53} -306.316 q^{56} -653.643 q^{58} -474.280 q^{59} +319.211 q^{61} -435.239 q^{62} +528.402 q^{64} -266.197 q^{67} +15.4319 q^{68} +624.757 q^{71} -680.763 q^{73} +448.221 q^{74} +29.5931 q^{76} -209.603 q^{77} +1028.80 q^{79} +703.698 q^{82} -428.681 q^{83} +739.553 q^{86} +361.986 q^{88} +507.722 q^{89} +451.598 q^{91} -27.6376 q^{92} +1039.73 q^{94} -1469.08 q^{97} +460.336 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 5 q^{2} + 25 q^{4} - 28 q^{7} + 96 q^{8} + 77 q^{11} - 137 q^{13} + 173 q^{14} + 73 q^{16} + 349 q^{17} + 34 q^{19} - 208 q^{22} + 131 q^{23} + 118 q^{26} - 254 q^{28} + 157 q^{29} - 261 q^{31} + 573 q^{32}+ \cdots - 807 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.77928 −0.982624 −0.491312 − 0.870984i \(-0.663482\pi\)
−0.491312 + 0.870984i \(0.663482\pi\)
\(3\) 0 0
\(4\) −0.275604 −0.0344505
\(5\) 0 0
\(6\) 0 0
\(7\) −13.3180 −0.719103 −0.359552 − 0.933125i \(-0.617070\pi\)
−0.359552 + 0.933125i \(0.617070\pi\)
\(8\) 23.0002 1.01648
\(9\) 0 0
\(10\) 0 0
\(11\) 15.7384 0.431391 0.215695 − 0.976461i \(-0.430798\pi\)
0.215695 + 0.976461i \(0.430798\pi\)
\(12\) 0 0
\(13\) −33.9089 −0.723434 −0.361717 − 0.932288i \(-0.617809\pi\)
−0.361717 + 0.932288i \(0.617809\pi\)
\(14\) 37.0144 0.706608
\(15\) 0 0
\(16\) −61.7192 −0.964363
\(17\) −55.9930 −0.798841 −0.399420 − 0.916768i \(-0.630789\pi\)
−0.399420 + 0.916768i \(0.630789\pi\)
\(18\) 0 0
\(19\) −107.375 −1.29651 −0.648253 − 0.761425i \(-0.724500\pi\)
−0.648253 + 0.761425i \(0.724500\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −43.7414 −0.423895
\(23\) 100.280 0.909126 0.454563 − 0.890715i \(-0.349795\pi\)
0.454563 + 0.890715i \(0.349795\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 94.2424 0.710863
\(27\) 0 0
\(28\) 3.67048 0.0247734
\(29\) 235.184 1.50595 0.752977 − 0.658047i \(-0.228617\pi\)
0.752977 + 0.658047i \(0.228617\pi\)
\(30\) 0 0
\(31\) 156.601 0.907304 0.453652 − 0.891179i \(-0.350121\pi\)
0.453652 + 0.891179i \(0.350121\pi\)
\(32\) −12.4668 −0.0688699
\(33\) 0 0
\(34\) 155.620 0.784960
\(35\) 0 0
\(36\) 0 0
\(37\) −161.272 −0.716567 −0.358284 − 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(38\) 298.426 1.27398
\(39\) 0 0
\(40\) 0 0
\(41\) −253.195 −0.964447 −0.482224 − 0.876048i \(-0.660171\pi\)
−0.482224 + 0.876048i \(0.660171\pi\)
\(42\) 0 0
\(43\) −266.095 −0.943701 −0.471850 − 0.881679i \(-0.656414\pi\)
−0.471850 + 0.881679i \(0.656414\pi\)
\(44\) −4.33756 −0.0148616
\(45\) 0 0
\(46\) −278.707 −0.893329
\(47\) −374.102 −1.16103 −0.580515 − 0.814250i \(-0.697149\pi\)
−0.580515 + 0.814250i \(0.697149\pi\)
\(48\) 0 0
\(49\) −165.631 −0.482891
\(50\) 0 0
\(51\) 0 0
\(52\) 9.34543 0.0249226
\(53\) 222.885 0.577653 0.288826 − 0.957381i \(-0.406735\pi\)
0.288826 + 0.957381i \(0.406735\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −306.316 −0.730951
\(57\) 0 0
\(58\) −653.643 −1.47979
\(59\) −474.280 −1.04654 −0.523272 − 0.852166i \(-0.675289\pi\)
−0.523272 + 0.852166i \(0.675289\pi\)
\(60\) 0 0
\(61\) 319.211 0.670012 0.335006 − 0.942216i \(-0.391262\pi\)
0.335006 + 0.942216i \(0.391262\pi\)
\(62\) −435.239 −0.891539
\(63\) 0 0
\(64\) 528.402 1.03204
\(65\) 0 0
\(66\) 0 0
\(67\) −266.197 −0.485391 −0.242695 − 0.970103i \(-0.578032\pi\)
−0.242695 + 0.970103i \(0.578032\pi\)
\(68\) 15.4319 0.0275204
\(69\) 0 0
\(70\) 0 0
\(71\) 624.757 1.04430 0.522148 − 0.852855i \(-0.325131\pi\)
0.522148 + 0.852855i \(0.325131\pi\)
\(72\) 0 0
\(73\) −680.763 −1.09147 −0.545735 − 0.837958i \(-0.683749\pi\)
−0.545735 + 0.837958i \(0.683749\pi\)
\(74\) 448.221 0.704116
\(75\) 0 0
\(76\) 29.5931 0.0446652
\(77\) −209.603 −0.310215
\(78\) 0 0
\(79\) 1028.80 1.46518 0.732589 − 0.680671i \(-0.238312\pi\)
0.732589 + 0.680671i \(0.238312\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 703.698 0.947689
\(83\) −428.681 −0.566914 −0.283457 − 0.958985i \(-0.591481\pi\)
−0.283457 + 0.958985i \(0.591481\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 739.553 0.927303
\(87\) 0 0
\(88\) 361.986 0.438498
\(89\) 507.722 0.604702 0.302351 − 0.953197i \(-0.402229\pi\)
0.302351 + 0.953197i \(0.402229\pi\)
\(90\) 0 0
\(91\) 451.598 0.520224
\(92\) −27.6376 −0.0313198
\(93\) 0 0
\(94\) 1039.73 1.14085
\(95\) 0 0
\(96\) 0 0
\(97\) −1469.08 −1.53776 −0.768879 − 0.639394i \(-0.779185\pi\)
−0.768879 + 0.639394i \(0.779185\pi\)
\(98\) 460.336 0.474500
\(99\) 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.j.1.2 6
3.2 odd 2 375.4.a.c.1.5 ✓ 6
5.4 even 2 1125.4.a.g.1.5 6
15.2 even 4 375.4.b.c.124.9 12
15.8 even 4 375.4.b.c.124.4 12
15.14 odd 2 375.4.a.f.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
375.4.a.c.1.5 ✓ 6 3.2 odd 2
375.4.a.f.1.2 yes 6 15.14 odd 2
375.4.b.c.124.4 12 15.8 even 4
375.4.b.c.124.9 12 15.2 even 4
1125.4.a.g.1.5 6 5.4 even 2
1125.4.a.j.1.2 6 1.1 even 1 trivial