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Properties

Label 2160.4.a.bo.1.3
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
Dimension $3$
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] = [2160,4,Mod(1,2160)] 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("2160.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,15,0,-6,0,0,0,-12] 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: \(127.444125612\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.985.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.09376\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.00000 q^{5} +12.8657 q^{7} -18.2595 q^{11} +20.2595 q^{13} +6.65336 q^{17} -150.972 q^{19} +88.1068 q^{23} +25.0000 q^{25} -201.867 q^{29} +268.330 q^{31} +64.3283 q^{35} -123.539 q^{37} +275.174 q^{41} -488.679 q^{43} -436.725 q^{47} -177.475 q^{49} +340.573 q^{53} -91.2975 q^{55} -548.360 q^{59} -206.793 q^{61} +101.298 q^{65} -499.621 q^{67} +460.900 q^{71} -416.818 q^{73} -234.921 q^{77} +289.912 q^{79} -909.471 q^{83} +33.2668 q^{85} +186.814 q^{89} +260.652 q^{91} -754.862 q^{95} +648.440 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} - 6 q^{7} - 12 q^{11} + 18 q^{13} - 21 q^{17} - 57 q^{19} - 87 q^{23} + 75 q^{25} + 138 q^{29} - 117 q^{31} - 30 q^{35} + 150 q^{37} - 180 q^{43} - 684 q^{47} - 81 q^{49} + 87 q^{53} - 60 q^{55}+ \cdots - 1080 q^{97}+O(q^{100}) \) 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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 12.8657 0.694680 0.347340 − 0.937739i \(-0.387085\pi\)
0.347340 + 0.937739i \(0.387085\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −18.2595 −0.500495 −0.250248 − 0.968182i \(-0.580512\pi\)
−0.250248 + 0.968182i \(0.580512\pi\)
\(12\) 0 0
\(13\) 20.2595 0.432229 0.216114 − 0.976368i \(-0.430662\pi\)
0.216114 + 0.976368i \(0.430662\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.65336 0.0949222 0.0474611 − 0.998873i \(-0.484887\pi\)
0.0474611 + 0.998873i \(0.484887\pi\)
\(18\) 0 0
\(19\) −150.972 −1.82292 −0.911459 − 0.411390i \(-0.865043\pi\)
−0.911459 + 0.411390i \(0.865043\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 88.1068 0.798762 0.399381 − 0.916785i \(-0.369225\pi\)
0.399381 + 0.916785i \(0.369225\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −201.867 −1.29261 −0.646306 − 0.763078i \(-0.723687\pi\)
−0.646306 + 0.763078i \(0.723687\pi\)
\(30\) 0 0
\(31\) 268.330 1.55463 0.777313 − 0.629114i \(-0.216582\pi\)
0.777313 + 0.629114i \(0.216582\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 64.3283 0.310670
\(36\) 0 0
\(37\) −123.539 −0.548909 −0.274455 − 0.961600i \(-0.588497\pi\)
−0.274455 + 0.961600i \(0.588497\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 275.174 1.04817 0.524084 − 0.851666i \(-0.324408\pi\)
0.524084 + 0.851666i \(0.324408\pi\)
\(42\) 0 0
\(43\) −488.679 −1.73309 −0.866545 − 0.499098i \(-0.833665\pi\)
−0.866545 + 0.499098i \(0.833665\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −436.725 −1.35538 −0.677691 − 0.735347i \(-0.737019\pi\)
−0.677691 + 0.735347i \(0.737019\pi\)
\(48\) 0 0
\(49\) −177.475 −0.517420
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 340.573 0.882665 0.441332 − 0.897344i \(-0.354506\pi\)
0.441332 + 0.897344i \(0.354506\pi\)
\(54\) 0 0
\(55\) −91.2975 −0.223828
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −548.360 −1.21001 −0.605004 − 0.796223i \(-0.706828\pi\)
−0.605004 + 0.796223i \(0.706828\pi\)
\(60\) 0 0
\(61\) −206.793 −0.434051 −0.217025 − 0.976166i \(-0.569635\pi\)
−0.217025 + 0.976166i \(0.569635\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 101.298 0.193299
\(66\) 0 0
\(67\) −499.621 −0.911021 −0.455511 − 0.890230i \(-0.650543\pi\)
−0.455511 + 0.890230i \(0.650543\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 460.900 0.770406 0.385203 − 0.922832i \(-0.374131\pi\)
0.385203 + 0.922832i \(0.374131\pi\)
\(72\) 0 0
\(73\) −416.818 −0.668285 −0.334143 − 0.942522i \(-0.608447\pi\)
−0.334143 + 0.942522i \(0.608447\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −234.921 −0.347684
\(78\) 0 0
\(79\) 289.912 0.412882 0.206441 − 0.978459i \(-0.433812\pi\)
0.206441 + 0.978459i \(0.433812\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −909.471 −1.20274 −0.601370 − 0.798971i \(-0.705378\pi\)
−0.601370 + 0.798971i \(0.705378\pi\)
\(84\) 0 0
\(85\) 33.2668 0.0424505
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 186.814 0.222497 0.111249 − 0.993793i \(-0.464515\pi\)
0.111249 + 0.993793i \(0.464515\pi\)
\(90\) 0 0
\(91\) 260.652 0.300261
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −754.862 −0.815234
\(96\) 0 0
\(97\) 648.440 0.678754 0.339377 − 0.940651i \(-0.389784\pi\)
0.339377 + 0.940651i \(0.389784\pi\)
\(98\) 0 0
\(99\) 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 2160.4.a.bo.1.3 3
3.2 odd 2 2160.4.a.bg.1.3 3
4.3 odd 2 1080.4.a.m.1.1 yes 3
12.11 even 2 1080.4.a.g.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.g.1.1 ✓ 3 12.11 even 2
1080.4.a.m.1.1 yes 3 4.3 odd 2
2160.4.a.bg.1.3 3 3.2 odd 2
2160.4.a.bo.1.3 3 1.1 even 1 trivial