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Properties

Label 1080.4.a.c.1.2
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $0$
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] = [1080,4,Mod(1,1080)] 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("1080.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-15,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.7220628062\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.697.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 7x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.94883\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} -14.4806 q^{7} +13.9443 q^{11} -38.8276 q^{13} -126.063 q^{17} +60.5995 q^{19} +50.3522 q^{23} +25.0000 q^{25} -71.3302 q^{29} -28.7087 q^{31} +72.4028 q^{35} +76.6164 q^{37} +396.151 q^{41} -351.146 q^{43} -4.16809 q^{47} -133.313 q^{49} +171.983 q^{53} -69.7213 q^{55} +116.634 q^{59} +523.955 q^{61} +194.138 q^{65} -872.421 q^{67} +802.371 q^{71} -13.5264 q^{73} -201.921 q^{77} +597.959 q^{79} +75.9439 q^{83} +630.316 q^{85} +799.744 q^{89} +562.246 q^{91} -302.998 q^{95} +445.123 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 24 q^{7} - 6 q^{11} + 48 q^{13} + 27 q^{17} - 195 q^{19} + 27 q^{23} + 75 q^{25} - 60 q^{29} - 279 q^{31} + 120 q^{35} - 138 q^{37} - 66 q^{41} + 222 q^{43} + 264 q^{47} - 237 q^{49} + 507 q^{53}+ \cdots - 336 q^{97}+O(q^{100}) \) 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
\(7\) −14.4806 −0.781877 −0.390938 − 0.920417i \(-0.627849\pi\)
−0.390938 + 0.920417i \(0.627849\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 13.9443 0.382214 0.191107 − 0.981569i \(-0.438792\pi\)
0.191107 + 0.981569i \(0.438792\pi\)
\(12\) 0 0
\(13\) −38.8276 −0.828373 −0.414186 − 0.910192i \(-0.635934\pi\)
−0.414186 + 0.910192i \(0.635934\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −126.063 −1.79852 −0.899259 − 0.437416i \(-0.855894\pi\)
−0.899259 + 0.437416i \(0.855894\pi\)
\(18\) 0 0
\(19\) 60.5995 0.731709 0.365855 − 0.930672i \(-0.380777\pi\)
0.365855 + 0.930672i \(0.380777\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 50.3522 0.456485 0.228243 − 0.973604i \(-0.426702\pi\)
0.228243 + 0.973604i \(0.426702\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −71.3302 −0.456748 −0.228374 − 0.973574i \(-0.573341\pi\)
−0.228374 + 0.973574i \(0.573341\pi\)
\(30\) 0 0
\(31\) −28.7087 −0.166330 −0.0831650 − 0.996536i \(-0.526503\pi\)
−0.0831650 + 0.996536i \(0.526503\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 72.4028 0.349666
\(36\) 0 0
\(37\) 76.6164 0.340423 0.170212 − 0.985408i \(-0.445555\pi\)
0.170212 + 0.985408i \(0.445555\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 396.151 1.50898 0.754492 − 0.656309i \(-0.227883\pi\)
0.754492 + 0.656309i \(0.227883\pi\)
\(42\) 0 0
\(43\) −351.146 −1.24533 −0.622665 − 0.782488i \(-0.713950\pi\)
−0.622665 + 0.782488i \(0.713950\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.16809 −0.0129357 −0.00646786 − 0.999979i \(-0.502059\pi\)
−0.00646786 + 0.999979i \(0.502059\pi\)
\(48\) 0 0
\(49\) −133.313 −0.388669
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 171.983 0.445730 0.222865 − 0.974849i \(-0.428459\pi\)
0.222865 + 0.974849i \(0.428459\pi\)
\(54\) 0 0
\(55\) −69.7213 −0.170931
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 116.634 0.257363 0.128682 − 0.991686i \(-0.458926\pi\)
0.128682 + 0.991686i \(0.458926\pi\)
\(60\) 0 0
\(61\) 523.955 1.09976 0.549882 − 0.835242i \(-0.314673\pi\)
0.549882 + 0.835242i \(0.314673\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 194.138 0.370460
\(66\) 0 0
\(67\) −872.421 −1.59079 −0.795397 − 0.606088i \(-0.792738\pi\)
−0.795397 + 0.606088i \(0.792738\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 802.371 1.34118 0.670591 − 0.741827i \(-0.266040\pi\)
0.670591 + 0.741827i \(0.266040\pi\)
\(72\) 0 0
\(73\) −13.5264 −0.0216869 −0.0108435 − 0.999941i \(-0.503452\pi\)
−0.0108435 + 0.999941i \(0.503452\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −201.921 −0.298844
\(78\) 0 0
\(79\) 597.959 0.851590 0.425795 − 0.904820i \(-0.359994\pi\)
0.425795 + 0.904820i \(0.359994\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 75.9439 0.100433 0.0502164 − 0.998738i \(-0.484009\pi\)
0.0502164 + 0.998738i \(0.484009\pi\)
\(84\) 0 0
\(85\) 630.316 0.804322
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 799.744 0.952502 0.476251 − 0.879309i \(-0.341995\pi\)
0.476251 + 0.879309i \(0.341995\pi\)
\(90\) 0 0
\(91\) 562.246 0.647685
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −302.998 −0.327230
\(96\) 0 0
\(97\) 445.123 0.465932 0.232966 − 0.972485i \(-0.425157\pi\)
0.232966 + 0.972485i \(0.425157\pi\)
\(98\) 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 1080.4.a.c.1.2 ✓ 3
3.2 odd 2 1080.4.a.i.1.2 yes 3
4.3 odd 2 2160.4.a.bl.1.2 3
12.11 even 2 2160.4.a.bt.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.c.1.2 ✓ 3 1.1 even 1 trivial
1080.4.a.i.1.2 yes 3 3.2 odd 2
2160.4.a.bl.1.2 3 4.3 odd 2
2160.4.a.bt.1.2 3 12.11 even 2