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Properties

Label 1080.4.a.a.1.1
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $1$
Dimension $2$
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 = [2,0,0,0,-10,0,-5] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.26209\) 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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} -10.2621 q^{7} -5.26209 q^{11} +34.5725 q^{13} +74.8346 q^{17} -50.8830 q^{19} -81.4555 q^{23} +25.0000 q^{25} +152.407 q^{29} +93.9797 q^{31} +51.3104 q^{35} -167.310 q^{37} +18.6209 q^{41} +133.552 q^{43} +46.8346 q^{47} -237.690 q^{49} -105.104 q^{53} +26.3104 q^{55} -77.7099 q^{59} -48.6896 q^{61} -172.863 q^{65} -667.132 q^{67} -344.056 q^{71} -1074.70 q^{73} +54.0000 q^{77} -420.863 q^{79} -296.565 q^{83} -374.173 q^{85} +950.596 q^{89} -354.786 q^{91} +254.415 q^{95} -135.916 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} - 5 q^{7} + 5 q^{11} - 24 q^{13} + 41 q^{17} + 69 q^{19} + 101 q^{23} + 50 q^{25} + 103 q^{29} - 107 q^{31} + 25 q^{35} - 257 q^{37} - 118 q^{41} - 121 q^{43} - 15 q^{47} - 553 q^{49} + 566 q^{53}+ \cdots - 1343 q^{97}+O(q^{100}) \) 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
\(7\) −10.2621 −0.554101 −0.277050 − 0.960855i \(-0.589357\pi\)
−0.277050 + 0.960855i \(0.589357\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.26209 −0.144234 −0.0721172 − 0.997396i \(-0.522976\pi\)
−0.0721172 + 0.997396i \(0.522976\pi\)
\(12\) 0 0
\(13\) 34.5725 0.737592 0.368796 − 0.929510i \(-0.379770\pi\)
0.368796 + 0.929510i \(0.379770\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 74.8346 1.06765 0.533825 − 0.845595i \(-0.320754\pi\)
0.533825 + 0.845595i \(0.320754\pi\)
\(18\) 0 0
\(19\) −50.8830 −0.614387 −0.307193 − 0.951647i \(-0.599390\pi\)
−0.307193 + 0.951647i \(0.599390\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −81.4555 −0.738463 −0.369231 − 0.929338i \(-0.620379\pi\)
−0.369231 + 0.929338i \(0.620379\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 152.407 0.975907 0.487953 − 0.872870i \(-0.337744\pi\)
0.487953 + 0.872870i \(0.337744\pi\)
\(30\) 0 0
\(31\) 93.9797 0.544492 0.272246 − 0.962228i \(-0.412234\pi\)
0.272246 + 0.962228i \(0.412234\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 51.3104 0.247801
\(36\) 0 0
\(37\) −167.310 −0.743396 −0.371698 − 0.928354i \(-0.621224\pi\)
−0.371698 + 0.928354i \(0.621224\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18.6209 0.0709291 0.0354645 − 0.999371i \(-0.488709\pi\)
0.0354645 + 0.999371i \(0.488709\pi\)
\(42\) 0 0
\(43\) 133.552 0.473640 0.236820 − 0.971554i \(-0.423895\pi\)
0.236820 + 0.971554i \(0.423895\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 46.8346 0.145352 0.0726759 − 0.997356i \(-0.476846\pi\)
0.0726759 + 0.997356i \(0.476846\pi\)
\(48\) 0 0
\(49\) −237.690 −0.692972
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −105.104 −0.272400 −0.136200 − 0.990681i \(-0.543489\pi\)
−0.136200 + 0.990681i \(0.543489\pi\)
\(54\) 0 0
\(55\) 26.3104 0.0645036
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −77.7099 −0.171474 −0.0857370 − 0.996318i \(-0.527324\pi\)
−0.0857370 + 0.996318i \(0.527324\pi\)
\(60\) 0 0
\(61\) −48.6896 −0.102198 −0.0510989 − 0.998694i \(-0.516272\pi\)
−0.0510989 + 0.998694i \(0.516272\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −172.863 −0.329861
\(66\) 0 0
\(67\) −667.132 −1.21647 −0.608233 − 0.793759i \(-0.708121\pi\)
−0.608233 + 0.793759i \(0.708121\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −344.056 −0.575098 −0.287549 − 0.957766i \(-0.592840\pi\)
−0.287549 + 0.957766i \(0.592840\pi\)
\(72\) 0 0
\(73\) −1074.70 −1.72308 −0.861539 − 0.507691i \(-0.830499\pi\)
−0.861539 + 0.507691i \(0.830499\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 54.0000 0.0799204
\(78\) 0 0
\(79\) −420.863 −0.599377 −0.299688 − 0.954037i \(-0.596883\pi\)
−0.299688 + 0.954037i \(0.596883\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −296.565 −0.392195 −0.196098 − 0.980584i \(-0.562827\pi\)
−0.196098 + 0.980584i \(0.562827\pi\)
\(84\) 0 0
\(85\) −374.173 −0.477468
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 950.596 1.13217 0.566084 − 0.824348i \(-0.308458\pi\)
0.566084 + 0.824348i \(0.308458\pi\)
\(90\) 0 0
\(91\) −354.786 −0.408700
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 254.415 0.274762
\(96\) 0 0
\(97\) −135.916 −0.142270 −0.0711349 − 0.997467i \(-0.522662\pi\)
−0.0711349 + 0.997467i \(0.522662\pi\)
\(98\) 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.a.1.1 ✓ 2
3.2 odd 2 1080.4.a.b.1.1 yes 2
4.3 odd 2 2160.4.a.y.1.2 2
12.11 even 2 2160.4.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.a.1.1 ✓ 2 1.1 even 1 trivial
1080.4.a.b.1.1 yes 2 3.2 odd 2
2160.4.a.y.1.2 2 4.3 odd 2
2160.4.a.bd.1.2 2 12.11 even 2