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Properties

Label 1160.4.a.g.1.4
Level $1160$
Weight $4$
Character 1160.1
Self dual yes
Analytic conductor $68.442$
Analytic rank $0$
Dimension $11$
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] = [1160,4,Mod(1,1160)] 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("1160.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1160.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [11,0,2,0,-55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.4422156067\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 2 x^{10} - 179 x^{9} + 370 x^{8} + 10353 x^{7} - 19394 x^{6} - 210392 x^{5} + 267796 x^{4} + \cdots + 567808 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-2.40564\) of defining polynomial
Character \(\chi\) \(=\) 1160.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.40564 q^{3} -5.00000 q^{5} -2.91555 q^{7} -21.2129 q^{9} +32.3782 q^{11} -78.7456 q^{13} +12.0282 q^{15} -100.818 q^{17} -73.6273 q^{19} +7.01378 q^{21} +148.015 q^{23} +25.0000 q^{25} +115.983 q^{27} -29.0000 q^{29} -171.441 q^{31} -77.8904 q^{33} +14.5778 q^{35} -91.5959 q^{37} +189.434 q^{39} +155.623 q^{41} +109.640 q^{43} +106.064 q^{45} +93.2416 q^{47} -334.500 q^{49} +242.532 q^{51} +10.3898 q^{53} -161.891 q^{55} +177.121 q^{57} -576.665 q^{59} +118.041 q^{61} +61.8473 q^{63} +393.728 q^{65} -263.320 q^{67} -356.072 q^{69} -640.898 q^{71} +314.329 q^{73} -60.1411 q^{75} -94.4003 q^{77} -826.141 q^{79} +293.734 q^{81} +743.332 q^{83} +504.090 q^{85} +69.7637 q^{87} +1501.88 q^{89} +229.587 q^{91} +412.427 q^{93} +368.137 q^{95} -1090.15 q^{97} -686.835 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) \) 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
\(3\) −2.40564 −0.462966 −0.231483 − 0.972839i \(-0.574358\pi\)
−0.231483 + 0.972839i \(0.574358\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −2.91555 −0.157425 −0.0787125 − 0.996897i \(-0.525081\pi\)
−0.0787125 + 0.996897i \(0.525081\pi\)
\(8\) 0 0
\(9\) −21.2129 −0.785662
\(10\) 0 0
\(11\) 32.3782 0.887490 0.443745 − 0.896153i \(-0.353650\pi\)
0.443745 + 0.896153i \(0.353650\pi\)
\(12\) 0 0
\(13\) −78.7456 −1.68001 −0.840004 − 0.542580i \(-0.817447\pi\)
−0.840004 + 0.542580i \(0.817447\pi\)
\(14\) 0 0
\(15\) 12.0282 0.207045
\(16\) 0 0
\(17\) −100.818 −1.43835 −0.719175 − 0.694829i \(-0.755480\pi\)
−0.719175 + 0.694829i \(0.755480\pi\)
\(18\) 0 0
\(19\) −73.6273 −0.889014 −0.444507 − 0.895775i \(-0.646621\pi\)
−0.444507 + 0.895775i \(0.646621\pi\)
\(20\) 0 0
\(21\) 7.01378 0.0728825
\(22\) 0 0
\(23\) 148.015 1.34188 0.670941 − 0.741511i \(-0.265890\pi\)
0.670941 + 0.741511i \(0.265890\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 115.983 0.826701
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) −171.441 −0.993283 −0.496641 − 0.867956i \(-0.665434\pi\)
−0.496641 + 0.867956i \(0.665434\pi\)
\(32\) 0 0
\(33\) −77.8904 −0.410878
\(34\) 0 0
\(35\) 14.5778 0.0704026
\(36\) 0 0
\(37\) −91.5959 −0.406980 −0.203490 − 0.979077i \(-0.565228\pi\)
−0.203490 + 0.979077i \(0.565228\pi\)
\(38\) 0 0
\(39\) 189.434 0.777787
\(40\) 0 0
\(41\) 155.623 0.592787 0.296393 − 0.955066i \(-0.404216\pi\)
0.296393 + 0.955066i \(0.404216\pi\)
\(42\) 0 0
\(43\) 109.640 0.388837 0.194418 − 0.980919i \(-0.437718\pi\)
0.194418 + 0.980919i \(0.437718\pi\)
\(44\) 0 0
\(45\) 106.064 0.351359
\(46\) 0 0
\(47\) 93.2416 0.289376 0.144688 − 0.989477i \(-0.453782\pi\)
0.144688 + 0.989477i \(0.453782\pi\)
\(48\) 0 0
\(49\) −334.500 −0.975217
\(50\) 0 0
\(51\) 242.532 0.665907
\(52\) 0 0
\(53\) 10.3898 0.0269273 0.0134636 − 0.999909i \(-0.495714\pi\)
0.0134636 + 0.999909i \(0.495714\pi\)
\(54\) 0 0
\(55\) −161.891 −0.396898
\(56\) 0 0
\(57\) 177.121 0.411584
\(58\) 0 0
\(59\) −576.665 −1.27246 −0.636232 − 0.771498i \(-0.719508\pi\)
−0.636232 + 0.771498i \(0.719508\pi\)
\(60\) 0 0
\(61\) 118.041 0.247763 0.123882 − 0.992297i \(-0.460466\pi\)
0.123882 + 0.992297i \(0.460466\pi\)
\(62\) 0 0
\(63\) 61.8473 0.123683
\(64\) 0 0
\(65\) 393.728 0.751322
\(66\) 0 0
\(67\) −263.320 −0.480144 −0.240072 − 0.970755i \(-0.577171\pi\)
−0.240072 + 0.970755i \(0.577171\pi\)
\(68\) 0 0
\(69\) −356.072 −0.621246
\(70\) 0 0
\(71\) −640.898 −1.07128 −0.535638 − 0.844448i \(-0.679929\pi\)
−0.535638 + 0.844448i \(0.679929\pi\)
\(72\) 0 0
\(73\) 314.329 0.503964 0.251982 − 0.967732i \(-0.418918\pi\)
0.251982 + 0.967732i \(0.418918\pi\)
\(74\) 0 0
\(75\) −60.1411 −0.0925932
\(76\) 0 0
\(77\) −94.4003 −0.139713
\(78\) 0 0
\(79\) −826.141 −1.17656 −0.588279 − 0.808658i \(-0.700194\pi\)
−0.588279 + 0.808658i \(0.700194\pi\)
\(80\) 0 0
\(81\) 293.734 0.402927
\(82\) 0 0
\(83\) 743.332 0.983027 0.491514 − 0.870870i \(-0.336444\pi\)
0.491514 + 0.870870i \(0.336444\pi\)
\(84\) 0 0
\(85\) 504.090 0.643250
\(86\) 0 0
\(87\) 69.7637 0.0859707
\(88\) 0 0
\(89\) 1501.88 1.78875 0.894373 − 0.447321i \(-0.147622\pi\)
0.894373 + 0.447321i \(0.147622\pi\)
\(90\) 0 0
\(91\) 229.587 0.264475
\(92\) 0 0
\(93\) 412.427 0.459856
\(94\) 0 0
\(95\) 368.137 0.397579
\(96\) 0 0
\(97\) −1090.15 −1.14112 −0.570558 − 0.821258i \(-0.693273\pi\)
−0.570558 + 0.821258i \(0.693273\pi\)
\(98\) 0 0
\(99\) −686.835 −0.697268
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 1160.4.a.g.1.4 ✓ 11
4.3 odd 2 2320.4.a.bb.1.8 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.4.a.g.1.4 ✓ 11 1.1 even 1 trivial
2320.4.a.bb.1.8 11 4.3 odd 2