↑

Properties

Label 1160.4.a.g.1.3
Level $1160$
Weight $4$
Character 1160.1
Self dual yes
Analytic conductor $68.442$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
Root \(-4.74161\) 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)
 
\(f(q)\) \(=\) \(q-4.74161 q^{3} -5.00000 q^{5} +33.7088 q^{7} -4.51718 q^{9} +40.4441 q^{11} +87.7907 q^{13} +23.7080 q^{15} -94.3494 q^{17} +142.234 q^{19} -159.834 q^{21} +55.3271 q^{23} +25.0000 q^{25} +149.442 q^{27} -29.0000 q^{29} +108.224 q^{31} -191.770 q^{33} -168.544 q^{35} +99.3841 q^{37} -416.269 q^{39} -51.2021 q^{41} -435.763 q^{43} +22.5859 q^{45} -55.0193 q^{47} +793.285 q^{49} +447.368 q^{51} -159.312 q^{53} -202.221 q^{55} -674.419 q^{57} -487.994 q^{59} -298.707 q^{61} -152.269 q^{63} -438.953 q^{65} +26.3304 q^{67} -262.339 q^{69} -507.027 q^{71} -114.654 q^{73} -118.540 q^{75} +1363.32 q^{77} +1237.46 q^{79} -586.631 q^{81} +493.794 q^{83} +471.747 q^{85} +137.507 q^{87} +73.1811 q^{89} +2959.32 q^{91} -513.158 q^{93} -711.172 q^{95} +860.328 q^{97} -182.693 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) \) 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
\(3\) −4.74161 −0.912522 −0.456261 − 0.889846i \(-0.650812\pi\)
−0.456261 + 0.889846i \(0.650812\pi\)
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 33.7088 1.82011 0.910053 − 0.414492i \(-0.136041\pi\)
0.910053 + 0.414492i \(0.136041\pi\)
\(8\) 0 0
\(9\) −4.51718 −0.167303
\(10\) 0 0
\(11\) 40.4441 1.10858 0.554289 − 0.832324i \(-0.312990\pi\)
0.554289 + 0.832324i \(0.312990\pi\)
\(12\) 0 0
\(13\) 87.7907 1.87298 0.936490 − 0.350693i \(-0.114054\pi\)
0.936490 + 0.350693i \(0.114054\pi\)
\(14\) 0 0
\(15\) 23.7080 0.408092
\(16\) 0 0
\(17\) −94.3494 −1.34606 −0.673032 − 0.739613i \(-0.735008\pi\)
−0.673032 + 0.739613i \(0.735008\pi\)
\(18\) 0 0
\(19\) 142.234 1.71741 0.858705 − 0.512470i \(-0.171270\pi\)
0.858705 + 0.512470i \(0.171270\pi\)
\(20\) 0 0
\(21\) −159.834 −1.66089
\(22\) 0 0
\(23\) 55.3271 0.501587 0.250793 − 0.968041i \(-0.419309\pi\)
0.250793 + 0.968041i \(0.419309\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 149.442 1.06519
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 108.224 0.627022 0.313511 − 0.949585i \(-0.398495\pi\)
0.313511 + 0.949585i \(0.398495\pi\)
\(32\) 0 0
\(33\) −191.770 −1.01160
\(34\) 0 0
\(35\) −168.544 −0.813976
\(36\) 0 0
\(37\) 99.3841 0.441585 0.220792 − 0.975321i \(-0.429136\pi\)
0.220792 + 0.975321i \(0.429136\pi\)
\(38\) 0 0
\(39\) −416.269 −1.70914
\(40\) 0 0
\(41\) −51.2021 −0.195035 −0.0975174 − 0.995234i \(-0.531090\pi\)
−0.0975174 + 0.995234i \(0.531090\pi\)
\(42\) 0 0
\(43\) −435.763 −1.54542 −0.772712 − 0.634757i \(-0.781100\pi\)
−0.772712 + 0.634757i \(0.781100\pi\)
\(44\) 0 0
\(45\) 22.5859 0.0748201
\(46\) 0 0
\(47\) −55.0193 −0.170753 −0.0853765 − 0.996349i \(-0.527209\pi\)
−0.0853765 + 0.996349i \(0.527209\pi\)
\(48\) 0 0
\(49\) 793.285 2.31279
\(50\) 0 0
\(51\) 447.368 1.22831
\(52\) 0 0
\(53\) −159.312 −0.412889 −0.206445 − 0.978458i \(-0.566189\pi\)
−0.206445 + 0.978458i \(0.566189\pi\)
\(54\) 0 0
\(55\) −202.221 −0.495771
\(56\) 0 0
\(57\) −674.419 −1.56718
\(58\) 0 0
\(59\) −487.994 −1.07680 −0.538402 − 0.842688i \(-0.680972\pi\)
−0.538402 + 0.842688i \(0.680972\pi\)
\(60\) 0 0
\(61\) −298.707 −0.626976 −0.313488 − 0.949592i \(-0.601498\pi\)
−0.313488 + 0.949592i \(0.601498\pi\)
\(62\) 0 0
\(63\) −152.269 −0.304509
\(64\) 0 0
\(65\) −438.953 −0.837623
\(66\) 0 0
\(67\) 26.3304 0.0480114 0.0240057 − 0.999712i \(-0.492358\pi\)
0.0240057 + 0.999712i \(0.492358\pi\)
\(68\) 0 0
\(69\) −262.339 −0.457709
\(70\) 0 0
\(71\) −507.027 −0.847508 −0.423754 − 0.905777i \(-0.639288\pi\)
−0.423754 + 0.905777i \(0.639288\pi\)
\(72\) 0 0
\(73\) −114.654 −0.183824 −0.0919122 − 0.995767i \(-0.529298\pi\)
−0.0919122 + 0.995767i \(0.529298\pi\)
\(74\) 0 0
\(75\) −118.540 −0.182504
\(76\) 0 0
\(77\) 1363.32 2.01773
\(78\) 0 0
\(79\) 1237.46 1.76234 0.881170 − 0.472800i \(-0.156757\pi\)
0.881170 + 0.472800i \(0.156757\pi\)
\(80\) 0 0
\(81\) −586.631 −0.804707
\(82\) 0 0
\(83\) 493.794 0.653024 0.326512 − 0.945193i \(-0.394127\pi\)
0.326512 + 0.945193i \(0.394127\pi\)
\(84\) 0 0
\(85\) 471.747 0.601978
\(86\) 0 0
\(87\) 137.507 0.169451
\(88\) 0 0
\(89\) 73.1811 0.0871593 0.0435797 − 0.999050i \(-0.486124\pi\)
0.0435797 + 0.999050i \(0.486124\pi\)
\(90\) 0 0
\(91\) 2959.32 3.40902
\(92\) 0 0
\(93\) −513.158 −0.572172
\(94\) 0 0
\(95\) −711.172 −0.768049
\(96\) 0 0
\(97\) 860.328 0.900547 0.450274 − 0.892891i \(-0.351326\pi\)
0.450274 + 0.892891i \(0.351326\pi\)
\(98\) 0 0
\(99\) −182.693 −0.185468
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.1.3 ✓ 11
4.3 odd 2 2320.4.a.bb.1.9 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.4.a.g.1.3 ✓ 11 1.1 even 1 trivial
2320.4.a.bb.1.9 11 4.3 odd 2