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Properties

Label 1156.4.b.h.577.13
Level $1156$
Weight $4$
Character 1156.577
Analytic conductor $68.206$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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] = [1156,4,Mod(577,1156)] 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("1156.577"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1156.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,0,0,0,-216] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(68.2062079666\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.13
Character \(\chi\) \(=\) 1156.577
Dual form 1156.4.b.h.577.12

$q$-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)
 
\(f(q)\) \(=\) \(q+1.14445i q^{3} -4.44584i q^{5} +23.8731i q^{7} +25.6902 q^{9} +35.8559i q^{11} -4.00455 q^{13} +5.08803 q^{15} +144.209 q^{19} -27.3215 q^{21} -146.305i q^{23} +105.235 q^{25} +60.3013i q^{27} -44.3405i q^{29} +90.5083i q^{31} -41.0353 q^{33} +106.136 q^{35} +142.148i q^{37} -4.58301i q^{39} +167.813i q^{41} -109.737 q^{43} -114.215i q^{45} -237.183 q^{47} -226.923 q^{49} -612.843 q^{53} +159.410 q^{55} +165.040i q^{57} +420.210 q^{59} -346.867i q^{61} +613.305i q^{63} +17.8036i q^{65} +279.957 q^{67} +167.439 q^{69} -501.325i q^{71} +1086.85i q^{73} +120.436i q^{75} -855.991 q^{77} -598.857i q^{79} +624.625 q^{81} +842.938 q^{83} +50.7454 q^{87} -1343.43 q^{89} -95.6010i q^{91} -103.582 q^{93} -641.131i q^{95} +562.857i q^{97} +921.148i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 216 q^{9} - 144 q^{13} + 276 q^{15} - 276 q^{19} - 84 q^{21} - 888 q^{25} - 780 q^{33} + 1356 q^{35} - 1308 q^{43} - 1164 q^{47} - 1308 q^{49} + 3540 q^{53} - 144 q^{55} + 492 q^{59} - 1596 q^{67}+ \cdots - 2040 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 1.14445i 0.220249i 0.993918 + 0.110125i \(0.0351250\pi\)
−0.993918 + 0.110125i \(0.964875\pi\)
\(4\) 0 0
\(5\) − 4.44584i − 0.397648i −0.980035 − 0.198824i \(-0.936288\pi\)
0.980035 − 0.198824i \(-0.0637121\pi\)
\(6\) 0 0
\(7\) 23.8731i 1.28902i 0.764594 + 0.644512i \(0.222939\pi\)
−0.764594 + 0.644512i \(0.777061\pi\)
\(8\) 0 0
\(9\) 25.6902 0.951490
\(10\) 0 0
\(11\) 35.8559i 0.982816i 0.870930 + 0.491408i \(0.163518\pi\)
−0.870930 + 0.491408i \(0.836482\pi\)
\(12\) 0 0
\(13\) −4.00455 −0.0854357 −0.0427178 − 0.999087i \(-0.513602\pi\)
−0.0427178 + 0.999087i \(0.513602\pi\)
\(14\) 0 0
\(15\) 5.08803 0.0875816
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 144.209 1.74126 0.870629 − 0.491941i \(-0.163712\pi\)
0.870629 + 0.491941i \(0.163712\pi\)
\(20\) 0 0
\(21\) −27.3215 −0.283907
\(22\) 0 0
\(23\) − 146.305i − 1.32638i −0.748452 − 0.663189i \(-0.769202\pi\)
0.748452 − 0.663189i \(-0.230798\pi\)
\(24\) 0 0
\(25\) 105.235 0.841876
\(26\) 0 0
\(27\) 60.3013i 0.429814i
\(28\) 0 0
\(29\) − 44.3405i − 0.283925i −0.989872 − 0.141962i \(-0.954659\pi\)
0.989872 − 0.141962i \(-0.0453412\pi\)
\(30\) 0 0
\(31\) 90.5083i 0.524380i 0.965016 + 0.262190i \(0.0844447\pi\)
−0.965016 + 0.262190i \(0.915555\pi\)
\(32\) 0 0
\(33\) −41.0353 −0.216464
\(34\) 0 0
\(35\) 106.136 0.512577
\(36\) 0 0
\(37\) 142.148i 0.631596i 0.948826 + 0.315798i \(0.102272\pi\)
−0.948826 + 0.315798i \(0.897728\pi\)
\(38\) 0 0
\(39\) − 4.58301i − 0.0188171i
\(40\) 0 0
\(41\) 167.813i 0.639218i 0.947550 + 0.319609i \(0.103552\pi\)
−0.947550 + 0.319609i \(0.896448\pi\)
\(42\) 0 0
\(43\) −109.737 −0.389179 −0.194590 − 0.980885i \(-0.562338\pi\)
−0.194590 + 0.980885i \(0.562338\pi\)
\(44\) 0 0
\(45\) − 114.215i − 0.378358i
\(46\) 0 0
\(47\) −237.183 −0.736100 −0.368050 − 0.929806i \(-0.619975\pi\)
−0.368050 + 0.929806i \(0.619975\pi\)
\(48\) 0 0
\(49\) −226.923 −0.661583
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −612.843 −1.58831 −0.794156 − 0.607715i \(-0.792086\pi\)
−0.794156 + 0.607715i \(0.792086\pi\)
\(54\) 0 0
\(55\) 159.410 0.390814
\(56\) 0 0
\(57\) 165.040i 0.383511i
\(58\) 0 0
\(59\) 420.210 0.927233 0.463616 − 0.886036i \(-0.346552\pi\)
0.463616 + 0.886036i \(0.346552\pi\)
\(60\) 0 0
\(61\) − 346.867i − 0.728062i −0.931387 − 0.364031i \(-0.881400\pi\)
0.931387 − 0.364031i \(-0.118600\pi\)
\(62\) 0 0
\(63\) 613.305i 1.22649i
\(64\) 0 0
\(65\) 17.8036i 0.0339733i
\(66\) 0 0
\(67\) 279.957 0.510480 0.255240 − 0.966878i \(-0.417846\pi\)
0.255240 + 0.966878i \(0.417846\pi\)
\(68\) 0 0
\(69\) 167.439 0.292134
\(70\) 0 0
\(71\) − 501.325i − 0.837976i −0.907992 − 0.418988i \(-0.862385\pi\)
0.907992 − 0.418988i \(-0.137615\pi\)
\(72\) 0 0
\(73\) 1086.85i 1.74255i 0.490798 + 0.871273i \(0.336705\pi\)
−0.490798 + 0.871273i \(0.663295\pi\)
\(74\) 0 0
\(75\) 120.436i 0.185423i
\(76\) 0 0
\(77\) −855.991 −1.26687
\(78\) 0 0
\(79\) − 598.857i − 0.852870i −0.904518 − 0.426435i \(-0.859769\pi\)
0.904518 − 0.426435i \(-0.140231\pi\)
\(80\) 0 0
\(81\) 624.625 0.856824
\(82\) 0 0
\(83\) 842.938 1.11475 0.557376 − 0.830260i \(-0.311808\pi\)
0.557376 + 0.830260i \(0.311808\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 50.7454 0.0625342
\(88\) 0 0
\(89\) −1343.43 −1.60003 −0.800017 − 0.599978i \(-0.795176\pi\)
−0.800017 + 0.599978i \(0.795176\pi\)
\(90\) 0 0
\(91\) − 95.6010i − 0.110129i
\(92\) 0 0
\(93\) −103.582 −0.115494
\(94\) 0 0
\(95\) − 641.131i − 0.692407i
\(96\) 0 0
\(97\) 562.857i 0.589170i 0.955625 + 0.294585i \(0.0951814\pi\)
−0.955625 + 0.294585i \(0.904819\pi\)
\(98\) 0 0
\(99\) 921.148i 0.935140i
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 1156.4.b.h.577.13 24
17.4 even 4 1156.4.a.k.1.7 yes 12
17.13 even 4 1156.4.a.j.1.6 ✓ 12
17.16 even 2 inner 1156.4.b.h.577.12 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1156.4.a.j.1.6 ✓ 12 17.13 even 4
1156.4.a.k.1.7 yes 12 17.4 even 4
1156.4.b.h.577.12 24 17.16 even 2 inner
1156.4.b.h.577.13 24 1.1 even 1 trivial