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Properties

Label 360.4.k.b.181.2
Level $360$
Weight $4$
Character 360.181
Analytic conductor $21.241$
Analytic rank $0$
Dimension $8$
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] = [360,4,Mod(181,360)] 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("360.181"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.2406876021\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.55839580416.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 7x^{6} - 6x^{5} + 18x^{4} - 24x^{3} + 112x^{2} - 128x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.2
Root \(0.694547 - 1.87553i\) of defining polynomial
Character \(\chi\) \(=\) 360.181
Dual form 360.4.k.b.181.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.57007 + 1.18098i) q^{2} +(5.21057 - 6.07042i) q^{4} +5.00000i q^{5} +1.05849 q^{7} +(-6.22250 + 21.7550i) q^{8} +(-5.90490 - 12.8504i) q^{10} +12.0304i q^{11} +1.58299i q^{13} +(-2.72039 + 1.25005i) q^{14} +(-9.69996 - 63.2607i) q^{16} +5.63366 q^{17} +26.2836i q^{19} +(30.3521 + 26.0528i) q^{20} +(-14.2077 - 30.9190i) q^{22} -104.197 q^{23} -25.0000 q^{25} +(-1.86948 - 4.06840i) q^{26} +(5.51531 - 6.42545i) q^{28} -32.7778i q^{29} +250.045 q^{31} +(99.6393 + 151.129i) q^{32} +(-14.4789 + 6.65325i) q^{34} +5.29243i q^{35} +235.185i q^{37} +(-31.0404 - 67.5508i) q^{38} +(-108.775 - 31.1125i) q^{40} -138.027 q^{41} +355.886i q^{43} +(73.0296 + 62.6852i) q^{44} +(267.793 - 123.054i) q^{46} -462.599 q^{47} -341.880 q^{49} +(64.2519 - 29.5245i) q^{50} +(9.60941 + 8.24828i) q^{52} -5.63429i q^{53} -60.1520 q^{55} +(-6.58643 + 23.0274i) q^{56} +(38.7100 + 84.2415i) q^{58} -46.7356i q^{59} -340.563i q^{61} +(-642.635 + 295.299i) q^{62} +(-434.561 - 270.741i) q^{64} -7.91495 q^{65} +790.372i q^{67} +(29.3546 - 34.1987i) q^{68} +(-6.25026 - 13.6019i) q^{70} -971.879 q^{71} -1026.48 q^{73} +(-277.749 - 604.442i) q^{74} +(159.552 + 136.952i) q^{76} +12.7340i q^{77} -693.490 q^{79} +(316.303 - 48.4998i) q^{80} +(354.741 - 163.008i) q^{82} -117.875i q^{83} +28.1683i q^{85} +(-420.294 - 914.653i) q^{86} +(-261.721 - 74.8592i) q^{88} -106.132 q^{89} +1.67557i q^{91} +(-542.923 + 632.517i) q^{92} +(1188.91 - 546.320i) q^{94} -131.418 q^{95} -1266.82 q^{97} +(878.656 - 403.753i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 24 q^{4} - 80 q^{7} - 44 q^{8} + 10 q^{10} - 76 q^{14} + 88 q^{16} + 216 q^{17} + 100 q^{20} + 272 q^{22} + 32 q^{23} - 200 q^{25} + 264 q^{26} + 432 q^{28} - 136 q^{31} + 8 q^{32} + 84 q^{34}+ \cdots + 2022 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.57007 + 1.18098i −0.908659 + 0.417540i
\(3\) 0 0
\(4\) 5.21057 − 6.07042i 0.651321 − 0.758802i
\(5\) 5.00000i 0.447214i
\(6\) 0 0
\(7\) 1.05849 0.0571529 0.0285764 − 0.999592i \(-0.490903\pi\)
0.0285764 + 0.999592i \(0.490903\pi\)
\(8\) −6.22250 + 21.7550i −0.274998 + 0.961445i
\(9\) 0 0
\(10\) −5.90490 − 12.8504i −0.186729 − 0.406364i
\(11\) 12.0304i 0.329755i 0.986314 + 0.164877i \(0.0527228\pi\)
−0.986314 + 0.164877i \(0.947277\pi\)
\(12\) 0 0
\(13\) 1.58299i 0.0337725i 0.999857 + 0.0168863i \(0.00537532\pi\)
−0.999857 + 0.0168863i \(0.994625\pi\)
\(14\) −2.72039 + 1.25005i −0.0519325 + 0.0238636i
\(15\) 0 0
\(16\) −9.69996 − 63.2607i −0.151562 − 0.988448i
\(17\) 5.63366 0.0803744 0.0401872 − 0.999192i \(-0.487205\pi\)
0.0401872 + 0.999192i \(0.487205\pi\)
\(18\) 0 0
\(19\) 26.2836i 0.317361i 0.987330 + 0.158681i \(0.0507241\pi\)
−0.987330 + 0.158681i \(0.949276\pi\)
\(20\) 30.3521 + 26.0528i 0.339347 + 0.291280i
\(21\) 0 0
\(22\) −14.2077 − 30.9190i −0.137686 − 0.299634i
\(23\) −104.197 −0.944630 −0.472315 − 0.881430i \(-0.656581\pi\)
−0.472315 + 0.881430i \(0.656581\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) −1.86948 − 4.06840i −0.0141014 − 0.0306877i
\(27\) 0 0
\(28\) 5.51531 − 6.42545i 0.0372249 − 0.0433677i
\(29\) − 32.7778i − 0.209886i −0.994478 − 0.104943i \(-0.966534\pi\)
0.994478 − 0.104943i \(-0.0334660\pi\)
\(30\) 0 0
\(31\) 250.045 1.44869 0.724346 − 0.689436i \(-0.242142\pi\)
0.724346 + 0.689436i \(0.242142\pi\)
\(32\) 99.6393 + 151.129i 0.550434 + 0.834878i
\(33\) 0 0
\(34\) −14.4789 + 6.65325i −0.0730328 + 0.0335595i
\(35\) 5.29243i 0.0255595i
\(36\) 0 0
\(37\) 235.185i 1.04498i 0.852646 + 0.522488i \(0.174996\pi\)
−0.852646 + 0.522488i \(0.825004\pi\)
\(38\) −31.0404 − 67.5508i −0.132511 − 0.288373i
\(39\) 0 0
\(40\) −108.775 − 31.1125i −0.429971 − 0.122983i
\(41\) −138.027 −0.525763 −0.262881 − 0.964828i \(-0.584673\pi\)
−0.262881 + 0.964828i \(0.584673\pi\)
\(42\) 0 0
\(43\) 355.886i 1.26214i 0.775726 + 0.631070i \(0.217384\pi\)
−0.775726 + 0.631070i \(0.782616\pi\)
\(44\) 73.0296 + 62.6852i 0.250219 + 0.214776i
\(45\) 0 0
\(46\) 267.793 − 123.054i 0.858346 − 0.394421i
\(47\) −462.599 −1.43568 −0.717840 − 0.696208i \(-0.754869\pi\)
−0.717840 + 0.696208i \(0.754869\pi\)
\(48\) 0 0
\(49\) −341.880 −0.996734
\(50\) 64.2519 − 29.5245i 0.181732 − 0.0835080i
\(51\) 0 0
\(52\) 9.60941 + 8.24828i 0.0256267 + 0.0219967i
\(53\) − 5.63429i − 0.0146024i −0.999973 − 0.00730122i \(-0.997676\pi\)
0.999973 − 0.00730122i \(-0.00232407\pi\)
\(54\) 0 0
\(55\) −60.1520 −0.147471
\(56\) −6.58643 + 23.0274i −0.0157169 + 0.0549493i
\(57\) 0 0
\(58\) 38.7100 + 84.2415i 0.0876357 + 0.190715i
\(59\) − 46.7356i − 0.103126i −0.998670 − 0.0515632i \(-0.983580\pi\)
0.998670 − 0.0515632i \(-0.0164203\pi\)
\(60\) 0 0
\(61\) − 340.563i − 0.714830i −0.933946 − 0.357415i \(-0.883658\pi\)
0.933946 − 0.357415i \(-0.116342\pi\)
\(62\) −642.635 + 295.299i −1.31637 + 0.604887i
\(63\) 0 0
\(64\) −434.561 − 270.741i −0.848752 − 0.528791i
\(65\) −7.91495 −0.0151035
\(66\) 0 0
\(67\) 790.372i 1.44118i 0.693360 + 0.720592i \(0.256130\pi\)
−0.693360 + 0.720592i \(0.743870\pi\)
\(68\) 29.3546 − 34.1987i 0.0523495 − 0.0609882i
\(69\) 0 0
\(70\) −6.25026 − 13.6019i −0.0106721 − 0.0232249i
\(71\) −971.879 −1.62452 −0.812259 − 0.583296i \(-0.801763\pi\)
−0.812259 + 0.583296i \(0.801763\pi\)
\(72\) 0 0
\(73\) −1026.48 −1.64575 −0.822876 − 0.568220i \(-0.807632\pi\)
−0.822876 + 0.568220i \(0.807632\pi\)
\(74\) −277.749 − 604.442i −0.436319 − 0.949527i
\(75\) 0 0
\(76\) 159.552 + 136.952i 0.240815 + 0.206704i
\(77\) 12.7340i 0.0188464i
\(78\) 0 0
\(79\) −693.490 −0.987642 −0.493821 − 0.869563i \(-0.664400\pi\)
−0.493821 + 0.869563i \(0.664400\pi\)
\(80\) 316.303 − 48.4998i 0.442047 − 0.0677806i
\(81\) 0 0
\(82\) 354.741 − 163.008i 0.477739 − 0.219527i
\(83\) − 117.875i − 0.155885i −0.996958 − 0.0779423i \(-0.975165\pi\)
0.996958 − 0.0779423i \(-0.0248350\pi\)
\(84\) 0 0
\(85\) 28.1683i 0.0359445i
\(86\) −420.294 − 914.653i −0.526994 − 1.14685i
\(87\) 0 0
\(88\) −261.721 − 74.8592i −0.317041 − 0.0906820i
\(89\) −106.132 −0.126404 −0.0632021 − 0.998001i \(-0.520131\pi\)
−0.0632021 + 0.998001i \(0.520131\pi\)
\(90\) 0 0
\(91\) 1.67557i 0.00193020i
\(92\) −542.923 + 632.517i −0.615257 + 0.716787i
\(93\) 0 0
\(94\) 1188.91 − 546.320i 1.30454 − 0.599453i
\(95\) −131.418 −0.141928
\(96\) 0 0
\(97\) −1266.82 −1.32604 −0.663021 − 0.748601i \(-0.730726\pi\)
−0.663021 + 0.748601i \(0.730726\pi\)
\(98\) 878.656 − 403.753i 0.905691 − 0.416176i
\(99\) 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 360.4.k.b.181.2 8
3.2 odd 2 120.4.k.b.61.7 ✓ 8
4.3 odd 2 1440.4.k.b.721.5 8
8.3 odd 2 1440.4.k.b.721.1 8
8.5 even 2 inner 360.4.k.b.181.1 8
12.11 even 2 480.4.k.b.241.5 8
24.5 odd 2 120.4.k.b.61.8 yes 8
24.11 even 2 480.4.k.b.241.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.k.b.61.7 ✓ 8 3.2 odd 2
120.4.k.b.61.8 yes 8 24.5 odd 2
360.4.k.b.181.1 8 8.5 even 2 inner
360.4.k.b.181.2 8 1.1 even 1 trivial
480.4.k.b.241.1 8 24.11 even 2
480.4.k.b.241.5 8 12.11 even 2
1440.4.k.b.721.1 8 8.3 odd 2
1440.4.k.b.721.5 8 4.3 odd 2