Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [360,2,Mod(163,360)] 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("360.163"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 163.4
Root \(0.951057 - 0.309017i\) of defining polynomial
Character \(\chi\) \(=\) 360.163
Dual form 360.2.w.c.307.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.39680 - 0.221232i) q^{2} +(1.90211 - 0.618034i) q^{4} +(-1.17557 - 1.90211i) q^{5} +(-1.90211 - 1.90211i) q^{7} +(2.52015 - 1.28408i) q^{8} +(-2.06285 - 2.39680i) q^{10} +3.23607 q^{11} +(0.726543 - 0.726543i) q^{13} +(-3.07768 - 2.23607i) q^{14} +(3.23607 - 2.35114i) q^{16} +(1.00000 - 1.00000i) q^{17} +2.00000i q^{19} +(-3.41164 - 2.89149i) q^{20} +(4.52015 - 0.715921i) q^{22} +(-4.25325 + 4.25325i) q^{23} +(-2.23607 + 4.47214i) q^{25} +(0.854102 - 1.17557i) q^{26} +(-4.79360 - 2.44246i) q^{28} +6.15537 q^{29} +8.50651i q^{31} +(4.00000 - 4.00000i) q^{32} +(1.17557 - 1.61803i) q^{34} +(-1.38197 + 5.85410i) q^{35} +(-0.726543 - 0.726543i) q^{37} +(0.442463 + 2.79360i) q^{38} +(-5.40507 - 3.28408i) q^{40} -5.70820 q^{41} +(4.61803 + 4.61803i) q^{43} +(6.15537 - 2.00000i) q^{44} +(-5.00000 + 6.88191i) q^{46} +(-3.35520 - 3.35520i) q^{47} +0.236068i q^{49} +(-2.13397 + 6.74138i) q^{50} +(0.932938 - 1.83099i) q^{52} +(3.07768 - 3.07768i) q^{53} +(-3.80423 - 6.15537i) q^{55} +(-7.23607 - 2.35114i) q^{56} +(8.59783 - 1.36176i) q^{58} -0.472136i q^{59} -0.898056i q^{61} +(1.88191 + 11.8819i) q^{62} +(4.70228 - 6.47214i) q^{64} +(-2.23607 - 0.527864i) q^{65} +(-4.61803 + 4.61803i) q^{67} +(1.28408 - 2.52015i) q^{68} +(-0.635220 + 8.48276i) q^{70} +11.4127i q^{71} +(-4.70820 - 4.70820i) q^{73} +(-1.17557 - 0.854102i) q^{74} +(1.23607 + 3.80423i) q^{76} +(-6.15537 - 6.15537i) q^{77} +2.90617 q^{79} +(-8.27636 - 3.39144i) q^{80} +(-7.97323 + 1.26284i) q^{82} +(6.61803 + 6.61803i) q^{83} +(-3.07768 - 0.726543i) q^{85} +(7.47214 + 5.42882i) q^{86} +(8.15537 - 4.15537i) q^{88} +2.47214i q^{89} -2.76393 q^{91} +(-5.46151 + 10.7188i) q^{92} +(-5.42882 - 3.94427i) q^{94} +(3.80423 - 2.35114i) q^{95} +(4.23607 - 4.23607i) q^{97} +(0.0522257 + 0.329740i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 4 q^{8} - 10 q^{10} + 8 q^{11} + 8 q^{16} + 8 q^{17} + 12 q^{22} - 20 q^{26} - 20 q^{28} + 32 q^{32} - 20 q^{35} + 4 q^{38} - 20 q^{40} + 8 q^{41} + 28 q^{43} - 40 q^{46} + 10 q^{50} + 20 q^{52}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.39680 − 0.221232i 0.987688 − 0.156434i
\(3\) 0 0
\(4\) 1.90211 − 0.618034i 0.951057 − 0.309017i
\(5\) −1.17557 − 1.90211i −0.525731 − 0.850651i
\(6\) 0 0
\(7\) −1.90211 − 1.90211i −0.718931 − 0.718931i 0.249455 − 0.968386i \(-0.419748\pi\)
−0.968386 + 0.249455i \(0.919748\pi\)
\(8\) 2.52015 − 1.28408i 0.891007 − 0.453990i
\(9\) 0 0
\(10\) −2.06285 − 2.39680i −0.652330 − 0.757935i
\(11\) 3.23607 0.975711 0.487856 − 0.872924i \(-0.337779\pi\)
0.487856 + 0.872924i \(0.337779\pi\)
\(12\) 0 0
\(13\) 0.726543 − 0.726543i 0.201507 − 0.201507i −0.599139 − 0.800645i \(-0.704490\pi\)
0.800645 + 0.599139i \(0.204490\pi\)
\(14\) −3.07768 − 2.23607i −0.822546 − 0.597614i
\(15\) 0 0
\(16\) 3.23607 − 2.35114i 0.809017 − 0.587785i
\(17\) 1.00000 − 1.00000i 0.242536 − 0.242536i −0.575363 − 0.817898i \(-0.695139\pi\)
0.817898 + 0.575363i \(0.195139\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) −3.41164 − 2.89149i −0.762866 − 0.646557i
\(21\) 0 0
\(22\) 4.52015 − 0.715921i 0.963699 − 0.152635i
\(23\) −4.25325 + 4.25325i −0.886865 + 0.886865i −0.994221 − 0.107356i \(-0.965762\pi\)
0.107356 + 0.994221i \(0.465762\pi\)
\(24\) 0 0
\(25\) −2.23607 + 4.47214i −0.447214 + 0.894427i
\(26\) 0.854102 − 1.17557i 0.167503 − 0.230548i
\(27\) 0 0
\(28\) −4.79360 − 2.44246i −0.905906 − 0.461582i
\(29\) 6.15537 1.14302 0.571511 − 0.820594i \(-0.306357\pi\)
0.571511 + 0.820594i \(0.306357\pi\)
\(30\) 0 0
\(31\) 8.50651i 1.52781i 0.645326 + 0.763907i \(0.276721\pi\)
−0.645326 + 0.763907i \(0.723279\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) 0 0
\(34\) 1.17557 − 1.61803i 0.201609 − 0.277491i
\(35\) −1.38197 + 5.85410i −0.233595 + 0.989524i
\(36\) 0 0
\(37\) −0.726543 − 0.726543i −0.119443 − 0.119443i 0.644859 − 0.764302i \(-0.276916\pi\)
−0.764302 + 0.644859i \(0.776916\pi\)
\(38\) 0.442463 + 2.79360i 0.0717771 + 0.453182i
\(39\) 0 0
\(40\) −5.40507 − 3.28408i −0.854617 − 0.519258i
\(41\) −5.70820 −0.891472 −0.445736 − 0.895165i \(-0.647058\pi\)
−0.445736 + 0.895165i \(0.647058\pi\)
\(42\) 0 0
\(43\) 4.61803 + 4.61803i 0.704244 + 0.704244i 0.965319 − 0.261075i \(-0.0840770\pi\)
−0.261075 + 0.965319i \(0.584077\pi\)
\(44\) 6.15537 − 2.00000i 0.927957 − 0.301511i
\(45\) 0 0
\(46\) −5.00000 + 6.88191i −0.737210 + 1.01468i
\(47\) −3.35520 − 3.35520i −0.489406 − 0.489406i 0.418713 − 0.908119i \(-0.362481\pi\)
−0.908119 + 0.418713i \(0.862481\pi\)
\(48\) 0 0
\(49\) 0.236068i 0.0337240i
\(50\) −2.13397 + 6.74138i −0.301788 + 0.953375i
\(51\) 0 0
\(52\) 0.932938 − 1.83099i 0.129375 − 0.253913i
\(53\) 3.07768 − 3.07768i 0.422752 − 0.422752i −0.463398 − 0.886150i \(-0.653370\pi\)
0.886150 + 0.463398i \(0.153370\pi\)
\(54\) 0 0
\(55\) −3.80423 − 6.15537i −0.512962 − 0.829990i
\(56\) −7.23607 − 2.35114i −0.966960 − 0.314184i
\(57\) 0 0
\(58\) 8.59783 − 1.36176i 1.12895 − 0.178808i
\(59\) − 0.472136i − 0.0614669i −0.999528 − 0.0307334i \(-0.990216\pi\)
0.999528 − 0.0307334i \(-0.00978430\pi\)
\(60\) 0 0
\(61\) − 0.898056i − 0.114984i −0.998346 − 0.0574921i \(-0.981690\pi\)
0.998346 − 0.0574921i \(-0.0183104\pi\)
\(62\) 1.88191 + 11.8819i 0.239003 + 1.50900i
\(63\) 0 0
\(64\) 4.70228 − 6.47214i 0.587785 − 0.809017i
\(65\) −2.23607 − 0.527864i −0.277350 − 0.0654735i
\(66\) 0 0
\(67\) −4.61803 + 4.61803i −0.564183 + 0.564183i −0.930493 − 0.366310i \(-0.880621\pi\)
0.366310 + 0.930493i \(0.380621\pi\)
\(68\) 1.28408 − 2.52015i 0.155717 − 0.305613i
\(69\) 0 0
\(70\) −0.635220 + 8.48276i −0.0759233 + 1.01388i
\(71\) 11.4127i 1.35444i 0.735783 + 0.677218i \(0.236815\pi\)
−0.735783 + 0.677218i \(0.763185\pi\)
\(72\) 0 0
\(73\) −4.70820 − 4.70820i −0.551054 − 0.551054i 0.375691 − 0.926745i \(-0.377405\pi\)
−0.926745 + 0.375691i \(0.877405\pi\)
\(74\) −1.17557 − 0.854102i −0.136657 − 0.0992873i
\(75\) 0 0
\(76\) 1.23607 + 3.80423i 0.141787 + 0.436375i
\(77\) −6.15537 − 6.15537i −0.701469 − 0.701469i
\(78\) 0 0
\(79\) 2.90617 0.326970 0.163485 − 0.986546i \(-0.447727\pi\)
0.163485 + 0.986546i \(0.447727\pi\)
\(80\) −8.27636 − 3.39144i −0.925325 − 0.379174i
\(81\) 0 0
\(82\) −7.97323 + 1.26284i −0.880496 + 0.139457i
\(83\) 6.61803 + 6.61803i 0.726424 + 0.726424i 0.969905 − 0.243482i \(-0.0782896\pi\)
−0.243482 + 0.969905i \(0.578290\pi\)
\(84\) 0 0
\(85\) −3.07768 − 0.726543i −0.333822 − 0.0788046i
\(86\) 7.47214 + 5.42882i 0.805741 + 0.585405i
\(87\) 0 0
\(88\) 8.15537 − 4.15537i 0.869365 − 0.442964i
\(89\) 2.47214i 0.262046i 0.991379 + 0.131023i \(0.0418262\pi\)
−0.991379 + 0.131023i \(0.958174\pi\)
\(90\) 0 0
\(91\) −2.76393 −0.289739
\(92\) −5.46151 + 10.7188i −0.569402 + 1.11751i
\(93\) 0 0
\(94\) −5.42882 − 3.94427i −0.559940 − 0.406821i
\(95\) 3.80423 − 2.35114i 0.390305 − 0.241222i
\(96\) 0 0
\(97\) 4.23607 − 4.23607i 0.430108 − 0.430108i −0.458557 − 0.888665i \(-0.651634\pi\)
0.888665 + 0.458557i \(0.151634\pi\)
\(98\) 0.0522257 + 0.329740i 0.00527560 + 0.0333088i
\(99\) 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 360.2.w.c.163.4 8
3.2 odd 2 40.2.k.a.3.1 ✓ 8
4.3 odd 2 1440.2.bi.c.1423.2 8
5.2 odd 4 inner 360.2.w.c.307.2 8
8.3 odd 2 inner 360.2.w.c.163.2 8
8.5 even 2 1440.2.bi.c.1423.3 8
12.11 even 2 160.2.o.a.143.2 8
15.2 even 4 40.2.k.a.27.3 yes 8
15.8 even 4 200.2.k.h.107.2 8
15.14 odd 2 200.2.k.h.43.4 8
20.7 even 4 1440.2.bi.c.847.3 8
24.5 odd 2 160.2.o.a.143.1 8
24.11 even 2 40.2.k.a.3.3 yes 8
40.27 even 4 inner 360.2.w.c.307.4 8
40.37 odd 4 1440.2.bi.c.847.2 8
48.5 odd 4 1280.2.n.m.1023.3 8
48.11 even 4 1280.2.n.q.1023.1 8
48.29 odd 4 1280.2.n.q.1023.2 8
48.35 even 4 1280.2.n.m.1023.4 8
60.23 odd 4 800.2.o.g.207.4 8
60.47 odd 4 160.2.o.a.47.1 8
60.59 even 2 800.2.o.g.143.3 8
120.29 odd 2 800.2.o.g.143.4 8
120.53 even 4 800.2.o.g.207.3 8
120.59 even 2 200.2.k.h.43.2 8
120.77 even 4 160.2.o.a.47.2 8
120.83 odd 4 200.2.k.h.107.4 8
120.107 odd 4 40.2.k.a.27.1 yes 8
240.77 even 4 1280.2.n.m.767.4 8
240.107 odd 4 1280.2.n.m.767.3 8
240.197 even 4 1280.2.n.q.767.1 8
240.227 odd 4 1280.2.n.q.767.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.k.a.3.1 ✓ 8 3.2 odd 2
40.2.k.a.3.3 yes 8 24.11 even 2
40.2.k.a.27.1 yes 8 120.107 odd 4
40.2.k.a.27.3 yes 8 15.2 even 4
160.2.o.a.47.1 8 60.47 odd 4
160.2.o.a.47.2 8 120.77 even 4
160.2.o.a.143.1 8 24.5 odd 2
160.2.o.a.143.2 8 12.11 even 2
200.2.k.h.43.2 8 120.59 even 2
200.2.k.h.43.4 8 15.14 odd 2
200.2.k.h.107.2 8 15.8 even 4
200.2.k.h.107.4 8 120.83 odd 4
360.2.w.c.163.2 8 8.3 odd 2 inner
360.2.w.c.163.4 8 1.1 even 1 trivial
360.2.w.c.307.2 8 5.2 odd 4 inner
360.2.w.c.307.4 8 40.27 even 4 inner
800.2.o.g.143.3 8 60.59 even 2
800.2.o.g.143.4 8 120.29 odd 2
800.2.o.g.207.3 8 120.53 even 4
800.2.o.g.207.4 8 60.23 odd 4
1280.2.n.m.767.3 8 240.107 odd 4
1280.2.n.m.767.4 8 240.77 even 4
1280.2.n.m.1023.3 8 48.5 odd 4
1280.2.n.m.1023.4 8 48.35 even 4
1280.2.n.q.767.1 8 240.197 even 4
1280.2.n.q.767.2 8 240.227 odd 4
1280.2.n.q.1023.1 8 48.11 even 4
1280.2.n.q.1023.2 8 48.29 odd 4
1440.2.bi.c.847.2 8 40.37 odd 4
1440.2.bi.c.847.3 8 20.7 even 4
1440.2.bi.c.1423.2 8 4.3 odd 2
1440.2.bi.c.1423.3 8 8.5 even 2