Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [360,5,Mod(73,360)] 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("360.73"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 360.v (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,4] 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: no
Analytic conductor: \(37.2131867102\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.313431616.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 27x^{4} + 145x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 73.2
Root \(-1.13432i\) of defining polynomial
Character \(\chi\) \(=\) 360.73
Dual form 360.5.v.c.217.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.04169 + 24.9783i) q^{5} +(26.0617 - 26.0617i) q^{7} -75.9566 q^{11} +(-99.1428 - 99.1428i) q^{13} +(-77.9171 + 77.9171i) q^{17} +640.445i q^{19} +(-117.134 - 117.134i) q^{23} +(-622.830 + 52.0390i) q^{25} -752.534i q^{29} +708.205 q^{31} +(678.124 + 623.828i) q^{35} +(1170.51 - 1170.51i) q^{37} -1822.21 q^{41} +(-2372.12 - 2372.12i) q^{43} +(-2458.79 + 2458.79i) q^{47} +1042.58i q^{49} +(-2118.26 - 2118.26i) q^{53} +(-79.1229 - 1897.27i) q^{55} -5251.96i q^{59} -1312.89 q^{61} +(2373.14 - 2579.69i) q^{65} +(-5569.35 + 5569.35i) q^{67} -7554.66 q^{71} +(1206.35 + 1206.35i) q^{73} +(-1979.55 + 1979.55i) q^{77} +3397.97i q^{79} +(-928.421 - 928.421i) q^{83} +(-2027.40 - 1865.07i) q^{85} +11981.9i q^{89} -5167.65 q^{91} +(-15997.2 + 667.143i) q^{95} +(4683.76 - 4683.76i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{5} - 40 q^{7} - 72 q^{11} - 62 q^{13} - 418 q^{17} + 1760 q^{23} - 2974 q^{25} + 2864 q^{31} + 4224 q^{35} + 5366 q^{37} - 1656 q^{41} - 2760 q^{43} + 432 q^{47} - 534 q^{53} - 136 q^{55} + 216 q^{61}+ \cdots - 9418 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.04169 + 24.9783i 0.0416674 + 0.999132i
\(6\) 0 0
\(7\) 26.0617 − 26.0617i 0.531871 − 0.531871i −0.389258 − 0.921129i \(-0.627269\pi\)
0.921129 + 0.389258i \(0.127269\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −75.9566 −0.627740 −0.313870 − 0.949466i \(-0.601626\pi\)
−0.313870 + 0.949466i \(0.601626\pi\)
\(12\) 0 0
\(13\) −99.1428 − 99.1428i −0.586644 − 0.586644i 0.350077 − 0.936721i \(-0.386155\pi\)
−0.936721 + 0.350077i \(0.886155\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −77.9171 + 77.9171i −0.269609 + 0.269609i −0.828943 − 0.559333i \(-0.811057\pi\)
0.559333 + 0.828943i \(0.311057\pi\)
\(18\) 0 0
\(19\) 640.445i 1.77409i 0.461686 + 0.887044i \(0.347245\pi\)
−0.461686 + 0.887044i \(0.652755\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −117.134 − 117.134i −0.221426 − 0.221426i 0.587673 − 0.809099i \(-0.300044\pi\)
−0.809099 + 0.587673i \(0.800044\pi\)
\(24\) 0 0
\(25\) −622.830 + 52.0390i −0.996528 + 0.0832625i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 752.534i − 0.894809i −0.894332 − 0.447404i \(-0.852348\pi\)
0.894332 − 0.447404i \(-0.147652\pi\)
\(30\) 0 0
\(31\) 708.205 0.736946 0.368473 − 0.929638i \(-0.379881\pi\)
0.368473 + 0.929638i \(0.379881\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 678.124 + 623.828i 0.553570 + 0.509247i
\(36\) 0 0
\(37\) 1170.51 − 1170.51i 0.855013 − 0.855013i −0.135733 − 0.990745i \(-0.543339\pi\)
0.990745 + 0.135733i \(0.0433389\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1822.21 −1.08400 −0.542001 − 0.840378i \(-0.682333\pi\)
−0.542001 + 0.840378i \(0.682333\pi\)
\(42\) 0 0
\(43\) −2372.12 − 2372.12i −1.28292 − 1.28292i −0.938998 − 0.343921i \(-0.888245\pi\)
−0.343921 − 0.938998i \(-0.611755\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2458.79 + 2458.79i −1.11308 + 1.11308i −0.120344 + 0.992732i \(0.538400\pi\)
−0.992732 + 0.120344i \(0.961600\pi\)
\(48\) 0 0
\(49\) 1042.58i 0.434227i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2118.26 − 2118.26i −0.754097 − 0.754097i 0.221144 − 0.975241i \(-0.429021\pi\)
−0.975241 + 0.221144i \(0.929021\pi\)
\(54\) 0 0
\(55\) −79.1229 − 1897.27i −0.0261563 − 0.627195i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5251.96i − 1.50875i −0.656443 − 0.754376i \(-0.727940\pi\)
0.656443 − 0.754376i \(-0.272060\pi\)
\(60\) 0 0
\(61\) −1312.89 −0.352831 −0.176416 − 0.984316i \(-0.556450\pi\)
−0.176416 + 0.984316i \(0.556450\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2373.14 − 2579.69i 0.561691 − 0.610578i
\(66\) 0 0
\(67\) −5569.35 + 5569.35i −1.24067 + 1.24067i −0.280940 + 0.959725i \(0.590646\pi\)
−0.959725 + 0.280940i \(0.909354\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7554.66 −1.49864 −0.749322 − 0.662206i \(-0.769620\pi\)
−0.749322 + 0.662206i \(0.769620\pi\)
\(72\) 0 0
\(73\) 1206.35 + 1206.35i 0.226374 + 0.226374i 0.811176 − 0.584802i \(-0.198828\pi\)
−0.584802 + 0.811176i \(0.698828\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1979.55 + 1979.55i −0.333877 + 0.333877i
\(78\) 0 0
\(79\) 3397.97i 0.544460i 0.962232 + 0.272230i \(0.0877612\pi\)
−0.962232 + 0.272230i \(0.912239\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −928.421 − 928.421i −0.134769 − 0.134769i 0.636504 − 0.771273i \(-0.280380\pi\)
−0.771273 + 0.636504i \(0.780380\pi\)
\(84\) 0 0
\(85\) −2027.40 − 1865.07i −0.280609 − 0.258141i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11981.9i 1.51268i 0.654180 + 0.756339i \(0.273014\pi\)
−0.654180 + 0.756339i \(0.726986\pi\)
\(90\) 0 0
\(91\) −5167.65 −0.624037
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15997.2 + 667.143i −1.77255 + 0.0739216i
\(96\) 0 0
\(97\) 4683.76 − 4683.76i 0.497796 − 0.497796i −0.412955 − 0.910751i \(-0.635503\pi\)
0.910751 + 0.412955i \(0.135503\pi\)
\(98\) 0 0
\(99\) 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 360.5.v.c.73.2 6
3.2 odd 2 40.5.l.c.33.2 yes 6
5.2 odd 4 inner 360.5.v.c.217.2 6
12.11 even 2 80.5.p.g.33.2 6
15.2 even 4 40.5.l.c.17.2 ✓ 6
15.8 even 4 200.5.l.e.57.2 6
15.14 odd 2 200.5.l.e.193.2 6
24.5 odd 2 320.5.p.p.193.2 6
24.11 even 2 320.5.p.o.193.2 6
60.23 odd 4 400.5.p.o.257.2 6
60.47 odd 4 80.5.p.g.17.2 6
60.59 even 2 400.5.p.o.193.2 6
120.77 even 4 320.5.p.p.257.2 6
120.107 odd 4 320.5.p.o.257.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.c.17.2 ✓ 6 15.2 even 4
40.5.l.c.33.2 yes 6 3.2 odd 2
80.5.p.g.17.2 6 60.47 odd 4
80.5.p.g.33.2 6 12.11 even 2
200.5.l.e.57.2 6 15.8 even 4
200.5.l.e.193.2 6 15.14 odd 2
320.5.p.o.193.2 6 24.11 even 2
320.5.p.o.257.2 6 120.107 odd 4
320.5.p.p.193.2 6 24.5 odd 2
320.5.p.p.257.2 6 120.77 even 4
360.5.v.c.73.2 6 1.1 even 1 trivial
360.5.v.c.217.2 6 5.2 odd 4 inner
400.5.p.o.193.2 6 60.59 even 2
400.5.p.o.257.2 6 60.23 odd 4