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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 = [4,0,0,0,12] 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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{29})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 217.1
Root \(3.19258i\) of defining polynomial
Character \(\chi\) \(=\) 360.217
Dual form 360.5.v.b.73.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-18.5407 + 16.7703i) q^{5} +(64.8516 + 64.8516i) q^{7} -33.7033 q^{11} +(132.852 - 132.852i) q^{13} +(270.407 + 270.407i) q^{17} +295.703i q^{19} +(-1.44506 + 1.44506i) q^{23} +(62.5121 - 621.866i) q^{25} -297.484i q^{29} -1228.52 q^{31} +(-2289.98 - 114.809i) q^{35} +(402.478 + 402.478i) q^{37} -860.659 q^{41} +(753.736 - 753.736i) q^{43} +(2228.48 + 2228.48i) q^{47} +6010.47i q^{49} +(-2538.77 + 2538.77i) q^{53} +(624.881 - 565.215i) q^{55} +3421.02i q^{59} -6691.91 q^{61} +(-235.191 + 4691.12i) q^{65} +(-318.264 - 318.264i) q^{67} +3427.02 q^{71} +(3052.86 - 3052.86i) q^{73} +(-2185.71 - 2185.71i) q^{77} +2639.38i q^{79} +(-225.132 + 225.132i) q^{83} +(-9548.32 - 478.709i) q^{85} +4989.60i q^{89} +17231.3 q^{91} +(-4959.04 - 5482.53i) q^{95} +(4891.66 + 4891.66i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{5} + 44 q^{7} + 296 q^{11} + 316 q^{13} + 220 q^{17} - 652 q^{23} + 1284 q^{25} - 2760 q^{31} - 7092 q^{35} - 2052 q^{37} + 4312 q^{41} - 1724 q^{43} + 5252 q^{47} - 6924 q^{53} + 10168 q^{55}+ \cdots + 11812 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −18.5407 + 16.7703i −0.741626 + 0.670813i
\(6\) 0 0
\(7\) 64.8516 + 64.8516i 1.32350 + 1.32350i 0.910920 + 0.412583i \(0.135373\pi\)
0.412583 + 0.910920i \(0.364627\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −33.7033 −0.278540 −0.139270 − 0.990254i \(-0.544476\pi\)
−0.139270 + 0.990254i \(0.544476\pi\)
\(12\) 0 0
\(13\) 132.852 − 132.852i 0.786104 − 0.786104i −0.194749 − 0.980853i \(-0.562389\pi\)
0.980853 + 0.194749i \(0.0623891\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 270.407 + 270.407i 0.935663 + 0.935663i 0.998052 − 0.0623890i \(-0.0198719\pi\)
−0.0623890 + 0.998052i \(0.519872\pi\)
\(18\) 0 0
\(19\) 295.703i 0.819123i 0.912283 + 0.409561i \(0.134318\pi\)
−0.912283 + 0.409561i \(0.865682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.44506 + 1.44506i −0.00273167 + 0.00273167i −0.708471 − 0.705740i \(-0.750615\pi\)
0.705740 + 0.708471i \(0.250615\pi\)
\(24\) 0 0
\(25\) 62.5121 − 621.866i 0.100019 − 0.994985i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 297.484i − 0.353726i −0.984235 − 0.176863i \(-0.943405\pi\)
0.984235 − 0.176863i \(-0.0565949\pi\)
\(30\) 0 0
\(31\) −1228.52 −1.27837 −0.639187 − 0.769052i \(-0.720729\pi\)
−0.639187 + 0.769052i \(0.720729\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2289.98 − 114.809i −1.86937 − 0.0937215i
\(36\) 0 0
\(37\) 402.478 + 402.478i 0.293994 + 0.293994i 0.838656 − 0.544662i \(-0.183342\pi\)
−0.544662 + 0.838656i \(0.683342\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −860.659 −0.511992 −0.255996 − 0.966678i \(-0.582403\pi\)
−0.255996 + 0.966678i \(0.582403\pi\)
\(42\) 0 0
\(43\) 753.736 − 753.736i 0.407645 − 0.407645i −0.473271 − 0.880917i \(-0.656927\pi\)
0.880917 + 0.473271i \(0.156927\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2228.48 + 2228.48i 1.00882 + 1.00882i 0.999961 + 0.00885679i \(0.00281924\pi\)
0.00885679 + 0.999961i \(0.497181\pi\)
\(48\) 0 0
\(49\) 6010.47i 2.50332i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2538.77 + 2538.77i −0.903800 + 0.903800i −0.995762 − 0.0919623i \(-0.970686\pi\)
0.0919623 + 0.995762i \(0.470686\pi\)
\(54\) 0 0
\(55\) 624.881 − 565.215i 0.206572 − 0.186848i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3421.02i 0.982770i 0.870943 + 0.491385i \(0.163509\pi\)
−0.870943 + 0.491385i \(0.836491\pi\)
\(60\) 0 0
\(61\) −6691.91 −1.79842 −0.899209 − 0.437520i \(-0.855857\pi\)
−0.899209 + 0.437520i \(0.855857\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −235.191 + 4691.12i −0.0556666 + 1.11032i
\(66\) 0 0
\(67\) −318.264 − 318.264i −0.0708986 − 0.0708986i 0.670768 − 0.741667i \(-0.265964\pi\)
−0.741667 + 0.670768i \(0.765964\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3427.02 0.679830 0.339915 − 0.940456i \(-0.389602\pi\)
0.339915 + 0.940456i \(0.389602\pi\)
\(72\) 0 0
\(73\) 3052.86 − 3052.86i 0.572876 − 0.572876i −0.360055 − 0.932931i \(-0.617242\pi\)
0.932931 + 0.360055i \(0.117242\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2185.71 − 2185.71i −0.368648 − 0.368648i
\(78\) 0 0
\(79\) 2639.38i 0.422911i 0.977388 + 0.211455i \(0.0678203\pi\)
−0.977388 + 0.211455i \(0.932180\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −225.132 + 225.132i −0.0326799 + 0.0326799i −0.723258 − 0.690578i \(-0.757356\pi\)
0.690578 + 0.723258i \(0.257356\pi\)
\(84\) 0 0
\(85\) −9548.32 − 478.709i −1.32157 − 0.0662573i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4989.60i 0.629921i 0.949105 + 0.314961i \(0.101991\pi\)
−0.949105 + 0.314961i \(0.898009\pi\)
\(90\) 0 0
\(91\) 17231.3 2.08082
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4959.04 − 5482.53i −0.549478 − 0.607483i
\(96\) 0 0
\(97\) 4891.66 + 4891.66i 0.519892 + 0.519892i 0.917538 − 0.397647i \(-0.130173\pi\)
−0.397647 + 0.917538i \(0.630173\pi\)
\(98\) 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.b.217.1 4
3.2 odd 2 40.5.l.b.17.2 ✓ 4
5.3 odd 4 inner 360.5.v.b.73.1 4
12.11 even 2 80.5.p.f.17.2 4
15.2 even 4 200.5.l.c.193.1 4
15.8 even 4 40.5.l.b.33.2 yes 4
15.14 odd 2 200.5.l.c.57.1 4
24.5 odd 2 320.5.p.n.257.1 4
24.11 even 2 320.5.p.k.257.1 4
60.23 odd 4 80.5.p.f.33.2 4
60.47 odd 4 400.5.p.g.193.2 4
60.59 even 2 400.5.p.g.257.2 4
120.53 even 4 320.5.p.n.193.1 4
120.83 odd 4 320.5.p.k.193.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.b.17.2 ✓ 4 3.2 odd 2
40.5.l.b.33.2 yes 4 15.8 even 4
80.5.p.f.17.2 4 12.11 even 2
80.5.p.f.33.2 4 60.23 odd 4
200.5.l.c.57.1 4 15.14 odd 2
200.5.l.c.193.1 4 15.2 even 4
320.5.p.k.193.1 4 120.83 odd 4
320.5.p.k.257.1 4 24.11 even 2
320.5.p.n.193.1 4 120.53 even 4
320.5.p.n.257.1 4 24.5 odd 2
360.5.v.b.73.1 4 5.3 odd 4 inner
360.5.v.b.217.1 4 1.1 even 1 trivial
400.5.p.g.193.2 4 60.47 odd 4
400.5.p.g.257.2 4 60.59 even 2