Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,5,Mod(449,450)] 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("450.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,64,0,0,0,0,0,0,0,0,0,0,0,512,0,0,-2368] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.5164833877\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(i, \sqrt{2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.6
Root \(-1.14412 - 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 450.449
Dual form 450.5.b.c.449.7

$q$-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)
 
\(f(q)\) \(=\) \(q+2.82843 q^{2} +8.00000 q^{4} -15.4342i q^{7} +22.6274 q^{8} +205.489i q^{11} +23.6975i q^{13} -43.6544i q^{14} +64.0000 q^{16} -336.210 q^{17} -390.868 q^{19} +581.210i q^{22} +174.190 q^{23} +67.0267i q^{26} -123.473i q^{28} -233.587i q^{29} -241.395 q^{31} +181.019 q^{32} -950.947 q^{34} +215.619i q^{37} -1105.54 q^{38} +3035.57i q^{41} +3021.44i q^{43} +1643.91i q^{44} +492.683 q^{46} +2295.68 q^{47} +2162.79 q^{49} +189.580i q^{52} -3854.40 q^{53} -349.235i q^{56} -660.683i q^{58} +195.403i q^{59} -6598.02 q^{61} -682.768 q^{62} +512.000 q^{64} +6174.42i q^{67} -2689.68 q^{68} +2099.85i q^{71} -4819.23i q^{73} +609.863i q^{74} -3126.95 q^{76} +3171.55 q^{77} -10330.9 q^{79} +8585.89i q^{82} +224.935 q^{83} +8545.94i q^{86} +4649.68i q^{88} -1661.59i q^{89} +365.751 q^{91} +1393.52 q^{92} +6493.15 q^{94} +1987.66i q^{97} +6117.28 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{4} + 512 q^{16} - 2368 q^{19} - 4208 q^{31} - 1536 q^{34} - 3648 q^{46} - 6984 q^{49} - 12560 q^{61} + 4096 q^{64} - 18944 q^{76} - 24208 q^{79} - 96496 q^{91} + 12480 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843 0.707107
\(3\) 0 0
\(4\) 8.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 15.4342i − 0.314983i −0.987520 − 0.157491i \(-0.949659\pi\)
0.987520 − 0.157491i \(-0.0503407\pi\)
\(8\) 22.6274 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 205.489i 1.69825i 0.528189 + 0.849127i \(0.322871\pi\)
−0.528189 + 0.849127i \(0.677129\pi\)
\(12\) 0 0
\(13\) 23.6975i 0.140222i 0.997539 + 0.0701110i \(0.0223353\pi\)
−0.997539 + 0.0701110i \(0.977665\pi\)
\(14\) − 43.6544i − 0.222727i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) −336.210 −1.16336 −0.581679 − 0.813418i \(-0.697604\pi\)
−0.581679 + 0.813418i \(0.697604\pi\)
\(18\) 0 0
\(19\) −390.868 −1.08274 −0.541369 − 0.840785i \(-0.682094\pi\)
−0.541369 + 0.840785i \(0.682094\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 581.210i 1.20085i
\(23\) 174.190 0.329281 0.164641 − 0.986354i \(-0.447354\pi\)
0.164641 + 0.986354i \(0.447354\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 67.0267i 0.0991519i
\(27\) 0 0
\(28\) − 123.473i − 0.157491i
\(29\) − 233.587i − 0.277749i −0.990310 − 0.138874i \(-0.955652\pi\)
0.990310 − 0.138874i \(-0.0443484\pi\)
\(30\) 0 0
\(31\) −241.395 −0.251191 −0.125596 − 0.992082i \(-0.540084\pi\)
−0.125596 + 0.992082i \(0.540084\pi\)
\(32\) 181.019 0.176777
\(33\) 0 0
\(34\) −950.947 −0.822618
\(35\) 0 0
\(36\) 0 0
\(37\) 215.619i 0.157501i 0.996894 + 0.0787506i \(0.0250931\pi\)
−0.996894 + 0.0787506i \(0.974907\pi\)
\(38\) −1105.54 −0.765611
\(39\) 0 0
\(40\) 0 0
\(41\) 3035.57i 1.80581i 0.429837 + 0.902907i \(0.358571\pi\)
−0.429837 + 0.902907i \(0.641429\pi\)
\(42\) 0 0
\(43\) 3021.44i 1.63410i 0.576569 + 0.817048i \(0.304391\pi\)
−0.576569 + 0.817048i \(0.695609\pi\)
\(44\) 1643.91i 0.849127i
\(45\) 0 0
\(46\) 492.683 0.232837
\(47\) 2295.68 1.03924 0.519619 − 0.854398i \(-0.326074\pi\)
0.519619 + 0.854398i \(0.326074\pi\)
\(48\) 0 0
\(49\) 2162.79 0.900786
\(50\) 0 0
\(51\) 0 0
\(52\) 189.580i 0.0701110i
\(53\) −3854.40 −1.37216 −0.686081 − 0.727525i \(-0.740670\pi\)
−0.686081 + 0.727525i \(0.740670\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 349.235i − 0.111363i
\(57\) 0 0
\(58\) − 660.683i − 0.196398i
\(59\) 195.403i 0.0561342i 0.999606 + 0.0280671i \(0.00893520\pi\)
−0.999606 + 0.0280671i \(0.991065\pi\)
\(60\) 0 0
\(61\) −6598.02 −1.77319 −0.886593 − 0.462551i \(-0.846934\pi\)
−0.886593 + 0.462551i \(0.846934\pi\)
\(62\) −682.768 −0.177619
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 6174.42i 1.37546i 0.725969 + 0.687728i \(0.241392\pi\)
−0.725969 + 0.687728i \(0.758608\pi\)
\(68\) −2689.68 −0.581679
\(69\) 0 0
\(70\) 0 0
\(71\) 2099.85i 0.416554i 0.978070 + 0.208277i \(0.0667856\pi\)
−0.978070 + 0.208277i \(0.933214\pi\)
\(72\) 0 0
\(73\) − 4819.23i − 0.904341i −0.891932 − 0.452170i \(-0.850650\pi\)
0.891932 − 0.452170i \(-0.149350\pi\)
\(74\) 609.863i 0.111370i
\(75\) 0 0
\(76\) −3126.95 −0.541369
\(77\) 3171.55 0.534921
\(78\) 0 0
\(79\) −10330.9 −1.65532 −0.827661 − 0.561229i \(-0.810329\pi\)
−0.827661 + 0.561229i \(0.810329\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8585.89i 1.27690i
\(83\) 224.935 0.0326514 0.0163257 − 0.999867i \(-0.494803\pi\)
0.0163257 + 0.999867i \(0.494803\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8545.94i 1.15548i
\(87\) 0 0
\(88\) 4649.68i 0.600424i
\(89\) − 1661.59i − 0.209770i −0.994484 − 0.104885i \(-0.966553\pi\)
0.994484 − 0.104885i \(-0.0334475\pi\)
\(90\) 0 0
\(91\) 365.751 0.0441675
\(92\) 1393.52 0.164641
\(93\) 0 0
\(94\) 6493.15 0.734852
\(95\) 0 0
\(96\) 0 0
\(97\) 1987.66i 0.211251i 0.994406 + 0.105625i \(0.0336844\pi\)
−0.994406 + 0.105625i \(0.966316\pi\)
\(98\) 6117.28 0.636952
\(99\) 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 450.5.b.c.449.6 8
3.2 odd 2 inner 450.5.b.c.449.2 8
5.2 odd 4 450.5.d.e.251.4 4
5.3 odd 4 90.5.d.b.71.2 ✓ 4
5.4 even 2 inner 450.5.b.c.449.3 8
15.2 even 4 450.5.d.e.251.2 4
15.8 even 4 90.5.d.b.71.3 yes 4
15.14 odd 2 inner 450.5.b.c.449.7 8
20.3 even 4 720.5.l.a.161.4 4
60.23 odd 4 720.5.l.a.161.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.5.d.b.71.2 ✓ 4 5.3 odd 4
90.5.d.b.71.3 yes 4 15.8 even 4
450.5.b.c.449.2 8 3.2 odd 2 inner
450.5.b.c.449.3 8 5.4 even 2 inner
450.5.b.c.449.6 8 1.1 even 1 trivial
450.5.b.c.449.7 8 15.14 odd 2 inner
450.5.d.e.251.2 4 15.2 even 4
450.5.d.e.251.4 4 5.2 odd 4
720.5.l.a.161.2 4 60.23 odd 4
720.5.l.a.161.4 4 20.3 even 4