↑

Properties

Label 450.2.h.e.271.1
Level $450$
Weight $2$
Character 450.271
Analytic conductor $3.593$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,2,Mod(91,450)] 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("450.91"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,0,-2,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.58140625.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 271.1
Root \(1.17421 - 0.0566033i\) of defining polynomial
Character \(\chi\) \(=\) 450.271
Dual form 450.2.h.e.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)
 
\(f(q)\) \(=\) \(q+(-0.309017 + 0.951057i) q^{2} +(-0.809017 - 0.587785i) q^{4} +(-2.15743 - 0.587785i) q^{5} +1.83337 q^{7} +(0.809017 - 0.587785i) q^{8} +(1.22570 - 1.87020i) q^{10} +(0.566541 - 1.74363i) q^{11} +(-0.747156 - 2.29951i) q^{13} +(-0.566541 + 1.74363i) q^{14} +(0.309017 + 0.951057i) q^{16} +(2.25284 - 1.63679i) q^{17} +(1.35294 - 0.982966i) q^{19} +(1.39991 + 1.74363i) q^{20} +(1.48322 + 1.07763i) q^{22} +(2.39991 - 7.38615i) q^{23} +(4.30902 + 2.53621i) q^{25} +2.41785 q^{26} +(-1.48322 - 1.07763i) q^{28} +(6.13597 + 4.45805i) q^{29} +(4.28304 - 3.11181i) q^{31} -1.00000 q^{32} +(0.860510 + 2.64838i) q^{34} +(-3.95536 - 1.07763i) q^{35} +(-0.406315 - 1.25051i) q^{37} +(0.516776 + 1.59047i) q^{38} +(-2.09089 + 0.792578i) q^{40} +(-1.08621 - 3.34301i) q^{41} -4.30550 q^{43} +(-1.48322 + 1.07763i) q^{44} +(6.28304 + 4.56489i) q^{46} +(1.48322 + 1.07763i) q^{47} -3.63877 q^{49} +(-3.74364 + 3.31439i) q^{50} +(-0.747156 + 2.29951i) q^{52} +(-5.27267 - 3.83082i) q^{53} +(-2.24716 + 3.42877i) q^{55} +(1.48322 - 1.07763i) q^{56} +(-6.13597 + 4.45805i) q^{58} +(2.79981 + 8.61694i) q^{59} +(0.799717 - 2.46127i) q^{61} +(1.63597 + 5.03501i) q^{62} +(0.309017 - 0.951057i) q^{64} +(0.260320 + 5.40020i) q^{65} +(-7.68574 + 5.58402i) q^{67} -2.78467 q^{68} +(2.24716 - 3.42877i) q^{70} +(0.247156 + 0.179569i) q^{71} +(4.61920 - 14.2164i) q^{73} +1.31486 q^{74} -1.67232 q^{76} +(1.03868 - 3.19672i) q^{77} +(2.79981 + 2.03418i) q^{79} +(-0.107666 - 2.23347i) q^{80} +3.51505 q^{82} +(-5.15555 + 3.74572i) q^{83} +(-5.82243 + 2.20707i) q^{85} +(1.33047 - 4.09478i) q^{86} +(-0.566541 - 1.74363i) q^{88} +(1.02608 - 3.15794i) q^{89} +(-1.36981 - 4.21584i) q^{91} +(-6.28304 + 4.56489i) q^{92} +(-1.48322 + 1.07763i) q^{94} +(-3.49664 + 1.32545i) q^{95} +(-8.97214 - 6.51864i) q^{97} +(1.12444 - 3.46068i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 2 q^{4} + 4 q^{7} + 2 q^{8} - q^{11} - 13 q^{13} + q^{14} - 2 q^{16} + 11 q^{17} + 20 q^{19} - 5 q^{20} + q^{22} + 3 q^{23} + 30 q^{25} - 22 q^{26} - q^{28} + 15 q^{29} - 9 q^{31} - 8 q^{32}+ \cdots + 19 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.309017 + 0.951057i −0.218508 + 0.672499i
\(3\) 0 0
\(4\) −0.809017 − 0.587785i −0.404508 − 0.293893i
\(5\) −2.15743 − 0.587785i −0.964832 − 0.262866i
\(6\) 0 0
\(7\) 1.83337 0.692947 0.346474 − 0.938060i \(-0.387379\pi\)
0.346474 + 0.938060i \(0.387379\pi\)
\(8\) 0.809017 − 0.587785i 0.286031 − 0.207813i
\(9\) 0 0
\(10\) 1.22570 − 1.87020i 0.387600 − 0.591410i
\(11\) 0.566541 − 1.74363i 0.170819 − 0.525726i −0.828599 − 0.559842i \(-0.810862\pi\)
0.999418 + 0.0341166i \(0.0108618\pi\)
\(12\) 0 0
\(13\) −0.747156 − 2.29951i −0.207224 − 0.637769i −0.999615 − 0.0277557i \(-0.991164\pi\)
0.792391 − 0.610014i \(-0.208836\pi\)
\(14\) −0.566541 + 1.74363i −0.151414 + 0.466006i
\(15\) 0 0
\(16\) 0.309017 + 0.951057i 0.0772542 + 0.237764i
\(17\) 2.25284 − 1.63679i 0.546395 − 0.396979i −0.280060 − 0.959983i \(-0.590354\pi\)
0.826455 + 0.563003i \(0.190354\pi\)
\(18\) 0 0
\(19\) 1.35294 − 0.982966i 0.310385 − 0.225508i −0.421677 − 0.906746i \(-0.638558\pi\)
0.732062 + 0.681238i \(0.238558\pi\)
\(20\) 1.39991 + 1.74363i 0.313029 + 0.389889i
\(21\) 0 0
\(22\) 1.48322 + 1.07763i 0.316224 + 0.229750i
\(23\) 2.39991 − 7.38615i 0.500415 − 1.54012i −0.307929 − 0.951409i \(-0.599636\pi\)
0.808344 − 0.588710i \(-0.200364\pi\)
\(24\) 0 0
\(25\) 4.30902 + 2.53621i 0.861803 + 0.507242i
\(26\) 2.41785 0.474179
\(27\) 0 0
\(28\) −1.48322 − 1.07763i −0.280303 − 0.203652i
\(29\) 6.13597 + 4.45805i 1.13942 + 0.827838i 0.987039 − 0.160483i \(-0.0513052\pi\)
0.152383 + 0.988321i \(0.451305\pi\)
\(30\) 0 0
\(31\) 4.28304 − 3.11181i 0.769256 − 0.558897i −0.132479 − 0.991186i \(-0.542294\pi\)
0.901735 + 0.432288i \(0.142294\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.860510 + 2.64838i 0.147576 + 0.454193i
\(35\) −3.95536 − 1.07763i −0.668578 − 0.182152i
\(36\) 0 0
\(37\) −0.406315 − 1.25051i −0.0667977 − 0.205582i 0.912086 − 0.409998i \(-0.134471\pi\)
−0.978884 + 0.204416i \(0.934471\pi\)
\(38\) 0.516776 + 1.59047i 0.0838321 + 0.258009i
\(39\) 0 0
\(40\) −2.09089 + 0.792578i −0.330599 + 0.125318i
\(41\) −1.08621 − 3.34301i −0.169637 − 0.522090i 0.829711 − 0.558194i \(-0.188505\pi\)
−0.999348 + 0.0361034i \(0.988505\pi\)
\(42\) 0 0
\(43\) −4.30550 −0.656583 −0.328291 − 0.944576i \(-0.606473\pi\)
−0.328291 + 0.944576i \(0.606473\pi\)
\(44\) −1.48322 + 1.07763i −0.223604 + 0.162458i
\(45\) 0 0
\(46\) 6.28304 + 4.56489i 0.926383 + 0.673057i
\(47\) 1.48322 + 1.07763i 0.216350 + 0.157188i 0.690682 − 0.723158i \(-0.257310\pi\)
−0.474332 + 0.880346i \(0.657310\pi\)
\(48\) 0 0
\(49\) −3.63877 −0.519824
\(50\) −3.74364 + 3.31439i −0.529431 + 0.468725i
\(51\) 0 0
\(52\) −0.747156 + 2.29951i −0.103612 + 0.318885i
\(53\) −5.27267 − 3.83082i −0.724257 − 0.526203i 0.163485 − 0.986546i \(-0.447727\pi\)
−0.887741 + 0.460342i \(0.847727\pi\)
\(54\) 0 0
\(55\) −2.24716 + 3.42877i −0.303006 + 0.462335i
\(56\) 1.48322 − 1.07763i 0.198204 − 0.144004i
\(57\) 0 0
\(58\) −6.13597 + 4.45805i −0.805693 + 0.585370i
\(59\) 2.79981 + 8.61694i 0.364505 + 1.12183i 0.950291 + 0.311364i \(0.100786\pi\)
−0.585786 + 0.810466i \(0.699214\pi\)
\(60\) 0 0
\(61\) 0.799717 − 2.46127i 0.102393 − 0.315134i −0.886717 − 0.462313i \(-0.847019\pi\)
0.989110 + 0.147180i \(0.0470195\pi\)
\(62\) 1.63597 + 5.03501i 0.207769 + 0.639447i
\(63\) 0 0
\(64\) 0.309017 − 0.951057i 0.0386271 − 0.118882i
\(65\) 0.260320 + 5.40020i 0.0322887 + 0.669812i
\(66\) 0 0
\(67\) −7.68574 + 5.58402i −0.938963 + 0.682196i −0.948171 − 0.317761i \(-0.897069\pi\)
0.00920814 + 0.999958i \(0.497069\pi\)
\(68\) −2.78467 −0.337691
\(69\) 0 0
\(70\) 2.24716 − 3.42877i 0.268587 − 0.409816i
\(71\) 0.247156 + 0.179569i 0.0293320 + 0.0213110i 0.602355 − 0.798229i \(-0.294229\pi\)
−0.573023 + 0.819540i \(0.694229\pi\)
\(72\) 0 0
\(73\) 4.61920 − 14.2164i 0.540636 − 1.66391i −0.190509 − 0.981685i \(-0.561014\pi\)
0.731145 − 0.682222i \(-0.238986\pi\)
\(74\) 1.31486 0.152850
\(75\) 0 0
\(76\) −1.67232 −0.191829
\(77\) 1.03868 − 3.19672i 0.118368 − 0.364300i
\(78\) 0 0
\(79\) 2.79981 + 2.03418i 0.315004 + 0.228864i 0.734041 − 0.679106i \(-0.237632\pi\)
−0.419037 + 0.907969i \(0.637632\pi\)
\(80\) −0.107666 − 2.23347i −0.0120374 − 0.249710i
\(81\) 0 0
\(82\) 3.51505 0.388172
\(83\) −5.15555 + 3.74572i −0.565895 + 0.411147i −0.833612 − 0.552351i \(-0.813731\pi\)
0.267717 + 0.963498i \(0.413731\pi\)
\(84\) 0 0
\(85\) −5.82243 + 2.20707i −0.631532 + 0.239390i
\(86\) 1.33047 − 4.09478i 0.143469 − 0.441551i
\(87\) 0 0
\(88\) −0.566541 − 1.74363i −0.0603935 − 0.185872i
\(89\) 1.02608 − 3.15794i 0.108764 − 0.334741i −0.881832 − 0.471565i \(-0.843689\pi\)
0.990595 + 0.136824i \(0.0436894\pi\)
\(90\) 0 0
\(91\) −1.36981 − 4.21584i −0.143595 − 0.441940i
\(92\) −6.28304 + 4.56489i −0.655052 + 0.475923i
\(93\) 0 0
\(94\) −1.48322 + 1.07763i −0.152983 + 0.111149i
\(95\) −3.49664 + 1.32545i −0.358748 + 0.135988i
\(96\) 0 0
\(97\) −8.97214 − 6.51864i −0.910982 − 0.661867i 0.0302807 − 0.999541i \(-0.490360\pi\)
−0.941263 + 0.337674i \(0.890360\pi\)
\(98\) 1.12444 − 3.46068i 0.113586 − 0.349581i
\(99\) 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 450.2.h.e.271.1 8
3.2 odd 2 50.2.d.b.21.1 ✓ 8
12.11 even 2 400.2.u.d.321.2 8
15.2 even 4 250.2.e.c.149.3 16
15.8 even 4 250.2.e.c.149.2 16
15.14 odd 2 250.2.d.d.101.2 8
25.6 even 5 inner 450.2.h.e.181.1 8
75.8 even 20 250.2.e.c.99.3 16
75.17 even 20 250.2.e.c.99.2 16
75.38 even 20 1250.2.b.e.1249.4 8
75.41 odd 10 1250.2.a.l.1.4 4
75.44 odd 10 250.2.d.d.151.2 8
75.56 odd 10 50.2.d.b.31.1 yes 8
75.59 odd 10 1250.2.a.f.1.1 4
75.62 even 20 1250.2.b.e.1249.5 8
300.59 even 10 10000.2.a.x.1.4 4
300.131 even 10 400.2.u.d.81.2 8
300.191 even 10 10000.2.a.t.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.2.d.b.21.1 ✓ 8 3.2 odd 2
50.2.d.b.31.1 yes 8 75.56 odd 10
250.2.d.d.101.2 8 15.14 odd 2
250.2.d.d.151.2 8 75.44 odd 10
250.2.e.c.99.2 16 75.17 even 20
250.2.e.c.99.3 16 75.8 even 20
250.2.e.c.149.2 16 15.8 even 4
250.2.e.c.149.3 16 15.2 even 4
400.2.u.d.81.2 8 300.131 even 10
400.2.u.d.321.2 8 12.11 even 2
450.2.h.e.181.1 8 25.6 even 5 inner
450.2.h.e.271.1 8 1.1 even 1 trivial
1250.2.a.f.1.1 4 75.59 odd 10
1250.2.a.l.1.4 4 75.41 odd 10
1250.2.b.e.1249.4 8 75.38 even 20
1250.2.b.e.1249.5 8 75.62 even 20
10000.2.a.t.1.1 4 300.191 even 10
10000.2.a.x.1.4 4 300.59 even 10