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Properties

Label 450.2.e.i.151.1
Level $450$
Weight $2$
Character 450.151
Analytic conductor $3.593$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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(151,450)] 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("450.151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,3,-1,0,3,2] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 151.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 450.151
Dual form 450.2.e.i.301.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(1.50000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(1.50000 - 0.866025i) q^{6} +(1.00000 - 1.73205i) q^{7} -1.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +(1.50000 - 2.59808i) q^{11} -1.73205i q^{12} +(1.00000 + 1.73205i) q^{13} +(-1.00000 - 1.73205i) q^{14} +(-0.500000 + 0.866025i) q^{16} +3.00000 q^{17} +3.00000 q^{18} -1.00000 q^{19} +(3.00000 - 1.73205i) q^{21} +(-1.50000 - 2.59808i) q^{22} +(-3.00000 - 5.19615i) q^{23} +(-1.50000 - 0.866025i) q^{24} +2.00000 q^{26} +5.19615i q^{27} -2.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(2.00000 + 3.46410i) q^{31} +(0.500000 + 0.866025i) q^{32} +(4.50000 - 2.59808i) q^{33} +(1.50000 - 2.59808i) q^{34} +(1.50000 - 2.59808i) q^{36} +4.00000 q^{37} +(-0.500000 + 0.866025i) q^{38} +3.46410i q^{39} +(-4.50000 - 7.79423i) q^{41} -3.46410i q^{42} +(-0.500000 + 0.866025i) q^{43} -3.00000 q^{44} -6.00000 q^{46} +(-3.00000 + 5.19615i) q^{47} +(-1.50000 + 0.866025i) q^{48} +(1.50000 + 2.59808i) q^{49} +(4.50000 + 2.59808i) q^{51} +(1.00000 - 1.73205i) q^{52} -12.0000 q^{53} +(4.50000 + 2.59808i) q^{54} +(-1.00000 + 1.73205i) q^{56} +(-1.50000 - 0.866025i) q^{57} +(3.00000 + 5.19615i) q^{58} +(-1.50000 - 2.59808i) q^{59} +(-4.00000 + 6.92820i) q^{61} +4.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -5.19615i q^{66} +(2.50000 + 4.33013i) q^{67} +(-1.50000 - 2.59808i) q^{68} -10.3923i q^{69} -12.0000 q^{71} +(-1.50000 - 2.59808i) q^{72} -11.0000 q^{73} +(2.00000 - 3.46410i) q^{74} +(0.500000 + 0.866025i) q^{76} +(-3.00000 - 5.19615i) q^{77} +(3.00000 + 1.73205i) q^{78} +(2.00000 - 3.46410i) q^{79} +(-4.50000 + 7.79423i) q^{81} -9.00000 q^{82} +(6.00000 - 10.3923i) q^{83} +(-3.00000 - 1.73205i) q^{84} +(0.500000 + 0.866025i) q^{86} +(-9.00000 + 5.19615i) q^{87} +(-1.50000 + 2.59808i) q^{88} +6.00000 q^{89} +4.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +6.92820i q^{93} +(3.00000 + 5.19615i) q^{94} +1.73205i q^{96} +(2.50000 - 4.33013i) q^{97} +3.00000 q^{98} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} + 3 q^{6} + 2 q^{7} - 2 q^{8} + 3 q^{9} + 3 q^{11} + 2 q^{13} - 2 q^{14} - q^{16} + 6 q^{17} + 6 q^{18} - 2 q^{19} + 6 q^{21} - 3 q^{22} - 6 q^{23} - 3 q^{24} + 4 q^{26}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0
\(6\) 1.50000 − 0.866025i 0.612372 − 0.353553i
\(7\) 1.00000 − 1.73205i 0.377964 − 0.654654i −0.612801 − 0.790237i \(-0.709957\pi\)
0.990766 + 0.135583i \(0.0432908\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 1.50000 − 2.59808i 0.452267 − 0.783349i −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) − 1.73205i − 0.500000i
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 − 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −1.00000 − 1.73205i −0.267261 − 0.462910i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 3.00000 0.707107
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 3.00000 − 1.73205i 0.654654 − 0.377964i
\(22\) −1.50000 − 2.59808i −0.319801 − 0.553912i
\(23\) −3.00000 − 5.19615i −0.625543 − 1.08347i −0.988436 − 0.151642i \(-0.951544\pi\)
0.362892 − 0.931831i \(-0.381789\pi\)
\(24\) −1.50000 − 0.866025i −0.306186 − 0.176777i
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 5.19615i 1.00000i
\(28\) −2.00000 −0.377964
\(29\) −3.00000 + 5.19615i −0.557086 + 0.964901i 0.440652 + 0.897678i \(0.354747\pi\)
−0.997738 + 0.0672232i \(0.978586\pi\)
\(30\) 0 0
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 − 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 4.50000 − 2.59808i 0.783349 − 0.452267i
\(34\) 1.50000 − 2.59808i 0.257248 − 0.445566i
\(35\) 0 0
\(36\) 1.50000 − 2.59808i 0.250000 − 0.433013i
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −0.500000 + 0.866025i −0.0811107 + 0.140488i
\(39\) 3.46410i 0.554700i
\(40\) 0 0
\(41\) −4.50000 − 7.79423i −0.702782 − 1.21725i −0.967486 − 0.252924i \(-0.918608\pi\)
0.264704 − 0.964330i \(-0.414726\pi\)
\(42\) − 3.46410i − 0.534522i
\(43\) −0.500000 + 0.866025i −0.0762493 + 0.132068i −0.901629 − 0.432511i \(-0.857628\pi\)
0.825380 + 0.564578i \(0.190961\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 − 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) −1.50000 + 0.866025i −0.216506 + 0.125000i
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) 0 0
\(51\) 4.50000 + 2.59808i 0.630126 + 0.363803i
\(52\) 1.00000 − 1.73205i 0.138675 − 0.240192i
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 4.50000 + 2.59808i 0.612372 + 0.353553i
\(55\) 0 0
\(56\) −1.00000 + 1.73205i −0.133631 + 0.231455i
\(57\) −1.50000 − 0.866025i −0.198680 − 0.114708i
\(58\) 3.00000 + 5.19615i 0.393919 + 0.682288i
\(59\) −1.50000 − 2.59808i −0.195283 − 0.338241i 0.751710 − 0.659494i \(-0.229229\pi\)
−0.946993 + 0.321253i \(0.895896\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 4.00000 0.508001
\(63\) 6.00000 0.755929
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) − 5.19615i − 0.639602i
\(67\) 2.50000 + 4.33013i 0.305424 + 0.529009i 0.977356 − 0.211604i \(-0.0678686\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) −1.50000 − 2.59808i −0.181902 − 0.315063i
\(69\) − 10.3923i − 1.25109i
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.50000 − 2.59808i −0.176777 − 0.306186i
\(73\) −11.0000 −1.28745 −0.643726 − 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 2.00000 − 3.46410i 0.232495 − 0.402694i
\(75\) 0 0
\(76\) 0.500000 + 0.866025i 0.0573539 + 0.0993399i
\(77\) −3.00000 − 5.19615i −0.341882 − 0.592157i
\(78\) 3.00000 + 1.73205i 0.339683 + 0.196116i
\(79\) 2.00000 − 3.46410i 0.225018 − 0.389742i −0.731307 − 0.682048i \(-0.761089\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) −9.00000 −0.993884
\(83\) 6.00000 − 10.3923i 0.658586 − 1.14070i −0.322396 − 0.946605i \(-0.604488\pi\)
0.980982 − 0.194099i \(-0.0621783\pi\)
\(84\) −3.00000 − 1.73205i −0.327327 − 0.188982i
\(85\) 0 0
\(86\) 0.500000 + 0.866025i 0.0539164 + 0.0933859i
\(87\) −9.00000 + 5.19615i −0.964901 + 0.557086i
\(88\) −1.50000 + 2.59808i −0.159901 + 0.276956i
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −3.00000 + 5.19615i −0.312772 + 0.541736i
\(93\) 6.92820i 0.718421i
\(94\) 3.00000 + 5.19615i 0.309426 + 0.535942i
\(95\) 0 0
\(96\) 1.73205i 0.176777i
\(97\) 2.50000 − 4.33013i 0.253837 − 0.439658i −0.710742 − 0.703452i \(-0.751641\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 3.00000 0.303046
\(99\) 9.00000 0.904534
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.e.i.151.1 2
3.2 odd 2 1350.2.e.c.451.1 2
5.2 odd 4 450.2.j.e.349.2 4
5.3 odd 4 450.2.j.e.349.1 4
5.4 even 2 18.2.c.a.7.1 ✓ 2
9.2 odd 6 4050.2.a.v.1.1 1
9.4 even 3 inner 450.2.e.i.301.1 2
9.5 odd 6 1350.2.e.c.901.1 2
9.7 even 3 4050.2.a.c.1.1 1
15.2 even 4 1350.2.j.a.1099.1 4
15.8 even 4 1350.2.j.a.1099.2 4
15.14 odd 2 54.2.c.a.19.1 2
20.19 odd 2 144.2.i.c.97.1 2
35.4 even 6 882.2.h.c.79.1 2
35.9 even 6 882.2.e.i.655.1 2
35.19 odd 6 882.2.e.g.655.1 2
35.24 odd 6 882.2.h.b.79.1 2
35.34 odd 2 882.2.f.d.295.1 2
40.19 odd 2 576.2.i.a.385.1 2
40.29 even 2 576.2.i.g.385.1 2
45.2 even 12 4050.2.c.r.649.2 2
45.4 even 6 18.2.c.a.13.1 yes 2
45.7 odd 12 4050.2.c.c.649.1 2
45.13 odd 12 450.2.j.e.49.2 4
45.14 odd 6 54.2.c.a.37.1 2
45.22 odd 12 450.2.j.e.49.1 4
45.23 even 12 1350.2.j.a.199.1 4
45.29 odd 6 162.2.a.b.1.1 1
45.32 even 12 1350.2.j.a.199.2 4
45.34 even 6 162.2.a.c.1.1 1
45.38 even 12 4050.2.c.r.649.1 2
45.43 odd 12 4050.2.c.c.649.2 2
60.59 even 2 432.2.i.b.289.1 2
105.44 odd 6 2646.2.e.b.2125.1 2
105.59 even 6 2646.2.h.i.667.1 2
105.74 odd 6 2646.2.h.h.667.1 2
105.89 even 6 2646.2.e.c.2125.1 2
105.104 even 2 2646.2.f.g.883.1 2
120.29 odd 2 1728.2.i.e.1153.1 2
120.59 even 2 1728.2.i.f.1153.1 2
180.59 even 6 432.2.i.b.145.1 2
180.79 odd 6 1296.2.a.g.1.1 1
180.119 even 6 1296.2.a.f.1.1 1
180.139 odd 6 144.2.i.c.49.1 2
315.4 even 6 882.2.e.i.373.1 2
315.34 odd 6 7938.2.a.x.1.1 1
315.59 even 6 2646.2.e.c.1549.1 2
315.94 odd 6 882.2.e.g.373.1 2
315.104 even 6 2646.2.f.g.1765.1 2
315.139 odd 6 882.2.f.d.589.1 2
315.149 odd 6 2646.2.h.h.361.1 2
315.184 even 6 882.2.h.c.67.1 2
315.194 even 6 2646.2.h.i.361.1 2
315.209 even 6 7938.2.a.i.1.1 1
315.229 odd 6 882.2.h.b.67.1 2
315.284 odd 6 2646.2.e.b.1549.1 2
360.29 odd 6 5184.2.a.q.1.1 1
360.59 even 6 1728.2.i.f.577.1 2
360.139 odd 6 576.2.i.a.193.1 2
360.149 odd 6 1728.2.i.e.577.1 2
360.229 even 6 576.2.i.g.193.1 2
360.259 odd 6 5184.2.a.o.1.1 1
360.299 even 6 5184.2.a.p.1.1 1
360.349 even 6 5184.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.2.c.a.7.1 ✓ 2 5.4 even 2
18.2.c.a.13.1 yes 2 45.4 even 6
54.2.c.a.19.1 2 15.14 odd 2
54.2.c.a.37.1 2 45.14 odd 6
144.2.i.c.49.1 2 180.139 odd 6
144.2.i.c.97.1 2 20.19 odd 2
162.2.a.b.1.1 1 45.29 odd 6
162.2.a.c.1.1 1 45.34 even 6
432.2.i.b.145.1 2 180.59 even 6
432.2.i.b.289.1 2 60.59 even 2
450.2.e.i.151.1 2 1.1 even 1 trivial
450.2.e.i.301.1 2 9.4 even 3 inner
450.2.j.e.49.1 4 45.22 odd 12
450.2.j.e.49.2 4 45.13 odd 12
450.2.j.e.349.1 4 5.3 odd 4
450.2.j.e.349.2 4 5.2 odd 4
576.2.i.a.193.1 2 360.139 odd 6
576.2.i.a.385.1 2 40.19 odd 2
576.2.i.g.193.1 2 360.229 even 6
576.2.i.g.385.1 2 40.29 even 2
882.2.e.g.373.1 2 315.94 odd 6
882.2.e.g.655.1 2 35.19 odd 6
882.2.e.i.373.1 2 315.4 even 6
882.2.e.i.655.1 2 35.9 even 6
882.2.f.d.295.1 2 35.34 odd 2
882.2.f.d.589.1 2 315.139 odd 6
882.2.h.b.67.1 2 315.229 odd 6
882.2.h.b.79.1 2 35.24 odd 6
882.2.h.c.67.1 2 315.184 even 6
882.2.h.c.79.1 2 35.4 even 6
1296.2.a.f.1.1 1 180.119 even 6
1296.2.a.g.1.1 1 180.79 odd 6
1350.2.e.c.451.1 2 3.2 odd 2
1350.2.e.c.901.1 2 9.5 odd 6
1350.2.j.a.199.1 4 45.23 even 12
1350.2.j.a.199.2 4 45.32 even 12
1350.2.j.a.1099.1 4 15.2 even 4
1350.2.j.a.1099.2 4 15.8 even 4
1728.2.i.e.577.1 2 360.149 odd 6
1728.2.i.e.1153.1 2 120.29 odd 2
1728.2.i.f.577.1 2 360.59 even 6
1728.2.i.f.1153.1 2 120.59 even 2
2646.2.e.b.1549.1 2 315.284 odd 6
2646.2.e.b.2125.1 2 105.44 odd 6
2646.2.e.c.1549.1 2 315.59 even 6
2646.2.e.c.2125.1 2 105.89 even 6
2646.2.f.g.883.1 2 105.104 even 2
2646.2.f.g.1765.1 2 315.104 even 6
2646.2.h.h.361.1 2 315.149 odd 6
2646.2.h.h.667.1 2 105.74 odd 6
2646.2.h.i.361.1 2 315.194 even 6
2646.2.h.i.667.1 2 105.59 even 6
4050.2.a.c.1.1 1 9.7 even 3
4050.2.a.v.1.1 1 9.2 odd 6
4050.2.c.c.649.1 2 45.7 odd 12
4050.2.c.c.649.2 2 45.43 odd 12
4050.2.c.r.649.1 2 45.38 even 12
4050.2.c.r.649.2 2 45.2 even 12
5184.2.a.o.1.1 1 360.259 odd 6
5184.2.a.p.1.1 1 360.299 even 6
5184.2.a.q.1.1 1 360.29 odd 6
5184.2.a.r.1.1 1 360.349 even 6
7938.2.a.i.1.1 1 315.209 even 6
7938.2.a.x.1.1 1 315.34 odd 6