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Properties

Label 450.2.e.b.301.1
Level $450$
Weight $2$
Character 450.301
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,-1,0,0,1] 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 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 301.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 450.301
Dual form 450.2.e.b.151.1

$q$-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} +1.73205i q^{6} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{9} +(1.00000 + 1.73205i) q^{11} +(1.50000 - 0.866025i) q^{12} +(3.00000 - 5.19615i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -2.00000 q^{17} +(1.50000 - 2.59808i) q^{18} +6.00000 q^{19} -1.73205i q^{21} +(1.00000 - 1.73205i) q^{22} +(-0.500000 + 0.866025i) q^{23} +(-1.50000 - 0.866025i) q^{24} -6.00000 q^{26} -5.19615i q^{27} -1.00000 q^{28} +(-4.50000 - 7.79423i) q^{29} +(1.00000 - 1.73205i) q^{31} +(-0.500000 + 0.866025i) q^{32} -3.46410i q^{33} +(1.00000 + 1.73205i) q^{34} -3.00000 q^{36} +2.00000 q^{37} +(-3.00000 - 5.19615i) q^{38} +(-9.00000 + 5.19615i) q^{39} +(5.50000 - 9.52628i) q^{41} +(-1.50000 + 0.866025i) q^{42} +(2.00000 + 3.46410i) q^{43} -2.00000 q^{44} +1.00000 q^{46} +(3.50000 + 6.06218i) q^{47} +1.73205i q^{48} +(3.00000 - 5.19615i) q^{49} +(3.00000 + 1.73205i) q^{51} +(3.00000 + 5.19615i) q^{52} +(-4.50000 + 2.59808i) q^{54} +(0.500000 + 0.866025i) q^{56} +(-9.00000 - 5.19615i) q^{57} +(-4.50000 + 7.79423i) q^{58} +(2.00000 - 3.46410i) q^{59} +(3.50000 + 6.06218i) q^{61} -2.00000 q^{62} +(-1.50000 + 2.59808i) q^{63} +1.00000 q^{64} +(-3.00000 + 1.73205i) q^{66} +(5.50000 - 9.52628i) q^{67} +(1.00000 - 1.73205i) q^{68} +(1.50000 - 0.866025i) q^{69} -6.00000 q^{71} +(1.50000 + 2.59808i) q^{72} -4.00000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(-3.00000 + 5.19615i) q^{76} +(-1.00000 + 1.73205i) q^{77} +(9.00000 + 5.19615i) q^{78} +(6.00000 + 10.3923i) q^{79} +(-4.50000 + 7.79423i) q^{81} -11.0000 q^{82} +(5.50000 + 9.52628i) q^{83} +(1.50000 + 0.866025i) q^{84} +(2.00000 - 3.46410i) q^{86} +15.5885i q^{87} +(1.00000 + 1.73205i) q^{88} +1.00000 q^{89} +6.00000 q^{91} +(-0.500000 - 0.866025i) q^{92} +(-3.00000 + 1.73205i) q^{93} +(3.50000 - 6.06218i) q^{94} +(1.50000 - 0.866025i) q^{96} +(4.00000 + 6.92820i) q^{97} -6.00000 q^{98} +(-3.00000 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 3 q^{3} - q^{4} + q^{7} + 2 q^{8} + 3 q^{9} + 2 q^{11} + 3 q^{12} + 6 q^{13} + q^{14} - q^{16} - 4 q^{17} + 3 q^{18} + 12 q^{19} + 2 q^{22} - q^{23} - 3 q^{24} - 12 q^{26} - 2 q^{28}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(6\) 1.73205i 0.707107i
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i 0.944911 − 0.327327i \(-0.106148\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 − 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 1.50000 − 0.866025i 0.433013 − 0.250000i
\(13\) 3.00000 − 5.19615i 0.832050 − 1.44115i −0.0643593 − 0.997927i \(-0.520500\pi\)
0.896410 − 0.443227i \(-0.146166\pi\)
\(14\) 0.500000 − 0.866025i 0.133631 − 0.231455i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.50000 − 2.59808i 0.353553 − 0.612372i
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) − 1.73205i − 0.377964i
\(22\) 1.00000 − 1.73205i 0.213201 − 0.369274i
\(23\) −0.500000 + 0.866025i −0.104257 + 0.180579i −0.913434 − 0.406986i \(-0.866580\pi\)
0.809177 + 0.587565i \(0.199913\pi\)
\(24\) −1.50000 − 0.866025i −0.306186 − 0.176777i
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) − 5.19615i − 1.00000i
\(28\) −1.00000 −0.188982
\(29\) −4.50000 − 7.79423i −0.835629 − 1.44735i −0.893517 − 0.449029i \(-0.851770\pi\)
0.0578882 − 0.998323i \(-0.481563\pi\)
\(30\) 0 0
\(31\) 1.00000 − 1.73205i 0.179605 − 0.311086i −0.762140 − 0.647412i \(-0.775851\pi\)
0.941745 + 0.336327i \(0.109185\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) − 3.46410i − 0.603023i
\(34\) 1.00000 + 1.73205i 0.171499 + 0.297044i
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −3.00000 − 5.19615i −0.486664 − 0.842927i
\(39\) −9.00000 + 5.19615i −1.44115 + 0.832050i
\(40\) 0 0
\(41\) 5.50000 − 9.52628i 0.858956 − 1.48775i −0.0139704 − 0.999902i \(-0.504447\pi\)
0.872926 − 0.487852i \(-0.162220\pi\)
\(42\) −1.50000 + 0.866025i −0.231455 + 0.133631i
\(43\) 2.00000 + 3.46410i 0.304997 + 0.528271i 0.977261 − 0.212041i \(-0.0680112\pi\)
−0.672264 + 0.740312i \(0.734678\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.50000 + 6.06218i 0.510527 + 0.884260i 0.999926 + 0.0121990i \(0.00388317\pi\)
−0.489398 + 0.872060i \(0.662783\pi\)
\(48\) 1.73205i 0.250000i
\(49\) 3.00000 − 5.19615i 0.428571 − 0.742307i
\(50\) 0 0
\(51\) 3.00000 + 1.73205i 0.420084 + 0.242536i
\(52\) 3.00000 + 5.19615i 0.416025 + 0.720577i
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −4.50000 + 2.59808i −0.612372 + 0.353553i
\(55\) 0 0
\(56\) 0.500000 + 0.866025i 0.0668153 + 0.115728i
\(57\) −9.00000 − 5.19615i −1.19208 − 0.688247i
\(58\) −4.50000 + 7.79423i −0.590879 + 1.02343i
\(59\) 2.00000 − 3.46410i 0.260378 − 0.450988i −0.705965 − 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) 3.50000 + 6.06218i 0.448129 + 0.776182i 0.998264 − 0.0588933i \(-0.0187572\pi\)
−0.550135 + 0.835076i \(0.685424\pi\)
\(62\) −2.00000 −0.254000
\(63\) −1.50000 + 2.59808i −0.188982 + 0.327327i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 + 1.73205i −0.369274 + 0.213201i
\(67\) 5.50000 − 9.52628i 0.671932 − 1.16382i −0.305424 − 0.952217i \(-0.598798\pi\)
0.977356 − 0.211604i \(-0.0678686\pi\)
\(68\) 1.00000 − 1.73205i 0.121268 − 0.210042i
\(69\) 1.50000 − 0.866025i 0.180579 − 0.104257i
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.50000 + 2.59808i 0.176777 + 0.306186i
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −1.00000 − 1.73205i −0.116248 − 0.201347i
\(75\) 0 0
\(76\) −3.00000 + 5.19615i −0.344124 + 0.596040i
\(77\) −1.00000 + 1.73205i −0.113961 + 0.197386i
\(78\) 9.00000 + 5.19615i 1.01905 + 0.588348i
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) −11.0000 −1.21475
\(83\) 5.50000 + 9.52628i 0.603703 + 1.04565i 0.992255 + 0.124218i \(0.0396422\pi\)
−0.388552 + 0.921427i \(0.627024\pi\)
\(84\) 1.50000 + 0.866025i 0.163663 + 0.0944911i
\(85\) 0 0
\(86\) 2.00000 − 3.46410i 0.215666 − 0.373544i
\(87\) 15.5885i 1.67126i
\(88\) 1.00000 + 1.73205i 0.106600 + 0.184637i
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −0.500000 − 0.866025i −0.0521286 − 0.0902894i
\(93\) −3.00000 + 1.73205i −0.311086 + 0.179605i
\(94\) 3.50000 − 6.06218i 0.360997 − 0.625266i
\(95\) 0 0
\(96\) 1.50000 − 0.866025i 0.153093 − 0.0883883i
\(97\) 4.00000 + 6.92820i 0.406138 + 0.703452i 0.994453 − 0.105180i \(-0.0335417\pi\)
−0.588315 + 0.808632i \(0.700208\pi\)
\(98\) −6.00000 −0.606092
\(99\) −3.00000 + 5.19615i −0.301511 + 0.522233i
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.b.301.1 2
3.2 odd 2 1350.2.e.i.901.1 2
5.2 odd 4 90.2.i.a.49.2 yes 4
5.3 odd 4 90.2.i.a.49.1 ✓ 4
5.4 even 2 450.2.e.g.301.1 2
9.2 odd 6 1350.2.e.i.451.1 2
9.4 even 3 4050.2.a.x.1.1 1
9.5 odd 6 4050.2.a.g.1.1 1
9.7 even 3 inner 450.2.e.b.151.1 2
15.2 even 4 270.2.i.a.199.1 4
15.8 even 4 270.2.i.a.199.2 4
15.14 odd 2 1350.2.e.a.901.1 2
20.3 even 4 720.2.by.a.49.1 4
20.7 even 4 720.2.by.a.49.2 4
45.2 even 12 270.2.i.a.19.2 4
45.4 even 6 4050.2.a.j.1.1 1
45.7 odd 12 90.2.i.a.79.1 yes 4
45.13 odd 12 810.2.c.b.649.1 2
45.14 odd 6 4050.2.a.be.1.1 1
45.22 odd 12 810.2.c.b.649.2 2
45.23 even 12 810.2.c.c.649.2 2
45.29 odd 6 1350.2.e.a.451.1 2
45.32 even 12 810.2.c.c.649.1 2
45.34 even 6 450.2.e.g.151.1 2
45.38 even 12 270.2.i.a.19.1 4
45.43 odd 12 90.2.i.a.79.2 yes 4
60.23 odd 4 2160.2.by.b.1009.2 4
60.47 odd 4 2160.2.by.b.1009.1 4
180.7 even 12 720.2.by.a.529.1 4
180.43 even 12 720.2.by.a.529.2 4
180.47 odd 12 2160.2.by.b.289.2 4
180.83 odd 12 2160.2.by.b.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.a.49.1 ✓ 4 5.3 odd 4
90.2.i.a.49.2 yes 4 5.2 odd 4
90.2.i.a.79.1 yes 4 45.7 odd 12
90.2.i.a.79.2 yes 4 45.43 odd 12
270.2.i.a.19.1 4 45.38 even 12
270.2.i.a.19.2 4 45.2 even 12
270.2.i.a.199.1 4 15.2 even 4
270.2.i.a.199.2 4 15.8 even 4
450.2.e.b.151.1 2 9.7 even 3 inner
450.2.e.b.301.1 2 1.1 even 1 trivial
450.2.e.g.151.1 2 45.34 even 6
450.2.e.g.301.1 2 5.4 even 2
720.2.by.a.49.1 4 20.3 even 4
720.2.by.a.49.2 4 20.7 even 4
720.2.by.a.529.1 4 180.7 even 12
720.2.by.a.529.2 4 180.43 even 12
810.2.c.b.649.1 2 45.13 odd 12
810.2.c.b.649.2 2 45.22 odd 12
810.2.c.c.649.1 2 45.32 even 12
810.2.c.c.649.2 2 45.23 even 12
1350.2.e.a.451.1 2 45.29 odd 6
1350.2.e.a.901.1 2 15.14 odd 2
1350.2.e.i.451.1 2 9.2 odd 6
1350.2.e.i.901.1 2 3.2 odd 2
2160.2.by.b.289.1 4 180.83 odd 12
2160.2.by.b.289.2 4 180.47 odd 12
2160.2.by.b.1009.1 4 60.47 odd 4
2160.2.by.b.1009.2 4 60.23 odd 4
4050.2.a.g.1.1 1 9.5 odd 6
4050.2.a.j.1.1 1 45.4 even 6
4050.2.a.x.1.1 1 9.4 even 3
4050.2.a.be.1.1 1 45.14 odd 6