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Properties

Label 450.2.e.k.301.1
Level $450$
Weight $2$
Character 450.301
Analytic conductor $3.593$
Analytic rank $0$
Dimension $4$
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 = [4,-2,2,-2,0,2,4] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 450.301
Dual form 450.2.e.k.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} +(-0.724745 + 1.57313i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(1.72474 - 0.158919i) q^{6} +(2.22474 + 3.85337i) q^{7} +1.00000 q^{8} +(-1.94949 - 2.28024i) q^{9} +(-0.724745 - 1.25529i) q^{11} +(-1.00000 - 1.41421i) q^{12} +(-1.22474 + 2.12132i) q^{13} +(2.22474 - 3.85337i) q^{14} +(-0.500000 - 0.866025i) q^{16} -3.89898 q^{17} +(-1.00000 + 2.82843i) q^{18} -0.550510 q^{19} +(-7.67423 + 0.707107i) q^{21} +(-0.724745 + 1.25529i) q^{22} +(-1.44949 + 2.51059i) q^{23} +(-0.724745 + 1.57313i) q^{24} +2.44949 q^{26} +(5.00000 - 1.41421i) q^{27} -4.44949 q^{28} +(3.00000 + 5.19615i) q^{29} +(-3.22474 + 5.58542i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(2.50000 - 0.230351i) q^{33} +(1.94949 + 3.37662i) q^{34} +(2.94949 - 0.548188i) q^{36} -8.00000 q^{37} +(0.275255 + 0.476756i) q^{38} +(-2.44949 - 3.46410i) q^{39} +(0.500000 - 0.866025i) q^{41} +(4.44949 + 6.29253i) q^{42} +(3.72474 + 6.45145i) q^{43} +1.44949 q^{44} +2.89898 q^{46} +(0.224745 + 0.389270i) q^{47} +(1.72474 - 0.158919i) q^{48} +(-6.39898 + 11.0834i) q^{49} +(2.82577 - 6.13361i) q^{51} +(-1.22474 - 2.12132i) q^{52} -8.44949 q^{53} +(-3.72474 - 3.62302i) q^{54} +(2.22474 + 3.85337i) q^{56} +(0.398979 - 0.866025i) q^{57} +(3.00000 - 5.19615i) q^{58} +(5.62372 - 9.74058i) q^{59} +(0.224745 + 0.389270i) q^{61} +6.44949 q^{62} +(4.44949 - 12.5851i) q^{63} +1.00000 q^{64} +(-1.44949 - 2.04989i) q^{66} +(4.72474 - 8.18350i) q^{67} +(1.94949 - 3.37662i) q^{68} +(-2.89898 - 4.09978i) q^{69} +2.44949 q^{71} +(-1.94949 - 2.28024i) q^{72} +4.79796 q^{73} +(4.00000 + 6.92820i) q^{74} +(0.275255 - 0.476756i) q^{76} +(3.22474 - 5.58542i) q^{77} +(-1.77526 + 3.85337i) q^{78} +(3.67423 + 6.36396i) q^{79} +(-1.39898 + 8.89060i) q^{81} -1.00000 q^{82} +(-2.00000 - 3.46410i) q^{83} +(3.22474 - 6.99964i) q^{84} +(3.72474 - 6.45145i) q^{86} +(-10.3485 + 0.953512i) q^{87} +(-0.724745 - 1.25529i) q^{88} +12.8990 q^{89} -10.8990 q^{91} +(-1.44949 - 2.51059i) q^{92} +(-6.44949 - 9.12096i) q^{93} +(0.224745 - 0.389270i) q^{94} +(-1.00000 - 1.41421i) q^{96} +(6.50000 + 11.2583i) q^{97} +12.7980 q^{98} +(-1.44949 + 4.09978i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 2 q^{9} + 2 q^{11} - 4 q^{12} + 4 q^{14} - 2 q^{16} + 4 q^{17} - 4 q^{18} - 12 q^{19} - 16 q^{21} + 2 q^{22} + 4 q^{23} + 2 q^{24} + 20 q^{27}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) −0.724745 + 1.57313i −0.418432 + 0.908248i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0 0
\(6\) 1.72474 − 0.158919i 0.704124 − 0.0648783i
\(7\) 2.22474 + 3.85337i 0.840875 + 1.45644i 0.889156 + 0.457604i \(0.151292\pi\)
−0.0482818 + 0.998834i \(0.515375\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.94949 − 2.28024i −0.649830 − 0.760080i
\(10\) 0 0
\(11\) −0.724745 − 1.25529i −0.218519 − 0.378486i 0.735837 − 0.677159i \(-0.236789\pi\)
−0.954355 + 0.298674i \(0.903456\pi\)
\(12\) −1.00000 − 1.41421i −0.288675 − 0.408248i
\(13\) −1.22474 + 2.12132i −0.339683 + 0.588348i −0.984373 − 0.176096i \(-0.943653\pi\)
0.644690 + 0.764444i \(0.276986\pi\)
\(14\) 2.22474 − 3.85337i 0.594588 − 1.02986i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −3.89898 −0.945641 −0.472821 − 0.881159i \(-0.656764\pi\)
−0.472821 + 0.881159i \(0.656764\pi\)
\(18\) −1.00000 + 2.82843i −0.235702 + 0.666667i
\(19\) −0.550510 −0.126296 −0.0631479 − 0.998004i \(-0.520114\pi\)
−0.0631479 + 0.998004i \(0.520114\pi\)
\(20\) 0 0
\(21\) −7.67423 + 0.707107i −1.67466 + 0.154303i
\(22\) −0.724745 + 1.25529i −0.154516 + 0.267630i
\(23\) −1.44949 + 2.51059i −0.302240 + 0.523494i −0.976643 − 0.214869i \(-0.931068\pi\)
0.674403 + 0.738363i \(0.264401\pi\)
\(24\) −0.724745 + 1.57313i −0.147938 + 0.321114i
\(25\) 0 0
\(26\) 2.44949 0.480384
\(27\) 5.00000 − 1.41421i 0.962250 − 0.272166i
\(28\) −4.44949 −0.840875
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −3.22474 + 5.58542i −0.579181 + 1.00317i 0.416392 + 0.909185i \(0.363294\pi\)
−0.995573 + 0.0939863i \(0.970039\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 2.50000 − 0.230351i 0.435194 − 0.0400989i
\(34\) 1.94949 + 3.37662i 0.334335 + 0.579085i
\(35\) 0 0
\(36\) 2.94949 − 0.548188i 0.491582 − 0.0913647i
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0.275255 + 0.476756i 0.0446523 + 0.0773400i
\(39\) −2.44949 − 3.46410i −0.392232 − 0.554700i
\(40\) 0 0
\(41\) 0.500000 − 0.866025i 0.0780869 − 0.135250i −0.824338 − 0.566099i \(-0.808452\pi\)
0.902424 + 0.430848i \(0.141786\pi\)
\(42\) 4.44949 + 6.29253i 0.686571 + 0.970958i
\(43\) 3.72474 + 6.45145i 0.568018 + 0.983836i 0.996762 + 0.0804103i \(0.0256230\pi\)
−0.428744 + 0.903426i \(0.641044\pi\)
\(44\) 1.44949 0.218519
\(45\) 0 0
\(46\) 2.89898 0.427431
\(47\) 0.224745 + 0.389270i 0.0327824 + 0.0567808i 0.881951 − 0.471341i \(-0.156230\pi\)
−0.849169 + 0.528122i \(0.822896\pi\)
\(48\) 1.72474 − 0.158919i 0.248945 − 0.0229379i
\(49\) −6.39898 + 11.0834i −0.914140 + 1.58334i
\(50\) 0 0
\(51\) 2.82577 − 6.13361i 0.395686 − 0.858877i
\(52\) −1.22474 − 2.12132i −0.169842 − 0.294174i
\(53\) −8.44949 −1.16063 −0.580313 − 0.814393i \(-0.697070\pi\)
−0.580313 + 0.814393i \(0.697070\pi\)
\(54\) −3.72474 − 3.62302i −0.506874 − 0.493031i
\(55\) 0 0
\(56\) 2.22474 + 3.85337i 0.297294 + 0.514928i
\(57\) 0.398979 − 0.866025i 0.0528461 − 0.114708i
\(58\) 3.00000 − 5.19615i 0.393919 − 0.682288i
\(59\) 5.62372 − 9.74058i 0.732147 − 1.26812i −0.223817 − 0.974631i \(-0.571852\pi\)
0.955964 − 0.293484i \(-0.0948147\pi\)
\(60\) 0 0
\(61\) 0.224745 + 0.389270i 0.0287756 + 0.0498409i 0.880055 − 0.474873i \(-0.157506\pi\)
−0.851279 + 0.524713i \(0.824173\pi\)
\(62\) 6.44949 0.819086
\(63\) 4.44949 − 12.5851i 0.560583 − 1.58557i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.44949 − 2.04989i −0.178420 − 0.252324i
\(67\) 4.72474 − 8.18350i 0.577219 − 0.999773i −0.418577 − 0.908181i \(-0.637471\pi\)
0.995797 − 0.0915922i \(-0.0291956\pi\)
\(68\) 1.94949 − 3.37662i 0.236410 − 0.409475i
\(69\) −2.89898 − 4.09978i −0.348996 − 0.493555i
\(70\) 0 0
\(71\) 2.44949 0.290701 0.145350 − 0.989380i \(-0.453569\pi\)
0.145350 + 0.989380i \(0.453569\pi\)
\(72\) −1.94949 − 2.28024i −0.229750 − 0.268729i
\(73\) 4.79796 0.561559 0.280779 − 0.959772i \(-0.409407\pi\)
0.280779 + 0.959772i \(0.409407\pi\)
\(74\) 4.00000 + 6.92820i 0.464991 + 0.805387i
\(75\) 0 0
\(76\) 0.275255 − 0.476756i 0.0315739 − 0.0546876i
\(77\) 3.22474 − 5.58542i 0.367494 − 0.636518i
\(78\) −1.77526 + 3.85337i −0.201008 + 0.436308i
\(79\) 3.67423 + 6.36396i 0.413384 + 0.716002i 0.995257 − 0.0972777i \(-0.0310135\pi\)
−0.581874 + 0.813279i \(0.697680\pi\)
\(80\) 0 0
\(81\) −1.39898 + 8.89060i −0.155442 + 0.987845i
\(82\) −1.00000 −0.110432
\(83\) −2.00000 − 3.46410i −0.219529 − 0.380235i 0.735135 − 0.677920i \(-0.237119\pi\)
−0.954664 + 0.297686i \(0.903785\pi\)
\(84\) 3.22474 − 6.99964i 0.351849 − 0.763723i
\(85\) 0 0
\(86\) 3.72474 − 6.45145i 0.401650 − 0.695677i
\(87\) −10.3485 + 0.953512i −1.10947 + 0.102227i
\(88\) −0.724745 − 1.25529i −0.0772581 − 0.133815i
\(89\) 12.8990 1.36729 0.683645 − 0.729815i \(-0.260394\pi\)
0.683645 + 0.729815i \(0.260394\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) −1.44949 − 2.51059i −0.151120 − 0.261747i
\(93\) −6.44949 − 9.12096i −0.668781 − 0.945799i
\(94\) 0.224745 − 0.389270i 0.0231807 − 0.0401501i
\(95\) 0 0
\(96\) −1.00000 − 1.41421i −0.102062 − 0.144338i
\(97\) 6.50000 + 11.2583i 0.659975 + 1.14311i 0.980622 + 0.195911i \(0.0627665\pi\)
−0.320647 + 0.947199i \(0.603900\pi\)
\(98\) 12.7980 1.29279
\(99\) −1.44949 + 4.09978i −0.145679 + 0.412043i
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.k.301.1 4
3.2 odd 2 1350.2.e.m.901.2 4
5.2 odd 4 90.2.i.b.49.4 yes 8
5.3 odd 4 90.2.i.b.49.1 ✓ 8
5.4 even 2 450.2.e.n.301.2 4
9.2 odd 6 1350.2.e.m.451.2 4
9.4 even 3 4050.2.a.bs.1.1 2
9.5 odd 6 4050.2.a.bm.1.1 2
9.7 even 3 inner 450.2.e.k.151.1 4
15.2 even 4 270.2.i.b.199.2 8
15.8 even 4 270.2.i.b.199.3 8
15.14 odd 2 1350.2.e.j.901.1 4
20.3 even 4 720.2.by.c.49.4 8
20.7 even 4 720.2.by.c.49.1 8
45.2 even 12 270.2.i.b.19.3 8
45.4 even 6 4050.2.a.bq.1.2 2
45.7 odd 12 90.2.i.b.79.1 yes 8
45.13 odd 12 810.2.c.f.649.1 4
45.14 odd 6 4050.2.a.bz.1.2 2
45.22 odd 12 810.2.c.f.649.3 4
45.23 even 12 810.2.c.e.649.4 4
45.29 odd 6 1350.2.e.j.451.1 4
45.32 even 12 810.2.c.e.649.2 4
45.34 even 6 450.2.e.n.151.2 4
45.38 even 12 270.2.i.b.19.2 8
45.43 odd 12 90.2.i.b.79.4 yes 8
60.23 odd 4 2160.2.by.d.1009.1 8
60.47 odd 4 2160.2.by.d.1009.4 8
180.7 even 12 720.2.by.c.529.4 8
180.43 even 12 720.2.by.c.529.1 8
180.47 odd 12 2160.2.by.d.289.1 8
180.83 odd 12 2160.2.by.d.289.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.b.49.1 ✓ 8 5.3 odd 4
90.2.i.b.49.4 yes 8 5.2 odd 4
90.2.i.b.79.1 yes 8 45.7 odd 12
90.2.i.b.79.4 yes 8 45.43 odd 12
270.2.i.b.19.2 8 45.38 even 12
270.2.i.b.19.3 8 45.2 even 12
270.2.i.b.199.2 8 15.2 even 4
270.2.i.b.199.3 8 15.8 even 4
450.2.e.k.151.1 4 9.7 even 3 inner
450.2.e.k.301.1 4 1.1 even 1 trivial
450.2.e.n.151.2 4 45.34 even 6
450.2.e.n.301.2 4 5.4 even 2
720.2.by.c.49.1 8 20.7 even 4
720.2.by.c.49.4 8 20.3 even 4
720.2.by.c.529.1 8 180.43 even 12
720.2.by.c.529.4 8 180.7 even 12
810.2.c.e.649.2 4 45.32 even 12
810.2.c.e.649.4 4 45.23 even 12
810.2.c.f.649.1 4 45.13 odd 12
810.2.c.f.649.3 4 45.22 odd 12
1350.2.e.j.451.1 4 45.29 odd 6
1350.2.e.j.901.1 4 15.14 odd 2
1350.2.e.m.451.2 4 9.2 odd 6
1350.2.e.m.901.2 4 3.2 odd 2
2160.2.by.d.289.1 8 180.47 odd 12
2160.2.by.d.289.4 8 180.83 odd 12
2160.2.by.d.1009.1 8 60.23 odd 4
2160.2.by.d.1009.4 8 60.47 odd 4
4050.2.a.bm.1.1 2 9.5 odd 6
4050.2.a.bq.1.2 2 45.4 even 6
4050.2.a.bs.1.1 2 9.4 even 3
4050.2.a.bz.1.2 2 45.14 odd 6