Properties

Label 450.8.a.l.1.1
Level $450$
Weight $8$
Character 450.1
Self dual yes
Analytic conductor $140.573$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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] = [450,8,Mod(1,450)] 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("450.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-8,0,64,0,0,1174] 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: yes
Analytic conductor: \(140.573261468\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 450.1

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +64.0000 q^{4} +1174.00 q^{7} -512.000 q^{8} +7563.00 q^{11} -5372.00 q^{13} -9392.00 q^{14} +4096.00 q^{16} +24021.0 q^{17} -51235.0 q^{19} -60504.0 q^{22} -57618.0 q^{23} +42976.0 q^{26} +75136.0 q^{28} -47040.0 q^{29} -192358. q^{31} -32768.0 q^{32} -192168. q^{34} -197066. q^{37} +409880. q^{38} +237723. q^{41} -653012. q^{43} +484032. q^{44} +460944. q^{46} -826884. q^{47} +554733. q^{49} -343808. q^{52} +569022. q^{53} -601088. q^{56} +376320. q^{58} -1.50108e6 q^{59} -2.06892e6 q^{61} +1.53886e6 q^{62} +262144. q^{64} +3.44435e6 q^{67} +1.53734e6 q^{68} -4.12105e6 q^{71} +83653.0 q^{73} +1.57653e6 q^{74} -3.27904e6 q^{76} +8.87896e6 q^{77} +1.45403e6 q^{79} -1.90178e6 q^{82} +1.62657e6 q^{83} +5.22410e6 q^{86} -3.87226e6 q^{88} -6.00434e6 q^{89} -6.30673e6 q^{91} -3.68755e6 q^{92} +6.61507e6 q^{94} -3.41175e6 q^{97} -4.43786e6 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1174.00 1.29367 0.646837 − 0.762628i \(-0.276091\pi\)
0.646837 + 0.762628i \(0.276091\pi\)
\(8\) −512.000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 7563.00 1.71325 0.856623 − 0.515943i \(-0.172558\pi\)
0.856623 + 0.515943i \(0.172558\pi\)
\(12\) 0 0
\(13\) −5372.00 −0.678163 −0.339082 − 0.940757i \(-0.610116\pi\)
−0.339082 + 0.940757i \(0.610116\pi\)
\(14\) −9392.00 −0.914766
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 24021.0 1.18582 0.592911 − 0.805268i \(-0.297978\pi\)
0.592911 + 0.805268i \(0.297978\pi\)
\(18\) 0 0
\(19\) −51235.0 −1.71368 −0.856839 − 0.515584i \(-0.827575\pi\)
−0.856839 + 0.515584i \(0.827575\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −60504.0 −1.21145
\(23\) −57618.0 −0.987440 −0.493720 − 0.869621i \(-0.664363\pi\)
−0.493720 + 0.869621i \(0.664363\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 42976.0 0.479534
\(27\) 0 0
\(28\) 75136.0 0.646837
\(29\) −47040.0 −0.358158 −0.179079 − 0.983835i \(-0.557312\pi\)
−0.179079 + 0.983835i \(0.557312\pi\)
\(30\) 0 0
\(31\) −192358. −1.15970 −0.579848 − 0.814725i \(-0.696888\pi\)
−0.579848 + 0.814725i \(0.696888\pi\)
\(32\) −32768.0 −0.176777
\(33\) 0 0
\(34\) −192168. −0.838503
\(35\) 0 0
\(36\) 0 0
\(37\) −197066. −0.639596 −0.319798 − 0.947486i \(-0.603615\pi\)
−0.319798 + 0.947486i \(0.603615\pi\)
\(38\) 409880. 1.21175
\(39\) 0 0
\(40\) 0 0
\(41\) 237723. 0.538676 0.269338 − 0.963046i \(-0.413195\pi\)
0.269338 + 0.963046i \(0.413195\pi\)
\(42\) 0 0
\(43\) −653012. −1.25251 −0.626256 − 0.779618i \(-0.715413\pi\)
−0.626256 + 0.779618i \(0.715413\pi\)
\(44\) 484032. 0.856623
\(45\) 0 0
\(46\) 460944. 0.698226
\(47\) −826884. −1.16172 −0.580861 − 0.814003i \(-0.697284\pi\)
−0.580861 + 0.814003i \(0.697284\pi\)
\(48\) 0 0
\(49\) 554733. 0.673593
\(50\) 0 0
\(51\) 0 0
\(52\) −343808. −0.339082
\(53\) 569022. 0.525005 0.262503 − 0.964931i \(-0.415452\pi\)
0.262503 + 0.964931i \(0.415452\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −601088. −0.457383
\(57\) 0 0
\(58\) 376320. 0.253256
\(59\) −1.50108e6 −0.951528 −0.475764 − 0.879573i \(-0.657828\pi\)
−0.475764 + 0.879573i \(0.657828\pi\)
\(60\) 0 0
\(61\) −2.06892e6 −1.16705 −0.583524 − 0.812096i \(-0.698327\pi\)
−0.583524 + 0.812096i \(0.698327\pi\)
\(62\) 1.53886e6 0.820029
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.44435e6 1.39909 0.699544 − 0.714589i \(-0.253386\pi\)
0.699544 + 0.714589i \(0.253386\pi\)
\(68\) 1.53734e6 0.592911
\(69\) 0 0
\(70\) 0 0
\(71\) −4.12105e6 −1.36648 −0.683241 − 0.730193i \(-0.739430\pi\)
−0.683241 + 0.730193i \(0.739430\pi\)
\(72\) 0 0
\(73\) 83653.0 0.0251682 0.0125841 − 0.999921i \(-0.495994\pi\)
0.0125841 + 0.999921i \(0.495994\pi\)
\(74\) 1.57653e6 0.452263
\(75\) 0 0
\(76\) −3.27904e6 −0.856839
\(77\) 8.87896e6 2.21638
\(78\) 0 0
\(79\) 1.45403e6 0.331802 0.165901 − 0.986142i \(-0.446947\pi\)
0.165901 + 0.986142i \(0.446947\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.90178e6 −0.380902
\(83\) 1.62657e6 0.312247 0.156124 − 0.987738i \(-0.450100\pi\)
0.156124 + 0.987738i \(0.450100\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.22410e6 0.885659
\(87\) 0 0
\(88\) −3.87226e6 −0.605724
\(89\) −6.00434e6 −0.902817 −0.451409 − 0.892317i \(-0.649078\pi\)
−0.451409 + 0.892317i \(0.649078\pi\)
\(90\) 0 0
\(91\) −6.30673e6 −0.877322
\(92\) −3.68755e6 −0.493720
\(93\) 0 0
\(94\) 6.61507e6 0.821461
\(95\) 0 0
\(96\) 0 0
\(97\) −3.41175e6 −0.379556 −0.189778 − 0.981827i \(-0.560777\pi\)
−0.189778 + 0.981827i \(0.560777\pi\)
\(98\) −4.43786e6 −0.476302
\(99\) 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 450.8.a.l.1.1 1
3.2 odd 2 50.8.a.f.1.1 yes 1
5.2 odd 4 450.8.c.q.199.1 2
5.3 odd 4 450.8.c.q.199.2 2
5.4 even 2 450.8.a.p.1.1 1
12.11 even 2 400.8.a.q.1.1 1
15.2 even 4 50.8.b.b.49.2 2
15.8 even 4 50.8.b.b.49.1 2
15.14 odd 2 50.8.a.c.1.1 ✓ 1
60.23 odd 4 400.8.c.d.49.2 2
60.47 odd 4 400.8.c.d.49.1 2
60.59 even 2 400.8.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.c.1.1 ✓ 1 15.14 odd 2
50.8.a.f.1.1 yes 1 3.2 odd 2
50.8.b.b.49.1 2 15.8 even 4
50.8.b.b.49.2 2 15.2 even 4
400.8.a.d.1.1 1 60.59 even 2
400.8.a.q.1.1 1 12.11 even 2
400.8.c.d.49.1 2 60.47 odd 4
400.8.c.d.49.2 2 60.23 odd 4
450.8.a.l.1.1 1 1.1 even 1 trivial
450.8.a.p.1.1 1 5.4 even 2
450.8.c.q.199.1 2 5.2 odd 4
450.8.c.q.199.2 2 5.3 odd 4