Properties

Label 450.8.a.z.1.1
Level $450$
Weight $8$
Character 450.1
Self dual yes
Analytic conductor $140.573$
Analytic rank $0$
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 = [1,8,0,64,0,0,1366] 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: \(0\)
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)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +64.0000 q^{4} +1366.00 q^{7} +512.000 q^{8} +1083.00 q^{11} -5468.00 q^{13} +10928.0 q^{14} +4096.00 q^{16} +25269.0 q^{17} +33485.0 q^{19} +8664.00 q^{22} +5838.00 q^{23} -43744.0 q^{26} +87424.0 q^{28} -125280. q^{29} -73798.0 q^{31} +32768.0 q^{32} +202152. q^{34} +395926. q^{37} +267880. q^{38} +22683.0 q^{41} -100148. q^{43} +69312.0 q^{44} +46704.0 q^{46} +1.14524e6 q^{47} +1.04241e6 q^{49} -349952. q^{52} -354882. q^{53} +699392. q^{56} -1.00224e6 q^{58} -1.09836e6 q^{59} -422998. q^{61} -590384. q^{62} +262144. q^{64} -2.55858e6 q^{67} +1.61722e6 q^{68} +2.28743e6 q^{71} -6.37244e6 q^{73} +3.16741e6 q^{74} +2.14304e6 q^{76} +1.47938e6 q^{77} -2.01925e6 q^{79} +181464. q^{82} +7.97298e6 q^{83} -801184. q^{86} +554496. q^{88} -2.18594e6 q^{89} -7.46929e6 q^{91} +373632. q^{92} +9.16195e6 q^{94} +5.82365e6 q^{97} +8.33930e6 q^{98} +O(q^{100})\)

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
\(7\) 1366.00 1.50525 0.752623 − 0.658452i \(-0.228788\pi\)
0.752623 + 0.658452i \(0.228788\pi\)
\(8\) 512.000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1083.00 0.245332 0.122666 − 0.992448i \(-0.460856\pi\)
0.122666 + 0.992448i \(0.460856\pi\)
\(12\) 0 0
\(13\) −5468.00 −0.690282 −0.345141 − 0.938551i \(-0.612169\pi\)
−0.345141 + 0.938551i \(0.612169\pi\)
\(14\) 10928.0 1.06437
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 25269.0 1.24743 0.623716 − 0.781651i \(-0.285622\pi\)
0.623716 + 0.781651i \(0.285622\pi\)
\(18\) 0 0
\(19\) 33485.0 1.11999 0.559993 − 0.828497i \(-0.310804\pi\)
0.559993 + 0.828497i \(0.310804\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8664.00 0.173476
\(23\) 5838.00 0.100050 0.0500250 − 0.998748i \(-0.484070\pi\)
0.0500250 + 0.998748i \(0.484070\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −43744.0 −0.488103
\(27\) 0 0
\(28\) 87424.0 0.752623
\(29\) −125280. −0.953869 −0.476935 − 0.878939i \(-0.658252\pi\)
−0.476935 + 0.878939i \(0.658252\pi\)
\(30\) 0 0
\(31\) −73798.0 −0.444917 −0.222458 − 0.974942i \(-0.571408\pi\)
−0.222458 + 0.974942i \(0.571408\pi\)
\(32\) 32768.0 0.176777
\(33\) 0 0
\(34\) 202152. 0.882068
\(35\) 0 0
\(36\) 0 0
\(37\) 395926. 1.28501 0.642507 − 0.766280i \(-0.277894\pi\)
0.642507 + 0.766280i \(0.277894\pi\)
\(38\) 267880. 0.791950
\(39\) 0 0
\(40\) 0 0
\(41\) 22683.0 0.0513993 0.0256996 − 0.999670i \(-0.491819\pi\)
0.0256996 + 0.999670i \(0.491819\pi\)
\(42\) 0 0
\(43\) −100148. −0.192089 −0.0960445 − 0.995377i \(-0.530619\pi\)
−0.0960445 + 0.995377i \(0.530619\pi\)
\(44\) 69312.0 0.122666
\(45\) 0 0
\(46\) 46704.0 0.0707460
\(47\) 1.14524e6 1.60900 0.804499 − 0.593954i \(-0.202434\pi\)
0.804499 + 0.593954i \(0.202434\pi\)
\(48\) 0 0
\(49\) 1.04241e6 1.26577
\(50\) 0 0
\(51\) 0 0
\(52\) −349952. −0.345141
\(53\) −354882. −0.327430 −0.163715 − 0.986508i \(-0.552348\pi\)
−0.163715 + 0.986508i \(0.552348\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 699392. 0.532185
\(57\) 0 0
\(58\) −1.00224e6 −0.674487
\(59\) −1.09836e6 −0.696246 −0.348123 − 0.937449i \(-0.613181\pi\)
−0.348123 + 0.937449i \(0.613181\pi\)
\(60\) 0 0
\(61\) −422998. −0.238607 −0.119304 − 0.992858i \(-0.538066\pi\)
−0.119304 + 0.992858i \(0.538066\pi\)
\(62\) −590384. −0.314604
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.55858e6 −1.03929 −0.519645 − 0.854382i \(-0.673936\pi\)
−0.519645 + 0.854382i \(0.673936\pi\)
\(68\) 1.61722e6 0.623716
\(69\) 0 0
\(70\) 0 0
\(71\) 2.28743e6 0.758478 0.379239 − 0.925299i \(-0.376186\pi\)
0.379239 + 0.925299i \(0.376186\pi\)
\(72\) 0 0
\(73\) −6.37244e6 −1.91724 −0.958619 − 0.284693i \(-0.908108\pi\)
−0.958619 + 0.284693i \(0.908108\pi\)
\(74\) 3.16741e6 0.908642
\(75\) 0 0
\(76\) 2.14304e6 0.559993
\(77\) 1.47938e6 0.369285
\(78\) 0 0
\(79\) −2.01925e6 −0.460782 −0.230391 − 0.973098i \(-0.574000\pi\)
−0.230391 + 0.973098i \(0.574000\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 181464. 0.0363448
\(83\) 7.97298e6 1.53055 0.765275 − 0.643703i \(-0.222603\pi\)
0.765275 + 0.643703i \(0.222603\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −801184. −0.135827
\(87\) 0 0
\(88\) 554496. 0.0867379
\(89\) −2.18594e6 −0.328679 −0.164340 − 0.986404i \(-0.552549\pi\)
−0.164340 + 0.986404i \(0.552549\pi\)
\(90\) 0 0
\(91\) −7.46929e6 −1.03904
\(92\) 373632. 0.0500250
\(93\) 0 0
\(94\) 9.16195e6 1.13773
\(95\) 0 0
\(96\) 0 0
\(97\) 5.82365e6 0.647879 0.323939 − 0.946078i \(-0.394993\pi\)
0.323939 + 0.946078i \(0.394993\pi\)
\(98\) 8.33930e6 0.895032
\(99\) 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.z.1.1 1
3.2 odd 2 50.8.a.d.1.1 ✓ 1
5.2 odd 4 450.8.c.l.199.2 2
5.3 odd 4 450.8.c.l.199.1 2
5.4 even 2 450.8.a.a.1.1 1
12.11 even 2 400.8.a.a.1.1 1
15.2 even 4 50.8.b.a.49.1 2
15.8 even 4 50.8.b.a.49.2 2
15.14 odd 2 50.8.a.e.1.1 yes 1
60.23 odd 4 400.8.c.a.49.1 2
60.47 odd 4 400.8.c.a.49.2 2
60.59 even 2 400.8.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.d.1.1 ✓ 1 3.2 odd 2
50.8.a.e.1.1 yes 1 15.14 odd 2
50.8.b.a.49.1 2 15.2 even 4
50.8.b.a.49.2 2 15.8 even 4
400.8.a.a.1.1 1 12.11 even 2
400.8.a.s.1.1 1 60.59 even 2
400.8.c.a.49.1 2 60.23 odd 4
400.8.c.a.49.2 2 60.47 odd 4
450.8.a.a.1.1 1 5.4 even 2
450.8.a.z.1.1 1 1.1 even 1 trivial
450.8.c.l.199.1 2 5.3 odd 4
450.8.c.l.199.2 2 5.2 odd 4