Properties

Label 900.4.a.b.1.1
Level $900$
Weight $4$
Character 900.1
Self dual yes
Analytic conductor $53.102$
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] = [900,4,Mod(1,900)] 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("900.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,-32,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(53.1017190052\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 900.1

$q$-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)
 
\(f(q)\) \(=\) \(q-32.0000 q^{7} -36.0000 q^{11} +10.0000 q^{13} -78.0000 q^{17} +140.000 q^{19} -192.000 q^{23} -6.00000 q^{29} -16.0000 q^{31} +34.0000 q^{37} +390.000 q^{41} +52.0000 q^{43} +408.000 q^{47} +681.000 q^{49} -114.000 q^{53} -516.000 q^{59} -58.0000 q^{61} +892.000 q^{67} +120.000 q^{71} +646.000 q^{73} +1152.00 q^{77} -1168.00 q^{79} -732.000 q^{83} +1590.00 q^{89} -320.000 q^{91} -194.000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −32.0000 −1.72784 −0.863919 − 0.503631i \(-0.831997\pi\)
−0.863919 + 0.503631i \(0.831997\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 − 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) 10.0000 0.213346 0.106673 − 0.994294i \(-0.465980\pi\)
0.106673 + 0.994294i \(0.465980\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −78.0000 −1.11281 −0.556405 − 0.830911i \(-0.687820\pi\)
−0.556405 + 0.830911i \(0.687820\pi\)
\(18\) 0 0
\(19\) 140.000 1.69043 0.845216 − 0.534425i \(-0.179472\pi\)
0.845216 + 0.534425i \(0.179472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −192.000 −1.74064 −0.870321 − 0.492485i \(-0.836089\pi\)
−0.870321 + 0.492485i \(0.836089\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −0.0384197 −0.0192099 − 0.999815i \(-0.506115\pi\)
−0.0192099 + 0.999815i \(0.506115\pi\)
\(30\) 0 0
\(31\) −16.0000 −0.0926995 −0.0463498 − 0.998925i \(-0.514759\pi\)
−0.0463498 + 0.998925i \(0.514759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 34.0000 0.151069 0.0755347 − 0.997143i \(-0.475934\pi\)
0.0755347 + 0.997143i \(0.475934\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 390.000 1.48556 0.742778 − 0.669538i \(-0.233508\pi\)
0.742778 + 0.669538i \(0.233508\pi\)
\(42\) 0 0
\(43\) 52.0000 0.184417 0.0922084 − 0.995740i \(-0.470607\pi\)
0.0922084 + 0.995740i \(0.470607\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 408.000 1.26623 0.633116 − 0.774057i \(-0.281776\pi\)
0.633116 + 0.774057i \(0.281776\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −114.000 −0.295455 −0.147727 − 0.989028i \(-0.547196\pi\)
−0.147727 + 0.989028i \(0.547196\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −516.000 −1.13860 −0.569301 − 0.822129i \(-0.692786\pi\)
−0.569301 + 0.822129i \(0.692786\pi\)
\(60\) 0 0
\(61\) −58.0000 −0.121740 −0.0608700 − 0.998146i \(-0.519388\pi\)
−0.0608700 + 0.998146i \(0.519388\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 892.000 1.62649 0.813247 − 0.581918i \(-0.197698\pi\)
0.813247 + 0.581918i \(0.197698\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 − 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) 646.000 1.03573 0.517867 − 0.855461i \(-0.326726\pi\)
0.517867 + 0.855461i \(0.326726\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1152.00 1.70497
\(78\) 0 0
\(79\) −1168.00 −1.66342 −0.831711 − 0.555209i \(-0.812638\pi\)
−0.831711 + 0.555209i \(0.812638\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −732.000 −0.968041 −0.484021 − 0.875057i \(-0.660824\pi\)
−0.484021 + 0.875057i \(0.660824\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1590.00 1.89370 0.946852 − 0.321669i \(-0.104244\pi\)
0.946852 + 0.321669i \(0.104244\pi\)
\(90\) 0 0
\(91\) −320.000 −0.368628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −194.000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 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 900.4.a.b.1.1 1
3.2 odd 2 300.4.a.e.1.1 1
5.2 odd 4 900.4.d.b.649.1 2
5.3 odd 4 900.4.d.b.649.2 2
5.4 even 2 180.4.a.c.1.1 1
12.11 even 2 1200.4.a.s.1.1 1
15.2 even 4 300.4.d.d.49.1 2
15.8 even 4 300.4.d.d.49.2 2
15.14 odd 2 60.4.a.b.1.1 ✓ 1
20.19 odd 2 720.4.a.c.1.1 1
45.4 even 6 1620.4.i.g.541.1 2
45.14 odd 6 1620.4.i.a.541.1 2
45.29 odd 6 1620.4.i.a.1081.1 2
45.34 even 6 1620.4.i.g.1081.1 2
60.23 odd 4 1200.4.f.e.49.1 2
60.47 odd 4 1200.4.f.e.49.2 2
60.59 even 2 240.4.a.j.1.1 1
120.29 odd 2 960.4.a.bb.1.1 1
120.59 even 2 960.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.b.1.1 ✓ 1 15.14 odd 2
180.4.a.c.1.1 1 5.4 even 2
240.4.a.j.1.1 1 60.59 even 2
300.4.a.e.1.1 1 3.2 odd 2
300.4.d.d.49.1 2 15.2 even 4
300.4.d.d.49.2 2 15.8 even 4
720.4.a.c.1.1 1 20.19 odd 2
900.4.a.b.1.1 1 1.1 even 1 trivial
900.4.d.b.649.1 2 5.2 odd 4
900.4.d.b.649.2 2 5.3 odd 4
960.4.a.a.1.1 1 120.59 even 2
960.4.a.bb.1.1 1 120.29 odd 2
1200.4.a.s.1.1 1 12.11 even 2
1200.4.f.e.49.1 2 60.23 odd 4
1200.4.f.e.49.2 2 60.47 odd 4
1620.4.i.a.541.1 2 45.14 odd 6
1620.4.i.a.1081.1 2 45.29 odd 6
1620.4.i.g.541.1 2 45.4 even 6
1620.4.i.g.1081.1 2 45.34 even 6