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Properties

Label 9200.2.a.ct.1.5
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
Dimension $5$
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] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,0,0,0,0,0,-4,0,3,0,0] 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: \(73.4623698596\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.521397.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} - 3x^{2} + 18x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.29544\) of defining polynomial
Character \(\chi\) \(=\) 9200.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.29544 q^{3} +1.61758 q^{7} +2.26905 q^{9} +2.79255 q^{11} +4.88663 q^{13} +3.54388 q^{17} +2.79255 q^{19} +3.71306 q^{21} +1.00000 q^{23} -1.67786 q^{27} -3.36282 q^{29} +4.84564 q^{31} +6.41013 q^{33} +3.87204 q^{37} +11.2170 q^{39} -0.327923 q^{41} +5.38343 q^{43} -0.0121087 q^{47} -4.38343 q^{49} +8.13476 q^{51} -0.866254 q^{53} +6.41013 q^{57} -2.96752 q^{59} -7.29816 q^{61} +3.67037 q^{63} -6.94082 q^{67} +2.29544 q^{69} +2.16864 q^{71} -6.02888 q^{73} +4.51718 q^{77} -6.50561 q^{79} -10.6586 q^{81} +7.88030 q^{83} -7.71915 q^{87} -9.71986 q^{89} +7.90452 q^{91} +11.1229 q^{93} +11.9952 q^{97} +6.33643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.29544 1.32527 0.662637 − 0.748941i \(-0.269437\pi\)
0.662637 + 0.748941i \(0.269437\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.61758 0.611388 0.305694 − 0.952130i \(-0.401111\pi\)
0.305694 + 0.952130i \(0.401111\pi\)
\(8\) 0 0
\(9\) 2.26905 0.756349
\(10\) 0 0
\(11\) 2.79255 0.841985 0.420993 − 0.907064i \(-0.361682\pi\)
0.420993 + 0.907064i \(0.361682\pi\)
\(12\) 0 0
\(13\) 4.88663 1.35531 0.677653 − 0.735381i \(-0.262997\pi\)
0.677653 + 0.735381i \(0.262997\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.54388 0.859516 0.429758 − 0.902944i \(-0.358599\pi\)
0.429758 + 0.902944i \(0.358599\pi\)
\(18\) 0 0
\(19\) 2.79255 0.640655 0.320327 − 0.947307i \(-0.396207\pi\)
0.320327 + 0.947307i \(0.396207\pi\)
\(20\) 0 0
\(21\) 3.71306 0.810257
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.67786 −0.322904
\(28\) 0 0
\(29\) −3.36282 −0.624460 −0.312230 − 0.950007i \(-0.601076\pi\)
−0.312230 + 0.950007i \(0.601076\pi\)
\(30\) 0 0
\(31\) 4.84564 0.870303 0.435152 − 0.900357i \(-0.356695\pi\)
0.435152 + 0.900357i \(0.356695\pi\)
\(32\) 0 0
\(33\) 6.41013 1.11586
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.87204 0.636559 0.318279 − 0.947997i \(-0.396895\pi\)
0.318279 + 0.947997i \(0.396895\pi\)
\(38\) 0 0
\(39\) 11.2170 1.79615
\(40\) 0 0
\(41\) −0.327923 −0.0512130 −0.0256065 − 0.999672i \(-0.508152\pi\)
−0.0256065 + 0.999672i \(0.508152\pi\)
\(42\) 0 0
\(43\) 5.38343 0.820965 0.410483 − 0.911868i \(-0.365360\pi\)
0.410483 + 0.911868i \(0.365360\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.0121087 −0.00176623 −0.000883117 − 1.00000i \(-0.500281\pi\)
−0.000883117 1.00000i \(0.500281\pi\)
\(48\) 0 0
\(49\) −4.38343 −0.626204
\(50\) 0 0
\(51\) 8.13476 1.13909
\(52\) 0 0
\(53\) −0.866254 −0.118989 −0.0594946 − 0.998229i \(-0.518949\pi\)
−0.0594946 + 0.998229i \(0.518949\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.41013 0.849043
\(58\) 0 0
\(59\) −2.96752 −0.386338 −0.193169 − 0.981166i \(-0.561877\pi\)
−0.193169 + 0.981166i \(0.561877\pi\)
\(60\) 0 0
\(61\) −7.29816 −0.934434 −0.467217 − 0.884143i \(-0.654743\pi\)
−0.467217 + 0.884143i \(0.654743\pi\)
\(62\) 0 0
\(63\) 3.67037 0.462423
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.94082 −0.847956 −0.423978 − 0.905673i \(-0.639367\pi\)
−0.423978 + 0.905673i \(0.639367\pi\)
\(68\) 0 0
\(69\) 2.29544 0.276339
\(70\) 0 0
\(71\) 2.16864 0.257370 0.128685 − 0.991685i \(-0.458924\pi\)
0.128685 + 0.991685i \(0.458924\pi\)
\(72\) 0 0
\(73\) −6.02888 −0.705627 −0.352813 − 0.935694i \(-0.614775\pi\)
−0.352813 + 0.935694i \(0.614775\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.51718 0.514780
\(78\) 0 0
\(79\) −6.50561 −0.731938 −0.365969 − 0.930627i \(-0.619262\pi\)
−0.365969 + 0.930627i \(0.619262\pi\)
\(80\) 0 0
\(81\) −10.6586 −1.18429
\(82\) 0 0
\(83\) 7.88030 0.864976 0.432488 − 0.901640i \(-0.357636\pi\)
0.432488 + 0.901640i \(0.357636\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.71915 −0.827580
\(88\) 0 0
\(89\) −9.71986 −1.03030 −0.515151 − 0.857099i \(-0.672264\pi\)
−0.515151 + 0.857099i \(0.672264\pi\)
\(90\) 0 0
\(91\) 7.90452 0.828619
\(92\) 0 0
\(93\) 11.1229 1.15339
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 11.9952 1.21793 0.608966 − 0.793197i \(-0.291585\pi\)
0.608966 + 0.793197i \(0.291585\pi\)
\(98\) 0 0
\(99\) 6.33643 0.636835
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 9200.2.a.ct.1.5 5
4.3 odd 2 4600.2.a.bf.1.1 yes 5
5.4 even 2 9200.2.a.cv.1.1 5
20.3 even 4 4600.2.e.w.4049.2 10
20.7 even 4 4600.2.e.w.4049.9 10
20.19 odd 2 4600.2.a.bd.1.5 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.5 ✓ 5 20.19 odd 2
4600.2.a.bf.1.1 yes 5 4.3 odd 2
4600.2.e.w.4049.2 10 20.3 even 4
4600.2.e.w.4049.9 10 20.7 even 4
9200.2.a.ct.1.5 5 1.1 even 1 trivial
9200.2.a.cv.1.1 5 5.4 even 2