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Properties

Label 9200.2.a.ci.1.2
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $3$
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 = [3,0,2,0,0,0,7,0,3,0,-3] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) 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 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) 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)
 
\(f(q)\) \(=\) \(q+1.46260 q^{3} +1.53740 q^{7} -0.860806 q^{9} +0.860806 q^{11} -0.139194 q^{13} -5.50761 q^{17} -5.25901 q^{19} +2.24860 q^{21} -1.00000 q^{23} -5.64681 q^{27} +9.76663 q^{29} -6.78600 q^{31} +1.25901 q^{33} +12.0900 q^{37} -0.203585 q^{39} -9.98062 q^{41} +11.4432 q^{43} -2.32340 q^{47} -4.63640 q^{49} -8.05543 q^{51} -0.149606 q^{53} -7.69182 q^{57} +11.0152 q^{59} +4.43281 q^{61} -1.32340 q^{63} -10.4972 q^{67} -1.46260 q^{69} -7.31299 q^{71} -7.11982 q^{73} +1.32340 q^{77} -6.79641 q^{79} -5.67660 q^{81} +10.6468 q^{83} +14.2847 q^{87} -17.2936 q^{89} -0.213997 q^{91} -9.92520 q^{93} -1.93561 q^{97} -0.740987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 7 q^{7} + 3 q^{9} - 3 q^{11} - 6 q^{13} + 5 q^{17} - 7 q^{19} - 6 q^{21} - 3 q^{23} - q^{27} - q^{29} - 10 q^{31} - 5 q^{33} - 2 q^{37} - 7 q^{39} - 10 q^{41} + 12 q^{43} + q^{47} + 6 q^{49}+ \cdots - 11 q^{99}+O(q^{100}) \) 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
\(3\) 1.46260 0.844432 0.422216 − 0.906495i \(-0.361252\pi\)
0.422216 + 0.906495i \(0.361252\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.53740 0.581083 0.290542 − 0.956862i \(-0.406165\pi\)
0.290542 + 0.956862i \(0.406165\pi\)
\(8\) 0 0
\(9\) −0.860806 −0.286935
\(10\) 0 0
\(11\) 0.860806 0.259543 0.129771 − 0.991544i \(-0.458576\pi\)
0.129771 + 0.991544i \(0.458576\pi\)
\(12\) 0 0
\(13\) −0.139194 −0.0386055 −0.0193028 − 0.999814i \(-0.506145\pi\)
−0.0193028 + 0.999814i \(0.506145\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.50761 −1.33579 −0.667896 − 0.744254i \(-0.732805\pi\)
−0.667896 + 0.744254i \(0.732805\pi\)
\(18\) 0 0
\(19\) −5.25901 −1.20650 −0.603250 − 0.797552i \(-0.706128\pi\)
−0.603250 + 0.797552i \(0.706128\pi\)
\(20\) 0 0
\(21\) 2.24860 0.490685
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.64681 −1.08673
\(28\) 0 0
\(29\) 9.76663 1.81362 0.906809 − 0.421543i \(-0.138511\pi\)
0.906809 + 0.421543i \(0.138511\pi\)
\(30\) 0 0
\(31\) −6.78600 −1.21880 −0.609401 − 0.792862i \(-0.708590\pi\)
−0.609401 + 0.792862i \(0.708590\pi\)
\(32\) 0 0
\(33\) 1.25901 0.219166
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12.0900 1.98759 0.993795 − 0.111232i \(-0.0354796\pi\)
0.993795 + 0.111232i \(0.0354796\pi\)
\(38\) 0 0
\(39\) −0.203585 −0.0325997
\(40\) 0 0
\(41\) −9.98062 −1.55871 −0.779356 − 0.626582i \(-0.784454\pi\)
−0.779356 + 0.626582i \(0.784454\pi\)
\(42\) 0 0
\(43\) 11.4432 1.74508 0.872538 − 0.488547i \(-0.162473\pi\)
0.872538 + 0.488547i \(0.162473\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.32340 −0.338903 −0.169452 − 0.985538i \(-0.554200\pi\)
−0.169452 + 0.985538i \(0.554200\pi\)
\(48\) 0 0
\(49\) −4.63640 −0.662342
\(50\) 0 0
\(51\) −8.05543 −1.12799
\(52\) 0 0
\(53\) −0.149606 −0.0205500 −0.0102750 − 0.999947i \(-0.503271\pi\)
−0.0102750 + 0.999947i \(0.503271\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.69182 −1.01881
\(58\) 0 0
\(59\) 11.0152 1.43406 0.717030 − 0.697042i \(-0.245501\pi\)
0.717030 + 0.697042i \(0.245501\pi\)
\(60\) 0 0
\(61\) 4.43281 0.567563 0.283782 − 0.958889i \(-0.408411\pi\)
0.283782 + 0.958889i \(0.408411\pi\)
\(62\) 0 0
\(63\) −1.32340 −0.166733
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.4972 −1.28244 −0.641219 − 0.767358i \(-0.721571\pi\)
−0.641219 + 0.767358i \(0.721571\pi\)
\(68\) 0 0
\(69\) −1.46260 −0.176076
\(70\) 0 0
\(71\) −7.31299 −0.867892 −0.433946 − 0.900939i \(-0.642879\pi\)
−0.433946 + 0.900939i \(0.642879\pi\)
\(72\) 0 0
\(73\) −7.11982 −0.833312 −0.416656 − 0.909064i \(-0.636798\pi\)
−0.416656 + 0.909064i \(0.636798\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.32340 0.150816
\(78\) 0 0
\(79\) −6.79641 −0.764656 −0.382328 − 0.924027i \(-0.624878\pi\)
−0.382328 + 0.924027i \(0.624878\pi\)
\(80\) 0 0
\(81\) −5.67660 −0.630733
\(82\) 0 0
\(83\) 10.6468 1.16864 0.584320 − 0.811524i \(-0.301361\pi\)
0.584320 + 0.811524i \(0.301361\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 14.2847 1.53148
\(88\) 0 0
\(89\) −17.2936 −1.83312 −0.916560 − 0.399897i \(-0.869046\pi\)
−0.916560 + 0.399897i \(0.869046\pi\)
\(90\) 0 0
\(91\) −0.213997 −0.0224330
\(92\) 0 0
\(93\) −9.92520 −1.02919
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.93561 −0.196531 −0.0982657 − 0.995160i \(-0.531329\pi\)
−0.0982657 + 0.995160i \(0.531329\pi\)
\(98\) 0 0
\(99\) −0.740987 −0.0744720
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.ci.1.2 3
4.3 odd 2 4600.2.a.v.1.2 3
5.4 even 2 1840.2.a.q.1.2 3
20.3 even 4 4600.2.e.q.4049.3 6
20.7 even 4 4600.2.e.q.4049.4 6
20.19 odd 2 920.2.a.i.1.2 ✓ 3
40.19 odd 2 7360.2.a.bw.1.2 3
40.29 even 2 7360.2.a.cf.1.2 3
60.59 even 2 8280.2.a.bl.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.2 ✓ 3 20.19 odd 2
1840.2.a.q.1.2 3 5.4 even 2
4600.2.a.v.1.2 3 4.3 odd 2
4600.2.e.q.4049.3 6 20.3 even 4
4600.2.e.q.4049.4 6 20.7 even 4
7360.2.a.bw.1.2 3 40.19 odd 2
7360.2.a.cf.1.2 3 40.29 even 2
8280.2.a.bl.1.2 3 60.59 even 2
9200.2.a.ci.1.2 3 1.1 even 1 trivial