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Properties

Label 9200.2.a.ch.1.3
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 = [3,0,2,0,0,0,4,0,-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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 4600)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) 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)
 
\(f(q)\) \(=\) \(q+2.24698 q^{3} +4.49396 q^{7} +2.04892 q^{9} -3.38404 q^{11} -3.04892 q^{13} -4.49396 q^{17} -7.20775 q^{19} +10.0978 q^{21} -1.00000 q^{23} -2.13706 q^{27} -5.51573 q^{29} +1.29590 q^{31} -7.60388 q^{33} +5.82371 q^{37} -6.85086 q^{39} +3.63102 q^{41} -10.7138 q^{43} +2.06100 q^{47} +13.1957 q^{49} -10.0978 q^{51} +2.98792 q^{53} -16.1957 q^{57} +9.31767 q^{59} -13.3056 q^{61} +9.20775 q^{63} -8.19567 q^{67} -2.24698 q^{69} -3.48858 q^{71} +8.72348 q^{73} -15.2078 q^{77} +9.92154 q^{79} -10.9487 q^{81} -15.7560 q^{83} -12.3937 q^{87} -0.121998 q^{89} -13.7017 q^{91} +2.91185 q^{93} -15.6039 q^{97} -6.93362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 4 q^{7} - 3 q^{9} - 4 q^{17} - 4 q^{19} + 12 q^{21} - 3 q^{23} - q^{27} - 4 q^{29} - 10 q^{31} - 14 q^{33} + 10 q^{37} - 7 q^{39} - 4 q^{41} - 24 q^{43} + 16 q^{47} + 3 q^{49} - 12 q^{51}+ \cdots - 14 q^{99}+O(q^{100}) \) 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
\(3\) 2.24698 1.29729 0.648647 − 0.761089i \(-0.275335\pi\)
0.648647 + 0.761089i \(0.275335\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.49396 1.69856 0.849278 − 0.527945i \(-0.177037\pi\)
0.849278 + 0.527945i \(0.177037\pi\)
\(8\) 0 0
\(9\) 2.04892 0.682972
\(10\) 0 0
\(11\) −3.38404 −1.02033 −0.510164 − 0.860077i \(-0.670415\pi\)
−0.510164 + 0.860077i \(0.670415\pi\)
\(12\) 0 0
\(13\) −3.04892 −0.845618 −0.422809 − 0.906219i \(-0.638956\pi\)
−0.422809 + 0.906219i \(0.638956\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.49396 −1.08995 −0.544973 − 0.838454i \(-0.683460\pi\)
−0.544973 + 0.838454i \(0.683460\pi\)
\(18\) 0 0
\(19\) −7.20775 −1.65357 −0.826786 − 0.562517i \(-0.809833\pi\)
−0.826786 + 0.562517i \(0.809833\pi\)
\(20\) 0 0
\(21\) 10.0978 2.20353
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.13706 −0.411278
\(28\) 0 0
\(29\) −5.51573 −1.02425 −0.512123 − 0.858912i \(-0.671141\pi\)
−0.512123 + 0.858912i \(0.671141\pi\)
\(30\) 0 0
\(31\) 1.29590 0.232750 0.116375 − 0.993205i \(-0.462873\pi\)
0.116375 + 0.993205i \(0.462873\pi\)
\(32\) 0 0
\(33\) −7.60388 −1.32366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.82371 0.957412 0.478706 − 0.877975i \(-0.341106\pi\)
0.478706 + 0.877975i \(0.341106\pi\)
\(38\) 0 0
\(39\) −6.85086 −1.09701
\(40\) 0 0
\(41\) 3.63102 0.567070 0.283535 − 0.958962i \(-0.408493\pi\)
0.283535 + 0.958962i \(0.408493\pi\)
\(42\) 0 0
\(43\) −10.7138 −1.63384 −0.816919 − 0.576752i \(-0.804320\pi\)
−0.816919 + 0.576752i \(0.804320\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.06100 0.300628 0.150314 − 0.988638i \(-0.451972\pi\)
0.150314 + 0.988638i \(0.451972\pi\)
\(48\) 0 0
\(49\) 13.1957 1.88510
\(50\) 0 0
\(51\) −10.0978 −1.41398
\(52\) 0 0
\(53\) 2.98792 0.410422 0.205211 − 0.978718i \(-0.434212\pi\)
0.205211 + 0.978718i \(0.434212\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −16.1957 −2.14517
\(58\) 0 0
\(59\) 9.31767 1.21306 0.606528 − 0.795062i \(-0.292562\pi\)
0.606528 + 0.795062i \(0.292562\pi\)
\(60\) 0 0
\(61\) −13.3056 −1.70361 −0.851803 − 0.523863i \(-0.824491\pi\)
−0.851803 + 0.523863i \(0.824491\pi\)
\(62\) 0 0
\(63\) 9.20775 1.16007
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.19567 −1.00126 −0.500630 − 0.865661i \(-0.666898\pi\)
−0.500630 + 0.865661i \(0.666898\pi\)
\(68\) 0 0
\(69\) −2.24698 −0.270505
\(70\) 0 0
\(71\) −3.48858 −0.414019 −0.207009 − 0.978339i \(-0.566373\pi\)
−0.207009 + 0.978339i \(0.566373\pi\)
\(72\) 0 0
\(73\) 8.72348 1.02101 0.510503 − 0.859876i \(-0.329459\pi\)
0.510503 + 0.859876i \(0.329459\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.2078 −1.73308
\(78\) 0 0
\(79\) 9.92154 1.11626 0.558130 − 0.829753i \(-0.311519\pi\)
0.558130 + 0.829753i \(0.311519\pi\)
\(80\) 0 0
\(81\) −10.9487 −1.21652
\(82\) 0 0
\(83\) −15.7560 −1.72945 −0.864723 − 0.502249i \(-0.832506\pi\)
−0.864723 + 0.502249i \(0.832506\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −12.3937 −1.32875
\(88\) 0 0
\(89\) −0.121998 −0.0129317 −0.00646587 − 0.999979i \(-0.502058\pi\)
−0.00646587 + 0.999979i \(0.502058\pi\)
\(90\) 0 0
\(91\) −13.7017 −1.43633
\(92\) 0 0
\(93\) 2.91185 0.301945
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.6039 −1.58433 −0.792167 − 0.610305i \(-0.791047\pi\)
−0.792167 + 0.610305i \(0.791047\pi\)
\(98\) 0 0
\(99\) −6.93362 −0.696855
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.ch.1.3 3
4.3 odd 2 4600.2.a.w.1.1 ✓ 3
5.4 even 2 9200.2.a.cb.1.1 3
20.3 even 4 4600.2.e.s.4049.1 6
20.7 even 4 4600.2.e.s.4049.6 6
20.19 odd 2 4600.2.a.z.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.w.1.1 ✓ 3 4.3 odd 2
4600.2.a.z.1.3 yes 3 20.19 odd 2
4600.2.e.s.4049.1 6 20.3 even 4
4600.2.e.s.4049.6 6 20.7 even 4
9200.2.a.cb.1.1 3 5.4 even 2
9200.2.a.ch.1.3 3 1.1 even 1 trivial