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Properties

Label 9200.2.a.bz.1.1
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $2$
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 = [2,0,2,0,0,0,-4,0,6,0,-2] 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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 1.1
Root \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q-1.23607 q^{3} +0.236068 q^{7} -1.47214 q^{9} -1.00000 q^{11} -2.23607 q^{13} +2.47214 q^{17} -1.00000 q^{19} -0.291796 q^{21} -1.00000 q^{23} +5.52786 q^{27} -6.23607 q^{29} +8.47214 q^{31} +1.23607 q^{33} -6.76393 q^{37} +2.76393 q^{39} +11.9443 q^{41} +11.4721 q^{43} -1.70820 q^{47} -6.94427 q^{49} -3.05573 q^{51} +3.23607 q^{53} +1.23607 q^{57} +1.23607 q^{59} -2.76393 q^{61} -0.347524 q^{63} -4.94427 q^{67} +1.23607 q^{69} -10.0000 q^{71} +0.527864 q^{73} -0.236068 q^{77} +7.18034 q^{79} -2.41641 q^{81} +9.00000 q^{83} +7.70820 q^{87} +2.00000 q^{89} -0.527864 q^{91} -10.4721 q^{93} +16.1803 q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{7} + 6 q^{9} - 2 q^{11} - 4 q^{17} - 2 q^{19} - 14 q^{21} - 2 q^{23} + 20 q^{27} - 8 q^{29} + 8 q^{31} - 2 q^{33} - 18 q^{37} + 10 q^{39} + 6 q^{41} + 14 q^{43} + 10 q^{47} + 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.23607 −0.713644 −0.356822 − 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.236068 0.0892253 0.0446127 − 0.999004i \(-0.485795\pi\)
0.0446127 + 0.999004i \(0.485795\pi\)
\(8\) 0 0
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −2.23607 −0.620174 −0.310087 − 0.950708i \(-0.600358\pi\)
−0.310087 + 0.950708i \(0.600358\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.47214 0.599581 0.299791 − 0.954005i \(-0.403083\pi\)
0.299791 + 0.954005i \(0.403083\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −0.291796 −0.0636751
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.52786 1.06384
\(28\) 0 0
\(29\) −6.23607 −1.15801 −0.579004 − 0.815324i \(-0.696559\pi\)
−0.579004 + 0.815324i \(0.696559\pi\)
\(30\) 0 0
\(31\) 8.47214 1.52164 0.760820 − 0.648963i \(-0.224797\pi\)
0.760820 + 0.648963i \(0.224797\pi\)
\(32\) 0 0
\(33\) 1.23607 0.215172
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.76393 −1.11198 −0.555992 − 0.831188i \(-0.687661\pi\)
−0.555992 + 0.831188i \(0.687661\pi\)
\(38\) 0 0
\(39\) 2.76393 0.442583
\(40\) 0 0
\(41\) 11.9443 1.86538 0.932691 − 0.360677i \(-0.117454\pi\)
0.932691 + 0.360677i \(0.117454\pi\)
\(42\) 0 0
\(43\) 11.4721 1.74948 0.874742 − 0.484589i \(-0.161031\pi\)
0.874742 + 0.484589i \(0.161031\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.70820 −0.249167 −0.124584 − 0.992209i \(-0.539759\pi\)
−0.124584 + 0.992209i \(0.539759\pi\)
\(48\) 0 0
\(49\) −6.94427 −0.992039
\(50\) 0 0
\(51\) −3.05573 −0.427888
\(52\) 0 0
\(53\) 3.23607 0.444508 0.222254 − 0.974989i \(-0.428659\pi\)
0.222254 + 0.974989i \(0.428659\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.23607 0.163721
\(58\) 0 0
\(59\) 1.23607 0.160922 0.0804612 − 0.996758i \(-0.474361\pi\)
0.0804612 + 0.996758i \(0.474361\pi\)
\(60\) 0 0
\(61\) −2.76393 −0.353885 −0.176943 − 0.984221i \(-0.556621\pi\)
−0.176943 + 0.984221i \(0.556621\pi\)
\(62\) 0 0
\(63\) −0.347524 −0.0437839
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.94427 −0.604039 −0.302019 − 0.953302i \(-0.597661\pi\)
−0.302019 + 0.953302i \(0.597661\pi\)
\(68\) 0 0
\(69\) 1.23607 0.148805
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 − 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 0.527864 0.0617818 0.0308909 − 0.999523i \(-0.490166\pi\)
0.0308909 + 0.999523i \(0.490166\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.236068 −0.0269024
\(78\) 0 0
\(79\) 7.18034 0.807851 0.403926 − 0.914792i \(-0.367645\pi\)
0.403926 + 0.914792i \(0.367645\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.70820 0.826406
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 − 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −0.527864 −0.0553352
\(92\) 0 0
\(93\) −10.4721 −1.08591
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.1803 1.64286 0.821432 − 0.570306i \(-0.193175\pi\)
0.821432 + 0.570306i \(0.193175\pi\)
\(98\) 0 0
\(99\) 1.47214 0.147955
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.bz.1.1 2
4.3 odd 2 4600.2.a.q.1.2 ✓ 2
5.4 even 2 9200.2.a.bn.1.2 2
20.3 even 4 4600.2.e.l.4049.3 4
20.7 even 4 4600.2.e.l.4049.2 4
20.19 odd 2 4600.2.a.u.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.q.1.2 ✓ 2 4.3 odd 2
4600.2.a.u.1.1 yes 2 20.19 odd 2
4600.2.e.l.4049.2 4 20.7 even 4
4600.2.e.l.4049.3 4 20.3 even 4
9200.2.a.bn.1.2 2 5.4 even 2
9200.2.a.bz.1.1 2 1.1 even 1 trivial