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Properties

Label 9200.2.a.ca.1.1
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
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 = [2,0,3,0,0,0,3,0,5,0,7] 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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) 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)
 
\(f(q)\) \(=\) \(q-0.302776 q^{3} +3.30278 q^{7} -2.90833 q^{9} +1.69722 q^{11} -3.30278 q^{13} -6.90833 q^{17} -5.90833 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.78890 q^{27} -2.60555 q^{29} +7.90833 q^{31} -0.513878 q^{33} -8.00000 q^{37} +1.00000 q^{39} +0.908327 q^{41} -9.21110 q^{43} -2.60555 q^{47} +3.90833 q^{49} +2.09167 q^{51} +11.2111 q^{53} +1.78890 q^{57} +3.39445 q^{59} +11.5139 q^{61} -9.60555 q^{63} -4.00000 q^{67} +0.302776 q^{69} +16.3028 q^{71} +5.81665 q^{73} +5.60555 q^{77} +14.4222 q^{79} +8.18335 q^{81} +11.2111 q^{83} +0.788897 q^{87} -10.9083 q^{91} -2.39445 q^{93} -6.30278 q^{97} -4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 3 q^{7} + 5 q^{9} + 7 q^{11} - 3 q^{13} - 3 q^{17} - q^{19} - 2 q^{21} - 2 q^{23} + 18 q^{27} + 2 q^{29} + 5 q^{31} + 17 q^{33} - 16 q^{37} + 2 q^{39} - 9 q^{41} - 4 q^{43} + 2 q^{47} - 3 q^{49}+ \cdots + 37 q^{99}+O(q^{100}) \) 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
\(3\) −0.302776 −0.174808 −0.0874038 − 0.996173i \(-0.527857\pi\)
−0.0874038 + 0.996173i \(0.527857\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.30278 1.24833 0.624166 − 0.781292i \(-0.285439\pi\)
0.624166 + 0.781292i \(0.285439\pi\)
\(8\) 0 0
\(9\) −2.90833 −0.969442
\(10\) 0 0
\(11\) 1.69722 0.511732 0.255866 − 0.966712i \(-0.417639\pi\)
0.255866 + 0.966712i \(0.417639\pi\)
\(12\) 0 0
\(13\) −3.30278 −0.916025 −0.458013 − 0.888946i \(-0.651439\pi\)
−0.458013 + 0.888946i \(0.651439\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.90833 −1.67552 −0.837758 − 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) 0 0
\(19\) −5.90833 −1.35546 −0.677732 − 0.735309i \(-0.737037\pi\)
−0.677732 + 0.735309i \(0.737037\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.78890 0.344273
\(28\) 0 0
\(29\) −2.60555 −0.483839 −0.241919 − 0.970296i \(-0.577777\pi\)
−0.241919 + 0.970296i \(0.577777\pi\)
\(30\) 0 0
\(31\) 7.90833 1.42038 0.710189 − 0.704011i \(-0.248610\pi\)
0.710189 + 0.704011i \(0.248610\pi\)
\(32\) 0 0
\(33\) −0.513878 −0.0894547
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0.908327 0.141857 0.0709284 − 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 0 0
\(43\) −9.21110 −1.40468 −0.702340 − 0.711842i \(-0.747861\pi\)
−0.702340 + 0.711842i \(0.747861\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.60555 −0.380059 −0.190029 − 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(48\) 0 0
\(49\) 3.90833 0.558332
\(50\) 0 0
\(51\) 2.09167 0.292893
\(52\) 0 0
\(53\) 11.2111 1.53996 0.769982 − 0.638066i \(-0.220265\pi\)
0.769982 + 0.638066i \(0.220265\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.78890 0.236945
\(58\) 0 0
\(59\) 3.39445 0.441920 0.220960 − 0.975283i \(-0.429081\pi\)
0.220960 + 0.975283i \(0.429081\pi\)
\(60\) 0 0
\(61\) 11.5139 1.47420 0.737101 − 0.675783i \(-0.236194\pi\)
0.737101 + 0.675783i \(0.236194\pi\)
\(62\) 0 0
\(63\) −9.60555 −1.21019
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0.302776 0.0364499
\(70\) 0 0
\(71\) 16.3028 1.93478 0.967392 − 0.253285i \(-0.0815110\pi\)
0.967392 + 0.253285i \(0.0815110\pi\)
\(72\) 0 0
\(73\) 5.81665 0.680788 0.340394 − 0.940283i \(-0.389440\pi\)
0.340394 + 0.940283i \(0.389440\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.60555 0.638812
\(78\) 0 0
\(79\) 14.4222 1.62262 0.811312 − 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) 0 0
\(83\) 11.2111 1.23058 0.615289 − 0.788301i \(-0.289039\pi\)
0.615289 + 0.788301i \(0.289039\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.788897 0.0845787
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −10.9083 −1.14350
\(92\) 0 0
\(93\) −2.39445 −0.248293
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.30278 −0.639950 −0.319975 − 0.947426i \(-0.603675\pi\)
−0.319975 + 0.947426i \(0.603675\pi\)
\(98\) 0 0
\(99\) −4.93608 −0.496095
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.ca.1.1 2
4.3 odd 2 1150.2.a.m.1.2 2
5.4 even 2 1840.2.a.j.1.2 2
20.3 even 4 1150.2.b.f.599.2 4
20.7 even 4 1150.2.b.f.599.3 4
20.19 odd 2 230.2.a.b.1.1 ✓ 2
40.19 odd 2 7360.2.a.bc.1.2 2
40.29 even 2 7360.2.a.bu.1.1 2
60.59 even 2 2070.2.a.w.1.2 2
460.459 even 2 5290.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 ✓ 2 20.19 odd 2
1150.2.a.m.1.2 2 4.3 odd 2
1150.2.b.f.599.2 4 20.3 even 4
1150.2.b.f.599.3 4 20.7 even 4
1840.2.a.j.1.2 2 5.4 even 2
2070.2.a.w.1.2 2 60.59 even 2
5290.2.a.j.1.1 2 460.459 even 2
7360.2.a.bc.1.2 2 40.19 odd 2
7360.2.a.bu.1.1 2 40.29 even 2
9200.2.a.ca.1.1 2 1.1 even 1 trivial