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Properties

Label 9200.2.a.cs.1.5
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
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,-3,0,0,0,1,0,4,0,4] 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: \(5\)
Coefficient field: 5.5.791953.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 7x^{3} + 7x^{2} + 9x + 2 \) 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 1.5
Root \(3.21042\) 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)
 
\(f(q)\) \(=\) \(q+2.21042 q^{3} -2.22487 q^{7} +1.88594 q^{9} +1.57300 q^{11} -3.96189 q^{13} -0.294907 q^{17} +7.76572 q^{19} -4.91788 q^{21} -1.00000 q^{23} -2.46253 q^{27} -9.29233 q^{29} -9.18913 q^{31} +3.47698 q^{33} +10.5425 q^{37} -8.75744 q^{39} -2.34251 q^{41} +6.67460 q^{43} -1.38007 q^{47} -2.04998 q^{49} -0.651868 q^{51} +11.0395 q^{53} +17.1655 q^{57} +5.09378 q^{59} -8.91788 q^{61} -4.19597 q^{63} -1.12002 q^{67} -2.21042 q^{69} -7.60168 q^{71} -12.8549 q^{73} -3.49971 q^{77} -11.0211 q^{79} -11.1010 q^{81} -13.5257 q^{83} -20.5399 q^{87} +14.3475 q^{89} +8.81468 q^{91} -20.3118 q^{93} +0.199218 q^{97} +2.96658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} + q^{7} + 4 q^{9} + 4 q^{11} - q^{13} + 5 q^{17} - 4 q^{19} - 6 q^{21} - 5 q^{23} - 6 q^{27} - 11 q^{29} - 4 q^{31} + 13 q^{33} + 6 q^{37} - 31 q^{39} - 8 q^{41} - 3 q^{43} - 2 q^{47} - 2 q^{49}+ \cdots - 37 q^{99}+O(q^{100}) \) 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
\(3\) 2.21042 1.27618 0.638092 − 0.769960i \(-0.279724\pi\)
0.638092 + 0.769960i \(0.279724\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.22487 −0.840920 −0.420460 − 0.907311i \(-0.638131\pi\)
−0.420460 + 0.907311i \(0.638131\pi\)
\(8\) 0 0
\(9\) 1.88594 0.628648
\(10\) 0 0
\(11\) 1.57300 0.474276 0.237138 − 0.971476i \(-0.423791\pi\)
0.237138 + 0.971476i \(0.423791\pi\)
\(12\) 0 0
\(13\) −3.96189 −1.09883 −0.549416 − 0.835549i \(-0.685150\pi\)
−0.549416 + 0.835549i \(0.685150\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.294907 −0.0715256 −0.0357628 − 0.999360i \(-0.511386\pi\)
−0.0357628 + 0.999360i \(0.511386\pi\)
\(18\) 0 0
\(19\) 7.76572 1.78158 0.890789 − 0.454418i \(-0.150153\pi\)
0.890789 + 0.454418i \(0.150153\pi\)
\(20\) 0 0
\(21\) −4.91788 −1.07317
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.46253 −0.473914
\(28\) 0 0
\(29\) −9.29233 −1.72554 −0.862771 − 0.505595i \(-0.831273\pi\)
−0.862771 + 0.505595i \(0.831273\pi\)
\(30\) 0 0
\(31\) −9.18913 −1.65042 −0.825208 − 0.564829i \(-0.808942\pi\)
−0.825208 + 0.564829i \(0.808942\pi\)
\(32\) 0 0
\(33\) 3.47698 0.605264
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.5425 1.73318 0.866588 − 0.499024i \(-0.166308\pi\)
0.866588 + 0.499024i \(0.166308\pi\)
\(38\) 0 0
\(39\) −8.75744 −1.40231
\(40\) 0 0
\(41\) −2.34251 −0.365839 −0.182920 − 0.983128i \(-0.558555\pi\)
−0.182920 + 0.983128i \(0.558555\pi\)
\(42\) 0 0
\(43\) 6.67460 1.01787 0.508933 − 0.860806i \(-0.330040\pi\)
0.508933 + 0.860806i \(0.330040\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.38007 −0.201304 −0.100652 − 0.994922i \(-0.532093\pi\)
−0.100652 + 0.994922i \(0.532093\pi\)
\(48\) 0 0
\(49\) −2.04998 −0.292854
\(50\) 0 0
\(51\) −0.651868 −0.0912798
\(52\) 0 0
\(53\) 11.0395 1.51640 0.758199 − 0.652023i \(-0.226080\pi\)
0.758199 + 0.652023i \(0.226080\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 17.1655 2.27362
\(58\) 0 0
\(59\) 5.09378 0.663154 0.331577 − 0.943428i \(-0.392419\pi\)
0.331577 + 0.943428i \(0.392419\pi\)
\(60\) 0 0
\(61\) −8.91788 −1.14182 −0.570909 − 0.821014i \(-0.693409\pi\)
−0.570909 + 0.821014i \(0.693409\pi\)
\(62\) 0 0
\(63\) −4.19597 −0.528642
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.12002 −0.136832 −0.0684160 − 0.997657i \(-0.521795\pi\)
−0.0684160 + 0.997657i \(0.521795\pi\)
\(68\) 0 0
\(69\) −2.21042 −0.266103
\(70\) 0 0
\(71\) −7.60168 −0.902154 −0.451077 − 0.892485i \(-0.648960\pi\)
−0.451077 + 0.892485i \(0.648960\pi\)
\(72\) 0 0
\(73\) −12.8549 −1.50455 −0.752276 − 0.658848i \(-0.771044\pi\)
−0.752276 + 0.658848i \(0.771044\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.49971 −0.398828
\(78\) 0 0
\(79\) −11.0211 −1.23997 −0.619984 − 0.784614i \(-0.712861\pi\)
−0.619984 + 0.784614i \(0.712861\pi\)
\(80\) 0 0
\(81\) −11.1010 −1.23345
\(82\) 0 0
\(83\) −13.5257 −1.48464 −0.742318 − 0.670048i \(-0.766274\pi\)
−0.742318 + 0.670048i \(0.766274\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −20.5399 −2.20211
\(88\) 0 0
\(89\) 14.3475 1.52084 0.760418 − 0.649434i \(-0.224994\pi\)
0.760418 + 0.649434i \(0.224994\pi\)
\(90\) 0 0
\(91\) 8.81468 0.924030
\(92\) 0 0
\(93\) −20.3118 −2.10624
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.199218 0.0202275 0.0101137 − 0.999949i \(-0.496781\pi\)
0.0101137 + 0.999949i \(0.496781\pi\)
\(98\) 0 0
\(99\) 2.96658 0.298153
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.cs.1.5 5
4.3 odd 2 4600.2.a.bg.1.1 yes 5
5.4 even 2 9200.2.a.cw.1.1 5
20.3 even 4 4600.2.e.v.4049.2 10
20.7 even 4 4600.2.e.v.4049.9 10
20.19 odd 2 4600.2.a.bc.1.5 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.5 ✓ 5 20.19 odd 2
4600.2.a.bg.1.1 yes 5 4.3 odd 2
4600.2.e.v.4049.2 10 20.3 even 4
4600.2.e.v.4049.9 10 20.7 even 4
9200.2.a.cs.1.5 5 1.1 even 1 trivial
9200.2.a.cw.1.1 5 5.4 even 2