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Properties

Label 9200.2.a.dd.1.3
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $8$
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 = [8,0,-3,0,0,0,-7,0,11,0,-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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.69755\) 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)
 
\(f(q)\) \(=\) \(q-1.69755 q^{3} +4.22860 q^{7} -0.118308 q^{9} -4.59157 q^{11} +0.978254 q^{13} +3.04005 q^{17} -1.91463 q^{19} -7.17829 q^{21} +1.00000 q^{23} +5.29350 q^{27} -0.737607 q^{29} -2.97939 q^{31} +7.79445 q^{33} -8.93637 q^{37} -1.66064 q^{39} +9.08240 q^{41} -6.97873 q^{43} +2.58560 q^{47} +10.8811 q^{49} -5.16065 q^{51} +2.71400 q^{53} +3.25019 q^{57} -7.13514 q^{59} -0.731629 q^{61} -0.500277 q^{63} -7.16679 q^{67} -1.69755 q^{69} +6.08775 q^{71} +5.96311 q^{73} -19.4159 q^{77} +2.06100 q^{79} -8.63108 q^{81} +4.39408 q^{83} +1.25213 q^{87} -7.25034 q^{89} +4.13665 q^{91} +5.05767 q^{93} +7.04560 q^{97} +0.543219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} - 7 q^{11} - 11 q^{13} + 7 q^{17} - 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} - 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} + 19 q^{51}+ \cdots - 61 q^{99}+O(q^{100}) \) 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
\(3\) −1.69755 −0.980084 −0.490042 − 0.871699i \(-0.663019\pi\)
−0.490042 + 0.871699i \(0.663019\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.22860 1.59826 0.799131 − 0.601157i \(-0.205293\pi\)
0.799131 + 0.601157i \(0.205293\pi\)
\(8\) 0 0
\(9\) −0.118308 −0.0394359
\(10\) 0 0
\(11\) −4.59157 −1.38441 −0.692206 − 0.721700i \(-0.743361\pi\)
−0.692206 + 0.721700i \(0.743361\pi\)
\(12\) 0 0
\(13\) 0.978254 0.271319 0.135659 − 0.990756i \(-0.456685\pi\)
0.135659 + 0.990756i \(0.456685\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.04005 0.737321 0.368660 − 0.929564i \(-0.379817\pi\)
0.368660 + 0.929564i \(0.379817\pi\)
\(18\) 0 0
\(19\) −1.91463 −0.439247 −0.219623 − 0.975585i \(-0.570483\pi\)
−0.219623 + 0.975585i \(0.570483\pi\)
\(20\) 0 0
\(21\) −7.17829 −1.56643
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.29350 1.01873
\(28\) 0 0
\(29\) −0.737607 −0.136970 −0.0684851 − 0.997652i \(-0.521817\pi\)
−0.0684851 + 0.997652i \(0.521817\pi\)
\(30\) 0 0
\(31\) −2.97939 −0.535114 −0.267557 − 0.963542i \(-0.586216\pi\)
−0.267557 + 0.963542i \(0.586216\pi\)
\(32\) 0 0
\(33\) 7.79445 1.35684
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.93637 −1.46913 −0.734565 − 0.678539i \(-0.762614\pi\)
−0.734565 + 0.678539i \(0.762614\pi\)
\(38\) 0 0
\(39\) −1.66064 −0.265915
\(40\) 0 0
\(41\) 9.08240 1.41843 0.709216 − 0.704991i \(-0.249049\pi\)
0.709216 + 0.704991i \(0.249049\pi\)
\(42\) 0 0
\(43\) −6.97873 −1.06425 −0.532123 − 0.846667i \(-0.678606\pi\)
−0.532123 + 0.846667i \(0.678606\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.58560 0.377148 0.188574 − 0.982059i \(-0.439613\pi\)
0.188574 + 0.982059i \(0.439613\pi\)
\(48\) 0 0
\(49\) 10.8811 1.55444
\(50\) 0 0
\(51\) −5.16065 −0.722636
\(52\) 0 0
\(53\) 2.71400 0.372796 0.186398 − 0.982474i \(-0.440319\pi\)
0.186398 + 0.982474i \(0.440319\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.25019 0.430499
\(58\) 0 0
\(59\) −7.13514 −0.928916 −0.464458 − 0.885595i \(-0.653751\pi\)
−0.464458 + 0.885595i \(0.653751\pi\)
\(60\) 0 0
\(61\) −0.731629 −0.0936755 −0.0468377 − 0.998903i \(-0.514914\pi\)
−0.0468377 + 0.998903i \(0.514914\pi\)
\(62\) 0 0
\(63\) −0.500277 −0.0630290
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.16679 −0.875563 −0.437781 − 0.899081i \(-0.644236\pi\)
−0.437781 + 0.899081i \(0.644236\pi\)
\(68\) 0 0
\(69\) −1.69755 −0.204362
\(70\) 0 0
\(71\) 6.08775 0.722483 0.361241 − 0.932472i \(-0.382353\pi\)
0.361241 + 0.932472i \(0.382353\pi\)
\(72\) 0 0
\(73\) 5.96311 0.697930 0.348965 − 0.937136i \(-0.386533\pi\)
0.348965 + 0.937136i \(0.386533\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −19.4159 −2.21265
\(78\) 0 0
\(79\) 2.06100 0.231880 0.115940 − 0.993256i \(-0.463012\pi\)
0.115940 + 0.993256i \(0.463012\pi\)
\(80\) 0 0
\(81\) −8.63108 −0.959009
\(82\) 0 0
\(83\) 4.39408 0.482312 0.241156 − 0.970486i \(-0.422473\pi\)
0.241156 + 0.970486i \(0.422473\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.25213 0.134242
\(88\) 0 0
\(89\) −7.25034 −0.768534 −0.384267 − 0.923222i \(-0.625546\pi\)
−0.384267 + 0.923222i \(0.625546\pi\)
\(90\) 0 0
\(91\) 4.13665 0.433638
\(92\) 0 0
\(93\) 5.05767 0.524456
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.04560 0.715372 0.357686 − 0.933842i \(-0.383566\pi\)
0.357686 + 0.933842i \(0.383566\pi\)
\(98\) 0 0
\(99\) 0.543219 0.0545956
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.dd.1.3 8
4.3 odd 2 4600.2.a.bk.1.6 8
5.2 odd 4 1840.2.e.h.369.12 16
5.3 odd 4 1840.2.e.h.369.5 16
5.4 even 2 9200.2.a.de.1.6 8
20.3 even 4 920.2.e.c.369.12 yes 16
20.7 even 4 920.2.e.c.369.5 ✓ 16
20.19 odd 2 4600.2.a.bj.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.e.c.369.5 ✓ 16 20.7 even 4
920.2.e.c.369.12 yes 16 20.3 even 4
1840.2.e.h.369.5 16 5.3 odd 4
1840.2.e.h.369.12 16 5.2 odd 4
4600.2.a.bj.1.3 8 20.19 odd 2
4600.2.a.bk.1.6 8 4.3 odd 2
9200.2.a.dd.1.3 8 1.1 even 1 trivial
9200.2.a.de.1.6 8 5.4 even 2