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Properties

Label 9200.2.a.dd.1.4
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 1.4
Root \(0.493532\) 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)
 
\(f(q)\) \(=\) \(q-0.493532 q^{3} -4.54439 q^{7} -2.75643 q^{9} +4.61297 q^{11} -5.54274 q^{13} +7.16998 q^{17} -1.35966 q^{19} +2.24280 q^{21} +1.00000 q^{23} +2.84098 q^{27} -3.66351 q^{29} +4.46616 q^{31} -2.27665 q^{33} +3.32432 q^{37} +2.73552 q^{39} +8.95216 q^{41} -8.68458 q^{43} -9.59028 q^{47} +13.6515 q^{49} -3.53861 q^{51} +10.4676 q^{53} +0.671035 q^{57} +2.08543 q^{59} -0.686197 q^{61} +12.5263 q^{63} -8.84324 q^{67} -0.493532 q^{69} -15.9040 q^{71} +4.44647 q^{73} -20.9631 q^{77} +8.65227 q^{79} +6.86717 q^{81} +4.43591 q^{83} +1.80806 q^{87} +13.2284 q^{89} +25.1884 q^{91} -2.20419 q^{93} -1.21785 q^{97} -12.7153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} - 7 q^{11} - 11 q^{13} + 7 q^{17} - 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} - 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} + 19 q^{51}+ \cdots - 61 q^{99}+O(q^{100}) \) 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
\(3\) −0.493532 −0.284941 −0.142470 − 0.989799i \(-0.545505\pi\)
−0.142470 + 0.989799i \(0.545505\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.54439 −1.71762 −0.858809 − 0.512297i \(-0.828795\pi\)
−0.858809 + 0.512297i \(0.828795\pi\)
\(8\) 0 0
\(9\) −2.75643 −0.918809
\(10\) 0 0
\(11\) 4.61297 1.39086 0.695431 − 0.718593i \(-0.255213\pi\)
0.695431 + 0.718593i \(0.255213\pi\)
\(12\) 0 0
\(13\) −5.54274 −1.53728 −0.768640 − 0.639682i \(-0.779066\pi\)
−0.768640 + 0.639682i \(0.779066\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.16998 1.73897 0.869487 − 0.493955i \(-0.164449\pi\)
0.869487 + 0.493955i \(0.164449\pi\)
\(18\) 0 0
\(19\) −1.35966 −0.311928 −0.155964 − 0.987763i \(-0.549848\pi\)
−0.155964 + 0.987763i \(0.549848\pi\)
\(20\) 0 0
\(21\) 2.24280 0.489419
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.84098 0.546747
\(28\) 0 0
\(29\) −3.66351 −0.680297 −0.340148 − 0.940372i \(-0.610477\pi\)
−0.340148 + 0.940372i \(0.610477\pi\)
\(30\) 0 0
\(31\) 4.46616 0.802146 0.401073 − 0.916046i \(-0.368637\pi\)
0.401073 + 0.916046i \(0.368637\pi\)
\(32\) 0 0
\(33\) −2.27665 −0.396313
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.32432 0.546514 0.273257 − 0.961941i \(-0.411899\pi\)
0.273257 + 0.961941i \(0.411899\pi\)
\(38\) 0 0
\(39\) 2.73552 0.438033
\(40\) 0 0
\(41\) 8.95216 1.39809 0.699046 − 0.715076i \(-0.253608\pi\)
0.699046 + 0.715076i \(0.253608\pi\)
\(42\) 0 0
\(43\) −8.68458 −1.32439 −0.662193 − 0.749333i \(-0.730374\pi\)
−0.662193 + 0.749333i \(0.730374\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.59028 −1.39889 −0.699443 − 0.714688i \(-0.746569\pi\)
−0.699443 + 0.714688i \(0.746569\pi\)
\(48\) 0 0
\(49\) 13.6515 1.95021
\(50\) 0 0
\(51\) −3.53861 −0.495505
\(52\) 0 0
\(53\) 10.4676 1.43784 0.718918 − 0.695095i \(-0.244638\pi\)
0.718918 + 0.695095i \(0.244638\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.671035 0.0888808
\(58\) 0 0
\(59\) 2.08543 0.271499 0.135750 − 0.990743i \(-0.456656\pi\)
0.135750 + 0.990743i \(0.456656\pi\)
\(60\) 0 0
\(61\) −0.686197 −0.0878586 −0.0439293 − 0.999035i \(-0.513988\pi\)
−0.0439293 + 0.999035i \(0.513988\pi\)
\(62\) 0 0
\(63\) 12.5263 1.57816
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.84324 −1.08037 −0.540187 − 0.841545i \(-0.681646\pi\)
−0.540187 + 0.841545i \(0.681646\pi\)
\(68\) 0 0
\(69\) −0.493532 −0.0594142
\(70\) 0 0
\(71\) −15.9040 −1.88746 −0.943729 − 0.330721i \(-0.892708\pi\)
−0.943729 + 0.330721i \(0.892708\pi\)
\(72\) 0 0
\(73\) 4.44647 0.520420 0.260210 − 0.965552i \(-0.416208\pi\)
0.260210 + 0.965552i \(0.416208\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −20.9631 −2.38897
\(78\) 0 0
\(79\) 8.65227 0.973457 0.486728 − 0.873553i \(-0.338190\pi\)
0.486728 + 0.873553i \(0.338190\pi\)
\(80\) 0 0
\(81\) 6.86717 0.763019
\(82\) 0 0
\(83\) 4.43591 0.486904 0.243452 − 0.969913i \(-0.421720\pi\)
0.243452 + 0.969913i \(0.421720\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.80806 0.193844
\(88\) 0 0
\(89\) 13.2284 1.40220 0.701102 − 0.713061i \(-0.252692\pi\)
0.701102 + 0.713061i \(0.252692\pi\)
\(90\) 0 0
\(91\) 25.1884 2.64046
\(92\) 0 0
\(93\) −2.20419 −0.228564
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.21785 −0.123654 −0.0618270 − 0.998087i \(-0.519693\pi\)
−0.0618270 + 0.998087i \(0.519693\pi\)
\(98\) 0 0
\(99\) −12.7153 −1.27794
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.1.4 8
4.3 odd 2 4600.2.a.bk.1.5 8
5.2 odd 4 1840.2.e.h.369.10 16
5.3 odd 4 1840.2.e.h.369.7 16
5.4 even 2 9200.2.a.de.1.5 8
20.3 even 4 920.2.e.c.369.10 yes 16
20.7 even 4 920.2.e.c.369.7 ✓ 16
20.19 odd 2 4600.2.a.bj.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.e.c.369.7 ✓ 16 20.7 even 4
920.2.e.c.369.10 yes 16 20.3 even 4
1840.2.e.h.369.7 16 5.3 odd 4
1840.2.e.h.369.10 16 5.2 odd 4
4600.2.a.bj.1.4 8 20.19 odd 2
4600.2.a.bk.1.5 8 4.3 odd 2
9200.2.a.dd.1.4 8 1.1 even 1 trivial
9200.2.a.de.1.5 8 5.4 even 2