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Properties

Label 9200.2.a.ct.1.2
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
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 = [5,0,0,0,0,0,-4,0,3,0,0] 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: \(5\)
Coefficient field: 5.5.521397.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 9x^{3} - 3x^{2} + 18x + 12 \) 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 1.2
Root \(1.83957\) 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)
 
\(f(q)\) \(=\) \(q-1.83957 q^{3} +3.97272 q^{7} +0.384010 q^{9} -2.10339 q^{11} +5.35673 q^{13} +1.29567 q^{17} -2.10339 q^{19} -7.30809 q^{21} +1.00000 q^{23} +4.81229 q^{27} +6.03130 q^{29} +8.32489 q^{31} +3.86933 q^{33} +5.10131 q^{37} -9.85407 q^{39} -8.33994 q^{41} -7.78253 q^{43} -11.3964 q^{47} +8.78253 q^{49} -2.38347 q^{51} -0.573664 q^{53} +3.86933 q^{57} +9.17951 q^{59} +13.5149 q^{61} +1.52557 q^{63} +15.8314 q^{67} -1.83957 q^{69} -14.9449 q^{71} +8.36459 q^{73} -8.35619 q^{77} +9.41149 q^{79} -10.0046 q^{81} -1.51206 q^{83} -11.0950 q^{87} +10.5903 q^{89} +21.2808 q^{91} -15.3142 q^{93} -0.337451 q^{97} -0.807724 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.83957 −1.06208 −0.531038 − 0.847348i \(-0.678198\pi\)
−0.531038 + 0.847348i \(0.678198\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.97272 1.50155 0.750774 − 0.660559i \(-0.229681\pi\)
0.750774 + 0.660559i \(0.229681\pi\)
\(8\) 0 0
\(9\) 0.384010 0.128003
\(10\) 0 0
\(11\) −2.10339 −0.634196 −0.317098 − 0.948393i \(-0.602708\pi\)
−0.317098 + 0.948393i \(0.602708\pi\)
\(12\) 0 0
\(13\) 5.35673 1.48569 0.742845 − 0.669463i \(-0.233476\pi\)
0.742845 + 0.669463i \(0.233476\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.29567 0.314246 0.157123 − 0.987579i \(-0.449778\pi\)
0.157123 + 0.987579i \(0.449778\pi\)
\(18\) 0 0
\(19\) −2.10339 −0.482551 −0.241276 − 0.970457i \(-0.577566\pi\)
−0.241276 + 0.970457i \(0.577566\pi\)
\(20\) 0 0
\(21\) −7.30809 −1.59476
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.81229 0.926126
\(28\) 0 0
\(29\) 6.03130 1.11998 0.559992 − 0.828498i \(-0.310804\pi\)
0.559992 + 0.828498i \(0.310804\pi\)
\(30\) 0 0
\(31\) 8.32489 1.49519 0.747597 − 0.664153i \(-0.231207\pi\)
0.747597 + 0.664153i \(0.231207\pi\)
\(32\) 0 0
\(33\) 3.86933 0.673564
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.10131 0.838650 0.419325 − 0.907836i \(-0.362267\pi\)
0.419325 + 0.907836i \(0.362267\pi\)
\(38\) 0 0
\(39\) −9.85407 −1.57791
\(40\) 0 0
\(41\) −8.33994 −1.30248 −0.651240 − 0.758872i \(-0.725751\pi\)
−0.651240 + 0.758872i \(0.725751\pi\)
\(42\) 0 0
\(43\) −7.78253 −1.18682 −0.593412 − 0.804899i \(-0.702220\pi\)
−0.593412 + 0.804899i \(0.702220\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.3964 −1.66234 −0.831171 − 0.556018i \(-0.812329\pi\)
−0.831171 + 0.556018i \(0.812329\pi\)
\(48\) 0 0
\(49\) 8.78253 1.25465
\(50\) 0 0
\(51\) −2.38347 −0.333752
\(52\) 0 0
\(53\) −0.573664 −0.0787988 −0.0393994 − 0.999224i \(-0.512544\pi\)
−0.0393994 + 0.999224i \(0.512544\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.86933 0.512505
\(58\) 0 0
\(59\) 9.17951 1.19507 0.597535 − 0.801843i \(-0.296147\pi\)
0.597535 + 0.801843i \(0.296147\pi\)
\(60\) 0 0
\(61\) 13.5149 1.73040 0.865201 − 0.501425i \(-0.167191\pi\)
0.865201 + 0.501425i \(0.167191\pi\)
\(62\) 0 0
\(63\) 1.52557 0.192203
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.8314 1.93411 0.967055 − 0.254569i \(-0.0819337\pi\)
0.967055 + 0.254569i \(0.0819337\pi\)
\(68\) 0 0
\(69\) −1.83957 −0.221458
\(70\) 0 0
\(71\) −14.9449 −1.77363 −0.886817 − 0.462121i \(-0.847089\pi\)
−0.886817 + 0.462121i \(0.847089\pi\)
\(72\) 0 0
\(73\) 8.36459 0.979002 0.489501 − 0.872003i \(-0.337179\pi\)
0.489501 + 0.872003i \(0.337179\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.35619 −0.952276
\(78\) 0 0
\(79\) 9.41149 1.05887 0.529437 − 0.848349i \(-0.322403\pi\)
0.529437 + 0.848349i \(0.322403\pi\)
\(80\) 0 0
\(81\) −10.0046 −1.11162
\(82\) 0 0
\(83\) −1.51206 −0.165970 −0.0829849 − 0.996551i \(-0.526445\pi\)
−0.0829849 + 0.996551i \(0.526445\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −11.0950 −1.18951
\(88\) 0 0
\(89\) 10.5903 1.12256 0.561282 − 0.827624i \(-0.310308\pi\)
0.561282 + 0.827624i \(0.310308\pi\)
\(90\) 0 0
\(91\) 21.2808 2.23084
\(92\) 0 0
\(93\) −15.3142 −1.58801
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.337451 −0.0342630 −0.0171315 − 0.999853i \(-0.505453\pi\)
−0.0171315 + 0.999853i \(0.505453\pi\)
\(98\) 0 0
\(99\) −0.807724 −0.0811793
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.ct.1.2 5
4.3 odd 2 4600.2.a.bf.1.4 yes 5
5.4 even 2 9200.2.a.cv.1.4 5
20.3 even 4 4600.2.e.w.4049.8 10
20.7 even 4 4600.2.e.w.4049.3 10
20.19 odd 2 4600.2.a.bd.1.2 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.2 ✓ 5 20.19 odd 2
4600.2.a.bf.1.4 yes 5 4.3 odd 2
4600.2.e.w.4049.3 10 20.7 even 4
4600.2.e.w.4049.8 10 20.3 even 4
9200.2.a.ct.1.2 5 1.1 even 1 trivial
9200.2.a.cv.1.4 5 5.4 even 2