↑

Properties

Label 9200.2.a.ch.1.2
Level $9200$
Weight $2$
Character 9200.1
Self dual yes
Analytic conductor $73.462$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 = [3,0,2,0,0,0,4,0,-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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 1.2
Root \(0.445042\) 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)
 
\(f(q)\) \(=\) \(q+0.554958 q^{3} +1.10992 q^{7} -2.69202 q^{9} -2.71379 q^{11} +1.69202 q^{13} -1.10992 q^{17} +4.98792 q^{19} +0.615957 q^{21} -1.00000 q^{23} -3.15883 q^{27} +6.34481 q^{29} -5.13706 q^{31} -1.50604 q^{33} -5.70171 q^{37} +0.939001 q^{39} +1.26875 q^{41} -1.90217 q^{43} +4.08815 q^{47} -5.76809 q^{49} -0.615957 q^{51} -3.78017 q^{53} +2.76809 q^{57} -5.59179 q^{59} +8.37196 q^{61} -2.98792 q^{63} +10.7681 q^{67} -0.554958 q^{69} +12.1075 q^{71} -15.3327 q^{73} -3.01208 q^{77} -11.0858 q^{79} +6.32304 q^{81} -7.64742 q^{83} +3.52111 q^{87} -4.17629 q^{89} +1.87800 q^{91} -2.85086 q^{93} -9.50604 q^{97} +7.30559 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 4 q^{7} - 3 q^{9} - 4 q^{17} - 4 q^{19} + 12 q^{21} - 3 q^{23} - q^{27} - 4 q^{29} - 10 q^{31} - 14 q^{33} + 10 q^{37} - 7 q^{39} - 4 q^{41} - 24 q^{43} + 16 q^{47} + 3 q^{49} - 12 q^{51}+ \cdots - 14 q^{99}+O(q^{100}) \) 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
\(3\) 0.554958 0.320405 0.160203 − 0.987084i \(-0.448785\pi\)
0.160203 + 0.987084i \(0.448785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.10992 0.419509 0.209754 − 0.977754i \(-0.432734\pi\)
0.209754 + 0.977754i \(0.432734\pi\)
\(8\) 0 0
\(9\) −2.69202 −0.897340
\(10\) 0 0
\(11\) −2.71379 −0.818239 −0.409119 − 0.912481i \(-0.634164\pi\)
−0.409119 + 0.912481i \(0.634164\pi\)
\(12\) 0 0
\(13\) 1.69202 0.469282 0.234641 − 0.972082i \(-0.424608\pi\)
0.234641 + 0.972082i \(0.424608\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.10992 −0.269194 −0.134597 − 0.990900i \(-0.542974\pi\)
−0.134597 + 0.990900i \(0.542974\pi\)
\(18\) 0 0
\(19\) 4.98792 1.14431 0.572153 − 0.820147i \(-0.306108\pi\)
0.572153 + 0.820147i \(0.306108\pi\)
\(20\) 0 0
\(21\) 0.615957 0.134413
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3.15883 −0.607918
\(28\) 0 0
\(29\) 6.34481 1.17820 0.589101 − 0.808059i \(-0.299482\pi\)
0.589101 + 0.808059i \(0.299482\pi\)
\(30\) 0 0
\(31\) −5.13706 −0.922644 −0.461322 − 0.887233i \(-0.652625\pi\)
−0.461322 + 0.887233i \(0.652625\pi\)
\(32\) 0 0
\(33\) −1.50604 −0.262168
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.70171 −0.937355 −0.468678 − 0.883369i \(-0.655269\pi\)
−0.468678 + 0.883369i \(0.655269\pi\)
\(38\) 0 0
\(39\) 0.939001 0.150361
\(40\) 0 0
\(41\) 1.26875 0.198145 0.0990727 − 0.995080i \(-0.468412\pi\)
0.0990727 + 0.995080i \(0.468412\pi\)
\(42\) 0 0
\(43\) −1.90217 −0.290077 −0.145039 − 0.989426i \(-0.546331\pi\)
−0.145039 + 0.989426i \(0.546331\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.08815 0.596317 0.298159 − 0.954516i \(-0.403628\pi\)
0.298159 + 0.954516i \(0.403628\pi\)
\(48\) 0 0
\(49\) −5.76809 −0.824012
\(50\) 0 0
\(51\) −0.615957 −0.0862512
\(52\) 0 0
\(53\) −3.78017 −0.519246 −0.259623 − 0.965710i \(-0.583598\pi\)
−0.259623 + 0.965710i \(0.583598\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.76809 0.366642
\(58\) 0 0
\(59\) −5.59179 −0.727990 −0.363995 − 0.931401i \(-0.618587\pi\)
−0.363995 + 0.931401i \(0.618587\pi\)
\(60\) 0 0
\(61\) 8.37196 1.07192 0.535960 − 0.844243i \(-0.319950\pi\)
0.535960 + 0.844243i \(0.319950\pi\)
\(62\) 0 0
\(63\) −2.98792 −0.376442
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.7681 1.31553 0.657766 − 0.753223i \(-0.271502\pi\)
0.657766 + 0.753223i \(0.271502\pi\)
\(68\) 0 0
\(69\) −0.554958 −0.0668091
\(70\) 0 0
\(71\) 12.1075 1.43690 0.718449 − 0.695579i \(-0.244852\pi\)
0.718449 + 0.695579i \(0.244852\pi\)
\(72\) 0 0
\(73\) −15.3327 −1.79456 −0.897280 − 0.441461i \(-0.854460\pi\)
−0.897280 + 0.441461i \(0.854460\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.01208 −0.343259
\(78\) 0 0
\(79\) −11.0858 −1.24724 −0.623622 − 0.781726i \(-0.714340\pi\)
−0.623622 + 0.781726i \(0.714340\pi\)
\(80\) 0 0
\(81\) 6.32304 0.702560
\(82\) 0 0
\(83\) −7.64742 −0.839413 −0.419706 − 0.907660i \(-0.637867\pi\)
−0.419706 + 0.907660i \(0.637867\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.52111 0.377502
\(88\) 0 0
\(89\) −4.17629 −0.442686 −0.221343 − 0.975196i \(-0.571044\pi\)
−0.221343 + 0.975196i \(0.571044\pi\)
\(90\) 0 0
\(91\) 1.87800 0.196868
\(92\) 0 0
\(93\) −2.85086 −0.295620
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.50604 −0.965192 −0.482596 − 0.875843i \(-0.660306\pi\)
−0.482596 + 0.875843i \(0.660306\pi\)
\(98\) 0 0
\(99\) 7.30559 0.734239
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.ch.1.2 3
4.3 odd 2 4600.2.a.w.1.2 ✓ 3
5.4 even 2 9200.2.a.cb.1.2 3
20.3 even 4 4600.2.e.s.4049.3 6
20.7 even 4 4600.2.e.s.4049.4 6
20.19 odd 2 4600.2.a.z.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.w.1.2 ✓ 3 4.3 odd 2
4600.2.a.z.1.2 yes 3 20.19 odd 2
4600.2.e.s.4049.3 6 20.3 even 4
4600.2.e.s.4049.4 6 20.7 even 4
9200.2.a.cb.1.2 3 5.4 even 2
9200.2.a.ch.1.2 3 1.1 even 1 trivial