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 1.3
Root \(-0.794805\) 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)
 
\(f(q)\) \(=\) \(q+0.794805 q^{3} -2.47193 q^{7} -2.36829 q^{9} +2.29993 q^{11} -3.84022 q^{13} -7.74682 q^{17} +2.29993 q^{19} -1.96471 q^{21} +1.00000 q^{23} -4.26674 q^{27} +5.28380 q^{29} +6.40148 q^{31} +1.82800 q^{33} +8.56457 q^{37} -3.05223 q^{39} +4.27699 q^{41} +1.88954 q^{43} -12.3432 q^{47} -0.889540 q^{49} -6.15721 q^{51} -7.57482 q^{53} +1.82800 q^{57} -6.07180 q^{59} -0.635155 q^{61} +5.85425 q^{63} -11.1333 q^{67} +0.794805 q^{69} -8.58163 q^{71} +16.5849 q^{73} -5.68528 q^{77} -0.335225 q^{79} +3.71363 q^{81} -15.1937 q^{83} +4.19959 q^{87} +5.55735 q^{89} +9.49277 q^{91} +5.08792 q^{93} +6.42786 q^{97} -5.44689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{7} + 3 q^{9} + 4 q^{13} + 6 q^{17} + 5 q^{23} - 9 q^{27} + 12 q^{29} + 18 q^{31} + 6 q^{33} + 10 q^{37} - 9 q^{39} - 6 q^{41} - 10 q^{43} - 22 q^{47} + 15 q^{49} + 6 q^{51} + 10 q^{53} + 6 q^{57}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) 0.794805 0.458881 0.229440 − 0.973323i \(-0.426310\pi\)
0.229440 + 0.973323i \(0.426310\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −2.47193 −0.934303 −0.467152 − 0.884177i \(-0.654720\pi\)
−0.467152 + 0.884177i \(0.654720\pi\)
\(8\) 0 0
\(9\) −2.36829 −0.789428
\(10\) 0 0
\(11\) 2.29993 0.693455 0.346728 − 0.937966i \(-0.387293\pi\)
0.346728 + 0.937966i \(0.387293\pi\)
\(12\) 0 0
\(13\) −3.84022 −1.06509 −0.532543 − 0.846403i \(-0.678763\pi\)
−0.532543 + 0.846403i \(0.678763\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.74682 −1.87888 −0.939440 − 0.342713i \(-0.888654\pi\)
−0.939440 + 0.342713i \(0.888654\pi\)
\(18\) 0 0
\(19\) 2.29993 0.527640 0.263820 − 0.964572i \(-0.415017\pi\)
0.263820 + 0.964572i \(0.415017\pi\)
\(20\) 0 0
\(21\) −1.96471 −0.428734
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.26674 −0.821134
\(28\) 0 0
\(29\) 5.28380 0.981177 0.490589 − 0.871391i \(-0.336782\pi\)
0.490589 + 0.871391i \(0.336782\pi\)
\(30\) 0 0
\(31\) 6.40148 1.14974 0.574870 − 0.818245i \(-0.305053\pi\)
0.574870 + 0.818245i \(0.305053\pi\)
\(32\) 0 0
\(33\) 1.82800 0.318213
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.56457 1.40801 0.704003 − 0.710197i \(-0.251394\pi\)
0.704003 + 0.710197i \(0.251394\pi\)
\(38\) 0 0
\(39\) −3.05223 −0.488747
\(40\) 0 0
\(41\) 4.27699 0.667954 0.333977 − 0.942581i \(-0.391609\pi\)
0.333977 + 0.942581i \(0.391609\pi\)
\(42\) 0 0
\(43\) 1.88954 0.288152 0.144076 − 0.989567i \(-0.453979\pi\)
0.144076 + 0.989567i \(0.453979\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.3432 −1.80045 −0.900223 − 0.435428i \(-0.856597\pi\)
−0.900223 + 0.435428i \(0.856597\pi\)
\(48\) 0 0
\(49\) −0.889540 −0.127077
\(50\) 0 0
\(51\) −6.15721 −0.862182
\(52\) 0 0
\(53\) −7.57482 −1.04048 −0.520241 − 0.854020i \(-0.674158\pi\)
−0.520241 + 0.854020i \(0.674158\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.82800 0.242124
\(58\) 0 0
\(59\) −6.07180 −0.790480 −0.395240 − 0.918578i \(-0.629339\pi\)
−0.395240 + 0.918578i \(0.629339\pi\)
\(60\) 0 0
\(61\) −0.635155 −0.0813233 −0.0406616 − 0.999173i \(-0.512947\pi\)
−0.0406616 + 0.999173i \(0.512947\pi\)
\(62\) 0 0
\(63\) 5.85425 0.737566
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.1333 −1.36015 −0.680077 − 0.733141i \(-0.738054\pi\)
−0.680077 + 0.733141i \(0.738054\pi\)
\(68\) 0 0
\(69\) 0.794805 0.0956833
\(70\) 0 0
\(71\) −8.58163 −1.01845 −0.509226 − 0.860633i \(-0.670068\pi\)
−0.509226 + 0.860633i \(0.670068\pi\)
\(72\) 0 0
\(73\) 16.5849 1.94112 0.970560 − 0.240859i \(-0.0774292\pi\)
0.970560 + 0.240859i \(0.0774292\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.68528 −0.647897
\(78\) 0 0
\(79\) −0.335225 −0.0377157 −0.0188579 − 0.999822i \(-0.506003\pi\)
−0.0188579 + 0.999822i \(0.506003\pi\)
\(80\) 0 0
\(81\) 3.71363 0.412626
\(82\) 0 0
\(83\) −15.1937 −1.66773 −0.833863 − 0.551971i \(-0.813876\pi\)
−0.833863 + 0.551971i \(0.813876\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.19959 0.450243
\(88\) 0 0
\(89\) 5.55735 0.589078 0.294539 − 0.955639i \(-0.404834\pi\)
0.294539 + 0.955639i \(0.404834\pi\)
\(90\) 0 0
\(91\) 9.49277 0.995113
\(92\) 0 0
\(93\) 5.08792 0.527593
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.42786 0.652650 0.326325 − 0.945258i \(-0.394190\pi\)
0.326325 + 0.945258i \(0.394190\pi\)
\(98\) 0 0
\(99\) −5.44689 −0.547433
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.1.3 5
4.3 odd 2 4600.2.a.bf.1.3 yes 5
5.4 even 2 9200.2.a.cv.1.3 5
20.3 even 4 4600.2.e.w.4049.5 10
20.7 even 4 4600.2.e.w.4049.6 10
20.19 odd 2 4600.2.a.bd.1.3 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bd.1.3 ✓ 5 20.19 odd 2
4600.2.a.bf.1.3 yes 5 4.3 odd 2
4600.2.e.w.4049.5 10 20.3 even 4
4600.2.e.w.4049.6 10 20.7 even 4
9200.2.a.ct.1.3 5 1.1 even 1 trivial
9200.2.a.cv.1.3 5 5.4 even 2