Properties

Label 2200.2.a.t.1.1
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
Analytic rank $0$
Dimension $3$
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] = [2200,2,Mod(1,2200)] 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("2200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,0,0,-3,0,6,0,-3] 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: \(17.5670884447\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.36147\) of defining polynomial
Character \(\chi\) \(=\) 2200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.36147 q^{3} +0.576535 q^{7} +8.29947 q^{9} -1.00000 q^{11} -3.72294 q^{13} +6.51454 q^{17} +1.00000 q^{19} -1.93800 q^{21} -4.36147 q^{23} -17.8140 q^{27} +2.42347 q^{29} +2.15307 q^{31} +3.36147 q^{33} -3.21507 q^{37} +12.5145 q^{39} +5.93800 q^{41} +4.93800 q^{43} -12.6609 q^{47} -6.66761 q^{49} -21.8984 q^{51} -8.02241 q^{53} -3.36147 q^{57} -0.660941 q^{59} -10.4525 q^{61} +4.78493 q^{63} +5.87601 q^{67} +14.6609 q^{69} -15.2308 q^{71} +12.5765 q^{73} -0.576535 q^{77} +12.0844 q^{79} +34.9828 q^{81} +7.45254 q^{83} -8.14640 q^{87} +11.4525 q^{89} -2.14640 q^{91} -7.23748 q^{93} +14.6543 q^{97} -8.29947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} + 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} - 3 q^{31} + 3 q^{33} - 12 q^{37} + 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) −3.36147 −1.94074 −0.970372 − 0.241614i \(-0.922323\pi\)
−0.970372 + 0.241614i \(0.922323\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.576535 0.217910 0.108955 − 0.994047i \(-0.465250\pi\)
0.108955 + 0.994047i \(0.465250\pi\)
\(8\) 0 0
\(9\) 8.29947 2.76649
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.72294 −1.03256 −0.516279 − 0.856421i \(-0.672683\pi\)
−0.516279 + 0.856421i \(0.672683\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.51454 1.58001 0.790004 − 0.613102i \(-0.210079\pi\)
0.790004 + 0.613102i \(0.210079\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −1.93800 −0.422907
\(22\) 0 0
\(23\) −4.36147 −0.909429 −0.454715 − 0.890637i \(-0.650259\pi\)
−0.454715 + 0.890637i \(0.650259\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −17.8140 −3.42831
\(28\) 0 0
\(29\) 2.42347 0.450026 0.225013 − 0.974356i \(-0.427758\pi\)
0.225013 + 0.974356i \(0.427758\pi\)
\(30\) 0 0
\(31\) 2.15307 0.386703 0.193351 − 0.981130i \(-0.438064\pi\)
0.193351 + 0.981130i \(0.438064\pi\)
\(32\) 0 0
\(33\) 3.36147 0.585157
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.21507 −0.528554 −0.264277 − 0.964447i \(-0.585133\pi\)
−0.264277 + 0.964447i \(0.585133\pi\)
\(38\) 0 0
\(39\) 12.5145 2.00393
\(40\) 0 0
\(41\) 5.93800 0.927360 0.463680 − 0.886003i \(-0.346529\pi\)
0.463680 + 0.886003i \(0.346529\pi\)
\(42\) 0 0
\(43\) 4.93800 0.753038 0.376519 − 0.926409i \(-0.377121\pi\)
0.376519 + 0.926409i \(0.377121\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.6609 −1.84679 −0.923394 − 0.383853i \(-0.874597\pi\)
−0.923394 + 0.383853i \(0.874597\pi\)
\(48\) 0 0
\(49\) −6.66761 −0.952515
\(50\) 0 0
\(51\) −21.8984 −3.06639
\(52\) 0 0
\(53\) −8.02241 −1.10196 −0.550981 − 0.834518i \(-0.685746\pi\)
−0.550981 + 0.834518i \(0.685746\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −3.36147 −0.445237
\(58\) 0 0
\(59\) −0.660941 −0.0860472 −0.0430236 − 0.999074i \(-0.513699\pi\)
−0.0430236 + 0.999074i \(0.513699\pi\)
\(60\) 0 0
\(61\) −10.4525 −1.33831 −0.669155 − 0.743122i \(-0.733344\pi\)
−0.669155 + 0.743122i \(0.733344\pi\)
\(62\) 0 0
\(63\) 4.78493 0.602845
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.87601 0.717869 0.358934 − 0.933363i \(-0.383140\pi\)
0.358934 + 0.933363i \(0.383140\pi\)
\(68\) 0 0
\(69\) 14.6609 1.76497
\(70\) 0 0
\(71\) −15.2308 −1.80756 −0.903782 − 0.427993i \(-0.859221\pi\)
−0.903782 + 0.427993i \(0.859221\pi\)
\(72\) 0 0
\(73\) 12.5765 1.47197 0.735986 − 0.676997i \(-0.236719\pi\)
0.735986 + 0.676997i \(0.236719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.576535 −0.0657022
\(78\) 0 0
\(79\) 12.0844 1.35960 0.679801 − 0.733397i \(-0.262066\pi\)
0.679801 + 0.733397i \(0.262066\pi\)
\(80\) 0 0
\(81\) 34.9828 3.88698
\(82\) 0 0
\(83\) 7.45254 0.818023 0.409011 − 0.912529i \(-0.365874\pi\)
0.409011 + 0.912529i \(0.365874\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.14640 −0.873386
\(88\) 0 0
\(89\) 11.4525 1.21397 0.606983 − 0.794714i \(-0.292379\pi\)
0.606983 + 0.794714i \(0.292379\pi\)
\(90\) 0 0
\(91\) −2.14640 −0.225004
\(92\) 0 0
\(93\) −7.23748 −0.750491
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.6543 1.48792 0.743958 − 0.668226i \(-0.232946\pi\)
0.743958 + 0.668226i \(0.232946\pi\)
\(98\) 0 0
\(99\) −8.29947 −0.834128
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 2200.2.a.t.1.1 ✓ 3
4.3 odd 2 4400.2.a.ca.1.3 3
5.2 odd 4 2200.2.b.l.1849.6 6
5.3 odd 4 2200.2.b.l.1849.1 6
5.4 even 2 2200.2.a.w.1.3 yes 3
20.3 even 4 4400.2.b.bc.4049.6 6
20.7 even 4 4400.2.b.bc.4049.1 6
20.19 odd 2 4400.2.a.bx.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.t.1.1 ✓ 3 1.1 even 1 trivial
2200.2.a.w.1.3 yes 3 5.4 even 2
2200.2.b.l.1849.1 6 5.3 odd 4
2200.2.b.l.1849.6 6 5.2 odd 4
4400.2.a.bx.1.1 3 20.19 odd 2
4400.2.a.ca.1.3 3 4.3 odd 2
4400.2.b.bc.4049.1 6 20.7 even 4
4400.2.b.bc.4049.6 6 20.3 even 4