Properties

Label 2200.2.a.t.1.3
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 1.3
Root \(2.52892\) 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)
 
\(f(q)\) \(=\) \(q+1.52892 q^{3} +1.39543 q^{7} -0.662410 q^{9} -1.00000 q^{11} +6.05784 q^{13} +3.26193 q^{17} +1.00000 q^{19} +2.13349 q^{21} +0.528918 q^{23} -5.59952 q^{27} +1.60457 q^{29} +3.79085 q^{31} -1.52892 q^{33} -8.92434 q^{37} +9.26193 q^{39} +1.86651 q^{41} +0.866508 q^{43} +1.19133 q^{47} -5.05279 q^{49} +4.98723 q^{51} +10.7202 q^{53} +1.52892 q^{57} +13.1913 q^{59} -3.12844 q^{61} -0.924344 q^{63} -2.26698 q^{67} +0.808672 q^{69} +10.0400 q^{71} +13.3954 q^{73} -1.39543 q^{77} -2.58675 q^{79} -6.57398 q^{81} +0.128442 q^{83} +2.45326 q^{87} +4.12844 q^{89} +8.45326 q^{91} +5.79590 q^{93} -11.4354 q^{97} +0.662410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} - 3 q^{11} + 3 q^{13} + 3 q^{17} + 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} - 3 q^{31} + 3 q^{33} - 12 q^{37} + 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.52892 0.882721 0.441361 − 0.897330i \(-0.354496\pi\)
0.441361 + 0.897330i \(0.354496\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.39543 0.527421 0.263711 − 0.964602i \(-0.415054\pi\)
0.263711 + 0.964602i \(0.415054\pi\)
\(8\) 0 0
\(9\) −0.662410 −0.220803
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 6.05784 1.68014 0.840071 − 0.542477i \(-0.182513\pi\)
0.840071 + 0.542477i \(0.182513\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.26193 0.791135 0.395568 − 0.918437i \(-0.370548\pi\)
0.395568 + 0.918437i \(0.370548\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 2.13349 0.465566
\(22\) 0 0
\(23\) 0.528918 0.110287 0.0551435 − 0.998478i \(-0.482438\pi\)
0.0551435 + 0.998478i \(0.482438\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.59952 −1.07763
\(28\) 0 0
\(29\) 1.60457 0.297962 0.148981 − 0.988840i \(-0.452401\pi\)
0.148981 + 0.988840i \(0.452401\pi\)
\(30\) 0 0
\(31\) 3.79085 0.680857 0.340429 − 0.940270i \(-0.389428\pi\)
0.340429 + 0.940270i \(0.389428\pi\)
\(32\) 0 0
\(33\) −1.52892 −0.266150
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.92434 −1.46715 −0.733577 − 0.679607i \(-0.762150\pi\)
−0.733577 + 0.679607i \(0.762150\pi\)
\(38\) 0 0
\(39\) 9.26193 1.48310
\(40\) 0 0
\(41\) 1.86651 0.291500 0.145750 − 0.989321i \(-0.453441\pi\)
0.145750 + 0.989321i \(0.453441\pi\)
\(42\) 0 0
\(43\) 0.866508 0.132141 0.0660706 − 0.997815i \(-0.478954\pi\)
0.0660706 + 0.997815i \(0.478954\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.19133 0.173773 0.0868865 − 0.996218i \(-0.472308\pi\)
0.0868865 + 0.996218i \(0.472308\pi\)
\(48\) 0 0
\(49\) −5.05279 −0.721827
\(50\) 0 0
\(51\) 4.98723 0.698352
\(52\) 0 0
\(53\) 10.7202 1.47254 0.736270 − 0.676688i \(-0.236586\pi\)
0.736270 + 0.676688i \(0.236586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.52892 0.202510
\(58\) 0 0
\(59\) 13.1913 1.71736 0.858682 − 0.512508i \(-0.171284\pi\)
0.858682 + 0.512508i \(0.171284\pi\)
\(60\) 0 0
\(61\) −3.12844 −0.400556 −0.200278 − 0.979739i \(-0.564185\pi\)
−0.200278 + 0.979739i \(0.564185\pi\)
\(62\) 0 0
\(63\) −0.924344 −0.116456
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.26698 −0.276956 −0.138478 − 0.990365i \(-0.544221\pi\)
−0.138478 + 0.990365i \(0.544221\pi\)
\(68\) 0 0
\(69\) 0.808672 0.0973527
\(70\) 0 0
\(71\) 10.0400 1.19153 0.595765 − 0.803159i \(-0.296849\pi\)
0.595765 + 0.803159i \(0.296849\pi\)
\(72\) 0 0
\(73\) 13.3954 1.56782 0.783908 − 0.620877i \(-0.213223\pi\)
0.783908 + 0.620877i \(0.213223\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.39543 −0.159024
\(78\) 0 0
\(79\) −2.58675 −0.291033 −0.145516 − 0.989356i \(-0.546484\pi\)
−0.145516 + 0.989356i \(0.546484\pi\)
\(80\) 0 0
\(81\) −6.57398 −0.730443
\(82\) 0 0
\(83\) 0.128442 0.0140984 0.00704918 − 0.999975i \(-0.497756\pi\)
0.00704918 + 0.999975i \(0.497756\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.45326 0.263017
\(88\) 0 0
\(89\) 4.12844 0.437614 0.218807 − 0.975768i \(-0.429783\pi\)
0.218807 + 0.975768i \(0.429783\pi\)
\(90\) 0 0
\(91\) 8.45326 0.886143
\(92\) 0 0
\(93\) 5.79590 0.601007
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.4354 −1.16109 −0.580547 − 0.814227i \(-0.697161\pi\)
−0.580547 + 0.814227i \(0.697161\pi\)
\(98\) 0 0
\(99\) 0.662410 0.0665747
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.1.3 ✓ 3
4.3 odd 2 4400.2.a.ca.1.1 3
5.2 odd 4 2200.2.b.l.1849.2 6
5.3 odd 4 2200.2.b.l.1849.5 6
5.4 even 2 2200.2.a.w.1.1 yes 3
20.3 even 4 4400.2.b.bc.4049.2 6
20.7 even 4 4400.2.b.bc.4049.5 6
20.19 odd 2 4400.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.t.1.3 ✓ 3 1.1 even 1 trivial
2200.2.a.w.1.1 yes 3 5.4 even 2
2200.2.b.l.1849.2 6 5.2 odd 4
2200.2.b.l.1849.5 6 5.3 odd 4
4400.2.a.bx.1.3 3 20.19 odd 2
4400.2.a.ca.1.1 3 4.3 odd 2
4400.2.b.bc.4049.2 6 20.3 even 4
4400.2.b.bc.4049.5 6 20.7 even 4