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Properties

Label 4650.2.a.cp.1.3
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)] 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("4650.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,3,3,0,3,6,3,3,0,8,3,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1708.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.21018\) of defining polynomial
Character \(\chi\) \(=\) 4650.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.21018 q^{11} +1.00000 q^{12} +0.789816 q^{13} +2.00000 q^{14} +1.00000 q^{16} -0.115086 q^{17} +1.00000 q^{18} +0.115086 q^{19} +2.00000 q^{21} +5.21018 q^{22} -4.42037 q^{23} +1.00000 q^{24} +0.789816 q^{26} +1.00000 q^{27} +2.00000 q^{28} +4.42037 q^{29} -1.00000 q^{31} +1.00000 q^{32} +5.21018 q^{33} -0.115086 q^{34} +1.00000 q^{36} +6.61056 q^{37} +0.115086 q^{38} +0.789816 q^{39} +2.00000 q^{41} +2.00000 q^{42} -8.61056 q^{43} +5.21018 q^{44} -4.42037 q^{46} +0.115086 q^{47} +1.00000 q^{48} -3.00000 q^{49} -0.115086 q^{51} +0.789816 q^{52} -0.190196 q^{53} +1.00000 q^{54} +2.00000 q^{56} +0.115086 q^{57} +4.42037 q^{58} -4.19020 q^{59} +12.4955 q^{61} -1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +5.21018 q^{66} +5.82075 q^{67} -0.115086 q^{68} -4.42037 q^{69} -13.8207 q^{71} +1.00000 q^{72} -10.8407 q^{73} +6.61056 q^{74} +0.115086 q^{76} +10.4204 q^{77} +0.789816 q^{78} +2.30528 q^{79} +1.00000 q^{81} +2.00000 q^{82} +15.1460 q^{83} +2.00000 q^{84} -8.61056 q^{86} +4.42037 q^{87} +5.21018 q^{88} -2.23017 q^{89} +1.57963 q^{91} -4.42037 q^{92} -1.00000 q^{93} +0.115086 q^{94} +1.00000 q^{96} -9.21018 q^{97} -3.00000 q^{98} +5.21018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{6} + 6 q^{7} + 3 q^{8} + 3 q^{9} + 8 q^{11} + 3 q^{12} + 10 q^{13} + 6 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{18} - 2 q^{19} + 6 q^{21} + 8 q^{22} + 2 q^{23} + 3 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.21018 1.57093 0.785465 − 0.618906i \(-0.212424\pi\)
0.785465 + 0.618906i \(0.212424\pi\)
\(12\) 1.00000 0.288675
\(13\) 0.789816 0.219056 0.109528 − 0.993984i \(-0.465066\pi\)
0.109528 + 0.993984i \(0.465066\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.115086 −0.0279125 −0.0139562 − 0.999903i \(-0.504443\pi\)
−0.0139562 + 0.999903i \(0.504443\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.115086 0.0264026 0.0132013 − 0.999913i \(-0.495798\pi\)
0.0132013 + 0.999913i \(0.495798\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 5.21018 1.11081
\(23\) −4.42037 −0.921710 −0.460855 − 0.887475i \(-0.652457\pi\)
−0.460855 + 0.887475i \(0.652457\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 0.789816 0.154896
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 4.42037 0.820842 0.410421 − 0.911896i \(-0.365382\pi\)
0.410421 + 0.911896i \(0.365382\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) 5.21018 0.906977
\(34\) −0.115086 −0.0197371
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.61056 1.08677 0.543385 − 0.839484i \(-0.317142\pi\)
0.543385 + 0.839484i \(0.317142\pi\)
\(38\) 0.115086 0.0186694
\(39\) 0.789816 0.126472
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 2.00000 0.308607
\(43\) −8.61056 −1.31310 −0.656549 − 0.754283i \(-0.727985\pi\)
−0.656549 + 0.754283i \(0.727985\pi\)
\(44\) 5.21018 0.785465
\(45\) 0 0
\(46\) −4.42037 −0.651748
\(47\) 0.115086 0.0167870 0.00839352 − 0.999965i \(-0.497328\pi\)
0.00839352 + 0.999965i \(0.497328\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −0.115086 −0.0161153
\(52\) 0.789816 0.109528
\(53\) −0.190196 −0.0261254 −0.0130627 − 0.999915i \(-0.504158\pi\)
−0.0130627 + 0.999915i \(0.504158\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 0.115086 0.0152435
\(58\) 4.42037 0.580423
\(59\) −4.19020 −0.545517 −0.272759 − 0.962083i \(-0.587936\pi\)
−0.272759 + 0.962083i \(0.587936\pi\)
\(60\) 0 0
\(61\) 12.4955 1.59988 0.799941 − 0.600079i \(-0.204864\pi\)
0.799941 + 0.600079i \(0.204864\pi\)
\(62\) −1.00000 −0.127000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.21018 0.641329
\(67\) 5.82075 0.711118 0.355559 − 0.934654i \(-0.384291\pi\)
0.355559 + 0.934654i \(0.384291\pi\)
\(68\) −0.115086 −0.0139562
\(69\) −4.42037 −0.532150
\(70\) 0 0
\(71\) −13.8207 −1.64022 −0.820111 − 0.572205i \(-0.806088\pi\)
−0.820111 + 0.572205i \(0.806088\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.8407 −1.26881 −0.634406 − 0.773000i \(-0.718755\pi\)
−0.634406 + 0.773000i \(0.718755\pi\)
\(74\) 6.61056 0.768462
\(75\) 0 0
\(76\) 0.115086 0.0132013
\(77\) 10.4204 1.18751
\(78\) 0.789816 0.0894290
\(79\) 2.30528 0.259365 0.129682 − 0.991556i \(-0.458604\pi\)
0.129682 + 0.991556i \(0.458604\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 15.1460 1.66249 0.831246 − 0.555905i \(-0.187628\pi\)
0.831246 + 0.555905i \(0.187628\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −8.61056 −0.928501
\(87\) 4.42037 0.473913
\(88\) 5.21018 0.555407
\(89\) −2.23017 −0.236398 −0.118199 − 0.992990i \(-0.537712\pi\)
−0.118199 + 0.992990i \(0.537712\pi\)
\(90\) 0 0
\(91\) 1.57963 0.165590
\(92\) −4.42037 −0.460855
\(93\) −1.00000 −0.103695
\(94\) 0.115086 0.0118702
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −9.21018 −0.935153 −0.467576 − 0.883953i \(-0.654873\pi\)
−0.467576 + 0.883953i \(0.654873\pi\)
\(98\) −3.00000 −0.303046
\(99\) 5.21018 0.523643
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 4650.2.a.cp.1.3 3
5.2 odd 4 930.2.d.i.559.6 yes 6
5.3 odd 4 930.2.d.i.559.3 ✓ 6
5.4 even 2 4650.2.a.ci.1.3 3
15.2 even 4 2790.2.d.j.559.1 6
15.8 even 4 2790.2.d.j.559.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.i.559.3 ✓ 6 5.3 odd 4
930.2.d.i.559.6 yes 6 5.2 odd 4
2790.2.d.j.559.1 6 15.2 even 4
2790.2.d.j.559.4 6 15.8 even 4
4650.2.a.ci.1.3 3 5.4 even 2
4650.2.a.cp.1.3 3 1.1 even 1 trivial