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Properties

Label 4650.2.a.ci.1.2
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 1.2
Root \(-0.260711\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.26071 q^{11} -1.00000 q^{12} -2.73929 q^{13} +2.00000 q^{14} +1.00000 q^{16} +4.93203 q^{17} -1.00000 q^{18} +4.93203 q^{19} +2.00000 q^{21} -3.26071 q^{22} +0.521423 q^{23} +1.00000 q^{24} +2.73929 q^{26} -1.00000 q^{27} -2.00000 q^{28} +0.521423 q^{29} -1.00000 q^{31} -1.00000 q^{32} -3.26071 q^{33} -4.93203 q^{34} +1.00000 q^{36} +10.8212 q^{37} -4.93203 q^{38} +2.73929 q^{39} +2.00000 q^{41} -2.00000 q^{42} -8.82121 q^{43} +3.26071 q^{44} -0.521423 q^{46} -4.93203 q^{47} -1.00000 q^{48} -3.00000 q^{49} -4.93203 q^{51} -2.73929 q^{52} -13.3426 q^{53} +1.00000 q^{54} +2.00000 q^{56} -4.93203 q^{57} -0.521423 q^{58} +9.34264 q^{59} -9.75324 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +3.26071 q^{66} +13.5605 q^{67} +4.93203 q^{68} -0.521423 q^{69} +5.56050 q^{71} -1.00000 q^{72} +3.04285 q^{73} -10.8212 q^{74} +4.93203 q^{76} -6.52142 q^{77} -2.73929 q^{78} -6.41061 q^{79} +1.00000 q^{81} -2.00000 q^{82} +1.36776 q^{83} +2.00000 q^{84} +8.82121 q^{86} -0.521423 q^{87} -3.26071 q^{88} -11.8641 q^{89} +5.47858 q^{91} +0.521423 q^{92} +1.00000 q^{93} +4.93203 q^{94} +1.00000 q^{96} +7.26071 q^{97} +3.00000 q^{98} +3.26071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 6 q^{7} - 3 q^{8} + 3 q^{9} + 8 q^{11} - 3 q^{12} - 10 q^{13} + 6 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} - 2 q^{19} + 6 q^{21} - 8 q^{22} - 2 q^{23} + 3 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 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 \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.26071 0.983141 0.491571 − 0.870838i \(-0.336423\pi\)
0.491571 + 0.870838i \(0.336423\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.73929 −0.759742 −0.379871 − 0.925039i \(-0.624032\pi\)
−0.379871 + 0.925039i \(0.624032\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.93203 1.19619 0.598096 − 0.801424i \(-0.295924\pi\)
0.598096 + 0.801424i \(0.295924\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.93203 1.13149 0.565743 − 0.824582i \(-0.308590\pi\)
0.565743 + 0.824582i \(0.308590\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −3.26071 −0.695186
\(23\) 0.521423 0.108724 0.0543621 − 0.998521i \(-0.482687\pi\)
0.0543621 + 0.998521i \(0.482687\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.73929 0.537219
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) 0.521423 0.0968258 0.0484129 − 0.998827i \(-0.484584\pi\)
0.0484129 + 0.998827i \(0.484584\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −3.26071 −0.567617
\(34\) −4.93203 −0.845836
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.8212 1.77900 0.889498 − 0.456939i \(-0.151054\pi\)
0.889498 + 0.456939i \(0.151054\pi\)
\(38\) −4.93203 −0.800081
\(39\) 2.73929 0.438637
\(40\) 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
\(43\) −8.82121 −1.34522 −0.672611 − 0.739996i \(-0.734827\pi\)
−0.672611 + 0.739996i \(0.734827\pi\)
\(44\) 3.26071 0.491571
\(45\) 0 0
\(46\) −0.521423 −0.0768796
\(47\) −4.93203 −0.719410 −0.359705 − 0.933066i \(-0.617123\pi\)
−0.359705 + 0.933066i \(0.617123\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −4.93203 −0.690622
\(52\) −2.73929 −0.379871
\(53\) −13.3426 −1.83275 −0.916376 − 0.400319i \(-0.868899\pi\)
−0.916376 + 0.400319i \(0.868899\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −4.93203 −0.653263
\(58\) −0.521423 −0.0684662
\(59\) 9.34264 1.21631 0.608154 − 0.793819i \(-0.291910\pi\)
0.608154 + 0.793819i \(0.291910\pi\)
\(60\) 0 0
\(61\) −9.75324 −1.24877 −0.624387 − 0.781115i \(-0.714651\pi\)
−0.624387 + 0.781115i \(0.714651\pi\)
\(62\) 1.00000 0.127000
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.26071 0.401366
\(67\) 13.5605 1.65668 0.828340 − 0.560226i \(-0.189286\pi\)
0.828340 + 0.560226i \(0.189286\pi\)
\(68\) 4.93203 0.598096
\(69\) −0.521423 −0.0627719
\(70\) 0 0
\(71\) 5.56050 0.659910 0.329955 − 0.943997i \(-0.392966\pi\)
0.329955 + 0.943997i \(0.392966\pi\)
\(72\) −1.00000 −0.117851
\(73\) 3.04285 0.356138 0.178069 − 0.984018i \(-0.443015\pi\)
0.178069 + 0.984018i \(0.443015\pi\)
\(74\) −10.8212 −1.25794
\(75\) 0 0
\(76\) 4.93203 0.565743
\(77\) −6.52142 −0.743185
\(78\) −2.73929 −0.310163
\(79\) −6.41061 −0.721250 −0.360625 − 0.932711i \(-0.617437\pi\)
−0.360625 + 0.932711i \(0.617437\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 1.36776 0.150131 0.0750657 − 0.997179i \(-0.476083\pi\)
0.0750657 + 0.997179i \(0.476083\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 8.82121 0.951216
\(87\) −0.521423 −0.0559024
\(88\) −3.26071 −0.347593
\(89\) −11.8641 −1.25759 −0.628794 − 0.777572i \(-0.716451\pi\)
−0.628794 + 0.777572i \(0.716451\pi\)
\(90\) 0 0
\(91\) 5.47858 0.574311
\(92\) 0.521423 0.0543621
\(93\) 1.00000 0.103695
\(94\) 4.93203 0.508700
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 7.26071 0.737214 0.368607 − 0.929585i \(-0.379835\pi\)
0.368607 + 0.929585i \(0.379835\pi\)
\(98\) 3.00000 0.303046
\(99\) 3.26071 0.327714
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.ci.1.2 3
5.2 odd 4 930.2.d.i.559.2 ✓ 6
5.3 odd 4 930.2.d.i.559.5 yes 6
5.4 even 2 4650.2.a.cp.1.2 3
15.2 even 4 2790.2.d.j.559.5 6
15.8 even 4 2790.2.d.j.559.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.i.559.2 ✓ 6 5.2 odd 4
930.2.d.i.559.5 yes 6 5.3 odd 4
2790.2.d.j.559.2 6 15.8 even 4
2790.2.d.j.559.5 6 15.2 even 4
4650.2.a.ci.1.2 3 1.1 even 1 trivial
4650.2.a.cp.1.2 3 5.4 even 2