Properties

Label 483.2.a.i.1.3
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $0$
Dimension $4$
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] = [483,2,Mod(1,483)] 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("483.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(483, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,4,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.24197.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.700017\) of defining polynomial
Character \(\chi\) \(=\) 483.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.700017 q^{2} -1.00000 q^{3} -1.50998 q^{4} -1.15706 q^{5} -0.700017 q^{6} +1.00000 q^{7} -2.45704 q^{8} +1.00000 q^{9} -0.809960 q^{10} +3.66704 q^{11} +1.50998 q^{12} +2.34710 q^{13} +0.700017 q^{14} +1.15706 q^{15} +1.29998 q^{16} +4.80996 q^{17} +0.700017 q^{18} +7.06707 q^{19} +1.74713 q^{20} -1.00000 q^{21} +2.56699 q^{22} -1.00000 q^{23} +2.45704 q^{24} -3.66122 q^{25} +1.64301 q^{26} -1.00000 q^{27} -1.50998 q^{28} -6.52411 q^{29} +0.809960 q^{30} +2.80996 q^{31} +5.82409 q^{32} -3.66704 q^{33} +3.36705 q^{34} -1.15706 q^{35} -1.50998 q^{36} +5.40993 q^{37} +4.94707 q^{38} -2.34710 q^{39} +2.84294 q^{40} +0.647082 q^{41} -0.700017 q^{42} -4.76709 q^{43} -5.53714 q^{44} -1.15706 q^{45} -0.700017 q^{46} +4.85708 q^{47} -1.29998 q^{48} +1.00000 q^{49} -2.56291 q^{50} -4.80996 q^{51} -3.54406 q^{52} +7.63405 q^{53} -0.700017 q^{54} -4.24297 q^{55} -2.45704 q^{56} -7.06707 q^{57} -4.56699 q^{58} -10.2142 q^{59} -1.74713 q^{60} +9.91001 q^{61} +1.96702 q^{62} +1.00000 q^{63} +1.47700 q^{64} -2.71573 q^{65} -2.56699 q^{66} +6.07114 q^{67} -7.26293 q^{68} +1.00000 q^{69} -0.809960 q^{70} -3.65290 q^{71} -2.45704 q^{72} +11.8183 q^{73} +3.78704 q^{74} +3.66122 q^{75} -10.6711 q^{76} +3.66704 q^{77} -1.64301 q^{78} -9.12408 q^{79} -1.50416 q^{80} +1.00000 q^{81} +0.452968 q^{82} -15.8582 q^{83} +1.50998 q^{84} -5.56541 q^{85} -3.33704 q^{86} +6.52411 q^{87} -9.01006 q^{88} +4.66122 q^{89} -0.809960 q^{90} +2.34710 q^{91} +1.50998 q^{92} -2.80996 q^{93} +3.40003 q^{94} -8.17701 q^{95} -5.82409 q^{96} -4.90419 q^{97} +0.700017 q^{98} +3.66704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8} + 4 q^{9} + 4 q^{10} - 5 q^{11} - 4 q^{12} + 7 q^{13} - 5 q^{15} + 8 q^{16} + 12 q^{17} + 3 q^{19} - q^{20} - 4 q^{21} - q^{22} - 4 q^{23} + 3 q^{24}+ \cdots - 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.700017 0.494986 0.247493 − 0.968890i \(-0.420393\pi\)
0.247493 + 0.968890i \(0.420393\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.50998 −0.754988
\(5\) −1.15706 −0.517452 −0.258726 − 0.965951i \(-0.583303\pi\)
−0.258726 + 0.965951i \(0.583303\pi\)
\(6\) −0.700017 −0.285781
\(7\) 1.00000 0.377964
\(8\) −2.45704 −0.868696
\(9\) 1.00000 0.333333
\(10\) −0.809960 −0.256132
\(11\) 3.66704 1.10565 0.552826 − 0.833296i \(-0.313549\pi\)
0.552826 + 0.833296i \(0.313549\pi\)
\(12\) 1.50998 0.435893
\(13\) 2.34710 0.650968 0.325484 − 0.945548i \(-0.394473\pi\)
0.325484 + 0.945548i \(0.394473\pi\)
\(14\) 0.700017 0.187087
\(15\) 1.15706 0.298751
\(16\) 1.29998 0.324996
\(17\) 4.80996 1.16659 0.583293 − 0.812262i \(-0.301764\pi\)
0.583293 + 0.812262i \(0.301764\pi\)
\(18\) 0.700017 0.164995
\(19\) 7.06707 1.62130 0.810648 − 0.585533i \(-0.199115\pi\)
0.810648 + 0.585533i \(0.199115\pi\)
\(20\) 1.74713 0.390670
\(21\) −1.00000 −0.218218
\(22\) 2.56699 0.547283
\(23\) −1.00000 −0.208514
\(24\) 2.45704 0.501542
\(25\) −3.66122 −0.732243
\(26\) 1.64301 0.322220
\(27\) −1.00000 −0.192450
\(28\) −1.50998 −0.285359
\(29\) −6.52411 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(30\) 0.809960 0.147878
\(31\) 2.80996 0.504684 0.252342 − 0.967638i \(-0.418799\pi\)
0.252342 + 0.967638i \(0.418799\pi\)
\(32\) 5.82409 1.02956
\(33\) −3.66704 −0.638349
\(34\) 3.36705 0.577445
\(35\) −1.15706 −0.195579
\(36\) −1.50998 −0.251663
\(37\) 5.40993 0.889387 0.444693 − 0.895683i \(-0.353313\pi\)
0.444693 + 0.895683i \(0.353313\pi\)
\(38\) 4.94707 0.802520
\(39\) −2.34710 −0.375837
\(40\) 2.84294 0.449509
\(41\) 0.647082 0.101057 0.0505286 − 0.998723i \(-0.483909\pi\)
0.0505286 + 0.998723i \(0.483909\pi\)
\(42\) −0.700017 −0.108015
\(43\) −4.76709 −0.726974 −0.363487 − 0.931599i \(-0.618414\pi\)
−0.363487 + 0.931599i \(0.618414\pi\)
\(44\) −5.53714 −0.834755
\(45\) −1.15706 −0.172484
\(46\) −0.700017 −0.103212
\(47\) 4.85708 0.708477 0.354239 − 0.935155i \(-0.384740\pi\)
0.354239 + 0.935155i \(0.384740\pi\)
\(48\) −1.29998 −0.187636
\(49\) 1.00000 0.142857
\(50\) −2.56291 −0.362450
\(51\) −4.80996 −0.673529
\(52\) −3.54406 −0.491473
\(53\) 7.63405 1.04862 0.524309 − 0.851528i \(-0.324324\pi\)
0.524309 + 0.851528i \(0.324324\pi\)
\(54\) −0.700017 −0.0952602
\(55\) −4.24297 −0.572123
\(56\) −2.45704 −0.328336
\(57\) −7.06707 −0.936056
\(58\) −4.56699 −0.599675
\(59\) −10.2142 −1.32978 −0.664890 − 0.746941i \(-0.731522\pi\)
−0.664890 + 0.746941i \(0.731522\pi\)
\(60\) −1.74713 −0.225554
\(61\) 9.91001 1.26885 0.634423 − 0.772986i \(-0.281238\pi\)
0.634423 + 0.772986i \(0.281238\pi\)
\(62\) 1.96702 0.249812
\(63\) 1.00000 0.125988
\(64\) 1.47700 0.184624
\(65\) −2.71573 −0.336845
\(66\) −2.56699 −0.315974
\(67\) 6.07114 0.741708 0.370854 − 0.928691i \(-0.379065\pi\)
0.370854 + 0.928691i \(0.379065\pi\)
\(68\) −7.26293 −0.880759
\(69\) 1.00000 0.120386
\(70\) −0.809960 −0.0968088
\(71\) −3.65290 −0.433520 −0.216760 − 0.976225i \(-0.569549\pi\)
−0.216760 + 0.976225i \(0.569549\pi\)
\(72\) −2.45704 −0.289565
\(73\) 11.8183 1.38322 0.691612 − 0.722269i \(-0.256901\pi\)
0.691612 + 0.722269i \(0.256901\pi\)
\(74\) 3.78704 0.440234
\(75\) 3.66122 0.422761
\(76\) −10.6711 −1.22406
\(77\) 3.66704 0.417897
\(78\) −1.64301 −0.186034
\(79\) −9.12408 −1.02654 −0.513269 − 0.858228i \(-0.671566\pi\)
−0.513269 + 0.858228i \(0.671566\pi\)
\(80\) −1.50416 −0.168170
\(81\) 1.00000 0.111111
\(82\) 0.452968 0.0500219
\(83\) −15.8582 −1.74066 −0.870331 − 0.492468i \(-0.836095\pi\)
−0.870331 + 0.492468i \(0.836095\pi\)
\(84\) 1.50998 0.164752
\(85\) −5.56541 −0.603653
\(86\) −3.33704 −0.359842
\(87\) 6.52411 0.699458
\(88\) −9.01006 −0.960476
\(89\) 4.66122 0.494088 0.247044 − 0.969004i \(-0.420541\pi\)
0.247044 + 0.969004i \(0.420541\pi\)
\(90\) −0.809960 −0.0853773
\(91\) 2.34710 0.246043
\(92\) 1.50998 0.157426
\(93\) −2.80996 −0.291379
\(94\) 3.40003 0.350687
\(95\) −8.17701 −0.838944
\(96\) −5.82409 −0.594419
\(97\) −4.90419 −0.497945 −0.248973 − 0.968511i \(-0.580093\pi\)
−0.248973 + 0.968511i \(0.580093\pi\)
\(98\) 0.700017 0.0707124
\(99\) 3.66704 0.368551
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 483.2.a.i.1.3 ✓ 4
3.2 odd 2 1449.2.a.p.1.2 4
4.3 odd 2 7728.2.a.cd.1.1 4
7.6 odd 2 3381.2.a.w.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.i.1.3 ✓ 4 1.1 even 1 trivial
1449.2.a.p.1.2 4 3.2 odd 2
3381.2.a.w.1.3 4 7.6 odd 2
7728.2.a.cd.1.1 4 4.3 odd 2