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Properties

Label 483.2.a.i.1.2
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 1.2
Root \(0.509552\) 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)
 
\(f(q)\) \(=\) \(q-0.509552 q^{2} -1.00000 q^{3} -1.74036 q^{4} +4.41546 q^{5} +0.509552 q^{6} +1.00000 q^{7} +1.90591 q^{8} +1.00000 q^{9} -2.24991 q^{10} -1.67510 q^{11} +1.74036 q^{12} -4.66537 q^{13} -0.509552 q^{14} -4.41546 q^{15} +2.50955 q^{16} +6.24991 q^{17} -0.509552 q^{18} -0.694209 q^{19} -7.68447 q^{20} -1.00000 q^{21} +0.853553 q^{22} -1.00000 q^{23} -1.90591 q^{24} +14.4963 q^{25} +2.37725 q^{26} -1.00000 q^{27} -1.74036 q^{28} +5.60012 q^{29} +2.24991 q^{30} +4.24991 q^{31} -5.09056 q^{32} +1.67510 q^{33} -3.18466 q^{34} +4.41546 q^{35} -1.74036 q^{36} +9.26901 q^{37} +0.353736 q^{38} +4.66537 q^{39} +8.41546 q^{40} -5.15582 q^{41} +0.509552 q^{42} +4.20376 q^{43} +2.91528 q^{44} +4.41546 q^{45} +0.509552 q^{46} -1.92501 q^{47} -2.50955 q^{48} +1.00000 q^{49} -7.38662 q^{50} -6.24991 q^{51} +8.11940 q^{52} -1.84066 q^{53} +0.509552 q^{54} -7.39636 q^{55} +1.90591 q^{56} +0.694209 q^{57} -2.85355 q^{58} +9.39779 q^{59} +7.68447 q^{60} +7.72125 q^{61} -2.16555 q^{62} +1.00000 q^{63} -2.42520 q^{64} -20.5998 q^{65} -0.853553 q^{66} -8.22728 q^{67} -10.8771 q^{68} +1.00000 q^{69} -2.24991 q^{70} -10.6654 q^{71} +1.90591 q^{72} -11.9117 q^{73} -4.72305 q^{74} -14.4963 q^{75} +1.20817 q^{76} -1.67510 q^{77} -2.37725 q^{78} +0.581012 q^{79} +11.0808 q^{80} +1.00000 q^{81} +2.62716 q^{82} +6.95032 q^{83} +1.74036 q^{84} +27.5962 q^{85} -2.14204 q^{86} -5.60012 q^{87} -3.19259 q^{88} -13.4963 q^{89} -2.24991 q^{90} -4.66537 q^{91} +1.74036 q^{92} -4.24991 q^{93} +0.980895 q^{94} -3.06525 q^{95} +5.09056 q^{96} +10.0999 q^{97} -0.509552 q^{98} -1.67510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8} + 4 q^{9} + 4 q^{10} - 5 q^{11} - 4 q^{12} + 7 q^{13} - 5 q^{15} + 8 q^{16} + 12 q^{17} + 3 q^{19} - q^{20} - 4 q^{21} - q^{22} - 4 q^{23} + 3 q^{24}+ \cdots - 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.509552 −0.360308 −0.180154 − 0.983638i \(-0.557660\pi\)
−0.180154 + 0.983638i \(0.557660\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.74036 −0.870178
\(5\) 4.41546 1.97465 0.987327 − 0.158699i \(-0.0507301\pi\)
0.987327 + 0.158699i \(0.0507301\pi\)
\(6\) 0.509552 0.208024
\(7\) 1.00000 0.377964
\(8\) 1.90591 0.673840
\(9\) 1.00000 0.333333
\(10\) −2.24991 −0.711484
\(11\) −1.67510 −0.505063 −0.252531 − 0.967589i \(-0.581263\pi\)
−0.252531 + 0.967589i \(0.581263\pi\)
\(12\) 1.74036 0.502398
\(13\) −4.66537 −1.29394 −0.646970 − 0.762515i \(-0.723964\pi\)
−0.646970 + 0.762515i \(0.723964\pi\)
\(14\) −0.509552 −0.136184
\(15\) −4.41546 −1.14007
\(16\) 2.50955 0.627388
\(17\) 6.24991 1.51583 0.757913 − 0.652356i \(-0.226219\pi\)
0.757913 + 0.652356i \(0.226219\pi\)
\(18\) −0.509552 −0.120103
\(19\) −0.694209 −0.159262 −0.0796312 − 0.996824i \(-0.525374\pi\)
−0.0796312 + 0.996824i \(0.525374\pi\)
\(20\) −7.68447 −1.71830
\(21\) −1.00000 −0.218218
\(22\) 0.853553 0.181978
\(23\) −1.00000 −0.208514
\(24\) −1.90591 −0.389042
\(25\) 14.4963 2.89926
\(26\) 2.37725 0.466217
\(27\) −1.00000 −0.192450
\(28\) −1.74036 −0.328896
\(29\) 5.60012 1.03992 0.519958 − 0.854192i \(-0.325948\pi\)
0.519958 + 0.854192i \(0.325948\pi\)
\(30\) 2.24991 0.410775
\(31\) 4.24991 0.763306 0.381653 − 0.924306i \(-0.375355\pi\)
0.381653 + 0.924306i \(0.375355\pi\)
\(32\) −5.09056 −0.899893
\(33\) 1.67510 0.291598
\(34\) −3.18466 −0.546164
\(35\) 4.41546 0.746349
\(36\) −1.74036 −0.290059
\(37\) 9.26901 1.52382 0.761908 − 0.647685i \(-0.224263\pi\)
0.761908 + 0.647685i \(0.224263\pi\)
\(38\) 0.353736 0.0573835
\(39\) 4.66537 0.747057
\(40\) 8.41546 1.33060
\(41\) −5.15582 −0.805203 −0.402602 − 0.915375i \(-0.631894\pi\)
−0.402602 + 0.915375i \(0.631894\pi\)
\(42\) 0.509552 0.0786257
\(43\) 4.20376 0.641068 0.320534 − 0.947237i \(-0.396138\pi\)
0.320534 + 0.947237i \(0.396138\pi\)
\(44\) 2.91528 0.439495
\(45\) 4.41546 0.658218
\(46\) 0.509552 0.0751294
\(47\) −1.92501 −0.280792 −0.140396 − 0.990095i \(-0.544838\pi\)
−0.140396 + 0.990095i \(0.544838\pi\)
\(48\) −2.50955 −0.362223
\(49\) 1.00000 0.142857
\(50\) −7.38662 −1.04463
\(51\) −6.24991 −0.875162
\(52\) 8.11940 1.12596
\(53\) −1.84066 −0.252833 −0.126417 − 0.991977i \(-0.540348\pi\)
−0.126417 + 0.991977i \(0.540348\pi\)
\(54\) 0.509552 0.0693413
\(55\) −7.39636 −0.997324
\(56\) 1.90591 0.254688
\(57\) 0.694209 0.0919502
\(58\) −2.85355 −0.374690
\(59\) 9.39779 1.22349 0.611744 − 0.791056i \(-0.290468\pi\)
0.611744 + 0.791056i \(0.290468\pi\)
\(60\) 7.68447 0.992061
\(61\) 7.72125 0.988605 0.494302 − 0.869290i \(-0.335424\pi\)
0.494302 + 0.869290i \(0.335424\pi\)
\(62\) −2.16555 −0.275025
\(63\) 1.00000 0.125988
\(64\) −2.42520 −0.303149
\(65\) −20.5998 −2.55508
\(66\) −0.853553 −0.105065
\(67\) −8.22728 −1.00512 −0.502561 − 0.864542i \(-0.667609\pi\)
−0.502561 + 0.864542i \(0.667609\pi\)
\(68\) −10.8771 −1.31904
\(69\) 1.00000 0.120386
\(70\) −2.24991 −0.268916
\(71\) −10.6654 −1.26575 −0.632873 − 0.774255i \(-0.718125\pi\)
−0.632873 + 0.774255i \(0.718125\pi\)
\(72\) 1.90591 0.224613
\(73\) −11.9117 −1.39416 −0.697082 − 0.716991i \(-0.745519\pi\)
−0.697082 + 0.716991i \(0.745519\pi\)
\(74\) −4.72305 −0.549043
\(75\) −14.4963 −1.67389
\(76\) 1.20817 0.138587
\(77\) −1.67510 −0.190896
\(78\) −2.37725 −0.269171
\(79\) 0.581012 0.0653689 0.0326845 − 0.999466i \(-0.489594\pi\)
0.0326845 + 0.999466i \(0.489594\pi\)
\(80\) 11.0808 1.23887
\(81\) 1.00000 0.111111
\(82\) 2.62716 0.290121
\(83\) 6.95032 0.762897 0.381449 − 0.924390i \(-0.375425\pi\)
0.381449 + 0.924390i \(0.375425\pi\)
\(84\) 1.74036 0.189888
\(85\) 27.5962 2.99323
\(86\) −2.14204 −0.230982
\(87\) −5.60012 −0.600396
\(88\) −3.19259 −0.340332
\(89\) −13.4963 −1.43060 −0.715302 − 0.698816i \(-0.753711\pi\)
−0.715302 + 0.698816i \(0.753711\pi\)
\(90\) −2.24991 −0.237161
\(91\) −4.66537 −0.489064
\(92\) 1.74036 0.181445
\(93\) −4.24991 −0.440695
\(94\) 0.980895 0.101172
\(95\) −3.06525 −0.314488
\(96\) 5.09056 0.519554
\(97\) 10.0999 1.02549 0.512746 − 0.858540i \(-0.328628\pi\)
0.512746 + 0.858540i \(0.328628\pi\)
\(98\) −0.509552 −0.0514726
\(99\) −1.67510 −0.168354
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.1.2 ✓ 4
3.2 odd 2 1449.2.a.p.1.3 4
4.3 odd 2 7728.2.a.cd.1.4 4
7.6 odd 2 3381.2.a.w.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.i.1.2 ✓ 4 1.1 even 1 trivial
1449.2.a.p.1.3 4 3.2 odd 2
3381.2.a.w.1.2 4 7.6 odd 2
7728.2.a.cd.1.4 4 4.3 odd 2