Properties

Label 483.2.a.c.1.2
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 = [2,-3,2,3,-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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -1.38197 q^{5} -0.381966 q^{6} +1.00000 q^{7} +1.47214 q^{8} +1.00000 q^{9} +0.527864 q^{10} -5.47214 q^{11} -1.85410 q^{12} -2.38197 q^{13} -0.381966 q^{14} -1.38197 q^{15} +3.14590 q^{16} -1.00000 q^{17} -0.381966 q^{18} -3.00000 q^{19} +2.56231 q^{20} +1.00000 q^{21} +2.09017 q^{22} -1.00000 q^{23} +1.47214 q^{24} -3.09017 q^{25} +0.909830 q^{26} +1.00000 q^{27} -1.85410 q^{28} -7.47214 q^{29} +0.527864 q^{30} -3.76393 q^{31} -4.14590 q^{32} -5.47214 q^{33} +0.381966 q^{34} -1.38197 q^{35} -1.85410 q^{36} +1.47214 q^{37} +1.14590 q^{38} -2.38197 q^{39} -2.03444 q^{40} -4.70820 q^{41} -0.381966 q^{42} +8.09017 q^{43} +10.1459 q^{44} -1.38197 q^{45} +0.381966 q^{46} +1.70820 q^{47} +3.14590 q^{48} +1.00000 q^{49} +1.18034 q^{50} -1.00000 q^{51} +4.41641 q^{52} -3.38197 q^{53} -0.381966 q^{54} +7.56231 q^{55} +1.47214 q^{56} -3.00000 q^{57} +2.85410 q^{58} -6.14590 q^{59} +2.56231 q^{60} +13.7984 q^{61} +1.43769 q^{62} +1.00000 q^{63} -4.70820 q^{64} +3.29180 q^{65} +2.09017 q^{66} +4.14590 q^{67} +1.85410 q^{68} -1.00000 q^{69} +0.527864 q^{70} -3.90983 q^{71} +1.47214 q^{72} +2.70820 q^{73} -0.562306 q^{74} -3.09017 q^{75} +5.56231 q^{76} -5.47214 q^{77} +0.909830 q^{78} +0.527864 q^{79} -4.34752 q^{80} +1.00000 q^{81} +1.79837 q^{82} -3.00000 q^{83} -1.85410 q^{84} +1.38197 q^{85} -3.09017 q^{86} -7.47214 q^{87} -8.05573 q^{88} +3.14590 q^{89} +0.527864 q^{90} -2.38197 q^{91} +1.85410 q^{92} -3.76393 q^{93} -0.652476 q^{94} +4.14590 q^{95} -4.14590 q^{96} +5.00000 q^{97} -0.381966 q^{98} -5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 5 q^{5} - 3 q^{6} + 2 q^{7} - 6 q^{8} + 2 q^{9} + 10 q^{10} - 2 q^{11} + 3 q^{12} - 7 q^{13} - 3 q^{14} - 5 q^{15} + 13 q^{16} - 2 q^{17} - 3 q^{18} - 6 q^{19} - 15 q^{20}+ \cdots - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.381966 −0.270091 −0.135045 − 0.990839i \(-0.543118\pi\)
−0.135045 + 0.990839i \(0.543118\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.85410 −0.927051
\(5\) −1.38197 −0.618034 −0.309017 − 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(6\) −0.381966 −0.155937
\(7\) 1.00000 0.377964
\(8\) 1.47214 0.520479
\(9\) 1.00000 0.333333
\(10\) 0.527864 0.166925
\(11\) −5.47214 −1.64991 −0.824956 − 0.565198i \(-0.808800\pi\)
−0.824956 + 0.565198i \(0.808800\pi\)
\(12\) −1.85410 −0.535233
\(13\) −2.38197 −0.660639 −0.330319 − 0.943869i \(-0.607156\pi\)
−0.330319 + 0.943869i \(0.607156\pi\)
\(14\) −0.381966 −0.102085
\(15\) −1.38197 −0.356822
\(16\) 3.14590 0.786475
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) −0.381966 −0.0900303
\(19\) −3.00000 −0.688247 −0.344124 − 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 2.56231 0.572949
\(21\) 1.00000 0.218218
\(22\) 2.09017 0.445626
\(23\) −1.00000 −0.208514
\(24\) 1.47214 0.300498
\(25\) −3.09017 −0.618034
\(26\) 0.909830 0.178432
\(27\) 1.00000 0.192450
\(28\) −1.85410 −0.350392
\(29\) −7.47214 −1.38754 −0.693770 − 0.720196i \(-0.744052\pi\)
−0.693770 + 0.720196i \(0.744052\pi\)
\(30\) 0.527864 0.0963743
\(31\) −3.76393 −0.676022 −0.338011 − 0.941142i \(-0.609754\pi\)
−0.338011 + 0.941142i \(0.609754\pi\)
\(32\) −4.14590 −0.732898
\(33\) −5.47214 −0.952577
\(34\) 0.381966 0.0655066
\(35\) −1.38197 −0.233595
\(36\) −1.85410 −0.309017
\(37\) 1.47214 0.242018 0.121009 − 0.992651i \(-0.461387\pi\)
0.121009 + 0.992651i \(0.461387\pi\)
\(38\) 1.14590 0.185889
\(39\) −2.38197 −0.381420
\(40\) −2.03444 −0.321674
\(41\) −4.70820 −0.735298 −0.367649 − 0.929965i \(-0.619837\pi\)
−0.367649 + 0.929965i \(0.619837\pi\)
\(42\) −0.381966 −0.0589386
\(43\) 8.09017 1.23374 0.616870 − 0.787065i \(-0.288401\pi\)
0.616870 + 0.787065i \(0.288401\pi\)
\(44\) 10.1459 1.52955
\(45\) −1.38197 −0.206011
\(46\) 0.381966 0.0563178
\(47\) 1.70820 0.249167 0.124584 − 0.992209i \(-0.460241\pi\)
0.124584 + 0.992209i \(0.460241\pi\)
\(48\) 3.14590 0.454071
\(49\) 1.00000 0.142857
\(50\) 1.18034 0.166925
\(51\) −1.00000 −0.140028
\(52\) 4.41641 0.612446
\(53\) −3.38197 −0.464549 −0.232274 − 0.972650i \(-0.574617\pi\)
−0.232274 + 0.972650i \(0.574617\pi\)
\(54\) −0.381966 −0.0519790
\(55\) 7.56231 1.01970
\(56\) 1.47214 0.196722
\(57\) −3.00000 −0.397360
\(58\) 2.85410 0.374762
\(59\) −6.14590 −0.800128 −0.400064 − 0.916487i \(-0.631012\pi\)
−0.400064 + 0.916487i \(0.631012\pi\)
\(60\) 2.56231 0.330792
\(61\) 13.7984 1.76670 0.883350 − 0.468713i \(-0.155282\pi\)
0.883350 + 0.468713i \(0.155282\pi\)
\(62\) 1.43769 0.182587
\(63\) 1.00000 0.125988
\(64\) −4.70820 −0.588525
\(65\) 3.29180 0.408297
\(66\) 2.09017 0.257282
\(67\) 4.14590 0.506502 0.253251 − 0.967401i \(-0.418500\pi\)
0.253251 + 0.967401i \(0.418500\pi\)
\(68\) 1.85410 0.224843
\(69\) −1.00000 −0.120386
\(70\) 0.527864 0.0630918
\(71\) −3.90983 −0.464011 −0.232006 − 0.972714i \(-0.574529\pi\)
−0.232006 + 0.972714i \(0.574529\pi\)
\(72\) 1.47214 0.173493
\(73\) 2.70820 0.316971 0.158486 − 0.987361i \(-0.449339\pi\)
0.158486 + 0.987361i \(0.449339\pi\)
\(74\) −0.562306 −0.0653667
\(75\) −3.09017 −0.356822
\(76\) 5.56231 0.638040
\(77\) −5.47214 −0.623608
\(78\) 0.909830 0.103018
\(79\) 0.527864 0.0593893 0.0296947 − 0.999559i \(-0.490547\pi\)
0.0296947 + 0.999559i \(0.490547\pi\)
\(80\) −4.34752 −0.486068
\(81\) 1.00000 0.111111
\(82\) 1.79837 0.198597
\(83\) −3.00000 −0.329293 −0.164646 − 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) −1.85410 −0.202299
\(85\) 1.38197 0.149895
\(86\) −3.09017 −0.333222
\(87\) −7.47214 −0.801097
\(88\) −8.05573 −0.858743
\(89\) 3.14590 0.333465 0.166732 − 0.986002i \(-0.446678\pi\)
0.166732 + 0.986002i \(0.446678\pi\)
\(90\) 0.527864 0.0556418
\(91\) −2.38197 −0.249698
\(92\) 1.85410 0.193303
\(93\) −3.76393 −0.390302
\(94\) −0.652476 −0.0672977
\(95\) 4.14590 0.425360
\(96\) −4.14590 −0.423139
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) −0.381966 −0.0385844
\(99\) −5.47214 −0.549970
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.c.1.2 ✓ 2
3.2 odd 2 1449.2.a.k.1.1 2
4.3 odd 2 7728.2.a.v.1.2 2
7.6 odd 2 3381.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.c.1.2 ✓ 2 1.1 even 1 trivial
1449.2.a.k.1.1 2 3.2 odd 2
3381.2.a.n.1.2 2 7.6 odd 2
7728.2.a.v.1.2 2 4.3 odd 2