Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(1,513)] 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("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.09632562369\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3 + \sqrt{6}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 3 \) 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 \(0.741964\) of defining polynomial
Character \(\chi\) \(=\) 513.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.741964 q^{2} -1.44949 q^{4} -3.30136 q^{5} +4.44949 q^{7} -2.55940 q^{8} -2.44949 q^{10} +4.04332 q^{11} +2.00000 q^{13} +3.30136 q^{14} +1.00000 q^{16} +6.60272 q^{17} +1.00000 q^{19} +4.78529 q^{20} +3.00000 q^{22} -4.78529 q^{23} +5.89898 q^{25} +1.48393 q^{26} -6.44949 q^{28} -2.55940 q^{29} +7.44949 q^{31} +5.86076 q^{32} +4.89898 q^{34} -14.6894 q^{35} +4.44949 q^{37} +0.741964 q^{38} +8.44949 q^{40} -3.70982 q^{41} -5.34847 q^{43} -5.86076 q^{44} -3.55051 q^{46} -0.741964 q^{47} +12.7980 q^{49} +4.37683 q^{50} -2.89898 q^{52} +10.9795 q^{53} -13.3485 q^{55} -11.3880 q^{56} -1.89898 q^{58} -11.3880 q^{59} -8.34847 q^{61} +5.52725 q^{62} +2.34847 q^{64} -6.60272 q^{65} -2.34847 q^{67} -9.57058 q^{68} -10.8990 q^{70} +10.2376 q^{71} -8.34847 q^{73} +3.30136 q^{74} -1.44949 q^{76} +17.9907 q^{77} +5.00000 q^{79} -3.30136 q^{80} -2.75255 q^{82} -5.86076 q^{83} -21.7980 q^{85} -3.96837 q^{86} -10.3485 q^{88} +5.52725 q^{89} +8.89898 q^{91} +6.93623 q^{92} -0.550510 q^{94} -3.30136 q^{95} +5.55051 q^{97} +9.49562 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 8 q^{7} + 8 q^{13} + 4 q^{16} + 4 q^{19} + 12 q^{22} + 4 q^{25} - 16 q^{28} + 20 q^{31} + 8 q^{37} + 24 q^{40} + 8 q^{43} - 24 q^{46} + 12 q^{49} + 8 q^{52} - 24 q^{55} + 12 q^{58} - 4 q^{61}+ \cdots + 32 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.741964 0.524648 0.262324 − 0.964980i \(-0.415511\pi\)
0.262324 + 0.964980i \(0.415511\pi\)
\(3\) 0 0
\(4\) −1.44949 −0.724745
\(5\) −3.30136 −1.47641 −0.738207 − 0.674575i \(-0.764327\pi\)
−0.738207 + 0.674575i \(0.764327\pi\)
\(6\) 0 0
\(7\) 4.44949 1.68175 0.840875 − 0.541230i \(-0.182041\pi\)
0.840875 + 0.541230i \(0.182041\pi\)
\(8\) −2.55940 −0.904883
\(9\) 0 0
\(10\) −2.44949 −0.774597
\(11\) 4.04332 1.21911 0.609554 − 0.792745i \(-0.291349\pi\)
0.609554 + 0.792745i \(0.291349\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 3.30136 0.882326
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.60272 1.60139 0.800697 − 0.599069i \(-0.204462\pi\)
0.800697 + 0.599069i \(0.204462\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 4.78529 1.07002
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) −4.78529 −0.997801 −0.498901 − 0.866659i \(-0.666263\pi\)
−0.498901 + 0.866659i \(0.666263\pi\)
\(24\) 0 0
\(25\) 5.89898 1.17980
\(26\) 1.48393 0.291022
\(27\) 0 0
\(28\) −6.44949 −1.21884
\(29\) −2.55940 −0.475268 −0.237634 − 0.971355i \(-0.576372\pi\)
−0.237634 + 0.971355i \(0.576372\pi\)
\(30\) 0 0
\(31\) 7.44949 1.33797 0.668984 − 0.743277i \(-0.266729\pi\)
0.668984 + 0.743277i \(0.266729\pi\)
\(32\) 5.86076 1.03605
\(33\) 0 0
\(34\) 4.89898 0.840168
\(35\) −14.6894 −2.48296
\(36\) 0 0
\(37\) 4.44949 0.731492 0.365746 − 0.930715i \(-0.380814\pi\)
0.365746 + 0.930715i \(0.380814\pi\)
\(38\) 0.741964 0.120362
\(39\) 0 0
\(40\) 8.44949 1.33598
\(41\) −3.70982 −0.579376 −0.289688 − 0.957121i \(-0.593552\pi\)
−0.289688 + 0.957121i \(0.593552\pi\)
\(42\) 0 0
\(43\) −5.34847 −0.815634 −0.407817 − 0.913064i \(-0.633710\pi\)
−0.407817 + 0.913064i \(0.633710\pi\)
\(44\) −5.86076 −0.883542
\(45\) 0 0
\(46\) −3.55051 −0.523494
\(47\) −0.741964 −0.108227 −0.0541133 − 0.998535i \(-0.517233\pi\)
−0.0541133 + 0.998535i \(0.517233\pi\)
\(48\) 0 0
\(49\) 12.7980 1.82828
\(50\) 4.37683 0.618977
\(51\) 0 0
\(52\) −2.89898 −0.402016
\(53\) 10.9795 1.50816 0.754079 − 0.656784i \(-0.228084\pi\)
0.754079 + 0.656784i \(0.228084\pi\)
\(54\) 0 0
\(55\) −13.3485 −1.79991
\(56\) −11.3880 −1.52179
\(57\) 0 0
\(58\) −1.89898 −0.249348
\(59\) −11.3880 −1.48259 −0.741296 − 0.671178i \(-0.765789\pi\)
−0.741296 + 0.671178i \(0.765789\pi\)
\(60\) 0 0
\(61\) −8.34847 −1.06891 −0.534456 − 0.845196i \(-0.679483\pi\)
−0.534456 + 0.845196i \(0.679483\pi\)
\(62\) 5.52725 0.701962
\(63\) 0 0
\(64\) 2.34847 0.293559
\(65\) −6.60272 −0.818967
\(66\) 0 0
\(67\) −2.34847 −0.286911 −0.143456 − 0.989657i \(-0.545821\pi\)
−0.143456 + 0.989657i \(0.545821\pi\)
\(68\) −9.57058 −1.16060
\(69\) 0 0
\(70\) −10.8990 −1.30268
\(71\) 10.2376 1.21498 0.607489 − 0.794328i \(-0.292177\pi\)
0.607489 + 0.794328i \(0.292177\pi\)
\(72\) 0 0
\(73\) −8.34847 −0.977114 −0.488557 − 0.872532i \(-0.662477\pi\)
−0.488557 + 0.872532i \(0.662477\pi\)
\(74\) 3.30136 0.383775
\(75\) 0 0
\(76\) −1.44949 −0.166268
\(77\) 17.9907 2.05023
\(78\) 0 0
\(79\) 5.00000 0.562544 0.281272 − 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −3.30136 −0.369103
\(81\) 0 0
\(82\) −2.75255 −0.303968
\(83\) −5.86076 −0.643302 −0.321651 − 0.946858i \(-0.604238\pi\)
−0.321651 + 0.946858i \(0.604238\pi\)
\(84\) 0 0
\(85\) −21.7980 −2.36432
\(86\) −3.96837 −0.427920
\(87\) 0 0
\(88\) −10.3485 −1.10315
\(89\) 5.52725 0.585887 0.292944 − 0.956130i \(-0.405365\pi\)
0.292944 + 0.956130i \(0.405365\pi\)
\(90\) 0 0
\(91\) 8.89898 0.932867
\(92\) 6.93623 0.723152
\(93\) 0 0
\(94\) −0.550510 −0.0567808
\(95\) −3.30136 −0.338712
\(96\) 0 0
\(97\) 5.55051 0.563569 0.281784 − 0.959478i \(-0.409074\pi\)
0.281784 + 0.959478i \(0.409074\pi\)
\(98\) 9.49562 0.959203
\(99\) 0 0
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 513.2.a.h.1.3 yes 4
3.2 odd 2 inner 513.2.a.h.1.2 ✓ 4
4.3 odd 2 8208.2.a.bu.1.1 4
12.11 even 2 8208.2.a.bu.1.4 4
19.18 odd 2 9747.2.a.be.1.2 4
57.56 even 2 9747.2.a.be.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
513.2.a.h.1.2 ✓ 4 3.2 odd 2 inner
513.2.a.h.1.3 yes 4 1.1 even 1 trivial
8208.2.a.bu.1.1 4 4.3 odd 2
8208.2.a.bu.1.4 4 12.11 even 2
9747.2.a.be.1.2 4 19.18 odd 2
9747.2.a.be.1.3 4 57.56 even 2