Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [736,2,Mod(1,736)] 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("736.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 736.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.87698958877\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 736.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +2.73205 q^{5} +1.26795 q^{7} +4.73205 q^{11} +0.464102 q^{13} +4.73205 q^{15} -6.19615 q^{17} -3.46410 q^{19} +2.19615 q^{21} -1.00000 q^{23} +2.46410 q^{25} -5.19615 q^{27} +1.53590 q^{29} +7.73205 q^{31} +8.19615 q^{33} +3.46410 q^{35} -4.19615 q^{37} +0.803848 q^{39} -8.46410 q^{41} +6.92820 q^{43} +0.803848 q^{47} -5.39230 q^{49} -10.7321 q^{51} -4.92820 q^{53} +12.9282 q^{55} -6.00000 q^{57} +2.53590 q^{59} +2.00000 q^{61} +1.26795 q^{65} -14.1962 q^{67} -1.73205 q^{69} +13.7321 q^{71} +6.46410 q^{73} +4.26795 q^{75} +6.00000 q^{77} +10.3923 q^{79} -9.00000 q^{81} -8.19615 q^{83} -16.9282 q^{85} +2.66025 q^{87} +17.8564 q^{89} +0.588457 q^{91} +13.3923 q^{93} -9.46410 q^{95} -4.73205 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{7} + 6 q^{11} - 6 q^{13} + 6 q^{15} - 2 q^{17} - 6 q^{21} - 2 q^{23} - 2 q^{25} + 10 q^{29} + 12 q^{31} + 6 q^{33} + 2 q^{37} + 12 q^{39} - 10 q^{41} + 12 q^{47} + 10 q^{49} - 18 q^{51}+ \cdots - 6 q^{97}+O(q^{100}) \) 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 0
\(3\) 1.73205 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 0 0
\(5\) 2.73205 1.22181 0.610905 − 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 − 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.73205 1.42677 0.713384 − 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 0 0
\(13\) 0.464102 0.128719 0.0643593 − 0.997927i \(-0.479500\pi\)
0.0643593 + 0.997927i \(0.479500\pi\)
\(14\) 0 0
\(15\) 4.73205 1.22181
\(16\) 0 0
\(17\) −6.19615 −1.50279 −0.751394 − 0.659854i \(-0.770618\pi\)
−0.751394 + 0.659854i \(0.770618\pi\)
\(18\) 0 0
\(19\) −3.46410 −0.794719 −0.397360 − 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) 2.19615 0.479240
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 1.53590 0.285209 0.142605 − 0.989780i \(-0.454452\pi\)
0.142605 + 0.989780i \(0.454452\pi\)
\(30\) 0 0
\(31\) 7.73205 1.38872 0.694359 − 0.719629i \(-0.255688\pi\)
0.694359 + 0.719629i \(0.255688\pi\)
\(32\) 0 0
\(33\) 8.19615 1.42677
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) −4.19615 −0.689843 −0.344922 − 0.938631i \(-0.612095\pi\)
−0.344922 + 0.938631i \(0.612095\pi\)
\(38\) 0 0
\(39\) 0.803848 0.128719
\(40\) 0 0
\(41\) −8.46410 −1.32187 −0.660935 − 0.750443i \(-0.729840\pi\)
−0.660935 + 0.750443i \(0.729840\pi\)
\(42\) 0 0
\(43\) 6.92820 1.05654 0.528271 − 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.803848 0.117253 0.0586266 − 0.998280i \(-0.481328\pi\)
0.0586266 + 0.998280i \(0.481328\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) −10.7321 −1.50279
\(52\) 0 0
\(53\) −4.92820 −0.676941 −0.338470 − 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(54\) 0 0
\(55\) 12.9282 1.74324
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 2.53590 0.330146 0.165073 − 0.986281i \(-0.447214\pi\)
0.165073 + 0.986281i \(0.447214\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.26795 0.157270
\(66\) 0 0
\(67\) −14.1962 −1.73434 −0.867168 − 0.498016i \(-0.834062\pi\)
−0.867168 + 0.498016i \(0.834062\pi\)
\(68\) 0 0
\(69\) −1.73205 −0.208514
\(70\) 0 0
\(71\) 13.7321 1.62969 0.814847 − 0.579676i \(-0.196821\pi\)
0.814847 + 0.579676i \(0.196821\pi\)
\(72\) 0 0
\(73\) 6.46410 0.756566 0.378283 − 0.925690i \(-0.376515\pi\)
0.378283 + 0.925690i \(0.376515\pi\)
\(74\) 0 0
\(75\) 4.26795 0.492820
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 10.3923 1.16923 0.584613 − 0.811312i \(-0.301246\pi\)
0.584613 + 0.811312i \(0.301246\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −8.19615 −0.899645 −0.449822 − 0.893118i \(-0.648513\pi\)
−0.449822 + 0.893118i \(0.648513\pi\)
\(84\) 0 0
\(85\) −16.9282 −1.83612
\(86\) 0 0
\(87\) 2.66025 0.285209
\(88\) 0 0
\(89\) 17.8564 1.89278 0.946388 − 0.323033i \(-0.104703\pi\)
0.946388 + 0.323033i \(0.104703\pi\)
\(90\) 0 0
\(91\) 0.588457 0.0616871
\(92\) 0 0
\(93\) 13.3923 1.38872
\(94\) 0 0
\(95\) −9.46410 −0.970996
\(96\) 0 0
\(97\) −4.73205 −0.480467 −0.240233 − 0.970715i \(-0.577224\pi\)
−0.240233 + 0.970715i \(0.577224\pi\)
\(98\) 0 0
\(99\) 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 736.2.a.c.1.2 yes 2
3.2 odd 2 6624.2.a.n.1.1 2
4.3 odd 2 736.2.a.b.1.1 ✓ 2
8.3 odd 2 1472.2.a.q.1.2 2
8.5 even 2 1472.2.a.r.1.1 2
12.11 even 2 6624.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.2.a.b.1.1 ✓ 2 4.3 odd 2
736.2.a.c.1.2 yes 2 1.1 even 1 trivial
1472.2.a.q.1.2 2 8.3 odd 2
1472.2.a.r.1.1 2 8.5 even 2
6624.2.a.k.1.1 2 12.11 even 2
6624.2.a.n.1.1 2 3.2 odd 2