Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4640,2,Mod(1,4640)] 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("4640.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4640.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,1,0,-2,0,-1,0,-3,0,2,0,1,0,-1,0,-5,0,-10,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.0505865379\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 4640.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.618034 q^{3} -1.00000 q^{5} +2.85410 q^{7} -2.61803 q^{9} +3.23607 q^{11} -5.09017 q^{13} +0.618034 q^{15} -3.61803 q^{17} -2.76393 q^{19} -1.76393 q^{21} -5.85410 q^{23} +1.00000 q^{25} +3.47214 q^{27} +1.00000 q^{29} +1.61803 q^{31} -2.00000 q^{33} -2.85410 q^{35} +9.23607 q^{37} +3.14590 q^{39} -3.70820 q^{41} +7.61803 q^{43} +2.61803 q^{45} +8.00000 q^{47} +1.14590 q^{49} +2.23607 q^{51} +9.32624 q^{53} -3.23607 q^{55} +1.70820 q^{57} +9.38197 q^{59} +2.14590 q^{61} -7.47214 q^{63} +5.09017 q^{65} +2.47214 q^{67} +3.61803 q^{69} -12.9443 q^{71} +1.90983 q^{73} -0.618034 q^{75} +9.23607 q^{77} +2.90983 q^{79} +5.70820 q^{81} -15.7082 q^{83} +3.61803 q^{85} -0.618034 q^{87} -1.70820 q^{89} -14.5279 q^{91} -1.00000 q^{93} +2.76393 q^{95} +8.79837 q^{97} -8.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 2 q^{5} - q^{7} - 3 q^{9} + 2 q^{11} + q^{13} - q^{15} - 5 q^{17} - 10 q^{19} - 8 q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} + q^{31} - 4 q^{33} + q^{35} + 14 q^{37} + 13 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) \) 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 0
\(3\) −0.618034 −0.356822 −0.178411 − 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.85410 1.07875 0.539375 − 0.842066i \(-0.318661\pi\)
0.539375 + 0.842066i \(0.318661\pi\)
\(8\) 0 0
\(9\) −2.61803 −0.872678
\(10\) 0 0
\(11\) 3.23607 0.975711 0.487856 − 0.872924i \(-0.337779\pi\)
0.487856 + 0.872924i \(0.337779\pi\)
\(12\) 0 0
\(13\) −5.09017 −1.41176 −0.705880 − 0.708332i \(-0.749448\pi\)
−0.705880 + 0.708332i \(0.749448\pi\)
\(14\) 0 0
\(15\) 0.618034 0.159576
\(16\) 0 0
\(17\) −3.61803 −0.877502 −0.438751 − 0.898609i \(-0.644579\pi\)
−0.438751 + 0.898609i \(0.644579\pi\)
\(18\) 0 0
\(19\) −2.76393 −0.634089 −0.317045 − 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 0 0
\(21\) −1.76393 −0.384922
\(22\) 0 0
\(23\) −5.85410 −1.22066 −0.610332 − 0.792145i \(-0.708964\pi\)
−0.610332 + 0.792145i \(0.708964\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.47214 0.668213
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 1.61803 0.290607 0.145304 − 0.989387i \(-0.453584\pi\)
0.145304 + 0.989387i \(0.453584\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) −2.85410 −0.482431
\(36\) 0 0
\(37\) 9.23607 1.51840 0.759200 − 0.650857i \(-0.225590\pi\)
0.759200 + 0.650857i \(0.225590\pi\)
\(38\) 0 0
\(39\) 3.14590 0.503747
\(40\) 0 0
\(41\) −3.70820 −0.579124 −0.289562 − 0.957159i \(-0.593510\pi\)
−0.289562 + 0.957159i \(0.593510\pi\)
\(42\) 0 0
\(43\) 7.61803 1.16174 0.580870 − 0.813997i \(-0.302713\pi\)
0.580870 + 0.813997i \(0.302713\pi\)
\(44\) 0 0
\(45\) 2.61803 0.390273
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.14590 0.163700
\(50\) 0 0
\(51\) 2.23607 0.313112
\(52\) 0 0
\(53\) 9.32624 1.28106 0.640529 − 0.767934i \(-0.278715\pi\)
0.640529 + 0.767934i \(0.278715\pi\)
\(54\) 0 0
\(55\) −3.23607 −0.436351
\(56\) 0 0
\(57\) 1.70820 0.226257
\(58\) 0 0
\(59\) 9.38197 1.22143 0.610714 − 0.791851i \(-0.290883\pi\)
0.610714 + 0.791851i \(0.290883\pi\)
\(60\) 0 0
\(61\) 2.14590 0.274754 0.137377 − 0.990519i \(-0.456133\pi\)
0.137377 + 0.990519i \(0.456133\pi\)
\(62\) 0 0
\(63\) −7.47214 −0.941401
\(64\) 0 0
\(65\) 5.09017 0.631358
\(66\) 0 0
\(67\) 2.47214 0.302019 0.151010 − 0.988532i \(-0.451748\pi\)
0.151010 + 0.988532i \(0.451748\pi\)
\(68\) 0 0
\(69\) 3.61803 0.435560
\(70\) 0 0
\(71\) −12.9443 −1.53620 −0.768101 − 0.640328i \(-0.778798\pi\)
−0.768101 + 0.640328i \(0.778798\pi\)
\(72\) 0 0
\(73\) 1.90983 0.223529 0.111764 − 0.993735i \(-0.464350\pi\)
0.111764 + 0.993735i \(0.464350\pi\)
\(74\) 0 0
\(75\) −0.618034 −0.0713644
\(76\) 0 0
\(77\) 9.23607 1.05255
\(78\) 0 0
\(79\) 2.90983 0.327381 0.163691 − 0.986512i \(-0.447660\pi\)
0.163691 + 0.986512i \(0.447660\pi\)
\(80\) 0 0
\(81\) 5.70820 0.634245
\(82\) 0 0
\(83\) −15.7082 −1.72420 −0.862100 − 0.506739i \(-0.830851\pi\)
−0.862100 + 0.506739i \(0.830851\pi\)
\(84\) 0 0
\(85\) 3.61803 0.392431
\(86\) 0 0
\(87\) −0.618034 −0.0662602
\(88\) 0 0
\(89\) −1.70820 −0.181069 −0.0905346 − 0.995893i \(-0.528858\pi\)
−0.0905346 + 0.995893i \(0.528858\pi\)
\(90\) 0 0
\(91\) −14.5279 −1.52293
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 2.76393 0.283573
\(96\) 0 0
\(97\) 8.79837 0.893340 0.446670 − 0.894699i \(-0.352610\pi\)
0.446670 + 0.894699i \(0.352610\pi\)
\(98\) 0 0
\(99\) −8.47214 −0.851482
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 4640.2.a.i.1.1 yes 2
4.3 odd 2 4640.2.a.g.1.2 ✓ 2
8.3 odd 2 9280.2.a.bd.1.1 2
8.5 even 2 9280.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.g.1.2 ✓ 2 4.3 odd 2
4640.2.a.i.1.1 yes 2 1.1 even 1 trivial
9280.2.a.y.1.2 2 8.5 even 2
9280.2.a.bd.1.1 2 8.3 odd 2