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 = [8,0,0,0,8,0,0,0,6,0,0,0,6,0,0,0,2,0,0,0,24] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 15x^{6} + 57x^{4} - 40x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.346147\) 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)
 
\(f(q)\) \(=\) \(q+0.346147 q^{3} +1.00000 q^{5} -1.09203 q^{7} -2.88018 q^{9} +5.77789 q^{11} +0.497822 q^{13} +0.346147 q^{15} +4.15482 q^{17} -4.48837 q^{19} -0.378004 q^{21} +0.346147 q^{23} +1.00000 q^{25} -2.03541 q^{27} -1.00000 q^{29} +9.77145 q^{31} +2.00000 q^{33} -1.09203 q^{35} +2.27464 q^{37} +0.172320 q^{39} +0.446365 q^{41} -5.25791 q^{43} -2.88018 q^{45} -2.20924 q^{47} -5.80746 q^{49} +1.43818 q^{51} +13.1942 q^{53} +5.77789 q^{55} -1.55364 q^{57} +1.26586 q^{59} -3.81182 q^{61} +3.14525 q^{63} +0.497822 q^{65} -11.1860 q^{67} +0.119818 q^{69} +4.16588 q^{71} +13.4688 q^{73} +0.346147 q^{75} -6.30964 q^{77} -3.22251 q^{79} +7.93600 q^{81} -1.26435 q^{83} +4.15482 q^{85} -0.346147 q^{87} +13.6236 q^{89} -0.543638 q^{91} +3.38236 q^{93} -4.48837 q^{95} -7.09082 q^{97} -16.6414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 6 q^{9} + 6 q^{13} + 2 q^{17} + 24 q^{21} + 8 q^{25} - 8 q^{29} + 16 q^{33} + 16 q^{37} + 12 q^{41} + 6 q^{45} + 14 q^{49} + 10 q^{53} - 4 q^{57} + 34 q^{61} + 6 q^{65} + 30 q^{69} + 10 q^{73}+ \cdots + 14 q^{97}+O(q^{100}) \) 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
\(3\) 0.346147 0.199848 0.0999241 − 0.994995i \(-0.468140\pi\)
0.0999241 + 0.994995i \(0.468140\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.09203 −0.412749 −0.206375 − 0.978473i \(-0.566167\pi\)
−0.206375 + 0.978473i \(0.566167\pi\)
\(8\) 0 0
\(9\) −2.88018 −0.960061
\(10\) 0 0
\(11\) 5.77789 1.74210 0.871050 − 0.491195i \(-0.163440\pi\)
0.871050 + 0.491195i \(0.163440\pi\)
\(12\) 0 0
\(13\) 0.497822 0.138071 0.0690355 − 0.997614i \(-0.478008\pi\)
0.0690355 + 0.997614i \(0.478008\pi\)
\(14\) 0 0
\(15\) 0.346147 0.0893748
\(16\) 0 0
\(17\) 4.15482 1.00769 0.503846 − 0.863793i \(-0.331918\pi\)
0.503846 + 0.863793i \(0.331918\pi\)
\(18\) 0 0
\(19\) −4.48837 −1.02970 −0.514851 − 0.857280i \(-0.672153\pi\)
−0.514851 + 0.857280i \(0.672153\pi\)
\(20\) 0 0
\(21\) −0.378004 −0.0824872
\(22\) 0 0
\(23\) 0.346147 0.0721767 0.0360883 − 0.999349i \(-0.488510\pi\)
0.0360883 + 0.999349i \(0.488510\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.03541 −0.391715
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 9.77145 1.75500 0.877502 − 0.479572i \(-0.159208\pi\)
0.877502 + 0.479572i \(0.159208\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −1.09203 −0.184587
\(36\) 0 0
\(37\) 2.27464 0.373948 0.186974 − 0.982365i \(-0.440132\pi\)
0.186974 + 0.982365i \(0.440132\pi\)
\(38\) 0 0
\(39\) 0.172320 0.0275932
\(40\) 0 0
\(41\) 0.446365 0.0697105 0.0348552 − 0.999392i \(-0.488903\pi\)
0.0348552 + 0.999392i \(0.488903\pi\)
\(42\) 0 0
\(43\) −5.25791 −0.801824 −0.400912 − 0.916116i \(-0.631307\pi\)
−0.400912 + 0.916116i \(0.631307\pi\)
\(44\) 0 0
\(45\) −2.88018 −0.429352
\(46\) 0 0
\(47\) −2.20924 −0.322250 −0.161125 − 0.986934i \(-0.551512\pi\)
−0.161125 + 0.986934i \(0.551512\pi\)
\(48\) 0 0
\(49\) −5.80746 −0.829638
\(50\) 0 0
\(51\) 1.43818 0.201385
\(52\) 0 0
\(53\) 13.1942 1.81236 0.906180 − 0.422892i \(-0.138985\pi\)
0.906180 + 0.422892i \(0.138985\pi\)
\(54\) 0 0
\(55\) 5.77789 0.779090
\(56\) 0 0
\(57\) −1.55364 −0.205784
\(58\) 0 0
\(59\) 1.26586 0.164801 0.0824005 − 0.996599i \(-0.473741\pi\)
0.0824005 + 0.996599i \(0.473741\pi\)
\(60\) 0 0
\(61\) −3.81182 −0.488054 −0.244027 − 0.969768i \(-0.578469\pi\)
−0.244027 + 0.969768i \(0.578469\pi\)
\(62\) 0 0
\(63\) 3.14525 0.396265
\(64\) 0 0
\(65\) 0.497822 0.0617472
\(66\) 0 0
\(67\) −11.1860 −1.36658 −0.683292 − 0.730145i \(-0.739452\pi\)
−0.683292 + 0.730145i \(0.739452\pi\)
\(68\) 0 0
\(69\) 0.119818 0.0144244
\(70\) 0 0
\(71\) 4.16588 0.494399 0.247200 − 0.968965i \(-0.420490\pi\)
0.247200 + 0.968965i \(0.420490\pi\)
\(72\) 0 0
\(73\) 13.4688 1.57641 0.788203 − 0.615415i \(-0.211012\pi\)
0.788203 + 0.615415i \(0.211012\pi\)
\(74\) 0 0
\(75\) 0.346147 0.0399696
\(76\) 0 0
\(77\) −6.30964 −0.719051
\(78\) 0 0
\(79\) −3.22251 −0.362560 −0.181280 − 0.983431i \(-0.558024\pi\)
−0.181280 + 0.983431i \(0.558024\pi\)
\(80\) 0 0
\(81\) 7.93600 0.881777
\(82\) 0 0
\(83\) −1.26435 −0.138781 −0.0693903 − 0.997590i \(-0.522105\pi\)
−0.0693903 + 0.997590i \(0.522105\pi\)
\(84\) 0 0
\(85\) 4.15482 0.450654
\(86\) 0 0
\(87\) −0.346147 −0.0371109
\(88\) 0 0
\(89\) 13.6236 1.44410 0.722052 − 0.691839i \(-0.243199\pi\)
0.722052 + 0.691839i \(0.243199\pi\)
\(90\) 0 0
\(91\) −0.543638 −0.0569887
\(92\) 0 0
\(93\) 3.38236 0.350734
\(94\) 0 0
\(95\) −4.48837 −0.460497
\(96\) 0 0
\(97\) −7.09082 −0.719963 −0.359982 − 0.932959i \(-0.617217\pi\)
−0.359982 + 0.932959i \(0.617217\pi\)
\(98\) 0 0
\(99\) −16.6414 −1.67252
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.y.1.5 yes 8
4.3 odd 2 inner 4640.2.a.y.1.4 ✓ 8
8.3 odd 2 9280.2.a.cs.1.5 8
8.5 even 2 9280.2.a.cs.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4640.2.a.y.1.4 ✓ 8 4.3 odd 2 inner
4640.2.a.y.1.5 yes 8 1.1 even 1 trivial
9280.2.a.cs.1.4 8 8.5 even 2
9280.2.a.cs.1.5 8 8.3 odd 2