Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6664,2,Mod(1,6664)] 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("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1,0,1] 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: \(53.2123079070\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.30091192.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 9x^{4} + 10x^{3} + 18x^{2} - 23x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.83437\) of defining polynomial
Character \(\chi\) \(=\) 6664.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.83437 q^{3} +0.128742 q^{5} +0.364902 q^{9} +1.16500 q^{11} +3.63447 q^{13} -0.236160 q^{15} -1.00000 q^{17} -3.24769 q^{19} -0.484483 q^{23} -4.98343 q^{25} +4.83374 q^{27} -1.87449 q^{29} +3.83853 q^{31} -2.13704 q^{33} +0.822581 q^{37} -6.66695 q^{39} -2.40154 q^{41} -4.60873 q^{43} +0.0469782 q^{45} +12.9692 q^{47} +1.83437 q^{51} -8.90813 q^{53} +0.149985 q^{55} +5.95746 q^{57} +7.29049 q^{59} -7.37795 q^{61} +0.467909 q^{65} -0.967251 q^{67} +0.888720 q^{69} -4.43543 q^{71} -6.94475 q^{73} +9.14143 q^{75} +6.21017 q^{79} -9.96155 q^{81} -8.30112 q^{83} -0.128742 q^{85} +3.43850 q^{87} -0.544484 q^{89} -7.04126 q^{93} -0.418114 q^{95} -4.01855 q^{97} +0.425112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + q^{5} + q^{9} - 6 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 5 q^{23} + q^{25} - 8 q^{27} - 15 q^{29} + 8 q^{31} + 11 q^{33} - 16 q^{37} - 2 q^{39} + 4 q^{41} + 24 q^{43} + 2 q^{45} - q^{47}+ \cdots + 2 q^{99}+O(q^{100}) \) 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 0
\(3\) −1.83437 −1.05907 −0.529536 − 0.848287i \(-0.677634\pi\)
−0.529536 + 0.848287i \(0.677634\pi\)
\(4\) 0 0
\(5\) 0.128742 0.0575752 0.0287876 − 0.999586i \(-0.490835\pi\)
0.0287876 + 0.999586i \(0.490835\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.364902 0.121634
\(10\) 0 0
\(11\) 1.16500 0.351262 0.175631 − 0.984456i \(-0.443804\pi\)
0.175631 + 0.984456i \(0.443804\pi\)
\(12\) 0 0
\(13\) 3.63447 1.00802 0.504010 − 0.863698i \(-0.331857\pi\)
0.504010 + 0.863698i \(0.331857\pi\)
\(14\) 0 0
\(15\) −0.236160 −0.0609762
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −3.24769 −0.745072 −0.372536 − 0.928018i \(-0.621512\pi\)
−0.372536 + 0.928018i \(0.621512\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.484483 −0.101022 −0.0505108 − 0.998724i \(-0.516085\pi\)
−0.0505108 + 0.998724i \(0.516085\pi\)
\(24\) 0 0
\(25\) −4.98343 −0.996685
\(26\) 0 0
\(27\) 4.83374 0.930253
\(28\) 0 0
\(29\) −1.87449 −0.348084 −0.174042 − 0.984738i \(-0.555683\pi\)
−0.174042 + 0.984738i \(0.555683\pi\)
\(30\) 0 0
\(31\) 3.83853 0.689419 0.344710 − 0.938709i \(-0.387977\pi\)
0.344710 + 0.938709i \(0.387977\pi\)
\(32\) 0 0
\(33\) −2.13704 −0.372011
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.822581 0.135231 0.0676157 − 0.997711i \(-0.478461\pi\)
0.0676157 + 0.997711i \(0.478461\pi\)
\(38\) 0 0
\(39\) −6.66695 −1.06757
\(40\) 0 0
\(41\) −2.40154 −0.375058 −0.187529 − 0.982259i \(-0.560048\pi\)
−0.187529 + 0.982259i \(0.560048\pi\)
\(42\) 0 0
\(43\) −4.60873 −0.702825 −0.351413 − 0.936221i \(-0.614299\pi\)
−0.351413 + 0.936221i \(0.614299\pi\)
\(44\) 0 0
\(45\) 0.0469782 0.00700310
\(46\) 0 0
\(47\) 12.9692 1.89176 0.945878 − 0.324523i \(-0.105204\pi\)
0.945878 + 0.324523i \(0.105204\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.83437 0.256863
\(52\) 0 0
\(53\) −8.90813 −1.22363 −0.611813 − 0.791003i \(-0.709559\pi\)
−0.611813 + 0.791003i \(0.709559\pi\)
\(54\) 0 0
\(55\) 0.149985 0.0202239
\(56\) 0 0
\(57\) 5.95746 0.789085
\(58\) 0 0
\(59\) 7.29049 0.949141 0.474570 − 0.880218i \(-0.342603\pi\)
0.474570 + 0.880218i \(0.342603\pi\)
\(60\) 0 0
\(61\) −7.37795 −0.944650 −0.472325 − 0.881424i \(-0.656585\pi\)
−0.472325 + 0.881424i \(0.656585\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.467909 0.0580369
\(66\) 0 0
\(67\) −0.967251 −0.118169 −0.0590843 − 0.998253i \(-0.518818\pi\)
−0.0590843 + 0.998253i \(0.518818\pi\)
\(68\) 0 0
\(69\) 0.888720 0.106989
\(70\) 0 0
\(71\) −4.43543 −0.526389 −0.263194 − 0.964743i \(-0.584776\pi\)
−0.263194 + 0.964743i \(0.584776\pi\)
\(72\) 0 0
\(73\) −6.94475 −0.812821 −0.406411 − 0.913691i \(-0.633220\pi\)
−0.406411 + 0.913691i \(0.633220\pi\)
\(74\) 0 0
\(75\) 9.14143 1.05556
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.21017 0.698699 0.349349 − 0.936993i \(-0.386403\pi\)
0.349349 + 0.936993i \(0.386403\pi\)
\(80\) 0 0
\(81\) −9.96155 −1.10684
\(82\) 0 0
\(83\) −8.30112 −0.911166 −0.455583 − 0.890193i \(-0.650569\pi\)
−0.455583 + 0.890193i \(0.650569\pi\)
\(84\) 0 0
\(85\) −0.128742 −0.0139640
\(86\) 0 0
\(87\) 3.43850 0.368646
\(88\) 0 0
\(89\) −0.544484 −0.0577152 −0.0288576 − 0.999584i \(-0.509187\pi\)
−0.0288576 + 0.999584i \(0.509187\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −7.04126 −0.730145
\(94\) 0 0
\(95\) −0.418114 −0.0428976
\(96\) 0 0
\(97\) −4.01855 −0.408022 −0.204011 − 0.978969i \(-0.565398\pi\)
−0.204011 + 0.978969i \(0.565398\pi\)
\(98\) 0 0
\(99\) 0.425112 0.0427253
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 6664.2.a.s.1.2 6
7.2 even 3 952.2.q.c.137.5 ✓ 12
7.4 even 3 952.2.q.c.681.5 yes 12
7.6 odd 2 6664.2.a.r.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.q.c.137.5 ✓ 12 7.2 even 3
952.2.q.c.681.5 yes 12 7.4 even 3
6664.2.a.r.1.5 6 7.6 odd 2
6664.2.a.s.1.2 6 1.1 even 1 trivial