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 = [7,0,3,0,-2] 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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 8x^{5} + 21x^{4} + 25x^{3} - 41x^{2} - 28x + 19 \) 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 1.3
Root \(-1.31731\) 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)
 
\(f(q)\) \(=\) \(q-1.31731 q^{3} -1.06660 q^{5} -1.26468 q^{9} +1.51266 q^{11} +1.38743 q^{13} +1.40505 q^{15} -1.00000 q^{17} +4.63775 q^{19} -5.81951 q^{23} -3.86235 q^{25} +5.61793 q^{27} +9.67008 q^{29} -1.77350 q^{31} -1.99265 q^{33} -8.93647 q^{37} -1.82768 q^{39} -6.52095 q^{41} -7.24599 q^{43} +1.34892 q^{45} +4.80622 q^{47} +1.31731 q^{51} +1.41889 q^{53} -1.61341 q^{55} -6.10937 q^{57} -1.04178 q^{59} +13.8495 q^{61} -1.47983 q^{65} +12.2619 q^{67} +7.66613 q^{69} -1.86001 q^{71} +9.05795 q^{73} +5.08794 q^{75} -4.84533 q^{79} -3.60653 q^{81} +8.41568 q^{83} +1.06660 q^{85} -12.7385 q^{87} -8.60625 q^{89} +2.33626 q^{93} -4.94664 q^{95} +19.5711 q^{97} -1.91304 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} - 2 q^{5} + 4 q^{9} - 12 q^{11} - 2 q^{13} - 6 q^{15} - 7 q^{17} + 3 q^{19} - 18 q^{23} + 15 q^{25} + 18 q^{27} + 5 q^{29} + 10 q^{31} - 21 q^{33} + 11 q^{37} - 12 q^{39} - 15 q^{43} - 13 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 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
\(3\) −1.31731 −0.760552 −0.380276 − 0.924873i \(-0.624171\pi\)
−0.380276 + 0.924873i \(0.624171\pi\)
\(4\) 0 0
\(5\) −1.06660 −0.477000 −0.238500 − 0.971142i \(-0.576656\pi\)
−0.238500 + 0.971142i \(0.576656\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.26468 −0.421561
\(10\) 0 0
\(11\) 1.51266 0.456085 0.228042 − 0.973651i \(-0.426767\pi\)
0.228042 + 0.973651i \(0.426767\pi\)
\(12\) 0 0
\(13\) 1.38743 0.384802 0.192401 − 0.981316i \(-0.438372\pi\)
0.192401 + 0.981316i \(0.438372\pi\)
\(14\) 0 0
\(15\) 1.40505 0.362783
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 4.63775 1.06397 0.531986 − 0.846753i \(-0.321446\pi\)
0.531986 + 0.846753i \(0.321446\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.81951 −1.21345 −0.606726 − 0.794911i \(-0.707517\pi\)
−0.606726 + 0.794911i \(0.707517\pi\)
\(24\) 0 0
\(25\) −3.86235 −0.772471
\(26\) 0 0
\(27\) 5.61793 1.08117
\(28\) 0 0
\(29\) 9.67008 1.79569 0.897845 − 0.440312i \(-0.145132\pi\)
0.897845 + 0.440312i \(0.145132\pi\)
\(30\) 0 0
\(31\) −1.77350 −0.318530 −0.159265 − 0.987236i \(-0.550913\pi\)
−0.159265 + 0.987236i \(0.550913\pi\)
\(32\) 0 0
\(33\) −1.99265 −0.346876
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.93647 −1.46915 −0.734573 − 0.678529i \(-0.762618\pi\)
−0.734573 + 0.678529i \(0.762618\pi\)
\(38\) 0 0
\(39\) −1.82768 −0.292662
\(40\) 0 0
\(41\) −6.52095 −1.01840 −0.509201 − 0.860648i \(-0.670059\pi\)
−0.509201 + 0.860648i \(0.670059\pi\)
\(42\) 0 0
\(43\) −7.24599 −1.10500 −0.552501 − 0.833512i \(-0.686327\pi\)
−0.552501 + 0.833512i \(0.686327\pi\)
\(44\) 0 0
\(45\) 1.34892 0.201084
\(46\) 0 0
\(47\) 4.80622 0.701059 0.350529 − 0.936552i \(-0.386002\pi\)
0.350529 + 0.936552i \(0.386002\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.31731 0.184461
\(52\) 0 0
\(53\) 1.41889 0.194900 0.0974498 − 0.995240i \(-0.468931\pi\)
0.0974498 + 0.995240i \(0.468931\pi\)
\(54\) 0 0
\(55\) −1.61341 −0.217553
\(56\) 0 0
\(57\) −6.10937 −0.809206
\(58\) 0 0
\(59\) −1.04178 −0.135628 −0.0678141 − 0.997698i \(-0.521602\pi\)
−0.0678141 + 0.997698i \(0.521602\pi\)
\(60\) 0 0
\(61\) 13.8495 1.77324 0.886621 − 0.462496i \(-0.153046\pi\)
0.886621 + 0.462496i \(0.153046\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.47983 −0.183551
\(66\) 0 0
\(67\) 12.2619 1.49803 0.749013 − 0.662556i \(-0.230528\pi\)
0.749013 + 0.662556i \(0.230528\pi\)
\(68\) 0 0
\(69\) 7.66613 0.922893
\(70\) 0 0
\(71\) −1.86001 −0.220743 −0.110372 − 0.993890i \(-0.535204\pi\)
−0.110372 + 0.993890i \(0.535204\pi\)
\(72\) 0 0
\(73\) 9.05795 1.06015 0.530076 − 0.847950i \(-0.322163\pi\)
0.530076 + 0.847950i \(0.322163\pi\)
\(74\) 0 0
\(75\) 5.08794 0.587504
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.84533 −0.545142 −0.272571 − 0.962136i \(-0.587874\pi\)
−0.272571 + 0.962136i \(0.587874\pi\)
\(80\) 0 0
\(81\) −3.60653 −0.400726
\(82\) 0 0
\(83\) 8.41568 0.923741 0.461871 − 0.886947i \(-0.347178\pi\)
0.461871 + 0.886947i \(0.347178\pi\)
\(84\) 0 0
\(85\) 1.06660 0.115690
\(86\) 0 0
\(87\) −12.7385 −1.36572
\(88\) 0 0
\(89\) −8.60625 −0.912261 −0.456130 − 0.889913i \(-0.650765\pi\)
−0.456130 + 0.889913i \(0.650765\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.33626 0.242259
\(94\) 0 0
\(95\) −4.94664 −0.507515
\(96\) 0 0
\(97\) 19.5711 1.98714 0.993572 − 0.113201i \(-0.0361105\pi\)
0.993572 + 0.113201i \(0.0361105\pi\)
\(98\) 0 0
\(99\) −1.91304 −0.192267
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.y.1.3 7
7.3 odd 6 952.2.q.e.681.3 yes 14
7.5 odd 6 952.2.q.e.137.3 ✓ 14
7.6 odd 2 6664.2.a.v.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.q.e.137.3 ✓ 14 7.5 odd 6
952.2.q.e.681.3 yes 14 7.3 odd 6
6664.2.a.v.1.5 7 7.6 odd 2
6664.2.a.y.1.3 7 1.1 even 1 trivial