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 1.5
Root \(1.49411\) 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)
 
\(f(q)\) \(=\) \(q+1.49411 q^{3} +1.55355 q^{5} -0.767628 q^{9} -2.64103 q^{11} -2.36752 q^{13} +2.32118 q^{15} -1.00000 q^{17} +3.07851 q^{19} +1.26454 q^{23} -2.58649 q^{25} -5.62926 q^{27} +3.70334 q^{29} -8.82748 q^{31} -3.94600 q^{33} -7.10585 q^{37} -3.53734 q^{39} +5.19614 q^{41} +5.83150 q^{43} -1.19255 q^{45} +2.36517 q^{47} -1.49411 q^{51} +4.45919 q^{53} -4.10298 q^{55} +4.59965 q^{57} -4.78390 q^{59} -9.63959 q^{61} -3.67806 q^{65} -8.78526 q^{67} +1.88937 q^{69} +2.58292 q^{71} -6.17149 q^{73} -3.86450 q^{75} -9.15423 q^{79} -6.10786 q^{81} -11.4885 q^{83} -1.55355 q^{85} +5.53321 q^{87} -2.57873 q^{89} -13.1892 q^{93} +4.78262 q^{95} +12.0066 q^{97} +2.02733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + q^{5} + q^{9} - 6 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 5 q^{23} + q^{25} - 8 q^{27} - 15 q^{29} + 8 q^{31} + 11 q^{33} - 16 q^{37} - 2 q^{39} + 4 q^{41} + 24 q^{43} + 2 q^{45} - q^{47}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(3\) 1.49411 0.862626 0.431313 − 0.902202i \(-0.358050\pi\)
0.431313 + 0.902202i \(0.358050\pi\)
\(4\) 0 0
\(5\) 1.55355 0.694768 0.347384 − 0.937723i \(-0.387070\pi\)
0.347384 + 0.937723i \(0.387070\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.767628 −0.255876
\(10\) 0 0
\(11\) −2.64103 −0.796302 −0.398151 − 0.917320i \(-0.630348\pi\)
−0.398151 + 0.917320i \(0.630348\pi\)
\(12\) 0 0
\(13\) −2.36752 −0.656632 −0.328316 − 0.944568i \(-0.606481\pi\)
−0.328316 + 0.944568i \(0.606481\pi\)
\(14\) 0 0
\(15\) 2.32118 0.599325
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 3.07851 0.706260 0.353130 − 0.935574i \(-0.385117\pi\)
0.353130 + 0.935574i \(0.385117\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.26454 0.263676 0.131838 − 0.991271i \(-0.457912\pi\)
0.131838 + 0.991271i \(0.457912\pi\)
\(24\) 0 0
\(25\) −2.58649 −0.517297
\(26\) 0 0
\(27\) −5.62926 −1.08335
\(28\) 0 0
\(29\) 3.70334 0.687693 0.343847 − 0.939026i \(-0.388270\pi\)
0.343847 + 0.939026i \(0.388270\pi\)
\(30\) 0 0
\(31\) −8.82748 −1.58546 −0.792731 − 0.609572i \(-0.791342\pi\)
−0.792731 + 0.609572i \(0.791342\pi\)
\(32\) 0 0
\(33\) −3.94600 −0.686911
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.10585 −1.16819 −0.584097 − 0.811684i \(-0.698551\pi\)
−0.584097 + 0.811684i \(0.698551\pi\)
\(38\) 0 0
\(39\) −3.53734 −0.566428
\(40\) 0 0
\(41\) 5.19614 0.811500 0.405750 − 0.913984i \(-0.367010\pi\)
0.405750 + 0.913984i \(0.367010\pi\)
\(42\) 0 0
\(43\) 5.83150 0.889295 0.444647 − 0.895706i \(-0.353329\pi\)
0.444647 + 0.895706i \(0.353329\pi\)
\(44\) 0 0
\(45\) −1.19255 −0.177774
\(46\) 0 0
\(47\) 2.36517 0.344996 0.172498 − 0.985010i \(-0.444816\pi\)
0.172498 + 0.985010i \(0.444816\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.49411 −0.209218
\(52\) 0 0
\(53\) 4.45919 0.612517 0.306259 − 0.951948i \(-0.400923\pi\)
0.306259 + 0.951948i \(0.400923\pi\)
\(54\) 0 0
\(55\) −4.10298 −0.553245
\(56\) 0 0
\(57\) 4.59965 0.609238
\(58\) 0 0
\(59\) −4.78390 −0.622811 −0.311406 − 0.950277i \(-0.600800\pi\)
−0.311406 + 0.950277i \(0.600800\pi\)
\(60\) 0 0
\(61\) −9.63959 −1.23422 −0.617111 − 0.786876i \(-0.711697\pi\)
−0.617111 + 0.786876i \(0.711697\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.67806 −0.456207
\(66\) 0 0
\(67\) −8.78526 −1.07329 −0.536645 − 0.843808i \(-0.680309\pi\)
−0.536645 + 0.843808i \(0.680309\pi\)
\(68\) 0 0
\(69\) 1.88937 0.227453
\(70\) 0 0
\(71\) 2.58292 0.306536 0.153268 − 0.988185i \(-0.451020\pi\)
0.153268 + 0.988185i \(0.451020\pi\)
\(72\) 0 0
\(73\) −6.17149 −0.722318 −0.361159 − 0.932504i \(-0.617619\pi\)
−0.361159 + 0.932504i \(0.617619\pi\)
\(74\) 0 0
\(75\) −3.86450 −0.446234
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −9.15423 −1.02993 −0.514965 − 0.857211i \(-0.672195\pi\)
−0.514965 + 0.857211i \(0.672195\pi\)
\(80\) 0 0
\(81\) −6.10786 −0.678652
\(82\) 0 0
\(83\) −11.4885 −1.26103 −0.630516 − 0.776176i \(-0.717157\pi\)
−0.630516 + 0.776176i \(0.717157\pi\)
\(84\) 0 0
\(85\) −1.55355 −0.168506
\(86\) 0 0
\(87\) 5.53321 0.593222
\(88\) 0 0
\(89\) −2.57873 −0.273345 −0.136672 − 0.990616i \(-0.543641\pi\)
−0.136672 + 0.990616i \(0.543641\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −13.1892 −1.36766
\(94\) 0 0
\(95\) 4.78262 0.490687
\(96\) 0 0
\(97\) 12.0066 1.21908 0.609541 − 0.792754i \(-0.291354\pi\)
0.609541 + 0.792754i \(0.291354\pi\)
\(98\) 0 0
\(99\) 2.02733 0.203755
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.1.5 6
7.2 even 3 952.2.q.c.137.2 ✓ 12
7.4 even 3 952.2.q.c.681.2 yes 12
7.6 odd 2 6664.2.a.r.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.q.c.137.2 ✓ 12 7.2 even 3
952.2.q.c.681.2 yes 12 7.4 even 3
6664.2.a.r.1.2 6 7.6 odd 2
6664.2.a.s.1.5 6 1.1 even 1 trivial