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 1.3
Root \(0.0942397\) 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)
 
\(f(q)\) \(=\) \(q+0.0942397 q^{3} -3.30233 q^{5} -2.99112 q^{9} -0.376122 q^{11} +3.52076 q^{13} -0.311211 q^{15} -1.00000 q^{17} +1.50289 q^{19} +0.721321 q^{23} +5.90538 q^{25} -0.564601 q^{27} +0.849549 q^{29} +5.80710 q^{31} -0.0354456 q^{33} -4.62792 q^{37} +0.331795 q^{39} +3.94233 q^{41} +0.812495 q^{43} +9.87766 q^{45} -8.05274 q^{47} -0.0942397 q^{51} +1.02915 q^{53} +1.24208 q^{55} +0.141632 q^{57} -10.2546 q^{59} +12.4746 q^{61} -11.6267 q^{65} +6.96013 q^{67} +0.0679771 q^{69} +0.774148 q^{71} +12.4453 q^{73} +0.556521 q^{75} -7.36982 q^{79} +8.92015 q^{81} +5.39389 q^{83} +3.30233 q^{85} +0.0800613 q^{87} -0.902695 q^{89} +0.547259 q^{93} -4.96305 q^{95} -1.81278 q^{97} +1.12503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + q^{5} + q^{9} - 6 q^{11} - 2 q^{13} - 6 q^{17} + 2 q^{19} - 5 q^{23} + q^{25} - 8 q^{27} - 15 q^{29} + 8 q^{31} + 11 q^{33} - 16 q^{37} - 2 q^{39} + 4 q^{41} + 24 q^{43} + 2 q^{45} - q^{47}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(3\) 0.0942397 0.0544093 0.0272047 − 0.999630i \(-0.491339\pi\)
0.0272047 + 0.999630i \(0.491339\pi\)
\(4\) 0 0
\(5\) −3.30233 −1.47685 −0.738423 − 0.674337i \(-0.764429\pi\)
−0.738423 + 0.674337i \(0.764429\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.99112 −0.997040
\(10\) 0 0
\(11\) −0.376122 −0.113405 −0.0567025 − 0.998391i \(-0.518059\pi\)
−0.0567025 + 0.998391i \(0.518059\pi\)
\(12\) 0 0
\(13\) 3.52076 0.976482 0.488241 − 0.872709i \(-0.337639\pi\)
0.488241 + 0.872709i \(0.337639\pi\)
\(14\) 0 0
\(15\) −0.311211 −0.0803542
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 1.50289 0.344787 0.172394 − 0.985028i \(-0.444850\pi\)
0.172394 + 0.985028i \(0.444850\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.721321 0.150406 0.0752029 − 0.997168i \(-0.476040\pi\)
0.0752029 + 0.997168i \(0.476040\pi\)
\(24\) 0 0
\(25\) 5.90538 1.18108
\(26\) 0 0
\(27\) −0.564601 −0.108658
\(28\) 0 0
\(29\) 0.849549 0.157757 0.0788787 − 0.996884i \(-0.474866\pi\)
0.0788787 + 0.996884i \(0.474866\pi\)
\(30\) 0 0
\(31\) 5.80710 1.04299 0.521493 − 0.853256i \(-0.325375\pi\)
0.521493 + 0.853256i \(0.325375\pi\)
\(32\) 0 0
\(33\) −0.0354456 −0.00617029
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.62792 −0.760825 −0.380413 − 0.924817i \(-0.624218\pi\)
−0.380413 + 0.924817i \(0.624218\pi\)
\(38\) 0 0
\(39\) 0.331795 0.0531297
\(40\) 0 0
\(41\) 3.94233 0.615689 0.307844 − 0.951437i \(-0.400392\pi\)
0.307844 + 0.951437i \(0.400392\pi\)
\(42\) 0 0
\(43\) 0.812495 0.123904 0.0619522 − 0.998079i \(-0.480267\pi\)
0.0619522 + 0.998079i \(0.480267\pi\)
\(44\) 0 0
\(45\) 9.87766 1.47247
\(46\) 0 0
\(47\) −8.05274 −1.17461 −0.587306 − 0.809365i \(-0.699812\pi\)
−0.587306 + 0.809365i \(0.699812\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.0942397 −0.0131962
\(52\) 0 0
\(53\) 1.02915 0.141364 0.0706821 − 0.997499i \(-0.477482\pi\)
0.0706821 + 0.997499i \(0.477482\pi\)
\(54\) 0 0
\(55\) 1.24208 0.167482
\(56\) 0 0
\(57\) 0.141632 0.0187596
\(58\) 0 0
\(59\) −10.2546 −1.33504 −0.667519 − 0.744593i \(-0.732644\pi\)
−0.667519 + 0.744593i \(0.732644\pi\)
\(60\) 0 0
\(61\) 12.4746 1.59721 0.798604 − 0.601857i \(-0.205572\pi\)
0.798604 + 0.601857i \(0.205572\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −11.6267 −1.44211
\(66\) 0 0
\(67\) 6.96013 0.850316 0.425158 − 0.905119i \(-0.360219\pi\)
0.425158 + 0.905119i \(0.360219\pi\)
\(68\) 0 0
\(69\) 0.0679771 0.00818348
\(70\) 0 0
\(71\) 0.774148 0.0918745 0.0459372 − 0.998944i \(-0.485373\pi\)
0.0459372 + 0.998944i \(0.485373\pi\)
\(72\) 0 0
\(73\) 12.4453 1.45661 0.728304 − 0.685254i \(-0.240309\pi\)
0.728304 + 0.685254i \(0.240309\pi\)
\(74\) 0 0
\(75\) 0.556521 0.0642615
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.36982 −0.829170 −0.414585 − 0.910011i \(-0.636073\pi\)
−0.414585 + 0.910011i \(0.636073\pi\)
\(80\) 0 0
\(81\) 8.92015 0.991128
\(82\) 0 0
\(83\) 5.39389 0.592057 0.296028 − 0.955179i \(-0.404338\pi\)
0.296028 + 0.955179i \(0.404338\pi\)
\(84\) 0 0
\(85\) 3.30233 0.358188
\(86\) 0 0
\(87\) 0.0800613 0.00858347
\(88\) 0 0
\(89\) −0.902695 −0.0956854 −0.0478427 − 0.998855i \(-0.515235\pi\)
−0.0478427 + 0.998855i \(0.515235\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.547259 0.0567482
\(94\) 0 0
\(95\) −4.96305 −0.509198
\(96\) 0 0
\(97\) −1.81278 −0.184060 −0.0920299 − 0.995756i \(-0.529336\pi\)
−0.0920299 + 0.995756i \(0.529336\pi\)
\(98\) 0 0
\(99\) 1.12503 0.113069
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.1.3 6
7.2 even 3 952.2.q.c.137.4 ✓ 12
7.4 even 3 952.2.q.c.681.4 yes 12
7.6 odd 2 6664.2.a.r.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.q.c.137.4 ✓ 12 7.2 even 3
952.2.q.c.681.4 yes 12 7.4 even 3
6664.2.a.r.1.4 6 7.6 odd 2
6664.2.a.s.1.3 6 1.1 even 1 trivial