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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(289,504)] 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("504.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,1,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 504.289
Dual form 504.2.s.f.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-2.50000 - 4.33013i) q^{11} +2.00000 q^{13} +(-3.00000 - 5.19615i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(3.00000 - 5.19615i) q^{23} +(2.00000 + 3.46410i) q^{25} -3.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-2.00000 + 1.73205i) q^{35} +(1.00000 - 1.73205i) q^{37} +8.00000 q^{41} -4.00000 q^{43} +(-2.00000 + 3.46410i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-4.50000 - 7.79423i) q^{53} -5.00000 q^{55} +(1.50000 + 2.59808i) q^{59} +(6.00000 - 10.3923i) q^{61} +(1.00000 - 1.73205i) q^{65} +(-1.00000 - 1.73205i) q^{67} -8.00000 q^{71} +(7.00000 + 12.1244i) q^{73} +(2.50000 + 12.9904i) q^{77} +(-0.500000 + 0.866025i) q^{79} +17.0000 q^{83} -6.00000 q^{85} +(-9.00000 + 15.5885i) q^{89} +(-5.00000 - 1.73205i) q^{91} +(1.00000 + 1.73205i) q^{95} +3.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 5 q^{7} - 5 q^{11} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 6 q^{23} + 4 q^{25} - 6 q^{29} - 5 q^{31} - 4 q^{35} + 2 q^{37} + 16 q^{41} - 8 q^{43} - 4 q^{47} + 11 q^{49} - 9 q^{53} - 10 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i −0.732294 − 0.680989i \(-0.761550\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) −2.50000 − 0.866025i −0.944911 − 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.50000 − 4.33013i −0.753778 − 1.30558i −0.945979 − 0.324227i \(-0.894896\pi\)
0.192201 − 0.981356i \(-0.438437\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 − 5.19615i −0.727607 − 1.26025i −0.957892 − 0.287129i \(-0.907299\pi\)
0.230285 − 0.973123i \(-0.426034\pi\)
\(18\) 0 0
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 − 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 − 5.19615i 0.625543 − 1.08347i −0.362892 − 0.931831i \(-0.618211\pi\)
0.988436 − 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −2.50000 − 4.33013i −0.449013 − 0.777714i 0.549309 − 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 + 1.73205i −0.338062 + 0.292770i
\(36\) 0 0
\(37\) 1.00000 − 1.73205i 0.164399 − 0.284747i −0.772043 − 0.635571i \(-0.780765\pi\)
0.936442 + 0.350823i \(0.114098\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 + 3.46410i −0.291730 + 0.505291i −0.974219 − 0.225605i \(-0.927564\pi\)
0.682489 + 0.730896i \(0.260898\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 − 7.79423i −0.618123 − 1.07062i −0.989828 − 0.142269i \(-0.954560\pi\)
0.371706 − 0.928351i \(-0.378773\pi\)
\(54\) 0 0
\(55\) −5.00000 −0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 − 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 0 0
\(61\) 6.00000 − 10.3923i 0.768221 − 1.33060i −0.170305 − 0.985391i \(-0.554475\pi\)
0.938527 − 0.345207i \(-0.112191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 − 1.73205i 0.124035 − 0.214834i
\(66\) 0 0
\(67\) −1.00000 − 1.73205i −0.122169 − 0.211604i 0.798454 − 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 7.00000 + 12.1244i 0.819288 + 1.41905i 0.906208 + 0.422833i \(0.138964\pi\)
−0.0869195 + 0.996215i \(0.527702\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.50000 + 12.9904i 0.284901 + 1.48039i
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.0562544 + 0.0974355i −0.892781 − 0.450490i \(-0.851249\pi\)
0.836527 + 0.547926i \(0.184582\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.0000 1.86599 0.932996 − 0.359886i \(-0.117184\pi\)
0.932996 + 0.359886i \(0.117184\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 + 15.5885i −0.953998 + 1.65237i −0.217354 + 0.976093i \(0.569742\pi\)
−0.736644 + 0.676280i \(0.763591\pi\)
\(90\) 0 0
\(91\) −5.00000 − 1.73205i −0.524142 − 0.181568i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 + 1.73205i 0.102598 + 0.177705i
\(96\) 0 0
\(97\) 3.00000 0.304604 0.152302 − 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) 0 0
\(99\) 0 0
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 504.2.s.f.289.1 yes 2
3.2 odd 2 504.2.s.b.289.1 ✓ 2
4.3 odd 2 1008.2.s.l.289.1 2
7.2 even 3 3528.2.a.l.1.1 1
7.3 odd 6 3528.2.s.l.361.1 2
7.4 even 3 inner 504.2.s.f.361.1 yes 2
7.5 odd 6 3528.2.a.s.1.1 1
7.6 odd 2 3528.2.s.l.3313.1 2
12.11 even 2 1008.2.s.h.289.1 2
21.2 odd 6 3528.2.a.o.1.1 1
21.5 even 6 3528.2.a.h.1.1 1
21.11 odd 6 504.2.s.b.361.1 yes 2
21.17 even 6 3528.2.s.r.361.1 2
21.20 even 2 3528.2.s.r.3313.1 2
28.11 odd 6 1008.2.s.l.865.1 2
28.19 even 6 7056.2.a.bh.1.1 1
28.23 odd 6 7056.2.a.r.1.1 1
84.11 even 6 1008.2.s.h.865.1 2
84.23 even 6 7056.2.a.bm.1.1 1
84.47 odd 6 7056.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.s.b.289.1 ✓ 2 3.2 odd 2
504.2.s.b.361.1 yes 2 21.11 odd 6
504.2.s.f.289.1 yes 2 1.1 even 1 trivial
504.2.s.f.361.1 yes 2 7.4 even 3 inner
1008.2.s.h.289.1 2 12.11 even 2
1008.2.s.h.865.1 2 84.11 even 6
1008.2.s.l.289.1 2 4.3 odd 2
1008.2.s.l.865.1 2 28.11 odd 6
3528.2.a.h.1.1 1 21.5 even 6
3528.2.a.l.1.1 1 7.2 even 3
3528.2.a.o.1.1 1 21.2 odd 6
3528.2.a.s.1.1 1 7.5 odd 6
3528.2.s.l.361.1 2 7.3 odd 6
3528.2.s.l.3313.1 2 7.6 odd 2
3528.2.s.r.361.1 2 21.17 even 6
3528.2.s.r.3313.1 2 21.20 even 2
7056.2.a.r.1.1 1 28.23 odd 6
7056.2.a.v.1.1 1 84.47 odd 6
7056.2.a.bh.1.1 1 28.19 even 6
7056.2.a.bm.1.1 1 84.23 even 6