Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,2,Mod(49,288)] 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("288.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 288.49
Dual form 288.2.r.a.241.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 1.50000i) q^{3} +(1.73205 + 1.00000i) q^{5} +(2.00000 + 3.46410i) q^{7} +(-1.50000 + 2.59808i) q^{9} +(2.59808 - 1.50000i) q^{11} +(-1.73205 - 1.00000i) q^{13} -3.46410i q^{15} +5.00000 q^{17} -1.00000i q^{19} +(3.46410 - 6.00000i) q^{21} +(1.00000 - 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.19615 q^{27} +(-2.00000 + 3.46410i) q^{31} +(-4.50000 - 2.59808i) q^{33} +8.00000i q^{35} +2.00000i q^{37} +3.46410i q^{39} +(2.50000 - 4.33013i) q^{41} +(-9.52628 + 5.50000i) q^{43} +(-5.19615 + 3.00000i) q^{45} +(-3.00000 - 5.19615i) q^{47} +(-4.50000 + 7.79423i) q^{49} +(-4.33013 - 7.50000i) q^{51} +6.00000 q^{55} +(-1.50000 + 0.866025i) q^{57} +(-0.866025 - 0.500000i) q^{59} +(-10.3923 + 6.00000i) q^{61} -12.0000 q^{63} +(-2.00000 - 3.46410i) q^{65} +(2.59808 + 1.50000i) q^{67} -3.46410 q^{69} +6.00000 q^{71} +9.00000 q^{73} +(-0.866025 + 1.50000i) q^{75} +(10.3923 + 6.00000i) q^{77} +(-7.00000 - 12.1244i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(3.46410 - 2.00000i) q^{83} +(8.66025 + 5.00000i) q^{85} -14.0000 q^{89} -8.00000i q^{91} +6.92820 q^{93} +(1.00000 - 1.73205i) q^{95} +(-0.500000 - 0.866025i) q^{97} +9.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7} - 6 q^{9} + 20 q^{17} + 4 q^{23} - 2 q^{25} - 8 q^{31} - 18 q^{33} + 10 q^{41} - 12 q^{47} - 18 q^{49} + 24 q^{55} - 6 q^{57} - 48 q^{63} - 8 q^{65} + 24 q^{71} + 36 q^{73} - 28 q^{79} - 18 q^{81}+ \cdots - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) −0.866025 − 1.50000i −0.500000 − 0.866025i
\(4\) 0 0
\(5\) 1.73205 + 1.00000i 0.774597 + 0.447214i 0.834512 − 0.550990i \(-0.185750\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) 0 0
\(7\) 2.00000 + 3.46410i 0.755929 + 1.30931i 0.944911 + 0.327327i \(0.106148\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) −1.50000 + 2.59808i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 2.59808 − 1.50000i 0.783349 − 0.452267i −0.0542666 − 0.998526i \(-0.517282\pi\)
0.837616 + 0.546259i \(0.183949\pi\)
\(12\) 0 0
\(13\) −1.73205 − 1.00000i −0.480384 − 0.277350i 0.240192 − 0.970725i \(-0.422790\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) − 3.46410i − 0.894427i
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i −0.993399 − 0.114708i \(-0.963407\pi\)
0.993399 − 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) 3.46410 − 6.00000i 0.755929 − 1.30931i
\(22\) 0 0
\(23\) 1.00000 − 1.73205i 0.208514 − 0.361158i −0.742732 − 0.669588i \(-0.766471\pi\)
0.951247 + 0.308431i \(0.0998038\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 − 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) −4.50000 − 2.59808i −0.783349 − 0.452267i
\(34\) 0 0
\(35\) 8.00000i 1.35225i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 3.46410i 0.554700i
\(40\) 0 0
\(41\) 2.50000 − 4.33013i 0.390434 − 0.676252i −0.602072 − 0.798441i \(-0.705658\pi\)
0.992507 + 0.122189i \(0.0389915\pi\)
\(42\) 0 0
\(43\) −9.52628 + 5.50000i −1.45274 + 0.838742i −0.998636 − 0.0522047i \(-0.983375\pi\)
−0.454108 + 0.890947i \(0.650042\pi\)
\(44\) 0 0
\(45\) −5.19615 + 3.00000i −0.774597 + 0.447214i
\(46\) 0 0
\(47\) −3.00000 − 5.19615i −0.437595 − 0.757937i 0.559908 − 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) −4.50000 + 7.79423i −0.642857 + 1.11346i
\(50\) 0 0
\(51\) −4.33013 − 7.50000i −0.606339 − 1.05021i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) −1.50000 + 0.866025i −0.198680 + 0.114708i
\(58\) 0 0
\(59\) −0.866025 − 0.500000i −0.112747 − 0.0650945i 0.442566 − 0.896736i \(-0.354068\pi\)
−0.555313 + 0.831641i \(0.687402\pi\)
\(60\) 0 0
\(61\) −10.3923 + 6.00000i −1.33060 + 0.768221i −0.985391 − 0.170305i \(-0.945525\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 0 0
\(63\) −12.0000 −1.51186
\(64\) 0 0
\(65\) −2.00000 − 3.46410i −0.248069 − 0.429669i
\(66\) 0 0
\(67\) 2.59808 + 1.50000i 0.317406 + 0.183254i 0.650236 − 0.759733i \(-0.274670\pi\)
−0.332830 + 0.942987i \(0.608004\pi\)
\(68\) 0 0
\(69\) −3.46410 −0.417029
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 − 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) −0.866025 + 1.50000i −0.100000 + 0.173205i
\(76\) 0 0
\(77\) 10.3923 + 6.00000i 1.18431 + 0.683763i
\(78\) 0 0
\(79\) −7.00000 − 12.1244i −0.787562 − 1.36410i −0.927457 − 0.373930i \(-0.878010\pi\)
0.139895 − 0.990166i \(-0.455323\pi\)
\(80\) 0 0
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) 0 0
\(83\) 3.46410 − 2.00000i 0.380235 − 0.219529i −0.297686 − 0.954664i \(-0.596215\pi\)
0.677920 + 0.735135i \(0.262881\pi\)
\(84\) 0 0
\(85\) 8.66025 + 5.00000i 0.939336 + 0.542326i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) − 8.00000i − 0.838628i
\(92\) 0 0
\(93\) 6.92820 0.718421
\(94\) 0 0
\(95\) 1.00000 − 1.73205i 0.102598 − 0.177705i
\(96\) 0 0
\(97\) −0.500000 − 0.866025i −0.0507673 − 0.0879316i 0.839525 − 0.543321i \(-0.182833\pi\)
−0.890292 + 0.455389i \(0.849500\pi\)
\(98\) 0 0
\(99\) 9.00000i 0.904534i
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 288.2.r.a.49.1 4
3.2 odd 2 864.2.r.a.145.1 4
4.3 odd 2 72.2.n.a.13.2 yes 4
8.3 odd 2 72.2.n.a.13.1 ✓ 4
8.5 even 2 inner 288.2.r.a.49.2 4
9.2 odd 6 864.2.r.a.721.2 4
9.4 even 3 2592.2.d.b.1297.1 2
9.5 odd 6 2592.2.d.a.1297.2 2
9.7 even 3 inner 288.2.r.a.241.2 4
12.11 even 2 216.2.n.a.37.1 4
24.5 odd 2 864.2.r.a.145.2 4
24.11 even 2 216.2.n.a.37.2 4
36.7 odd 6 72.2.n.a.61.1 yes 4
36.11 even 6 216.2.n.a.181.2 4
36.23 even 6 648.2.d.a.325.1 2
36.31 odd 6 648.2.d.d.325.2 2
72.5 odd 6 2592.2.d.a.1297.1 2
72.11 even 6 216.2.n.a.181.1 4
72.13 even 6 2592.2.d.b.1297.2 2
72.29 odd 6 864.2.r.a.721.1 4
72.43 odd 6 72.2.n.a.61.2 yes 4
72.59 even 6 648.2.d.a.325.2 2
72.61 even 6 inner 288.2.r.a.241.1 4
72.67 odd 6 648.2.d.d.325.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.n.a.13.1 ✓ 4 8.3 odd 2
72.2.n.a.13.2 yes 4 4.3 odd 2
72.2.n.a.61.1 yes 4 36.7 odd 6
72.2.n.a.61.2 yes 4 72.43 odd 6
216.2.n.a.37.1 4 12.11 even 2
216.2.n.a.37.2 4 24.11 even 2
216.2.n.a.181.1 4 72.11 even 6
216.2.n.a.181.2 4 36.11 even 6
288.2.r.a.49.1 4 1.1 even 1 trivial
288.2.r.a.49.2 4 8.5 even 2 inner
288.2.r.a.241.1 4 72.61 even 6 inner
288.2.r.a.241.2 4 9.7 even 3 inner
648.2.d.a.325.1 2 36.23 even 6
648.2.d.a.325.2 2 72.59 even 6
648.2.d.d.325.1 2 72.67 odd 6
648.2.d.d.325.2 2 36.31 odd 6
864.2.r.a.145.1 4 3.2 odd 2
864.2.r.a.145.2 4 24.5 odd 2
864.2.r.a.721.1 4 72.29 odd 6
864.2.r.a.721.2 4 9.2 odd 6
2592.2.d.a.1297.1 2 72.5 odd 6
2592.2.d.a.1297.2 2 9.5 odd 6
2592.2.d.b.1297.1 2 9.4 even 3
2592.2.d.b.1297.2 2 72.13 even 6