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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,Mod(361,588)] 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("588.361"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-54,0,264] 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: no
Analytic conductor: \(183.682394985\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{3649})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 913x^{2} + 912x + 831744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.2
Root \(15.3517 + 26.5900i\) of defining polynomial
Character \(\chi\) \(=\) 588.373
Dual form 588.8.i.j.361.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-13.5000 - 23.3827i) q^{3} +(247.221 - 428.199i) q^{5} +(-364.500 + 631.333i) q^{9} +(-23.5460 - 40.7829i) q^{11} -3624.60 q^{13} -13349.9 q^{15} +(-7321.36 - 12681.0i) q^{17} +(-1697.98 + 2940.98i) q^{19} +(8991.24 - 15573.3i) q^{23} +(-83173.8 - 144061. i) q^{25} +19683.0 q^{27} +139916. q^{29} +(114444. + 198223. i) q^{31} +(-635.742 + 1101.14i) q^{33} +(-219131. + 379546. i) q^{37} +(48932.1 + 84752.9i) q^{39} -312110. q^{41} -556478. q^{43} +(180224. + 312157. i) q^{45} +(-397117. + 687826. i) q^{47} +(-197677. + 342386. i) q^{51} +(-1.02360e6 - 1.77293e6i) q^{53} -23284.3 q^{55} +91690.7 q^{57} +(1.28373e6 + 2.22348e6i) q^{59} +(-1.23054e6 + 2.13136e6i) q^{61} +(-896077. + 1.55205e6i) q^{65} +(-1.07740e6 - 1.86611e6i) q^{67} -485527. q^{69} -2.38723e6 q^{71} +(-986281. - 1.70829e6i) q^{73} +(-2.24569e6 + 3.88965e6i) q^{75} +(-58847.1 + 101926. i) q^{79} +(-265720. - 460241. i) q^{81} +509356. q^{83} -7.23997e6 q^{85} +(-1.88887e6 - 3.27161e6i) q^{87} +(-2.08016e6 + 3.60293e6i) q^{89} +(3.08999e6 - 5.35202e6i) q^{93} +(839550. + 1.45414e6i) q^{95} -8.40834e6 q^{97} +34330.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 54 q^{3} + 264 q^{5} - 1458 q^{9} + 4980 q^{11} + 20296 q^{13} - 14256 q^{15} + 17832 q^{17} + 6256 q^{19} - 14052 q^{23} - 141326 q^{25} + 78732 q^{27} + 487176 q^{29} + 470824 q^{31} + 134460 q^{33}+ \cdots - 7260840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −13.5000 − 23.3827i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 247.221 − 428.199i 0.884484 − 1.53197i 0.0381806 − 0.999271i \(-0.487844\pi\)
0.846304 − 0.532701i \(-0.178823\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −364.500 + 631.333i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −23.5460 − 40.7829i −0.00533388 − 0.00923855i 0.863346 − 0.504612i \(-0.168365\pi\)
−0.868680 + 0.495374i \(0.835031\pi\)
\(12\) 0 0
\(13\) −3624.60 −0.457571 −0.228786 − 0.973477i \(-0.573475\pi\)
−0.228786 + 0.973477i \(0.573475\pi\)
\(14\) 0 0
\(15\) −13349.9 −1.02131
\(16\) 0 0
\(17\) −7321.36 − 12681.0i −0.361427 − 0.626009i 0.626769 − 0.779205i \(-0.284377\pi\)
−0.988196 + 0.153196i \(0.951044\pi\)
\(18\) 0 0
\(19\) −1697.98 + 2940.98i −0.0567929 + 0.0983681i −0.893024 − 0.450009i \(-0.851421\pi\)
0.836231 + 0.548377i \(0.184754\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8991.24 − 15573.3i 0.154089 − 0.266890i −0.778638 − 0.627474i \(-0.784089\pi\)
0.932727 + 0.360583i \(0.117422\pi\)
\(24\) 0 0
\(25\) −83173.8 − 144061.i −1.06462 − 1.84398i
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 139916. 1.06531 0.532653 − 0.846334i \(-0.321195\pi\)
0.532653 + 0.846334i \(0.321195\pi\)
\(30\) 0 0
\(31\) 114444. + 198223.i 0.689965 + 1.19505i 0.971849 + 0.235606i \(0.0757075\pi\)
−0.281883 + 0.959449i \(0.590959\pi\)
\(32\) 0 0
\(33\) −635.742 + 1101.14i −0.00307952 + 0.00533388i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −219131. + 379546.i −0.711211 + 1.23185i 0.253192 + 0.967416i \(0.418520\pi\)
−0.964403 + 0.264437i \(0.914814\pi\)
\(38\) 0 0
\(39\) 48932.1 + 84752.9i 0.132089 + 0.228786i
\(40\) 0 0
\(41\) −312110. −0.707236 −0.353618 − 0.935390i \(-0.615049\pi\)
−0.353618 + 0.935390i \(0.615049\pi\)
\(42\) 0 0
\(43\) −556478. −1.06735 −0.533676 − 0.845689i \(-0.679190\pi\)
−0.533676 + 0.845689i \(0.679190\pi\)
\(44\) 0 0
\(45\) 180224. + 312157.i 0.294828 + 0.510657i
\(46\) 0 0
\(47\) −397117. + 687826.i −0.557925 + 0.966354i 0.439745 + 0.898123i \(0.355069\pi\)
−0.997670 + 0.0682314i \(0.978264\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −197677. + 342386.i −0.208670 + 0.361427i
\(52\) 0 0
\(53\) −1.02360e6 − 1.77293e6i −0.944421 − 1.63578i −0.756907 − 0.653523i \(-0.773290\pi\)
−0.187514 − 0.982262i \(-0.560043\pi\)
\(54\) 0 0
\(55\) −23284.3 −0.0188709
\(56\) 0 0
\(57\) 91690.7 0.0655788
\(58\) 0 0
\(59\) 1.28373e6 + 2.22348e6i 0.813750 + 1.40946i 0.910222 + 0.414121i \(0.135911\pi\)
−0.0964718 + 0.995336i \(0.530756\pi\)
\(60\) 0 0
\(61\) −1.23054e6 + 2.13136e6i −0.694130 + 1.20227i 0.276343 + 0.961059i \(0.410877\pi\)
−0.970473 + 0.241210i \(0.922456\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −896077. + 1.55205e6i −0.404714 + 0.700986i
\(66\) 0 0
\(67\) −1.07740e6 − 1.86611e6i −0.437638 − 0.758012i 0.559868 − 0.828582i \(-0.310852\pi\)
−0.997507 + 0.0705695i \(0.977518\pi\)
\(68\) 0 0
\(69\) −485527. −0.177927
\(70\) 0 0
\(71\) −2.38723e6 −0.791570 −0.395785 − 0.918343i \(-0.629528\pi\)
−0.395785 + 0.918343i \(0.629528\pi\)
\(72\) 0 0
\(73\) −986281. − 1.70829e6i −0.296736 − 0.513962i 0.678651 − 0.734461i \(-0.262565\pi\)
−0.975387 + 0.220499i \(0.929232\pi\)
\(74\) 0 0
\(75\) −2.24569e6 + 3.88965e6i −0.614661 + 1.06462i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −58847.1 + 101926.i −0.0134286 + 0.0232590i −0.872662 − 0.488325i \(-0.837608\pi\)
0.859233 + 0.511584i \(0.170941\pi\)
\(80\) 0 0
\(81\) −265720. − 460241.i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 509356. 0.0977796 0.0488898 − 0.998804i \(-0.484432\pi\)
0.0488898 + 0.998804i \(0.484432\pi\)
\(84\) 0 0
\(85\) −7.23997e6 −1.27871
\(86\) 0 0
\(87\) −1.88887e6 − 3.27161e6i −0.307528 − 0.532653i
\(88\) 0 0
\(89\) −2.08016e6 + 3.60293e6i −0.312774 + 0.541741i −0.978962 − 0.204043i \(-0.934592\pi\)
0.666188 + 0.745784i \(0.267925\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.08999e6 − 5.35202e6i 0.398352 − 0.689965i
\(94\) 0 0
\(95\) 839550. + 1.45414e6i 0.100465 + 0.174010i
\(96\) 0 0
\(97\) −8.40834e6 −0.935425 −0.467713 − 0.883881i \(-0.654922\pi\)
−0.467713 + 0.883881i \(0.654922\pi\)
\(98\) 0 0
\(99\) 34330.1 0.00355592
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 588.8.i.j.373.2 4
7.2 even 3 588.8.a.f.1.1 2
7.3 odd 6 588.8.i.k.361.1 4
7.4 even 3 inner 588.8.i.j.361.2 4
7.5 odd 6 84.8.a.c.1.2 ✓ 2
7.6 odd 2 588.8.i.k.373.1 4
21.5 even 6 252.8.a.c.1.1 2
28.19 even 6 336.8.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.c.1.2 ✓ 2 7.5 odd 6
252.8.a.c.1.1 2 21.5 even 6
336.8.a.q.1.2 2 28.19 even 6
588.8.a.f.1.1 2 7.2 even 3
588.8.i.j.361.2 4 7.4 even 3 inner
588.8.i.j.373.2 4 1.1 even 1 trivial
588.8.i.k.361.1 4 7.3 odd 6
588.8.i.k.373.1 4 7.6 odd 2