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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3420,2,Mod(1189,3420)] 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("3420.1189"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,0,0] 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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 1140)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1189.9
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.d.2629.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.549131 + 2.16759i) q^{5} +4.44246i q^{7} +0.476281 q^{11} +(-1.45776 + 0.841636i) q^{13} +(4.30084 + 2.48309i) q^{17} +(-3.82342 + 2.09319i) q^{19} +(-0.678558 + 0.391766i) q^{23} +(-4.39691 - 2.38059i) q^{25} +(4.10032 + 7.10196i) q^{29} -1.32785 q^{31} +(-9.62945 - 2.43950i) q^{35} +6.75800i q^{37} +(0.993841 - 1.72138i) q^{41} +(0.154258 + 0.0890607i) q^{43} +(7.98652 - 4.61102i) q^{47} -12.7355 q^{49} +(4.30260 - 2.48411i) q^{53} +(-0.261541 + 1.03238i) q^{55} +(2.15464 - 3.73195i) q^{59} +(5.07076 + 8.78281i) q^{61} +(-1.02382 - 3.62199i) q^{65} +(4.67223 - 2.69751i) q^{67} +(-3.99299 + 6.91606i) q^{71} +(-11.7990 - 6.81216i) q^{73} +2.11586i q^{77} +(-5.10049 + 8.83431i) q^{79} -7.90678i q^{83} +(-7.74405 + 7.95892i) q^{85} +(-2.38968 - 4.13904i) q^{89} +(-3.73894 - 6.47603i) q^{91} +(-2.43763 - 9.43705i) q^{95} +(-5.46076 - 3.15277i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{11} + 12 q^{19} - 12 q^{25} - 22 q^{29} - 56 q^{31} + 2 q^{35} + 16 q^{41} - 88 q^{49} - 14 q^{55} + 86 q^{59} + 6 q^{61} - 36 q^{65} - 10 q^{71} + 50 q^{79} + 30 q^{85} - 34 q^{89} + 4 q^{91}+ \cdots - 46 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.549131 + 2.16759i −0.245579 + 0.969377i
\(6\) 0 0
\(7\) 4.44246i 1.67909i 0.543288 + 0.839547i \(0.317179\pi\)
−0.543288 + 0.839547i \(0.682821\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.476281 0.143604 0.0718021 − 0.997419i \(-0.477125\pi\)
0.0718021 + 0.997419i \(0.477125\pi\)
\(12\) 0 0
\(13\) −1.45776 + 0.841636i −0.404309 + 0.233428i −0.688342 − 0.725387i \(-0.741661\pi\)
0.284033 + 0.958815i \(0.408328\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.30084 + 2.48309i 1.04311 + 0.602238i 0.920712 − 0.390243i \(-0.127609\pi\)
0.122395 + 0.992481i \(0.460942\pi\)
\(18\) 0 0
\(19\) −3.82342 + 2.09319i −0.877153 + 0.480212i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.678558 + 0.391766i −0.141489 + 0.0816888i −0.569074 − 0.822287i \(-0.692698\pi\)
0.427584 + 0.903975i \(0.359365\pi\)
\(24\) 0 0
\(25\) −4.39691 − 2.38059i −0.879382 − 0.476117i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.10032 + 7.10196i 0.761410 + 1.31880i 0.942124 + 0.335265i \(0.108826\pi\)
−0.180714 + 0.983536i \(0.557841\pi\)
\(30\) 0 0
\(31\) −1.32785 −0.238489 −0.119245 − 0.992865i \(-0.538047\pi\)
−0.119245 + 0.992865i \(0.538047\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.62945 − 2.43950i −1.62767 − 0.412350i
\(36\) 0 0
\(37\) 6.75800i 1.11101i 0.831514 + 0.555504i \(0.187475\pi\)
−0.831514 + 0.555504i \(0.812525\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.993841 − 1.72138i 0.155212 − 0.268835i −0.777924 − 0.628358i \(-0.783727\pi\)
0.933136 + 0.359523i \(0.117061\pi\)
\(42\) 0 0
\(43\) 0.154258 + 0.0890607i 0.0235241 + 0.0135816i 0.511716 − 0.859155i \(-0.329010\pi\)
−0.488192 + 0.872736i \(0.662343\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.98652 − 4.61102i 1.16495 − 0.672587i 0.212468 − 0.977168i \(-0.431850\pi\)
0.952486 + 0.304581i \(0.0985165\pi\)
\(48\) 0 0
\(49\) −12.7355 −1.81935
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.30260 − 2.48411i 0.591008 − 0.341218i −0.174488 − 0.984659i \(-0.555827\pi\)
0.765496 + 0.643441i \(0.222494\pi\)
\(54\) 0 0
\(55\) −0.261541 + 1.03238i −0.0352662 + 0.139207i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.15464 − 3.73195i 0.280511 − 0.485858i −0.691000 − 0.722855i \(-0.742830\pi\)
0.971511 + 0.236996i \(0.0761629\pi\)
\(60\) 0 0
\(61\) 5.07076 + 8.78281i 0.649244 + 1.12452i 0.983304 + 0.181971i \(0.0582478\pi\)
−0.334060 + 0.942552i \(0.608419\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.02382 − 3.62199i −0.126990 − 0.449253i
\(66\) 0 0
\(67\) 4.67223 − 2.69751i 0.570804 − 0.329554i −0.186667 − 0.982423i \(-0.559768\pi\)
0.757470 + 0.652870i \(0.226435\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.99299 + 6.91606i −0.473881 + 0.820785i −0.999553 − 0.0299019i \(-0.990480\pi\)
0.525672 + 0.850687i \(0.323814\pi\)
\(72\) 0 0
\(73\) −11.7990 − 6.81216i −1.38097 − 0.797303i −0.388695 − 0.921366i \(-0.627074\pi\)
−0.992274 + 0.124063i \(0.960407\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.11586i 0.241125i
\(78\) 0 0
\(79\) −5.10049 + 8.83431i −0.573850 + 0.993938i 0.422315 + 0.906449i \(0.361217\pi\)
−0.996166 + 0.0874886i \(0.972116\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 7.90678i − 0.867882i −0.900941 − 0.433941i \(-0.857123\pi\)
0.900941 − 0.433941i \(-0.142877\pi\)
\(84\) 0 0
\(85\) −7.74405 + 7.95892i −0.839961 + 0.863266i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.38968 − 4.13904i −0.253305 − 0.438738i 0.711129 − 0.703062i \(-0.248184\pi\)
−0.964434 + 0.264324i \(0.914851\pi\)
\(90\) 0 0
\(91\) −3.73894 − 6.47603i −0.391947 − 0.678873i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.43763 − 9.43705i −0.250096 − 0.968221i
\(96\) 0 0
\(97\) −5.46076 − 3.15277i −0.554456 − 0.320116i 0.196461 − 0.980512i \(-0.437055\pi\)
−0.750917 + 0.660396i \(0.770388\pi\)
\(98\) 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 3420.2.bj.d.1189.9 32
3.2 odd 2 1140.2.bg.c.49.1 ✓ 32
5.4 even 2 inner 3420.2.bj.d.1189.3 32
15.14 odd 2 1140.2.bg.c.49.16 yes 32
19.7 even 3 inner 3420.2.bj.d.2629.3 32
57.26 odd 6 1140.2.bg.c.349.16 yes 32
95.64 even 6 inner 3420.2.bj.d.2629.9 32
285.254 odd 6 1140.2.bg.c.349.1 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.c.49.1 ✓ 32 3.2 odd 2
1140.2.bg.c.49.16 yes 32 15.14 odd 2
1140.2.bg.c.349.1 yes 32 285.254 odd 6
1140.2.bg.c.349.16 yes 32 57.26 odd 6
3420.2.bj.d.1189.3 32 5.4 even 2 inner
3420.2.bj.d.1189.9 32 1.1 even 1 trivial
3420.2.bj.d.2629.3 32 19.7 even 3 inner
3420.2.bj.d.2629.9 32 95.64 even 6 inner