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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 1189.1
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.d.2629.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.04519 + 0.903992i) q^{5} +0.920170i q^{7} +5.86884 q^{11} +(3.76771 - 2.17529i) q^{13} +(-3.41965 - 1.97433i) q^{17} +(2.41685 - 3.62751i) q^{19} +(-4.01695 + 2.31919i) q^{23} +(3.36560 - 3.69767i) q^{25} +(0.0437716 + 0.0758147i) q^{29} -1.97858 q^{31} +(-0.831826 - 1.88192i) q^{35} -0.283119i q^{37} +(-1.09094 + 1.88956i) q^{41} +(-6.91634 - 3.99315i) q^{43} +(-8.50107 + 4.90809i) q^{47} +6.15329 q^{49} +(4.99259 - 2.88247i) q^{53} +(-12.0029 + 5.30538i) q^{55} +(6.24158 - 10.8107i) q^{59} +(-3.45996 - 5.99282i) q^{61} +(-5.73923 + 7.85485i) q^{65} +(10.9620 - 6.32892i) q^{67} +(-7.18657 + 12.4475i) q^{71} +(9.26982 + 5.35193i) q^{73} +5.40033i q^{77} +(7.47006 - 12.9385i) q^{79} -10.5081i q^{83} +(8.77861 + 0.946552i) q^{85} +(4.90137 + 8.48942i) q^{89} +(2.00163 + 3.46693i) q^{91} +(-1.66367 + 9.60376i) q^{95} +(-13.2715 - 7.66228i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{11} + 12 q^{19} - 12 q^{25} - 22 q^{29} - 56 q^{31} + 2 q^{35} + 16 q^{41} - 88 q^{49} - 14 q^{55} + 86 q^{59} + 6 q^{61} - 36 q^{65} - 10 q^{71} + 50 q^{79} + 30 q^{85} - 34 q^{89} + 4 q^{91}+ \cdots - 46 q^{95}+O(q^{100}) \) 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
\(5\) −2.04519 + 0.903992i −0.914636 + 0.404278i
\(6\) 0 0
\(7\) 0.920170i 0.347791i 0.984764 + 0.173896i \(0.0556356\pi\)
−0.984764 + 0.173896i \(0.944364\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.86884 1.76952 0.884761 − 0.466046i \(-0.154322\pi\)
0.884761 + 0.466046i \(0.154322\pi\)
\(12\) 0 0
\(13\) 3.76771 − 2.17529i 1.04497 − 0.603316i 0.123736 − 0.992315i \(-0.460512\pi\)
0.921238 + 0.388999i \(0.127179\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.41965 − 1.97433i −0.829386 − 0.478846i 0.0242562 − 0.999706i \(-0.492278\pi\)
−0.853642 + 0.520859i \(0.825612\pi\)
\(18\) 0 0
\(19\) 2.41685 − 3.62751i 0.554463 − 0.832208i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.01695 + 2.31919i −0.837593 + 0.483584i −0.856445 − 0.516238i \(-0.827332\pi\)
0.0188524 + 0.999822i \(0.493999\pi\)
\(24\) 0 0
\(25\) 3.36560 − 3.69767i 0.673119 − 0.739534i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.0437716 + 0.0758147i 0.00812819 + 0.0140784i 0.870061 − 0.492944i \(-0.164079\pi\)
−0.861933 + 0.507023i \(0.830746\pi\)
\(30\) 0 0
\(31\) −1.97858 −0.355364 −0.177682 − 0.984088i \(-0.556860\pi\)
−0.177682 + 0.984088i \(0.556860\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.831826 − 1.88192i −0.140604 − 0.318103i
\(36\) 0 0
\(37\) − 0.283119i − 0.0465445i −0.999729 − 0.0232722i \(-0.992592\pi\)
0.999729 − 0.0232722i \(-0.00740846\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.09094 + 1.88956i −0.170376 + 0.295099i −0.938551 − 0.345140i \(-0.887831\pi\)
0.768176 + 0.640239i \(0.221165\pi\)
\(42\) 0 0
\(43\) −6.91634 − 3.99315i −1.05473 − 0.608950i −0.130762 − 0.991414i \(-0.541742\pi\)
−0.923971 + 0.382464i \(0.875076\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.50107 + 4.90809i −1.24001 + 0.715919i −0.969096 − 0.246685i \(-0.920659\pi\)
−0.270912 + 0.962604i \(0.587325\pi\)
\(48\) 0 0
\(49\) 6.15329 0.879041
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.99259 − 2.88247i 0.685784 − 0.395938i −0.116247 − 0.993220i \(-0.537086\pi\)
0.802031 + 0.597283i \(0.203753\pi\)
\(54\) 0 0
\(55\) −12.0029 + 5.30538i −1.61847 + 0.715378i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.24158 − 10.8107i 0.812584 − 1.40744i −0.0984660 − 0.995140i \(-0.531394\pi\)
0.911050 − 0.412296i \(-0.135273\pi\)
\(60\) 0 0
\(61\) −3.45996 − 5.99282i −0.443002 − 0.767303i 0.554908 − 0.831911i \(-0.312753\pi\)
−0.997911 + 0.0646090i \(0.979420\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.73923 + 7.85485i −0.711864 + 0.974275i
\(66\) 0 0
\(67\) 10.9620 − 6.32892i 1.33922 − 0.773201i 0.352531 − 0.935800i \(-0.385321\pi\)
0.986692 + 0.162599i \(0.0519877\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.18657 + 12.4475i −0.852889 + 1.47725i 0.0257001 + 0.999670i \(0.491818\pi\)
−0.878589 + 0.477578i \(0.841515\pi\)
\(72\) 0 0
\(73\) 9.26982 + 5.35193i 1.08495 + 0.626396i 0.932227 − 0.361873i \(-0.117863\pi\)
0.152723 + 0.988269i \(0.451196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.40033i 0.615424i
\(78\) 0 0
\(79\) 7.47006 − 12.9385i 0.840447 − 1.45570i −0.0490697 − 0.998795i \(-0.515626\pi\)
0.889517 − 0.456902i \(-0.151041\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 10.5081i − 1.15342i −0.816950 − 0.576708i \(-0.804337\pi\)
0.816950 − 0.576708i \(-0.195663\pi\)
\(84\) 0 0
\(85\) 8.77861 + 0.946552i 0.952174 + 0.102668i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.90137 + 8.48942i 0.519544 + 0.899877i 0.999742 + 0.0227168i \(0.00723161\pi\)
−0.480198 + 0.877160i \(0.659435\pi\)
\(90\) 0 0
\(91\) 2.00163 + 3.46693i 0.209828 + 0.363433i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.66367 + 9.60376i −0.170689 + 0.985325i
\(96\) 0 0
\(97\) −13.2715 − 7.66228i −1.34751 − 0.777987i −0.359616 − 0.933100i \(-0.617092\pi\)
−0.987897 + 0.155114i \(0.950426\pi\)
\(98\) 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.1189.1 32
3.2 odd 2 1140.2.bg.c.49.3 ✓ 32
5.4 even 2 inner 3420.2.bj.d.1189.10 32
15.14 odd 2 1140.2.bg.c.49.14 yes 32
19.7 even 3 inner 3420.2.bj.d.2629.10 32
57.26 odd 6 1140.2.bg.c.349.13 yes 32
95.64 even 6 inner 3420.2.bj.d.2629.1 32
285.254 odd 6 1140.2.bg.c.349.4 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.c.49.3 ✓ 32 3.2 odd 2
1140.2.bg.c.49.14 yes 32 15.14 odd 2
1140.2.bg.c.349.4 yes 32 285.254 odd 6
1140.2.bg.c.349.13 yes 32 57.26 odd 6
3420.2.bj.d.1189.1 32 1.1 even 1 trivial
3420.2.bj.d.1189.10 32 5.4 even 2 inner
3420.2.bj.d.2629.1 32 95.64 even 6 inner
3420.2.bj.d.2629.10 32 19.7 even 3 inner