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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 1189.4
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.d.2629.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.54855 + 1.61307i) q^{5} +3.05973i q^{7} -3.49139 q^{11} +(-3.59012 + 2.07276i) q^{13} +(-1.23985 - 0.715830i) q^{17} +(3.07570 + 3.08870i) q^{19} +(0.806184 - 0.465451i) q^{23} +(-0.203983 - 4.99584i) q^{25} +(-1.08575 - 1.88057i) q^{29} -3.18759 q^{31} +(-4.93556 - 4.73815i) q^{35} +5.47838i q^{37} +(-2.30312 + 3.98912i) q^{41} +(0.989283 + 0.571163i) q^{43} +(-10.5708 + 6.10305i) q^{47} -2.36196 q^{49} +(8.05198 - 4.64881i) q^{53} +(5.40660 - 5.63186i) q^{55} +(-1.15350 + 1.99792i) q^{59} +(-4.25742 - 7.37406i) q^{61} +(2.21598 - 9.00088i) q^{65} +(8.69707 - 5.02126i) q^{67} +(6.70767 - 11.6180i) q^{71} +(1.46266 + 0.844468i) q^{73} -10.6827i q^{77} +(7.43077 - 12.8705i) q^{79} +8.57587i q^{83} +(3.07466 - 0.891471i) q^{85} +(-7.41018 - 12.8348i) q^{89} +(-6.34208 - 10.9848i) q^{91} +(-9.74516 + 0.178320i) q^{95} +(-3.79910 - 2.19341i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{11} + 12 q^{19} - 12 q^{25} - 22 q^{29} - 56 q^{31} + 2 q^{35} + 16 q^{41} - 88 q^{49} - 14 q^{55} + 86 q^{59} + 6 q^{61} - 36 q^{65} - 10 q^{71} + 50 q^{79} + 30 q^{85} - 34 q^{89} + 4 q^{91}+ \cdots - 46 q^{95}+O(q^{100}) \) 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
\(5\) −1.54855 + 1.61307i −0.692533 + 0.721386i
\(6\) 0 0
\(7\) 3.05973i 1.15647i 0.815870 + 0.578235i \(0.196258\pi\)
−0.815870 + 0.578235i \(0.803742\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.49139 −1.05269 −0.526347 − 0.850270i \(-0.676439\pi\)
−0.526347 + 0.850270i \(0.676439\pi\)
\(12\) 0 0
\(13\) −3.59012 + 2.07276i −0.995721 + 0.574880i −0.906979 − 0.421175i \(-0.861618\pi\)
−0.0887413 + 0.996055i \(0.528284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.23985 − 0.715830i −0.300709 − 0.173614i 0.342053 − 0.939681i \(-0.388878\pi\)
−0.642761 + 0.766067i \(0.722211\pi\)
\(18\) 0 0
\(19\) 3.07570 + 3.08870i 0.705615 + 0.708596i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.806184 − 0.465451i 0.168101 − 0.0970531i −0.413589 − 0.910464i \(-0.635725\pi\)
0.581690 + 0.813410i \(0.302392\pi\)
\(24\) 0 0
\(25\) −0.203983 − 4.99584i −0.0407967 − 0.999167i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.08575 − 1.88057i −0.201619 − 0.349214i 0.747431 − 0.664339i \(-0.231287\pi\)
−0.949050 + 0.315125i \(0.897954\pi\)
\(30\) 0 0
\(31\) −3.18759 −0.572507 −0.286254 − 0.958154i \(-0.592410\pi\)
−0.286254 + 0.958154i \(0.592410\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.93556 − 4.73815i −0.834262 − 0.800893i
\(36\) 0 0
\(37\) 5.47838i 0.900640i 0.892867 + 0.450320i \(0.148690\pi\)
−0.892867 + 0.450320i \(0.851310\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.30312 + 3.98912i −0.359687 + 0.622995i −0.987908 − 0.155039i \(-0.950450\pi\)
0.628222 + 0.778034i \(0.283783\pi\)
\(42\) 0 0
\(43\) 0.989283 + 0.571163i 0.150864 + 0.0871015i 0.573532 − 0.819183i \(-0.305573\pi\)
−0.422668 + 0.906285i \(0.638906\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.5708 + 6.10305i −1.54191 + 0.890221i −0.543189 + 0.839610i \(0.682783\pi\)
−0.998718 + 0.0506106i \(0.983883\pi\)
\(48\) 0 0
\(49\) −2.36196 −0.337423
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.05198 − 4.64881i 1.10603 − 0.638564i 0.168228 − 0.985748i \(-0.446195\pi\)
0.937797 + 0.347184i \(0.112862\pi\)
\(54\) 0 0
\(55\) 5.40660 − 5.63186i 0.729025 − 0.759399i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.15350 + 1.99792i −0.150173 + 0.260107i −0.931291 − 0.364277i \(-0.881316\pi\)
0.781118 + 0.624383i \(0.214650\pi\)
\(60\) 0 0
\(61\) −4.25742 − 7.37406i −0.545106 − 0.944152i −0.998600 − 0.0528923i \(-0.983156\pi\)
0.453494 − 0.891259i \(-0.350177\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.21598 − 9.00088i 0.274859 − 1.11642i
\(66\) 0 0
\(67\) 8.69707 − 5.02126i 1.06252 − 0.613444i 0.136389 − 0.990655i \(-0.456450\pi\)
0.926127 + 0.377211i \(0.123117\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.70767 − 11.6180i 0.796054 − 1.37881i −0.126113 − 0.992016i \(-0.540250\pi\)
0.922167 − 0.386791i \(-0.126416\pi\)
\(72\) 0 0
\(73\) 1.46266 + 0.844468i 0.171192 + 0.0988375i 0.583148 − 0.812366i \(-0.301821\pi\)
−0.411956 + 0.911204i \(0.635154\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 10.6827i − 1.21741i
\(78\) 0 0
\(79\) 7.43077 − 12.8705i 0.836027 − 1.44804i −0.0571645 − 0.998365i \(-0.518206\pi\)
0.893192 − 0.449676i \(-0.148461\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.57587i 0.941324i 0.882314 + 0.470662i \(0.155985\pi\)
−0.882314 + 0.470662i \(0.844015\pi\)
\(84\) 0 0
\(85\) 3.07466 − 0.891471i 0.333493 − 0.0966936i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.41018 − 12.8348i −0.785478 − 1.36049i −0.928713 − 0.370799i \(-0.879084\pi\)
0.143235 − 0.989689i \(-0.454249\pi\)
\(90\) 0 0
\(91\) −6.34208 − 10.9848i −0.664831 − 1.15152i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −9.74516 + 0.178320i −0.999833 + 0.0182953i
\(96\) 0 0
\(97\) −3.79910 − 2.19341i −0.385740 − 0.222707i 0.294573 − 0.955629i \(-0.404823\pi\)
−0.680313 + 0.732922i \(0.738156\pi\)
\(98\) 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.1189.4 32
3.2 odd 2 1140.2.bg.c.49.11 yes 32
5.4 even 2 inner 3420.2.bj.d.1189.7 32
15.14 odd 2 1140.2.bg.c.49.6 ✓ 32
19.7 even 3 inner 3420.2.bj.d.2629.7 32
57.26 odd 6 1140.2.bg.c.349.6 yes 32
95.64 even 6 inner 3420.2.bj.d.2629.4 32
285.254 odd 6 1140.2.bg.c.349.11 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.c.49.6 ✓ 32 15.14 odd 2
1140.2.bg.c.49.11 yes 32 3.2 odd 2
1140.2.bg.c.349.6 yes 32 57.26 odd 6
1140.2.bg.c.349.11 yes 32 285.254 odd 6
3420.2.bj.d.1189.4 32 1.1 even 1 trivial
3420.2.bj.d.1189.7 32 5.4 even 2 inner
3420.2.bj.d.2629.4 32 95.64 even 6 inner
3420.2.bj.d.2629.7 32 19.7 even 3 inner