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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 1189.7
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.d.2629.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.622684 + 2.14762i) q^{5} -3.05973i q^{7} -3.49139 q^{11} +(3.59012 - 2.07276i) q^{13} +(1.23985 + 0.715830i) q^{17} +(3.07570 + 3.08870i) q^{19} +(-0.806184 + 0.465451i) q^{23} +(-4.22453 - 2.67457i) q^{25} +(-1.08575 - 1.88057i) q^{29} -3.18759 q^{31} +(6.57114 + 1.90524i) q^{35} -5.47838i q^{37} +(-2.30312 + 3.98912i) q^{41} +(-0.989283 - 0.571163i) q^{43} +(10.5708 - 6.10305i) q^{47} -2.36196 q^{49} +(-8.05198 + 4.64881i) q^{53} +(2.17403 - 7.49818i) q^{55} +(-1.15350 + 1.99792i) q^{59} +(-4.25742 - 7.37406i) q^{61} +(2.21598 + 9.00088i) q^{65} +(-8.69707 + 5.02126i) q^{67} +(6.70767 - 11.6180i) q^{71} +(-1.46266 - 0.844468i) q^{73} +10.6827i q^{77} +(7.43077 - 12.8705i) q^{79} -8.57587i q^{83} +(-2.30937 + 2.21700i) q^{85} +(-7.41018 - 12.8348i) q^{89} +(-6.34208 - 10.9848i) q^{91} +(-8.54853 + 4.68216i) q^{95} +(3.79910 + 2.19341i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{11} + 12 q^{19} - 12 q^{25} - 22 q^{29} - 56 q^{31} + 2 q^{35} + 16 q^{41} - 88 q^{49} - 14 q^{55} + 86 q^{59} + 6 q^{61} - 36 q^{65} - 10 q^{71} + 50 q^{79} + 30 q^{85} - 34 q^{89} + 4 q^{91}+ \cdots - 46 q^{95}+O(q^{100}) \) 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
\(5\) −0.622684 + 2.14762i −0.278473 + 0.960444i
\(6\) 0 0
\(7\) − 3.05973i − 1.15647i −0.815870 − 0.578235i \(-0.803742\pi\)
0.815870 − 0.578235i \(-0.196258\pi\)
\(8\) 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 0.0887413 − 0.996055i \(-0.471716\pi\)
0.906979 + 0.421175i \(0.138382\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.23985 + 0.715830i 0.300709 + 0.173614i 0.642761 − 0.766067i \(-0.277789\pi\)
−0.342053 + 0.939681i \(0.611122\pi\)
\(18\) 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 −0.581690 − 0.813410i \(-0.697608\pi\)
0.413589 + 0.910464i \(0.364275\pi\)
\(24\) 0 0
\(25\) −4.22453 − 2.67457i −0.844906 − 0.534915i
\(26\) 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
\(35\) 6.57114 + 1.90524i 1.11072 + 0.322045i
\(36\) 0 0
\(37\) − 5.47838i − 0.900640i −0.892867 − 0.450320i \(-0.851310\pi\)
0.892867 − 0.450320i \(-0.148690\pi\)
\(38\) 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 0.422668 − 0.906285i \(-0.361094\pi\)
−0.573532 + 0.819183i \(0.694427\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.5708 − 6.10305i 1.54191 − 0.890221i 0.543189 − 0.839610i \(-0.317217\pi\)
0.998718 − 0.0506106i \(-0.0161168\pi\)
\(48\) 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 −0.937797 − 0.347184i \(-0.887138\pi\)
−0.168228 + 0.985748i \(0.553805\pi\)
\(54\) 0 0
\(55\) 2.17403 − 7.49818i 0.293146 − 1.01105i
\(56\) 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 −0.926127 − 0.377211i \(-0.876883\pi\)
−0.136389 + 0.990655i \(0.543550\pi\)
\(68\) 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 0.411956 − 0.911204i \(-0.364846\pi\)
−0.583148 + 0.812366i \(0.698179\pi\)
\(74\) 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 \(-0.844015\pi\)
0.882314 − 0.470662i \(-0.155985\pi\)
\(84\) 0 0
\(85\) −2.30937 + 2.21700i −0.250486 + 0.240467i
\(86\) 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
\(95\) −8.54853 + 4.68216i −0.877061 + 0.480379i
\(96\) 0 0
\(97\) 3.79910 + 2.19341i 0.385740 + 0.222707i 0.680313 − 0.732922i \(-0.261844\pi\)
−0.294573 + 0.955629i \(0.595177\pi\)
\(98\) 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.1189.7 32
3.2 odd 2 1140.2.bg.c.49.6 ✓ 32
5.4 even 2 inner 3420.2.bj.d.1189.4 32
15.14 odd 2 1140.2.bg.c.49.11 yes 32
19.7 even 3 inner 3420.2.bj.d.2629.4 32
57.26 odd 6 1140.2.bg.c.349.11 yes 32
95.64 even 6 inner 3420.2.bj.d.2629.7 32
285.254 odd 6 1140.2.bg.c.349.6 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.c.49.6 ✓ 32 3.2 odd 2
1140.2.bg.c.49.11 yes 32 15.14 odd 2
1140.2.bg.c.349.6 yes 32 285.254 odd 6
1140.2.bg.c.349.11 yes 32 57.26 odd 6
3420.2.bj.d.1189.4 32 5.4 even 2 inner
3420.2.bj.d.1189.7 32 1.1 even 1 trivial
3420.2.bj.d.2629.4 32 19.7 even 3 inner
3420.2.bj.d.2629.7 32 95.64 even 6 inner