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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 1189.8
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.d.2629.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.589297 - 2.15702i) q^{5} -4.17007i q^{7} +4.21474 q^{11} +(-4.75912 + 2.74768i) q^{13} +(-6.39543 - 3.69241i) q^{17} +(2.32390 + 3.68775i) q^{19} +(-4.48403 + 2.58886i) q^{23} +(-4.30546 + 2.54225i) q^{25} +(-1.79374 - 3.10684i) q^{29} -10.6964 q^{31} +(-8.99493 + 2.45741i) q^{35} +4.99964i q^{37} +(-0.263328 + 0.456098i) q^{41} +(10.6033 + 6.12182i) q^{43} +(7.73453 - 4.46553i) q^{47} -10.3895 q^{49} +(-2.75827 + 1.59249i) q^{53} +(-2.48373 - 9.09127i) q^{55} +(4.58141 - 7.93523i) q^{59} +(3.75876 + 6.51037i) q^{61} +(8.73132 + 8.64631i) q^{65} +(3.68875 - 2.12970i) q^{67} +(-0.781721 + 1.35398i) q^{71} +(-0.642648 - 0.371033i) q^{73} -17.5758i q^{77} +(-6.56411 + 11.3694i) q^{79} +2.31607i q^{83} +(-4.19578 + 15.9710i) q^{85} +(-4.93725 - 8.55157i) q^{89} +(11.4580 + 19.8459i) q^{91} +(6.58507 - 7.18588i) q^{95} +(-2.64404 - 1.52654i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 4 q^{11} + 12 q^{19} - 12 q^{25} - 22 q^{29} - 56 q^{31} + 2 q^{35} + 16 q^{41} - 88 q^{49} - 14 q^{55} + 86 q^{59} + 6 q^{61} - 36 q^{65} - 10 q^{71} + 50 q^{79} + 30 q^{85} - 34 q^{89} + 4 q^{91}+ \cdots - 46 q^{95}+O(q^{100}) \) 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
\(5\) −0.589297 − 2.15702i −0.263542 − 0.964648i
\(6\) 0 0
\(7\) − 4.17007i − 1.57614i −0.615586 − 0.788070i \(-0.711081\pi\)
0.615586 − 0.788070i \(-0.288919\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.21474 1.27079 0.635396 − 0.772187i \(-0.280837\pi\)
0.635396 + 0.772187i \(0.280837\pi\)
\(12\) 0 0
\(13\) −4.75912 + 2.74768i −1.31994 + 0.762069i −0.983719 − 0.179714i \(-0.942483\pi\)
−0.336223 + 0.941783i \(0.609149\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.39543 − 3.69241i −1.55112 − 0.895540i −0.998051 − 0.0624049i \(-0.980123\pi\)
−0.553070 − 0.833135i \(-0.686544\pi\)
\(18\) 0 0
\(19\) 2.32390 + 3.68775i 0.533140 + 0.846027i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.48403 + 2.58886i −0.934985 + 0.539814i −0.888385 − 0.459100i \(-0.848172\pi\)
−0.0466001 + 0.998914i \(0.514839\pi\)
\(24\) 0 0
\(25\) −4.30546 + 2.54225i −0.861092 + 0.508450i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.79374 − 3.10684i −0.333089 − 0.576926i 0.650027 − 0.759911i \(-0.274758\pi\)
−0.983116 + 0.182985i \(0.941424\pi\)
\(30\) 0 0
\(31\) −10.6964 −1.92112 −0.960562 − 0.278065i \(-0.910307\pi\)
−0.960562 + 0.278065i \(0.910307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.99493 + 2.45741i −1.52042 + 0.415378i
\(36\) 0 0
\(37\) 4.99964i 0.821936i 0.911650 + 0.410968i \(0.134809\pi\)
−0.911650 + 0.410968i \(0.865191\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.263328 + 0.456098i −0.0411250 + 0.0712305i −0.885855 − 0.463962i \(-0.846428\pi\)
0.844730 + 0.535192i \(0.179761\pi\)
\(42\) 0 0
\(43\) 10.6033 + 6.12182i 1.61699 + 0.933569i 0.987693 + 0.156404i \(0.0499903\pi\)
0.629297 + 0.777165i \(0.283343\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.73453 − 4.46553i 1.12820 − 0.651365i 0.184716 − 0.982792i \(-0.440863\pi\)
0.943481 + 0.331427i \(0.107530\pi\)
\(48\) 0 0
\(49\) −10.3895 −1.48422
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.75827 + 1.59249i −0.378877 + 0.218745i −0.677330 − 0.735680i \(-0.736863\pi\)
0.298453 + 0.954424i \(0.403530\pi\)
\(54\) 0 0
\(55\) −2.48373 − 9.09127i −0.334906 − 1.22587i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.58141 − 7.93523i 0.596448 − 1.03308i −0.396892 − 0.917865i \(-0.629911\pi\)
0.993341 − 0.115214i \(-0.0367553\pi\)
\(60\) 0 0
\(61\) 3.75876 + 6.51037i 0.481260 + 0.833568i 0.999769 − 0.0215050i \(-0.00684578\pi\)
−0.518508 + 0.855073i \(0.673512\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.73132 + 8.64631i 1.08299 + 1.07244i
\(66\) 0 0
\(67\) 3.68875 − 2.12970i 0.450653 − 0.260184i −0.257453 − 0.966291i \(-0.582883\pi\)
0.708106 + 0.706106i \(0.249550\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.781721 + 1.35398i −0.0927732 + 0.160688i −0.908677 − 0.417500i \(-0.862906\pi\)
0.815904 + 0.578188i \(0.196240\pi\)
\(72\) 0 0
\(73\) −0.642648 − 0.371033i −0.0752163 − 0.0434261i 0.461920 − 0.886921i \(-0.347161\pi\)
−0.537136 + 0.843495i \(0.680494\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 17.5758i − 2.00294i
\(78\) 0 0
\(79\) −6.56411 + 11.3694i −0.738520 + 1.27915i 0.214641 + 0.976693i \(0.431142\pi\)
−0.953162 + 0.302462i \(0.902192\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.31607i 0.254222i 0.991888 + 0.127111i \(0.0405705\pi\)
−0.991888 + 0.127111i \(0.959430\pi\)
\(84\) 0 0
\(85\) −4.19578 + 15.9710i −0.455096 + 1.73230i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.93725 − 8.55157i −0.523348 − 0.906465i −0.999631 − 0.0271728i \(-0.991350\pi\)
0.476283 − 0.879292i \(-0.341984\pi\)
\(90\) 0 0
\(91\) 11.4580 + 19.8459i 1.20113 + 2.08041i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.58507 − 7.18588i 0.675614 − 0.737256i
\(96\) 0 0
\(97\) −2.64404 − 1.52654i −0.268461 − 0.154996i 0.359727 − 0.933058i \(-0.382870\pi\)
−0.628188 + 0.778061i \(0.716203\pi\)
\(98\) 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.1189.8 32
3.2 odd 2 1140.2.bg.c.49.7 ✓ 32
5.4 even 2 inner 3420.2.bj.d.1189.14 32
15.14 odd 2 1140.2.bg.c.49.10 yes 32
19.7 even 3 inner 3420.2.bj.d.2629.14 32
57.26 odd 6 1140.2.bg.c.349.10 yes 32
95.64 even 6 inner 3420.2.bj.d.2629.8 32
285.254 odd 6 1140.2.bg.c.349.7 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.bg.c.49.7 ✓ 32 3.2 odd 2
1140.2.bg.c.49.10 yes 32 15.14 odd 2
1140.2.bg.c.349.7 yes 32 285.254 odd 6
1140.2.bg.c.349.10 yes 32 57.26 odd 6
3420.2.bj.d.1189.8 32 1.1 even 1 trivial
3420.2.bj.d.1189.14 32 5.4 even 2 inner
3420.2.bj.d.2629.8 32 95.64 even 6 inner
3420.2.bj.d.2629.14 32 19.7 even 3 inner