Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2205,4,Mod(1,2205)] 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("2205.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2205, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-8,0,20,-10,0,0,-48,0,40,14,0,-50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.41421 q^{2} +21.3137 q^{4} -5.00000 q^{5} -72.0833 q^{8} +27.0711 q^{10} +52.2548 q^{11} -30.6569 q^{13} +219.765 q^{16} +37.2254 q^{17} -80.2254 q^{19} -106.569 q^{20} -282.919 q^{22} -25.8335 q^{23} +25.0000 q^{25} +165.983 q^{26} -20.9411 q^{29} +314.558 q^{31} -613.186 q^{32} -201.546 q^{34} +197.147 q^{37} +434.357 q^{38} +360.416 q^{40} +11.3625 q^{41} -33.8335 q^{43} +1113.74 q^{44} +139.868 q^{46} -361.676 q^{47} -135.355 q^{50} -653.411 q^{52} -153.019 q^{53} -261.274 q^{55} +113.380 q^{58} -616.000 q^{59} -15.2649 q^{61} -1703.09 q^{62} +1561.80 q^{64} +153.284 q^{65} -166.510 q^{67} +793.411 q^{68} +952.000 q^{71} +148.489 q^{73} -1067.40 q^{74} -1709.90 q^{76} +857.725 q^{79} -1098.82 q^{80} -61.5189 q^{82} +660.528 q^{83} -186.127 q^{85} +183.182 q^{86} -3766.70 q^{88} -45.7746 q^{89} -550.607 q^{92} +1958.19 q^{94} +401.127 q^{95} -1682.13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 20 q^{4} - 10 q^{5} - 48 q^{8} + 40 q^{10} + 14 q^{11} - 50 q^{13} + 168 q^{16} - 50 q^{17} - 36 q^{19} - 100 q^{20} - 184 q^{22} - 244 q^{23} + 50 q^{25} + 216 q^{26} + 26 q^{29} + 120 q^{31}+ \cdots - 2742 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.41421 −1.91421 −0.957107 − 0.289735i \(-0.906433\pi\)
−0.957107 + 0.289735i \(0.906433\pi\)
\(3\) 0 0
\(4\) 21.3137 2.66421
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −72.0833 −3.18566
\(9\) 0 0
\(10\) 27.0711 0.856062
\(11\) 52.2548 1.43231 0.716156 − 0.697941i \(-0.245900\pi\)
0.716156 + 0.697941i \(0.245900\pi\)
\(12\) 0 0
\(13\) −30.6569 −0.654052 −0.327026 − 0.945015i \(-0.606047\pi\)
−0.327026 + 0.945015i \(0.606047\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 219.765 3.43382
\(17\) 37.2254 0.531087 0.265544 − 0.964099i \(-0.414449\pi\)
0.265544 + 0.964099i \(0.414449\pi\)
\(18\) 0 0
\(19\) −80.2254 −0.968683 −0.484341 − 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(20\) −106.569 −1.19147
\(21\) 0 0
\(22\) −282.919 −2.74175
\(23\) −25.8335 −0.234202 −0.117101 − 0.993120i \(-0.537360\pi\)
−0.117101 + 0.993120i \(0.537360\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 165.983 1.25200
\(27\) 0 0
\(28\) 0 0
\(29\) −20.9411 −0.134092 −0.0670460 − 0.997750i \(-0.521357\pi\)
−0.0670460 + 0.997750i \(0.521357\pi\)
\(30\) 0 0
\(31\) 314.558 1.82246 0.911232 − 0.411894i \(-0.135133\pi\)
0.911232 + 0.411894i \(0.135133\pi\)
\(32\) −613.186 −3.38741
\(33\) 0 0
\(34\) −201.546 −1.01661
\(35\) 0 0
\(36\) 0 0
\(37\) 197.147 0.875968 0.437984 − 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(38\) 434.357 1.85427
\(39\) 0 0
\(40\) 360.416 1.42467
\(41\) 11.3625 0.0432810 0.0216405 − 0.999766i \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) 0 0
\(43\) −33.8335 −0.119990 −0.0599948 − 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) 1113.74 3.81598
\(45\) 0 0
\(46\) 139.868 0.448313
\(47\) −361.676 −1.12247 −0.561233 − 0.827658i \(-0.689673\pi\)
−0.561233 + 0.827658i \(0.689673\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −135.355 −0.382843
\(51\) 0 0
\(52\) −653.411 −1.74254
\(53\) −153.019 −0.396582 −0.198291 − 0.980143i \(-0.563539\pi\)
−0.198291 + 0.980143i \(0.563539\pi\)
\(54\) 0 0
\(55\) −261.274 −0.640549
\(56\) 0 0
\(57\) 0 0
\(58\) 113.380 0.256681
\(59\) −616.000 −1.35926 −0.679630 − 0.733555i \(-0.737860\pi\)
−0.679630 + 0.733555i \(0.737860\pi\)
\(60\) 0 0
\(61\) −15.2649 −0.0320406 −0.0160203 − 0.999872i \(-0.505100\pi\)
−0.0160203 + 0.999872i \(0.505100\pi\)
\(62\) −1703.09 −3.48858
\(63\) 0 0
\(64\) 1561.80 3.05040
\(65\) 153.284 0.292501
\(66\) 0 0
\(67\) −166.510 −0.303618 −0.151809 − 0.988410i \(-0.548510\pi\)
−0.151809 + 0.988410i \(0.548510\pi\)
\(68\) 793.411 1.41493
\(69\) 0 0
\(70\) 0 0
\(71\) 952.000 1.59129 0.795645 − 0.605763i \(-0.207132\pi\)
0.795645 + 0.605763i \(0.207132\pi\)
\(72\) 0 0
\(73\) 148.489 0.238074 0.119037 − 0.992890i \(-0.462019\pi\)
0.119037 + 0.992890i \(0.462019\pi\)
\(74\) −1067.40 −1.67679
\(75\) 0 0
\(76\) −1709.90 −2.58078
\(77\) 0 0
\(78\) 0 0
\(79\) 857.725 1.22154 0.610770 − 0.791808i \(-0.290860\pi\)
0.610770 + 0.791808i \(0.290860\pi\)
\(80\) −1098.82 −1.53565
\(81\) 0 0
\(82\) −61.5189 −0.0828491
\(83\) 660.528 0.873523 0.436761 − 0.899577i \(-0.356125\pi\)
0.436761 + 0.899577i \(0.356125\pi\)
\(84\) 0 0
\(85\) −186.127 −0.237509
\(86\) 183.182 0.229686
\(87\) 0 0
\(88\) −3766.70 −4.56286
\(89\) −45.7746 −0.0545180 −0.0272590 − 0.999628i \(-0.508678\pi\)
−0.0272590 + 0.999628i \(0.508678\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −550.607 −0.623965
\(93\) 0 0
\(94\) 1958.19 2.14864
\(95\) 401.127 0.433208
\(96\) 0 0
\(97\) −1682.13 −1.76076 −0.880382 − 0.474265i \(-0.842714\pi\)
−0.880382 + 0.474265i \(0.842714\pi\)
\(98\) 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 2205.4.a.u.1.1 2
3.2 odd 2 245.4.a.k.1.2 2
7.6 odd 2 315.4.a.f.1.1 2
15.14 odd 2 1225.4.a.m.1.1 2
21.2 odd 6 245.4.e.i.116.1 4
21.5 even 6 245.4.e.h.116.1 4
21.11 odd 6 245.4.e.i.226.1 4
21.17 even 6 245.4.e.h.226.1 4
21.20 even 2 35.4.a.b.1.2 ✓ 2
35.34 odd 2 1575.4.a.z.1.2 2
84.83 odd 2 560.4.a.r.1.2 2
105.62 odd 4 175.4.b.c.99.4 4
105.83 odd 4 175.4.b.c.99.1 4
105.104 even 2 175.4.a.c.1.1 2
168.83 odd 2 2240.4.a.bo.1.1 2
168.125 even 2 2240.4.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 ✓ 2 21.20 even 2
175.4.a.c.1.1 2 105.104 even 2
175.4.b.c.99.1 4 105.83 odd 4
175.4.b.c.99.4 4 105.62 odd 4
245.4.a.k.1.2 2 3.2 odd 2
245.4.e.h.116.1 4 21.5 even 6
245.4.e.h.226.1 4 21.17 even 6
245.4.e.i.116.1 4 21.2 odd 6
245.4.e.i.226.1 4 21.11 odd 6
315.4.a.f.1.1 2 7.6 odd 2
560.4.a.r.1.2 2 84.83 odd 2
1225.4.a.m.1.1 2 15.14 odd 2
1575.4.a.z.1.2 2 35.34 odd 2
2205.4.a.u.1.1 2 1.1 even 1 trivial
2240.4.a.bn.1.2 2 168.125 even 2
2240.4.a.bo.1.1 2 168.83 odd 2