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Properties

Label 605.6.a.a.1.1
Level $605$
Weight $6$
Character 605.1
Self dual yes
Analytic conductor $97.032$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,6,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 605.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -4.00000 q^{3} -28.0000 q^{4} +25.0000 q^{5} +8.00000 q^{6} -192.000 q^{7} +120.000 q^{8} -227.000 q^{9} -50.0000 q^{10} +112.000 q^{12} -286.000 q^{13} +384.000 q^{14} -100.000 q^{15} +656.000 q^{16} +1678.00 q^{17} +454.000 q^{18} -1060.00 q^{19} -700.000 q^{20} +768.000 q^{21} +2976.00 q^{23} -480.000 q^{24} +625.000 q^{25} +572.000 q^{26} +1880.00 q^{27} +5376.00 q^{28} +3410.00 q^{29} +200.000 q^{30} -2448.00 q^{31} -5152.00 q^{32} -3356.00 q^{34} -4800.00 q^{35} +6356.00 q^{36} +182.000 q^{37} +2120.00 q^{38} +1144.00 q^{39} +3000.00 q^{40} +9398.00 q^{41} -1536.00 q^{42} +1244.00 q^{43} -5675.00 q^{45} -5952.00 q^{46} -12088.0 q^{47} -2624.00 q^{48} +20057.0 q^{49} -1250.00 q^{50} -6712.00 q^{51} +8008.00 q^{52} +23846.0 q^{53} -3760.00 q^{54} -23040.0 q^{56} +4240.00 q^{57} -6820.00 q^{58} -20020.0 q^{59} +2800.00 q^{60} -32302.0 q^{61} +4896.00 q^{62} +43584.0 q^{63} -10688.0 q^{64} -7150.00 q^{65} +60972.0 q^{67} -46984.0 q^{68} -11904.0 q^{69} +9600.00 q^{70} -32648.0 q^{71} -27240.0 q^{72} +38774.0 q^{73} -364.000 q^{74} -2500.00 q^{75} +29680.0 q^{76} -2288.00 q^{78} +33360.0 q^{79} +16400.0 q^{80} +47641.0 q^{81} -18796.0 q^{82} -16716.0 q^{83} -21504.0 q^{84} +41950.0 q^{85} -2488.00 q^{86} -13640.0 q^{87} +101370. q^{89} +11350.0 q^{90} +54912.0 q^{91} -83328.0 q^{92} +9792.00 q^{93} +24176.0 q^{94} -26500.0 q^{95} +20608.0 q^{96} -119038. q^{97} -40114.0 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.353553 −0.176777 − 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) −4.00000 −0.256600 −0.128300 − 0.991735i \(-0.540952\pi\)
−0.128300 + 0.991735i \(0.540952\pi\)
\(4\) −28.0000 −0.875000
\(5\) 25.0000 0.447214
\(6\) 8.00000 0.0907218
\(7\) −192.000 −1.48100 −0.740502 − 0.672054i \(-0.765412\pi\)
−0.740502 + 0.672054i \(0.765412\pi\)
\(8\) 120.000 0.662913
\(9\) −227.000 −0.934156
\(10\) −50.0000 −0.158114
\(11\) 0 0
\(12\) 112.000 0.224525
\(13\) −286.000 −0.469362 −0.234681 − 0.972072i \(-0.575405\pi\)
−0.234681 + 0.972072i \(0.575405\pi\)
\(14\) 384.000 0.523614
\(15\) −100.000 −0.114755
\(16\) 656.000 0.640625
\(17\) 1678.00 1.40822 0.704109 − 0.710092i \(-0.251347\pi\)
0.704109 + 0.710092i \(0.251347\pi\)
\(18\) 454.000 0.330274
\(19\) −1060.00 −0.673631 −0.336815 − 0.941571i \(-0.609350\pi\)
−0.336815 + 0.941571i \(0.609350\pi\)
\(20\) −700.000 −0.391312
\(21\) 768.000 0.380026
\(22\) 0 0
\(23\) 2976.00 1.17304 0.586521 − 0.809934i \(-0.300497\pi\)
0.586521 + 0.809934i \(0.300497\pi\)
\(24\) −480.000 −0.170103
\(25\) 625.000 0.200000
\(26\) 572.000 0.165944
\(27\) 1880.00 0.496305
\(28\) 5376.00 1.29588
\(29\) 3410.00 0.752938 0.376469 − 0.926429i \(-0.377138\pi\)
0.376469 + 0.926429i \(0.377138\pi\)
\(30\) 200.000 0.0405720
\(31\) −2448.00 −0.457517 −0.228758 − 0.973483i \(-0.573467\pi\)
−0.228758 + 0.973483i \(0.573467\pi\)
\(32\) −5152.00 −0.889408
\(33\) 0 0
\(34\) −3356.00 −0.497880
\(35\) −4800.00 −0.662325
\(36\) 6356.00 0.817387
\(37\) 182.000 0.0218558 0.0109279 − 0.999940i \(-0.496521\pi\)
0.0109279 + 0.999940i \(0.496521\pi\)
\(38\) 2120.00 0.238164
\(39\) 1144.00 0.120438
\(40\) 3000.00 0.296464
\(41\) 9398.00 0.873124 0.436562 − 0.899674i \(-0.356196\pi\)
0.436562 + 0.899674i \(0.356196\pi\)
\(42\) −1536.00 −0.134359
\(43\) 1244.00 0.102600 0.0513002 − 0.998683i \(-0.483663\pi\)
0.0513002 + 0.998683i \(0.483663\pi\)
\(44\) 0 0
\(45\) −5675.00 −0.417767
\(46\) −5952.00 −0.414733
\(47\) −12088.0 −0.798196 −0.399098 − 0.916908i \(-0.630677\pi\)
−0.399098 + 0.916908i \(0.630677\pi\)
\(48\) −2624.00 −0.164384
\(49\) 20057.0 1.19337
\(50\) −1250.00 −0.0707107
\(51\) −6712.00 −0.361349
\(52\) 8008.00 0.410691
\(53\) 23846.0 1.16607 0.583037 − 0.812446i \(-0.301864\pi\)
0.583037 + 0.812446i \(0.301864\pi\)
\(54\) −3760.00 −0.175470
\(55\) 0 0
\(56\) −23040.0 −0.981776
\(57\) 4240.00 0.172854
\(58\) −6820.00 −0.266204
\(59\) −20020.0 −0.748745 −0.374373 − 0.927278i \(-0.622142\pi\)
−0.374373 + 0.927278i \(0.622142\pi\)
\(60\) 2800.00 0.100411
\(61\) −32302.0 −1.11149 −0.555744 − 0.831353i \(-0.687567\pi\)
−0.555744 + 0.831353i \(0.687567\pi\)
\(62\) 4896.00 0.161757
\(63\) 43584.0 1.38349
\(64\) −10688.0 −0.326172
\(65\) −7150.00 −0.209905
\(66\) 0 0
\(67\) 60972.0 1.65937 0.829685 − 0.558231i \(-0.188520\pi\)
0.829685 + 0.558231i \(0.188520\pi\)
\(68\) −46984.0 −1.23219
\(69\) −11904.0 −0.301003
\(70\) 9600.00 0.234167
\(71\) −32648.0 −0.768618 −0.384309 − 0.923204i \(-0.625560\pi\)
−0.384309 + 0.923204i \(0.625560\pi\)
\(72\) −27240.0 −0.619264
\(73\) 38774.0 0.851596 0.425798 − 0.904818i \(-0.359993\pi\)
0.425798 + 0.904818i \(0.359993\pi\)
\(74\) −364.000 −0.00772720
\(75\) −2500.00 −0.0513200
\(76\) 29680.0 0.589427
\(77\) 0 0
\(78\) −2288.00 −0.0425814
\(79\) 33360.0 0.601393 0.300696 − 0.953720i \(-0.402781\pi\)
0.300696 + 0.953720i \(0.402781\pi\)
\(80\) 16400.0 0.286496
\(81\) 47641.0 0.806805
\(82\) −18796.0 −0.308696
\(83\) −16716.0 −0.266340 −0.133170 − 0.991093i \(-0.542516\pi\)
−0.133170 + 0.991093i \(0.542516\pi\)
\(84\) −21504.0 −0.332522
\(85\) 41950.0 0.629774
\(86\) −2488.00 −0.0362747
\(87\) −13640.0 −0.193204
\(88\) 0 0
\(89\) 101370. 1.35655 0.678273 − 0.734810i \(-0.262729\pi\)
0.678273 + 0.734810i \(0.262729\pi\)
\(90\) 11350.0 0.147703
\(91\) 54912.0 0.695126
\(92\) −83328.0 −1.02641
\(93\) 9792.00 0.117399
\(94\) 24176.0 0.282205
\(95\) −26500.0 −0.301257
\(96\) 20608.0 0.228222
\(97\) −119038. −1.28457 −0.642283 − 0.766468i \(-0.722013\pi\)
−0.642283 + 0.766468i \(0.722013\pi\)
\(98\) −40114.0 −0.421921
\(99\) 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 605.6.a.a.1.1 1
11.10 odd 2 5.6.a.a.1.1 ✓ 1
33.32 even 2 45.6.a.b.1.1 1
44.43 even 2 80.6.a.e.1.1 1
55.32 even 4 25.6.b.a.24.2 2
55.43 even 4 25.6.b.a.24.1 2
55.54 odd 2 25.6.a.a.1.1 1
77.76 even 2 245.6.a.b.1.1 1
88.21 odd 2 320.6.a.j.1.1 1
88.43 even 2 320.6.a.g.1.1 1
132.131 odd 2 720.6.a.a.1.1 1
143.142 odd 2 845.6.a.b.1.1 1
165.32 odd 4 225.6.b.e.199.1 2
165.98 odd 4 225.6.b.e.199.2 2
165.164 even 2 225.6.a.f.1.1 1
220.43 odd 4 400.6.c.j.49.2 2
220.87 odd 4 400.6.c.j.49.1 2
220.219 even 2 400.6.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 ✓ 1 11.10 odd 2
25.6.a.a.1.1 1 55.54 odd 2
25.6.b.a.24.1 2 55.43 even 4
25.6.b.a.24.2 2 55.32 even 4
45.6.a.b.1.1 1 33.32 even 2
80.6.a.e.1.1 1 44.43 even 2
225.6.a.f.1.1 1 165.164 even 2
225.6.b.e.199.1 2 165.32 odd 4
225.6.b.e.199.2 2 165.98 odd 4
245.6.a.b.1.1 1 77.76 even 2
320.6.a.g.1.1 1 88.43 even 2
320.6.a.j.1.1 1 88.21 odd 2
400.6.a.g.1.1 1 220.219 even 2
400.6.c.j.49.1 2 220.87 odd 4
400.6.c.j.49.2 2 220.43 odd 4
605.6.a.a.1.1 1 1.1 even 1 trivial
720.6.a.a.1.1 1 132.131 odd 2
845.6.a.b.1.1 1 143.142 odd 2