Properties

Label 15.4.a.a.1.1
Level $15$
Weight $4$
Character 15.1
Self dual yes
Analytic conductor $0.885$
Analytic rank $0$
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] = [15,4,Mod(1,15)] 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("15.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(15, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 15.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1] 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: \(0.885028650086\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 15.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} -7.00000 q^{4} +5.00000 q^{5} +3.00000 q^{6} -24.0000 q^{7} -15.0000 q^{8} +9.00000 q^{9} +5.00000 q^{10} +52.0000 q^{11} -21.0000 q^{12} +22.0000 q^{13} -24.0000 q^{14} +15.0000 q^{15} +41.0000 q^{16} -14.0000 q^{17} +9.00000 q^{18} -20.0000 q^{19} -35.0000 q^{20} -72.0000 q^{21} +52.0000 q^{22} -168.000 q^{23} -45.0000 q^{24} +25.0000 q^{25} +22.0000 q^{26} +27.0000 q^{27} +168.000 q^{28} +230.000 q^{29} +15.0000 q^{30} -288.000 q^{31} +161.000 q^{32} +156.000 q^{33} -14.0000 q^{34} -120.000 q^{35} -63.0000 q^{36} -34.0000 q^{37} -20.0000 q^{38} +66.0000 q^{39} -75.0000 q^{40} +122.000 q^{41} -72.0000 q^{42} -188.000 q^{43} -364.000 q^{44} +45.0000 q^{45} -168.000 q^{46} +256.000 q^{47} +123.000 q^{48} +233.000 q^{49} +25.0000 q^{50} -42.0000 q^{51} -154.000 q^{52} -338.000 q^{53} +27.0000 q^{54} +260.000 q^{55} +360.000 q^{56} -60.0000 q^{57} +230.000 q^{58} +100.000 q^{59} -105.000 q^{60} +742.000 q^{61} -288.000 q^{62} -216.000 q^{63} -167.000 q^{64} +110.000 q^{65} +156.000 q^{66} -84.0000 q^{67} +98.0000 q^{68} -504.000 q^{69} -120.000 q^{70} -328.000 q^{71} -135.000 q^{72} -38.0000 q^{73} -34.0000 q^{74} +75.0000 q^{75} +140.000 q^{76} -1248.00 q^{77} +66.0000 q^{78} -240.000 q^{79} +205.000 q^{80} +81.0000 q^{81} +122.000 q^{82} +1212.00 q^{83} +504.000 q^{84} -70.0000 q^{85} -188.000 q^{86} +690.000 q^{87} -780.000 q^{88} +330.000 q^{89} +45.0000 q^{90} -528.000 q^{91} +1176.00 q^{92} -864.000 q^{93} +256.000 q^{94} -100.000 q^{95} +483.000 q^{96} +866.000 q^{97} +233.000 q^{98} +468.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.353553 0.176777 − 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.00000 −0.875000
\(5\) 5.00000 0.447214
\(6\) 3.00000 0.204124
\(7\) −24.0000 −1.29588 −0.647939 − 0.761692i \(-0.724369\pi\)
−0.647939 + 0.761692i \(0.724369\pi\)
\(8\) −15.0000 −0.662913
\(9\) 9.00000 0.333333
\(10\) 5.00000 0.158114
\(11\) 52.0000 1.42533 0.712663 − 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) −21.0000 −0.505181
\(13\) 22.0000 0.469362 0.234681 − 0.972072i \(-0.424595\pi\)
0.234681 + 0.972072i \(0.424595\pi\)
\(14\) −24.0000 −0.458162
\(15\) 15.0000 0.258199
\(16\) 41.0000 0.640625
\(17\) −14.0000 −0.199735 −0.0998676 − 0.995001i \(-0.531842\pi\)
−0.0998676 + 0.995001i \(0.531842\pi\)
\(18\) 9.00000 0.117851
\(19\) −20.0000 −0.241490 −0.120745 − 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) −35.0000 −0.391312
\(21\) −72.0000 −0.748176
\(22\) 52.0000 0.503929
\(23\) −168.000 −1.52306 −0.761531 − 0.648129i \(-0.775552\pi\)
−0.761531 + 0.648129i \(0.775552\pi\)
\(24\) −45.0000 −0.382733
\(25\) 25.0000 0.200000
\(26\) 22.0000 0.165944
\(27\) 27.0000 0.192450
\(28\) 168.000 1.13389
\(29\) 230.000 1.47276 0.736378 − 0.676570i \(-0.236535\pi\)
0.736378 + 0.676570i \(0.236535\pi\)
\(30\) 15.0000 0.0912871
\(31\) −288.000 −1.66859 −0.834296 − 0.551317i \(-0.814125\pi\)
−0.834296 + 0.551317i \(0.814125\pi\)
\(32\) 161.000 0.889408
\(33\) 156.000 0.822913
\(34\) −14.0000 −0.0706171
\(35\) −120.000 −0.579534
\(36\) −63.0000 −0.291667
\(37\) −34.0000 −0.151069 −0.0755347 − 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) −20.0000 −0.0853797
\(39\) 66.0000 0.270986
\(40\) −75.0000 −0.296464
\(41\) 122.000 0.464712 0.232356 − 0.972631i \(-0.425357\pi\)
0.232356 + 0.972631i \(0.425357\pi\)
\(42\) −72.0000 −0.264520
\(43\) −188.000 −0.666738 −0.333369 − 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) −364.000 −1.24716
\(45\) 45.0000 0.149071
\(46\) −168.000 −0.538484
\(47\) 256.000 0.794499 0.397249 − 0.917711i \(-0.369965\pi\)
0.397249 + 0.917711i \(0.369965\pi\)
\(48\) 123.000 0.369865
\(49\) 233.000 0.679300
\(50\) 25.0000 0.0707107
\(51\) −42.0000 −0.115317
\(52\) −154.000 −0.410691
\(53\) −338.000 −0.875998 −0.437999 − 0.898976i \(-0.644313\pi\)
−0.437999 + 0.898976i \(0.644313\pi\)
\(54\) 27.0000 0.0680414
\(55\) 260.000 0.637425
\(56\) 360.000 0.859054
\(57\) −60.0000 −0.139424
\(58\) 230.000 0.520698
\(59\) 100.000 0.220659 0.110330 − 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) −105.000 −0.225924
\(61\) 742.000 1.55743 0.778716 − 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) −288.000 −0.589936
\(63\) −216.000 −0.431959
\(64\) −167.000 −0.326172
\(65\) 110.000 0.209905
\(66\) 156.000 0.290944
\(67\) −84.0000 −0.153168 −0.0765838 − 0.997063i \(-0.524401\pi\)
−0.0765838 + 0.997063i \(0.524401\pi\)
\(68\) 98.0000 0.174768
\(69\) −504.000 −0.879340
\(70\) −120.000 −0.204896
\(71\) −328.000 −0.548260 −0.274130 − 0.961693i \(-0.588390\pi\)
−0.274130 + 0.961693i \(0.588390\pi\)
\(72\) −135.000 −0.220971
\(73\) −38.0000 −0.0609255 −0.0304628 − 0.999536i \(-0.509698\pi\)
−0.0304628 + 0.999536i \(0.509698\pi\)
\(74\) −34.0000 −0.0534111
\(75\) 75.0000 0.115470
\(76\) 140.000 0.211304
\(77\) −1248.00 −1.84705
\(78\) 66.0000 0.0958081
\(79\) −240.000 −0.341799 −0.170899 − 0.985288i \(-0.554667\pi\)
−0.170899 + 0.985288i \(0.554667\pi\)
\(80\) 205.000 0.286496
\(81\) 81.0000 0.111111
\(82\) 122.000 0.164301
\(83\) 1212.00 1.60282 0.801411 − 0.598114i \(-0.204083\pi\)
0.801411 + 0.598114i \(0.204083\pi\)
\(84\) 504.000 0.654654
\(85\) −70.0000 −0.0893243
\(86\) −188.000 −0.235727
\(87\) 690.000 0.850296
\(88\) −780.000 −0.944867
\(89\) 330.000 0.393033 0.196516 − 0.980501i \(-0.437037\pi\)
0.196516 + 0.980501i \(0.437037\pi\)
\(90\) 45.0000 0.0527046
\(91\) −528.000 −0.608236
\(92\) 1176.00 1.33268
\(93\) −864.000 −0.963362
\(94\) 256.000 0.280898
\(95\) −100.000 −0.107998
\(96\) 483.000 0.513500
\(97\) 866.000 0.906484 0.453242 − 0.891387i \(-0.350267\pi\)
0.453242 + 0.891387i \(0.350267\pi\)
\(98\) 233.000 0.240169
\(99\) 468.000 0.475109
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 15.4.a.a.1.1 ✓ 1
3.2 odd 2 45.4.a.c.1.1 1
4.3 odd 2 240.4.a.e.1.1 1
5.2 odd 4 75.4.b.b.49.2 2
5.3 odd 4 75.4.b.b.49.1 2
5.4 even 2 75.4.a.b.1.1 1
7.6 odd 2 735.4.a.e.1.1 1
8.3 odd 2 960.4.a.ba.1.1 1
8.5 even 2 960.4.a.b.1.1 1
9.2 odd 6 405.4.e.i.271.1 2
9.4 even 3 405.4.e.g.136.1 2
9.5 odd 6 405.4.e.i.136.1 2
9.7 even 3 405.4.e.g.271.1 2
11.10 odd 2 1815.4.a.e.1.1 1
12.11 even 2 720.4.a.n.1.1 1
15.2 even 4 225.4.b.e.199.1 2
15.8 even 4 225.4.b.e.199.2 2
15.14 odd 2 225.4.a.f.1.1 1
20.3 even 4 1200.4.f.b.49.1 2
20.7 even 4 1200.4.f.b.49.2 2
20.19 odd 2 1200.4.a.t.1.1 1
21.20 even 2 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 ✓ 1 1.1 even 1 trivial
45.4.a.c.1.1 1 3.2 odd 2
75.4.a.b.1.1 1 5.4 even 2
75.4.b.b.49.1 2 5.3 odd 4
75.4.b.b.49.2 2 5.2 odd 4
225.4.a.f.1.1 1 15.14 odd 2
225.4.b.e.199.1 2 15.2 even 4
225.4.b.e.199.2 2 15.8 even 4
240.4.a.e.1.1 1 4.3 odd 2
405.4.e.g.136.1 2 9.4 even 3
405.4.e.g.271.1 2 9.7 even 3
405.4.e.i.136.1 2 9.5 odd 6
405.4.e.i.271.1 2 9.2 odd 6
720.4.a.n.1.1 1 12.11 even 2
735.4.a.e.1.1 1 7.6 odd 2
960.4.a.b.1.1 1 8.5 even 2
960.4.a.ba.1.1 1 8.3 odd 2
1200.4.a.t.1.1 1 20.19 odd 2
1200.4.f.b.49.1 2 20.3 even 4
1200.4.f.b.49.2 2 20.7 even 4
1815.4.a.e.1.1 1 11.10 odd 2
2205.4.a.l.1.1 1 21.20 even 2