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Properties

Label 1665.2.a.p.1.2
Level $1665$
Weight $2$
Character 1665.1
Self dual yes
Analytic conductor $13.295$
Analytic rank $0$
Dimension $5$
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] = [1665,2,Mod(1,1665)] 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("1665.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1665, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1665 = 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1665.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-2,0,10,5,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(13.2950919365\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.383115\) of defining polynomial
Character \(\chi\) \(=\) 1665.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.15510 q^{2} +2.64446 q^{4} +1.00000 q^{5} -2.62521 q^{7} -1.38887 q^{8} -2.15510 q^{10} +1.64446 q^{11} +2.44254 q^{13} +5.65759 q^{14} -2.29576 q^{16} +0.578749 q^{17} +5.20156 q^{19} +2.64446 q^{20} -3.54397 q^{22} -8.22913 q^{23} +1.00000 q^{25} -5.26391 q^{26} -6.94225 q^{28} -0.766229 q^{29} +4.21452 q^{31} +7.72533 q^{32} -1.24726 q^{34} -2.62521 q^{35} +1.00000 q^{37} -11.2099 q^{38} -1.38887 q^{40} +1.64446 q^{41} -1.91893 q^{43} +4.34870 q^{44} +17.7346 q^{46} +9.56543 q^{47} -0.108279 q^{49} -2.15510 q^{50} +6.45918 q^{52} -7.74217 q^{53} +1.64446 q^{55} +3.64608 q^{56} +1.65130 q^{58} +13.0359 q^{59} -3.86379 q^{61} -9.08272 q^{62} -12.0574 q^{64} +2.44254 q^{65} +11.4566 q^{67} +1.53048 q^{68} +5.65759 q^{70} +2.54690 q^{71} -9.79732 q^{73} -2.15510 q^{74} +13.7553 q^{76} -4.31704 q^{77} -1.81364 q^{79} -2.29576 q^{80} -3.54397 q^{82} -10.9822 q^{83} +0.578749 q^{85} +4.13549 q^{86} -2.28394 q^{88} +8.85915 q^{89} -6.41217 q^{91} -21.7616 q^{92} -20.6145 q^{94} +5.20156 q^{95} -10.5605 q^{97} +0.233352 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 10 q^{4} + 5 q^{5} + 11 q^{7} - 6 q^{8} - 2 q^{10} + 5 q^{11} + 4 q^{13} + 8 q^{14} + 16 q^{16} - 4 q^{19} + 10 q^{20} - 8 q^{22} - 4 q^{23} + 5 q^{25} + 4 q^{26} + 28 q^{28} + 4 q^{29}+ \cdots + 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.15510 −1.52389 −0.761943 − 0.647644i \(-0.775754\pi\)
−0.761943 + 0.647644i \(0.775754\pi\)
\(3\) 0 0
\(4\) 2.64446 1.32223
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.62521 −0.992236 −0.496118 − 0.868255i \(-0.665242\pi\)
−0.496118 + 0.868255i \(0.665242\pi\)
\(8\) −1.38887 −0.491040
\(9\) 0 0
\(10\) −2.15510 −0.681503
\(11\) 1.64446 0.495823 0.247911 − 0.968783i \(-0.420256\pi\)
0.247911 + 0.968783i \(0.420256\pi\)
\(12\) 0 0
\(13\) 2.44254 0.677438 0.338719 − 0.940888i \(-0.390006\pi\)
0.338719 + 0.940888i \(0.390006\pi\)
\(14\) 5.65759 1.51205
\(15\) 0 0
\(16\) −2.29576 −0.573940
\(17\) 0.578749 0.140367 0.0701837 − 0.997534i \(-0.477641\pi\)
0.0701837 + 0.997534i \(0.477641\pi\)
\(18\) 0 0
\(19\) 5.20156 1.19332 0.596660 − 0.802494i \(-0.296494\pi\)
0.596660 + 0.802494i \(0.296494\pi\)
\(20\) 2.64446 0.591319
\(21\) 0 0
\(22\) −3.54397 −0.755577
\(23\) −8.22913 −1.71589 −0.857946 − 0.513739i \(-0.828260\pi\)
−0.857946 + 0.513739i \(0.828260\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.26391 −1.03234
\(27\) 0 0
\(28\) −6.94225 −1.31196
\(29\) −0.766229 −0.142285 −0.0711426 − 0.997466i \(-0.522665\pi\)
−0.0711426 + 0.997466i \(0.522665\pi\)
\(30\) 0 0
\(31\) 4.21452 0.756950 0.378475 − 0.925611i \(-0.376449\pi\)
0.378475 + 0.925611i \(0.376449\pi\)
\(32\) 7.72533 1.36566
\(33\) 0 0
\(34\) −1.24726 −0.213904
\(35\) −2.62521 −0.443741
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −11.2099 −1.81848
\(39\) 0 0
\(40\) −1.38887 −0.219600
\(41\) 1.64446 0.256821 0.128411 − 0.991721i \(-0.459012\pi\)
0.128411 + 0.991721i \(0.459012\pi\)
\(42\) 0 0
\(43\) −1.91893 −0.292634 −0.146317 − 0.989238i \(-0.546742\pi\)
−0.146317 + 0.989238i \(0.546742\pi\)
\(44\) 4.34870 0.655591
\(45\) 0 0
\(46\) 17.7346 2.61482
\(47\) 9.56543 1.39526 0.697630 − 0.716458i \(-0.254238\pi\)
0.697630 + 0.716458i \(0.254238\pi\)
\(48\) 0 0
\(49\) −0.108279 −0.0154684
\(50\) −2.15510 −0.304777
\(51\) 0 0
\(52\) 6.45918 0.895728
\(53\) −7.74217 −1.06347 −0.531735 − 0.846911i \(-0.678460\pi\)
−0.531735 + 0.846911i \(0.678460\pi\)
\(54\) 0 0
\(55\) 1.64446 0.221739
\(56\) 3.64608 0.487227
\(57\) 0 0
\(58\) 1.65130 0.216827
\(59\) 13.0359 1.69713 0.848565 − 0.529092i \(-0.177467\pi\)
0.848565 + 0.529092i \(0.177467\pi\)
\(60\) 0 0
\(61\) −3.86379 −0.494707 −0.247354 − 0.968925i \(-0.579561\pi\)
−0.247354 + 0.968925i \(0.579561\pi\)
\(62\) −9.08272 −1.15351
\(63\) 0 0
\(64\) −12.0574 −1.50717
\(65\) 2.44254 0.302959
\(66\) 0 0
\(67\) 11.4566 1.39965 0.699824 − 0.714315i \(-0.253262\pi\)
0.699824 + 0.714315i \(0.253262\pi\)
\(68\) 1.53048 0.185598
\(69\) 0 0
\(70\) 5.65759 0.676211
\(71\) 2.54690 0.302261 0.151131 − 0.988514i \(-0.451709\pi\)
0.151131 + 0.988514i \(0.451709\pi\)
\(72\) 0 0
\(73\) −9.79732 −1.14669 −0.573345 − 0.819314i \(-0.694354\pi\)
−0.573345 + 0.819314i \(0.694354\pi\)
\(74\) −2.15510 −0.250525
\(75\) 0 0
\(76\) 13.7553 1.57784
\(77\) −4.31704 −0.491973
\(78\) 0 0
\(79\) −1.81364 −0.204050 −0.102025 − 0.994782i \(-0.532532\pi\)
−0.102025 + 0.994782i \(0.532532\pi\)
\(80\) −2.29576 −0.256674
\(81\) 0 0
\(82\) −3.54397 −0.391366
\(83\) −10.9822 −1.20546 −0.602728 − 0.797947i \(-0.705920\pi\)
−0.602728 + 0.797947i \(0.705920\pi\)
\(84\) 0 0
\(85\) 0.578749 0.0627742
\(86\) 4.13549 0.445941
\(87\) 0 0
\(88\) −2.28394 −0.243469
\(89\) 8.85915 0.939068 0.469534 − 0.882914i \(-0.344422\pi\)
0.469534 + 0.882914i \(0.344422\pi\)
\(90\) 0 0
\(91\) −6.41217 −0.672178
\(92\) −21.7616 −2.26880
\(93\) 0 0
\(94\) −20.6145 −2.12622
\(95\) 5.20156 0.533669
\(96\) 0 0
\(97\) −10.5605 −1.07225 −0.536126 − 0.844138i \(-0.680113\pi\)
−0.536126 + 0.844138i \(0.680113\pi\)
\(98\) 0.233352 0.0235721
\(99\) 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 1665.2.a.p.1.2 5
3.2 odd 2 185.2.a.e.1.4 ✓ 5
5.4 even 2 8325.2.a.ch.1.4 5
12.11 even 2 2960.2.a.w.1.3 5
15.2 even 4 925.2.b.f.149.8 10
15.8 even 4 925.2.b.f.149.3 10
15.14 odd 2 925.2.a.f.1.2 5
21.20 even 2 9065.2.a.k.1.4 5
111.110 odd 2 6845.2.a.f.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.4 ✓ 5 3.2 odd 2
925.2.a.f.1.2 5 15.14 odd 2
925.2.b.f.149.3 10 15.8 even 4
925.2.b.f.149.8 10 15.2 even 4
1665.2.a.p.1.2 5 1.1 even 1 trivial
2960.2.a.w.1.3 5 12.11 even 2
6845.2.a.f.1.2 5 111.110 odd 2
8325.2.a.ch.1.4 5 5.4 even 2
9065.2.a.k.1.4 5 21.20 even 2