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Properties

Label 185.2.a.e.1.4
Level $185$
Weight $2$
Character 185.1
Self dual yes
Analytic conductor $1.477$
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] = [185,2,Mod(1,185)] 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("185.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(185, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 185.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2] 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: \(1.47723243739\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.383115\) of defining polynomial
Character \(\chi\) \(=\) 185.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.15510 q^{2} +1.38311 q^{3} +2.64446 q^{4} -1.00000 q^{5} +2.98075 q^{6} -2.62521 q^{7} +1.38887 q^{8} -1.08699 q^{9} -2.15510 q^{10} -1.64446 q^{11} +3.65759 q^{12} +2.44254 q^{13} -5.65759 q^{14} -1.38311 q^{15} -2.29576 q^{16} -0.578749 q^{17} -2.34258 q^{18} +5.20156 q^{19} -2.64446 q^{20} -3.63096 q^{21} -3.54397 q^{22} +8.22913 q^{23} +1.92097 q^{24} +1.00000 q^{25} +5.26391 q^{26} -5.65278 q^{27} -6.94225 q^{28} +0.766229 q^{29} -2.98075 q^{30} +4.21452 q^{31} -7.72533 q^{32} -2.27447 q^{33} -1.24726 q^{34} +2.62521 q^{35} -2.87451 q^{36} +1.00000 q^{37} +11.2099 q^{38} +3.37831 q^{39} -1.38887 q^{40} -1.64446 q^{41} -7.82509 q^{42} -1.91893 q^{43} -4.34870 q^{44} +1.08699 q^{45} +17.7346 q^{46} -9.56543 q^{47} -3.17530 q^{48} -0.108279 q^{49} +2.15510 q^{50} -0.800477 q^{51} +6.45918 q^{52} +7.74217 q^{53} -12.1823 q^{54} +1.64446 q^{55} -3.64608 q^{56} +7.19435 q^{57} +1.65130 q^{58} -13.0359 q^{59} -3.65759 q^{60} -3.86379 q^{61} +9.08272 q^{62} +2.85359 q^{63} -12.0574 q^{64} -2.44254 q^{65} -4.90172 q^{66} +11.4566 q^{67} -1.53048 q^{68} +11.3818 q^{69} +5.65759 q^{70} -2.54690 q^{71} -1.50969 q^{72} -9.79732 q^{73} +2.15510 q^{74} +1.38311 q^{75} +13.7553 q^{76} +4.31704 q^{77} +7.28059 q^{78} -1.81364 q^{79} +2.29576 q^{80} -4.55746 q^{81} -3.54397 q^{82} +10.9822 q^{83} -9.60193 q^{84} +0.578749 q^{85} -4.13549 q^{86} +1.05978 q^{87} -2.28394 q^{88} -8.85915 q^{89} +2.34258 q^{90} -6.41217 q^{91} +21.7616 q^{92} +5.82917 q^{93} -20.6145 q^{94} -5.20156 q^{95} -10.6850 q^{96} -10.5605 q^{97} -0.233352 q^{98} +1.78751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 3 q^{3} + 10 q^{4} - 5 q^{5} - 6 q^{6} + 11 q^{7} + 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} - 8 q^{14} - 3 q^{15} + 16 q^{16} + 2 q^{18} - 4 q^{19} - 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100}) \) 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 \(-0.224246\pi\)
0.761943 + 0.647644i \(0.224246\pi\)
\(3\) 1.38311 0.798542 0.399271 − 0.916833i \(-0.369263\pi\)
0.399271 + 0.916833i \(0.369263\pi\)
\(4\) 2.64446 1.32223
\(5\) −1.00000 −0.447214
\(6\) 2.98075 1.21689
\(7\) −2.62521 −0.992236 −0.496118 − 0.868255i \(-0.665242\pi\)
−0.496118 + 0.868255i \(0.665242\pi\)
\(8\) 1.38887 0.491040
\(9\) −1.08699 −0.362331
\(10\) −2.15510 −0.681503
\(11\) −1.64446 −0.495823 −0.247911 − 0.968783i \(-0.579744\pi\)
−0.247911 + 0.968783i \(0.579744\pi\)
\(12\) 3.65759 1.05585
\(13\) 2.44254 0.677438 0.338719 − 0.940888i \(-0.390006\pi\)
0.338719 + 0.940888i \(0.390006\pi\)
\(14\) −5.65759 −1.51205
\(15\) −1.38311 −0.357119
\(16\) −2.29576 −0.573940
\(17\) −0.578749 −0.140367 −0.0701837 − 0.997534i \(-0.522359\pi\)
−0.0701837 + 0.997534i \(0.522359\pi\)
\(18\) −2.34258 −0.552151
\(19\) 5.20156 1.19332 0.596660 − 0.802494i \(-0.296494\pi\)
0.596660 + 0.802494i \(0.296494\pi\)
\(20\) −2.64446 −0.591319
\(21\) −3.63096 −0.792341
\(22\) −3.54397 −0.755577
\(23\) 8.22913 1.71589 0.857946 − 0.513739i \(-0.171740\pi\)
0.857946 + 0.513739i \(0.171740\pi\)
\(24\) 1.92097 0.392116
\(25\) 1.00000 0.200000
\(26\) 5.26391 1.03234
\(27\) −5.65278 −1.08788
\(28\) −6.94225 −1.31196
\(29\) 0.766229 0.142285 0.0711426 − 0.997466i \(-0.477335\pi\)
0.0711426 + 0.997466i \(0.477335\pi\)
\(30\) −2.98075 −0.544208
\(31\) 4.21452 0.756950 0.378475 − 0.925611i \(-0.376449\pi\)
0.378475 + 0.925611i \(0.376449\pi\)
\(32\) −7.72533 −1.36566
\(33\) −2.27447 −0.395935
\(34\) −1.24726 −0.213904
\(35\) 2.62521 0.443741
\(36\) −2.87451 −0.479085
\(37\) 1.00000 0.164399
\(38\) 11.2099 1.81848
\(39\) 3.37831 0.540962
\(40\) −1.38887 −0.219600
\(41\) −1.64446 −0.256821 −0.128411 − 0.991721i \(-0.540988\pi\)
−0.128411 + 0.991721i \(0.540988\pi\)
\(42\) −7.82509 −1.20744
\(43\) −1.91893 −0.292634 −0.146317 − 0.989238i \(-0.546742\pi\)
−0.146317 + 0.989238i \(0.546742\pi\)
\(44\) −4.34870 −0.655591
\(45\) 1.08699 0.162039
\(46\) 17.7346 2.61482
\(47\) −9.56543 −1.39526 −0.697630 − 0.716458i \(-0.745762\pi\)
−0.697630 + 0.716458i \(0.745762\pi\)
\(48\) −3.17530 −0.458315
\(49\) −0.108279 −0.0154684
\(50\) 2.15510 0.304777
\(51\) −0.800477 −0.112089
\(52\) 6.45918 0.895728
\(53\) 7.74217 1.06347 0.531735 − 0.846911i \(-0.321540\pi\)
0.531735 + 0.846911i \(0.321540\pi\)
\(54\) −12.1823 −1.65780
\(55\) 1.64446 0.221739
\(56\) −3.64608 −0.487227
\(57\) 7.19435 0.952915
\(58\) 1.65130 0.216827
\(59\) −13.0359 −1.69713 −0.848565 − 0.529092i \(-0.822533\pi\)
−0.848565 + 0.529092i \(0.822533\pi\)
\(60\) −3.65759 −0.472193
\(61\) −3.86379 −0.494707 −0.247354 − 0.968925i \(-0.579561\pi\)
−0.247354 + 0.968925i \(0.579561\pi\)
\(62\) 9.08272 1.15351
\(63\) 2.85359 0.359518
\(64\) −12.0574 −1.50717
\(65\) −2.44254 −0.302959
\(66\) −4.90172 −0.603360
\(67\) 11.4566 1.39965 0.699824 − 0.714315i \(-0.253262\pi\)
0.699824 + 0.714315i \(0.253262\pi\)
\(68\) −1.53048 −0.185598
\(69\) 11.3818 1.37021
\(70\) 5.65759 0.676211
\(71\) −2.54690 −0.302261 −0.151131 − 0.988514i \(-0.548291\pi\)
−0.151131 + 0.988514i \(0.548291\pi\)
\(72\) −1.50969 −0.177919
\(73\) −9.79732 −1.14669 −0.573345 − 0.819314i \(-0.694354\pi\)
−0.573345 + 0.819314i \(0.694354\pi\)
\(74\) 2.15510 0.250525
\(75\) 1.38311 0.159708
\(76\) 13.7553 1.57784
\(77\) 4.31704 0.491973
\(78\) 7.28059 0.824365
\(79\) −1.81364 −0.204050 −0.102025 − 0.994782i \(-0.532532\pi\)
−0.102025 + 0.994782i \(0.532532\pi\)
\(80\) 2.29576 0.256674
\(81\) −4.55746 −0.506385
\(82\) −3.54397 −0.391366
\(83\) 10.9822 1.20546 0.602728 − 0.797947i \(-0.294080\pi\)
0.602728 + 0.797947i \(0.294080\pi\)
\(84\) −9.60193 −1.04766
\(85\) 0.578749 0.0627742
\(86\) −4.13549 −0.445941
\(87\) 1.05978 0.113621
\(88\) −2.28394 −0.243469
\(89\) −8.85915 −0.939068 −0.469534 − 0.882914i \(-0.655578\pi\)
−0.469534 + 0.882914i \(0.655578\pi\)
\(90\) 2.34258 0.246930
\(91\) −6.41217 −0.672178
\(92\) 21.7616 2.26880
\(93\) 5.82917 0.604456
\(94\) −20.6145 −2.12622
\(95\) −5.20156 −0.533669
\(96\) −10.6850 −1.09054
\(97\) −10.5605 −1.07225 −0.536126 − 0.844138i \(-0.680113\pi\)
−0.536126 + 0.844138i \(0.680113\pi\)
\(98\) −0.233352 −0.0235721
\(99\) 1.78751 0.179652
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 185.2.a.e.1.4 ✓ 5
3.2 odd 2 1665.2.a.p.1.2 5
4.3 odd 2 2960.2.a.w.1.3 5
5.2 odd 4 925.2.b.f.149.8 10
5.3 odd 4 925.2.b.f.149.3 10
5.4 even 2 925.2.a.f.1.2 5
7.6 odd 2 9065.2.a.k.1.4 5
15.14 odd 2 8325.2.a.ch.1.4 5
37.36 even 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 1.1 even 1 trivial
925.2.a.f.1.2 5 5.4 even 2
925.2.b.f.149.3 10 5.3 odd 4
925.2.b.f.149.8 10 5.2 odd 4
1665.2.a.p.1.2 5 3.2 odd 2
2960.2.a.w.1.3 5 4.3 odd 2
6845.2.a.f.1.2 5 37.36 even 2
8325.2.a.ch.1.4 5 15.14 odd 2
9065.2.a.k.1.4 5 7.6 odd 2