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Properties

Label 1665.2.a.p.1.3
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 1.3
Root \(-1.62871\) 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)
 
\(f(q)\) \(=\) \(q-0.728950 q^{2} -1.46863 q^{4} +1.00000 q^{5} +2.55244 q^{7} +2.52846 q^{8} -0.728950 q^{10} -2.46863 q^{11} +1.55854 q^{13} -1.86060 q^{14} +1.09414 q^{16} +6.83662 q^{17} -7.66011 q^{19} -1.46863 q^{20} +1.79951 q^{22} +7.50003 q^{23} +1.00000 q^{25} -1.13610 q^{26} -3.74859 q^{28} -3.25741 q^{29} +0.658785 q^{31} -5.85449 q^{32} -4.98356 q^{34} +2.55244 q^{35} +1.00000 q^{37} +5.58384 q^{38} +2.52846 q^{40} -2.46863 q^{41} +10.9579 q^{43} +3.62551 q^{44} -5.46715 q^{46} -3.11521 q^{47} -0.485072 q^{49} -0.728950 q^{50} -2.28892 q^{52} -8.64184 q^{53} -2.46863 q^{55} +6.45373 q^{56} +2.37449 q^{58} +6.23634 q^{59} +3.27808 q^{61} -0.480222 q^{62} +2.07935 q^{64} +1.55854 q^{65} +1.47764 q^{67} -10.0405 q^{68} -1.86060 q^{70} +8.06686 q^{71} -4.96199 q^{73} -0.728950 q^{74} +11.2499 q^{76} -6.30102 q^{77} +12.8206 q^{79} +1.09414 q^{80} +1.79951 q^{82} -1.14934 q^{83} +6.83662 q^{85} -7.98779 q^{86} -6.24184 q^{88} -11.5207 q^{89} +3.97807 q^{91} -11.0148 q^{92} +2.27083 q^{94} -7.66011 q^{95} +17.2929 q^{97} +0.353594 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 10 q^{4} + 5 q^{5} + 11 q^{7} - 6 q^{8} - 2 q^{10} + 5 q^{11} + 4 q^{13} + 8 q^{14} + 16 q^{16} - 4 q^{19} + 10 q^{20} - 8 q^{22} - 4 q^{23} + 5 q^{25} + 4 q^{26} + 28 q^{28} + 4 q^{29}+ \cdots + 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.728950 −0.515446 −0.257723 − 0.966219i \(-0.582972\pi\)
−0.257723 + 0.966219i \(0.582972\pi\)
\(3\) 0 0
\(4\) −1.46863 −0.734316
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.55244 0.964730 0.482365 − 0.875970i \(-0.339778\pi\)
0.482365 + 0.875970i \(0.339778\pi\)
\(8\) 2.52846 0.893946
\(9\) 0 0
\(10\) −0.728950 −0.230514
\(11\) −2.46863 −0.744320 −0.372160 − 0.928169i \(-0.621383\pi\)
−0.372160 + 0.928169i \(0.621383\pi\)
\(12\) 0 0
\(13\) 1.55854 0.432261 0.216131 − 0.976364i \(-0.430656\pi\)
0.216131 + 0.976364i \(0.430656\pi\)
\(14\) −1.86060 −0.497266
\(15\) 0 0
\(16\) 1.09414 0.273535
\(17\) 6.83662 1.65812 0.829062 − 0.559156i \(-0.188875\pi\)
0.829062 + 0.559156i \(0.188875\pi\)
\(18\) 0 0
\(19\) −7.66011 −1.75735 −0.878675 − 0.477421i \(-0.841572\pi\)
−0.878675 + 0.477421i \(0.841572\pi\)
\(20\) −1.46863 −0.328396
\(21\) 0 0
\(22\) 1.79951 0.383657
\(23\) 7.50003 1.56387 0.781933 − 0.623363i \(-0.214234\pi\)
0.781933 + 0.623363i \(0.214234\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.13610 −0.222807
\(27\) 0 0
\(28\) −3.74859 −0.708416
\(29\) −3.25741 −0.604886 −0.302443 − 0.953167i \(-0.597802\pi\)
−0.302443 + 0.953167i \(0.597802\pi\)
\(30\) 0 0
\(31\) 0.658785 0.118321 0.0591607 − 0.998248i \(-0.481158\pi\)
0.0591607 + 0.998248i \(0.481158\pi\)
\(32\) −5.85449 −1.03494
\(33\) 0 0
\(34\) −4.98356 −0.854673
\(35\) 2.55244 0.431440
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 5.58384 0.905818
\(39\) 0 0
\(40\) 2.52846 0.399785
\(41\) −2.46863 −0.385535 −0.192768 − 0.981244i \(-0.561746\pi\)
−0.192768 + 0.981244i \(0.561746\pi\)
\(42\) 0 0
\(43\) 10.9579 1.67107 0.835535 − 0.549438i \(-0.185158\pi\)
0.835535 + 0.549438i \(0.185158\pi\)
\(44\) 3.62551 0.546566
\(45\) 0 0
\(46\) −5.46715 −0.806088
\(47\) −3.11521 −0.454400 −0.227200 − 0.973848i \(-0.572957\pi\)
−0.227200 + 0.973848i \(0.572957\pi\)
\(48\) 0 0
\(49\) −0.485072 −0.0692960
\(50\) −0.728950 −0.103089
\(51\) 0 0
\(52\) −2.28892 −0.317416
\(53\) −8.64184 −1.18705 −0.593524 − 0.804816i \(-0.702264\pi\)
−0.593524 + 0.804816i \(0.702264\pi\)
\(54\) 0 0
\(55\) −2.46863 −0.332870
\(56\) 6.45373 0.862416
\(57\) 0 0
\(58\) 2.37449 0.311786
\(59\) 6.23634 0.811903 0.405951 − 0.913895i \(-0.366940\pi\)
0.405951 + 0.913895i \(0.366940\pi\)
\(60\) 0 0
\(61\) 3.27808 0.419716 0.209858 − 0.977732i \(-0.432700\pi\)
0.209858 + 0.977732i \(0.432700\pi\)
\(62\) −0.480222 −0.0609882
\(63\) 0 0
\(64\) 2.07935 0.259919
\(65\) 1.55854 0.193313
\(66\) 0 0
\(67\) 1.47764 0.180523 0.0902615 − 0.995918i \(-0.471230\pi\)
0.0902615 + 0.995918i \(0.471230\pi\)
\(68\) −10.0405 −1.21759
\(69\) 0 0
\(70\) −1.86060 −0.222384
\(71\) 8.06686 0.957360 0.478680 − 0.877989i \(-0.341115\pi\)
0.478680 + 0.877989i \(0.341115\pi\)
\(72\) 0 0
\(73\) −4.96199 −0.580757 −0.290379 − 0.956912i \(-0.593781\pi\)
−0.290379 + 0.956912i \(0.593781\pi\)
\(74\) −0.728950 −0.0847388
\(75\) 0 0
\(76\) 11.2499 1.29045
\(77\) −6.30102 −0.718068
\(78\) 0 0
\(79\) 12.8206 1.44243 0.721214 − 0.692713i \(-0.243585\pi\)
0.721214 + 0.692713i \(0.243585\pi\)
\(80\) 1.09414 0.122329
\(81\) 0 0
\(82\) 1.79951 0.198723
\(83\) −1.14934 −0.126157 −0.0630784 − 0.998009i \(-0.520092\pi\)
−0.0630784 + 0.998009i \(0.520092\pi\)
\(84\) 0 0
\(85\) 6.83662 0.741536
\(86\) −7.98779 −0.861346
\(87\) 0 0
\(88\) −6.24184 −0.665382
\(89\) −11.5207 −1.22119 −0.610596 − 0.791942i \(-0.709070\pi\)
−0.610596 + 0.791942i \(0.709070\pi\)
\(90\) 0 0
\(91\) 3.97807 0.417015
\(92\) −11.0148 −1.14837
\(93\) 0 0
\(94\) 2.27083 0.234218
\(95\) −7.66011 −0.785911
\(96\) 0 0
\(97\) 17.2929 1.75583 0.877916 − 0.478815i \(-0.158933\pi\)
0.877916 + 0.478815i \(0.158933\pi\)
\(98\) 0.353594 0.0357183
\(99\) 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.1.3 5
3.2 odd 2 185.2.a.e.1.3 ✓ 5
5.4 even 2 8325.2.a.ch.1.3 5
12.11 even 2 2960.2.a.w.1.1 5
15.2 even 4 925.2.b.f.149.6 10
15.8 even 4 925.2.b.f.149.5 10
15.14 odd 2 925.2.a.f.1.3 5
21.20 even 2 9065.2.a.k.1.3 5
111.110 odd 2 6845.2.a.f.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.3 ✓ 5 3.2 odd 2
925.2.a.f.1.3 5 15.14 odd 2
925.2.b.f.149.5 10 15.8 even 4
925.2.b.f.149.6 10 15.2 even 4
1665.2.a.p.1.3 5 1.1 even 1 trivial
2960.2.a.w.1.1 5 12.11 even 2
6845.2.a.f.1.3 5 111.110 odd 2
8325.2.a.ch.1.3 5 5.4 even 2
9065.2.a.k.1.3 5 21.20 even 2