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Properties

Label 9065.2.a.k.1.3
Level $9065$
Weight $2$
Character 9065.1
Self dual yes
Analytic conductor $72.384$
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] = [9065,2,Mod(1,9065)] 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("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9065.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,-3,10,5,6,0,6,6,2,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.3843894323\)
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: 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 \(\chi\) \(=\) 9065.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.728950 q^{2} -2.62871 q^{3} -1.46863 q^{4} +1.00000 q^{5} -1.91620 q^{6} -2.52846 q^{8} +3.91009 q^{9} +0.728950 q^{10} +2.46863 q^{11} +3.86060 q^{12} -1.55854 q^{13} -2.62871 q^{15} +1.09414 q^{16} +6.83662 q^{17} +2.85026 q^{18} +7.66011 q^{19} -1.46863 q^{20} +1.79951 q^{22} -7.50003 q^{23} +6.64658 q^{24} +1.00000 q^{25} -1.13610 q^{26} -2.39236 q^{27} +3.25741 q^{29} -1.91620 q^{30} -0.658785 q^{31} +5.85449 q^{32} -6.48930 q^{33} +4.98356 q^{34} -5.74248 q^{36} +1.00000 q^{37} +5.58384 q^{38} +4.09694 q^{39} -2.52846 q^{40} -2.46863 q^{41} +10.9579 q^{43} -3.62551 q^{44} +3.91009 q^{45} -5.46715 q^{46} -3.11521 q^{47} -2.87617 q^{48} +0.728950 q^{50} -17.9715 q^{51} +2.28892 q^{52} +8.64184 q^{53} -1.74391 q^{54} +2.46863 q^{55} -20.1362 q^{57} +2.37449 q^{58} +6.23634 q^{59} +3.86060 q^{60} -3.27808 q^{61} -0.480222 q^{62} +2.07935 q^{64} -1.55854 q^{65} -4.73038 q^{66} +1.47764 q^{67} -10.0405 q^{68} +19.7154 q^{69} -8.06686 q^{71} -9.88651 q^{72} +4.96199 q^{73} +0.728950 q^{74} -2.62871 q^{75} -11.2499 q^{76} +2.98647 q^{78} +12.8206 q^{79} +1.09414 q^{80} -5.44146 q^{81} -1.79951 q^{82} -1.14934 q^{83} +6.83662 q^{85} +7.98779 q^{86} -8.56277 q^{87} -6.24184 q^{88} -11.5207 q^{89} +2.85026 q^{90} +11.0148 q^{92} +1.73175 q^{93} -2.27083 q^{94} +7.66011 q^{95} -15.3897 q^{96} -17.2929 q^{97} +9.65257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} - 3 q^{3} + 10 q^{4} + 5 q^{5} + 6 q^{6} + 6 q^{8} + 6 q^{9} + 2 q^{10} - 5 q^{11} + 2 q^{12} - 4 q^{13} - 3 q^{15} + 16 q^{16} + 2 q^{18} + 4 q^{19} + 10 q^{20} - 8 q^{22} + 4 q^{23} + 42 q^{24}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.728950 0.515446 0.257723 − 0.966219i \(-0.417028\pi\)
0.257723 + 0.966219i \(0.417028\pi\)
\(3\) −2.62871 −1.51768 −0.758842 − 0.651275i \(-0.774234\pi\)
−0.758842 + 0.651275i \(0.774234\pi\)
\(4\) −1.46863 −0.734316
\(5\) 1.00000 0.447214
\(6\) −1.91620 −0.782284
\(7\) 0 0
\(8\) −2.52846 −0.893946
\(9\) 3.91009 1.30336
\(10\) 0.728950 0.230514
\(11\) 2.46863 0.744320 0.372160 − 0.928169i \(-0.378617\pi\)
0.372160 + 0.928169i \(0.378617\pi\)
\(12\) 3.86060 1.11446
\(13\) −1.55854 −0.432261 −0.216131 − 0.976364i \(-0.569344\pi\)
−0.216131 + 0.976364i \(0.569344\pi\)
\(14\) 0 0
\(15\) −2.62871 −0.678729
\(16\) 1.09414 0.273535
\(17\) 6.83662 1.65812 0.829062 − 0.559156i \(-0.188875\pi\)
0.829062 + 0.559156i \(0.188875\pi\)
\(18\) 2.85026 0.671813
\(19\) 7.66011 1.75735 0.878675 − 0.477421i \(-0.158428\pi\)
0.878675 + 0.477421i \(0.158428\pi\)
\(20\) −1.46863 −0.328396
\(21\) 0 0
\(22\) 1.79951 0.383657
\(23\) −7.50003 −1.56387 −0.781933 − 0.623363i \(-0.785766\pi\)
−0.781933 + 0.623363i \(0.785766\pi\)
\(24\) 6.64658 1.35673
\(25\) 1.00000 0.200000
\(26\) −1.13610 −0.222807
\(27\) −2.39236 −0.460410
\(28\) 0 0
\(29\) 3.25741 0.604886 0.302443 − 0.953167i \(-0.402198\pi\)
0.302443 + 0.953167i \(0.402198\pi\)
\(30\) −1.91620 −0.349848
\(31\) −0.658785 −0.118321 −0.0591607 − 0.998248i \(-0.518842\pi\)
−0.0591607 + 0.998248i \(0.518842\pi\)
\(32\) 5.85449 1.03494
\(33\) −6.48930 −1.12964
\(34\) 4.98356 0.854673
\(35\) 0 0
\(36\) −5.74248 −0.957080
\(37\) 1.00000 0.164399
\(38\) 5.58384 0.905818
\(39\) 4.09694 0.656036
\(40\) −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
\(45\) 3.91009 0.582882
\(46\) −5.46715 −0.806088
\(47\) −3.11521 −0.454400 −0.227200 − 0.973848i \(-0.572957\pi\)
−0.227200 + 0.973848i \(0.572957\pi\)
\(48\) −2.87617 −0.415140
\(49\) 0 0
\(50\) 0.728950 0.103089
\(51\) −17.9715 −2.51651
\(52\) 2.28892 0.317416
\(53\) 8.64184 1.18705 0.593524 − 0.804816i \(-0.297736\pi\)
0.593524 + 0.804816i \(0.297736\pi\)
\(54\) −1.74391 −0.237317
\(55\) 2.46863 0.332870
\(56\) 0 0
\(57\) −20.1362 −2.66710
\(58\) 2.37449 0.311786
\(59\) 6.23634 0.811903 0.405951 − 0.913895i \(-0.366940\pi\)
0.405951 + 0.913895i \(0.366940\pi\)
\(60\) 3.86060 0.498401
\(61\) −3.27808 −0.419716 −0.209858 − 0.977732i \(-0.567300\pi\)
−0.209858 + 0.977732i \(0.567300\pi\)
\(62\) −0.480222 −0.0609882
\(63\) 0 0
\(64\) 2.07935 0.259919
\(65\) −1.55854 −0.193313
\(66\) −4.73038 −0.582270
\(67\) 1.47764 0.180523 0.0902615 − 0.995918i \(-0.471230\pi\)
0.0902615 + 0.995918i \(0.471230\pi\)
\(68\) −10.0405 −1.21759
\(69\) 19.7154 2.37345
\(70\) 0 0
\(71\) −8.06686 −0.957360 −0.478680 − 0.877989i \(-0.658885\pi\)
−0.478680 + 0.877989i \(0.658885\pi\)
\(72\) −9.88651 −1.16514
\(73\) 4.96199 0.580757 0.290379 − 0.956912i \(-0.406219\pi\)
0.290379 + 0.956912i \(0.406219\pi\)
\(74\) 0.728950 0.0847388
\(75\) −2.62871 −0.303537
\(76\) −11.2499 −1.29045
\(77\) 0 0
\(78\) 2.98647 0.338151
\(79\) 12.8206 1.44243 0.721214 − 0.692713i \(-0.243585\pi\)
0.721214 + 0.692713i \(0.243585\pi\)
\(80\) 1.09414 0.122329
\(81\) −5.44146 −0.604607
\(82\) −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
\(87\) −8.56277 −0.918026
\(88\) −6.24184 −0.665382
\(89\) −11.5207 −1.22119 −0.610596 − 0.791942i \(-0.709070\pi\)
−0.610596 + 0.791942i \(0.709070\pi\)
\(90\) 2.85026 0.300444
\(91\) 0 0
\(92\) 11.0148 1.14837
\(93\) 1.73175 0.179574
\(94\) −2.27083 −0.234218
\(95\) 7.66011 0.785911
\(96\) −15.3897 −1.57071
\(97\) −17.2929 −1.75583 −0.877916 − 0.478815i \(-0.841067\pi\)
−0.877916 + 0.478815i \(0.841067\pi\)
\(98\) 0 0
\(99\) 9.65257 0.970120
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 9065.2.a.k.1.3 5
7.6 odd 2 185.2.a.e.1.3 ✓ 5
21.20 even 2 1665.2.a.p.1.3 5
28.27 even 2 2960.2.a.w.1.1 5
35.13 even 4 925.2.b.f.149.5 10
35.27 even 4 925.2.b.f.149.6 10
35.34 odd 2 925.2.a.f.1.3 5
105.104 even 2 8325.2.a.ch.1.3 5
259.258 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 7.6 odd 2
925.2.a.f.1.3 5 35.34 odd 2
925.2.b.f.149.5 10 35.13 even 4
925.2.b.f.149.6 10 35.27 even 4
1665.2.a.p.1.3 5 21.20 even 2
2960.2.a.w.1.1 5 28.27 even 2
6845.2.a.f.1.3 5 259.258 odd 2
8325.2.a.ch.1.3 5 105.104 even 2
9065.2.a.k.1.3 5 1.1 even 1 trivial