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Properties

Label 9065.2.a.k.1.2
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 1.2
Root \(2.10563\) 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)
 
\(f(q)\) \(=\) \(q-1.13359 q^{2} +1.10563 q^{3} -0.714970 q^{4} +1.00000 q^{5} -1.25333 q^{6} +3.07767 q^{8} -1.77758 q^{9} -1.13359 q^{10} +1.71497 q^{11} -0.790492 q^{12} -6.49255 q^{13} +1.10563 q^{15} -2.05888 q^{16} -3.32980 q^{17} +2.01505 q^{18} -0.734568 q^{19} -0.714970 q^{20} -1.94408 q^{22} -2.08603 q^{23} +3.40276 q^{24} +1.00000 q^{25} +7.35990 q^{26} -5.28224 q^{27} -4.21126 q^{29} -1.25333 q^{30} -7.46459 q^{31} -3.82141 q^{32} +1.89612 q^{33} +3.77463 q^{34} +1.27092 q^{36} +1.00000 q^{37} +0.832700 q^{38} -7.17836 q^{39} +3.07767 q^{40} -1.71497 q^{41} +1.81885 q^{43} -1.22615 q^{44} -1.77758 q^{45} +2.36471 q^{46} +0.882270 q^{47} -2.27636 q^{48} -1.13359 q^{50} -3.68152 q^{51} +4.64198 q^{52} -7.03066 q^{53} +5.98790 q^{54} +1.71497 q^{55} -0.812159 q^{57} +4.77385 q^{58} -0.387867 q^{59} -0.790492 q^{60} +11.8224 q^{61} +8.46180 q^{62} +8.44967 q^{64} -6.49255 q^{65} -2.14943 q^{66} +12.1086 q^{67} +2.38070 q^{68} -2.30638 q^{69} +13.7486 q^{71} -5.47081 q^{72} -16.6719 q^{73} -1.13359 q^{74} +1.10563 q^{75} +0.525194 q^{76} +8.13733 q^{78} +8.23253 q^{79} -2.05888 q^{80} -0.507447 q^{81} +1.94408 q^{82} +4.80275 q^{83} -3.32980 q^{85} -2.06183 q^{86} -4.65609 q^{87} +5.27811 q^{88} +1.52506 q^{89} +2.01505 q^{90} +1.49145 q^{92} -8.25307 q^{93} -1.00013 q^{94} -0.734568 q^{95} -4.22506 q^{96} +18.1588 q^{97} -3.04850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} - 3 q^{3} + 10 q^{4} + 5 q^{5} + 6 q^{6} + 6 q^{8} + 6 q^{9} + 2 q^{10} - 5 q^{11} + 2 q^{12} - 4 q^{13} - 3 q^{15} + 16 q^{16} + 2 q^{18} + 4 q^{19} + 10 q^{20} - 8 q^{22} + 4 q^{23} + 42 q^{24}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13359 −0.801570 −0.400785 − 0.916172i \(-0.631263\pi\)
−0.400785 + 0.916172i \(0.631263\pi\)
\(3\) 1.10563 0.638335 0.319168 − 0.947698i \(-0.396597\pi\)
0.319168 + 0.947698i \(0.396597\pi\)
\(4\) −0.714970 −0.357485
\(5\) 1.00000 0.447214
\(6\) −1.25333 −0.511671
\(7\) 0 0
\(8\) 3.07767 1.08812
\(9\) −1.77758 −0.592528
\(10\) −1.13359 −0.358473
\(11\) 1.71497 0.517083 0.258541 − 0.966000i \(-0.416758\pi\)
0.258541 + 0.966000i \(0.416758\pi\)
\(12\) −0.790492 −0.228195
\(13\) −6.49255 −1.80071 −0.900355 − 0.435156i \(-0.856693\pi\)
−0.900355 + 0.435156i \(0.856693\pi\)
\(14\) 0 0
\(15\) 1.10563 0.285472
\(16\) −2.05888 −0.514719
\(17\) −3.32980 −0.807594 −0.403797 − 0.914849i \(-0.632310\pi\)
−0.403797 + 0.914849i \(0.632310\pi\)
\(18\) 2.01505 0.474953
\(19\) −0.734568 −0.168521 −0.0842607 − 0.996444i \(-0.526853\pi\)
−0.0842607 + 0.996444i \(0.526853\pi\)
\(20\) −0.714970 −0.159872
\(21\) 0 0
\(22\) −1.94408 −0.414478
\(23\) −2.08603 −0.434968 −0.217484 − 0.976064i \(-0.569785\pi\)
−0.217484 + 0.976064i \(0.569785\pi\)
\(24\) 3.40276 0.694585
\(25\) 1.00000 0.200000
\(26\) 7.35990 1.44340
\(27\) −5.28224 −1.01657
\(28\) 0 0
\(29\) −4.21126 −0.782011 −0.391006 − 0.920388i \(-0.627873\pi\)
−0.391006 + 0.920388i \(0.627873\pi\)
\(30\) −1.25333 −0.228826
\(31\) −7.46459 −1.34068 −0.670340 − 0.742054i \(-0.733852\pi\)
−0.670340 + 0.742054i \(0.733852\pi\)
\(32\) −3.82141 −0.675536
\(33\) 1.89612 0.330072
\(34\) 3.77463 0.647344
\(35\) 0 0
\(36\) 1.27092 0.211820
\(37\) 1.00000 0.164399
\(38\) 0.832700 0.135082
\(39\) −7.17836 −1.14946
\(40\) 3.07767 0.486622
\(41\) −1.71497 −0.267833 −0.133917 − 0.990993i \(-0.542755\pi\)
−0.133917 + 0.990993i \(0.542755\pi\)
\(42\) 0 0
\(43\) 1.81885 0.277372 0.138686 − 0.990336i \(-0.455712\pi\)
0.138686 + 0.990336i \(0.455712\pi\)
\(44\) −1.22615 −0.184849
\(45\) −1.77758 −0.264986
\(46\) 2.36471 0.348657
\(47\) 0.882270 0.128692 0.0643462 − 0.997928i \(-0.479504\pi\)
0.0643462 + 0.997928i \(0.479504\pi\)
\(48\) −2.27636 −0.328564
\(49\) 0 0
\(50\) −1.13359 −0.160314
\(51\) −3.68152 −0.515516
\(52\) 4.64198 0.643727
\(53\) −7.03066 −0.965735 −0.482867 − 0.875693i \(-0.660405\pi\)
−0.482867 + 0.875693i \(0.660405\pi\)
\(54\) 5.98790 0.814850
\(55\) 1.71497 0.231247
\(56\) 0 0
\(57\) −0.812159 −0.107573
\(58\) 4.77385 0.626837
\(59\) −0.387867 −0.0504959 −0.0252480 − 0.999681i \(-0.508038\pi\)
−0.0252480 + 0.999681i \(0.508038\pi\)
\(60\) −0.790492 −0.102052
\(61\) 11.8224 1.51370 0.756848 − 0.653590i \(-0.226738\pi\)
0.756848 + 0.653590i \(0.226738\pi\)
\(62\) 8.46180 1.07465
\(63\) 0 0
\(64\) 8.44967 1.05621
\(65\) −6.49255 −0.805302
\(66\) −2.14943 −0.264576
\(67\) 12.1086 1.47930 0.739649 − 0.672992i \(-0.234991\pi\)
0.739649 + 0.672992i \(0.234991\pi\)
\(68\) 2.38070 0.288703
\(69\) −2.30638 −0.277655
\(70\) 0 0
\(71\) 13.7486 1.63166 0.815828 − 0.578295i \(-0.196282\pi\)
0.815828 + 0.578295i \(0.196282\pi\)
\(72\) −5.47081 −0.644741
\(73\) −16.6719 −1.95129 −0.975646 − 0.219349i \(-0.929607\pi\)
−0.975646 + 0.219349i \(0.929607\pi\)
\(74\) −1.13359 −0.131777
\(75\) 1.10563 0.127667
\(76\) 0.525194 0.0602439
\(77\) 0 0
\(78\) 8.13733 0.921371
\(79\) 8.23253 0.926232 0.463116 − 0.886298i \(-0.346731\pi\)
0.463116 + 0.886298i \(0.346731\pi\)
\(80\) −2.05888 −0.230190
\(81\) −0.507447 −0.0563829
\(82\) 1.94408 0.214687
\(83\) 4.80275 0.527171 0.263585 − 0.964636i \(-0.415095\pi\)
0.263585 + 0.964636i \(0.415095\pi\)
\(84\) 0 0
\(85\) −3.32980 −0.361167
\(86\) −2.06183 −0.222333
\(87\) −4.65609 −0.499185
\(88\) 5.27811 0.562648
\(89\) 1.52506 0.161656 0.0808280 − 0.996728i \(-0.474244\pi\)
0.0808280 + 0.996728i \(0.474244\pi\)
\(90\) 2.01505 0.212405
\(91\) 0 0
\(92\) 1.49145 0.155494
\(93\) −8.25307 −0.855804
\(94\) −1.00013 −0.103156
\(95\) −0.734568 −0.0753650
\(96\) −4.22506 −0.431218
\(97\) 18.1588 1.84375 0.921875 − 0.387487i \(-0.126657\pi\)
0.921875 + 0.387487i \(0.126657\pi\)
\(98\) 0 0
\(99\) −3.04850 −0.306386
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.1.2 5
7.6 odd 2 185.2.a.e.1.2 ✓ 5
21.20 even 2 1665.2.a.p.1.4 5
28.27 even 2 2960.2.a.w.1.4 5
35.13 even 4 925.2.b.f.149.7 10
35.27 even 4 925.2.b.f.149.4 10
35.34 odd 2 925.2.a.f.1.4 5
105.104 even 2 8325.2.a.ch.1.2 5
259.258 odd 2 6845.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.2 ✓ 5 7.6 odd 2
925.2.a.f.1.4 5 35.34 odd 2
925.2.b.f.149.4 10 35.27 even 4
925.2.b.f.149.7 10 35.13 even 4
1665.2.a.p.1.4 5 21.20 even 2
2960.2.a.w.1.4 5 28.27 even 2
6845.2.a.f.1.4 5 259.258 odd 2
8325.2.a.ch.1.2 5 105.104 even 2
9065.2.a.k.1.2 5 1.1 even 1 trivial