Properties

Label 209.2.a.d.1.2
Level $209$
Weight $2$
Character 209.1
Self dual yes
Analytic conductor $1.669$
Analytic rank $0$
Dimension $7$
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] = [209,2,Mod(1,209)] 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("209.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.66887340224\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) 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 1.2
Root \(2.61330\) of defining polynomial
Character \(\chi\) \(=\) 209.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.61330 q^{2} +1.19599 q^{3} +4.82936 q^{4} +4.07680 q^{5} -3.12549 q^{6} +3.61829 q^{7} -7.39397 q^{8} -1.56960 q^{9} -10.6539 q^{10} -1.00000 q^{11} +5.77587 q^{12} -1.47857 q^{13} -9.45570 q^{14} +4.87582 q^{15} +9.66398 q^{16} -3.27003 q^{17} +4.10185 q^{18} +1.00000 q^{19} +19.6883 q^{20} +4.32745 q^{21} +2.61330 q^{22} -7.45793 q^{23} -8.84313 q^{24} +11.6203 q^{25} +3.86395 q^{26} -5.46521 q^{27} +17.4740 q^{28} +1.02535 q^{29} -12.7420 q^{30} +1.64921 q^{31} -10.4670 q^{32} -1.19599 q^{33} +8.54558 q^{34} +14.7511 q^{35} -7.58018 q^{36} -6.71293 q^{37} -2.61330 q^{38} -1.76836 q^{39} -30.1438 q^{40} -3.92451 q^{41} -11.3089 q^{42} +5.38113 q^{43} -4.82936 q^{44} -6.39896 q^{45} +19.4898 q^{46} -3.71597 q^{47} +11.5580 q^{48} +6.09205 q^{49} -30.3674 q^{50} -3.91093 q^{51} -7.14054 q^{52} -0.102902 q^{53} +14.2823 q^{54} -4.07680 q^{55} -26.7536 q^{56} +1.19599 q^{57} -2.67955 q^{58} +13.2986 q^{59} +23.5471 q^{60} -6.49664 q^{61} -4.30989 q^{62} -5.67929 q^{63} +8.02543 q^{64} -6.02783 q^{65} +3.12549 q^{66} -3.70989 q^{67} -15.7921 q^{68} -8.91962 q^{69} -38.5490 q^{70} +6.32968 q^{71} +11.6056 q^{72} -1.37759 q^{73} +17.5429 q^{74} +13.8978 q^{75} +4.82936 q^{76} -3.61829 q^{77} +4.62125 q^{78} +13.6725 q^{79} +39.3981 q^{80} -1.82753 q^{81} +10.2559 q^{82} +5.44061 q^{83} +20.8988 q^{84} -13.3313 q^{85} -14.0625 q^{86} +1.22631 q^{87} +7.39397 q^{88} +12.1357 q^{89} +16.7224 q^{90} -5.34990 q^{91} -36.0170 q^{92} +1.97244 q^{93} +9.71096 q^{94} +4.07680 q^{95} -12.5184 q^{96} -13.7910 q^{97} -15.9204 q^{98} +1.56960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} - 9 q^{8} + 11 q^{9} - 6 q^{10} - 7 q^{11} - 16 q^{12} - 4 q^{13} + 6 q^{14} + 12 q^{15} + 27 q^{16} + 2 q^{17} + 9 q^{18} + 7 q^{19} - 4 q^{20}+ \cdots - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61330 −1.84789 −0.923943 − 0.382531i \(-0.875052\pi\)
−0.923943 + 0.382531i \(0.875052\pi\)
\(3\) 1.19599 0.690506 0.345253 − 0.938510i \(-0.387793\pi\)
0.345253 + 0.938510i \(0.387793\pi\)
\(4\) 4.82936 2.41468
\(5\) 4.07680 1.82320 0.911600 − 0.411078i \(-0.134847\pi\)
0.911600 + 0.411078i \(0.134847\pi\)
\(6\) −3.12549 −1.27598
\(7\) 3.61829 1.36759 0.683793 − 0.729676i \(-0.260329\pi\)
0.683793 + 0.729676i \(0.260329\pi\)
\(8\) −7.39397 −2.61416
\(9\) −1.56960 −0.523201
\(10\) −10.6539 −3.36907
\(11\) −1.00000 −0.301511
\(12\) 5.77587 1.66735
\(13\) −1.47857 −0.410081 −0.205041 − 0.978753i \(-0.565733\pi\)
−0.205041 + 0.978753i \(0.565733\pi\)
\(14\) −9.45570 −2.52714
\(15\) 4.87582 1.25893
\(16\) 9.66398 2.41600
\(17\) −3.27003 −0.793099 −0.396549 − 0.918013i \(-0.629792\pi\)
−0.396549 + 0.918013i \(0.629792\pi\)
\(18\) 4.10185 0.966816
\(19\) 1.00000 0.229416
\(20\) 19.6883 4.40244
\(21\) 4.32745 0.944327
\(22\) 2.61330 0.557158
\(23\) −7.45793 −1.55509 −0.777543 − 0.628830i \(-0.783534\pi\)
−0.777543 + 0.628830i \(0.783534\pi\)
\(24\) −8.84313 −1.80510
\(25\) 11.6203 2.32406
\(26\) 3.86395 0.757783
\(27\) −5.46521 −1.05178
\(28\) 17.4740 3.30228
\(29\) 1.02535 0.190403 0.0952013 − 0.995458i \(-0.469651\pi\)
0.0952013 + 0.995458i \(0.469651\pi\)
\(30\) −12.7420 −2.32636
\(31\) 1.64921 0.296207 0.148104 − 0.988972i \(-0.452683\pi\)
0.148104 + 0.988972i \(0.452683\pi\)
\(32\) −10.4670 −1.85032
\(33\) −1.19599 −0.208195
\(34\) 8.54558 1.46556
\(35\) 14.7511 2.49338
\(36\) −7.58018 −1.26336
\(37\) −6.71293 −1.10360 −0.551799 − 0.833977i \(-0.686059\pi\)
−0.551799 + 0.833977i \(0.686059\pi\)
\(38\) −2.61330 −0.423934
\(39\) −1.76836 −0.283164
\(40\) −30.1438 −4.76615
\(41\) −3.92451 −0.612905 −0.306453 − 0.951886i \(-0.599142\pi\)
−0.306453 + 0.951886i \(0.599142\pi\)
\(42\) −11.3089 −1.74501
\(43\) 5.38113 0.820614 0.410307 − 0.911947i \(-0.365422\pi\)
0.410307 + 0.911947i \(0.365422\pi\)
\(44\) −4.82936 −0.728053
\(45\) −6.39896 −0.953901
\(46\) 19.4898 2.87362
\(47\) −3.71597 −0.542030 −0.271015 − 0.962575i \(-0.587359\pi\)
−0.271015 + 0.962575i \(0.587359\pi\)
\(48\) 11.5580 1.66826
\(49\) 6.09205 0.870292
\(50\) −30.3674 −4.29460
\(51\) −3.91093 −0.547640
\(52\) −7.14054 −0.990215
\(53\) −0.102902 −0.0141347 −0.00706733 − 0.999975i \(-0.502250\pi\)
−0.00706733 + 0.999975i \(0.502250\pi\)
\(54\) 14.2823 1.94357
\(55\) −4.07680 −0.549716
\(56\) −26.7536 −3.57509
\(57\) 1.19599 0.158413
\(58\) −2.67955 −0.351842
\(59\) 13.2986 1.73134 0.865668 − 0.500619i \(-0.166894\pi\)
0.865668 + 0.500619i \(0.166894\pi\)
\(60\) 23.5471 3.03991
\(61\) −6.49664 −0.831809 −0.415905 − 0.909408i \(-0.636535\pi\)
−0.415905 + 0.909408i \(0.636535\pi\)
\(62\) −4.30989 −0.547357
\(63\) −5.67929 −0.715523
\(64\) 8.02543 1.00318
\(65\) −6.02783 −0.747661
\(66\) 3.12549 0.384721
\(67\) −3.70989 −0.453235 −0.226618 − 0.973984i \(-0.572767\pi\)
−0.226618 + 0.973984i \(0.572767\pi\)
\(68\) −15.7921 −1.91508
\(69\) −8.91962 −1.07380
\(70\) −38.5490 −4.60749
\(71\) 6.32968 0.751194 0.375597 − 0.926783i \(-0.377438\pi\)
0.375597 + 0.926783i \(0.377438\pi\)
\(72\) 11.6056 1.36773
\(73\) −1.37759 −0.161235 −0.0806173 − 0.996745i \(-0.525689\pi\)
−0.0806173 + 0.996745i \(0.525689\pi\)
\(74\) 17.5429 2.03932
\(75\) 13.8978 1.60478
\(76\) 4.82936 0.553965
\(77\) −3.61829 −0.412343
\(78\) 4.62125 0.523254
\(79\) 13.6725 1.53828 0.769141 − 0.639079i \(-0.220684\pi\)
0.769141 + 0.639079i \(0.220684\pi\)
\(80\) 39.3981 4.40484
\(81\) −1.82753 −0.203059
\(82\) 10.2559 1.13258
\(83\) 5.44061 0.597184 0.298592 − 0.954381i \(-0.403483\pi\)
0.298592 + 0.954381i \(0.403483\pi\)
\(84\) 20.8988 2.28025
\(85\) −13.3313 −1.44598
\(86\) −14.0625 −1.51640
\(87\) 1.22631 0.131474
\(88\) 7.39397 0.788200
\(89\) 12.1357 1.28638 0.643191 − 0.765706i \(-0.277610\pi\)
0.643191 + 0.765706i \(0.277610\pi\)
\(90\) 16.7224 1.76270
\(91\) −5.34990 −0.560822
\(92\) −36.0170 −3.75503
\(93\) 1.97244 0.204533
\(94\) 9.71096 1.00161
\(95\) 4.07680 0.418271
\(96\) −12.5184 −1.27766
\(97\) −13.7910 −1.40026 −0.700131 − 0.714014i \(-0.746875\pi\)
−0.700131 + 0.714014i \(0.746875\pi\)
\(98\) −15.9204 −1.60820
\(99\) 1.56960 0.157751
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 209.2.a.d.1.2 ✓ 7
3.2 odd 2 1881.2.a.p.1.6 7
4.3 odd 2 3344.2.a.ba.1.4 7
5.4 even 2 5225.2.a.n.1.6 7
11.10 odd 2 2299.2.a.q.1.6 7
19.18 odd 2 3971.2.a.i.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 ✓ 7 1.1 even 1 trivial
1881.2.a.p.1.6 7 3.2 odd 2
2299.2.a.q.1.6 7 11.10 odd 2
3344.2.a.ba.1.4 7 4.3 odd 2
3971.2.a.i.1.6 7 19.18 odd 2
5225.2.a.n.1.6 7 5.4 even 2