Properties

Label 209.2.a.c.1.2
Level $209$
Weight $2$
Character 209.1
Self dual yes
Analytic conductor $1.669$
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] = [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 = [5] 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: \(5\)
Coefficient field: 5.5.246832.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) 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 \(1.71250\) 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)
 
\(f(q)\) \(=\) \(q-0.779856 q^{2} -2.98063 q^{3} -1.39182 q^{4} -3.49235 q^{5} +2.32446 q^{6} +1.06736 q^{7} +2.64513 q^{8} +5.88418 q^{9} +2.72353 q^{10} +1.00000 q^{11} +4.14852 q^{12} -0.0563258 q^{13} -0.832387 q^{14} +10.4094 q^{15} +0.720827 q^{16} -4.53628 q^{17} -4.58881 q^{18} -1.00000 q^{19} +4.86074 q^{20} -3.18141 q^{21} -0.779856 q^{22} -1.07949 q^{23} -7.88418 q^{24} +7.19651 q^{25} +0.0439260 q^{26} -8.59667 q^{27} -1.48558 q^{28} +0.299905 q^{29} -8.11784 q^{30} +9.18548 q^{31} -5.85241 q^{32} -2.98063 q^{33} +3.53764 q^{34} -3.72760 q^{35} -8.18974 q^{36} +4.50448 q^{37} +0.779856 q^{38} +0.167887 q^{39} -9.23774 q^{40} +12.0009 q^{41} +2.48104 q^{42} +10.7260 q^{43} -1.39182 q^{44} -20.5496 q^{45} +0.841844 q^{46} +2.89630 q^{47} -2.14852 q^{48} -5.86074 q^{49} -5.61224 q^{50} +13.5210 q^{51} +0.0783957 q^{52} -12.3213 q^{53} +6.70416 q^{54} -3.49235 q^{55} +2.82331 q^{56} +2.98063 q^{57} -0.233882 q^{58} -1.14582 q^{59} -14.4881 q^{60} -8.09599 q^{61} -7.16335 q^{62} +6.28054 q^{63} +3.12238 q^{64} +0.196709 q^{65} +2.32446 q^{66} +11.2733 q^{67} +6.31370 q^{68} +3.21755 q^{69} +2.90699 q^{70} +13.4948 q^{71} +15.5644 q^{72} -11.1470 q^{73} -3.51284 q^{74} -21.4502 q^{75} +1.39182 q^{76} +1.06736 q^{77} -0.130927 q^{78} +11.4250 q^{79} -2.51738 q^{80} +7.97100 q^{81} -9.35899 q^{82} -13.9802 q^{83} +4.42797 q^{84} +15.8423 q^{85} -8.36475 q^{86} -0.893906 q^{87} +2.64513 q^{88} -0.183185 q^{89} +16.0257 q^{90} -0.0601200 q^{91} +1.50246 q^{92} -27.3785 q^{93} -2.25870 q^{94} +3.49235 q^{95} +17.4439 q^{96} +5.66263 q^{97} +4.57053 q^{98} +5.88418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9} + 12 q^{10} + 5 q^{11} + 6 q^{12} + 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} - 5 q^{19} - 8 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.779856 −0.551441 −0.275721 − 0.961238i \(-0.588916\pi\)
−0.275721 + 0.961238i \(0.588916\pi\)
\(3\) −2.98063 −1.72087 −0.860435 − 0.509561i \(-0.829808\pi\)
−0.860435 + 0.509561i \(0.829808\pi\)
\(4\) −1.39182 −0.695912
\(5\) −3.49235 −1.56183 −0.780913 − 0.624639i \(-0.785246\pi\)
−0.780913 + 0.624639i \(0.785246\pi\)
\(6\) 2.32446 0.948959
\(7\) 1.06736 0.403424 0.201712 − 0.979445i \(-0.435349\pi\)
0.201712 + 0.979445i \(0.435349\pi\)
\(8\) 2.64513 0.935196
\(9\) 5.88418 1.96139
\(10\) 2.72353 0.861256
\(11\) 1.00000 0.301511
\(12\) 4.14852 1.19757
\(13\) −0.0563258 −0.0156220 −0.00781098 − 0.999969i \(-0.502486\pi\)
−0.00781098 + 0.999969i \(0.502486\pi\)
\(14\) −0.832387 −0.222465
\(15\) 10.4094 2.68770
\(16\) 0.720827 0.180207
\(17\) −4.53628 −1.10021 −0.550104 − 0.835096i \(-0.685412\pi\)
−0.550104 + 0.835096i \(0.685412\pi\)
\(18\) −4.58881 −1.08159
\(19\) −1.00000 −0.229416
\(20\) 4.86074 1.08689
\(21\) −3.18141 −0.694241
\(22\) −0.779856 −0.166266
\(23\) −1.07949 −0.225088 −0.112544 − 0.993647i \(-0.535900\pi\)
−0.112544 + 0.993647i \(0.535900\pi\)
\(24\) −7.88418 −1.60935
\(25\) 7.19651 1.43930
\(26\) 0.0439260 0.00861460
\(27\) −8.59667 −1.65443
\(28\) −1.48558 −0.280748
\(29\) 0.299905 0.0556909 0.0278455 − 0.999612i \(-0.491135\pi\)
0.0278455 + 0.999612i \(0.491135\pi\)
\(30\) −8.11784 −1.48211
\(31\) 9.18548 1.64976 0.824880 − 0.565307i \(-0.191242\pi\)
0.824880 + 0.565307i \(0.191242\pi\)
\(32\) −5.85241 −1.03457
\(33\) −2.98063 −0.518862
\(34\) 3.53764 0.606701
\(35\) −3.72760 −0.630079
\(36\) −8.18974 −1.36496
\(37\) 4.50448 0.740531 0.370266 − 0.928926i \(-0.379267\pi\)
0.370266 + 0.928926i \(0.379267\pi\)
\(38\) 0.779856 0.126509
\(39\) 0.167887 0.0268834
\(40\) −9.23774 −1.46061
\(41\) 12.0009 1.87423 0.937115 − 0.349020i \(-0.113486\pi\)
0.937115 + 0.349020i \(0.113486\pi\)
\(42\) 2.48104 0.382833
\(43\) 10.7260 1.63570 0.817851 − 0.575430i \(-0.195165\pi\)
0.817851 + 0.575430i \(0.195165\pi\)
\(44\) −1.39182 −0.209826
\(45\) −20.5496 −3.06335
\(46\) 0.841844 0.124123
\(47\) 2.89630 0.422469 0.211235 − 0.977435i \(-0.432252\pi\)
0.211235 + 0.977435i \(0.432252\pi\)
\(48\) −2.14852 −0.310112
\(49\) −5.86074 −0.837249
\(50\) −5.61224 −0.793691
\(51\) 13.5210 1.89332
\(52\) 0.0783957 0.0108715
\(53\) −12.3213 −1.69246 −0.846230 − 0.532818i \(-0.821133\pi\)
−0.846230 + 0.532818i \(0.821133\pi\)
\(54\) 6.70416 0.912321
\(55\) −3.49235 −0.470908
\(56\) 2.82331 0.377281
\(57\) 2.98063 0.394795
\(58\) −0.233882 −0.0307103
\(59\) −1.14582 −0.149173 −0.0745863 − 0.997215i \(-0.523764\pi\)
−0.0745863 + 0.997215i \(0.523764\pi\)
\(60\) −14.4881 −1.87040
\(61\) −8.09599 −1.03659 −0.518293 − 0.855203i \(-0.673432\pi\)
−0.518293 + 0.855203i \(0.673432\pi\)
\(62\) −7.16335 −0.909746
\(63\) 6.28054 0.791273
\(64\) 3.12238 0.390298
\(65\) 0.196709 0.0243988
\(66\) 2.32446 0.286122
\(67\) 11.2733 1.37726 0.688628 − 0.725115i \(-0.258213\pi\)
0.688628 + 0.725115i \(0.258213\pi\)
\(68\) 6.31370 0.765649
\(69\) 3.21755 0.387348
\(70\) 2.90699 0.347452
\(71\) 13.4948 1.60154 0.800771 − 0.598970i \(-0.204423\pi\)
0.800771 + 0.598970i \(0.204423\pi\)
\(72\) 15.5644 1.83429
\(73\) −11.1470 −1.30466 −0.652330 − 0.757935i \(-0.726208\pi\)
−0.652330 + 0.757935i \(0.726208\pi\)
\(74\) −3.51284 −0.408360
\(75\) −21.4502 −2.47685
\(76\) 1.39182 0.159653
\(77\) 1.06736 0.121637
\(78\) −0.130927 −0.0148246
\(79\) 11.4250 1.28541 0.642706 − 0.766113i \(-0.277812\pi\)
0.642706 + 0.766113i \(0.277812\pi\)
\(80\) −2.51738 −0.281452
\(81\) 7.97100 0.885666
\(82\) −9.35899 −1.03353
\(83\) −13.9802 −1.53452 −0.767261 − 0.641335i \(-0.778381\pi\)
−0.767261 + 0.641335i \(0.778381\pi\)
\(84\) 4.42797 0.483131
\(85\) 15.8423 1.71834
\(86\) −8.36475 −0.901994
\(87\) −0.893906 −0.0958368
\(88\) 2.64513 0.281972
\(89\) −0.183185 −0.0194176 −0.00970878 − 0.999953i \(-0.503090\pi\)
−0.00970878 + 0.999953i \(0.503090\pi\)
\(90\) 16.0257 1.68926
\(91\) −0.0601200 −0.00630228
\(92\) 1.50246 0.156642
\(93\) −27.3785 −2.83902
\(94\) −2.25870 −0.232967
\(95\) 3.49235 0.358308
\(96\) 17.4439 1.78036
\(97\) 5.66263 0.574953 0.287477 − 0.957788i \(-0.407184\pi\)
0.287477 + 0.957788i \(0.407184\pi\)
\(98\) 4.57053 0.461694
\(99\) 5.88418 0.591382
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.c.1.2 ✓ 5
3.2 odd 2 1881.2.a.k.1.4 5
4.3 odd 2 3344.2.a.t.1.5 5
5.4 even 2 5225.2.a.h.1.4 5
11.10 odd 2 2299.2.a.n.1.4 5
19.18 odd 2 3971.2.a.h.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.2 ✓ 5 1.1 even 1 trivial
1881.2.a.k.1.4 5 3.2 odd 2
2299.2.a.n.1.4 5 11.10 odd 2
3344.2.a.t.1.5 5 4.3 odd 2
3971.2.a.h.1.4 5 19.18 odd 2
5225.2.a.h.1.4 5 5.4 even 2