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Properties

Label 164.3.n.a.43.1
Level $164$
Weight $3$
Character 164.43
Analytic conductor $4.469$
Analytic rank $0$
Dimension $8$
CM discriminant -4
Inner twists $4$

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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] = [164,3,Mod(39,164)] 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("164.39"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 164.n (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,8,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.46867633551\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

Embedding invariants

Embedding label 43.1
Root \(-0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 164.43
Dual form 164.3.n.a.103.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.90211 + 0.618034i) q^{2} +(3.23607 - 2.35114i) q^{4} +(5.56195 + 7.65537i) q^{5} +(-4.70228 + 6.47214i) q^{8} -9.00000i q^{9} +(-15.3107 - 11.1239i) q^{10} +(5.45515 + 2.77954i) q^{13} +(4.94427 - 15.2169i) q^{16} +(5.31784 + 33.5755i) q^{17} +(5.56231 + 17.1190i) q^{18} +(35.9977 + 11.6964i) q^{20} +(-19.9439 + 61.3811i) q^{25} +(-12.0942 - 1.91553i) q^{26} +(-0.221232 + 1.39680i) q^{29} +32.0000i q^{32} +(-30.8659 - 60.5778i) q^{34} +(-21.1603 - 29.1246i) q^{36} +(17.5788 - 12.7717i) q^{37} -75.7004 q^{40} +(40.8234 - 3.80117i) q^{41} +(68.8983 - 50.0575i) q^{45} +(-28.8015 + 39.6418i) q^{49} -129.080i q^{50} +(24.1883 - 3.83105i) q^{52} +(14.1973 - 89.6382i) q^{53} +(-0.442463 - 2.79360i) q^{58} +(-80.0321 - 26.0040i) q^{61} +(-19.7771 - 60.8676i) q^{64} +(9.06289 + 57.2208i) q^{65} +(96.1497 + 96.1497i) q^{68} +(58.2492 + 42.3205i) q^{72} -145.419i q^{73} +(-25.5435 + 35.1575i) q^{74} +(143.991 - 46.7854i) q^{80} -81.0000 q^{81} +(-75.3015 + 32.4605i) q^{82} +(-227.455 + 227.455i) q^{85} +(-76.4027 - 149.949i) q^{89} +(-100.115 + 137.797i) q^{90} +(160.562 + 25.4306i) q^{97} +(30.2837 - 93.2035i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 24 q^{10} - 14 q^{13} - 32 q^{16} + 14 q^{17} - 36 q^{18} + 160 q^{20} + 22 q^{25} - 128 q^{26} - 2 q^{29} - 188 q^{34} - 72 q^{37} + 96 q^{40} - 18 q^{41} + 108 q^{45} + 256 q^{52} - 34 q^{53}+ \cdots - 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90211 + 0.618034i −0.951057 + 0.309017i
\(3\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 3.23607 − 2.35114i 0.809017 − 0.587785i
\(5\) 5.56195 + 7.65537i 1.11239 + 1.53107i 0.817848 + 0.575435i \(0.195167\pi\)
0.294542 + 0.955638i \(0.404833\pi\)
\(6\) 0 0
\(7\) 0 0 −0.453990 − 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(8\) −4.70228 + 6.47214i −0.587785 + 0.809017i
\(9\) − 9.00000i − 1.00000i
\(10\) −15.3107 − 11.1239i −1.53107 − 1.11239i
\(11\) 0 0 0.987688 − 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(12\) 0 0
\(13\) 5.45515 + 2.77954i 0.419627 + 0.213811i 0.651037 − 0.759046i \(-0.274334\pi\)
−0.231411 + 0.972856i \(0.574334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.94427 − 15.2169i 0.309017 − 0.951057i
\(17\) 5.31784 + 33.5755i 0.312814 + 1.97503i 0.174894 + 0.984587i \(0.444042\pi\)
0.137920 + 0.990443i \(0.455958\pi\)
\(18\) 5.56231 + 17.1190i 0.309017 + 0.951057i
\(19\) 0 0 0.891007 − 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(20\) 35.9977 + 11.6964i 1.79988 + 0.584818i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(24\) 0 0
\(25\) −19.9439 + 61.3811i −0.797757 + 2.45524i
\(26\) −12.0942 − 1.91553i −0.465160 − 0.0736741i
\(27\) 0 0
\(28\) 0 0
\(29\) −0.221232 + 1.39680i −0.00762868 + 0.0481656i −0.991209 − 0.132305i \(-0.957762\pi\)
0.983580 + 0.180471i \(0.0577621\pi\)
\(30\) 0 0
\(31\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(32\) 32.0000i 1.00000i
\(33\) 0 0
\(34\) −30.8659 − 60.5778i −0.907822 − 1.78170i
\(35\) 0 0
\(36\) −21.1603 − 29.1246i −0.587785 − 0.809017i
\(37\) 17.5788 − 12.7717i 0.475102 − 0.345182i −0.324324 − 0.945946i \(-0.605137\pi\)
0.799426 + 0.600764i \(0.205137\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −75.7004 −1.89251
\(41\) 40.8234 − 3.80117i 0.995693 − 0.0927115i
\(42\) 0 0
\(43\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(44\) 0 0
\(45\) 68.8983 − 50.0575i 1.53107 − 1.11239i
\(46\) 0 0
\(47\) 0 0 0.453990 − 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(48\) 0 0
\(49\) −28.8015 + 39.6418i −0.587785 + 0.809017i
\(50\) − 129.080i − 2.58160i
\(51\) 0 0
\(52\) 24.1883 − 3.83105i 0.465160 − 0.0736741i
\(53\) 14.1973 − 89.6382i 0.267873 − 1.69129i −0.376373 − 0.926468i \(-0.622829\pi\)
0.644247 − 0.764818i \(-0.277171\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.442463 − 2.79360i −0.00762868 − 0.0481656i
\(59\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) 0 0
\(61\) −80.0321 − 26.0040i −1.31200 − 0.426295i −0.432261 − 0.901748i \(-0.642284\pi\)
−0.879741 + 0.475453i \(0.842284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −19.7771 − 60.8676i −0.309017 − 0.951057i
\(65\) 9.06289 + 57.2208i 0.139429 + 0.880320i
\(66\) 0 0
\(67\) 0 0 −0.987688 − 0.156434i \(-0.950000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(68\) 96.1497 + 96.1497i 1.41397 + 1.41397i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.987688 − 0.156434i \(-0.0500000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(72\) 58.2492 + 42.3205i 0.809017 + 0.587785i
\(73\) − 145.419i − 1.99204i −0.0891045 − 0.996022i \(-0.528401\pi\)
0.0891045 − 0.996022i \(-0.471599\pi\)
\(74\) −25.5435 + 35.1575i −0.345182 + 0.475102i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) 143.991 − 46.7854i 1.79988 − 0.584818i
\(81\) −81.0000 −1.00000
\(82\) −75.3015 + 32.4605i −0.918311 + 0.395860i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −227.455 + 227.455i −2.67595 + 2.67595i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −76.4027 − 149.949i −0.858457 − 1.68482i −0.719470 − 0.694523i \(-0.755615\pi\)
−0.138987 − 0.990294i \(-0.544385\pi\)
\(90\) −100.115 + 137.797i −1.11239 + 1.53107i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 160.562 + 25.4306i 1.65528 + 0.262171i 0.913012 − 0.407932i \(-0.133750\pi\)
0.742268 + 0.670103i \(0.233750\pi\)
\(98\) 30.2837 − 93.2035i 0.309017 − 0.951057i
\(99\) 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 164.3.n.a.43.1 ✓ 8
4.3 odd 2 CM 164.3.n.a.43.1 ✓ 8
41.21 even 20 inner 164.3.n.a.103.1 yes 8
164.103 odd 20 inner 164.3.n.a.103.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.3.n.a.43.1 ✓ 8 1.1 even 1 trivial
164.3.n.a.43.1 ✓ 8 4.3 odd 2 CM
164.3.n.a.103.1 yes 8 41.21 even 20 inner
164.3.n.a.103.1 yes 8 164.103 odd 20 inner