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Properties

Label 164.6.b.a.81.13
Level $164$
Weight $6$
Character 164.81
Analytic conductor $26.303$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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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] = [164,6,Mod(81,164)] 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("164.81"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 164.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.3029464493\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 3198 x^{16} + 4145894 x^{14} + 2811630520 x^{12} + 1078698643464 x^{10} + 238264942196960 x^{8} + \cdots + 89\!\cdots\!32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{57}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 81.13
Root \(11.2815i\) of defining polynomial
Character \(\chi\) \(=\) 164.81
Dual form 164.6.b.a.81.6

$q$-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)
 
\(f(q)\) \(=\) \(q+11.2815i q^{3} +35.2361 q^{5} +248.823i q^{7} +115.729 q^{9} -198.962i q^{11} +463.506i q^{13} +397.515i q^{15} +60.6006i q^{17} -956.649i q^{19} -2807.09 q^{21} +3112.83 q^{23} -1883.42 q^{25} +4046.98i q^{27} +2985.17i q^{29} +4648.41 q^{31} +2244.58 q^{33} +8767.57i q^{35} -13718.6 q^{37} -5229.03 q^{39} +(-6925.22 - 8240.00i) q^{41} -12551.7 q^{43} +4077.83 q^{45} +9971.71i q^{47} -45106.1 q^{49} -683.663 q^{51} +35423.0i q^{53} -7010.63i q^{55} +10792.4 q^{57} +8638.80 q^{59} +37476.2 q^{61} +28796.0i q^{63} +16332.2i q^{65} -781.059i q^{67} +35117.3i q^{69} -8153.12i q^{71} -40488.1 q^{73} -21247.7i q^{75} +49506.3 q^{77} +23281.6i q^{79} -17533.8 q^{81} +13713.0 q^{83} +2135.33i q^{85} -33677.1 q^{87} -92410.7i q^{89} -115331. q^{91} +52440.9i q^{93} -33708.6i q^{95} +34444.7i q^{97} -23025.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 72 q^{5} - 2022 q^{9} - 5892 q^{21} + 2680 q^{23} + 11406 q^{25} - 2536 q^{31} - 3412 q^{33} + 2672 q^{37} - 29896 q^{39} - 5078 q^{41} + 12352 q^{43} + 13824 q^{45} + 2202 q^{49} + 42176 q^{51}+ \cdots - 288904 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\) \(129\)
\(\chi(n)\) \(1\) \(-1\)

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 0
\(3\) 11.2815i 0.723706i 0.932235 + 0.361853i \(0.117856\pi\)
−0.932235 + 0.361853i \(0.882144\pi\)
\(4\) 0 0
\(5\) 35.2361 0.630322 0.315161 − 0.949038i \(-0.397941\pi\)
0.315161 + 0.949038i \(0.397941\pi\)
\(6\) 0 0
\(7\) 248.823i 1.91931i 0.281172 + 0.959657i \(0.409277\pi\)
−0.281172 + 0.959657i \(0.590723\pi\)
\(8\) 0 0
\(9\) 115.729 0.476250
\(10\) 0 0
\(11\) − 198.962i − 0.495779i −0.968788 − 0.247889i \(-0.920263\pi\)
0.968788 − 0.247889i \(-0.0797369\pi\)
\(12\) 0 0
\(13\) 463.506i 0.760672i 0.924848 + 0.380336i \(0.124192\pi\)
−0.924848 + 0.380336i \(0.875808\pi\)
\(14\) 0 0
\(15\) 397.515i 0.456168i
\(16\) 0 0
\(17\) 60.6006i 0.0508574i 0.999677 + 0.0254287i \(0.00809508\pi\)
−0.999677 + 0.0254287i \(0.991905\pi\)
\(18\) 0 0
\(19\) − 956.649i − 0.607951i −0.952680 − 0.303975i \(-0.901686\pi\)
0.952680 − 0.303975i \(-0.0983141\pi\)
\(20\) 0 0
\(21\) −2807.09 −1.38902
\(22\) 0 0
\(23\) 3112.83 1.22698 0.613488 − 0.789704i \(-0.289766\pi\)
0.613488 + 0.789704i \(0.289766\pi\)
\(24\) 0 0
\(25\) −1883.42 −0.602694
\(26\) 0 0
\(27\) 4046.98i 1.06837i
\(28\) 0 0
\(29\) 2985.17i 0.659135i 0.944132 + 0.329567i \(0.106903\pi\)
−0.944132 + 0.329567i \(0.893097\pi\)
\(30\) 0 0
\(31\) 4648.41 0.868761 0.434381 − 0.900729i \(-0.356967\pi\)
0.434381 + 0.900729i \(0.356967\pi\)
\(32\) 0 0
\(33\) 2244.58 0.358798
\(34\) 0 0
\(35\) 8767.57i 1.20979i
\(36\) 0 0
\(37\) −13718.6 −1.64742 −0.823710 − 0.567011i \(-0.808100\pi\)
−0.823710 + 0.567011i \(0.808100\pi\)
\(38\) 0 0
\(39\) −5229.03 −0.550503
\(40\) 0 0
\(41\) −6925.22 − 8240.00i −0.643389 − 0.765539i
\(42\) 0 0
\(43\) −12551.7 −1.03522 −0.517608 − 0.855618i \(-0.673177\pi\)
−0.517608 + 0.855618i \(0.673177\pi\)
\(44\) 0 0
\(45\) 4077.83 0.300191
\(46\) 0 0
\(47\) 9971.71i 0.658454i 0.944251 + 0.329227i \(0.106788\pi\)
−0.944251 + 0.329227i \(0.893212\pi\)
\(48\) 0 0
\(49\) −45106.1 −2.68377
\(50\) 0 0
\(51\) −683.663 −0.0368058
\(52\) 0 0
\(53\) 35423.0i 1.73219i 0.499881 + 0.866094i \(0.333377\pi\)
−0.499881 + 0.866094i \(0.666623\pi\)
\(54\) 0 0
\(55\) − 7010.63i − 0.312500i
\(56\) 0 0
\(57\) 10792.4 0.439978
\(58\) 0 0
\(59\) 8638.80 0.323090 0.161545 − 0.986865i \(-0.448352\pi\)
0.161545 + 0.986865i \(0.448352\pi\)
\(60\) 0 0
\(61\) 37476.2 1.28953 0.644764 − 0.764382i \(-0.276956\pi\)
0.644764 + 0.764382i \(0.276956\pi\)
\(62\) 0 0
\(63\) 28796.0i 0.914073i
\(64\) 0 0
\(65\) 16332.2i 0.479469i
\(66\) 0 0
\(67\) − 781.059i − 0.0212567i −0.999944 − 0.0106284i \(-0.996617\pi\)
0.999944 − 0.0106284i \(-0.00338318\pi\)
\(68\) 0 0
\(69\) 35117.3i 0.887970i
\(70\) 0 0
\(71\) − 8153.12i − 0.191946i −0.995384 − 0.0959728i \(-0.969404\pi\)
0.995384 − 0.0959728i \(-0.0305962\pi\)
\(72\) 0 0
\(73\) −40488.1 −0.889243 −0.444621 − 0.895719i \(-0.646662\pi\)
−0.444621 + 0.895719i \(0.646662\pi\)
\(74\) 0 0
\(75\) − 21247.7i − 0.436173i
\(76\) 0 0
\(77\) 49506.3 0.951555
\(78\) 0 0
\(79\) 23281.6i 0.419706i 0.977733 + 0.209853i \(0.0672986\pi\)
−0.977733 + 0.209853i \(0.932701\pi\)
\(80\) 0 0
\(81\) −17533.8 −0.296937
\(82\) 0 0
\(83\) 13713.0 0.218494 0.109247 − 0.994015i \(-0.465156\pi\)
0.109247 + 0.994015i \(0.465156\pi\)
\(84\) 0 0
\(85\) 2135.33i 0.0320566i
\(86\) 0 0
\(87\) −33677.1 −0.477020
\(88\) 0 0
\(89\) − 92410.7i − 1.23665i −0.785922 − 0.618326i \(-0.787811\pi\)
0.785922 − 0.618326i \(-0.212189\pi\)
\(90\) 0 0
\(91\) −115331. −1.45997
\(92\) 0 0
\(93\) 52440.9i 0.628728i
\(94\) 0 0
\(95\) − 33708.6i − 0.383205i
\(96\) 0 0
\(97\) 34444.7i 0.371700i 0.982578 + 0.185850i \(0.0595039\pi\)
−0.982578 + 0.185850i \(0.940496\pi\)
\(98\) 0 0
\(99\) − 23025.6i − 0.236114i
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 164.6.b.a.81.13 yes 18
41.40 even 2 inner 164.6.b.a.81.6 ✓ 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
164.6.b.a.81.6 ✓ 18 41.40 even 2 inner
164.6.b.a.81.13 yes 18 1.1 even 1 trivial