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Properties

Label 162.8.c.h.109.1
Level $162$
Weight $8$
Character 162.109
Analytic conductor $50.606$
Analytic rank $0$
Dimension $2$
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] = [162,8,Mod(55,162)] 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("162.55"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,8,0,-64,-165] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.6063741284\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 162.109
Dual form 162.8.c.h.55.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.00000 + 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(-82.5000 + 142.894i) q^{5} +(254.000 + 439.941i) q^{7} -512.000 q^{8} -1320.00 q^{10} +(1512.00 + 2618.86i) q^{11} +(-2519.50 + 4363.90i) q^{13} +(-2032.00 + 3519.53i) q^{14} +(-2048.00 - 3547.24i) q^{16} +3189.00 q^{17} +1508.00 q^{19} +(-5280.00 - 9145.23i) q^{20} +(-12096.0 + 20950.9i) q^{22} +(-37800.0 + 65471.5i) q^{23} +(25450.0 + 44080.7i) q^{25} -40312.0 q^{26} -32512.0 q^{28} +(-41332.5 - 71590.0i) q^{29} +(87446.0 - 151461. i) q^{31} +(16384.0 - 28377.9i) q^{32} +(12756.0 + 22094.0i) q^{34} -83820.0 q^{35} -323569. q^{37} +(6032.00 + 10447.7i) q^{38} +(42240.0 - 73161.8i) q^{40} +(-154059. + 266838. i) q^{41} +(-168340. - 291573. i) q^{43} -193536. q^{44} -604800. q^{46} +(-191598. - 331857. i) q^{47} +(282740. - 489719. i) q^{49} +(-203600. + 352646. i) q^{50} +(-161248. - 279290. i) q^{52} -760206. q^{53} -498960. q^{55} +(-130048. - 225250. i) q^{56} +(330660. - 572720. i) q^{58} +(-1.11283e6 + 1.92748e6i) q^{59} +(-1.12241e6 - 1.94407e6i) q^{61} +1.39914e6 q^{62} +262144. q^{64} +(-415718. - 720044. i) q^{65} +(-736594. + 1.27582e6i) q^{67} +(-102048. + 176752. i) q^{68} +(-335280. - 580722. i) q^{70} +5.00689e6 q^{71} -5.89830e6 q^{73} +(-1.29428e6 - 2.24175e6i) q^{74} +(-48256.0 + 83581.8i) q^{76} +(-768096. + 1.33038e6i) q^{77} +(-3.51438e6 - 6.08709e6i) q^{79} +675840. q^{80} -2.46494e6 q^{82} +(-1.32560e6 - 2.29600e6i) q^{83} +(-263092. + 455690. i) q^{85} +(1.34672e6 - 2.33259e6i) q^{86} +(-774144. - 1.34086e6i) q^{88} +6.77090e6 q^{89} -2.55981e6 q^{91} +(-2.41920e6 - 4.19018e6i) q^{92} +(1.53278e6 - 2.65486e6i) q^{94} +(-124410. + 215484. i) q^{95} +(-8.08819e6 - 1.40092e7i) q^{97} +4.52383e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 64 q^{4} - 165 q^{5} + 508 q^{7} - 1024 q^{8} - 2640 q^{10} + 3024 q^{11} - 5039 q^{13} - 4064 q^{14} - 4096 q^{16} + 6378 q^{17} + 3016 q^{19} - 10560 q^{20} - 24192 q^{22} - 75600 q^{23}+ \cdots + 9047664 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 + 6.92820i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −32.0000 + 55.4256i −0.250000 + 0.433013i
\(5\) −82.5000 + 142.894i −0.295161 + 0.511234i −0.975022 − 0.222107i \(-0.928707\pi\)
0.679861 + 0.733341i \(0.262040\pi\)
\(6\) 0 0
\(7\) 254.000 + 439.941i 0.279892 + 0.484787i 0.971358 − 0.237622i \(-0.0763680\pi\)
−0.691466 + 0.722409i \(0.743035\pi\)
\(8\) −512.000 −0.353553
\(9\) 0 0
\(10\) −1320.00 −0.417421
\(11\) 1512.00 + 2618.86i 0.342513 + 0.593250i 0.984899 − 0.173131i \(-0.0553885\pi\)
−0.642385 + 0.766382i \(0.722055\pi\)
\(12\) 0 0
\(13\) −2519.50 + 4363.90i −0.318063 + 0.550901i −0.980084 − 0.198585i \(-0.936366\pi\)
0.662021 + 0.749485i \(0.269699\pi\)
\(14\) −2032.00 + 3519.53i −0.197914 + 0.342796i
\(15\) 0 0
\(16\) −2048.00 − 3547.24i −0.125000 − 0.216506i
\(17\) 3189.00 0.157428 0.0787142 − 0.996897i \(-0.474919\pi\)
0.0787142 + 0.996897i \(0.474919\pi\)
\(18\) 0 0
\(19\) 1508.00 0.0504387 0.0252193 − 0.999682i \(-0.491972\pi\)
0.0252193 + 0.999682i \(0.491972\pi\)
\(20\) −5280.00 − 9145.23i −0.147580 − 0.255617i
\(21\) 0 0
\(22\) −12096.0 + 20950.9i −0.242193 + 0.419491i
\(23\) −37800.0 + 65471.5i −0.647805 + 1.12203i 0.335841 + 0.941919i \(0.390980\pi\)
−0.983646 + 0.180113i \(0.942354\pi\)
\(24\) 0 0
\(25\) 25450.0 + 44080.7i 0.325760 + 0.564233i
\(26\) −40312.0 −0.449808
\(27\) 0 0
\(28\) −32512.0 −0.279892
\(29\) −41332.5 − 71590.0i −0.314701 − 0.545079i 0.664673 − 0.747135i \(-0.268571\pi\)
−0.979374 + 0.202056i \(0.935238\pi\)
\(30\) 0 0
\(31\) 87446.0 − 151461.i 0.527198 − 0.913134i −0.472299 − 0.881438i \(-0.656576\pi\)
0.999498 − 0.0316960i \(-0.0100908\pi\)
\(32\) 16384.0 − 28377.9i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 12756.0 + 22094.0i 0.0556594 + 0.0964049i
\(35\) −83820.0 −0.330453
\(36\) 0 0
\(37\) −323569. −1.05017 −0.525087 − 0.851049i \(-0.675967\pi\)
−0.525087 + 0.851049i \(0.675967\pi\)
\(38\) 6032.00 + 10447.7i 0.0178328 + 0.0308873i
\(39\) 0 0
\(40\) 42240.0 − 73161.8i 0.104355 − 0.180748i
\(41\) −154059. + 266838.i −0.349095 + 0.604650i −0.986089 − 0.166219i \(-0.946844\pi\)
0.636994 + 0.770869i \(0.280178\pi\)
\(42\) 0 0
\(43\) −168340. − 291573.i −0.322885 − 0.559253i 0.658197 − 0.752846i \(-0.271319\pi\)
−0.981082 + 0.193593i \(0.937986\pi\)
\(44\) −193536. −0.342513
\(45\) 0 0
\(46\) −604800. −0.916135
\(47\) −191598. − 331857.i −0.269184 − 0.466240i 0.699468 − 0.714664i \(-0.253420\pi\)
−0.968651 + 0.248425i \(0.920087\pi\)
\(48\) 0 0
\(49\) 282740. − 489719.i 0.343321 − 0.594649i
\(50\) −203600. + 352646.i −0.230347 + 0.398973i
\(51\) 0 0
\(52\) −161248. − 279290.i −0.159031 − 0.275450i
\(53\) −760206. −0.701400 −0.350700 − 0.936488i \(-0.614056\pi\)
−0.350700 + 0.936488i \(0.614056\pi\)
\(54\) 0 0
\(55\) −498960. −0.404386
\(56\) −130048. − 225250.i −0.0989568 − 0.171398i
\(57\) 0 0
\(58\) 330660. − 572720.i 0.222527 − 0.385429i
\(59\) −1.11283e6 + 1.92748e6i −0.705420 + 1.22182i 0.261120 + 0.965306i \(0.415908\pi\)
−0.966540 + 0.256516i \(0.917425\pi\)
\(60\) 0 0
\(61\) −1.12241e6 − 1.94407e6i −0.633135 − 1.09662i −0.986907 − 0.161290i \(-0.948435\pi\)
0.353772 − 0.935332i \(-0.384899\pi\)
\(62\) 1.39914e6 0.745571
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −415718. − 720044.i −0.187759 − 0.325209i
\(66\) 0 0
\(67\) −736594. + 1.27582e6i −0.299203 + 0.518235i −0.975954 − 0.217977i \(-0.930054\pi\)
0.676751 + 0.736212i \(0.263388\pi\)
\(68\) −102048. + 176752.i −0.0393571 + 0.0681685i
\(69\) 0 0
\(70\) −335280. − 580722.i −0.116833 − 0.202360i
\(71\) 5.00689e6 1.66021 0.830107 − 0.557604i \(-0.188279\pi\)
0.830107 + 0.557604i \(0.188279\pi\)
\(72\) 0 0
\(73\) −5.89830e6 −1.77459 −0.887293 − 0.461207i \(-0.847417\pi\)
−0.887293 + 0.461207i \(0.847417\pi\)
\(74\) −1.29428e6 − 2.24175e6i −0.371292 − 0.643097i
\(75\) 0 0
\(76\) −48256.0 + 83581.8i −0.0126097 + 0.0218406i
\(77\) −768096. + 1.33038e6i −0.191733 + 0.332092i
\(78\) 0 0
\(79\) −3.51438e6 − 6.08709e6i −0.801963 − 1.38904i −0.918322 − 0.395834i \(-0.870456\pi\)
0.116359 − 0.993207i \(-0.462878\pi\)
\(80\) 675840. 0.147580
\(81\) 0 0
\(82\) −2.46494e6 −0.493695
\(83\) −1.32560e6 − 2.29600e6i −0.254471 − 0.440757i 0.710281 − 0.703919i \(-0.248568\pi\)
−0.964752 + 0.263162i \(0.915235\pi\)
\(84\) 0 0
\(85\) −263092. + 455690.i −0.0464667 + 0.0804828i
\(86\) 1.34672e6 − 2.33259e6i 0.228314 − 0.395452i
\(87\) 0 0
\(88\) −774144. − 1.34086e6i −0.121097 − 0.209746i
\(89\) 6.77090e6 1.01808 0.509039 − 0.860743i \(-0.330001\pi\)
0.509039 + 0.860743i \(0.330001\pi\)
\(90\) 0 0
\(91\) −2.55981e6 −0.356093
\(92\) −2.41920e6 − 4.19018e6i −0.323903 − 0.561016i
\(93\) 0 0
\(94\) 1.53278e6 − 2.65486e6i 0.190341 − 0.329681i
\(95\) −124410. + 215484.i −0.0148875 + 0.0257860i
\(96\) 0 0
\(97\) −8.08819e6 − 1.40092e7i −0.899809 − 1.55852i −0.827737 − 0.561116i \(-0.810372\pi\)
−0.0720719 − 0.997399i \(-0.522961\pi\)
\(98\) 4.52383e6 0.485529
\(99\) 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 162.8.c.h.109.1 2
3.2 odd 2 162.8.c.e.109.1 2
9.2 odd 6 162.8.c.e.55.1 2
9.4 even 3 162.8.a.a.1.1 ✓ 1
9.5 odd 6 162.8.a.b.1.1 yes 1
9.7 even 3 inner 162.8.c.h.55.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.8.a.a.1.1 ✓ 1 9.4 even 3
162.8.a.b.1.1 yes 1 9.5 odd 6
162.8.c.e.55.1 2 9.2 odd 6
162.8.c.e.109.1 2 3.2 odd 2
162.8.c.h.55.1 2 9.7 even 3 inner
162.8.c.h.109.1 2 1.1 even 1 trivial