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Properties

Label 162.4.c.c.55.1
Level $162$
Weight $4$
Character 162.55
Analytic conductor $9.558$
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] = [162,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,-4,6] 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: \(9.55830942093\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 55.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 162.55
Dual form 162.4.c.c.109.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(3.00000 + 5.19615i) q^{5} +(8.00000 - 13.8564i) q^{7} +8.00000 q^{8} -12.0000 q^{10} +(6.00000 - 10.3923i) q^{11} +(-19.0000 - 32.9090i) q^{13} +(16.0000 + 27.7128i) q^{14} +(-8.00000 + 13.8564i) q^{16} +126.000 q^{17} +20.0000 q^{19} +(12.0000 - 20.7846i) q^{20} +(12.0000 + 20.7846i) q^{22} +(84.0000 + 145.492i) q^{23} +(44.5000 - 77.0763i) q^{25} +76.0000 q^{26} -64.0000 q^{28} +(15.0000 - 25.9808i) q^{29} +(44.0000 + 76.2102i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(-126.000 + 218.238i) q^{34} +96.0000 q^{35} +254.000 q^{37} +(-20.0000 + 34.6410i) q^{38} +(24.0000 + 41.5692i) q^{40} +(21.0000 + 36.3731i) q^{41} +(26.0000 - 45.0333i) q^{43} -48.0000 q^{44} -336.000 q^{46} +(-48.0000 + 83.1384i) q^{47} +(43.5000 + 75.3442i) q^{49} +(89.0000 + 154.153i) q^{50} +(-76.0000 + 131.636i) q^{52} -198.000 q^{53} +72.0000 q^{55} +(64.0000 - 110.851i) q^{56} +(30.0000 + 51.9615i) q^{58} +(-330.000 - 571.577i) q^{59} +(269.000 - 465.922i) q^{61} -176.000 q^{62} +64.0000 q^{64} +(114.000 - 197.454i) q^{65} +(-442.000 - 765.566i) q^{67} +(-252.000 - 436.477i) q^{68} +(-96.0000 + 166.277i) q^{70} -792.000 q^{71} +218.000 q^{73} +(-254.000 + 439.941i) q^{74} +(-40.0000 - 69.2820i) q^{76} +(-96.0000 - 166.277i) q^{77} +(260.000 - 450.333i) q^{79} -96.0000 q^{80} -84.0000 q^{82} +(-246.000 + 426.084i) q^{83} +(378.000 + 654.715i) q^{85} +(52.0000 + 90.0666i) q^{86} +(48.0000 - 83.1384i) q^{88} -810.000 q^{89} -608.000 q^{91} +(336.000 - 581.969i) q^{92} +(-96.0000 - 166.277i) q^{94} +(60.0000 + 103.923i) q^{95} +(-577.000 + 999.393i) q^{97} -174.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 6 q^{5} + 16 q^{7} + 16 q^{8} - 24 q^{10} + 12 q^{11} - 38 q^{13} + 32 q^{14} - 16 q^{16} + 252 q^{17} + 40 q^{19} + 24 q^{20} + 24 q^{22} + 168 q^{23} + 89 q^{25} + 152 q^{26}+ \cdots - 348 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.73205i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −2.00000 − 3.46410i −0.250000 − 0.433013i
\(5\) 3.00000 + 5.19615i 0.268328 + 0.464758i 0.968430 − 0.249285i \(-0.0801955\pi\)
−0.700102 + 0.714043i \(0.746862\pi\)
\(6\) 0 0
\(7\) 8.00000 − 13.8564i 0.431959 − 0.748176i −0.565083 − 0.825034i \(-0.691156\pi\)
0.997042 + 0.0768587i \(0.0244890\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −12.0000 −0.379473
\(11\) 6.00000 − 10.3923i 0.164461 − 0.284854i −0.772003 − 0.635619i \(-0.780745\pi\)
0.936464 + 0.350765i \(0.114078\pi\)
\(12\) 0 0
\(13\) −19.0000 − 32.9090i −0.405358 − 0.702100i 0.589005 − 0.808129i \(-0.299520\pi\)
−0.994363 + 0.106029i \(0.966186\pi\)
\(14\) 16.0000 + 27.7128i 0.305441 + 0.529040i
\(15\) 0 0
\(16\) −8.00000 + 13.8564i −0.125000 + 0.216506i
\(17\) 126.000 1.79762 0.898808 − 0.438342i \(-0.144434\pi\)
0.898808 + 0.438342i \(0.144434\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 − 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 12.0000 − 20.7846i 0.134164 − 0.232379i
\(21\) 0 0
\(22\) 12.0000 + 20.7846i 0.116291 + 0.201422i
\(23\) 84.0000 + 145.492i 0.761531 + 1.31901i 0.942061 + 0.335441i \(0.108885\pi\)
−0.180530 + 0.983569i \(0.557781\pi\)
\(24\) 0 0
\(25\) 44.5000 − 77.0763i 0.356000 − 0.616610i
\(26\) 76.0000 0.573263
\(27\) 0 0
\(28\) −64.0000 −0.431959
\(29\) 15.0000 − 25.9808i 0.0960493 − 0.166362i −0.813997 − 0.580869i \(-0.802713\pi\)
0.910046 + 0.414507i \(0.136046\pi\)
\(30\) 0 0
\(31\) 44.0000 + 76.2102i 0.254924 + 0.441541i 0.964875 − 0.262710i \(-0.0846163\pi\)
−0.709951 + 0.704251i \(0.751283\pi\)
\(32\) −16.0000 − 27.7128i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −126.000 + 218.238i −0.635554 + 1.10081i
\(35\) 96.0000 0.463627
\(36\) 0 0
\(37\) 254.000 1.12858 0.564288 − 0.825578i \(-0.309151\pi\)
0.564288 + 0.825578i \(0.309151\pi\)
\(38\) −20.0000 + 34.6410i −0.0853797 + 0.147882i
\(39\) 0 0
\(40\) 24.0000 + 41.5692i 0.0948683 + 0.164317i
\(41\) 21.0000 + 36.3731i 0.0799914 + 0.138549i 0.903246 − 0.429123i \(-0.141177\pi\)
−0.823255 + 0.567672i \(0.807844\pi\)
\(42\) 0 0
\(43\) 26.0000 − 45.0333i 0.0922084 − 0.159710i −0.816232 − 0.577725i \(-0.803941\pi\)
0.908440 + 0.418015i \(0.137274\pi\)
\(44\) −48.0000 −0.164461
\(45\) 0 0
\(46\) −336.000 −1.07697
\(47\) −48.0000 + 83.1384i −0.148969 + 0.258021i −0.930846 − 0.365410i \(-0.880929\pi\)
0.781878 + 0.623431i \(0.214262\pi\)
\(48\) 0 0
\(49\) 43.5000 + 75.3442i 0.126822 + 0.219662i
\(50\) 89.0000 + 154.153i 0.251730 + 0.436009i
\(51\) 0 0
\(52\) −76.0000 + 131.636i −0.202679 + 0.351050i
\(53\) −198.000 −0.513158 −0.256579 − 0.966523i \(-0.582595\pi\)
−0.256579 + 0.966523i \(0.582595\pi\)
\(54\) 0 0
\(55\) 72.0000 0.176518
\(56\) 64.0000 − 110.851i 0.152721 − 0.264520i
\(57\) 0 0
\(58\) 30.0000 + 51.9615i 0.0679171 + 0.117636i
\(59\) −330.000 − 571.577i −0.728175 − 1.26124i −0.957654 − 0.287923i \(-0.907035\pi\)
0.229478 − 0.973314i \(-0.426298\pi\)
\(60\) 0 0
\(61\) 269.000 − 465.922i 0.564622 − 0.977953i −0.432463 − 0.901652i \(-0.642355\pi\)
0.997085 − 0.0763018i \(-0.0243112\pi\)
\(62\) −176.000 −0.360516
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 114.000 − 197.454i 0.217538 − 0.376787i
\(66\) 0 0
\(67\) −442.000 − 765.566i −0.805954 − 1.39595i −0.915645 − 0.401987i \(-0.868320\pi\)
0.109692 − 0.993966i \(-0.465014\pi\)
\(68\) −252.000 − 436.477i −0.449404 − 0.778391i
\(69\) 0 0
\(70\) −96.0000 + 166.277i −0.163917 + 0.283913i
\(71\) −792.000 −1.32385 −0.661923 − 0.749572i \(-0.730260\pi\)
−0.661923 + 0.749572i \(0.730260\pi\)
\(72\) 0 0
\(73\) 218.000 0.349520 0.174760 − 0.984611i \(-0.444085\pi\)
0.174760 + 0.984611i \(0.444085\pi\)
\(74\) −254.000 + 439.941i −0.399012 + 0.691109i
\(75\) 0 0
\(76\) −40.0000 − 69.2820i −0.0603726 − 0.104568i
\(77\) −96.0000 − 166.277i −0.142081 − 0.246091i
\(78\) 0 0
\(79\) 260.000 − 450.333i 0.370282 − 0.641347i −0.619327 − 0.785133i \(-0.712594\pi\)
0.989609 + 0.143786i \(0.0459277\pi\)
\(80\) −96.0000 −0.134164
\(81\) 0 0
\(82\) −84.0000 −0.113125
\(83\) −246.000 + 426.084i −0.325325 + 0.563480i −0.981578 − 0.191061i \(-0.938807\pi\)
0.656253 + 0.754541i \(0.272141\pi\)
\(84\) 0 0
\(85\) 378.000 + 654.715i 0.482351 + 0.835457i
\(86\) 52.0000 + 90.0666i 0.0652012 + 0.112932i
\(87\) 0 0
\(88\) 48.0000 − 83.1384i 0.0581456 − 0.100711i
\(89\) −810.000 −0.964717 −0.482359 − 0.875974i \(-0.660220\pi\)
−0.482359 + 0.875974i \(0.660220\pi\)
\(90\) 0 0
\(91\) −608.000 −0.700393
\(92\) 336.000 − 581.969i 0.380765 − 0.659505i
\(93\) 0 0
\(94\) −96.0000 − 166.277i −0.105337 − 0.182448i
\(95\) 60.0000 + 103.923i 0.0647986 + 0.112235i
\(96\) 0 0
\(97\) −577.000 + 999.393i −0.603974 + 1.04611i 0.388239 + 0.921559i \(0.373084\pi\)
−0.992213 + 0.124555i \(0.960250\pi\)
\(98\) −174.000 −0.179354
\(99\) 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 162.4.c.c.55.1 2
3.2 odd 2 162.4.c.f.55.1 2
9.2 odd 6 6.4.a.a.1.1 ✓ 1
9.4 even 3 inner 162.4.c.c.109.1 2
9.5 odd 6 162.4.c.f.109.1 2
9.7 even 3 18.4.a.a.1.1 1
36.7 odd 6 144.4.a.c.1.1 1
36.11 even 6 48.4.a.c.1.1 1
45.2 even 12 150.4.c.d.49.1 2
45.7 odd 12 450.4.c.e.199.2 2
45.29 odd 6 150.4.a.i.1.1 1
45.34 even 6 450.4.a.h.1.1 1
45.38 even 12 150.4.c.d.49.2 2
45.43 odd 12 450.4.c.e.199.1 2
63.2 odd 6 294.4.e.h.67.1 2
63.11 odd 6 294.4.e.h.79.1 2
63.16 even 3 882.4.g.i.361.1 2
63.20 even 6 294.4.a.e.1.1 1
63.25 even 3 882.4.g.i.667.1 2
63.34 odd 6 882.4.a.n.1.1 1
63.38 even 6 294.4.e.g.79.1 2
63.47 even 6 294.4.e.g.67.1 2
63.52 odd 6 882.4.g.f.667.1 2
63.61 odd 6 882.4.g.f.361.1 2
72.11 even 6 192.4.a.c.1.1 1
72.29 odd 6 192.4.a.i.1.1 1
72.43 odd 6 576.4.a.r.1.1 1
72.61 even 6 576.4.a.q.1.1 1
99.43 odd 6 2178.4.a.e.1.1 1
99.65 even 6 726.4.a.f.1.1 1
117.38 odd 6 1014.4.a.g.1.1 1
117.47 even 12 1014.4.b.d.337.1 2
117.83 even 12 1014.4.b.d.337.2 2
144.11 even 12 768.4.d.c.385.1 2
144.29 odd 12 768.4.d.n.385.1 2
144.83 even 12 768.4.d.c.385.2 2
144.101 odd 12 768.4.d.n.385.2 2
153.101 odd 6 1734.4.a.d.1.1 1
171.56 even 6 2166.4.a.i.1.1 1
180.47 odd 12 1200.4.f.j.49.1 2
180.83 odd 12 1200.4.f.j.49.2 2
180.119 even 6 1200.4.a.b.1.1 1
252.83 odd 6 2352.4.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.4.a.a.1.1 ✓ 1 9.2 odd 6
18.4.a.a.1.1 1 9.7 even 3
48.4.a.c.1.1 1 36.11 even 6
144.4.a.c.1.1 1 36.7 odd 6
150.4.a.i.1.1 1 45.29 odd 6
150.4.c.d.49.1 2 45.2 even 12
150.4.c.d.49.2 2 45.38 even 12
162.4.c.c.55.1 2 1.1 even 1 trivial
162.4.c.c.109.1 2 9.4 even 3 inner
162.4.c.f.55.1 2 3.2 odd 2
162.4.c.f.109.1 2 9.5 odd 6
192.4.a.c.1.1 1 72.11 even 6
192.4.a.i.1.1 1 72.29 odd 6
294.4.a.e.1.1 1 63.20 even 6
294.4.e.g.67.1 2 63.47 even 6
294.4.e.g.79.1 2 63.38 even 6
294.4.e.h.67.1 2 63.2 odd 6
294.4.e.h.79.1 2 63.11 odd 6
450.4.a.h.1.1 1 45.34 even 6
450.4.c.e.199.1 2 45.43 odd 12
450.4.c.e.199.2 2 45.7 odd 12
576.4.a.q.1.1 1 72.61 even 6
576.4.a.r.1.1 1 72.43 odd 6
726.4.a.f.1.1 1 99.65 even 6
768.4.d.c.385.1 2 144.11 even 12
768.4.d.c.385.2 2 144.83 even 12
768.4.d.n.385.1 2 144.29 odd 12
768.4.d.n.385.2 2 144.101 odd 12
882.4.a.n.1.1 1 63.34 odd 6
882.4.g.f.361.1 2 63.61 odd 6
882.4.g.f.667.1 2 63.52 odd 6
882.4.g.i.361.1 2 63.16 even 3
882.4.g.i.667.1 2 63.25 even 3
1014.4.a.g.1.1 1 117.38 odd 6
1014.4.b.d.337.1 2 117.47 even 12
1014.4.b.d.337.2 2 117.83 even 12
1200.4.a.b.1.1 1 180.119 even 6
1200.4.f.j.49.1 2 180.47 odd 12
1200.4.f.j.49.2 2 180.83 odd 12
1734.4.a.d.1.1 1 153.101 odd 6
2166.4.a.i.1.1 1 171.56 even 6
2178.4.a.e.1.1 1 99.43 odd 6
2352.4.a.e.1.1 1 252.83 odd 6