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Properties

Label 162.4.c.c.109.1
Level $162$
Weight $4$
Character 162.109
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 109.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 162.109
Dual form 162.4.c.c.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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(3.00000 - 5.19615i) q^{5} +(8.00000 + 13.8564i) q^{7} +8.00000 q^{8} -12.0000 q^{10} +(6.00000 + 10.3923i) q^{11} +(-19.0000 + 32.9090i) q^{13} +(16.0000 - 27.7128i) q^{14} +(-8.00000 - 13.8564i) q^{16} +126.000 q^{17} +20.0000 q^{19} +(12.0000 + 20.7846i) q^{20} +(12.0000 - 20.7846i) q^{22} +(84.0000 - 145.492i) q^{23} +(44.5000 + 77.0763i) q^{25} +76.0000 q^{26} -64.0000 q^{28} +(15.0000 + 25.9808i) q^{29} +(44.0000 - 76.2102i) q^{31} +(-16.0000 + 27.7128i) q^{32} +(-126.000 - 218.238i) q^{34} +96.0000 q^{35} +254.000 q^{37} +(-20.0000 - 34.6410i) q^{38} +(24.0000 - 41.5692i) q^{40} +(21.0000 - 36.3731i) q^{41} +(26.0000 + 45.0333i) q^{43} -48.0000 q^{44} -336.000 q^{46} +(-48.0000 - 83.1384i) q^{47} +(43.5000 - 75.3442i) q^{49} +(89.0000 - 154.153i) q^{50} +(-76.0000 - 131.636i) q^{52} -198.000 q^{53} +72.0000 q^{55} +(64.0000 + 110.851i) q^{56} +(30.0000 - 51.9615i) q^{58} +(-330.000 + 571.577i) q^{59} +(269.000 + 465.922i) q^{61} -176.000 q^{62} +64.0000 q^{64} +(114.000 + 197.454i) q^{65} +(-442.000 + 765.566i) q^{67} +(-252.000 + 436.477i) q^{68} +(-96.0000 - 166.277i) q^{70} -792.000 q^{71} +218.000 q^{73} +(-254.000 - 439.941i) q^{74} +(-40.0000 + 69.2820i) q^{76} +(-96.0000 + 166.277i) q^{77} +(260.000 + 450.333i) q^{79} -96.0000 q^{80} -84.0000 q^{82} +(-246.000 - 426.084i) q^{83} +(378.000 - 654.715i) q^{85} +(52.0000 - 90.0666i) q^{86} +(48.0000 + 83.1384i) q^{88} -810.000 q^{89} -608.000 q^{91} +(336.000 + 581.969i) q^{92} +(-96.0000 + 166.277i) q^{94} +(60.0000 - 103.923i) q^{95} +(-577.000 - 999.393i) q^{97} -174.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 6 q^{5} + 16 q^{7} + 16 q^{8} - 24 q^{10} + 12 q^{11} - 38 q^{13} + 32 q^{14} - 16 q^{16} + 252 q^{17} + 40 q^{19} + 24 q^{20} + 24 q^{22} + 168 q^{23} + 89 q^{25} + 152 q^{26}+ \cdots - 348 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −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 −0.700102 − 0.714043i \(-0.746862\pi\)
0.968430 + 0.249285i \(0.0801955\pi\)
\(6\) 0 0
\(7\) 8.00000 + 13.8564i 0.431959 + 0.748176i 0.997042 − 0.0768587i \(-0.0244890\pi\)
−0.565083 + 0.825034i \(0.691156\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) −12.0000 −0.379473
\(11\) 6.00000 + 10.3923i 0.164461 + 0.284854i 0.936464 − 0.350765i \(-0.114078\pi\)
−0.772003 + 0.635619i \(0.780745\pi\)
\(12\) 0 0
\(13\) −19.0000 + 32.9090i −0.405358 + 0.702100i −0.994363 − 0.106029i \(-0.966186\pi\)
0.589005 + 0.808129i \(0.299520\pi\)
\(14\) 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 −0.180530 − 0.983569i \(-0.557781\pi\)
0.942061 − 0.335441i \(-0.108885\pi\)
\(24\) 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 0.910046 − 0.414507i \(-0.136046\pi\)
−0.813997 + 0.580869i \(0.802713\pi\)
\(30\) 0 0
\(31\) 44.0000 − 76.2102i 0.254924 − 0.441541i −0.709951 − 0.704251i \(-0.751283\pi\)
0.964875 + 0.262710i \(0.0846163\pi\)
\(32\) −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 −0.823255 − 0.567672i \(-0.807844\pi\)
0.903246 + 0.429123i \(0.141177\pi\)
\(42\) 0 0
\(43\) 26.0000 + 45.0333i 0.0922084 + 0.159710i 0.908440 − 0.418015i \(-0.137274\pi\)
−0.816232 + 0.577725i \(0.803941\pi\)
\(44\) −48.0000 −0.164461
\(45\) 0 0
\(46\) −336.000 −1.07697
\(47\) −48.0000 − 83.1384i −0.148969 − 0.258021i 0.781878 − 0.623431i \(-0.214262\pi\)
−0.930846 + 0.365410i \(0.880929\pi\)
\(48\) 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 0.229478 + 0.973314i \(0.426298\pi\)
−0.957654 + 0.287923i \(0.907035\pi\)
\(60\) 0 0
\(61\) 269.000 + 465.922i 0.564622 + 0.977953i 0.997085 + 0.0763018i \(0.0243112\pi\)
−0.432463 + 0.901652i \(0.642355\pi\)
\(62\) −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 0.109692 + 0.993966i \(0.465014\pi\)
−0.915645 + 0.401987i \(0.868320\pi\)
\(68\) −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 0.989609 − 0.143786i \(-0.0459277\pi\)
−0.619327 + 0.785133i \(0.712594\pi\)
\(80\) −96.0000 −0.134164
\(81\) 0 0
\(82\) −84.0000 −0.113125
\(83\) −246.000 − 426.084i −0.325325 − 0.563480i 0.656253 − 0.754541i \(-0.272141\pi\)
−0.981578 + 0.191061i \(0.938807\pi\)
\(84\) 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 −0.992213 − 0.124555i \(-0.960250\pi\)
0.388239 − 0.921559i \(-0.373084\pi\)
\(98\) −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.109.1 2
3.2 odd 2 162.4.c.f.109.1 2
9.2 odd 6 162.4.c.f.55.1 2
9.4 even 3 18.4.a.a.1.1 1
9.5 odd 6 6.4.a.a.1.1 ✓ 1
9.7 even 3 inner 162.4.c.c.55.1 2
36.23 even 6 48.4.a.c.1.1 1
36.31 odd 6 144.4.a.c.1.1 1
45.4 even 6 450.4.a.h.1.1 1
45.13 odd 12 450.4.c.e.199.1 2
45.14 odd 6 150.4.a.i.1.1 1
45.22 odd 12 450.4.c.e.199.2 2
45.23 even 12 150.4.c.d.49.2 2
45.32 even 12 150.4.c.d.49.1 2
63.4 even 3 882.4.g.i.667.1 2
63.5 even 6 294.4.e.g.67.1 2
63.13 odd 6 882.4.a.n.1.1 1
63.23 odd 6 294.4.e.h.67.1 2
63.31 odd 6 882.4.g.f.667.1 2
63.32 odd 6 294.4.e.h.79.1 2
63.40 odd 6 882.4.g.f.361.1 2
63.41 even 6 294.4.a.e.1.1 1
63.58 even 3 882.4.g.i.361.1 2
63.59 even 6 294.4.e.g.79.1 2
72.5 odd 6 192.4.a.i.1.1 1
72.13 even 6 576.4.a.q.1.1 1
72.59 even 6 192.4.a.c.1.1 1
72.67 odd 6 576.4.a.r.1.1 1
99.32 even 6 726.4.a.f.1.1 1
99.76 odd 6 2178.4.a.e.1.1 1
117.5 even 12 1014.4.b.d.337.2 2
117.77 odd 6 1014.4.a.g.1.1 1
117.86 even 12 1014.4.b.d.337.1 2
144.5 odd 12 768.4.d.n.385.2 2
144.59 even 12 768.4.d.c.385.1 2
144.77 odd 12 768.4.d.n.385.1 2
144.131 even 12 768.4.d.c.385.2 2
153.50 odd 6 1734.4.a.d.1.1 1
171.113 even 6 2166.4.a.i.1.1 1
180.23 odd 12 1200.4.f.j.49.2 2
180.59 even 6 1200.4.a.b.1.1 1
180.167 odd 12 1200.4.f.j.49.1 2
252.167 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 9.5 odd 6
18.4.a.a.1.1 1 9.4 even 3
48.4.a.c.1.1 1 36.23 even 6
144.4.a.c.1.1 1 36.31 odd 6
150.4.a.i.1.1 1 45.14 odd 6
150.4.c.d.49.1 2 45.32 even 12
150.4.c.d.49.2 2 45.23 even 12
162.4.c.c.55.1 2 9.7 even 3 inner
162.4.c.c.109.1 2 1.1 even 1 trivial
162.4.c.f.55.1 2 9.2 odd 6
162.4.c.f.109.1 2 3.2 odd 2
192.4.a.c.1.1 1 72.59 even 6
192.4.a.i.1.1 1 72.5 odd 6
294.4.a.e.1.1 1 63.41 even 6
294.4.e.g.67.1 2 63.5 even 6
294.4.e.g.79.1 2 63.59 even 6
294.4.e.h.67.1 2 63.23 odd 6
294.4.e.h.79.1 2 63.32 odd 6
450.4.a.h.1.1 1 45.4 even 6
450.4.c.e.199.1 2 45.13 odd 12
450.4.c.e.199.2 2 45.22 odd 12
576.4.a.q.1.1 1 72.13 even 6
576.4.a.r.1.1 1 72.67 odd 6
726.4.a.f.1.1 1 99.32 even 6
768.4.d.c.385.1 2 144.59 even 12
768.4.d.c.385.2 2 144.131 even 12
768.4.d.n.385.1 2 144.77 odd 12
768.4.d.n.385.2 2 144.5 odd 12
882.4.a.n.1.1 1 63.13 odd 6
882.4.g.f.361.1 2 63.40 odd 6
882.4.g.f.667.1 2 63.31 odd 6
882.4.g.i.361.1 2 63.58 even 3
882.4.g.i.667.1 2 63.4 even 3
1014.4.a.g.1.1 1 117.77 odd 6
1014.4.b.d.337.1 2 117.86 even 12
1014.4.b.d.337.2 2 117.5 even 12
1200.4.a.b.1.1 1 180.59 even 6
1200.4.f.j.49.1 2 180.167 odd 12
1200.4.f.j.49.2 2 180.23 odd 12
1734.4.a.d.1.1 1 153.50 odd 6
2166.4.a.i.1.1 1 171.113 even 6
2178.4.a.e.1.1 1 99.76 odd 6
2352.4.a.e.1.1 1 252.167 odd 6