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Properties

Label 169.2.c.a.22.2
Level $169$
Weight $2$
Character 169.22
Analytic conductor $1.349$
Analytic rank $0$
Dimension $4$
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] = [169,2,Mod(22,169)] 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("169.22"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 22.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 169.22
Dual form 169.2.c.a.146.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 1.50000i) q^{2} +(-1.00000 + 1.73205i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.73205 q^{5} +(1.73205 + 3.00000i) q^{6} +1.73205 q^{8} +(-0.500000 - 0.866025i) q^{9} +(1.50000 - 2.59808i) q^{10} +2.00000 q^{12} +(-1.73205 + 3.00000i) q^{15} +(2.50000 - 4.33013i) q^{16} +(-1.50000 - 2.59808i) q^{17} -1.73205 q^{18} +(-1.73205 - 3.00000i) q^{19} +(-0.866025 - 1.50000i) q^{20} +(-3.00000 + 5.19615i) q^{23} +(-1.73205 + 3.00000i) q^{24} -2.00000 q^{25} -4.00000 q^{27} +(-1.50000 + 2.59808i) q^{29} +(3.00000 + 5.19615i) q^{30} -3.46410 q^{31} +(-2.59808 - 4.50000i) q^{32} -5.19615 q^{34} +(-0.500000 + 0.866025i) q^{36} +(4.33013 - 7.50000i) q^{37} -6.00000 q^{38} +3.00000 q^{40} +(2.59808 - 4.50000i) q^{41} +(4.00000 + 6.92820i) q^{43} +(-0.866025 - 1.50000i) q^{45} +(5.19615 + 9.00000i) q^{46} -3.46410 q^{47} +(5.00000 + 8.66025i) q^{48} +(3.50000 - 6.06218i) q^{49} +(-1.73205 + 3.00000i) q^{50} +6.00000 q^{51} -3.00000 q^{53} +(-3.46410 + 6.00000i) q^{54} +6.92820 q^{57} +(2.59808 + 4.50000i) q^{58} +(-3.46410 - 6.00000i) q^{59} +3.46410 q^{60} +(-0.500000 - 0.866025i) q^{61} +(-3.00000 + 5.19615i) q^{62} +1.00000 q^{64} +(-1.73205 + 3.00000i) q^{67} +(-1.50000 + 2.59808i) q^{68} +(-6.00000 - 10.3923i) q^{69} +(1.73205 + 3.00000i) q^{71} +(-0.866025 - 1.50000i) q^{72} +1.73205 q^{73} +(-7.50000 - 12.9904i) q^{74} +(2.00000 - 3.46410i) q^{75} +(-1.73205 + 3.00000i) q^{76} +4.00000 q^{79} +(4.33013 - 7.50000i) q^{80} +(5.50000 - 9.52628i) q^{81} +(-4.50000 - 7.79423i) q^{82} -13.8564 q^{83} +(-2.59808 - 4.50000i) q^{85} +13.8564 q^{86} +(-3.00000 - 5.19615i) q^{87} +(-3.46410 + 6.00000i) q^{89} -3.00000 q^{90} +6.00000 q^{92} +(3.46410 - 6.00000i) q^{93} +(-3.00000 + 5.19615i) q^{94} +(-3.00000 - 5.19615i) q^{95} +10.3923 q^{96} +(3.46410 + 6.00000i) q^{97} +(-6.06218 - 10.5000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 2 q^{4} - 2 q^{9} + 6 q^{10} + 8 q^{12} + 10 q^{16} - 6 q^{17} - 12 q^{23} - 8 q^{25} - 16 q^{27} - 6 q^{29} + 12 q^{30} - 2 q^{36} - 24 q^{38} + 12 q^{40} + 16 q^{43} + 20 q^{48} + 14 q^{49}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 1.50000i 0.612372 − 1.06066i −0.378467 − 0.925615i \(-0.623549\pi\)
0.990839 − 0.135045i \(-0.0431180\pi\)
\(3\) −1.00000 + 1.73205i −0.577350 + 1.00000i 0.418432 + 0.908248i \(0.362580\pi\)
−0.995782 + 0.0917517i \(0.970753\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 1.73205 0.774597 0.387298 − 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) 1.73205 + 3.00000i 0.707107 + 1.22474i
\(7\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 1.73205 0.612372
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 1.50000 − 2.59808i 0.474342 − 0.821584i
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) −1.73205 + 3.00000i −0.447214 + 0.774597i
\(16\) 2.50000 − 4.33013i 0.625000 − 1.08253i
\(17\) −1.50000 − 2.59808i −0.363803 − 0.630126i 0.624780 − 0.780801i \(-0.285189\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(18\) −1.73205 −0.408248
\(19\) −1.73205 − 3.00000i −0.397360 − 0.688247i 0.596040 − 0.802955i \(-0.296740\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) −0.866025 − 1.50000i −0.193649 − 0.335410i
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) −1.73205 + 3.00000i −0.353553 + 0.612372i
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −1.50000 + 2.59808i −0.278543 + 0.482451i −0.971023 − 0.238987i \(-0.923185\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(30\) 3.00000 + 5.19615i 0.547723 + 0.948683i
\(31\) −3.46410 −0.622171 −0.311086 − 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) −2.59808 − 4.50000i −0.459279 − 0.795495i
\(33\) 0 0
\(34\) −5.19615 −0.891133
\(35\) 0 0
\(36\) −0.500000 + 0.866025i −0.0833333 + 0.144338i
\(37\) 4.33013 − 7.50000i 0.711868 − 1.23299i −0.252286 − 0.967653i \(-0.581183\pi\)
0.964155 − 0.265340i \(-0.0854841\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) 2.59808 − 4.50000i 0.405751 − 0.702782i −0.588657 − 0.808383i \(-0.700343\pi\)
0.994409 + 0.105601i \(0.0336766\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) −0.866025 − 1.50000i −0.129099 − 0.223607i
\(46\) 5.19615 + 9.00000i 0.766131 + 1.32698i
\(47\) −3.46410 −0.505291 −0.252646 − 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 5.00000 + 8.66025i 0.721688 + 1.25000i
\(49\) 3.50000 − 6.06218i 0.500000 − 0.866025i
\(50\) −1.73205 + 3.00000i −0.244949 + 0.424264i
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −3.46410 + 6.00000i −0.471405 + 0.816497i
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) 2.59808 + 4.50000i 0.341144 + 0.590879i
\(59\) −3.46410 − 6.00000i −0.450988 − 0.781133i 0.547460 − 0.836832i \(-0.315595\pi\)
−0.998448 + 0.0556984i \(0.982261\pi\)
\(60\) 3.46410 0.447214
\(61\) −0.500000 − 0.866025i −0.0640184 − 0.110883i 0.832240 − 0.554416i \(-0.187058\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −3.00000 + 5.19615i −0.381000 + 0.659912i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.73205 + 3.00000i −0.211604 + 0.366508i −0.952217 − 0.305424i \(-0.901202\pi\)
0.740613 + 0.671932i \(0.234535\pi\)
\(68\) −1.50000 + 2.59808i −0.181902 + 0.315063i
\(69\) −6.00000 − 10.3923i −0.722315 − 1.25109i
\(70\) 0 0
\(71\) 1.73205 + 3.00000i 0.205557 + 0.356034i 0.950310 − 0.311305i \(-0.100766\pi\)
−0.744753 + 0.667340i \(0.767433\pi\)
\(72\) −0.866025 − 1.50000i −0.102062 − 0.176777i
\(73\) 1.73205 0.202721 0.101361 − 0.994850i \(-0.467680\pi\)
0.101361 + 0.994850i \(0.467680\pi\)
\(74\) −7.50000 − 12.9904i −0.871857 − 1.51010i
\(75\) 2.00000 − 3.46410i 0.230940 − 0.400000i
\(76\) −1.73205 + 3.00000i −0.198680 + 0.344124i
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 4.33013 − 7.50000i 0.484123 − 0.838525i
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) −4.50000 − 7.79423i −0.496942 − 0.860729i
\(83\) −13.8564 −1.52094 −0.760469 − 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(84\) 0 0
\(85\) −2.59808 − 4.50000i −0.281801 − 0.488094i
\(86\) 13.8564 1.49417
\(87\) −3.00000 − 5.19615i −0.321634 − 0.557086i
\(88\) 0 0
\(89\) −3.46410 + 6.00000i −0.367194 + 0.635999i −0.989126 − 0.147073i \(-0.953015\pi\)
0.621932 + 0.783072i \(0.286348\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 3.46410 − 6.00000i 0.359211 − 0.622171i
\(94\) −3.00000 + 5.19615i −0.309426 + 0.535942i
\(95\) −3.00000 − 5.19615i −0.307794 − 0.533114i
\(96\) 10.3923 1.06066
\(97\) 3.46410 + 6.00000i 0.351726 + 0.609208i 0.986552 − 0.163448i \(-0.0522615\pi\)
−0.634826 + 0.772655i \(0.718928\pi\)
\(98\) −6.06218 − 10.5000i −0.612372 − 1.06066i
\(99\) 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 169.2.c.a.22.2 4
13.2 odd 12 169.2.e.a.23.1 2
13.3 even 3 inner 169.2.c.a.146.2 4
13.4 even 6 169.2.a.a.1.2 2
13.5 odd 4 13.2.e.a.4.1 ✓ 2
13.6 odd 12 169.2.b.a.168.2 2
13.7 odd 12 169.2.b.a.168.1 2
13.8 odd 4 169.2.e.a.147.1 2
13.9 even 3 169.2.a.a.1.1 2
13.10 even 6 inner 169.2.c.a.146.1 4
13.11 odd 12 13.2.e.a.10.1 yes 2
13.12 even 2 inner 169.2.c.a.22.1 4
39.5 even 4 117.2.q.c.82.1 2
39.11 even 12 117.2.q.c.10.1 2
39.17 odd 6 1521.2.a.k.1.1 2
39.20 even 12 1521.2.b.a.1351.2 2
39.32 even 12 1521.2.b.a.1351.1 2
39.35 odd 6 1521.2.a.k.1.2 2
52.7 even 12 2704.2.f.b.337.2 2
52.11 even 12 208.2.w.b.49.1 2
52.19 even 12 2704.2.f.b.337.1 2
52.31 even 4 208.2.w.b.17.1 2
52.35 odd 6 2704.2.a.o.1.2 2
52.43 odd 6 2704.2.a.o.1.1 2
65.4 even 6 4225.2.a.v.1.1 2
65.9 even 6 4225.2.a.v.1.2 2
65.18 even 4 325.2.m.a.199.1 4
65.24 odd 12 325.2.n.a.101.1 2
65.37 even 12 325.2.m.a.49.1 4
65.44 odd 4 325.2.n.a.251.1 2
65.57 even 4 325.2.m.a.199.2 4
65.63 even 12 325.2.m.a.49.2 4
91.5 even 12 637.2.k.c.459.1 2
91.11 odd 12 637.2.k.a.569.1 2
91.18 odd 12 637.2.u.c.30.1 2
91.24 even 12 637.2.k.c.569.1 2
91.31 even 12 637.2.u.b.30.1 2
91.37 odd 12 637.2.u.c.361.1 2
91.44 odd 12 637.2.k.a.459.1 2
91.48 odd 6 8281.2.a.q.1.1 2
91.69 odd 6 8281.2.a.q.1.2 2
91.76 even 12 637.2.q.a.491.1 2
91.83 even 4 637.2.q.a.589.1 2
91.89 even 12 637.2.u.b.361.1 2
104.5 odd 4 832.2.w.d.641.1 2
104.11 even 12 832.2.w.a.257.1 2
104.37 odd 12 832.2.w.d.257.1 2
104.83 even 4 832.2.w.a.641.1 2
156.11 odd 12 1872.2.by.d.1297.1 2
156.83 odd 4 1872.2.by.d.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 ✓ 2 13.5 odd 4
13.2.e.a.10.1 yes 2 13.11 odd 12
117.2.q.c.10.1 2 39.11 even 12
117.2.q.c.82.1 2 39.5 even 4
169.2.a.a.1.1 2 13.9 even 3
169.2.a.a.1.2 2 13.4 even 6
169.2.b.a.168.1 2 13.7 odd 12
169.2.b.a.168.2 2 13.6 odd 12
169.2.c.a.22.1 4 13.12 even 2 inner
169.2.c.a.22.2 4 1.1 even 1 trivial
169.2.c.a.146.1 4 13.10 even 6 inner
169.2.c.a.146.2 4 13.3 even 3 inner
169.2.e.a.23.1 2 13.2 odd 12
169.2.e.a.147.1 2 13.8 odd 4
208.2.w.b.17.1 2 52.31 even 4
208.2.w.b.49.1 2 52.11 even 12
325.2.m.a.49.1 4 65.37 even 12
325.2.m.a.49.2 4 65.63 even 12
325.2.m.a.199.1 4 65.18 even 4
325.2.m.a.199.2 4 65.57 even 4
325.2.n.a.101.1 2 65.24 odd 12
325.2.n.a.251.1 2 65.44 odd 4
637.2.k.a.459.1 2 91.44 odd 12
637.2.k.a.569.1 2 91.11 odd 12
637.2.k.c.459.1 2 91.5 even 12
637.2.k.c.569.1 2 91.24 even 12
637.2.q.a.491.1 2 91.76 even 12
637.2.q.a.589.1 2 91.83 even 4
637.2.u.b.30.1 2 91.31 even 12
637.2.u.b.361.1 2 91.89 even 12
637.2.u.c.30.1 2 91.18 odd 12
637.2.u.c.361.1 2 91.37 odd 12
832.2.w.a.257.1 2 104.11 even 12
832.2.w.a.641.1 2 104.83 even 4
832.2.w.d.257.1 2 104.37 odd 12
832.2.w.d.641.1 2 104.5 odd 4
1521.2.a.k.1.1 2 39.17 odd 6
1521.2.a.k.1.2 2 39.35 odd 6
1521.2.b.a.1351.1 2 39.32 even 12
1521.2.b.a.1351.2 2 39.20 even 12
1872.2.by.d.433.1 2 156.83 odd 4
1872.2.by.d.1297.1 2 156.11 odd 12
2704.2.a.o.1.1 2 52.43 odd 6
2704.2.a.o.1.2 2 52.35 odd 6
2704.2.f.b.337.1 2 52.19 even 12
2704.2.f.b.337.2 2 52.7 even 12
4225.2.a.v.1.1 2 65.4 even 6
4225.2.a.v.1.2 2 65.9 even 6
8281.2.a.q.1.1 2 91.48 odd 6
8281.2.a.q.1.2 2 91.69 odd 6