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Properties

Label 13.2.e.a.4.1
Level $13$
Weight $2$
Character 13.4
Analytic conductor $0.104$
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] = [13,2,Mod(4,13)] 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("13.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\), degree \(2\), 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: \(0.103805522628\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 4.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 13.4
Dual form 13.2.e.a.10.1

$q$-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+(-1.50000 - 0.866025i) q^{2} +(-1.00000 + 1.73205i) q^{3} +(0.500000 + 0.866025i) q^{4} -1.73205i q^{5} +(3.00000 - 1.73205i) q^{6} +1.73205i q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 + 2.59808i) q^{10} -2.00000 q^{12} +(-2.50000 - 2.59808i) q^{13} +(3.00000 + 1.73205i) q^{15} +(2.50000 - 4.33013i) q^{16} +(1.50000 + 2.59808i) q^{17} +1.73205i q^{18} +(-3.00000 + 1.73205i) q^{19} +(1.50000 - 0.866025i) q^{20} +(3.00000 - 5.19615i) q^{23} +(-3.00000 - 1.73205i) q^{24} +2.00000 q^{25} +(1.50000 + 6.06218i) q^{26} -4.00000 q^{27} +(-1.50000 + 2.59808i) q^{29} +(-3.00000 - 5.19615i) q^{30} +3.46410i q^{31} +(-4.50000 + 2.59808i) q^{32} -5.19615i q^{34} +(0.500000 - 0.866025i) q^{36} +(7.50000 + 4.33013i) q^{37} +6.00000 q^{38} +(7.00000 - 1.73205i) q^{39} +3.00000 q^{40} +(-4.50000 - 2.59808i) q^{41} +(-4.00000 - 6.92820i) q^{43} +(-1.50000 + 0.866025i) q^{45} +(-9.00000 + 5.19615i) q^{46} -3.46410i q^{47} +(5.00000 + 8.66025i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(-3.00000 - 1.73205i) q^{50} -6.00000 q^{51} +(1.00000 - 3.46410i) q^{52} -3.00000 q^{53} +(6.00000 + 3.46410i) q^{54} -6.92820i q^{57} +(4.50000 - 2.59808i) q^{58} +(6.00000 - 3.46410i) q^{59} +3.46410i q^{60} +(-0.500000 - 0.866025i) q^{61} +(3.00000 - 5.19615i) q^{62} -1.00000 q^{64} +(-4.50000 + 4.33013i) q^{65} +(3.00000 + 1.73205i) q^{67} +(-1.50000 + 2.59808i) q^{68} +(6.00000 + 10.3923i) q^{69} +(3.00000 - 1.73205i) q^{71} +(1.50000 - 0.866025i) q^{72} +1.73205i q^{73} +(-7.50000 - 12.9904i) q^{74} +(-2.00000 + 3.46410i) q^{75} +(-3.00000 - 1.73205i) q^{76} +(-12.0000 - 3.46410i) q^{78} +4.00000 q^{79} +(-7.50000 - 4.33013i) q^{80} +(5.50000 - 9.52628i) q^{81} +(4.50000 + 7.79423i) q^{82} +13.8564i q^{83} +(4.50000 - 2.59808i) q^{85} +13.8564i q^{86} +(-3.00000 - 5.19615i) q^{87} +(-6.00000 - 3.46410i) q^{89} +3.00000 q^{90} +6.00000 q^{92} +(-6.00000 - 3.46410i) q^{93} +(-3.00000 + 5.19615i) q^{94} +(3.00000 + 5.19615i) q^{95} -10.3923i q^{96} +(6.00000 - 3.46410i) q^{97} +(10.5000 - 6.06218i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 2 q^{3} + q^{4} + 6 q^{6} - q^{9} - 3 q^{10} - 4 q^{12} - 5 q^{13} + 6 q^{15} + 5 q^{16} + 3 q^{17} - 6 q^{19} + 3 q^{20} + 6 q^{23} - 6 q^{24} + 4 q^{25} + 3 q^{26} - 8 q^{27} - 3 q^{29}+ \cdots + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.50000 − 0.866025i −1.06066 − 0.612372i −0.135045 − 0.990839i \(-0.543118\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) −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
\(5\) − 1.73205i − 0.774597i −0.921954 − 0.387298i \(-0.873408\pi\)
0.921954 − 0.387298i \(-0.126592\pi\)
\(6\) 3.00000 − 1.73205i 1.22474 − 0.707107i
\(7\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 1.73205i 0.612372i
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −1.50000 + 2.59808i −0.474342 + 0.821584i
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −2.00000 −0.577350
\(13\) −2.50000 − 2.59808i −0.693375 − 0.720577i
\(14\) 0 0
\(15\) 3.00000 + 1.73205i 0.774597 + 0.447214i
\(16\) 2.50000 − 4.33013i 0.625000 − 1.08253i
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 1.73205i 0.408248i
\(19\) −3.00000 + 1.73205i −0.688247 + 0.397360i −0.802955 − 0.596040i \(-0.796740\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 1.50000 − 0.866025i 0.335410 − 0.193649i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 − 5.19615i 0.625543 − 1.08347i −0.362892 − 0.931831i \(-0.618211\pi\)
0.988436 − 0.151642i \(-0.0484560\pi\)
\(24\) −3.00000 − 1.73205i −0.612372 − 0.353553i
\(25\) 2.00000 0.400000
\(26\) 1.50000 + 6.06218i 0.294174 + 1.18889i
\(27\) −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
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) −4.50000 + 2.59808i −0.795495 + 0.459279i
\(33\) 0 0
\(34\) − 5.19615i − 0.891133i
\(35\) 0 0
\(36\) 0.500000 − 0.866025i 0.0833333 − 0.144338i
\(37\) 7.50000 + 4.33013i 1.23299 + 0.711868i 0.967653 − 0.252286i \(-0.0811825\pi\)
0.265340 + 0.964155i \(0.414516\pi\)
\(38\) 6.00000 0.973329
\(39\) 7.00000 − 1.73205i 1.12090 − 0.277350i
\(40\) 3.00000 0.474342
\(41\) −4.50000 − 2.59808i −0.702782 − 0.405751i 0.105601 − 0.994409i \(-0.466323\pi\)
−0.808383 + 0.588657i \(0.799657\pi\)
\(42\) 0 0
\(43\) −4.00000 − 6.92820i −0.609994 − 1.05654i −0.991241 − 0.132068i \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) 0 0
\(45\) −1.50000 + 0.866025i −0.223607 + 0.129099i
\(46\) −9.00000 + 5.19615i −1.32698 + 0.766131i
\(47\) − 3.46410i − 0.505291i −0.967559 − 0.252646i \(-0.918699\pi\)
0.967559 − 0.252646i \(-0.0813007\pi\)
\(48\) 5.00000 + 8.66025i 0.721688 + 1.25000i
\(49\) −3.50000 + 6.06218i −0.500000 + 0.866025i
\(50\) −3.00000 − 1.73205i −0.424264 − 0.244949i
\(51\) −6.00000 −0.840168
\(52\) 1.00000 − 3.46410i 0.138675 − 0.480384i
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 6.00000 + 3.46410i 0.816497 + 0.471405i
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.92820i − 0.917663i
\(58\) 4.50000 − 2.59808i 0.590879 − 0.341144i
\(59\) 6.00000 − 3.46410i 0.781133 − 0.450988i −0.0556984 − 0.998448i \(-0.517739\pi\)
0.836832 + 0.547460i \(0.184405\pi\)
\(60\) 3.46410i 0.447214i
\(61\) −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
\(65\) −4.50000 + 4.33013i −0.558156 + 0.537086i
\(66\) 0 0
\(67\) 3.00000 + 1.73205i 0.366508 + 0.211604i 0.671932 − 0.740613i \(-0.265465\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) −1.50000 + 2.59808i −0.181902 + 0.315063i
\(69\) 6.00000 + 10.3923i 0.722315 + 1.25109i
\(70\) 0 0
\(71\) 3.00000 − 1.73205i 0.356034 − 0.205557i −0.311305 − 0.950310i \(-0.600766\pi\)
0.667340 + 0.744753i \(0.267433\pi\)
\(72\) 1.50000 − 0.866025i 0.176777 − 0.102062i
\(73\) 1.73205i 0.202721i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) −7.50000 − 12.9904i −0.871857 − 1.51010i
\(75\) −2.00000 + 3.46410i −0.230940 + 0.400000i
\(76\) −3.00000 − 1.73205i −0.344124 − 0.198680i
\(77\) 0 0
\(78\) −12.0000 − 3.46410i −1.35873 − 0.392232i
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −7.50000 − 4.33013i −0.838525 − 0.484123i
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) 4.50000 + 7.79423i 0.496942 + 0.860729i
\(83\) 13.8564i 1.52094i 0.649374 + 0.760469i \(0.275031\pi\)
−0.649374 + 0.760469i \(0.724969\pi\)
\(84\) 0 0
\(85\) 4.50000 − 2.59808i 0.488094 − 0.281801i
\(86\) 13.8564i 1.49417i
\(87\) −3.00000 − 5.19615i −0.321634 − 0.557086i
\(88\) 0 0
\(89\) −6.00000 − 3.46410i −0.635999 − 0.367194i 0.147073 − 0.989126i \(-0.453015\pi\)
−0.783072 + 0.621932i \(0.786348\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −6.00000 − 3.46410i −0.622171 − 0.359211i
\(94\) −3.00000 + 5.19615i −0.309426 + 0.535942i
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) − 10.3923i − 1.06066i
\(97\) 6.00000 − 3.46410i 0.609208 − 0.351726i −0.163448 − 0.986552i \(-0.552261\pi\)
0.772655 + 0.634826i \(0.218928\pi\)
\(98\) 10.5000 − 6.06218i 1.06066 − 0.612372i
\(99\) 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 13.2.e.a.4.1 ✓ 2
3.2 odd 2 117.2.q.c.82.1 2
4.3 odd 2 208.2.w.b.17.1 2
5.2 odd 4 325.2.m.a.199.2 4
5.3 odd 4 325.2.m.a.199.1 4
5.4 even 2 325.2.n.a.251.1 2
7.2 even 3 637.2.k.a.459.1 2
7.3 odd 6 637.2.u.b.30.1 2
7.4 even 3 637.2.u.c.30.1 2
7.5 odd 6 637.2.k.c.459.1 2
7.6 odd 2 637.2.q.a.589.1 2
8.3 odd 2 832.2.w.a.641.1 2
8.5 even 2 832.2.w.d.641.1 2
12.11 even 2 1872.2.by.d.433.1 2
13.2 odd 12 169.2.c.a.146.1 4
13.3 even 3 169.2.e.a.23.1 2
13.4 even 6 169.2.b.a.168.1 2
13.5 odd 4 169.2.c.a.22.1 4
13.6 odd 12 169.2.a.a.1.2 2
13.7 odd 12 169.2.a.a.1.1 2
13.8 odd 4 169.2.c.a.22.2 4
13.9 even 3 169.2.b.a.168.2 2
13.10 even 6 inner 13.2.e.a.10.1 yes 2
13.11 odd 12 169.2.c.a.146.2 4
13.12 even 2 169.2.e.a.147.1 2
39.17 odd 6 1521.2.b.a.1351.2 2
39.20 even 12 1521.2.a.k.1.2 2
39.23 odd 6 117.2.q.c.10.1 2
39.32 even 12 1521.2.a.k.1.1 2
39.35 odd 6 1521.2.b.a.1351.1 2
52.7 even 12 2704.2.a.o.1.2 2
52.19 even 12 2704.2.a.o.1.1 2
52.23 odd 6 208.2.w.b.49.1 2
52.35 odd 6 2704.2.f.b.337.1 2
52.43 odd 6 2704.2.f.b.337.2 2
65.19 odd 12 4225.2.a.v.1.1 2
65.23 odd 12 325.2.m.a.49.2 4
65.49 even 6 325.2.n.a.101.1 2
65.59 odd 12 4225.2.a.v.1.2 2
65.62 odd 12 325.2.m.a.49.1 4
91.6 even 12 8281.2.a.q.1.2 2
91.10 odd 6 637.2.k.c.569.1 2
91.20 even 12 8281.2.a.q.1.1 2
91.23 even 6 637.2.u.c.361.1 2
91.62 odd 6 637.2.q.a.491.1 2
91.75 odd 6 637.2.u.b.361.1 2
91.88 even 6 637.2.k.a.569.1 2
104.75 odd 6 832.2.w.a.257.1 2
104.101 even 6 832.2.w.d.257.1 2
156.23 even 6 1872.2.by.d.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 ✓ 2 1.1 even 1 trivial
13.2.e.a.10.1 yes 2 13.10 even 6 inner
117.2.q.c.10.1 2 39.23 odd 6
117.2.q.c.82.1 2 3.2 odd 2
169.2.a.a.1.1 2 13.7 odd 12
169.2.a.a.1.2 2 13.6 odd 12
169.2.b.a.168.1 2 13.4 even 6
169.2.b.a.168.2 2 13.9 even 3
169.2.c.a.22.1 4 13.5 odd 4
169.2.c.a.22.2 4 13.8 odd 4
169.2.c.a.146.1 4 13.2 odd 12
169.2.c.a.146.2 4 13.11 odd 12
169.2.e.a.23.1 2 13.3 even 3
169.2.e.a.147.1 2 13.12 even 2
208.2.w.b.17.1 2 4.3 odd 2
208.2.w.b.49.1 2 52.23 odd 6
325.2.m.a.49.1 4 65.62 odd 12
325.2.m.a.49.2 4 65.23 odd 12
325.2.m.a.199.1 4 5.3 odd 4
325.2.m.a.199.2 4 5.2 odd 4
325.2.n.a.101.1 2 65.49 even 6
325.2.n.a.251.1 2 5.4 even 2
637.2.k.a.459.1 2 7.2 even 3
637.2.k.a.569.1 2 91.88 even 6
637.2.k.c.459.1 2 7.5 odd 6
637.2.k.c.569.1 2 91.10 odd 6
637.2.q.a.491.1 2 91.62 odd 6
637.2.q.a.589.1 2 7.6 odd 2
637.2.u.b.30.1 2 7.3 odd 6
637.2.u.b.361.1 2 91.75 odd 6
637.2.u.c.30.1 2 7.4 even 3
637.2.u.c.361.1 2 91.23 even 6
832.2.w.a.257.1 2 104.75 odd 6
832.2.w.a.641.1 2 8.3 odd 2
832.2.w.d.257.1 2 104.101 even 6
832.2.w.d.641.1 2 8.5 even 2
1521.2.a.k.1.1 2 39.32 even 12
1521.2.a.k.1.2 2 39.20 even 12
1521.2.b.a.1351.1 2 39.35 odd 6
1521.2.b.a.1351.2 2 39.17 odd 6
1872.2.by.d.433.1 2 12.11 even 2
1872.2.by.d.1297.1 2 156.23 even 6
2704.2.a.o.1.1 2 52.19 even 12
2704.2.a.o.1.2 2 52.7 even 12
2704.2.f.b.337.1 2 52.35 odd 6
2704.2.f.b.337.2 2 52.43 odd 6
4225.2.a.v.1.1 2 65.19 odd 12
4225.2.a.v.1.2 2 65.59 odd 12
8281.2.a.q.1.1 2 91.20 even 12
8281.2.a.q.1.2 2 91.6 even 12