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Properties

Label 13.4.c.a.9.1
Level $13$
Weight $4$
Character 13.9
Analytic conductor $0.767$
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] = [13,4,Mod(3,13)] 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("13.3"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 13.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.767024830075\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 9.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 13.9
Dual form 13.4.c.a.3.1

$q$-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+(-2.00000 + 3.46410i) q^{2} +(-1.00000 + 1.73205i) q^{3} +(-4.00000 - 6.92820i) q^{4} +17.0000 q^{5} +(-4.00000 - 6.92820i) q^{6} +(-10.0000 - 17.3205i) q^{7} +(11.5000 + 19.9186i) q^{9} +(-34.0000 + 58.8897i) q^{10} +(16.0000 - 27.7128i) q^{11} +16.0000 q^{12} +(-45.5000 - 11.2583i) q^{13} +80.0000 q^{14} +(-17.0000 + 29.4449i) q^{15} +(32.0000 - 55.4256i) q^{16} +(6.50000 + 11.2583i) q^{17} -92.0000 q^{18} +(-15.0000 - 25.9808i) q^{19} +(-68.0000 - 117.779i) q^{20} +40.0000 q^{21} +(64.0000 + 110.851i) q^{22} +(-39.0000 + 67.5500i) q^{23} +164.000 q^{25} +(130.000 - 135.100i) q^{26} -100.000 q^{27} +(-80.0000 + 138.564i) q^{28} +(-98.5000 + 170.607i) q^{29} +(-68.0000 - 117.779i) q^{30} -74.0000 q^{31} +(128.000 + 221.703i) q^{32} +(32.0000 + 55.4256i) q^{33} -52.0000 q^{34} +(-170.000 - 294.449i) q^{35} +(92.0000 - 159.349i) q^{36} +(113.500 - 196.588i) q^{37} +120.000 q^{38} +(65.0000 - 67.5500i) q^{39} +(82.5000 - 142.894i) q^{41} +(-80.0000 + 138.564i) q^{42} +(78.0000 + 135.100i) q^{43} -256.000 q^{44} +(195.500 + 338.616i) q^{45} +(-156.000 - 270.200i) q^{46} -162.000 q^{47} +(64.0000 + 110.851i) q^{48} +(-28.5000 + 49.3634i) q^{49} +(-328.000 + 568.113i) q^{50} -26.0000 q^{51} +(104.000 + 360.267i) q^{52} +93.0000 q^{53} +(200.000 - 346.410i) q^{54} +(272.000 - 471.118i) q^{55} +60.0000 q^{57} +(-394.000 - 682.428i) q^{58} +(432.000 + 748.246i) q^{59} +272.000 q^{60} +(-72.5000 - 125.574i) q^{61} +(148.000 - 256.344i) q^{62} +(230.000 - 398.372i) q^{63} -512.000 q^{64} +(-773.500 - 191.392i) q^{65} -256.000 q^{66} +(-431.000 + 746.514i) q^{67} +(52.0000 - 90.0666i) q^{68} +(-78.0000 - 135.100i) q^{69} +1360.00 q^{70} +(-327.000 - 566.381i) q^{71} +215.000 q^{73} +(454.000 + 786.351i) q^{74} +(-164.000 + 284.056i) q^{75} +(-120.000 + 207.846i) q^{76} -640.000 q^{77} +(104.000 + 360.267i) q^{78} -76.0000 q^{79} +(544.000 - 942.236i) q^{80} +(-210.500 + 364.597i) q^{81} +(330.000 + 571.577i) q^{82} +628.000 q^{83} +(-160.000 - 277.128i) q^{84} +(110.500 + 191.392i) q^{85} -624.000 q^{86} +(-197.000 - 341.214i) q^{87} +(133.000 - 230.363i) q^{89} -1564.00 q^{90} +(260.000 + 900.666i) q^{91} +624.000 q^{92} +(74.0000 - 128.172i) q^{93} +(324.000 - 561.184i) q^{94} +(-255.000 - 441.673i) q^{95} -512.000 q^{96} +(-119.000 - 206.114i) q^{97} +(-114.000 - 197.454i) q^{98} +736.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9} - 68 q^{10} + 32 q^{11} + 32 q^{12} - 91 q^{13} + 160 q^{14} - 34 q^{15} + 64 q^{16} + 13 q^{17} - 184 q^{18} - 30 q^{19} - 136 q^{20}+ \cdots + 1472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 + 3.46410i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) −1.00000 + 1.73205i −0.192450 + 0.333333i −0.946062 − 0.323987i \(-0.894977\pi\)
0.753612 + 0.657320i \(0.228310\pi\)
\(4\) −4.00000 − 6.92820i −0.500000 − 0.866025i
\(5\) 17.0000 1.52053 0.760263 − 0.649615i \(-0.225070\pi\)
0.760263 + 0.649615i \(0.225070\pi\)
\(6\) −4.00000 − 6.92820i −0.272166 − 0.471405i
\(7\) −10.0000 − 17.3205i −0.539949 − 0.935220i −0.998906 − 0.0467610i \(-0.985110\pi\)
0.458957 − 0.888459i \(-0.348223\pi\)
\(8\) 0 0
\(9\) 11.5000 + 19.9186i 0.425926 + 0.737725i
\(10\) −34.0000 + 58.8897i −1.07517 + 1.86226i
\(11\) 16.0000 − 27.7128i 0.438562 − 0.759612i −0.559017 − 0.829156i \(-0.688821\pi\)
0.997579 + 0.0695447i \(0.0221546\pi\)
\(12\) 16.0000 0.384900
\(13\) −45.5000 − 11.2583i −0.970725 − 0.240192i
\(14\) 80.0000 1.52721
\(15\) −17.0000 + 29.4449i −0.292625 + 0.506842i
\(16\) 32.0000 − 55.4256i 0.500000 − 0.866025i
\(17\) 6.50000 + 11.2583i 0.0927342 + 0.160620i 0.908661 − 0.417535i \(-0.137106\pi\)
−0.815927 + 0.578156i \(0.803773\pi\)
\(18\) −92.0000 −1.20470
\(19\) −15.0000 − 25.9808i −0.181118 − 0.313705i 0.761144 − 0.648583i \(-0.224638\pi\)
−0.942261 + 0.334878i \(0.891305\pi\)
\(20\) −68.0000 − 117.779i −0.760263 − 1.31681i
\(21\) 40.0000 0.415653
\(22\) 64.0000 + 110.851i 0.620220 + 1.07425i
\(23\) −39.0000 + 67.5500i −0.353568 + 0.612398i −0.986872 − 0.161506i \(-0.948365\pi\)
0.633304 + 0.773903i \(0.281698\pi\)
\(24\) 0 0
\(25\) 164.000 1.31200
\(26\) 130.000 − 135.100i 0.980581 − 1.01905i
\(27\) −100.000 −0.712778
\(28\) −80.0000 + 138.564i −0.539949 + 0.935220i
\(29\) −98.5000 + 170.607i −0.630724 + 1.09245i 0.356680 + 0.934227i \(0.383909\pi\)
−0.987404 + 0.158219i \(0.949425\pi\)
\(30\) −68.0000 − 117.779i −0.413835 − 0.716783i
\(31\) −74.0000 −0.428735 −0.214368 − 0.976753i \(-0.568769\pi\)
−0.214368 + 0.976753i \(0.568769\pi\)
\(32\) 128.000 + 221.703i 0.707107 + 1.22474i
\(33\) 32.0000 + 55.4256i 0.168803 + 0.292375i
\(34\) −52.0000 −0.262292
\(35\) −170.000 − 294.449i −0.821007 − 1.42203i
\(36\) 92.0000 − 159.349i 0.425926 − 0.737725i
\(37\) 113.500 − 196.588i 0.504305 − 0.873482i −0.495683 − 0.868504i \(-0.665082\pi\)
0.999988 − 0.00497814i \(-0.00158460\pi\)
\(38\) 120.000 0.512278
\(39\) 65.0000 − 67.5500i 0.266880 − 0.277350i
\(40\) 0 0
\(41\) 82.5000 − 142.894i 0.314252 − 0.544301i −0.665026 − 0.746820i \(-0.731580\pi\)
0.979278 + 0.202520i \(0.0649130\pi\)
\(42\) −80.0000 + 138.564i −0.293911 + 0.509069i
\(43\) 78.0000 + 135.100i 0.276625 + 0.479129i 0.970544 − 0.240924i \(-0.0774506\pi\)
−0.693919 + 0.720053i \(0.744117\pi\)
\(44\) −256.000 −0.877124
\(45\) 195.500 + 338.616i 0.647632 + 1.12173i
\(46\) −156.000 − 270.200i −0.500021 − 0.866061i
\(47\) −162.000 −0.502769 −0.251384 − 0.967887i \(-0.580886\pi\)
−0.251384 + 0.967887i \(0.580886\pi\)
\(48\) 64.0000 + 110.851i 0.192450 + 0.333333i
\(49\) −28.5000 + 49.3634i −0.0830904 + 0.143917i
\(50\) −328.000 + 568.113i −0.927724 + 1.60687i
\(51\) −26.0000 −0.0713868
\(52\) 104.000 + 360.267i 0.277350 + 0.960769i
\(53\) 93.0000 0.241029 0.120514 − 0.992712i \(-0.461546\pi\)
0.120514 + 0.992712i \(0.461546\pi\)
\(54\) 200.000 − 346.410i 0.504010 − 0.872971i
\(55\) 272.000 − 471.118i 0.666845 − 1.15501i
\(56\) 0 0
\(57\) 60.0000 0.139424
\(58\) −394.000 − 682.428i −0.891978 − 1.54495i
\(59\) 432.000 + 748.246i 0.953248 + 1.65107i 0.738328 + 0.674442i \(0.235616\pi\)
0.214919 + 0.976632i \(0.431051\pi\)
\(60\) 272.000 0.585251
\(61\) −72.5000 − 125.574i −0.152175 − 0.263575i 0.779852 − 0.625964i \(-0.215294\pi\)
−0.932027 + 0.362389i \(0.881961\pi\)
\(62\) 148.000 − 256.344i 0.303162 − 0.525091i
\(63\) 230.000 − 398.372i 0.459957 − 0.796668i
\(64\) −512.000 −1.00000
\(65\) −773.500 − 191.392i −1.47601 − 0.365219i
\(66\) −256.000 −0.477446
\(67\) −431.000 + 746.514i −0.785896 + 1.36121i 0.142566 + 0.989785i \(0.454465\pi\)
−0.928462 + 0.371427i \(0.878869\pi\)
\(68\) 52.0000 − 90.0666i 0.0927342 − 0.160620i
\(69\) −78.0000 − 135.100i −0.136088 − 0.235712i
\(70\) 1360.00 2.32216
\(71\) −327.000 − 566.381i −0.546588 − 0.946718i −0.998505 − 0.0546585i \(-0.982593\pi\)
0.451917 − 0.892060i \(-0.350740\pi\)
\(72\) 0 0
\(73\) 215.000 0.344710 0.172355 − 0.985035i \(-0.444862\pi\)
0.172355 + 0.985035i \(0.444862\pi\)
\(74\) 454.000 + 786.351i 0.713195 + 1.23529i
\(75\) −164.000 + 284.056i −0.252495 + 0.437333i
\(76\) −120.000 + 207.846i −0.181118 + 0.313705i
\(77\) −640.000 −0.947205
\(78\) 104.000 + 360.267i 0.150970 + 0.522976i
\(79\) −76.0000 −0.108236 −0.0541182 − 0.998535i \(-0.517235\pi\)
−0.0541182 + 0.998535i \(0.517235\pi\)
\(80\) 544.000 − 942.236i 0.760263 − 1.31681i
\(81\) −210.500 + 364.597i −0.288752 + 0.500133i
\(82\) 330.000 + 571.577i 0.444420 + 0.769757i
\(83\) 628.000 0.830505 0.415253 − 0.909706i \(-0.363693\pi\)
0.415253 + 0.909706i \(0.363693\pi\)
\(84\) −160.000 − 277.128i −0.207827 − 0.359966i
\(85\) 110.500 + 191.392i 0.141005 + 0.244227i
\(86\) −624.000 −0.782415
\(87\) −197.000 − 341.214i −0.242766 − 0.420483i
\(88\) 0 0
\(89\) 133.000 − 230.363i 0.158404 − 0.274364i −0.775889 − 0.630869i \(-0.782698\pi\)
0.934293 + 0.356505i \(0.116032\pi\)
\(90\) −1564.00 −1.83178
\(91\) 260.000 + 900.666i 0.299510 + 1.03753i
\(92\) 624.000 0.707136
\(93\) 74.0000 − 128.172i 0.0825101 − 0.142912i
\(94\) 324.000 − 561.184i 0.355511 − 0.615763i
\(95\) −255.000 − 441.673i −0.275394 − 0.476997i
\(96\) −512.000 −0.544331
\(97\) −119.000 − 206.114i −0.124563 − 0.215750i 0.796999 − 0.603981i \(-0.206420\pi\)
−0.921562 + 0.388231i \(0.873086\pi\)
\(98\) −114.000 − 197.454i −0.117508 − 0.203529i
\(99\) 736.000 0.747180
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 13.4.c.a.9.1 yes 2
3.2 odd 2 117.4.g.c.100.1 2
4.3 odd 2 208.4.i.b.113.1 2
13.2 odd 12 169.4.e.c.23.1 4
13.3 even 3 inner 13.4.c.a.3.1 ✓ 2
13.4 even 6 169.4.a.a.1.1 1
13.5 odd 4 169.4.e.c.147.2 4
13.6 odd 12 169.4.b.c.168.1 2
13.7 odd 12 169.4.b.c.168.2 2
13.8 odd 4 169.4.e.c.147.1 4
13.9 even 3 169.4.a.d.1.1 1
13.10 even 6 169.4.c.d.146.1 2
13.11 odd 12 169.4.e.c.23.2 4
13.12 even 2 169.4.c.d.22.1 2
39.17 odd 6 1521.4.a.k.1.1 1
39.29 odd 6 117.4.g.c.55.1 2
39.35 odd 6 1521.4.a.b.1.1 1
52.3 odd 6 208.4.i.b.81.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.a.3.1 ✓ 2 13.3 even 3 inner
13.4.c.a.9.1 yes 2 1.1 even 1 trivial
117.4.g.c.55.1 2 39.29 odd 6
117.4.g.c.100.1 2 3.2 odd 2
169.4.a.a.1.1 1 13.4 even 6
169.4.a.d.1.1 1 13.9 even 3
169.4.b.c.168.1 2 13.6 odd 12
169.4.b.c.168.2 2 13.7 odd 12
169.4.c.d.22.1 2 13.12 even 2
169.4.c.d.146.1 2 13.10 even 6
169.4.e.c.23.1 4 13.2 odd 12
169.4.e.c.23.2 4 13.11 odd 12
169.4.e.c.147.1 4 13.8 odd 4
169.4.e.c.147.2 4 13.5 odd 4
208.4.i.b.81.1 2 52.3 odd 6
208.4.i.b.113.1 2 4.3 odd 2
1521.4.a.b.1.1 1 39.35 odd 6
1521.4.a.k.1.1 1 39.17 odd 6