↑

Properties

Label 13.6.a.a.1.1
Level $13$
Weight $6$
Character 13.1
Self dual yes
Analytic conductor $2.085$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [13,6,Mod(1,13)] 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("13.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 13.a (trivial)

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: yes
Analytic conductor: \(2.08498965757\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 13.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.56155 q^{2} -1.63068 q^{3} -11.1922 q^{4} -103.462 q^{5} +7.43845 q^{6} +126.309 q^{7} +197.024 q^{8} -240.341 q^{9} +471.948 q^{10} -14.8296 q^{11} +18.2510 q^{12} -169.000 q^{13} -576.164 q^{14} +168.714 q^{15} -540.582 q^{16} -1051.12 q^{17} +1096.33 q^{18} -213.723 q^{19} +1157.97 q^{20} -205.969 q^{21} +67.6458 q^{22} -4231.16 q^{23} -321.283 q^{24} +7579.41 q^{25} +770.902 q^{26} +788.176 q^{27} -1413.68 q^{28} -504.955 q^{29} -769.597 q^{30} +4783.58 q^{31} -3838.86 q^{32} +24.1823 q^{33} +4794.75 q^{34} -13068.2 q^{35} +2689.95 q^{36} -4635.74 q^{37} +974.911 q^{38} +275.585 q^{39} -20384.5 q^{40} +7944.15 q^{41} +939.541 q^{42} -8516.41 q^{43} +165.976 q^{44} +24866.2 q^{45} +19300.7 q^{46} +24921.2 q^{47} +881.518 q^{48} -853.113 q^{49} -34573.9 q^{50} +1714.05 q^{51} +1891.49 q^{52} -7808.46 q^{53} -3595.31 q^{54} +1534.30 q^{55} +24885.8 q^{56} +348.515 q^{57} +2303.38 q^{58} -37337.5 q^{59} -1888.29 q^{60} -18172.2 q^{61} -21820.5 q^{62} -30357.1 q^{63} +34809.8 q^{64} +17485.1 q^{65} -110.309 q^{66} -34559.9 q^{67} +11764.4 q^{68} +6899.68 q^{69} +59611.1 q^{70} +41255.7 q^{71} -47352.8 q^{72} -1056.42 q^{73} +21146.2 q^{74} -12359.6 q^{75} +2392.04 q^{76} -1873.10 q^{77} -1257.10 q^{78} -47719.3 q^{79} +55929.8 q^{80} +57117.6 q^{81} -36237.7 q^{82} -74799.0 q^{83} +2305.26 q^{84} +108751. q^{85} +38848.0 q^{86} +823.421 q^{87} -2921.78 q^{88} +9799.26 q^{89} -113428. q^{90} -21346.2 q^{91} +47356.1 q^{92} -7800.50 q^{93} -113679. q^{94} +22112.3 q^{95} +6259.97 q^{96} -138432. q^{97} +3891.52 q^{98} +3564.15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} - 28 q^{3} - 43 q^{4} - 42 q^{5} + 19 q^{6} - 36 q^{7} + 225 q^{8} + 212 q^{9} + 445 q^{10} - 376 q^{11} + 857 q^{12} - 338 q^{13} - 505 q^{14} - 1452 q^{15} + 465 q^{16} - 2630 q^{17} + 898 q^{18}+ \cdots - 159808 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.56155 −0.806376 −0.403188 − 0.915117i \(-0.632098\pi\)
−0.403188 + 0.915117i \(0.632098\pi\)
\(3\) −1.63068 −0.104608 −0.0523042 − 0.998631i \(-0.516657\pi\)
−0.0523042 + 0.998631i \(0.516657\pi\)
\(4\) −11.1922 −0.349757
\(5\) −103.462 −1.85079 −0.925393 − 0.379008i \(-0.876265\pi\)
−0.925393 + 0.379008i \(0.876265\pi\)
\(6\) 7.43845 0.0843537
\(7\) 126.309 0.974290 0.487145 − 0.873321i \(-0.338038\pi\)
0.487145 + 0.873321i \(0.338038\pi\)
\(8\) 197.024 1.08841
\(9\) −240.341 −0.989057
\(10\) 471.948 1.49243
\(11\) −14.8296 −0.0369527 −0.0184764 − 0.999829i \(-0.505882\pi\)
−0.0184764 + 0.999829i \(0.505882\pi\)
\(12\) 18.2510 0.0365875
\(13\) −169.000 −0.277350
\(14\) −576.164 −0.785644
\(15\) 168.714 0.193608
\(16\) −540.582 −0.527912
\(17\) −1051.12 −0.882126 −0.441063 − 0.897476i \(-0.645398\pi\)
−0.441063 + 0.897476i \(0.645398\pi\)
\(18\) 1096.33 0.797552
\(19\) −213.723 −0.135821 −0.0679107 − 0.997691i \(-0.521633\pi\)
−0.0679107 + 0.997691i \(0.521633\pi\)
\(20\) 1157.97 0.647326
\(21\) −205.969 −0.101919
\(22\) 67.6458 0.0297978
\(23\) −4231.16 −1.66778 −0.833892 − 0.551928i \(-0.813892\pi\)
−0.833892 + 0.551928i \(0.813892\pi\)
\(24\) −321.283 −0.113857
\(25\) 7579.41 2.42541
\(26\) 770.902 0.223649
\(27\) 788.176 0.208072
\(28\) −1413.68 −0.340765
\(29\) −504.955 −0.111495 −0.0557477 − 0.998445i \(-0.517754\pi\)
−0.0557477 + 0.998445i \(0.517754\pi\)
\(30\) −769.597 −0.156121
\(31\) 4783.58 0.894022 0.447011 − 0.894528i \(-0.352488\pi\)
0.447011 + 0.894528i \(0.352488\pi\)
\(32\) −3838.86 −0.662716
\(33\) 24.1823 0.00386557
\(34\) 4794.75 0.711325
\(35\) −13068.2 −1.80320
\(36\) 2689.95 0.345930
\(37\) −4635.74 −0.556692 −0.278346 − 0.960481i \(-0.589786\pi\)
−0.278346 + 0.960481i \(0.589786\pi\)
\(38\) 974.911 0.109523
\(39\) 275.585 0.0290131
\(40\) −20384.5 −2.01442
\(41\) 7944.15 0.738054 0.369027 − 0.929419i \(-0.379691\pi\)
0.369027 + 0.929419i \(0.379691\pi\)
\(42\) 939.541 0.0821850
\(43\) −8516.41 −0.702401 −0.351201 − 0.936300i \(-0.614226\pi\)
−0.351201 + 0.936300i \(0.614226\pi\)
\(44\) 165.976 0.0129245
\(45\) 24866.2 1.83053
\(46\) 19300.7 1.34486
\(47\) 24921.2 1.64560 0.822801 − 0.568330i \(-0.192410\pi\)
0.822801 + 0.568330i \(0.192410\pi\)
\(48\) 881.518 0.0552241
\(49\) −853.113 −0.0507594
\(50\) −34573.9 −1.95579
\(51\) 1714.05 0.0922777
\(52\) 1891.49 0.0970052
\(53\) −7808.46 −0.381835 −0.190917 − 0.981606i \(-0.561146\pi\)
−0.190917 + 0.981606i \(0.561146\pi\)
\(54\) −3595.31 −0.167784
\(55\) 1534.30 0.0683916
\(56\) 24885.8 1.06043
\(57\) 348.515 0.0142081
\(58\) 2303.38 0.0899073
\(59\) −37337.5 −1.39642 −0.698209 − 0.715894i \(-0.746020\pi\)
−0.698209 + 0.715894i \(0.746020\pi\)
\(60\) −1888.29 −0.0677157
\(61\) −18172.2 −0.625292 −0.312646 − 0.949870i \(-0.601215\pi\)
−0.312646 + 0.949870i \(0.601215\pi\)
\(62\) −21820.5 −0.720918
\(63\) −30357.1 −0.963628
\(64\) 34809.8 1.06231
\(65\) 17485.1 0.513316
\(66\) −110.309 −0.00311710
\(67\) −34559.9 −0.940559 −0.470279 − 0.882518i \(-0.655847\pi\)
−0.470279 + 0.882518i \(0.655847\pi\)
\(68\) 11764.4 0.308530
\(69\) 6899.68 0.174464
\(70\) 59611.1 1.45406
\(71\) 41255.7 0.971265 0.485632 − 0.874163i \(-0.338589\pi\)
0.485632 + 0.874163i \(0.338589\pi\)
\(72\) −47352.8 −1.07650
\(73\) −1056.42 −0.0232022 −0.0116011 − 0.999933i \(-0.503693\pi\)
−0.0116011 + 0.999933i \(0.503693\pi\)
\(74\) 21146.2 0.448903
\(75\) −12359.6 −0.253718
\(76\) 2392.04 0.0475045
\(77\) −1873.10 −0.0360027
\(78\) −1257.10 −0.0233955
\(79\) −47719.3 −0.860253 −0.430126 − 0.902769i \(-0.641531\pi\)
−0.430126 + 0.902769i \(0.641531\pi\)
\(80\) 55929.8 0.977053
\(81\) 57117.6 0.967291
\(82\) −36237.7 −0.595149
\(83\) −74799.0 −1.19179 −0.595896 − 0.803061i \(-0.703203\pi\)
−0.595896 + 0.803061i \(0.703203\pi\)
\(84\) 2305.26 0.0356469
\(85\) 108751. 1.63263
\(86\) 38848.0 0.566400
\(87\) 823.421 0.0116634
\(88\) −2921.78 −0.0402198
\(89\) 9799.26 0.131135 0.0655675 − 0.997848i \(-0.479114\pi\)
0.0655675 + 0.997848i \(0.479114\pi\)
\(90\) −113428. −1.47610
\(91\) −21346.2 −0.270219
\(92\) 47356.1 0.583320
\(93\) −7800.50 −0.0935222
\(94\) −113679. −1.32697
\(95\) 22112.3 0.251376
\(96\) 6259.97 0.0693257
\(97\) −138432. −1.49385 −0.746927 − 0.664906i \(-0.768472\pi\)
−0.746927 + 0.664906i \(0.768472\pi\)
\(98\) 3891.52 0.0409312
\(99\) 3564.15 0.0365484
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 13.6.a.a.1.1 ✓ 2
3.2 odd 2 117.6.a.c.1.2 2
4.3 odd 2 208.6.a.h.1.1 2
5.2 odd 4 325.6.b.b.274.1 4
5.3 odd 4 325.6.b.b.274.4 4
5.4 even 2 325.6.a.b.1.2 2
7.6 odd 2 637.6.a.a.1.1 2
8.3 odd 2 832.6.a.i.1.2 2
8.5 even 2 832.6.a.p.1.1 2
13.5 odd 4 169.6.b.a.168.4 4
13.8 odd 4 169.6.b.a.168.1 4
13.12 even 2 169.6.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.6.a.a.1.1 ✓ 2 1.1 even 1 trivial
117.6.a.c.1.2 2 3.2 odd 2
169.6.a.a.1.2 2 13.12 even 2
169.6.b.a.168.1 4 13.8 odd 4
169.6.b.a.168.4 4 13.5 odd 4
208.6.a.h.1.1 2 4.3 odd 2
325.6.a.b.1.2 2 5.4 even 2
325.6.b.b.274.1 4 5.2 odd 4
325.6.b.b.274.4 4 5.3 odd 4
637.6.a.a.1.1 2 7.6 odd 2
832.6.a.i.1.2 2 8.3 odd 2
832.6.a.p.1.1 2 8.5 even 2