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Properties

Label 13.6.a.a.1.2
Level $13$
Weight $6$
Character 13.1
Self dual yes
Analytic conductor $2.085$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [13,6,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 1.2
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q-0.438447 q^{2} -26.3693 q^{3} -31.8078 q^{4} +61.4621 q^{5} +11.5616 q^{6} -162.309 q^{7} +27.9763 q^{8} +452.341 q^{9} -26.9479 q^{10} -361.170 q^{11} +838.749 q^{12} -169.000 q^{13} +71.1638 q^{14} -1620.71 q^{15} +1005.58 q^{16} -1578.88 q^{17} -198.328 q^{18} -98.2765 q^{19} -1954.97 q^{20} +4279.97 q^{21} +158.354 q^{22} +1607.16 q^{23} -737.717 q^{24} +652.591 q^{25} +74.0976 q^{26} -5520.18 q^{27} +5162.68 q^{28} -307.045 q^{29} +710.597 q^{30} +2936.42 q^{31} -1336.14 q^{32} +9523.82 q^{33} +692.255 q^{34} -9975.84 q^{35} -14388.0 q^{36} -12222.3 q^{37} +43.0891 q^{38} +4456.41 q^{39} +1719.48 q^{40} -104.151 q^{41} -1876.54 q^{42} +10936.4 q^{43} +11488.0 q^{44} +27801.8 q^{45} -704.654 q^{46} -14949.2 q^{47} -26516.5 q^{48} +9537.11 q^{49} -286.127 q^{50} +41634.0 q^{51} +5375.51 q^{52} -35911.5 q^{53} +2420.31 q^{54} -22198.3 q^{55} -4540.80 q^{56} +2591.48 q^{57} +134.623 q^{58} -1598.46 q^{59} +51551.3 q^{60} +20156.2 q^{61} -1287.47 q^{62} -73418.9 q^{63} -31592.8 q^{64} -10387.1 q^{65} -4175.69 q^{66} -35368.1 q^{67} +50220.6 q^{68} -42379.7 q^{69} +4373.88 q^{70} +26140.3 q^{71} +12654.8 q^{72} +75468.4 q^{73} +5358.81 q^{74} -17208.4 q^{75} +3125.96 q^{76} +58621.1 q^{77} -1953.90 q^{78} -7576.72 q^{79} +61805.2 q^{80} +35644.4 q^{81} +45.6648 q^{82} -912.974 q^{83} -136136. q^{84} -97041.2 q^{85} -4795.04 q^{86} +8096.58 q^{87} -10104.2 q^{88} +106709. q^{89} -12189.6 q^{90} +27430.2 q^{91} -51120.1 q^{92} -77431.5 q^{93} +6554.44 q^{94} -6040.28 q^{95} +35233.0 q^{96} +103676. q^{97} -4181.52 q^{98} -163372. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{2} - 28 q^{3} - 43 q^{4} - 42 q^{5} + 19 q^{6} - 36 q^{7} + 225 q^{8} + 212 q^{9} + 445 q^{10} - 376 q^{11} + 857 q^{12} - 338 q^{13} - 505 q^{14} - 1452 q^{15} + 465 q^{16} - 2630 q^{17} + 898 q^{18}+ \cdots - 159808 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.438447 −0.0775072 −0.0387536 − 0.999249i \(-0.512339\pi\)
−0.0387536 + 0.999249i \(0.512339\pi\)
\(3\) −26.3693 −1.69159 −0.845796 − 0.533506i \(-0.820874\pi\)
−0.845796 + 0.533506i \(0.820874\pi\)
\(4\) −31.8078 −0.993993
\(5\) 61.4621 1.09947 0.549734 − 0.835340i \(-0.314729\pi\)
0.549734 + 0.835340i \(0.314729\pi\)
\(6\) 11.5616 0.131111
\(7\) −162.309 −1.25198 −0.625989 − 0.779832i \(-0.715305\pi\)
−0.625989 + 0.779832i \(0.715305\pi\)
\(8\) 27.9763 0.154549
\(9\) 452.341 1.86149
\(10\) −26.9479 −0.0852167
\(11\) −361.170 −0.899975 −0.449988 − 0.893035i \(-0.648572\pi\)
−0.449988 + 0.893035i \(0.648572\pi\)
\(12\) 838.749 1.68143
\(13\) −169.000 −0.277350
\(14\) 71.1638 0.0970374
\(15\) −1620.71 −1.85985
\(16\) 1005.58 0.982014
\(17\) −1578.88 −1.32503 −0.662516 − 0.749048i \(-0.730511\pi\)
−0.662516 + 0.749048i \(0.730511\pi\)
\(18\) −198.328 −0.144279
\(19\) −98.2765 −0.0624548 −0.0312274 − 0.999512i \(-0.509942\pi\)
−0.0312274 + 0.999512i \(0.509942\pi\)
\(20\) −1954.97 −1.09286
\(21\) 4279.97 2.11784
\(22\) 158.354 0.0697546
\(23\) 1607.16 0.633489 0.316745 − 0.948511i \(-0.397410\pi\)
0.316745 + 0.948511i \(0.397410\pi\)
\(24\) −737.717 −0.261434
\(25\) 652.591 0.208829
\(26\) 74.0976 0.0214966
\(27\) −5520.18 −1.45728
\(28\) 5162.68 1.24446
\(29\) −307.045 −0.0677966 −0.0338983 − 0.999425i \(-0.510792\pi\)
−0.0338983 + 0.999425i \(0.510792\pi\)
\(30\) 710.597 0.144152
\(31\) 2936.42 0.548801 0.274400 − 0.961616i \(-0.411521\pi\)
0.274400 + 0.961616i \(0.411521\pi\)
\(32\) −1336.14 −0.230662
\(33\) 9523.82 1.52239
\(34\) 692.255 0.102700
\(35\) −9975.84 −1.37651
\(36\) −14388.0 −1.85030
\(37\) −12222.3 −1.46773 −0.733867 − 0.679294i \(-0.762286\pi\)
−0.733867 + 0.679294i \(0.762286\pi\)
\(38\) 43.0891 0.00484070
\(39\) 4456.41 0.469163
\(40\) 1719.48 0.169921
\(41\) −104.151 −0.00967619 −0.00483809 − 0.999988i \(-0.501540\pi\)
−0.00483809 + 0.999988i \(0.501540\pi\)
\(42\) −1876.54 −0.164148
\(43\) 10936.4 0.901994 0.450997 − 0.892526i \(-0.351069\pi\)
0.450997 + 0.892526i \(0.351069\pi\)
\(44\) 11488.0 0.894569
\(45\) 27801.8 2.04664
\(46\) −704.654 −0.0491000
\(47\) −14949.2 −0.987129 −0.493564 − 0.869709i \(-0.664306\pi\)
−0.493564 + 0.869709i \(0.664306\pi\)
\(48\) −26516.5 −1.66117
\(49\) 9537.11 0.567449
\(50\) −286.127 −0.0161858
\(51\) 41634.0 2.24141
\(52\) 5375.51 0.275684
\(53\) −35911.5 −1.75608 −0.878040 − 0.478587i \(-0.841149\pi\)
−0.878040 + 0.478587i \(0.841149\pi\)
\(54\) 2420.31 0.112950
\(55\) −22198.3 −0.989494
\(56\) −4540.80 −0.193492
\(57\) 2591.48 0.105648
\(58\) 134.623 0.00525472
\(59\) −1598.46 −0.0597822 −0.0298911 − 0.999553i \(-0.509516\pi\)
−0.0298911 + 0.999553i \(0.509516\pi\)
\(60\) 51551.3 1.84868
\(61\) 20156.2 0.693560 0.346780 − 0.937947i \(-0.387275\pi\)
0.346780 + 0.937947i \(0.387275\pi\)
\(62\) −1287.47 −0.0425360
\(63\) −73418.9 −2.33054
\(64\) −31592.8 −0.964136
\(65\) −10387.1 −0.304937
\(66\) −4175.69 −0.117996
\(67\) −35368.1 −0.962552 −0.481276 − 0.876569i \(-0.659827\pi\)
−0.481276 + 0.876569i \(0.659827\pi\)
\(68\) 50220.6 1.31707
\(69\) −42379.7 −1.07161
\(70\) 4373.88 0.106689
\(71\) 26140.3 0.615411 0.307706 − 0.951482i \(-0.400439\pi\)
0.307706 + 0.951482i \(0.400439\pi\)
\(72\) 12654.8 0.287690
\(73\) 75468.4 1.65752 0.828759 − 0.559606i \(-0.189048\pi\)
0.828759 + 0.559606i \(0.189048\pi\)
\(74\) 5358.81 0.113760
\(75\) −17208.4 −0.353254
\(76\) 3125.96 0.0620796
\(77\) 58621.1 1.12675
\(78\) −1953.90 −0.0363636
\(79\) −7576.72 −0.136588 −0.0682942 − 0.997665i \(-0.521756\pi\)
−0.0682942 + 0.997665i \(0.521756\pi\)
\(80\) 61805.2 1.07969
\(81\) 35644.4 0.603642
\(82\) 45.6648 0.000749974 0
\(83\) −912.974 −0.0145466 −0.00727332 − 0.999974i \(-0.502315\pi\)
−0.00727332 + 0.999974i \(0.502315\pi\)
\(84\) −136136. −2.10511
\(85\) −97041.2 −1.45683
\(86\) −4795.04 −0.0699110
\(87\) 8096.58 0.114684
\(88\) −10104.2 −0.139090
\(89\) 106709. 1.42799 0.713995 − 0.700151i \(-0.246884\pi\)
0.713995 + 0.700151i \(0.246884\pi\)
\(90\) −12189.6 −0.158630
\(91\) 27430.2 0.347236
\(92\) −51120.1 −0.629684
\(93\) −77431.5 −0.928347
\(94\) 6554.44 0.0765096
\(95\) −6040.28 −0.0686670
\(96\) 35233.0 0.390186
\(97\) 103676. 1.11879 0.559397 − 0.828900i \(-0.311033\pi\)
0.559397 + 0.828900i \(0.311033\pi\)
\(98\) −4181.52 −0.0439814
\(99\) −163372. −1.67529
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.1.2 ✓ 2
3.2 odd 2 117.6.a.c.1.1 2
4.3 odd 2 208.6.a.h.1.2 2
5.2 odd 4 325.6.b.b.274.2 4
5.3 odd 4 325.6.b.b.274.3 4
5.4 even 2 325.6.a.b.1.1 2
7.6 odd 2 637.6.a.a.1.2 2
8.3 odd 2 832.6.a.i.1.1 2
8.5 even 2 832.6.a.p.1.2 2
13.5 odd 4 169.6.b.a.168.3 4
13.8 odd 4 169.6.b.a.168.2 4
13.12 even 2 169.6.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.6.a.a.1.2 ✓ 2 1.1 even 1 trivial
117.6.a.c.1.1 2 3.2 odd 2
169.6.a.a.1.1 2 13.12 even 2
169.6.b.a.168.2 4 13.8 odd 4
169.6.b.a.168.3 4 13.5 odd 4
208.6.a.h.1.2 2 4.3 odd 2
325.6.a.b.1.1 2 5.4 even 2
325.6.b.b.274.2 4 5.2 odd 4
325.6.b.b.274.3 4 5.3 odd 4
637.6.a.a.1.2 2 7.6 odd 2
832.6.a.i.1.1 2 8.3 odd 2
832.6.a.p.1.2 2 8.5 even 2