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Properties

Label 117.6.a.c.1.1
Level $117$
Weight $6$
Character 117.1
Self dual yes
Analytic conductor $18.765$
Analytic rank $0$
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] = [117,6,Mod(1,117)] 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("117.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,5] 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: yes
Analytic conductor: \(18.7649069181\)
Analytic rank: \(0\)
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: no (minimal twist has level 13)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 117.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.438447 q^{2} -31.8078 q^{4} -61.4621 q^{5} -162.309 q^{7} -27.9763 q^{8} -26.9479 q^{10} +361.170 q^{11} -169.000 q^{13} -71.1638 q^{14} +1005.58 q^{16} +1578.88 q^{17} -98.2765 q^{19} +1954.97 q^{20} +158.354 q^{22} -1607.16 q^{23} +652.591 q^{25} -74.0976 q^{26} +5162.68 q^{28} +307.045 q^{29} +2936.42 q^{31} +1336.14 q^{32} +692.255 q^{34} +9975.84 q^{35} -12222.3 q^{37} -43.0891 q^{38} +1719.48 q^{40} +104.151 q^{41} +10936.4 q^{43} -11488.0 q^{44} -704.654 q^{46} +14949.2 q^{47} +9537.11 q^{49} +286.127 q^{50} +5375.51 q^{52} +35911.5 q^{53} -22198.3 q^{55} +4540.80 q^{56} +134.623 q^{58} +1598.46 q^{59} +20156.2 q^{61} +1287.47 q^{62} -31592.8 q^{64} +10387.1 q^{65} -35368.1 q^{67} -50220.6 q^{68} +4373.88 q^{70} -26140.3 q^{71} +75468.4 q^{73} -5358.81 q^{74} +3125.96 q^{76} -58621.1 q^{77} -7576.72 q^{79} -61805.2 q^{80} +45.6648 q^{82} +912.974 q^{83} -97041.2 q^{85} +4795.04 q^{86} -10104.2 q^{88} -106709. q^{89} +27430.2 q^{91} +51120.1 q^{92} +6554.44 q^{94} +6040.28 q^{95} +103676. q^{97} +4181.52 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} - 43 q^{4} + 42 q^{5} - 36 q^{7} - 225 q^{8} + 445 q^{10} + 376 q^{11} - 338 q^{13} + 505 q^{14} + 465 q^{16} + 2630 q^{17} - 312 q^{19} + 797 q^{20} + 226 q^{22} + 2624 q^{23} + 8232 q^{25}+ \cdots + 290 q^{98}+O(q^{100}) \) 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 \(-0.487661\pi\)
0.0387536 + 0.999249i \(0.487661\pi\)
\(3\) 0 0
\(4\) −31.8078 −0.993993
\(5\) −61.4621 −1.09947 −0.549734 − 0.835340i \(-0.685271\pi\)
−0.549734 + 0.835340i \(0.685271\pi\)
\(6\) 0 0
\(7\) −162.309 −1.25198 −0.625989 − 0.779832i \(-0.715305\pi\)
−0.625989 + 0.779832i \(0.715305\pi\)
\(8\) −27.9763 −0.154549
\(9\) 0 0
\(10\) −26.9479 −0.0852167
\(11\) 361.170 0.899975 0.449988 − 0.893035i \(-0.351428\pi\)
0.449988 + 0.893035i \(0.351428\pi\)
\(12\) 0 0
\(13\) −169.000 −0.277350
\(14\) −71.1638 −0.0970374
\(15\) 0 0
\(16\) 1005.58 0.982014
\(17\) 1578.88 1.32503 0.662516 − 0.749048i \(-0.269489\pi\)
0.662516 + 0.749048i \(0.269489\pi\)
\(18\) 0 0
\(19\) −98.2765 −0.0624548 −0.0312274 − 0.999512i \(-0.509942\pi\)
−0.0312274 + 0.999512i \(0.509942\pi\)
\(20\) 1954.97 1.09286
\(21\) 0 0
\(22\) 158.354 0.0697546
\(23\) −1607.16 −0.633489 −0.316745 − 0.948511i \(-0.602590\pi\)
−0.316745 + 0.948511i \(0.602590\pi\)
\(24\) 0 0
\(25\) 652.591 0.208829
\(26\) −74.0976 −0.0214966
\(27\) 0 0
\(28\) 5162.68 1.24446
\(29\) 307.045 0.0677966 0.0338983 − 0.999425i \(-0.489208\pi\)
0.0338983 + 0.999425i \(0.489208\pi\)
\(30\) 0 0
\(31\) 2936.42 0.548801 0.274400 − 0.961616i \(-0.411521\pi\)
0.274400 + 0.961616i \(0.411521\pi\)
\(32\) 1336.14 0.230662
\(33\) 0 0
\(34\) 692.255 0.102700
\(35\) 9975.84 1.37651
\(36\) 0 0
\(37\) −12222.3 −1.46773 −0.733867 − 0.679294i \(-0.762286\pi\)
−0.733867 + 0.679294i \(0.762286\pi\)
\(38\) −43.0891 −0.00484070
\(39\) 0 0
\(40\) 1719.48 0.169921
\(41\) 104.151 0.00967619 0.00483809 − 0.999988i \(-0.498460\pi\)
0.00483809 + 0.999988i \(0.498460\pi\)
\(42\) 0 0
\(43\) 10936.4 0.901994 0.450997 − 0.892526i \(-0.351069\pi\)
0.450997 + 0.892526i \(0.351069\pi\)
\(44\) −11488.0 −0.894569
\(45\) 0 0
\(46\) −704.654 −0.0491000
\(47\) 14949.2 0.987129 0.493564 − 0.869709i \(-0.335694\pi\)
0.493564 + 0.869709i \(0.335694\pi\)
\(48\) 0 0
\(49\) 9537.11 0.567449
\(50\) 286.127 0.0161858
\(51\) 0 0
\(52\) 5375.51 0.275684
\(53\) 35911.5 1.75608 0.878040 − 0.478587i \(-0.158851\pi\)
0.878040 + 0.478587i \(0.158851\pi\)
\(54\) 0 0
\(55\) −22198.3 −0.989494
\(56\) 4540.80 0.193492
\(57\) 0 0
\(58\) 134.623 0.00525472
\(59\) 1598.46 0.0597822 0.0298911 − 0.999553i \(-0.490484\pi\)
0.0298911 + 0.999553i \(0.490484\pi\)
\(60\) 0 0
\(61\) 20156.2 0.693560 0.346780 − 0.937947i \(-0.387275\pi\)
0.346780 + 0.937947i \(0.387275\pi\)
\(62\) 1287.47 0.0425360
\(63\) 0 0
\(64\) −31592.8 −0.964136
\(65\) 10387.1 0.304937
\(66\) 0 0
\(67\) −35368.1 −0.962552 −0.481276 − 0.876569i \(-0.659827\pi\)
−0.481276 + 0.876569i \(0.659827\pi\)
\(68\) −50220.6 −1.31707
\(69\) 0 0
\(70\) 4373.88 0.106689
\(71\) −26140.3 −0.615411 −0.307706 − 0.951482i \(-0.599561\pi\)
−0.307706 + 0.951482i \(0.599561\pi\)
\(72\) 0 0
\(73\) 75468.4 1.65752 0.828759 − 0.559606i \(-0.189048\pi\)
0.828759 + 0.559606i \(0.189048\pi\)
\(74\) −5358.81 −0.113760
\(75\) 0 0
\(76\) 3125.96 0.0620796
\(77\) −58621.1 −1.12675
\(78\) 0 0
\(79\) −7576.72 −0.136588 −0.0682942 − 0.997665i \(-0.521756\pi\)
−0.0682942 + 0.997665i \(0.521756\pi\)
\(80\) −61805.2 −1.07969
\(81\) 0 0
\(82\) 45.6648 0.000749974 0
\(83\) 912.974 0.0145466 0.00727332 − 0.999974i \(-0.497685\pi\)
0.00727332 + 0.999974i \(0.497685\pi\)
\(84\) 0 0
\(85\) −97041.2 −1.45683
\(86\) 4795.04 0.0699110
\(87\) 0 0
\(88\) −10104.2 −0.139090
\(89\) −106709. −1.42799 −0.713995 − 0.700151i \(-0.753116\pi\)
−0.713995 + 0.700151i \(0.753116\pi\)
\(90\) 0 0
\(91\) 27430.2 0.347236
\(92\) 51120.1 0.629684
\(93\) 0 0
\(94\) 6554.44 0.0765096
\(95\) 6040.28 0.0686670
\(96\) 0 0
\(97\) 103676. 1.11879 0.559397 − 0.828900i \(-0.311033\pi\)
0.559397 + 0.828900i \(0.311033\pi\)
\(98\) 4181.52 0.0439814
\(99\) 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 117.6.a.c.1.1 2
3.2 odd 2 13.6.a.a.1.2 ✓ 2
12.11 even 2 208.6.a.h.1.2 2
15.2 even 4 325.6.b.b.274.2 4
15.8 even 4 325.6.b.b.274.3 4
15.14 odd 2 325.6.a.b.1.1 2
21.20 even 2 637.6.a.a.1.2 2
24.5 odd 2 832.6.a.p.1.2 2
24.11 even 2 832.6.a.i.1.1 2
39.5 even 4 169.6.b.a.168.3 4
39.8 even 4 169.6.b.a.168.2 4
39.38 odd 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 3.2 odd 2
117.6.a.c.1.1 2 1.1 even 1 trivial
169.6.a.a.1.1 2 39.38 odd 2
169.6.b.a.168.2 4 39.8 even 4
169.6.b.a.168.3 4 39.5 even 4
208.6.a.h.1.2 2 12.11 even 2
325.6.a.b.1.1 2 15.14 odd 2
325.6.b.b.274.2 4 15.2 even 4
325.6.b.b.274.3 4 15.8 even 4
637.6.a.a.1.2 2 21.20 even 2
832.6.a.i.1.1 2 24.11 even 2
832.6.a.p.1.2 2 24.5 odd 2