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Properties

Label 117.4.a.c.1.2
Level $117$
Weight $4$
Character 117.1
Self dual yes
Analytic conductor $6.903$
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] = [117,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2] 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: \(6.90322347067\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) 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)
 
\(f(q)\) \(=\) \(q+2.74166 q^{2} -0.483315 q^{4} -19.4833 q^{5} +7.48331 q^{7} -23.2583 q^{8} -53.4166 q^{10} -22.8999 q^{11} -13.0000 q^{13} +20.5167 q^{14} -59.8999 q^{16} -67.0334 q^{17} +16.5167 q^{19} +9.41657 q^{20} -62.7836 q^{22} +175.600 q^{23} +254.600 q^{25} -35.6415 q^{26} -3.61680 q^{28} -291.800 q^{29} +117.283 q^{31} +21.8418 q^{32} -183.783 q^{34} -145.800 q^{35} -154.766 q^{37} +45.2831 q^{38} +453.150 q^{40} +251.716 q^{41} -502.566 q^{43} +11.0679 q^{44} +481.434 q^{46} +281.733 q^{47} -287.000 q^{49} +698.025 q^{50} +6.28309 q^{52} -366.999 q^{53} +446.166 q^{55} -174.049 q^{56} -800.015 q^{58} +79.6663 q^{59} -194.865 q^{61} +321.550 q^{62} +539.082 q^{64} +253.283 q^{65} +400.082 q^{67} +32.3982 q^{68} -399.733 q^{70} -528.299 q^{71} -734.366 q^{73} -424.316 q^{74} -7.98276 q^{76} -171.367 q^{77} +113.266 q^{79} +1167.05 q^{80} +690.118 q^{82} +933.466 q^{83} +1306.03 q^{85} -1377.86 q^{86} +532.613 q^{88} -1190.91 q^{89} -97.2831 q^{91} -84.8699 q^{92} +772.415 q^{94} -321.800 q^{95} +557.165 q^{97} -786.856 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28}+ \cdots + 574 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.74166 0.969322 0.484661 − 0.874702i \(-0.338943\pi\)
0.484661 + 0.874702i \(0.338943\pi\)
\(3\) 0 0
\(4\) −0.483315 −0.0604143
\(5\) −19.4833 −1.74264 −0.871320 − 0.490715i \(-0.836736\pi\)
−0.871320 + 0.490715i \(0.836736\pi\)
\(6\) 0 0
\(7\) 7.48331 0.404061 0.202031 − 0.979379i \(-0.435246\pi\)
0.202031 + 0.979379i \(0.435246\pi\)
\(8\) −23.2583 −1.02788
\(9\) 0 0
\(10\) −53.4166 −1.68918
\(11\) −22.8999 −0.627689 −0.313844 − 0.949474i \(-0.601617\pi\)
−0.313844 + 0.949474i \(0.601617\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 20.5167 0.391665
\(15\) 0 0
\(16\) −59.8999 −0.935936
\(17\) −67.0334 −0.956352 −0.478176 − 0.878264i \(-0.658702\pi\)
−0.478176 + 0.878264i \(0.658702\pi\)
\(18\) 0 0
\(19\) 16.5167 0.199431 0.0997155 − 0.995016i \(-0.468207\pi\)
0.0997155 + 0.995016i \(0.468207\pi\)
\(20\) 9.41657 0.105280
\(21\) 0 0
\(22\) −62.7836 −0.608433
\(23\) 175.600 1.59196 0.795979 − 0.605324i \(-0.206956\pi\)
0.795979 + 0.605324i \(0.206956\pi\)
\(24\) 0 0
\(25\) 254.600 2.03680
\(26\) −35.6415 −0.268842
\(27\) 0 0
\(28\) −3.61680 −0.0244111
\(29\) −291.800 −1.86848 −0.934239 − 0.356648i \(-0.883920\pi\)
−0.934239 + 0.356648i \(0.883920\pi\)
\(30\) 0 0
\(31\) 117.283 0.679505 0.339753 − 0.940515i \(-0.389657\pi\)
0.339753 + 0.940515i \(0.389657\pi\)
\(32\) 21.8418 0.120660
\(33\) 0 0
\(34\) −183.783 −0.927013
\(35\) −145.800 −0.704133
\(36\) 0 0
\(37\) −154.766 −0.687661 −0.343830 − 0.939032i \(-0.611724\pi\)
−0.343830 + 0.939032i \(0.611724\pi\)
\(38\) 45.2831 0.193313
\(39\) 0 0
\(40\) 453.150 1.79123
\(41\) 251.716 0.958815 0.479407 − 0.877592i \(-0.340852\pi\)
0.479407 + 0.877592i \(0.340852\pi\)
\(42\) 0 0
\(43\) −502.566 −1.78234 −0.891170 − 0.453669i \(-0.850115\pi\)
−0.891170 + 0.453669i \(0.850115\pi\)
\(44\) 11.0679 0.0379214
\(45\) 0 0
\(46\) 481.434 1.54312
\(47\) 281.733 0.874361 0.437181 − 0.899374i \(-0.355977\pi\)
0.437181 + 0.899374i \(0.355977\pi\)
\(48\) 0 0
\(49\) −287.000 −0.836735
\(50\) 698.025 1.97431
\(51\) 0 0
\(52\) 6.28309 0.0167559
\(53\) −366.999 −0.951154 −0.475577 − 0.879674i \(-0.657761\pi\)
−0.475577 + 0.879674i \(0.657761\pi\)
\(54\) 0 0
\(55\) 446.166 1.09384
\(56\) −174.049 −0.415328
\(57\) 0 0
\(58\) −800.015 −1.81116
\(59\) 79.6663 0.175791 0.0878955 − 0.996130i \(-0.471986\pi\)
0.0878955 + 0.996130i \(0.471986\pi\)
\(60\) 0 0
\(61\) −194.865 −0.409016 −0.204508 − 0.978865i \(-0.565559\pi\)
−0.204508 + 0.978865i \(0.565559\pi\)
\(62\) 321.550 0.658660
\(63\) 0 0
\(64\) 539.082 1.05289
\(65\) 253.283 0.483322
\(66\) 0 0
\(67\) 400.082 0.729519 0.364759 − 0.931102i \(-0.381151\pi\)
0.364759 + 0.931102i \(0.381151\pi\)
\(68\) 32.3982 0.0577774
\(69\) 0 0
\(70\) −399.733 −0.682532
\(71\) −528.299 −0.883065 −0.441532 − 0.897245i \(-0.645565\pi\)
−0.441532 + 0.897245i \(0.645565\pi\)
\(72\) 0 0
\(73\) −734.366 −1.17741 −0.588706 − 0.808347i \(-0.700362\pi\)
−0.588706 + 0.808347i \(0.700362\pi\)
\(74\) −424.316 −0.666565
\(75\) 0 0
\(76\) −7.98276 −0.0120485
\(77\) −171.367 −0.253625
\(78\) 0 0
\(79\) 113.266 0.161309 0.0806545 − 0.996742i \(-0.474299\pi\)
0.0806545 + 0.996742i \(0.474299\pi\)
\(80\) 1167.05 1.63100
\(81\) 0 0
\(82\) 690.118 0.929400
\(83\) 933.466 1.23447 0.617236 − 0.786778i \(-0.288252\pi\)
0.617236 + 0.786778i \(0.288252\pi\)
\(84\) 0 0
\(85\) 1306.03 1.66658
\(86\) −1377.86 −1.72766
\(87\) 0 0
\(88\) 532.613 0.645191
\(89\) −1190.91 −1.41839 −0.709195 − 0.705012i \(-0.750941\pi\)
−0.709195 + 0.705012i \(0.750941\pi\)
\(90\) 0 0
\(91\) −97.2831 −0.112066
\(92\) −84.8699 −0.0961771
\(93\) 0 0
\(94\) 772.415 0.847538
\(95\) −321.800 −0.347536
\(96\) 0 0
\(97\) 557.165 0.583211 0.291606 − 0.956539i \(-0.405811\pi\)
0.291606 + 0.956539i \(0.405811\pi\)
\(98\) −786.856 −0.811066
\(99\) 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 117.4.a.c.1.2 2
3.2 odd 2 39.4.a.b.1.1 ✓ 2
4.3 odd 2 1872.4.a.t.1.1 2
12.11 even 2 624.4.a.r.1.2 2
13.12 even 2 1521.4.a.s.1.1 2
15.14 odd 2 975.4.a.j.1.2 2
21.20 even 2 1911.4.a.h.1.1 2
24.5 odd 2 2496.4.a.bc.1.1 2
24.11 even 2 2496.4.a.s.1.1 2
39.5 even 4 507.4.b.f.337.3 4
39.8 even 4 507.4.b.f.337.2 4
39.38 odd 2 507.4.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 ✓ 2 3.2 odd 2
117.4.a.c.1.2 2 1.1 even 1 trivial
507.4.a.f.1.2 2 39.38 odd 2
507.4.b.f.337.2 4 39.8 even 4
507.4.b.f.337.3 4 39.5 even 4
624.4.a.r.1.2 2 12.11 even 2
975.4.a.j.1.2 2 15.14 odd 2
1521.4.a.s.1.1 2 13.12 even 2
1872.4.a.t.1.1 2 4.3 odd 2
1911.4.a.h.1.1 2 21.20 even 2
2496.4.a.s.1.1 2 24.11 even 2
2496.4.a.bc.1.1 2 24.5 odd 2