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Properties

Label 117.6.a.c.1.2
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 1.2
Root \(-1.56155\) 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)
 
\(f(q)\) \(=\) \(q+4.56155 q^{2} -11.1922 q^{4} +103.462 q^{5} +126.309 q^{7} -197.024 q^{8} +471.948 q^{10} +14.8296 q^{11} -169.000 q^{13} +576.164 q^{14} -540.582 q^{16} +1051.12 q^{17} -213.723 q^{19} -1157.97 q^{20} +67.6458 q^{22} +4231.16 q^{23} +7579.41 q^{25} -770.902 q^{26} -1413.68 q^{28} +504.955 q^{29} +4783.58 q^{31} +3838.86 q^{32} +4794.75 q^{34} +13068.2 q^{35} -4635.74 q^{37} -974.911 q^{38} -20384.5 q^{40} -7944.15 q^{41} -8516.41 q^{43} -165.976 q^{44} +19300.7 q^{46} -24921.2 q^{47} -853.113 q^{49} +34573.9 q^{50} +1891.49 q^{52} +7808.46 q^{53} +1534.30 q^{55} -24885.8 q^{56} +2303.38 q^{58} +37337.5 q^{59} -18172.2 q^{61} +21820.5 q^{62} +34809.8 q^{64} -17485.1 q^{65} -34559.9 q^{67} -11764.4 q^{68} +59611.1 q^{70} -41255.7 q^{71} -1056.42 q^{73} -21146.2 q^{74} +2392.04 q^{76} +1873.10 q^{77} -47719.3 q^{79} -55929.8 q^{80} -36237.7 q^{82} +74799.0 q^{83} +108751. q^{85} -38848.0 q^{86} -2921.78 q^{88} -9799.26 q^{89} -21346.2 q^{91} -47356.1 q^{92} -113679. q^{94} -22112.3 q^{95} -138432. q^{97} -3891.52 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} - 43 q^{4} + 42 q^{5} - 36 q^{7} - 225 q^{8} + 445 q^{10} + 376 q^{11} - 338 q^{13} + 505 q^{14} + 465 q^{16} + 2630 q^{17} - 312 q^{19} + 797 q^{20} + 226 q^{22} + 2624 q^{23} + 8232 q^{25}+ \cdots + 290 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.56155 0.806376 0.403188 − 0.915117i \(-0.367902\pi\)
0.403188 + 0.915117i \(0.367902\pi\)
\(3\) 0 0
\(4\) −11.1922 −0.349757
\(5\) 103.462 1.85079 0.925393 − 0.379008i \(-0.123735\pi\)
0.925393 + 0.379008i \(0.123735\pi\)
\(6\) 0 0
\(7\) 126.309 0.974290 0.487145 − 0.873321i \(-0.338038\pi\)
0.487145 + 0.873321i \(0.338038\pi\)
\(8\) −197.024 −1.08841
\(9\) 0 0
\(10\) 471.948 1.49243
\(11\) 14.8296 0.0369527 0.0184764 − 0.999829i \(-0.494118\pi\)
0.0184764 + 0.999829i \(0.494118\pi\)
\(12\) 0 0
\(13\) −169.000 −0.277350
\(14\) 576.164 0.785644
\(15\) 0 0
\(16\) −540.582 −0.527912
\(17\) 1051.12 0.882126 0.441063 − 0.897476i \(-0.354602\pi\)
0.441063 + 0.897476i \(0.354602\pi\)
\(18\) 0 0
\(19\) −213.723 −0.135821 −0.0679107 − 0.997691i \(-0.521633\pi\)
−0.0679107 + 0.997691i \(0.521633\pi\)
\(20\) −1157.97 −0.647326
\(21\) 0 0
\(22\) 67.6458 0.0297978
\(23\) 4231.16 1.66778 0.833892 − 0.551928i \(-0.186108\pi\)
0.833892 + 0.551928i \(0.186108\pi\)
\(24\) 0 0
\(25\) 7579.41 2.42541
\(26\) −770.902 −0.223649
\(27\) 0 0
\(28\) −1413.68 −0.340765
\(29\) 504.955 0.111495 0.0557477 − 0.998445i \(-0.482246\pi\)
0.0557477 + 0.998445i \(0.482246\pi\)
\(30\) 0 0
\(31\) 4783.58 0.894022 0.447011 − 0.894528i \(-0.352488\pi\)
0.447011 + 0.894528i \(0.352488\pi\)
\(32\) 3838.86 0.662716
\(33\) 0 0
\(34\) 4794.75 0.711325
\(35\) 13068.2 1.80320
\(36\) 0 0
\(37\) −4635.74 −0.556692 −0.278346 − 0.960481i \(-0.589786\pi\)
−0.278346 + 0.960481i \(0.589786\pi\)
\(38\) −974.911 −0.109523
\(39\) 0 0
\(40\) −20384.5 −2.01442
\(41\) −7944.15 −0.738054 −0.369027 − 0.929419i \(-0.620309\pi\)
−0.369027 + 0.929419i \(0.620309\pi\)
\(42\) 0 0
\(43\) −8516.41 −0.702401 −0.351201 − 0.936300i \(-0.614226\pi\)
−0.351201 + 0.936300i \(0.614226\pi\)
\(44\) −165.976 −0.0129245
\(45\) 0 0
\(46\) 19300.7 1.34486
\(47\) −24921.2 −1.64560 −0.822801 − 0.568330i \(-0.807590\pi\)
−0.822801 + 0.568330i \(0.807590\pi\)
\(48\) 0 0
\(49\) −853.113 −0.0507594
\(50\) 34573.9 1.95579
\(51\) 0 0
\(52\) 1891.49 0.0970052
\(53\) 7808.46 0.381835 0.190917 − 0.981606i \(-0.438854\pi\)
0.190917 + 0.981606i \(0.438854\pi\)
\(54\) 0 0
\(55\) 1534.30 0.0683916
\(56\) −24885.8 −1.06043
\(57\) 0 0
\(58\) 2303.38 0.0899073
\(59\) 37337.5 1.39642 0.698209 − 0.715894i \(-0.253980\pi\)
0.698209 + 0.715894i \(0.253980\pi\)
\(60\) 0 0
\(61\) −18172.2 −0.625292 −0.312646 − 0.949870i \(-0.601215\pi\)
−0.312646 + 0.949870i \(0.601215\pi\)
\(62\) 21820.5 0.720918
\(63\) 0 0
\(64\) 34809.8 1.06231
\(65\) −17485.1 −0.513316
\(66\) 0 0
\(67\) −34559.9 −0.940559 −0.470279 − 0.882518i \(-0.655847\pi\)
−0.470279 + 0.882518i \(0.655847\pi\)
\(68\) −11764.4 −0.308530
\(69\) 0 0
\(70\) 59611.1 1.45406
\(71\) −41255.7 −0.971265 −0.485632 − 0.874163i \(-0.661411\pi\)
−0.485632 + 0.874163i \(0.661411\pi\)
\(72\) 0 0
\(73\) −1056.42 −0.0232022 −0.0116011 − 0.999933i \(-0.503693\pi\)
−0.0116011 + 0.999933i \(0.503693\pi\)
\(74\) −21146.2 −0.448903
\(75\) 0 0
\(76\) 2392.04 0.0475045
\(77\) 1873.10 0.0360027
\(78\) 0 0
\(79\) −47719.3 −0.860253 −0.430126 − 0.902769i \(-0.641531\pi\)
−0.430126 + 0.902769i \(0.641531\pi\)
\(80\) −55929.8 −0.977053
\(81\) 0 0
\(82\) −36237.7 −0.595149
\(83\) 74799.0 1.19179 0.595896 − 0.803061i \(-0.296797\pi\)
0.595896 + 0.803061i \(0.296797\pi\)
\(84\) 0 0
\(85\) 108751. 1.63263
\(86\) −38848.0 −0.566400
\(87\) 0 0
\(88\) −2921.78 −0.0402198
\(89\) −9799.26 −0.131135 −0.0655675 − 0.997848i \(-0.520886\pi\)
−0.0655675 + 0.997848i \(0.520886\pi\)
\(90\) 0 0
\(91\) −21346.2 −0.270219
\(92\) −47356.1 −0.583320
\(93\) 0 0
\(94\) −113679. −1.32697
\(95\) −22112.3 −0.251376
\(96\) 0 0
\(97\) −138432. −1.49385 −0.746927 − 0.664906i \(-0.768472\pi\)
−0.746927 + 0.664906i \(0.768472\pi\)
\(98\) −3891.52 −0.0409312
\(99\) 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.1.2 2
3.2 odd 2 13.6.a.a.1.1 ✓ 2
12.11 even 2 208.6.a.h.1.1 2
15.2 even 4 325.6.b.b.274.1 4
15.8 even 4 325.6.b.b.274.4 4
15.14 odd 2 325.6.a.b.1.2 2
21.20 even 2 637.6.a.a.1.1 2
24.5 odd 2 832.6.a.p.1.1 2
24.11 even 2 832.6.a.i.1.2 2
39.5 even 4 169.6.b.a.168.4 4
39.8 even 4 169.6.b.a.168.1 4
39.38 odd 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 3.2 odd 2
117.6.a.c.1.2 2 1.1 even 1 trivial
169.6.a.a.1.2 2 39.38 odd 2
169.6.b.a.168.1 4 39.8 even 4
169.6.b.a.168.4 4 39.5 even 4
208.6.a.h.1.1 2 12.11 even 2
325.6.a.b.1.2 2 15.14 odd 2
325.6.b.b.274.1 4 15.2 even 4
325.6.b.b.274.4 4 15.8 even 4
637.6.a.a.1.1 2 21.20 even 2
832.6.a.i.1.2 2 24.11 even 2
832.6.a.p.1.1 2 24.5 odd 2