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Properties

Label 117.4.q.e.10.1
Level $117$
Weight $4$
Character 117.10
Analytic conductor $6.903$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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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(10,117)] 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("117.10"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0] 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: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 10.1
Root \(5.04537i\) of defining polynomial
Character \(\chi\) \(=\) 117.10
Dual form 117.4.q.e.82.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.36942 + 2.52268i) q^{2} +(8.72787 - 15.1171i) q^{4} -20.1174i q^{5} +(-13.3609 - 7.71395i) q^{7} +47.7076i q^{8} +(50.7498 + 87.9013i) q^{10} +(-23.3283 + 13.4686i) q^{11} +(-3.96071 + 46.7045i) q^{13} +77.8394 q^{14} +(-50.5283 - 87.5177i) q^{16} +(-11.6167 + 20.1207i) q^{17} +(-39.0399 - 22.5397i) q^{19} +(-304.117 - 175.582i) q^{20} +(67.9540 - 117.700i) q^{22} +(71.0050 + 122.984i) q^{23} -279.710 q^{25} +(-100.515 - 214.063i) q^{26} +(-233.225 + 134.653i) q^{28} +(1.14534 + 1.98379i) q^{29} +37.7740i q^{31} +(111.031 + 64.1035i) q^{32} -117.221i q^{34} +(-155.185 + 268.787i) q^{35} +(271.793 - 156.920i) q^{37} +227.442 q^{38} +959.753 q^{40} +(-5.08201 + 2.93410i) q^{41} +(-180.449 + 312.547i) q^{43} +470.209i q^{44} +(-620.501 - 358.246i) q^{46} +209.748i q^{47} +(-52.4900 - 90.9154i) q^{49} +(1222.17 - 705.619i) q^{50} +(671.469 + 467.505i) q^{52} -276.886 q^{53} +(270.953 + 469.305i) q^{55} +(368.014 - 637.419i) q^{56} +(-10.0089 - 5.77866i) q^{58} +(-470.415 - 271.594i) q^{59} +(-102.894 + 178.218i) q^{61} +(-95.2917 - 165.050i) q^{62} +161.602 q^{64} +(939.573 + 79.6791i) q^{65} +(-426.585 + 246.289i) q^{67} +(202.778 + 351.222i) q^{68} -1565.93i q^{70} +(-716.081 - 413.430i) q^{71} +66.1205i q^{73} +(-791.718 + 1371.30i) q^{74} +(-681.470 + 393.447i) q^{76} +415.584 q^{77} +317.642 q^{79} +(-1760.63 + 1016.50i) q^{80} +(14.8036 - 25.6406i) q^{82} +141.450i q^{83} +(404.777 + 233.698i) q^{85} -1820.86i q^{86} +(-642.555 - 1112.94i) q^{88} +(-555.399 + 320.660i) q^{89} +(413.195 - 593.464i) q^{91} +2478.89 q^{92} +(-529.129 - 916.478i) q^{94} +(-453.440 + 785.381i) q^{95} +(-965.551 - 557.461i) q^{97} +(458.702 + 264.832i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{4} + 30 q^{7} + 40 q^{10} - 60 q^{11} + 25 q^{13} + 60 q^{14} - 250 q^{16} - 105 q^{17} + 180 q^{19} - 510 q^{20} - 290 q^{22} + 60 q^{23} - 960 q^{25} + 30 q^{26} + 150 q^{28} + 495 q^{29}+ \cdots - 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.36942 + 2.52268i −1.54482 + 0.891903i −0.546298 + 0.837591i \(0.683963\pi\)
−0.998524 + 0.0543124i \(0.982703\pi\)
\(3\) 0 0
\(4\) 8.72787 − 15.1171i 1.09098 − 1.88964i
\(5\) − 20.1174i − 1.79935i −0.436556 − 0.899677i \(-0.643802\pi\)
0.436556 − 0.899677i \(-0.356198\pi\)
\(6\) 0 0
\(7\) −13.3609 − 7.71395i −0.721423 − 0.416514i 0.0938530 − 0.995586i \(-0.470082\pi\)
−0.815276 + 0.579072i \(0.803415\pi\)
\(8\) 47.7076i 2.10840i
\(9\) 0 0
\(10\) 50.7498 + 87.9013i 1.60485 + 2.77968i
\(11\) −23.3283 + 13.4686i −0.639432 + 0.369176i −0.784396 − 0.620261i \(-0.787027\pi\)
0.144964 + 0.989437i \(0.453693\pi\)
\(12\) 0 0
\(13\) −3.96071 + 46.7045i −0.0845002 + 0.996423i
\(14\) 77.8394 1.48596
\(15\) 0 0
\(16\) −50.5283 − 87.5177i −0.789505 − 1.36746i
\(17\) −11.6167 + 20.1207i −0.165733 + 0.287059i −0.936915 − 0.349556i \(-0.886332\pi\)
0.771182 + 0.636615i \(0.219666\pi\)
\(18\) 0 0
\(19\) −39.0399 − 22.5397i −0.471388 − 0.272156i 0.245433 − 0.969414i \(-0.421070\pi\)
−0.716821 + 0.697258i \(0.754403\pi\)
\(20\) −304.117 − 175.582i −3.40013 − 1.96307i
\(21\) 0 0
\(22\) 67.9540 − 117.700i 0.658539 − 1.14062i
\(23\) 71.0050 + 122.984i 0.643720 + 1.11496i 0.984595 + 0.174848i \(0.0559434\pi\)
−0.340875 + 0.940109i \(0.610723\pi\)
\(24\) 0 0
\(25\) −279.710 −2.23768
\(26\) −100.515 − 214.063i −0.758176 − 1.61466i
\(27\) 0 0
\(28\) −233.225 + 134.653i −1.57412 + 0.908820i
\(29\) 1.14534 + 1.98379i 0.00733394 + 0.0127028i 0.869669 − 0.493635i \(-0.164332\pi\)
−0.862335 + 0.506338i \(0.830999\pi\)
\(30\) 0 0
\(31\) 37.7740i 0.218852i 0.993995 + 0.109426i \(0.0349012\pi\)
−0.993995 + 0.109426i \(0.965099\pi\)
\(32\) 111.031 + 64.1035i 0.613363 + 0.354125i
\(33\) 0 0
\(34\) − 117.221i − 0.591273i
\(35\) −155.185 + 268.787i −0.749456 + 1.29810i
\(36\) 0 0
\(37\) 271.793 − 156.920i 1.20764 − 0.697228i 0.245393 − 0.969424i \(-0.421083\pi\)
0.962242 + 0.272195i \(0.0877497\pi\)
\(38\) 227.442 0.970947
\(39\) 0 0
\(40\) 959.753 3.79376
\(41\) −5.08201 + 2.93410i −0.0193580 + 0.0111763i −0.509648 − 0.860383i \(-0.670224\pi\)
0.490290 + 0.871559i \(0.336891\pi\)
\(42\) 0 0
\(43\) −180.449 + 312.547i −0.639958 + 1.10844i 0.345483 + 0.938425i \(0.387715\pi\)
−0.985441 + 0.170015i \(0.945618\pi\)
\(44\) 470.209i 1.61106i
\(45\) 0 0
\(46\) −620.501 − 358.246i −1.98887 − 1.14827i
\(47\) 209.748i 0.650956i 0.945550 + 0.325478i \(0.105525\pi\)
−0.945550 + 0.325478i \(0.894475\pi\)
\(48\) 0 0
\(49\) −52.4900 − 90.9154i −0.153032 − 0.265060i
\(50\) 1222.17 − 705.619i 3.45681 − 1.99579i
\(51\) 0 0
\(52\) 671.469 + 467.505i 1.79069 + 1.24676i
\(53\) −276.886 −0.717609 −0.358804 − 0.933413i \(-0.616815\pi\)
−0.358804 + 0.933413i \(0.616815\pi\)
\(54\) 0 0
\(55\) 270.953 + 469.305i 0.664278 + 1.15056i
\(56\) 368.014 − 637.419i 0.878178 − 1.52105i
\(57\) 0 0
\(58\) −10.0089 − 5.77866i −0.0226593 − 0.0130823i
\(59\) −470.415 − 271.594i −1.03801 − 0.599298i −0.118744 − 0.992925i \(-0.537887\pi\)
−0.919270 + 0.393627i \(0.871220\pi\)
\(60\) 0 0
\(61\) −102.894 + 178.218i −0.215971 + 0.374073i −0.953573 − 0.301163i \(-0.902625\pi\)
0.737601 + 0.675236i \(0.235958\pi\)
\(62\) −95.2917 − 165.050i −0.195195 − 0.338087i
\(63\) 0 0
\(64\) 161.602 0.315629
\(65\) 939.573 + 79.6791i 1.79292 + 0.152046i
\(66\) 0 0
\(67\) −426.585 + 246.289i −0.777846 + 0.449090i −0.835666 − 0.549237i \(-0.814918\pi\)
0.0578203 + 0.998327i \(0.481585\pi\)
\(68\) 202.778 + 351.222i 0.361625 + 0.626352i
\(69\) 0 0
\(70\) − 1565.93i − 2.67377i
\(71\) −716.081 − 413.430i −1.19695 − 0.691057i −0.237073 − 0.971492i \(-0.576188\pi\)
−0.959873 + 0.280435i \(0.909521\pi\)
\(72\) 0 0
\(73\) 66.1205i 0.106011i 0.998594 + 0.0530056i \(0.0168801\pi\)
−0.998594 + 0.0530056i \(0.983120\pi\)
\(74\) −791.718 + 1371.30i −1.24372 + 2.15419i
\(75\) 0 0
\(76\) −681.470 + 393.447i −1.02855 + 0.593835i
\(77\) 415.584 0.615068
\(78\) 0 0
\(79\) 317.642 0.452374 0.226187 − 0.974084i \(-0.427374\pi\)
0.226187 + 0.974084i \(0.427374\pi\)
\(80\) −1760.63 + 1016.50i −2.46055 + 1.42060i
\(81\) 0 0
\(82\) 14.8036 − 25.6406i 0.0199364 − 0.0345309i
\(83\) 141.450i 0.187063i 0.995616 + 0.0935313i \(0.0298155\pi\)
−0.995616 + 0.0935313i \(0.970184\pi\)
\(84\) 0 0
\(85\) 404.777 + 233.698i 0.516520 + 0.298213i
\(86\) − 1820.86i − 2.28312i
\(87\) 0 0
\(88\) −642.555 − 1112.94i −0.778371 − 1.34818i
\(89\) −555.399 + 320.660i −0.661486 + 0.381909i −0.792843 − 0.609426i \(-0.791400\pi\)
0.131357 + 0.991335i \(0.458067\pi\)
\(90\) 0 0
\(91\) 413.195 − 593.464i 0.475985 − 0.683648i
\(92\) 2478.89 2.80915
\(93\) 0 0
\(94\) −529.129 − 916.478i −0.580590 − 1.00561i
\(95\) −453.440 + 785.381i −0.489705 + 0.848194i
\(96\) 0 0
\(97\) −965.551 − 557.461i −1.01069 − 0.583522i −0.0992962 − 0.995058i \(-0.531659\pi\)
−0.911394 + 0.411536i \(0.864992\pi\)
\(98\) 458.702 + 264.832i 0.472815 + 0.272980i
\(99\) 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.q.e.10.1 10
3.2 odd 2 39.4.j.c.10.5 yes 10
12.11 even 2 624.4.bv.h.49.5 10
13.2 odd 12 1521.4.a.bk.1.2 10
13.4 even 6 inner 117.4.q.e.82.1 10
13.11 odd 12 1521.4.a.bk.1.9 10
39.2 even 12 507.4.a.r.1.9 10
39.11 even 12 507.4.a.r.1.2 10
39.17 odd 6 39.4.j.c.4.5 ✓ 10
39.23 odd 6 507.4.b.i.337.2 10
39.29 odd 6 507.4.b.i.337.9 10
156.95 even 6 624.4.bv.h.433.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.5 ✓ 10 39.17 odd 6
39.4.j.c.10.5 yes 10 3.2 odd 2
117.4.q.e.10.1 10 1.1 even 1 trivial
117.4.q.e.82.1 10 13.4 even 6 inner
507.4.a.r.1.2 10 39.11 even 12
507.4.a.r.1.9 10 39.2 even 12
507.4.b.i.337.2 10 39.23 odd 6
507.4.b.i.337.9 10 39.29 odd 6
624.4.bv.h.49.5 10 12.11 even 2
624.4.bv.h.433.1 10 156.95 even 6
1521.4.a.bk.1.2 10 13.2 odd 12
1521.4.a.bk.1.9 10 13.11 odd 12