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Properties

Label 117.4.q.e.10.3
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 10.3
Root \(-0.917374i\) of defining polynomial
Character \(\chi\) \(=\) 117.10
Dual form 117.4.q.e.82.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.794469 - 0.458687i) q^{2} +(-3.57921 + 6.19938i) q^{4} -15.4704i q^{5} +(17.8257 + 10.2917i) q^{7} +13.9059i q^{8} +(-7.09608 - 12.2908i) q^{10} +(57.0209 - 32.9210i) q^{11} +(19.2429 - 42.7400i) q^{13} +18.8826 q^{14} +(-22.2552 - 38.5472i) q^{16} +(-22.1478 + 38.3611i) q^{17} +(127.352 + 73.5266i) q^{19} +(95.9069 + 55.3719i) q^{20} +(30.2009 - 52.3094i) q^{22} +(-26.5793 - 46.0367i) q^{23} -114.334 q^{25} +(-4.31639 - 42.7821i) q^{26} +(-127.604 + 73.6721i) q^{28} +(-19.3128 - 33.4508i) q^{29} -88.3894i q^{31} +(-131.705 - 76.0401i) q^{32} +40.6357i q^{34} +(159.216 - 275.771i) q^{35} +(68.3803 - 39.4794i) q^{37} +134.903 q^{38} +215.131 q^{40} +(-307.410 + 177.483i) q^{41} +(-203.923 + 353.205i) q^{43} +471.325i q^{44} +(-42.2329 - 24.3832i) q^{46} +67.9674i q^{47} +(40.3369 + 69.8656i) q^{49} +(-90.8345 + 52.4433i) q^{50} +(196.087 + 272.270i) q^{52} -226.572 q^{53} +(-509.302 - 882.136i) q^{55} +(-143.115 + 247.883i) q^{56} +(-30.6869 - 17.7171i) q^{58} +(123.002 + 71.0154i) q^{59} +(-133.416 + 231.083i) q^{61} +(-40.5431 - 70.2227i) q^{62} +216.569 q^{64} +(-661.206 - 297.696i) q^{65} +(356.098 - 205.593i) q^{67} +(-158.543 - 274.605i) q^{68} -292.122i q^{70} +(79.2458 + 45.7526i) q^{71} +63.1328i q^{73} +(36.2173 - 62.7303i) q^{74} +(-911.638 + 526.335i) q^{76} +1355.25 q^{77} -287.115 q^{79} +(-596.341 + 344.298i) q^{80} +(-162.818 + 282.010i) q^{82} -373.812i q^{83} +(593.463 + 342.636i) q^{85} +374.147i q^{86} +(457.798 + 792.929i) q^{88} +(-103.406 + 59.7013i) q^{89} +(782.885 - 563.829i) q^{91} +380.532 q^{92} +(31.1758 + 53.9980i) q^{94} +(1137.49 - 1970.18i) q^{95} +(480.341 + 277.325i) q^{97} +(64.0928 + 37.0040i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 30 q^{4} + 30 q^{7} + 40 q^{10} - 60 q^{11} + 25 q^{13} + 60 q^{14} - 250 q^{16} - 105 q^{17} + 180 q^{19} - 510 q^{20} - 290 q^{22} + 60 q^{23} - 960 q^{25} + 30 q^{26} + 150 q^{28} + 495 q^{29}+ \cdots - 180 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.794469 − 0.458687i 0.280887 − 0.162170i −0.352938 − 0.935647i \(-0.614817\pi\)
0.633825 + 0.773476i \(0.281484\pi\)
\(3\) 0 0
\(4\) −3.57921 + 6.19938i −0.447402 + 0.774922i
\(5\) − 15.4704i − 1.38372i −0.722034 − 0.691858i \(-0.756792\pi\)
0.722034 − 0.691858i \(-0.243208\pi\)
\(6\) 0 0
\(7\) 17.8257 + 10.2917i 0.962497 + 0.555698i 0.896941 − 0.442151i \(-0.145784\pi\)
0.0655563 + 0.997849i \(0.479118\pi\)
\(8\) 13.9059i 0.614562i
\(9\) 0 0
\(10\) −7.09608 − 12.2908i −0.224398 − 0.388668i
\(11\) 57.0209 − 32.9210i 1.56295 − 0.902369i 0.565992 − 0.824411i \(-0.308493\pi\)
0.996957 − 0.0779583i \(-0.0248401\pi\)
\(12\) 0 0
\(13\) 19.2429 − 42.7400i 0.410540 − 0.911842i
\(14\) 18.8826 0.360471
\(15\) 0 0
\(16\) −22.2552 − 38.5472i −0.347738 − 0.602300i
\(17\) −22.1478 + 38.3611i −0.315979 + 0.547291i −0.979645 − 0.200738i \(-0.935666\pi\)
0.663666 + 0.748029i \(0.268999\pi\)
\(18\) 0 0
\(19\) 127.352 + 73.5266i 1.53771 + 0.887798i 0.998972 + 0.0453247i \(0.0144322\pi\)
0.538738 + 0.842473i \(0.318901\pi\)
\(20\) 95.9069 + 55.3719i 1.07227 + 0.619077i
\(21\) 0 0
\(22\) 30.2009 − 52.3094i 0.292675 − 0.506928i
\(23\) −26.5793 − 46.0367i −0.240964 − 0.417362i 0.720025 − 0.693948i \(-0.244130\pi\)
−0.960989 + 0.276586i \(0.910797\pi\)
\(24\) 0 0
\(25\) −114.334 −0.914669
\(26\) −4.31639 − 42.7821i −0.0325582 − 0.322702i
\(27\) 0 0
\(28\) −127.604 + 73.6721i −0.861245 + 0.497240i
\(29\) −19.3128 − 33.4508i −0.123666 − 0.214195i 0.797545 − 0.603260i \(-0.206132\pi\)
−0.921211 + 0.389064i \(0.872798\pi\)
\(30\) 0 0
\(31\) − 88.3894i − 0.512104i −0.966663 − 0.256052i \(-0.917578\pi\)
0.966663 − 0.256052i \(-0.0824218\pi\)
\(32\) −131.705 − 76.0401i −0.727576 − 0.420066i
\(33\) 0 0
\(34\) 40.6357i 0.204969i
\(35\) 159.216 − 275.771i 0.768928 − 1.33182i
\(36\) 0 0
\(37\) 68.3803 − 39.4794i 0.303828 − 0.175415i −0.340333 − 0.940305i \(-0.610540\pi\)
0.644161 + 0.764890i \(0.277206\pi\)
\(38\) 134.903 0.575898
\(39\) 0 0
\(40\) 215.131 0.850379
\(41\) −307.410 + 177.483i −1.17096 + 0.676054i −0.953906 − 0.300106i \(-0.902978\pi\)
−0.217053 + 0.976160i \(0.569645\pi\)
\(42\) 0 0
\(43\) −203.923 + 353.205i −0.723208 + 1.25263i 0.236499 + 0.971632i \(0.424000\pi\)
−0.959707 + 0.281002i \(0.909333\pi\)
\(44\) 471.325i 1.61489i
\(45\) 0 0
\(46\) −42.2329 − 24.3832i −0.135367 − 0.0781544i
\(47\) 67.9674i 0.210938i 0.994423 + 0.105469i \(0.0336343\pi\)
−0.994423 + 0.105469i \(0.966366\pi\)
\(48\) 0 0
\(49\) 40.3369 + 69.8656i 0.117600 + 0.203690i
\(50\) −90.8345 + 52.4433i −0.256919 + 0.148332i
\(51\) 0 0
\(52\) 196.087 + 272.270i 0.522931 + 0.726097i
\(53\) −226.572 −0.587209 −0.293604 − 0.955927i \(-0.594855\pi\)
−0.293604 + 0.955927i \(0.594855\pi\)
\(54\) 0 0
\(55\) −509.302 − 882.136i −1.24862 − 2.16268i
\(56\) −143.115 + 247.883i −0.341511 + 0.591514i
\(57\) 0 0
\(58\) −30.6869 − 17.7171i −0.0694722 − 0.0401098i
\(59\) 123.002 + 71.0154i 0.271416 + 0.156702i 0.629531 − 0.776976i \(-0.283247\pi\)
−0.358115 + 0.933677i \(0.616580\pi\)
\(60\) 0 0
\(61\) −133.416 + 231.083i −0.280035 + 0.485034i −0.971393 − 0.237478i \(-0.923679\pi\)
0.691358 + 0.722512i \(0.257013\pi\)
\(62\) −40.5431 − 70.2227i −0.0830480 − 0.143843i
\(63\) 0 0
\(64\) 216.569 0.422987
\(65\) −661.206 − 297.696i −1.26173 − 0.568071i
\(66\) 0 0
\(67\) 356.098 − 205.593i 0.649318 − 0.374884i −0.138877 − 0.990310i \(-0.544349\pi\)
0.788195 + 0.615426i \(0.211016\pi\)
\(68\) −158.543 − 274.605i −0.282739 − 0.489718i
\(69\) 0 0
\(70\) − 292.122i − 0.498789i
\(71\) 79.2458 + 45.7526i 0.132461 + 0.0764765i 0.564766 − 0.825251i \(-0.308966\pi\)
−0.432305 + 0.901727i \(0.642300\pi\)
\(72\) 0 0
\(73\) 63.1328i 0.101221i 0.998718 + 0.0506105i \(0.0161167\pi\)
−0.998718 + 0.0506105i \(0.983883\pi\)
\(74\) 36.2173 − 62.7303i 0.0568943 − 0.0985439i
\(75\) 0 0
\(76\) −911.638 + 526.335i −1.37595 + 0.794404i
\(77\) 1355.25 2.00578
\(78\) 0 0
\(79\) −287.115 −0.408899 −0.204449 − 0.978877i \(-0.565540\pi\)
−0.204449 + 0.978877i \(0.565540\pi\)
\(80\) −596.341 + 344.298i −0.833412 + 0.481170i
\(81\) 0 0
\(82\) −162.818 + 282.010i −0.219272 + 0.379790i
\(83\) − 373.812i − 0.494352i −0.968971 − 0.247176i \(-0.920497\pi\)
0.968971 − 0.247176i \(-0.0795026\pi\)
\(84\) 0 0
\(85\) 593.463 + 342.636i 0.757295 + 0.437224i
\(86\) 374.147i 0.469132i
\(87\) 0 0
\(88\) 457.798 + 792.929i 0.554561 + 0.960528i
\(89\) −103.406 + 59.7013i −0.123157 + 0.0711047i −0.560313 − 0.828281i \(-0.689319\pi\)
0.437156 + 0.899386i \(0.355986\pi\)
\(90\) 0 0
\(91\) 782.885 − 563.829i 0.901853 − 0.649509i
\(92\) 380.532 0.431231
\(93\) 0 0
\(94\) 31.1758 + 53.9980i 0.0342078 + 0.0592497i
\(95\) 1137.49 − 1970.18i 1.22846 − 2.12775i
\(96\) 0 0
\(97\) 480.341 + 277.325i 0.502796 + 0.290290i 0.729868 − 0.683589i \(-0.239582\pi\)
−0.227071 + 0.973878i \(0.572915\pi\)
\(98\) 64.0928 + 37.0040i 0.0660648 + 0.0381426i
\(99\) 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.10.3 10
3.2 odd 2 39.4.j.c.10.3 yes 10
12.11 even 2 624.4.bv.h.49.4 10
13.2 odd 12 1521.4.a.bk.1.6 10
13.4 even 6 inner 117.4.q.e.82.3 10
13.11 odd 12 1521.4.a.bk.1.5 10
39.2 even 12 507.4.a.r.1.5 10
39.11 even 12 507.4.a.r.1.6 10
39.17 odd 6 39.4.j.c.4.3 ✓ 10
39.23 odd 6 507.4.b.i.337.6 10
39.29 odd 6 507.4.b.i.337.5 10
156.95 even 6 624.4.bv.h.433.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 ✓ 10 39.17 odd 6
39.4.j.c.10.3 yes 10 3.2 odd 2
117.4.q.e.10.3 10 1.1 even 1 trivial
117.4.q.e.82.3 10 13.4 even 6 inner
507.4.a.r.1.5 10 39.2 even 12
507.4.a.r.1.6 10 39.11 even 12
507.4.b.i.337.5 10 39.29 odd 6
507.4.b.i.337.6 10 39.23 odd 6
624.4.bv.h.49.4 10 12.11 even 2
624.4.bv.h.433.2 10 156.95 even 6
1521.4.a.bk.1.5 10 13.11 odd 12
1521.4.a.bk.1.6 10 13.2 odd 12