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Properties

Label 245.3.i.c.129.4
Level $245$
Weight $3$
Character 245.129
Analytic conductor $6.676$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [245,3,Mod(19,245)] 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("245.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(i, \sqrt{3}, \sqrt{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 129.4
Root \(-0.578737 + 2.15988i\) of defining polynomial
Character \(\chi\) \(=\) 245.129
Dual form 245.3.i.c.19.4

$q$-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+(2.59808 - 1.50000i) q^{2} +(1.58114 - 2.73861i) q^{3} +(2.50000 - 4.33013i) q^{4} +(3.31735 - 3.74101i) q^{5} -9.48683i q^{6} -3.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +(3.00721 - 14.6955i) q^{10} +(-7.00000 + 12.1244i) q^{11} +(-7.90569 - 13.6931i) q^{12} +3.16228 q^{13} +(-5.00000 - 15.0000i) q^{15} +(5.50000 + 9.52628i) q^{16} +(3.16228 - 5.47723i) q^{17} +(-2.59808 - 1.50000i) q^{18} +(-24.6475 + 14.2302i) q^{19} +(-7.90569 - 23.7171i) q^{20} +42.0000i q^{22} +(-10.3923 + 6.00000i) q^{23} +(-8.21584 - 4.74342i) q^{24} +(-2.99038 - 24.8205i) q^{25} +(8.21584 - 4.74342i) q^{26} +25.2982 q^{27} +14.0000 q^{29} +(-35.4904 - 31.4711i) q^{30} +(-32.8634 - 18.9737i) q^{31} +(38.9711 + 22.5000i) q^{32} +(22.1359 + 38.3406i) q^{33} -18.9737i q^{34} -5.00000 q^{36} +(15.5885 - 9.00000i) q^{37} +(-42.6907 + 73.9425i) q^{38} +(5.00000 - 8.66025i) q^{39} +(-11.2230 - 9.95205i) q^{40} -18.9737i q^{41} -42.0000i q^{43} +(35.0000 + 60.6218i) q^{44} +(-4.89849 - 1.00240i) q^{45} +(-18.0000 + 31.1769i) q^{46} +(22.1359 + 38.3406i) q^{47} +34.7851 q^{48} +(-45.0000 - 60.0000i) q^{50} +(-10.0000 - 17.3205i) q^{51} +(7.90569 - 13.6931i) q^{52} +(-46.7654 - 27.0000i) q^{53} +(65.7267 - 37.9473i) q^{54} +(22.1359 + 66.4078i) q^{55} +90.0000i q^{57} +(36.3731 - 21.0000i) q^{58} +(-8.21584 - 4.74342i) q^{59} +(-77.4519 - 15.8494i) q^{60} +(-57.5109 + 33.2039i) q^{61} -113.842 q^{62} +91.0000 q^{64} +(10.4904 - 11.8301i) q^{65} +(115.022 + 66.4078i) q^{66} +(88.3346 + 51.0000i) q^{67} +(-15.8114 - 27.3861i) q^{68} +37.9473i q^{69} -16.0000 q^{71} +(-2.59808 + 1.50000i) q^{72} +(31.6228 - 54.7723i) q^{73} +(27.0000 - 46.7654i) q^{74} +(-72.7020 - 31.0552i) q^{75} +142.302i q^{76} -30.0000i q^{78} +(38.0000 + 65.8179i) q^{79} +(53.8834 + 11.0264i) q^{80} +(44.5000 - 77.0763i) q^{81} +(-28.4605 - 49.2950i) q^{82} -72.7324 q^{83} +(-10.0000 - 30.0000i) q^{85} +(-63.0000 - 109.119i) q^{86} +(22.1359 - 38.3406i) q^{87} +(36.3731 + 21.0000i) q^{88} +(49.2950 - 28.4605i) q^{89} +(-14.2302 + 4.74342i) q^{90} +60.0000i q^{92} +(-103.923 + 60.0000i) q^{93} +(115.022 + 66.4078i) q^{94} +(-28.5289 + 139.413i) q^{95} +(123.238 - 71.1512i) q^{96} +69.5701 q^{97} +14.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{4} - 4 q^{9} - 56 q^{11} - 40 q^{15} + 44 q^{16} + 80 q^{25} + 112 q^{29} - 180 q^{30} - 40 q^{36} + 40 q^{39} + 280 q^{44} - 144 q^{46} - 360 q^{50} - 80 q^{51} - 100 q^{60} + 728 q^{64} - 20 q^{65}+ \cdots + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.59808 − 1.50000i 1.29904 − 0.750000i 0.318800 − 0.947822i \(-0.396720\pi\)
0.980238 + 0.197822i \(0.0633868\pi\)
\(3\) 1.58114 − 2.73861i 0.527046 − 0.912871i −0.472457 − 0.881354i \(-0.656633\pi\)
0.999503 − 0.0315172i \(-0.0100339\pi\)
\(4\) 2.50000 − 4.33013i 0.625000 − 1.08253i
\(5\) 3.31735 − 3.74101i 0.663470 − 0.748203i
\(6\) − 9.48683i − 1.58114i
\(7\) 0 0
\(8\) − 3.00000i − 0.375000i
\(9\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(10\) 3.00721 − 14.6955i 0.300721 − 1.46955i
\(11\) −7.00000 + 12.1244i −0.636364 + 1.10221i 0.349861 + 0.936802i \(0.386229\pi\)
−0.986224 + 0.165412i \(0.947104\pi\)
\(12\) −7.90569 − 13.6931i −0.658808 − 1.14109i
\(13\) 3.16228 0.243252 0.121626 − 0.992576i \(-0.461189\pi\)
0.121626 + 0.992576i \(0.461189\pi\)
\(14\) 0 0
\(15\) −5.00000 − 15.0000i −0.333333 − 1.00000i
\(16\) 5.50000 + 9.52628i 0.343750 + 0.595392i
\(17\) 3.16228 − 5.47723i 0.186016 − 0.322190i −0.757902 − 0.652368i \(-0.773776\pi\)
0.943919 + 0.330178i \(0.107109\pi\)
\(18\) −2.59808 − 1.50000i −0.144338 − 0.0833333i
\(19\) −24.6475 + 14.2302i −1.29724 + 0.748960i −0.979926 − 0.199361i \(-0.936113\pi\)
−0.317311 + 0.948321i \(0.602780\pi\)
\(20\) −7.90569 − 23.7171i −0.395285 − 1.18585i
\(21\) 0 0
\(22\) 42.0000i 1.90909i
\(23\) −10.3923 + 6.00000i −0.451839 + 0.260870i −0.708607 − 0.705604i \(-0.750676\pi\)
0.256767 + 0.966473i \(0.417343\pi\)
\(24\) −8.21584 − 4.74342i −0.342327 − 0.197642i
\(25\) −2.99038 − 24.8205i −0.119615 − 0.992820i
\(26\) 8.21584 − 4.74342i 0.315994 − 0.182439i
\(27\) 25.2982 0.936971
\(28\) 0 0
\(29\) 14.0000 0.482759 0.241379 − 0.970431i \(-0.422400\pi\)
0.241379 + 0.970431i \(0.422400\pi\)
\(30\) −35.4904 − 31.4711i −1.18301 − 1.04904i
\(31\) −32.8634 − 18.9737i −1.06011 − 0.612054i −0.134646 − 0.990894i \(-0.542990\pi\)
−0.925462 + 0.378840i \(0.876323\pi\)
\(32\) 38.9711 + 22.5000i 1.21785 + 0.703125i
\(33\) 22.1359 + 38.3406i 0.670786 + 1.16184i
\(34\) − 18.9737i − 0.558049i
\(35\) 0 0
\(36\) −5.00000 −0.138889
\(37\) 15.5885 − 9.00000i 0.421310 − 0.243243i −0.274328 − 0.961636i \(-0.588455\pi\)
0.695637 + 0.718393i \(0.255122\pi\)
\(38\) −42.6907 + 73.9425i −1.12344 + 1.94586i
\(39\) 5.00000 − 8.66025i 0.128205 − 0.222058i
\(40\) −11.2230 − 9.95205i −0.280576 − 0.248801i
\(41\) − 18.9737i − 0.462772i −0.972862 − 0.231386i \(-0.925674\pi\)
0.972862 − 0.231386i \(-0.0743261\pi\)
\(42\) 0 0
\(43\) − 42.0000i − 0.976744i −0.872635 − 0.488372i \(-0.837591\pi\)
0.872635 − 0.488372i \(-0.162409\pi\)
\(44\) 35.0000 + 60.6218i 0.795455 + 1.37777i
\(45\) −4.89849 − 1.00240i −0.108855 − 0.0222756i
\(46\) −18.0000 + 31.1769i −0.391304 + 0.677759i
\(47\) 22.1359 + 38.3406i 0.470978 + 0.815757i 0.999449 − 0.0331941i \(-0.0105680\pi\)
−0.528471 + 0.848951i \(0.677235\pi\)
\(48\) 34.7851 0.724689
\(49\) 0 0
\(50\) −45.0000 − 60.0000i −0.900000 − 1.20000i
\(51\) −10.0000 − 17.3205i −0.196078 − 0.339618i
\(52\) 7.90569 − 13.6931i 0.152033 − 0.263328i
\(53\) −46.7654 − 27.0000i −0.882366 − 0.509434i −0.0109279 − 0.999940i \(-0.503479\pi\)
−0.871438 + 0.490506i \(0.836812\pi\)
\(54\) 65.7267 − 37.9473i 1.21716 − 0.702728i
\(55\) 22.1359 + 66.4078i 0.402472 + 1.20742i
\(56\) 0 0
\(57\) 90.0000i 1.57895i
\(58\) 36.3731 − 21.0000i 0.627122 − 0.362069i
\(59\) −8.21584 − 4.74342i −0.139251 − 0.0803969i 0.428756 − 0.903420i \(-0.358952\pi\)
−0.568007 + 0.823024i \(0.692285\pi\)
\(60\) −77.4519 − 15.8494i −1.29087 − 0.264156i
\(61\) −57.5109 + 33.2039i −0.942801 + 0.544326i −0.890837 − 0.454322i \(-0.849881\pi\)
−0.0519638 + 0.998649i \(0.516548\pi\)
\(62\) −113.842 −1.83616
\(63\) 0 0
\(64\) 91.0000 1.42188
\(65\) 10.4904 − 11.8301i 0.161390 − 0.182002i
\(66\) 115.022 + 66.4078i 1.74275 + 1.00618i
\(67\) 88.3346 + 51.0000i 1.31843 + 0.761194i 0.983475 − 0.181041i \(-0.0579468\pi\)
0.334951 + 0.942235i \(0.391280\pi\)
\(68\) −15.8114 − 27.3861i −0.232520 − 0.402737i
\(69\) 37.9473i 0.549961i
\(70\) 0 0
\(71\) −16.0000 −0.225352 −0.112676 − 0.993632i \(-0.535942\pi\)
−0.112676 + 0.993632i \(0.535942\pi\)
\(72\) −2.59808 + 1.50000i −0.0360844 + 0.0208333i
\(73\) 31.6228 − 54.7723i 0.433189 − 0.750305i −0.563957 − 0.825804i \(-0.690722\pi\)
0.997146 + 0.0754992i \(0.0240550\pi\)
\(74\) 27.0000 − 46.7654i 0.364865 − 0.631964i
\(75\) −72.7020 − 31.0552i −0.969360 − 0.414069i
\(76\) 142.302i 1.87240i
\(77\) 0 0
\(78\) − 30.0000i − 0.384615i
\(79\) 38.0000 + 65.8179i 0.481013 + 0.833138i 0.999763 − 0.0217876i \(-0.00693577\pi\)
−0.518750 + 0.854926i \(0.673602\pi\)
\(80\) 53.8834 + 11.0264i 0.673542 + 0.137830i
\(81\) 44.5000 − 77.0763i 0.549383 − 0.951559i
\(82\) −28.4605 − 49.2950i −0.347079 − 0.601159i
\(83\) −72.7324 −0.876294 −0.438147 − 0.898903i \(-0.644365\pi\)
−0.438147 + 0.898903i \(0.644365\pi\)
\(84\) 0 0
\(85\) −10.0000 − 30.0000i −0.117647 − 0.352941i
\(86\) −63.0000 − 109.119i −0.732558 − 1.26883i
\(87\) 22.1359 − 38.3406i 0.254436 − 0.440696i
\(88\) 36.3731 + 21.0000i 0.413330 + 0.238636i
\(89\) 49.2950 − 28.4605i 0.553877 − 0.319781i −0.196807 − 0.980442i \(-0.563057\pi\)
0.750684 + 0.660661i \(0.229724\pi\)
\(90\) −14.2302 + 4.74342i −0.158114 + 0.0527046i
\(91\) 0 0
\(92\) 60.0000i 0.652174i
\(93\) −103.923 + 60.0000i −1.11745 + 0.645161i
\(94\) 115.022 + 66.4078i 1.22364 + 0.706466i
\(95\) −28.5289 + 139.413i −0.300304 + 1.46751i
\(96\) 123.238 − 71.1512i 1.28372 − 0.741159i
\(97\) 69.5701 0.717218 0.358609 − 0.933488i \(-0.383251\pi\)
0.358609 + 0.933488i \(0.383251\pi\)
\(98\) 0 0
\(99\) 14.0000 0.141414
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 245.3.i.c.129.4 8
5.4 even 2 inner 245.3.i.c.129.1 8
7.2 even 3 inner 245.3.i.c.19.2 8
7.3 odd 6 35.3.c.c.34.4 yes 4
7.4 even 3 35.3.c.c.34.3 yes 4
7.5 odd 6 inner 245.3.i.c.19.1 8
7.6 odd 2 inner 245.3.i.c.129.3 8
21.11 odd 6 315.3.e.c.244.1 4
21.17 even 6 315.3.e.c.244.2 4
28.3 even 6 560.3.p.f.209.1 4
28.11 odd 6 560.3.p.f.209.4 4
35.3 even 12 175.3.d.h.76.2 2
35.4 even 6 35.3.c.c.34.2 yes 4
35.9 even 6 inner 245.3.i.c.19.3 8
35.17 even 12 175.3.d.b.76.1 2
35.18 odd 12 175.3.d.h.76.1 2
35.19 odd 6 inner 245.3.i.c.19.4 8
35.24 odd 6 35.3.c.c.34.1 ✓ 4
35.32 odd 12 175.3.d.b.76.2 2
35.34 odd 2 inner 245.3.i.c.129.2 8
105.59 even 6 315.3.e.c.244.3 4
105.74 odd 6 315.3.e.c.244.4 4
140.39 odd 6 560.3.p.f.209.2 4
140.59 even 6 560.3.p.f.209.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.c.c.34.1 ✓ 4 35.24 odd 6
35.3.c.c.34.2 yes 4 35.4 even 6
35.3.c.c.34.3 yes 4 7.4 even 3
35.3.c.c.34.4 yes 4 7.3 odd 6
175.3.d.b.76.1 2 35.17 even 12
175.3.d.b.76.2 2 35.32 odd 12
175.3.d.h.76.1 2 35.18 odd 12
175.3.d.h.76.2 2 35.3 even 12
245.3.i.c.19.1 8 7.5 odd 6 inner
245.3.i.c.19.2 8 7.2 even 3 inner
245.3.i.c.19.3 8 35.9 even 6 inner
245.3.i.c.19.4 8 35.19 odd 6 inner
245.3.i.c.129.1 8 5.4 even 2 inner
245.3.i.c.129.2 8 35.34 odd 2 inner
245.3.i.c.129.3 8 7.6 odd 2 inner
245.3.i.c.129.4 8 1.1 even 1 trivial
315.3.e.c.244.1 4 21.11 odd 6
315.3.e.c.244.2 4 21.17 even 6
315.3.e.c.244.3 4 105.59 even 6
315.3.e.c.244.4 4 105.74 odd 6
560.3.p.f.209.1 4 28.3 even 6
560.3.p.f.209.2 4 140.39 odd 6
560.3.p.f.209.3 4 140.59 even 6
560.3.p.f.209.4 4 28.11 odd 6