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Properties

Label 245.3.i.c.19.3
Level $245$
Weight $3$
Character 245.19
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 19.3
Root \(0.578737 + 2.15988i\) of defining polynomial
Character \(\chi\) \(=\) 245.19
Dual form 245.3.i.c.129.3

$q$-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+(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 \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.59808 + 1.50000i 1.29904 + 0.750000i 0.980238 − 0.197822i \(-0.0633868\pi\)
0.318800 + 0.947822i \(0.396720\pi\)
\(3\) −1.58114 − 2.73861i −0.527046 − 0.912871i −0.999503 − 0.0315172i \(-0.989966\pi\)
0.472457 − 0.881354i \(-0.343367\pi\)
\(4\) 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 −0.986224 − 0.165412i \(-0.947104\pi\)
0.349861 − 0.936802i \(-0.386229\pi\)
\(12\) 7.90569 − 13.6931i 0.658808 − 1.14109i
\(13\) −3.16228 −0.243252 −0.121626 − 0.992576i \(-0.538811\pi\)
−0.121626 + 0.992576i \(0.538811\pi\)
\(14\) 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 \(-0.226224\pi\)
−0.943919 + 0.330178i \(0.892891\pi\)
\(18\) −2.59808 + 1.50000i −0.144338 + 0.0833333i
\(19\) 24.6475 + 14.2302i 1.29724 + 0.748960i 0.979926 − 0.199361i \(-0.0638866\pi\)
0.317311 + 0.948321i \(0.397220\pi\)
\(20\) 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 0.256767 − 0.966473i \(-0.417343\pi\)
−0.708607 + 0.705604i \(0.750676\pi\)
\(24\) 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 \(-0.457010\pi\)
0.925462 + 0.378840i \(0.123677\pi\)
\(32\) 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 0.695637 − 0.718393i \(-0.255122\pi\)
−0.274328 + 0.961636i \(0.588455\pi\)
\(38\) 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 \(0.162409\pi\)
−0.872635 + 0.488372i \(0.837591\pi\)
\(44\) 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 \(-0.989432\pi\)
0.528471 + 0.848951i \(0.322765\pi\)
\(48\) −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 −0.871438 − 0.490506i \(-0.836812\pi\)
−0.0109279 + 0.999940i \(0.503479\pi\)
\(54\) −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 \(-0.641048\pi\)
0.568007 + 0.823024i \(0.307715\pi\)
\(60\) −77.4519 + 15.8494i −1.29087 + 0.264156i
\(61\) 57.5109 + 33.2039i 0.942801 + 0.544326i 0.890837 − 0.454322i \(-0.150119\pi\)
0.0519638 + 0.998649i \(0.483452\pi\)
\(62\) 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 0.334951 − 0.942235i \(-0.391280\pi\)
0.983475 + 0.181041i \(0.0579468\pi\)
\(68\) 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 \(-0.309278\pi\)
−0.997146 + 0.0754992i \(0.975945\pi\)
\(74\) 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 −0.518750 − 0.854926i \(-0.673602\pi\)
0.999763 + 0.0217876i \(0.00693577\pi\)
\(80\) −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 \(-0.355635\pi\)
0.438147 + 0.898903i \(0.355635\pi\)
\(84\) 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 \(-0.436943\pi\)
−0.750684 + 0.660661i \(0.770276\pi\)
\(90\) 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 \(-0.616749\pi\)
−0.358609 + 0.933488i \(0.616749\pi\)
\(98\) 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.19.3 8
5.4 even 2 inner 245.3.i.c.19.2 8
7.2 even 3 35.3.c.c.34.2 yes 4
7.3 odd 6 inner 245.3.i.c.129.2 8
7.4 even 3 inner 245.3.i.c.129.1 8
7.5 odd 6 35.3.c.c.34.1 ✓ 4
7.6 odd 2 inner 245.3.i.c.19.4 8
21.2 odd 6 315.3.e.c.244.4 4
21.5 even 6 315.3.e.c.244.3 4
28.19 even 6 560.3.p.f.209.3 4
28.23 odd 6 560.3.p.f.209.2 4
35.2 odd 12 175.3.d.h.76.1 2
35.4 even 6 inner 245.3.i.c.129.4 8
35.9 even 6 35.3.c.c.34.3 yes 4
35.12 even 12 175.3.d.h.76.2 2
35.19 odd 6 35.3.c.c.34.4 yes 4
35.23 odd 12 175.3.d.b.76.2 2
35.24 odd 6 inner 245.3.i.c.129.3 8
35.33 even 12 175.3.d.b.76.1 2
35.34 odd 2 inner 245.3.i.c.19.1 8
105.44 odd 6 315.3.e.c.244.1 4
105.89 even 6 315.3.e.c.244.2 4
140.19 even 6 560.3.p.f.209.1 4
140.79 odd 6 560.3.p.f.209.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.c.c.34.1 ✓ 4 7.5 odd 6
35.3.c.c.34.2 yes 4 7.2 even 3
35.3.c.c.34.3 yes 4 35.9 even 6
35.3.c.c.34.4 yes 4 35.19 odd 6
175.3.d.b.76.1 2 35.33 even 12
175.3.d.b.76.2 2 35.23 odd 12
175.3.d.h.76.1 2 35.2 odd 12
175.3.d.h.76.2 2 35.12 even 12
245.3.i.c.19.1 8 35.34 odd 2 inner
245.3.i.c.19.2 8 5.4 even 2 inner
245.3.i.c.19.3 8 1.1 even 1 trivial
245.3.i.c.19.4 8 7.6 odd 2 inner
245.3.i.c.129.1 8 7.4 even 3 inner
245.3.i.c.129.2 8 7.3 odd 6 inner
245.3.i.c.129.3 8 35.24 odd 6 inner
245.3.i.c.129.4 8 35.4 even 6 inner
315.3.e.c.244.1 4 105.44 odd 6
315.3.e.c.244.2 4 105.89 even 6
315.3.e.c.244.3 4 21.5 even 6
315.3.e.c.244.4 4 21.2 odd 6
560.3.p.f.209.1 4 140.19 even 6
560.3.p.f.209.2 4 28.23 odd 6
560.3.p.f.209.3 4 28.19 even 6
560.3.p.f.209.4 4 140.79 odd 6