Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [245,4,Mod(116,245)] 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("245.116"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-8,2] 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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 116.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 245.116
Dual form 245.4.e.i.226.1

$q$-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+(-2.70711 - 4.68885i) q^{2} +(-2.32843 + 4.03295i) q^{3} +(-10.6569 + 18.4582i) q^{4} +(-2.50000 - 4.33013i) q^{5} +25.2132 q^{6} +72.0833 q^{8} +(2.65685 + 4.60181i) q^{9} +(-13.5355 + 23.4442i) q^{10} +(26.1274 - 45.2540i) q^{11} +(-49.6274 - 85.9572i) q^{12} -30.6569 q^{13} +23.2843 q^{15} +(-109.882 - 190.322i) q^{16} +(18.6127 - 32.2381i) q^{17} +(14.3848 - 24.9152i) q^{18} +(40.1127 + 69.4772i) q^{19} +106.569 q^{20} -282.919 q^{22} +(-12.9167 - 22.3724i) q^{23} +(-167.841 + 290.708i) q^{24} +(-12.5000 + 21.6506i) q^{25} +(82.9914 + 143.745i) q^{26} -150.480 q^{27} +20.9411 q^{29} +(-63.0330 - 109.176i) q^{30} +(-157.279 + 272.416i) q^{31} +(-306.593 + 531.035i) q^{32} +(121.672 + 210.741i) q^{33} -201.546 q^{34} -113.255 q^{36} +(-98.5736 - 170.734i) q^{37} +(217.179 - 376.165i) q^{38} +(71.3823 - 123.638i) q^{39} +(-180.208 - 312.130i) q^{40} -11.3625 q^{41} -33.8335 q^{43} +(556.872 + 964.531i) q^{44} +(13.2843 - 23.0090i) q^{45} +(-69.9340 + 121.129i) q^{46} +(-180.838 - 313.221i) q^{47} +1023.41 q^{48} +135.355 q^{50} +(86.6766 + 150.128i) q^{51} +(326.706 - 565.871i) q^{52} +(-76.5097 + 132.519i) q^{53} +(407.366 + 705.579i) q^{54} -261.274 q^{55} -373.598 q^{57} +(-56.6899 - 98.1897i) q^{58} +(-308.000 + 533.472i) q^{59} +(-248.137 + 429.786i) q^{60} +(7.63247 + 13.2198i) q^{61} +1703.09 q^{62} +1561.80 q^{64} +(76.6421 + 132.748i) q^{65} +(658.756 - 1141.00i) q^{66} +(83.2548 - 144.202i) q^{67} +(396.706 + 687.114i) q^{68} +120.303 q^{69} -952.000 q^{71} +(191.515 + 331.713i) q^{72} +(-74.2447 + 128.596i) q^{73} +(-533.698 + 924.393i) q^{74} +(-58.2107 - 100.824i) q^{75} -1709.90 q^{76} -772.958 q^{78} +(-428.862 - 742.812i) q^{79} +(-549.411 + 951.608i) q^{80} +(278.647 - 482.631i) q^{81} +(30.7595 + 53.2769i) q^{82} -660.528 q^{83} -186.127 q^{85} +(91.5908 + 158.640i) q^{86} +(-48.7599 + 84.4546i) q^{87} +(1883.35 - 3262.06i) q^{88} +(-22.8873 - 39.6420i) q^{89} -143.848 q^{90} +550.607 q^{92} +(-732.426 - 1268.60i) q^{93} +(-979.096 + 1695.84i) q^{94} +(200.563 - 347.386i) q^{95} +(-1427.76 - 2472.95i) q^{96} -1682.13 q^{97} +277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 2 q^{3} - 20 q^{4} - 10 q^{5} + 16 q^{6} + 96 q^{8} - 12 q^{9} - 40 q^{10} + 14 q^{11} - 108 q^{12} - 100 q^{13} - 20 q^{15} - 168 q^{16} - 50 q^{17} - 16 q^{18} + 36 q^{19} + 200 q^{20}+ \cdots + 1880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.70711 − 4.68885i −0.957107 − 1.65776i −0.729472 − 0.684011i \(-0.760234\pi\)
−0.227635 − 0.973746i \(-0.573099\pi\)
\(3\) −2.32843 + 4.03295i −0.448106 + 0.776142i −0.998263 − 0.0589190i \(-0.981235\pi\)
0.550157 + 0.835061i \(0.314568\pi\)
\(4\) −10.6569 + 18.4582i −1.33211 + 2.30728i
\(5\) −2.50000 − 4.33013i −0.223607 − 0.387298i
\(6\) 25.2132 1.71554
\(7\) 0 0
\(8\) 72.0833 3.18566
\(9\) 2.65685 + 4.60181i 0.0984020 + 0.170437i
\(10\) −13.5355 + 23.4442i −0.428031 + 0.741372i
\(11\) 26.1274 − 45.2540i 0.716156 − 1.24042i −0.246356 − 0.969179i \(-0.579233\pi\)
0.962512 − 0.271239i \(-0.0874333\pi\)
\(12\) −49.6274 − 85.9572i −1.19385 − 2.06781i
\(13\) −30.6569 −0.654052 −0.327026 − 0.945015i \(-0.606047\pi\)
−0.327026 + 0.945015i \(0.606047\pi\)
\(14\) 0 0
\(15\) 23.2843 0.400798
\(16\) −109.882 − 190.322i −1.71691 − 2.97378i
\(17\) 18.6127 − 32.2381i 0.265544 − 0.459935i −0.702162 − 0.712017i \(-0.747782\pi\)
0.967706 + 0.252082i \(0.0811152\pi\)
\(18\) 14.3848 − 24.9152i 0.188362 − 0.326253i
\(19\) 40.1127 + 69.4772i 0.484341 + 0.838904i 0.999838 − 0.0179877i \(-0.00572598\pi\)
−0.515497 + 0.856891i \(0.672393\pi\)
\(20\) 106.569 1.19147
\(21\) 0 0
\(22\) −282.919 −2.74175
\(23\) −12.9167 − 22.3724i −0.117101 − 0.202825i 0.801517 − 0.597973i \(-0.204027\pi\)
−0.918618 + 0.395147i \(0.870694\pi\)
\(24\) −167.841 + 290.708i −1.42751 + 2.47253i
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 82.9914 + 143.745i 0.625998 + 1.08426i
\(27\) −150.480 −1.07259
\(28\) 0 0
\(29\) 20.9411 0.134092 0.0670460 − 0.997750i \(-0.478643\pi\)
0.0670460 + 0.997750i \(0.478643\pi\)
\(30\) −63.0330 − 109.176i −0.383607 − 0.664426i
\(31\) −157.279 + 272.416i −0.911232 + 1.57830i −0.0989050 + 0.995097i \(0.531534\pi\)
−0.812327 + 0.583203i \(0.801799\pi\)
\(32\) −306.593 + 531.035i −1.69370 + 2.93358i
\(33\) 121.672 + 210.741i 0.641827 + 1.11168i
\(34\) −201.546 −1.01661
\(35\) 0 0
\(36\) −113.255 −0.524328
\(37\) −98.5736 − 170.734i −0.437984 − 0.758610i 0.559550 − 0.828797i \(-0.310974\pi\)
−0.997534 + 0.0701864i \(0.977641\pi\)
\(38\) 217.179 − 376.165i 0.927133 − 1.60584i
\(39\) 71.3823 − 123.638i 0.293085 − 0.507638i
\(40\) −180.208 − 312.130i −0.712335 − 1.23380i
\(41\) −11.3625 −0.0432810 −0.0216405 − 0.999766i \(-0.506889\pi\)
−0.0216405 + 0.999766i \(0.506889\pi\)
\(42\) 0 0
\(43\) −33.8335 −0.119990 −0.0599948 − 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) 556.872 + 964.531i 1.90799 + 3.30474i
\(45\) 13.2843 − 23.0090i 0.0440067 − 0.0762219i
\(46\) −69.9340 + 121.129i −0.224157 + 0.388251i
\(47\) −180.838 − 313.221i −0.561233 − 0.972084i −0.997389 − 0.0722130i \(-0.976994\pi\)
0.436156 − 0.899871i \(-0.356339\pi\)
\(48\) 1023.41 3.07743
\(49\) 0 0
\(50\) 135.355 0.382843
\(51\) 86.6766 + 150.128i 0.237983 + 0.412199i
\(52\) 326.706 − 565.871i 0.871268 − 1.50908i
\(53\) −76.5097 + 132.519i −0.198291 + 0.343450i −0.947974 − 0.318347i \(-0.896872\pi\)
0.749684 + 0.661796i \(0.230206\pi\)
\(54\) 407.366 + 705.579i 1.02658 + 1.77809i
\(55\) −261.274 −0.640549
\(56\) 0 0
\(57\) −373.598 −0.868145
\(58\) −56.6899 − 98.1897i −0.128340 − 0.222292i
\(59\) −308.000 + 533.472i −0.679630 + 1.17715i 0.295462 + 0.955354i \(0.404526\pi\)
−0.975092 + 0.221800i \(0.928807\pi\)
\(60\) −248.137 + 429.786i −0.533906 + 0.924752i
\(61\) 7.63247 + 13.2198i 0.0160203 + 0.0277479i 0.873924 − 0.486062i \(-0.161567\pi\)
−0.857904 + 0.513810i \(0.828234\pi\)
\(62\) 1703.09 3.48858
\(63\) 0 0
\(64\) 1561.80 3.05040
\(65\) 76.6421 + 132.748i 0.146251 + 0.253313i
\(66\) 658.756 − 1141.00i 1.22859 − 2.12799i
\(67\) 83.2548 − 144.202i 0.151809 − 0.262941i −0.780084 − 0.625675i \(-0.784824\pi\)
0.931893 + 0.362735i \(0.118157\pi\)
\(68\) 396.706 + 687.114i 0.707465 + 1.22537i
\(69\) 120.303 0.209895
\(70\) 0 0
\(71\) −952.000 −1.59129 −0.795645 − 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) 191.515 + 331.713i 0.313475 + 0.542955i
\(73\) −74.2447 + 128.596i −0.119037 + 0.206178i −0.919386 − 0.393356i \(-0.871314\pi\)
0.800349 + 0.599534i \(0.204647\pi\)
\(74\) −533.698 + 924.393i −0.838394 + 1.45214i
\(75\) −58.2107 − 100.824i −0.0896212 − 0.155228i
\(76\) −1709.90 −2.58078
\(77\) 0 0
\(78\) −772.958 −1.12205
\(79\) −428.862 − 742.812i −0.610770 − 1.05788i −0.991111 − 0.133038i \(-0.957527\pi\)
0.380341 − 0.924846i \(-0.375807\pi\)
\(80\) −549.411 + 951.608i −0.767826 + 1.32991i
\(81\) 278.647 − 482.631i 0.382232 − 0.662045i
\(82\) 30.7595 + 53.2769i 0.0414246 + 0.0717494i
\(83\) −660.528 −0.873523 −0.436761 − 0.899577i \(-0.643875\pi\)
−0.436761 + 0.899577i \(0.643875\pi\)
\(84\) 0 0
\(85\) −186.127 −0.237509
\(86\) 91.5908 + 158.640i 0.114843 + 0.198914i
\(87\) −48.7599 + 84.4546i −0.0600875 + 0.104075i
\(88\) 1883.35 − 3262.06i 2.28143 − 3.95155i
\(89\) −22.8873 − 39.6420i −0.0272590 − 0.0472139i 0.852074 − 0.523421i \(-0.175345\pi\)
−0.879333 + 0.476207i \(0.842011\pi\)
\(90\) −143.848 −0.168477
\(91\) 0 0
\(92\) 550.607 0.623965
\(93\) −732.426 − 1268.60i −0.816657 − 1.41449i
\(94\) −979.096 + 1695.84i −1.07432 + 1.86078i
\(95\) 200.563 − 347.386i 0.216604 − 0.375169i
\(96\) −1427.76 − 2472.95i −1.51792 − 2.62911i
\(97\) −1682.13 −1.76076 −0.880382 − 0.474265i \(-0.842714\pi\)
−0.880382 + 0.474265i \(0.842714\pi\)
\(98\) 0 0
\(99\) 277.667 0.281885
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 245.4.e.i.116.1 4
7.2 even 3 inner 245.4.e.i.226.1 4
7.3 odd 6 35.4.a.b.1.2 ✓ 2
7.4 even 3 245.4.a.k.1.2 2
7.5 odd 6 245.4.e.h.226.1 4
7.6 odd 2 245.4.e.h.116.1 4
21.11 odd 6 2205.4.a.u.1.1 2
21.17 even 6 315.4.a.f.1.1 2
28.3 even 6 560.4.a.r.1.2 2
35.3 even 12 175.4.b.c.99.1 4
35.4 even 6 1225.4.a.m.1.1 2
35.17 even 12 175.4.b.c.99.4 4
35.24 odd 6 175.4.a.c.1.1 2
56.3 even 6 2240.4.a.bo.1.1 2
56.45 odd 6 2240.4.a.bn.1.2 2
105.59 even 6 1575.4.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 ✓ 2 7.3 odd 6
175.4.a.c.1.1 2 35.24 odd 6
175.4.b.c.99.1 4 35.3 even 12
175.4.b.c.99.4 4 35.17 even 12
245.4.a.k.1.2 2 7.4 even 3
245.4.e.h.116.1 4 7.6 odd 2
245.4.e.h.226.1 4 7.5 odd 6
245.4.e.i.116.1 4 1.1 even 1 trivial
245.4.e.i.226.1 4 7.2 even 3 inner
315.4.a.f.1.1 2 21.17 even 6
560.4.a.r.1.2 2 28.3 even 6
1225.4.a.m.1.1 2 35.4 even 6
1575.4.a.z.1.2 2 105.59 even 6
2205.4.a.u.1.1 2 21.11 odd 6
2240.4.a.bn.1.2 2 56.45 odd 6
2240.4.a.bo.1.1 2 56.3 even 6