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 226.1
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 245.226
Dual form 245.4.e.h.116.1

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

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

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 −0.227635 + 0.973746i \(0.573099\pi\)
−0.729472 + 0.684011i \(0.760234\pi\)
\(3\) 2.32843 + 4.03295i 0.448106 + 0.776142i 0.998263 − 0.0589190i \(-0.0187654\pi\)
−0.550157 + 0.835061i \(0.685432\pi\)
\(4\) −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 0.962512 + 0.271239i \(0.0874333\pi\)
−0.246356 + 0.969179i \(0.579233\pi\)
\(12\) 49.6274 − 85.9572i 1.19385 − 2.06781i
\(13\) 30.6569 0.654052 0.327026 − 0.945015i \(-0.393953\pi\)
0.327026 + 0.945015i \(0.393953\pi\)
\(14\) 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 \(-0.252218\pi\)
−0.967706 + 0.252082i \(0.918885\pi\)
\(18\) 14.3848 + 24.9152i 0.188362 + 0.326253i
\(19\) −40.1127 + 69.4772i −0.484341 + 0.838904i −0.999838 − 0.0179877i \(-0.994274\pi\)
0.515497 + 0.856891i \(0.327607\pi\)
\(20\) −106.569 −1.19147
\(21\) 0 0
\(22\) −282.919 −2.74175
\(23\) −12.9167 + 22.3724i −0.117101 + 0.202825i −0.918618 − 0.395147i \(-0.870694\pi\)
0.801517 + 0.597973i \(0.204027\pi\)
\(24\) 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 0.812327 + 0.583203i \(0.198201\pi\)
0.0989050 + 0.995097i \(0.468466\pi\)
\(32\) −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 −0.997534 − 0.0701864i \(-0.977641\pi\)
0.559550 + 0.828797i \(0.310974\pi\)
\(38\) −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 \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) 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 −0.436156 − 0.899871i \(-0.643661\pi\)
0.997389 − 0.0722130i \(-0.0230061\pi\)
\(48\) −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 0.749684 − 0.661796i \(-0.230206\pi\)
−0.947974 + 0.318347i \(0.896872\pi\)
\(54\) −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 0.975092 + 0.221800i \(0.0711931\pi\)
−0.295462 + 0.955354i \(0.595474\pi\)
\(60\) −248.137 − 429.786i −0.533906 − 0.924752i
\(61\) −7.63247 + 13.2198i −0.0160203 + 0.0277479i −0.873924 − 0.486062i \(-0.838433\pi\)
0.857904 + 0.513810i \(0.171766\pi\)
\(62\) −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 0.931893 − 0.362735i \(-0.118157\pi\)
−0.780084 + 0.625675i \(0.784824\pi\)
\(68\) −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 \(-0.128686\pi\)
−0.800349 + 0.599534i \(0.795353\pi\)
\(74\) −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 0.380341 + 0.924846i \(0.375807\pi\)
−0.991111 + 0.133038i \(0.957527\pi\)
\(80\) 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 \(-0.356125\pi\)
0.436761 + 0.899577i \(0.356125\pi\)
\(84\) 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 \(-0.824655\pi\)
0.879333 + 0.476207i \(0.157989\pi\)
\(90\) 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 \(-0.157286\pi\)
0.880382 + 0.474265i \(0.157286\pi\)
\(98\) 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.h.226.1 4
7.2 even 3 35.4.a.b.1.2 ✓ 2
7.3 odd 6 245.4.e.i.116.1 4
7.4 even 3 inner 245.4.e.h.116.1 4
7.5 odd 6 245.4.a.k.1.2 2
7.6 odd 2 245.4.e.i.226.1 4
21.2 odd 6 315.4.a.f.1.1 2
21.5 even 6 2205.4.a.u.1.1 2
28.23 odd 6 560.4.a.r.1.2 2
35.2 odd 12 175.4.b.c.99.4 4
35.9 even 6 175.4.a.c.1.1 2
35.19 odd 6 1225.4.a.m.1.1 2
35.23 odd 12 175.4.b.c.99.1 4
56.37 even 6 2240.4.a.bn.1.2 2
56.51 odd 6 2240.4.a.bo.1.1 2
105.44 odd 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 7.2 even 3
175.4.a.c.1.1 2 35.9 even 6
175.4.b.c.99.1 4 35.23 odd 12
175.4.b.c.99.4 4 35.2 odd 12
245.4.a.k.1.2 2 7.5 odd 6
245.4.e.h.116.1 4 7.4 even 3 inner
245.4.e.h.226.1 4 1.1 even 1 trivial
245.4.e.i.116.1 4 7.3 odd 6
245.4.e.i.226.1 4 7.6 odd 2
315.4.a.f.1.1 2 21.2 odd 6
560.4.a.r.1.2 2 28.23 odd 6
1225.4.a.m.1.1 2 35.19 odd 6
1575.4.a.z.1.2 2 105.44 odd 6
2205.4.a.u.1.1 2 21.5 even 6
2240.4.a.bn.1.2 2 56.37 even 6
2240.4.a.bo.1.1 2 56.51 odd 6