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(18,245)] 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("245.18"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 8])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 245.m (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-2,2] 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: \(6.67576647683\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 67.1
Character \(\chi\) \(=\) 245.67
Dual form 245.3.m.b.128.1

$q$-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+(-0.792728 + 2.95850i) q^{2} +(-0.633552 - 2.36445i) q^{3} +(-4.66022 - 2.69058i) q^{4} +(0.307372 + 4.99054i) q^{5} +7.49746 q^{6} +(2.99127 - 2.99127i) q^{8} +(2.60501 - 1.50400i) q^{9} +(-15.0082 - 3.04678i) q^{10} +(-3.12159 + 5.40675i) q^{11} +(-3.40924 + 12.7235i) q^{12} +(-11.7345 + 11.7345i) q^{13} +(11.6051 - 3.88853i) q^{15} +(-4.28389 - 7.41992i) q^{16} +(4.56805 - 1.22400i) q^{17} +(2.38453 + 8.89918i) q^{18} +(-11.4064 + 6.58549i) q^{19} +(11.9950 - 24.0840i) q^{20} +(-13.5213 - 13.5213i) q^{22} +(-34.8444 - 9.33654i) q^{23} +(-8.96783 - 5.17758i) q^{24} +(-24.8110 + 3.06791i) q^{25} +(-25.4144 - 44.0190i) q^{26} +(-20.7846 - 20.7846i) q^{27} -28.5683i q^{29} +(2.30451 + 37.4164i) q^{30} +(-10.1948 + 17.6579i) q^{31} +(41.6924 - 11.1715i) q^{32} +(14.7617 + 3.95538i) q^{33} +14.4849i q^{34} -16.1865 q^{36} +(-6.06849 + 22.6479i) q^{37} +(-10.4410 - 38.9664i) q^{38} +(35.1801 + 20.3113i) q^{39} +(15.8475 + 14.0086i) q^{40} +45.1077 q^{41} +(-21.9185 + 21.9185i) q^{43} +(29.0946 - 16.7978i) q^{44} +(8.30649 + 12.5381i) q^{45} +(55.2443 - 95.6860i) q^{46} +(-2.79810 + 10.4426i) q^{47} +(-14.8299 + 14.8299i) q^{48} +(10.5920 - 75.8356i) q^{50} +(-5.78819 - 10.0254i) q^{51} +(86.2582 - 23.1128i) q^{52} +(19.2094 + 71.6904i) q^{53} +(77.9678 - 45.0147i) q^{54} +(-27.9421 - 13.9165i) q^{55} +(22.7976 + 22.7976i) q^{57} +(84.5194 + 22.6469i) q^{58} +(-14.6115 - 8.43595i) q^{59} +(-64.5449 - 13.1031i) q^{60} +(-16.7743 - 29.0540i) q^{61} +(-44.1593 - 44.1593i) q^{62} +97.9319i q^{64} +(-62.1686 - 54.9548i) q^{65} +(-23.4040 + 40.5369i) q^{66} +(-95.8731 + 25.6891i) q^{67} +(-24.5814 - 6.58656i) q^{68} +88.3030i q^{69} +66.2415 q^{71} +(3.29341 - 12.2912i) q^{72} +(26.7635 + 99.8826i) q^{73} +(-62.1933 - 35.9073i) q^{74} +(22.9730 + 56.7207i) q^{75} +70.8751 q^{76} +(-87.9792 + 87.9792i) q^{78} +(36.1253 - 20.8569i) q^{79} +(35.7127 - 23.6596i) q^{80} +(-22.4400 + 38.8671i) q^{81} +(-35.7582 + 133.451i) q^{82} +(-7.39450 + 7.39450i) q^{83} +(7.51254 + 22.4208i) q^{85} +(-47.4706 - 82.2215i) q^{86} +(-67.5483 + 18.0995i) q^{87} +(6.83554 + 25.5106i) q^{88} +(-19.2257 + 11.0999i) q^{89} +(-43.6788 + 14.6355i) q^{90} +(137.262 + 137.262i) q^{92} +(48.2102 + 12.9179i) q^{93} +(-28.6764 - 16.5563i) q^{94} +(-36.3712 - 54.9000i) q^{95} +(-52.8286 - 91.5018i) q^{96} +(73.6717 + 73.6717i) q^{97} +18.7795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} + 2 q^{3} + 4 q^{5} - 36 q^{8} - 14 q^{10} - 24 q^{11} + 46 q^{12} + 8 q^{13} + 52 q^{15} + 20 q^{16} + 48 q^{17} - 4 q^{18} + 72 q^{20} + 104 q^{22} - 86 q^{23} - 16 q^{25} - 140 q^{26}+ \cdots + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.792728 + 2.95850i −0.396364 + 1.47925i 0.423080 + 0.906092i \(0.360949\pi\)
−0.819444 + 0.573159i \(0.805718\pi\)
\(3\) −0.633552 − 2.36445i −0.211184 − 0.788149i −0.987475 − 0.157775i \(-0.949568\pi\)
0.776291 − 0.630374i \(-0.217099\pi\)
\(4\) −4.66022 − 2.69058i −1.16505 − 0.672645i
\(5\) 0.307372 + 4.99054i 0.0614745 + 0.998109i
\(6\) 7.49746 1.24958
\(7\) 0 0
\(8\) 2.99127 − 2.99127i 0.373909 − 0.373909i
\(9\) 2.60501 − 1.50400i 0.289445 − 0.167111i
\(10\) −15.0082 − 3.04678i −1.50082 − 0.304678i
\(11\) −3.12159 + 5.40675i −0.283781 + 0.491523i −0.972313 − 0.233683i \(-0.924922\pi\)
0.688532 + 0.725206i \(0.258256\pi\)
\(12\) −3.40924 + 12.7235i −0.284103 + 1.06029i
\(13\) −11.7345 + 11.7345i −0.902657 + 0.902657i −0.995665 − 0.0930085i \(-0.970352\pi\)
0.0930085 + 0.995665i \(0.470352\pi\)
\(14\) 0 0
\(15\) 11.6051 − 3.88853i 0.773676 − 0.259236i
\(16\) −4.28389 − 7.41992i −0.267743 − 0.463745i
\(17\) 4.56805 − 1.22400i 0.268709 − 0.0720003i −0.121949 − 0.992536i \(-0.538914\pi\)
0.390657 + 0.920536i \(0.372248\pi\)
\(18\) 2.38453 + 8.89918i 0.132474 + 0.494399i
\(19\) −11.4064 + 6.58549i −0.600337 + 0.346605i −0.769174 − 0.639039i \(-0.779332\pi\)
0.168837 + 0.985644i \(0.445999\pi\)
\(20\) 11.9950 − 24.0840i 0.599751 − 1.20420i
\(21\) 0 0
\(22\) −13.5213 − 13.5213i −0.614606 − 0.614606i
\(23\) −34.8444 − 9.33654i −1.51498 − 0.405936i −0.596891 − 0.802322i \(-0.703598\pi\)
−0.918084 + 0.396386i \(0.870264\pi\)
\(24\) −8.96783 − 5.17758i −0.373660 − 0.215732i
\(25\) −24.8110 + 3.06791i −0.992442 + 0.122716i
\(26\) −25.4144 − 44.0190i −0.977475 − 1.69304i
\(27\) −20.7846 − 20.7846i −0.769800 − 0.769800i
\(28\) 0 0
\(29\) − 28.5683i − 0.985114i −0.870280 − 0.492557i \(-0.836062\pi\)
0.870280 − 0.492557i \(-0.163938\pi\)
\(30\) 2.30451 + 37.4164i 0.0768171 + 1.24721i
\(31\) −10.1948 + 17.6579i −0.328865 + 0.569611i −0.982287 − 0.187383i \(-0.939999\pi\)
0.653422 + 0.756994i \(0.273333\pi\)
\(32\) 41.6924 − 11.1715i 1.30289 − 0.349108i
\(33\) 14.7617 + 3.95538i 0.447323 + 0.119860i
\(34\) 14.4849i 0.426026i
\(35\) 0 0
\(36\) −16.1865 −0.449626
\(37\) −6.06849 + 22.6479i −0.164013 + 0.612106i 0.834151 + 0.551537i \(0.185958\pi\)
−0.998164 + 0.0605694i \(0.980708\pi\)
\(38\) −10.4410 − 38.9664i −0.274763 − 1.02543i
\(39\) 35.1801 + 20.3113i 0.902055 + 0.520802i
\(40\) 15.8475 + 14.0086i 0.396188 + 0.350216i
\(41\) 45.1077 1.10019 0.550094 − 0.835103i \(-0.314592\pi\)
0.550094 + 0.835103i \(0.314592\pi\)
\(42\) 0 0
\(43\) −21.9185 + 21.9185i −0.509733 + 0.509733i −0.914445 − 0.404711i \(-0.867372\pi\)
0.404711 + 0.914445i \(0.367372\pi\)
\(44\) 29.0946 − 16.7978i 0.661241 − 0.381767i
\(45\) 8.30649 + 12.5381i 0.184589 + 0.278625i
\(46\) 55.2443 − 95.6860i 1.20096 − 2.08013i
\(47\) −2.79810 + 10.4426i −0.0595339 + 0.222184i −0.989283 − 0.146010i \(-0.953357\pi\)
0.929749 + 0.368193i \(0.120024\pi\)
\(48\) −14.8299 + 14.8299i −0.308957 + 0.308957i
\(49\) 0 0
\(50\) 10.5920 − 75.8356i 0.211840 − 1.51671i
\(51\) −5.78819 − 10.0254i −0.113494 − 0.196577i
\(52\) 86.2582 − 23.1128i 1.65881 − 0.444477i
\(53\) 19.2094 + 71.6904i 0.362441 + 1.35265i 0.870856 + 0.491537i \(0.163565\pi\)
−0.508415 + 0.861112i \(0.669768\pi\)
\(54\) 77.9678 − 45.0147i 1.44385 − 0.833606i
\(55\) −27.9421 − 13.9165i −0.508039 − 0.253028i
\(56\) 0 0
\(57\) 22.7976 + 22.7976i 0.399958 + 0.399958i
\(58\) 84.5194 + 22.6469i 1.45723 + 0.390464i
\(59\) −14.6115 − 8.43595i −0.247652 − 0.142982i 0.371036 − 0.928618i \(-0.379002\pi\)
−0.618689 + 0.785636i \(0.712336\pi\)
\(60\) −64.5449 − 13.1031i −1.07575 − 0.218385i
\(61\) −16.7743 − 29.0540i −0.274989 − 0.476295i 0.695143 − 0.718871i \(-0.255341\pi\)
−0.970132 + 0.242576i \(0.922008\pi\)
\(62\) −44.1593 − 44.1593i −0.712247 − 0.712247i
\(63\) 0 0
\(64\) 97.9319i 1.53019i
\(65\) −62.1686 − 54.9548i −0.956440 − 0.845459i
\(66\) −23.4040 + 40.5369i −0.354606 + 0.614196i
\(67\) −95.8731 + 25.6891i −1.43094 + 0.383420i −0.889352 − 0.457223i \(-0.848844\pi\)
−0.541590 + 0.840643i \(0.682177\pi\)
\(68\) −24.5814 − 6.58656i −0.361491 − 0.0968612i
\(69\) 88.3030i 1.27975i
\(70\) 0 0
\(71\) 66.2415 0.932979 0.466489 − 0.884527i \(-0.345519\pi\)
0.466489 + 0.884527i \(0.345519\pi\)
\(72\) 3.29341 − 12.2912i 0.0457417 − 0.170711i
\(73\) 26.7635 + 99.8826i 0.366623 + 1.36825i 0.865208 + 0.501414i \(0.167187\pi\)
−0.498585 + 0.866841i \(0.666147\pi\)
\(74\) −62.1933 − 35.9073i −0.840450 − 0.485234i
\(75\) 22.9730 + 56.7207i 0.306307 + 0.756276i
\(76\) 70.8751 0.932567
\(77\) 0 0
\(78\) −87.9792 + 87.9792i −1.12794 + 1.12794i
\(79\) 36.1253 − 20.8569i 0.457282 − 0.264012i −0.253619 − 0.967304i \(-0.581621\pi\)
0.710901 + 0.703292i \(0.248288\pi\)
\(80\) 35.7127 − 23.6596i 0.446408 − 0.295745i
\(81\) −22.4400 + 38.8671i −0.277036 + 0.479841i
\(82\) −35.7582 + 133.451i −0.436075 + 1.62745i
\(83\) −7.39450 + 7.39450i −0.0890904 + 0.0890904i −0.750247 − 0.661157i \(-0.770066\pi\)
0.661157 + 0.750247i \(0.270066\pi\)
\(84\) 0 0
\(85\) 7.51254 + 22.4208i 0.0883828 + 0.263774i
\(86\) −47.4706 − 82.2215i −0.551984 − 0.956064i
\(87\) −67.5483 + 18.0995i −0.776417 + 0.208040i
\(88\) 6.83554 + 25.5106i 0.0776766 + 0.289893i
\(89\) −19.2257 + 11.0999i −0.216019 + 0.124718i −0.604105 − 0.796904i \(-0.706469\pi\)
0.388087 + 0.921623i \(0.373136\pi\)
\(90\) −43.6788 + 14.6355i −0.485320 + 0.162616i
\(91\) 0 0
\(92\) 137.262 + 137.262i 1.49198 + 1.49198i
\(93\) 48.2102 + 12.9179i 0.518389 + 0.138902i
\(94\) −28.6764 − 16.5563i −0.305068 − 0.176131i
\(95\) −36.3712 − 54.9000i −0.382855 − 0.577894i
\(96\) −52.8286 − 91.5018i −0.550298 − 0.953144i
\(97\) 73.6717 + 73.6717i 0.759502 + 0.759502i 0.976232 − 0.216730i \(-0.0695390\pi\)
−0.216730 + 0.976232i \(0.569539\pi\)
\(98\) 0 0
\(99\) 18.7795i 0.189692i
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.m.b.67.1 24
5.3 odd 4 inner 245.3.m.b.18.6 24
7.2 even 3 inner 245.3.m.b.177.6 24
7.3 odd 6 245.3.g.c.197.1 12
7.4 even 3 245.3.g.b.197.1 12
7.5 odd 6 35.3.l.a.2.6 ✓ 24
7.6 odd 2 35.3.l.a.32.1 yes 24
21.5 even 6 315.3.ca.a.37.1 24
21.20 even 2 315.3.ca.a.172.6 24
35.3 even 12 245.3.g.c.148.1 12
35.12 even 12 175.3.p.c.93.6 24
35.13 even 4 35.3.l.a.18.6 yes 24
35.18 odd 12 245.3.g.b.148.1 12
35.19 odd 6 175.3.p.c.107.1 24
35.23 odd 12 inner 245.3.m.b.128.1 24
35.27 even 4 175.3.p.c.18.1 24
35.33 even 12 35.3.l.a.23.1 yes 24
35.34 odd 2 175.3.p.c.32.6 24
105.68 odd 12 315.3.ca.a.163.6 24
105.83 odd 4 315.3.ca.a.298.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.6 ✓ 24 7.5 odd 6
35.3.l.a.18.6 yes 24 35.13 even 4
35.3.l.a.23.1 yes 24 35.33 even 12
35.3.l.a.32.1 yes 24 7.6 odd 2
175.3.p.c.18.1 24 35.27 even 4
175.3.p.c.32.6 24 35.34 odd 2
175.3.p.c.93.6 24 35.12 even 12
175.3.p.c.107.1 24 35.19 odd 6
245.3.g.b.148.1 12 35.18 odd 12
245.3.g.b.197.1 12 7.4 even 3
245.3.g.c.148.1 12 35.3 even 12
245.3.g.c.197.1 12 7.3 odd 6
245.3.m.b.18.6 24 5.3 odd 4 inner
245.3.m.b.67.1 24 1.1 even 1 trivial
245.3.m.b.128.1 24 35.23 odd 12 inner
245.3.m.b.177.6 24 7.2 even 3 inner
315.3.ca.a.37.1 24 21.5 even 6
315.3.ca.a.163.6 24 105.68 odd 12
315.3.ca.a.172.6 24 21.20 even 2
315.3.ca.a.298.1 24 105.83 odd 4