Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [35,3,Mod(2,35)] 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("35.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 18.6
Character \(\chi\) \(=\) 35.18
Dual form 35.3.l.a.2.6

$q$-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+(2.95850 + 0.792728i) q^{2} +(-2.36445 + 0.633552i) q^{3} +(4.66022 + 2.69058i) q^{4} +(-4.16825 - 2.76146i) q^{5} -7.49746 q^{6} +(5.78419 - 3.94249i) q^{7} +(2.99127 + 2.99127i) q^{8} +(-2.60501 + 1.50400i) q^{9} +(-10.1427 - 11.4741i) q^{10} +(-3.12159 + 5.40675i) q^{11} +(-12.7235 - 3.40924i) q^{12} +(11.7345 + 11.7345i) q^{13} +(20.2378 - 7.07857i) q^{14} +(11.6051 + 3.88853i) q^{15} +(-4.28389 - 7.41992i) q^{16} +(1.22400 + 4.56805i) q^{17} +(-8.89918 + 2.38453i) q^{18} +(-11.4064 + 6.58549i) q^{19} +(-11.9950 - 24.0840i) q^{20} +(-11.1786 + 12.9864i) q^{21} +(-13.5213 + 13.5213i) q^{22} +(9.33654 - 34.8444i) q^{23} +(-8.96783 - 5.17758i) q^{24} +(9.74863 + 23.0210i) q^{25} +(25.4144 + 44.0190i) q^{26} +(20.7846 - 20.7846i) q^{27} +(37.5631 - 2.81005i) q^{28} +28.5683i q^{29} +(31.2513 + 20.7040i) q^{30} +(10.1948 - 17.6579i) q^{31} +(-11.1715 - 41.6924i) q^{32} +(3.95538 - 14.7617i) q^{33} +14.4849i q^{34} +(-34.9970 + 0.460456i) q^{35} -16.1865 q^{36} +(22.6479 + 6.06849i) q^{37} +(-38.9664 + 10.4410i) q^{38} +(-35.1801 - 20.3113i) q^{39} +(-4.20808 - 20.7287i) q^{40} -45.1077 q^{41} +(-43.3667 + 29.5586i) q^{42} +(-21.9185 - 21.9185i) q^{43} +(-29.0946 + 16.7978i) q^{44} +(15.0116 + 0.924577i) q^{45} +(55.2443 - 95.6860i) q^{46} +(-10.4426 - 2.79810i) q^{47} +(14.8299 + 14.8299i) q^{48} +(17.9136 - 45.6081i) q^{49} +(10.5920 + 75.8356i) q^{50} +(-5.78819 - 10.0254i) q^{51} +(23.1128 + 86.2582i) q^{52} +(-71.6904 + 19.2094i) q^{53} +(77.9678 - 45.0147i) q^{54} +(27.9421 - 13.9165i) q^{55} +(29.0951 + 5.50902i) q^{56} +(22.7976 - 22.7976i) q^{57} +(-22.6469 + 84.5194i) q^{58} +(-14.6115 - 8.43595i) q^{59} +(43.6201 + 49.3459i) q^{60} +(16.7743 + 29.0540i) q^{61} +(44.1593 - 44.1593i) q^{62} +(-9.13833 + 18.9696i) q^{63} -97.9319i q^{64} +(-16.5080 - 81.3170i) q^{65} +(23.4040 - 40.5369i) q^{66} +(25.6891 + 95.8731i) q^{67} +(-6.58656 + 24.5814i) q^{68} +88.3030i q^{69} +(-103.904 - 26.3808i) q^{70} +66.2415 q^{71} +(-12.2912 - 3.29341i) q^{72} +(99.8826 - 26.7635i) q^{73} +(62.1933 + 35.9073i) q^{74} +(-37.6351 - 48.2556i) q^{75} -70.8751 q^{76} +(3.26020 + 43.5805i) q^{77} +(-87.9792 - 87.9792i) q^{78} +(-36.1253 + 20.8569i) q^{79} +(-2.63350 + 42.7579i) q^{80} +(-22.4400 + 38.8671i) q^{81} +(-133.451 - 35.7582i) q^{82} +(7.39450 + 7.39450i) q^{83} +(-87.0357 + 30.4424i) q^{84} +(7.51254 - 22.4208i) q^{85} +(-47.4706 - 82.2215i) q^{86} +(-18.0995 - 67.5483i) q^{87} +(-25.5106 + 6.83554i) q^{88} +(-19.2257 + 11.0999i) q^{89} +(43.6788 + 14.6355i) q^{90} +(114.138 + 21.6115i) q^{91} +(137.262 - 137.262i) q^{92} +(-12.9179 + 48.2102i) q^{93} +(-28.6764 - 16.5563i) q^{94} +(65.7303 + 4.04840i) q^{95} +(52.8286 + 91.5018i) q^{96} +(-73.6717 + 73.6717i) q^{97} +(89.1523 - 120.731i) q^{98} -18.7795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - 6 q^{7} - 36 q^{8} + 14 q^{10} - 24 q^{11} - 46 q^{12} - 8 q^{13} + 52 q^{15} + 20 q^{16} - 48 q^{17} - 4 q^{18} - 72 q^{20} + 56 q^{21} + 104 q^{22} - 86 q^{23} - 16 q^{25}+ \cdots + 482 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.95850 + 0.792728i 1.47925 + 0.396364i 0.906092 − 0.423080i \(-0.139051\pi\)
0.573159 + 0.819444i \(0.305718\pi\)
\(3\) −2.36445 + 0.633552i −0.788149 + 0.211184i −0.630374 − 0.776291i \(-0.717099\pi\)
−0.157775 + 0.987475i \(0.550432\pi\)
\(4\) 4.66022 + 2.69058i 1.16505 + 0.672645i
\(5\) −4.16825 − 2.76146i −0.833650 − 0.552293i
\(6\) −7.49746 −1.24958
\(7\) 5.78419 − 3.94249i 0.826312 − 0.563212i
\(8\) 2.99127 + 2.99127i 0.373909 + 0.373909i
\(9\) −2.60501 + 1.50400i −0.289445 + 0.167111i
\(10\) −10.1427 − 11.4741i −1.01427 − 1.14741i
\(11\) −3.12159 + 5.40675i −0.283781 + 0.491523i −0.972313 − 0.233683i \(-0.924922\pi\)
0.688532 + 0.725206i \(0.258256\pi\)
\(12\) −12.7235 − 3.40924i −1.06029 − 0.284103i
\(13\) 11.7345 + 11.7345i 0.902657 + 0.902657i 0.995665 − 0.0930085i \(-0.0296484\pi\)
−0.0930085 + 0.995665i \(0.529648\pi\)
\(14\) 20.2378 − 7.07857i 1.44556 − 0.505612i
\(15\) 11.6051 + 3.88853i 0.773676 + 0.259236i
\(16\) −4.28389 − 7.41992i −0.267743 − 0.463745i
\(17\) 1.22400 + 4.56805i 0.0720003 + 0.268709i 0.992536 − 0.121949i \(-0.0389144\pi\)
−0.920536 + 0.390657i \(0.872248\pi\)
\(18\) −8.89918 + 2.38453i −0.494399 + 0.132474i
\(19\) −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
\(21\) −11.1786 + 12.9864i −0.532316 + 0.618399i
\(22\) −13.5213 + 13.5213i −0.614606 + 0.614606i
\(23\) 9.33654 − 34.8444i 0.405936 − 1.51498i −0.396386 − 0.918084i \(-0.629736\pi\)
0.802322 − 0.596891i \(-0.203598\pi\)
\(24\) −8.96783 − 5.17758i −0.373660 − 0.215732i
\(25\) 9.74863 + 23.0210i 0.389945 + 0.920838i
\(26\) 25.4144 + 44.0190i 0.977475 + 1.69304i
\(27\) 20.7846 − 20.7846i 0.769800 − 0.769800i
\(28\) 37.5631 − 2.81005i 1.34154 − 0.100359i
\(29\) 28.5683i 0.985114i 0.870280 + 0.492557i \(0.163938\pi\)
−0.870280 + 0.492557i \(0.836062\pi\)
\(30\) 31.2513 + 20.7040i 1.04171 + 0.690132i
\(31\) 10.1948 − 17.6579i 0.328865 − 0.569611i −0.653422 − 0.756994i \(-0.726667\pi\)
0.982287 + 0.187383i \(0.0600006\pi\)
\(32\) −11.1715 − 41.6924i −0.349108 − 1.30289i
\(33\) 3.95538 − 14.7617i 0.119860 − 0.447323i
\(34\) 14.4849i 0.426026i
\(35\) −34.9970 + 0.460456i −0.999913 + 0.0131559i
\(36\) −16.1865 −0.449626
\(37\) 22.6479 + 6.06849i 0.612106 + 0.164013i 0.551537 − 0.834151i \(-0.314042\pi\)
0.0605694 + 0.998164i \(0.480708\pi\)
\(38\) −38.9664 + 10.4410i −1.02543 + 0.274763i
\(39\) −35.1801 − 20.3113i −0.902055 − 0.520802i
\(40\) −4.20808 − 20.7287i −0.105202 − 0.518217i
\(41\) −45.1077 −1.10019 −0.550094 − 0.835103i \(-0.685408\pi\)
−0.550094 + 0.835103i \(0.685408\pi\)
\(42\) −43.3667 + 29.5586i −1.03254 + 0.703777i
\(43\) −21.9185 − 21.9185i −0.509733 − 0.509733i 0.404711 − 0.914445i \(-0.367372\pi\)
−0.914445 + 0.404711i \(0.867372\pi\)
\(44\) −29.0946 + 16.7978i −0.661241 + 0.381767i
\(45\) 15.0116 + 0.924577i 0.333590 + 0.0205461i
\(46\) 55.2443 − 95.6860i 1.20096 − 2.08013i
\(47\) −10.4426 − 2.79810i −0.222184 − 0.0595339i 0.146010 − 0.989283i \(-0.453357\pi\)
−0.368193 + 0.929749i \(0.620024\pi\)
\(48\) 14.8299 + 14.8299i 0.308957 + 0.308957i
\(49\) 17.9136 − 45.6081i 0.365584 − 0.930779i
\(50\) 10.5920 + 75.8356i 0.211840 + 1.51671i
\(51\) −5.78819 − 10.0254i −0.113494 − 0.196577i
\(52\) 23.1128 + 86.2582i 0.444477 + 1.65881i
\(53\) −71.6904 + 19.2094i −1.35265 + 0.362441i −0.861112 − 0.508415i \(-0.830232\pi\)
−0.491537 + 0.870856i \(0.663565\pi\)
\(54\) 77.9678 − 45.0147i 1.44385 − 0.833606i
\(55\) 27.9421 − 13.9165i 0.508039 − 0.253028i
\(56\) 29.0951 + 5.50902i 0.519556 + 0.0983754i
\(57\) 22.7976 − 22.7976i 0.399958 − 0.399958i
\(58\) −22.6469 + 84.5194i −0.390464 + 1.45723i
\(59\) −14.6115 − 8.43595i −0.247652 − 0.142982i 0.371036 − 0.928618i \(-0.379002\pi\)
−0.618689 + 0.785636i \(0.712336\pi\)
\(60\) 43.6201 + 49.3459i 0.727001 + 0.822432i
\(61\) 16.7743 + 29.0540i 0.274989 + 0.476295i 0.970132 − 0.242576i \(-0.0779924\pi\)
−0.695143 + 0.718871i \(0.744659\pi\)
\(62\) 44.1593 − 44.1593i 0.712247 − 0.712247i
\(63\) −9.13833 + 18.9696i −0.145053 + 0.301105i
\(64\) − 97.9319i − 1.53019i
\(65\) −16.5080 − 81.3170i −0.253969 − 1.25103i
\(66\) 23.4040 − 40.5369i 0.354606 − 0.614196i
\(67\) 25.6891 + 95.8731i 0.383420 + 1.43094i 0.840643 + 0.541590i \(0.182177\pi\)
−0.457223 + 0.889352i \(0.651156\pi\)
\(68\) −6.58656 + 24.5814i −0.0968612 + 0.361491i
\(69\) 88.3030i 1.27975i
\(70\) −103.904 − 26.3808i −1.48434 − 0.376869i
\(71\) 66.2415 0.932979 0.466489 − 0.884527i \(-0.345519\pi\)
0.466489 + 0.884527i \(0.345519\pi\)
\(72\) −12.2912 − 3.29341i −0.170711 − 0.0457417i
\(73\) 99.8826 − 26.7635i 1.36825 − 0.366623i 0.501414 − 0.865208i \(-0.332813\pi\)
0.866841 + 0.498585i \(0.166147\pi\)
\(74\) 62.1933 + 35.9073i 0.840450 + 0.485234i
\(75\) −37.6351 − 48.2556i −0.501801 − 0.643407i
\(76\) −70.8751 −0.932567
\(77\) 3.26020 + 43.5805i 0.0423402 + 0.565980i
\(78\) −87.9792 − 87.9792i −1.12794 − 1.12794i
\(79\) −36.1253 + 20.8569i −0.457282 + 0.264012i −0.710901 − 0.703292i \(-0.751712\pi\)
0.253619 + 0.967304i \(0.418379\pi\)
\(80\) −2.63350 + 42.7579i −0.0329188 + 0.534474i
\(81\) −22.4400 + 38.8671i −0.277036 + 0.479841i
\(82\) −133.451 − 35.7582i −1.62745 − 0.436075i
\(83\) 7.39450 + 7.39450i 0.0890904 + 0.0890904i 0.750247 − 0.661157i \(-0.229934\pi\)
−0.661157 + 0.750247i \(0.729934\pi\)
\(84\) −87.0357 + 30.4424i −1.03614 + 0.362409i
\(85\) 7.51254 − 22.4208i 0.0883828 − 0.263774i
\(86\) −47.4706 − 82.2215i −0.551984 − 0.956064i
\(87\) −18.0995 − 67.5483i −0.208040 − 0.776417i
\(88\) −25.5106 + 6.83554i −0.289893 + 0.0776766i
\(89\) −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
\(91\) 114.138 + 21.6115i 1.25426 + 0.237489i
\(92\) 137.262 − 137.262i 1.49198 − 1.49198i
\(93\) −12.9179 + 48.2102i −0.138902 + 0.518389i
\(94\) −28.6764 − 16.5563i −0.305068 − 0.176131i
\(95\) 65.7303 + 4.04840i 0.691898 + 0.0426147i
\(96\) 52.8286 + 91.5018i 0.550298 + 0.953144i
\(97\) −73.6717 + 73.6717i −0.759502 + 0.759502i −0.976232 − 0.216730i \(-0.930461\pi\)
0.216730 + 0.976232i \(0.430461\pi\)
\(98\) 89.1523 − 120.731i 0.909717 − 1.23195i
\(99\) − 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 35.3.l.a.18.6 yes 24
3.2 odd 2 315.3.ca.a.298.1 24
5.2 odd 4 inner 35.3.l.a.32.1 yes 24
5.3 odd 4 175.3.p.c.32.6 24
5.4 even 2 175.3.p.c.18.1 24
7.2 even 3 inner 35.3.l.a.23.1 yes 24
7.3 odd 6 245.3.g.b.148.1 12
7.4 even 3 245.3.g.c.148.1 12
7.5 odd 6 245.3.m.b.128.1 24
7.6 odd 2 245.3.m.b.18.6 24
15.2 even 4 315.3.ca.a.172.6 24
21.2 odd 6 315.3.ca.a.163.6 24
35.2 odd 12 inner 35.3.l.a.2.6 ✓ 24
35.9 even 6 175.3.p.c.93.6 24
35.12 even 12 245.3.m.b.177.6 24
35.17 even 12 245.3.g.b.197.1 12
35.23 odd 12 175.3.p.c.107.1 24
35.27 even 4 245.3.m.b.67.1 24
35.32 odd 12 245.3.g.c.197.1 12
105.2 even 12 315.3.ca.a.37.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.6 ✓ 24 35.2 odd 12 inner
35.3.l.a.18.6 yes 24 1.1 even 1 trivial
35.3.l.a.23.1 yes 24 7.2 even 3 inner
35.3.l.a.32.1 yes 24 5.2 odd 4 inner
175.3.p.c.18.1 24 5.4 even 2
175.3.p.c.32.6 24 5.3 odd 4
175.3.p.c.93.6 24 35.9 even 6
175.3.p.c.107.1 24 35.23 odd 12
245.3.g.b.148.1 12 7.3 odd 6
245.3.g.b.197.1 12 35.17 even 12
245.3.g.c.148.1 12 7.4 even 3
245.3.g.c.197.1 12 35.32 odd 12
245.3.m.b.18.6 24 7.6 odd 2
245.3.m.b.67.1 24 35.27 even 4
245.3.m.b.128.1 24 7.5 odd 6
245.3.m.b.177.6 24 35.12 even 12
315.3.ca.a.37.1 24 105.2 even 12
315.3.ca.a.163.6 24 21.2 odd 6
315.3.ca.a.172.6 24 15.2 even 4
315.3.ca.a.298.1 24 3.2 odd 2