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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,3,Mod(37,315)] 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("315.37"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.ca (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,2,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.58312832735\)
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 298.1
Character \(\chi\) \(=\) 315.298
Dual form 315.3.ca.a.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.95850 - 0.792728i) q^{2} +(4.66022 + 2.69058i) q^{4} +(4.16825 + 2.76146i) q^{5} +(5.78419 - 3.94249i) q^{7} +(-2.99127 - 2.99127i) q^{8} +(-10.1427 - 11.4741i) q^{10} +(3.12159 - 5.40675i) q^{11} +(11.7345 + 11.7345i) q^{13} +(-20.2378 + 7.07857i) q^{14} +(-4.28389 - 7.41992i) q^{16} +(-1.22400 - 4.56805i) q^{17} +(-11.4064 + 6.58549i) q^{19} +(11.9950 + 24.0840i) q^{20} +(-13.5213 + 13.5213i) q^{22} +(-9.33654 + 34.8444i) q^{23} +(9.74863 + 23.0210i) q^{25} +(-25.4144 - 44.0190i) q^{26} +(37.5631 - 2.81005i) q^{28} -28.5683i q^{29} +(10.1948 - 17.6579i) q^{31} +(11.1715 + 41.6924i) q^{32} +14.4849i q^{34} +(34.9970 - 0.460456i) q^{35} +(22.6479 + 6.06849i) q^{37} +(38.9664 - 10.4410i) q^{38} +(-4.20808 - 20.7287i) q^{40} +45.1077 q^{41} +(-21.9185 - 21.9185i) q^{43} +(29.0946 - 16.7978i) q^{44} +(55.2443 - 95.6860i) q^{46} +(10.4426 + 2.79810i) q^{47} +(17.9136 - 45.6081i) q^{49} +(-10.5920 - 75.8356i) q^{50} +(23.1128 + 86.2582i) q^{52} +(71.6904 - 19.2094i) q^{53} +(27.9421 - 13.9165i) q^{55} +(-29.0951 - 5.50902i) q^{56} +(-22.6469 + 84.5194i) q^{58} +(14.6115 + 8.43595i) q^{59} +(16.7743 + 29.0540i) q^{61} +(-44.1593 + 44.1593i) q^{62} -97.9319i q^{64} +(16.5080 + 81.3170i) q^{65} +(25.6891 + 95.8731i) q^{67} +(6.58656 - 24.5814i) q^{68} +(-103.904 - 26.3808i) q^{70} -66.2415 q^{71} +(99.8826 - 26.7635i) q^{73} +(-62.1933 - 35.9073i) q^{74} -70.8751 q^{76} +(-3.26020 - 43.5805i) q^{77} +(-36.1253 + 20.8569i) q^{79} +(2.63350 - 42.7579i) q^{80} +(-133.451 - 35.7582i) q^{82} +(-7.39450 - 7.39450i) q^{83} +(7.51254 - 22.4208i) q^{85} +(47.4706 + 82.2215i) q^{86} +(-25.5106 + 6.83554i) q^{88} +(19.2257 - 11.0999i) q^{89} +(114.138 + 21.6115i) q^{91} +(-137.262 + 137.262i) q^{92} +(-28.6764 - 16.5563i) q^{94} +(-65.7303 - 4.04840i) q^{95} +(-73.6717 + 73.6717i) q^{97} +(-89.1523 + 120.731i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 4 q^{5} - 6 q^{7} + 36 q^{8} + 14 q^{10} + 24 q^{11} - 8 q^{13} + 20 q^{16} + 48 q^{17} + 72 q^{20} + 104 q^{22} + 86 q^{23} - 16 q^{25} - 140 q^{26} + 186 q^{28} + 120 q^{31} - 130 q^{32}+ \cdots - 482 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.95850 − 0.792728i −1.47925 − 0.396364i −0.573159 − 0.819444i \(-0.694282\pi\)
−0.906092 + 0.423080i \(0.860949\pi\)
\(3\) 0 0
\(4\) 4.66022 + 2.69058i 1.16505 + 0.672645i
\(5\) 4.16825 + 2.76146i 0.833650 + 0.552293i
\(6\) 0 0
\(7\) 5.78419 − 3.94249i 0.826312 − 0.563212i
\(8\) −2.99127 − 2.99127i −0.373909 − 0.373909i
\(9\) 0 0
\(10\) −10.1427 − 11.4741i −1.01427 − 1.14741i
\(11\) 3.12159 − 5.40675i 0.283781 − 0.491523i −0.688532 − 0.725206i \(-0.741744\pi\)
0.972313 + 0.233683i \(0.0750778\pi\)
\(12\) 0 0
\(13\) 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
\(15\) 0 0
\(16\) −4.28389 − 7.41992i −0.267743 − 0.463745i
\(17\) −1.22400 − 4.56805i −0.0720003 − 0.268709i 0.920536 − 0.390657i \(-0.127752\pi\)
−0.992536 + 0.121949i \(0.961086\pi\)
\(18\) 0 0
\(19\) −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
\(23\) −9.33654 + 34.8444i −0.405936 + 1.51498i 0.396386 + 0.918084i \(0.370264\pi\)
−0.802322 + 0.596891i \(0.796402\pi\)
\(24\) 0 0
\(25\) 9.74863 + 23.0210i 0.389945 + 0.920838i
\(26\) −25.4144 − 44.0190i −0.977475 − 1.69304i
\(27\) 0 0
\(28\) 37.5631 − 2.81005i 1.34154 − 0.100359i
\(29\) − 28.5683i − 0.985114i −0.870280 − 0.492557i \(-0.836062\pi\)
0.870280 − 0.492557i \(-0.163938\pi\)
\(30\) 0 0
\(31\) 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
\(33\) 0 0
\(34\) 14.4849i 0.426026i
\(35\) 34.9970 − 0.460456i 0.999913 − 0.0131559i
\(36\) 0 0
\(37\) 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
\(39\) 0 0
\(40\) −4.20808 − 20.7287i −0.105202 − 0.518217i
\(41\) 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 0.404711 − 0.914445i \(-0.367372\pi\)
−0.914445 + 0.404711i \(0.867372\pi\)
\(44\) 29.0946 − 16.7978i 0.661241 − 0.381767i
\(45\) 0 0
\(46\) 55.2443 − 95.6860i 1.20096 − 2.08013i
\(47\) 10.4426 + 2.79810i 0.222184 + 0.0595339i 0.368193 − 0.929749i \(-0.379976\pi\)
−0.146010 + 0.989283i \(0.546643\pi\)
\(48\) 0 0
\(49\) 17.9136 − 45.6081i 0.365584 − 0.930779i
\(50\) −10.5920 − 75.8356i −0.211840 − 1.51671i
\(51\) 0 0
\(52\) 23.1128 + 86.2582i 0.444477 + 1.65881i
\(53\) 71.6904 − 19.2094i 1.35265 − 0.362441i 0.491537 − 0.870856i \(-0.336435\pi\)
0.861112 + 0.508415i \(0.169768\pi\)
\(54\) 0 0
\(55\) 27.9421 − 13.9165i 0.508039 − 0.253028i
\(56\) −29.0951 − 5.50902i −0.519556 − 0.0983754i
\(57\) 0 0
\(58\) −22.6469 + 84.5194i −0.390464 + 1.45723i
\(59\) 14.6115 + 8.43595i 0.247652 + 0.142982i 0.618689 − 0.785636i \(-0.287664\pi\)
−0.371036 + 0.928618i \(0.620998\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) − 97.9319i − 1.53019i
\(65\) 16.5080 + 81.3170i 0.253969 + 1.25103i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) −103.904 − 26.3808i −1.48434 − 0.376869i
\(71\) −66.2415 −0.932979 −0.466489 − 0.884527i \(-0.654481\pi\)
−0.466489 + 0.884527i \(0.654481\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) −70.8751 −0.932567
\(77\) −3.26020 − 43.5805i −0.0423402 − 0.565980i
\(78\) 0 0
\(79\) −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
\(81\) 0 0
\(82\) −133.451 − 35.7582i −1.62745 − 0.436075i
\(83\) −7.39450 − 7.39450i −0.0890904 − 0.0890904i 0.661157 − 0.750247i \(-0.270066\pi\)
−0.750247 + 0.661157i \(0.770066\pi\)
\(84\) 0 0
\(85\) 7.51254 − 22.4208i 0.0883828 − 0.263774i
\(86\) 47.4706 + 82.2215i 0.551984 + 0.956064i
\(87\) 0 0
\(88\) −25.5106 + 6.83554i −0.289893 + 0.0776766i
\(89\) 19.2257 − 11.0999i 0.216019 − 0.124718i −0.388087 − 0.921623i \(-0.626864\pi\)
0.604105 + 0.796904i \(0.293531\pi\)
\(90\) 0 0
\(91\) 114.138 + 21.6115i 1.25426 + 0.237489i
\(92\) −137.262 + 137.262i −1.49198 + 1.49198i
\(93\) 0 0
\(94\) −28.6764 − 16.5563i −0.305068 − 0.176131i
\(95\) −65.7303 − 4.04840i −0.691898 − 0.0426147i
\(96\) 0 0
\(97\) −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
\(99\) 0 0
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 315.3.ca.a.298.1 24
3.2 odd 2 35.3.l.a.18.6 yes 24
5.2 odd 4 inner 315.3.ca.a.172.6 24
7.2 even 3 inner 315.3.ca.a.163.6 24
15.2 even 4 35.3.l.a.32.1 yes 24
15.8 even 4 175.3.p.c.32.6 24
15.14 odd 2 175.3.p.c.18.1 24
21.2 odd 6 35.3.l.a.23.1 yes 24
21.5 even 6 245.3.m.b.128.1 24
21.11 odd 6 245.3.g.c.148.1 12
21.17 even 6 245.3.g.b.148.1 12
21.20 even 2 245.3.m.b.18.6 24
35.2 odd 12 inner 315.3.ca.a.37.1 24
105.2 even 12 35.3.l.a.2.6 ✓ 24
105.17 odd 12 245.3.g.b.197.1 12
105.23 even 12 175.3.p.c.107.1 24
105.32 even 12 245.3.g.c.197.1 12
105.44 odd 6 175.3.p.c.93.6 24
105.47 odd 12 245.3.m.b.177.6 24
105.62 odd 4 245.3.m.b.67.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.6 ✓ 24 105.2 even 12
35.3.l.a.18.6 yes 24 3.2 odd 2
35.3.l.a.23.1 yes 24 21.2 odd 6
35.3.l.a.32.1 yes 24 15.2 even 4
175.3.p.c.18.1 24 15.14 odd 2
175.3.p.c.32.6 24 15.8 even 4
175.3.p.c.93.6 24 105.44 odd 6
175.3.p.c.107.1 24 105.23 even 12
245.3.g.b.148.1 12 21.17 even 6
245.3.g.b.197.1 12 105.17 odd 12
245.3.g.c.148.1 12 21.11 odd 6
245.3.g.c.197.1 12 105.32 even 12
245.3.m.b.18.6 24 21.20 even 2
245.3.m.b.67.1 24 105.62 odd 4
245.3.m.b.128.1 24 21.5 even 6
245.3.m.b.177.6 24 105.47 odd 12
315.3.ca.a.37.1 24 35.2 odd 12 inner
315.3.ca.a.163.6 24 7.2 even 3 inner
315.3.ca.a.172.6 24 5.2 odd 4 inner
315.3.ca.a.298.1 24 1.1 even 1 trivial