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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 163.6
Character \(\chi\) \(=\) 315.163
Dual form 315.3.ca.a.172.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.792728 + 2.95850i) q^{2} +(-4.66022 + 2.69058i) q^{4} +(0.307372 - 4.99054i) q^{5} +(3.94249 - 5.78419i) q^{7} +(-2.99127 - 2.99127i) q^{8} +(15.0082 - 3.04678i) q^{10} +(3.12159 + 5.40675i) q^{11} +(11.7345 + 11.7345i) q^{13} +(20.2378 + 7.07857i) q^{14} +(-4.28389 + 7.41992i) q^{16} +(4.56805 + 1.22400i) q^{17} +(11.4064 + 6.58549i) q^{19} +(11.9950 + 24.0840i) q^{20} +(-13.5213 + 13.5213i) q^{22} +(34.8444 - 9.33654i) q^{23} +(-24.8110 - 3.06791i) q^{25} +(-25.4144 + 44.0190i) q^{26} +(-2.81005 + 37.5631i) q^{28} -28.5683i q^{29} +(10.1948 + 17.6579i) q^{31} +(-41.6924 - 11.1715i) q^{32} +14.4849i q^{34} +(-27.6544 - 21.4530i) q^{35} +(-6.06849 - 22.6479i) q^{37} +(-10.4410 + 38.9664i) q^{38} +(-15.8475 + 14.0086i) q^{40} +45.1077 q^{41} +(-21.9185 - 21.9185i) q^{43} +(-29.0946 - 16.7978i) q^{44} +(55.2443 + 95.6860i) q^{46} +(-2.79810 - 10.4426i) q^{47} +(-17.9136 - 45.6081i) q^{49} +(-10.5920 - 75.8356i) q^{50} +(-86.2582 - 23.1128i) q^{52} +(-19.2094 + 71.6904i) q^{53} +(27.9421 - 13.9165i) q^{55} +(-29.0951 + 5.50902i) q^{56} +(84.5194 - 22.6469i) q^{58} +(-14.6115 + 8.43595i) q^{59} +(16.7743 - 29.0540i) q^{61} +(-44.1593 + 44.1593i) q^{62} -97.9319i q^{64} +(62.1686 - 54.9548i) q^{65} +(-95.8731 - 25.6891i) q^{67} +(-24.5814 + 6.58656i) q^{68} +(41.5465 - 98.8221i) q^{70} -66.2415 q^{71} +(-26.7635 + 99.8826i) q^{73} +(62.1933 - 35.9073i) q^{74} -70.8751 q^{76} +(43.5805 + 3.26020i) q^{77} +(36.1253 + 20.8569i) q^{79} +(35.7127 + 23.6596i) q^{80} +(35.7582 + 133.451i) q^{82} +(-7.39450 - 7.39450i) q^{83} +(7.51254 - 22.4208i) q^{85} +(47.4706 - 82.2215i) q^{86} +(6.83554 - 25.5106i) q^{88} +(-19.2257 - 11.0999i) q^{89} +(114.138 - 21.6115i) q^{91} +(-137.262 + 137.262i) q^{92} +(28.6764 - 16.5563i) q^{94} +(36.3712 - 54.9000i) q^{95} +(-73.6717 + 73.6717i) q^{97} +(120.731 - 89.1523i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 4 q^{5} - 6 q^{7} + 36 q^{8} + 14 q^{10} + 24 q^{11} - 8 q^{13} + 20 q^{16} + 48 q^{17} + 72 q^{20} + 104 q^{22} + 86 q^{23} - 16 q^{25} - 140 q^{26} + 186 q^{28} + 120 q^{31} - 130 q^{32}+ \cdots - 482 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.792728 + 2.95850i 0.396364 + 1.47925i 0.819444 + 0.573159i \(0.194282\pi\)
−0.423080 + 0.906092i \(0.639051\pi\)
\(3\) 0 0
\(4\) −4.66022 + 2.69058i −1.16505 + 0.672645i
\(5\) 0.307372 − 4.99054i 0.0614745 − 0.998109i
\(6\) 0 0
\(7\) 3.94249 − 5.78419i 0.563212 − 0.826312i
\(8\) −2.99127 − 2.99127i −0.373909 − 0.373909i
\(9\) 0 0
\(10\) 15.0082 − 3.04678i 1.50082 − 0.304678i
\(11\) 3.12159 + 5.40675i 0.283781 + 0.491523i 0.972313 − 0.233683i \(-0.0750778\pi\)
−0.688532 + 0.725206i \(0.741744\pi\)
\(12\) 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
\(17\) 4.56805 + 1.22400i 0.268709 + 0.0720003i 0.390657 − 0.920536i \(-0.372248\pi\)
−0.121949 + 0.992536i \(0.538914\pi\)
\(18\) 0 0
\(19\) 11.4064 + 6.58549i 0.600337 + 0.346605i 0.769174 − 0.639039i \(-0.220668\pi\)
−0.168837 + 0.985644i \(0.554001\pi\)
\(20\) 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 \(-0.296402\pi\)
0.918084 + 0.396386i \(0.129736\pi\)
\(24\) 0 0
\(25\) −24.8110 − 3.06791i −0.992442 − 0.122716i
\(26\) −25.4144 + 44.0190i −0.977475 + 1.69304i
\(27\) 0 0
\(28\) −2.81005 + 37.5631i −0.100359 + 1.34154i
\(29\) − 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 0.982287 − 0.187383i \(-0.0600006\pi\)
−0.653422 + 0.756994i \(0.726667\pi\)
\(32\) −41.6924 − 11.1715i −1.30289 − 0.349108i
\(33\) 0 0
\(34\) 14.4849i 0.426026i
\(35\) −27.6544 − 21.4530i −0.790126 − 0.612944i
\(36\) 0 0
\(37\) −6.06849 − 22.6479i −0.164013 − 0.612106i −0.998164 − 0.0605694i \(-0.980708\pi\)
0.834151 − 0.551537i \(-0.185958\pi\)
\(38\) −10.4410 + 38.9664i −0.274763 + 1.02543i
\(39\) 0 0
\(40\) −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 0.404711 − 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
\(47\) −2.79810 − 10.4426i −0.0595339 − 0.222184i 0.929749 − 0.368193i \(-0.120024\pi\)
−0.989283 + 0.146010i \(0.953357\pi\)
\(48\) 0 0
\(49\) −17.9136 − 45.6081i −0.365584 − 0.930779i
\(50\) −10.5920 − 75.8356i −0.211840 − 1.51671i
\(51\) 0 0
\(52\) −86.2582 − 23.1128i −1.65881 − 0.444477i
\(53\) −19.2094 + 71.6904i −0.362441 + 1.35265i 0.508415 + 0.861112i \(0.330232\pi\)
−0.870856 + 0.491537i \(0.836435\pi\)
\(54\) 0 0
\(55\) 27.9421 − 13.9165i 0.508039 − 0.253028i
\(56\) −29.0951 + 5.50902i −0.519556 + 0.0983754i
\(57\) 0 0
\(58\) 84.5194 − 22.6469i 1.45723 − 0.390464i
\(59\) −14.6115 + 8.43595i −0.247652 + 0.142982i −0.618689 − 0.785636i \(-0.712336\pi\)
0.371036 + 0.928618i \(0.379002\pi\)
\(60\) 0 0
\(61\) 16.7743 − 29.0540i 0.274989 − 0.476295i −0.695143 − 0.718871i \(-0.744659\pi\)
0.970132 + 0.242576i \(0.0779924\pi\)
\(62\) −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
\(66\) 0 0
\(67\) −95.8731 − 25.6891i −1.43094 − 0.383420i −0.541590 − 0.840643i \(-0.682177\pi\)
−0.889352 + 0.457223i \(0.848844\pi\)
\(68\) −24.5814 + 6.58656i −0.361491 + 0.0968612i
\(69\) 0 0
\(70\) 41.5465 − 98.8221i 0.593521 − 1.41174i
\(71\) −66.2415 −0.932979 −0.466489 − 0.884527i \(-0.654481\pi\)
−0.466489 + 0.884527i \(0.654481\pi\)
\(72\) 0 0
\(73\) −26.7635 + 99.8826i −0.366623 + 1.36825i 0.498585 + 0.866841i \(0.333853\pi\)
−0.865208 + 0.501414i \(0.832813\pi\)
\(74\) 62.1933 − 35.9073i 0.840450 − 0.485234i
\(75\) 0 0
\(76\) −70.8751 −0.932567
\(77\) 43.5805 + 3.26020i 0.565980 + 0.0423402i
\(78\) 0 0
\(79\) 36.1253 + 20.8569i 0.457282 + 0.264012i 0.710901 − 0.703292i \(-0.248288\pi\)
−0.253619 + 0.967304i \(0.581621\pi\)
\(80\) 35.7127 + 23.6596i 0.446408 + 0.295745i
\(81\) 0 0
\(82\) 35.7582 + 133.451i 0.436075 + 1.62745i
\(83\) −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
\(88\) 6.83554 − 25.5106i 0.0776766 − 0.289893i
\(89\) −19.2257 − 11.0999i −0.216019 − 0.124718i 0.388087 − 0.921623i \(-0.373136\pi\)
−0.604105 + 0.796904i \(0.706469\pi\)
\(90\) 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
\(95\) 36.3712 − 54.9000i 0.382855 − 0.577894i
\(96\) 0 0
\(97\) −73.6717 + 73.6717i −0.759502 + 0.759502i −0.976232 − 0.216730i \(-0.930461\pi\)
0.216730 + 0.976232i \(0.430461\pi\)
\(98\) 120.731 − 89.1523i 1.23195 − 0.909717i
\(99\) 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.163.6 24
3.2 odd 2 35.3.l.a.23.1 yes 24
5.2 odd 4 inner 315.3.ca.a.37.1 24
7.4 even 3 inner 315.3.ca.a.298.1 24
15.2 even 4 35.3.l.a.2.6 ✓ 24
15.8 even 4 175.3.p.c.107.1 24
15.14 odd 2 175.3.p.c.93.6 24
21.2 odd 6 245.3.g.c.148.1 12
21.5 even 6 245.3.g.b.148.1 12
21.11 odd 6 35.3.l.a.18.6 yes 24
21.17 even 6 245.3.m.b.18.6 24
21.20 even 2 245.3.m.b.128.1 24
35.32 odd 12 inner 315.3.ca.a.172.6 24
105.2 even 12 245.3.g.c.197.1 12
105.17 odd 12 245.3.m.b.67.1 24
105.32 even 12 35.3.l.a.32.1 yes 24
105.47 odd 12 245.3.g.b.197.1 12
105.53 even 12 175.3.p.c.32.6 24
105.62 odd 4 245.3.m.b.177.6 24
105.74 odd 6 175.3.p.c.18.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.l.a.2.6 ✓ 24 15.2 even 4
35.3.l.a.18.6 yes 24 21.11 odd 6
35.3.l.a.23.1 yes 24 3.2 odd 2
35.3.l.a.32.1 yes 24 105.32 even 12
175.3.p.c.18.1 24 105.74 odd 6
175.3.p.c.32.6 24 105.53 even 12
175.3.p.c.93.6 24 15.14 odd 2
175.3.p.c.107.1 24 15.8 even 4
245.3.g.b.148.1 12 21.5 even 6
245.3.g.b.197.1 12 105.47 odd 12
245.3.g.c.148.1 12 21.2 odd 6
245.3.g.c.197.1 12 105.2 even 12
245.3.m.b.18.6 24 21.17 even 6
245.3.m.b.67.1 24 105.17 odd 12
245.3.m.b.128.1 24 21.20 even 2
245.3.m.b.177.6 24 105.62 odd 4
315.3.ca.a.37.1 24 5.2 odd 4 inner
315.3.ca.a.163.6 24 1.1 even 1 trivial
315.3.ca.a.172.6 24 35.32 odd 12 inner
315.3.ca.a.298.1 24 7.4 even 3 inner