Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,10,Mod(1,315)] 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("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(162.236288392\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 1253x^{2} - 1039x + 42996 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.41896\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.41896 q^{2} -500.311 q^{4} +625.000 q^{5} +2401.00 q^{7} +3461.05 q^{8} -2136.85 q^{10} -42324.4 q^{11} -65379.6 q^{13} -8208.93 q^{14} +244326. q^{16} -361401. q^{17} -280552. q^{19} -312694. q^{20} +144706. q^{22} -80488.4 q^{23} +390625. q^{25} +223531. q^{26} -1.20125e6 q^{28} +6.86593e6 q^{29} -2.38755e6 q^{31} -2.60740e6 q^{32} +1.23562e6 q^{34} +1.50062e6 q^{35} -5.05576e6 q^{37} +959196. q^{38} +2.16316e6 q^{40} +1.33202e7 q^{41} -2.61112e7 q^{43} +2.11753e7 q^{44} +275187. q^{46} +8.24496e6 q^{47} +5.76480e6 q^{49} -1.33553e6 q^{50} +3.27101e7 q^{52} -1.87123e7 q^{53} -2.64527e7 q^{55} +8.30999e6 q^{56} -2.34744e7 q^{58} -8.65949e7 q^{59} -8.16988e6 q^{61} +8.16296e6 q^{62} -1.16180e8 q^{64} -4.08623e7 q^{65} -2.71486e8 q^{67} +1.80813e8 q^{68} -5.13058e6 q^{70} -8.00533e7 q^{71} -3.35309e8 q^{73} +1.72855e7 q^{74} +1.40363e8 q^{76} -1.01621e8 q^{77} -1.79224e8 q^{79} +1.52704e8 q^{80} -4.55413e7 q^{82} +5.62335e8 q^{83} -2.25876e8 q^{85} +8.92734e7 q^{86} -1.46487e8 q^{88} +2.46154e8 q^{89} -1.56976e8 q^{91} +4.02692e7 q^{92} -2.81892e7 q^{94} -1.75345e8 q^{95} +7.58371e8 q^{97} -1.97096e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.41896 −0.151098 −0.0755491 − 0.997142i \(-0.524071\pi\)
−0.0755491 + 0.997142i \(0.524071\pi\)
\(3\) 0 0
\(4\) −500.311 −0.977169
\(5\) 625.000 0.447214
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) 3461.05 0.298747
\(9\) 0 0
\(10\) −2136.85 −0.0675732
\(11\) −42324.4 −0.871613 −0.435806 − 0.900040i \(-0.643537\pi\)
−0.435806 + 0.900040i \(0.643537\pi\)
\(12\) 0 0
\(13\) −65379.6 −0.634888 −0.317444 − 0.948277i \(-0.602825\pi\)
−0.317444 + 0.948277i \(0.602825\pi\)
\(14\) −8208.93 −0.0571098
\(15\) 0 0
\(16\) 244326. 0.932029
\(17\) −361401. −1.04947 −0.524734 − 0.851266i \(-0.675835\pi\)
−0.524734 + 0.851266i \(0.675835\pi\)
\(18\) 0 0
\(19\) −280552. −0.493880 −0.246940 − 0.969031i \(-0.579425\pi\)
−0.246940 + 0.969031i \(0.579425\pi\)
\(20\) −312694. −0.437003
\(21\) 0 0
\(22\) 144706. 0.131699
\(23\) −80488.4 −0.0599733 −0.0299867 − 0.999550i \(-0.509546\pi\)
−0.0299867 + 0.999550i \(0.509546\pi\)
\(24\) 0 0
\(25\) 390625. 0.200000
\(26\) 223531. 0.0959305
\(27\) 0 0
\(28\) −1.20125e6 −0.369335
\(29\) 6.86593e6 1.80264 0.901319 − 0.433156i \(-0.142600\pi\)
0.901319 + 0.433156i \(0.142600\pi\)
\(30\) 0 0
\(31\) −2.38755e6 −0.464329 −0.232164 − 0.972677i \(-0.574581\pi\)
−0.232164 + 0.972677i \(0.574581\pi\)
\(32\) −2.60740e6 −0.439575
\(33\) 0 0
\(34\) 1.23562e6 0.158573
\(35\) 1.50062e6 0.169031
\(36\) 0 0
\(37\) −5.05576e6 −0.443485 −0.221742 − 0.975105i \(-0.571174\pi\)
−0.221742 + 0.975105i \(0.571174\pi\)
\(38\) 959196. 0.0746244
\(39\) 0 0
\(40\) 2.16316e6 0.133604
\(41\) 1.33202e7 0.736180 0.368090 − 0.929790i \(-0.380012\pi\)
0.368090 + 0.929790i \(0.380012\pi\)
\(42\) 0 0
\(43\) −2.61112e7 −1.16471 −0.582357 − 0.812933i \(-0.697870\pi\)
−0.582357 + 0.812933i \(0.697870\pi\)
\(44\) 2.11753e7 0.851713
\(45\) 0 0
\(46\) 275187. 0.00906186
\(47\) 8.24496e6 0.246461 0.123230 − 0.992378i \(-0.460675\pi\)
0.123230 + 0.992378i \(0.460675\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) −1.33553e6 −0.0302197
\(51\) 0 0
\(52\) 3.27101e7 0.620393
\(53\) −1.87123e7 −0.325751 −0.162875 − 0.986647i \(-0.552077\pi\)
−0.162875 + 0.986647i \(0.552077\pi\)
\(54\) 0 0
\(55\) −2.64527e7 −0.389797
\(56\) 8.30999e6 0.112916
\(57\) 0 0
\(58\) −2.34744e7 −0.272375
\(59\) −8.65949e7 −0.930375 −0.465188 − 0.885212i \(-0.654013\pi\)
−0.465188 + 0.885212i \(0.654013\pi\)
\(60\) 0 0
\(61\) −8.16988e6 −0.0755495 −0.0377747 − 0.999286i \(-0.512027\pi\)
−0.0377747 + 0.999286i \(0.512027\pi\)
\(62\) 8.16296e6 0.0701593
\(63\) 0 0
\(64\) −1.16180e8 −0.865610
\(65\) −4.08623e7 −0.283931
\(66\) 0 0
\(67\) −2.71486e8 −1.64593 −0.822966 − 0.568091i \(-0.807682\pi\)
−0.822966 + 0.568091i \(0.807682\pi\)
\(68\) 1.80813e8 1.02551
\(69\) 0 0
\(70\) −5.13058e6 −0.0255403
\(71\) −8.00533e7 −0.373866 −0.186933 − 0.982373i \(-0.559855\pi\)
−0.186933 + 0.982373i \(0.559855\pi\)
\(72\) 0 0
\(73\) −3.35309e8 −1.38195 −0.690975 − 0.722879i \(-0.742818\pi\)
−0.690975 + 0.722879i \(0.742818\pi\)
\(74\) 1.72855e7 0.0670098
\(75\) 0 0
\(76\) 1.40363e8 0.482604
\(77\) −1.01621e8 −0.329439
\(78\) 0 0
\(79\) −1.79224e8 −0.517697 −0.258848 − 0.965918i \(-0.583343\pi\)
−0.258848 + 0.965918i \(0.583343\pi\)
\(80\) 1.52704e8 0.416816
\(81\) 0 0
\(82\) −4.55413e7 −0.111235
\(83\) 5.62335e8 1.30060 0.650300 − 0.759678i \(-0.274643\pi\)
0.650300 + 0.759678i \(0.274643\pi\)
\(84\) 0 0
\(85\) −2.25876e8 −0.469337
\(86\) 8.92734e7 0.175986
\(87\) 0 0
\(88\) −1.46487e8 −0.260392
\(89\) 2.46154e8 0.415864 0.207932 − 0.978143i \(-0.433327\pi\)
0.207932 + 0.978143i \(0.433327\pi\)
\(90\) 0 0
\(91\) −1.56976e8 −0.239965
\(92\) 4.02692e7 0.0586041
\(93\) 0 0
\(94\) −2.81892e7 −0.0372398
\(95\) −1.75345e8 −0.220870
\(96\) 0 0
\(97\) 7.58371e8 0.869779 0.434889 − 0.900484i \(-0.356787\pi\)
0.434889 + 0.900484i \(0.356787\pi\)
\(98\) −1.97096e7 −0.0215855
\(99\) 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 315.10.a.e.1.2 4
3.2 odd 2 105.10.a.d.1.3 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.10.a.d.1.3 ✓ 4 3.2 odd 2
315.10.a.e.1.2 4 1.1 even 1 trivial