Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1395,2,Mod(1,1395)] 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("1395.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1395, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1395 = 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1395.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,0,5,3] 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: yes
Analytic conductor: \(11.1391310820\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 1395.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.21432 q^{2} -0.525428 q^{4} +1.00000 q^{5} -1.59210 q^{7} -3.06668 q^{8} +1.21432 q^{10} -0.622216 q^{11} +0.214320 q^{13} -1.93332 q^{14} -2.67307 q^{16} -3.52543 q^{17} +1.80642 q^{19} -0.525428 q^{20} -0.755569 q^{22} -6.90321 q^{23} +1.00000 q^{25} +0.260253 q^{26} +0.836535 q^{28} -9.73975 q^{29} -1.00000 q^{31} +2.88739 q^{32} -4.28100 q^{34} -1.59210 q^{35} -4.83654 q^{37} +2.19358 q^{38} -3.06668 q^{40} +7.47949 q^{41} +8.23506 q^{43} +0.326929 q^{44} -8.38271 q^{46} -11.4652 q^{47} -4.46520 q^{49} +1.21432 q^{50} -0.112610 q^{52} -13.7605 q^{53} -0.622216 q^{55} +4.88247 q^{56} -11.8272 q^{58} +4.26025 q^{59} +2.85728 q^{61} -1.21432 q^{62} +8.85236 q^{64} +0.214320 q^{65} -2.08097 q^{67} +1.85236 q^{68} -1.93332 q^{70} -1.31111 q^{71} -1.65233 q^{73} -5.87310 q^{74} -0.949145 q^{76} +0.990632 q^{77} -5.19850 q^{79} -2.67307 q^{80} +9.08250 q^{82} +5.65878 q^{83} -3.52543 q^{85} +10.0000 q^{86} +1.90813 q^{88} +1.93332 q^{89} -0.341219 q^{91} +3.62714 q^{92} -13.9224 q^{94} +1.80642 q^{95} +6.91750 q^{97} -5.42219 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 2 q^{7} - 9 q^{8} - 3 q^{10} - 2 q^{11} - 6 q^{13} - 6 q^{14} + 5 q^{16} - 4 q^{17} - 8 q^{19} + 5 q^{20} - 2 q^{22} - 14 q^{23} + 3 q^{25} + 14 q^{26} - 4 q^{28} - 16 q^{29}+ \cdots - 17 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.21432 0.858654 0.429327 − 0.903149i \(-0.358751\pi\)
0.429327 + 0.903149i \(0.358751\pi\)
\(3\) 0 0
\(4\) −0.525428 −0.262714
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.59210 −0.601759 −0.300879 − 0.953662i \(-0.597280\pi\)
−0.300879 + 0.953662i \(0.597280\pi\)
\(8\) −3.06668 −1.08423
\(9\) 0 0
\(10\) 1.21432 0.384002
\(11\) −0.622216 −0.187605 −0.0938025 − 0.995591i \(-0.529902\pi\)
−0.0938025 + 0.995591i \(0.529902\pi\)
\(12\) 0 0
\(13\) 0.214320 0.0594416 0.0297208 − 0.999558i \(-0.490538\pi\)
0.0297208 + 0.999558i \(0.490538\pi\)
\(14\) −1.93332 −0.516702
\(15\) 0 0
\(16\) −2.67307 −0.668268
\(17\) −3.52543 −0.855042 −0.427521 − 0.904005i \(-0.640613\pi\)
−0.427521 + 0.904005i \(0.640613\pi\)
\(18\) 0 0
\(19\) 1.80642 0.414422 0.207211 − 0.978296i \(-0.433561\pi\)
0.207211 + 0.978296i \(0.433561\pi\)
\(20\) −0.525428 −0.117489
\(21\) 0 0
\(22\) −0.755569 −0.161088
\(23\) −6.90321 −1.43942 −0.719710 − 0.694275i \(-0.755725\pi\)
−0.719710 + 0.694275i \(0.755725\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.260253 0.0510398
\(27\) 0 0
\(28\) 0.836535 0.158090
\(29\) −9.73975 −1.80863 −0.904313 − 0.426870i \(-0.859616\pi\)
−0.904313 + 0.426870i \(0.859616\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 2.88739 0.510423
\(33\) 0 0
\(34\) −4.28100 −0.734185
\(35\) −1.59210 −0.269115
\(36\) 0 0
\(37\) −4.83654 −0.795122 −0.397561 − 0.917576i \(-0.630143\pi\)
−0.397561 + 0.917576i \(0.630143\pi\)
\(38\) 2.19358 0.355845
\(39\) 0 0
\(40\) −3.06668 −0.484884
\(41\) 7.47949 1.16810 0.584050 − 0.811717i \(-0.301467\pi\)
0.584050 + 0.811717i \(0.301467\pi\)
\(42\) 0 0
\(43\) 8.23506 1.25584 0.627918 − 0.778280i \(-0.283907\pi\)
0.627918 + 0.778280i \(0.283907\pi\)
\(44\) 0.326929 0.0492864
\(45\) 0 0
\(46\) −8.38271 −1.23596
\(47\) −11.4652 −1.67237 −0.836186 − 0.548446i \(-0.815220\pi\)
−0.836186 + 0.548446i \(0.815220\pi\)
\(48\) 0 0
\(49\) −4.46520 −0.637886
\(50\) 1.21432 0.171731
\(51\) 0 0
\(52\) −0.112610 −0.0156161
\(53\) −13.7605 −1.89015 −0.945074 − 0.326855i \(-0.894011\pi\)
−0.945074 + 0.326855i \(0.894011\pi\)
\(54\) 0 0
\(55\) −0.622216 −0.0838995
\(56\) 4.88247 0.652447
\(57\) 0 0
\(58\) −11.8272 −1.55298
\(59\) 4.26025 0.554638 0.277319 − 0.960778i \(-0.410554\pi\)
0.277319 + 0.960778i \(0.410554\pi\)
\(60\) 0 0
\(61\) 2.85728 0.365837 0.182919 − 0.983128i \(-0.441446\pi\)
0.182919 + 0.983128i \(0.441446\pi\)
\(62\) −1.21432 −0.154219
\(63\) 0 0
\(64\) 8.85236 1.10654
\(65\) 0.214320 0.0265831
\(66\) 0 0
\(67\) −2.08097 −0.254231 −0.127115 − 0.991888i \(-0.540572\pi\)
−0.127115 + 0.991888i \(0.540572\pi\)
\(68\) 1.85236 0.224631
\(69\) 0 0
\(70\) −1.93332 −0.231076
\(71\) −1.31111 −0.155600 −0.0777999 − 0.996969i \(-0.524790\pi\)
−0.0777999 + 0.996969i \(0.524790\pi\)
\(72\) 0 0
\(73\) −1.65233 −0.193390 −0.0966951 − 0.995314i \(-0.530827\pi\)
−0.0966951 + 0.995314i \(0.530827\pi\)
\(74\) −5.87310 −0.682734
\(75\) 0 0
\(76\) −0.949145 −0.108874
\(77\) 0.990632 0.112893
\(78\) 0 0
\(79\) −5.19850 −0.584877 −0.292438 − 0.956284i \(-0.594467\pi\)
−0.292438 + 0.956284i \(0.594467\pi\)
\(80\) −2.67307 −0.298858
\(81\) 0 0
\(82\) 9.08250 1.00299
\(83\) 5.65878 0.621132 0.310566 − 0.950552i \(-0.399481\pi\)
0.310566 + 0.950552i \(0.399481\pi\)
\(84\) 0 0
\(85\) −3.52543 −0.382386
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) 1.90813 0.203408
\(89\) 1.93332 0.204932 0.102466 − 0.994737i \(-0.467327\pi\)
0.102466 + 0.994737i \(0.467327\pi\)
\(90\) 0 0
\(91\) −0.341219 −0.0357695
\(92\) 3.62714 0.378155
\(93\) 0 0
\(94\) −13.9224 −1.43599
\(95\) 1.80642 0.185335
\(96\) 0 0
\(97\) 6.91750 0.702366 0.351183 − 0.936307i \(-0.385780\pi\)
0.351183 + 0.936307i \(0.385780\pi\)
\(98\) −5.42219 −0.547723
\(99\) 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 1395.2.a.h.1.3 3
3.2 odd 2 465.2.a.g.1.1 ✓ 3
5.4 even 2 6975.2.a.bi.1.1 3
12.11 even 2 7440.2.a.bm.1.3 3
15.2 even 4 2325.2.c.l.1024.3 6
15.8 even 4 2325.2.c.l.1024.4 6
15.14 odd 2 2325.2.a.p.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.1 ✓ 3 3.2 odd 2
1395.2.a.h.1.3 3 1.1 even 1 trivial
2325.2.a.p.1.3 3 15.14 odd 2
2325.2.c.l.1024.3 6 15.2 even 4
2325.2.c.l.1024.4 6 15.8 even 4
6975.2.a.bi.1.1 3 5.4 even 2
7440.2.a.bm.1.3 3 12.11 even 2