Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,-3,0,-2,0,3,0,-2,0,-6,0,3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.4086991038\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\chi\) \(=\) 7440.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.59210 q^{7} +1.00000 q^{9} -0.622216 q^{11} +0.214320 q^{13} +1.00000 q^{15} +3.52543 q^{17} -1.80642 q^{19} -1.59210 q^{21} -6.90321 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.73975 q^{29} +1.00000 q^{31} +0.622216 q^{33} -1.59210 q^{35} -4.83654 q^{37} -0.214320 q^{39} -7.47949 q^{41} -8.23506 q^{43} -1.00000 q^{45} -11.4652 q^{47} -4.46520 q^{49} -3.52543 q^{51} +13.7605 q^{53} +0.622216 q^{55} +1.80642 q^{57} +4.26025 q^{59} +2.85728 q^{61} +1.59210 q^{63} -0.214320 q^{65} +2.08097 q^{67} +6.90321 q^{69} -1.31111 q^{71} -1.65233 q^{73} -1.00000 q^{75} -0.990632 q^{77} +5.19850 q^{79} +1.00000 q^{81} +5.65878 q^{83} -3.52543 q^{85} -9.73975 q^{87} -1.93332 q^{89} +0.341219 q^{91} -1.00000 q^{93} +1.80642 q^{95} +6.91750 q^{97} -0.622216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.59210 0.601759 0.300879 − 0.953662i \(-0.402720\pi\)
0.300879 + 0.953662i \(0.402720\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −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 \(0.490538\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 3.52543 0.855042 0.427521 − 0.904005i \(-0.359387\pi\)
0.427521 + 0.904005i \(0.359387\pi\)
\(18\) 0 0
\(19\) −1.80642 −0.414422 −0.207211 − 0.978296i \(-0.566439\pi\)
−0.207211 + 0.978296i \(0.566439\pi\)
\(20\) 0 0
\(21\) −1.59210 −0.347426
\(22\) 0 0
\(23\) −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 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.73975 1.80863 0.904313 − 0.426870i \(-0.140384\pi\)
0.904313 + 0.426870i \(0.140384\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 0.622216 0.108314
\(34\) 0 0
\(35\) −1.59210 −0.269115
\(36\) 0 0
\(37\) −4.83654 −0.795122 −0.397561 − 0.917576i \(-0.630143\pi\)
−0.397561 + 0.917576i \(0.630143\pi\)
\(38\) 0 0
\(39\) −0.214320 −0.0343186
\(40\) 0 0
\(41\) −7.47949 −1.16810 −0.584050 − 0.811717i \(-0.698533\pi\)
−0.584050 + 0.811717i \(0.698533\pi\)
\(42\) 0 0
\(43\) −8.23506 −1.25584 −0.627918 − 0.778280i \(-0.716093\pi\)
−0.627918 + 0.778280i \(0.716093\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −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
\(50\) 0 0
\(51\) −3.52543 −0.493659
\(52\) 0 0
\(53\) 13.7605 1.89015 0.945074 − 0.326855i \(-0.105989\pi\)
0.945074 + 0.326855i \(0.105989\pi\)
\(54\) 0 0
\(55\) 0.622216 0.0838995
\(56\) 0 0
\(57\) 1.80642 0.239267
\(58\) 0 0
\(59\) 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 \(0.441446\pi\)
\(62\) 0 0
\(63\) 1.59210 0.200586
\(64\) 0 0
\(65\) −0.214320 −0.0265831
\(66\) 0 0
\(67\) 2.08097 0.254231 0.127115 − 0.991888i \(-0.459428\pi\)
0.127115 + 0.991888i \(0.459428\pi\)
\(68\) 0 0
\(69\) 6.90321 0.831049
\(70\) 0 0
\(71\) −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 \(0.530827\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −0.990632 −0.112893
\(78\) 0 0
\(79\) 5.19850 0.584877 0.292438 − 0.956284i \(-0.405533\pi\)
0.292438 + 0.956284i \(0.405533\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) −9.73975 −1.04421
\(88\) 0 0
\(89\) −1.93332 −0.204932 −0.102466 − 0.994737i \(-0.532673\pi\)
−0.102466 + 0.994737i \(0.532673\pi\)
\(90\) 0 0
\(91\) 0.341219 0.0357695
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 1.80642 0.185335
\(96\) 0 0
\(97\) 6.91750 0.702366 0.351183 − 0.936307i \(-0.385780\pi\)
0.351183 + 0.936307i \(0.385780\pi\)
\(98\) 0 0
\(99\) −0.622216 −0.0625350
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 7440.2.a.bm.1.3 3
4.3 odd 2 465.2.a.g.1.1 ✓ 3
12.11 even 2 1395.2.a.h.1.3 3
20.3 even 4 2325.2.c.l.1024.4 6
20.7 even 4 2325.2.c.l.1024.3 6
20.19 odd 2 2325.2.a.p.1.3 3
60.59 even 2 6975.2.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.1 ✓ 3 4.3 odd 2
1395.2.a.h.1.3 3 12.11 even 2
2325.2.a.p.1.3 3 20.19 odd 2
2325.2.c.l.1024.3 6 20.7 even 4
2325.2.c.l.1024.4 6 20.3 even 4
6975.2.a.bi.1.1 3 60.59 even 2
7440.2.a.bm.1.3 3 1.1 even 1 trivial