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 1.1
Root \(2.17009\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -4.87936 q^{7} +1.00000 q^{9} -4.34017 q^{11} -2.53919 q^{13} +1.00000 q^{15} +2.63090 q^{17} +7.41855 q^{19} +4.87936 q^{21} -2.29072 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.09171 q^{29} +1.00000 q^{31} +4.34017 q^{33} +4.87936 q^{35} -5.80098 q^{37} +2.53919 q^{39} -0.183417 q^{41} +6.49693 q^{43} -1.00000 q^{45} +9.80817 q^{47} +16.8082 q^{49} -2.63090 q^{51} -1.86603 q^{53} +4.34017 q^{55} -7.41855 q^{57} +7.90829 q^{59} -8.15676 q^{61} -4.87936 q^{63} +2.53919 q^{65} +10.4813 q^{67} +2.29072 q^{69} -3.17009 q^{71} -15.5597 q^{73} -1.00000 q^{75} +21.1773 q^{77} +6.23287 q^{79} +1.00000 q^{81} -6.38962 q^{83} -2.63090 q^{85} -6.09171 q^{87} -7.51026 q^{89} +12.3896 q^{91} -1.00000 q^{93} -7.41855 q^{95} +16.2823 q^{97} -4.34017 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(7\) −4.87936 −1.84423 −0.922113 − 0.386921i \(-0.873538\pi\)
−0.922113 + 0.386921i \(0.873538\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.34017 −1.30861 −0.654306 − 0.756230i \(-0.727039\pi\)
−0.654306 + 0.756230i \(0.727039\pi\)
\(12\) 0 0
\(13\) −2.53919 −0.704244 −0.352122 − 0.935954i \(-0.614540\pi\)
−0.352122 + 0.935954i \(0.614540\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.63090 0.638086 0.319043 − 0.947740i \(-0.396638\pi\)
0.319043 + 0.947740i \(0.396638\pi\)
\(18\) 0 0
\(19\) 7.41855 1.70193 0.850966 − 0.525221i \(-0.176017\pi\)
0.850966 + 0.525221i \(0.176017\pi\)
\(20\) 0 0
\(21\) 4.87936 1.06476
\(22\) 0 0
\(23\) −2.29072 −0.477649 −0.238825 − 0.971063i \(-0.576762\pi\)
−0.238825 + 0.971063i \(0.576762\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.09171 1.13120 0.565601 − 0.824679i \(-0.308644\pi\)
0.565601 + 0.824679i \(0.308644\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) 4.34017 0.755527
\(34\) 0 0
\(35\) 4.87936 0.824763
\(36\) 0 0
\(37\) −5.80098 −0.953676 −0.476838 − 0.878991i \(-0.658217\pi\)
−0.476838 + 0.878991i \(0.658217\pi\)
\(38\) 0 0
\(39\) 2.53919 0.406596
\(40\) 0 0
\(41\) −0.183417 −0.0286450 −0.0143225 − 0.999897i \(-0.504559\pi\)
−0.0143225 + 0.999897i \(0.504559\pi\)
\(42\) 0 0
\(43\) 6.49693 0.990772 0.495386 − 0.868673i \(-0.335027\pi\)
0.495386 + 0.868673i \(0.335027\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 9.80817 1.43067 0.715334 − 0.698782i \(-0.246274\pi\)
0.715334 + 0.698782i \(0.246274\pi\)
\(48\) 0 0
\(49\) 16.8082 2.40117
\(50\) 0 0
\(51\) −2.63090 −0.368399
\(52\) 0 0
\(53\) −1.86603 −0.256319 −0.128160 − 0.991754i \(-0.540907\pi\)
−0.128160 + 0.991754i \(0.540907\pi\)
\(54\) 0 0
\(55\) 4.34017 0.585229
\(56\) 0 0
\(57\) −7.41855 −0.982611
\(58\) 0 0
\(59\) 7.90829 1.02957 0.514786 − 0.857319i \(-0.327871\pi\)
0.514786 + 0.857319i \(0.327871\pi\)
\(60\) 0 0
\(61\) −8.15676 −1.04437 −0.522183 − 0.852834i \(-0.674882\pi\)
−0.522183 + 0.852834i \(0.674882\pi\)
\(62\) 0 0
\(63\) −4.87936 −0.614742
\(64\) 0 0
\(65\) 2.53919 0.314948
\(66\) 0 0
\(67\) 10.4813 1.28050 0.640249 − 0.768167i \(-0.278831\pi\)
0.640249 + 0.768167i \(0.278831\pi\)
\(68\) 0 0
\(69\) 2.29072 0.275771
\(70\) 0 0
\(71\) −3.17009 −0.376220 −0.188110 − 0.982148i \(-0.560236\pi\)
−0.188110 + 0.982148i \(0.560236\pi\)
\(72\) 0 0
\(73\) −15.5597 −1.82113 −0.910563 − 0.413370i \(-0.864352\pi\)
−0.910563 + 0.413370i \(0.864352\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 21.1773 2.41337
\(78\) 0 0
\(79\) 6.23287 0.701252 0.350626 − 0.936516i \(-0.385969\pi\)
0.350626 + 0.936516i \(0.385969\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.38962 −0.701352 −0.350676 − 0.936497i \(-0.614048\pi\)
−0.350676 + 0.936497i \(0.614048\pi\)
\(84\) 0 0
\(85\) −2.63090 −0.285361
\(86\) 0 0
\(87\) −6.09171 −0.653100
\(88\) 0 0
\(89\) −7.51026 −0.796086 −0.398043 − 0.917367i \(-0.630311\pi\)
−0.398043 + 0.917367i \(0.630311\pi\)
\(90\) 0 0
\(91\) 12.3896 1.29879
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −7.41855 −0.761127
\(96\) 0 0
\(97\) 16.2823 1.65322 0.826609 − 0.562776i \(-0.190267\pi\)
0.826609 + 0.562776i \(0.190267\pi\)
\(98\) 0 0
\(99\) −4.34017 −0.436204
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.1.1 3
4.3 odd 2 465.2.a.g.1.2 ✓ 3
12.11 even 2 1395.2.a.h.1.2 3
20.3 even 4 2325.2.c.l.1024.2 6
20.7 even 4 2325.2.c.l.1024.5 6
20.19 odd 2 2325.2.a.p.1.2 3
60.59 even 2 6975.2.a.bi.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.g.1.2 ✓ 3 4.3 odd 2
1395.2.a.h.1.2 3 12.11 even 2
2325.2.a.p.1.2 3 20.19 odd 2
2325.2.c.l.1024.2 6 20.3 even 4
2325.2.c.l.1024.5 6 20.7 even 4
6975.2.a.bi.1.2 3 60.59 even 2
7440.2.a.bm.1.1 3 1.1 even 1 trivial