Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7104,2,Mod(1,7104)] 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("7104.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7104.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,-2,0,5,0,3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.7257255959\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 7104.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.83221 q^{5} +3.19656 q^{7} +1.00000 q^{9} -3.48929 q^{11} -5.48929 q^{13} -3.83221 q^{15} -7.32150 q^{17} +3.48929 q^{19} -3.19656 q^{21} -4.05019 q^{23} +9.68585 q^{25} -1.00000 q^{27} -7.83221 q^{29} +6.97858 q^{31} +3.48929 q^{33} +12.2499 q^{35} +1.00000 q^{37} +5.48929 q^{39} +2.58546 q^{41} +4.00000 q^{43} +3.83221 q^{45} -6.39312 q^{47} +3.21798 q^{49} +7.32150 q^{51} -11.8824 q^{53} -13.3717 q^{55} -3.48929 q^{57} -12.8108 q^{59} -6.58546 q^{61} +3.19656 q^{63} -21.0361 q^{65} -3.37169 q^{67} +4.05019 q^{69} +13.3717 q^{71} +10.4679 q^{73} -9.68585 q^{75} -11.1537 q^{77} +5.27131 q^{79} +1.00000 q^{81} +9.88240 q^{83} -28.0575 q^{85} +7.83221 q^{87} -15.3215 q^{89} -17.5468 q^{91} -6.97858 q^{93} +13.3717 q^{95} -1.66442 q^{97} -3.48929 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 3 q^{11} - 9 q^{13} + 2 q^{15} - q^{17} + 3 q^{19} - 5 q^{21} - 9 q^{23} + 17 q^{25} - 3 q^{27} - 10 q^{29} + 6 q^{31} + 3 q^{33} + 4 q^{35} + 3 q^{37} + 9 q^{39}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(5\) 3.83221 1.71382 0.856909 − 0.515468i \(-0.172382\pi\)
0.856909 + 0.515468i \(0.172382\pi\)
\(6\) 0 0
\(7\) 3.19656 1.20819 0.604093 − 0.796914i \(-0.293536\pi\)
0.604093 + 0.796914i \(0.293536\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.48929 −1.05206 −0.526030 − 0.850466i \(-0.676320\pi\)
−0.526030 + 0.850466i \(0.676320\pi\)
\(12\) 0 0
\(13\) −5.48929 −1.52245 −0.761227 − 0.648485i \(-0.775403\pi\)
−0.761227 + 0.648485i \(0.775403\pi\)
\(14\) 0 0
\(15\) −3.83221 −0.989473
\(16\) 0 0
\(17\) −7.32150 −1.77572 −0.887862 − 0.460109i \(-0.847810\pi\)
−0.887862 + 0.460109i \(0.847810\pi\)
\(18\) 0 0
\(19\) 3.48929 0.800498 0.400249 − 0.916406i \(-0.368924\pi\)
0.400249 + 0.916406i \(0.368924\pi\)
\(20\) 0 0
\(21\) −3.19656 −0.697546
\(22\) 0 0
\(23\) −4.05019 −0.844523 −0.422262 − 0.906474i \(-0.638764\pi\)
−0.422262 + 0.906474i \(0.638764\pi\)
\(24\) 0 0
\(25\) 9.68585 1.93717
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.83221 −1.45441 −0.727203 − 0.686423i \(-0.759180\pi\)
−0.727203 + 0.686423i \(0.759180\pi\)
\(30\) 0 0
\(31\) 6.97858 1.25339 0.626695 − 0.779265i \(-0.284407\pi\)
0.626695 + 0.779265i \(0.284407\pi\)
\(32\) 0 0
\(33\) 3.48929 0.607407
\(34\) 0 0
\(35\) 12.2499 2.07061
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 5.48929 0.878990
\(40\) 0 0
\(41\) 2.58546 0.403781 0.201891 − 0.979408i \(-0.435291\pi\)
0.201891 + 0.979408i \(0.435291\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 3.83221 0.571272
\(46\) 0 0
\(47\) −6.39312 −0.932532 −0.466266 − 0.884645i \(-0.654401\pi\)
−0.466266 + 0.884645i \(0.654401\pi\)
\(48\) 0 0
\(49\) 3.21798 0.459711
\(50\) 0 0
\(51\) 7.32150 1.02522
\(52\) 0 0
\(53\) −11.8824 −1.63217 −0.816087 − 0.577929i \(-0.803861\pi\)
−0.816087 + 0.577929i \(0.803861\pi\)
\(54\) 0 0
\(55\) −13.3717 −1.80304
\(56\) 0 0
\(57\) −3.48929 −0.462168
\(58\) 0 0
\(59\) −12.8108 −1.66782 −0.833911 − 0.551898i \(-0.813904\pi\)
−0.833911 + 0.551898i \(0.813904\pi\)
\(60\) 0 0
\(61\) −6.58546 −0.843182 −0.421591 − 0.906786i \(-0.638528\pi\)
−0.421591 + 0.906786i \(0.638528\pi\)
\(62\) 0 0
\(63\) 3.19656 0.402728
\(64\) 0 0
\(65\) −21.0361 −2.60921
\(66\) 0 0
\(67\) −3.37169 −0.411918 −0.205959 − 0.978561i \(-0.566031\pi\)
−0.205959 + 0.978561i \(0.566031\pi\)
\(68\) 0 0
\(69\) 4.05019 0.487586
\(70\) 0 0
\(71\) 13.3717 1.58693 0.793464 − 0.608617i \(-0.208275\pi\)
0.793464 + 0.608617i \(0.208275\pi\)
\(72\) 0 0
\(73\) 10.4679 1.22517 0.612586 − 0.790404i \(-0.290130\pi\)
0.612586 + 0.790404i \(0.290130\pi\)
\(74\) 0 0
\(75\) −9.68585 −1.11843
\(76\) 0 0
\(77\) −11.1537 −1.27108
\(78\) 0 0
\(79\) 5.27131 0.593068 0.296534 − 0.955022i \(-0.404169\pi\)
0.296534 + 0.955022i \(0.404169\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.88240 1.08473 0.542367 − 0.840141i \(-0.317528\pi\)
0.542367 + 0.840141i \(0.317528\pi\)
\(84\) 0 0
\(85\) −28.0575 −3.04327
\(86\) 0 0
\(87\) 7.83221 0.839701
\(88\) 0 0
\(89\) −15.3215 −1.62408 −0.812038 − 0.583605i \(-0.801642\pi\)
−0.812038 + 0.583605i \(0.801642\pi\)
\(90\) 0 0
\(91\) −17.5468 −1.83941
\(92\) 0 0
\(93\) −6.97858 −0.723645
\(94\) 0 0
\(95\) 13.3717 1.37191
\(96\) 0 0
\(97\) −1.66442 −0.168997 −0.0844983 − 0.996424i \(-0.526929\pi\)
−0.0844983 + 0.996424i \(0.526929\pi\)
\(98\) 0 0
\(99\) −3.48929 −0.350687
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 7104.2.a.bq.1.3 3
4.3 odd 2 7104.2.a.bw.1.3 3
8.3 odd 2 888.2.a.i.1.1 ✓ 3
8.5 even 2 1776.2.a.s.1.1 3
24.5 odd 2 5328.2.a.bl.1.3 3
24.11 even 2 2664.2.a.m.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.i.1.1 ✓ 3 8.3 odd 2
1776.2.a.s.1.1 3 8.5 even 2
2664.2.a.m.1.3 3 24.11 even 2
5328.2.a.bl.1.3 3 24.5 odd 2
7104.2.a.bq.1.3 3 1.1 even 1 trivial
7104.2.a.bw.1.3 3 4.3 odd 2