Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2496,2,Mod(1,2496)] 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("2496.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2496.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,0,-2,0,0,0,3,0,-4,0,-3,0,2,0,6] 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: \(19.9306603445\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 2496.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.96239 q^{5} -3.35026 q^{7} +1.00000 q^{9} -1.61213 q^{11} -1.00000 q^{13} -2.96239 q^{15} +2.00000 q^{17} -3.35026 q^{19} +3.35026 q^{21} +6.70052 q^{23} +3.77575 q^{25} -1.00000 q^{27} -2.00000 q^{29} -6.57452 q^{31} +1.61213 q^{33} -9.92478 q^{35} -7.92478 q^{37} +1.00000 q^{39} +6.96239 q^{41} -0.775746 q^{43} +2.96239 q^{45} -2.38787 q^{47} +4.22425 q^{49} -2.00000 q^{51} -11.9248 q^{53} -4.77575 q^{55} +3.35026 q^{57} +0.312650 q^{59} -14.6253 q^{61} -3.35026 q^{63} -2.96239 q^{65} +8.12601 q^{67} -6.70052 q^{69} +4.31265 q^{71} +0.0752228 q^{73} -3.77575 q^{75} +5.40105 q^{77} -12.0000 q^{79} +1.00000 q^{81} -8.31265 q^{83} +5.92478 q^{85} +2.00000 q^{87} +8.88717 q^{89} +3.35026 q^{91} +6.57452 q^{93} -9.92478 q^{95} -7.92478 q^{97} -1.61213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 2 q^{5} + 3 q^{9} - 4 q^{11} - 3 q^{13} + 2 q^{15} + 6 q^{17} + 13 q^{25} - 3 q^{27} - 6 q^{29} - 8 q^{31} + 4 q^{33} - 8 q^{35} - 2 q^{37} + 3 q^{39} + 10 q^{41} - 4 q^{43} - 2 q^{45} - 8 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(5\) 2.96239 1.32482 0.662410 − 0.749141i \(-0.269534\pi\)
0.662410 + 0.749141i \(0.269534\pi\)
\(6\) 0 0
\(7\) −3.35026 −1.26628 −0.633140 − 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.61213 −0.486075 −0.243037 − 0.970017i \(-0.578144\pi\)
−0.243037 + 0.970017i \(0.578144\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −2.96239 −0.764885
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −3.35026 −0.768603 −0.384301 − 0.923208i \(-0.625558\pi\)
−0.384301 + 0.923208i \(0.625558\pi\)
\(20\) 0 0
\(21\) 3.35026 0.731087
\(22\) 0 0
\(23\) 6.70052 1.39716 0.698578 − 0.715534i \(-0.253817\pi\)
0.698578 + 0.715534i \(0.253817\pi\)
\(24\) 0 0
\(25\) 3.77575 0.755149
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −6.57452 −1.18082 −0.590409 − 0.807104i \(-0.701034\pi\)
−0.590409 + 0.807104i \(0.701034\pi\)
\(32\) 0 0
\(33\) 1.61213 0.280635
\(34\) 0 0
\(35\) −9.92478 −1.67759
\(36\) 0 0
\(37\) −7.92478 −1.30283 −0.651413 − 0.758724i \(-0.725823\pi\)
−0.651413 + 0.758724i \(0.725823\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 6.96239 1.08734 0.543671 − 0.839298i \(-0.317034\pi\)
0.543671 + 0.839298i \(0.317034\pi\)
\(42\) 0 0
\(43\) −0.775746 −0.118300 −0.0591501 − 0.998249i \(-0.518839\pi\)
−0.0591501 + 0.998249i \(0.518839\pi\)
\(44\) 0 0
\(45\) 2.96239 0.441607
\(46\) 0 0
\(47\) −2.38787 −0.348307 −0.174154 − 0.984719i \(-0.555719\pi\)
−0.174154 + 0.984719i \(0.555719\pi\)
\(48\) 0 0
\(49\) 4.22425 0.603465
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −11.9248 −1.63799 −0.818997 − 0.573798i \(-0.805470\pi\)
−0.818997 + 0.573798i \(0.805470\pi\)
\(54\) 0 0
\(55\) −4.77575 −0.643961
\(56\) 0 0
\(57\) 3.35026 0.443753
\(58\) 0 0
\(59\) 0.312650 0.0407036 0.0203518 − 0.999793i \(-0.493521\pi\)
0.0203518 + 0.999793i \(0.493521\pi\)
\(60\) 0 0
\(61\) −14.6253 −1.87258 −0.936289 − 0.351231i \(-0.885763\pi\)
−0.936289 + 0.351231i \(0.885763\pi\)
\(62\) 0 0
\(63\) −3.35026 −0.422093
\(64\) 0 0
\(65\) −2.96239 −0.367439
\(66\) 0 0
\(67\) 8.12601 0.992750 0.496375 − 0.868108i \(-0.334664\pi\)
0.496375 + 0.868108i \(0.334664\pi\)
\(68\) 0 0
\(69\) −6.70052 −0.806648
\(70\) 0 0
\(71\) 4.31265 0.511817 0.255909 − 0.966701i \(-0.417625\pi\)
0.255909 + 0.966701i \(0.417625\pi\)
\(72\) 0 0
\(73\) 0.0752228 0.00880416 0.00440208 − 0.999990i \(-0.498599\pi\)
0.00440208 + 0.999990i \(0.498599\pi\)
\(74\) 0 0
\(75\) −3.77575 −0.435986
\(76\) 0 0
\(77\) 5.40105 0.615506
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.31265 −0.912432 −0.456216 − 0.889869i \(-0.650796\pi\)
−0.456216 + 0.889869i \(0.650796\pi\)
\(84\) 0 0
\(85\) 5.92478 0.642632
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 8.88717 0.942038 0.471019 − 0.882123i \(-0.343886\pi\)
0.471019 + 0.882123i \(0.343886\pi\)
\(90\) 0 0
\(91\) 3.35026 0.351203
\(92\) 0 0
\(93\) 6.57452 0.681745
\(94\) 0 0
\(95\) −9.92478 −1.01826
\(96\) 0 0
\(97\) −7.92478 −0.804639 −0.402320 − 0.915499i \(-0.631796\pi\)
−0.402320 + 0.915499i \(0.631796\pi\)
\(98\) 0 0
\(99\) −1.61213 −0.162025
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 2496.2.a.bk.1.3 3
3.2 odd 2 7488.2.a.cy.1.1 3
4.3 odd 2 2496.2.a.bl.1.3 3
8.3 odd 2 1248.2.a.o.1.1 ✓ 3
8.5 even 2 1248.2.a.p.1.1 yes 3
12.11 even 2 7488.2.a.cx.1.1 3
24.5 odd 2 3744.2.a.z.1.3 3
24.11 even 2 3744.2.a.ba.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.2.a.o.1.1 ✓ 3 8.3 odd 2
1248.2.a.p.1.1 yes 3 8.5 even 2
2496.2.a.bk.1.3 3 1.1 even 1 trivial
2496.2.a.bl.1.3 3 4.3 odd 2
3744.2.a.z.1.3 3 24.5 odd 2
3744.2.a.ba.1.3 3 24.11 even 2
7488.2.a.cx.1.1 3 12.11 even 2
7488.2.a.cy.1.1 3 3.2 odd 2