Properties

Label 984.2.a.h.1.3
Level $984$
Weight $2$
Character 984.1
Self dual yes
Analytic conductor $7.857$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [984,2,Mod(1,984)] 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("984.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(984, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 984 = 2^{3} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 984.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,1] 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: \(7.85727955889\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.961.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 10x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.29707\) of defining polynomial
Character \(\chi\) \(=\) 984.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.29707 q^{5} +2.00000 q^{7} +1.00000 q^{9} -0.786802 q^{11} -1.29707 q^{13} +3.29707 q^{15} -4.08387 q^{17} +6.87067 q^{19} +2.00000 q^{21} +8.16774 q^{23} +5.87067 q^{25} +1.00000 q^{27} -5.80734 q^{29} -10.2516 q^{31} -0.786802 q^{33} +6.59414 q^{35} -6.95455 q^{37} -1.29707 q^{39} -1.00000 q^{41} -3.38094 q^{43} +3.29707 q^{45} +1.21320 q^{47} -3.00000 q^{49} -4.08387 q^{51} -5.02054 q^{53} -2.59414 q^{55} +6.87067 q^{57} +0.276533 q^{59} +9.38094 q^{61} +2.00000 q^{63} -4.27653 q^{65} +15.8912 q^{67} +8.16774 q^{69} -4.08387 q^{71} +14.8458 q^{73} +5.87067 q^{75} -1.57360 q^{77} +12.5941 q^{79} +1.00000 q^{81} -16.0590 q^{83} -13.4648 q^{85} -5.80734 q^{87} -2.27653 q^{89} -2.59414 q^{91} -10.2516 q^{93} +22.6531 q^{95} -11.5736 q^{97} -0.786802 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + q^{5} + 6 q^{7} + 3 q^{9} - q^{11} + 5 q^{13} + q^{15} - 2 q^{17} + 9 q^{19} + 6 q^{21} + 4 q^{23} + 6 q^{25} + 3 q^{27} - q^{29} - q^{33} + 2 q^{35} + q^{37} + 5 q^{39} - 3 q^{41}+ \cdots - q^{99}+O(q^{100}) \) 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
\(5\) 3.29707 1.47449 0.737247 − 0.675623i \(-0.236125\pi\)
0.737247 + 0.675623i \(0.236125\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.786802 −0.237230 −0.118615 − 0.992940i \(-0.537845\pi\)
−0.118615 + 0.992940i \(0.537845\pi\)
\(12\) 0 0
\(13\) −1.29707 −0.359743 −0.179871 − 0.983690i \(-0.557568\pi\)
−0.179871 + 0.983690i \(0.557568\pi\)
\(14\) 0 0
\(15\) 3.29707 0.851300
\(16\) 0 0
\(17\) −4.08387 −0.990485 −0.495242 − 0.868755i \(-0.664921\pi\)
−0.495242 + 0.868755i \(0.664921\pi\)
\(18\) 0 0
\(19\) 6.87067 1.57624 0.788120 − 0.615521i \(-0.211054\pi\)
0.788120 + 0.615521i \(0.211054\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 8.16774 1.70309 0.851546 − 0.524279i \(-0.175665\pi\)
0.851546 + 0.524279i \(0.175665\pi\)
\(24\) 0 0
\(25\) 5.87067 1.17413
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.80734 −1.07840 −0.539198 − 0.842179i \(-0.681273\pi\)
−0.539198 + 0.842179i \(0.681273\pi\)
\(30\) 0 0
\(31\) −10.2516 −1.84124 −0.920622 − 0.390454i \(-0.872318\pi\)
−0.920622 + 0.390454i \(0.872318\pi\)
\(32\) 0 0
\(33\) −0.786802 −0.136965
\(34\) 0 0
\(35\) 6.59414 1.11461
\(36\) 0 0
\(37\) −6.95455 −1.14332 −0.571660 − 0.820490i \(-0.693700\pi\)
−0.571660 + 0.820490i \(0.693700\pi\)
\(38\) 0 0
\(39\) −1.29707 −0.207698
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −3.38094 −0.515589 −0.257794 − 0.966200i \(-0.582996\pi\)
−0.257794 + 0.966200i \(0.582996\pi\)
\(44\) 0 0
\(45\) 3.29707 0.491498
\(46\) 0 0
\(47\) 1.21320 0.176963 0.0884816 − 0.996078i \(-0.471799\pi\)
0.0884816 + 0.996078i \(0.471799\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −4.08387 −0.571857
\(52\) 0 0
\(53\) −5.02054 −0.689624 −0.344812 − 0.938672i \(-0.612057\pi\)
−0.344812 + 0.938672i \(0.612057\pi\)
\(54\) 0 0
\(55\) −2.59414 −0.349794
\(56\) 0 0
\(57\) 6.87067 0.910043
\(58\) 0 0
\(59\) 0.276533 0.0360015 0.0180008 − 0.999838i \(-0.494270\pi\)
0.0180008 + 0.999838i \(0.494270\pi\)
\(60\) 0 0
\(61\) 9.38094 1.20111 0.600553 − 0.799585i \(-0.294947\pi\)
0.600553 + 0.799585i \(0.294947\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −4.27653 −0.530439
\(66\) 0 0
\(67\) 15.8912 1.94142 0.970710 − 0.240253i \(-0.0772305\pi\)
0.970710 + 0.240253i \(0.0772305\pi\)
\(68\) 0 0
\(69\) 8.16774 0.983281
\(70\) 0 0
\(71\) −4.08387 −0.484666 −0.242333 − 0.970193i \(-0.577913\pi\)
−0.242333 + 0.970193i \(0.577913\pi\)
\(72\) 0 0
\(73\) 14.8458 1.73756 0.868782 − 0.495194i \(-0.164903\pi\)
0.868782 + 0.495194i \(0.164903\pi\)
\(74\) 0 0
\(75\) 5.87067 0.677887
\(76\) 0 0
\(77\) −1.57360 −0.179329
\(78\) 0 0
\(79\) 12.5941 1.41695 0.708476 − 0.705735i \(-0.249383\pi\)
0.708476 + 0.705735i \(0.249383\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0590 −1.76270 −0.881350 − 0.472464i \(-0.843365\pi\)
−0.881350 + 0.472464i \(0.843365\pi\)
\(84\) 0 0
\(85\) −13.4648 −1.46046
\(86\) 0 0
\(87\) −5.80734 −0.622612
\(88\) 0 0
\(89\) −2.27653 −0.241312 −0.120656 − 0.992694i \(-0.538500\pi\)
−0.120656 + 0.992694i \(0.538500\pi\)
\(90\) 0 0
\(91\) −2.59414 −0.271940
\(92\) 0 0
\(93\) −10.2516 −1.06304
\(94\) 0 0
\(95\) 22.6531 2.32416
\(96\) 0 0
\(97\) −11.5736 −1.17512 −0.587561 − 0.809180i \(-0.699912\pi\)
−0.587561 + 0.809180i \(0.699912\pi\)
\(98\) 0 0
\(99\) −0.786802 −0.0790766
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 984.2.a.h.1.3 ✓ 3
3.2 odd 2 2952.2.a.m.1.1 3
4.3 odd 2 1968.2.a.t.1.3 3
8.3 odd 2 7872.2.a.bz.1.1 3
8.5 even 2 7872.2.a.bu.1.1 3
12.11 even 2 5904.2.a.bi.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
984.2.a.h.1.3 ✓ 3 1.1 even 1 trivial
1968.2.a.t.1.3 3 4.3 odd 2
2952.2.a.m.1.1 3 3.2 odd 2
5904.2.a.bi.1.1 3 12.11 even 2
7872.2.a.bu.1.1 3 8.5 even 2
7872.2.a.bz.1.1 3 8.3 odd 2