Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3800,2,Mod(3649,3800)] 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("3800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-12,0,6,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 194x^{8} + 618x^{6} + 733x^{4} + 286x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.7
Root \(0.185519i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.q.3649.6

$q$-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)
 
\(f(q)\) \(=\) \(q+0.185519i q^{3} -4.45651i q^{7} +2.96558 q^{9} +2.64623 q^{11} -1.30142i q^{13} -3.51716i q^{17} +1.00000 q^{19} +0.826767 q^{21} +6.52882i q^{23} +1.10673i q^{27} +5.20946 q^{29} +10.8219 q^{31} +0.490925i q^{33} +2.04607i q^{37} +0.241439 q^{39} -3.80044 q^{41} -4.77089i q^{43} +1.49093i q^{47} -12.8605 q^{49} +0.652501 q^{51} -0.225583i q^{53} +0.185519i q^{57} +2.86056 q^{59} -6.31449 q^{61} -13.2161i q^{63} -13.1831i q^{67} -1.21122 q^{69} +12.3310 q^{71} +5.42276i q^{73} -11.7929i q^{77} -14.9688 q^{79} +8.69143 q^{81} -3.84470i q^{83} +0.966455i q^{87} -1.67666 q^{89} -5.79981 q^{91} +2.00768i q^{93} +9.48523i q^{97} +7.84760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9} + 6 q^{11} + 12 q^{19} + 30 q^{21} - 18 q^{29} + 10 q^{31} - 24 q^{39} + 6 q^{41} - 44 q^{49} + 66 q^{51} + 18 q^{61} + 22 q^{69} + 38 q^{71} + 32 q^{79} + 52 q^{81} - 28 q^{89} + 84 q^{91}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.185519i 0.107109i 0.998565 + 0.0535547i \(0.0170552\pi\)
−0.998565 + 0.0535547i \(0.982945\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.45651i − 1.68440i −0.539164 − 0.842201i \(-0.681260\pi\)
0.539164 − 0.842201i \(-0.318740\pi\)
\(8\) 0 0
\(9\) 2.96558 0.988528
\(10\) 0 0
\(11\) 2.64623 0.797867 0.398934 − 0.916980i \(-0.369380\pi\)
0.398934 + 0.916980i \(0.369380\pi\)
\(12\) 0 0
\(13\) − 1.30142i − 0.360950i −0.983580 − 0.180475i \(-0.942236\pi\)
0.983580 − 0.180475i \(-0.0577635\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.51716i − 0.853037i −0.904479 − 0.426519i \(-0.859740\pi\)
0.904479 − 0.426519i \(-0.140260\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.826767 0.180415
\(22\) 0 0
\(23\) 6.52882i 1.36135i 0.732584 + 0.680677i \(0.238314\pi\)
−0.732584 + 0.680677i \(0.761686\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.10673i 0.212990i
\(28\) 0 0
\(29\) 5.20946 0.967373 0.483686 − 0.875241i \(-0.339298\pi\)
0.483686 + 0.875241i \(0.339298\pi\)
\(30\) 0 0
\(31\) 10.8219 1.94368 0.971839 − 0.235646i \(-0.0757207\pi\)
0.971839 + 0.235646i \(0.0757207\pi\)
\(32\) 0 0
\(33\) 0.490925i 0.0854591i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.04607i 0.336373i 0.985755 + 0.168186i \(0.0537910\pi\)
−0.985755 + 0.168186i \(0.946209\pi\)
\(38\) 0 0
\(39\) 0.241439 0.0386612
\(40\) 0 0
\(41\) −3.80044 −0.593529 −0.296764 − 0.954951i \(-0.595908\pi\)
−0.296764 + 0.954951i \(0.595908\pi\)
\(42\) 0 0
\(43\) − 4.77089i − 0.727554i −0.931486 − 0.363777i \(-0.881487\pi\)
0.931486 − 0.363777i \(-0.118513\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.49093i 0.217474i 0.994071 + 0.108737i \(0.0346806\pi\)
−0.994071 + 0.108737i \(0.965319\pi\)
\(48\) 0 0
\(49\) −12.8605 −1.83721
\(50\) 0 0
\(51\) 0.652501 0.0913684
\(52\) 0 0
\(53\) − 0.225583i − 0.0309862i −0.999880 − 0.0154931i \(-0.995068\pi\)
0.999880 − 0.0154931i \(-0.00493180\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.185519i 0.0245726i
\(58\) 0 0
\(59\) 2.86056 0.372413 0.186206 − 0.982511i \(-0.440381\pi\)
0.186206 + 0.982511i \(0.440381\pi\)
\(60\) 0 0
\(61\) −6.31449 −0.808488 −0.404244 − 0.914651i \(-0.632465\pi\)
−0.404244 + 0.914651i \(0.632465\pi\)
\(62\) 0 0
\(63\) − 13.2161i − 1.66508i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 13.1831i − 1.61058i −0.592884 − 0.805288i \(-0.702011\pi\)
0.592884 − 0.805288i \(-0.297989\pi\)
\(68\) 0 0
\(69\) −1.21122 −0.145814
\(70\) 0 0
\(71\) 12.3310 1.46342 0.731711 − 0.681615i \(-0.238722\pi\)
0.731711 + 0.681615i \(0.238722\pi\)
\(72\) 0 0
\(73\) 5.42276i 0.634686i 0.948311 + 0.317343i \(0.102791\pi\)
−0.948311 + 0.317343i \(0.897209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 11.7929i − 1.34393i
\(78\) 0 0
\(79\) −14.9688 −1.68413 −0.842063 − 0.539379i \(-0.818659\pi\)
−0.842063 + 0.539379i \(0.818659\pi\)
\(80\) 0 0
\(81\) 8.69143 0.965714
\(82\) 0 0
\(83\) − 3.84470i − 0.422011i −0.977485 − 0.211005i \(-0.932326\pi\)
0.977485 − 0.211005i \(-0.0676737\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.966455i 0.103615i
\(88\) 0 0
\(89\) −1.67666 −0.177726 −0.0888629 − 0.996044i \(-0.528323\pi\)
−0.0888629 + 0.996044i \(0.528323\pi\)
\(90\) 0 0
\(91\) −5.79981 −0.607985
\(92\) 0 0
\(93\) 2.00768i 0.208186i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.48523i 0.963079i 0.876424 + 0.481539i \(0.159922\pi\)
−0.876424 + 0.481539i \(0.840078\pi\)
\(98\) 0 0
\(99\) 7.84760 0.788714
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 3800.2.d.q.3649.7 12
5.2 odd 4 3800.2.a.ba.1.4 ✓ 6
5.3 odd 4 3800.2.a.bc.1.3 yes 6
5.4 even 2 inner 3800.2.d.q.3649.6 12
20.3 even 4 7600.2.a.ch.1.4 6
20.7 even 4 7600.2.a.cl.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.4 ✓ 6 5.2 odd 4
3800.2.a.bc.1.3 yes 6 5.3 odd 4
3800.2.d.q.3649.6 12 5.4 even 2 inner
3800.2.d.q.3649.7 12 1.1 even 1 trivial
7600.2.a.ch.1.4 6 20.3 even 4
7600.2.a.cl.1.3 6 20.7 even 4