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 = [6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6] 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: \(6\)
Coefficient field: 6.0.399424.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.2
Root \(1.40680 + 0.144584i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.n.3649.5

$q$-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)
 
\(f(q)\) \(=\) \(q-1.81361i q^{3} -4.91638i q^{7} -0.289169 q^{9} +0.578337 q^{11} +6.39194i q^{13} -0.710831i q^{17} -1.00000 q^{19} -8.91638 q^{21} +2.71083i q^{23} -4.91638i q^{27} -6.54359 q^{29} +1.42166 q^{31} -1.04888i q^{33} -9.10278i q^{37} +11.5925 q^{39} -11.0489 q^{41} -5.83276i q^{43} -1.15667i q^{47} -17.1708 q^{49} -1.28917 q^{51} -13.2736i q^{53} +1.81361i q^{57} -11.3869 q^{59} -9.04888 q^{61} +1.42166i q^{63} +2.97028i q^{67} +4.91638 q^{69} +9.38692i q^{73} -2.84333i q^{77} -4.37279 q^{79} -9.78389 q^{81} +0.372787i q^{83} +11.8675i q^{87} +16.6167 q^{89} +31.4252 q^{91} -2.57834i q^{93} +3.94610i q^{97} -0.167237 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) 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
\(3\) − 1.81361i − 1.04709i −0.851999 − 0.523543i \(-0.824610\pi\)
0.851999 − 0.523543i \(-0.175390\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.91638i − 1.85822i −0.369807 − 0.929109i \(-0.620576\pi\)
0.369807 − 0.929109i \(-0.379424\pi\)
\(8\) 0 0
\(9\) −0.289169 −0.0963895
\(10\) 0 0
\(11\) 0.578337 0.174375 0.0871876 − 0.996192i \(-0.472212\pi\)
0.0871876 + 0.996192i \(0.472212\pi\)
\(12\) 0 0
\(13\) 6.39194i 1.77281i 0.462914 + 0.886403i \(0.346804\pi\)
−0.462914 + 0.886403i \(0.653196\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.710831i − 0.172402i −0.996278 − 0.0862010i \(-0.972527\pi\)
0.996278 − 0.0862010i \(-0.0274727\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −8.91638 −1.94571
\(22\) 0 0
\(23\) 2.71083i 0.565247i 0.959231 + 0.282624i \(0.0912048\pi\)
−0.959231 + 0.282624i \(0.908795\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.91638i − 0.946158i
\(28\) 0 0
\(29\) −6.54359 −1.21512 −0.607558 − 0.794276i \(-0.707851\pi\)
−0.607558 + 0.794276i \(0.707851\pi\)
\(30\) 0 0
\(31\) 1.42166 0.255338 0.127669 − 0.991817i \(-0.459250\pi\)
0.127669 + 0.991817i \(0.459250\pi\)
\(32\) 0 0
\(33\) − 1.04888i − 0.182586i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.10278i − 1.49649i −0.663424 − 0.748243i \(-0.730897\pi\)
0.663424 − 0.748243i \(-0.269103\pi\)
\(38\) 0 0
\(39\) 11.5925 1.85628
\(40\) 0 0
\(41\) −11.0489 −1.72554 −0.862772 − 0.505593i \(-0.831274\pi\)
−0.862772 + 0.505593i \(0.831274\pi\)
\(42\) 0 0
\(43\) − 5.83276i − 0.889488i −0.895658 − 0.444744i \(-0.853295\pi\)
0.895658 − 0.444744i \(-0.146705\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.15667i − 0.168718i −0.996435 − 0.0843591i \(-0.973116\pi\)
0.996435 − 0.0843591i \(-0.0268843\pi\)
\(48\) 0 0
\(49\) −17.1708 −2.45297
\(50\) 0 0
\(51\) −1.28917 −0.180520
\(52\) 0 0
\(53\) − 13.2736i − 1.82327i −0.411004 − 0.911633i \(-0.634822\pi\)
0.411004 − 0.911633i \(-0.365178\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.81361i 0.240218i
\(58\) 0 0
\(59\) −11.3869 −1.48245 −0.741225 − 0.671256i \(-0.765755\pi\)
−0.741225 + 0.671256i \(0.765755\pi\)
\(60\) 0 0
\(61\) −9.04888 −1.15859 −0.579295 − 0.815118i \(-0.696672\pi\)
−0.579295 + 0.815118i \(0.696672\pi\)
\(62\) 0 0
\(63\) 1.42166i 0.179113i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.97028i 0.362878i 0.983402 + 0.181439i \(0.0580754\pi\)
−0.983402 + 0.181439i \(0.941925\pi\)
\(68\) 0 0
\(69\) 4.91638 0.591863
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.38692i 1.09866i 0.835607 + 0.549328i \(0.185116\pi\)
−0.835607 + 0.549328i \(0.814884\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.84333i − 0.324027i
\(78\) 0 0
\(79\) −4.37279 −0.491977 −0.245988 − 0.969273i \(-0.579113\pi\)
−0.245988 + 0.969273i \(0.579113\pi\)
\(80\) 0 0
\(81\) −9.78389 −1.08710
\(82\) 0 0
\(83\) 0.372787i 0.0409187i 0.999791 + 0.0204593i \(0.00651287\pi\)
−0.999791 + 0.0204593i \(0.993487\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.8675i 1.27233i
\(88\) 0 0
\(89\) 16.6167 1.76136 0.880681 − 0.473710i \(-0.157086\pi\)
0.880681 + 0.473710i \(0.157086\pi\)
\(90\) 0 0
\(91\) 31.4252 3.29426
\(92\) 0 0
\(93\) − 2.57834i − 0.267361i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.94610i 0.400666i 0.979728 + 0.200333i \(0.0642024\pi\)
−0.979728 + 0.200333i \(0.935798\pi\)
\(98\) 0 0
\(99\) −0.167237 −0.0168079
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.n.3649.2 6
5.2 odd 4 3800.2.a.w.1.1 3
5.3 odd 4 760.2.a.i.1.3 ✓ 3
5.4 even 2 inner 3800.2.d.n.3649.5 6
15.8 even 4 6840.2.a.bm.1.1 3
20.3 even 4 1520.2.a.q.1.1 3
20.7 even 4 7600.2.a.bp.1.3 3
40.3 even 4 6080.2.a.br.1.3 3
40.13 odd 4 6080.2.a.bx.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.3 ✓ 3 5.3 odd 4
1520.2.a.q.1.1 3 20.3 even 4
3800.2.a.w.1.1 3 5.2 odd 4
3800.2.d.n.3649.2 6 1.1 even 1 trivial
3800.2.d.n.3649.5 6 5.4 even 2 inner
6080.2.a.br.1.3 3 40.3 even 4
6080.2.a.bx.1.1 3 40.13 odd 4
6840.2.a.bm.1.1 3 15.8 even 4
7600.2.a.bp.1.3 3 20.7 even 4