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Properties

Label 3800.2.d.n.3649.6
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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 3649.6
Root \(-0.671462 + 1.24464i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.n.3649.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.34292i q^{3} +1.19656i q^{7} -2.48929 q^{9} +4.97858 q^{11} +6.63565i q^{13} +1.48929i q^{17} -1.00000 q^{19} -2.80344 q^{21} +0.510711i q^{23} +1.19656i q^{27} +7.88240 q^{29} -2.97858 q^{31} +11.6644i q^{33} -7.14637i q^{37} -15.5468 q^{39} +1.66442 q^{41} +6.39312i q^{43} -9.95715i q^{47} +5.56825 q^{49} -3.48929 q^{51} +11.4219i q^{53} -2.34292i q^{57} +11.8396 q^{59} +3.66442 q^{61} -2.97858i q^{63} +7.61423i q^{67} -1.19656 q^{69} -13.8396i q^{73} +5.95715i q^{77} -12.6858 q^{79} -10.2713 q^{81} +8.68585i q^{83} +18.4679i q^{87} +4.87819 q^{89} -7.93994 q^{91} -6.97858i q^{93} -6.81079i q^{97} -12.3931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100}) \) 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
\(3\) 2.34292i 1.35269i 0.736586 + 0.676344i \(0.236437\pi\)
−0.736586 + 0.676344i \(0.763563\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.19656i 0.452256i 0.974098 + 0.226128i \(0.0726068\pi\)
−0.974098 + 0.226128i \(0.927393\pi\)
\(8\) 0 0
\(9\) −2.48929 −0.829763
\(10\) 0 0
\(11\) 4.97858 1.50110 0.750549 − 0.660815i \(-0.229789\pi\)
0.750549 + 0.660815i \(0.229789\pi\)
\(12\) 0 0
\(13\) 6.63565i 1.84040i 0.391449 + 0.920200i \(0.371974\pi\)
−0.391449 + 0.920200i \(0.628026\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.48929i 0.361206i 0.983556 + 0.180603i \(0.0578048\pi\)
−0.983556 + 0.180603i \(0.942195\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.80344 −0.611761
\(22\) 0 0
\(23\) 0.510711i 0.106491i 0.998581 + 0.0532453i \(0.0169565\pi\)
−0.998581 + 0.0532453i \(0.983043\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.19656i 0.230278i
\(28\) 0 0
\(29\) 7.88240 1.46373 0.731863 − 0.681452i \(-0.238651\pi\)
0.731863 + 0.681452i \(0.238651\pi\)
\(30\) 0 0
\(31\) −2.97858 −0.534968 −0.267484 − 0.963562i \(-0.586192\pi\)
−0.267484 + 0.963562i \(0.586192\pi\)
\(32\) 0 0
\(33\) 11.6644i 2.03052i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.14637i − 1.17486i −0.809277 − 0.587428i \(-0.800141\pi\)
0.809277 − 0.587428i \(-0.199859\pi\)
\(38\) 0 0
\(39\) −15.5468 −2.48948
\(40\) 0 0
\(41\) 1.66442 0.259939 0.129970 − 0.991518i \(-0.458512\pi\)
0.129970 + 0.991518i \(0.458512\pi\)
\(42\) 0 0
\(43\) 6.39312i 0.974941i 0.873139 + 0.487470i \(0.162080\pi\)
−0.873139 + 0.487470i \(0.837920\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 9.95715i − 1.45240i −0.687483 − 0.726200i \(-0.741285\pi\)
0.687483 − 0.726200i \(-0.258715\pi\)
\(48\) 0 0
\(49\) 5.56825 0.795464
\(50\) 0 0
\(51\) −3.48929 −0.488598
\(52\) 0 0
\(53\) 11.4219i 1.56892i 0.620182 + 0.784458i \(0.287059\pi\)
−0.620182 + 0.784458i \(0.712941\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.34292i − 0.310328i
\(58\) 0 0
\(59\) 11.8396 1.54138 0.770690 − 0.637211i \(-0.219912\pi\)
0.770690 + 0.637211i \(0.219912\pi\)
\(60\) 0 0
\(61\) 3.66442 0.469181 0.234591 − 0.972094i \(-0.424625\pi\)
0.234591 + 0.972094i \(0.424625\pi\)
\(62\) 0 0
\(63\) − 2.97858i − 0.375265i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.61423i 0.930226i 0.885251 + 0.465113i \(0.153986\pi\)
−0.885251 + 0.465113i \(0.846014\pi\)
\(68\) 0 0
\(69\) −1.19656 −0.144049
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 13.8396i − 1.61980i −0.586570 − 0.809899i \(-0.699522\pi\)
0.586570 − 0.809899i \(-0.300478\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.95715i 0.678881i
\(78\) 0 0
\(79\) −12.6858 −1.42727 −0.713635 − 0.700518i \(-0.752952\pi\)
−0.713635 + 0.700518i \(0.752952\pi\)
\(80\) 0 0
\(81\) −10.2713 −1.14126
\(82\) 0 0
\(83\) 8.68585i 0.953395i 0.879067 + 0.476698i \(0.158166\pi\)
−0.879067 + 0.476698i \(0.841834\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 18.4679i 1.97996i
\(88\) 0 0
\(89\) 4.87819 0.517087 0.258544 − 0.966000i \(-0.416757\pi\)
0.258544 + 0.966000i \(0.416757\pi\)
\(90\) 0 0
\(91\) −7.93994 −0.832332
\(92\) 0 0
\(93\) − 6.97858i − 0.723645i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.81079i − 0.691531i −0.938321 − 0.345765i \(-0.887619\pi\)
0.938321 − 0.345765i \(-0.112381\pi\)
\(98\) 0 0
\(99\) −12.3931 −1.24555
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.3649.6 6
5.2 odd 4 3800.2.a.w.1.3 3
5.3 odd 4 760.2.a.i.1.1 ✓ 3
5.4 even 2 inner 3800.2.d.n.3649.1 6
15.8 even 4 6840.2.a.bm.1.2 3
20.3 even 4 1520.2.a.q.1.3 3
20.7 even 4 7600.2.a.bp.1.1 3
40.3 even 4 6080.2.a.br.1.1 3
40.13 odd 4 6080.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.1 ✓ 3 5.3 odd 4
1520.2.a.q.1.3 3 20.3 even 4
3800.2.a.w.1.3 3 5.2 odd 4
3800.2.d.n.3649.1 6 5.4 even 2 inner
3800.2.d.n.3649.6 6 1.1 even 1 trivial
6080.2.a.br.1.1 3 40.3 even 4
6080.2.a.bx.1.3 3 40.13 odd 4
6840.2.a.bm.1.2 3 15.8 even 4
7600.2.a.bp.1.1 3 20.7 even 4