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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [200,2,Mod(43,200)] 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("200.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 107.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 200.107
Dual form 200.2.k.d.43.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.00000i) q^{2} +(2.00000 - 2.00000i) q^{3} +2.00000i q^{4} +4.00000 q^{6} +(-2.00000 + 2.00000i) q^{8} -5.00000i q^{9} -6.00000 q^{11} +(4.00000 + 4.00000i) q^{12} -4.00000 q^{16} +(4.00000 + 4.00000i) q^{17} +(5.00000 - 5.00000i) q^{18} -2.00000i q^{19} +(-6.00000 - 6.00000i) q^{22} +8.00000i q^{24} +(-4.00000 - 4.00000i) q^{27} +(-4.00000 - 4.00000i) q^{32} +(-12.0000 + 12.0000i) q^{33} +8.00000i q^{34} +10.0000 q^{36} +(2.00000 - 2.00000i) q^{38} -6.00000 q^{41} +(6.00000 - 6.00000i) q^{43} -12.0000i q^{44} +(-8.00000 + 8.00000i) q^{48} +7.00000i q^{49} +16.0000 q^{51} -8.00000i q^{54} +(-4.00000 - 4.00000i) q^{57} -6.00000i q^{59} -8.00000i q^{64} -24.0000 q^{66} +(-6.00000 - 6.00000i) q^{67} +(-8.00000 + 8.00000i) q^{68} +(10.0000 + 10.0000i) q^{72} +(12.0000 - 12.0000i) q^{73} +4.00000 q^{76} -1.00000 q^{81} +(-6.00000 - 6.00000i) q^{82} +(2.00000 - 2.00000i) q^{83} +12.0000 q^{86} +(12.0000 - 12.0000i) q^{88} +18.0000i q^{89} -16.0000 q^{96} +(12.0000 + 12.0000i) q^{97} +(-7.00000 + 7.00000i) q^{98} +30.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 8 q^{6} - 4 q^{8} - 12 q^{11} + 8 q^{12} - 8 q^{16} + 8 q^{17} + 10 q^{18} - 12 q^{22} - 8 q^{27} - 8 q^{32} - 24 q^{33} + 20 q^{36} + 4 q^{38} - 12 q^{41} + 12 q^{43} - 16 q^{48}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 + 1.00000i 0.707107 + 0.707107i
\(3\) 2.00000 − 2.00000i 1.15470 − 1.15470i 0.169102 − 0.985599i \(-0.445913\pi\)
0.985599 − 0.169102i \(-0.0540867\pi\)
\(4\) 2.00000i 1.00000i
\(5\) 0 0
\(6\) 4.00000 1.63299
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −2.00000 + 2.00000i −0.707107 + 0.707107i
\(9\) − 5.00000i − 1.66667i
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 4.00000 + 4.00000i 1.15470 + 1.15470i
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 4.00000 + 4.00000i 0.970143 + 0.970143i 0.999567 − 0.0294245i \(-0.00936746\pi\)
−0.0294245 + 0.999567i \(0.509367\pi\)
\(18\) 5.00000 − 5.00000i 1.17851 − 1.17851i
\(19\) − 2.00000i − 0.458831i −0.973329 − 0.229416i \(-0.926318\pi\)
0.973329 − 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 − 6.00000i −1.27920 − 1.27920i
\(23\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 8.00000i 1.63299i
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 − 4.00000i −0.769800 − 0.769800i
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.00000 − 4.00000i −0.707107 − 0.707107i
\(33\) −12.0000 + 12.0000i −2.08893 + 2.08893i
\(34\) 8.00000i 1.37199i
\(35\) 0 0
\(36\) 10.0000 1.66667
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) 2.00000 − 2.00000i 0.324443 − 0.324443i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 − 6.00000i 0.914991 − 0.914991i −0.0816682 − 0.996660i \(-0.526025\pi\)
0.996660 + 0.0816682i \(0.0260248\pi\)
\(44\) − 12.0000i − 1.80907i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −8.00000 + 8.00000i −1.15470 + 1.15470i
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 16.0000 2.24045
\(52\) 0 0
\(53\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) − 8.00000i − 1.08866i
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 − 4.00000i −0.529813 − 0.529813i
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 − 0.390567i \(-0.872279\pi\)
0.920575 − 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) − 8.00000i − 1.00000i
\(65\) 0 0
\(66\) −24.0000 −2.95420
\(67\) −6.00000 − 6.00000i −0.733017 − 0.733017i 0.238200 − 0.971216i \(-0.423443\pi\)
−0.971216 + 0.238200i \(0.923443\pi\)
\(68\) −8.00000 + 8.00000i −0.970143 + 0.970143i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 10.0000 + 10.0000i 1.17851 + 1.17851i
\(73\) 12.0000 − 12.0000i 1.40449 − 1.40449i 0.619486 − 0.785007i \(-0.287341\pi\)
0.785007 − 0.619486i \(-0.212659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) −6.00000 − 6.00000i −0.662589 − 0.662589i
\(83\) 2.00000 − 2.00000i 0.219529 − 0.219529i −0.588771 − 0.808300i \(-0.700388\pi\)
0.808300 + 0.588771i \(0.200388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 12.0000 − 12.0000i 1.27920 − 1.27920i
\(89\) 18.0000i 1.90800i 0.299813 + 0.953998i \(0.403076\pi\)
−0.299813 + 0.953998i \(0.596924\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −16.0000 −1.63299
\(97\) 12.0000 + 12.0000i 1.21842 + 1.21842i 0.968187 + 0.250229i \(0.0805058\pi\)
0.250229 + 0.968187i \(0.419494\pi\)
\(98\) −7.00000 + 7.00000i −0.707107 + 0.707107i
\(99\) 30.0000i 3.01511i
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 200.2.k.d.107.1 yes 2
4.3 odd 2 800.2.o.a.207.1 2
5.2 odd 4 200.2.k.a.43.1 ✓ 2
5.3 odd 4 inner 200.2.k.d.43.1 yes 2
5.4 even 2 200.2.k.a.107.1 yes 2
8.3 odd 2 CM 200.2.k.d.107.1 yes 2
8.5 even 2 800.2.o.a.207.1 2
20.3 even 4 800.2.o.a.143.1 2
20.7 even 4 800.2.o.d.143.1 2
20.19 odd 2 800.2.o.d.207.1 2
40.3 even 4 inner 200.2.k.d.43.1 yes 2
40.13 odd 4 800.2.o.a.143.1 2
40.19 odd 2 200.2.k.a.107.1 yes 2
40.27 even 4 200.2.k.a.43.1 ✓ 2
40.29 even 2 800.2.o.d.207.1 2
40.37 odd 4 800.2.o.d.143.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.2.k.a.43.1 ✓ 2 5.2 odd 4
200.2.k.a.43.1 ✓ 2 40.27 even 4
200.2.k.a.107.1 yes 2 5.4 even 2
200.2.k.a.107.1 yes 2 40.19 odd 2
200.2.k.d.43.1 yes 2 5.3 odd 4 inner
200.2.k.d.43.1 yes 2 40.3 even 4 inner
200.2.k.d.107.1 yes 2 1.1 even 1 trivial
200.2.k.d.107.1 yes 2 8.3 odd 2 CM
800.2.o.a.143.1 2 20.3 even 4
800.2.o.a.143.1 2 40.13 odd 4
800.2.o.a.207.1 2 4.3 odd 2
800.2.o.a.207.1 2 8.5 even 2
800.2.o.d.143.1 2 20.7 even 4
800.2.o.d.143.1 2 40.37 odd 4
800.2.o.d.207.1 2 20.19 odd 2
800.2.o.d.207.1 2 40.29 even 2