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Properties

Label 200.2.d.f.101.4
Level $200$
Weight $2$
Character 200.101
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
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] = [200,2,Mod(101,200)] 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("200.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.4
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 200.101
Dual form 200.2.d.f.101.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.36603 + 0.366025i) q^{2} -2.73205i q^{3} +(1.73205 + 1.00000i) q^{4} +(1.00000 - 3.73205i) q^{6} -0.732051 q^{7} +(2.00000 + 2.00000i) q^{8} -4.46410 q^{9} -2.00000i q^{11} +(2.73205 - 4.73205i) q^{12} +3.46410i q^{13} +(-1.00000 - 0.267949i) q^{14} +(2.00000 + 3.46410i) q^{16} -3.46410 q^{17} +(-6.09808 - 1.63397i) q^{18} +0.535898i q^{19} +2.00000i q^{21} +(0.732051 - 2.73205i) q^{22} +6.19615 q^{23} +(5.46410 - 5.46410i) q^{24} +(-1.26795 + 4.73205i) q^{26} +4.00000i q^{27} +(-1.26795 - 0.732051i) q^{28} +6.92820i q^{29} -5.46410 q^{31} +(1.46410 + 5.46410i) q^{32} -5.46410 q^{33} +(-4.73205 - 1.26795i) q^{34} +(-7.73205 - 4.46410i) q^{36} +2.00000i q^{37} +(-0.196152 + 0.732051i) q^{38} +9.46410 q^{39} +1.46410 q^{41} +(-0.732051 + 2.73205i) q^{42} +5.26795i q^{43} +(2.00000 - 3.46410i) q^{44} +(8.46410 + 2.26795i) q^{46} -3.26795 q^{47} +(9.46410 - 5.46410i) q^{48} -6.46410 q^{49} +9.46410i q^{51} +(-3.46410 + 6.00000i) q^{52} -11.4641i q^{53} +(-1.46410 + 5.46410i) q^{54} +(-1.46410 - 1.46410i) q^{56} +1.46410 q^{57} +(-2.53590 + 9.46410i) q^{58} -7.46410i q^{59} -8.92820i q^{61} +(-7.46410 - 2.00000i) q^{62} +3.26795 q^{63} +8.00000i q^{64} +(-7.46410 - 2.00000i) q^{66} -10.7321i q^{67} +(-6.00000 - 3.46410i) q^{68} -16.9282i q^{69} +5.46410 q^{71} +(-8.92820 - 8.92820i) q^{72} -7.46410 q^{73} +(-0.732051 + 2.73205i) q^{74} +(-0.535898 + 0.928203i) q^{76} +1.46410i q^{77} +(12.9282 + 3.46410i) q^{78} -1.07180 q^{79} -2.46410 q^{81} +(2.00000 + 0.535898i) q^{82} -1.26795i q^{83} +(-2.00000 + 3.46410i) q^{84} +(-1.92820 + 7.19615i) q^{86} +18.9282 q^{87} +(4.00000 - 4.00000i) q^{88} +8.92820 q^{89} -2.53590i q^{91} +(10.7321 + 6.19615i) q^{92} +14.9282i q^{93} +(-4.46410 - 1.19615i) q^{94} +(14.9282 - 4.00000i) q^{96} +14.3923 q^{97} +(-8.83013 - 2.36603i) q^{98} +8.92820i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{6} + 4 q^{7} + 8 q^{8} - 4 q^{9} + 4 q^{12} - 4 q^{14} + 8 q^{16} - 14 q^{18} - 4 q^{22} + 4 q^{23} + 8 q^{24} - 12 q^{26} - 12 q^{28} - 8 q^{31} - 8 q^{32} - 8 q^{33} - 12 q^{34} - 24 q^{36}+ \cdots - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36603 + 0.366025i 0.965926 + 0.258819i
\(3\) − 2.73205i − 1.57735i −0.614810 − 0.788675i \(-0.710767\pi\)
0.614810 − 0.788675i \(-0.289233\pi\)
\(4\) 1.73205 + 1.00000i 0.866025 + 0.500000i
\(5\) 0 0
\(6\) 1.00000 − 3.73205i 0.408248 − 1.52360i
\(7\) −0.732051 −0.276689 −0.138345 − 0.990384i \(-0.544178\pi\)
−0.138345 + 0.990384i \(0.544178\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) −4.46410 −1.48803
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 2.73205 − 4.73205i 0.788675 − 1.36603i
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) −1.00000 − 0.267949i −0.267261 − 0.0716124i
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) −6.09808 − 1.63397i −1.43733 − 0.385132i
\(19\) 0.535898i 0.122944i 0.998109 + 0.0614718i \(0.0195794\pi\)
−0.998109 + 0.0614718i \(0.980421\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0.732051 − 2.73205i 0.156074 − 0.582475i
\(23\) 6.19615 1.29199 0.645994 − 0.763343i \(-0.276443\pi\)
0.645994 + 0.763343i \(0.276443\pi\)
\(24\) 5.46410 − 5.46410i 1.11536 − 1.11536i
\(25\) 0 0
\(26\) −1.26795 + 4.73205i −0.248665 + 0.928032i
\(27\) 4.00000i 0.769800i
\(28\) −1.26795 − 0.732051i −0.239620 − 0.138345i
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 − 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 1.46410 + 5.46410i 0.258819 + 0.965926i
\(33\) −5.46410 −0.951178
\(34\) −4.73205 − 1.26795i −0.811540 − 0.217451i
\(35\) 0 0
\(36\) −7.73205 − 4.46410i −1.28868 − 0.744017i
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) −0.196152 + 0.732051i −0.0318201 + 0.118754i
\(39\) 9.46410 1.51547
\(40\) 0 0
\(41\) 1.46410 0.228654 0.114327 − 0.993443i \(-0.463529\pi\)
0.114327 + 0.993443i \(0.463529\pi\)
\(42\) −0.732051 + 2.73205i −0.112958 + 0.421565i
\(43\) 5.26795i 0.803355i 0.915781 + 0.401677i \(0.131573\pi\)
−0.915781 + 0.401677i \(0.868427\pi\)
\(44\) 2.00000 − 3.46410i 0.301511 − 0.522233i
\(45\) 0 0
\(46\) 8.46410 + 2.26795i 1.24796 + 0.334391i
\(47\) −3.26795 −0.476679 −0.238340 − 0.971182i \(-0.576603\pi\)
−0.238340 + 0.971182i \(0.576603\pi\)
\(48\) 9.46410 − 5.46410i 1.36603 − 0.788675i
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) 9.46410i 1.32524i
\(52\) −3.46410 + 6.00000i −0.480384 + 0.832050i
\(53\) − 11.4641i − 1.57472i −0.616496 − 0.787358i \(-0.711449\pi\)
0.616496 − 0.787358i \(-0.288551\pi\)
\(54\) −1.46410 + 5.46410i −0.199239 + 0.743570i
\(55\) 0 0
\(56\) −1.46410 − 1.46410i −0.195649 − 0.195649i
\(57\) 1.46410 0.193925
\(58\) −2.53590 + 9.46410i −0.332980 + 1.24270i
\(59\) − 7.46410i − 0.971743i −0.874030 − 0.485872i \(-0.838502\pi\)
0.874030 − 0.485872i \(-0.161498\pi\)
\(60\) 0 0
\(61\) − 8.92820i − 1.14314i −0.820554 − 0.571570i \(-0.806335\pi\)
0.820554 − 0.571570i \(-0.193665\pi\)
\(62\) −7.46410 − 2.00000i −0.947942 − 0.254000i
\(63\) 3.26795 0.411723
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) −7.46410 − 2.00000i −0.918767 − 0.246183i
\(67\) − 10.7321i − 1.31113i −0.755139 − 0.655564i \(-0.772431\pi\)
0.755139 − 0.655564i \(-0.227569\pi\)
\(68\) −6.00000 − 3.46410i −0.727607 − 0.420084i
\(69\) − 16.9282i − 2.03792i
\(70\) 0 0
\(71\) 5.46410 0.648470 0.324235 − 0.945977i \(-0.394893\pi\)
0.324235 + 0.945977i \(0.394893\pi\)
\(72\) −8.92820 − 8.92820i −1.05220 − 1.05220i
\(73\) −7.46410 −0.873607 −0.436804 − 0.899557i \(-0.643889\pi\)
−0.436804 + 0.899557i \(0.643889\pi\)
\(74\) −0.732051 + 2.73205i −0.0850992 + 0.317594i
\(75\) 0 0
\(76\) −0.535898 + 0.928203i −0.0614718 + 0.106472i
\(77\) 1.46410i 0.166850i
\(78\) 12.9282 + 3.46410i 1.46383 + 0.392232i
\(79\) −1.07180 −0.120587 −0.0602933 − 0.998181i \(-0.519204\pi\)
−0.0602933 + 0.998181i \(0.519204\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 2.00000 + 0.535898i 0.220863 + 0.0591801i
\(83\) − 1.26795i − 0.139176i −0.997576 − 0.0695878i \(-0.977832\pi\)
0.997576 − 0.0695878i \(-0.0221684\pi\)
\(84\) −2.00000 + 3.46410i −0.218218 + 0.377964i
\(85\) 0 0
\(86\) −1.92820 + 7.19615i −0.207924 + 0.775981i
\(87\) 18.9282 2.02932
\(88\) 4.00000 − 4.00000i 0.426401 − 0.426401i
\(89\) 8.92820 0.946388 0.473194 − 0.880958i \(-0.343101\pi\)
0.473194 + 0.880958i \(0.343101\pi\)
\(90\) 0 0
\(91\) − 2.53590i − 0.265834i
\(92\) 10.7321 + 6.19615i 1.11889 + 0.645994i
\(93\) 14.9282i 1.54798i
\(94\) −4.46410 − 1.19615i −0.460437 − 0.123374i
\(95\) 0 0
\(96\) 14.9282 − 4.00000i 1.52360 − 0.408248i
\(97\) 14.3923 1.46132 0.730659 − 0.682743i \(-0.239213\pi\)
0.730659 + 0.682743i \(0.239213\pi\)
\(98\) −8.83013 − 2.36603i −0.891978 − 0.239005i
\(99\) 8.92820i 0.897318i
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.d.f.101.4 4
3.2 odd 2 1800.2.k.j.901.1 4
4.3 odd 2 800.2.d.e.401.4 4
5.2 odd 4 200.2.f.e.149.2 4
5.3 odd 4 200.2.f.c.149.3 4
5.4 even 2 40.2.d.a.21.1 ✓ 4
8.3 odd 2 800.2.d.e.401.1 4
8.5 even 2 inner 200.2.d.f.101.3 4
12.11 even 2 7200.2.k.j.3601.3 4
15.2 even 4 1800.2.d.l.1549.3 4
15.8 even 4 1800.2.d.p.1549.2 4
15.14 odd 2 360.2.k.e.181.4 4
16.3 odd 4 6400.2.a.be.1.1 2
16.5 even 4 6400.2.a.z.1.1 2
16.11 odd 4 6400.2.a.cj.1.2 2
16.13 even 4 6400.2.a.ce.1.2 2
20.3 even 4 800.2.f.c.49.1 4
20.7 even 4 800.2.f.e.49.4 4
20.19 odd 2 160.2.d.a.81.1 4
24.5 odd 2 1800.2.k.j.901.2 4
24.11 even 2 7200.2.k.j.3601.4 4
40.3 even 4 800.2.f.e.49.3 4
40.13 odd 4 200.2.f.e.149.1 4
40.19 odd 2 160.2.d.a.81.4 4
40.27 even 4 800.2.f.c.49.2 4
40.29 even 2 40.2.d.a.21.2 yes 4
40.37 odd 4 200.2.f.c.149.4 4
60.23 odd 4 7200.2.d.o.2449.2 4
60.47 odd 4 7200.2.d.n.2449.3 4
60.59 even 2 1440.2.k.e.721.3 4
80.19 odd 4 1280.2.a.n.1.2 2
80.29 even 4 1280.2.a.a.1.1 2
80.59 odd 4 1280.2.a.d.1.1 2
80.69 even 4 1280.2.a.o.1.2 2
120.29 odd 2 360.2.k.e.181.3 4
120.53 even 4 1800.2.d.l.1549.4 4
120.59 even 2 1440.2.k.e.721.1 4
120.77 even 4 1800.2.d.p.1549.1 4
120.83 odd 4 7200.2.d.n.2449.2 4
120.107 odd 4 7200.2.d.o.2449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 ✓ 4 5.4 even 2
40.2.d.a.21.2 yes 4 40.29 even 2
160.2.d.a.81.1 4 20.19 odd 2
160.2.d.a.81.4 4 40.19 odd 2
200.2.d.f.101.3 4 8.5 even 2 inner
200.2.d.f.101.4 4 1.1 even 1 trivial
200.2.f.c.149.3 4 5.3 odd 4
200.2.f.c.149.4 4 40.37 odd 4
200.2.f.e.149.1 4 40.13 odd 4
200.2.f.e.149.2 4 5.2 odd 4
360.2.k.e.181.3 4 120.29 odd 2
360.2.k.e.181.4 4 15.14 odd 2
800.2.d.e.401.1 4 8.3 odd 2
800.2.d.e.401.4 4 4.3 odd 2
800.2.f.c.49.1 4 20.3 even 4
800.2.f.c.49.2 4 40.27 even 4
800.2.f.e.49.3 4 40.3 even 4
800.2.f.e.49.4 4 20.7 even 4
1280.2.a.a.1.1 2 80.29 even 4
1280.2.a.d.1.1 2 80.59 odd 4
1280.2.a.n.1.2 2 80.19 odd 4
1280.2.a.o.1.2 2 80.69 even 4
1440.2.k.e.721.1 4 120.59 even 2
1440.2.k.e.721.3 4 60.59 even 2
1800.2.d.l.1549.3 4 15.2 even 4
1800.2.d.l.1549.4 4 120.53 even 4
1800.2.d.p.1549.1 4 120.77 even 4
1800.2.d.p.1549.2 4 15.8 even 4
1800.2.k.j.901.1 4 3.2 odd 2
1800.2.k.j.901.2 4 24.5 odd 2
6400.2.a.z.1.1 2 16.5 even 4
6400.2.a.be.1.1 2 16.3 odd 4
6400.2.a.ce.1.2 2 16.13 even 4
6400.2.a.cj.1.2 2 16.11 odd 4
7200.2.d.n.2449.2 4 120.83 odd 4
7200.2.d.n.2449.3 4 60.47 odd 4
7200.2.d.o.2449.2 4 60.23 odd 4
7200.2.d.o.2449.3 4 120.107 odd 4
7200.2.k.j.3601.3 4 12.11 even 2
7200.2.k.j.3601.4 4 24.11 even 2