Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1800,2,Mod(899,1800)] 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("1800.899"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,32,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(26)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 899.7
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1800.899
Dual form 1800.2.m.c.899.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.707107 + 1.22474i) q^{2} +(-1.00000 + 1.73205i) q^{4} -3.46410 q^{7} -2.82843 q^{8} +2.82843i q^{11} -3.46410 q^{13} +(-2.44949 - 4.24264i) q^{14} +(-2.00000 - 3.46410i) q^{16} +1.41421 q^{17} +4.00000 q^{19} +(-3.46410 + 2.00000i) q^{22} -4.89898i q^{23} +(-2.44949 - 4.24264i) q^{26} +(3.46410 - 6.00000i) q^{28} -2.44949 q^{29} -3.46410i q^{31} +(2.82843 - 4.89898i) q^{32} +(1.00000 + 1.73205i) q^{34} +(2.82843 + 4.89898i) q^{38} -1.41421i q^{41} -8.00000i q^{43} +(-4.89898 - 2.82843i) q^{44} +(6.00000 - 3.46410i) q^{46} -4.89898i q^{47} +5.00000 q^{49} +(3.46410 - 6.00000i) q^{52} -7.34847i q^{53} +9.79796 q^{56} +(-1.73205 - 3.00000i) q^{58} -11.3137i q^{59} +13.8564i q^{61} +(4.24264 - 2.44949i) q^{62} +8.00000 q^{64} -4.00000i q^{67} +(-1.41421 + 2.44949i) q^{68} -14.6969 q^{71} +4.00000i q^{73} +(-4.00000 + 6.92820i) q^{76} -9.79796i q^{77} -3.46410i q^{79} +(1.73205 - 1.00000i) q^{82} -14.1421 q^{83} +(9.79796 - 5.65685i) q^{86} -8.00000i q^{88} -7.07107i q^{89} +12.0000 q^{91} +(8.48528 + 4.89898i) q^{92} +(6.00000 - 3.46410i) q^{94} +8.00000i q^{97} +(3.53553 + 6.12372i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 16 q^{16} + 32 q^{19} + 8 q^{34} + 48 q^{46} + 40 q^{49} + 64 q^{64} - 32 q^{76} + 96 q^{91} + 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 + 1.22474i 0.500000 + 0.866025i
\(3\) 0 0
\(4\) −1.00000 + 1.73205i −0.500000 + 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 − 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 − 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) −2.44949 − 4.24264i −0.654654 − 1.13389i
\(15\) 0 0
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) 1.41421 0.342997 0.171499 − 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.46410 + 2.00000i −0.738549 + 0.426401i
\(23\) − 4.89898i − 1.02151i −0.859727 − 0.510754i \(-0.829366\pi\)
0.859727 − 0.510754i \(-0.170634\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.44949 − 4.24264i −0.480384 − 0.832050i
\(27\) 0 0
\(28\) 3.46410 − 6.00000i 0.654654 − 1.13389i
\(29\) −2.44949 −0.454859 −0.227429 − 0.973795i \(-0.573032\pi\)
−0.227429 + 0.973795i \(0.573032\pi\)
\(30\) 0 0
\(31\) − 3.46410i − 0.622171i −0.950382 − 0.311086i \(-0.899307\pi\)
0.950382 − 0.311086i \(-0.100693\pi\)
\(32\) 2.82843 − 4.89898i 0.500000 − 0.866025i
\(33\) 0 0
\(34\) 1.00000 + 1.73205i 0.171499 + 0.297044i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 2.82843 + 4.89898i 0.458831 + 0.794719i
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.41421i − 0.220863i −0.993884 − 0.110432i \(-0.964777\pi\)
0.993884 − 0.110432i \(-0.0352233\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) −4.89898 − 2.82843i −0.738549 − 0.426401i
\(45\) 0 0
\(46\) 6.00000 − 3.46410i 0.884652 − 0.510754i
\(47\) − 4.89898i − 0.714590i −0.933992 − 0.357295i \(-0.883699\pi\)
0.933992 − 0.357295i \(-0.116301\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 3.46410 − 6.00000i 0.480384 − 0.832050i
\(53\) − 7.34847i − 1.00939i −0.863298 − 0.504695i \(-0.831605\pi\)
0.863298 − 0.504695i \(-0.168395\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.79796 1.30931
\(57\) 0 0
\(58\) −1.73205 − 3.00000i −0.227429 − 0.393919i
\(59\) − 11.3137i − 1.47292i −0.676481 − 0.736460i \(-0.736496\pi\)
0.676481 − 0.736460i \(-0.263504\pi\)
\(60\) 0 0
\(61\) 13.8564i 1.77413i 0.461644 + 0.887066i \(0.347260\pi\)
−0.461644 + 0.887066i \(0.652740\pi\)
\(62\) 4.24264 − 2.44949i 0.538816 − 0.311086i
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) −1.41421 + 2.44949i −0.171499 + 0.297044i
\(69\) 0 0
\(70\) 0 0
\(71\) −14.6969 −1.74421 −0.872103 − 0.489323i \(-0.837244\pi\)
−0.872103 + 0.489323i \(0.837244\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.00000 + 6.92820i −0.458831 + 0.794719i
\(77\) − 9.79796i − 1.11658i
\(78\) 0 0
\(79\) − 3.46410i − 0.389742i −0.980829 − 0.194871i \(-0.937571\pi\)
0.980829 − 0.194871i \(-0.0624288\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.73205 − 1.00000i 0.191273 − 0.110432i
\(83\) −14.1421 −1.55230 −0.776151 − 0.630548i \(-0.782830\pi\)
−0.776151 + 0.630548i \(0.782830\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.79796 − 5.65685i 1.05654 − 0.609994i
\(87\) 0 0
\(88\) − 8.00000i − 0.852803i
\(89\) − 7.07107i − 0.749532i −0.927119 − 0.374766i \(-0.877723\pi\)
0.927119 − 0.374766i \(-0.122277\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 8.48528 + 4.89898i 0.884652 + 0.510754i
\(93\) 0 0
\(94\) 6.00000 − 3.46410i 0.618853 − 0.357295i
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 3.53553 + 6.12372i 0.357143 + 0.618590i
\(99\) 0 0
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 1800.2.m.c.899.7 8
3.2 odd 2 inner 1800.2.m.c.899.1 8
4.3 odd 2 7200.2.m.c.3599.6 8
5.2 odd 4 1800.2.b.c.251.2 4
5.3 odd 4 72.2.f.a.35.3 yes 4
5.4 even 2 inner 1800.2.m.c.899.2 8
8.3 odd 2 inner 1800.2.m.c.899.6 8
8.5 even 2 7200.2.m.c.3599.2 8
12.11 even 2 7200.2.m.c.3599.7 8
15.2 even 4 1800.2.b.c.251.3 4
15.8 even 4 72.2.f.a.35.2 yes 4
15.14 odd 2 inner 1800.2.m.c.899.8 8
20.3 even 4 288.2.f.a.143.1 4
20.7 even 4 7200.2.b.c.4751.3 4
20.19 odd 2 7200.2.m.c.3599.1 8
24.5 odd 2 7200.2.m.c.3599.3 8
24.11 even 2 inner 1800.2.m.c.899.4 8
40.3 even 4 72.2.f.a.35.1 ✓ 4
40.13 odd 4 288.2.f.a.143.4 4
40.19 odd 2 inner 1800.2.m.c.899.3 8
40.27 even 4 1800.2.b.c.251.4 4
40.29 even 2 7200.2.m.c.3599.5 8
40.37 odd 4 7200.2.b.c.4751.1 4
45.13 odd 12 648.2.l.a.107.2 4
45.23 even 12 648.2.l.a.107.1 4
45.38 even 12 648.2.l.c.539.2 4
45.43 odd 12 648.2.l.c.539.1 4
60.23 odd 4 288.2.f.a.143.3 4
60.47 odd 4 7200.2.b.c.4751.4 4
60.59 even 2 7200.2.m.c.3599.4 8
80.3 even 4 2304.2.c.i.2303.8 8
80.13 odd 4 2304.2.c.i.2303.6 8
80.43 even 4 2304.2.c.i.2303.3 8
80.53 odd 4 2304.2.c.i.2303.1 8
120.29 odd 2 7200.2.m.c.3599.8 8
120.53 even 4 288.2.f.a.143.2 4
120.59 even 2 inner 1800.2.m.c.899.5 8
120.77 even 4 7200.2.b.c.4751.2 4
120.83 odd 4 72.2.f.a.35.4 yes 4
120.107 odd 4 1800.2.b.c.251.1 4
180.23 odd 12 2592.2.p.c.431.1 4
180.43 even 12 2592.2.p.a.2159.2 4
180.83 odd 12 2592.2.p.a.2159.1 4
180.103 even 12 2592.2.p.c.431.2 4
240.53 even 4 2304.2.c.i.2303.5 8
240.83 odd 4 2304.2.c.i.2303.4 8
240.173 even 4 2304.2.c.i.2303.2 8
240.203 odd 4 2304.2.c.i.2303.7 8
360.13 odd 12 2592.2.p.a.431.1 4
360.43 even 12 648.2.l.a.539.2 4
360.83 odd 12 648.2.l.a.539.1 4
360.133 odd 12 2592.2.p.c.2159.1 4
360.173 even 12 2592.2.p.c.2159.2 4
360.203 odd 12 648.2.l.c.107.1 4
360.283 even 12 648.2.l.c.107.2 4
360.293 even 12 2592.2.p.a.431.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.f.a.35.1 ✓ 4 40.3 even 4
72.2.f.a.35.2 yes 4 15.8 even 4
72.2.f.a.35.3 yes 4 5.3 odd 4
72.2.f.a.35.4 yes 4 120.83 odd 4
288.2.f.a.143.1 4 20.3 even 4
288.2.f.a.143.2 4 120.53 even 4
288.2.f.a.143.3 4 60.23 odd 4
288.2.f.a.143.4 4 40.13 odd 4
648.2.l.a.107.1 4 45.23 even 12
648.2.l.a.107.2 4 45.13 odd 12
648.2.l.a.539.1 4 360.83 odd 12
648.2.l.a.539.2 4 360.43 even 12
648.2.l.c.107.1 4 360.203 odd 12
648.2.l.c.107.2 4 360.283 even 12
648.2.l.c.539.1 4 45.43 odd 12
648.2.l.c.539.2 4 45.38 even 12
1800.2.b.c.251.1 4 120.107 odd 4
1800.2.b.c.251.2 4 5.2 odd 4
1800.2.b.c.251.3 4 15.2 even 4
1800.2.b.c.251.4 4 40.27 even 4
1800.2.m.c.899.1 8 3.2 odd 2 inner
1800.2.m.c.899.2 8 5.4 even 2 inner
1800.2.m.c.899.3 8 40.19 odd 2 inner
1800.2.m.c.899.4 8 24.11 even 2 inner
1800.2.m.c.899.5 8 120.59 even 2 inner
1800.2.m.c.899.6 8 8.3 odd 2 inner
1800.2.m.c.899.7 8 1.1 even 1 trivial
1800.2.m.c.899.8 8 15.14 odd 2 inner
2304.2.c.i.2303.1 8 80.53 odd 4
2304.2.c.i.2303.2 8 240.173 even 4
2304.2.c.i.2303.3 8 80.43 even 4
2304.2.c.i.2303.4 8 240.83 odd 4
2304.2.c.i.2303.5 8 240.53 even 4
2304.2.c.i.2303.6 8 80.13 odd 4
2304.2.c.i.2303.7 8 240.203 odd 4
2304.2.c.i.2303.8 8 80.3 even 4
2592.2.p.a.431.1 4 360.13 odd 12
2592.2.p.a.431.2 4 360.293 even 12
2592.2.p.a.2159.1 4 180.83 odd 12
2592.2.p.a.2159.2 4 180.43 even 12
2592.2.p.c.431.1 4 180.23 odd 12
2592.2.p.c.431.2 4 180.103 even 12
2592.2.p.c.2159.1 4 360.133 odd 12
2592.2.p.c.2159.2 4 360.173 even 12
7200.2.b.c.4751.1 4 40.37 odd 4
7200.2.b.c.4751.2 4 120.77 even 4
7200.2.b.c.4751.3 4 20.7 even 4
7200.2.b.c.4751.4 4 60.47 odd 4
7200.2.m.c.3599.1 8 20.19 odd 2
7200.2.m.c.3599.2 8 8.5 even 2
7200.2.m.c.3599.3 8 24.5 odd 2
7200.2.m.c.3599.4 8 60.59 even 2
7200.2.m.c.3599.5 8 40.29 even 2
7200.2.m.c.3599.6 8 4.3 odd 2
7200.2.m.c.3599.7 8 12.11 even 2
7200.2.m.c.3599.8 8 120.29 odd 2