Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,2,Mod(35,72)] 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("72.35"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 35.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 72.35
Dual form 72.2.f.a.35.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.22474 - 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +2.44949 q^{5} -3.46410i q^{7} -2.82843i q^{8} +(-3.00000 - 1.73205i) q^{10} +2.82843i q^{11} +3.46410i q^{13} +(-2.44949 + 4.24264i) q^{14} +(-2.00000 + 3.46410i) q^{16} -1.41421i q^{17} -4.00000 q^{19} +(2.44949 + 4.24264i) q^{20} +(2.00000 - 3.46410i) q^{22} -4.89898 q^{23} +1.00000 q^{25} +(2.44949 - 4.24264i) q^{26} +(6.00000 - 3.46410i) q^{28} -2.44949 q^{29} +3.46410i q^{31} +(4.89898 - 2.82843i) q^{32} +(-1.00000 + 1.73205i) q^{34} -8.48528i q^{35} +(4.89898 + 2.82843i) q^{38} -6.92820i q^{40} -1.41421i q^{41} +8.00000 q^{43} +(-4.89898 + 2.82843i) q^{44} +(6.00000 + 3.46410i) q^{46} +4.89898 q^{47} -5.00000 q^{49} +(-1.22474 - 0.707107i) q^{50} +(-6.00000 + 3.46410i) q^{52} -7.34847 q^{53} +6.92820i q^{55} -9.79796 q^{56} +(3.00000 + 1.73205i) q^{58} +11.3137i q^{59} -13.8564i q^{61} +(2.44949 - 4.24264i) q^{62} -8.00000 q^{64} +8.48528i q^{65} -4.00000 q^{67} +(2.44949 - 1.41421i) q^{68} +(-6.00000 + 10.3923i) q^{70} +14.6969 q^{71} -4.00000 q^{73} +(-4.00000 - 6.92820i) q^{76} +9.79796 q^{77} -3.46410i q^{79} +(-4.89898 + 8.48528i) q^{80} +(-1.00000 + 1.73205i) q^{82} -14.1421i q^{83} -3.46410i q^{85} +(-9.79796 - 5.65685i) q^{86} +8.00000 q^{88} +7.07107i q^{89} +12.0000 q^{91} +(-4.89898 - 8.48528i) q^{92} +(-6.00000 - 3.46410i) q^{94} -9.79796 q^{95} +8.00000 q^{97} +(6.12372 + 3.53553i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 12 q^{10} - 8 q^{16} - 16 q^{19} + 8 q^{22} + 4 q^{25} + 24 q^{28} - 4 q^{34} + 32 q^{43} + 24 q^{46} - 20 q^{49} - 24 q^{52} + 12 q^{58} - 32 q^{64} - 16 q^{67} - 24 q^{70} - 16 q^{73}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22474 − 0.707107i −0.866025 − 0.500000i
\(3\) 0 0
\(4\) 1.00000 + 1.73205i 0.500000 + 0.866025i
\(5\) 2.44949 1.09545 0.547723 − 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) − 3.46410i − 1.30931i −0.755929 − 0.654654i \(-0.772814\pi\)
0.755929 − 0.654654i \(-0.227186\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) −3.00000 − 1.73205i −0.948683 − 0.547723i
\(11\) 2.82843i 0.852803i 0.904534 + 0.426401i \(0.140219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) −2.44949 + 4.24264i −0.654654 + 1.13389i
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) − 1.41421i − 0.342997i −0.985184 − 0.171499i \(-0.945139\pi\)
0.985184 − 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.44949 + 4.24264i 0.547723 + 0.948683i
\(21\) 0 0
\(22\) 2.00000 − 3.46410i 0.426401 − 0.738549i
\(23\) −4.89898 −1.02151 −0.510754 − 0.859727i \(-0.670634\pi\)
−0.510754 + 0.859727i \(0.670634\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.44949 − 4.24264i 0.480384 − 0.832050i
\(27\) 0 0
\(28\) 6.00000 − 3.46410i 1.13389 − 0.654654i
\(29\) −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 \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 4.89898 − 2.82843i 0.866025 − 0.500000i
\(33\) 0 0
\(34\) −1.00000 + 1.73205i −0.171499 + 0.297044i
\(35\) − 8.48528i − 1.43427i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 4.89898 + 2.82843i 0.794719 + 0.458831i
\(39\) 0 0
\(40\) − 6.92820i − 1.09545i
\(41\) − 1.41421i − 0.220863i −0.993884 − 0.110432i \(-0.964777\pi\)
0.993884 − 0.110432i \(-0.0352233\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.89898 + 2.82843i −0.738549 + 0.426401i
\(45\) 0 0
\(46\) 6.00000 + 3.46410i 0.884652 + 0.510754i
\(47\) 4.89898 0.714590 0.357295 − 0.933992i \(-0.383699\pi\)
0.357295 + 0.933992i \(0.383699\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) −1.22474 − 0.707107i −0.173205 − 0.100000i
\(51\) 0 0
\(52\) −6.00000 + 3.46410i −0.832050 + 0.480384i
\(53\) −7.34847 −1.00939 −0.504695 − 0.863298i \(-0.668395\pi\)
−0.504695 + 0.863298i \(0.668395\pi\)
\(54\) 0 0
\(55\) 6.92820i 0.934199i
\(56\) −9.79796 −1.30931
\(57\) 0 0
\(58\) 3.00000 + 1.73205i 0.393919 + 0.227429i
\(59\) 11.3137i 1.47292i 0.676481 + 0.736460i \(0.263504\pi\)
−0.676481 + 0.736460i \(0.736496\pi\)
\(60\) 0 0
\(61\) − 13.8564i − 1.77413i −0.461644 − 0.887066i \(-0.652740\pi\)
0.461644 − 0.887066i \(-0.347260\pi\)
\(62\) 2.44949 − 4.24264i 0.311086 − 0.538816i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 8.48528i 1.05247i
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.44949 − 1.41421i 0.297044 − 0.171499i
\(69\) 0 0
\(70\) −6.00000 + 10.3923i −0.717137 + 1.24212i
\(71\) 14.6969 1.74421 0.872103 − 0.489323i \(-0.162756\pi\)
0.872103 + 0.489323i \(0.162756\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.00000 − 6.92820i −0.458831 − 0.794719i
\(77\) 9.79796 1.11658
\(78\) 0 0
\(79\) − 3.46410i − 0.389742i −0.980829 − 0.194871i \(-0.937571\pi\)
0.980829 − 0.194871i \(-0.0624288\pi\)
\(80\) −4.89898 + 8.48528i −0.547723 + 0.948683i
\(81\) 0 0
\(82\) −1.00000 + 1.73205i −0.110432 + 0.191273i
\(83\) − 14.1421i − 1.55230i −0.630548 − 0.776151i \(-0.717170\pi\)
0.630548 − 0.776151i \(-0.282830\pi\)
\(84\) 0 0
\(85\) − 3.46410i − 0.375735i
\(86\) −9.79796 − 5.65685i −1.05654 − 0.609994i
\(87\) 0 0
\(88\) 8.00000 0.852803
\(89\) 7.07107i 0.749532i 0.927119 + 0.374766i \(0.122277\pi\)
−0.927119 + 0.374766i \(0.877723\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) −4.89898 − 8.48528i −0.510754 − 0.884652i
\(93\) 0 0
\(94\) −6.00000 − 3.46410i −0.618853 − 0.357295i
\(95\) −9.79796 −1.00525
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 6.12372 + 3.53553i 0.618590 + 0.357143i
\(99\) 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 72.2.f.a.35.1 ✓ 4
3.2 odd 2 inner 72.2.f.a.35.4 yes 4
4.3 odd 2 288.2.f.a.143.4 4
5.2 odd 4 1800.2.m.c.899.6 8
5.3 odd 4 1800.2.m.c.899.3 8
5.4 even 2 1800.2.b.c.251.4 4
8.3 odd 2 inner 72.2.f.a.35.3 yes 4
8.5 even 2 288.2.f.a.143.1 4
9.2 odd 6 648.2.l.a.539.1 4
9.4 even 3 648.2.l.c.107.2 4
9.5 odd 6 648.2.l.c.107.1 4
9.7 even 3 648.2.l.a.539.2 4
12.11 even 2 288.2.f.a.143.2 4
15.2 even 4 1800.2.m.c.899.4 8
15.8 even 4 1800.2.m.c.899.5 8
15.14 odd 2 1800.2.b.c.251.1 4
16.3 odd 4 2304.2.c.i.2303.1 8
16.5 even 4 2304.2.c.i.2303.8 8
16.11 odd 4 2304.2.c.i.2303.6 8
16.13 even 4 2304.2.c.i.2303.3 8
20.3 even 4 7200.2.m.c.3599.5 8
20.7 even 4 7200.2.m.c.3599.2 8
20.19 odd 2 7200.2.b.c.4751.1 4
24.5 odd 2 288.2.f.a.143.3 4
24.11 even 2 inner 72.2.f.a.35.2 yes 4
36.7 odd 6 2592.2.p.c.2159.1 4
36.11 even 6 2592.2.p.c.2159.2 4
36.23 even 6 2592.2.p.a.431.2 4
36.31 odd 6 2592.2.p.a.431.1 4
40.3 even 4 1800.2.m.c.899.2 8
40.13 odd 4 7200.2.m.c.3599.1 8
40.19 odd 2 1800.2.b.c.251.2 4
40.27 even 4 1800.2.m.c.899.7 8
40.29 even 2 7200.2.b.c.4751.3 4
40.37 odd 4 7200.2.m.c.3599.6 8
48.5 odd 4 2304.2.c.i.2303.4 8
48.11 even 4 2304.2.c.i.2303.2 8
48.29 odd 4 2304.2.c.i.2303.7 8
48.35 even 4 2304.2.c.i.2303.5 8
60.23 odd 4 7200.2.m.c.3599.8 8
60.47 odd 4 7200.2.m.c.3599.3 8
60.59 even 2 7200.2.b.c.4751.2 4
72.5 odd 6 2592.2.p.c.431.1 4
72.11 even 6 648.2.l.c.539.2 4
72.13 even 6 2592.2.p.c.431.2 4
72.29 odd 6 2592.2.p.a.2159.1 4
72.43 odd 6 648.2.l.c.539.1 4
72.59 even 6 648.2.l.a.107.1 4
72.61 even 6 2592.2.p.a.2159.2 4
72.67 odd 6 648.2.l.a.107.2 4
120.29 odd 2 7200.2.b.c.4751.4 4
120.53 even 4 7200.2.m.c.3599.4 8
120.59 even 2 1800.2.b.c.251.3 4
120.77 even 4 7200.2.m.c.3599.7 8
120.83 odd 4 1800.2.m.c.899.8 8
120.107 odd 4 1800.2.m.c.899.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.f.a.35.1 ✓ 4 1.1 even 1 trivial
72.2.f.a.35.2 yes 4 24.11 even 2 inner
72.2.f.a.35.3 yes 4 8.3 odd 2 inner
72.2.f.a.35.4 yes 4 3.2 odd 2 inner
288.2.f.a.143.1 4 8.5 even 2
288.2.f.a.143.2 4 12.11 even 2
288.2.f.a.143.3 4 24.5 odd 2
288.2.f.a.143.4 4 4.3 odd 2
648.2.l.a.107.1 4 72.59 even 6
648.2.l.a.107.2 4 72.67 odd 6
648.2.l.a.539.1 4 9.2 odd 6
648.2.l.a.539.2 4 9.7 even 3
648.2.l.c.107.1 4 9.5 odd 6
648.2.l.c.107.2 4 9.4 even 3
648.2.l.c.539.1 4 72.43 odd 6
648.2.l.c.539.2 4 72.11 even 6
1800.2.b.c.251.1 4 15.14 odd 2
1800.2.b.c.251.2 4 40.19 odd 2
1800.2.b.c.251.3 4 120.59 even 2
1800.2.b.c.251.4 4 5.4 even 2
1800.2.m.c.899.1 8 120.107 odd 4
1800.2.m.c.899.2 8 40.3 even 4
1800.2.m.c.899.3 8 5.3 odd 4
1800.2.m.c.899.4 8 15.2 even 4
1800.2.m.c.899.5 8 15.8 even 4
1800.2.m.c.899.6 8 5.2 odd 4
1800.2.m.c.899.7 8 40.27 even 4
1800.2.m.c.899.8 8 120.83 odd 4
2304.2.c.i.2303.1 8 16.3 odd 4
2304.2.c.i.2303.2 8 48.11 even 4
2304.2.c.i.2303.3 8 16.13 even 4
2304.2.c.i.2303.4 8 48.5 odd 4
2304.2.c.i.2303.5 8 48.35 even 4
2304.2.c.i.2303.6 8 16.11 odd 4
2304.2.c.i.2303.7 8 48.29 odd 4
2304.2.c.i.2303.8 8 16.5 even 4
2592.2.p.a.431.1 4 36.31 odd 6
2592.2.p.a.431.2 4 36.23 even 6
2592.2.p.a.2159.1 4 72.29 odd 6
2592.2.p.a.2159.2 4 72.61 even 6
2592.2.p.c.431.1 4 72.5 odd 6
2592.2.p.c.431.2 4 72.13 even 6
2592.2.p.c.2159.1 4 36.7 odd 6
2592.2.p.c.2159.2 4 36.11 even 6
7200.2.b.c.4751.1 4 20.19 odd 2
7200.2.b.c.4751.2 4 60.59 even 2
7200.2.b.c.4751.3 4 40.29 even 2
7200.2.b.c.4751.4 4 120.29 odd 2
7200.2.m.c.3599.1 8 40.13 odd 4
7200.2.m.c.3599.2 8 20.7 even 4
7200.2.m.c.3599.3 8 60.47 odd 4
7200.2.m.c.3599.4 8 120.53 even 4
7200.2.m.c.3599.5 8 20.3 even 4
7200.2.m.c.3599.6 8 40.37 odd 4
7200.2.m.c.3599.7 8 120.77 even 4
7200.2.m.c.3599.8 8 60.23 odd 4