Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1232,2,Mod(1,1232)] 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("1232.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.83756952902\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1232.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.23607 q^{3} +3.23607 q^{5} -1.00000 q^{7} +7.47214 q^{9} -1.00000 q^{11} +1.23607 q^{13} +10.4721 q^{15} -6.47214 q^{17} +2.76393 q^{19} -3.23607 q^{21} -4.00000 q^{23} +5.47214 q^{25} +14.4721 q^{27} -4.47214 q^{29} -2.00000 q^{31} -3.23607 q^{33} -3.23607 q^{35} -10.9443 q^{37} +4.00000 q^{39} +6.47214 q^{41} +1.52786 q^{43} +24.1803 q^{45} +2.00000 q^{47} +1.00000 q^{49} -20.9443 q^{51} -0.472136 q^{53} -3.23607 q^{55} +8.94427 q^{57} -7.23607 q^{59} -5.23607 q^{61} -7.47214 q^{63} +4.00000 q^{65} +15.4164 q^{67} -12.9443 q^{69} +2.47214 q^{71} -4.94427 q^{73} +17.7082 q^{75} +1.00000 q^{77} +24.4164 q^{81} -10.1803 q^{83} -20.9443 q^{85} -14.4721 q^{87} +10.0000 q^{89} -1.23607 q^{91} -6.47214 q^{93} +8.94427 q^{95} +3.52786 q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 6 q^{9} - 2 q^{11} - 2 q^{13} + 12 q^{15} - 4 q^{17} + 10 q^{19} - 2 q^{21} - 8 q^{23} + 2 q^{25} + 20 q^{27} - 4 q^{31} - 2 q^{33} - 2 q^{35} - 4 q^{37} + 8 q^{39} + 4 q^{41}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 3.23607 1.86834 0.934172 − 0.356822i \(-0.116140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) 0 0
\(5\) 3.23607 1.44721 0.723607 − 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 7.47214 2.49071
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.23607 0.342824 0.171412 − 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 0 0
\(15\) 10.4721 2.70389
\(16\) 0 0
\(17\) −6.47214 −1.56972 −0.784862 − 0.619671i \(-0.787266\pi\)
−0.784862 + 0.619671i \(0.787266\pi\)
\(18\) 0 0
\(19\) 2.76393 0.634089 0.317045 − 0.948411i \(-0.397309\pi\)
0.317045 + 0.948411i \(0.397309\pi\)
\(20\) 0 0
\(21\) −3.23607 −0.706168
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 14.4721 2.78516
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −3.23607 −0.563327
\(34\) 0 0
\(35\) −3.23607 −0.546995
\(36\) 0 0
\(37\) −10.9443 −1.79923 −0.899614 − 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 6.47214 1.01078 0.505389 − 0.862892i \(-0.331349\pi\)
0.505389 + 0.862892i \(0.331349\pi\)
\(42\) 0 0
\(43\) 1.52786 0.232997 0.116499 − 0.993191i \(-0.462833\pi\)
0.116499 + 0.993191i \(0.462833\pi\)
\(44\) 0 0
\(45\) 24.1803 3.60459
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −20.9443 −2.93278
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 − 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −3.23607 −0.436351
\(56\) 0 0
\(57\) 8.94427 1.18470
\(58\) 0 0
\(59\) −7.23607 −0.942056 −0.471028 − 0.882118i \(-0.656117\pi\)
−0.471028 + 0.882118i \(0.656117\pi\)
\(60\) 0 0
\(61\) −5.23607 −0.670410 −0.335205 − 0.942145i \(-0.608806\pi\)
−0.335205 + 0.942145i \(0.608806\pi\)
\(62\) 0 0
\(63\) −7.47214 −0.941401
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 15.4164 1.88341 0.941707 − 0.336434i \(-0.109221\pi\)
0.941707 + 0.336434i \(0.109221\pi\)
\(68\) 0 0
\(69\) −12.9443 −1.55831
\(70\) 0 0
\(71\) 2.47214 0.293389 0.146694 − 0.989182i \(-0.453137\pi\)
0.146694 + 0.989182i \(0.453137\pi\)
\(72\) 0 0
\(73\) −4.94427 −0.578683 −0.289342 − 0.957226i \(-0.593436\pi\)
−0.289342 + 0.957226i \(0.593436\pi\)
\(74\) 0 0
\(75\) 17.7082 2.04477
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) −10.1803 −1.11744 −0.558719 − 0.829357i \(-0.688707\pi\)
−0.558719 + 0.829357i \(0.688707\pi\)
\(84\) 0 0
\(85\) −20.9443 −2.27173
\(86\) 0 0
\(87\) −14.4721 −1.55158
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 0 0
\(93\) −6.47214 −0.671129
\(94\) 0 0
\(95\) 8.94427 0.917663
\(96\) 0 0
\(97\) 3.52786 0.358200 0.179100 − 0.983831i \(-0.442681\pi\)
0.179100 + 0.983831i \(0.442681\pi\)
\(98\) 0 0
\(99\) −7.47214 −0.750978
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 1232.2.a.p.1.2 2
4.3 odd 2 154.2.a.d.1.1 ✓ 2
7.6 odd 2 8624.2.a.bf.1.1 2
8.3 odd 2 4928.2.a.bt.1.2 2
8.5 even 2 4928.2.a.bk.1.1 2
12.11 even 2 1386.2.a.m.1.1 2
20.3 even 4 3850.2.c.q.1849.1 4
20.7 even 4 3850.2.c.q.1849.4 4
20.19 odd 2 3850.2.a.bj.1.2 2
28.3 even 6 1078.2.e.n.177.1 4
28.11 odd 6 1078.2.e.q.177.2 4
28.19 even 6 1078.2.e.n.67.1 4
28.23 odd 6 1078.2.e.q.67.2 4
28.27 even 2 1078.2.a.w.1.2 2
44.43 even 2 1694.2.a.l.1.1 2
84.83 odd 2 9702.2.a.cu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.1 ✓ 2 4.3 odd 2
1078.2.a.w.1.2 2 28.27 even 2
1078.2.e.n.67.1 4 28.19 even 6
1078.2.e.n.177.1 4 28.3 even 6
1078.2.e.q.67.2 4 28.23 odd 6
1078.2.e.q.177.2 4 28.11 odd 6
1232.2.a.p.1.2 2 1.1 even 1 trivial
1386.2.a.m.1.1 2 12.11 even 2
1694.2.a.l.1.1 2 44.43 even 2
3850.2.a.bj.1.2 2 20.19 odd 2
3850.2.c.q.1849.1 4 20.3 even 4
3850.2.c.q.1849.4 4 20.7 even 4
4928.2.a.bk.1.1 2 8.5 even 2
4928.2.a.bt.1.2 2 8.3 odd 2
8624.2.a.bf.1.1 2 7.6 odd 2
9702.2.a.cu.1.2 2 84.83 odd 2