Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1856,2,Mod(1217,1856)] 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("1856.1217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1856, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-2,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8202346151\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1217.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1856.1217
Dual form 1856.2.e.b.1217.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000 q^{5} -2.00000 q^{7} +2.00000 q^{9} +5.00000i q^{11} +1.00000 q^{13} +1.00000i q^{15} -2.00000i q^{17} -4.00000i q^{19} +2.00000i q^{21} -6.00000 q^{23} -4.00000 q^{25} -5.00000i q^{27} +(-5.00000 - 2.00000i) q^{29} +5.00000i q^{31} +5.00000 q^{33} +2.00000 q^{35} -8.00000i q^{37} -1.00000i q^{39} -10.0000i q^{41} +9.00000i q^{43} -2.00000 q^{45} +3.00000i q^{47} -3.00000 q^{49} -2.00000 q^{51} +1.00000 q^{53} -5.00000i q^{55} -4.00000 q^{57} -10.0000 q^{59} +10.0000i q^{61} -4.00000 q^{63} -1.00000 q^{65} -8.00000 q^{67} +6.00000i q^{69} -8.00000 q^{71} +16.0000i q^{73} +4.00000i q^{75} -10.0000i q^{77} -1.00000i q^{79} +1.00000 q^{81} -14.0000 q^{83} +2.00000i q^{85} +(-2.00000 + 5.00000i) q^{87} +14.0000i q^{89} -2.00000 q^{91} +5.00000 q^{93} +4.00000i q^{95} -2.00000i q^{97} +10.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{13} - 12 q^{23} - 8 q^{25} - 10 q^{29} + 10 q^{33} + 4 q^{35} - 4 q^{45} - 6 q^{49} - 4 q^{51} + 2 q^{53} - 8 q^{57} - 20 q^{59} - 8 q^{63} - 2 q^{65} - 16 q^{67}+ \cdots + 10 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i −0.957427 − 0.288675i \(-0.906785\pi\)
0.957427 − 0.288675i \(-0.0932147\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 − 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 5.00000i 1.50756i 0.657129 + 0.753778i \(0.271771\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 1.00000i 0.258199i
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) − 5.00000i − 0.962250i
\(28\) 0 0
\(29\) −5.00000 − 2.00000i −0.928477 − 0.371391i
\(30\) 0 0
\(31\) 5.00000i 0.898027i 0.893525 + 0.449013i \(0.148224\pi\)
−0.893525 + 0.449013i \(0.851776\pi\)
\(32\) 0 0
\(33\) 5.00000 0.870388
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) − 1.00000i − 0.160128i
\(40\) 0 0
\(41\) − 10.0000i − 1.56174i −0.624695 − 0.780869i \(-0.714777\pi\)
0.624695 − 0.780869i \(-0.285223\pi\)
\(42\) 0 0
\(43\) 9.00000i 1.37249i 0.727372 + 0.686244i \(0.240742\pi\)
−0.727372 + 0.686244i \(0.759258\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) − 5.00000i − 0.674200i
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i 0.768221 + 0.640184i \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 4.00000i 0.461880i
\(76\) 0 0
\(77\) − 10.0000i − 1.13961i
\(78\) 0 0
\(79\) − 1.00000i − 0.112509i −0.998416 − 0.0562544i \(-0.982084\pi\)
0.998416 − 0.0562544i \(-0.0179158\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 2.00000i 0.216930i
\(86\) 0 0
\(87\) −2.00000 + 5.00000i −0.214423 + 0.536056i
\(88\) 0 0
\(89\) 14.0000i 1.48400i 0.670402 + 0.741999i \(0.266122\pi\)
−0.670402 + 0.741999i \(0.733878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) 4.00000i 0.410391i
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 10.0000i 1.00504i
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 1856.2.e.b.1217.1 2
4.3 odd 2 1856.2.e.d.1217.2 2
8.3 odd 2 464.2.e.c.289.1 2
8.5 even 2 58.2.b.a.57.2 yes 2
24.5 odd 2 522.2.d.a.289.1 2
24.11 even 2 4176.2.o.d.289.1 2
29.28 even 2 inner 1856.2.e.b.1217.2 2
40.13 odd 4 1450.2.d.c.1449.2 2
40.29 even 2 1450.2.c.a.1101.1 2
40.37 odd 4 1450.2.d.b.1449.1 2
116.115 odd 2 1856.2.e.d.1217.1 2
232.115 odd 2 464.2.e.c.289.2 2
232.133 odd 4 1682.2.a.g.1.1 1
232.157 odd 4 1682.2.a.c.1.1 1
232.173 even 2 58.2.b.a.57.1 ✓ 2
696.173 odd 2 522.2.d.a.289.2 2
696.347 even 2 4176.2.o.d.289.2 2
1160.173 odd 4 1450.2.d.b.1449.2 2
1160.637 odd 4 1450.2.d.c.1449.1 2
1160.869 even 2 1450.2.c.a.1101.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.b.a.57.1 ✓ 2 232.173 even 2
58.2.b.a.57.2 yes 2 8.5 even 2
464.2.e.c.289.1 2 8.3 odd 2
464.2.e.c.289.2 2 232.115 odd 2
522.2.d.a.289.1 2 24.5 odd 2
522.2.d.a.289.2 2 696.173 odd 2
1450.2.c.a.1101.1 2 40.29 even 2
1450.2.c.a.1101.2 2 1160.869 even 2
1450.2.d.b.1449.1 2 40.37 odd 4
1450.2.d.b.1449.2 2 1160.173 odd 4
1450.2.d.c.1449.1 2 1160.637 odd 4
1450.2.d.c.1449.2 2 40.13 odd 4
1682.2.a.c.1.1 1 232.157 odd 4
1682.2.a.g.1.1 1 232.133 odd 4
1856.2.e.b.1217.1 2 1.1 even 1 trivial
1856.2.e.b.1217.2 2 29.28 even 2 inner
1856.2.e.d.1217.1 2 116.115 odd 2
1856.2.e.d.1217.2 2 4.3 odd 2
4176.2.o.d.289.1 2 24.11 even 2
4176.2.o.d.289.2 2 696.347 even 2