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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1792,2,Mod(897,1792)] 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("1792.897"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-2,0,6,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 897.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1792.897
Dual form 1792.2.b.d.897.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{5} -1.00000 q^{7} +3.00000 q^{9} +4.00000i q^{11} -2.00000i q^{13} -6.00000 q^{17} +8.00000i q^{19} +1.00000 q^{25} -6.00000i q^{29} -8.00000 q^{31} -2.00000i q^{35} -2.00000i q^{37} -2.00000 q^{41} +4.00000i q^{43} +6.00000i q^{45} +8.00000 q^{47} +1.00000 q^{49} +6.00000i q^{53} -8.00000 q^{55} +6.00000i q^{61} -3.00000 q^{63} +4.00000 q^{65} -4.00000i q^{67} -8.00000 q^{71} -10.0000 q^{73} -4.00000i q^{77} -16.0000 q^{79} +9.00000 q^{81} +8.00000i q^{83} -12.0000i q^{85} +6.00000 q^{89} +2.00000i q^{91} -16.0000 q^{95} -6.00000 q^{97} +12.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} + 6 q^{9} - 12 q^{17} + 2 q^{25} - 16 q^{31} - 4 q^{41} + 16 q^{47} + 2 q^{49} - 16 q^{55} - 6 q^{63} + 8 q^{65} - 16 q^{71} - 20 q^{73} - 32 q^{79} + 18 q^{81} + 12 q^{89} - 32 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 − 0.557086i \(-0.811919\pi\)
0.830455 − 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.00000i − 0.338062i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 6.00000i 0.894427i
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 6.00000i 0.768221i 0.923287 + 0.384111i \(0.125492\pi\)
−0.923287 + 0.384111i \(0.874508\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.00000i − 0.455842i
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 − 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 0 0
\(85\) − 12.0000i − 1.30158i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000i 0.209657i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.0000 −1.64157
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 12.0000i 1.20605i
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 1792.2.b.d.897.2 2
4.3 odd 2 1792.2.b.i.897.2 2
8.3 odd 2 1792.2.b.i.897.1 2
8.5 even 2 inner 1792.2.b.d.897.1 2
16.3 odd 4 56.2.a.a.1.1 ✓ 1
16.5 even 4 448.2.a.e.1.1 1
16.11 odd 4 448.2.a.d.1.1 1
16.13 even 4 112.2.a.b.1.1 1
48.5 odd 4 4032.2.a.bk.1.1 1
48.11 even 4 4032.2.a.bb.1.1 1
48.29 odd 4 1008.2.a.d.1.1 1
48.35 even 4 504.2.a.c.1.1 1
80.3 even 4 1400.2.g.g.449.2 2
80.13 odd 4 2800.2.g.p.449.1 2
80.19 odd 4 1400.2.a.g.1.1 1
80.29 even 4 2800.2.a.p.1.1 1
80.67 even 4 1400.2.g.g.449.1 2
80.77 odd 4 2800.2.g.p.449.2 2
112.3 even 12 392.2.i.d.177.1 2
112.13 odd 4 784.2.a.e.1.1 1
112.19 even 12 392.2.i.d.361.1 2
112.27 even 4 3136.2.a.q.1.1 1
112.45 odd 12 784.2.i.g.177.1 2
112.51 odd 12 392.2.i.c.361.1 2
112.61 odd 12 784.2.i.g.753.1 2
112.67 odd 12 392.2.i.c.177.1 2
112.69 odd 4 3136.2.a.p.1.1 1
112.83 even 4 392.2.a.d.1.1 1
112.93 even 12 784.2.i.e.753.1 2
112.109 even 12 784.2.i.e.177.1 2
176.131 even 4 6776.2.a.g.1.1 1
208.51 odd 4 9464.2.a.c.1.1 1
336.83 odd 4 3528.2.a.x.1.1 1
336.125 even 4 7056.2.a.bo.1.1 1
336.131 odd 12 3528.2.s.e.361.1 2
336.179 even 12 3528.2.s.t.3313.1 2
336.227 odd 12 3528.2.s.e.3313.1 2
336.275 even 12 3528.2.s.t.361.1 2
560.419 even 4 9800.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.a.1.1 ✓ 1 16.3 odd 4
112.2.a.b.1.1 1 16.13 even 4
392.2.a.d.1.1 1 112.83 even 4
392.2.i.c.177.1 2 112.67 odd 12
392.2.i.c.361.1 2 112.51 odd 12
392.2.i.d.177.1 2 112.3 even 12
392.2.i.d.361.1 2 112.19 even 12
448.2.a.d.1.1 1 16.11 odd 4
448.2.a.e.1.1 1 16.5 even 4
504.2.a.c.1.1 1 48.35 even 4
784.2.a.e.1.1 1 112.13 odd 4
784.2.i.e.177.1 2 112.109 even 12
784.2.i.e.753.1 2 112.93 even 12
784.2.i.g.177.1 2 112.45 odd 12
784.2.i.g.753.1 2 112.61 odd 12
1008.2.a.d.1.1 1 48.29 odd 4
1400.2.a.g.1.1 1 80.19 odd 4
1400.2.g.g.449.1 2 80.67 even 4
1400.2.g.g.449.2 2 80.3 even 4
1792.2.b.d.897.1 2 8.5 even 2 inner
1792.2.b.d.897.2 2 1.1 even 1 trivial
1792.2.b.i.897.1 2 8.3 odd 2
1792.2.b.i.897.2 2 4.3 odd 2
2800.2.a.p.1.1 1 80.29 even 4
2800.2.g.p.449.1 2 80.13 odd 4
2800.2.g.p.449.2 2 80.77 odd 4
3136.2.a.p.1.1 1 112.69 odd 4
3136.2.a.q.1.1 1 112.27 even 4
3528.2.a.x.1.1 1 336.83 odd 4
3528.2.s.e.361.1 2 336.131 odd 12
3528.2.s.e.3313.1 2 336.227 odd 12
3528.2.s.t.361.1 2 336.275 even 12
3528.2.s.t.3313.1 2 336.179 even 12
4032.2.a.bb.1.1 1 48.11 even 4
4032.2.a.bk.1.1 1 48.5 odd 4
6776.2.a.g.1.1 1 176.131 even 4
7056.2.a.bo.1.1 1 336.125 even 4
9464.2.a.c.1.1 1 208.51 odd 4
9800.2.a.u.1.1 1 560.419 even 4