Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2800,2,Mod(449,2800)] 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("2800.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,6,0,8,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.3581125660\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2800.449
Dual form 2800.2.g.p.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{7} +3.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} -6.00000i q^{17} +8.00000 q^{19} -6.00000 q^{29} -8.00000 q^{31} -2.00000i q^{37} +2.00000 q^{41} -4.00000i q^{43} +8.00000i q^{47} -1.00000 q^{49} -6.00000i q^{53} -6.00000 q^{61} +3.00000i q^{63} +4.00000i q^{67} +8.00000 q^{71} -10.0000i q^{73} +4.00000i q^{77} +16.0000 q^{79} +9.00000 q^{81} +8.00000i q^{83} +6.00000 q^{89} +2.00000 q^{91} -6.00000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9} + 8 q^{11} + 16 q^{19} - 12 q^{29} - 16 q^{31} + 4 q^{41} - 2 q^{49} - 12 q^{61} + 16 q^{71} + 32 q^{79} + 18 q^{81} + 12 q^{89} + 4 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 0
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 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
\(17\) − 6.00000i − 1.45521i −0.685994 − 0.727607i \(-0.740633\pi\)
0.685994 − 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 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
\(35\) 0 0
\(36\) 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 \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 − 0.304997i \(-0.901344\pi\)
0.952353 − 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 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 \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 3.00000i 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 − 0.585206i \(-0.801014\pi\)
0.810885 − 0.585206i \(-0.198986\pi\)
\(74\) 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 \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 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
\(85\) 0 0
\(86\) 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
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.00000i − 0.609208i −0.952479 − 0.304604i \(-0.901476\pi\)
0.952479 − 0.304604i \(-0.0985241\pi\)
\(98\) 0 0
\(99\) 12.0000 1.20605
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 2800.2.g.p.449.2 2
4.3 odd 2 1400.2.g.g.449.1 2
5.2 odd 4 2800.2.a.p.1.1 1
5.3 odd 4 112.2.a.b.1.1 1
5.4 even 2 inner 2800.2.g.p.449.1 2
15.8 even 4 1008.2.a.d.1.1 1
20.3 even 4 56.2.a.a.1.1 ✓ 1
20.7 even 4 1400.2.a.g.1.1 1
20.19 odd 2 1400.2.g.g.449.2 2
35.3 even 12 784.2.i.g.177.1 2
35.13 even 4 784.2.a.e.1.1 1
35.18 odd 12 784.2.i.e.177.1 2
35.23 odd 12 784.2.i.e.753.1 2
35.33 even 12 784.2.i.g.753.1 2
40.3 even 4 448.2.a.d.1.1 1
40.13 odd 4 448.2.a.e.1.1 1
60.23 odd 4 504.2.a.c.1.1 1
80.3 even 4 1792.2.b.i.897.1 2
80.13 odd 4 1792.2.b.d.897.1 2
80.43 even 4 1792.2.b.i.897.2 2
80.53 odd 4 1792.2.b.d.897.2 2
105.83 odd 4 7056.2.a.bo.1.1 1
120.53 even 4 4032.2.a.bk.1.1 1
120.83 odd 4 4032.2.a.bb.1.1 1
140.3 odd 12 392.2.i.d.177.1 2
140.23 even 12 392.2.i.c.361.1 2
140.27 odd 4 9800.2.a.u.1.1 1
140.83 odd 4 392.2.a.d.1.1 1
140.103 odd 12 392.2.i.d.361.1 2
140.123 even 12 392.2.i.c.177.1 2
220.43 odd 4 6776.2.a.g.1.1 1
260.103 even 4 9464.2.a.c.1.1 1
280.13 even 4 3136.2.a.p.1.1 1
280.83 odd 4 3136.2.a.q.1.1 1
420.23 odd 12 3528.2.s.t.361.1 2
420.83 even 4 3528.2.a.x.1.1 1
420.143 even 12 3528.2.s.e.3313.1 2
420.263 odd 12 3528.2.s.t.3313.1 2
420.383 even 12 3528.2.s.e.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.a.1.1 ✓ 1 20.3 even 4
112.2.a.b.1.1 1 5.3 odd 4
392.2.a.d.1.1 1 140.83 odd 4
392.2.i.c.177.1 2 140.123 even 12
392.2.i.c.361.1 2 140.23 even 12
392.2.i.d.177.1 2 140.3 odd 12
392.2.i.d.361.1 2 140.103 odd 12
448.2.a.d.1.1 1 40.3 even 4
448.2.a.e.1.1 1 40.13 odd 4
504.2.a.c.1.1 1 60.23 odd 4
784.2.a.e.1.1 1 35.13 even 4
784.2.i.e.177.1 2 35.18 odd 12
784.2.i.e.753.1 2 35.23 odd 12
784.2.i.g.177.1 2 35.3 even 12
784.2.i.g.753.1 2 35.33 even 12
1008.2.a.d.1.1 1 15.8 even 4
1400.2.a.g.1.1 1 20.7 even 4
1400.2.g.g.449.1 2 4.3 odd 2
1400.2.g.g.449.2 2 20.19 odd 2
1792.2.b.d.897.1 2 80.13 odd 4
1792.2.b.d.897.2 2 80.53 odd 4
1792.2.b.i.897.1 2 80.3 even 4
1792.2.b.i.897.2 2 80.43 even 4
2800.2.a.p.1.1 1 5.2 odd 4
2800.2.g.p.449.1 2 5.4 even 2 inner
2800.2.g.p.449.2 2 1.1 even 1 trivial
3136.2.a.p.1.1 1 280.13 even 4
3136.2.a.q.1.1 1 280.83 odd 4
3528.2.a.x.1.1 1 420.83 even 4
3528.2.s.e.361.1 2 420.383 even 12
3528.2.s.e.3313.1 2 420.143 even 12
3528.2.s.t.361.1 2 420.23 odd 12
3528.2.s.t.3313.1 2 420.263 odd 12
4032.2.a.bb.1.1 1 120.83 odd 4
4032.2.a.bk.1.1 1 120.53 even 4
6776.2.a.g.1.1 1 220.43 odd 4
7056.2.a.bo.1.1 1 105.83 odd 4
9464.2.a.c.1.1 1 260.103 even 4
9800.2.a.u.1.1 1 140.27 odd 4