Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(881,882)] 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("882.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.5
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 882.881
Dual form 882.2.d.a.881.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -4.18154 q^{5} -1.00000i q^{8} -4.18154i q^{10} +3.00000i q^{11} -2.44949i q^{13} +1.00000 q^{16} +1.01461 q^{17} -1.01461i q^{19} +4.18154 q^{20} -3.00000 q^{22} -4.24264i q^{23} +12.4853 q^{25} +2.44949 q^{26} -1.24264i q^{29} -5.61642i q^{31} +1.00000i q^{32} +1.01461i q^{34} +8.24264 q^{37} +1.01461 q^{38} +4.18154i q^{40} +2.02922 q^{41} +8.24264 q^{43} -3.00000i q^{44} +4.24264 q^{46} +1.01461 q^{47} +12.4853i q^{50} +2.44949i q^{52} -1.24264i q^{53} -12.5446i q^{55} +1.24264 q^{58} +11.5300 q^{59} +5.91359i q^{61} +5.61642 q^{62} -1.00000 q^{64} +10.2426i q^{65} -10.0000 q^{67} -1.01461 q^{68} -10.2426i q^{71} -8.36308i q^{73} +8.24264i q^{74} +1.01461i q^{76} -11.2426 q^{79} -4.18154 q^{80} +2.02922i q^{82} -3.16693 q^{83} -4.24264 q^{85} +8.24264i q^{86} +3.00000 q^{88} -10.3923 q^{89} +4.24264i q^{92} +1.01461i q^{94} +4.24264i q^{95} -3.76127i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 8 q^{16} - 24 q^{22} + 32 q^{25} + 32 q^{37} + 32 q^{43} - 24 q^{58} - 8 q^{64} - 80 q^{67} - 56 q^{79} + 24 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −4.18154 −1.87004 −0.935021 − 0.354593i \(-0.884620\pi\)
−0.935021 + 0.354593i \(0.884620\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) − 4.18154i − 1.32232i
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) − 2.44949i − 0.679366i −0.940540 − 0.339683i \(-0.889680\pi\)
0.940540 − 0.339683i \(-0.110320\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.01461 0.246080 0.123040 − 0.992402i \(-0.460736\pi\)
0.123040 + 0.992402i \(0.460736\pi\)
\(18\) 0 0
\(19\) − 1.01461i − 0.232768i −0.993204 − 0.116384i \(-0.962870\pi\)
0.993204 − 0.116384i \(-0.0371303\pi\)
\(20\) 4.18154 0.935021
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) − 4.24264i − 0.884652i −0.896854 − 0.442326i \(-0.854153\pi\)
0.896854 − 0.442326i \(-0.145847\pi\)
\(24\) 0 0
\(25\) 12.4853 2.49706
\(26\) 2.44949 0.480384
\(27\) 0 0
\(28\) 0 0
\(29\) − 1.24264i − 0.230753i −0.993322 − 0.115376i \(-0.963193\pi\)
0.993322 − 0.115376i \(-0.0368074\pi\)
\(30\) 0 0
\(31\) − 5.61642i − 1.00874i −0.863488 − 0.504369i \(-0.831725\pi\)
0.863488 − 0.504369i \(-0.168275\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 1.01461i 0.174005i
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24264 1.35508 0.677541 − 0.735485i \(-0.263046\pi\)
0.677541 + 0.735485i \(0.263046\pi\)
\(38\) 1.01461 0.164592
\(39\) 0 0
\(40\) 4.18154i 0.661160i
\(41\) 2.02922 0.316912 0.158456 − 0.987366i \(-0.449348\pi\)
0.158456 + 0.987366i \(0.449348\pi\)
\(42\) 0 0
\(43\) 8.24264 1.25699 0.628495 − 0.777813i \(-0.283671\pi\)
0.628495 + 0.777813i \(0.283671\pi\)
\(44\) − 3.00000i − 0.452267i
\(45\) 0 0
\(46\) 4.24264 0.625543
\(47\) 1.01461 0.147996 0.0739982 − 0.997258i \(-0.476424\pi\)
0.0739982 + 0.997258i \(0.476424\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 12.4853i 1.76569i
\(51\) 0 0
\(52\) 2.44949i 0.339683i
\(53\) − 1.24264i − 0.170690i −0.996351 − 0.0853449i \(-0.972801\pi\)
0.996351 − 0.0853449i \(-0.0271992\pi\)
\(54\) 0 0
\(55\) − 12.5446i − 1.69152i
\(56\) 0 0
\(57\) 0 0
\(58\) 1.24264 0.163167
\(59\) 11.5300 1.50108 0.750540 − 0.660825i \(-0.229794\pi\)
0.750540 + 0.660825i \(0.229794\pi\)
\(60\) 0 0
\(61\) 5.91359i 0.757158i 0.925569 + 0.378579i \(0.123587\pi\)
−0.925569 + 0.378579i \(0.876413\pi\)
\(62\) 5.61642 0.713286
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 10.2426i 1.27044i
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −1.01461 −0.123040
\(69\) 0 0
\(70\) 0 0
\(71\) − 10.2426i − 1.21558i −0.794099 − 0.607789i \(-0.792057\pi\)
0.794099 − 0.607789i \(-0.207943\pi\)
\(72\) 0 0
\(73\) − 8.36308i − 0.978825i −0.872053 − 0.489412i \(-0.837211\pi\)
0.872053 − 0.489412i \(-0.162789\pi\)
\(74\) 8.24264i 0.958188i
\(75\) 0 0
\(76\) 1.01461i 0.116384i
\(77\) 0 0
\(78\) 0 0
\(79\) −11.2426 −1.26490 −0.632448 − 0.774603i \(-0.717950\pi\)
−0.632448 + 0.774603i \(0.717950\pi\)
\(80\) −4.18154 −0.467510
\(81\) 0 0
\(82\) 2.02922i 0.224090i
\(83\) −3.16693 −0.347616 −0.173808 − 0.984780i \(-0.555607\pi\)
−0.173808 + 0.984780i \(0.555607\pi\)
\(84\) 0 0
\(85\) −4.24264 −0.460179
\(86\) 8.24264i 0.888827i
\(87\) 0 0
\(88\) 3.00000 0.319801
\(89\) −10.3923 −1.10158 −0.550791 − 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.24264i 0.442326i
\(93\) 0 0
\(94\) 1.01461i 0.104649i
\(95\) 4.24264i 0.435286i
\(96\) 0 0
\(97\) − 3.76127i − 0.381900i −0.981600 − 0.190950i \(-0.938843\pi\)
0.981600 − 0.190950i \(-0.0611568\pi\)
\(98\) 0 0
\(99\) 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 882.2.d.a.881.5 8
3.2 odd 2 inner 882.2.d.a.881.4 8
4.3 odd 2 7056.2.k.f.881.1 8
7.2 even 3 126.2.k.a.17.4 yes 8
7.3 odd 6 126.2.k.a.89.1 yes 8
7.4 even 3 882.2.k.a.215.2 8
7.5 odd 6 882.2.k.a.521.3 8
7.6 odd 2 inner 882.2.d.a.881.8 8
12.11 even 2 7056.2.k.f.881.8 8
21.2 odd 6 126.2.k.a.17.1 ✓ 8
21.5 even 6 882.2.k.a.521.2 8
21.11 odd 6 882.2.k.a.215.3 8
21.17 even 6 126.2.k.a.89.4 yes 8
21.20 even 2 inner 882.2.d.a.881.1 8
28.3 even 6 1008.2.bt.c.593.1 8
28.23 odd 6 1008.2.bt.c.17.4 8
28.27 even 2 7056.2.k.f.881.7 8
35.2 odd 12 3150.2.bp.e.899.4 8
35.3 even 12 3150.2.bp.b.1349.4 8
35.9 even 6 3150.2.bf.a.1151.2 8
35.17 even 12 3150.2.bp.e.1349.1 8
35.23 odd 12 3150.2.bp.b.899.1 8
35.24 odd 6 3150.2.bf.a.1601.4 8
63.2 odd 6 1134.2.t.e.1025.4 8
63.16 even 3 1134.2.t.e.1025.1 8
63.23 odd 6 1134.2.l.f.269.1 8
63.31 odd 6 1134.2.t.e.593.4 8
63.38 even 6 1134.2.l.f.215.2 8
63.52 odd 6 1134.2.l.f.215.3 8
63.58 even 3 1134.2.l.f.269.4 8
63.59 even 6 1134.2.t.e.593.1 8
84.23 even 6 1008.2.bt.c.17.1 8
84.59 odd 6 1008.2.bt.c.593.4 8
84.83 odd 2 7056.2.k.f.881.2 8
105.2 even 12 3150.2.bp.b.899.4 8
105.17 odd 12 3150.2.bp.b.1349.1 8
105.23 even 12 3150.2.bp.e.899.1 8
105.38 odd 12 3150.2.bp.e.1349.4 8
105.44 odd 6 3150.2.bf.a.1151.4 8
105.59 even 6 3150.2.bf.a.1601.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 21.2 odd 6
126.2.k.a.17.4 yes 8 7.2 even 3
126.2.k.a.89.1 yes 8 7.3 odd 6
126.2.k.a.89.4 yes 8 21.17 even 6
882.2.d.a.881.1 8 21.20 even 2 inner
882.2.d.a.881.4 8 3.2 odd 2 inner
882.2.d.a.881.5 8 1.1 even 1 trivial
882.2.d.a.881.8 8 7.6 odd 2 inner
882.2.k.a.215.2 8 7.4 even 3
882.2.k.a.215.3 8 21.11 odd 6
882.2.k.a.521.2 8 21.5 even 6
882.2.k.a.521.3 8 7.5 odd 6
1008.2.bt.c.17.1 8 84.23 even 6
1008.2.bt.c.17.4 8 28.23 odd 6
1008.2.bt.c.593.1 8 28.3 even 6
1008.2.bt.c.593.4 8 84.59 odd 6
1134.2.l.f.215.2 8 63.38 even 6
1134.2.l.f.215.3 8 63.52 odd 6
1134.2.l.f.269.1 8 63.23 odd 6
1134.2.l.f.269.4 8 63.58 even 3
1134.2.t.e.593.1 8 63.59 even 6
1134.2.t.e.593.4 8 63.31 odd 6
1134.2.t.e.1025.1 8 63.16 even 3
1134.2.t.e.1025.4 8 63.2 odd 6
3150.2.bf.a.1151.2 8 35.9 even 6
3150.2.bf.a.1151.4 8 105.44 odd 6
3150.2.bf.a.1601.2 8 105.59 even 6
3150.2.bf.a.1601.4 8 35.24 odd 6
3150.2.bp.b.899.1 8 35.23 odd 12
3150.2.bp.b.899.4 8 105.2 even 12
3150.2.bp.b.1349.1 8 105.17 odd 12
3150.2.bp.b.1349.4 8 35.3 even 12
3150.2.bp.e.899.1 8 105.23 even 12
3150.2.bp.e.899.4 8 35.2 odd 12
3150.2.bp.e.1349.1 8 35.17 even 12
3150.2.bp.e.1349.4 8 105.38 odd 12
7056.2.k.f.881.1 8 4.3 odd 2
7056.2.k.f.881.2 8 84.83 odd 2
7056.2.k.f.881.7 8 28.27 even 2
7056.2.k.f.881.8 8 12.11 even 2