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Properties

Label 7056.2.k.f.881.7
Level $7056$
Weight $2$
Character 7056.881
Analytic conductor $56.342$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7056,2,Mod(881,7056)] 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("7056.881"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0, 0,0,0,0,0,0,32,0,0,0,0,0,-32,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,80,0,0,0,0,0,0,0,0,0,0,0,56,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,80,0,0,0,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(121)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(56.3424436662\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.7
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 7056.881
Dual form 7056.2.k.f.881.8

$q$-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)
 
\(f(q)\) \(=\) \(q+4.18154 q^{5} -3.00000i q^{11} +2.44949i q^{13} -1.01461 q^{17} -1.01461i q^{19} +4.24264i q^{23} +12.4853 q^{25} -1.24264i q^{29} -5.61642i q^{31} +8.24264 q^{37} -2.02922 q^{41} -8.24264 q^{43} +1.01461 q^{47} -1.24264i q^{53} -12.5446i q^{55} +11.5300 q^{59} -5.91359i q^{61} +10.2426i q^{65} +10.0000 q^{67} +10.2426i q^{71} +8.36308i q^{73} +11.2426 q^{79} -3.16693 q^{83} -4.24264 q^{85} +10.3923 q^{89} -4.24264i q^{95} +3.76127i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{25} + 32 q^{37} - 32 q^{43} + 80 q^{67} + 56 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(4\) 0 0
\(5\) 4.18154 1.87004 0.935021 − 0.354593i \(-0.115380\pi\)
0.935021 + 0.354593i \(0.115380\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.00000i − 0.904534i −0.891883 − 0.452267i \(-0.850615\pi\)
0.891883 − 0.452267i \(-0.149385\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.01461 −0.246080 −0.123040 − 0.992402i \(-0.539264\pi\)
−0.123040 + 0.992402i \(0.539264\pi\)
\(18\) 0 0
\(19\) − 1.01461i − 0.232768i −0.993204 − 0.116384i \(-0.962870\pi\)
0.993204 − 0.116384i \(-0.0371303\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.24264i 0.884652i 0.896854 + 0.442326i \(0.145847\pi\)
−0.896854 + 0.442326i \(0.854153\pi\)
\(24\) 0 0
\(25\) 12.4853 2.49706
\(26\) 0 0
\(27\) 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 \(-0.168275\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.24264 1.35508 0.677541 − 0.735485i \(-0.263046\pi\)
0.677541 + 0.735485i \(0.263046\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.02922 −0.316912 −0.158456 − 0.987366i \(-0.550652\pi\)
−0.158456 + 0.987366i \(0.550652\pi\)
\(42\) 0 0
\(43\) −8.24264 −1.25699 −0.628495 − 0.777813i \(-0.716329\pi\)
−0.628495 + 0.777813i \(0.716329\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.01461 0.147996 0.0739982 − 0.997258i \(-0.476424\pi\)
0.0739982 + 0.997258i \(0.476424\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 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
\(58\) 0 0
\(59\) 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 \(-0.876413\pi\)
0.925569 − 0.378579i \(-0.123587\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.2426i 1.27044i
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2426i 1.21558i 0.794099 + 0.607789i \(0.207943\pi\)
−0.794099 + 0.607789i \(0.792057\pi\)
\(72\) 0 0
\(73\) 8.36308i 0.978825i 0.872053 + 0.489412i \(0.162789\pi\)
−0.872053 + 0.489412i \(0.837211\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.2426 1.26490 0.632448 − 0.774603i \(-0.282050\pi\)
0.632448 + 0.774603i \(0.282050\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −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
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.3923 1.10158 0.550791 − 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 4.24264i − 0.435286i
\(96\) 0 0
\(97\) 3.76127i 0.381900i 0.981600 + 0.190950i \(0.0611568\pi\)
−0.981600 + 0.190950i \(0.938843\pi\)
\(98\) 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 7056.2.k.f.881.7 8
3.2 odd 2 inner 7056.2.k.f.881.2 8
4.3 odd 2 882.2.d.a.881.8 8
7.4 even 3 1008.2.bt.c.593.1 8
7.5 odd 6 1008.2.bt.c.17.4 8
7.6 odd 2 inner 7056.2.k.f.881.1 8
12.11 even 2 882.2.d.a.881.1 8
21.5 even 6 1008.2.bt.c.17.1 8
21.11 odd 6 1008.2.bt.c.593.4 8
21.20 even 2 inner 7056.2.k.f.881.8 8
28.3 even 6 882.2.k.a.215.2 8
28.11 odd 6 126.2.k.a.89.1 yes 8
28.19 even 6 126.2.k.a.17.4 yes 8
28.23 odd 6 882.2.k.a.521.3 8
28.27 even 2 882.2.d.a.881.5 8
84.11 even 6 126.2.k.a.89.4 yes 8
84.23 even 6 882.2.k.a.521.2 8
84.47 odd 6 126.2.k.a.17.1 ✓ 8
84.59 odd 6 882.2.k.a.215.3 8
84.83 odd 2 882.2.d.a.881.4 8
140.19 even 6 3150.2.bf.a.1151.2 8
140.39 odd 6 3150.2.bf.a.1601.4 8
140.47 odd 12 3150.2.bp.e.899.4 8
140.67 even 12 3150.2.bp.e.1349.1 8
140.103 odd 12 3150.2.bp.b.899.1 8
140.123 even 12 3150.2.bp.b.1349.4 8
252.11 even 6 1134.2.l.f.215.2 8
252.47 odd 6 1134.2.t.e.1025.4 8
252.67 odd 6 1134.2.t.e.593.4 8
252.95 even 6 1134.2.t.e.593.1 8
252.103 even 6 1134.2.l.f.269.4 8
252.131 odd 6 1134.2.l.f.269.1 8
252.151 odd 6 1134.2.l.f.215.3 8
252.187 even 6 1134.2.t.e.1025.1 8
420.47 even 12 3150.2.bp.b.899.4 8
420.179 even 6 3150.2.bf.a.1601.2 8
420.263 odd 12 3150.2.bp.e.1349.4 8
420.299 odd 6 3150.2.bf.a.1151.4 8
420.347 odd 12 3150.2.bp.b.1349.1 8
420.383 even 12 3150.2.bp.e.899.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.k.a.17.1 ✓ 8 84.47 odd 6
126.2.k.a.17.4 yes 8 28.19 even 6
126.2.k.a.89.1 yes 8 28.11 odd 6
126.2.k.a.89.4 yes 8 84.11 even 6
882.2.d.a.881.1 8 12.11 even 2
882.2.d.a.881.4 8 84.83 odd 2
882.2.d.a.881.5 8 28.27 even 2
882.2.d.a.881.8 8 4.3 odd 2
882.2.k.a.215.2 8 28.3 even 6
882.2.k.a.215.3 8 84.59 odd 6
882.2.k.a.521.2 8 84.23 even 6
882.2.k.a.521.3 8 28.23 odd 6
1008.2.bt.c.17.1 8 21.5 even 6
1008.2.bt.c.17.4 8 7.5 odd 6
1008.2.bt.c.593.1 8 7.4 even 3
1008.2.bt.c.593.4 8 21.11 odd 6
1134.2.l.f.215.2 8 252.11 even 6
1134.2.l.f.215.3 8 252.151 odd 6
1134.2.l.f.269.1 8 252.131 odd 6
1134.2.l.f.269.4 8 252.103 even 6
1134.2.t.e.593.1 8 252.95 even 6
1134.2.t.e.593.4 8 252.67 odd 6
1134.2.t.e.1025.1 8 252.187 even 6
1134.2.t.e.1025.4 8 252.47 odd 6
3150.2.bf.a.1151.2 8 140.19 even 6
3150.2.bf.a.1151.4 8 420.299 odd 6
3150.2.bf.a.1601.2 8 420.179 even 6
3150.2.bf.a.1601.4 8 140.39 odd 6
3150.2.bp.b.899.1 8 140.103 odd 12
3150.2.bp.b.899.4 8 420.47 even 12
3150.2.bp.b.1349.1 8 420.347 odd 12
3150.2.bp.b.1349.4 8 140.123 even 12
3150.2.bp.e.899.1 8 420.383 even 12
3150.2.bp.e.899.4 8 140.47 odd 12
3150.2.bp.e.1349.1 8 140.67 even 12
3150.2.bp.e.1349.4 8 420.263 odd 12
7056.2.k.f.881.1 8 7.6 odd 2 inner
7056.2.k.f.881.2 8 3.2 odd 2 inner
7056.2.k.f.881.7 8 1.1 even 1 trivial
7056.2.k.f.881.8 8 21.20 even 2 inner