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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [484,2,Mod(9,484)] 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("484.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(484, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 484 = 2^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 484.e (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-1,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.86475945783\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.18530015625.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 9x^{6} - 17x^{5} + 89x^{4} + 136x^{3} + 576x^{2} + 512x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{5}]$

Embedding invariants

Embedding label 245.1
Root \(0.733075 + 2.25617i\) of defining polynomial
Character \(\chi\) \(=\) 484.245
Dual form 484.2.e.g.81.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.733075 - 2.25617i) q^{3} +(-3.53725 - 2.56996i) q^{5} +(-2.12587 + 1.54453i) q^{9} +(-3.20521 + 9.86463i) q^{15} +1.62772 q^{23} +(4.36234 + 13.4259i) q^{25} +(-0.714488 - 0.519106i) q^{27} +(-8.99372 + 6.53432i) q^{31} +(1.58119 - 4.86641i) q^{37} +11.4891 q^{45} +(-3.70820 - 11.4127i) q^{47} +(5.66312 + 4.11450i) q^{49} +(-4.85410 + 3.52671i) q^{53} +(3.20521 - 9.86463i) q^{59} -15.1168 q^{67} +(-1.19324 - 3.67242i) q^{69} +(-12.8321 - 9.32310i) q^{71} +(27.0933 - 19.6844i) q^{75} +(-3.08345 + 9.48988i) q^{81} -9.86141 q^{89} +(21.3356 + 15.5012i) q^{93} +(13.8478 - 10.0610i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 3 q^{5} - 11 q^{9} + 15 q^{15} + 36 q^{23} - 11 q^{25} - 19 q^{27} - 5 q^{31} + 7 q^{37} + 24 q^{47} + 14 q^{49} - 12 q^{53} - 15 q^{59} - 52 q^{67} - 21 q^{69} - 3 q^{71} + 44 q^{75} - 26 q^{81}+ \cdots + 17 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).

\(n\) \(243\) \(365\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

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
\(3\) −0.733075 − 2.25617i −0.423241 − 1.30260i −0.904668 − 0.426116i \(-0.859881\pi\)
0.481427 − 0.876486i \(-0.340119\pi\)
\(4\) 0 0
\(5\) −3.53725 − 2.56996i −1.58191 − 1.14932i −0.914469 − 0.404656i \(-0.867391\pi\)
−0.667437 − 0.744666i \(-0.732609\pi\)
\(6\) 0 0
\(7\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(8\) 0 0
\(9\) −2.12587 + 1.54453i −0.708623 + 0.514845i
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(14\) 0 0
\(15\) −3.20521 + 9.86463i −0.827582 + 2.54704i
\(16\) 0 0
\(17\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(18\) 0 0
\(19\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.62772 0.339403 0.169701 − 0.985496i \(-0.445720\pi\)
0.169701 + 0.985496i \(0.445720\pi\)
\(24\) 0 0
\(25\) 4.36234 + 13.4259i 0.872469 + 2.68518i
\(26\) 0 0
\(27\) −0.714488 − 0.519106i −0.137503 − 0.0999020i
\(28\) 0 0
\(29\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(30\) 0 0
\(31\) −8.99372 + 6.53432i −1.61532 + 1.17360i −0.773623 + 0.633646i \(0.781558\pi\)
−0.841696 + 0.539952i \(0.818442\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.58119 − 4.86641i 0.259946 − 0.800033i −0.732868 − 0.680370i \(-0.761819\pi\)
0.992815 − 0.119662i \(-0.0381811\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 11.4891 1.71270
\(46\) 0 0
\(47\) −3.70820 − 11.4127i −0.540897 − 1.66471i −0.730550 − 0.682859i \(-0.760736\pi\)
0.189653 − 0.981851i \(-0.439264\pi\)
\(48\) 0 0
\(49\) 5.66312 + 4.11450i 0.809017 + 0.587785i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.85410 + 3.52671i −0.666762 + 0.484431i −0.868940 − 0.494918i \(-0.835198\pi\)
0.202178 + 0.979349i \(0.435198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.20521 − 9.86463i 0.417283 − 1.28426i −0.492910 − 0.870080i \(-0.664067\pi\)
0.910193 − 0.414185i \(-0.135933\pi\)
\(60\) 0 0
\(61\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −15.1168 −1.84682 −0.923408 − 0.383819i \(-0.874609\pi\)
−0.923408 + 0.383819i \(0.874609\pi\)
\(68\) 0 0
\(69\) −1.19324 − 3.67242i −0.143649 − 0.442107i
\(70\) 0 0
\(71\) −12.8321 − 9.32310i −1.52290 − 1.10645i −0.960029 − 0.279901i \(-0.909698\pi\)
−0.562867 − 0.826548i \(-0.690302\pi\)
\(72\) 0 0
\(73\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(74\) 0 0
\(75\) 27.0933 − 19.6844i 3.12846 − 2.27296i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(80\) 0 0
\(81\) −3.08345 + 9.48988i −0.342605 + 1.05443i
\(82\) 0 0
\(83\) 0 0 0.587785 − 0.809017i \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.86141 −1.04531 −0.522654 − 0.852545i \(-0.675058\pi\)
−0.522654 + 0.852545i \(0.675058\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 21.3356 + 15.5012i 2.21240 + 1.60740i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.8478 − 10.0610i 1.40603 − 1.02154i 0.412149 − 0.911117i \(-0.364778\pi\)
0.993884 − 0.110426i \(-0.0352215\pi\)
\(98\) 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 484.2.e.g.245.1 8
11.2 odd 10 484.2.a.d.1.1 ✓ 2
11.3 even 5 inner 484.2.e.g.269.2 8
11.4 even 5 inner 484.2.e.g.81.1 8
11.5 even 5 inner 484.2.e.g.9.2 8
11.6 odd 10 inner 484.2.e.g.9.2 8
11.7 odd 10 inner 484.2.e.g.81.1 8
11.8 odd 10 inner 484.2.e.g.269.2 8
11.9 even 5 484.2.a.d.1.1 ✓ 2
11.10 odd 2 CM 484.2.e.g.245.1 8
33.2 even 10 4356.2.a.n.1.1 2
33.20 odd 10 4356.2.a.n.1.1 2
44.31 odd 10 1936.2.a.s.1.2 2
44.35 even 10 1936.2.a.s.1.2 2
88.13 odd 10 7744.2.a.bx.1.2 2
88.35 even 10 7744.2.a.cn.1.1 2
88.53 even 10 7744.2.a.bx.1.2 2
88.75 odd 10 7744.2.a.cn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
484.2.a.d.1.1 ✓ 2 11.2 odd 10
484.2.a.d.1.1 ✓ 2 11.9 even 5
484.2.e.g.9.2 8 11.5 even 5 inner
484.2.e.g.9.2 8 11.6 odd 10 inner
484.2.e.g.81.1 8 11.4 even 5 inner
484.2.e.g.81.1 8 11.7 odd 10 inner
484.2.e.g.245.1 8 1.1 even 1 trivial
484.2.e.g.245.1 8 11.10 odd 2 CM
484.2.e.g.269.2 8 11.3 even 5 inner
484.2.e.g.269.2 8 11.8 odd 10 inner
1936.2.a.s.1.2 2 44.31 odd 10
1936.2.a.s.1.2 2 44.35 even 10
4356.2.a.n.1.1 2 33.2 even 10
4356.2.a.n.1.1 2 33.20 odd 10
7744.2.a.bx.1.2 2 88.13 odd 10
7744.2.a.bx.1.2 2 88.53 even 10
7744.2.a.cn.1.1 2 88.35 even 10
7744.2.a.cn.1.1 2 88.75 odd 10