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Properties

Label 448.2.m.c.337.1
Level $448$
Weight $2$
Character 448.337
Analytic conductor $3.577$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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] = [448,2,Mod(113,448)] 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("448.113"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.m (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.214798336.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 337.1
Root \(-1.08003 - 0.912978i\) of defining polynomial
Character \(\chi\) \(=\) 448.337
Dual form 448.2.m.c.113.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.47209 - 1.47209i) q^{3} +(0.353863 - 0.353863i) q^{5} -1.00000i q^{7} +1.33411i q^{9} +(-0.334112 + 0.334112i) q^{11} +(-3.17982 - 3.17982i) q^{13} -1.04184 q^{15} +1.29227 q^{17} +(-2.63216 - 2.63216i) q^{19} +(-1.47209 + 1.47209i) q^{21} +2.61007i q^{23} +4.74956i q^{25} +(-2.45234 + 2.45234i) q^{27} +(-6.98602 - 6.98602i) q^{29} -8.35964 q^{31} +0.983687 q^{33} +(-0.353863 - 0.353863i) q^{35} +(-3.32013 + 3.32013i) q^{37} +9.36197i q^{39} -9.93254i q^{41} +(7.61241 - 7.61241i) q^{43} +(0.472092 + 0.472092i) q^{45} +4.59609 q^{47} -1.00000 q^{49} +(-1.90235 - 1.90235i) q^{51} +(-3.94418 + 3.94418i) q^{53} +0.236459i q^{55} +7.74956i q^{57} +(6.39570 - 6.39570i) q^{59} +(4.35386 + 4.35386i) q^{61} +1.33411 q^{63} -2.25044 q^{65} +(9.97204 + 9.97204i) q^{67} +(3.84227 - 3.84227i) q^{69} -7.62395i q^{71} +0.556593i q^{73} +(6.99179 - 6.99179i) q^{75} +(0.334112 + 0.334112i) q^{77} +1.48923 q^{79} +11.2225 q^{81} +(-1.47209 - 1.47209i) q^{83} +(0.457288 - 0.457288i) q^{85} +20.5681i q^{87} -12.2922i q^{89} +(-3.17982 + 3.17982i) q^{91} +(12.3062 + 12.3062i) q^{93} -1.86285 q^{95} +5.50078 q^{97} +(-0.445742 - 0.445742i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + 8 q^{15} + 24 q^{17} + 12 q^{19} - 12 q^{27} - 16 q^{29} - 16 q^{31} - 24 q^{33} + 4 q^{35} + 16 q^{37} + 32 q^{43} - 8 q^{45} - 24 q^{47} - 8 q^{49} - 8 q^{51} - 8 q^{53} + 28 q^{59} + 28 q^{61}+ \cdots - 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −1.47209 − 1.47209i −0.849913 − 0.849913i 0.140209 − 0.990122i \(-0.455223\pi\)
−0.990122 + 0.140209i \(0.955223\pi\)
\(4\) 0 0
\(5\) 0.353863 − 0.353863i 0.158252 − 0.158252i −0.623540 − 0.781792i \(-0.714306\pi\)
0.781792 + 0.623540i \(0.214306\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 1.33411i 0.444704i
\(10\) 0 0
\(11\) −0.334112 + 0.334112i −0.100738 + 0.100738i −0.755680 − 0.654941i \(-0.772693\pi\)
0.654941 + 0.755680i \(0.272693\pi\)
\(12\) 0 0
\(13\) −3.17982 − 3.17982i −0.881923 − 0.881923i 0.111807 − 0.993730i \(-0.464336\pi\)
−0.993730 + 0.111807i \(0.964336\pi\)
\(14\) 0 0
\(15\) −1.04184 −0.269001
\(16\) 0 0
\(17\) 1.29227 0.313423 0.156711 − 0.987644i \(-0.449911\pi\)
0.156711 + 0.987644i \(0.449911\pi\)
\(18\) 0 0
\(19\) −2.63216 − 2.63216i −0.603859 − 0.603859i 0.337476 − 0.941334i \(-0.390427\pi\)
−0.941334 + 0.337476i \(0.890427\pi\)
\(20\) 0 0
\(21\) −1.47209 + 1.47209i −0.321237 + 0.321237i
\(22\) 0 0
\(23\) 2.61007i 0.544238i 0.962264 + 0.272119i \(0.0877244\pi\)
−0.962264 + 0.272119i \(0.912276\pi\)
\(24\) 0 0
\(25\) 4.74956i 0.949912i
\(26\) 0 0
\(27\) −2.45234 + 2.45234i −0.471953 + 0.471953i
\(28\) 0 0
\(29\) −6.98602 − 6.98602i −1.29727 − 1.29727i −0.930188 − 0.367084i \(-0.880356\pi\)
−0.367084 − 0.930188i \(-0.619644\pi\)
\(30\) 0 0
\(31\) −8.35964 −1.50143 −0.750717 − 0.660623i \(-0.770292\pi\)
−0.750717 + 0.660623i \(0.770292\pi\)
\(32\) 0 0
\(33\) 0.983687 0.171238
\(34\) 0 0
\(35\) −0.353863 − 0.353863i −0.0598137 − 0.0598137i
\(36\) 0 0
\(37\) −3.32013 + 3.32013i −0.545827 + 0.545827i −0.925231 − 0.379404i \(-0.876129\pi\)
0.379404 + 0.925231i \(0.376129\pi\)
\(38\) 0 0
\(39\) 9.36197i 1.49912i
\(40\) 0 0
\(41\) − 9.93254i − 1.55120i −0.631223 − 0.775601i \(-0.717447\pi\)
0.631223 − 0.775601i \(-0.282553\pi\)
\(42\) 0 0
\(43\) 7.61241 − 7.61241i 1.16088 − 1.16088i 0.176598 − 0.984283i \(-0.443491\pi\)
0.984283 − 0.176598i \(-0.0565094\pi\)
\(44\) 0 0
\(45\) 0.472092 + 0.472092i 0.0703754 + 0.0703754i
\(46\) 0 0
\(47\) 4.59609 0.670409 0.335205 − 0.942145i \(-0.391195\pi\)
0.335205 + 0.942145i \(0.391195\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −1.90235 − 1.90235i −0.266382 − 0.266382i
\(52\) 0 0
\(53\) −3.94418 + 3.94418i −0.541775 + 0.541775i −0.924049 − 0.382274i \(-0.875141\pi\)
0.382274 + 0.924049i \(0.375141\pi\)
\(54\) 0 0
\(55\) 0.236459i 0.0318842i
\(56\) 0 0
\(57\) 7.74956i 1.02645i
\(58\) 0 0
\(59\) 6.39570 − 6.39570i 0.832649 − 0.832649i −0.155229 − 0.987878i \(-0.549612\pi\)
0.987878 + 0.155229i \(0.0496116\pi\)
\(60\) 0 0
\(61\) 4.35386 + 4.35386i 0.557455 + 0.557455i 0.928582 − 0.371127i \(-0.121029\pi\)
−0.371127 + 0.928582i \(0.621029\pi\)
\(62\) 0 0
\(63\) 1.33411 0.168082
\(64\) 0 0
\(65\) −2.25044 −0.279132
\(66\) 0 0
\(67\) 9.97204 + 9.97204i 1.21828 + 1.21828i 0.968234 + 0.250045i \(0.0804453\pi\)
0.250045 + 0.968234i \(0.419555\pi\)
\(68\) 0 0
\(69\) 3.84227 − 3.84227i 0.462555 − 0.462555i
\(70\) 0 0
\(71\) − 7.62395i − 0.904797i −0.891816 − 0.452398i \(-0.850569\pi\)
0.891816 − 0.452398i \(-0.149431\pi\)
\(72\) 0 0
\(73\) 0.556593i 0.0651443i 0.999469 + 0.0325721i \(0.0103699\pi\)
−0.999469 + 0.0325721i \(0.989630\pi\)
\(74\) 0 0
\(75\) 6.99179 − 6.99179i 0.807343 − 0.807343i
\(76\) 0 0
\(77\) 0.334112 + 0.334112i 0.0380756 + 0.0380756i
\(78\) 0 0
\(79\) 1.48923 0.167552 0.0837758 − 0.996485i \(-0.473302\pi\)
0.0837758 + 0.996485i \(0.473302\pi\)
\(80\) 0 0
\(81\) 11.2225 1.24694
\(82\) 0 0
\(83\) −1.47209 − 1.47209i −0.161583 − 0.161583i 0.621685 − 0.783268i \(-0.286449\pi\)
−0.783268 + 0.621685i \(0.786449\pi\)
\(84\) 0 0
\(85\) 0.457288 − 0.457288i 0.0495998 − 0.0495998i
\(86\) 0 0
\(87\) 20.5681i 2.20514i
\(88\) 0 0
\(89\) − 12.2922i − 1.30297i −0.758662 − 0.651484i \(-0.774147\pi\)
0.758662 − 0.651484i \(-0.225853\pi\)
\(90\) 0 0
\(91\) −3.17982 + 3.17982i −0.333335 + 0.333335i
\(92\) 0 0
\(93\) 12.3062 + 12.3062i 1.27609 + 1.27609i
\(94\) 0 0
\(95\) −1.86285 −0.191124
\(96\) 0 0
\(97\) 5.50078 0.558519 0.279260 − 0.960216i \(-0.409911\pi\)
0.279260 + 0.960216i \(0.409911\pi\)
\(98\) 0 0
\(99\) −0.445742 − 0.445742i −0.0447988 − 0.0447988i
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 448.2.m.c.337.1 8
4.3 odd 2 112.2.m.c.29.2 ✓ 8
8.3 odd 2 896.2.m.e.673.1 8
8.5 even 2 896.2.m.f.673.4 8
16.3 odd 4 896.2.m.e.225.1 8
16.5 even 4 inner 448.2.m.c.113.1 8
16.11 odd 4 112.2.m.c.85.2 yes 8
16.13 even 4 896.2.m.f.225.4 8
28.3 even 6 784.2.x.j.765.1 16
28.11 odd 6 784.2.x.k.765.1 16
28.19 even 6 784.2.x.j.557.4 16
28.23 odd 6 784.2.x.k.557.4 16
28.27 even 2 784.2.m.g.589.2 8
32.5 even 8 7168.2.a.bd.1.7 8
32.11 odd 8 7168.2.a.bc.1.7 8
32.21 even 8 7168.2.a.bd.1.2 8
32.27 odd 8 7168.2.a.bc.1.2 8
112.11 odd 12 784.2.x.k.373.4 16
112.27 even 4 784.2.m.g.197.2 8
112.59 even 12 784.2.x.j.373.4 16
112.75 even 12 784.2.x.j.165.1 16
112.107 odd 12 784.2.x.k.165.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.c.29.2 ✓ 8 4.3 odd 2
112.2.m.c.85.2 yes 8 16.11 odd 4
448.2.m.c.113.1 8 16.5 even 4 inner
448.2.m.c.337.1 8 1.1 even 1 trivial
784.2.m.g.197.2 8 112.27 even 4
784.2.m.g.589.2 8 28.27 even 2
784.2.x.j.165.1 16 112.75 even 12
784.2.x.j.373.4 16 112.59 even 12
784.2.x.j.557.4 16 28.19 even 6
784.2.x.j.765.1 16 28.3 even 6
784.2.x.k.165.1 16 112.107 odd 12
784.2.x.k.373.4 16 112.11 odd 12
784.2.x.k.557.4 16 28.23 odd 6
784.2.x.k.765.1 16 28.11 odd 6
896.2.m.e.225.1 8 16.3 odd 4
896.2.m.e.673.1 8 8.3 odd 2
896.2.m.f.225.4 8 16.13 even 4
896.2.m.f.673.4 8 8.5 even 2
7168.2.a.bc.1.2 8 32.27 odd 8
7168.2.a.bc.1.7 8 32.11 odd 8
7168.2.a.bd.1.2 8 32.21 even 8
7168.2.a.bd.1.7 8 32.5 even 8