Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,8,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-14,0,-42,0,-686] 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: yes
Analytic conductor: \(139.948491417\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3529}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 882 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(30.2027\) of defining polynomial
Character \(\chi\) \(=\) 448.1

$q$-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)
 
\(f(q)\) \(=\) \(q-66.4054 q^{3} +157.216 q^{5} -343.000 q^{7} +2222.68 q^{9} +6209.03 q^{11} -5380.35 q^{13} -10440.0 q^{15} -10994.9 q^{17} +11703.1 q^{19} +22777.0 q^{21} -106141. q^{23} -53408.1 q^{25} -2369.04 q^{27} +51562.7 q^{29} +247557. q^{31} -412313. q^{33} -53925.1 q^{35} -433678. q^{37} +357284. q^{39} +322819. q^{41} +878703. q^{43} +349440. q^{45} -655126. q^{47} +117649. q^{49} +730121. q^{51} +444837. q^{53} +976159. q^{55} -777146. q^{57} +2.14545e6 q^{59} -592902. q^{61} -762378. q^{63} -845878. q^{65} +1.72866e6 q^{67} +7.04833e6 q^{69} -1.58060e6 q^{71} -4.33164e6 q^{73} +3.54658e6 q^{75} -2.12970e6 q^{77} +6.08518e6 q^{79} -4.70367e6 q^{81} -8.10357e6 q^{83} -1.72858e6 q^{85} -3.42404e6 q^{87} +9.86032e6 q^{89} +1.84546e6 q^{91} -1.64391e7 q^{93} +1.83991e6 q^{95} -171786. q^{97} +1.38007e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} - 42 q^{5} - 686 q^{7} + 2782 q^{9} + 7428 q^{11} - 11830 q^{13} - 20880 q^{15} + 15792 q^{17} + 26614 q^{19} + 4802 q^{21} - 32640 q^{23} - 91846 q^{25} - 87668 q^{27} + 158016 q^{29} + 180740 q^{31}+ \cdots + 14482452 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −66.4054 −1.41997 −0.709985 − 0.704217i \(-0.751298\pi\)
−0.709985 + 0.704217i \(0.751298\pi\)
\(4\) 0 0
\(5\) 157.216 0.562474 0.281237 − 0.959638i \(-0.409255\pi\)
0.281237 + 0.959638i \(0.409255\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 2222.68 1.01631
\(10\) 0 0
\(11\) 6209.03 1.40653 0.703265 − 0.710928i \(-0.251725\pi\)
0.703265 + 0.710928i \(0.251725\pi\)
\(12\) 0 0
\(13\) −5380.35 −0.679218 −0.339609 − 0.940567i \(-0.610295\pi\)
−0.339609 + 0.940567i \(0.610295\pi\)
\(14\) 0 0
\(15\) −10440.0 −0.798695
\(16\) 0 0
\(17\) −10994.9 −0.542776 −0.271388 − 0.962470i \(-0.587483\pi\)
−0.271388 + 0.962470i \(0.587483\pi\)
\(18\) 0 0
\(19\) 11703.1 0.391437 0.195718 − 0.980660i \(-0.437296\pi\)
0.195718 + 0.980660i \(0.437296\pi\)
\(20\) 0 0
\(21\) 22777.0 0.536698
\(22\) 0 0
\(23\) −106141. −1.81901 −0.909506 − 0.415691i \(-0.863540\pi\)
−0.909506 + 0.415691i \(0.863540\pi\)
\(24\) 0 0
\(25\) −53408.1 −0.683623
\(26\) 0 0
\(27\) −2369.04 −0.0231632
\(28\) 0 0
\(29\) 51562.7 0.392593 0.196297 − 0.980545i \(-0.437108\pi\)
0.196297 + 0.980545i \(0.437108\pi\)
\(30\) 0 0
\(31\) 247557. 1.49248 0.746240 − 0.665677i \(-0.231857\pi\)
0.746240 + 0.665677i \(0.231857\pi\)
\(32\) 0 0
\(33\) −412313. −1.99723
\(34\) 0 0
\(35\) −53925.1 −0.212595
\(36\) 0 0
\(37\) −433678. −1.40754 −0.703771 − 0.710427i \(-0.748502\pi\)
−0.703771 + 0.710427i \(0.748502\pi\)
\(38\) 0 0
\(39\) 357284. 0.964468
\(40\) 0 0
\(41\) 322819. 0.731501 0.365751 − 0.930713i \(-0.380812\pi\)
0.365751 + 0.930713i \(0.380812\pi\)
\(42\) 0 0
\(43\) 878703. 1.68540 0.842699 − 0.538385i \(-0.180965\pi\)
0.842699 + 0.538385i \(0.180965\pi\)
\(44\) 0 0
\(45\) 349440. 0.571649
\(46\) 0 0
\(47\) −655126. −0.920413 −0.460206 − 0.887812i \(-0.652225\pi\)
−0.460206 + 0.887812i \(0.652225\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 730121. 0.770725
\(52\) 0 0
\(53\) 444837. 0.410426 0.205213 − 0.978717i \(-0.434211\pi\)
0.205213 + 0.978717i \(0.434211\pi\)
\(54\) 0 0
\(55\) 976159. 0.791136
\(56\) 0 0
\(57\) −777146. −0.555828
\(58\) 0 0
\(59\) 2.14545e6 1.35999 0.679996 − 0.733216i \(-0.261981\pi\)
0.679996 + 0.733216i \(0.261981\pi\)
\(60\) 0 0
\(61\) −592902. −0.334448 −0.167224 − 0.985919i \(-0.553480\pi\)
−0.167224 + 0.985919i \(0.553480\pi\)
\(62\) 0 0
\(63\) −762378. −0.384130
\(64\) 0 0
\(65\) −845878. −0.382042
\(66\) 0 0
\(67\) 1.72866e6 0.702180 0.351090 − 0.936342i \(-0.385811\pi\)
0.351090 + 0.936342i \(0.385811\pi\)
\(68\) 0 0
\(69\) 7.04833e6 2.58294
\(70\) 0 0
\(71\) −1.58060e6 −0.524105 −0.262052 − 0.965054i \(-0.584399\pi\)
−0.262052 + 0.965054i \(0.584399\pi\)
\(72\) 0 0
\(73\) −4.33164e6 −1.30323 −0.651617 − 0.758548i \(-0.725909\pi\)
−0.651617 + 0.758548i \(0.725909\pi\)
\(74\) 0 0
\(75\) 3.54658e6 0.970724
\(76\) 0 0
\(77\) −2.12970e6 −0.531618
\(78\) 0 0
\(79\) 6.08518e6 1.38860 0.694302 − 0.719684i \(-0.255713\pi\)
0.694302 + 0.719684i \(0.255713\pi\)
\(80\) 0 0
\(81\) −4.70367e6 −0.983421
\(82\) 0 0
\(83\) −8.10357e6 −1.55562 −0.777809 − 0.628500i \(-0.783669\pi\)
−0.777809 + 0.628500i \(0.783669\pi\)
\(84\) 0 0
\(85\) −1.72858e6 −0.305297
\(86\) 0 0
\(87\) −3.42404e6 −0.557470
\(88\) 0 0
\(89\) 9.86032e6 1.48261 0.741303 − 0.671170i \(-0.234208\pi\)
0.741303 + 0.671170i \(0.234208\pi\)
\(90\) 0 0
\(91\) 1.84546e6 0.256720
\(92\) 0 0
\(93\) −1.64391e7 −2.11928
\(94\) 0 0
\(95\) 1.83991e6 0.220173
\(96\) 0 0
\(97\) −171786. −0.0191111 −0.00955555 − 0.999954i \(-0.503042\pi\)
−0.00955555 + 0.999954i \(0.503042\pi\)
\(98\) 0 0
\(99\) 1.38007e7 1.42947
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 448.8.a.n.1.1 2
4.3 odd 2 448.8.a.p.1.2 2
8.3 odd 2 28.8.a.a.1.1 ✓ 2
8.5 even 2 112.8.a.i.1.2 2
24.11 even 2 252.8.a.e.1.2 2
56.3 even 6 196.8.e.a.177.1 4
56.11 odd 6 196.8.e.d.177.2 4
56.19 even 6 196.8.e.a.165.1 4
56.27 even 2 196.8.a.b.1.2 2
56.51 odd 6 196.8.e.d.165.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.8.a.a.1.1 ✓ 2 8.3 odd 2
112.8.a.i.1.2 2 8.5 even 2
196.8.a.b.1.2 2 56.27 even 2
196.8.e.a.165.1 4 56.19 even 6
196.8.e.a.177.1 4 56.3 even 6
196.8.e.d.165.2 4 56.51 odd 6
196.8.e.d.177.2 4 56.11 odd 6
252.8.a.e.1.2 2 24.11 even 2
448.8.a.n.1.1 2 1.1 even 1 trivial
448.8.a.p.1.2 2 4.3 odd 2