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Properties

Label 448.8.a.l.1.1
Level $448$
Weight $8$
Character 448.1
Self dual yes
Analytic conductor $139.948$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [448,8,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,-70,0,-126,0,686] 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: \(\Q(\sqrt{1969}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 492 \) 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 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(22.6867\) 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)
 
\(f(q)\) \(=\) \(q-79.3734 q^{3} +336.361 q^{5} +343.000 q^{7} +4113.14 q^{9} +7301.05 q^{11} -5187.58 q^{13} -26698.1 q^{15} -23229.6 q^{17} +10896.3 q^{19} -27225.1 q^{21} +33781.9 q^{23} +35013.5 q^{25} -152884. q^{27} -186018. q^{29} -98163.8 q^{31} -579509. q^{33} +115372. q^{35} -283413. q^{37} +411756. q^{39} -241537. q^{41} -747209. q^{43} +1.38350e6 q^{45} +1.01668e6 q^{47} +117649. q^{49} +1.84381e6 q^{51} -217877. q^{53} +2.45579e6 q^{55} -864873. q^{57} +2.04271e6 q^{59} +1.16948e6 q^{61} +1.41081e6 q^{63} -1.74490e6 q^{65} -1.32624e6 q^{67} -2.68138e6 q^{69} -1.28248e6 q^{71} +2.59761e6 q^{73} -2.77914e6 q^{75} +2.50426e6 q^{77} -2.40172e6 q^{79} +3.13951e6 q^{81} +5.37289e6 q^{83} -7.81353e6 q^{85} +1.47649e7 q^{87} -1.19114e7 q^{89} -1.77934e6 q^{91} +7.79160e6 q^{93} +3.66507e6 q^{95} +1.35717e6 q^{97} +3.00302e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 70 q^{3} - 126 q^{5} + 686 q^{7} + 2014 q^{9} + 3420 q^{11} + 6398 q^{13} - 31032 q^{15} - 38472 q^{17} + 43358 q^{19} - 24010 q^{21} + 89928 q^{23} + 170666 q^{25} - 193060 q^{27} - 159576 q^{29}+ \cdots + 38177100 q^{99}+O(q^{100}) \) 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
\(3\) −79.3734 −1.69727 −0.848634 − 0.528980i \(-0.822575\pi\)
−0.848634 + 0.528980i \(0.822575\pi\)
\(4\) 0 0
\(5\) 336.361 1.20340 0.601700 − 0.798722i \(-0.294490\pi\)
0.601700 + 0.798722i \(0.294490\pi\)
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) 0 0
\(9\) 4113.14 1.88072
\(10\) 0 0
\(11\) 7301.05 1.65391 0.826953 − 0.562271i \(-0.190072\pi\)
0.826953 + 0.562271i \(0.190072\pi\)
\(12\) 0 0
\(13\) −5187.58 −0.654881 −0.327441 − 0.944872i \(-0.606186\pi\)
−0.327441 + 0.944872i \(0.606186\pi\)
\(14\) 0 0
\(15\) −26698.1 −2.04249
\(16\) 0 0
\(17\) −23229.6 −1.14676 −0.573378 − 0.819291i \(-0.694367\pi\)
−0.573378 + 0.819291i \(0.694367\pi\)
\(18\) 0 0
\(19\) 10896.3 0.364452 0.182226 − 0.983257i \(-0.441670\pi\)
0.182226 + 0.983257i \(0.441670\pi\)
\(20\) 0 0
\(21\) −27225.1 −0.641507
\(22\) 0 0
\(23\) 33781.9 0.578944 0.289472 − 0.957186i \(-0.406520\pi\)
0.289472 + 0.957186i \(0.406520\pi\)
\(24\) 0 0
\(25\) 35013.5 0.448173
\(26\) 0 0
\(27\) −152884. −1.49482
\(28\) 0 0
\(29\) −186018. −1.41632 −0.708161 − 0.706051i \(-0.750475\pi\)
−0.708161 + 0.706051i \(0.750475\pi\)
\(30\) 0 0
\(31\) −98163.8 −0.591814 −0.295907 − 0.955217i \(-0.595622\pi\)
−0.295907 + 0.955217i \(0.595622\pi\)
\(32\) 0 0
\(33\) −579509. −2.80712
\(34\) 0 0
\(35\) 115372. 0.454843
\(36\) 0 0
\(37\) −283413. −0.919842 −0.459921 − 0.887960i \(-0.652122\pi\)
−0.459921 + 0.887960i \(0.652122\pi\)
\(38\) 0 0
\(39\) 411756. 1.11151
\(40\) 0 0
\(41\) −241537. −0.547320 −0.273660 − 0.961827i \(-0.588234\pi\)
−0.273660 + 0.961827i \(0.588234\pi\)
\(42\) 0 0
\(43\) −747209. −1.43319 −0.716593 − 0.697492i \(-0.754299\pi\)
−0.716593 + 0.697492i \(0.754299\pi\)
\(44\) 0 0
\(45\) 1.38350e6 2.26326
\(46\) 0 0
\(47\) 1.01668e6 1.42837 0.714186 − 0.699956i \(-0.246797\pi\)
0.714186 + 0.699956i \(0.246797\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 1.84381e6 1.94635
\(52\) 0 0
\(53\) −217877. −0.201023 −0.100511 − 0.994936i \(-0.532048\pi\)
−0.100511 + 0.994936i \(0.532048\pi\)
\(54\) 0 0
\(55\) 2.45579e6 1.99031
\(56\) 0 0
\(57\) −864873. −0.618572
\(58\) 0 0
\(59\) 2.04271e6 1.29487 0.647433 − 0.762123i \(-0.275843\pi\)
0.647433 + 0.762123i \(0.275843\pi\)
\(60\) 0 0
\(61\) 1.16948e6 0.659689 0.329844 − 0.944035i \(-0.393004\pi\)
0.329844 + 0.944035i \(0.393004\pi\)
\(62\) 0 0
\(63\) 1.41081e6 0.710846
\(64\) 0 0
\(65\) −1.74490e6 −0.788085
\(66\) 0 0
\(67\) −1.32624e6 −0.538717 −0.269358 − 0.963040i \(-0.586812\pi\)
−0.269358 + 0.963040i \(0.586812\pi\)
\(68\) 0 0
\(69\) −2.68138e6 −0.982624
\(70\) 0 0
\(71\) −1.28248e6 −0.425252 −0.212626 − 0.977134i \(-0.568202\pi\)
−0.212626 + 0.977134i \(0.568202\pi\)
\(72\) 0 0
\(73\) 2.59761e6 0.781527 0.390763 − 0.920491i \(-0.372211\pi\)
0.390763 + 0.920491i \(0.372211\pi\)
\(74\) 0 0
\(75\) −2.77914e6 −0.760671
\(76\) 0 0
\(77\) 2.50426e6 0.625118
\(78\) 0 0
\(79\) −2.40172e6 −0.548059 −0.274030 − 0.961721i \(-0.588357\pi\)
−0.274030 + 0.961721i \(0.588357\pi\)
\(80\) 0 0
\(81\) 3.13951e6 0.656393
\(82\) 0 0
\(83\) 5.37289e6 1.03142 0.515708 − 0.856764i \(-0.327529\pi\)
0.515708 + 0.856764i \(0.327529\pi\)
\(84\) 0 0
\(85\) −7.81353e6 −1.38001
\(86\) 0 0
\(87\) 1.47649e7 2.40388
\(88\) 0 0
\(89\) −1.19114e7 −1.79102 −0.895508 − 0.445046i \(-0.853187\pi\)
−0.895508 + 0.445046i \(0.853187\pi\)
\(90\) 0 0
\(91\) −1.77934e6 −0.247522
\(92\) 0 0
\(93\) 7.79160e6 1.00447
\(94\) 0 0
\(95\) 3.66507e6 0.438581
\(96\) 0 0
\(97\) 1.35717e6 0.150984 0.0754922 − 0.997146i \(-0.475947\pi\)
0.0754922 + 0.997146i \(0.475947\pi\)
\(98\) 0 0
\(99\) 3.00302e7 3.11054
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.l.1.1 2
4.3 odd 2 448.8.a.s.1.2 2
8.3 odd 2 112.8.a.g.1.1 2
8.5 even 2 14.8.a.c.1.2 ✓ 2
24.5 odd 2 126.8.a.i.1.2 2
40.13 odd 4 350.8.c.k.99.2 4
40.29 even 2 350.8.a.j.1.1 2
40.37 odd 4 350.8.c.k.99.3 4
56.5 odd 6 98.8.c.k.67.2 4
56.13 odd 2 98.8.a.g.1.1 2
56.37 even 6 98.8.c.g.67.1 4
56.45 odd 6 98.8.c.k.79.2 4
56.53 even 6 98.8.c.g.79.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.8.a.c.1.2 ✓ 2 8.5 even 2
98.8.a.g.1.1 2 56.13 odd 2
98.8.c.g.67.1 4 56.37 even 6
98.8.c.g.79.1 4 56.53 even 6
98.8.c.k.67.2 4 56.5 odd 6
98.8.c.k.79.2 4 56.45 odd 6
112.8.a.g.1.1 2 8.3 odd 2
126.8.a.i.1.2 2 24.5 odd 2
350.8.a.j.1.1 2 40.29 even 2
350.8.c.k.99.2 4 40.13 odd 4
350.8.c.k.99.3 4 40.37 odd 4
448.8.a.l.1.1 2 1.1 even 1 trivial
448.8.a.s.1.2 2 4.3 odd 2