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Properties

Label 224.4.a.f.1.3
Level $224$
Weight $4$
Character 224.1
Self dual yes
Analytic conductor $13.216$
Analytic rank $1$
Dimension $3$
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] = [224,4,Mod(1,224)] 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("224.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(224, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-6,0,-21] 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: \(13.2164278413\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.66908\) of defining polynomial
Character \(\chi\) \(=\) 224.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.24797 q^{3} -18.9243 q^{5} -7.00000 q^{7} +12.0371 q^{9} +49.7423 q^{11} -38.6987 q^{13} -118.238 q^{15} -83.9969 q^{17} -136.741 q^{19} -43.7358 q^{21} -149.729 q^{23} +233.128 q^{25} -93.4878 q^{27} +63.8486 q^{29} -6.03564 q^{31} +310.789 q^{33} +132.470 q^{35} +37.1749 q^{37} -241.788 q^{39} -361.465 q^{41} +99.2433 q^{43} -227.793 q^{45} +343.368 q^{47} +49.0000 q^{49} -524.810 q^{51} +625.497 q^{53} -941.338 q^{55} -854.352 q^{57} +363.618 q^{59} +120.927 q^{61} -84.2596 q^{63} +732.345 q^{65} +173.794 q^{67} -935.503 q^{69} -468.385 q^{71} -710.075 q^{73} +1456.58 q^{75} -348.196 q^{77} -1079.70 q^{79} -909.110 q^{81} +1375.18 q^{83} +1589.58 q^{85} +398.924 q^{87} +462.708 q^{89} +270.891 q^{91} -37.7105 q^{93} +2587.72 q^{95} +456.619 q^{97} +598.753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{5} - 21 q^{7} + 15 q^{9} - 6 q^{13} - 168 q^{15} - 66 q^{17} - 168 q^{19} - 336 q^{23} + 69 q^{25} - 168 q^{27} + 90 q^{29} - 504 q^{31} + 120 q^{33} + 42 q^{35} + 18 q^{37} - 840 q^{39} - 450 q^{41}+ \cdots + 2688 q^{99}+O(q^{100}) \) 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
\(3\) 6.24797 1.20242 0.601211 − 0.799090i \(-0.294685\pi\)
0.601211 + 0.799090i \(0.294685\pi\)
\(4\) 0 0
\(5\) −18.9243 −1.69264 −0.846320 − 0.532675i \(-0.821187\pi\)
−0.846320 + 0.532675i \(0.821187\pi\)
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 0 0
\(9\) 12.0371 0.445818
\(10\) 0 0
\(11\) 49.7423 1.36344 0.681722 − 0.731611i \(-0.261231\pi\)
0.681722 + 0.731611i \(0.261231\pi\)
\(12\) 0 0
\(13\) −38.6987 −0.825622 −0.412811 − 0.910817i \(-0.635453\pi\)
−0.412811 + 0.910817i \(0.635453\pi\)
\(14\) 0 0
\(15\) −118.238 −2.03527
\(16\) 0 0
\(17\) −83.9969 −1.19837 −0.599184 − 0.800612i \(-0.704508\pi\)
−0.599184 + 0.800612i \(0.704508\pi\)
\(18\) 0 0
\(19\) −136.741 −1.65108 −0.825539 − 0.564345i \(-0.809129\pi\)
−0.825539 + 0.564345i \(0.809129\pi\)
\(20\) 0 0
\(21\) −43.7358 −0.454473
\(22\) 0 0
\(23\) −149.729 −1.35742 −0.678711 − 0.734406i \(-0.737461\pi\)
−0.678711 + 0.734406i \(0.737461\pi\)
\(24\) 0 0
\(25\) 233.128 1.86503
\(26\) 0 0
\(27\) −93.4878 −0.666361
\(28\) 0 0
\(29\) 63.8486 0.408841 0.204420 − 0.978883i \(-0.434469\pi\)
0.204420 + 0.978883i \(0.434469\pi\)
\(30\) 0 0
\(31\) −6.03564 −0.0349688 −0.0174844 − 0.999847i \(-0.505566\pi\)
−0.0174844 + 0.999847i \(0.505566\pi\)
\(32\) 0 0
\(33\) 310.789 1.63943
\(34\) 0 0
\(35\) 132.470 0.639758
\(36\) 0 0
\(37\) 37.1749 0.165176 0.0825880 − 0.996584i \(-0.473681\pi\)
0.0825880 + 0.996584i \(0.473681\pi\)
\(38\) 0 0
\(39\) −241.788 −0.992746
\(40\) 0 0
\(41\) −361.465 −1.37686 −0.688432 − 0.725301i \(-0.741701\pi\)
−0.688432 + 0.725301i \(0.741701\pi\)
\(42\) 0 0
\(43\) 99.2433 0.351964 0.175982 − 0.984393i \(-0.443690\pi\)
0.175982 + 0.984393i \(0.443690\pi\)
\(44\) 0 0
\(45\) −227.793 −0.754609
\(46\) 0 0
\(47\) 343.368 1.06565 0.532823 − 0.846227i \(-0.321131\pi\)
0.532823 + 0.846227i \(0.321131\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −524.810 −1.44094
\(52\) 0 0
\(53\) 625.497 1.62111 0.810553 − 0.585666i \(-0.199167\pi\)
0.810553 + 0.585666i \(0.199167\pi\)
\(54\) 0 0
\(55\) −941.338 −2.30782
\(56\) 0 0
\(57\) −854.352 −1.98529
\(58\) 0 0
\(59\) 363.618 0.802356 0.401178 − 0.916000i \(-0.368601\pi\)
0.401178 + 0.916000i \(0.368601\pi\)
\(60\) 0 0
\(61\) 120.927 0.253822 0.126911 − 0.991914i \(-0.459494\pi\)
0.126911 + 0.991914i \(0.459494\pi\)
\(62\) 0 0
\(63\) −84.2596 −0.168503
\(64\) 0 0
\(65\) 732.345 1.39748
\(66\) 0 0
\(67\) 173.794 0.316901 0.158450 − 0.987367i \(-0.449350\pi\)
0.158450 + 0.987367i \(0.449350\pi\)
\(68\) 0 0
\(69\) −935.503 −1.63219
\(70\) 0 0
\(71\) −468.385 −0.782917 −0.391458 − 0.920196i \(-0.628029\pi\)
−0.391458 + 0.920196i \(0.628029\pi\)
\(72\) 0 0
\(73\) −710.075 −1.13847 −0.569233 − 0.822176i \(-0.692760\pi\)
−0.569233 + 0.822176i \(0.692760\pi\)
\(74\) 0 0
\(75\) 1456.58 2.24255
\(76\) 0 0
\(77\) −348.196 −0.515333
\(78\) 0 0
\(79\) −1079.70 −1.53767 −0.768834 − 0.639448i \(-0.779163\pi\)
−0.768834 + 0.639448i \(0.779163\pi\)
\(80\) 0 0
\(81\) −909.110 −1.24706
\(82\) 0 0
\(83\) 1375.18 1.81862 0.909312 − 0.416115i \(-0.136609\pi\)
0.909312 + 0.416115i \(0.136609\pi\)
\(84\) 0 0
\(85\) 1589.58 2.02840
\(86\) 0 0
\(87\) 398.924 0.491599
\(88\) 0 0
\(89\) 462.708 0.551090 0.275545 − 0.961288i \(-0.411142\pi\)
0.275545 + 0.961288i \(0.411142\pi\)
\(90\) 0 0
\(91\) 270.891 0.312056
\(92\) 0 0
\(93\) −37.7105 −0.0420472
\(94\) 0 0
\(95\) 2587.72 2.79468
\(96\) 0 0
\(97\) 456.619 0.477966 0.238983 − 0.971024i \(-0.423186\pi\)
0.238983 + 0.971024i \(0.423186\pi\)
\(98\) 0 0
\(99\) 598.753 0.607848
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 224.4.a.f.1.3 ✓ 3
3.2 odd 2 2016.4.a.w.1.3 3
4.3 odd 2 224.4.a.g.1.1 yes 3
7.6 odd 2 1568.4.a.w.1.1 3
8.3 odd 2 448.4.a.w.1.3 3
8.5 even 2 448.4.a.v.1.1 3
12.11 even 2 2016.4.a.x.1.3 3
28.27 even 2 1568.4.a.x.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.4.a.f.1.3 ✓ 3 1.1 even 1 trivial
224.4.a.g.1.1 yes 3 4.3 odd 2
448.4.a.v.1.1 3 8.5 even 2
448.4.a.w.1.3 3 8.3 odd 2
1568.4.a.w.1.1 3 7.6 odd 2
1568.4.a.x.1.3 3 28.27 even 2
2016.4.a.w.1.3 3 3.2 odd 2
2016.4.a.x.1.3 3 12.11 even 2