Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [176,4,Mod(1,176)] 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("176.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 176.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.3843361610\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.11109.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.56976\) of defining polynomial
Character \(\chi\) \(=\) 176.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82655 q^{3} +17.4525 q^{5} +4.62596 q^{7} -19.0106 q^{9} -11.0000 q^{11} +85.1298 q^{13} -49.3304 q^{15} -6.08908 q^{17} -52.0891 q^{19} -13.0755 q^{21} +183.964 q^{23} +179.590 q^{25} +130.051 q^{27} +140.498 q^{29} +250.685 q^{31} +31.0920 q^{33} +80.7345 q^{35} -203.522 q^{37} -240.624 q^{39} -22.3275 q^{41} +117.409 q^{43} -331.783 q^{45} +275.744 q^{47} -321.601 q^{49} +17.2111 q^{51} +26.6548 q^{53} -191.978 q^{55} +147.232 q^{57} -515.922 q^{59} -693.071 q^{61} -87.9423 q^{63} +1485.73 q^{65} -341.545 q^{67} -519.983 q^{69} -831.425 q^{71} +251.591 q^{73} -507.620 q^{75} -50.8855 q^{77} +917.036 q^{79} +145.690 q^{81} +456.749 q^{83} -106.270 q^{85} -397.124 q^{87} +91.4974 q^{89} +393.807 q^{91} -708.574 q^{93} -909.085 q^{95} +146.569 q^{97} +209.117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 8 q^{5} - 24 q^{7} + 89 q^{9} - 33 q^{11} + 66 q^{13} + 10 q^{15} + 210 q^{17} + 72 q^{19} + 200 q^{21} + 50 q^{23} - q^{25} + 286 q^{27} - 50 q^{29} + 298 q^{31} + 22 q^{33} + 304 q^{35}+ \cdots - 979 q^{99}+O(q^{100}) \) 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
\(3\) −2.82655 −0.543970 −0.271985 − 0.962302i \(-0.587680\pi\)
−0.271985 + 0.962302i \(0.587680\pi\)
\(4\) 0 0
\(5\) 17.4525 1.56100 0.780500 − 0.625156i \(-0.214965\pi\)
0.780500 + 0.625156i \(0.214965\pi\)
\(6\) 0 0
\(7\) 4.62596 0.249778 0.124889 − 0.992171i \(-0.460142\pi\)
0.124889 + 0.992171i \(0.460142\pi\)
\(8\) 0 0
\(9\) −19.0106 −0.704097
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) 85.1298 1.81621 0.908106 − 0.418741i \(-0.137528\pi\)
0.908106 + 0.418741i \(0.137528\pi\)
\(14\) 0 0
\(15\) −49.3304 −0.849137
\(16\) 0 0
\(17\) −6.08908 −0.0868717 −0.0434358 − 0.999056i \(-0.513830\pi\)
−0.0434358 + 0.999056i \(0.513830\pi\)
\(18\) 0 0
\(19\) −52.0891 −0.628950 −0.314475 − 0.949266i \(-0.601828\pi\)
−0.314475 + 0.949266i \(0.601828\pi\)
\(20\) 0 0
\(21\) −13.0755 −0.135872
\(22\) 0 0
\(23\) 183.964 1.66779 0.833894 − 0.551924i \(-0.186106\pi\)
0.833894 + 0.551924i \(0.186106\pi\)
\(24\) 0 0
\(25\) 179.590 1.43672
\(26\) 0 0
\(27\) 130.051 0.926977
\(28\) 0 0
\(29\) 140.498 0.899649 0.449824 − 0.893117i \(-0.351487\pi\)
0.449824 + 0.893117i \(0.351487\pi\)
\(30\) 0 0
\(31\) 250.685 1.45240 0.726199 − 0.687484i \(-0.241285\pi\)
0.726199 + 0.687484i \(0.241285\pi\)
\(32\) 0 0
\(33\) 31.0920 0.164013
\(34\) 0 0
\(35\) 80.7345 0.389904
\(36\) 0 0
\(37\) −203.522 −0.904293 −0.452146 − 0.891944i \(-0.649342\pi\)
−0.452146 + 0.891944i \(0.649342\pi\)
\(38\) 0 0
\(39\) −240.624 −0.987964
\(40\) 0 0
\(41\) −22.3275 −0.0850479 −0.0425239 − 0.999095i \(-0.513540\pi\)
−0.0425239 + 0.999095i \(0.513540\pi\)
\(42\) 0 0
\(43\) 117.409 0.416388 0.208194 − 0.978088i \(-0.433241\pi\)
0.208194 + 0.978088i \(0.433241\pi\)
\(44\) 0 0
\(45\) −331.783 −1.09910
\(46\) 0 0
\(47\) 275.744 0.855774 0.427887 − 0.903832i \(-0.359258\pi\)
0.427887 + 0.903832i \(0.359258\pi\)
\(48\) 0 0
\(49\) −321.601 −0.937611
\(50\) 0 0
\(51\) 17.2111 0.0472556
\(52\) 0 0
\(53\) 26.6548 0.0690815 0.0345408 − 0.999403i \(-0.489003\pi\)
0.0345408 + 0.999403i \(0.489003\pi\)
\(54\) 0 0
\(55\) −191.978 −0.470659
\(56\) 0 0
\(57\) 147.232 0.342130
\(58\) 0 0
\(59\) −515.922 −1.13843 −0.569214 − 0.822189i \(-0.692752\pi\)
−0.569214 + 0.822189i \(0.692752\pi\)
\(60\) 0 0
\(61\) −693.071 −1.45473 −0.727366 − 0.686249i \(-0.759256\pi\)
−0.727366 + 0.686249i \(0.759256\pi\)
\(62\) 0 0
\(63\) −87.9423 −0.175868
\(64\) 0 0
\(65\) 1485.73 2.83511
\(66\) 0 0
\(67\) −341.545 −0.622782 −0.311391 − 0.950282i \(-0.600795\pi\)
−0.311391 + 0.950282i \(0.600795\pi\)
\(68\) 0 0
\(69\) −519.983 −0.907227
\(70\) 0 0
\(71\) −831.425 −1.38975 −0.694873 − 0.719133i \(-0.744539\pi\)
−0.694873 + 0.719133i \(0.744539\pi\)
\(72\) 0 0
\(73\) 251.591 0.403377 0.201688 − 0.979450i \(-0.435357\pi\)
0.201688 + 0.979450i \(0.435357\pi\)
\(74\) 0 0
\(75\) −507.620 −0.781532
\(76\) 0 0
\(77\) −50.8855 −0.0753109
\(78\) 0 0
\(79\) 917.036 1.30601 0.653004 − 0.757355i \(-0.273509\pi\)
0.653004 + 0.757355i \(0.273509\pi\)
\(80\) 0 0
\(81\) 145.690 0.199849
\(82\) 0 0
\(83\) 456.749 0.604033 0.302017 − 0.953303i \(-0.402340\pi\)
0.302017 + 0.953303i \(0.402340\pi\)
\(84\) 0 0
\(85\) −106.270 −0.135607
\(86\) 0 0
\(87\) −397.124 −0.489382
\(88\) 0 0
\(89\) 91.4974 0.108974 0.0544871 − 0.998514i \(-0.482648\pi\)
0.0544871 + 0.998514i \(0.482648\pi\)
\(90\) 0 0
\(91\) 393.807 0.453650
\(92\) 0 0
\(93\) −708.574 −0.790061
\(94\) 0 0
\(95\) −909.085 −0.981791
\(96\) 0 0
\(97\) 146.569 0.153421 0.0767104 − 0.997053i \(-0.475558\pi\)
0.0767104 + 0.997053i \(0.475558\pi\)
\(98\) 0 0
\(99\) 209.117 0.212293
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 176.4.a.j.1.2 3
3.2 odd 2 1584.4.a.bl.1.1 3
4.3 odd 2 88.4.a.d.1.2 ✓ 3
8.3 odd 2 704.4.a.t.1.2 3
8.5 even 2 704.4.a.u.1.2 3
11.10 odd 2 1936.4.a.bh.1.2 3
12.11 even 2 792.4.a.l.1.1 3
20.19 odd 2 2200.4.a.m.1.2 3
44.43 even 2 968.4.a.i.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.4.a.d.1.2 ✓ 3 4.3 odd 2
176.4.a.j.1.2 3 1.1 even 1 trivial
704.4.a.t.1.2 3 8.3 odd 2
704.4.a.u.1.2 3 8.5 even 2
792.4.a.l.1.1 3 12.11 even 2
968.4.a.i.1.2 3 44.43 even 2
1584.4.a.bl.1.1 3 3.2 odd 2
1936.4.a.bh.1.2 3 11.10 odd 2
2200.4.a.m.1.2 3 20.19 odd 2