Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,4,Mod(170,171)] 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("171.170"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 171.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.0893266110\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 20x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 170.4
Root \(3.78931i\) of defining polynomial
Character \(\chi\) \(=\) 171.170
Dual form 171.4.d.a.170.1

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{4} +22.2283i q^{5} -8.71780 q^{7} -71.4352i q^{11} +64.0000 q^{16} -88.7311i q^{17} -82.8191 q^{19} -177.827i q^{20} +106.756i q^{23} -369.098 q^{25} +69.7424 q^{28} -193.782i q^{35} +128.000 q^{43} +571.481i q^{44} -633.416i q^{47} -267.000 q^{49} +1587.88 q^{55} -714.859 q^{61} -512.000 q^{64} +709.849i q^{68} -1078.00 q^{73} +662.553 q^{76} +622.757i q^{77} +1422.61i q^{80} +987.248i q^{83} +1972.34 q^{85} -854.046i q^{92} -1840.93i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 32 q^{4} + 256 q^{16} - 500 q^{25} + 512 q^{43} - 1068 q^{49} + 3248 q^{55} - 2048 q^{64} - 4312 q^{73} + 4472 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(-1\) \(-1\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −8.00000 −1.00000
\(5\) 22.2283i 1.98816i 0.108643 + 0.994081i \(0.465349\pi\)
−0.108643 + 0.994081i \(0.534651\pi\)
\(6\) 0 0
\(7\) −8.71780 −0.470717 −0.235358 − 0.971909i \(-0.575626\pi\)
−0.235358 + 0.971909i \(0.575626\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 71.4352i − 1.95805i −0.203748 − 0.979023i \(-0.565312\pi\)
0.203748 − 0.979023i \(-0.434688\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 1.00000
\(17\) − 88.7311i − 1.26591i −0.774189 − 0.632955i \(-0.781842\pi\)
0.774189 − 0.632955i \(-0.218158\pi\)
\(18\) 0 0
\(19\) −82.8191 −1.00000
\(20\) − 177.827i − 1.98816i
\(21\) 0 0
\(22\) 0 0
\(23\) 106.756i 0.967831i 0.875115 + 0.483915i \(0.160786\pi\)
−0.875115 + 0.483915i \(0.839214\pi\)
\(24\) 0 0
\(25\) −369.098 −2.95279
\(26\) 0 0
\(27\) 0 0
\(28\) 69.7424 0.470717
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 193.782i − 0.935861i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 128.000 0.453949 0.226975 − 0.973901i \(-0.427117\pi\)
0.226975 + 0.973901i \(0.427117\pi\)
\(44\) 571.481i 1.95805i
\(45\) 0 0
\(46\) 0 0
\(47\) − 633.416i − 1.96581i −0.184104 − 0.982907i \(-0.558938\pi\)
0.184104 − 0.982907i \(-0.441062\pi\)
\(48\) 0 0
\(49\) −267.000 −0.778426
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 1587.88 3.89291
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −714.859 −1.50047 −0.750233 − 0.661174i \(-0.770058\pi\)
−0.750233 + 0.661174i \(0.770058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 709.849i 1.26591i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1078.00 −1.72836 −0.864181 − 0.503182i \(-0.832163\pi\)
−0.864181 + 0.503182i \(0.832163\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 662.553 1.00000
\(77\) 622.757i 0.921686i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1422.61i 1.98816i
\(81\) 0 0
\(82\) 0 0
\(83\) 987.248i 1.30560i 0.757532 + 0.652798i \(0.226405\pi\)
−0.757532 + 0.652798i \(0.773595\pi\)
\(84\) 0 0
\(85\) 1972.34 2.51683
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 854.046i − 0.967831i
\(93\) 0 0
\(94\) 0 0
\(95\) − 1840.93i − 1.98816i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
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 171.4.d.a.170.4 yes 4
3.2 odd 2 inner 171.4.d.a.170.1 ✓ 4
19.18 odd 2 CM 171.4.d.a.170.4 yes 4
57.56 even 2 inner 171.4.d.a.170.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.4.d.a.170.1 ✓ 4 3.2 odd 2 inner
171.4.d.a.170.1 ✓ 4 57.56 even 2 inner
171.4.d.a.170.4 yes 4 1.1 even 1 trivial
171.4.d.a.170.4 yes 4 19.18 odd 2 CM