Properties

Label 171.11.c.b.37.1
Level $171$
Weight $11$
Character 171.37
Self dual yes
Analytic conductor $108.646$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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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] = [171,11,Mod(37,171)] 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("171.37"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 171.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: yes
Analytic conductor: \(108.646090207\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 37.1
Root \(4.35890\) of defining polynomial
Character \(\chi\) \(=\) 171.37

$q$-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)
 
\(f(q)\) \(=\) \(q+1024.00 q^{4} -4842.74 q^{5} +32525.0 q^{7} +249656. q^{11} +1.04858e6 q^{16} +1.85908e6 q^{17} +2.47610e6 q^{19} -4.95896e6 q^{20} -1.18025e7 q^{23} +1.36865e7 q^{25} +3.33056e7 q^{28} -1.57510e8 q^{35} -2.12458e8 q^{43} +2.55648e8 q^{44} +4.28715e7 q^{47} +7.75400e8 q^{49} -1.20902e9 q^{55} +1.60684e9 q^{61} +1.07374e9 q^{64} +1.90370e9 q^{68} -3.14322e9 q^{73} +2.53553e9 q^{76} +8.12006e9 q^{77} -5.07798e9 q^{80} -7.57882e9 q^{83} -9.00305e9 q^{85} -1.20857e10 q^{92} -1.19911e10 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2048 q^{4} + 65050 q^{7} + 2097152 q^{16} + 4952198 q^{19} + 27372948 q^{25} + 66611200 q^{28} - 424915850 q^{43} + 1550800752 q^{49} - 2418035950 q^{55} + 3213673954 q^{61} + 2147483648 q^{64} - 6286435250 q^{73}+ \cdots - 18006103654 q^{85}+O(q^{100}) \) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 1024.00 1.00000
\(5\) −4842.74 −1.54968 −0.774838 − 0.632160i \(-0.782169\pi\)
−0.774838 + 0.632160i \(0.782169\pi\)
\(6\) 0 0
\(7\) 32525.0 1.93521 0.967603 − 0.252477i \(-0.0812453\pi\)
0.967603 + 0.252477i \(0.0812453\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 249656. 1.55017 0.775083 − 0.631859i \(-0.217708\pi\)
0.775083 + 0.631859i \(0.217708\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) 1.85908e6 1.30935 0.654673 − 0.755912i \(-0.272806\pi\)
0.654673 + 0.755912i \(0.272806\pi\)
\(18\) 0 0
\(19\) 2.47610e6 1.00000
\(20\) −4.95896e6 −1.54968
\(21\) 0 0
\(22\) 0 0
\(23\) −1.18025e7 −1.83372 −0.916861 − 0.399206i \(-0.869286\pi\)
−0.916861 + 0.399206i \(0.869286\pi\)
\(24\) 0 0
\(25\) 1.36865e7 1.40149
\(26\) 0 0
\(27\) 0 0
\(28\) 3.33056e7 1.93521
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.57510e8 −2.99894
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.12458e8 −1.44521 −0.722605 − 0.691262i \(-0.757055\pi\)
−0.722605 + 0.691262i \(0.757055\pi\)
\(44\) 2.55648e8 1.55017
\(45\) 0 0
\(46\) 0 0
\(47\) 4.28715e7 0.186930 0.0934651 − 0.995623i \(-0.470206\pi\)
0.0934651 + 0.995623i \(0.470206\pi\)
\(48\) 0 0
\(49\) 7.75400e8 2.74502
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −1.20902e9 −2.40226
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.60684e9 1.90249 0.951246 − 0.308435i \(-0.0998051\pi\)
0.951246 + 0.308435i \(0.0998051\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.07374e9 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 1.90370e9 1.30935
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −3.14322e9 −1.51621 −0.758106 − 0.652131i \(-0.773875\pi\)
−0.758106 + 0.652131i \(0.773875\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.53553e9 1.00000
\(77\) 8.12006e9 2.99989
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −5.07798e9 −1.54968
\(81\) 0 0
\(82\) 0 0
\(83\) −7.57882e9 −1.92403 −0.962013 − 0.273003i \(-0.911983\pi\)
−0.962013 + 0.273003i \(0.911983\pi\)
\(84\) 0 0
\(85\) −9.00305e9 −2.02906
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.20857e10 −1.83372
\(93\) 0 0
\(94\) 0 0
\(95\) −1.19911e10 −1.54968
\(96\) 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 171.11.c.b.37.1 ✓ 2
3.2 odd 2 inner 171.11.c.b.37.2 yes 2
19.18 odd 2 CM 171.11.c.b.37.1 ✓ 2
57.56 even 2 inner 171.11.c.b.37.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.11.c.b.37.1 ✓ 2 1.1 even 1 trivial
171.11.c.b.37.1 ✓ 2 19.18 odd 2 CM
171.11.c.b.37.2 yes 2 3.2 odd 2 inner
171.11.c.b.37.2 yes 2 57.56 even 2 inner