Properties

Label 369.2.d.b.163.1
Level $369$
Weight $2$
Character 369.163
Analytic conductor $2.946$
Analytic rank $0$
Dimension $2$
CM discriminant -123
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [369,2,Mod(163,369)] 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("369.163"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(369, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.94647983459\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-41}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 41 \) 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 163.1
Root \(-6.40312i\) of defining polynomial
Character \(\chi\) \(=\) 369.163
Dual form 369.2.d.b.163.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -6.40312i q^{11} +4.00000 q^{16} -6.40312i q^{17} -5.00000 q^{25} -6.40312i q^{29} -1.00000 q^{31} +5.00000 q^{37} +6.40312i q^{41} -7.00000 q^{43} +12.8062i q^{44} -6.40312i q^{47} +7.00000 q^{49} +12.8062i q^{53} +11.0000 q^{61} -8.00000 q^{64} +12.8062i q^{68} -6.40312i q^{71} -13.0000 q^{73} +12.8062i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 8 q^{16} - 10 q^{25} - 2 q^{31} + 10 q^{37} - 14 q^{43} + 14 q^{49} + 22 q^{61} - 16 q^{64} - 26 q^{73}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/369\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(334\)
\(\chi(n)\) \(1\) \(-1\)

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
\(4\) −2.00000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 6.40312i − 1.93061i −0.261116 − 0.965307i \(-0.584091\pi\)
0.261116 − 0.965307i \(-0.415909\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) − 6.40312i − 1.55299i −0.630126 − 0.776493i \(-0.716997\pi\)
0.630126 − 0.776493i \(-0.283003\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.40312i − 1.18903i −0.804084 − 0.594515i \(-0.797344\pi\)
0.804084 − 0.594515i \(-0.202656\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.40312i 1.00000i
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 − 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 12.8062i 1.93061i
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.40312i − 0.933992i −0.884260 − 0.466996i \(-0.845336\pi\)
0.884260 − 0.466996i \(-0.154664\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.8062i 1.75907i 0.475831 + 0.879537i \(0.342147\pi\)
−0.475831 + 0.879537i \(0.657853\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 − 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 12.8062i 1.55299i
\(69\) 0 0
\(70\) 0 0
\(71\) − 6.40312i − 0.759911i −0.925005 − 0.379956i \(-0.875939\pi\)
0.925005 − 0.379956i \(-0.124061\pi\)
\(72\) 0 0
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.8062i 1.35746i 0.734388 + 0.678730i \(0.237469\pi\)
−0.734388 + 0.678730i \(0.762531\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 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 369.2.d.b.163.1 ✓ 2
3.2 odd 2 inner 369.2.d.b.163.2 yes 2
41.40 even 2 inner 369.2.d.b.163.2 yes 2
123.122 odd 2 CM 369.2.d.b.163.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
369.2.d.b.163.1 ✓ 2 1.1 even 1 trivial
369.2.d.b.163.1 ✓ 2 123.122 odd 2 CM
369.2.d.b.163.2 yes 2 3.2 odd 2 inner
369.2.d.b.163.2 yes 2 41.40 even 2 inner