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Properties

Label 2312.1.x.a.171.1
Level $2312$
Weight $1$
Character 2312.171
Analytic conductor $1.154$
Analytic rank $0$
Dimension $16$
Projective image $D_{17}$
CM discriminant -8
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] = [2312,1,Mod(35,2312)] 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("2312.35"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([17, 17, 14])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2312.x (of order \(34\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.15383830921\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{34})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{17}\)
Projective field: Galois closure of 17.1.39726964294200827244451673677113820402444926976.1

Embedding invariants

Embedding label 171.1
Root \(0.850217 + 0.526432i\) of defining polynomial
Character \(\chi\) \(=\) 2312.171
Dual form 2312.1.x.a.987.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.850217 - 0.526432i) q^{2} +(1.67148 + 0.312454i) q^{3} +(0.445738 + 0.895163i) q^{4} +(-1.25664 - 1.14558i) q^{6} +(0.0922684 - 0.995734i) q^{8} +(1.76375 + 0.683280i) q^{9} +(0.891477 + 1.79033i) q^{11} +(0.465346 + 1.63552i) q^{12} +(-0.602635 + 0.798017i) q^{16} +(-0.850217 - 0.526432i) q^{17} +(-1.13987 - 1.50943i) q^{18} +(0.465346 - 0.288130i) q^{19} +(0.184537 - 1.99147i) q^{22} +(0.465346 - 1.63552i) q^{24} +(-0.982973 + 0.183750i) q^{25} +(1.28884 + 0.798017i) q^{27} +(0.932472 - 0.361242i) q^{32} +(0.930692 + 3.27104i) q^{33} +(0.445738 + 0.895163i) q^{34} +(0.174523 + 1.88341i) q^{36} -0.547326 q^{38} +(-1.83319 - 0.342683i) q^{41} +(-0.537235 - 0.711414i) q^{43} +(-1.20527 + 1.59603i) q^{44} +(-1.25664 + 1.14558i) q^{48} +(0.739009 + 0.673696i) q^{49} +(0.932472 + 0.361242i) q^{50} +(-1.25664 - 1.14558i) q^{51} +(-0.675694 - 1.35698i) q^{54} +(0.867844 - 0.336205i) q^{57} +(0.136374 - 0.124322i) q^{59} +(-0.982973 - 0.183750i) q^{64} +(0.930692 - 3.27104i) q^{66} +(1.44574 - 0.895163i) q^{67} +(0.0922684 - 0.995734i) q^{68} +(0.843104 - 1.69318i) q^{72} +(1.18475 - 1.56886i) q^{73} -1.70043 q^{75} +(0.465346 + 0.288130i) q^{76} +(0.507113 + 0.462295i) q^{81} +(1.37821 + 1.25640i) q^{82} +(-1.83319 + 0.342683i) q^{83} +(0.0822551 + 0.887674i) q^{86} +(1.86494 - 0.722483i) q^{88} +(-0.111208 - 1.20013i) q^{89} +(1.67148 - 0.312454i) q^{96} +(1.37821 + 0.533922i) q^{97} +(-0.273663 - 0.961826i) q^{98} +(0.349047 + 3.76682i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - q^{17} - 3 q^{18} - 2 q^{19} - 2 q^{22} - 2 q^{24} - q^{25} + 13 q^{27} - q^{32} - 4 q^{33} - q^{34} - 3 q^{36}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{17}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.850217 − 0.526432i −0.850217 − 0.526432i
\(3\) 1.67148 + 0.312454i 1.67148 + 0.312454i 0.932472 − 0.361242i \(-0.117647\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(4\) 0.445738 + 0.895163i 0.445738 + 0.895163i
\(5\) 0 0 −0.0922684 − 0.995734i \(-0.529412\pi\)
0.0922684 + 0.995734i \(0.470588\pi\)
\(6\) −1.25664 − 1.14558i −1.25664 − 1.14558i
\(7\) 0 0 −0.932472 − 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(8\) 0.0922684 − 0.995734i 0.0922684 − 0.995734i
\(9\) 1.76375 + 0.683280i 1.76375 + 0.683280i
\(10\) 0 0
\(11\) 0.891477 + 1.79033i 0.891477 + 1.79033i 0.445738 + 0.895163i \(0.352941\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(12\) 0.465346 + 1.63552i 0.465346 + 1.63552i
\(13\) 0 0 0.0922684 − 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.602635 + 0.798017i −0.602635 + 0.798017i
\(17\) −0.850217 − 0.526432i −0.850217 − 0.526432i
\(18\) −1.13987 − 1.50943i −1.13987 − 1.50943i
\(19\) 0.465346 − 0.288130i 0.465346 − 0.288130i −0.273663 − 0.961826i \(-0.588235\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.184537 − 1.99147i 0.184537 − 1.99147i
\(23\) 0 0 −0.932472 − 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(24\) 0.465346 − 1.63552i 0.465346 − 1.63552i
\(25\) −0.982973 + 0.183750i −0.982973 + 0.183750i
\(26\) 0 0
\(27\) 1.28884 + 0.798017i 1.28884 + 0.798017i
\(28\) 0 0
\(29\) 0 0 −0.445738 − 0.895163i \(-0.647059\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(30\) 0 0
\(31\) 0 0 0.0922684 − 0.995734i \(-0.470588\pi\)
−0.0922684 + 0.995734i \(0.529412\pi\)
\(32\) 0.932472 − 0.361242i 0.932472 − 0.361242i
\(33\) 0.930692 + 3.27104i 0.930692 + 3.27104i
\(34\) 0.445738 + 0.895163i 0.445738 + 0.895163i
\(35\) 0 0
\(36\) 0.174523 + 1.88341i 0.174523 + 1.88341i
\(37\) 0 0 0.273663 − 0.961826i \(-0.411765\pi\)
−0.273663 + 0.961826i \(0.588235\pi\)
\(38\) −0.547326 −0.547326
\(39\) 0 0
\(40\) 0 0
\(41\) −1.83319 − 0.342683i −1.83319 − 0.342683i −0.850217 − 0.526432i \(-0.823529\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(42\) 0 0
\(43\) −0.537235 − 0.711414i −0.537235 − 0.711414i 0.445738 − 0.895163i \(-0.352941\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(44\) −1.20527 + 1.59603i −1.20527 + 1.59603i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.932472 − 0.361242i \(-0.117647\pi\)
−0.932472 + 0.361242i \(0.882353\pi\)
\(48\) −1.25664 + 1.14558i −1.25664 + 1.14558i
\(49\) 0.739009 + 0.673696i 0.739009 + 0.673696i
\(50\) 0.932472 + 0.361242i 0.932472 + 0.361242i
\(51\) −1.25664 − 1.14558i −1.25664 − 1.14558i
\(52\) 0 0
\(53\) 0 0 −0.932472 − 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(54\) −0.675694 − 1.35698i −0.675694 − 1.35698i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.867844 − 0.336205i 0.867844 − 0.336205i
\(58\) 0 0
\(59\) 0.136374 − 0.124322i 0.136374 − 0.124322i −0.602635 − 0.798017i \(-0.705882\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(60\) 0 0
\(61\) 0 0 −0.739009 − 0.673696i \(-0.764706\pi\)
0.739009 + 0.673696i \(0.235294\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.982973 − 0.183750i −0.982973 − 0.183750i
\(65\) 0 0
\(66\) 0.930692 − 3.27104i 0.930692 − 3.27104i
\(67\) 1.44574 − 0.895163i 1.44574 − 0.895163i 0.445738 − 0.895163i \(-0.352941\pi\)
1.00000 \(0\)
\(68\) 0.0922684 − 0.995734i 0.0922684 − 0.995734i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.932472 − 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(72\) 0.843104 − 1.69318i 0.843104 − 1.69318i
\(73\) 1.18475 − 1.56886i 1.18475 − 1.56886i 0.445738 − 0.895163i \(-0.352941\pi\)
0.739009 − 0.673696i \(-0.235294\pi\)
\(74\) 0 0
\(75\) −1.70043 −1.70043
\(76\) 0.465346 + 0.288130i 0.465346 + 0.288130i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.850217 − 0.526432i \(-0.176471\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(80\) 0 0
\(81\) 0.507113 + 0.462295i 0.507113 + 0.462295i
\(82\) 1.37821 + 1.25640i 1.37821 + 1.25640i
\(83\) −1.83319 + 0.342683i −1.83319 + 0.342683i −0.982973 − 0.183750i \(-0.941176\pi\)
−0.850217 + 0.526432i \(0.823529\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.0822551 + 0.887674i 0.0822551 + 0.887674i
\(87\) 0 0
\(88\) 1.86494 − 0.722483i 1.86494 − 0.722483i
\(89\) −0.111208 − 1.20013i −0.111208 − 1.20013i −0.850217 − 0.526432i \(-0.823529\pi\)
0.739009 − 0.673696i \(-0.235294\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.67148 − 0.312454i 1.67148 − 0.312454i
\(97\) 1.37821 + 0.533922i 1.37821 + 0.533922i 0.932472 − 0.361242i \(-0.117647\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(98\) −0.273663 − 0.961826i −0.273663 − 0.961826i
\(99\) 0.349047 + 3.76682i 0.349047 + 3.76682i
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 2312.1.x.a.171.1 ✓ 16
8.3 odd 2 CM 2312.1.x.a.171.1 ✓ 16
289.120 even 17 inner 2312.1.x.a.987.1 yes 16
2312.987 odd 34 inner 2312.1.x.a.987.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2312.1.x.a.171.1 ✓ 16 1.1 even 1 trivial
2312.1.x.a.171.1 ✓ 16 8.3 odd 2 CM
2312.1.x.a.987.1 yes 16 289.120 even 17 inner
2312.1.x.a.987.1 yes 16 2312.987 odd 34 inner