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Properties

Label 1156.1.o.a.319.1
Level $1156$
Weight $1$
Character 1156.319
Analytic conductor $0.577$
Analytic rank $0$
Dimension $32$
Projective image $D_{68}$
CM discriminant -4
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] = [1156,1,Mod(47,1156)] 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("1156.47"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(68)) chi = DirichletCharacter(H, H._module([34, 25])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1156.o (of order \(68\), degree \(32\), 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: \(0.576919154604\)
Analytic rank: \(0\)
Dimension: \(32\)
Coefficient field: \(\Q(\zeta_{68})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{68}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{68} - \cdots)\)

Embedding invariants

Embedding label 319.1
Root \(-0.183750 + 0.982973i\) of defining polynomial
Character \(\chi\) \(=\) 1156.319
Dual form 1156.1.o.a.395.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.361242 + 0.932472i) q^{2} +(-0.739009 + 0.673696i) q^{4} +(0.602635 - 1.79802i) q^{5} +(-0.895163 - 0.445738i) q^{8} +(-0.961826 - 0.273663i) q^{9} +(1.89430 - 0.0875787i) q^{10} +(0.857445 - 1.72198i) q^{13} +(0.0922684 - 0.995734i) q^{16} +(-0.0922684 + 0.995734i) q^{17} +(-0.0922684 - 0.995734i) q^{18} +(0.765964 + 1.73474i) q^{20} +(-2.07168 - 1.56446i) q^{25} +(1.91545 + 0.177492i) q^{26} +(0.972171 + 0.0449462i) q^{29} +(0.961826 - 0.273663i) q^{32} +(-0.961826 + 0.273663i) q^{34} +(0.895163 - 0.445738i) q^{36} +(0.666468 + 0.456541i) q^{37} +(-1.34090 + 1.34090i) q^{40} +(0.629488 + 0.0878098i) q^{41} +(-1.07168 + 1.56446i) q^{45} +(-0.526432 + 0.850217i) q^{49} +(0.710439 - 2.49693i) q^{50} +(0.526432 + 1.85022i) q^{52} +(1.01267 + 0.288130i) q^{53} +(0.309277 + 0.922758i) q^{58} +(-1.92821 + 0.453510i) q^{61} +(0.602635 + 0.798017i) q^{64} +(-2.57943 - 2.57943i) q^{65} +(-0.602635 - 0.798017i) q^{68} +(0.739009 + 0.673696i) q^{72} +(-1.34421 + 1.11622i) q^{73} +(-0.184956 + 0.786384i) q^{74} +(-1.73474 - 0.765964i) q^{80} +(0.850217 + 0.526432i) q^{81} +(0.145517 + 0.618701i) q^{82} +(1.73474 + 0.765964i) q^{85} +(0.322039 + 0.646741i) q^{89} +(-1.84595 - 0.434164i) q^{90} +(0.0806938 - 0.0449462i) q^{97} +(-0.982973 - 0.183750i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 2 q^{4} + 2 q^{5} - 2 q^{10} - 2 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{20} - 34 q^{25} - 2 q^{29} + 2 q^{37} + 2 q^{40} - 2 q^{41} - 2 q^{45} + 2 q^{50} + 2 q^{58} + 2 q^{61} + 2 q^{64} - 2 q^{68}+ \cdots - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(-1\) \(e\left(\frac{37}{68}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.361242 + 0.932472i 0.361242 + 0.932472i
\(3\) 0 0 0.138156 − 0.990410i \(-0.455882\pi\)
−0.138156 + 0.990410i \(0.544118\pi\)
\(4\) −0.739009 + 0.673696i −0.739009 + 0.673696i
\(5\) 0.602635 − 1.79802i 0.602635 − 1.79802i − 1.00000i \(-0.5\pi\)
0.602635 − 0.798017i \(-0.294118\pi\)
\(6\) 0 0
\(7\) 0 0 −0.486604 − 0.873622i \(-0.661765\pi\)
0.486604 + 0.873622i \(0.338235\pi\)
\(8\) −0.895163 − 0.445738i −0.895163 − 0.445738i
\(9\) −0.961826 − 0.273663i −0.961826 − 0.273663i
\(10\) 1.89430 − 0.0875787i 1.89430 − 0.0875787i
\(11\) 0 0 0.0461835 − 0.998933i \(-0.485294\pi\)
−0.0461835 + 0.998933i \(0.514706\pi\)
\(12\) 0 0
\(13\) 0.857445 − 1.72198i 0.857445 − 1.72198i 0.183750 − 0.982973i \(-0.441176\pi\)
0.673696 − 0.739009i \(-0.264706\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.0922684 − 0.995734i 0.0922684 − 0.995734i
\(17\) −0.0922684 + 0.995734i −0.0922684 + 0.995734i
\(18\) −0.0922684 − 0.995734i −0.0922684 − 0.995734i
\(19\) 0 0 0.361242 − 0.932472i \(-0.382353\pi\)
−0.361242 + 0.932472i \(0.617647\pi\)
\(20\) 0.765964 + 1.73474i 0.765964 + 1.73474i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.486604 − 0.873622i \(-0.661765\pi\)
0.486604 + 0.873622i \(0.338235\pi\)
\(24\) 0 0
\(25\) −2.07168 − 1.56446i −2.07168 − 1.56446i
\(26\) 1.91545 + 0.177492i 1.91545 + 0.177492i
\(27\) 0 0
\(28\) 0 0
\(29\) 0.972171 + 0.0449462i 0.972171 + 0.0449462i 0.526432 − 0.850217i \(-0.323529\pi\)
0.445738 + 0.895163i \(0.352941\pi\)
\(30\) 0 0
\(31\) 0 0 0.948161 − 0.317791i \(-0.102941\pi\)
−0.948161 + 0.317791i \(0.897059\pi\)
\(32\) 0.961826 − 0.273663i 0.961826 − 0.273663i
\(33\) 0 0
\(34\) −0.961826 + 0.273663i −0.961826 + 0.273663i
\(35\) 0 0
\(36\) 0.895163 − 0.445738i 0.895163 − 0.445738i
\(37\) 0.666468 + 0.456541i 0.666468 + 0.456541i 0.850217 − 0.526432i \(-0.176471\pi\)
−0.183750 + 0.982973i \(0.558824\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.34090 + 1.34090i −1.34090 + 1.34090i
\(41\) 0.629488 + 0.0878098i 0.629488 + 0.0878098i 0.445738 − 0.895163i \(-0.352941\pi\)
0.183750 + 0.982973i \(0.441176\pi\)
\(42\) 0 0
\(43\) 0 0 0.995734 − 0.0922684i \(-0.0294118\pi\)
−0.995734 + 0.0922684i \(0.970588\pi\)
\(44\) 0 0
\(45\) −1.07168 + 1.56446i −1.07168 + 1.56446i
\(46\) 0 0
\(47\) 0 0 −0.273663 − 0.961826i \(-0.588235\pi\)
0.273663 + 0.961826i \(0.411765\pi\)
\(48\) 0 0
\(49\) −0.526432 + 0.850217i −0.526432 + 0.850217i
\(50\) 0.710439 − 2.49693i 0.710439 − 2.49693i
\(51\) 0 0
\(52\) 0.526432 + 1.85022i 0.526432 + 1.85022i
\(53\) 1.01267 + 0.288130i 1.01267 + 0.288130i 0.739009 − 0.673696i \(-0.235294\pi\)
0.273663 + 0.961826i \(0.411765\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.309277 + 0.922758i 0.309277 + 0.922758i
\(59\) 0 0 −0.526432 − 0.850217i \(-0.676471\pi\)
0.526432 + 0.850217i \(0.323529\pi\)
\(60\) 0 0
\(61\) −1.92821 + 0.453510i −1.92821 + 0.453510i −0.932472 + 0.361242i \(0.882353\pi\)
−0.995734 + 0.0922684i \(0.970588\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.602635 + 0.798017i 0.602635 + 0.798017i
\(65\) −2.57943 − 2.57943i −2.57943 − 2.57943i
\(66\) 0 0
\(67\) 0 0 −0.932472 − 0.361242i \(-0.882353\pi\)
0.932472 + 0.361242i \(0.117647\pi\)
\(68\) −0.602635 − 0.798017i −0.602635 − 0.798017i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.873622 − 0.486604i \(-0.161765\pi\)
−0.873622 + 0.486604i \(0.838235\pi\)
\(72\) 0.739009 + 0.673696i 0.739009 + 0.673696i
\(73\) −1.34421 + 1.11622i −1.34421 + 1.11622i −0.361242 + 0.932472i \(0.617647\pi\)
−0.982973 + 0.183750i \(0.941176\pi\)
\(74\) −0.184956 + 0.786384i −0.184956 + 0.786384i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.914794 − 0.403921i \(-0.132353\pi\)
−0.914794 + 0.403921i \(0.867647\pi\)
\(80\) −1.73474 − 0.765964i −1.73474 − 0.765964i
\(81\) 0.850217 + 0.526432i 0.850217 + 0.526432i
\(82\) 0.145517 + 0.618701i 0.145517 + 0.618701i
\(83\) 0 0 −0.798017 − 0.602635i \(-0.794118\pi\)
0.798017 + 0.602635i \(0.205882\pi\)
\(84\) 0 0
\(85\) 1.73474 + 0.765964i 1.73474 + 0.765964i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.322039 + 0.646741i 0.322039 + 0.646741i 0.995734 − 0.0922684i \(-0.0294118\pi\)
−0.673696 + 0.739009i \(0.735294\pi\)
\(90\) −1.84595 − 0.434164i −1.84595 − 0.434164i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.0806938 − 0.0449462i 0.0806938 − 0.0449462i −0.445738 − 0.895163i \(-0.647059\pi\)
0.526432 + 0.850217i \(0.323529\pi\)
\(98\) −0.982973 − 0.183750i −0.982973 − 0.183750i
\(99\) 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 1156.1.o.a.319.1 ✓ 32
4.3 odd 2 CM 1156.1.o.a.319.1 ✓ 32
289.106 even 68 inner 1156.1.o.a.395.1 yes 32
1156.395 odd 68 inner 1156.1.o.a.395.1 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1156.1.o.a.319.1 ✓ 32 1.1 even 1 trivial
1156.1.o.a.319.1 ✓ 32 4.3 odd 2 CM
1156.1.o.a.395.1 yes 32 289.106 even 68 inner
1156.1.o.a.395.1 yes 32 1156.395 odd 68 inner