Properties

Label 736.1.p.b.413.2
Level $736$
Weight $1$
Character 736.413
Analytic conductor $0.367$
Analytic rank $0$
Dimension $8$
Projective image $D_{24}$
CM discriminant -23
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] = [736,1,Mod(45,736)] 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("736.45"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 736 = 2^{5} \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 736.p (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(0.367311849298\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} + \cdots)\)

Embedding invariants

Embedding label 413.2
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 736.413
Dual form 736.1.p.b.597.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.965926 - 0.258819i) q^{2} +(0.241181 + 0.0999004i) q^{3} +(0.866025 - 0.500000i) q^{4} +(0.258819 + 0.0340742i) q^{6} +(0.707107 - 0.707107i) q^{8} +(-0.658919 - 0.658919i) q^{9} +(0.258819 - 0.0340742i) q^{12} +(-0.607206 + 1.46593i) q^{13} +(0.500000 - 0.866025i) q^{16} +(-0.807007 - 0.465926i) q^{18} +(-0.707107 - 0.707107i) q^{23} +(0.241181 - 0.0999004i) q^{24} +(-0.707107 + 0.707107i) q^{25} +(-0.207107 + 1.57313i) q^{26} +(-0.192993 - 0.465926i) q^{27} +(1.83195 + 0.758819i) q^{29} -1.93185 q^{31} +(0.258819 - 0.965926i) q^{32} +(-0.900100 - 0.241181i) q^{36} +(-0.292893 + 0.292893i) q^{39} +(0.707107 + 0.707107i) q^{41} +(-0.866025 - 0.500000i) q^{46} +1.73205i q^{47} +(0.207107 - 0.158919i) q^{48} -1.00000i q^{49} +(-0.500000 + 0.866025i) q^{50} +(0.207107 + 1.57313i) q^{52} +(-0.307007 - 0.400100i) q^{54} +(1.96593 + 0.258819i) q^{58} +(-0.707107 - 1.70711i) q^{59} +(-1.86603 + 0.500000i) q^{62} -1.00000i q^{64} +(-0.0999004 - 0.241181i) q^{69} +(0.366025 - 0.366025i) q^{71} -0.931852 q^{72} +(-1.22474 - 1.22474i) q^{73} +(-0.241181 + 0.0999004i) q^{75} +(-0.207107 + 0.358719i) q^{78} +0.800199i q^{81} +(0.866025 + 0.500000i) q^{82} +(0.366025 + 0.366025i) q^{87} +(-0.965926 - 0.258819i) q^{92} +(-0.465926 - 0.192993i) q^{93} +(0.448288 + 1.67303i) q^{94} +(0.158919 - 0.207107i) q^{96} +(-0.258819 - 0.965926i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{9} + 4 q^{16} + 4 q^{24} + 4 q^{26} - 8 q^{27} - 8 q^{36} - 8 q^{39} - 4 q^{48} - 4 q^{50} - 4 q^{52} + 4 q^{54} + 8 q^{58} - 8 q^{62} - 4 q^{71} + 8 q^{72} - 4 q^{75} + 4 q^{78} - 4 q^{87}+ \cdots + 4 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(415\) \(645\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.965926 − 0.258819i 0.965926 − 0.258819i
\(3\) 0.241181 + 0.0999004i 0.241181 + 0.0999004i 0.500000 − 0.866025i \(-0.333333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(4\) 0.866025 − 0.500000i 0.866025 − 0.500000i
\(5\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(6\) 0.258819 + 0.0340742i 0.258819 + 0.0340742i
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(9\) −0.658919 − 0.658919i −0.658919 − 0.658919i
\(10\) 0 0
\(11\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(12\) 0.258819 − 0.0340742i 0.258819 − 0.0340742i
\(13\) −0.607206 + 1.46593i −0.607206 + 1.46593i 0.258819 + 0.965926i \(0.416667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −0.807007 − 0.465926i −0.807007 − 0.465926i
\(19\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(24\) 0.241181 − 0.0999004i 0.241181 − 0.0999004i
\(25\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(26\) −0.207107 + 1.57313i −0.207107 + 1.57313i
\(27\) −0.192993 − 0.465926i −0.192993 − 0.465926i
\(28\) 0 0
\(29\) 1.83195 + 0.758819i 1.83195 + 0.758819i 0.965926 + 0.258819i \(0.0833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) −1.93185 −1.93185 −0.965926 − 0.258819i \(-0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(32\) 0.258819 − 0.965926i 0.258819 − 0.965926i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.900100 − 0.241181i −0.900100 − 0.241181i
\(37\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(38\) 0 0
\(39\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(40\) 0 0
\(41\) 0.707107 + 0.707107i 0.707107 + 0.707107i 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(42\) 0 0
\(43\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.866025 − 0.500000i −0.866025 − 0.500000i
\(47\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0.207107 − 0.158919i 0.207107 − 0.158919i
\(49\) − 1.00000i − 1.00000i
\(50\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(51\) 0 0
\(52\) 0.207107 + 1.57313i 0.207107 + 1.57313i
\(53\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(54\) −0.307007 − 0.400100i −0.307007 − 0.400100i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.96593 + 0.258819i 1.96593 + 0.258819i
\(59\) −0.707107 − 1.70711i −0.707107 − 1.70711i −0.707107 − 0.707107i \(-0.750000\pi\)
− 1.00000i \(-0.5\pi\)
\(60\) 0 0
\(61\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(62\) −1.86603 + 0.500000i −1.86603 + 0.500000i
\(63\) 0 0
\(64\) − 1.00000i − 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(68\) 0 0
\(69\) −0.0999004 − 0.241181i −0.0999004 − 0.241181i
\(70\) 0 0
\(71\) 0.366025 − 0.366025i 0.366025 − 0.366025i −0.500000 − 0.866025i \(-0.666667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(72\) −0.931852 −0.931852
\(73\) −1.22474 − 1.22474i −1.22474 − 1.22474i −0.965926 − 0.258819i \(-0.916667\pi\)
−0.258819 − 0.965926i \(-0.583333\pi\)
\(74\) 0 0
\(75\) −0.241181 + 0.0999004i −0.241181 + 0.0999004i
\(76\) 0 0
\(77\) 0 0
\(78\) −0.207107 + 0.358719i −0.207107 + 0.358719i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0.800199i 0.800199i
\(82\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(83\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.366025 + 0.366025i 0.366025 + 0.366025i
\(88\) 0 0
\(89\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.965926 − 0.258819i −0.965926 − 0.258819i
\(93\) −0.465926 − 0.192993i −0.465926 − 0.192993i
\(94\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(95\) 0 0
\(96\) 0.158919 − 0.207107i 0.158919 − 0.207107i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.258819 − 0.965926i −0.258819 − 0.965926i
\(99\) 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 736.1.p.b.413.2 ✓ 8
4.3 odd 2 2944.1.p.b.2897.2 8
23.22 odd 2 CM 736.1.p.b.413.2 ✓ 8
32.11 odd 8 2944.1.p.b.689.2 8
32.21 even 8 inner 736.1.p.b.597.2 yes 8
92.91 even 2 2944.1.p.b.2897.2 8
736.459 even 8 2944.1.p.b.689.2 8
736.597 odd 8 inner 736.1.p.b.597.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
736.1.p.b.413.2 ✓ 8 1.1 even 1 trivial
736.1.p.b.413.2 ✓ 8 23.22 odd 2 CM
736.1.p.b.597.2 yes 8 32.21 even 8 inner
736.1.p.b.597.2 yes 8 736.597 odd 8 inner
2944.1.p.b.689.2 8 32.11 odd 8
2944.1.p.b.689.2 8 736.459 even 8
2944.1.p.b.2897.2 8 4.3 odd 2
2944.1.p.b.2897.2 8 92.91 even 2