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Properties

Label 799.1.e.b.140.1
Level $799$
Weight $1$
Character 799.140
Analytic conductor $0.399$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
CM discriminant -47
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] = [799,1,Mod(140,799)] 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("799.140"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(799, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 799.e (of order \(4\), degree \(2\), 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: \(0.398752945094\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

Embedding invariants

Embedding label 140.1
Root \(0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 799.140
Dual form 799.1.e.b.234.3

$q$-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)
 
\(f(q)\) \(=\) \(q-0.618034i q^{2} +(-1.39680 + 1.39680i) q^{3} +0.618034 q^{4} +(0.863271 + 0.863271i) q^{6} +(-1.26007 - 1.26007i) q^{7} -1.00000i q^{8} -2.90211i q^{9} +(-0.863271 + 0.863271i) q^{12} +(-0.778768 + 0.778768i) q^{14} +(-0.809017 - 0.587785i) q^{17} -1.79360 q^{18} +3.52015 q^{21} +(1.39680 + 1.39680i) q^{24} -1.00000i q^{25} +(2.65688 + 2.65688i) q^{27} +(-0.778768 - 0.778768i) q^{28} -1.00000i q^{32} +(-0.363271 + 0.500000i) q^{34} -1.79360i q^{36} +(1.39680 - 1.39680i) q^{37} -2.17557i q^{42} -1.00000 q^{47} +2.17557i q^{49} -0.618034 q^{50} +(1.95106 - 0.309017i) q^{51} +1.90211i q^{53} +(1.64204 - 1.64204i) q^{54} +(-1.26007 + 1.26007i) q^{56} -0.618034i q^{59} +(-0.642040 - 0.642040i) q^{61} +(-3.65688 + 3.65688i) q^{63} -0.618034 q^{64} +(-0.500000 - 0.363271i) q^{68} +(-0.221232 + 0.221232i) q^{71} -2.90211 q^{72} +(-0.863271 - 0.863271i) q^{74} +(1.39680 + 1.39680i) q^{75} +(-0.221232 - 0.221232i) q^{79} -4.52015 q^{81} +2.17557 q^{84} +0.618034 q^{89} +0.618034i q^{94} +(1.39680 + 1.39680i) q^{96} +(0.642040 - 0.642040i) q^{97} +1.34458 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} - 4 q^{4} + 4 q^{6} + 2 q^{7} - 4 q^{12} - 6 q^{14} - 2 q^{17} + 4 q^{18} + 4 q^{21} + 2 q^{24} - 6 q^{28} + 2 q^{37} - 8 q^{47} + 4 q^{50} + 8 q^{51} + 10 q^{54} + 2 q^{56} - 2 q^{61} - 8 q^{63}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/799\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.618034i − 0.618034i −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 − 0.309017i \(-0.100000\pi\)
\(3\) −1.39680 + 1.39680i −1.39680 + 1.39680i −0.587785 + 0.809017i \(0.700000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(4\) 0.618034 0.618034
\(5\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0.863271 + 0.863271i 0.863271 + 0.863271i
\(7\) −1.26007 − 1.26007i −1.26007 − 1.26007i −0.951057 − 0.309017i \(-0.900000\pi\)
−0.309017 − 0.951057i \(-0.600000\pi\)
\(8\) − 1.00000i − 1.00000i
\(9\) − 2.90211i − 2.90211i
\(10\) 0 0
\(11\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(12\) −0.863271 + 0.863271i −0.863271 + 0.863271i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −0.778768 + 0.778768i −0.778768 + 0.778768i
\(15\) 0 0
\(16\) 0 0
\(17\) −0.809017 − 0.587785i −0.809017 − 0.587785i
\(18\) −1.79360 −1.79360
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.52015 3.52015
\(22\) 0 0
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(25\) − 1.00000i − 1.00000i
\(26\) 0 0
\(27\) 2.65688 + 2.65688i 2.65688 + 2.65688i
\(28\) −0.778768 − 0.778768i −0.778768 − 0.778768i
\(29\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 0 0
\(34\) −0.363271 + 0.500000i −0.363271 + 0.500000i
\(35\) 0 0
\(36\) − 1.79360i − 1.79360i
\(37\) 1.39680 − 1.39680i 1.39680 − 1.39680i 0.587785 − 0.809017i \(-0.300000\pi\)
0.809017 − 0.587785i \(-0.200000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(42\) − 2.17557i − 2.17557i
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 −1.00000
\(48\) 0 0
\(49\) 2.17557i 2.17557i
\(50\) −0.618034 −0.618034
\(51\) 1.95106 − 0.309017i 1.95106 − 0.309017i
\(52\) 0 0
\(53\) 1.90211i 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(54\) 1.64204 − 1.64204i 1.64204 − 1.64204i
\(55\) 0 0
\(56\) −1.26007 + 1.26007i −1.26007 + 1.26007i
\(57\) 0 0
\(58\) 0 0
\(59\) − 0.618034i − 0.618034i −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 − 0.309017i \(-0.100000\pi\)
\(60\) 0 0
\(61\) −0.642040 − 0.642040i −0.642040 − 0.642040i 0.309017 − 0.951057i \(-0.400000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(62\) 0 0
\(63\) −3.65688 + 3.65688i −3.65688 + 3.65688i
\(64\) −0.618034 −0.618034
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −0.500000 − 0.363271i −0.500000 − 0.363271i
\(69\) 0 0
\(70\) 0 0
\(71\) −0.221232 + 0.221232i −0.221232 + 0.221232i −0.809017 − 0.587785i \(-0.800000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(72\) −2.90211 −2.90211
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) −0.863271 − 0.863271i −0.863271 − 0.863271i
\(75\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.221232 − 0.221232i −0.221232 − 0.221232i 0.587785 − 0.809017i \(-0.300000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(80\) 0 0
\(81\) −4.52015 −4.52015
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 2.17557 2.17557
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.618034 0.618034 0.309017 − 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0.618034i 0.618034i
\(95\) 0 0
\(96\) 1.39680 + 1.39680i 1.39680 + 1.39680i
\(97\) 0.642040 − 0.642040i 0.642040 − 0.642040i −0.309017 − 0.951057i \(-0.600000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(98\) 1.34458 1.34458
\(99\) 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 799.1.e.b.140.1 ✓ 8
17.13 even 4 inner 799.1.e.b.234.3 yes 8
47.46 odd 2 CM 799.1.e.b.140.1 ✓ 8
799.234 odd 4 inner 799.1.e.b.234.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
799.1.e.b.140.1 ✓ 8 1.1 even 1 trivial
799.1.e.b.140.1 ✓ 8 47.46 odd 2 CM
799.1.e.b.234.3 yes 8 17.13 even 4 inner
799.1.e.b.234.3 yes 8 799.234 odd 4 inner