Properties

Label 799.1.h.b.281.3
Level $799$
Weight $1$
Character 799.281
Analytic conductor $0.399$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -47
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [799,1,Mod(93,799)] 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("799.93"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(799, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([5, 4])) 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.h (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{40})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

Embedding invariants

Embedding label 281.3
Root \(0.987688 - 0.156434i\) of defining polynomial
Character \(\chi\) \(=\) 799.281
Dual form 799.1.h.b.563.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.831254 - 0.831254i) q^{2} +(-0.144974 - 0.0600500i) q^{3} -0.381966i q^{4} +(-0.170427 + 0.0705930i) q^{6} +(0.744220 + 1.79671i) q^{7} +(0.513743 + 0.513743i) q^{8} +(-0.689695 - 0.689695i) q^{9} +(-0.0229371 + 0.0553750i) q^{12} +(2.11215 + 0.874883i) q^{14} +1.23607 q^{16} +(-0.309017 - 0.951057i) q^{17} -1.14662 q^{18} -0.305165i q^{21} +(-0.0436289 - 0.105329i) q^{24} +(-0.707107 - 0.707107i) q^{25} +(0.118621 + 0.286377i) q^{27} +(0.686280 - 0.284267i) q^{28} +(0.513743 - 0.513743i) q^{32} +(-1.04744 - 0.533698i) q^{34} +(-0.263440 + 0.263440i) q^{36} +(-1.84206 - 0.763007i) q^{37} +(-0.253670 - 0.253670i) q^{42} -1.00000i q^{47} +(-0.179197 - 0.0742259i) q^{48} +(-1.96718 + 1.96718i) q^{49} -1.17557 q^{50} +(-0.0123117 + 0.156434i) q^{51} +(1.39680 - 1.39680i) q^{53} +(0.336657 + 0.139448i) q^{54} +(-0.540707 + 1.30538i) q^{56} +(-1.14412 - 1.14412i) q^{59} +(0.652583 + 1.57547i) q^{61} +(0.725895 - 1.75246i) q^{63} +0.381966i q^{64} +(-0.363271 + 0.118034i) q^{68} +(1.20002 + 0.497066i) q^{71} -0.708653i q^{72} +(-2.16547 + 0.896969i) q^{74} +(0.0600500 + 0.144974i) q^{75} +(1.20002 - 0.497066i) q^{79} +0.926736i q^{81} +(-1.00000 + 1.00000i) q^{83} -0.116563 q^{84} +1.17557i q^{89} +(-0.831254 - 0.831254i) q^{94} +(-0.105329 + 0.0436289i) q^{96} +(-0.399903 + 0.965451i) q^{97} +3.27045i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.831254 − 0.831254i 0.831254 − 0.831254i −0.156434 − 0.987688i \(-0.550000\pi\)
0.987688 + 0.156434i \(0.0500000\pi\)
\(3\) −0.144974 − 0.0600500i −0.144974 − 0.0600500i 0.309017 − 0.951057i \(-0.400000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(4\) − 0.381966i − 0.381966i
\(5\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(6\) −0.170427 + 0.0705930i −0.170427 + 0.0705930i
\(7\) 0.744220 + 1.79671i 0.744220 + 1.79671i 0.587785 + 0.809017i \(0.300000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(8\) 0.513743 + 0.513743i 0.513743 + 0.513743i
\(9\) −0.689695 − 0.689695i −0.689695 − 0.689695i
\(10\) 0 0
\(11\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(12\) −0.0229371 + 0.0553750i −0.0229371 + 0.0553750i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.11215 + 0.874883i 2.11215 + 0.874883i
\(15\) 0 0
\(16\) 1.23607 1.23607
\(17\) −0.309017 − 0.951057i −0.309017 − 0.951057i
\(18\) −1.14662 −1.14662
\(19\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0 0
\(21\) − 0.305165i − 0.305165i
\(22\) 0 0
\(23\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(24\) −0.0436289 − 0.105329i −0.0436289 − 0.105329i
\(25\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(26\) 0 0
\(27\) 0.118621 + 0.286377i 0.118621 + 0.286377i
\(28\) 0.686280 − 0.284267i 0.686280 − 0.284267i
\(29\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(32\) 0.513743 − 0.513743i 0.513743 − 0.513743i
\(33\) 0 0
\(34\) −1.04744 − 0.533698i −1.04744 − 0.533698i
\(35\) 0 0
\(36\) −0.263440 + 0.263440i −0.263440 + 0.263440i
\(37\) −1.84206 − 0.763007i −1.84206 − 0.763007i −0.951057 − 0.309017i \(-0.900000\pi\)
−0.891007 − 0.453990i \(-0.850000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(42\) −0.253670 − 0.253670i −0.253670 − 0.253670i
\(43\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.00000i − 1.00000i
\(48\) −0.179197 − 0.0742259i −0.179197 − 0.0742259i
\(49\) −1.96718 + 1.96718i −1.96718 + 1.96718i
\(50\) −1.17557 −1.17557
\(51\) −0.0123117 + 0.156434i −0.0123117 + 0.156434i
\(52\) 0 0
\(53\) 1.39680 − 1.39680i 1.39680 − 1.39680i 0.587785 − 0.809017i \(-0.300000\pi\)
0.809017 − 0.587785i \(-0.200000\pi\)
\(54\) 0.336657 + 0.139448i 0.336657 + 0.139448i
\(55\) 0 0
\(56\) −0.540707 + 1.30538i −0.540707 + 1.30538i
\(57\) 0 0
\(58\) 0 0
\(59\) −1.14412 − 1.14412i −1.14412 − 1.14412i −0.987688 − 0.156434i \(-0.950000\pi\)
−0.156434 − 0.987688i \(-0.550000\pi\)
\(60\) 0 0
\(61\) 0.652583 + 1.57547i 0.652583 + 1.57547i 0.809017 + 0.587785i \(0.200000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(62\) 0 0
\(63\) 0.725895 − 1.75246i 0.725895 − 1.75246i
\(64\) 0.381966i 0.381966i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −0.363271 + 0.118034i −0.363271 + 0.118034i
\(69\) 0 0
\(70\) 0 0
\(71\) 1.20002 + 0.497066i 1.20002 + 0.497066i 0.891007 − 0.453990i \(-0.150000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(72\) − 0.708653i − 0.708653i
\(73\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(74\) −2.16547 + 0.896969i −2.16547 + 0.896969i
\(75\) 0.0600500 + 0.144974i 0.0600500 + 0.144974i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.20002 − 0.497066i 1.20002 − 0.497066i 0.309017 − 0.951057i \(-0.400000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(80\) 0 0
\(81\) 0.926736i 0.926736i
\(82\) 0 0
\(83\) −1.00000 + 1.00000i −1.00000 + 1.00000i 1.00000i \(0.5\pi\)
−1.00000 \(\pi\)
\(84\) −0.116563 −0.116563
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.17557i 1.17557i 0.809017 + 0.587785i \(0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −0.831254 − 0.831254i −0.831254 − 0.831254i
\(95\) 0 0
\(96\) −0.105329 + 0.0436289i −0.105329 + 0.0436289i
\(97\) −0.399903 + 0.965451i −0.399903 + 0.965451i 0.587785 + 0.809017i \(0.300000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(98\) 3.27045i 3.27045i
\(99\) 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.h.b.281.3 ✓ 16
17.2 even 8 inner 799.1.h.b.563.3 yes 16
47.46 odd 2 CM 799.1.h.b.281.3 ✓ 16
799.563 odd 8 inner 799.1.h.b.563.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
799.1.h.b.281.3 ✓ 16 1.1 even 1 trivial
799.1.h.b.281.3 ✓ 16 47.46 odd 2 CM
799.1.h.b.563.3 yes 16 17.2 even 8 inner
799.1.h.b.563.3 yes 16 799.563 odd 8 inner