Properties

Label 799.1.h.b.610.3
Level $799$
Weight $1$
Character 799.610
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 610.3
Root \(-0.156434 - 0.987688i\) of defining polynomial
Character \(\chi\) \(=\) 799.610
Dual form 799.1.h.b.93.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.831254 + 0.831254i) q^{2} +(-0.581990 - 1.40505i) q^{3} +0.381966i q^{4} +(0.684170 - 1.65173i) q^{6} +(-1.57547 - 0.652583i) q^{7} +(0.513743 - 0.513743i) q^{8} +(-0.928339 + 0.928339i) q^{9} +(0.536680 - 0.222300i) q^{12} +(-0.767157 - 1.85208i) q^{14} +1.23607 q^{16} +(-0.309017 - 0.951057i) q^{17} -1.54337 q^{18} +2.59341i q^{21} +(-1.02083 - 0.422840i) q^{24} +(0.707107 - 0.707107i) q^{25} +(0.439596 + 0.182086i) q^{27} +(0.249264 - 0.601777i) q^{28} +(0.513743 + 0.513743i) q^{32} +(0.533698 - 1.04744i) q^{34} +(-0.354594 - 0.354594i) q^{36} +(0.497066 + 1.20002i) q^{37} +(-2.15578 + 2.15578i) q^{42} +1.00000i q^{47} +(-0.719379 - 1.73673i) q^{48} +(1.34915 + 1.34915i) q^{49} +1.17557 q^{50} +(-1.15643 + 0.987688i) q^{51} +(0.221232 + 0.221232i) q^{53} +(0.214055 + 0.516776i) q^{54} +(-1.14465 + 0.474129i) q^{56} +(1.14412 - 1.14412i) q^{59} +(1.79671 + 0.744220i) q^{61} +(2.06839 - 0.856755i) q^{63} -0.381966i q^{64} +(0.363271 - 0.118034i) q^{68} +(0.763007 + 1.84206i) q^{71} +0.953855i q^{72} +(-0.584336 + 1.41071i) q^{74} +(-1.40505 - 0.581990i) q^{75} +(0.763007 - 1.84206i) q^{79} +0.589244i q^{81} +(-1.00000 - 1.00000i) q^{83} -0.990595 q^{84} +1.17557i q^{89} +(-0.831254 + 0.831254i) q^{94} +(0.422840 - 1.02083i) q^{96} +(-0.431351 + 0.178671i) q^{97} +2.24297i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 4 q^{9} - 16 q^{16} + 4 q^{17} - 20 q^{24} - 4 q^{27} - 4 q^{28} + 4 q^{36} - 20 q^{42} + 24 q^{48} + 4 q^{49} - 16 q^{51} + 4 q^{53} + 20 q^{54} + 20 q^{56} + 4 q^{61} - 4 q^{63} - 4 q^{71}+ \cdots + 32 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(377\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.831254 + 0.831254i 0.831254 + 0.831254i 0.987688 − 0.156434i \(-0.0500000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(3\) −0.581990 − 1.40505i −0.581990 − 1.40505i −0.891007 − 0.453990i \(-0.850000\pi\)
0.309017 − 0.951057i \(-0.400000\pi\)
\(4\) 0.381966i 0.381966i
\(5\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(6\) 0.684170 − 1.65173i 0.684170 − 1.65173i
\(7\) −1.57547 − 0.652583i −1.57547 − 0.652583i −0.587785 − 0.809017i \(-0.700000\pi\)
−0.987688 + 0.156434i \(0.950000\pi\)
\(8\) 0.513743 − 0.513743i 0.513743 − 0.513743i
\(9\) −0.928339 + 0.928339i −0.928339 + 0.928339i
\(10\) 0 0
\(11\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(12\) 0.536680 − 0.222300i 0.536680 − 0.222300i
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −0.767157 − 1.85208i −0.767157 − 1.85208i
\(15\) 0 0
\(16\) 1.23607 1.23607
\(17\) −0.309017 − 0.951057i −0.309017 − 0.951057i
\(18\) −1.54337 −1.54337
\(19\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 0 0
\(21\) 2.59341i 2.59341i
\(22\) 0 0
\(23\) 0 0 0.382683 − 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(24\) −1.02083 − 0.422840i −1.02083 − 0.422840i
\(25\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(26\) 0 0
\(27\) 0.439596 + 0.182086i 0.439596 + 0.182086i
\(28\) 0.249264 − 0.601777i 0.249264 − 0.601777i
\(29\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.382683 − 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(32\) 0.513743 + 0.513743i 0.513743 + 0.513743i
\(33\) 0 0
\(34\) 0.533698 − 1.04744i 0.533698 − 1.04744i
\(35\) 0 0
\(36\) −0.354594 − 0.354594i −0.354594 − 0.354594i
\(37\) 0.497066 + 1.20002i 0.497066 + 1.20002i 0.951057 + 0.309017i \(0.100000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.923880 − 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(42\) −2.15578 + 2.15578i −2.15578 + 2.15578i
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000i 1.00000i
\(48\) −0.719379 − 1.73673i −0.719379 − 1.73673i
\(49\) 1.34915 + 1.34915i 1.34915 + 1.34915i
\(50\) 1.17557 1.17557
\(51\) −1.15643 + 0.987688i −1.15643 + 0.987688i
\(52\) 0 0
\(53\) 0.221232 + 0.221232i 0.221232 + 0.221232i 0.809017 − 0.587785i \(-0.200000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(54\) 0.214055 + 0.516776i 0.214055 + 0.516776i
\(55\) 0 0
\(56\) −1.14465 + 0.474129i −1.14465 + 0.474129i
\(57\) 0 0
\(58\) 0 0
\(59\) 1.14412 − 1.14412i 1.14412 − 1.14412i 0.156434 − 0.987688i \(-0.450000\pi\)
0.987688 − 0.156434i \(-0.0500000\pi\)
\(60\) 0 0
\(61\) 1.79671 + 0.744220i 1.79671 + 0.744220i 0.987688 + 0.156434i \(0.0500000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(62\) 0 0
\(63\) 2.06839 − 0.856755i 2.06839 − 0.856755i
\(64\) − 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
\(71\) 0.763007 + 1.84206i 0.763007 + 1.84206i 0.453990 + 0.891007i \(0.350000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(72\) 0.953855i 0.953855i
\(73\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(74\) −0.584336 + 1.41071i −0.584336 + 1.41071i
\(75\) −1.40505 − 0.581990i −1.40505 − 0.581990i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.763007 − 1.84206i 0.763007 − 1.84206i 0.309017 − 0.951057i \(-0.400000\pi\)
0.453990 − 0.891007i \(-0.350000\pi\)
\(80\) 0 0
\(81\) 0.589244i 0.589244i
\(82\) 0 0
\(83\) −1.00000 − 1.00000i −1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(84\) −0.990595 −0.990595
\(85\) 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
\(96\) 0.422840 − 1.02083i 0.422840 − 1.02083i
\(97\) −0.431351 + 0.178671i −0.431351 + 0.178671i −0.587785 − 0.809017i \(-0.700000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(98\) 2.24297i 2.24297i
\(99\) 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.610.3 yes 16
17.8 even 8 inner 799.1.h.b.93.3 ✓ 16
47.46 odd 2 CM 799.1.h.b.610.3 yes 16
799.93 odd 8 inner 799.1.h.b.93.3 ✓ 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
799.1.h.b.93.3 ✓ 16 17.8 even 8 inner
799.1.h.b.93.3 ✓ 16 799.93 odd 8 inner
799.1.h.b.610.3 yes 16 1.1 even 1 trivial
799.1.h.b.610.3 yes 16 47.46 odd 2 CM