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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(703,1408)] 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("1408.703"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2429366046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 9x^{2} - 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 703.1
Root \(0.500000 - 2.73709i\) of defining polynomial
Character \(\chi\) \(=\) 1408.703
Dual form 1408.2.g.a.703.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.64575i q^{5} -3.74166 q^{7} -2.00000 q^{9} +(3.00000 + 1.41421i) q^{11} -3.74166 q^{13} +2.64575i q^{15} -4.24264i q^{17} +4.24264i q^{19} +3.74166 q^{21} -2.64575i q^{23} -2.00000 q^{25} +5.00000 q^{27} +7.93725i q^{31} +(-3.00000 - 1.41421i) q^{33} +9.89949i q^{35} +7.93725i q^{37} +3.74166 q^{39} -1.41421i q^{41} +8.48528i q^{43} +5.29150i q^{45} -10.5830i q^{47} +7.00000 q^{49} +4.24264i q^{51} -5.29150i q^{53} +(3.74166 - 7.93725i) q^{55} -4.24264i q^{57} +9.00000 q^{59} +14.9666 q^{61} +7.48331 q^{63} +9.89949i q^{65} -3.00000 q^{67} +2.64575i q^{69} +2.64575i q^{71} +12.7279i q^{73} +2.00000 q^{75} +(-11.2250 - 5.29150i) q^{77} -7.48331 q^{79} +1.00000 q^{81} +8.48528i q^{83} -11.2250 q^{85} +3.00000 q^{89} +14.0000 q^{91} -7.93725i q^{93} +11.2250 q^{95} -3.00000 q^{97} +(-6.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{9} + 12 q^{11} - 8 q^{25} + 20 q^{27} - 12 q^{33} + 28 q^{49} + 36 q^{59} - 12 q^{67} + 8 q^{75} + 4 q^{81} + 12 q^{89} + 56 q^{91} - 12 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1408\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(639\) \(1025\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 0
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) − 2.64575i − 1.18322i −0.806226 − 0.591608i \(-0.798493\pi\)
0.806226 − 0.591608i \(-0.201507\pi\)
\(6\) 0 0
\(7\) −3.74166 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 3.00000 + 1.41421i 0.904534 + 0.426401i
\(12\) 0 0
\(13\) −3.74166 −1.03775 −0.518875 − 0.854850i \(-0.673649\pi\)
−0.518875 + 0.854850i \(0.673649\pi\)
\(14\) 0 0
\(15\) 2.64575i 0.683130i
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 − 0.514496i \(-0.827979\pi\)
0.857493 − 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) 4.24264i 0.973329i 0.873589 + 0.486664i \(0.161786\pi\)
−0.873589 + 0.486664i \(0.838214\pi\)
\(20\) 0 0
\(21\) 3.74166 0.816497
\(22\) 0 0
\(23\) − 2.64575i − 0.551677i −0.961204 − 0.275839i \(-0.911044\pi\)
0.961204 − 0.275839i \(-0.0889555\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7.93725i 1.42557i 0.701381 + 0.712786i \(0.252567\pi\)
−0.701381 + 0.712786i \(0.747433\pi\)
\(32\) 0 0
\(33\) −3.00000 − 1.41421i −0.522233 − 0.246183i
\(34\) 0 0
\(35\) 9.89949i 1.67332i
\(36\) 0 0
\(37\) 7.93725i 1.30488i 0.757842 + 0.652438i \(0.226254\pi\)
−0.757842 + 0.652438i \(0.773746\pi\)
\(38\) 0 0
\(39\) 3.74166 0.599145
\(40\) 0 0
\(41\) − 1.41421i − 0.220863i −0.993884 − 0.110432i \(-0.964777\pi\)
0.993884 − 0.110432i \(-0.0352233\pi\)
\(42\) 0 0
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 5.29150i 0.788811i
\(46\) 0 0
\(47\) − 10.5830i − 1.54369i −0.635811 − 0.771845i \(-0.719334\pi\)
0.635811 − 0.771845i \(-0.280666\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.24264i 0.594089i
\(52\) 0 0
\(53\) − 5.29150i − 0.726844i −0.931625 − 0.363422i \(-0.881608\pi\)
0.931625 − 0.363422i \(-0.118392\pi\)
\(54\) 0 0
\(55\) 3.74166 − 7.93725i 0.504525 − 1.07026i
\(56\) 0 0
\(57\) − 4.24264i − 0.561951i
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 − 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) 14.9666 1.91628 0.958140 − 0.286299i \(-0.0924251\pi\)
0.958140 + 0.286299i \(0.0924251\pi\)
\(62\) 0 0
\(63\) 7.48331 0.942809
\(64\) 0 0
\(65\) 9.89949i 1.22788i
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 − 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) 2.64575i 0.318511i
\(70\) 0 0
\(71\) 2.64575i 0.313993i 0.987599 + 0.156996i \(0.0501811\pi\)
−0.987599 + 0.156996i \(0.949819\pi\)
\(72\) 0 0
\(73\) 12.7279i 1.48969i 0.667237 + 0.744845i \(0.267477\pi\)
−0.667237 + 0.744845i \(0.732523\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) −11.2250 − 5.29150i −1.27920 − 0.603023i
\(78\) 0 0
\(79\) −7.48331 −0.841939 −0.420969 − 0.907075i \(-0.638310\pi\)
−0.420969 + 0.907075i \(0.638310\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.48528i 0.931381i 0.884948 + 0.465690i \(0.154194\pi\)
−0.884948 + 0.465690i \(0.845806\pi\)
\(84\) 0 0
\(85\) −11.2250 −1.21752
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 14.0000 1.46760
\(92\) 0 0
\(93\) − 7.93725i − 0.823055i
\(94\) 0 0
\(95\) 11.2250 1.15166
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 − 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 0 0
\(99\) −6.00000 − 2.82843i −0.603023 − 0.284268i
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 1408.2.g.a.703.1 ✓ 4
4.3 odd 2 1408.2.g.d.703.2 yes 4
8.3 odd 2 inner 1408.2.g.a.703.4 yes 4
8.5 even 2 1408.2.g.d.703.3 yes 4
11.10 odd 2 inner 1408.2.g.a.703.2 yes 4
16.3 odd 4 2816.2.e.n.2815.5 8
16.5 even 4 2816.2.e.n.2815.8 8
16.11 odd 4 2816.2.e.n.2815.3 8
16.13 even 4 2816.2.e.n.2815.2 8
44.43 even 2 1408.2.g.d.703.1 yes 4
88.21 odd 2 1408.2.g.d.703.4 yes 4
88.43 even 2 inner 1408.2.g.a.703.3 yes 4
176.21 odd 4 2816.2.e.n.2815.7 8
176.43 even 4 2816.2.e.n.2815.4 8
176.109 odd 4 2816.2.e.n.2815.1 8
176.131 even 4 2816.2.e.n.2815.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1408.2.g.a.703.1 ✓ 4 1.1 even 1 trivial
1408.2.g.a.703.2 yes 4 11.10 odd 2 inner
1408.2.g.a.703.3 yes 4 88.43 even 2 inner
1408.2.g.a.703.4 yes 4 8.3 odd 2 inner
1408.2.g.d.703.1 yes 4 44.43 even 2
1408.2.g.d.703.2 yes 4 4.3 odd 2
1408.2.g.d.703.3 yes 4 8.5 even 2
1408.2.g.d.703.4 yes 4 88.21 odd 2
2816.2.e.n.2815.1 8 176.109 odd 4
2816.2.e.n.2815.2 8 16.13 even 4
2816.2.e.n.2815.3 8 16.11 odd 4
2816.2.e.n.2815.4 8 176.43 even 4
2816.2.e.n.2815.5 8 16.3 odd 4
2816.2.e.n.2815.6 8 176.131 even 4
2816.2.e.n.2815.7 8 176.21 odd 4
2816.2.e.n.2815.8 8 16.5 even 4