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Properties

Label 784.2.a.m.1.1
Level $784$
Weight $2$
Character 784.1
Self dual yes
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [784,2,Mod(1,784)] 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("784.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,10,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} -4.00000 q^{11} -4.24264 q^{13} +4.00000 q^{15} -1.41421 q^{17} +2.82843 q^{19} +4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +8.00000 q^{29} +11.3137 q^{33} -8.00000 q^{37} +12.0000 q^{39} +7.07107 q^{41} +4.00000 q^{43} -7.07107 q^{45} +5.65685 q^{47} +4.00000 q^{51} +10.0000 q^{53} +5.65685 q^{55} -8.00000 q^{57} +14.1421 q^{59} +7.07107 q^{61} +6.00000 q^{65} -11.3137 q^{69} +7.07107 q^{73} +8.48528 q^{75} -8.00000 q^{79} +1.00000 q^{81} -14.1421 q^{83} +2.00000 q^{85} -22.6274 q^{87} -7.07107 q^{89} -4.00000 q^{95} +1.41421 q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{9} - 8 q^{11} + 8 q^{15} + 8 q^{23} - 6 q^{25} + 16 q^{29} - 16 q^{37} + 24 q^{39} + 8 q^{43} + 8 q^{51} + 20 q^{53} - 16 q^{57} + 12 q^{65} - 16 q^{79} + 2 q^{81} + 4 q^{85} - 8 q^{95} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.82843 −1.63299 −0.816497 − 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) −1.41421 −0.632456 −0.316228 − 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 − 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −1.41421 −0.342997 −0.171499 − 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 11.3137 1.96946
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 12.0000 1.92154
\(40\) 0 0
\(41\) 7.07107 1.10432 0.552158 − 0.833740i \(-0.313805\pi\)
0.552158 + 0.833740i \(0.313805\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −7.07107 −1.05409
\(46\) 0 0
\(47\) 5.65685 0.825137 0.412568 − 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 14.1421 1.84115 0.920575 − 0.390567i \(-0.127721\pi\)
0.920575 + 0.390567i \(0.127721\pi\)
\(60\) 0 0
\(61\) 7.07107 0.905357 0.452679 − 0.891674i \(-0.350468\pi\)
0.452679 + 0.891674i \(0.350468\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −11.3137 −1.36201
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 7.07107 0.827606 0.413803 − 0.910366i \(-0.364200\pi\)
0.413803 + 0.910366i \(0.364200\pi\)
\(74\) 0 0
\(75\) 8.48528 0.979796
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.1421 −1.55230 −0.776151 − 0.630548i \(-0.782830\pi\)
−0.776151 + 0.630548i \(0.782830\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −22.6274 −2.42591
\(88\) 0 0
\(89\) −7.07107 −0.749532 −0.374766 − 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 1.41421 0.143592 0.0717958 − 0.997419i \(-0.477127\pi\)
0.0717958 + 0.997419i \(0.477127\pi\)
\(98\) 0 0
\(99\) −20.0000 −2.01008
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 784.2.a.m.1.1 2
3.2 odd 2 7056.2.a.cr.1.2 2
4.3 odd 2 196.2.a.c.1.2 yes 2
7.2 even 3 784.2.i.l.753.2 4
7.3 odd 6 784.2.i.l.177.1 4
7.4 even 3 784.2.i.l.177.2 4
7.5 odd 6 784.2.i.l.753.1 4
7.6 odd 2 inner 784.2.a.m.1.2 2
8.3 odd 2 3136.2.a.br.1.1 2
8.5 even 2 3136.2.a.bs.1.2 2
12.11 even 2 1764.2.a.l.1.2 2
20.3 even 4 4900.2.e.p.2549.4 4
20.7 even 4 4900.2.e.p.2549.2 4
20.19 odd 2 4900.2.a.y.1.1 2
21.20 even 2 7056.2.a.cr.1.1 2
28.3 even 6 196.2.e.b.177.2 4
28.11 odd 6 196.2.e.b.177.1 4
28.19 even 6 196.2.e.b.165.2 4
28.23 odd 6 196.2.e.b.165.1 4
28.27 even 2 196.2.a.c.1.1 ✓ 2
56.13 odd 2 3136.2.a.bs.1.1 2
56.27 even 2 3136.2.a.br.1.2 2
84.11 even 6 1764.2.k.l.1549.1 4
84.23 even 6 1764.2.k.l.361.1 4
84.47 odd 6 1764.2.k.l.361.2 4
84.59 odd 6 1764.2.k.l.1549.2 4
84.83 odd 2 1764.2.a.l.1.1 2
140.27 odd 4 4900.2.e.p.2549.3 4
140.83 odd 4 4900.2.e.p.2549.1 4
140.139 even 2 4900.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.2.a.c.1.1 ✓ 2 28.27 even 2
196.2.a.c.1.2 yes 2 4.3 odd 2
196.2.e.b.165.1 4 28.23 odd 6
196.2.e.b.165.2 4 28.19 even 6
196.2.e.b.177.1 4 28.11 odd 6
196.2.e.b.177.2 4 28.3 even 6
784.2.a.m.1.1 2 1.1 even 1 trivial
784.2.a.m.1.2 2 7.6 odd 2 inner
784.2.i.l.177.1 4 7.3 odd 6
784.2.i.l.177.2 4 7.4 even 3
784.2.i.l.753.1 4 7.5 odd 6
784.2.i.l.753.2 4 7.2 even 3
1764.2.a.l.1.1 2 84.83 odd 2
1764.2.a.l.1.2 2 12.11 even 2
1764.2.k.l.361.1 4 84.23 even 6
1764.2.k.l.361.2 4 84.47 odd 6
1764.2.k.l.1549.1 4 84.11 even 6
1764.2.k.l.1549.2 4 84.59 odd 6
3136.2.a.br.1.1 2 8.3 odd 2
3136.2.a.br.1.2 2 56.27 even 2
3136.2.a.bs.1.1 2 56.13 odd 2
3136.2.a.bs.1.2 2 8.5 even 2
4900.2.a.y.1.1 2 20.19 odd 2
4900.2.a.y.1.2 2 140.139 even 2
4900.2.e.p.2549.1 4 140.83 odd 4
4900.2.e.p.2549.2 4 20.7 even 4
4900.2.e.p.2549.3 4 140.27 odd 4
4900.2.e.p.2549.4 4 20.3 even 4
7056.2.a.cr.1.1 2 21.20 even 2
7056.2.a.cr.1.2 2 3.2 odd 2