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Properties

Label 7168.2.a.b.1.1
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
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] = [7168,2,Mod(1,7168)] 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("7168.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7168 = 2^{10} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7168.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-2,0,-6,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.2367681689\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 7168.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843 q^{5} -1.00000 q^{7} -3.00000 q^{9} +1.41421 q^{11} +2.00000 q^{17} -2.82843 q^{19} +6.00000 q^{23} +3.00000 q^{25} +9.89949 q^{29} -8.00000 q^{31} +2.82843 q^{35} +7.07107 q^{37} -10.0000 q^{41} -1.41421 q^{43} +8.48528 q^{45} +12.0000 q^{47} +1.00000 q^{49} -1.41421 q^{53} -4.00000 q^{55} -11.3137 q^{59} +8.48528 q^{61} +3.00000 q^{63} +4.24264 q^{67} +6.00000 q^{73} -1.41421 q^{77} -10.0000 q^{79} +9.00000 q^{81} +14.1421 q^{83} -5.65685 q^{85} -14.0000 q^{89} +8.00000 q^{95} -2.00000 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 6 q^{9} + 4 q^{17} + 12 q^{23} + 6 q^{25} - 16 q^{31} - 20 q^{41} + 24 q^{47} + 2 q^{49} - 8 q^{55} + 6 q^{63} + 12 q^{73} - 20 q^{79} + 18 q^{81} - 28 q^{89} + 16 q^{95} - 4 q^{97}+O(q^{100}) \) 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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −2.82843 −1.26491 −0.632456 − 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 − 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 − 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.89949 1.83829 0.919145 − 0.393919i \(-0.128881\pi\)
0.919145 + 0.393919i \(0.128881\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.82843 0.478091
\(36\) 0 0
\(37\) 7.07107 1.16248 0.581238 − 0.813733i \(-0.302568\pi\)
0.581238 + 0.813733i \(0.302568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −1.41421 −0.215666 −0.107833 − 0.994169i \(-0.534391\pi\)
−0.107833 + 0.994169i \(0.534391\pi\)
\(44\) 0 0
\(45\) 8.48528 1.26491
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41421 −0.194257 −0.0971286 − 0.995272i \(-0.530966\pi\)
−0.0971286 + 0.995272i \(0.530966\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 − 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 8.48528 1.08643 0.543214 − 0.839594i \(-0.317207\pi\)
0.543214 + 0.839594i \(0.317207\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 − 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.41421 −0.161165
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 14.1421 1.55230 0.776151 − 0.630548i \(-0.217170\pi\)
0.776151 + 0.630548i \(0.217170\pi\)
\(84\) 0 0
\(85\) −5.65685 −0.613572
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −4.24264 −0.426401
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 7168.2.a.b.1.1 2
4.3 odd 2 7168.2.a.k.1.1 2
8.3 odd 2 7168.2.a.k.1.2 2
8.5 even 2 inner 7168.2.a.b.1.2 2
32.3 odd 8 448.2.m.a.337.1 2
32.5 even 8 896.2.m.b.225.1 2
32.11 odd 8 448.2.m.a.113.1 2
32.13 even 8 896.2.m.b.673.1 2
32.19 odd 8 896.2.m.c.673.1 2
32.21 even 8 112.2.m.b.85.1 yes 2
32.27 odd 8 896.2.m.c.225.1 2
32.29 even 8 112.2.m.b.29.1 ✓ 2
224.53 even 24 784.2.x.d.373.1 4
224.61 odd 24 784.2.x.e.557.1 4
224.93 even 24 784.2.x.d.557.1 4
224.117 odd 24 784.2.x.e.165.1 4
224.125 odd 8 784.2.m.a.589.1 2
224.149 even 24 784.2.x.d.165.1 4
224.157 odd 24 784.2.x.e.765.1 4
224.181 odd 8 784.2.m.a.197.1 2
224.213 odd 24 784.2.x.e.373.1 4
224.221 even 24 784.2.x.d.765.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.b.29.1 ✓ 2 32.29 even 8
112.2.m.b.85.1 yes 2 32.21 even 8
448.2.m.a.113.1 2 32.11 odd 8
448.2.m.a.337.1 2 32.3 odd 8
784.2.m.a.197.1 2 224.181 odd 8
784.2.m.a.589.1 2 224.125 odd 8
784.2.x.d.165.1 4 224.149 even 24
784.2.x.d.373.1 4 224.53 even 24
784.2.x.d.557.1 4 224.93 even 24
784.2.x.d.765.1 4 224.221 even 24
784.2.x.e.165.1 4 224.117 odd 24
784.2.x.e.373.1 4 224.213 odd 24
784.2.x.e.557.1 4 224.61 odd 24
784.2.x.e.765.1 4 224.157 odd 24
896.2.m.b.225.1 2 32.5 even 8
896.2.m.b.673.1 2 32.13 even 8
896.2.m.c.225.1 2 32.27 odd 8
896.2.m.c.673.1 2 32.19 odd 8
7168.2.a.b.1.1 2 1.1 even 1 trivial
7168.2.a.b.1.2 2 8.5 even 2 inner
7168.2.a.k.1.1 2 4.3 odd 2
7168.2.a.k.1.2 2 8.3 odd 2