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Properties

Label 7168.2.a.bd.1.2
Level $7168$
Weight $2$
Character 7168.1
Self dual yes
Analytic conductor $57.237$
Analytic rank $1$
Dimension $8$
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 = [8,0,0,0,0,0,8,0,8,0,0,0,0,0,-8,0,-24] 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: \(8\)
Coefficient field: 8.8.13747093504.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 38x^{4} - 20x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 112)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.236253\) 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)
 
\(f(q)\) \(=\) \(q-2.08185 q^{3} -0.500437 q^{5} +1.00000 q^{7} +1.33411 q^{9} -0.472505 q^{11} +4.49694 q^{13} +1.04184 q^{15} -1.29227 q^{17} +3.72244 q^{19} -2.08185 q^{21} +2.61007 q^{23} -4.74956 q^{25} +3.46813 q^{27} -9.87973 q^{29} -8.35964 q^{31} +0.983687 q^{33} -0.500437 q^{35} +4.69538 q^{37} -9.36197 q^{39} -9.93254 q^{41} +10.7656 q^{43} -0.667639 q^{45} -4.59609 q^{47} +1.00000 q^{49} +2.69033 q^{51} -5.57792 q^{53} +0.236459 q^{55} -7.74956 q^{57} -9.04489 q^{59} +6.15729 q^{61} +1.33411 q^{63} -2.25044 q^{65} +14.1026 q^{67} -5.43379 q^{69} +7.62395 q^{71} +0.556593 q^{73} +9.88789 q^{75} -0.472505 q^{77} -1.48923 q^{79} -11.2225 q^{81} +2.08185 q^{83} +0.646703 q^{85} +20.5681 q^{87} +12.2922 q^{89} +4.49694 q^{91} +17.4035 q^{93} -1.86285 q^{95} +5.50078 q^{97} -0.630375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7} + 8 q^{9} - 8 q^{15} - 24 q^{17} - 8 q^{25} - 16 q^{31} - 24 q^{33} - 16 q^{39} - 8 q^{41} + 24 q^{47} + 8 q^{49} - 8 q^{55} - 32 q^{57} + 8 q^{63} - 48 q^{65} - 56 q^{71} - 48 q^{73} - 24 q^{79}+ \cdots - 32 q^{97}+O(q^{100}) \) 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
\(3\) −2.08185 −1.20196 −0.600979 − 0.799265i \(-0.705223\pi\)
−0.600979 + 0.799265i \(0.705223\pi\)
\(4\) 0 0
\(5\) −0.500437 −0.223802 −0.111901 − 0.993719i \(-0.535694\pi\)
−0.111901 + 0.993719i \(0.535694\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.33411 0.444704
\(10\) 0 0
\(11\) −0.472505 −0.142466 −0.0712329 − 0.997460i \(-0.522693\pi\)
−0.0712329 + 0.997460i \(0.522693\pi\)
\(12\) 0 0
\(13\) 4.49694 1.24723 0.623614 − 0.781733i \(-0.285664\pi\)
0.623614 + 0.781733i \(0.285664\pi\)
\(14\) 0 0
\(15\) 1.04184 0.269001
\(16\) 0 0
\(17\) −1.29227 −0.313423 −0.156711 − 0.987644i \(-0.550089\pi\)
−0.156711 + 0.987644i \(0.550089\pi\)
\(18\) 0 0
\(19\) 3.72244 0.853985 0.426993 − 0.904255i \(-0.359573\pi\)
0.426993 + 0.904255i \(0.359573\pi\)
\(20\) 0 0
\(21\) −2.08185 −0.454298
\(22\) 0 0
\(23\) 2.61007 0.544238 0.272119 − 0.962264i \(-0.412276\pi\)
0.272119 + 0.962264i \(0.412276\pi\)
\(24\) 0 0
\(25\) −4.74956 −0.949912
\(26\) 0 0
\(27\) 3.46813 0.667443
\(28\) 0 0
\(29\) −9.87973 −1.83462 −0.917310 − 0.398175i \(-0.869644\pi\)
−0.917310 + 0.398175i \(0.869644\pi\)
\(30\) 0 0
\(31\) −8.35964 −1.50143 −0.750717 − 0.660623i \(-0.770292\pi\)
−0.750717 + 0.660623i \(0.770292\pi\)
\(32\) 0 0
\(33\) 0.983687 0.171238
\(34\) 0 0
\(35\) −0.500437 −0.0845894
\(36\) 0 0
\(37\) 4.69538 0.771915 0.385958 − 0.922516i \(-0.373871\pi\)
0.385958 + 0.922516i \(0.373871\pi\)
\(38\) 0 0
\(39\) −9.36197 −1.49912
\(40\) 0 0
\(41\) −9.93254 −1.55120 −0.775601 − 0.631223i \(-0.782553\pi\)
−0.775601 + 0.631223i \(0.782553\pi\)
\(42\) 0 0
\(43\) 10.7656 1.64173 0.820867 − 0.571119i \(-0.193491\pi\)
0.820867 + 0.571119i \(0.193491\pi\)
\(44\) 0 0
\(45\) −0.667639 −0.0995258
\(46\) 0 0
\(47\) −4.59609 −0.670409 −0.335205 − 0.942145i \(-0.608805\pi\)
−0.335205 + 0.942145i \(0.608805\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.69033 0.376721
\(52\) 0 0
\(53\) −5.57792 −0.766186 −0.383093 − 0.923710i \(-0.625141\pi\)
−0.383093 + 0.923710i \(0.625141\pi\)
\(54\) 0 0
\(55\) 0.236459 0.0318842
\(56\) 0 0
\(57\) −7.74956 −1.02645
\(58\) 0 0
\(59\) −9.04489 −1.17754 −0.588772 − 0.808299i \(-0.700388\pi\)
−0.588772 + 0.808299i \(0.700388\pi\)
\(60\) 0 0
\(61\) 6.15729 0.788360 0.394180 − 0.919033i \(-0.371029\pi\)
0.394180 + 0.919033i \(0.371029\pi\)
\(62\) 0 0
\(63\) 1.33411 0.168082
\(64\) 0 0
\(65\) −2.25044 −0.279132
\(66\) 0 0
\(67\) 14.1026 1.72291 0.861453 − 0.507837i \(-0.169555\pi\)
0.861453 + 0.507837i \(0.169555\pi\)
\(68\) 0 0
\(69\) −5.43379 −0.654151
\(70\) 0 0
\(71\) 7.62395 0.904797 0.452398 − 0.891816i \(-0.350569\pi\)
0.452398 + 0.891816i \(0.350569\pi\)
\(72\) 0 0
\(73\) 0.556593 0.0651443 0.0325721 − 0.999469i \(-0.489630\pi\)
0.0325721 + 0.999469i \(0.489630\pi\)
\(74\) 0 0
\(75\) 9.88789 1.14176
\(76\) 0 0
\(77\) −0.472505 −0.0538470
\(78\) 0 0
\(79\) −1.48923 −0.167552 −0.0837758 − 0.996485i \(-0.526698\pi\)
−0.0837758 + 0.996485i \(0.526698\pi\)
\(80\) 0 0
\(81\) −11.2225 −1.24694
\(82\) 0 0
\(83\) 2.08185 0.228513 0.114257 − 0.993451i \(-0.463551\pi\)
0.114257 + 0.993451i \(0.463551\pi\)
\(84\) 0 0
\(85\) 0.646703 0.0701447
\(86\) 0 0
\(87\) 20.5681 2.20514
\(88\) 0 0
\(89\) 12.2922 1.30297 0.651484 − 0.758662i \(-0.274147\pi\)
0.651484 + 0.758662i \(0.274147\pi\)
\(90\) 0 0
\(91\) 4.49694 0.471408
\(92\) 0 0
\(93\) 17.4035 1.80466
\(94\) 0 0
\(95\) −1.86285 −0.191124
\(96\) 0 0
\(97\) 5.50078 0.558519 0.279260 − 0.960216i \(-0.409911\pi\)
0.279260 + 0.960216i \(0.409911\pi\)
\(98\) 0 0
\(99\) −0.630375 −0.0633551
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.bd.1.2 8
4.3 odd 2 7168.2.a.bc.1.7 8
8.3 odd 2 7168.2.a.bc.1.2 8
8.5 even 2 inner 7168.2.a.bd.1.7 8
32.3 odd 8 112.2.m.c.29.2 ✓ 8
32.5 even 8 896.2.m.f.225.4 8
32.11 odd 8 112.2.m.c.85.2 yes 8
32.13 even 8 896.2.m.f.673.4 8
32.19 odd 8 896.2.m.e.673.1 8
32.21 even 8 448.2.m.c.113.1 8
32.27 odd 8 896.2.m.e.225.1 8
32.29 even 8 448.2.m.c.337.1 8
224.3 even 24 784.2.x.j.765.1 16
224.11 odd 24 784.2.x.k.373.4 16
224.67 odd 24 784.2.x.k.765.1 16
224.75 even 24 784.2.x.j.165.1 16
224.107 odd 24 784.2.x.k.165.1 16
224.131 even 24 784.2.x.j.557.4 16
224.139 even 8 784.2.m.g.197.2 8
224.163 odd 24 784.2.x.k.557.4 16
224.171 even 24 784.2.x.j.373.4 16
224.195 even 8 784.2.m.g.589.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.c.29.2 ✓ 8 32.3 odd 8
112.2.m.c.85.2 yes 8 32.11 odd 8
448.2.m.c.113.1 8 32.21 even 8
448.2.m.c.337.1 8 32.29 even 8
784.2.m.g.197.2 8 224.139 even 8
784.2.m.g.589.2 8 224.195 even 8
784.2.x.j.165.1 16 224.75 even 24
784.2.x.j.373.4 16 224.171 even 24
784.2.x.j.557.4 16 224.131 even 24
784.2.x.j.765.1 16 224.3 even 24
784.2.x.k.165.1 16 224.107 odd 24
784.2.x.k.373.4 16 224.11 odd 24
784.2.x.k.557.4 16 224.163 odd 24
784.2.x.k.765.1 16 224.67 odd 24
896.2.m.e.225.1 8 32.27 odd 8
896.2.m.e.673.1 8 32.19 odd 8
896.2.m.f.225.4 8 32.5 even 8
896.2.m.f.673.4 8 32.13 even 8
7168.2.a.bc.1.2 8 8.3 odd 2
7168.2.a.bc.1.7 8 4.3 odd 2
7168.2.a.bd.1.2 8 1.1 even 1 trivial
7168.2.a.bd.1.7 8 8.5 even 2 inner