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Properties

Label 7168.2.a.bc.1.4
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 1.4
Root \(-2.10489\) 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)
 
\(f(q)\) \(=\) \(q-1.01155 q^{3} -1.22714 q^{5} -1.00000 q^{7} -1.97676 q^{9} -4.20978 q^{11} +2.85695 q^{13} +1.24132 q^{15} -0.264559 q^{17} +6.41995 q^{19} +1.01155 q^{21} -1.54621 q^{23} -3.49412 q^{25} +5.03426 q^{27} -0.464044 q^{29} +6.04033 q^{31} +4.25841 q^{33} +1.22714 q^{35} -9.40259 q^{37} -2.88995 q^{39} +11.0327 q^{41} +4.78580 q^{43} +2.42577 q^{45} -3.12566 q^{47} +1.00000 q^{49} +0.267615 q^{51} +0.608892 q^{53} +5.16599 q^{55} -6.49412 q^{57} +6.54272 q^{59} +6.88400 q^{61} +1.97676 q^{63} -3.50588 q^{65} +4.72877 q^{67} +1.56407 q^{69} +9.03885 q^{71} -14.8146 q^{73} +3.53449 q^{75} +4.20978 q^{77} -12.5904 q^{79} +0.837864 q^{81} +1.01155 q^{83} +0.324651 q^{85} +0.469405 q^{87} -10.9924 q^{89} -2.85695 q^{91} -6.11011 q^{93} -7.87820 q^{95} -14.2452 q^{97} +8.32172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} + 8 q^{9} + 8 q^{15} - 24 q^{17} - 8 q^{25} + 16 q^{31} - 24 q^{33} + 16 q^{39} - 8 q^{41} - 24 q^{47} + 8 q^{49} + 8 q^{55} - 32 q^{57} - 8 q^{63} - 48 q^{65} + 56 q^{71} - 48 q^{73} + 24 q^{79}+ \cdots - 32 q^{97}+O(q^{100}) \) 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
\(3\) −1.01155 −0.584020 −0.292010 − 0.956415i \(-0.594324\pi\)
−0.292010 + 0.956415i \(0.594324\pi\)
\(4\) 0 0
\(5\) −1.22714 −0.548795 −0.274397 − 0.961616i \(-0.588478\pi\)
−0.274397 + 0.961616i \(0.588478\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.97676 −0.658920
\(10\) 0 0
\(11\) −4.20978 −1.26929 −0.634647 − 0.772802i \(-0.718855\pi\)
−0.634647 + 0.772802i \(0.718855\pi\)
\(12\) 0 0
\(13\) 2.85695 0.792374 0.396187 − 0.918170i \(-0.370333\pi\)
0.396187 + 0.918170i \(0.370333\pi\)
\(14\) 0 0
\(15\) 1.24132 0.320507
\(16\) 0 0
\(17\) −0.264559 −0.0641649 −0.0320825 − 0.999485i \(-0.510214\pi\)
−0.0320825 + 0.999485i \(0.510214\pi\)
\(18\) 0 0
\(19\) 6.41995 1.47284 0.736419 − 0.676526i \(-0.236515\pi\)
0.736419 + 0.676526i \(0.236515\pi\)
\(20\) 0 0
\(21\) 1.01155 0.220739
\(22\) 0 0
\(23\) −1.54621 −0.322407 −0.161203 − 0.986921i \(-0.551538\pi\)
−0.161203 + 0.986921i \(0.551538\pi\)
\(24\) 0 0
\(25\) −3.49412 −0.698824
\(26\) 0 0
\(27\) 5.03426 0.968843
\(28\) 0 0
\(29\) −0.464044 −0.0861708 −0.0430854 − 0.999071i \(-0.513719\pi\)
−0.0430854 + 0.999071i \(0.513719\pi\)
\(30\) 0 0
\(31\) 6.04033 1.08488 0.542438 − 0.840096i \(-0.317501\pi\)
0.542438 + 0.840096i \(0.317501\pi\)
\(32\) 0 0
\(33\) 4.25841 0.741294
\(34\) 0 0
\(35\) 1.22714 0.207425
\(36\) 0 0
\(37\) −9.40259 −1.54578 −0.772888 − 0.634543i \(-0.781188\pi\)
−0.772888 + 0.634543i \(0.781188\pi\)
\(38\) 0 0
\(39\) −2.88995 −0.462763
\(40\) 0 0
\(41\) 11.0327 1.72302 0.861510 − 0.507741i \(-0.169519\pi\)
0.861510 + 0.507741i \(0.169519\pi\)
\(42\) 0 0
\(43\) 4.78580 0.729828 0.364914 − 0.931041i \(-0.381098\pi\)
0.364914 + 0.931041i \(0.381098\pi\)
\(44\) 0 0
\(45\) 2.42577 0.361612
\(46\) 0 0
\(47\) −3.12566 −0.455925 −0.227962 − 0.973670i \(-0.573206\pi\)
−0.227962 + 0.973670i \(0.573206\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.267615 0.0374736
\(52\) 0 0
\(53\) 0.608892 0.0836378 0.0418189 − 0.999125i \(-0.486685\pi\)
0.0418189 + 0.999125i \(0.486685\pi\)
\(54\) 0 0
\(55\) 5.16599 0.696582
\(56\) 0 0
\(57\) −6.49412 −0.860167
\(58\) 0 0
\(59\) 6.54272 0.851790 0.425895 − 0.904773i \(-0.359959\pi\)
0.425895 + 0.904773i \(0.359959\pi\)
\(60\) 0 0
\(61\) 6.88400 0.881405 0.440703 − 0.897653i \(-0.354729\pi\)
0.440703 + 0.897653i \(0.354729\pi\)
\(62\) 0 0
\(63\) 1.97676 0.249048
\(64\) 0 0
\(65\) −3.50588 −0.434851
\(66\) 0 0
\(67\) 4.72877 0.577711 0.288855 − 0.957373i \(-0.406725\pi\)
0.288855 + 0.957373i \(0.406725\pi\)
\(68\) 0 0
\(69\) 1.56407 0.188292
\(70\) 0 0
\(71\) 9.03885 1.07271 0.536357 − 0.843991i \(-0.319800\pi\)
0.536357 + 0.843991i \(0.319800\pi\)
\(72\) 0 0
\(73\) −14.8146 −1.73392 −0.866960 − 0.498377i \(-0.833930\pi\)
−0.866960 + 0.498377i \(0.833930\pi\)
\(74\) 0 0
\(75\) 3.53449 0.408128
\(76\) 0 0
\(77\) 4.20978 0.479748
\(78\) 0 0
\(79\) −12.5904 −1.41653 −0.708265 − 0.705947i \(-0.750522\pi\)
−0.708265 + 0.705947i \(0.750522\pi\)
\(80\) 0 0
\(81\) 0.837864 0.0930960
\(82\) 0 0
\(83\) 1.01155 0.111032 0.0555162 − 0.998458i \(-0.482320\pi\)
0.0555162 + 0.998458i \(0.482320\pi\)
\(84\) 0 0
\(85\) 0.324651 0.0352134
\(86\) 0 0
\(87\) 0.469405 0.0503255
\(88\) 0 0
\(89\) −10.9924 −1.16519 −0.582595 − 0.812763i \(-0.697962\pi\)
−0.582595 + 0.812763i \(0.697962\pi\)
\(90\) 0 0
\(91\) −2.85695 −0.299489
\(92\) 0 0
\(93\) −6.11011 −0.633589
\(94\) 0 0
\(95\) −7.87820 −0.808286
\(96\) 0 0
\(97\) −14.2452 −1.44638 −0.723189 − 0.690650i \(-0.757325\pi\)
−0.723189 + 0.690650i \(0.757325\pi\)
\(98\) 0 0
\(99\) 8.32172 0.836364
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.bc.1.4 8
4.3 odd 2 7168.2.a.bd.1.5 8
8.3 odd 2 7168.2.a.bd.1.4 8
8.5 even 2 inner 7168.2.a.bc.1.5 8
32.3 odd 8 448.2.m.c.337.3 8
32.5 even 8 896.2.m.e.225.3 8
32.11 odd 8 448.2.m.c.113.3 8
32.13 even 8 896.2.m.e.673.3 8
32.19 odd 8 896.2.m.f.673.2 8
32.21 even 8 112.2.m.c.85.4 yes 8
32.27 odd 8 896.2.m.f.225.2 8
32.29 even 8 112.2.m.c.29.4 ✓ 8
224.53 even 24 784.2.x.k.373.2 16
224.61 odd 24 784.2.x.j.557.2 16
224.93 even 24 784.2.x.k.557.2 16
224.117 odd 24 784.2.x.j.165.2 16
224.125 odd 8 784.2.m.g.589.4 8
224.149 even 24 784.2.x.k.165.2 16
224.157 odd 24 784.2.x.j.765.2 16
224.181 odd 8 784.2.m.g.197.4 8
224.213 odd 24 784.2.x.j.373.2 16
224.221 even 24 784.2.x.k.765.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.2.m.c.29.4 ✓ 8 32.29 even 8
112.2.m.c.85.4 yes 8 32.21 even 8
448.2.m.c.113.3 8 32.11 odd 8
448.2.m.c.337.3 8 32.3 odd 8
784.2.m.g.197.4 8 224.181 odd 8
784.2.m.g.589.4 8 224.125 odd 8
784.2.x.j.165.2 16 224.117 odd 24
784.2.x.j.373.2 16 224.213 odd 24
784.2.x.j.557.2 16 224.61 odd 24
784.2.x.j.765.2 16 224.157 odd 24
784.2.x.k.165.2 16 224.149 even 24
784.2.x.k.373.2 16 224.53 even 24
784.2.x.k.557.2 16 224.93 even 24
784.2.x.k.765.2 16 224.221 even 24
896.2.m.e.225.3 8 32.5 even 8
896.2.m.e.673.3 8 32.13 even 8
896.2.m.f.225.2 8 32.27 odd 8
896.2.m.f.673.2 8 32.19 odd 8
7168.2.a.bc.1.4 8 1.1 even 1 trivial
7168.2.a.bc.1.5 8 8.5 even 2 inner
7168.2.a.bd.1.4 8 8.3 odd 2
7168.2.a.bd.1.5 8 4.3 odd 2