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Properties

Label 8624.2.a.cs.1.1
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $4$
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] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,4,0,0,0,-12,0,0,0,0,0,0,0,0,0,4,0,0,0,8, 0,0,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2156)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.93185\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.93185 q^{3} +3.34607 q^{5} +0.732051 q^{9} +1.00000 q^{11} +4.89898 q^{13} -6.46410 q^{15} -1.03528 q^{17} -6.69213 q^{19} +5.19615 q^{23} +6.19615 q^{25} +4.38134 q^{27} -4.92820 q^{29} -7.86611 q^{31} -1.93185 q^{33} -1.92820 q^{37} -9.46410 q^{39} -3.86370 q^{41} -4.00000 q^{43} +2.44949 q^{45} -6.03579 q^{47} +2.00000 q^{51} -10.9282 q^{53} +3.34607 q^{55} +12.9282 q^{57} +7.20977 q^{59} +14.4195 q^{61} +16.3923 q^{65} -9.73205 q^{67} -10.0382 q^{69} +1.19615 q^{71} -11.5911 q^{73} -11.9700 q^{75} -7.46410 q^{79} -10.6603 q^{81} -9.79796 q^{83} -3.46410 q^{85} +9.52056 q^{87} +1.27551 q^{89} +15.1962 q^{93} -22.3923 q^{95} -14.6598 q^{97} +0.732051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 4 q^{11} - 12 q^{15} + 4 q^{25} + 8 q^{29} + 20 q^{37} - 24 q^{39} - 16 q^{43} + 8 q^{51} - 16 q^{53} + 24 q^{57} + 24 q^{65} - 32 q^{67} - 16 q^{71} - 16 q^{79} - 8 q^{81} + 40 q^{93} - 48 q^{95}+ \cdots - 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.93185 −1.11536 −0.557678 − 0.830058i \(-0.688307\pi\)
−0.557678 + 0.830058i \(0.688307\pi\)
\(4\) 0 0
\(5\) 3.34607 1.49641 0.748203 − 0.663470i \(-0.230917\pi\)
0.748203 + 0.663470i \(0.230917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.732051 0.244017
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.89898 1.35873 0.679366 − 0.733799i \(-0.262255\pi\)
0.679366 + 0.733799i \(0.262255\pi\)
\(14\) 0 0
\(15\) −6.46410 −1.66902
\(16\) 0 0
\(17\) −1.03528 −0.251091 −0.125546 − 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(18\) 0 0
\(19\) −6.69213 −1.53528 −0.767640 − 0.640881i \(-0.778569\pi\)
−0.767640 + 0.640881i \(0.778569\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.19615 1.08347 0.541736 − 0.840548i \(-0.317767\pi\)
0.541736 + 0.840548i \(0.317767\pi\)
\(24\) 0 0
\(25\) 6.19615 1.23923
\(26\) 0 0
\(27\) 4.38134 0.843190
\(28\) 0 0
\(29\) −4.92820 −0.915144 −0.457572 − 0.889172i \(-0.651281\pi\)
−0.457572 + 0.889172i \(0.651281\pi\)
\(30\) 0 0
\(31\) −7.86611 −1.41279 −0.706397 − 0.707816i \(-0.749681\pi\)
−0.706397 + 0.707816i \(0.749681\pi\)
\(32\) 0 0
\(33\) −1.93185 −0.336292
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.92820 −0.316995 −0.158497 − 0.987359i \(-0.550665\pi\)
−0.158497 + 0.987359i \(0.550665\pi\)
\(38\) 0 0
\(39\) −9.46410 −1.51547
\(40\) 0 0
\(41\) −3.86370 −0.603409 −0.301705 − 0.953401i \(-0.597556\pi\)
−0.301705 + 0.953401i \(0.597556\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.44949 0.365148
\(46\) 0 0
\(47\) −6.03579 −0.880411 −0.440205 − 0.897897i \(-0.645094\pi\)
−0.440205 + 0.897897i \(0.645094\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −10.9282 −1.50110 −0.750552 − 0.660811i \(-0.770212\pi\)
−0.750552 + 0.660811i \(0.770212\pi\)
\(54\) 0 0
\(55\) 3.34607 0.451183
\(56\) 0 0
\(57\) 12.9282 1.71238
\(58\) 0 0
\(59\) 7.20977 0.938632 0.469316 − 0.883030i \(-0.344501\pi\)
0.469316 + 0.883030i \(0.344501\pi\)
\(60\) 0 0
\(61\) 14.4195 1.84623 0.923116 − 0.384521i \(-0.125633\pi\)
0.923116 + 0.384521i \(0.125633\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.3923 2.03322
\(66\) 0 0
\(67\) −9.73205 −1.18896 −0.594480 − 0.804111i \(-0.702642\pi\)
−0.594480 + 0.804111i \(0.702642\pi\)
\(68\) 0 0
\(69\) −10.0382 −1.20846
\(70\) 0 0
\(71\) 1.19615 0.141957 0.0709786 − 0.997478i \(-0.477388\pi\)
0.0709786 + 0.997478i \(0.477388\pi\)
\(72\) 0 0
\(73\) −11.5911 −1.35664 −0.678318 − 0.734768i \(-0.737291\pi\)
−0.678318 + 0.734768i \(0.737291\pi\)
\(74\) 0 0
\(75\) −11.9700 −1.38218
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −7.46410 −0.839777 −0.419889 − 0.907576i \(-0.637931\pi\)
−0.419889 + 0.907576i \(0.637931\pi\)
\(80\) 0 0
\(81\) −10.6603 −1.18447
\(82\) 0 0
\(83\) −9.79796 −1.07547 −0.537733 − 0.843115i \(-0.680719\pi\)
−0.537733 + 0.843115i \(0.680719\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) 9.52056 1.02071
\(88\) 0 0
\(89\) 1.27551 0.135204 0.0676020 − 0.997712i \(-0.478465\pi\)
0.0676020 + 0.997712i \(0.478465\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 15.1962 1.57577
\(94\) 0 0
\(95\) −22.3923 −2.29740
\(96\) 0 0
\(97\) −14.6598 −1.48847 −0.744237 − 0.667915i \(-0.767187\pi\)
−0.744237 + 0.667915i \(0.767187\pi\)
\(98\) 0 0
\(99\) 0.732051 0.0735739
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 8624.2.a.cs.1.1 4
4.3 odd 2 2156.2.a.l.1.4 yes 4
7.6 odd 2 inner 8624.2.a.cs.1.4 4
28.3 even 6 2156.2.i.o.177.4 8
28.11 odd 6 2156.2.i.o.177.1 8
28.19 even 6 2156.2.i.o.1145.4 8
28.23 odd 6 2156.2.i.o.1145.1 8
28.27 even 2 2156.2.a.l.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2156.2.a.l.1.1 ✓ 4 28.27 even 2
2156.2.a.l.1.4 yes 4 4.3 odd 2
2156.2.i.o.177.1 8 28.11 odd 6
2156.2.i.o.177.4 8 28.3 even 6
2156.2.i.o.1145.1 8 28.23 odd 6
2156.2.i.o.1145.4 8 28.19 even 6
8624.2.a.cs.1.1 4 1.1 even 1 trivial
8624.2.a.cs.1.4 4 7.6 odd 2 inner