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Properties

Label 7728.2.a.cd.1.4
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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] = [7728,2,Mod(1,7728)] 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("7728.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,5,0,-4,0,4,0,5,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.24197.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.509552\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.41546 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.67510 q^{11} -4.66537 q^{13} +4.41546 q^{15} +6.24991 q^{17} +0.694209 q^{19} -1.00000 q^{21} +1.00000 q^{23} +14.4963 q^{25} +1.00000 q^{27} +5.60012 q^{29} -4.24991 q^{31} +1.67510 q^{33} -4.41546 q^{35} +9.26901 q^{37} -4.66537 q^{39} -5.15582 q^{41} -4.20376 q^{43} +4.41546 q^{45} +1.92501 q^{47} +1.00000 q^{49} +6.24991 q^{51} -1.84066 q^{53} +7.39636 q^{55} +0.694209 q^{57} -9.39779 q^{59} +7.72125 q^{61} -1.00000 q^{63} -20.5998 q^{65} +8.22728 q^{67} +1.00000 q^{69} +10.6654 q^{71} -11.9117 q^{73} +14.4963 q^{75} -1.67510 q^{77} -0.581012 q^{79} +1.00000 q^{81} -6.95032 q^{83} +27.5962 q^{85} +5.60012 q^{87} -13.4963 q^{89} +4.66537 q^{91} -4.24991 q^{93} +3.06525 q^{95} +10.0999 q^{97} +1.67510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 4.41546 1.97465 0.987327 − 0.158699i \(-0.0507301\pi\)
0.987327 + 0.158699i \(0.0507301\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.67510 0.505063 0.252531 − 0.967589i \(-0.418737\pi\)
0.252531 + 0.967589i \(0.418737\pi\)
\(12\) 0 0
\(13\) −4.66537 −1.29394 −0.646970 − 0.762515i \(-0.723964\pi\)
−0.646970 + 0.762515i \(0.723964\pi\)
\(14\) 0 0
\(15\) 4.41546 1.14007
\(16\) 0 0
\(17\) 6.24991 1.51583 0.757913 − 0.652356i \(-0.226219\pi\)
0.757913 + 0.652356i \(0.226219\pi\)
\(18\) 0 0
\(19\) 0.694209 0.159262 0.0796312 − 0.996824i \(-0.474626\pi\)
0.0796312 + 0.996824i \(0.474626\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 14.4963 2.89926
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.60012 1.03992 0.519958 − 0.854192i \(-0.325948\pi\)
0.519958 + 0.854192i \(0.325948\pi\)
\(30\) 0 0
\(31\) −4.24991 −0.763306 −0.381653 − 0.924306i \(-0.624645\pi\)
−0.381653 + 0.924306i \(0.624645\pi\)
\(32\) 0 0
\(33\) 1.67510 0.291598
\(34\) 0 0
\(35\) −4.41546 −0.746349
\(36\) 0 0
\(37\) 9.26901 1.52382 0.761908 − 0.647685i \(-0.224263\pi\)
0.761908 + 0.647685i \(0.224263\pi\)
\(38\) 0 0
\(39\) −4.66537 −0.747057
\(40\) 0 0
\(41\) −5.15582 −0.805203 −0.402602 − 0.915375i \(-0.631894\pi\)
−0.402602 + 0.915375i \(0.631894\pi\)
\(42\) 0 0
\(43\) −4.20376 −0.641068 −0.320534 − 0.947237i \(-0.603862\pi\)
−0.320534 + 0.947237i \(0.603862\pi\)
\(44\) 0 0
\(45\) 4.41546 0.658218
\(46\) 0 0
\(47\) 1.92501 0.280792 0.140396 − 0.990095i \(-0.455162\pi\)
0.140396 + 0.990095i \(0.455162\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.24991 0.875162
\(52\) 0 0
\(53\) −1.84066 −0.252833 −0.126417 − 0.991977i \(-0.540348\pi\)
−0.126417 + 0.991977i \(0.540348\pi\)
\(54\) 0 0
\(55\) 7.39636 0.997324
\(56\) 0 0
\(57\) 0.694209 0.0919502
\(58\) 0 0
\(59\) −9.39779 −1.22349 −0.611744 − 0.791056i \(-0.709532\pi\)
−0.611744 + 0.791056i \(0.709532\pi\)
\(60\) 0 0
\(61\) 7.72125 0.988605 0.494302 − 0.869290i \(-0.335424\pi\)
0.494302 + 0.869290i \(0.335424\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −20.5998 −2.55508
\(66\) 0 0
\(67\) 8.22728 1.00512 0.502561 − 0.864542i \(-0.332391\pi\)
0.502561 + 0.864542i \(0.332391\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.6654 1.26575 0.632873 − 0.774255i \(-0.281875\pi\)
0.632873 + 0.774255i \(0.281875\pi\)
\(72\) 0 0
\(73\) −11.9117 −1.39416 −0.697082 − 0.716991i \(-0.745519\pi\)
−0.697082 + 0.716991i \(0.745519\pi\)
\(74\) 0 0
\(75\) 14.4963 1.67389
\(76\) 0 0
\(77\) −1.67510 −0.190896
\(78\) 0 0
\(79\) −0.581012 −0.0653689 −0.0326845 − 0.999466i \(-0.510406\pi\)
−0.0326845 + 0.999466i \(0.510406\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.95032 −0.762897 −0.381449 − 0.924390i \(-0.624575\pi\)
−0.381449 + 0.924390i \(0.624575\pi\)
\(84\) 0 0
\(85\) 27.5962 2.99323
\(86\) 0 0
\(87\) 5.60012 0.600396
\(88\) 0 0
\(89\) −13.4963 −1.43060 −0.715302 − 0.698816i \(-0.753711\pi\)
−0.715302 + 0.698816i \(0.753711\pi\)
\(90\) 0 0
\(91\) 4.66537 0.489064
\(92\) 0 0
\(93\) −4.24991 −0.440695
\(94\) 0 0
\(95\) 3.06525 0.314488
\(96\) 0 0
\(97\) 10.0999 1.02549 0.512746 − 0.858540i \(-0.328628\pi\)
0.512746 + 0.858540i \(0.328628\pi\)
\(98\) 0 0
\(99\) 1.67510 0.168354
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 7728.2.a.cd.1.4 4
4.3 odd 2 483.2.a.i.1.2 ✓ 4
12.11 even 2 1449.2.a.p.1.3 4
28.27 even 2 3381.2.a.w.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.i.1.2 ✓ 4 4.3 odd 2
1449.2.a.p.1.3 4 12.11 even 2
3381.2.a.w.1.2 4 28.27 even 2
7728.2.a.cd.1.4 4 1.1 even 1 trivial