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Properties

Label 7938.2.a.ci.1.3
Level $7938$
Weight $2$
Character 7938.1
Self dual yes
Analytic conductor $63.385$
Analytic rank $0$
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] = [7938,2,Mod(1,7938)] 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("7938.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7938, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-4,0,4,0,0,0,-4,0,0,8,0,0,0,0,4,0,0,0,0,0,-8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3852491245\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 882)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 7938.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.03528 q^{5} -1.00000 q^{8} -1.03528 q^{10} +0.267949 q^{11} +1.79315 q^{13} +1.00000 q^{16} +6.83083 q^{17} +4.38134 q^{19} +1.03528 q^{20} -0.267949 q^{22} -5.46410 q^{23} -3.92820 q^{25} -1.79315 q^{26} +4.00000 q^{29} +6.69213 q^{31} -1.00000 q^{32} -6.83083 q^{34} +7.46410 q^{37} -4.38134 q^{38} -1.03528 q^{40} +8.62398 q^{41} +0.267949 q^{43} +0.267949 q^{44} +5.46410 q^{46} -0.757875 q^{47} +3.92820 q^{50} +1.79315 q^{52} -10.9282 q^{53} +0.277401 q^{55} -4.00000 q^{58} +1.27551 q^{59} -12.6264 q^{61} -6.69213 q^{62} +1.00000 q^{64} +1.85641 q^{65} +12.4641 q^{67} +6.83083 q^{68} -9.46410 q^{71} +5.41662 q^{73} -7.46410 q^{74} +4.38134 q^{76} +8.92820 q^{79} +1.03528 q^{80} -8.62398 q^{82} -6.59059 q^{83} +7.07180 q^{85} -0.267949 q^{86} -0.267949 q^{88} -7.07107 q^{89} -5.46410 q^{92} +0.757875 q^{94} +4.53590 q^{95} -18.1445 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 8 q^{11} + 4 q^{16} - 8 q^{22} - 8 q^{23} + 12 q^{25} + 16 q^{29} - 4 q^{32} + 16 q^{37} + 8 q^{43} + 8 q^{44} + 8 q^{46} - 12 q^{50} - 16 q^{53} - 16 q^{58} + 4 q^{64}+ \cdots + 32 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.03528 0.462990 0.231495 − 0.972836i \(-0.425638\pi\)
0.231495 + 0.972836i \(0.425638\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.03528 −0.327383
\(11\) 0.267949 0.0807897 0.0403949 − 0.999184i \(-0.487138\pi\)
0.0403949 + 0.999184i \(0.487138\pi\)
\(12\) 0 0
\(13\) 1.79315 0.497331 0.248665 − 0.968589i \(-0.420008\pi\)
0.248665 + 0.968589i \(0.420008\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.83083 1.65672 0.828360 − 0.560196i \(-0.189274\pi\)
0.828360 + 0.560196i \(0.189274\pi\)
\(18\) 0 0
\(19\) 4.38134 1.00515 0.502574 − 0.864534i \(-0.332386\pi\)
0.502574 + 0.864534i \(0.332386\pi\)
\(20\) 1.03528 0.231495
\(21\) 0 0
\(22\) −0.267949 −0.0571270
\(23\) −5.46410 −1.13934 −0.569672 − 0.821872i \(-0.692930\pi\)
−0.569672 + 0.821872i \(0.692930\pi\)
\(24\) 0 0
\(25\) −3.92820 −0.785641
\(26\) −1.79315 −0.351666
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 6.69213 1.20194 0.600971 − 0.799271i \(-0.294781\pi\)
0.600971 + 0.799271i \(0.294781\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.83083 −1.17148
\(35\) 0 0
\(36\) 0 0
\(37\) 7.46410 1.22709 0.613545 − 0.789659i \(-0.289743\pi\)
0.613545 + 0.789659i \(0.289743\pi\)
\(38\) −4.38134 −0.710747
\(39\) 0 0
\(40\) −1.03528 −0.163692
\(41\) 8.62398 1.34684 0.673420 − 0.739260i \(-0.264825\pi\)
0.673420 + 0.739260i \(0.264825\pi\)
\(42\) 0 0
\(43\) 0.267949 0.0408619 0.0204309 − 0.999791i \(-0.493496\pi\)
0.0204309 + 0.999791i \(0.493496\pi\)
\(44\) 0.267949 0.0403949
\(45\) 0 0
\(46\) 5.46410 0.805638
\(47\) −0.757875 −0.110547 −0.0552737 − 0.998471i \(-0.517603\pi\)
−0.0552737 + 0.998471i \(0.517603\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 3.92820 0.555532
\(51\) 0 0
\(52\) 1.79315 0.248665
\(53\) −10.9282 −1.50110 −0.750552 − 0.660811i \(-0.770212\pi\)
−0.750552 + 0.660811i \(0.770212\pi\)
\(54\) 0 0
\(55\) 0.277401 0.0374048
\(56\) 0 0
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 1.27551 0.166058 0.0830288 − 0.996547i \(-0.473541\pi\)
0.0830288 + 0.996547i \(0.473541\pi\)
\(60\) 0 0
\(61\) −12.6264 −1.61664 −0.808322 − 0.588741i \(-0.799624\pi\)
−0.808322 + 0.588741i \(0.799624\pi\)
\(62\) −6.69213 −0.849901
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.85641 0.230259
\(66\) 0 0
\(67\) 12.4641 1.52273 0.761366 − 0.648322i \(-0.224529\pi\)
0.761366 + 0.648322i \(0.224529\pi\)
\(68\) 6.83083 0.828360
\(69\) 0 0
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 − 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) 5.41662 0.633967 0.316984 − 0.948431i \(-0.397330\pi\)
0.316984 + 0.948431i \(0.397330\pi\)
\(74\) −7.46410 −0.867684
\(75\) 0 0
\(76\) 4.38134 0.502574
\(77\) 0 0
\(78\) 0 0
\(79\) 8.92820 1.00450 0.502251 − 0.864722i \(-0.332505\pi\)
0.502251 + 0.864722i \(0.332505\pi\)
\(80\) 1.03528 0.115747
\(81\) 0 0
\(82\) −8.62398 −0.952360
\(83\) −6.59059 −0.723412 −0.361706 − 0.932292i \(-0.617806\pi\)
−0.361706 + 0.932292i \(0.617806\pi\)
\(84\) 0 0
\(85\) 7.07180 0.767044
\(86\) −0.267949 −0.0288937
\(87\) 0 0
\(88\) −0.267949 −0.0285635
\(89\) −7.07107 −0.749532 −0.374766 − 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.46410 −0.569672
\(93\) 0 0
\(94\) 0.757875 0.0781688
\(95\) 4.53590 0.465373
\(96\) 0 0
\(97\) −18.1445 −1.84230 −0.921149 − 0.389209i \(-0.872748\pi\)
−0.921149 + 0.389209i \(0.872748\pi\)
\(98\) 0 0
\(99\) 0 0
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 7938.2.a.ci.1.3 4
3.2 odd 2 7938.2.a.cp.1.2 4
7.6 odd 2 inner 7938.2.a.ci.1.2 4
9.2 odd 6 882.2.f.q.589.1 yes 8
9.4 even 3 2646.2.f.r.883.2 8
9.5 odd 6 882.2.f.q.295.1 ✓ 8
9.7 even 3 2646.2.f.r.1765.2 8
21.20 even 2 7938.2.a.cp.1.3 4
63.2 odd 6 882.2.h.q.67.3 8
63.4 even 3 2646.2.h.t.667.3 8
63.5 even 6 882.2.e.s.655.2 8
63.11 odd 6 882.2.e.s.373.3 8
63.13 odd 6 2646.2.f.r.883.3 8
63.16 even 3 2646.2.h.t.361.3 8
63.20 even 6 882.2.f.q.589.4 yes 8
63.23 odd 6 882.2.e.s.655.3 8
63.25 even 3 2646.2.e.q.1549.2 8
63.31 odd 6 2646.2.h.t.667.2 8
63.32 odd 6 882.2.h.q.79.4 8
63.34 odd 6 2646.2.f.r.1765.3 8
63.38 even 6 882.2.e.s.373.2 8
63.40 odd 6 2646.2.e.q.2125.3 8
63.41 even 6 882.2.f.q.295.4 yes 8
63.47 even 6 882.2.h.q.67.2 8
63.52 odd 6 2646.2.e.q.1549.3 8
63.58 even 3 2646.2.e.q.2125.2 8
63.59 even 6 882.2.h.q.79.1 8
63.61 odd 6 2646.2.h.t.361.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.e.s.373.2 8 63.38 even 6
882.2.e.s.373.3 8 63.11 odd 6
882.2.e.s.655.2 8 63.5 even 6
882.2.e.s.655.3 8 63.23 odd 6
882.2.f.q.295.1 ✓ 8 9.5 odd 6
882.2.f.q.295.4 yes 8 63.41 even 6
882.2.f.q.589.1 yes 8 9.2 odd 6
882.2.f.q.589.4 yes 8 63.20 even 6
882.2.h.q.67.2 8 63.47 even 6
882.2.h.q.67.3 8 63.2 odd 6
882.2.h.q.79.1 8 63.59 even 6
882.2.h.q.79.4 8 63.32 odd 6
2646.2.e.q.1549.2 8 63.25 even 3
2646.2.e.q.1549.3 8 63.52 odd 6
2646.2.e.q.2125.2 8 63.58 even 3
2646.2.e.q.2125.3 8 63.40 odd 6
2646.2.f.r.883.2 8 9.4 even 3
2646.2.f.r.883.3 8 63.13 odd 6
2646.2.f.r.1765.2 8 9.7 even 3
2646.2.f.r.1765.3 8 63.34 odd 6
2646.2.h.t.361.2 8 63.61 odd 6
2646.2.h.t.361.3 8 63.16 even 3
2646.2.h.t.667.2 8 63.31 odd 6
2646.2.h.t.667.3 8 63.4 even 3
7938.2.a.ci.1.2 4 7.6 odd 2 inner
7938.2.a.ci.1.3 4 1.1 even 1 trivial
7938.2.a.cp.1.2 4 3.2 odd 2
7938.2.a.cp.1.3 4 21.20 even 2