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 = [3,3,0,3,-1,0,0,3,0,-1,-1,0,8,0,0,3,4,0,-3,-1,0,-1,-7] 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: \(3\)
Coefficient field: 3.3.321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.69963\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.58836 q^{5} +1.00000 q^{8} +1.58836 q^{10} +1.58836 q^{11} +4.81089 q^{13} +1.00000 q^{16} +5.39926 q^{17} -7.09888 q^{19} +1.58836 q^{20} +1.58836 q^{22} -0.300372 q^{23} -2.47710 q^{25} +4.81089 q^{26} +8.27561 q^{29} +2.71201 q^{31} +1.00000 q^{32} +5.39926 q^{34} -1.00000 q^{37} -7.09888 q^{38} +1.58836 q^{40} -5.87636 q^{41} +1.66621 q^{43} +1.58836 q^{44} -0.300372 q^{46} +2.66621 q^{47} -2.47710 q^{50} +4.81089 q^{52} +4.88874 q^{53} +2.52290 q^{55} +8.27561 q^{58} +6.47710 q^{59} +4.47710 q^{61} +2.71201 q^{62} +1.00000 q^{64} +7.64145 q^{65} -10.0531 q^{67} +5.39926 q^{68} -12.7207 q^{71} +16.0531 q^{73} -1.00000 q^{74} -7.09888 q^{76} +8.38688 q^{79} +1.58836 q^{80} -5.87636 q^{82} -2.36584 q^{83} +8.57598 q^{85} +1.66621 q^{86} +1.58836 q^{88} -3.21015 q^{89} -0.300372 q^{92} +2.66621 q^{94} -11.2756 q^{95} +1.42402 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - q^{5} + 3 q^{8} - q^{10} - q^{11} + 8 q^{13} + 3 q^{16} + 4 q^{17} - 3 q^{19} - q^{20} - q^{22} - 7 q^{23} - 2 q^{25} + 8 q^{26} - 5 q^{29} + 20 q^{31} + 3 q^{32} + 4 q^{34}+ \cdots + 28 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.58836 0.710338 0.355169 − 0.934802i \(-0.384423\pi\)
0.355169 + 0.934802i \(0.384423\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.58836 0.502285
\(11\) 1.58836 0.478910 0.239455 − 0.970907i \(-0.423031\pi\)
0.239455 + 0.970907i \(0.423031\pi\)
\(12\) 0 0
\(13\) 4.81089 1.33430 0.667151 − 0.744923i \(-0.267514\pi\)
0.667151 + 0.744923i \(0.267514\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.39926 1.30951 0.654756 − 0.755840i \(-0.272771\pi\)
0.654756 + 0.755840i \(0.272771\pi\)
\(18\) 0 0
\(19\) −7.09888 −1.62860 −0.814298 − 0.580447i \(-0.802878\pi\)
−0.814298 + 0.580447i \(0.802878\pi\)
\(20\) 1.58836 0.355169
\(21\) 0 0
\(22\) 1.58836 0.338640
\(23\) −0.300372 −0.0626319 −0.0313159 − 0.999510i \(-0.509970\pi\)
−0.0313159 + 0.999510i \(0.509970\pi\)
\(24\) 0 0
\(25\) −2.47710 −0.495420
\(26\) 4.81089 0.943494
\(27\) 0 0
\(28\) 0 0
\(29\) 8.27561 1.53674 0.768371 − 0.640004i \(-0.221067\pi\)
0.768371 + 0.640004i \(0.221067\pi\)
\(30\) 0 0
\(31\) 2.71201 0.487091 0.243545 − 0.969889i \(-0.421689\pi\)
0.243545 + 0.969889i \(0.421689\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.39926 0.925965
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −7.09888 −1.15159
\(39\) 0 0
\(40\) 1.58836 0.251142
\(41\) −5.87636 −0.917733 −0.458866 − 0.888505i \(-0.651744\pi\)
−0.458866 + 0.888505i \(0.651744\pi\)
\(42\) 0 0
\(43\) 1.66621 0.254094 0.127047 − 0.991897i \(-0.459450\pi\)
0.127047 + 0.991897i \(0.459450\pi\)
\(44\) 1.58836 0.239455
\(45\) 0 0
\(46\) −0.300372 −0.0442874
\(47\) 2.66621 0.388906 0.194453 − 0.980912i \(-0.437707\pi\)
0.194453 + 0.980912i \(0.437707\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.47710 −0.350315
\(51\) 0 0
\(52\) 4.81089 0.667151
\(53\) 4.88874 0.671520 0.335760 − 0.941948i \(-0.391007\pi\)
0.335760 + 0.941948i \(0.391007\pi\)
\(54\) 0 0
\(55\) 2.52290 0.340188
\(56\) 0 0
\(57\) 0 0
\(58\) 8.27561 1.08664
\(59\) 6.47710 0.843247 0.421623 − 0.906771i \(-0.361460\pi\)
0.421623 + 0.906771i \(0.361460\pi\)
\(60\) 0 0
\(61\) 4.47710 0.573234 0.286617 − 0.958045i \(-0.407469\pi\)
0.286617 + 0.958045i \(0.407469\pi\)
\(62\) 2.71201 0.344425
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.64145 0.947805
\(66\) 0 0
\(67\) −10.0531 −1.22818 −0.614090 − 0.789236i \(-0.710477\pi\)
−0.614090 + 0.789236i \(0.710477\pi\)
\(68\) 5.39926 0.654756
\(69\) 0 0
\(70\) 0 0
\(71\) −12.7207 −1.50967 −0.754833 − 0.655917i \(-0.772282\pi\)
−0.754833 + 0.655917i \(0.772282\pi\)
\(72\) 0 0
\(73\) 16.0531 1.87887 0.939436 − 0.342725i \(-0.111350\pi\)
0.939436 + 0.342725i \(0.111350\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −7.09888 −0.814298
\(77\) 0 0
\(78\) 0 0
\(79\) 8.38688 0.943597 0.471799 − 0.881706i \(-0.343605\pi\)
0.471799 + 0.881706i \(0.343605\pi\)
\(80\) 1.58836 0.177584
\(81\) 0 0
\(82\) −5.87636 −0.648935
\(83\) −2.36584 −0.259684 −0.129842 − 0.991535i \(-0.541447\pi\)
−0.129842 + 0.991535i \(0.541447\pi\)
\(84\) 0 0
\(85\) 8.57598 0.930196
\(86\) 1.66621 0.179672
\(87\) 0 0
\(88\) 1.58836 0.169320
\(89\) −3.21015 −0.340275 −0.170138 − 0.985420i \(-0.554421\pi\)
−0.170138 + 0.985420i \(0.554421\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.300372 −0.0313159
\(93\) 0 0
\(94\) 2.66621 0.274998
\(95\) −11.2756 −1.15685
\(96\) 0 0
\(97\) 1.42402 0.144587 0.0722934 − 0.997383i \(-0.476968\pi\)
0.0722934 + 0.997383i \(0.476968\pi\)
\(98\) 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.bz.1.3 3
3.2 odd 2 7938.2.a.bw.1.1 3
7.3 odd 6 1134.2.g.l.163.3 6
7.5 odd 6 1134.2.g.l.487.3 6
7.6 odd 2 7938.2.a.ca.1.1 3
9.2 odd 6 882.2.f.o.589.1 6
9.4 even 3 2646.2.f.m.883.1 6
9.5 odd 6 882.2.f.o.295.1 6
9.7 even 3 2646.2.f.m.1765.1 6
21.5 even 6 1134.2.g.m.487.1 6
21.17 even 6 1134.2.g.m.163.1 6
21.20 even 2 7938.2.a.bv.1.3 3
63.2 odd 6 882.2.h.p.67.1 6
63.4 even 3 2646.2.h.o.667.3 6
63.5 even 6 126.2.e.c.25.1 ✓ 6
63.11 odd 6 882.2.e.o.373.3 6
63.13 odd 6 2646.2.f.l.883.3 6
63.16 even 3 2646.2.h.o.361.3 6
63.20 even 6 882.2.f.n.589.3 6
63.23 odd 6 882.2.e.o.655.3 6
63.25 even 3 2646.2.e.p.1549.1 6
63.31 odd 6 378.2.h.c.289.1 6
63.32 odd 6 882.2.h.p.79.1 6
63.34 odd 6 2646.2.f.l.1765.3 6
63.38 even 6 126.2.e.c.121.1 yes 6
63.40 odd 6 378.2.e.d.235.3 6
63.41 even 6 882.2.f.n.295.3 6
63.47 even 6 126.2.h.d.67.3 yes 6
63.52 odd 6 378.2.e.d.37.3 6
63.58 even 3 2646.2.e.p.2125.1 6
63.59 even 6 126.2.h.d.79.3 yes 6
63.61 odd 6 378.2.h.c.361.1 6
252.31 even 6 3024.2.t.h.289.1 6
252.47 odd 6 1008.2.t.h.193.1 6
252.59 odd 6 1008.2.t.h.961.1 6
252.103 even 6 3024.2.q.g.2881.3 6
252.115 even 6 3024.2.q.g.2305.3 6
252.131 odd 6 1008.2.q.g.529.3 6
252.187 even 6 3024.2.t.h.1873.1 6
252.227 odd 6 1008.2.q.g.625.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.1 ✓ 6 63.5 even 6
126.2.e.c.121.1 yes 6 63.38 even 6
126.2.h.d.67.3 yes 6 63.47 even 6
126.2.h.d.79.3 yes 6 63.59 even 6
378.2.e.d.37.3 6 63.52 odd 6
378.2.e.d.235.3 6 63.40 odd 6
378.2.h.c.289.1 6 63.31 odd 6
378.2.h.c.361.1 6 63.61 odd 6
882.2.e.o.373.3 6 63.11 odd 6
882.2.e.o.655.3 6 63.23 odd 6
882.2.f.n.295.3 6 63.41 even 6
882.2.f.n.589.3 6 63.20 even 6
882.2.f.o.295.1 6 9.5 odd 6
882.2.f.o.589.1 6 9.2 odd 6
882.2.h.p.67.1 6 63.2 odd 6
882.2.h.p.79.1 6 63.32 odd 6
1008.2.q.g.529.3 6 252.131 odd 6
1008.2.q.g.625.3 6 252.227 odd 6
1008.2.t.h.193.1 6 252.47 odd 6
1008.2.t.h.961.1 6 252.59 odd 6
1134.2.g.l.163.3 6 7.3 odd 6
1134.2.g.l.487.3 6 7.5 odd 6
1134.2.g.m.163.1 6 21.17 even 6
1134.2.g.m.487.1 6 21.5 even 6
2646.2.e.p.1549.1 6 63.25 even 3
2646.2.e.p.2125.1 6 63.58 even 3
2646.2.f.l.883.3 6 63.13 odd 6
2646.2.f.l.1765.3 6 63.34 odd 6
2646.2.f.m.883.1 6 9.4 even 3
2646.2.f.m.1765.1 6 9.7 even 3
2646.2.h.o.361.3 6 63.16 even 3
2646.2.h.o.667.3 6 63.4 even 3
3024.2.q.g.2305.3 6 252.115 even 6
3024.2.q.g.2881.3 6 252.103 even 6
3024.2.t.h.289.1 6 252.31 even 6
3024.2.t.h.1873.1 6 252.187 even 6
7938.2.a.bv.1.3 3 21.20 even 2
7938.2.a.bw.1.1 3 3.2 odd 2
7938.2.a.bz.1.3 3 1.1 even 1 trivial
7938.2.a.ca.1.1 3 7.6 odd 2