Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7942,2,Mod(1,7942)] 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("7942.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7942, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12,3,12,-9,-3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 41 x^{9} + 90 x^{8} - 198 x^{7} - 285 x^{6} + 396 x^{5} + 486 x^{4} + \cdots - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.60684\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.60684 q^{3} +1.00000 q^{4} +0.823492 q^{5} -1.60684 q^{6} +0.0824650 q^{7} -1.00000 q^{8} -0.418066 q^{9} -0.823492 q^{10} -1.00000 q^{11} +1.60684 q^{12} -0.855674 q^{13} -0.0824650 q^{14} +1.32322 q^{15} +1.00000 q^{16} +0.454424 q^{17} +0.418066 q^{18} +0.823492 q^{20} +0.132508 q^{21} +1.00000 q^{22} +4.06078 q^{23} -1.60684 q^{24} -4.32186 q^{25} +0.855674 q^{26} -5.49228 q^{27} +0.0824650 q^{28} +8.34273 q^{29} -1.32322 q^{30} -2.85839 q^{31} -1.00000 q^{32} -1.60684 q^{33} -0.454424 q^{34} +0.0679093 q^{35} -0.418066 q^{36} +5.05599 q^{37} -1.37493 q^{39} -0.823492 q^{40} -9.27821 q^{41} -0.132508 q^{42} -11.4632 q^{43} -1.00000 q^{44} -0.344274 q^{45} -4.06078 q^{46} -4.20763 q^{47} +1.60684 q^{48} -6.99320 q^{49} +4.32186 q^{50} +0.730187 q^{51} -0.855674 q^{52} -5.20678 q^{53} +5.49228 q^{54} -0.823492 q^{55} -0.0824650 q^{56} -8.34273 q^{58} -0.507301 q^{59} +1.32322 q^{60} +5.40019 q^{61} +2.85839 q^{62} -0.0344758 q^{63} +1.00000 q^{64} -0.704641 q^{65} +1.60684 q^{66} -4.13017 q^{67} +0.454424 q^{68} +6.52502 q^{69} -0.0679093 q^{70} +1.51855 q^{71} +0.418066 q^{72} +2.03491 q^{73} -5.05599 q^{74} -6.94454 q^{75} -0.0824650 q^{77} +1.37493 q^{78} +7.33950 q^{79} +0.823492 q^{80} -7.57102 q^{81} +9.27821 q^{82} +1.00311 q^{83} +0.132508 q^{84} +0.374215 q^{85} +11.4632 q^{86} +13.4054 q^{87} +1.00000 q^{88} +7.89670 q^{89} +0.344274 q^{90} -0.0705631 q^{91} +4.06078 q^{92} -4.59298 q^{93} +4.20763 q^{94} -1.60684 q^{96} -12.4191 q^{97} +6.99320 q^{98} +0.418066 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} - 12 q^{8} + 3 q^{9} + 9 q^{10} - 12 q^{11} + 3 q^{12} - 9 q^{15} + 12 q^{16} - 9 q^{17} - 3 q^{18} - 9 q^{20} + 9 q^{21} + 12 q^{22} - 18 q^{23}+ \cdots - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.60684 0.927709 0.463855 − 0.885911i \(-0.346466\pi\)
0.463855 + 0.885911i \(0.346466\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.823492 0.368277 0.184138 − 0.982900i \(-0.441051\pi\)
0.184138 + 0.982900i \(0.441051\pi\)
\(6\) −1.60684 −0.655990
\(7\) 0.0824650 0.0311688 0.0155844 − 0.999879i \(-0.495039\pi\)
0.0155844 + 0.999879i \(0.495039\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.418066 −0.139355
\(10\) −0.823492 −0.260411
\(11\) −1.00000 −0.301511
\(12\) 1.60684 0.463855
\(13\) −0.855674 −0.237321 −0.118661 − 0.992935i \(-0.537860\pi\)
−0.118661 + 0.992935i \(0.537860\pi\)
\(14\) −0.0824650 −0.0220397
\(15\) 1.32322 0.341654
\(16\) 1.00000 0.250000
\(17\) 0.454424 0.110214 0.0551071 − 0.998480i \(-0.482450\pi\)
0.0551071 + 0.998480i \(0.482450\pi\)
\(18\) 0.418066 0.0985391
\(19\) 0 0
\(20\) 0.823492 0.184138
\(21\) 0.132508 0.0289156
\(22\) 1.00000 0.213201
\(23\) 4.06078 0.846730 0.423365 − 0.905959i \(-0.360849\pi\)
0.423365 + 0.905959i \(0.360849\pi\)
\(24\) −1.60684 −0.327995
\(25\) −4.32186 −0.864372
\(26\) 0.855674 0.167811
\(27\) −5.49228 −1.05699
\(28\) 0.0824650 0.0155844
\(29\) 8.34273 1.54921 0.774603 − 0.632447i \(-0.217950\pi\)
0.774603 + 0.632447i \(0.217950\pi\)
\(30\) −1.32322 −0.241586
\(31\) −2.85839 −0.513383 −0.256691 − 0.966493i \(-0.582632\pi\)
−0.256691 + 0.966493i \(0.582632\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.60684 −0.279715
\(34\) −0.454424 −0.0779332
\(35\) 0.0679093 0.0114788
\(36\) −0.418066 −0.0696777
\(37\) 5.05599 0.831200 0.415600 − 0.909548i \(-0.363572\pi\)
0.415600 + 0.909548i \(0.363572\pi\)
\(38\) 0 0
\(39\) −1.37493 −0.220165
\(40\) −0.823492 −0.130206
\(41\) −9.27821 −1.44901 −0.724507 − 0.689268i \(-0.757932\pi\)
−0.724507 + 0.689268i \(0.757932\pi\)
\(42\) −0.132508 −0.0204464
\(43\) −11.4632 −1.74812 −0.874059 − 0.485820i \(-0.838521\pi\)
−0.874059 + 0.485820i \(0.838521\pi\)
\(44\) −1.00000 −0.150756
\(45\) −0.344274 −0.0513214
\(46\) −4.06078 −0.598729
\(47\) −4.20763 −0.613746 −0.306873 − 0.951751i \(-0.599283\pi\)
−0.306873 + 0.951751i \(0.599283\pi\)
\(48\) 1.60684 0.231927
\(49\) −6.99320 −0.999029
\(50\) 4.32186 0.611203
\(51\) 0.730187 0.102247
\(52\) −0.855674 −0.118661
\(53\) −5.20678 −0.715206 −0.357603 − 0.933874i \(-0.616406\pi\)
−0.357603 + 0.933874i \(0.616406\pi\)
\(54\) 5.49228 0.747405
\(55\) −0.823492 −0.111040
\(56\) −0.0824650 −0.0110198
\(57\) 0 0
\(58\) −8.34273 −1.09545
\(59\) −0.507301 −0.0660450 −0.0330225 − 0.999455i \(-0.510513\pi\)
−0.0330225 + 0.999455i \(0.510513\pi\)
\(60\) 1.32322 0.170827
\(61\) 5.40019 0.691423 0.345712 − 0.938341i \(-0.387638\pi\)
0.345712 + 0.938341i \(0.387638\pi\)
\(62\) 2.85839 0.363016
\(63\) −0.0344758 −0.00434355
\(64\) 1.00000 0.125000
\(65\) −0.704641 −0.0873999
\(66\) 1.60684 0.197788
\(67\) −4.13017 −0.504581 −0.252290 − 0.967652i \(-0.581184\pi\)
−0.252290 + 0.967652i \(0.581184\pi\)
\(68\) 0.454424 0.0551071
\(69\) 6.52502 0.785520
\(70\) −0.0679093 −0.00811671
\(71\) 1.51855 0.180218 0.0901091 − 0.995932i \(-0.471278\pi\)
0.0901091 + 0.995932i \(0.471278\pi\)
\(72\) 0.418066 0.0492696
\(73\) 2.03491 0.238168 0.119084 − 0.992884i \(-0.462004\pi\)
0.119084 + 0.992884i \(0.462004\pi\)
\(74\) −5.05599 −0.587747
\(75\) −6.94454 −0.801886
\(76\) 0 0
\(77\) −0.0824650 −0.00939776
\(78\) 1.37493 0.155680
\(79\) 7.33950 0.825758 0.412879 − 0.910786i \(-0.364523\pi\)
0.412879 + 0.910786i \(0.364523\pi\)
\(80\) 0.823492 0.0920692
\(81\) −7.57102 −0.841225
\(82\) 9.27821 1.02461
\(83\) 1.00311 0.110105 0.0550527 − 0.998483i \(-0.482467\pi\)
0.0550527 + 0.998483i \(0.482467\pi\)
\(84\) 0.132508 0.0144578
\(85\) 0.374215 0.0405893
\(86\) 11.4632 1.23611
\(87\) 13.4054 1.43721
\(88\) 1.00000 0.106600
\(89\) 7.89670 0.837048 0.418524 − 0.908206i \(-0.362548\pi\)
0.418524 + 0.908206i \(0.362548\pi\)
\(90\) 0.344274 0.0362897
\(91\) −0.0705631 −0.00739703
\(92\) 4.06078 0.423365
\(93\) −4.59298 −0.476270
\(94\) 4.20763 0.433984
\(95\) 0 0
\(96\) −1.60684 −0.163997
\(97\) −12.4191 −1.26097 −0.630484 − 0.776202i \(-0.717143\pi\)
−0.630484 + 0.776202i \(0.717143\pi\)
\(98\) 6.99320 0.706420
\(99\) 0.418066 0.0420172
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 7942.2.a.bt.1.8 12
19.4 even 9 418.2.j.a.111.2 ✓ 24
19.5 even 9 418.2.j.a.177.2 yes 24
19.18 odd 2 7942.2.a.bx.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.j.a.111.2 ✓ 24 19.4 even 9
418.2.j.a.177.2 yes 24 19.5 even 9
7942.2.a.bt.1.8 12 1.1 even 1 trivial
7942.2.a.bx.1.5 12 19.18 odd 2