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 1.7
Root \(-0.0391340\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.0391340 q^{3} +1.00000 q^{4} +2.72238 q^{5} +0.0391340 q^{6} +0.967809 q^{7} -1.00000 q^{8} -2.99847 q^{9} -2.72238 q^{10} -1.00000 q^{11} -0.0391340 q^{12} +6.51127 q^{13} -0.967809 q^{14} -0.106538 q^{15} +1.00000 q^{16} -3.44101 q^{17} +2.99847 q^{18} +2.72238 q^{20} -0.0378742 q^{21} +1.00000 q^{22} -3.07415 q^{23} +0.0391340 q^{24} +2.41138 q^{25} -6.51127 q^{26} +0.234744 q^{27} +0.967809 q^{28} -5.67043 q^{29} +0.106538 q^{30} +4.67796 q^{31} -1.00000 q^{32} +0.0391340 q^{33} +3.44101 q^{34} +2.63475 q^{35} -2.99847 q^{36} -4.12456 q^{37} -0.254812 q^{39} -2.72238 q^{40} -5.68427 q^{41} +0.0378742 q^{42} +0.357012 q^{43} -1.00000 q^{44} -8.16298 q^{45} +3.07415 q^{46} -7.98739 q^{47} -0.0391340 q^{48} -6.06335 q^{49} -2.41138 q^{50} +0.134660 q^{51} +6.51127 q^{52} +1.24271 q^{53} -0.234744 q^{54} -2.72238 q^{55} -0.967809 q^{56} +5.67043 q^{58} -10.1715 q^{59} -0.106538 q^{60} -5.13342 q^{61} -4.67796 q^{62} -2.90195 q^{63} +1.00000 q^{64} +17.7262 q^{65} -0.0391340 q^{66} +3.72764 q^{67} -3.44101 q^{68} +0.120304 q^{69} -2.63475 q^{70} -6.73892 q^{71} +2.99847 q^{72} -6.21976 q^{73} +4.12456 q^{74} -0.0943668 q^{75} -0.967809 q^{77} +0.254812 q^{78} +3.14563 q^{79} +2.72238 q^{80} +8.98622 q^{81} +5.68427 q^{82} -1.53008 q^{83} -0.0378742 q^{84} -9.36774 q^{85} -0.357012 q^{86} +0.221907 q^{87} +1.00000 q^{88} +7.67784 q^{89} +8.16298 q^{90} +6.30166 q^{91} -3.07415 q^{92} -0.183067 q^{93} +7.98739 q^{94} +0.0391340 q^{96} +15.9219 q^{97} +6.06335 q^{98} +2.99847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} - 12 q^{8} + 3 q^{9} + 9 q^{10} - 12 q^{11} + 3 q^{12} - 9 q^{15} + 12 q^{16} - 9 q^{17} - 3 q^{18} - 9 q^{20} + 9 q^{21} + 12 q^{22} - 18 q^{23}+ \cdots - 3 q^{99}+O(q^{100}) \) 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
\(3\) −0.0391340 −0.0225940 −0.0112970 − 0.999936i \(-0.503596\pi\)
−0.0112970 + 0.999936i \(0.503596\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.72238 1.21749 0.608744 − 0.793367i \(-0.291674\pi\)
0.608744 + 0.793367i \(0.291674\pi\)
\(6\) 0.0391340 0.0159764
\(7\) 0.967809 0.365798 0.182899 − 0.983132i \(-0.441452\pi\)
0.182899 + 0.983132i \(0.441452\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.99847 −0.999490
\(10\) −2.72238 −0.860893
\(11\) −1.00000 −0.301511
\(12\) −0.0391340 −0.0112970
\(13\) 6.51127 1.80590 0.902950 − 0.429745i \(-0.141397\pi\)
0.902950 + 0.429745i \(0.141397\pi\)
\(14\) −0.967809 −0.258658
\(15\) −0.106538 −0.0275079
\(16\) 1.00000 0.250000
\(17\) −3.44101 −0.834567 −0.417283 − 0.908776i \(-0.637018\pi\)
−0.417283 + 0.908776i \(0.637018\pi\)
\(18\) 2.99847 0.706746
\(19\) 0 0
\(20\) 2.72238 0.608744
\(21\) −0.0378742 −0.00826484
\(22\) 1.00000 0.213201
\(23\) −3.07415 −0.641004 −0.320502 − 0.947248i \(-0.603852\pi\)
−0.320502 + 0.947248i \(0.603852\pi\)
\(24\) 0.0391340 0.00798819
\(25\) 2.41138 0.482275
\(26\) −6.51127 −1.27696
\(27\) 0.234744 0.0451765
\(28\) 0.967809 0.182899
\(29\) −5.67043 −1.05297 −0.526486 − 0.850184i \(-0.676491\pi\)
−0.526486 + 0.850184i \(0.676491\pi\)
\(30\) 0.106538 0.0194510
\(31\) 4.67796 0.840186 0.420093 − 0.907481i \(-0.361997\pi\)
0.420093 + 0.907481i \(0.361997\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.0391340 0.00681235
\(34\) 3.44101 0.590128
\(35\) 2.63475 0.445354
\(36\) −2.99847 −0.499745
\(37\) −4.12456 −0.678074 −0.339037 − 0.940773i \(-0.610101\pi\)
−0.339037 + 0.940773i \(0.610101\pi\)
\(38\) 0 0
\(39\) −0.254812 −0.0408026
\(40\) −2.72238 −0.430447
\(41\) −5.68427 −0.887734 −0.443867 − 0.896093i \(-0.646394\pi\)
−0.443867 + 0.896093i \(0.646394\pi\)
\(42\) 0.0378742 0.00584412
\(43\) 0.357012 0.0544438 0.0272219 − 0.999629i \(-0.491334\pi\)
0.0272219 + 0.999629i \(0.491334\pi\)
\(44\) −1.00000 −0.150756
\(45\) −8.16298 −1.21687
\(46\) 3.07415 0.453258
\(47\) −7.98739 −1.16508 −0.582541 − 0.812802i \(-0.697941\pi\)
−0.582541 + 0.812802i \(0.697941\pi\)
\(48\) −0.0391340 −0.00564851
\(49\) −6.06335 −0.866192
\(50\) −2.41138 −0.341020
\(51\) 0.134660 0.0188562
\(52\) 6.51127 0.902950
\(53\) 1.24271 0.170699 0.0853494 − 0.996351i \(-0.472799\pi\)
0.0853494 + 0.996351i \(0.472799\pi\)
\(54\) −0.234744 −0.0319446
\(55\) −2.72238 −0.367086
\(56\) −0.967809 −0.129329
\(57\) 0 0
\(58\) 5.67043 0.744564
\(59\) −10.1715 −1.32422 −0.662110 − 0.749406i \(-0.730339\pi\)
−0.662110 + 0.749406i \(0.730339\pi\)
\(60\) −0.106538 −0.0137540
\(61\) −5.13342 −0.657267 −0.328633 − 0.944458i \(-0.606588\pi\)
−0.328633 + 0.944458i \(0.606588\pi\)
\(62\) −4.67796 −0.594101
\(63\) −2.90195 −0.365611
\(64\) 1.00000 0.125000
\(65\) 17.7262 2.19866
\(66\) −0.0391340 −0.00481706
\(67\) 3.72764 0.455404 0.227702 − 0.973731i \(-0.426879\pi\)
0.227702 + 0.973731i \(0.426879\pi\)
\(68\) −3.44101 −0.417283
\(69\) 0.120304 0.0144829
\(70\) −2.63475 −0.314913
\(71\) −6.73892 −0.799763 −0.399882 − 0.916567i \(-0.630949\pi\)
−0.399882 + 0.916567i \(0.630949\pi\)
\(72\) 2.99847 0.353373
\(73\) −6.21976 −0.727968 −0.363984 − 0.931405i \(-0.618584\pi\)
−0.363984 + 0.931405i \(0.618584\pi\)
\(74\) 4.12456 0.479471
\(75\) −0.0943668 −0.0108965
\(76\) 0 0
\(77\) −0.967809 −0.110292
\(78\) 0.254812 0.0288518
\(79\) 3.14563 0.353911 0.176956 − 0.984219i \(-0.443375\pi\)
0.176956 + 0.984219i \(0.443375\pi\)
\(80\) 2.72238 0.304372
\(81\) 8.98622 0.998469
\(82\) 5.68427 0.627723
\(83\) −1.53008 −0.167948 −0.0839738 − 0.996468i \(-0.526761\pi\)
−0.0839738 + 0.996468i \(0.526761\pi\)
\(84\) −0.0378742 −0.00413242
\(85\) −9.36774 −1.01607
\(86\) −0.357012 −0.0384976
\(87\) 0.221907 0.0237909
\(88\) 1.00000 0.106600
\(89\) 7.67784 0.813849 0.406925 − 0.913462i \(-0.366601\pi\)
0.406925 + 0.913462i \(0.366601\pi\)
\(90\) 8.16298 0.860454
\(91\) 6.30166 0.660594
\(92\) −3.07415 −0.320502
\(93\) −0.183067 −0.0189832
\(94\) 7.98739 0.823837
\(95\) 0 0
\(96\) 0.0391340 0.00399410
\(97\) 15.9219 1.61662 0.808310 − 0.588758i \(-0.200383\pi\)
0.808310 + 0.588758i \(0.200383\pi\)
\(98\) 6.06335 0.612490
\(99\) 2.99847 0.301357
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.1.7 12
19.6 even 9 418.2.j.a.397.3 yes 24
19.16 even 9 418.2.j.a.199.3 ✓ 24
19.18 odd 2 7942.2.a.bx.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.j.a.199.3 ✓ 24 19.16 even 9
418.2.j.a.397.3 yes 24 19.6 even 9
7942.2.a.bt.1.7 12 1.1 even 1 trivial
7942.2.a.bx.1.6 12 19.18 odd 2