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 = [8,8,-5,8,1,-5,-4] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 14x^{6} + 29x^{5} + 64x^{4} - 50x^{3} - 36x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(3.33202\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.33202 q^{3} +1.00000 q^{4} -3.95006 q^{5} +2.33202 q^{6} -0.471110 q^{7} +1.00000 q^{8} +2.43832 q^{9} -3.95006 q^{10} -1.00000 q^{11} +2.33202 q^{12} +0.121931 q^{13} -0.471110 q^{14} -9.21161 q^{15} +1.00000 q^{16} +0.909196 q^{17} +2.43832 q^{18} -3.95006 q^{20} -1.09864 q^{21} -1.00000 q^{22} +2.24103 q^{23} +2.33202 q^{24} +10.6029 q^{25} +0.121931 q^{26} -1.30984 q^{27} -0.471110 q^{28} -4.53735 q^{29} -9.21161 q^{30} +3.59334 q^{31} +1.00000 q^{32} -2.33202 q^{33} +0.909196 q^{34} +1.86091 q^{35} +2.43832 q^{36} +4.43922 q^{37} +0.284346 q^{39} -3.95006 q^{40} -10.7764 q^{41} -1.09864 q^{42} +2.24398 q^{43} -1.00000 q^{44} -9.63151 q^{45} +2.24103 q^{46} -9.47918 q^{47} +2.33202 q^{48} -6.77806 q^{49} +10.6029 q^{50} +2.12026 q^{51} +0.121931 q^{52} -12.8798 q^{53} -1.30984 q^{54} +3.95006 q^{55} -0.471110 q^{56} -4.53735 q^{58} -2.61973 q^{59} -9.21161 q^{60} -4.68127 q^{61} +3.59334 q^{62} -1.14872 q^{63} +1.00000 q^{64} -0.481635 q^{65} -2.33202 q^{66} +3.11452 q^{67} +0.909196 q^{68} +5.22614 q^{69} +1.86091 q^{70} +8.69874 q^{71} +2.43832 q^{72} -13.1535 q^{73} +4.43922 q^{74} +24.7263 q^{75} +0.471110 q^{77} +0.284346 q^{78} +6.36354 q^{79} -3.95006 q^{80} -10.3695 q^{81} -10.7764 q^{82} -2.37575 q^{83} -1.09864 q^{84} -3.59137 q^{85} +2.24398 q^{86} -10.5812 q^{87} -1.00000 q^{88} +3.88474 q^{89} -9.63151 q^{90} -0.0574430 q^{91} +2.24103 q^{92} +8.37975 q^{93} -9.47918 q^{94} +2.33202 q^{96} -18.7664 q^{97} -6.77806 q^{98} -2.43832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 5 q^{3} + 8 q^{4} + q^{5} - 5 q^{6} - 4 q^{7} + 8 q^{8} + 15 q^{9} + q^{10} - 8 q^{11} - 5 q^{12} - 3 q^{13} - 4 q^{14} - 34 q^{15} + 8 q^{16} - 6 q^{17} + 15 q^{18} + q^{20} + 11 q^{21}+ \cdots - 15 q^{99}+O(q^{100}) \) 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
\(3\) 2.33202 1.34639 0.673197 − 0.739464i \(-0.264921\pi\)
0.673197 + 0.739464i \(0.264921\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.95006 −1.76652 −0.883259 − 0.468885i \(-0.844656\pi\)
−0.883259 + 0.468885i \(0.844656\pi\)
\(6\) 2.33202 0.952044
\(7\) −0.471110 −0.178063 −0.0890314 − 0.996029i \(-0.528377\pi\)
−0.0890314 + 0.996029i \(0.528377\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.43832 0.812774
\(10\) −3.95006 −1.24912
\(11\) −1.00000 −0.301511
\(12\) 2.33202 0.673197
\(13\) 0.121931 0.0338176 0.0169088 − 0.999857i \(-0.494618\pi\)
0.0169088 + 0.999857i \(0.494618\pi\)
\(14\) −0.471110 −0.125909
\(15\) −9.21161 −2.37843
\(16\) 1.00000 0.250000
\(17\) 0.909196 0.220512 0.110256 − 0.993903i \(-0.464833\pi\)
0.110256 + 0.993903i \(0.464833\pi\)
\(18\) 2.43832 0.574718
\(19\) 0 0
\(20\) −3.95006 −0.883259
\(21\) −1.09864 −0.239743
\(22\) −1.00000 −0.213201
\(23\) 2.24103 0.467288 0.233644 − 0.972322i \(-0.424935\pi\)
0.233644 + 0.972322i \(0.424935\pi\)
\(24\) 2.33202 0.476022
\(25\) 10.6029 2.12059
\(26\) 0.121931 0.0239127
\(27\) −1.30984 −0.252079
\(28\) −0.471110 −0.0890314
\(29\) −4.53735 −0.842564 −0.421282 − 0.906930i \(-0.638420\pi\)
−0.421282 + 0.906930i \(0.638420\pi\)
\(30\) −9.21161 −1.68180
\(31\) 3.59334 0.645383 0.322692 − 0.946504i \(-0.395412\pi\)
0.322692 + 0.946504i \(0.395412\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.33202 −0.405953
\(34\) 0.909196 0.155926
\(35\) 1.86091 0.314551
\(36\) 2.43832 0.406387
\(37\) 4.43922 0.729804 0.364902 − 0.931046i \(-0.381102\pi\)
0.364902 + 0.931046i \(0.381102\pi\)
\(38\) 0 0
\(39\) 0.284346 0.0455318
\(40\) −3.95006 −0.624559
\(41\) −10.7764 −1.68298 −0.841492 − 0.540269i \(-0.818322\pi\)
−0.841492 + 0.540269i \(0.818322\pi\)
\(42\) −1.09864 −0.169524
\(43\) 2.24398 0.342204 0.171102 − 0.985253i \(-0.445267\pi\)
0.171102 + 0.985253i \(0.445267\pi\)
\(44\) −1.00000 −0.150756
\(45\) −9.63151 −1.43578
\(46\) 2.24103 0.330422
\(47\) −9.47918 −1.38268 −0.691341 − 0.722529i \(-0.742980\pi\)
−0.691341 + 0.722529i \(0.742980\pi\)
\(48\) 2.33202 0.336598
\(49\) −6.77806 −0.968294
\(50\) 10.6029 1.49948
\(51\) 2.12026 0.296896
\(52\) 0.121931 0.0169088
\(53\) −12.8798 −1.76917 −0.884587 − 0.466376i \(-0.845559\pi\)
−0.884587 + 0.466376i \(0.845559\pi\)
\(54\) −1.30984 −0.178247
\(55\) 3.95006 0.532625
\(56\) −0.471110 −0.0629547
\(57\) 0 0
\(58\) −4.53735 −0.595783
\(59\) −2.61973 −0.341060 −0.170530 − 0.985352i \(-0.554548\pi\)
−0.170530 + 0.985352i \(0.554548\pi\)
\(60\) −9.21161 −1.18921
\(61\) −4.68127 −0.599376 −0.299688 − 0.954037i \(-0.596883\pi\)
−0.299688 + 0.954037i \(0.596883\pi\)
\(62\) 3.59334 0.456355
\(63\) −1.14872 −0.144725
\(64\) 1.00000 0.125000
\(65\) −0.481635 −0.0597395
\(66\) −2.33202 −0.287052
\(67\) 3.11452 0.380499 0.190250 − 0.981736i \(-0.439070\pi\)
0.190250 + 0.981736i \(0.439070\pi\)
\(68\) 0.909196 0.110256
\(69\) 5.22614 0.629153
\(70\) 1.86091 0.222421
\(71\) 8.69874 1.03235 0.516175 − 0.856483i \(-0.327355\pi\)
0.516175 + 0.856483i \(0.327355\pi\)
\(72\) 2.43832 0.287359
\(73\) −13.1535 −1.53950 −0.769752 − 0.638343i \(-0.779620\pi\)
−0.769752 + 0.638343i \(0.779620\pi\)
\(74\) 4.43922 0.516049
\(75\) 24.7263 2.85514
\(76\) 0 0
\(77\) 0.471110 0.0536880
\(78\) 0.284346 0.0321959
\(79\) 6.36354 0.715954 0.357977 − 0.933730i \(-0.383467\pi\)
0.357977 + 0.933730i \(0.383467\pi\)
\(80\) −3.95006 −0.441630
\(81\) −10.3695 −1.15217
\(82\) −10.7764 −1.19005
\(83\) −2.37575 −0.260773 −0.130386 − 0.991463i \(-0.541622\pi\)
−0.130386 + 0.991463i \(0.541622\pi\)
\(84\) −1.09864 −0.119871
\(85\) −3.59137 −0.389539
\(86\) 2.24398 0.241975
\(87\) −10.5812 −1.13442
\(88\) −1.00000 −0.106600
\(89\) 3.88474 0.411782 0.205891 − 0.978575i \(-0.433991\pi\)
0.205891 + 0.978575i \(0.433991\pi\)
\(90\) −9.63151 −1.01525
\(91\) −0.0574430 −0.00602166
\(92\) 2.24103 0.233644
\(93\) 8.37975 0.868939
\(94\) −9.47918 −0.977703
\(95\) 0 0
\(96\) 2.33202 0.238011
\(97\) −18.7664 −1.90544 −0.952718 − 0.303855i \(-0.901726\pi\)
−0.952718 + 0.303855i \(0.901726\pi\)
\(98\) −6.77806 −0.684687
\(99\) −2.43832 −0.245061
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.bp.1.7 yes 8
19.18 odd 2 7942.2.a.bo.1.2 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7942.2.a.bo.1.2 ✓ 8 19.18 odd 2
7942.2.a.bp.1.7 yes 8 1.1 even 1 trivial