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,-12] 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: \( x^{12} - 3 x^{11} - 21 x^{10} + 59 x^{9} + 162 x^{8} - 408 x^{7} - 581 x^{6} + 1236 x^{5} + 972 x^{4} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.9
Root \(1.87195\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.87195 q^{3} +1.00000 q^{4} +3.81859 q^{5} -1.87195 q^{6} -2.96733 q^{7} -1.00000 q^{8} +0.504212 q^{9} -3.81859 q^{10} +1.00000 q^{11} +1.87195 q^{12} -1.08537 q^{13} +2.96733 q^{14} +7.14822 q^{15} +1.00000 q^{16} -2.29885 q^{17} -0.504212 q^{18} +3.81859 q^{20} -5.55471 q^{21} -1.00000 q^{22} -7.37385 q^{23} -1.87195 q^{24} +9.58162 q^{25} +1.08537 q^{26} -4.67200 q^{27} -2.96733 q^{28} -7.32445 q^{29} -7.14822 q^{30} +8.37319 q^{31} -1.00000 q^{32} +1.87195 q^{33} +2.29885 q^{34} -11.3310 q^{35} +0.504212 q^{36} +9.10622 q^{37} -2.03176 q^{39} -3.81859 q^{40} -9.41379 q^{41} +5.55471 q^{42} -8.07120 q^{43} +1.00000 q^{44} +1.92538 q^{45} +7.37385 q^{46} -12.2614 q^{47} +1.87195 q^{48} +1.80506 q^{49} -9.58162 q^{50} -4.30334 q^{51} -1.08537 q^{52} -3.29423 q^{53} +4.67200 q^{54} +3.81859 q^{55} +2.96733 q^{56} +7.32445 q^{58} +0.00737709 q^{59} +7.14822 q^{60} +1.17369 q^{61} -8.37319 q^{62} -1.49616 q^{63} +1.00000 q^{64} -4.14457 q^{65} -1.87195 q^{66} -4.51665 q^{67} -2.29885 q^{68} -13.8035 q^{69} +11.3310 q^{70} -6.26636 q^{71} -0.504212 q^{72} -9.46560 q^{73} -9.10622 q^{74} +17.9363 q^{75} -2.96733 q^{77} +2.03176 q^{78} -0.124845 q^{79} +3.81859 q^{80} -10.2584 q^{81} +9.41379 q^{82} +10.9086 q^{83} -5.55471 q^{84} -8.77835 q^{85} +8.07120 q^{86} -13.7110 q^{87} -1.00000 q^{88} -4.14468 q^{89} -1.92538 q^{90} +3.22064 q^{91} -7.37385 q^{92} +15.6742 q^{93} +12.2614 q^{94} -1.87195 q^{96} +1.40370 q^{97} -1.80506 q^{98} +0.504212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} - 12 q^{7} - 12 q^{8} + 15 q^{9} + 9 q^{10} + 12 q^{11} + 3 q^{12} + 12 q^{14} + 21 q^{15} + 12 q^{16} - 33 q^{17} - 15 q^{18} - 9 q^{20} - 9 q^{21}+ \cdots + 15 q^{99}+O(q^{100}) \) 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
\(3\) 1.87195 1.08077 0.540387 − 0.841417i \(-0.318278\pi\)
0.540387 + 0.841417i \(0.318278\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.81859 1.70772 0.853862 − 0.520499i \(-0.174254\pi\)
0.853862 + 0.520499i \(0.174254\pi\)
\(6\) −1.87195 −0.764222
\(7\) −2.96733 −1.12155 −0.560773 − 0.827970i \(-0.689496\pi\)
−0.560773 + 0.827970i \(0.689496\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0.504212 0.168071
\(10\) −3.81859 −1.20754
\(11\) 1.00000 0.301511
\(12\) 1.87195 0.540387
\(13\) −1.08537 −0.301026 −0.150513 − 0.988608i \(-0.548093\pi\)
−0.150513 + 0.988608i \(0.548093\pi\)
\(14\) 2.96733 0.793053
\(15\) 7.14822 1.84566
\(16\) 1.00000 0.250000
\(17\) −2.29885 −0.557552 −0.278776 − 0.960356i \(-0.589929\pi\)
−0.278776 + 0.960356i \(0.589929\pi\)
\(18\) −0.504212 −0.118844
\(19\) 0 0
\(20\) 3.81859 0.853862
\(21\) −5.55471 −1.21214
\(22\) −1.00000 −0.213201
\(23\) −7.37385 −1.53755 −0.768777 − 0.639517i \(-0.779134\pi\)
−0.768777 + 0.639517i \(0.779134\pi\)
\(24\) −1.87195 −0.382111
\(25\) 9.58162 1.91632
\(26\) 1.08537 0.212858
\(27\) −4.67200 −0.899127
\(28\) −2.96733 −0.560773
\(29\) −7.32445 −1.36012 −0.680058 − 0.733158i \(-0.738045\pi\)
−0.680058 + 0.733158i \(0.738045\pi\)
\(30\) −7.14822 −1.30508
\(31\) 8.37319 1.50387 0.751935 − 0.659238i \(-0.229121\pi\)
0.751935 + 0.659238i \(0.229121\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.87195 0.325865
\(34\) 2.29885 0.394249
\(35\) −11.3310 −1.91529
\(36\) 0.504212 0.0840353
\(37\) 9.10622 1.49705 0.748526 − 0.663105i \(-0.230762\pi\)
0.748526 + 0.663105i \(0.230762\pi\)
\(38\) 0 0
\(39\) −2.03176 −0.325341
\(40\) −3.81859 −0.603772
\(41\) −9.41379 −1.47019 −0.735094 − 0.677966i \(-0.762862\pi\)
−0.735094 + 0.677966i \(0.762862\pi\)
\(42\) 5.55471 0.857110
\(43\) −8.07120 −1.23085 −0.615423 − 0.788197i \(-0.711015\pi\)
−0.615423 + 0.788197i \(0.711015\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.92538 0.287018
\(46\) 7.37385 1.08721
\(47\) −12.2614 −1.78850 −0.894252 − 0.447564i \(-0.852292\pi\)
−0.894252 + 0.447564i \(0.852292\pi\)
\(48\) 1.87195 0.270193
\(49\) 1.80506 0.257866
\(50\) −9.58162 −1.35505
\(51\) −4.30334 −0.602588
\(52\) −1.08537 −0.150513
\(53\) −3.29423 −0.452498 −0.226249 − 0.974070i \(-0.572646\pi\)
−0.226249 + 0.974070i \(0.572646\pi\)
\(54\) 4.67200 0.635779
\(55\) 3.81859 0.514898
\(56\) 2.96733 0.396526
\(57\) 0 0
\(58\) 7.32445 0.961747
\(59\) 0.00737709 0.000960415 0 0.000480207 − 1.00000i \(-0.499847\pi\)
0.000480207 1.00000i \(0.499847\pi\)
\(60\) 7.14822 0.922831
\(61\) 1.17369 0.150276 0.0751378 − 0.997173i \(-0.476060\pi\)
0.0751378 + 0.997173i \(0.476060\pi\)
\(62\) −8.37319 −1.06340
\(63\) −1.49616 −0.188499
\(64\) 1.00000 0.125000
\(65\) −4.14457 −0.514070
\(66\) −1.87195 −0.230422
\(67\) −4.51665 −0.551797 −0.275898 − 0.961187i \(-0.588975\pi\)
−0.275898 + 0.961187i \(0.588975\pi\)
\(68\) −2.29885 −0.278776
\(69\) −13.8035 −1.66175
\(70\) 11.3310 1.35432
\(71\) −6.26636 −0.743680 −0.371840 − 0.928297i \(-0.621273\pi\)
−0.371840 + 0.928297i \(0.621273\pi\)
\(72\) −0.504212 −0.0594219
\(73\) −9.46560 −1.10787 −0.553933 − 0.832562i \(-0.686873\pi\)
−0.553933 + 0.832562i \(0.686873\pi\)
\(74\) −9.10622 −1.05858
\(75\) 17.9363 2.07111
\(76\) 0 0
\(77\) −2.96733 −0.338159
\(78\) 2.03176 0.230051
\(79\) −0.124845 −0.0140461 −0.00702306 − 0.999975i \(-0.502236\pi\)
−0.00702306 + 0.999975i \(0.502236\pi\)
\(80\) 3.81859 0.426931
\(81\) −10.2584 −1.13982
\(82\) 9.41379 1.03958
\(83\) 10.9086 1.19738 0.598689 − 0.800982i \(-0.295689\pi\)
0.598689 + 0.800982i \(0.295689\pi\)
\(84\) −5.55471 −0.606069
\(85\) −8.77835 −0.952146
\(86\) 8.07120 0.870339
\(87\) −13.7110 −1.46998
\(88\) −1.00000 −0.106600
\(89\) −4.14468 −0.439336 −0.219668 − 0.975575i \(-0.570497\pi\)
−0.219668 + 0.975575i \(0.570497\pi\)
\(90\) −1.92538 −0.202953
\(91\) 3.22064 0.337615
\(92\) −7.37385 −0.768777
\(93\) 15.6742 1.62534
\(94\) 12.2614 1.26466
\(95\) 0 0
\(96\) −1.87195 −0.191056
\(97\) 1.40370 0.142524 0.0712620 − 0.997458i \(-0.477297\pi\)
0.0712620 + 0.997458i \(0.477297\pi\)
\(98\) −1.80506 −0.182339
\(99\) 0.504212 0.0506752
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.bs.1.9 12
19.14 odd 18 418.2.j.b.177.3 yes 24
19.15 odd 18 418.2.j.b.111.3 ✓ 24
19.18 odd 2 7942.2.a.bw.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.j.b.111.3 ✓ 24 19.15 odd 18
418.2.j.b.177.3 yes 24 19.14 odd 18
7942.2.a.bs.1.9 12 1.1 even 1 trivial
7942.2.a.bw.1.4 12 19.18 odd 2