Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3971,2,Mod(1,3971)] 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("3971.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3971, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3971 = 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3971.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.7085946427\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.61330\) of defining polynomial
Character \(\chi\) \(=\) 3971.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.61330 q^{2} -1.19599 q^{3} +4.82936 q^{4} +4.07680 q^{5} -3.12549 q^{6} +3.61829 q^{7} +7.39397 q^{8} -1.56960 q^{9} +10.6539 q^{10} -1.00000 q^{11} -5.77587 q^{12} +1.47857 q^{13} +9.45570 q^{14} -4.87582 q^{15} +9.66398 q^{16} -3.27003 q^{17} -4.10185 q^{18} +19.6883 q^{20} -4.32745 q^{21} -2.61330 q^{22} -7.45793 q^{23} -8.84313 q^{24} +11.6203 q^{25} +3.86395 q^{26} +5.46521 q^{27} +17.4740 q^{28} -1.02535 q^{29} -12.7420 q^{30} -1.64921 q^{31} +10.4670 q^{32} +1.19599 q^{33} -8.54558 q^{34} +14.7511 q^{35} -7.58018 q^{36} +6.71293 q^{37} -1.76836 q^{39} +30.1438 q^{40} +3.92451 q^{41} -11.3089 q^{42} +5.38113 q^{43} -4.82936 q^{44} -6.39896 q^{45} -19.4898 q^{46} -3.71597 q^{47} -11.5580 q^{48} +6.09205 q^{49} +30.3674 q^{50} +3.91093 q^{51} +7.14054 q^{52} +0.102902 q^{53} +14.2823 q^{54} -4.07680 q^{55} +26.7536 q^{56} -2.67955 q^{58} -13.2986 q^{59} -23.5471 q^{60} -6.49664 q^{61} -4.30989 q^{62} -5.67929 q^{63} +8.02543 q^{64} +6.02783 q^{65} +3.12549 q^{66} +3.70989 q^{67} -15.7921 q^{68} +8.91962 q^{69} +38.5490 q^{70} -6.32968 q^{71} -11.6056 q^{72} -1.37759 q^{73} +17.5429 q^{74} -13.8978 q^{75} -3.61829 q^{77} -4.62125 q^{78} -13.6725 q^{79} +39.3981 q^{80} -1.82753 q^{81} +10.2559 q^{82} +5.44061 q^{83} -20.8988 q^{84} -13.3313 q^{85} +14.0625 q^{86} +1.22631 q^{87} -7.39397 q^{88} -12.1357 q^{89} -16.7224 q^{90} +5.34990 q^{91} -36.0170 q^{92} +1.97244 q^{93} -9.71096 q^{94} -12.5184 q^{96} +13.7910 q^{97} +15.9204 q^{98} +1.56960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} + 9 q^{8} + 11 q^{9} + 6 q^{10} - 7 q^{11} + 16 q^{12} + 4 q^{13} - 6 q^{14} - 12 q^{15} + 27 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{20} + 14 q^{21}+ \cdots - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61330 1.84789 0.923943 − 0.382531i \(-0.124948\pi\)
0.923943 + 0.382531i \(0.124948\pi\)
\(3\) −1.19599 −0.690506 −0.345253 − 0.938510i \(-0.612207\pi\)
−0.345253 + 0.938510i \(0.612207\pi\)
\(4\) 4.82936 2.41468
\(5\) 4.07680 1.82320 0.911600 − 0.411078i \(-0.134847\pi\)
0.911600 + 0.411078i \(0.134847\pi\)
\(6\) −3.12549 −1.27598
\(7\) 3.61829 1.36759 0.683793 − 0.729676i \(-0.260329\pi\)
0.683793 + 0.729676i \(0.260329\pi\)
\(8\) 7.39397 2.61416
\(9\) −1.56960 −0.523201
\(10\) 10.6539 3.36907
\(11\) −1.00000 −0.301511
\(12\) −5.77587 −1.66735
\(13\) 1.47857 0.410081 0.205041 − 0.978753i \(-0.434267\pi\)
0.205041 + 0.978753i \(0.434267\pi\)
\(14\) 9.45570 2.52714
\(15\) −4.87582 −1.25893
\(16\) 9.66398 2.41600
\(17\) −3.27003 −0.793099 −0.396549 − 0.918013i \(-0.629792\pi\)
−0.396549 + 0.918013i \(0.629792\pi\)
\(18\) −4.10185 −0.966816
\(19\) 0 0
\(20\) 19.6883 4.40244
\(21\) −4.32745 −0.944327
\(22\) −2.61330 −0.557158
\(23\) −7.45793 −1.55509 −0.777543 − 0.628830i \(-0.783534\pi\)
−0.777543 + 0.628830i \(0.783534\pi\)
\(24\) −8.84313 −1.80510
\(25\) 11.6203 2.32406
\(26\) 3.86395 0.757783
\(27\) 5.46521 1.05178
\(28\) 17.4740 3.30228
\(29\) −1.02535 −0.190403 −0.0952013 − 0.995458i \(-0.530349\pi\)
−0.0952013 + 0.995458i \(0.530349\pi\)
\(30\) −12.7420 −2.32636
\(31\) −1.64921 −0.296207 −0.148104 − 0.988972i \(-0.547317\pi\)
−0.148104 + 0.988972i \(0.547317\pi\)
\(32\) 10.4670 1.85032
\(33\) 1.19599 0.208195
\(34\) −8.54558 −1.46556
\(35\) 14.7511 2.49338
\(36\) −7.58018 −1.26336
\(37\) 6.71293 1.10360 0.551799 − 0.833977i \(-0.313941\pi\)
0.551799 + 0.833977i \(0.313941\pi\)
\(38\) 0 0
\(39\) −1.76836 −0.283164
\(40\) 30.1438 4.76615
\(41\) 3.92451 0.612905 0.306453 − 0.951886i \(-0.400858\pi\)
0.306453 + 0.951886i \(0.400858\pi\)
\(42\) −11.3089 −1.74501
\(43\) 5.38113 0.820614 0.410307 − 0.911947i \(-0.365422\pi\)
0.410307 + 0.911947i \(0.365422\pi\)
\(44\) −4.82936 −0.728053
\(45\) −6.39896 −0.953901
\(46\) −19.4898 −2.87362
\(47\) −3.71597 −0.542030 −0.271015 − 0.962575i \(-0.587359\pi\)
−0.271015 + 0.962575i \(0.587359\pi\)
\(48\) −11.5580 −1.66826
\(49\) 6.09205 0.870292
\(50\) 30.3674 4.29460
\(51\) 3.91093 0.547640
\(52\) 7.14054 0.990215
\(53\) 0.102902 0.0141347 0.00706733 − 0.999975i \(-0.497750\pi\)
0.00706733 + 0.999975i \(0.497750\pi\)
\(54\) 14.2823 1.94357
\(55\) −4.07680 −0.549716
\(56\) 26.7536 3.57509
\(57\) 0 0
\(58\) −2.67955 −0.351842
\(59\) −13.2986 −1.73134 −0.865668 − 0.500619i \(-0.833106\pi\)
−0.865668 + 0.500619i \(0.833106\pi\)
\(60\) −23.5471 −3.03991
\(61\) −6.49664 −0.831809 −0.415905 − 0.909408i \(-0.636535\pi\)
−0.415905 + 0.909408i \(0.636535\pi\)
\(62\) −4.30989 −0.547357
\(63\) −5.67929 −0.715523
\(64\) 8.02543 1.00318
\(65\) 6.02783 0.747661
\(66\) 3.12549 0.384721
\(67\) 3.70989 0.453235 0.226618 − 0.973984i \(-0.427233\pi\)
0.226618 + 0.973984i \(0.427233\pi\)
\(68\) −15.7921 −1.91508
\(69\) 8.91962 1.07380
\(70\) 38.5490 4.60749
\(71\) −6.32968 −0.751194 −0.375597 − 0.926783i \(-0.622562\pi\)
−0.375597 + 0.926783i \(0.622562\pi\)
\(72\) −11.6056 −1.36773
\(73\) −1.37759 −0.161235 −0.0806173 − 0.996745i \(-0.525689\pi\)
−0.0806173 + 0.996745i \(0.525689\pi\)
\(74\) 17.5429 2.03932
\(75\) −13.8978 −1.60478
\(76\) 0 0
\(77\) −3.61829 −0.412343
\(78\) −4.62125 −0.523254
\(79\) −13.6725 −1.53828 −0.769141 − 0.639079i \(-0.779316\pi\)
−0.769141 + 0.639079i \(0.779316\pi\)
\(80\) 39.3981 4.40484
\(81\) −1.82753 −0.203059
\(82\) 10.2559 1.13258
\(83\) 5.44061 0.597184 0.298592 − 0.954381i \(-0.403483\pi\)
0.298592 + 0.954381i \(0.403483\pi\)
\(84\) −20.8988 −2.28025
\(85\) −13.3313 −1.44598
\(86\) 14.0625 1.51640
\(87\) 1.22631 0.131474
\(88\) −7.39397 −0.788200
\(89\) −12.1357 −1.28638 −0.643191 − 0.765706i \(-0.722390\pi\)
−0.643191 + 0.765706i \(0.722390\pi\)
\(90\) −16.7224 −1.76270
\(91\) 5.34990 0.560822
\(92\) −36.0170 −3.75503
\(93\) 1.97244 0.204533
\(94\) −9.71096 −1.00161
\(95\) 0 0
\(96\) −12.5184 −1.27766
\(97\) 13.7910 1.40026 0.700131 − 0.714014i \(-0.253125\pi\)
0.700131 + 0.714014i \(0.253125\pi\)
\(98\) 15.9204 1.60820
\(99\) 1.56960 0.157751
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 3971.2.a.i.1.6 7
19.18 odd 2 209.2.a.d.1.2 ✓ 7
57.56 even 2 1881.2.a.p.1.6 7
76.75 even 2 3344.2.a.ba.1.4 7
95.94 odd 2 5225.2.a.n.1.6 7
209.208 even 2 2299.2.a.q.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 ✓ 7 19.18 odd 2
1881.2.a.p.1.6 7 57.56 even 2
2299.2.a.q.1.6 7 209.208 even 2
3344.2.a.ba.1.4 7 76.75 even 2
3971.2.a.i.1.6 7 1.1 even 1 trivial
5225.2.a.n.1.6 7 95.94 odd 2