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 1.5
Root \(1.19313\) 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)
 
\(f(q)\) \(=\) \(q+1.19313 q^{2} -3.16232 q^{3} -0.576442 q^{4} +1.08235 q^{5} -3.77306 q^{6} -1.30958 q^{7} -3.07403 q^{8} +7.00029 q^{9} +1.29138 q^{10} -1.00000 q^{11} +1.82290 q^{12} -2.53997 q^{13} -1.56249 q^{14} -3.42273 q^{15} -2.51483 q^{16} -5.43892 q^{17} +8.35226 q^{18} -0.623909 q^{20} +4.14130 q^{21} -1.19313 q^{22} +3.87095 q^{23} +9.72108 q^{24} -3.82853 q^{25} -3.03052 q^{26} -12.6502 q^{27} +0.754894 q^{28} +2.41412 q^{29} -4.08376 q^{30} -3.03647 q^{31} +3.14754 q^{32} +3.16232 q^{33} -6.48934 q^{34} -1.41741 q^{35} -4.03526 q^{36} -6.85067 q^{37} +8.03222 q^{39} -3.32716 q^{40} +6.11344 q^{41} +4.94111 q^{42} -2.95329 q^{43} +0.576442 q^{44} +7.57673 q^{45} +4.61854 q^{46} -12.0923 q^{47} +7.95271 q^{48} -5.28501 q^{49} -4.56793 q^{50} +17.1996 q^{51} +1.46415 q^{52} +0.992927 q^{53} -15.0933 q^{54} -1.08235 q^{55} +4.02567 q^{56} +2.88036 q^{58} +14.2251 q^{59} +1.97300 q^{60} -5.82518 q^{61} -3.62290 q^{62} -9.16741 q^{63} +8.78508 q^{64} -2.74913 q^{65} +3.77306 q^{66} -8.79718 q^{67} +3.13522 q^{68} -12.2412 q^{69} -1.69116 q^{70} +2.44975 q^{71} -21.5191 q^{72} +5.84063 q^{73} -8.17374 q^{74} +12.1070 q^{75} +1.30958 q^{77} +9.58348 q^{78} -17.0397 q^{79} -2.72192 q^{80} +19.0032 q^{81} +7.29413 q^{82} -7.01303 q^{83} -2.38722 q^{84} -5.88679 q^{85} -3.52366 q^{86} -7.63422 q^{87} +3.07403 q^{88} +9.13103 q^{89} +9.04003 q^{90} +3.32629 q^{91} -2.23138 q^{92} +9.60229 q^{93} -14.4277 q^{94} -9.95354 q^{96} +14.7950 q^{97} -6.30570 q^{98} -7.00029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} - 2 q^{3} + 15 q^{4} + 2 q^{5} - 2 q^{6} + 10 q^{7} + 9 q^{8} + 11 q^{9} + 6 q^{10} - 7 q^{11} + 16 q^{12} + 4 q^{13} - 6 q^{14} - 12 q^{15} + 27 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{20} + 14 q^{21}+ \cdots - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19313 0.843670 0.421835 − 0.906673i \(-0.361386\pi\)
0.421835 + 0.906673i \(0.361386\pi\)
\(3\) −3.16232 −1.82577 −0.912884 − 0.408218i \(-0.866150\pi\)
−0.912884 + 0.408218i \(0.866150\pi\)
\(4\) −0.576442 −0.288221
\(5\) 1.08235 0.484040 0.242020 − 0.970271i \(-0.422190\pi\)
0.242020 + 0.970271i \(0.422190\pi\)
\(6\) −3.77306 −1.54035
\(7\) −1.30958 −0.494973 −0.247487 − 0.968891i \(-0.579605\pi\)
−0.247487 + 0.968891i \(0.579605\pi\)
\(8\) −3.07403 −1.08683
\(9\) 7.00029 2.33343
\(10\) 1.29138 0.408370
\(11\) −1.00000 −0.301511
\(12\) 1.82290 0.526225
\(13\) −2.53997 −0.704462 −0.352231 − 0.935913i \(-0.614577\pi\)
−0.352231 + 0.935913i \(0.614577\pi\)
\(14\) −1.56249 −0.417594
\(15\) −3.42273 −0.883744
\(16\) −2.51483 −0.628708
\(17\) −5.43892 −1.31913 −0.659566 − 0.751646i \(-0.729260\pi\)
−0.659566 + 0.751646i \(0.729260\pi\)
\(18\) 8.35226 1.96865
\(19\) 0 0
\(20\) −0.623909 −0.139510
\(21\) 4.14130 0.903706
\(22\) −1.19313 −0.254376
\(23\) 3.87095 0.807148 0.403574 − 0.914947i \(-0.367768\pi\)
0.403574 + 0.914947i \(0.367768\pi\)
\(24\) 9.72108 1.98431
\(25\) −3.82853 −0.765706
\(26\) −3.03052 −0.594334
\(27\) −12.6502 −2.43454
\(28\) 0.754894 0.142662
\(29\) 2.41412 0.448290 0.224145 − 0.974556i \(-0.428041\pi\)
0.224145 + 0.974556i \(0.428041\pi\)
\(30\) −4.08376 −0.745588
\(31\) −3.03647 −0.545366 −0.272683 − 0.962104i \(-0.587911\pi\)
−0.272683 + 0.962104i \(0.587911\pi\)
\(32\) 3.14754 0.556412
\(33\) 3.16232 0.550490
\(34\) −6.48934 −1.11291
\(35\) −1.41741 −0.239587
\(36\) −4.03526 −0.672544
\(37\) −6.85067 −1.12624 −0.563122 − 0.826374i \(-0.690400\pi\)
−0.563122 + 0.826374i \(0.690400\pi\)
\(38\) 0 0
\(39\) 8.03222 1.28618
\(40\) −3.32716 −0.526070
\(41\) 6.11344 0.954759 0.477380 − 0.878697i \(-0.341587\pi\)
0.477380 + 0.878697i \(0.341587\pi\)
\(42\) 4.94111 0.762430
\(43\) −2.95329 −0.450373 −0.225186 − 0.974316i \(-0.572299\pi\)
−0.225186 + 0.974316i \(0.572299\pi\)
\(44\) 0.576442 0.0869019
\(45\) 7.57673 1.12947
\(46\) 4.61854 0.680966
\(47\) −12.0923 −1.76385 −0.881925 − 0.471391i \(-0.843752\pi\)
−0.881925 + 0.471391i \(0.843752\pi\)
\(48\) 7.95271 1.14787
\(49\) −5.28501 −0.755002
\(50\) −4.56793 −0.646003
\(51\) 17.1996 2.40843
\(52\) 1.46415 0.203041
\(53\) 0.992927 0.136389 0.0681945 − 0.997672i \(-0.478276\pi\)
0.0681945 + 0.997672i \(0.478276\pi\)
\(54\) −15.0933 −2.05394
\(55\) −1.08235 −0.145943
\(56\) 4.02567 0.537953
\(57\) 0 0
\(58\) 2.88036 0.378209
\(59\) 14.2251 1.85195 0.925977 − 0.377580i \(-0.123244\pi\)
0.925977 + 0.377580i \(0.123244\pi\)
\(60\) 1.97300 0.254714
\(61\) −5.82518 −0.745838 −0.372919 − 0.927864i \(-0.621643\pi\)
−0.372919 + 0.927864i \(0.621643\pi\)
\(62\) −3.62290 −0.460109
\(63\) −9.16741 −1.15499
\(64\) 8.78508 1.09814
\(65\) −2.74913 −0.340988
\(66\) 3.77306 0.464432
\(67\) −8.79718 −1.07475 −0.537373 − 0.843345i \(-0.680583\pi\)
−0.537373 + 0.843345i \(0.680583\pi\)
\(68\) 3.13522 0.380202
\(69\) −12.2412 −1.47367
\(70\) −1.69116 −0.202132
\(71\) 2.44975 0.290732 0.145366 − 0.989378i \(-0.453564\pi\)
0.145366 + 0.989378i \(0.453564\pi\)
\(72\) −21.5191 −2.53605
\(73\) 5.84063 0.683594 0.341797 − 0.939774i \(-0.388965\pi\)
0.341797 + 0.939774i \(0.388965\pi\)
\(74\) −8.17374 −0.950178
\(75\) 12.1070 1.39800
\(76\) 0 0
\(77\) 1.30958 0.149240
\(78\) 9.58348 1.08512
\(79\) −17.0397 −1.91711 −0.958557 − 0.284902i \(-0.908039\pi\)
−0.958557 + 0.284902i \(0.908039\pi\)
\(80\) −2.72192 −0.304319
\(81\) 19.0032 2.11147
\(82\) 7.29413 0.805502
\(83\) −7.01303 −0.769780 −0.384890 − 0.922963i \(-0.625761\pi\)
−0.384890 + 0.922963i \(0.625761\pi\)
\(84\) −2.38722 −0.260467
\(85\) −5.88679 −0.638512
\(86\) −3.52366 −0.379966
\(87\) −7.63422 −0.818475
\(88\) 3.07403 0.327693
\(89\) 9.13103 0.967887 0.483944 − 0.875099i \(-0.339204\pi\)
0.483944 + 0.875099i \(0.339204\pi\)
\(90\) 9.04003 0.952902
\(91\) 3.32629 0.348690
\(92\) −2.23138 −0.232637
\(93\) 9.60229 0.995712
\(94\) −14.4277 −1.48811
\(95\) 0 0
\(96\) −9.95354 −1.01588
\(97\) 14.7950 1.50220 0.751100 − 0.660188i \(-0.229524\pi\)
0.751100 + 0.660188i \(0.229524\pi\)
\(98\) −6.30570 −0.636972
\(99\) −7.00029 −0.703556
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.1.5 7
19.18 odd 2 209.2.a.d.1.3 ✓ 7
57.56 even 2 1881.2.a.p.1.5 7
76.75 even 2 3344.2.a.ba.1.1 7
95.94 odd 2 5225.2.a.n.1.5 7
209.208 even 2 2299.2.a.q.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.3 ✓ 7 19.18 odd 2
1881.2.a.p.1.5 7 57.56 even 2
2299.2.a.q.1.5 7 209.208 even 2
3344.2.a.ba.1.1 7 76.75 even 2
3971.2.a.i.1.5 7 1.1 even 1 trivial
5225.2.a.n.1.5 7 95.94 odd 2