Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2299,2,Mod(1,2299)] 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("2299.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2299.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,2,15,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.3576074247\)
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 \(\chi\) \(=\) 2299.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.19313 q^{2} +3.16232 q^{3} -0.576442 q^{4} +1.08235 q^{5} +3.77306 q^{6} +1.30958 q^{7} -3.07403 q^{8} +7.00029 q^{9} +1.29138 q^{10} -1.82290 q^{12} -2.53997 q^{13} +1.56249 q^{14} +3.42273 q^{15} -2.51483 q^{16} +5.43892 q^{17} +8.35226 q^{18} -1.00000 q^{19} -0.623909 q^{20} +4.14130 q^{21} +3.87095 q^{23} -9.72108 q^{24} -3.82853 q^{25} -3.03052 q^{26} +12.6502 q^{27} -0.754894 q^{28} +2.41412 q^{29} +4.08376 q^{30} +3.03647 q^{31} +3.14754 q^{32} +6.48934 q^{34} +1.41741 q^{35} -4.03526 q^{36} +6.85067 q^{37} -1.19313 q^{38} -8.03222 q^{39} -3.32716 q^{40} +6.11344 q^{41} +4.94111 q^{42} +2.95329 q^{43} +7.57673 q^{45} +4.61854 q^{46} -12.0923 q^{47} -7.95271 q^{48} -5.28501 q^{49} -4.56793 q^{50} +17.1996 q^{51} +1.46415 q^{52} -0.992927 q^{53} +15.0933 q^{54} -4.02567 q^{56} -3.16232 q^{57} +2.88036 q^{58} -14.2251 q^{59} -1.97300 q^{60} +5.82518 q^{61} +3.62290 q^{62} +9.16741 q^{63} +8.78508 q^{64} -2.74913 q^{65} +8.79718 q^{67} -3.13522 q^{68} +12.2412 q^{69} +1.69116 q^{70} -2.44975 q^{71} -21.5191 q^{72} -5.84063 q^{73} +8.17374 q^{74} -12.1070 q^{75} +0.576442 q^{76} -9.58348 q^{78} -17.0397 q^{79} -2.72192 q^{80} +19.0032 q^{81} +7.29413 q^{82} +7.01303 q^{83} -2.38722 q^{84} +5.88679 q^{85} +3.52366 q^{86} +7.63422 q^{87} -9.13103 q^{89} +9.04003 q^{90} -3.32629 q^{91} -2.23138 q^{92} +9.60229 q^{93} -14.4277 q^{94} -1.08235 q^{95} +9.95354 q^{96} -14.7950 q^{97} -6.30570 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 2 q^{3} + 15 q^{4} + 2 q^{5} + 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9} + 6 q^{10} - 16 q^{12} + 4 q^{13} + 6 q^{14} + 12 q^{15} + 27 q^{16} - 2 q^{17} - 9 q^{18} - 7 q^{19} - 4 q^{20} + 14 q^{21}+ \cdots - 19 q^{98}+O(q^{100}) \) 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 \(-0.133850\pi\)
0.912884 + 0.408218i \(0.133850\pi\)
\(4\) −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 \(-0.420395\pi\)
0.247487 + 0.968891i \(0.420395\pi\)
\(8\) −3.07403 −1.08683
\(9\) 7.00029 2.33343
\(10\) 1.29138 0.408370
\(11\) 0 0
\(12\) −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 \(-0.270740\pi\)
0.659566 + 0.751646i \(0.270740\pi\)
\(18\) 8.35226 1.96865
\(19\) −1.00000 −0.229416
\(20\) −0.623909 −0.139510
\(21\) 4.14130 0.903706
\(22\) 0 0
\(23\) 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 \(-0.412089\pi\)
0.272683 + 0.962104i \(0.412089\pi\)
\(32\) 3.14754 0.556412
\(33\) 0 0
\(34\) 6.48934 1.11291
\(35\) 1.41741 0.239587
\(36\) −4.03526 −0.672544
\(37\) 6.85067 1.12624 0.563122 − 0.826374i \(-0.309600\pi\)
0.563122 + 0.826374i \(0.309600\pi\)
\(38\) −1.19313 −0.193551
\(39\) −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 \(-0.427701\pi\)
0.225186 + 0.974316i \(0.427701\pi\)
\(44\) 0 0
\(45\) 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 \(-0.521724\pi\)
−0.0681945 + 0.997672i \(0.521724\pi\)
\(54\) 15.0933 2.05394
\(55\) 0 0
\(56\) −4.02567 −0.537953
\(57\) −3.16232 −0.418860
\(58\) 2.88036 0.378209
\(59\) −14.2251 −1.85195 −0.925977 − 0.377580i \(-0.876756\pi\)
−0.925977 + 0.377580i \(0.876756\pi\)
\(60\) −1.97300 −0.254714
\(61\) 5.82518 0.745838 0.372919 − 0.927864i \(-0.378357\pi\)
0.372919 + 0.927864i \(0.378357\pi\)
\(62\) 3.62290 0.460109
\(63\) 9.16741 1.15499
\(64\) 8.78508 1.09814
\(65\) −2.74913 −0.340988
\(66\) 0 0
\(67\) 8.79718 1.07475 0.537373 − 0.843345i \(-0.319417\pi\)
0.537373 + 0.843345i \(0.319417\pi\)
\(68\) −3.13522 −0.380202
\(69\) 12.2412 1.47367
\(70\) 1.69116 0.202132
\(71\) −2.44975 −0.290732 −0.145366 − 0.989378i \(-0.546436\pi\)
−0.145366 + 0.989378i \(0.546436\pi\)
\(72\) −21.5191 −2.53605
\(73\) −5.84063 −0.683594 −0.341797 − 0.939774i \(-0.611035\pi\)
−0.341797 + 0.939774i \(0.611035\pi\)
\(74\) 8.17374 0.950178
\(75\) −12.1070 −1.39800
\(76\) 0.576442 0.0661224
\(77\) 0 0
\(78\) −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 \(-0.374239\pi\)
0.384890 + 0.922963i \(0.374239\pi\)
\(84\) −2.38722 −0.260467
\(85\) 5.88679 0.638512
\(86\) 3.52366 0.379966
\(87\) 7.63422 0.818475
\(88\) 0 0
\(89\) −9.13103 −0.967887 −0.483944 − 0.875099i \(-0.660796\pi\)
−0.483944 + 0.875099i \(0.660796\pi\)
\(90\) 9.04003 0.952902
\(91\) −3.32629 −0.348690
\(92\) −2.23138 −0.232637
\(93\) 9.60229 0.995712
\(94\) −14.4277 −1.48811
\(95\) −1.08235 −0.111046
\(96\) 9.95354 1.01588
\(97\) −14.7950 −1.50220 −0.751100 − 0.660188i \(-0.770476\pi\)
−0.751100 + 0.660188i \(0.770476\pi\)
\(98\) −6.30570 −0.636972
\(99\) 0 0
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 2299.2.a.q.1.5 7
11.10 odd 2 209.2.a.d.1.3 ✓ 7
33.32 even 2 1881.2.a.p.1.5 7
44.43 even 2 3344.2.a.ba.1.1 7
55.54 odd 2 5225.2.a.n.1.5 7
209.208 even 2 3971.2.a.i.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.3 ✓ 7 11.10 odd 2
1881.2.a.p.1.5 7 33.32 even 2
2299.2.a.q.1.5 7 1.1 even 1 trivial
3344.2.a.ba.1.1 7 44.43 even 2
3971.2.a.i.1.5 7 209.208 even 2
5225.2.a.n.1.5 7 55.54 odd 2