Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8349,2,Mod(1,8349)] 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("8349.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8349, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8349 = 3 \cdot 11^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8349.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,6,-2,0,-2] 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: \(66.6671006476\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 8349.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -3.23607 q^{5} +2.23607 q^{6} +1.23607 q^{7} -2.23607 q^{8} +1.00000 q^{9} +7.23607 q^{10} -3.00000 q^{12} -4.47214 q^{13} -2.76393 q^{14} +3.23607 q^{15} -1.00000 q^{16} +7.23607 q^{17} -2.23607 q^{18} -2.76393 q^{19} -9.70820 q^{20} -1.23607 q^{21} +1.00000 q^{23} +2.23607 q^{24} +5.47214 q^{25} +10.0000 q^{26} -1.00000 q^{27} +3.70820 q^{28} +4.47214 q^{29} -7.23607 q^{30} +2.47214 q^{31} +6.70820 q^{32} -16.1803 q^{34} -4.00000 q^{35} +3.00000 q^{36} -4.47214 q^{37} +6.18034 q^{38} +4.47214 q^{39} +7.23607 q^{40} -6.94427 q^{41} +2.76393 q^{42} -7.70820 q^{43} -3.23607 q^{45} -2.23607 q^{46} -4.00000 q^{47} +1.00000 q^{48} -5.47214 q^{49} -12.2361 q^{50} -7.23607 q^{51} -13.4164 q^{52} -0.763932 q^{53} +2.23607 q^{54} -2.76393 q^{56} +2.76393 q^{57} -10.0000 q^{58} +12.9443 q^{59} +9.70820 q^{60} +4.47214 q^{61} -5.52786 q^{62} +1.23607 q^{63} -13.0000 q^{64} +14.4721 q^{65} +5.23607 q^{67} +21.7082 q^{68} -1.00000 q^{69} +8.94427 q^{70} -8.00000 q^{71} -2.23607 q^{72} +10.9443 q^{73} +10.0000 q^{74} -5.47214 q^{75} -8.29180 q^{76} -10.0000 q^{78} +3.70820 q^{79} +3.23607 q^{80} +1.00000 q^{81} +15.5279 q^{82} -4.00000 q^{83} -3.70820 q^{84} -23.4164 q^{85} +17.2361 q^{86} -4.47214 q^{87} +3.23607 q^{89} +7.23607 q^{90} -5.52786 q^{91} +3.00000 q^{92} -2.47214 q^{93} +8.94427 q^{94} +8.94427 q^{95} -6.70820 q^{96} -0.472136 q^{97} +12.2361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 10 q^{10} - 6 q^{12} - 10 q^{14} + 2 q^{15} - 2 q^{16} + 10 q^{17} - 10 q^{19} - 6 q^{20} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 20 q^{26} - 2 q^{27}+ \cdots + 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.23607 −1.58114 −0.790569 − 0.612372i \(-0.790215\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.00000 1.50000
\(5\) −3.23607 −1.44721 −0.723607 − 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 2.23607 0.912871
\(7\) 1.23607 0.467190 0.233595 − 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 7.23607 2.28825
\(11\) 0 0
\(12\) −3.00000 −0.866025
\(13\) −4.47214 −1.24035 −0.620174 − 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) −2.76393 −0.738692
\(15\) 3.23607 0.835549
\(16\) −1.00000 −0.250000
\(17\) 7.23607 1.75500 0.877502 − 0.479573i \(-0.159208\pi\)
0.877502 + 0.479573i \(0.159208\pi\)
\(18\) −2.23607 −0.527046
\(19\) −2.76393 −0.634089 −0.317045 − 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) −9.70820 −2.17082
\(21\) −1.23607 −0.269732
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 2.23607 0.456435
\(25\) 5.47214 1.09443
\(26\) 10.0000 1.96116
\(27\) −1.00000 −0.192450
\(28\) 3.70820 0.700785
\(29\) 4.47214 0.830455 0.415227 − 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) −7.23607 −1.32112
\(31\) 2.47214 0.444009 0.222004 − 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) 6.70820 1.18585
\(33\) 0 0
\(34\) −16.1803 −2.77491
\(35\) −4.00000 −0.676123
\(36\) 3.00000 0.500000
\(37\) −4.47214 −0.735215 −0.367607 − 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 6.18034 1.00258
\(39\) 4.47214 0.716115
\(40\) 7.23607 1.14412
\(41\) −6.94427 −1.08451 −0.542257 − 0.840213i \(-0.682430\pi\)
−0.542257 + 0.840213i \(0.682430\pi\)
\(42\) 2.76393 0.426484
\(43\) −7.70820 −1.17549 −0.587745 − 0.809046i \(-0.699984\pi\)
−0.587745 + 0.809046i \(0.699984\pi\)
\(44\) 0 0
\(45\) −3.23607 −0.482405
\(46\) −2.23607 −0.329690
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 1.00000 0.144338
\(49\) −5.47214 −0.781734
\(50\) −12.2361 −1.73044
\(51\) −7.23607 −1.01325
\(52\) −13.4164 −1.86052
\(53\) −0.763932 −0.104934 −0.0524671 − 0.998623i \(-0.516708\pi\)
−0.0524671 + 0.998623i \(0.516708\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) −2.76393 −0.369346
\(57\) 2.76393 0.366092
\(58\) −10.0000 −1.31306
\(59\) 12.9443 1.68520 0.842600 − 0.538539i \(-0.181024\pi\)
0.842600 + 0.538539i \(0.181024\pi\)
\(60\) 9.70820 1.25332
\(61\) 4.47214 0.572598 0.286299 − 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) −5.52786 −0.702039
\(63\) 1.23607 0.155730
\(64\) −13.0000 −1.62500
\(65\) 14.4721 1.79505
\(66\) 0 0
\(67\) 5.23607 0.639688 0.319844 − 0.947470i \(-0.396370\pi\)
0.319844 + 0.947470i \(0.396370\pi\)
\(68\) 21.7082 2.63251
\(69\) −1.00000 −0.120386
\(70\) 8.94427 1.06904
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −2.23607 −0.263523
\(73\) 10.9443 1.28093 0.640465 − 0.767987i \(-0.278742\pi\)
0.640465 + 0.767987i \(0.278742\pi\)
\(74\) 10.0000 1.16248
\(75\) −5.47214 −0.631868
\(76\) −8.29180 −0.951134
\(77\) 0 0
\(78\) −10.0000 −1.13228
\(79\) 3.70820 0.417206 0.208603 − 0.978000i \(-0.433108\pi\)
0.208603 + 0.978000i \(0.433108\pi\)
\(80\) 3.23607 0.361803
\(81\) 1.00000 0.111111
\(82\) 15.5279 1.71477
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −3.70820 −0.404598
\(85\) −23.4164 −2.53987
\(86\) 17.2361 1.85861
\(87\) −4.47214 −0.479463
\(88\) 0 0
\(89\) 3.23607 0.343023 0.171511 − 0.985182i \(-0.445135\pi\)
0.171511 + 0.985182i \(0.445135\pi\)
\(90\) 7.23607 0.762749
\(91\) −5.52786 −0.579478
\(92\) 3.00000 0.312772
\(93\) −2.47214 −0.256349
\(94\) 8.94427 0.922531
\(95\) 8.94427 0.917663
\(96\) −6.70820 −0.684653
\(97\) −0.472136 −0.0479381 −0.0239691 − 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) 12.2361 1.23603
\(99\) 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 8349.2.a.i.1.1 2
11.10 odd 2 69.2.a.b.1.2 ✓ 2
33.32 even 2 207.2.a.c.1.1 2
44.43 even 2 1104.2.a.m.1.1 2
55.32 even 4 1725.2.b.o.1174.4 4
55.43 even 4 1725.2.b.o.1174.1 4
55.54 odd 2 1725.2.a.ba.1.1 2
77.76 even 2 3381.2.a.t.1.2 2
88.21 odd 2 4416.2.a.bm.1.2 2
88.43 even 2 4416.2.a.bg.1.2 2
132.131 odd 2 3312.2.a.bb.1.2 2
165.164 even 2 5175.2.a.bk.1.2 2
253.252 even 2 1587.2.a.i.1.2 2
759.758 odd 2 4761.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.a.b.1.2 ✓ 2 11.10 odd 2
207.2.a.c.1.1 2 33.32 even 2
1104.2.a.m.1.1 2 44.43 even 2
1587.2.a.i.1.2 2 253.252 even 2
1725.2.a.ba.1.1 2 55.54 odd 2
1725.2.b.o.1174.1 4 55.43 even 4
1725.2.b.o.1174.4 4 55.32 even 4
3312.2.a.bb.1.2 2 132.131 odd 2
3381.2.a.t.1.2 2 77.76 even 2
4416.2.a.bg.1.2 2 88.43 even 2
4416.2.a.bm.1.2 2 88.21 odd 2
4761.2.a.v.1.1 2 759.758 odd 2
5175.2.a.bk.1.2 2 165.164 even 2
8349.2.a.i.1.1 2 1.1 even 1 trivial