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Properties

Label 6897.2.a.g.1.1
Level $6897$
Weight $2$
Character 6897.1
Self dual yes
Analytic conductor $55.073$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6897,2,Mod(1,6897)] 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("6897.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6897, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6897.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,1,2,1,2,-3,0,1,2,0,2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.0728222741\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6897.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -3.00000 q^{7} +1.00000 q^{9} +2.00000 q^{10} +2.00000 q^{12} +6.00000 q^{13} -6.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} -3.00000 q^{21} +4.00000 q^{23} -4.00000 q^{25} +12.0000 q^{26} +1.00000 q^{27} -6.00000 q^{28} +10.0000 q^{29} +2.00000 q^{30} +2.00000 q^{31} -8.00000 q^{32} -6.00000 q^{34} -3.00000 q^{35} +2.00000 q^{36} +8.00000 q^{37} +2.00000 q^{38} +6.00000 q^{39} +8.00000 q^{41} -6.00000 q^{42} +1.00000 q^{43} +1.00000 q^{45} +8.00000 q^{46} +3.00000 q^{47} -4.00000 q^{48} +2.00000 q^{49} -8.00000 q^{50} -3.00000 q^{51} +12.0000 q^{52} -6.00000 q^{53} +2.00000 q^{54} +1.00000 q^{57} +20.0000 q^{58} +2.00000 q^{60} -7.00000 q^{61} +4.00000 q^{62} -3.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} -6.00000 q^{70} +12.0000 q^{71} +11.0000 q^{73} +16.0000 q^{74} -4.00000 q^{75} +2.00000 q^{76} +12.0000 q^{78} -4.00000 q^{80} +1.00000 q^{81} +16.0000 q^{82} -4.00000 q^{83} -6.00000 q^{84} -3.00000 q^{85} +2.00000 q^{86} +10.0000 q^{87} +10.0000 q^{89} +2.00000 q^{90} -18.0000 q^{91} +8.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} +1.00000 q^{95} -8.00000 q^{96} -2.00000 q^{97} +4.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 2.00000 0.816497
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0
\(12\) 2.00000 0.577350
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −6.00000 −1.60357
\(15\) 1.00000 0.258199
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416
\(20\) 2.00000 0.447214
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 12.0000 2.35339
\(27\) 1.00000 0.192450
\(28\) −6.00000 −1.13389
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 2.00000 0.365148
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −3.00000 −0.507093
\(36\) 2.00000 0.333333
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 2.00000 0.324443
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) −6.00000 −0.925820
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −4.00000 −0.577350
\(49\) 2.00000 0.285714
\(50\) −8.00000 −1.13137
\(51\) −3.00000 −0.420084
\(52\) 12.0000 1.66410
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 20.0000 2.62613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 4.00000 0.508001
\(63\) −3.00000 −0.377964
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 4.00000 0.481543
\(70\) −6.00000 −0.717137
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 − 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 16.0000 1.85996
\(75\) −4.00000 −0.461880
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 12.0000 1.35873
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 16.0000 1.76690
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −6.00000 −0.654654
\(85\) −3.00000 −0.325396
\(86\) 2.00000 0.215666
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 2.00000 0.210819
\(91\) −18.0000 −1.88691
\(92\) 8.00000 0.834058
\(93\) 2.00000 0.207390
\(94\) 6.00000 0.618853
\(95\) 1.00000 0.102598
\(96\) −8.00000 −0.816497
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 4.00000 0.404061
\(99\) 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 6897.2.a.g.1.1 1
11.10 odd 2 57.2.a.b.1.1 ✓ 1
33.32 even 2 171.2.a.c.1.1 1
44.43 even 2 912.2.a.d.1.1 1
55.32 even 4 1425.2.c.a.799.1 2
55.43 even 4 1425.2.c.a.799.2 2
55.54 odd 2 1425.2.a.i.1.1 1
77.76 even 2 2793.2.a.a.1.1 1
88.21 odd 2 3648.2.a.h.1.1 1
88.43 even 2 3648.2.a.y.1.1 1
132.131 odd 2 2736.2.a.h.1.1 1
143.142 odd 2 9633.2.a.p.1.1 1
165.164 even 2 4275.2.a.a.1.1 1
209.208 even 2 1083.2.a.d.1.1 1
231.230 odd 2 8379.2.a.q.1.1 1
627.626 odd 2 3249.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 ✓ 1 11.10 odd 2
171.2.a.c.1.1 1 33.32 even 2
912.2.a.d.1.1 1 44.43 even 2
1083.2.a.d.1.1 1 209.208 even 2
1425.2.a.i.1.1 1 55.54 odd 2
1425.2.c.a.799.1 2 55.32 even 4
1425.2.c.a.799.2 2 55.43 even 4
2736.2.a.h.1.1 1 132.131 odd 2
2793.2.a.a.1.1 1 77.76 even 2
3249.2.a.a.1.1 1 627.626 odd 2
3648.2.a.h.1.1 1 88.21 odd 2
3648.2.a.y.1.1 1 88.43 even 2
4275.2.a.a.1.1 1 165.164 even 2
6897.2.a.g.1.1 1 1.1 even 1 trivial
8379.2.a.q.1.1 1 231.230 odd 2
9633.2.a.p.1.1 1 143.142 odd 2