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Properties

Label 1472.2.a.a.1.1
Level $1472$
Weight $2$
Character 1472.1
Self dual yes
Analytic conductor $11.754$
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] = [1472,2,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,-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: \(11.7539791775\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1472.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -2.00000 q^{7} +6.00000 q^{9} +5.00000 q^{13} -6.00000 q^{17} -6.00000 q^{19} +6.00000 q^{21} +1.00000 q^{23} -5.00000 q^{25} -9.00000 q^{27} -9.00000 q^{29} +3.00000 q^{31} +8.00000 q^{37} -15.0000 q^{39} +3.00000 q^{41} +8.00000 q^{43} +7.00000 q^{47} -3.00000 q^{49} +18.0000 q^{51} +2.00000 q^{53} +18.0000 q^{57} -4.00000 q^{59} +10.0000 q^{61} -12.0000 q^{63} -8.00000 q^{67} -3.00000 q^{69} +7.00000 q^{71} +9.00000 q^{73} +15.0000 q^{75} -6.00000 q^{79} +9.00000 q^{81} +14.0000 q^{83} +27.0000 q^{87} +16.0000 q^{89} -10.0000 q^{91} -9.00000 q^{93} +6.00000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −15.0000 −2.40192
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00000 1.02105 0.510527 − 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 18.0000 2.52050
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) −12.0000 −1.51186
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 − 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 − 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 15.0000 1.73205
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 27.0000 2.89470
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 − 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) −10.0000 −1.04828
\(92\) 0 0
\(93\) −9.00000 −0.933257
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 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 1472.2.a.a.1.1 1
4.3 odd 2 1472.2.a.m.1.1 1
8.3 odd 2 368.2.a.a.1.1 1
8.5 even 2 184.2.a.d.1.1 ✓ 1
24.5 odd 2 1656.2.a.c.1.1 1
24.11 even 2 3312.2.a.i.1.1 1
40.13 odd 4 4600.2.e.a.4049.2 2
40.19 odd 2 9200.2.a.bj.1.1 1
40.29 even 2 4600.2.a.a.1.1 1
40.37 odd 4 4600.2.e.a.4049.1 2
56.13 odd 2 9016.2.a.b.1.1 1
184.45 odd 2 4232.2.a.j.1.1 1
184.91 even 2 8464.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.d.1.1 ✓ 1 8.5 even 2
368.2.a.a.1.1 1 8.3 odd 2
1472.2.a.a.1.1 1 1.1 even 1 trivial
1472.2.a.m.1.1 1 4.3 odd 2
1656.2.a.c.1.1 1 24.5 odd 2
3312.2.a.i.1.1 1 24.11 even 2
4232.2.a.j.1.1 1 184.45 odd 2
4600.2.a.a.1.1 1 40.29 even 2
4600.2.e.a.4049.1 2 40.37 odd 4
4600.2.e.a.4049.2 2 40.13 odd 4
8464.2.a.b.1.1 1 184.91 even 2
9016.2.a.b.1.1 1 56.13 odd 2
9200.2.a.bj.1.1 1 40.19 odd 2