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Properties

Label 8100.2.a.ba.1.2
Level $8100$
Weight $2$
Character 8100.1
Self dual yes
Analytic conductor $64.679$
Analytic rank $1$
Dimension $4$
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] = [8100,2,Mod(1,8100)] 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("8100.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,1,0,0,0,3,0,-2,0,0,0,-9,0,-4,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.6788256372\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3981.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 900)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.785261\) of defining polynomial
Character \(\chi\) \(=\) 8100.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.680426 q^{7} +1.68043 q^{11} -5.14503 q^{13} -1.31957 q^{17} -0.324642 q^{19} +3.78924 q^{23} +8.64584 q^{29} -4.14503 q^{31} -1.35578 q^{37} +7.15009 q^{41} +7.29005 q^{43} -12.9654 q^{47} -6.53702 q^{49} -8.83052 q^{53} -8.81532 q^{59} +9.97048 q^{61} +4.17617 q^{67} +0.891185 q^{71} +7.82038 q^{73} -1.14341 q^{77} +9.65597 q^{79} +4.85659 q^{83} -17.4764 q^{89} +3.50081 q^{91} -8.93427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{7} + 3 q^{11} - 2 q^{13} - 9 q^{17} - 4 q^{19} + 3 q^{23} - 9 q^{29} + 2 q^{31} + q^{37} + 9 q^{41} - 8 q^{43} - 12 q^{47} + 9 q^{49} - 12 q^{53} - 15 q^{59} - q^{61} - 11 q^{67} + 12 q^{71}+ \cdots - 5 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.680426 −0.257177 −0.128588 − 0.991698i \(-0.541045\pi\)
−0.128588 + 0.991698i \(0.541045\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.68043 0.506668 0.253334 − 0.967379i \(-0.418473\pi\)
0.253334 + 0.967379i \(0.418473\pi\)
\(12\) 0 0
\(13\) −5.14503 −1.42697 −0.713487 − 0.700669i \(-0.752885\pi\)
−0.713487 + 0.700669i \(0.752885\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.31957 −0.320044 −0.160022 − 0.987113i \(-0.551156\pi\)
−0.160022 + 0.987113i \(0.551156\pi\)
\(18\) 0 0
\(19\) −0.324642 −0.0744779 −0.0372390 − 0.999306i \(-0.511856\pi\)
−0.0372390 + 0.999306i \(0.511856\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.78924 0.790111 0.395056 − 0.918657i \(-0.370725\pi\)
0.395056 + 0.918657i \(0.370725\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.64584 1.60549 0.802746 − 0.596322i \(-0.203372\pi\)
0.802746 + 0.596322i \(0.203372\pi\)
\(30\) 0 0
\(31\) −4.14503 −0.744469 −0.372234 − 0.928139i \(-0.621408\pi\)
−0.372234 + 0.928139i \(0.621408\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.35578 −0.222890 −0.111445 − 0.993771i \(-0.535548\pi\)
−0.111445 + 0.993771i \(0.535548\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.15009 1.11666 0.558328 − 0.829620i \(-0.311443\pi\)
0.558328 + 0.829620i \(0.311443\pi\)
\(42\) 0 0
\(43\) 7.29005 1.11172 0.555861 − 0.831275i \(-0.312389\pi\)
0.555861 + 0.831275i \(0.312389\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.9654 −1.89120 −0.945600 − 0.325333i \(-0.894524\pi\)
−0.945600 + 0.325333i \(0.894524\pi\)
\(48\) 0 0
\(49\) −6.53702 −0.933860
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.83052 −1.21297 −0.606483 − 0.795097i \(-0.707420\pi\)
−0.606483 + 0.795097i \(0.707420\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.81532 −1.14766 −0.573828 − 0.818976i \(-0.694542\pi\)
−0.573828 + 0.818976i \(0.694542\pi\)
\(60\) 0 0
\(61\) 9.97048 1.27659 0.638294 − 0.769792i \(-0.279640\pi\)
0.638294 + 0.769792i \(0.279640\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.17617 0.510200 0.255100 − 0.966915i \(-0.417892\pi\)
0.255100 + 0.966915i \(0.417892\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.891185 0.105764 0.0528821 − 0.998601i \(-0.483159\pi\)
0.0528821 + 0.998601i \(0.483159\pi\)
\(72\) 0 0
\(73\) 7.82038 0.915307 0.457653 − 0.889131i \(-0.348690\pi\)
0.457653 + 0.889131i \(0.348690\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.14341 −0.130303
\(78\) 0 0
\(79\) 9.65597 1.08638 0.543191 − 0.839609i \(-0.317216\pi\)
0.543191 + 0.839609i \(0.317216\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.85659 0.533080 0.266540 − 0.963824i \(-0.414119\pi\)
0.266540 + 0.963824i \(0.414119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −17.4764 −1.85249 −0.926245 − 0.376922i \(-0.876982\pi\)
−0.926245 + 0.376922i \(0.876982\pi\)
\(90\) 0 0
\(91\) 3.50081 0.366985
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.93427 −0.907137 −0.453569 − 0.891221i \(-0.649849\pi\)
−0.453569 + 0.891221i \(0.649849\pi\)
\(98\) 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 8100.2.a.ba.1.2 4
3.2 odd 2 8100.2.a.z.1.2 4
5.2 odd 4 8100.2.d.s.649.3 8
5.3 odd 4 8100.2.d.s.649.6 8
5.4 even 2 8100.2.a.y.1.3 4
9.2 odd 6 900.2.i.e.301.2 yes 8
9.4 even 3 2700.2.i.d.1801.3 8
9.5 odd 6 900.2.i.e.601.2 yes 8
9.7 even 3 2700.2.i.d.901.3 8
15.2 even 4 8100.2.d.q.649.3 8
15.8 even 4 8100.2.d.q.649.6 8
15.14 odd 2 8100.2.a.x.1.3 4
45.2 even 12 900.2.s.d.49.1 16
45.4 even 6 2700.2.i.e.1801.2 8
45.7 odd 12 2700.2.s.d.1549.3 16
45.13 odd 12 2700.2.s.d.2449.3 16
45.14 odd 6 900.2.i.d.601.3 yes 8
45.22 odd 12 2700.2.s.d.2449.6 16
45.23 even 12 900.2.s.d.349.1 16
45.29 odd 6 900.2.i.d.301.3 ✓ 8
45.32 even 12 900.2.s.d.349.8 16
45.34 even 6 2700.2.i.e.901.2 8
45.38 even 12 900.2.s.d.49.8 16
45.43 odd 12 2700.2.s.d.1549.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.3 ✓ 8 45.29 odd 6
900.2.i.d.601.3 yes 8 45.14 odd 6
900.2.i.e.301.2 yes 8 9.2 odd 6
900.2.i.e.601.2 yes 8 9.5 odd 6
900.2.s.d.49.1 16 45.2 even 12
900.2.s.d.49.8 16 45.38 even 12
900.2.s.d.349.1 16 45.23 even 12
900.2.s.d.349.8 16 45.32 even 12
2700.2.i.d.901.3 8 9.7 even 3
2700.2.i.d.1801.3 8 9.4 even 3
2700.2.i.e.901.2 8 45.34 even 6
2700.2.i.e.1801.2 8 45.4 even 6
2700.2.s.d.1549.3 16 45.7 odd 12
2700.2.s.d.1549.6 16 45.43 odd 12
2700.2.s.d.2449.3 16 45.13 odd 12
2700.2.s.d.2449.6 16 45.22 odd 12
8100.2.a.x.1.3 4 15.14 odd 2
8100.2.a.y.1.3 4 5.4 even 2
8100.2.a.z.1.2 4 3.2 odd 2
8100.2.a.ba.1.2 4 1.1 even 1 trivial
8100.2.d.q.649.3 8 15.2 even 4
8100.2.d.q.649.6 8 15.8 even 4
8100.2.d.s.649.3 8 5.2 odd 4
8100.2.d.s.649.6 8 5.3 odd 4