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Properties

Label 8100.2.a.y.1.4
Level $8100$
Weight $2$
Character 8100.1
Self dual yes
Analytic conductor $64.679$
Analytic rank $0$
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: \(0\)
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 1.4
Root \(-0.318459\) 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)
 
\(f(q)\) \(=\) \(q+3.40179 q^{7} +4.40179 q^{11} -2.06320 q^{13} -1.40179 q^{17} -6.35717 q^{19} +0.107826 q^{23} -9.08178 q^{29} +3.06320 q^{31} -1.95538 q^{37} +8.69575 q^{41} +7.12641 q^{43} -7.48357 q^{47} +4.57217 q^{49} +13.0975 q^{53} +13.1793 q^{59} -1.72462 q^{61} +12.3757 q^{67} +7.50961 q^{71} +5.42037 q^{73} +14.9740 q^{77} +9.43613 q^{79} +8.97396 q^{83} -4.01576 q^{89} -7.01858 q^{91} -2.17103 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{7} + 3 q^{11} + 2 q^{13} + 9 q^{17} - 4 q^{19} - 3 q^{23} - 9 q^{29} + 2 q^{31} - q^{37} + 9 q^{41} + 8 q^{43} + 12 q^{47} + 9 q^{49} + 12 q^{53} - 15 q^{59} - q^{61} + 11 q^{67} + 12 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) \) 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
\(7\) 3.40179 1.28576 0.642878 − 0.765969i \(-0.277740\pi\)
0.642878 + 0.765969i \(0.277740\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.40179 1.32719 0.663595 − 0.748092i \(-0.269030\pi\)
0.663595 + 0.748092i \(0.269030\pi\)
\(12\) 0 0
\(13\) −2.06320 −0.572229 −0.286115 − 0.958195i \(-0.592364\pi\)
−0.286115 + 0.958195i \(0.592364\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.40179 −0.339984 −0.169992 − 0.985445i \(-0.554374\pi\)
−0.169992 + 0.985445i \(0.554374\pi\)
\(18\) 0 0
\(19\) −6.35717 −1.45843 −0.729217 − 0.684283i \(-0.760115\pi\)
−0.729217 + 0.684283i \(0.760115\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.107826 0.0224832 0.0112416 − 0.999937i \(-0.496422\pi\)
0.0112416 + 0.999937i \(0.496422\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.08178 −1.68644 −0.843222 − 0.537565i \(-0.819344\pi\)
−0.843222 + 0.537565i \(0.819344\pi\)
\(30\) 0 0
\(31\) 3.06320 0.550167 0.275084 − 0.961420i \(-0.411294\pi\)
0.275084 + 0.961420i \(0.411294\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.95538 −0.321462 −0.160731 − 0.986998i \(-0.551385\pi\)
−0.160731 + 0.986998i \(0.551385\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.69575 1.35805 0.679024 − 0.734116i \(-0.262403\pi\)
0.679024 + 0.734116i \(0.262403\pi\)
\(42\) 0 0
\(43\) 7.12641 1.08677 0.543383 − 0.839485i \(-0.317143\pi\)
0.543383 + 0.839485i \(0.317143\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.48357 −1.09159 −0.545795 − 0.837918i \(-0.683772\pi\)
−0.545795 + 0.837918i \(0.683772\pi\)
\(48\) 0 0
\(49\) 4.57217 0.653167
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.0975 1.79909 0.899543 − 0.436833i \(-0.143900\pi\)
0.899543 + 0.436833i \(0.143900\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.1793 1.71580 0.857901 − 0.513815i \(-0.171768\pi\)
0.857901 + 0.513815i \(0.171768\pi\)
\(60\) 0 0
\(61\) −1.72462 −0.220814 −0.110407 − 0.993886i \(-0.535216\pi\)
−0.110407 + 0.993886i \(0.535216\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.3757 1.51194 0.755969 − 0.654608i \(-0.227166\pi\)
0.755969 + 0.654608i \(0.227166\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.50961 0.891227 0.445614 − 0.895225i \(-0.352986\pi\)
0.445614 + 0.895225i \(0.352986\pi\)
\(72\) 0 0
\(73\) 5.42037 0.634406 0.317203 − 0.948358i \(-0.397256\pi\)
0.317203 + 0.948358i \(0.397256\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.9740 1.70644
\(78\) 0 0
\(79\) 9.43613 1.06165 0.530824 − 0.847482i \(-0.321883\pi\)
0.530824 + 0.847482i \(0.321883\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.97396 0.985020 0.492510 − 0.870307i \(-0.336080\pi\)
0.492510 + 0.870307i \(0.336080\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.01576 −0.425670 −0.212835 − 0.977088i \(-0.568270\pi\)
−0.212835 + 0.977088i \(0.568270\pi\)
\(90\) 0 0
\(91\) −7.01858 −0.735747
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.17103 −0.220435 −0.110217 − 0.993908i \(-0.535155\pi\)
−0.110217 + 0.993908i \(0.535155\pi\)
\(98\) 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.y.1.4 4
3.2 odd 2 8100.2.a.x.1.4 4
5.2 odd 4 8100.2.d.s.649.7 8
5.3 odd 4 8100.2.d.s.649.2 8
5.4 even 2 8100.2.a.ba.1.1 4
9.2 odd 6 900.2.i.d.301.1 ✓ 8
9.4 even 3 2700.2.i.e.1801.1 8
9.5 odd 6 900.2.i.d.601.1 yes 8
9.7 even 3 2700.2.i.e.901.1 8
15.2 even 4 8100.2.d.q.649.7 8
15.8 even 4 8100.2.d.q.649.2 8
15.14 odd 2 8100.2.a.z.1.1 4
45.2 even 12 900.2.s.d.49.5 16
45.4 even 6 2700.2.i.d.1801.4 8
45.7 odd 12 2700.2.s.d.1549.7 16
45.13 odd 12 2700.2.s.d.2449.7 16
45.14 odd 6 900.2.i.e.601.4 yes 8
45.22 odd 12 2700.2.s.d.2449.2 16
45.23 even 12 900.2.s.d.349.5 16
45.29 odd 6 900.2.i.e.301.4 yes 8
45.32 even 12 900.2.s.d.349.4 16
45.34 even 6 2700.2.i.d.901.4 8
45.38 even 12 900.2.s.d.49.4 16
45.43 odd 12 2700.2.s.d.1549.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.1 ✓ 8 9.2 odd 6
900.2.i.d.601.1 yes 8 9.5 odd 6
900.2.i.e.301.4 yes 8 45.29 odd 6
900.2.i.e.601.4 yes 8 45.14 odd 6
900.2.s.d.49.4 16 45.38 even 12
900.2.s.d.49.5 16 45.2 even 12
900.2.s.d.349.4 16 45.32 even 12
900.2.s.d.349.5 16 45.23 even 12
2700.2.i.d.901.4 8 45.34 even 6
2700.2.i.d.1801.4 8 45.4 even 6
2700.2.i.e.901.1 8 9.7 even 3
2700.2.i.e.1801.1 8 9.4 even 3
2700.2.s.d.1549.2 16 45.43 odd 12
2700.2.s.d.1549.7 16 45.7 odd 12
2700.2.s.d.2449.2 16 45.22 odd 12
2700.2.s.d.2449.7 16 45.13 odd 12
8100.2.a.x.1.4 4 3.2 odd 2
8100.2.a.y.1.4 4 1.1 even 1 trivial
8100.2.a.z.1.1 4 15.14 odd 2
8100.2.a.ba.1.1 4 5.4 even 2
8100.2.d.q.649.2 8 15.8 even 4
8100.2.d.q.649.7 8 15.2 even 4
8100.2.d.s.649.2 8 5.3 odd 4
8100.2.d.s.649.7 8 5.2 odd 4