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Properties

Label 8100.2.a.y.1.2
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 1.2
Root \(2.28400\) 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)
 
\(f(q)\) \(=\) \(q-0.0864793 q^{7} +0.913521 q^{11} -2.62499 q^{13} +2.08648 q^{17} +4.93847 q^{19} +8.47698 q^{23} -2.39798 q^{29} +3.62499 q^{31} +5.85199 q^{37} -6.64994 q^{41} +8.24997 q^{43} +2.68850 q^{47} -6.99252 q^{49} -5.73642 q^{53} -12.3384 q^{59} -6.33645 q^{61} -6.16548 q^{67} +12.3905 q^{71} -5.31349 q^{73} -0.0790006 q^{77} -13.4479 q^{79} -6.07900 q^{83} +8.13440 q^{89} +0.227007 q^{91} -11.1020 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{7} + 3 q^{11} + 2 q^{13} + 9 q^{17} - 4 q^{19} - 3 q^{23} - 9 q^{29} + 2 q^{31} - q^{37} + 9 q^{41} + 8 q^{43} + 12 q^{47} + 9 q^{49} + 12 q^{53} - 15 q^{59} - q^{61} + 11 q^{67} + 12 q^{71}+ \cdots + 5 q^{97}+O(q^{100}) \) 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
\(7\) −0.0864793 −0.0326861 −0.0163431 − 0.999866i \(-0.505202\pi\)
−0.0163431 + 0.999866i \(0.505202\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.913521 0.275437 0.137718 − 0.990471i \(-0.456023\pi\)
0.137718 + 0.990471i \(0.456023\pi\)
\(12\) 0 0
\(13\) −2.62499 −0.728040 −0.364020 − 0.931391i \(-0.618596\pi\)
−0.364020 + 0.931391i \(0.618596\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.08648 0.506046 0.253023 − 0.967460i \(-0.418575\pi\)
0.253023 + 0.967460i \(0.418575\pi\)
\(18\) 0 0
\(19\) 4.93847 1.13296 0.566482 − 0.824074i \(-0.308304\pi\)
0.566482 + 0.824074i \(0.308304\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.47698 1.76757 0.883786 − 0.467891i \(-0.154986\pi\)
0.883786 + 0.467891i \(0.154986\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.39798 −0.445294 −0.222647 − 0.974899i \(-0.571470\pi\)
−0.222647 + 0.974899i \(0.571470\pi\)
\(30\) 0 0
\(31\) 3.62499 0.651067 0.325533 − 0.945531i \(-0.394456\pi\)
0.325533 + 0.945531i \(0.394456\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.85199 0.962062 0.481031 − 0.876704i \(-0.340262\pi\)
0.481031 + 0.876704i \(0.340262\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.64994 −1.03855 −0.519273 − 0.854608i \(-0.673797\pi\)
−0.519273 + 0.854608i \(0.673797\pi\)
\(42\) 0 0
\(43\) 8.24997 1.25811 0.629055 − 0.777361i \(-0.283442\pi\)
0.629055 + 0.777361i \(0.283442\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.68850 0.392158 0.196079 − 0.980588i \(-0.437179\pi\)
0.196079 + 0.980588i \(0.437179\pi\)
\(48\) 0 0
\(49\) −6.99252 −0.998932
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.73642 −0.787958 −0.393979 − 0.919120i \(-0.628902\pi\)
−0.393979 + 0.919120i \(0.628902\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.3384 −1.60633 −0.803164 − 0.595758i \(-0.796852\pi\)
−0.803164 + 0.595758i \(0.796852\pi\)
\(60\) 0 0
\(61\) −6.33645 −0.811300 −0.405650 − 0.914029i \(-0.632955\pi\)
−0.405650 + 0.914029i \(0.632955\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.16548 −0.753233 −0.376617 − 0.926369i \(-0.622913\pi\)
−0.376617 + 0.926369i \(0.622913\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.3905 1.47048 0.735241 − 0.677806i \(-0.237069\pi\)
0.735241 + 0.677806i \(0.237069\pi\)
\(72\) 0 0
\(73\) −5.31349 −0.621897 −0.310948 − 0.950427i \(-0.600647\pi\)
−0.310948 + 0.950427i \(0.600647\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.0790006 −0.00900296
\(78\) 0 0
\(79\) −13.4479 −1.51301 −0.756503 − 0.653991i \(-0.773094\pi\)
−0.756503 + 0.653991i \(0.773094\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.07900 −0.667257 −0.333629 − 0.942705i \(-0.608273\pi\)
−0.333629 + 0.942705i \(0.608273\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.13440 0.862244 0.431122 − 0.902294i \(-0.358118\pi\)
0.431122 + 0.902294i \(0.358118\pi\)
\(90\) 0 0
\(91\) 0.227007 0.0237968
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.1020 −1.12723 −0.563617 − 0.826036i \(-0.690591\pi\)
−0.563617 + 0.826036i \(0.690591\pi\)
\(98\) 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.1.2 4
3.2 odd 2 8100.2.a.x.1.2 4
5.2 odd 4 8100.2.d.s.649.4 8
5.3 odd 4 8100.2.d.s.649.5 8
5.4 even 2 8100.2.a.ba.1.3 4
9.2 odd 6 900.2.i.d.301.4 ✓ 8
9.4 even 3 2700.2.i.e.1801.3 8
9.5 odd 6 900.2.i.d.601.4 yes 8
9.7 even 3 2700.2.i.e.901.3 8
15.2 even 4 8100.2.d.q.649.4 8
15.8 even 4 8100.2.d.q.649.5 8
15.14 odd 2 8100.2.a.z.1.3 4
45.2 even 12 900.2.s.d.49.3 16
45.4 even 6 2700.2.i.d.1801.2 8
45.7 odd 12 2700.2.s.d.1549.4 16
45.13 odd 12 2700.2.s.d.2449.4 16
45.14 odd 6 900.2.i.e.601.1 yes 8
45.22 odd 12 2700.2.s.d.2449.5 16
45.23 even 12 900.2.s.d.349.3 16
45.29 odd 6 900.2.i.e.301.1 yes 8
45.32 even 12 900.2.s.d.349.6 16
45.34 even 6 2700.2.i.d.901.2 8
45.38 even 12 900.2.s.d.49.6 16
45.43 odd 12 2700.2.s.d.1549.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.4 ✓ 8 9.2 odd 6
900.2.i.d.601.4 yes 8 9.5 odd 6
900.2.i.e.301.1 yes 8 45.29 odd 6
900.2.i.e.601.1 yes 8 45.14 odd 6
900.2.s.d.49.3 16 45.2 even 12
900.2.s.d.49.6 16 45.38 even 12
900.2.s.d.349.3 16 45.23 even 12
900.2.s.d.349.6 16 45.32 even 12
2700.2.i.d.901.2 8 45.34 even 6
2700.2.i.d.1801.2 8 45.4 even 6
2700.2.i.e.901.3 8 9.7 even 3
2700.2.i.e.1801.3 8 9.4 even 3
2700.2.s.d.1549.4 16 45.7 odd 12
2700.2.s.d.1549.5 16 45.43 odd 12
2700.2.s.d.2449.4 16 45.13 odd 12
2700.2.s.d.2449.5 16 45.22 odd 12
8100.2.a.x.1.2 4 3.2 odd 2
8100.2.a.y.1.2 4 1.1 even 1 trivial
8100.2.a.z.1.3 4 15.14 odd 2
8100.2.a.ba.1.3 4 5.4 even 2
8100.2.d.q.649.4 8 15.2 even 4
8100.2.d.q.649.5 8 15.8 even 4
8100.2.d.s.649.4 8 5.2 odd 4
8100.2.d.s.649.5 8 5.3 odd 4