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Properties

Label 8100.2.d.s.649.4
Level $8100$
Weight $2$
Character 8100.649
Analytic conductor $64.679$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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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(649,8100)] 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("8100.649"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4057180416.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 22x^{4} + 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 900)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.4
Root \(-2.28400i\) of defining polynomial
Character \(\chi\) \(=\) 8100.649
Dual form 8100.2.d.s.649.5

$q$-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)
 
\(f(q)\) \(=\) \(q-0.0864793i q^{7} +0.913521 q^{11} +2.62499i q^{13} +2.08648i q^{17} -4.93847 q^{19} -8.47698i q^{23} +2.39798 q^{29} +3.62499 q^{31} +5.85199i q^{37} -6.64994 q^{41} -8.24997i q^{43} +2.68850i q^{47} +6.99252 q^{49} +5.73642i q^{53} +12.3384 q^{59} -6.33645 q^{61} -6.16548i q^{67} +12.3905 q^{71} +5.31349i q^{73} -0.0790006i q^{77} +13.4479 q^{79} +6.07900i q^{83} -8.13440 q^{89} +0.227007 q^{91} -11.1020i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{11} + 8 q^{19} + 18 q^{29} + 4 q^{31} + 18 q^{41} - 18 q^{49} + 30 q^{59} - 2 q^{61} + 24 q^{71} + 14 q^{79} + 6 q^{89} - 22 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(7\) − 0.0864793i − 0.0326861i −0.999866 − 0.0163431i \(-0.994798\pi\)
0.999866 − 0.0163431i \(-0.00520239\pi\)
\(8\) 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
\(13\) 2.62499i 0.728040i 0.931391 + 0.364020i \(0.118596\pi\)
−0.931391 + 0.364020i \(0.881404\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.08648i 0.506046i 0.967460 + 0.253023i \(0.0814248\pi\)
−0.967460 + 0.253023i \(0.918575\pi\)
\(18\) 0 0
\(19\) −4.93847 −1.13296 −0.566482 − 0.824074i \(-0.691696\pi\)
−0.566482 + 0.824074i \(0.691696\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 8.47698i − 1.76757i −0.467891 − 0.883786i \(-0.654986\pi\)
0.467891 − 0.883786i \(-0.345014\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.39798 0.445294 0.222647 − 0.974899i \(-0.428530\pi\)
0.222647 + 0.974899i \(0.428530\pi\)
\(30\) 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
\(37\) 5.85199i 0.962062i 0.876704 + 0.481031i \(0.159738\pi\)
−0.876704 + 0.481031i \(0.840262\pi\)
\(38\) 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
\(43\) − 8.24997i − 1.25811i −0.777361 − 0.629055i \(-0.783442\pi\)
0.777361 − 0.629055i \(-0.216558\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.68850i 0.392158i 0.980588 + 0.196079i \(0.0628209\pi\)
−0.980588 + 0.196079i \(0.937179\pi\)
\(48\) 0 0
\(49\) 6.99252 0.998932
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.73642i 0.787958i 0.919120 + 0.393979i \(0.128902\pi\)
−0.919120 + 0.393979i \(0.871098\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.3384 1.60633 0.803164 − 0.595758i \(-0.203148\pi\)
0.803164 + 0.595758i \(0.203148\pi\)
\(60\) 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
\(67\) − 6.16548i − 0.753233i −0.926369 − 0.376617i \(-0.877087\pi\)
0.926369 − 0.376617i \(-0.122913\pi\)
\(68\) 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
\(73\) 5.31349i 0.621897i 0.950427 + 0.310948i \(0.100647\pi\)
−0.950427 + 0.310948i \(0.899353\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.0790006i − 0.00900296i
\(78\) 0 0
\(79\) 13.4479 1.51301 0.756503 − 0.653991i \(-0.226906\pi\)
0.756503 + 0.653991i \(0.226906\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.07900i 0.667257i 0.942705 + 0.333629i \(0.108273\pi\)
−0.942705 + 0.333629i \(0.891727\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.13440 −0.862244 −0.431122 − 0.902294i \(-0.641882\pi\)
−0.431122 + 0.902294i \(0.641882\pi\)
\(90\) 0 0
\(91\) 0.227007 0.0237968
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 11.1020i − 1.12723i −0.826036 − 0.563617i \(-0.809409\pi\)
0.826036 − 0.563617i \(-0.190591\pi\)
\(98\) 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.d.s.649.4 8
3.2 odd 2 8100.2.d.q.649.4 8
5.2 odd 4 8100.2.a.ba.1.3 4
5.3 odd 4 8100.2.a.y.1.2 4
5.4 even 2 inner 8100.2.d.s.649.5 8
9.2 odd 6 900.2.s.d.49.3 16
9.4 even 3 2700.2.s.d.2449.5 16
9.5 odd 6 900.2.s.d.349.6 16
9.7 even 3 2700.2.s.d.1549.4 16
15.2 even 4 8100.2.a.z.1.3 4
15.8 even 4 8100.2.a.x.1.2 4
15.14 odd 2 8100.2.d.q.649.5 8
45.2 even 12 900.2.i.e.301.1 yes 8
45.4 even 6 2700.2.s.d.2449.4 16
45.7 odd 12 2700.2.i.d.901.2 8
45.13 odd 12 2700.2.i.e.1801.3 8
45.14 odd 6 900.2.s.d.349.3 16
45.22 odd 12 2700.2.i.d.1801.2 8
45.23 even 12 900.2.i.d.601.4 yes 8
45.29 odd 6 900.2.s.d.49.6 16
45.32 even 12 900.2.i.e.601.1 yes 8
45.34 even 6 2700.2.s.d.1549.5 16
45.38 even 12 900.2.i.d.301.4 ✓ 8
45.43 odd 12 2700.2.i.e.901.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
900.2.i.d.301.4 ✓ 8 45.38 even 12
900.2.i.d.601.4 yes 8 45.23 even 12
900.2.i.e.301.1 yes 8 45.2 even 12
900.2.i.e.601.1 yes 8 45.32 even 12
900.2.s.d.49.3 16 9.2 odd 6
900.2.s.d.49.6 16 45.29 odd 6
900.2.s.d.349.3 16 45.14 odd 6
900.2.s.d.349.6 16 9.5 odd 6
2700.2.i.d.901.2 8 45.7 odd 12
2700.2.i.d.1801.2 8 45.22 odd 12
2700.2.i.e.901.3 8 45.43 odd 12
2700.2.i.e.1801.3 8 45.13 odd 12
2700.2.s.d.1549.4 16 9.7 even 3
2700.2.s.d.1549.5 16 45.34 even 6
2700.2.s.d.2449.4 16 45.4 even 6
2700.2.s.d.2449.5 16 9.4 even 3
8100.2.a.x.1.2 4 15.8 even 4
8100.2.a.y.1.2 4 5.3 odd 4
8100.2.a.z.1.3 4 15.2 even 4
8100.2.a.ba.1.3 4 5.2 odd 4
8100.2.d.q.649.4 8 3.2 odd 2
8100.2.d.q.649.5 8 15.14 odd 2
8100.2.d.s.649.4 8 1.1 even 1 trivial
8100.2.d.s.649.5 8 5.4 even 2 inner