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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 = [6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,0,0,-6] 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: \(6\)
Coefficient field: 6.0.5089536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.33641 - 1.33641i\) of defining polynomial
Character \(\chi\) \(=\) 8100.649
Dual form 8100.2.d.o.649.6

$q$-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)
 
\(f(q)\) \(=\) \(q-4.10083i q^{7} +3.81681 q^{11} +5.81681i q^{13} +3.81681i q^{17} +1.81681 q^{19} -2.10083i q^{23} -7.20166 q^{29} -1.81681 q^{31} +6.01847i q^{37} -11.0185 q^{41} +5.81681i q^{43} +11.9176i q^{47} -9.81681 q^{49} -4.20166i q^{53} -4.20166 q^{59} -3.01847 q^{61} -3.71598i q^{67} +2.01847 q^{71} +8.00000i q^{73} -15.6521i q^{77} -2.00000 q^{79} +3.89917i q^{83} -3.00000 q^{89} +23.8538 q^{91} -12.2017i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{19} - 6 q^{29} + 12 q^{31} - 6 q^{41} - 36 q^{49} + 12 q^{59} + 42 q^{61} - 48 q^{71} - 12 q^{79} - 18 q^{89}+O(q^{100}) \) 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
\(7\) − 4.10083i − 1.54997i −0.631981 − 0.774984i \(-0.717758\pi\)
0.631981 − 0.774984i \(-0.282242\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.81681 1.15081 0.575406 − 0.817868i \(-0.304844\pi\)
0.575406 + 0.817868i \(0.304844\pi\)
\(12\) 0 0
\(13\) 5.81681i 1.61329i 0.591034 + 0.806646i \(0.298720\pi\)
−0.591034 + 0.806646i \(0.701280\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.81681i 0.925712i 0.886433 + 0.462856i \(0.153175\pi\)
−0.886433 + 0.462856i \(0.846825\pi\)
\(18\) 0 0
\(19\) 1.81681 0.416805 0.208402 − 0.978043i \(-0.433174\pi\)
0.208402 + 0.978043i \(0.433174\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.10083i − 0.438053i −0.975719 − 0.219027i \(-0.929712\pi\)
0.975719 − 0.219027i \(-0.0702882\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.20166 −1.33731 −0.668657 − 0.743571i \(-0.733131\pi\)
−0.668657 + 0.743571i \(0.733131\pi\)
\(30\) 0 0
\(31\) −1.81681 −0.326309 −0.163154 − 0.986601i \(-0.552167\pi\)
−0.163154 + 0.986601i \(0.552167\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.01847i 0.989431i 0.869055 + 0.494715i \(0.164728\pi\)
−0.869055 + 0.494715i \(0.835272\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.0185 −1.72080 −0.860398 − 0.509623i \(-0.829785\pi\)
−0.860398 + 0.509623i \(0.829785\pi\)
\(42\) 0 0
\(43\) 5.81681i 0.887055i 0.896261 + 0.443528i \(0.146273\pi\)
−0.896261 + 0.443528i \(0.853727\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 11.9176i 1.73837i 0.494490 + 0.869183i \(0.335355\pi\)
−0.494490 + 0.869183i \(0.664645\pi\)
\(48\) 0 0
\(49\) −9.81681 −1.40240
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 4.20166i − 0.577143i −0.957458 − 0.288571i \(-0.906820\pi\)
0.957458 − 0.288571i \(-0.0931802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.20166 −0.547010 −0.273505 − 0.961871i \(-0.588183\pi\)
−0.273505 + 0.961871i \(0.588183\pi\)
\(60\) 0 0
\(61\) −3.01847 −0.386476 −0.193238 − 0.981152i \(-0.561899\pi\)
−0.193238 + 0.981152i \(0.561899\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.71598i − 0.453979i −0.973897 − 0.226990i \(-0.927112\pi\)
0.973897 − 0.226990i \(-0.0728883\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.01847 0.239548 0.119774 − 0.992801i \(-0.461783\pi\)
0.119774 + 0.992801i \(0.461783\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 15.6521i − 1.78372i
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.89917i 0.427989i 0.976835 + 0.213995i \(0.0686475\pi\)
−0.976835 + 0.213995i \(0.931352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 23.8538 2.50055
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 12.2017i − 1.23889i −0.785040 − 0.619445i \(-0.787357\pi\)
0.785040 − 0.619445i \(-0.212643\pi\)
\(98\) 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.o.649.1 6
3.2 odd 2 8100.2.d.p.649.1 6
5.2 odd 4 1620.2.a.j.1.3 3
5.3 odd 4 8100.2.a.u.1.1 3
5.4 even 2 inner 8100.2.d.o.649.6 6
9.2 odd 6 900.2.s.c.49.5 12
9.4 even 3 2700.2.s.c.2449.6 12
9.5 odd 6 900.2.s.c.349.2 12
9.7 even 3 2700.2.s.c.1549.1 12
15.2 even 4 1620.2.a.i.1.3 3
15.8 even 4 8100.2.a.v.1.1 3
15.14 odd 2 8100.2.d.p.649.6 6
20.7 even 4 6480.2.a.bw.1.1 3
45.2 even 12 180.2.i.b.121.3 yes 6
45.4 even 6 2700.2.s.c.2449.1 12
45.7 odd 12 540.2.i.b.361.1 6
45.13 odd 12 2700.2.i.c.1801.3 6
45.14 odd 6 900.2.s.c.349.5 12
45.22 odd 12 540.2.i.b.181.1 6
45.23 even 12 900.2.i.c.601.1 6
45.29 odd 6 900.2.s.c.49.2 12
45.32 even 12 180.2.i.b.61.3 ✓ 6
45.34 even 6 2700.2.s.c.1549.6 12
45.38 even 12 900.2.i.c.301.1 6
45.43 odd 12 2700.2.i.c.901.3 6
60.47 odd 4 6480.2.a.bt.1.1 3
180.7 even 12 2160.2.q.i.1441.3 6
180.47 odd 12 720.2.q.k.481.1 6
180.67 even 12 2160.2.q.i.721.3 6
180.167 odd 12 720.2.q.k.241.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.i.b.61.3 ✓ 6 45.32 even 12
180.2.i.b.121.3 yes 6 45.2 even 12
540.2.i.b.181.1 6 45.22 odd 12
540.2.i.b.361.1 6 45.7 odd 12
720.2.q.k.241.1 6 180.167 odd 12
720.2.q.k.481.1 6 180.47 odd 12
900.2.i.c.301.1 6 45.38 even 12
900.2.i.c.601.1 6 45.23 even 12
900.2.s.c.49.2 12 45.29 odd 6
900.2.s.c.49.5 12 9.2 odd 6
900.2.s.c.349.2 12 9.5 odd 6
900.2.s.c.349.5 12 45.14 odd 6
1620.2.a.i.1.3 3 15.2 even 4
1620.2.a.j.1.3 3 5.2 odd 4
2160.2.q.i.721.3 6 180.67 even 12
2160.2.q.i.1441.3 6 180.7 even 12
2700.2.i.c.901.3 6 45.43 odd 12
2700.2.i.c.1801.3 6 45.13 odd 12
2700.2.s.c.1549.1 12 9.7 even 3
2700.2.s.c.1549.6 12 45.34 even 6
2700.2.s.c.2449.1 12 45.4 even 6
2700.2.s.c.2449.6 12 9.4 even 3
6480.2.a.bt.1.1 3 60.47 odd 4
6480.2.a.bw.1.1 3 20.7 even 4
8100.2.a.u.1.1 3 5.3 odd 4
8100.2.a.v.1.1 3 15.8 even 4
8100.2.d.o.649.1 6 1.1 even 1 trivial
8100.2.d.o.649.6 6 5.4 even 2 inner
8100.2.d.p.649.1 6 3.2 odd 2
8100.2.d.p.649.6 6 15.14 odd 2