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Properties

Label 800.6.a.m.1.1
Level $800$
Weight $6$
Character 800.1
Self dual yes
Analytic conductor $128.307$
Analytic rank $0$
Dimension $2$
CM discriminant -20
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] = [800,6,Mod(1,800)] 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("800.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,38,0,0,0,-366,0,486,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+7.81966 q^{3} -171.820 q^{7} -181.853 q^{9} -1343.57 q^{21} -230.741 q^{23} -3322.21 q^{27} -1686.00 q^{29} -21041.4 q^{41} +22019.5 q^{43} -30116.9 q^{47} +12715.0 q^{49} +52256.9 q^{61} +31245.9 q^{63} +36856.5 q^{67} -1804.31 q^{69} +18211.7 q^{81} +46781.9 q^{83} -13183.9 q^{87} +149286. q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 38 q^{3} - 366 q^{7} + 486 q^{9} - 7204 q^{21} + 4838 q^{23} + 9500 q^{27} - 3372 q^{29} + 11862 q^{43} - 33334 q^{47} + 33614 q^{49} - 98438 q^{63} + 100434 q^{67} + 151172 q^{69} + 242902 q^{81}+ \cdots + 298572 q^{89}+O(q^{100}) \) 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
\(3\) 7.81966 0.501631 0.250816 − 0.968035i \(-0.419301\pi\)
0.250816 + 0.968035i \(0.419301\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −171.820 −1.32534 −0.662671 − 0.748911i \(-0.730577\pi\)
−0.662671 + 0.748911i \(0.730577\pi\)
\(8\) 0 0
\(9\) −181.853 −0.748366
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −1343.57 −0.664833
\(22\) 0 0
\(23\) −230.741 −0.0909503 −0.0454752 − 0.998965i \(-0.514480\pi\)
−0.0454752 + 0.998965i \(0.514480\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −3322.21 −0.877035
\(28\) 0 0
\(29\) −1686.00 −0.372274 −0.186137 − 0.982524i \(-0.559597\pi\)
−0.186137 + 0.982524i \(0.559597\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −21041.4 −1.95486 −0.977428 − 0.211267i \(-0.932241\pi\)
−0.977428 + 0.211267i \(0.932241\pi\)
\(42\) 0 0
\(43\) 22019.5 1.81609 0.908043 − 0.418877i \(-0.137576\pi\)
0.908043 + 0.418877i \(0.137576\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −30116.9 −1.98869 −0.994343 − 0.106214i \(-0.966127\pi\)
−0.994343 + 0.106214i \(0.966127\pi\)
\(48\) 0 0
\(49\) 12715.0 0.756530
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 52256.9 1.79812 0.899061 − 0.437824i \(-0.144251\pi\)
0.899061 + 0.437824i \(0.144251\pi\)
\(62\) 0 0
\(63\) 31245.9 0.991840
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 36856.5 1.00306 0.501530 − 0.865140i \(-0.332771\pi\)
0.501530 + 0.865140i \(0.332771\pi\)
\(68\) 0 0
\(69\) −1804.31 −0.0456236
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 18211.7 0.308417
\(82\) 0 0
\(83\) 46781.9 0.745388 0.372694 − 0.927954i \(-0.378434\pi\)
0.372694 + 0.927954i \(0.378434\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −13183.9 −0.186744
\(88\) 0 0
\(89\) 149286. 1.99776 0.998882 − 0.0472789i \(-0.0150549\pi\)
0.998882 + 0.0472789i \(0.0150549\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 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 800.6.a.m.1.1 2
4.3 odd 2 800.6.a.f.1.2 2
5.2 odd 4 160.6.c.b.129.2 ✓ 4
5.3 odd 4 160.6.c.b.129.3 yes 4
5.4 even 2 800.6.a.f.1.2 2
20.3 even 4 160.6.c.b.129.2 ✓ 4
20.7 even 4 160.6.c.b.129.3 yes 4
20.19 odd 2 CM 800.6.a.m.1.1 2
40.3 even 4 320.6.c.h.129.3 4
40.13 odd 4 320.6.c.h.129.2 4
40.27 even 4 320.6.c.h.129.2 4
40.37 odd 4 320.6.c.h.129.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.c.b.129.2 ✓ 4 5.2 odd 4
160.6.c.b.129.2 ✓ 4 20.3 even 4
160.6.c.b.129.3 yes 4 5.3 odd 4
160.6.c.b.129.3 yes 4 20.7 even 4
320.6.c.h.129.2 4 40.13 odd 4
320.6.c.h.129.2 4 40.27 even 4
320.6.c.h.129.3 4 40.3 even 4
320.6.c.h.129.3 4 40.37 odd 4
800.6.a.f.1.2 2 4.3 odd 2
800.6.a.f.1.2 2 5.4 even 2
800.6.a.m.1.1 2 1.1 even 1 trivial
800.6.a.m.1.1 2 20.19 odd 2 CM