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Properties

Label 800.6.a.z.1.2
Level $800$
Weight $6$
Character 800.1
Self dual yes
Analytic conductor $128.307$
Analytic rank $1$
Dimension $6$
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 = [6,0,0,0,0,0,0,0,838,0,0,0,-1368] 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: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 82x^{4} - 112x^{3} + 1514x^{2} + 3695x - 767 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.32684\) 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)
 
\(f(q)\) \(=\) \(q-19.0114 q^{3} -91.0286 q^{7} +118.434 q^{9} -290.263 q^{11} -112.108 q^{13} -939.348 q^{17} +2055.29 q^{19} +1730.58 q^{21} +1684.18 q^{23} +2368.18 q^{27} -3960.68 q^{29} +7627.48 q^{31} +5518.31 q^{33} -9579.76 q^{37} +2131.33 q^{39} -281.653 q^{41} -6515.03 q^{43} -1395.46 q^{47} -8520.79 q^{49} +17858.3 q^{51} +18538.0 q^{53} -39074.0 q^{57} +51956.2 q^{59} +51541.5 q^{61} -10780.8 q^{63} -39074.2 q^{67} -32018.7 q^{69} -30435.9 q^{71} +49694.2 q^{73} +26422.2 q^{77} +98240.4 q^{79} -73801.8 q^{81} +94470.1 q^{83} +75298.0 q^{87} +98225.5 q^{89} +10205.0 q^{91} -145009. q^{93} -59010.4 q^{97} -34376.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 838 q^{9} - 1368 q^{13} - 1744 q^{17} + 344 q^{21} + 10652 q^{29} - 17952 q^{33} - 20072 q^{37} - 54836 q^{41} + 16174 q^{49} + 68696 q^{53} - 4000 q^{57} + 32276 q^{61} + 83176 q^{69} + 7552 q^{73}+ \cdots - 139888 q^{97}+O(q^{100}) \) 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
\(3\) −19.0114 −1.21958 −0.609791 − 0.792562i \(-0.708747\pi\)
−0.609791 + 0.792562i \(0.708747\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −91.0286 −0.702155 −0.351077 − 0.936346i \(-0.614185\pi\)
−0.351077 + 0.936346i \(0.614185\pi\)
\(8\) 0 0
\(9\) 118.434 0.487381
\(10\) 0 0
\(11\) −290.263 −0.723286 −0.361643 − 0.932317i \(-0.617784\pi\)
−0.361643 + 0.932317i \(0.617784\pi\)
\(12\) 0 0
\(13\) −112.108 −0.183983 −0.0919914 − 0.995760i \(-0.529323\pi\)
−0.0919914 + 0.995760i \(0.529323\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −939.348 −0.788323 −0.394161 − 0.919041i \(-0.628965\pi\)
−0.394161 + 0.919041i \(0.628965\pi\)
\(18\) 0 0
\(19\) 2055.29 1.30614 0.653070 − 0.757298i \(-0.273481\pi\)
0.653070 + 0.757298i \(0.273481\pi\)
\(20\) 0 0
\(21\) 1730.58 0.856336
\(22\) 0 0
\(23\) 1684.18 0.663850 0.331925 − 0.943306i \(-0.392302\pi\)
0.331925 + 0.943306i \(0.392302\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2368.18 0.625181
\(28\) 0 0
\(29\) −3960.68 −0.874529 −0.437265 − 0.899333i \(-0.644053\pi\)
−0.437265 + 0.899333i \(0.644053\pi\)
\(30\) 0 0
\(31\) 7627.48 1.42553 0.712765 − 0.701402i \(-0.247442\pi\)
0.712765 + 0.701402i \(0.247442\pi\)
\(32\) 0 0
\(33\) 5518.31 0.882106
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9579.76 −1.15040 −0.575202 − 0.818012i \(-0.695076\pi\)
−0.575202 + 0.818012i \(0.695076\pi\)
\(38\) 0 0
\(39\) 2131.33 0.224382
\(40\) 0 0
\(41\) −281.653 −0.0261671 −0.0130835 − 0.999914i \(-0.504165\pi\)
−0.0130835 + 0.999914i \(0.504165\pi\)
\(42\) 0 0
\(43\) −6515.03 −0.537335 −0.268668 − 0.963233i \(-0.586583\pi\)
−0.268668 + 0.963233i \(0.586583\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1395.46 −0.0921450 −0.0460725 − 0.998938i \(-0.514671\pi\)
−0.0460725 + 0.998938i \(0.514671\pi\)
\(48\) 0 0
\(49\) −8520.79 −0.506979
\(50\) 0 0
\(51\) 17858.3 0.961425
\(52\) 0 0
\(53\) 18538.0 0.906510 0.453255 − 0.891381i \(-0.350263\pi\)
0.453255 + 0.891381i \(0.350263\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −39074.0 −1.59294
\(58\) 0 0
\(59\) 51956.2 1.94315 0.971577 − 0.236723i \(-0.0760735\pi\)
0.971577 + 0.236723i \(0.0760735\pi\)
\(60\) 0 0
\(61\) 51541.5 1.77351 0.886753 − 0.462243i \(-0.152955\pi\)
0.886753 + 0.462243i \(0.152955\pi\)
\(62\) 0 0
\(63\) −10780.8 −0.342217
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −39074.2 −1.06342 −0.531708 − 0.846928i \(-0.678449\pi\)
−0.531708 + 0.846928i \(0.678449\pi\)
\(68\) 0 0
\(69\) −32018.7 −0.809620
\(70\) 0 0
\(71\) −30435.9 −0.716540 −0.358270 − 0.933618i \(-0.616633\pi\)
−0.358270 + 0.933618i \(0.616633\pi\)
\(72\) 0 0
\(73\) 49694.2 1.09144 0.545719 − 0.837968i \(-0.316257\pi\)
0.545719 + 0.837968i \(0.316257\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 26422.2 0.507859
\(78\) 0 0
\(79\) 98240.4 1.77102 0.885508 − 0.464624i \(-0.153811\pi\)
0.885508 + 0.464624i \(0.153811\pi\)
\(80\) 0 0
\(81\) −73801.8 −1.24984
\(82\) 0 0
\(83\) 94470.1 1.50522 0.752608 − 0.658468i \(-0.228795\pi\)
0.752608 + 0.658468i \(0.228795\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 75298.0 1.06656
\(88\) 0 0
\(89\) 98225.5 1.31447 0.657233 − 0.753688i \(-0.271727\pi\)
0.657233 + 0.753688i \(0.271727\pi\)
\(90\) 0 0
\(91\) 10205.0 0.129184
\(92\) 0 0
\(93\) −145009. −1.73855
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −59010.4 −0.636795 −0.318397 − 0.947957i \(-0.603145\pi\)
−0.318397 + 0.947957i \(0.603145\pi\)
\(98\) 0 0
\(99\) −34376.9 −0.352516
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.z.1.2 6
4.3 odd 2 inner 800.6.a.z.1.5 6
5.2 odd 4 160.6.c.c.129.10 yes 12
5.3 odd 4 160.6.c.c.129.3 ✓ 12
5.4 even 2 800.6.a.ba.1.5 6
20.3 even 4 160.6.c.c.129.9 yes 12
20.7 even 4 160.6.c.c.129.4 yes 12
20.19 odd 2 800.6.a.ba.1.2 6
40.3 even 4 320.6.c.k.129.4 12
40.13 odd 4 320.6.c.k.129.10 12
40.27 even 4 320.6.c.k.129.9 12
40.37 odd 4 320.6.c.k.129.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.6.c.c.129.3 ✓ 12 5.3 odd 4
160.6.c.c.129.4 yes 12 20.7 even 4
160.6.c.c.129.9 yes 12 20.3 even 4
160.6.c.c.129.10 yes 12 5.2 odd 4
320.6.c.k.129.3 12 40.37 odd 4
320.6.c.k.129.4 12 40.3 even 4
320.6.c.k.129.9 12 40.27 even 4
320.6.c.k.129.10 12 40.13 odd 4
800.6.a.z.1.2 6 1.1 even 1 trivial
800.6.a.z.1.5 6 4.3 odd 2 inner
800.6.a.ba.1.2 6 20.19 odd 2
800.6.a.ba.1.5 6 5.4 even 2