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Properties

Label 800.6.c.p.449.9
Level $800$
Weight $6$
Character 800.449
Analytic conductor $128.307$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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(449,800)] 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("800.449"); 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, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-456,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(128.307055850\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 423x^{10} + 57217x^{8} + 2682817x^{6} + 43719927x^{4} + 136382881x^{2} + 69355584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{34}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.9
Root \(5.29401i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.6.c.p.449.3

$q$-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)
 
\(f(q)\) \(=\) \(q+11.5880i q^{3} +89.6769i q^{7} +108.718 q^{9} -718.855 q^{11} +871.743i q^{13} -2090.64i q^{17} +476.947 q^{19} -1039.18 q^{21} +1330.77i q^{23} +4075.71i q^{27} -730.313 q^{29} -9788.53 q^{31} -8330.12i q^{33} +6219.08i q^{37} -10101.8 q^{39} +867.570 q^{41} -12946.8i q^{43} -26161.9i q^{47} +8765.06 q^{49} +24226.4 q^{51} +11148.4i q^{53} +5526.87i q^{57} -21407.2 q^{59} +32686.2 q^{61} +9749.45i q^{63} -37209.3i q^{67} -15420.9 q^{69} -35282.1 q^{71} +20909.5i q^{73} -64464.7i q^{77} +21560.4 q^{79} -20811.1 q^{81} +87640.0i q^{83} -8462.89i q^{87} -111553. q^{89} -78175.2 q^{91} -113430. i q^{93} -18353.2i q^{97} -78152.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 456 q^{9} + 1768 q^{21} - 8304 q^{29} + 20668 q^{41} - 2908 q^{49} + 51592 q^{61} - 11592 q^{69} + 293996 q^{81} - 166900 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 11.5880i 0.743372i 0.928358 + 0.371686i \(0.121220\pi\)
−0.928358 + 0.371686i \(0.878780\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 89.6769i 0.691728i 0.938285 + 0.345864i \(0.112414\pi\)
−0.938285 + 0.345864i \(0.887586\pi\)
\(8\) 0 0
\(9\) 108.718 0.447397
\(10\) 0 0
\(11\) −718.855 −1.79126 −0.895632 − 0.444795i \(-0.853276\pi\)
−0.895632 + 0.444795i \(0.853276\pi\)
\(12\) 0 0
\(13\) 871.743i 1.43064i 0.698797 + 0.715320i \(0.253719\pi\)
−0.698797 + 0.715320i \(0.746281\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2090.64i − 1.75451i −0.480021 − 0.877257i \(-0.659371\pi\)
0.480021 − 0.877257i \(-0.340629\pi\)
\(18\) 0 0
\(19\) 476.947 0.303100 0.151550 − 0.988450i \(-0.451574\pi\)
0.151550 + 0.988450i \(0.451574\pi\)
\(20\) 0 0
\(21\) −1039.18 −0.514212
\(22\) 0 0
\(23\) 1330.77i 0.524544i 0.964994 + 0.262272i \(0.0844718\pi\)
−0.964994 + 0.262272i \(0.915528\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4075.71i 1.07596i
\(28\) 0 0
\(29\) −730.313 −0.161255 −0.0806277 − 0.996744i \(-0.525692\pi\)
−0.0806277 + 0.996744i \(0.525692\pi\)
\(30\) 0 0
\(31\) −9788.53 −1.82942 −0.914710 − 0.404111i \(-0.867581\pi\)
−0.914710 + 0.404111i \(0.867581\pi\)
\(32\) 0 0
\(33\) − 8330.12i − 1.33158i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6219.08i 0.746830i 0.927664 + 0.373415i \(0.121813\pi\)
−0.927664 + 0.373415i \(0.878187\pi\)
\(38\) 0 0
\(39\) −10101.8 −1.06350
\(40\) 0 0
\(41\) 867.570 0.0806018 0.0403009 − 0.999188i \(-0.487168\pi\)
0.0403009 + 0.999188i \(0.487168\pi\)
\(42\) 0 0
\(43\) − 12946.8i − 1.06780i −0.845548 − 0.533900i \(-0.820726\pi\)
0.845548 − 0.533900i \(-0.179274\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 26161.9i − 1.72753i −0.503897 − 0.863764i \(-0.668101\pi\)
0.503897 − 0.863764i \(-0.331899\pi\)
\(48\) 0 0
\(49\) 8765.06 0.521512
\(50\) 0 0
\(51\) 24226.4 1.30426
\(52\) 0 0
\(53\) 11148.4i 0.545160i 0.962133 + 0.272580i \(0.0878770\pi\)
−0.962133 + 0.272580i \(0.912123\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5526.87i 0.225316i
\(58\) 0 0
\(59\) −21407.2 −0.800628 −0.400314 − 0.916378i \(-0.631099\pi\)
−0.400314 + 0.916378i \(0.631099\pi\)
\(60\) 0 0
\(61\) 32686.2 1.12471 0.562354 − 0.826896i \(-0.309896\pi\)
0.562354 + 0.826896i \(0.309896\pi\)
\(62\) 0 0
\(63\) 9749.45i 0.309477i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 37209.3i − 1.01266i −0.862339 − 0.506331i \(-0.831002\pi\)
0.862339 − 0.506331i \(-0.168998\pi\)
\(68\) 0 0
\(69\) −15420.9 −0.389932
\(70\) 0 0
\(71\) −35282.1 −0.830633 −0.415316 − 0.909677i \(-0.636329\pi\)
−0.415316 + 0.909677i \(0.636329\pi\)
\(72\) 0 0
\(73\) 20909.5i 0.459238i 0.973281 + 0.229619i \(0.0737479\pi\)
−0.973281 + 0.229619i \(0.926252\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 64464.7i − 1.23907i
\(78\) 0 0
\(79\) 21560.4 0.388678 0.194339 − 0.980934i \(-0.437744\pi\)
0.194339 + 0.980934i \(0.437744\pi\)
\(80\) 0 0
\(81\) −20811.1 −0.352438
\(82\) 0 0
\(83\) 87640.0i 1.39639i 0.715907 + 0.698195i \(0.246013\pi\)
−0.715907 + 0.698195i \(0.753987\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 8462.89i − 0.119873i
\(88\) 0 0
\(89\) −111553. −1.49282 −0.746410 − 0.665486i \(-0.768224\pi\)
−0.746410 + 0.665486i \(0.768224\pi\)
\(90\) 0 0
\(91\) −78175.2 −0.989613
\(92\) 0 0
\(93\) − 113430.i − 1.35994i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 18353.2i − 0.198053i −0.995085 − 0.0990266i \(-0.968427\pi\)
0.995085 − 0.0990266i \(-0.0315729\pi\)
\(98\) 0 0
\(99\) −78152.2 −0.801407
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.c.p.449.9 12
4.3 odd 2 inner 800.6.c.p.449.4 12
5.2 odd 4 800.6.a.x.1.5 yes 6
5.3 odd 4 800.6.a.y.1.2 yes 6
5.4 even 2 inner 800.6.c.p.449.3 12
20.3 even 4 800.6.a.y.1.5 yes 6
20.7 even 4 800.6.a.x.1.2 ✓ 6
20.19 odd 2 inner 800.6.c.p.449.10 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.6.a.x.1.2 ✓ 6 20.7 even 4
800.6.a.x.1.5 yes 6 5.2 odd 4
800.6.a.y.1.2 yes 6 5.3 odd 4
800.6.a.y.1.5 yes 6 20.3 even 4
800.6.c.p.449.3 12 5.4 even 2 inner
800.6.c.p.449.4 12 4.3 odd 2 inner
800.6.c.p.449.9 12 1.1 even 1 trivial
800.6.c.p.449.10 12 20.19 odd 2 inner