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Properties

Label 800.4.f.b.49.9
Level $800$
Weight $4$
Character 800.49
Analytic conductor $47.202$
Analytic rank $0$
Dimension $12$
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] = [800,4,Mod(49,800)] 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("800.49"); S:= CuspForms(chi, 4); 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, 1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.2015280046\)
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: \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{32} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.9
Root \(1.71681 + 1.02595i\) of defining polynomial
Character \(\chi\) \(=\) 800.49
Dual form 800.4.f.b.49.10

$q$-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)
 
\(f(q)\) \(=\) \(q+4.24443 q^{3} -14.6308i q^{7} -8.98481 q^{9} +14.9969i q^{11} +85.6955 q^{13} +91.9247i q^{17} +60.3737i q^{19} -62.0995i q^{21} -1.33730i q^{23} -152.735 q^{27} +25.5118i q^{29} -73.4354 q^{31} +63.6535i q^{33} +211.259 q^{37} +363.728 q^{39} +330.839 q^{41} +388.500 q^{43} -550.348i q^{47} +128.939 q^{49} +390.168i q^{51} +187.705 q^{53} +256.252i q^{57} -779.090i q^{59} +358.850i q^{61} +131.455i q^{63} +283.674 q^{67} -5.67606i q^{69} +534.811 q^{71} +1016.16i q^{73} +219.418 q^{77} +1119.59 q^{79} -405.683 q^{81} -1190.49 q^{83} +108.283i q^{87} +398.940 q^{89} -1253.80i q^{91} -311.691 q^{93} -278.022i q^{97} -134.745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 108 q^{9} - 432 q^{27} + 264 q^{31} + 136 q^{37} + 600 q^{39} + 40 q^{41} + 1204 q^{43} - 1308 q^{49} + 1056 q^{53} + 2412 q^{67} + 1592 q^{71} + 824 q^{77} + 2016 q^{79} + 2508 q^{81}+ \cdots + 2784 q^{93}+O(q^{100}) \) 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
\(3\) 4.24443 0.816841 0.408420 − 0.912794i \(-0.366080\pi\)
0.408420 + 0.912794i \(0.366080\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 14.6308i − 0.789990i −0.918683 − 0.394995i \(-0.870746\pi\)
0.918683 − 0.394995i \(-0.129254\pi\)
\(8\) 0 0
\(9\) −8.98481 −0.332771
\(10\) 0 0
\(11\) 14.9969i 0.411068i 0.978650 + 0.205534i \(0.0658931\pi\)
−0.978650 + 0.205534i \(0.934107\pi\)
\(12\) 0 0
\(13\) 85.6955 1.82828 0.914140 − 0.405398i \(-0.132867\pi\)
0.914140 + 0.405398i \(0.132867\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 91.9247i 1.31147i 0.754991 + 0.655735i \(0.227641\pi\)
−0.754991 + 0.655735i \(0.772359\pi\)
\(18\) 0 0
\(19\) 60.3737i 0.728983i 0.931207 + 0.364492i \(0.118757\pi\)
−0.931207 + 0.364492i \(0.881243\pi\)
\(20\) 0 0
\(21\) − 62.0995i − 0.645296i
\(22\) 0 0
\(23\) − 1.33730i − 0.0121237i −0.999982 − 0.00606186i \(-0.998070\pi\)
0.999982 − 0.00606186i \(-0.00192956\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −152.735 −1.08866
\(28\) 0 0
\(29\) 25.5118i 0.163360i 0.996659 + 0.0816798i \(0.0260285\pi\)
−0.996659 + 0.0816798i \(0.973972\pi\)
\(30\) 0 0
\(31\) −73.4354 −0.425464 −0.212732 − 0.977111i \(-0.568236\pi\)
−0.212732 + 0.977111i \(0.568236\pi\)
\(32\) 0 0
\(33\) 63.6535i 0.335777i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 211.259 0.938668 0.469334 − 0.883021i \(-0.344494\pi\)
0.469334 + 0.883021i \(0.344494\pi\)
\(38\) 0 0
\(39\) 363.728 1.49341
\(40\) 0 0
\(41\) 330.839 1.26020 0.630102 − 0.776512i \(-0.283013\pi\)
0.630102 + 0.776512i \(0.283013\pi\)
\(42\) 0 0
\(43\) 388.500 1.37781 0.688904 − 0.724853i \(-0.258092\pi\)
0.688904 + 0.724853i \(0.258092\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 550.348i − 1.70801i −0.520265 − 0.854005i \(-0.674167\pi\)
0.520265 − 0.854005i \(-0.325833\pi\)
\(48\) 0 0
\(49\) 128.939 0.375915
\(50\) 0 0
\(51\) 390.168i 1.07126i
\(52\) 0 0
\(53\) 187.705 0.486477 0.243239 − 0.969966i \(-0.421790\pi\)
0.243239 + 0.969966i \(0.421790\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 256.252i 0.595463i
\(58\) 0 0
\(59\) − 779.090i − 1.71913i −0.511023 − 0.859567i \(-0.670733\pi\)
0.511023 − 0.859567i \(-0.329267\pi\)
\(60\) 0 0
\(61\) 358.850i 0.753215i 0.926373 + 0.376607i \(0.122909\pi\)
−0.926373 + 0.376607i \(0.877091\pi\)
\(62\) 0 0
\(63\) 131.455i 0.262886i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 283.674 0.517258 0.258629 − 0.965977i \(-0.416729\pi\)
0.258629 + 0.965977i \(0.416729\pi\)
\(68\) 0 0
\(69\) − 5.67606i − 0.00990315i
\(70\) 0 0
\(71\) 534.811 0.893949 0.446975 − 0.894547i \(-0.352501\pi\)
0.446975 + 0.894547i \(0.352501\pi\)
\(72\) 0 0
\(73\) 1016.16i 1.62921i 0.580014 + 0.814607i \(0.303047\pi\)
−0.580014 + 0.814607i \(0.696953\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 219.418 0.324740
\(78\) 0 0
\(79\) 1119.59 1.59447 0.797237 − 0.603666i \(-0.206294\pi\)
0.797237 + 0.603666i \(0.206294\pi\)
\(80\) 0 0
\(81\) −405.683 −0.556493
\(82\) 0 0
\(83\) −1190.49 −1.57437 −0.787186 − 0.616716i \(-0.788463\pi\)
−0.787186 + 0.616716i \(0.788463\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 108.283i 0.133439i
\(88\) 0 0
\(89\) 398.940 0.475141 0.237571 − 0.971370i \(-0.423649\pi\)
0.237571 + 0.971370i \(0.423649\pi\)
\(90\) 0 0
\(91\) − 1253.80i − 1.44432i
\(92\) 0 0
\(93\) −311.691 −0.347536
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 278.022i − 0.291019i −0.989357 − 0.145510i \(-0.953518\pi\)
0.989357 − 0.145510i \(-0.0464822\pi\)
\(98\) 0 0
\(99\) − 134.745i − 0.136791i
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 800.4.f.b.49.9 12
4.3 odd 2 200.4.f.c.149.5 12
5.2 odd 4 160.4.d.a.81.4 12
5.3 odd 4 800.4.d.d.401.9 12
5.4 even 2 800.4.f.c.49.4 12
8.3 odd 2 200.4.f.b.149.7 12
8.5 even 2 800.4.f.c.49.3 12
15.2 even 4 1440.4.k.c.721.11 12
20.3 even 4 200.4.d.b.101.1 12
20.7 even 4 40.4.d.a.21.12 yes 12
20.19 odd 2 200.4.f.b.149.8 12
40.3 even 4 200.4.d.b.101.2 12
40.13 odd 4 800.4.d.d.401.4 12
40.19 odd 2 200.4.f.c.149.6 12
40.27 even 4 40.4.d.a.21.11 ✓ 12
40.29 even 2 inner 800.4.f.b.49.10 12
40.37 odd 4 160.4.d.a.81.9 12
60.47 odd 4 360.4.k.c.181.1 12
80.27 even 4 1280.4.a.bb.1.5 6
80.37 odd 4 1280.4.a.bd.1.2 6
80.67 even 4 1280.4.a.bc.1.2 6
80.77 odd 4 1280.4.a.ba.1.5 6
120.77 even 4 1440.4.k.c.721.5 12
120.107 odd 4 360.4.k.c.181.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.11 ✓ 12 40.27 even 4
40.4.d.a.21.12 yes 12 20.7 even 4
160.4.d.a.81.4 12 5.2 odd 4
160.4.d.a.81.9 12 40.37 odd 4
200.4.d.b.101.1 12 20.3 even 4
200.4.d.b.101.2 12 40.3 even 4
200.4.f.b.149.7 12 8.3 odd 2
200.4.f.b.149.8 12 20.19 odd 2
200.4.f.c.149.5 12 4.3 odd 2
200.4.f.c.149.6 12 40.19 odd 2
360.4.k.c.181.1 12 60.47 odd 4
360.4.k.c.181.2 12 120.107 odd 4
800.4.d.d.401.4 12 40.13 odd 4
800.4.d.d.401.9 12 5.3 odd 4
800.4.f.b.49.9 12 1.1 even 1 trivial
800.4.f.b.49.10 12 40.29 even 2 inner
800.4.f.c.49.3 12 8.5 even 2
800.4.f.c.49.4 12 5.4 even 2
1280.4.a.ba.1.5 6 80.77 odd 4
1280.4.a.bb.1.5 6 80.27 even 4
1280.4.a.bc.1.2 6 80.67 even 4
1280.4.a.bd.1.2 6 80.37 odd 4
1440.4.k.c.721.5 12 120.77 even 4
1440.4.k.c.721.11 12 15.2 even 4