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Properties

Label 800.2.ba.a.549.1
Level $800$
Weight $2$
Character 800.549
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
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] = [800,2,Mod(149,800)] 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("800.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 5, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.ba (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,8,0,4,4,0,4,0,-8,-8,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 549.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 800.549
Dual form 800.2.ba.a.749.1

$q$-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-1.41421 q^{2} +(-1.70711 + 0.707107i) q^{3} +2.00000 q^{4} +(2.41421 - 1.00000i) q^{6} +(1.00000 - 1.00000i) q^{7} -2.82843 q^{8} +(0.292893 - 0.292893i) q^{9} +(0.121320 - 0.292893i) q^{11} +(-3.41421 + 1.41421i) q^{12} +(0.707107 + 1.70711i) q^{13} +(-1.41421 + 1.41421i) q^{14} +4.00000 q^{16} -2.82843 q^{17} +(-0.414214 + 0.414214i) q^{18} +(5.53553 - 2.29289i) q^{19} +(-1.00000 + 2.41421i) q^{21} +(-0.171573 + 0.414214i) q^{22} +(0.171573 + 0.171573i) q^{23} +(4.82843 - 2.00000i) q^{24} +(-1.00000 - 2.41421i) q^{26} +(1.82843 - 4.41421i) q^{27} +(2.00000 - 2.00000i) q^{28} +(-1.12132 - 2.70711i) q^{29} -4.00000 q^{31} -5.65685 q^{32} +0.585786i q^{33} +4.00000 q^{34} +(0.585786 - 0.585786i) q^{36} +(0.707107 - 1.70711i) q^{37} +(-7.82843 + 3.24264i) q^{38} +(-2.41421 - 2.41421i) q^{39} +(-5.82843 + 5.82843i) q^{41} +(1.41421 - 3.41421i) q^{42} +(7.94975 + 3.29289i) q^{43} +(0.242641 - 0.585786i) q^{44} +(-0.242641 - 0.242641i) q^{46} +11.6569 q^{47} +(-6.82843 + 2.82843i) q^{48} +5.00000i q^{49} +(4.82843 - 2.00000i) q^{51} +(1.41421 + 3.41421i) q^{52} +(7.53553 + 3.12132i) q^{53} +(-2.58579 + 6.24264i) q^{54} +(-2.82843 + 2.82843i) q^{56} +(-7.82843 + 7.82843i) q^{57} +(1.58579 + 3.82843i) q^{58} +(6.12132 + 2.53553i) q^{59} +(0.292893 + 0.707107i) q^{61} +5.65685 q^{62} -0.585786i q^{63} +8.00000 q^{64} -0.828427i q^{66} +(3.70711 - 1.53553i) q^{67} -5.65685 q^{68} +(-0.414214 - 0.171573i) q^{69} +(-0.171573 - 0.171573i) q^{71} +(-0.828427 + 0.828427i) q^{72} +(7.00000 + 7.00000i) q^{73} +(-1.00000 + 2.41421i) q^{74} +(11.0711 - 4.58579i) q^{76} +(-0.171573 - 0.414214i) q^{77} +(3.41421 + 3.41421i) q^{78} +6.00000i q^{79} +10.0711i q^{81} +(8.24264 - 8.24264i) q^{82} +(2.53553 + 6.12132i) q^{83} +(-2.00000 + 4.82843i) q^{84} +(-11.2426 - 4.65685i) q^{86} +(3.82843 + 3.82843i) q^{87} +(-0.343146 + 0.828427i) q^{88} +(2.65685 + 2.65685i) q^{89} +(2.41421 + 1.00000i) q^{91} +(0.343146 + 0.343146i) q^{92} +(6.82843 - 2.82843i) q^{93} -16.4853 q^{94} +(9.65685 - 4.00000i) q^{96} +1.51472i q^{97} -7.07107i q^{98} +(-0.0502525 - 0.121320i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 8 q^{4} + 4 q^{6} + 4 q^{7} + 4 q^{9} - 8 q^{11} - 8 q^{12} + 16 q^{16} + 4 q^{18} + 8 q^{19} - 4 q^{21} - 12 q^{22} + 12 q^{23} + 8 q^{24} - 4 q^{26} - 4 q^{27} + 8 q^{28} + 4 q^{29} - 16 q^{31}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000
\(3\) −1.70711 + 0.707107i −0.985599 + 0.408248i −0.816497 − 0.577350i \(-0.804087\pi\)
−0.169102 + 0.985599i \(0.554087\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 2.41421 − 1.00000i 0.985599 − 0.408248i
\(7\) 1.00000 − 1.00000i 0.377964 − 0.377964i −0.492403 − 0.870367i \(-0.663881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) −2.82843 −1.00000
\(9\) 0.292893 − 0.292893i 0.0976311 − 0.0976311i
\(10\) 0 0
\(11\) 0.121320 − 0.292893i 0.0365795 − 0.0883106i −0.904534 − 0.426401i \(-0.859781\pi\)
0.941113 + 0.338091i \(0.109781\pi\)
\(12\) −3.41421 + 1.41421i −0.985599 + 0.408248i
\(13\) 0.707107 + 1.70711i 0.196116 + 0.473466i 0.991093 − 0.133174i \(-0.0425169\pi\)
−0.794977 + 0.606640i \(0.792517\pi\)
\(14\) −1.41421 + 1.41421i −0.377964 + 0.377964i
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −2.82843 −0.685994 −0.342997 − 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) −0.414214 + 0.414214i −0.0976311 + 0.0976311i
\(19\) 5.53553 − 2.29289i 1.26994 − 0.526026i 0.356993 − 0.934107i \(-0.383802\pi\)
0.912946 + 0.408081i \(0.133802\pi\)
\(20\) 0 0
\(21\) −1.00000 + 2.41421i −0.218218 + 0.526825i
\(22\) −0.171573 + 0.414214i −0.0365795 + 0.0883106i
\(23\) 0.171573 + 0.171573i 0.0357754 + 0.0357754i 0.724768 − 0.688993i \(-0.241947\pi\)
−0.688993 + 0.724768i \(0.741947\pi\)
\(24\) 4.82843 − 2.00000i 0.985599 − 0.408248i
\(25\) 0 0
\(26\) −1.00000 − 2.41421i −0.196116 − 0.473466i
\(27\) 1.82843 − 4.41421i 0.351881 − 0.849516i
\(28\) 2.00000 − 2.00000i 0.377964 − 0.377964i
\(29\) −1.12132 − 2.70711i −0.208224 − 0.502697i 0.784920 − 0.619598i \(-0.212704\pi\)
−0.993144 + 0.116900i \(0.962704\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.65685 −1.00000
\(33\) 0.585786i 0.101972i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 0.585786 − 0.585786i 0.0976311 − 0.0976311i
\(37\) 0.707107 − 1.70711i 0.116248 − 0.280647i −0.855037 − 0.518567i \(-0.826466\pi\)
0.971285 + 0.237920i \(0.0764657\pi\)
\(38\) −7.82843 + 3.24264i −1.26994 + 0.526026i
\(39\) −2.41421 − 2.41421i −0.386584 − 0.386584i
\(40\) 0 0
\(41\) −5.82843 + 5.82843i −0.910247 + 0.910247i −0.996291 − 0.0860440i \(-0.972577\pi\)
0.0860440 + 0.996291i \(0.472577\pi\)
\(42\) 1.41421 − 3.41421i 0.218218 − 0.526825i
\(43\) 7.94975 + 3.29289i 1.21233 + 0.502162i 0.894962 − 0.446143i \(-0.147203\pi\)
0.317363 + 0.948304i \(0.397203\pi\)
\(44\) 0.242641 − 0.585786i 0.0365795 − 0.0883106i
\(45\) 0 0
\(46\) −0.242641 − 0.242641i −0.0357754 − 0.0357754i
\(47\) 11.6569 1.70033 0.850163 − 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) −6.82843 + 2.82843i −0.985599 + 0.408248i
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) 4.82843 − 2.00000i 0.676115 − 0.280056i
\(52\) 1.41421 + 3.41421i 0.196116 + 0.473466i
\(53\) 7.53553 + 3.12132i 1.03509 + 0.428746i 0.834545 − 0.550939i \(-0.185730\pi\)
0.200540 + 0.979686i \(0.435730\pi\)
\(54\) −2.58579 + 6.24264i −0.351881 + 0.849516i
\(55\) 0 0
\(56\) −2.82843 + 2.82843i −0.377964 + 0.377964i
\(57\) −7.82843 + 7.82843i −1.03690 + 1.03690i
\(58\) 1.58579 + 3.82843i 0.208224 + 0.502697i
\(59\) 6.12132 + 2.53553i 0.796928 + 0.330098i 0.743725 − 0.668485i \(-0.233057\pi\)
0.0532027 + 0.998584i \(0.483057\pi\)
\(60\) 0 0
\(61\) 0.292893 + 0.707107i 0.0375011 + 0.0905357i 0.941520 − 0.336956i \(-0.109397\pi\)
−0.904019 + 0.427492i \(0.859397\pi\)
\(62\) 5.65685 0.718421
\(63\) − 0.585786i − 0.0738022i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) − 0.828427i − 0.101972i
\(67\) 3.70711 − 1.53553i 0.452895 − 0.187595i −0.144563 − 0.989496i \(-0.546178\pi\)
0.597458 + 0.801900i \(0.296178\pi\)
\(68\) −5.65685 −0.685994
\(69\) −0.414214 − 0.171573i −0.0498655 − 0.0206549i
\(70\) 0 0
\(71\) −0.171573 − 0.171573i −0.0203620 − 0.0203620i 0.696853 − 0.717214i \(-0.254583\pi\)
−0.717214 + 0.696853i \(0.754583\pi\)
\(72\) −0.828427 + 0.828427i −0.0976311 + 0.0976311i
\(73\) 7.00000 + 7.00000i 0.819288 + 0.819288i 0.986005 − 0.166717i \(-0.0533166\pi\)
−0.166717 + 0.986005i \(0.553317\pi\)
\(74\) −1.00000 + 2.41421i −0.116248 + 0.280647i
\(75\) 0 0
\(76\) 11.0711 − 4.58579i 1.26994 − 0.526026i
\(77\) −0.171573 − 0.414214i −0.0195525 − 0.0472040i
\(78\) 3.41421 + 3.41421i 0.386584 + 0.386584i
\(79\) 6.00000i 0.675053i 0.941316 + 0.337526i \(0.109590\pi\)
−0.941316 + 0.337526i \(0.890410\pi\)
\(80\) 0 0
\(81\) 10.0711i 1.11901i
\(82\) 8.24264 − 8.24264i 0.910247 − 0.910247i
\(83\) 2.53553 + 6.12132i 0.278311 + 0.671902i 0.999789 − 0.0205350i \(-0.00653696\pi\)
−0.721478 + 0.692437i \(0.756537\pi\)
\(84\) −2.00000 + 4.82843i −0.218218 + 0.526825i
\(85\) 0 0
\(86\) −11.2426 − 4.65685i −1.21233 − 0.502162i
\(87\) 3.82843 + 3.82843i 0.410450 + 0.410450i
\(88\) −0.343146 + 0.828427i −0.0365795 + 0.0883106i
\(89\) 2.65685 + 2.65685i 0.281626 + 0.281626i 0.833757 − 0.552131i \(-0.186185\pi\)
−0.552131 + 0.833757i \(0.686185\pi\)
\(90\) 0 0
\(91\) 2.41421 + 1.00000i 0.253078 + 0.104828i
\(92\) 0.343146 + 0.343146i 0.0357754 + 0.0357754i
\(93\) 6.82843 − 2.82843i 0.708075 − 0.293294i
\(94\) −16.4853 −1.70033
\(95\) 0 0
\(96\) 9.65685 − 4.00000i 0.985599 − 0.408248i
\(97\) 1.51472i 0.153796i 0.997039 + 0.0768982i \(0.0245016\pi\)
−0.997039 + 0.0768982i \(0.975498\pi\)
\(98\) − 7.07107i − 0.714286i
\(99\) −0.0502525 − 0.121320i −0.00505057 − 0.0121932i
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 800.2.ba.a.549.1 4
5.2 odd 4 32.2.g.a.5.1 ✓ 4
5.3 odd 4 800.2.y.a.101.1 4
5.4 even 2 800.2.ba.b.549.1 4
15.2 even 4 288.2.v.a.37.1 4
20.7 even 4 128.2.g.a.113.1 4
32.13 even 8 800.2.ba.b.749.1 4
40.27 even 4 256.2.g.a.225.1 4
40.37 odd 4 256.2.g.b.225.1 4
60.47 odd 4 1152.2.v.a.1009.1 4
80.27 even 4 512.2.g.b.193.1 4
80.37 odd 4 512.2.g.d.193.1 4
80.67 even 4 512.2.g.c.193.1 4
80.77 odd 4 512.2.g.a.193.1 4
160.13 odd 8 800.2.y.a.301.1 4
160.27 even 8 512.2.g.c.321.1 4
160.37 odd 8 512.2.g.a.321.1 4
160.67 even 8 256.2.g.a.33.1 4
160.77 odd 8 32.2.g.a.13.1 yes 4
160.107 even 8 512.2.g.b.321.1 4
160.109 even 8 inner 800.2.ba.a.749.1 4
160.117 odd 8 512.2.g.d.321.1 4
160.147 even 8 128.2.g.a.17.1 4
160.157 odd 8 256.2.g.b.33.1 4
320.77 odd 16 4096.2.a.e.1.4 4
320.147 even 16 4096.2.a.f.1.4 4
320.237 odd 16 4096.2.a.e.1.1 4
320.307 even 16 4096.2.a.f.1.1 4
480.77 even 8 288.2.v.a.109.1 4
480.467 odd 8 1152.2.v.a.145.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.2.g.a.5.1 ✓ 4 5.2 odd 4
32.2.g.a.13.1 yes 4 160.77 odd 8
128.2.g.a.17.1 4 160.147 even 8
128.2.g.a.113.1 4 20.7 even 4
256.2.g.a.33.1 4 160.67 even 8
256.2.g.a.225.1 4 40.27 even 4
256.2.g.b.33.1 4 160.157 odd 8
256.2.g.b.225.1 4 40.37 odd 4
288.2.v.a.37.1 4 15.2 even 4
288.2.v.a.109.1 4 480.77 even 8
512.2.g.a.193.1 4 80.77 odd 4
512.2.g.a.321.1 4 160.37 odd 8
512.2.g.b.193.1 4 80.27 even 4
512.2.g.b.321.1 4 160.107 even 8
512.2.g.c.193.1 4 80.67 even 4
512.2.g.c.321.1 4 160.27 even 8
512.2.g.d.193.1 4 80.37 odd 4
512.2.g.d.321.1 4 160.117 odd 8
800.2.y.a.101.1 4 5.3 odd 4
800.2.y.a.301.1 4 160.13 odd 8
800.2.ba.a.549.1 4 1.1 even 1 trivial
800.2.ba.a.749.1 4 160.109 even 8 inner
800.2.ba.b.549.1 4 5.4 even 2
800.2.ba.b.749.1 4 32.13 even 8
1152.2.v.a.145.1 4 480.467 odd 8
1152.2.v.a.1009.1 4 60.47 odd 4
4096.2.a.e.1.1 4 320.237 odd 16
4096.2.a.e.1.4 4 320.77 odd 16
4096.2.a.f.1.1 4 320.307 even 16
4096.2.a.f.1.4 4 320.147 even 16