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Properties

Label 512.2.g.c.321.1
Level $512$
Weight $2$
Character 512.321
Analytic conductor $4.088$
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] = [512,2,Mod(65,512)] 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("512.65"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(512, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.g (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.08834058349\)
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 321.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 512.321
Dual form 512.2.g.c.193.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.70711 + 0.707107i) q^{3} +(-1.29289 - 3.12132i) q^{5} +(1.00000 - 1.00000i) q^{7} +(0.292893 + 0.292893i) q^{9} +(0.292893 - 0.121320i) q^{11} +(0.707107 - 1.70711i) q^{13} -6.24264i q^{15} +2.82843i q^{17} +(2.29289 - 5.53553i) q^{19} +(2.41421 - 1.00000i) q^{21} +(0.171573 + 0.171573i) q^{23} +(-4.53553 + 4.53553i) q^{25} +(-1.82843 - 4.41421i) q^{27} +(-2.70711 - 1.12132i) q^{29} +4.00000 q^{31} +0.585786 q^{33} +(-4.41421 - 1.82843i) q^{35} +(0.707107 + 1.70711i) q^{37} +(2.41421 - 2.41421i) q^{39} +(5.82843 + 5.82843i) q^{41} +(7.94975 - 3.29289i) q^{43} +(0.535534 - 1.29289i) q^{45} +11.6569i q^{47} +5.00000i q^{49} +(-2.00000 + 4.82843i) q^{51} +(-7.53553 + 3.12132i) q^{53} +(-0.757359 - 0.757359i) q^{55} +(7.82843 - 7.82843i) q^{57} +(2.53553 + 6.12132i) q^{59} +(-0.707107 - 0.292893i) q^{61} +0.585786 q^{63} -6.24264 q^{65} +(3.70711 + 1.53553i) q^{67} +(0.171573 + 0.414214i) q^{69} +(-0.171573 + 0.171573i) q^{71} +(-7.00000 - 7.00000i) q^{73} +(-10.9497 + 4.53553i) q^{75} +(0.171573 - 0.414214i) q^{77} -6.00000i q^{79} -10.0711i q^{81} +(-2.53553 + 6.12132i) q^{83} +(8.82843 - 3.65685i) q^{85} +(-3.82843 - 3.82843i) q^{87} +(2.65685 - 2.65685i) q^{89} +(-1.00000 - 2.41421i) q^{91} +(6.82843 + 2.82843i) q^{93} -20.2426 q^{95} -1.51472 q^{97} +(0.121320 + 0.0502525i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 8 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} + 12 q^{19} + 4 q^{21} + 12 q^{23} - 4 q^{25} + 4 q^{27} - 8 q^{29} + 16 q^{31} + 8 q^{33} - 12 q^{35} + 4 q^{39} + 12 q^{41} + 12 q^{43} - 12 q^{45}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/512\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(511\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\) \(1\)

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
\(3\) 1.70711 + 0.707107i 0.985599 + 0.408248i 0.816497 − 0.577350i \(-0.195913\pi\)
0.169102 + 0.985599i \(0.445913\pi\)
\(4\) 0 0
\(5\) −1.29289 − 3.12132i −0.578199 − 1.39590i −0.894427 − 0.447214i \(-0.852416\pi\)
0.316228 − 0.948683i \(-0.397584\pi\)
\(6\) 0 0
\(7\) 1.00000 − 1.00000i 0.377964 − 0.377964i −0.492403 − 0.870367i \(-0.663881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) 0 0
\(9\) 0.292893 + 0.292893i 0.0976311 + 0.0976311i
\(10\) 0 0
\(11\) 0.292893 − 0.121320i 0.0883106 − 0.0365795i −0.338091 − 0.941113i \(-0.609781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 0.707107 − 1.70711i 0.196116 − 0.473466i −0.794977 − 0.606640i \(-0.792517\pi\)
0.991093 + 0.133174i \(0.0425169\pi\)
\(14\) 0 0
\(15\) − 6.24264i − 1.61184i
\(16\) 0 0
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) 0 0
\(19\) 2.29289 − 5.53553i 0.526026 − 1.26994i −0.408081 − 0.912946i \(-0.633802\pi\)
0.934107 − 0.356993i \(-0.116198\pi\)
\(20\) 0 0
\(21\) 2.41421 − 1.00000i 0.526825 − 0.218218i
\(22\) 0 0
\(23\) 0.171573 + 0.171573i 0.0357754 + 0.0357754i 0.724768 − 0.688993i \(-0.241947\pi\)
−0.688993 + 0.724768i \(0.741947\pi\)
\(24\) 0 0
\(25\) −4.53553 + 4.53553i −0.907107 + 0.907107i
\(26\) 0 0
\(27\) −1.82843 − 4.41421i −0.351881 − 0.849516i
\(28\) 0 0
\(29\) −2.70711 − 1.12132i −0.502697 − 0.208224i 0.116900 − 0.993144i \(-0.462704\pi\)
−0.619598 + 0.784920i \(0.712704\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0.585786 0.101972
\(34\) 0 0
\(35\) −4.41421 − 1.82843i −0.746138 − 0.309061i
\(36\) 0 0
\(37\) 0.707107 + 1.70711i 0.116248 + 0.280647i 0.971285 − 0.237920i \(-0.0764657\pi\)
−0.855037 + 0.518567i \(0.826466\pi\)
\(38\) 0 0
\(39\) 2.41421 − 2.41421i 0.386584 − 0.386584i
\(40\) 0 0
\(41\) 5.82843 + 5.82843i 0.910247 + 0.910247i 0.996291 − 0.0860440i \(-0.0274225\pi\)
−0.0860440 + 0.996291i \(0.527423\pi\)
\(42\) 0 0
\(43\) 7.94975 − 3.29289i 1.21233 − 0.502162i 0.317363 − 0.948304i \(-0.397203\pi\)
0.894962 + 0.446143i \(0.147203\pi\)
\(44\) 0 0
\(45\) 0.535534 − 1.29289i 0.0798327 − 0.192733i
\(46\) 0 0
\(47\) 11.6569i 1.70033i 0.526519 + 0.850163i \(0.323497\pi\)
−0.526519 + 0.850163i \(0.676503\pi\)
\(48\) 0 0
\(49\) 5.00000i 0.714286i
\(50\) 0 0
\(51\) −2.00000 + 4.82843i −0.280056 + 0.676115i
\(52\) 0 0
\(53\) −7.53553 + 3.12132i −1.03509 + 0.428746i −0.834545 − 0.550939i \(-0.814270\pi\)
−0.200540 + 0.979686i \(0.564270\pi\)
\(54\) 0 0
\(55\) −0.757359 − 0.757359i −0.102122 − 0.102122i
\(56\) 0 0
\(57\) 7.82843 − 7.82843i 1.03690 − 1.03690i
\(58\) 0 0
\(59\) 2.53553 + 6.12132i 0.330098 + 0.796928i 0.998584 + 0.0532027i \(0.0169429\pi\)
−0.668485 + 0.743725i \(0.733057\pi\)
\(60\) 0 0
\(61\) −0.707107 − 0.292893i −0.0905357 − 0.0375011i 0.336956 − 0.941520i \(-0.390603\pi\)
−0.427492 + 0.904019i \(0.640603\pi\)
\(62\) 0 0
\(63\) 0.585786 0.0738022
\(64\) 0 0
\(65\) −6.24264 −0.774304
\(66\) 0 0
\(67\) 3.70711 + 1.53553i 0.452895 + 0.187595i 0.597458 − 0.801900i \(-0.296178\pi\)
−0.144563 + 0.989496i \(0.546178\pi\)
\(68\) 0 0
\(69\) 0.171573 + 0.414214i 0.0206549 + 0.0498655i
\(70\) 0 0
\(71\) −0.171573 + 0.171573i −0.0203620 + 0.0203620i −0.717214 − 0.696853i \(-0.754583\pi\)
0.696853 + 0.717214i \(0.254583\pi\)
\(72\) 0 0
\(73\) −7.00000 − 7.00000i −0.819288 − 0.819288i 0.166717 − 0.986005i \(-0.446683\pi\)
−0.986005 + 0.166717i \(0.946683\pi\)
\(74\) 0 0
\(75\) −10.9497 + 4.53553i −1.26437 + 0.523718i
\(76\) 0 0
\(77\) 0.171573 − 0.414214i 0.0195525 − 0.0472040i
\(78\) 0 0
\(79\) − 6.00000i − 0.675053i −0.941316 − 0.337526i \(-0.890410\pi\)
0.941316 − 0.337526i \(-0.109590\pi\)
\(80\) 0 0
\(81\) − 10.0711i − 1.11901i
\(82\) 0 0
\(83\) −2.53553 + 6.12132i −0.278311 + 0.671902i −0.999789 − 0.0205350i \(-0.993463\pi\)
0.721478 + 0.692437i \(0.243463\pi\)
\(84\) 0 0
\(85\) 8.82843 − 3.65685i 0.957577 − 0.396642i
\(86\) 0 0
\(87\) −3.82843 − 3.82843i −0.410450 − 0.410450i
\(88\) 0 0
\(89\) 2.65685 − 2.65685i 0.281626 − 0.281626i −0.552131 − 0.833757i \(-0.686185\pi\)
0.833757 + 0.552131i \(0.186185\pi\)
\(90\) 0 0
\(91\) −1.00000 − 2.41421i −0.104828 − 0.253078i
\(92\) 0 0
\(93\) 6.82843 + 2.82843i 0.708075 + 0.293294i
\(94\) 0 0
\(95\) −20.2426 −2.07685
\(96\) 0 0
\(97\) −1.51472 −0.153796 −0.0768982 − 0.997039i \(-0.524502\pi\)
−0.0768982 + 0.997039i \(0.524502\pi\)
\(98\) 0 0
\(99\) 0.121320 + 0.0502525i 0.0121932 + 0.00505057i
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 512.2.g.c.321.1 4
4.3 odd 2 512.2.g.a.321.1 4
8.3 odd 2 512.2.g.d.321.1 4
8.5 even 2 512.2.g.b.321.1 4
16.3 odd 4 32.2.g.a.13.1 yes 4
16.5 even 4 256.2.g.a.33.1 4
16.11 odd 4 256.2.g.b.33.1 4
16.13 even 4 128.2.g.a.17.1 4
32.3 odd 8 256.2.g.b.225.1 4
32.5 even 8 512.2.g.b.193.1 4
32.11 odd 8 512.2.g.a.193.1 4
32.13 even 8 128.2.g.a.113.1 4
32.19 odd 8 32.2.g.a.5.1 ✓ 4
32.21 even 8 inner 512.2.g.c.193.1 4
32.27 odd 8 512.2.g.d.193.1 4
32.29 even 8 256.2.g.a.225.1 4
48.29 odd 4 1152.2.v.a.145.1 4
48.35 even 4 288.2.v.a.109.1 4
64.11 odd 16 4096.2.a.e.1.1 4
64.21 even 16 4096.2.a.f.1.1 4
64.43 odd 16 4096.2.a.e.1.4 4
64.53 even 16 4096.2.a.f.1.4 4
80.3 even 4 800.2.ba.b.749.1 4
80.19 odd 4 800.2.y.a.301.1 4
80.67 even 4 800.2.ba.a.749.1 4
96.77 odd 8 1152.2.v.a.1009.1 4
96.83 even 8 288.2.v.a.37.1 4
160.19 odd 8 800.2.y.a.101.1 4
160.83 even 8 800.2.ba.a.549.1 4
160.147 even 8 800.2.ba.b.549.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.2.g.a.5.1 ✓ 4 32.19 odd 8
32.2.g.a.13.1 yes 4 16.3 odd 4
128.2.g.a.17.1 4 16.13 even 4
128.2.g.a.113.1 4 32.13 even 8
256.2.g.a.33.1 4 16.5 even 4
256.2.g.a.225.1 4 32.29 even 8
256.2.g.b.33.1 4 16.11 odd 4
256.2.g.b.225.1 4 32.3 odd 8
288.2.v.a.37.1 4 96.83 even 8
288.2.v.a.109.1 4 48.35 even 4
512.2.g.a.193.1 4 32.11 odd 8
512.2.g.a.321.1 4 4.3 odd 2
512.2.g.b.193.1 4 32.5 even 8
512.2.g.b.321.1 4 8.5 even 2
512.2.g.c.193.1 4 32.21 even 8 inner
512.2.g.c.321.1 4 1.1 even 1 trivial
512.2.g.d.193.1 4 32.27 odd 8
512.2.g.d.321.1 4 8.3 odd 2
800.2.y.a.101.1 4 160.19 odd 8
800.2.y.a.301.1 4 80.19 odd 4
800.2.ba.a.549.1 4 160.83 even 8
800.2.ba.a.749.1 4 80.67 even 4
800.2.ba.b.549.1 4 160.147 even 8
800.2.ba.b.749.1 4 80.3 even 4
1152.2.v.a.145.1 4 48.29 odd 4
1152.2.v.a.1009.1 4 96.77 odd 8
4096.2.a.e.1.1 4 64.11 odd 16
4096.2.a.e.1.4 4 64.43 odd 16
4096.2.a.f.1.1 4 64.21 even 16
4096.2.a.f.1.4 4 64.53 even 16