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Properties

Label 32.2.g.a.13.1
Level $32$
Weight $2$
Character 32.13
Analytic conductor $0.256$
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] = [32,2,Mod(5,32)] 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("32.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.255521286468\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 13.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 32.13
Dual form 32.2.g.a.5.1

$q$-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+1.41421i q^{2} +(0.707107 - 1.70711i) q^{3} -2.00000 q^{4} +(-3.12132 + 1.29289i) q^{5} +(2.41421 + 1.00000i) q^{6} +(1.00000 - 1.00000i) q^{7} -2.82843i q^{8} +(-0.292893 - 0.292893i) q^{9} +(-1.82843 - 4.41421i) q^{10} +(0.121320 + 0.292893i) q^{11} +(-1.41421 + 3.41421i) q^{12} +(1.70711 + 0.707107i) q^{13} +(1.41421 + 1.41421i) q^{14} +6.24264i q^{15} +4.00000 q^{16} +2.82843i q^{17} +(0.414214 - 0.414214i) q^{18} +(-5.53553 - 2.29289i) q^{19} +(6.24264 - 2.58579i) q^{20} +(-1.00000 - 2.41421i) q^{21} +(-0.414214 + 0.171573i) q^{22} +(0.171573 + 0.171573i) q^{23} +(-4.82843 - 2.00000i) q^{24} +(4.53553 - 4.53553i) q^{25} +(-1.00000 + 2.41421i) q^{26} +(4.41421 - 1.82843i) q^{27} +(-2.00000 + 2.00000i) q^{28} +(1.12132 - 2.70711i) q^{29} -8.82843 q^{30} -4.00000 q^{31} +5.65685i q^{32} +0.585786 q^{33} -4.00000 q^{34} +(-1.82843 + 4.41421i) q^{35} +(0.585786 + 0.585786i) q^{36} +(1.70711 - 0.707107i) q^{37} +(3.24264 - 7.82843i) q^{38} +(2.41421 - 2.41421i) q^{39} +(3.65685 + 8.82843i) q^{40} +(-5.82843 - 5.82843i) q^{41} +(3.41421 - 1.41421i) q^{42} +(3.29289 + 7.94975i) q^{43} +(-0.242641 - 0.585786i) q^{44} +(1.29289 + 0.535534i) q^{45} +(-0.242641 + 0.242641i) q^{46} -11.6569i q^{47} +(2.82843 - 6.82843i) q^{48} +5.00000i q^{49} +(6.41421 + 6.41421i) q^{50} +(4.82843 + 2.00000i) q^{51} +(-3.41421 - 1.41421i) q^{52} +(3.12132 + 7.53553i) q^{53} +(2.58579 + 6.24264i) q^{54} +(-0.757359 - 0.757359i) q^{55} +(-2.82843 - 2.82843i) q^{56} +(-7.82843 + 7.82843i) q^{57} +(3.82843 + 1.58579i) q^{58} +(-6.12132 + 2.53553i) q^{59} -12.4853i q^{60} +(0.292893 - 0.707107i) q^{61} -5.65685i q^{62} -0.585786 q^{63} -8.00000 q^{64} -6.24264 q^{65} +0.828427i q^{66} +(1.53553 - 3.70711i) q^{67} -5.65685i q^{68} +(0.414214 - 0.171573i) q^{69} +(-6.24264 - 2.58579i) q^{70} +(-0.171573 + 0.171573i) q^{71} +(-0.828427 + 0.828427i) q^{72} +(7.00000 + 7.00000i) q^{73} +(1.00000 + 2.41421i) q^{74} +(-4.53553 - 10.9497i) q^{75} +(11.0711 + 4.58579i) q^{76} +(0.414214 + 0.171573i) q^{77} +(3.41421 + 3.41421i) q^{78} +6.00000i q^{79} +(-12.4853 + 5.17157i) q^{80} -10.0711i q^{81} +(8.24264 - 8.24264i) q^{82} +(6.12132 + 2.53553i) q^{83} +(2.00000 + 4.82843i) q^{84} +(-3.65685 - 8.82843i) q^{85} +(-11.2426 + 4.65685i) q^{86} +(-3.82843 - 3.82843i) q^{87} +(0.828427 - 0.343146i) q^{88} +(-2.65685 + 2.65685i) q^{89} +(-0.757359 + 1.82843i) q^{90} +(2.41421 - 1.00000i) q^{91} +(-0.343146 - 0.343146i) q^{92} +(-2.82843 + 6.82843i) q^{93} +16.4853 q^{94} +20.2426 q^{95} +(9.65685 + 4.00000i) q^{96} -1.51472 q^{97} -7.07107 q^{98} +(0.0502525 - 0.121320i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{7} - 4 q^{9} + 4 q^{10} - 8 q^{11} + 4 q^{13} + 16 q^{16} - 4 q^{18} - 8 q^{19} + 8 q^{20} - 4 q^{21} + 4 q^{22} + 12 q^{23} - 8 q^{24} + 4 q^{25} - 4 q^{26} + 12 q^{27}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0.707107 − 1.70711i 0.408248 − 0.985599i −0.577350 − 0.816497i \(-0.695913\pi\)
0.985599 − 0.169102i \(-0.0540867\pi\)
\(4\) −2.00000 −1.00000
\(5\) −3.12132 + 1.29289i −1.39590 + 0.578199i −0.948683 − 0.316228i \(-0.897584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 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 \(0.163881\pi\)
\(8\) − 2.82843i − 1.00000i
\(9\) −0.292893 − 0.292893i −0.0976311 − 0.0976311i
\(10\) −1.82843 − 4.41421i −0.578199 − 1.39590i
\(11\) 0.121320 + 0.292893i 0.0365795 + 0.0883106i 0.941113 − 0.338091i \(-0.109781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.41421 + 3.41421i −0.408248 + 0.985599i
\(13\) 1.70711 + 0.707107i 0.473466 + 0.196116i 0.606640 − 0.794977i \(-0.292517\pi\)
−0.133174 + 0.991093i \(0.542517\pi\)
\(14\) 1.41421 + 1.41421i 0.377964 + 0.377964i
\(15\) 6.24264i 1.61184i
\(16\) 4.00000 1.00000
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) 0.414214 − 0.414214i 0.0976311 − 0.0976311i
\(19\) −5.53553 − 2.29289i −1.26994 − 0.526026i −0.356993 − 0.934107i \(-0.616198\pi\)
−0.912946 + 0.408081i \(0.866198\pi\)
\(20\) 6.24264 − 2.58579i 1.39590 − 0.578199i
\(21\) −1.00000 − 2.41421i −0.218218 − 0.526825i
\(22\) −0.414214 + 0.171573i −0.0883106 + 0.0365795i
\(23\) 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
\(25\) 4.53553 − 4.53553i 0.907107 − 0.907107i
\(26\) −1.00000 + 2.41421i −0.196116 + 0.473466i
\(27\) 4.41421 − 1.82843i 0.849516 − 0.351881i
\(28\) −2.00000 + 2.00000i −0.377964 + 0.377964i
\(29\) 1.12132 − 2.70711i 0.208224 − 0.502697i −0.784920 − 0.619598i \(-0.787296\pi\)
0.993144 + 0.116900i \(0.0372958\pi\)
\(30\) −8.82843 −1.61184
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0.585786 0.101972
\(34\) −4.00000 −0.685994
\(35\) −1.82843 + 4.41421i −0.309061 + 0.746138i
\(36\) 0.585786 + 0.585786i 0.0976311 + 0.0976311i
\(37\) 1.70711 − 0.707107i 0.280647 − 0.116248i −0.237920 − 0.971285i \(-0.576466\pi\)
0.518567 + 0.855037i \(0.326466\pi\)
\(38\) 3.24264 − 7.82843i 0.526026 − 1.26994i
\(39\) 2.41421 − 2.41421i 0.386584 − 0.386584i
\(40\) 3.65685 + 8.82843i 0.578199 + 1.39590i
\(41\) −5.82843 − 5.82843i −0.910247 − 0.910247i 0.0860440 − 0.996291i \(-0.472577\pi\)
−0.996291 + 0.0860440i \(0.972577\pi\)
\(42\) 3.41421 − 1.41421i 0.526825 − 0.218218i
\(43\) 3.29289 + 7.94975i 0.502162 + 1.21233i 0.948304 + 0.317363i \(0.102797\pi\)
−0.446143 + 0.894962i \(0.647203\pi\)
\(44\) −0.242641 − 0.585786i −0.0365795 − 0.0883106i
\(45\) 1.29289 + 0.535534i 0.192733 + 0.0798327i
\(46\) −0.242641 + 0.242641i −0.0357754 + 0.0357754i
\(47\) − 11.6569i − 1.70033i −0.526519 − 0.850163i \(-0.676503\pi\)
0.526519 − 0.850163i \(-0.323497\pi\)
\(48\) 2.82843 − 6.82843i 0.408248 − 0.985599i
\(49\) 5.00000i 0.714286i
\(50\) 6.41421 + 6.41421i 0.907107 + 0.907107i
\(51\) 4.82843 + 2.00000i 0.676115 + 0.280056i
\(52\) −3.41421 − 1.41421i −0.473466 − 0.196116i
\(53\) 3.12132 + 7.53553i 0.428746 + 1.03509i 0.979686 + 0.200540i \(0.0642696\pi\)
−0.550939 + 0.834545i \(0.685730\pi\)
\(54\) 2.58579 + 6.24264i 0.351881 + 0.849516i
\(55\) −0.757359 − 0.757359i −0.102122 − 0.102122i
\(56\) −2.82843 − 2.82843i −0.377964 − 0.377964i
\(57\) −7.82843 + 7.82843i −1.03690 + 1.03690i
\(58\) 3.82843 + 1.58579i 0.502697 + 0.208224i
\(59\) −6.12132 + 2.53553i −0.796928 + 0.330098i −0.743725 − 0.668485i \(-0.766943\pi\)
−0.0532027 + 0.998584i \(0.516943\pi\)
\(60\) − 12.4853i − 1.61184i
\(61\) 0.292893 − 0.707107i 0.0375011 − 0.0905357i −0.904019 − 0.427492i \(-0.859397\pi\)
0.941520 + 0.336956i \(0.109397\pi\)
\(62\) − 5.65685i − 0.718421i
\(63\) −0.585786 −0.0738022
\(64\) −8.00000 −1.00000
\(65\) −6.24264 −0.774304
\(66\) 0.828427i 0.101972i
\(67\) 1.53553 − 3.70711i 0.187595 − 0.452895i −0.801900 − 0.597458i \(-0.796178\pi\)
0.989496 + 0.144563i \(0.0461775\pi\)
\(68\) − 5.65685i − 0.685994i
\(69\) 0.414214 − 0.171573i 0.0498655 − 0.0206549i
\(70\) −6.24264 − 2.58579i −0.746138 − 0.309061i
\(71\) −0.171573 + 0.171573i −0.0203620 + 0.0203620i −0.717214 − 0.696853i \(-0.754583\pi\)
0.696853 + 0.717214i \(0.254583\pi\)
\(72\) −0.828427 + 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
\(75\) −4.53553 − 10.9497i −0.523718 − 1.26437i
\(76\) 11.0711 + 4.58579i 1.26994 + 0.526026i
\(77\) 0.414214 + 0.171573i 0.0472040 + 0.0195525i
\(78\) 3.41421 + 3.41421i 0.386584 + 0.386584i
\(79\) 6.00000i 0.675053i 0.941316 + 0.337526i \(0.109590\pi\)
−0.941316 + 0.337526i \(0.890410\pi\)
\(80\) −12.4853 + 5.17157i −1.39590 + 0.578199i
\(81\) − 10.0711i − 1.11901i
\(82\) 8.24264 − 8.24264i 0.910247 − 0.910247i
\(83\) 6.12132 + 2.53553i 0.671902 + 0.278311i 0.692437 − 0.721478i \(-0.256537\pi\)
−0.0205350 + 0.999789i \(0.506537\pi\)
\(84\) 2.00000 + 4.82843i 0.218218 + 0.526825i
\(85\) −3.65685 − 8.82843i −0.396642 − 0.957577i
\(86\) −11.2426 + 4.65685i −1.21233 + 0.502162i
\(87\) −3.82843 − 3.82843i −0.410450 − 0.410450i
\(88\) 0.828427 − 0.343146i 0.0883106 − 0.0365795i
\(89\) −2.65685 + 2.65685i −0.281626 + 0.281626i −0.833757 − 0.552131i \(-0.813815\pi\)
0.552131 + 0.833757i \(0.313815\pi\)
\(90\) −0.757359 + 1.82843i −0.0798327 + 0.192733i
\(91\) 2.41421 − 1.00000i 0.253078 − 0.104828i
\(92\) −0.343146 − 0.343146i −0.0357754 − 0.0357754i
\(93\) −2.82843 + 6.82843i −0.293294 + 0.708075i
\(94\) 16.4853 1.70033
\(95\) 20.2426 2.07685
\(96\) 9.65685 + 4.00000i 0.985599 + 0.408248i
\(97\) −1.51472 −0.153796 −0.0768982 − 0.997039i \(-0.524502\pi\)
−0.0768982 + 0.997039i \(0.524502\pi\)
\(98\) −7.07107 −0.714286
\(99\) 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 32.2.g.a.13.1 yes 4
3.2 odd 2 288.2.v.a.109.1 4
4.3 odd 2 128.2.g.a.17.1 4
5.2 odd 4 800.2.ba.a.749.1 4
5.3 odd 4 800.2.ba.b.749.1 4
5.4 even 2 800.2.y.a.301.1 4
8.3 odd 2 256.2.g.a.33.1 4
8.5 even 2 256.2.g.b.33.1 4
12.11 even 2 1152.2.v.a.145.1 4
16.3 odd 4 512.2.g.b.321.1 4
16.5 even 4 512.2.g.a.321.1 4
16.11 odd 4 512.2.g.c.321.1 4
16.13 even 4 512.2.g.d.321.1 4
32.3 odd 8 512.2.g.b.193.1 4
32.5 even 8 inner 32.2.g.a.5.1 ✓ 4
32.11 odd 8 256.2.g.a.225.1 4
32.13 even 8 512.2.g.a.193.1 4
32.19 odd 8 512.2.g.c.193.1 4
32.21 even 8 256.2.g.b.225.1 4
32.27 odd 8 128.2.g.a.113.1 4
32.29 even 8 512.2.g.d.193.1 4
64.5 even 16 4096.2.a.e.1.4 4
64.27 odd 16 4096.2.a.f.1.4 4
64.37 even 16 4096.2.a.e.1.1 4
64.59 odd 16 4096.2.a.f.1.1 4
96.5 odd 8 288.2.v.a.37.1 4
96.59 even 8 1152.2.v.a.1009.1 4
160.37 odd 8 800.2.ba.b.549.1 4
160.69 even 8 800.2.y.a.101.1 4
160.133 odd 8 800.2.ba.a.549.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.2.g.a.5.1 ✓ 4 32.5 even 8 inner
32.2.g.a.13.1 yes 4 1.1 even 1 trivial
128.2.g.a.17.1 4 4.3 odd 2
128.2.g.a.113.1 4 32.27 odd 8
256.2.g.a.33.1 4 8.3 odd 2
256.2.g.a.225.1 4 32.11 odd 8
256.2.g.b.33.1 4 8.5 even 2
256.2.g.b.225.1 4 32.21 even 8
288.2.v.a.37.1 4 96.5 odd 8
288.2.v.a.109.1 4 3.2 odd 2
512.2.g.a.193.1 4 32.13 even 8
512.2.g.a.321.1 4 16.5 even 4
512.2.g.b.193.1 4 32.3 odd 8
512.2.g.b.321.1 4 16.3 odd 4
512.2.g.c.193.1 4 32.19 odd 8
512.2.g.c.321.1 4 16.11 odd 4
512.2.g.d.193.1 4 32.29 even 8
512.2.g.d.321.1 4 16.13 even 4
800.2.y.a.101.1 4 160.69 even 8
800.2.y.a.301.1 4 5.4 even 2
800.2.ba.a.549.1 4 160.133 odd 8
800.2.ba.a.749.1 4 5.2 odd 4
800.2.ba.b.549.1 4 160.37 odd 8
800.2.ba.b.749.1 4 5.3 odd 4
1152.2.v.a.145.1 4 12.11 even 2
1152.2.v.a.1009.1 4 96.59 even 8
4096.2.a.e.1.1 4 64.37 even 16
4096.2.a.e.1.4 4 64.5 even 16
4096.2.a.f.1.1 4 64.59 odd 16
4096.2.a.f.1.4 4 64.27 odd 16