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Properties

Label 272.2.v.a.145.1
Level $272$
Weight $2$
Character 272.145
Analytic conductor $2.172$
Analytic rank $1$
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] = [272,2,Mod(49,272)] 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("272.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(272, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 272.v (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: \(2.17193093498\)
Analytic rank: \(1\)
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: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 145.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 272.145
Dual form 272.2.v.a.257.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 0.414214i) q^{3} +(-1.29289 + 3.12132i) q^{5} +(-1.00000 - 2.41421i) q^{7} +(-1.29289 - 1.29289i) q^{9} +(-4.41421 + 1.82843i) q^{11} -5.41421i q^{13} +(2.58579 - 2.58579i) q^{15} +(-2.82843 + 3.00000i) q^{17} +(-2.58579 + 2.58579i) q^{19} +2.82843i q^{21} +(0.414214 - 0.171573i) q^{23} +(-4.53553 - 4.53553i) q^{25} +(2.00000 + 4.82843i) q^{27} +(-0.878680 + 2.12132i) q^{29} +(-2.41421 - 1.00000i) q^{31} +5.17157 q^{33} +8.82843 q^{35} +(0.121320 + 0.0502525i) q^{37} +(-2.24264 + 5.41421i) q^{39} +(-1.46447 - 3.53553i) q^{41} +(8.24264 + 8.24264i) q^{43} +(5.70711 - 2.36396i) q^{45} -6.82843i q^{47} +(0.121320 - 0.121320i) q^{49} +(4.07107 - 1.82843i) q^{51} +(0.171573 - 0.171573i) q^{53} -16.1421i q^{55} +(3.65685 - 1.51472i) q^{57} +(-9.07107 - 9.07107i) q^{59} +(-0.121320 - 0.292893i) q^{61} +(-1.82843 + 4.41421i) q^{63} +(16.8995 + 7.00000i) q^{65} -2.82843 q^{67} -0.485281 q^{69} +(13.4853 + 5.58579i) q^{71} +(-5.36396 + 12.9497i) q^{73} +(2.65685 + 6.41421i) q^{75} +(8.82843 + 8.82843i) q^{77} +(-4.41421 + 1.82843i) q^{79} -0.171573i q^{81} +(-5.07107 + 5.07107i) q^{83} +(-5.70711 - 12.7071i) q^{85} +(1.75736 - 1.75736i) q^{87} +4.24264i q^{89} +(-13.0711 + 5.41421i) q^{91} +(2.00000 + 2.00000i) q^{93} +(-4.72792 - 11.4142i) q^{95} +(0.121320 - 0.292893i) q^{97} +(8.07107 + 3.34315i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 8 q^{5} - 4 q^{7} - 8 q^{9} - 12 q^{11} + 16 q^{15} - 16 q^{19} - 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 4 q^{31} + 32 q^{33} + 24 q^{35} - 8 q^{37} + 8 q^{39} - 20 q^{41} + 16 q^{43}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{8}\right)\)

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
\(3\) −1.00000 − 0.414214i −0.577350 − 0.239146i 0.0748477 − 0.997195i \(-0.476153\pi\)
−0.652198 + 0.758049i \(0.726153\pi\)
\(4\) 0 0
\(5\) −1.29289 + 3.12132i −0.578199 + 1.39590i 0.316228 + 0.948683i \(0.397584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) −1.00000 − 2.41421i −0.377964 − 0.912487i −0.992347 − 0.123479i \(-0.960595\pi\)
0.614383 − 0.789008i \(-0.289405\pi\)
\(8\) 0 0
\(9\) −1.29289 − 1.29289i −0.430964 − 0.430964i
\(10\) 0 0
\(11\) −4.41421 + 1.82843i −1.33094 + 0.551292i −0.930922 − 0.365218i \(-0.880995\pi\)
−0.400013 + 0.916509i \(0.630995\pi\)
\(12\) 0 0
\(13\) − 5.41421i − 1.50163i −0.660511 − 0.750816i \(-0.729660\pi\)
0.660511 − 0.750816i \(-0.270340\pi\)
\(14\) 0 0
\(15\) 2.58579 − 2.58579i 0.667647 − 0.667647i
\(16\) 0 0
\(17\) −2.82843 + 3.00000i −0.685994 + 0.727607i
\(18\) 0 0
\(19\) −2.58579 + 2.58579i −0.593220 + 0.593220i −0.938500 − 0.345280i \(-0.887784\pi\)
0.345280 + 0.938500i \(0.387784\pi\)
\(20\) 0 0
\(21\) 2.82843i 0.617213i
\(22\) 0 0
\(23\) 0.414214 − 0.171573i 0.0863695 − 0.0357754i −0.339080 − 0.940757i \(-0.610116\pi\)
0.425450 + 0.904982i \(0.360116\pi\)
\(24\) 0 0
\(25\) −4.53553 − 4.53553i −0.907107 − 0.907107i
\(26\) 0 0
\(27\) 2.00000 + 4.82843i 0.384900 + 0.929231i
\(28\) 0 0
\(29\) −0.878680 + 2.12132i −0.163167 + 0.393919i −0.984224 − 0.176926i \(-0.943385\pi\)
0.821057 + 0.570846i \(0.193385\pi\)
\(30\) 0 0
\(31\) −2.41421 − 1.00000i −0.433606 − 0.179605i 0.155195 − 0.987884i \(-0.450400\pi\)
−0.588800 + 0.808279i \(0.700400\pi\)
\(32\) 0 0
\(33\) 5.17157 0.900255
\(34\) 0 0
\(35\) 8.82843 1.49228
\(36\) 0 0
\(37\) 0.121320 + 0.0502525i 0.0199449 + 0.00826147i 0.392634 − 0.919695i \(-0.371564\pi\)
−0.372689 + 0.927956i \(0.621564\pi\)
\(38\) 0 0
\(39\) −2.24264 + 5.41421i −0.359110 + 0.866968i
\(40\) 0 0
\(41\) −1.46447 − 3.53553i −0.228711 − 0.552158i 0.767310 − 0.641277i \(-0.221595\pi\)
−0.996021 + 0.0891190i \(0.971595\pi\)
\(42\) 0 0
\(43\) 8.24264 + 8.24264i 1.25699 + 1.25699i 0.952522 + 0.304469i \(0.0984788\pi\)
0.304469 + 0.952522i \(0.401521\pi\)
\(44\) 0 0
\(45\) 5.70711 − 2.36396i 0.850765 − 0.352399i
\(46\) 0 0
\(47\) − 6.82843i − 0.996028i −0.867169 − 0.498014i \(-0.834063\pi\)
0.867169 − 0.498014i \(-0.165937\pi\)
\(48\) 0 0
\(49\) 0.121320 − 0.121320i 0.0173315 − 0.0173315i
\(50\) 0 0
\(51\) 4.07107 − 1.82843i 0.570064 − 0.256031i
\(52\) 0 0
\(53\) 0.171573 − 0.171573i 0.0235673 − 0.0235673i −0.695225 − 0.718792i \(-0.744695\pi\)
0.718792 + 0.695225i \(0.244695\pi\)
\(54\) 0 0
\(55\) − 16.1421i − 2.17661i
\(56\) 0 0
\(57\) 3.65685 − 1.51472i 0.484362 − 0.200629i
\(58\) 0 0
\(59\) −9.07107 − 9.07107i −1.18095 − 1.18095i −0.979498 − 0.201455i \(-0.935433\pi\)
−0.201455 − 0.979498i \(-0.564567\pi\)
\(60\) 0 0
\(61\) −0.121320 − 0.292893i −0.0155335 − 0.0375011i 0.915922 − 0.401355i \(-0.131461\pi\)
−0.931456 + 0.363854i \(0.881461\pi\)
\(62\) 0 0
\(63\) −1.82843 + 4.41421i −0.230360 + 0.556139i
\(64\) 0 0
\(65\) 16.8995 + 7.00000i 2.09612 + 0.868243i
\(66\) 0 0
\(67\) −2.82843 −0.345547 −0.172774 − 0.984962i \(-0.555273\pi\)
−0.172774 + 0.984962i \(0.555273\pi\)
\(68\) 0 0
\(69\) −0.485281 −0.0584210
\(70\) 0 0
\(71\) 13.4853 + 5.58579i 1.60041 + 0.662911i 0.991473 − 0.130309i \(-0.0415970\pi\)
0.608935 + 0.793220i \(0.291597\pi\)
\(72\) 0 0
\(73\) −5.36396 + 12.9497i −0.627804 + 1.51565i 0.214541 + 0.976715i \(0.431174\pi\)
−0.842345 + 0.538938i \(0.818826\pi\)
\(74\) 0 0
\(75\) 2.65685 + 6.41421i 0.306787 + 0.740650i
\(76\) 0 0
\(77\) 8.82843 + 8.82843i 1.00609 + 1.00609i
\(78\) 0 0
\(79\) −4.41421 + 1.82843i −0.496638 + 0.205714i −0.616920 − 0.787026i \(-0.711620\pi\)
0.120283 + 0.992740i \(0.461620\pi\)
\(80\) 0 0
\(81\) − 0.171573i − 0.0190637i
\(82\) 0 0
\(83\) −5.07107 + 5.07107i −0.556622 + 0.556622i −0.928344 − 0.371722i \(-0.878767\pi\)
0.371722 + 0.928344i \(0.378767\pi\)
\(84\) 0 0
\(85\) −5.70711 − 12.7071i −0.619023 − 1.37828i
\(86\) 0 0
\(87\) 1.75736 − 1.75736i 0.188409 − 0.188409i
\(88\) 0 0
\(89\) 4.24264i 0.449719i 0.974391 + 0.224860i \(0.0721923\pi\)
−0.974391 + 0.224860i \(0.927808\pi\)
\(90\) 0 0
\(91\) −13.0711 + 5.41421i −1.37022 + 0.567564i
\(92\) 0 0
\(93\) 2.00000 + 2.00000i 0.207390 + 0.207390i
\(94\) 0 0
\(95\) −4.72792 − 11.4142i −0.485075 − 1.17107i
\(96\) 0 0
\(97\) 0.121320 − 0.292893i 0.0123182 − 0.0297388i −0.917601 − 0.397503i \(-0.869877\pi\)
0.929919 + 0.367764i \(0.119877\pi\)
\(98\) 0 0
\(99\) 8.07107 + 3.34315i 0.811173 + 0.335999i
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 272.2.v.a.145.1 4
4.3 odd 2 136.2.n.b.9.1 ✓ 4
12.11 even 2 1224.2.bq.b.145.1 4
17.2 even 8 inner 272.2.v.a.257.1 4
17.6 odd 16 4624.2.a.bo.1.2 4
17.11 odd 16 4624.2.a.bo.1.3 4
68.7 even 16 2312.2.b.i.577.2 4
68.11 even 16 2312.2.a.t.1.2 4
68.19 odd 8 136.2.n.b.121.1 yes 4
68.23 even 16 2312.2.a.t.1.3 4
68.27 even 16 2312.2.b.i.577.3 4
204.155 even 8 1224.2.bq.b.937.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.2.n.b.9.1 ✓ 4 4.3 odd 2
136.2.n.b.121.1 yes 4 68.19 odd 8
272.2.v.a.145.1 4 1.1 even 1 trivial
272.2.v.a.257.1 4 17.2 even 8 inner
1224.2.bq.b.145.1 4 12.11 even 2
1224.2.bq.b.937.1 4 204.155 even 8
2312.2.a.t.1.2 4 68.11 even 16
2312.2.a.t.1.3 4 68.23 even 16
2312.2.b.i.577.2 4 68.7 even 16
2312.2.b.i.577.3 4 68.27 even 16
4624.2.a.bo.1.2 4 17.6 odd 16
4624.2.a.bo.1.3 4 17.11 odd 16