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Properties

Label 270.2.i.b.199.1
Level $270$
Weight $2$
Character 270.199
Analytic conductor $2.156$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [270,2,Mod(19,270)] 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("270.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 199.1
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 270.199
Dual form 270.2.i.b.19.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.917738 - 2.03906i) q^{5} +(0.389270 - 0.224745i) q^{7} +1.00000i q^{8} +(0.224745 + 2.22474i) q^{10} +(-1.72474 - 2.98735i) q^{11} +(-2.12132 - 1.22474i) q^{13} +(-0.224745 + 0.389270i) q^{14} +(-0.500000 - 0.866025i) q^{16} -5.89898i q^{17} +5.44949 q^{19} +(-1.30701 - 1.81431i) q^{20} +(2.98735 + 1.72474i) q^{22} +(5.97469 + 3.44949i) q^{23} +(-3.31552 - 3.74264i) q^{25} +2.44949 q^{26} -0.449490i q^{28} +(3.00000 + 5.19615i) q^{29} +(-0.775255 + 1.34278i) q^{31} +(0.866025 + 0.500000i) q^{32} +(2.94949 + 5.10867i) q^{34} +(-0.101021 - 1.00000i) q^{35} -8.00000i q^{37} +(-4.71940 + 2.72474i) q^{38} +(2.03906 + 0.917738i) q^{40} +(-0.500000 + 0.866025i) q^{41} +(2.20881 - 1.27526i) q^{43} -3.44949 q^{44} -6.89898 q^{46} +(-3.85337 + 2.22474i) q^{47} +(-3.39898 + 5.88721i) q^{49} +(4.74264 + 1.58346i) q^{50} +(-2.12132 + 1.22474i) q^{52} -3.55051i q^{53} +(-7.67423 + 0.775255i) q^{55} +(0.224745 + 0.389270i) q^{56} +(-5.19615 - 3.00000i) q^{58} +(-6.62372 + 11.4726i) q^{59} +(-2.22474 - 3.85337i) q^{61} -1.55051i q^{62} -1.00000 q^{64} +(-4.44414 + 3.20150i) q^{65} +(3.94086 + 2.27526i) q^{67} +(-5.10867 - 2.94949i) q^{68} +(0.587486 + 0.815515i) q^{70} +2.44949 q^{71} +14.7980i q^{73} +(4.00000 + 6.92820i) q^{74} +(2.72474 - 4.71940i) q^{76} +(-1.34278 - 0.775255i) q^{77} +(3.67423 + 6.36396i) q^{79} +(-2.22474 + 0.224745i) q^{80} -1.00000i q^{82} +(3.46410 - 2.00000i) q^{83} +(-12.0284 - 5.41372i) q^{85} +(-1.27526 + 2.20881i) q^{86} +(2.98735 - 1.72474i) q^{88} +3.10102 q^{89} -1.10102 q^{91} +(5.97469 - 3.44949i) q^{92} +(2.22474 - 3.85337i) q^{94} +(5.00120 - 11.1118i) q^{95} +(-11.2583 + 6.50000i) q^{97} -6.79796i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{5} - 8 q^{10} - 4 q^{11} + 8 q^{14} - 4 q^{16} + 24 q^{19} - 4 q^{20} + 24 q^{29} - 16 q^{31} + 4 q^{34} - 40 q^{35} - 4 q^{40} - 4 q^{41} - 8 q^{44} - 16 q^{46} + 12 q^{49} + 4 q^{50}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 0.917738 − 2.03906i 0.410425 − 0.911894i
\(6\) 0 0
\(7\) 0.389270 − 0.224745i 0.147130 − 0.0849456i −0.424628 − 0.905368i \(-0.639595\pi\)
0.571758 + 0.820422i \(0.306262\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0.224745 + 2.22474i 0.0710706 + 0.703526i
\(11\) −1.72474 − 2.98735i −0.520030 − 0.900719i −0.999729 − 0.0232854i \(-0.992587\pi\)
0.479699 − 0.877433i \(-0.340746\pi\)
\(12\) 0 0
\(13\) −2.12132 − 1.22474i −0.588348 − 0.339683i 0.176096 − 0.984373i \(-0.443653\pi\)
−0.764444 + 0.644690i \(0.776986\pi\)
\(14\) −0.224745 + 0.389270i −0.0600656 + 0.104037i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) − 5.89898i − 1.43071i −0.698760 − 0.715356i \(-0.746264\pi\)
0.698760 − 0.715356i \(-0.253736\pi\)
\(18\) 0 0
\(19\) 5.44949 1.25020 0.625099 − 0.780545i \(-0.285058\pi\)
0.625099 + 0.780545i \(0.285058\pi\)
\(20\) −1.30701 − 1.81431i −0.292256 − 0.405693i
\(21\) 0 0
\(22\) 2.98735 + 1.72474i 0.636904 + 0.367717i
\(23\) 5.97469 + 3.44949i 1.24581 + 0.719268i 0.970271 − 0.242022i \(-0.0778105\pi\)
0.275538 + 0.961290i \(0.411144\pi\)
\(24\) 0 0
\(25\) −3.31552 − 3.74264i −0.663103 − 0.748528i
\(26\) 2.44949 0.480384
\(27\) 0 0
\(28\) − 0.449490i − 0.0849456i
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −0.775255 + 1.34278i −0.139240 + 0.241171i −0.927209 − 0.374544i \(-0.877799\pi\)
0.787969 + 0.615715i \(0.211133\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 2.94949 + 5.10867i 0.505833 + 0.876129i
\(35\) −0.101021 − 1.00000i −0.0170756 − 0.169031i
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) −4.71940 + 2.72474i −0.765587 + 0.442012i
\(39\) 0 0
\(40\) 2.03906 + 0.917738i 0.322403 + 0.145107i
\(41\) −0.500000 + 0.866025i −0.0780869 + 0.135250i −0.902424 − 0.430848i \(-0.858214\pi\)
0.824338 + 0.566099i \(0.191548\pi\)
\(42\) 0 0
\(43\) 2.20881 − 1.27526i 0.336840 − 0.194475i −0.322034 − 0.946728i \(-0.604366\pi\)
0.658874 + 0.752254i \(0.271033\pi\)
\(44\) −3.44949 −0.520030
\(45\) 0 0
\(46\) −6.89898 −1.01720
\(47\) −3.85337 + 2.22474i −0.562072 + 0.324512i −0.753977 − 0.656901i \(-0.771867\pi\)
0.191905 + 0.981414i \(0.438534\pi\)
\(48\) 0 0
\(49\) −3.39898 + 5.88721i −0.485568 + 0.841029i
\(50\) 4.74264 + 1.58346i 0.670711 + 0.223936i
\(51\) 0 0
\(52\) −2.12132 + 1.22474i −0.294174 + 0.169842i
\(53\) − 3.55051i − 0.487700i −0.969813 − 0.243850i \(-0.921590\pi\)
0.969813 − 0.243850i \(-0.0784105\pi\)
\(54\) 0 0
\(55\) −7.67423 + 0.775255i −1.03479 + 0.104535i
\(56\) 0.224745 + 0.389270i 0.0300328 + 0.0520183i
\(57\) 0 0
\(58\) −5.19615 − 3.00000i −0.682288 − 0.393919i
\(59\) −6.62372 + 11.4726i −0.862335 + 1.49361i 0.00733331 + 0.999973i \(0.497666\pi\)
−0.869669 + 0.493636i \(0.835668\pi\)
\(60\) 0 0
\(61\) −2.22474 − 3.85337i −0.284849 − 0.493374i 0.687723 − 0.725973i \(-0.258610\pi\)
−0.972573 + 0.232599i \(0.925277\pi\)
\(62\) − 1.55051i − 0.196915i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −4.44414 + 3.20150i −0.551228 + 0.397097i
\(66\) 0 0
\(67\) 3.94086 + 2.27526i 0.481452 + 0.277967i 0.721022 − 0.692913i \(-0.243673\pi\)
−0.239569 + 0.970879i \(0.577006\pi\)
\(68\) −5.10867 − 2.94949i −0.619517 − 0.357678i
\(69\) 0 0
\(70\) 0.587486 + 0.815515i 0.0702180 + 0.0974727i
\(71\) 2.44949 0.290701 0.145350 − 0.989380i \(-0.453569\pi\)
0.145350 + 0.989380i \(0.453569\pi\)
\(72\) 0 0
\(73\) 14.7980i 1.73197i 0.500070 + 0.865985i \(0.333308\pi\)
−0.500070 + 0.865985i \(0.666692\pi\)
\(74\) 4.00000 + 6.92820i 0.464991 + 0.805387i
\(75\) 0 0
\(76\) 2.72474 − 4.71940i 0.312550 − 0.541352i
\(77\) −1.34278 − 0.775255i −0.153024 − 0.0883485i
\(78\) 0 0
\(79\) 3.67423 + 6.36396i 0.413384 + 0.716002i 0.995257 − 0.0972777i \(-0.0310135\pi\)
−0.581874 + 0.813279i \(0.697680\pi\)
\(80\) −2.22474 + 0.224745i −0.248734 + 0.0251272i
\(81\) 0 0
\(82\) − 1.00000i − 0.110432i
\(83\) 3.46410 − 2.00000i 0.380235 − 0.219529i −0.297686 − 0.954664i \(-0.596215\pi\)
0.677920 + 0.735135i \(0.262881\pi\)
\(84\) 0 0
\(85\) −12.0284 − 5.41372i −1.30466 − 0.587200i
\(86\) −1.27526 + 2.20881i −0.137514 + 0.238182i
\(87\) 0 0
\(88\) 2.98735 − 1.72474i 0.318452 − 0.183858i
\(89\) 3.10102 0.328708 0.164354 − 0.986401i \(-0.447446\pi\)
0.164354 + 0.986401i \(0.447446\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) 5.97469 − 3.44949i 0.622905 − 0.359634i
\(93\) 0 0
\(94\) 2.22474 − 3.85337i 0.229465 − 0.397445i
\(95\) 5.00120 − 11.1118i 0.513112 − 1.14005i
\(96\) 0 0
\(97\) −11.2583 + 6.50000i −1.14311 + 0.659975i −0.947199 − 0.320647i \(-0.896100\pi\)
−0.195911 + 0.980622i \(0.562766\pi\)
\(98\) − 6.79796i − 0.686698i
\(99\) 0 0
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 270.2.i.b.199.1 8
3.2 odd 2 90.2.i.b.49.3 yes 8
4.3 odd 2 2160.2.by.d.1009.2 8
5.2 odd 4 1350.2.e.j.901.2 4
5.3 odd 4 1350.2.e.m.901.1 4
5.4 even 2 inner 270.2.i.b.199.4 8
9.2 odd 6 90.2.i.b.79.2 yes 8
9.4 even 3 810.2.c.e.649.1 4
9.5 odd 6 810.2.c.f.649.4 4
9.7 even 3 inner 270.2.i.b.19.4 8
12.11 even 2 720.2.by.c.49.2 8
15.2 even 4 450.2.e.n.301.1 4
15.8 even 4 450.2.e.k.301.2 4
15.14 odd 2 90.2.i.b.49.2 ✓ 8
20.19 odd 2 2160.2.by.d.1009.3 8
36.7 odd 6 2160.2.by.d.289.3 8
36.11 even 6 720.2.by.c.529.3 8
45.2 even 12 450.2.e.n.151.1 4
45.4 even 6 810.2.c.e.649.3 4
45.7 odd 12 1350.2.e.j.451.2 4
45.13 odd 12 4050.2.a.bm.1.2 2
45.14 odd 6 810.2.c.f.649.2 4
45.22 odd 12 4050.2.a.bz.1.1 2
45.23 even 12 4050.2.a.bs.1.2 2
45.29 odd 6 90.2.i.b.79.3 yes 8
45.32 even 12 4050.2.a.bq.1.1 2
45.34 even 6 inner 270.2.i.b.19.1 8
45.38 even 12 450.2.e.k.151.2 4
45.43 odd 12 1350.2.e.m.451.1 4
60.59 even 2 720.2.by.c.49.3 8
180.79 odd 6 2160.2.by.d.289.2 8
180.119 even 6 720.2.by.c.529.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.b.49.2 ✓ 8 15.14 odd 2
90.2.i.b.49.3 yes 8 3.2 odd 2
90.2.i.b.79.2 yes 8 9.2 odd 6
90.2.i.b.79.3 yes 8 45.29 odd 6
270.2.i.b.19.1 8 45.34 even 6 inner
270.2.i.b.19.4 8 9.7 even 3 inner
270.2.i.b.199.1 8 1.1 even 1 trivial
270.2.i.b.199.4 8 5.4 even 2 inner
450.2.e.k.151.2 4 45.38 even 12
450.2.e.k.301.2 4 15.8 even 4
450.2.e.n.151.1 4 45.2 even 12
450.2.e.n.301.1 4 15.2 even 4
720.2.by.c.49.2 8 12.11 even 2
720.2.by.c.49.3 8 60.59 even 2
720.2.by.c.529.2 8 180.119 even 6
720.2.by.c.529.3 8 36.11 even 6
810.2.c.e.649.1 4 9.4 even 3
810.2.c.e.649.3 4 45.4 even 6
810.2.c.f.649.2 4 45.14 odd 6
810.2.c.f.649.4 4 9.5 odd 6
1350.2.e.j.451.2 4 45.7 odd 12
1350.2.e.j.901.2 4 5.2 odd 4
1350.2.e.m.451.1 4 45.43 odd 12
1350.2.e.m.901.1 4 5.3 odd 4
2160.2.by.d.289.2 8 180.79 odd 6
2160.2.by.d.289.3 8 36.7 odd 6
2160.2.by.d.1009.2 8 4.3 odd 2
2160.2.by.d.1009.3 8 20.19 odd 2
4050.2.a.bm.1.2 2 45.13 odd 12
4050.2.a.bq.1.1 2 45.32 even 12
4050.2.a.bs.1.2 2 45.23 even 12
4050.2.a.bz.1.1 2 45.22 odd 12