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Properties

Label 270.2.i.b.19.3
Level $270$
Weight $2$
Character 270.19
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 19.3
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 270.19
Dual form 270.2.i.b.199.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-2.03906 + 0.917738i) q^{5} +(3.85337 + 2.22474i) q^{7} +1.00000i q^{8} +(-2.22474 - 0.224745i) q^{10} +(0.724745 - 1.25529i) q^{11} +(-2.12132 + 1.22474i) q^{13} +(2.22474 + 3.85337i) q^{14} +(-0.500000 + 0.866025i) q^{16} +3.89898i q^{17} +0.550510 q^{19} +(-1.81431 - 1.30701i) q^{20} +(1.25529 - 0.724745i) q^{22} +(2.51059 - 1.44949i) q^{23} +(3.31552 - 3.74264i) q^{25} -2.44949 q^{26} +4.44949i q^{28} +(3.00000 - 5.19615i) q^{29} +(-3.22474 - 5.58542i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(-1.94949 + 3.37662i) q^{34} +(-9.89898 - 1.00000i) q^{35} -8.00000i q^{37} +(0.476756 + 0.275255i) q^{38} +(-0.917738 - 2.03906i) q^{40} +(-0.500000 - 0.866025i) q^{41} +(-6.45145 - 3.72474i) q^{43} +1.44949 q^{44} +2.89898 q^{46} +(-0.389270 - 0.224745i) q^{47} +(6.39898 + 11.0834i) q^{49} +(4.74264 - 1.58346i) q^{50} +(-2.12132 - 1.22474i) q^{52} -8.44949i q^{53} +(-0.325765 + 3.22474i) q^{55} +(-2.22474 + 3.85337i) q^{56} +(5.19615 - 3.00000i) q^{58} +(5.62372 + 9.74058i) q^{59} +(0.224745 - 0.389270i) q^{61} -6.44949i q^{62} -1.00000 q^{64} +(3.20150 - 4.44414i) q^{65} +(-8.18350 + 4.72474i) q^{67} +(-3.37662 + 1.94949i) q^{68} +(-8.07277 - 5.81552i) q^{70} -2.44949 q^{71} -4.79796i q^{73} +(4.00000 - 6.92820i) q^{74} +(0.275255 + 0.476756i) q^{76} +(5.58542 - 3.22474i) q^{77} +(-3.67423 + 6.36396i) q^{79} +(0.224745 - 2.22474i) q^{80} -1.00000i q^{82} +(-3.46410 - 2.00000i) q^{83} +(-3.57824 - 7.95025i) q^{85} +(-3.72474 - 6.45145i) q^{86} +(1.25529 + 0.724745i) q^{88} +12.8990 q^{89} -10.8990 q^{91} +(2.51059 + 1.44949i) q^{92} +(-0.224745 - 0.389270i) q^{94} +(-1.12252 + 0.505224i) q^{95} +(11.2583 + 6.50000i) q^{97} +12.7980i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{5} - 8 q^{10} - 4 q^{11} + 8 q^{14} - 4 q^{16} + 24 q^{19} - 4 q^{20} + 24 q^{29} - 16 q^{31} + 4 q^{34} - 40 q^{35} - 4 q^{40} - 4 q^{41} - 8 q^{44} - 16 q^{46} + 12 q^{49} + 4 q^{50}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(-1\)

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
\(5\) −2.03906 + 0.917738i −0.911894 + 0.410425i
\(6\) 0 0
\(7\) 3.85337 + 2.22474i 1.45644 + 0.840875i 0.998834 − 0.0482818i \(-0.0153745\pi\)
0.457604 + 0.889156i \(0.348708\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −2.22474 − 0.224745i −0.703526 − 0.0710706i
\(11\) 0.724745 − 1.25529i 0.218519 − 0.378486i −0.735837 − 0.677159i \(-0.763211\pi\)
0.954355 + 0.298674i \(0.0965442\pi\)
\(12\) 0 0
\(13\) −2.12132 + 1.22474i −0.588348 + 0.339683i −0.764444 − 0.644690i \(-0.776986\pi\)
0.176096 + 0.984373i \(0.443653\pi\)
\(14\) 2.22474 + 3.85337i 0.594588 + 1.02986i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 3.89898i 0.945641i 0.881159 + 0.472821i \(0.156764\pi\)
−0.881159 + 0.472821i \(0.843236\pi\)
\(18\) 0 0
\(19\) 0.550510 0.126296 0.0631479 − 0.998004i \(-0.479886\pi\)
0.0631479 + 0.998004i \(0.479886\pi\)
\(20\) −1.81431 − 1.30701i −0.405693 − 0.292256i
\(21\) 0 0
\(22\) 1.25529 − 0.724745i 0.267630 − 0.154516i
\(23\) 2.51059 − 1.44949i 0.523494 − 0.302240i −0.214869 − 0.976643i \(-0.568932\pi\)
0.738363 + 0.674403i \(0.235599\pi\)
\(24\) 0 0
\(25\) 3.31552 − 3.74264i 0.663103 − 0.748528i
\(26\) −2.44949 −0.480384
\(27\) 0 0
\(28\) 4.44949i 0.840875i
\(29\) 3.00000 − 5.19615i 0.557086 − 0.964901i −0.440652 − 0.897678i \(-0.645253\pi\)
0.997738 − 0.0672232i \(-0.0214140\pi\)
\(30\) 0 0
\(31\) −3.22474 − 5.58542i −0.579181 − 1.00317i −0.995573 − 0.0939863i \(-0.970039\pi\)
0.416392 − 0.909185i \(-0.363294\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) −1.94949 + 3.37662i −0.334335 + 0.579085i
\(35\) −9.89898 − 1.00000i −1.67323 − 0.169031i
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 0.476756 + 0.275255i 0.0773400 + 0.0446523i
\(39\) 0 0
\(40\) −0.917738 − 2.03906i −0.145107 − 0.322403i
\(41\) −0.500000 − 0.866025i −0.0780869 − 0.135250i 0.824338 − 0.566099i \(-0.191548\pi\)
−0.902424 + 0.430848i \(0.858214\pi\)
\(42\) 0 0
\(43\) −6.45145 − 3.72474i −0.983836 − 0.568018i −0.0804103 − 0.996762i \(-0.525623\pi\)
−0.903426 + 0.428744i \(0.858956\pi\)
\(44\) 1.44949 0.218519
\(45\) 0 0
\(46\) 2.89898 0.427431
\(47\) −0.389270 − 0.224745i −0.0567808 − 0.0327824i 0.471341 − 0.881951i \(-0.343770\pi\)
−0.528122 + 0.849169i \(0.677104\pi\)
\(48\) 0 0
\(49\) 6.39898 + 11.0834i 0.914140 + 1.58334i
\(50\) 4.74264 − 1.58346i 0.670711 − 0.223936i
\(51\) 0 0
\(52\) −2.12132 − 1.22474i −0.294174 − 0.169842i
\(53\) − 8.44949i − 1.16063i −0.814393 − 0.580313i \(-0.802930\pi\)
0.814393 − 0.580313i \(-0.197070\pi\)
\(54\) 0 0
\(55\) −0.325765 + 3.22474i −0.0439262 + 0.434825i
\(56\) −2.22474 + 3.85337i −0.297294 + 0.514928i
\(57\) 0 0
\(58\) 5.19615 − 3.00000i 0.682288 − 0.393919i
\(59\) 5.62372 + 9.74058i 0.732147 + 1.26812i 0.955964 + 0.293484i \(0.0948147\pi\)
−0.223817 + 0.974631i \(0.571852\pi\)
\(60\) 0 0
\(61\) 0.224745 − 0.389270i 0.0287756 − 0.0498409i −0.851279 − 0.524713i \(-0.824173\pi\)
0.880055 + 0.474873i \(0.157506\pi\)
\(62\) − 6.44949i − 0.819086i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.20150 − 4.44414i 0.397097 − 0.551228i
\(66\) 0 0
\(67\) −8.18350 + 4.72474i −0.999773 + 0.577219i −0.908181 − 0.418577i \(-0.862529\pi\)
−0.0915922 + 0.995797i \(0.529196\pi\)
\(68\) −3.37662 + 1.94949i −0.409475 + 0.236410i
\(69\) 0 0
\(70\) −8.07277 − 5.81552i −0.964880 − 0.695087i
\(71\) −2.44949 −0.290701 −0.145350 − 0.989380i \(-0.546431\pi\)
−0.145350 + 0.989380i \(0.546431\pi\)
\(72\) 0 0
\(73\) − 4.79796i − 0.561559i −0.959772 − 0.280779i \(-0.909407\pi\)
0.959772 − 0.280779i \(-0.0905929\pi\)
\(74\) 4.00000 − 6.92820i 0.464991 − 0.805387i
\(75\) 0 0
\(76\) 0.275255 + 0.476756i 0.0315739 + 0.0546876i
\(77\) 5.58542 − 3.22474i 0.636518 − 0.367494i
\(78\) 0 0
\(79\) −3.67423 + 6.36396i −0.413384 + 0.716002i −0.995257 − 0.0972777i \(-0.968987\pi\)
0.581874 + 0.813279i \(0.302320\pi\)
\(80\) 0.224745 − 2.22474i 0.0251272 − 0.248734i
\(81\) 0 0
\(82\) − 1.00000i − 0.110432i
\(83\) −3.46410 − 2.00000i −0.380235 − 0.219529i 0.297686 − 0.954664i \(-0.403785\pi\)
−0.677920 + 0.735135i \(0.737119\pi\)
\(84\) 0 0
\(85\) −3.57824 − 7.95025i −0.388115 − 0.862325i
\(86\) −3.72474 − 6.45145i −0.401650 − 0.695677i
\(87\) 0 0
\(88\) 1.25529 + 0.724745i 0.133815 + 0.0772581i
\(89\) 12.8990 1.36729 0.683645 − 0.729815i \(-0.260394\pi\)
0.683645 + 0.729815i \(0.260394\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) 2.51059 + 1.44949i 0.261747 + 0.151120i
\(93\) 0 0
\(94\) −0.224745 − 0.389270i −0.0231807 − 0.0401501i
\(95\) −1.12252 + 0.505224i −0.115168 + 0.0518349i
\(96\) 0 0
\(97\) 11.2583 + 6.50000i 1.14311 + 0.659975i 0.947199 − 0.320647i \(-0.103900\pi\)
0.195911 + 0.980622i \(0.437234\pi\)
\(98\) 12.7980i 1.29279i
\(99\) 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.19.3 8
3.2 odd 2 90.2.i.b.79.1 yes 8
4.3 odd 2 2160.2.by.d.289.1 8
5.2 odd 4 1350.2.e.j.451.1 4
5.3 odd 4 1350.2.e.m.451.2 4
5.4 even 2 inner 270.2.i.b.19.2 8
9.2 odd 6 810.2.c.f.649.3 4
9.4 even 3 inner 270.2.i.b.199.2 8
9.5 odd 6 90.2.i.b.49.4 yes 8
9.7 even 3 810.2.c.e.649.2 4
12.11 even 2 720.2.by.c.529.4 8
15.2 even 4 450.2.e.n.151.2 4
15.8 even 4 450.2.e.k.151.1 4
15.14 odd 2 90.2.i.b.79.4 yes 8
20.19 odd 2 2160.2.by.d.289.4 8
36.23 even 6 720.2.by.c.49.1 8
36.31 odd 6 2160.2.by.d.1009.4 8
45.2 even 12 4050.2.a.bq.1.2 2
45.4 even 6 inner 270.2.i.b.199.3 8
45.7 odd 12 4050.2.a.bz.1.2 2
45.13 odd 12 1350.2.e.m.901.2 4
45.14 odd 6 90.2.i.b.49.1 ✓ 8
45.22 odd 12 1350.2.e.j.901.1 4
45.23 even 12 450.2.e.k.301.1 4
45.29 odd 6 810.2.c.f.649.1 4
45.32 even 12 450.2.e.n.301.2 4
45.34 even 6 810.2.c.e.649.4 4
45.38 even 12 4050.2.a.bs.1.1 2
45.43 odd 12 4050.2.a.bm.1.1 2
60.59 even 2 720.2.by.c.529.1 8
180.59 even 6 720.2.by.c.49.4 8
180.139 odd 6 2160.2.by.d.1009.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.b.49.1 ✓ 8 45.14 odd 6
90.2.i.b.49.4 yes 8 9.5 odd 6
90.2.i.b.79.1 yes 8 3.2 odd 2
90.2.i.b.79.4 yes 8 15.14 odd 2
270.2.i.b.19.2 8 5.4 even 2 inner
270.2.i.b.19.3 8 1.1 even 1 trivial
270.2.i.b.199.2 8 9.4 even 3 inner
270.2.i.b.199.3 8 45.4 even 6 inner
450.2.e.k.151.1 4 15.8 even 4
450.2.e.k.301.1 4 45.23 even 12
450.2.e.n.151.2 4 15.2 even 4
450.2.e.n.301.2 4 45.32 even 12
720.2.by.c.49.1 8 36.23 even 6
720.2.by.c.49.4 8 180.59 even 6
720.2.by.c.529.1 8 60.59 even 2
720.2.by.c.529.4 8 12.11 even 2
810.2.c.e.649.2 4 9.7 even 3
810.2.c.e.649.4 4 45.34 even 6
810.2.c.f.649.1 4 45.29 odd 6
810.2.c.f.649.3 4 9.2 odd 6
1350.2.e.j.451.1 4 5.2 odd 4
1350.2.e.j.901.1 4 45.22 odd 12
1350.2.e.m.451.2 4 5.3 odd 4
1350.2.e.m.901.2 4 45.13 odd 12
2160.2.by.d.289.1 8 4.3 odd 2
2160.2.by.d.289.4 8 20.19 odd 2
2160.2.by.d.1009.1 8 180.139 odd 6
2160.2.by.d.1009.4 8 36.31 odd 6
4050.2.a.bm.1.1 2 45.43 odd 12
4050.2.a.bq.1.2 2 45.2 even 12
4050.2.a.bs.1.1 2 45.38 even 12
4050.2.a.bz.1.2 2 45.7 odd 12