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Properties

Label 270.2.i.b.199.3
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 199.3
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 270.199
Dual form 270.2.i.b.19.3

$q$-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 \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

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

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
\(5\) −2.03906 − 0.917738i −0.911894 − 0.410425i
\(6\) 0 0
\(7\) 3.85337 − 2.22474i 1.45644 − 0.840875i 0.457604 − 0.889156i \(-0.348708\pi\)
0.998834 + 0.0482818i \(0.0153745\pi\)
\(8\) − 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 0.954355 − 0.298674i \(-0.0965442\pi\)
−0.735837 + 0.677159i \(0.763211\pi\)
\(12\) 0 0
\(13\) −2.12132 − 1.22474i −0.588348 − 0.339683i 0.176096 − 0.984373i \(-0.443653\pi\)
−0.764444 + 0.644690i \(0.776986\pi\)
\(14\) 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 \(-0.843236\pi\)
0.881159 − 0.472821i \(-0.156764\pi\)
\(18\) 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 0.738363 − 0.674403i \(-0.235599\pi\)
−0.214869 + 0.976643i \(0.568932\pi\)
\(24\) 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 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −3.22474 + 5.58542i −0.579181 + 1.00317i 0.416392 + 0.909185i \(0.363294\pi\)
−0.995573 + 0.0939863i \(0.970039\pi\)
\(32\) −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 \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 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 −0.902424 − 0.430848i \(-0.858214\pi\)
0.824338 + 0.566099i \(0.191548\pi\)
\(42\) 0 0
\(43\) −6.45145 + 3.72474i −0.983836 + 0.568018i −0.903426 − 0.428744i \(-0.858956\pi\)
−0.0804103 + 0.996762i \(0.525623\pi\)
\(44\) 1.44949 0.218519
\(45\) 0 0
\(46\) 2.89898 0.427431
\(47\) −0.389270 + 0.224745i −0.0567808 + 0.0327824i −0.528122 − 0.849169i \(-0.677104\pi\)
0.471341 + 0.881951i \(0.343770\pi\)
\(48\) 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 \(0.197070\pi\)
−0.814393 + 0.580313i \(0.802930\pi\)
\(54\) 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 −0.223817 − 0.974631i \(-0.571852\pi\)
0.955964 − 0.293484i \(-0.0948147\pi\)
\(60\) 0 0
\(61\) 0.224745 + 0.389270i 0.0287756 + 0.0498409i 0.880055 − 0.474873i \(-0.157506\pi\)
−0.851279 + 0.524713i \(0.824173\pi\)
\(62\) 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 −0.0915922 − 0.995797i \(-0.529196\pi\)
−0.908181 + 0.418577i \(0.862529\pi\)
\(68\) −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 \(0.0905929\pi\)
−0.959772 + 0.280779i \(0.909407\pi\)
\(74\) 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 0.581874 − 0.813279i \(-0.302320\pi\)
−0.995257 + 0.0972777i \(0.968987\pi\)
\(80\) 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 −0.677920 − 0.735135i \(-0.737119\pi\)
0.297686 + 0.954664i \(0.403785\pi\)
\(84\) 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 0.195911 − 0.980622i \(-0.437234\pi\)
0.947199 + 0.320647i \(0.103900\pi\)
\(98\) − 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.199.3 8
3.2 odd 2 90.2.i.b.49.1 ✓ 8
4.3 odd 2 2160.2.by.d.1009.1 8
5.2 odd 4 1350.2.e.m.901.2 4
5.3 odd 4 1350.2.e.j.901.1 4
5.4 even 2 inner 270.2.i.b.199.2 8
9.2 odd 6 90.2.i.b.79.4 yes 8
9.4 even 3 810.2.c.e.649.4 4
9.5 odd 6 810.2.c.f.649.1 4
9.7 even 3 inner 270.2.i.b.19.2 8
12.11 even 2 720.2.by.c.49.4 8
15.2 even 4 450.2.e.k.301.1 4
15.8 even 4 450.2.e.n.301.2 4
15.14 odd 2 90.2.i.b.49.4 yes 8
20.19 odd 2 2160.2.by.d.1009.4 8
36.7 odd 6 2160.2.by.d.289.4 8
36.11 even 6 720.2.by.c.529.1 8
45.2 even 12 450.2.e.k.151.1 4
45.4 even 6 810.2.c.e.649.2 4
45.7 odd 12 1350.2.e.m.451.2 4
45.13 odd 12 4050.2.a.bz.1.2 2
45.14 odd 6 810.2.c.f.649.3 4
45.22 odd 12 4050.2.a.bm.1.1 2
45.23 even 12 4050.2.a.bq.1.2 2
45.29 odd 6 90.2.i.b.79.1 yes 8
45.32 even 12 4050.2.a.bs.1.1 2
45.34 even 6 inner 270.2.i.b.19.3 8
45.38 even 12 450.2.e.n.151.2 4
45.43 odd 12 1350.2.e.j.451.1 4
60.59 even 2 720.2.by.c.49.1 8
180.79 odd 6 2160.2.by.d.289.1 8
180.119 even 6 720.2.by.c.529.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.b.49.1 ✓ 8 3.2 odd 2
90.2.i.b.49.4 yes 8 15.14 odd 2
90.2.i.b.79.1 yes 8 45.29 odd 6
90.2.i.b.79.4 yes 8 9.2 odd 6
270.2.i.b.19.2 8 9.7 even 3 inner
270.2.i.b.19.3 8 45.34 even 6 inner
270.2.i.b.199.2 8 5.4 even 2 inner
270.2.i.b.199.3 8 1.1 even 1 trivial
450.2.e.k.151.1 4 45.2 even 12
450.2.e.k.301.1 4 15.2 even 4
450.2.e.n.151.2 4 45.38 even 12
450.2.e.n.301.2 4 15.8 even 4
720.2.by.c.49.1 8 60.59 even 2
720.2.by.c.49.4 8 12.11 even 2
720.2.by.c.529.1 8 36.11 even 6
720.2.by.c.529.4 8 180.119 even 6
810.2.c.e.649.2 4 45.4 even 6
810.2.c.e.649.4 4 9.4 even 3
810.2.c.f.649.1 4 9.5 odd 6
810.2.c.f.649.3 4 45.14 odd 6
1350.2.e.j.451.1 4 45.43 odd 12
1350.2.e.j.901.1 4 5.3 odd 4
1350.2.e.m.451.2 4 45.7 odd 12
1350.2.e.m.901.2 4 5.2 odd 4
2160.2.by.d.289.1 8 180.79 odd 6
2160.2.by.d.289.4 8 36.7 odd 6
2160.2.by.d.1009.1 8 4.3 odd 2
2160.2.by.d.1009.4 8 20.19 odd 2
4050.2.a.bm.1.1 2 45.22 odd 12
4050.2.a.bq.1.2 2 45.23 even 12
4050.2.a.bs.1.1 2 45.32 even 12
4050.2.a.bz.1.2 2 45.13 odd 12