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Properties

Label 270.3.g.c.163.1
Level $270$
Weight $3$
Character 270.163
Analytic conductor $7.357$
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] = [270,3,Mod(163,270)] 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("270.163"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,0,-12] 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: \(7.35696713773\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 163.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 270.163
Dual form 270.3.g.c.217.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{2} -2.00000i q^{4} +(-4.22474 - 2.67423i) q^{5} +(3.89898 - 3.89898i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-6.89898 + 1.55051i) q^{10} -5.10102 q^{11} +(-6.55051 - 6.55051i) q^{13} -7.79796i q^{14} -4.00000 q^{16} +(-11.7753 + 11.7753i) q^{17} -15.4495i q^{19} +(-5.34847 + 8.44949i) q^{20} +(-5.10102 + 5.10102i) q^{22} +(-29.7196 - 29.7196i) q^{23} +(10.6969 + 22.5959i) q^{25} -13.1010 q^{26} +(-7.79796 - 7.79796i) q^{28} -22.9444i q^{29} +4.05561 q^{31} +(-4.00000 + 4.00000i) q^{32} +23.5505i q^{34} +(-26.8990 + 6.04541i) q^{35} +(41.6969 - 41.6969i) q^{37} +(-15.4495 - 15.4495i) q^{38} +(3.10102 + 13.7980i) q^{40} +38.4949 q^{41} +(50.9444 + 50.9444i) q^{43} +10.2020i q^{44} -59.4393 q^{46} +(2.69694 - 2.69694i) q^{47} +18.5959i q^{49} +(33.2929 + 11.8990i) q^{50} +(-13.1010 + 13.1010i) q^{52} +(3.32577 + 3.32577i) q^{53} +(21.5505 + 13.6413i) q^{55} -15.5959 q^{56} +(-22.9444 - 22.9444i) q^{58} -15.9546i q^{59} -23.8082 q^{61} +(4.05561 - 4.05561i) q^{62} +8.00000i q^{64} +(10.1566 + 45.1918i) q^{65} +(56.4393 - 56.4393i) q^{67} +(23.5505 + 23.5505i) q^{68} +(-20.8536 + 32.9444i) q^{70} +58.9898 q^{71} +(-99.7321 - 99.7321i) q^{73} -83.3939i q^{74} -30.8990 q^{76} +(-19.8888 + 19.8888i) q^{77} +112.126i q^{79} +(16.8990 + 10.6969i) q^{80} +(38.4949 - 38.4949i) q^{82} +(65.2702 + 65.2702i) q^{83} +(81.2372 - 18.2577i) q^{85} +101.889 q^{86} +(10.2020 + 10.2020i) q^{88} +103.394i q^{89} -51.0806 q^{91} +(-59.4393 + 59.4393i) q^{92} -5.39388i q^{94} +(-41.3156 + 65.2702i) q^{95} +(89.7775 - 89.7775i) q^{97} +(18.5959 + 18.5959i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 12 q^{5} - 4 q^{7} - 8 q^{8} - 8 q^{10} - 40 q^{11} - 36 q^{13} - 16 q^{16} - 52 q^{17} + 8 q^{20} - 40 q^{22} - 16 q^{23} - 16 q^{25} - 72 q^{26} + 8 q^{28} + 124 q^{31} - 16 q^{32} - 88 q^{35}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 − 1.00000i 0.500000 − 0.500000i
\(3\) 0 0
\(4\) − 2.00000i − 0.500000i
\(5\) −4.22474 − 2.67423i −0.844949 − 0.534847i
\(6\) 0 0
\(7\) 3.89898 − 3.89898i 0.556997 − 0.556997i −0.371454 − 0.928451i \(-0.621141\pi\)
0.928451 + 0.371454i \(0.121141\pi\)
\(8\) −2.00000 − 2.00000i −0.250000 − 0.250000i
\(9\) 0 0
\(10\) −6.89898 + 1.55051i −0.689898 + 0.155051i
\(11\) −5.10102 −0.463729 −0.231865 − 0.972748i \(-0.574483\pi\)
−0.231865 + 0.972748i \(0.574483\pi\)
\(12\) 0 0
\(13\) −6.55051 − 6.55051i −0.503885 − 0.503885i 0.408758 − 0.912643i \(-0.365962\pi\)
−0.912643 + 0.408758i \(0.865962\pi\)
\(14\) − 7.79796i − 0.556997i
\(15\) 0 0
\(16\) −4.00000 −0.250000
\(17\) −11.7753 + 11.7753i −0.692662 + 0.692662i −0.962817 − 0.270155i \(-0.912925\pi\)
0.270155 + 0.962817i \(0.412925\pi\)
\(18\) 0 0
\(19\) − 15.4495i − 0.813131i −0.913622 − 0.406566i \(-0.866726\pi\)
0.913622 − 0.406566i \(-0.133274\pi\)
\(20\) −5.34847 + 8.44949i −0.267423 + 0.422474i
\(21\) 0 0
\(22\) −5.10102 + 5.10102i −0.231865 + 0.231865i
\(23\) −29.7196 − 29.7196i −1.29216 − 1.29216i −0.933449 − 0.358709i \(-0.883217\pi\)
−0.358709 − 0.933449i \(-0.616783\pi\)
\(24\) 0 0
\(25\) 10.6969 + 22.5959i 0.427878 + 0.903837i
\(26\) −13.1010 −0.503885
\(27\) 0 0
\(28\) −7.79796 − 7.79796i −0.278499 − 0.278499i
\(29\) − 22.9444i − 0.791186i −0.918426 − 0.395593i \(-0.870539\pi\)
0.918426 − 0.395593i \(-0.129461\pi\)
\(30\) 0 0
\(31\) 4.05561 0.130826 0.0654131 − 0.997858i \(-0.479163\pi\)
0.0654131 + 0.997858i \(0.479163\pi\)
\(32\) −4.00000 + 4.00000i −0.125000 + 0.125000i
\(33\) 0 0
\(34\) 23.5505i 0.692662i
\(35\) −26.8990 + 6.04541i −0.768542 + 0.172726i
\(36\) 0 0
\(37\) 41.6969 − 41.6969i 1.12694 − 1.12694i 0.136273 − 0.990671i \(-0.456488\pi\)
0.990671 − 0.136273i \(-0.0435124\pi\)
\(38\) −15.4495 − 15.4495i −0.406566 − 0.406566i
\(39\) 0 0
\(40\) 3.10102 + 13.7980i 0.0775255 + 0.344949i
\(41\) 38.4949 0.938900 0.469450 − 0.882959i \(-0.344452\pi\)
0.469450 + 0.882959i \(0.344452\pi\)
\(42\) 0 0
\(43\) 50.9444 + 50.9444i 1.18475 + 1.18475i 0.978498 + 0.206255i \(0.0661276\pi\)
0.206255 + 0.978498i \(0.433872\pi\)
\(44\) 10.2020i 0.231865i
\(45\) 0 0
\(46\) −59.4393 −1.29216
\(47\) 2.69694 − 2.69694i 0.0573817 − 0.0573817i −0.677834 − 0.735215i \(-0.737081\pi\)
0.735215 + 0.677834i \(0.237081\pi\)
\(48\) 0 0
\(49\) 18.5959i 0.379509i
\(50\) 33.2929 + 11.8990i 0.665857 + 0.237980i
\(51\) 0 0
\(52\) −13.1010 + 13.1010i −0.251943 + 0.251943i
\(53\) 3.32577 + 3.32577i 0.0627503 + 0.0627503i 0.737786 − 0.675035i \(-0.235872\pi\)
−0.675035 + 0.737786i \(0.735872\pi\)
\(54\) 0 0
\(55\) 21.5505 + 13.6413i 0.391827 + 0.248024i
\(56\) −15.5959 −0.278499
\(57\) 0 0
\(58\) −22.9444 − 22.9444i −0.395593 − 0.395593i
\(59\) − 15.9546i − 0.270417i −0.990817 − 0.135208i \(-0.956830\pi\)
0.990817 − 0.135208i \(-0.0431704\pi\)
\(60\) 0 0
\(61\) −23.8082 −0.390298 −0.195149 − 0.980774i \(-0.562519\pi\)
−0.195149 + 0.980774i \(0.562519\pi\)
\(62\) 4.05561 − 4.05561i 0.0654131 − 0.0654131i
\(63\) 0 0
\(64\) 8.00000i 0.125000i
\(65\) 10.1566 + 45.1918i 0.156256 + 0.695259i
\(66\) 0 0
\(67\) 56.4393 − 56.4393i 0.842377 − 0.842377i −0.146790 − 0.989168i \(-0.546894\pi\)
0.989168 + 0.146790i \(0.0468942\pi\)
\(68\) 23.5505 + 23.5505i 0.346331 + 0.346331i
\(69\) 0 0
\(70\) −20.8536 + 32.9444i −0.297908 + 0.470634i
\(71\) 58.9898 0.830842 0.415421 − 0.909629i \(-0.363634\pi\)
0.415421 + 0.909629i \(0.363634\pi\)
\(72\) 0 0
\(73\) −99.7321 − 99.7321i −1.36619 − 1.36619i −0.865795 − 0.500399i \(-0.833187\pi\)
−0.500399 − 0.865795i \(-0.666813\pi\)
\(74\) − 83.3939i − 1.12694i
\(75\) 0 0
\(76\) −30.8990 −0.406566
\(77\) −19.8888 + 19.8888i −0.258296 + 0.258296i
\(78\) 0 0
\(79\) 112.126i 1.41932i 0.704546 + 0.709658i \(0.251151\pi\)
−0.704546 + 0.709658i \(0.748849\pi\)
\(80\) 16.8990 + 10.6969i 0.211237 + 0.133712i
\(81\) 0 0
\(82\) 38.4949 − 38.4949i 0.469450 − 0.469450i
\(83\) 65.2702 + 65.2702i 0.786387 + 0.786387i 0.980900 − 0.194513i \(-0.0623125\pi\)
−0.194513 + 0.980900i \(0.562313\pi\)
\(84\) 0 0
\(85\) 81.2372 − 18.2577i 0.955732 − 0.214796i
\(86\) 101.889 1.18475
\(87\) 0 0
\(88\) 10.2020 + 10.2020i 0.115932 + 0.115932i
\(89\) 103.394i 1.16173i 0.814000 + 0.580864i \(0.197285\pi\)
−0.814000 + 0.580864i \(0.802715\pi\)
\(90\) 0 0
\(91\) −51.0806 −0.561325
\(92\) −59.4393 + 59.4393i −0.646079 + 0.646079i
\(93\) 0 0
\(94\) − 5.39388i − 0.0573817i
\(95\) −41.3156 + 65.2702i −0.434901 + 0.687054i
\(96\) 0 0
\(97\) 89.7775 − 89.7775i 0.925542 − 0.925542i −0.0718721 − 0.997414i \(-0.522897\pi\)
0.997414 + 0.0718721i \(0.0228973\pi\)
\(98\) 18.5959 + 18.5959i 0.189754 + 0.189754i
\(99\) 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 270.3.g.c.163.1 yes 4
3.2 odd 2 270.3.g.b.163.2 ✓ 4
5.2 odd 4 inner 270.3.g.c.217.1 yes 4
5.3 odd 4 1350.3.g.e.757.1 4
5.4 even 2 1350.3.g.e.1243.1 4
15.2 even 4 270.3.g.b.217.2 yes 4
15.8 even 4 1350.3.g.k.757.1 4
15.14 odd 2 1350.3.g.k.1243.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.3.g.b.163.2 ✓ 4 3.2 odd 2
270.3.g.b.217.2 yes 4 15.2 even 4
270.3.g.c.163.1 yes 4 1.1 even 1 trivial
270.3.g.c.217.1 yes 4 5.2 odd 4 inner
1350.3.g.e.757.1 4 5.3 odd 4
1350.3.g.e.1243.1 4 5.4 even 2
1350.3.g.k.757.1 4 15.8 even 4
1350.3.g.k.1243.1 4 15.14 odd 2