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Properties

Label 270.3.g.b.217.1
Level $270$
Weight $3$
Character 270.217
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 217.1
Root \(-1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 270.217
Dual form 270.3.g.b.163.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(1.77526 + 4.67423i) q^{5} +(-5.89898 - 5.89898i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.89898 - 6.44949i) q^{10} +14.8990 q^{11} +(-11.4495 + 11.4495i) q^{13} +11.7980i q^{14} -4.00000 q^{16} +(14.2247 + 14.2247i) q^{17} +10.5505i q^{19} +(-9.34847 + 3.55051i) q^{20} +(-14.8990 - 14.8990i) q^{22} +(-21.7196 + 21.7196i) q^{23} +(-18.6969 + 16.5959i) q^{25} +22.8990 q^{26} +(11.7980 - 11.7980i) q^{28} +30.9444i q^{29} +57.9444 q^{31} +(4.00000 + 4.00000i) q^{32} -28.4495i q^{34} +(17.1010 - 38.0454i) q^{35} +(12.3031 + 12.3031i) q^{37} +(10.5505 - 10.5505i) q^{38} +(12.8990 + 5.79796i) q^{40} +10.4949 q^{41} +(-2.94439 + 2.94439i) q^{43} +29.7980i q^{44} +43.4393 q^{46} +(26.6969 + 26.6969i) q^{47} +20.5959i q^{49} +(35.2929 + 2.10102i) q^{50} +(-22.8990 - 22.8990i) q^{52} +(-10.6742 + 10.6742i) q^{53} +(26.4495 + 69.6413i) q^{55} -23.5959 q^{56} +(30.9444 - 30.9444i) q^{58} -60.0454i q^{59} -102.192 q^{61} +(-57.9444 - 57.9444i) q^{62} -8.00000i q^{64} +(-73.8434 - 33.1918i) q^{65} +(-46.4393 - 46.4393i) q^{67} +(-28.4495 + 28.4495i) q^{68} +(-55.1464 + 20.9444i) q^{70} +38.9898 q^{71} +(71.7321 - 71.7321i) q^{73} -24.6061i q^{74} -21.1010 q^{76} +(-87.8888 - 87.8888i) q^{77} +118.126i q^{79} +(-7.10102 - 18.6969i) q^{80} +(-10.4949 - 10.4949i) q^{82} +(-18.7298 + 18.7298i) q^{83} +(-41.2372 + 91.7423i) q^{85} +5.88877 q^{86} +(29.7980 - 29.7980i) q^{88} +44.6061i q^{89} +135.081 q^{91} +(-43.4393 - 43.4393i) q^{92} -53.3939i q^{94} +(-49.3156 + 18.7298i) q^{95} +(-125.778 - 125.778i) q^{97} +(20.5959 - 20.5959i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 12 q^{5} - 4 q^{7} + 8 q^{8} - 8 q^{10} + 40 q^{11} - 36 q^{13} - 16 q^{16} + 52 q^{17} - 8 q^{20} - 40 q^{22} + 16 q^{23} - 16 q^{25} + 72 q^{26} + 8 q^{28} + 124 q^{31} + 16 q^{32} + 88 q^{35}+ \cdots + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 1.77526 + 4.67423i 0.355051 + 0.934847i
\(6\) 0 0
\(7\) −5.89898 − 5.89898i −0.842711 − 0.842711i 0.146499 − 0.989211i \(-0.453199\pi\)
−0.989211 + 0.146499i \(0.953199\pi\)
\(8\) 2.00000 − 2.00000i 0.250000 − 0.250000i
\(9\) 0 0
\(10\) 2.89898 − 6.44949i 0.289898 − 0.644949i
\(11\) 14.8990 1.35445 0.677226 − 0.735775i \(-0.263182\pi\)
0.677226 + 0.735775i \(0.263182\pi\)
\(12\) 0 0
\(13\) −11.4495 + 11.4495i −0.880730 + 0.880730i −0.993609 − 0.112879i \(-0.963993\pi\)
0.112879 + 0.993609i \(0.463993\pi\)
\(14\) 11.7980i 0.842711i
\(15\) 0 0
\(16\) −4.00000 −0.250000
\(17\) 14.2247 + 14.2247i 0.836750 + 0.836750i 0.988430 − 0.151680i \(-0.0484683\pi\)
−0.151680 + 0.988430i \(0.548468\pi\)
\(18\) 0 0
\(19\) 10.5505i 0.555290i 0.960684 + 0.277645i \(0.0895539\pi\)
−0.960684 + 0.277645i \(0.910446\pi\)
\(20\) −9.34847 + 3.55051i −0.467423 + 0.177526i
\(21\) 0 0
\(22\) −14.8990 − 14.8990i −0.677226 − 0.677226i
\(23\) −21.7196 + 21.7196i −0.944332 + 0.944332i −0.998530 − 0.0541979i \(-0.982740\pi\)
0.0541979 + 0.998530i \(0.482740\pi\)
\(24\) 0 0
\(25\) −18.6969 + 16.5959i −0.747878 + 0.663837i
\(26\) 22.8990 0.880730
\(27\) 0 0
\(28\) 11.7980 − 11.7980i 0.421356 − 0.421356i
\(29\) 30.9444i 1.06705i 0.845785 + 0.533524i \(0.179133\pi\)
−0.845785 + 0.533524i \(0.820867\pi\)
\(30\) 0 0
\(31\) 57.9444 1.86917 0.934587 − 0.355735i \(-0.115769\pi\)
0.934587 + 0.355735i \(0.115769\pi\)
\(32\) 4.00000 + 4.00000i 0.125000 + 0.125000i
\(33\) 0 0
\(34\) − 28.4495i − 0.836750i
\(35\) 17.1010 − 38.0454i 0.488601 − 1.08701i
\(36\) 0 0
\(37\) 12.3031 + 12.3031i 0.332515 + 0.332515i 0.853541 − 0.521026i \(-0.174450\pi\)
−0.521026 + 0.853541i \(0.674450\pi\)
\(38\) 10.5505 − 10.5505i 0.277645 − 0.277645i
\(39\) 0 0
\(40\) 12.8990 + 5.79796i 0.322474 + 0.144949i
\(41\) 10.4949 0.255973 0.127987 − 0.991776i \(-0.459149\pi\)
0.127987 + 0.991776i \(0.459149\pi\)
\(42\) 0 0
\(43\) −2.94439 + 2.94439i −0.0684741 + 0.0684741i −0.740515 − 0.672040i \(-0.765418\pi\)
0.672040 + 0.740515i \(0.265418\pi\)
\(44\) 29.7980i 0.677226i
\(45\) 0 0
\(46\) 43.4393 0.944332
\(47\) 26.6969 + 26.6969i 0.568020 + 0.568020i 0.931573 − 0.363553i \(-0.118437\pi\)
−0.363553 + 0.931573i \(0.618437\pi\)
\(48\) 0 0
\(49\) 20.5959i 0.420325i
\(50\) 35.2929 + 2.10102i 0.705857 + 0.0420204i
\(51\) 0 0
\(52\) −22.8990 − 22.8990i −0.440365 − 0.440365i
\(53\) −10.6742 + 10.6742i −0.201401 + 0.201401i −0.800600 − 0.599199i \(-0.795486\pi\)
0.599199 + 0.800600i \(0.295486\pi\)
\(54\) 0 0
\(55\) 26.4495 + 69.6413i 0.480900 + 1.26621i
\(56\) −23.5959 −0.421356
\(57\) 0 0
\(58\) 30.9444 − 30.9444i 0.533524 − 0.533524i
\(59\) − 60.0454i − 1.01772i −0.860850 − 0.508859i \(-0.830067\pi\)
0.860850 − 0.508859i \(-0.169933\pi\)
\(60\) 0 0
\(61\) −102.192 −1.67528 −0.837638 − 0.546226i \(-0.816064\pi\)
−0.837638 + 0.546226i \(0.816064\pi\)
\(62\) −57.9444 − 57.9444i −0.934587 − 0.934587i
\(63\) 0 0
\(64\) − 8.00000i − 0.125000i
\(65\) −73.8434 − 33.1918i −1.13605 − 0.510644i
\(66\) 0 0
\(67\) −46.4393 − 46.4393i −0.693124 − 0.693124i 0.269794 − 0.962918i \(-0.413044\pi\)
−0.962918 + 0.269794i \(0.913044\pi\)
\(68\) −28.4495 + 28.4495i −0.418375 + 0.418375i
\(69\) 0 0
\(70\) −55.1464 + 20.9444i −0.787806 + 0.299206i
\(71\) 38.9898 0.549152 0.274576 − 0.961565i \(-0.411463\pi\)
0.274576 + 0.961565i \(0.411463\pi\)
\(72\) 0 0
\(73\) 71.7321 − 71.7321i 0.982632 − 0.982632i −0.0172197 − 0.999852i \(-0.505481\pi\)
0.999852 + 0.0172197i \(0.00548146\pi\)
\(74\) − 24.6061i − 0.332515i
\(75\) 0 0
\(76\) −21.1010 −0.277645
\(77\) −87.8888 − 87.8888i −1.14141 − 1.14141i
\(78\) 0 0
\(79\) 118.126i 1.49527i 0.664112 + 0.747633i \(0.268810\pi\)
−0.664112 + 0.747633i \(0.731190\pi\)
\(80\) −7.10102 − 18.6969i −0.0887628 − 0.233712i
\(81\) 0 0
\(82\) −10.4949 − 10.4949i −0.127987 − 0.127987i
\(83\) −18.7298 + 18.7298i −0.225661 + 0.225661i −0.810877 − 0.585216i \(-0.801010\pi\)
0.585216 + 0.810877i \(0.301010\pi\)
\(84\) 0 0
\(85\) −41.2372 + 91.7423i −0.485144 + 1.07932i
\(86\) 5.88877 0.0684741
\(87\) 0 0
\(88\) 29.7980 − 29.7980i 0.338613 − 0.338613i
\(89\) 44.6061i 0.501192i 0.968092 + 0.250596i \(0.0806266\pi\)
−0.968092 + 0.250596i \(0.919373\pi\)
\(90\) 0 0
\(91\) 135.081 1.48440
\(92\) −43.4393 − 43.4393i −0.472166 − 0.472166i
\(93\) 0 0
\(94\) − 53.3939i − 0.568020i
\(95\) −49.3156 + 18.7298i −0.519111 + 0.197156i
\(96\) 0 0
\(97\) −125.778 − 125.778i −1.29668 − 1.29668i −0.930575 − 0.366100i \(-0.880693\pi\)
−0.366100 − 0.930575i \(-0.619307\pi\)
\(98\) 20.5959 − 20.5959i 0.210162 − 0.210162i
\(99\) 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.b.217.1 yes 4
3.2 odd 2 270.3.g.c.217.2 yes 4
5.2 odd 4 1350.3.g.k.1243.2 4
5.3 odd 4 inner 270.3.g.b.163.1 ✓ 4
5.4 even 2 1350.3.g.k.757.2 4
15.2 even 4 1350.3.g.e.1243.2 4
15.8 even 4 270.3.g.c.163.2 yes 4
15.14 odd 2 1350.3.g.e.757.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.3.g.b.163.1 ✓ 4 5.3 odd 4 inner
270.3.g.b.217.1 yes 4 1.1 even 1 trivial
270.3.g.c.163.2 yes 4 15.8 even 4
270.3.g.c.217.2 yes 4 3.2 odd 2
1350.3.g.e.757.2 4 15.14 odd 2
1350.3.g.e.1243.2 4 15.2 even 4
1350.3.g.k.757.2 4 5.4 even 2
1350.3.g.k.1243.2 4 5.2 odd 4