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Properties

Label 1386.2.k.t.991.2
Level $1386$
Weight $2$
Character 1386.991
Analytic conductor $11.067$
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] = [1386,2,Mod(793,1386)] 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("1386.793"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1386, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,-4,0,-2,-4,0,4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 991.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1386.991
Dual form 1386.2.k.t.793.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.292893 - 0.507306i) q^{5} +(1.62132 - 2.09077i) q^{7} -1.00000 q^{8} +(0.292893 - 0.507306i) q^{10} +(0.500000 - 0.866025i) q^{11} -3.82843 q^{13} +(2.62132 + 0.358719i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.82843 - 3.16693i) q^{17} +(0.292893 + 0.507306i) q^{19} +0.585786 q^{20} +1.00000 q^{22} +(-3.12132 - 5.40629i) q^{23} +(2.32843 - 4.03295i) q^{25} +(-1.91421 - 3.31552i) q^{26} +(1.00000 + 2.44949i) q^{28} -2.65685 q^{29} +(2.00000 - 3.46410i) q^{31} +(0.500000 - 0.866025i) q^{32} +3.65685 q^{34} +(-1.53553 - 0.210133i) q^{35} +(4.70711 + 8.15295i) q^{37} +(-0.292893 + 0.507306i) q^{38} +(0.292893 + 0.507306i) q^{40} +5.41421 q^{41} -5.65685 q^{43} +(0.500000 + 0.866025i) q^{44} +(3.12132 - 5.40629i) q^{46} +(-5.24264 - 9.08052i) q^{47} +(-1.74264 - 6.77962i) q^{49} +4.65685 q^{50} +(1.91421 - 3.31552i) q^{52} +(3.94975 - 6.84116i) q^{53} -0.585786 q^{55} +(-1.62132 + 2.09077i) q^{56} +(-1.32843 - 2.30090i) q^{58} +(-2.79289 + 4.83743i) q^{59} +(-5.91421 - 10.2437i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(1.12132 + 1.94218i) q^{65} +(-1.37868 + 2.38794i) q^{67} +(1.82843 + 3.16693i) q^{68} +(-0.585786 - 1.43488i) q^{70} +11.0711 q^{71} +(4.70711 - 8.15295i) q^{73} +(-4.70711 + 8.15295i) q^{74} -0.585786 q^{76} +(-1.00000 - 2.44949i) q^{77} +(6.62132 + 11.4685i) q^{79} +(-0.292893 + 0.507306i) q^{80} +(2.70711 + 4.68885i) q^{82} +12.1421 q^{83} -2.14214 q^{85} +(-2.82843 - 4.89898i) q^{86} +(-0.500000 + 0.866025i) q^{88} +(6.24264 + 10.8126i) q^{89} +(-6.20711 + 8.00436i) q^{91} +6.24264 q^{92} +(5.24264 - 9.08052i) q^{94} +(0.171573 - 0.297173i) q^{95} -3.82843 q^{97} +(5.00000 - 4.89898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 2 q^{7} - 4 q^{8} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 2 q^{14} - 2 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} - 4 q^{23} - 2 q^{25} - 2 q^{26} + 4 q^{28} + 12 q^{29}+ \cdots + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.292893 − 0.507306i −0.130986 − 0.226874i 0.793071 − 0.609129i \(-0.208481\pi\)
−0.924057 + 0.382255i \(0.875148\pi\)
\(6\) 0 0
\(7\) 1.62132 − 2.09077i 0.612801 − 0.790237i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.292893 − 0.507306i 0.0926210 − 0.160424i
\(11\) 0.500000 − 0.866025i 0.150756 − 0.261116i
\(12\) 0 0
\(13\) −3.82843 −1.06181 −0.530907 − 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 2.62132 + 0.358719i 0.700577 + 0.0958718i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.82843 − 3.16693i 0.443459 − 0.768093i −0.554485 − 0.832194i \(-0.687085\pi\)
0.997943 + 0.0641009i \(0.0204179\pi\)
\(18\) 0 0
\(19\) 0.292893 + 0.507306i 0.0671943 + 0.116384i 0.897665 − 0.440678i \(-0.145262\pi\)
−0.830471 + 0.557062i \(0.811929\pi\)
\(20\) 0.585786 0.130986
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −3.12132 − 5.40629i −0.650840 − 1.12729i −0.982919 − 0.184037i \(-0.941083\pi\)
0.332079 − 0.943252i \(-0.392250\pi\)
\(24\) 0 0
\(25\) 2.32843 − 4.03295i 0.465685 − 0.806591i
\(26\) −1.91421 − 3.31552i −0.375408 − 0.650226i
\(27\) 0 0
\(28\) 1.00000 + 2.44949i 0.188982 + 0.462910i
\(29\) −2.65685 −0.493365 −0.246683 − 0.969096i \(-0.579341\pi\)
−0.246683 + 0.969096i \(0.579341\pi\)
\(30\) 0 0
\(31\) 2.00000 − 3.46410i 0.359211 − 0.622171i −0.628619 − 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 3.65685 0.627145
\(35\) −1.53553 − 0.210133i −0.259553 − 0.0355190i
\(36\) 0 0
\(37\) 4.70711 + 8.15295i 0.773844 + 1.34034i 0.935442 + 0.353480i \(0.115002\pi\)
−0.161599 + 0.986857i \(0.551665\pi\)
\(38\) −0.292893 + 0.507306i −0.0475136 + 0.0822959i
\(39\) 0 0
\(40\) 0.292893 + 0.507306i 0.0463105 + 0.0802121i
\(41\) 5.41421 0.845558 0.422779 − 0.906233i \(-0.361055\pi\)
0.422779 + 0.906233i \(0.361055\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 − 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0.500000 + 0.866025i 0.0753778 + 0.130558i
\(45\) 0 0
\(46\) 3.12132 − 5.40629i 0.460214 − 0.797113i
\(47\) −5.24264 − 9.08052i −0.764718 − 1.32453i −0.940396 − 0.340082i \(-0.889545\pi\)
0.175678 − 0.984448i \(-0.443788\pi\)
\(48\) 0 0
\(49\) −1.74264 − 6.77962i −0.248949 − 0.968517i
\(50\) 4.65685 0.658579
\(51\) 0 0
\(52\) 1.91421 − 3.31552i 0.265454 − 0.459779i
\(53\) 3.94975 − 6.84116i 0.542540 − 0.939706i −0.456218 − 0.889868i \(-0.650796\pi\)
0.998757 − 0.0498379i \(-0.0158705\pi\)
\(54\) 0 0
\(55\) −0.585786 −0.0789874
\(56\) −1.62132 + 2.09077i −0.216658 + 0.279391i
\(57\) 0 0
\(58\) −1.32843 − 2.30090i −0.174431 − 0.302123i
\(59\) −2.79289 + 4.83743i −0.363604 + 0.629780i −0.988551 − 0.150887i \(-0.951787\pi\)
0.624947 + 0.780667i \(0.285120\pi\)
\(60\) 0 0
\(61\) −5.91421 − 10.2437i −0.757237 − 1.31157i −0.944254 − 0.329217i \(-0.893215\pi\)
0.187017 − 0.982357i \(-0.440118\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.12132 + 1.94218i 0.139083 + 0.240898i
\(66\) 0 0
\(67\) −1.37868 + 2.38794i −0.168433 + 0.291734i −0.937869 − 0.346990i \(-0.887204\pi\)
0.769436 + 0.638723i \(0.220537\pi\)
\(68\) 1.82843 + 3.16693i 0.221729 + 0.384047i
\(69\) 0 0
\(70\) −0.585786 − 1.43488i −0.0700149 − 0.171501i
\(71\) 11.0711 1.31389 0.656947 − 0.753937i \(-0.271848\pi\)
0.656947 + 0.753937i \(0.271848\pi\)
\(72\) 0 0
\(73\) 4.70711 − 8.15295i 0.550925 − 0.954230i −0.447283 − 0.894393i \(-0.647608\pi\)
0.998208 − 0.0598379i \(-0.0190584\pi\)
\(74\) −4.70711 + 8.15295i −0.547190 + 0.947761i
\(75\) 0 0
\(76\) −0.585786 −0.0671943
\(77\) −1.00000 − 2.44949i −0.113961 − 0.279145i
\(78\) 0 0
\(79\) 6.62132 + 11.4685i 0.744957 + 1.29030i 0.950215 + 0.311595i \(0.100863\pi\)
−0.205258 + 0.978708i \(0.565803\pi\)
\(80\) −0.292893 + 0.507306i −0.0327465 + 0.0567185i
\(81\) 0 0
\(82\) 2.70711 + 4.68885i 0.298950 + 0.517796i
\(83\) 12.1421 1.33277 0.666386 − 0.745607i \(-0.267840\pi\)
0.666386 + 0.745607i \(0.267840\pi\)
\(84\) 0 0
\(85\) −2.14214 −0.232347
\(86\) −2.82843 − 4.89898i −0.304997 − 0.528271i
\(87\) 0 0
\(88\) −0.500000 + 0.866025i −0.0533002 + 0.0923186i
\(89\) 6.24264 + 10.8126i 0.661719 + 1.14613i 0.980164 + 0.198189i \(0.0635060\pi\)
−0.318445 + 0.947941i \(0.603161\pi\)
\(90\) 0 0
\(91\) −6.20711 + 8.00436i −0.650682 + 0.839085i
\(92\) 6.24264 0.650840
\(93\) 0 0
\(94\) 5.24264 − 9.08052i 0.540737 − 0.936584i
\(95\) 0.171573 − 0.297173i 0.0176030 − 0.0304893i
\(96\) 0 0
\(97\) −3.82843 −0.388718 −0.194359 − 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(98\) 5.00000 − 4.89898i 0.505076 − 0.494872i
\(99\) 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 1386.2.k.t.991.2 4
3.2 odd 2 154.2.e.e.67.2 yes 4
7.2 even 3 inner 1386.2.k.t.793.2 4
7.3 odd 6 9702.2.a.ch.1.2 2
7.4 even 3 9702.2.a.cx.1.1 2
12.11 even 2 1232.2.q.f.529.1 4
21.2 odd 6 154.2.e.e.23.2 ✓ 4
21.5 even 6 1078.2.e.m.177.1 4
21.11 odd 6 1078.2.a.t.1.1 2
21.17 even 6 1078.2.a.x.1.2 2
21.20 even 2 1078.2.e.m.67.1 4
84.11 even 6 8624.2.a.cc.1.2 2
84.23 even 6 1232.2.q.f.177.1 4
84.59 odd 6 8624.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.2 ✓ 4 21.2 odd 6
154.2.e.e.67.2 yes 4 3.2 odd 2
1078.2.a.t.1.1 2 21.11 odd 6
1078.2.a.x.1.2 2 21.17 even 6
1078.2.e.m.67.1 4 21.20 even 2
1078.2.e.m.177.1 4 21.5 even 6
1232.2.q.f.177.1 4 84.23 even 6
1232.2.q.f.529.1 4 12.11 even 2
1386.2.k.t.793.2 4 7.2 even 3 inner
1386.2.k.t.991.2 4 1.1 even 1 trivial
8624.2.a.bh.1.1 2 84.59 odd 6
8624.2.a.cc.1.2 2 84.11 even 6
9702.2.a.ch.1.2 2 7.3 odd 6
9702.2.a.cx.1.1 2 7.4 even 3