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Properties

Label 1782.2.b.c.1781.5
Level $1782$
Weight $2$
Character 1782.1781
Analytic conductor $14.229$
Analytic rank $0$
Dimension $8$
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] = [1782,2,Mod(1781,1782)] 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("1782.1781"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1782, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1782 = 2 \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1782.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.2293416402\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3057647616.5
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{5} + 10x^{4} + 12x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1781.5
Root \(2.90421 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1781
Dual form 1782.2.b.c.1781.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.243476i q^{5} +2.17533i q^{7} -1.00000 q^{8} -0.243476i q^{10} +(-3.27024 + 0.552749i) q^{11} -2.87120i q^{13} -2.17533i q^{14} +1.00000 q^{16} -0.907738 q^{17} -2.91074i q^{19} +0.243476i q^{20} +(3.27024 - 0.552749i) q^{22} -5.58663i q^{23} +4.94072 q^{25} +2.87120i q^{26} +2.17533i q^{28} +3.45448 q^{29} +2.30172 q^{31} -1.00000 q^{32} +0.907738 q^{34} -0.529640 q^{35} +8.37105 q^{37} +2.91074i q^{38} -0.243476i q^{40} +2.47999 q^{41} +5.09968i q^{43} +(-3.27024 + 0.552749i) q^{44} +5.58663i q^{46} +0.757875i q^{47} +2.26795 q^{49} -4.94072 q^{50} -2.87120i q^{52} +9.07140i q^{53} +(-0.134581 - 0.796225i) q^{55} -2.17533i q^{56} -3.45448 q^{58} -12.2306i q^{59} +12.2135i q^{61} -2.30172 q^{62} +1.00000 q^{64} +0.699069 q^{65} +10.6506 q^{67} -0.907738 q^{68} +0.529640 q^{70} +9.05778i q^{71} +0.803890i q^{73} -8.37105 q^{74} -2.91074i q^{76} +(-1.20241 - 7.11384i) q^{77} +1.71408i q^{79} +0.243476i q^{80} -2.47999 q^{82} +4.36221 q^{83} -0.221012i q^{85} -5.09968i q^{86} +(3.27024 - 0.552749i) q^{88} +9.65926i q^{89} +6.24581 q^{91} -5.58663i q^{92} -0.757875i q^{94} +0.708695 q^{95} -0.0433964 q^{97} -2.26795 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 24 q^{29} + 16 q^{31} - 8 q^{32} - 24 q^{35} + 16 q^{37} + 24 q^{41} + 32 q^{49} + 40 q^{55} + 24 q^{58} - 16 q^{62} + 8 q^{64} + 24 q^{65} - 8 q^{67} + 24 q^{70}+ \cdots - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1782\mathbb{Z}\right)^\times\).

\(n\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0.243476i 0.108886i 0.998517 + 0.0544429i \(0.0173383\pi\)
−0.998517 + 0.0544429i \(0.982662\pi\)
\(6\) 0 0
\(7\) 2.17533i 0.822197i 0.911591 + 0.411098i \(0.134855\pi\)
−0.911591 + 0.411098i \(0.865145\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) − 0.243476i − 0.0769939i
\(11\) −3.27024 + 0.552749i −0.986014 + 0.166660i
\(12\) 0 0
\(13\) − 2.87120i − 0.796328i −0.917314 − 0.398164i \(-0.869647\pi\)
0.917314 − 0.398164i \(-0.130353\pi\)
\(14\) − 2.17533i − 0.581381i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.907738 −0.220159 −0.110079 − 0.993923i \(-0.535111\pi\)
−0.110079 + 0.993923i \(0.535111\pi\)
\(18\) 0 0
\(19\) − 2.91074i − 0.667769i −0.942614 − 0.333885i \(-0.891640\pi\)
0.942614 − 0.333885i \(-0.108360\pi\)
\(20\) 0.243476i 0.0544429i
\(21\) 0 0
\(22\) 3.27024 − 0.552749i 0.697217 − 0.117847i
\(23\) − 5.58663i − 1.16489i −0.812869 − 0.582447i \(-0.802095\pi\)
0.812869 − 0.582447i \(-0.197905\pi\)
\(24\) 0 0
\(25\) 4.94072 0.988144
\(26\) 2.87120i 0.563089i
\(27\) 0 0
\(28\) 2.17533i 0.411098i
\(29\) 3.45448 0.641480 0.320740 − 0.947167i \(-0.396068\pi\)
0.320740 + 0.947167i \(0.396068\pi\)
\(30\) 0 0
\(31\) 2.30172 0.413401 0.206701 − 0.978404i \(-0.433727\pi\)
0.206701 + 0.978404i \(0.433727\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.907738 0.155676
\(35\) −0.529640 −0.0895256
\(36\) 0 0
\(37\) 8.37105 1.37619 0.688096 − 0.725620i \(-0.258447\pi\)
0.688096 + 0.725620i \(0.258447\pi\)
\(38\) 2.91074i 0.472184i
\(39\) 0 0
\(40\) − 0.243476i − 0.0384970i
\(41\) 2.47999 0.387309 0.193654 − 0.981070i \(-0.437966\pi\)
0.193654 + 0.981070i \(0.437966\pi\)
\(42\) 0 0
\(43\) 5.09968i 0.777694i 0.921302 + 0.388847i \(0.127127\pi\)
−0.921302 + 0.388847i \(0.872873\pi\)
\(44\) −3.27024 + 0.552749i −0.493007 + 0.0833301i
\(45\) 0 0
\(46\) 5.58663i 0.823704i
\(47\) 0.757875i 0.110547i 0.998471 + 0.0552737i \(0.0176031\pi\)
−0.998471 + 0.0552737i \(0.982397\pi\)
\(48\) 0 0
\(49\) 2.26795 0.323993
\(50\) −4.94072 −0.698723
\(51\) 0 0
\(52\) − 2.87120i − 0.398164i
\(53\) 9.07140i 1.24605i 0.782201 + 0.623026i \(0.214097\pi\)
−0.782201 + 0.623026i \(0.785903\pi\)
\(54\) 0 0
\(55\) −0.134581 − 0.796225i −0.0181469 − 0.107363i
\(56\) − 2.17533i − 0.290690i
\(57\) 0 0
\(58\) −3.45448 −0.453595
\(59\) − 12.2306i − 1.59229i −0.605107 − 0.796144i \(-0.706870\pi\)
0.605107 − 0.796144i \(-0.293130\pi\)
\(60\) 0 0
\(61\) 12.2135i 1.56378i 0.623416 + 0.781891i \(0.285745\pi\)
−0.623416 + 0.781891i \(0.714255\pi\)
\(62\) −2.30172 −0.292319
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.699069 0.0867089
\(66\) 0 0
\(67\) 10.6506 1.30118 0.650591 − 0.759429i \(-0.274521\pi\)
0.650591 + 0.759429i \(0.274521\pi\)
\(68\) −0.907738 −0.110079
\(69\) 0 0
\(70\) 0.529640 0.0633041
\(71\) 9.05778i 1.07496i 0.843276 + 0.537481i \(0.180624\pi\)
−0.843276 + 0.537481i \(0.819376\pi\)
\(72\) 0 0
\(73\) 0.803890i 0.0940882i 0.998893 + 0.0470441i \(0.0149801\pi\)
−0.998893 + 0.0470441i \(0.985020\pi\)
\(74\) −8.37105 −0.973115
\(75\) 0 0
\(76\) − 2.91074i − 0.333885i
\(77\) −1.20241 − 7.11384i −0.137027 − 0.810698i
\(78\) 0 0
\(79\) 1.71408i 0.192849i 0.995340 + 0.0964245i \(0.0307406\pi\)
−0.995340 + 0.0964245i \(0.969259\pi\)
\(80\) 0.243476i 0.0272215i
\(81\) 0 0
\(82\) −2.47999 −0.273869
\(83\) 4.36221 0.478815 0.239408 − 0.970919i \(-0.423047\pi\)
0.239408 + 0.970919i \(0.423047\pi\)
\(84\) 0 0
\(85\) − 0.221012i − 0.0239722i
\(86\) − 5.09968i − 0.549913i
\(87\) 0 0
\(88\) 3.27024 − 0.552749i 0.348609 − 0.0589233i
\(89\) 9.65926i 1.02388i 0.859021 + 0.511940i \(0.171073\pi\)
−0.859021 + 0.511940i \(0.828927\pi\)
\(90\) 0 0
\(91\) 6.24581 0.654738
\(92\) − 5.58663i − 0.582447i
\(93\) 0 0
\(94\) − 0.757875i − 0.0781688i
\(95\) 0.708695 0.0727106
\(96\) 0 0
\(97\) −0.0433964 −0.00440624 −0.00220312 − 0.999998i \(-0.500701\pi\)
−0.00220312 + 0.999998i \(0.500701\pi\)
\(98\) −2.26795 −0.229097
\(99\) 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 1782.2.b.c.1781.5 yes 8
3.2 odd 2 1782.2.b.d.1781.4 yes 8
9.2 odd 6 1782.2.i.m.1187.5 16
9.4 even 3 1782.2.i.n.593.5 16
9.5 odd 6 1782.2.i.m.593.4 16
9.7 even 3 1782.2.i.n.1187.4 16
11.10 odd 2 1782.2.b.d.1781.5 yes 8
33.32 even 2 inner 1782.2.b.c.1781.4 ✓ 8
99.32 even 6 1782.2.i.n.593.4 16
99.43 odd 6 1782.2.i.m.1187.4 16
99.65 even 6 1782.2.i.n.1187.5 16
99.76 odd 6 1782.2.i.m.593.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1782.2.b.c.1781.4 ✓ 8 33.32 even 2 inner
1782.2.b.c.1781.5 yes 8 1.1 even 1 trivial
1782.2.b.d.1781.4 yes 8 3.2 odd 2
1782.2.b.d.1781.5 yes 8 11.10 odd 2
1782.2.i.m.593.4 16 9.5 odd 6
1782.2.i.m.593.5 16 99.76 odd 6
1782.2.i.m.1187.4 16 99.43 odd 6
1782.2.i.m.1187.5 16 9.2 odd 6
1782.2.i.n.593.4 16 99.32 even 6
1782.2.i.n.593.5 16 9.4 even 3
1782.2.i.n.1187.4 16 9.7 even 3
1782.2.i.n.1187.5 16 99.65 even 6