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Properties

Label 1782.2.i.n.1187.4
Level $1782$
Weight $2$
Character 1782.1187
Analytic conductor $14.229$
Analytic rank $0$
Dimension $16$
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] = [1782,2,Mod(593,1782)] 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("1782.593"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1782, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1782 = 2 \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1782.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,8,0,-8,0,0,0,-16,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.2293416402\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.9349208943630483456.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1187.4
Root \(0.500000 + 1.33108i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1187
Dual form 1782.2.i.n.593.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.210856 - 0.121738i) q^{5} +(1.88389 - 1.08766i) q^{7} -1.00000 q^{8} -0.243476i q^{10} +(2.11381 + 2.55574i) q^{11} +(2.48653 + 1.43560i) q^{13} +(1.88389 + 1.08766i) q^{14} +(-0.500000 - 0.866025i) q^{16} -0.907738 q^{17} -2.91074i q^{19} +(0.210856 - 0.121738i) q^{20} +(-1.15643 + 3.10849i) q^{22} +(4.83817 + 2.79332i) q^{23} +(-2.47036 - 4.27879i) q^{25} +2.87120i q^{26} +2.17533i q^{28} +(-1.72724 - 2.99166i) q^{29} +(-1.15086 + 1.99335i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-0.453869 - 0.786124i) q^{34} -0.529640 q^{35} +8.37105 q^{37} +(2.52077 - 1.45537i) q^{38} +(0.210856 + 0.121738i) q^{40} +(-1.23999 + 2.14773i) q^{41} +(4.41645 - 2.54984i) q^{43} +(-3.27024 + 0.552749i) q^{44} +5.58663i q^{46} +(0.656339 - 0.378937i) q^{47} +(-1.13397 + 1.96410i) q^{49} +(2.47036 - 4.27879i) q^{50} +(-2.48653 + 1.43560i) q^{52} +9.07140i q^{53} +(-0.134581 - 0.796225i) q^{55} +(-1.88389 + 1.08766i) q^{56} +(1.72724 - 2.99166i) q^{58} +(10.5920 + 6.11530i) q^{59} +(10.5772 - 6.10676i) q^{61} -2.30172 q^{62} +1.00000 q^{64} +(-0.349535 - 0.605412i) q^{65} +(-5.32531 + 9.22371i) q^{67} +(0.453869 - 0.786124i) q^{68} +(-0.264820 - 0.458682i) q^{70} +9.05778i q^{71} +0.803890i q^{73} +(4.18553 + 7.24954i) q^{74} +(2.52077 + 1.45537i) q^{76} +(6.76197 + 2.51560i) q^{77} +(1.48444 - 0.857039i) q^{79} +0.243476i q^{80} -2.47999 q^{82} +(-2.18111 - 3.77779i) q^{83} +(0.191402 + 0.110506i) q^{85} +(4.41645 + 2.54984i) q^{86} +(-2.11381 - 2.55574i) q^{88} +9.65926i q^{89} +6.24581 q^{91} +(-4.83817 + 2.79332i) q^{92} +(0.656339 + 0.378937i) q^{94} +(-0.354348 + 0.613748i) q^{95} +(0.0216982 + 0.0375824i) q^{97} -2.26795 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 8 q^{4} - 16 q^{8} - 8 q^{16} + 24 q^{29} - 16 q^{31} + 8 q^{32} - 48 q^{35} + 32 q^{37} - 24 q^{41} - 32 q^{49} + 80 q^{55} - 24 q^{58} - 32 q^{62} + 16 q^{64} - 24 q^{65} + 8 q^{67} - 24 q^{70}+ \cdots - 64 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1135\) \(1541\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −0.210856 − 0.121738i −0.0942979 − 0.0544429i 0.452109 − 0.891962i \(-0.350672\pi\)
−0.546407 + 0.837520i \(0.684005\pi\)
\(6\) 0 0
\(7\) 1.88389 − 1.08766i 0.712043 − 0.411098i −0.0997739 − 0.995010i \(-0.531812\pi\)
0.811817 + 0.583912i \(0.198479\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) − 0.243476i − 0.0769939i
\(11\) 2.11381 + 2.55574i 0.637339 + 0.770583i
\(12\) 0 0
\(13\) 2.48653 + 1.43560i 0.689641 + 0.398164i 0.803477 − 0.595335i \(-0.202981\pi\)
−0.113837 + 0.993499i \(0.536314\pi\)
\(14\) 1.88389 + 1.08766i 0.503491 + 0.290690i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −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 \(-0.108360\pi\)
\(20\) 0.210856 − 0.121738i 0.0471489 − 0.0272215i
\(21\) 0 0
\(22\) −1.15643 + 3.10849i −0.246551 + 0.662731i
\(23\) 4.83817 + 2.79332i 1.00883 + 0.582447i 0.910848 − 0.412742i \(-0.135429\pi\)
0.0979792 + 0.995188i \(0.468762\pi\)
\(24\) 0 0
\(25\) −2.47036 − 4.27879i −0.494072 − 0.855758i
\(26\) 2.87120i 0.563089i
\(27\) 0 0
\(28\) 2.17533i 0.411098i
\(29\) −1.72724 − 2.99166i −0.320740 − 0.555538i 0.659901 − 0.751353i \(-0.270598\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(30\) 0 0
\(31\) −1.15086 + 1.99335i −0.206701 + 0.358016i −0.950673 − 0.310194i \(-0.899606\pi\)
0.743973 + 0.668210i \(0.232939\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −0.453869 − 0.786124i −0.0778379 − 0.134819i
\(35\) −0.529640 −0.0895256
\(36\) 0 0
\(37\) 8.37105 1.37619 0.688096 − 0.725620i \(-0.258447\pi\)
0.688096 + 0.725620i \(0.258447\pi\)
\(38\) 2.52077 − 1.45537i 0.408924 − 0.236092i
\(39\) 0 0
\(40\) 0.210856 + 0.121738i 0.0333393 + 0.0192485i
\(41\) −1.23999 + 2.14773i −0.193654 + 0.335419i −0.946459 − 0.322825i \(-0.895367\pi\)
0.752804 + 0.658245i \(0.228701\pi\)
\(42\) 0 0
\(43\) 4.41645 − 2.54984i 0.673503 − 0.388847i −0.123900 − 0.992295i \(-0.539540\pi\)
0.797403 + 0.603448i \(0.206207\pi\)
\(44\) −3.27024 + 0.552749i −0.493007 + 0.0833301i
\(45\) 0 0
\(46\) 5.58663i 0.823704i
\(47\) 0.656339 − 0.378937i 0.0957369 − 0.0552737i −0.451367 − 0.892338i \(-0.649064\pi\)
0.547104 + 0.837065i \(0.315730\pi\)
\(48\) 0 0
\(49\) −1.13397 + 1.96410i −0.161996 + 0.280586i
\(50\) 2.47036 − 4.27879i 0.349362 − 0.605112i
\(51\) 0 0
\(52\) −2.48653 + 1.43560i −0.344820 + 0.199082i
\(53\) 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
\(56\) −1.88389 + 1.08766i −0.251745 + 0.145345i
\(57\) 0 0
\(58\) 1.72724 − 2.99166i 0.226797 − 0.392825i
\(59\) 10.5920 + 6.11530i 1.37896 + 0.796144i 0.992035 − 0.125966i \(-0.0402031\pi\)
0.386927 + 0.922110i \(0.373536\pi\)
\(60\) 0 0
\(61\) 10.5772 − 6.10676i 1.35427 − 0.781891i 0.365429 − 0.930839i \(-0.380922\pi\)
0.988845 + 0.148948i \(0.0475888\pi\)
\(62\) −2.30172 −0.292319
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.349535 − 0.605412i −0.0433544 − 0.0750921i
\(66\) 0 0
\(67\) −5.32531 + 9.22371i −0.650591 + 1.12686i 0.332389 + 0.943142i \(0.392145\pi\)
−0.982980 + 0.183714i \(0.941188\pi\)
\(68\) 0.453869 − 0.786124i 0.0550397 − 0.0953315i
\(69\) 0 0
\(70\) −0.264820 − 0.458682i −0.0316521 − 0.0548230i
\(71\) 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 \(0.985020\pi\)
\(74\) 4.18553 + 7.24954i 0.486557 + 0.842742i
\(75\) 0 0
\(76\) 2.52077 + 1.45537i 0.289153 + 0.166942i
\(77\) 6.76197 + 2.51560i 0.770598 + 0.286680i
\(78\) 0 0
\(79\) 1.48444 − 0.857039i 0.167012 − 0.0964245i −0.414164 − 0.910202i \(-0.635926\pi\)
0.581176 + 0.813778i \(0.302593\pi\)
\(80\) 0.243476i 0.0272215i
\(81\) 0 0
\(82\) −2.47999 −0.273869
\(83\) −2.18111 − 3.77779i −0.239408 − 0.414666i 0.721137 − 0.692793i \(-0.243620\pi\)
−0.960544 + 0.278127i \(0.910287\pi\)
\(84\) 0 0
\(85\) 0.191402 + 0.110506i 0.0207605 + 0.0119861i
\(86\) 4.41645 + 2.54984i 0.476238 + 0.274956i
\(87\) 0 0
\(88\) −2.11381 − 2.55574i −0.225333 − 0.272442i
\(89\) 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
\(92\) −4.83817 + 2.79332i −0.504414 + 0.291223i
\(93\) 0 0
\(94\) 0.656339 + 0.378937i 0.0676962 + 0.0390844i
\(95\) −0.354348 + 0.613748i −0.0363553 + 0.0629692i
\(96\) 0 0
\(97\) 0.0216982 + 0.0375824i 0.00220312 + 0.00381591i 0.867125 − 0.498091i \(-0.165965\pi\)
−0.864922 + 0.501907i \(0.832632\pi\)
\(98\) −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.i.n.1187.4 16
3.2 odd 2 1782.2.i.m.1187.5 16
9.2 odd 6 1782.2.i.m.593.4 16
9.4 even 3 1782.2.b.c.1781.5 yes 8
9.5 odd 6 1782.2.b.d.1781.4 yes 8
9.7 even 3 inner 1782.2.i.n.593.5 16
11.10 odd 2 1782.2.i.m.1187.4 16
33.32 even 2 inner 1782.2.i.n.1187.5 16
99.32 even 6 1782.2.b.c.1781.4 ✓ 8
99.43 odd 6 1782.2.i.m.593.5 16
99.65 even 6 inner 1782.2.i.n.593.4 16
99.76 odd 6 1782.2.b.d.1781.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1782.2.b.c.1781.4 ✓ 8 99.32 even 6
1782.2.b.c.1781.5 yes 8 9.4 even 3
1782.2.b.d.1781.4 yes 8 9.5 odd 6
1782.2.b.d.1781.5 yes 8 99.76 odd 6
1782.2.i.m.593.4 16 9.2 odd 6
1782.2.i.m.593.5 16 99.43 odd 6
1782.2.i.m.1187.4 16 11.10 odd 2
1782.2.i.m.1187.5 16 3.2 odd 2
1782.2.i.n.593.4 16 99.65 even 6 inner
1782.2.i.n.593.5 16 9.7 even 3 inner
1782.2.i.n.1187.4 16 1.1 even 1 trivial
1782.2.i.n.1187.5 16 33.32 even 2 inner