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Properties

Label 1782.2.i.n.1187.3
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 1187.3
Root \(0.500000 + 1.00333i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1187
Dual form 1782.2.i.n.593.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.526885 - 0.304197i) q^{5} +(0.975173 - 0.563016i) q^{7} -1.00000 q^{8} -0.608394i q^{10} +(-3.12313 - 1.11626i) q^{11} +(1.20873 + 0.697860i) q^{13} +(0.975173 + 0.563016i) q^{14} +(-0.500000 - 0.866025i) q^{16} -4.21106 q^{17} +7.47621i q^{19} +(0.526885 - 0.304197i) q^{20} +(-0.594858 - 3.26284i) q^{22} +(2.16508 + 1.25001i) q^{23} +(-2.31493 - 4.00957i) q^{25} +1.39572i q^{26} +1.12603i q^{28} +(-0.966766 - 1.67449i) q^{29} +(-5.05922 + 8.76282i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-2.10553 - 3.64688i) q^{34} -0.685072 q^{35} -3.22063 q^{37} +(-6.47459 + 3.73811i) q^{38} +(0.526885 + 0.304197i) q^{40} +(1.54135 - 2.66970i) q^{41} +(1.11131 - 0.641614i) q^{43} +(2.52828 - 2.14658i) q^{44} +2.50002i q^{46} +(-9.14162 + 5.27792i) q^{47} +(-2.86603 + 4.96410i) q^{49} +(2.31493 - 4.00957i) q^{50} +(-1.20873 + 0.697860i) q^{52} +8.81321i q^{53} +(1.30597 + 1.53819i) q^{55} +(-0.975173 + 0.563016i) q^{56} +(0.966766 - 1.67449i) q^{58} +(-7.34566 - 4.24102i) q^{59} +(-1.35420 + 0.781847i) q^{61} -10.1184 q^{62} +1.00000 q^{64} +(-0.424574 - 0.735384i) q^{65} +(0.631310 - 1.09346i) q^{67} +(2.10553 - 3.64688i) q^{68} +(-0.342536 - 0.593290i) q^{70} +1.17980i q^{71} +6.76439i q^{73} +(-1.61031 - 2.78915i) q^{74} +(-6.47459 - 3.73811i) q^{76} +(-3.67407 + 0.669829i) q^{77} +(9.57108 - 5.52587i) q^{79} +0.608394i q^{80} +3.08271 q^{82} +(-3.07229 - 5.32137i) q^{83} +(2.21874 + 1.28099i) q^{85} +(1.11131 + 0.641614i) q^{86} +(3.12313 + 1.11626i) q^{88} +2.58819i q^{89} +1.57163 q^{91} +(-2.16508 + 1.25001i) q^{92} +(-9.14162 - 5.27792i) q^{94} +(2.27424 - 3.93910i) q^{95} +(-0.505626 - 0.875770i) q^{97} -5.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 8 q^{4} - 16 q^{8} - 8 q^{16} + 24 q^{29} - 16 q^{31} + 8 q^{32} - 48 q^{35} + 32 q^{37} - 24 q^{41} - 32 q^{49} + 80 q^{55} - 24 q^{58} - 32 q^{62} + 16 q^{64} - 24 q^{65} + 8 q^{67} - 24 q^{70}+ \cdots - 64 q^{98}+O(q^{100}) \) 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
\(5\) −0.526885 − 0.304197i −0.235630 − 0.136041i 0.377537 − 0.925995i \(-0.376771\pi\)
−0.613167 + 0.789954i \(0.710105\pi\)
\(6\) 0 0
\(7\) 0.975173 − 0.563016i 0.368581 − 0.212800i −0.304258 − 0.952590i \(-0.598408\pi\)
0.672838 + 0.739790i \(0.265075\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) − 0.608394i − 0.192391i
\(11\) −3.12313 − 1.11626i −0.941660 − 0.336565i
\(12\) 0 0
\(13\) 1.20873 + 0.697860i 0.335241 + 0.193551i 0.658166 − 0.752873i \(-0.271333\pi\)
−0.322925 + 0.946425i \(0.604666\pi\)
\(14\) 0.975173 + 0.563016i 0.260626 + 0.150472i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −4.21106 −1.02133 −0.510665 − 0.859780i \(-0.670601\pi\)
−0.510665 + 0.859780i \(0.670601\pi\)
\(18\) 0 0
\(19\) 7.47621i 1.71516i 0.514350 + 0.857580i \(0.328033\pi\)
−0.514350 + 0.857580i \(0.671967\pi\)
\(20\) 0.526885 − 0.304197i 0.117815 − 0.0680206i
\(21\) 0 0
\(22\) −0.594858 − 3.26284i −0.126824 − 0.695640i
\(23\) 2.16508 + 1.25001i 0.451450 + 0.260645i 0.708442 − 0.705769i \(-0.249398\pi\)
−0.256992 + 0.966413i \(0.582732\pi\)
\(24\) 0 0
\(25\) −2.31493 − 4.00957i −0.462986 − 0.801915i
\(26\) 1.39572i 0.273723i
\(27\) 0 0
\(28\) 1.12603i 0.212800i
\(29\) −0.966766 − 1.67449i −0.179524 − 0.310945i 0.762194 − 0.647349i \(-0.224122\pi\)
−0.941718 + 0.336404i \(0.890789\pi\)
\(30\) 0 0
\(31\) −5.05922 + 8.76282i −0.908662 + 1.57385i −0.0927377 + 0.995691i \(0.529562\pi\)
−0.815924 + 0.578158i \(0.803772\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2.10553 − 3.64688i −0.361095 − 0.625435i
\(35\) −0.685072 −0.115798
\(36\) 0 0
\(37\) −3.22063 −0.529468 −0.264734 − 0.964321i \(-0.585284\pi\)
−0.264734 + 0.964321i \(0.585284\pi\)
\(38\) −6.47459 + 3.73811i −1.05032 + 0.606401i
\(39\) 0 0
\(40\) 0.526885 + 0.304197i 0.0833078 + 0.0480978i
\(41\) 1.54135 − 2.66970i 0.240719 − 0.416937i −0.720200 − 0.693766i \(-0.755950\pi\)
0.960919 + 0.276829i \(0.0892835\pi\)
\(42\) 0 0
\(43\) 1.11131 − 0.641614i 0.169473 − 0.0978451i −0.412864 − 0.910792i \(-0.635472\pi\)
0.582337 + 0.812947i \(0.302138\pi\)
\(44\) 2.52828 − 2.14658i 0.381152 − 0.323610i
\(45\) 0 0
\(46\) 2.50002i 0.368607i
\(47\) −9.14162 + 5.27792i −1.33344 + 0.769863i −0.985826 − 0.167773i \(-0.946342\pi\)
−0.347617 + 0.937637i \(0.613009\pi\)
\(48\) 0 0
\(49\) −2.86603 + 4.96410i −0.409432 + 0.709157i
\(50\) 2.31493 − 4.00957i 0.327380 − 0.567039i
\(51\) 0 0
\(52\) −1.20873 + 0.697860i −0.167621 + 0.0967757i
\(53\) 8.81321i 1.21059i 0.796002 + 0.605294i \(0.206944\pi\)
−0.796002 + 0.605294i \(0.793056\pi\)
\(54\) 0 0
\(55\) 1.30597 + 1.53819i 0.176097 + 0.207409i
\(56\) −0.975173 + 0.563016i −0.130313 + 0.0752362i
\(57\) 0 0
\(58\) 0.966766 − 1.67449i 0.126943 − 0.219871i
\(59\) −7.34566 − 4.24102i −0.956324 − 0.552134i −0.0612840 − 0.998120i \(-0.519520\pi\)
−0.895040 + 0.445987i \(0.852853\pi\)
\(60\) 0 0
\(61\) −1.35420 + 0.781847i −0.173387 + 0.100105i −0.584182 − 0.811623i \(-0.698585\pi\)
0.410795 + 0.911728i \(0.365251\pi\)
\(62\) −10.1184 −1.28504
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.424574 − 0.735384i −0.0526619 − 0.0912131i
\(66\) 0 0
\(67\) 0.631310 − 1.09346i 0.0771268 − 0.133587i −0.824882 − 0.565304i \(-0.808759\pi\)
0.902009 + 0.431717i \(0.142092\pi\)
\(68\) 2.10553 − 3.64688i 0.255333 − 0.442249i
\(69\) 0 0
\(70\) −0.342536 − 0.593290i −0.0409409 − 0.0709117i
\(71\) 1.17980i 0.140017i 0.997546 + 0.0700083i \(0.0223026\pi\)
−0.997546 + 0.0700083i \(0.977697\pi\)
\(72\) 0 0
\(73\) 6.76439i 0.791712i 0.918313 + 0.395856i \(0.129552\pi\)
−0.918313 + 0.395856i \(0.870448\pi\)
\(74\) −1.61031 − 2.78915i −0.187195 − 0.324232i
\(75\) 0 0
\(76\) −6.47459 − 3.73811i −0.742686 − 0.428790i
\(77\) −3.67407 + 0.669829i −0.418699 + 0.0763341i
\(78\) 0 0
\(79\) 9.57108 − 5.52587i 1.07683 − 0.621709i 0.146791 − 0.989167i \(-0.453105\pi\)
0.930040 + 0.367459i \(0.119772\pi\)
\(80\) 0.608394i 0.0680206i
\(81\) 0 0
\(82\) 3.08271 0.340428
\(83\) −3.07229 − 5.32137i −0.337228 − 0.584096i 0.646682 − 0.762760i \(-0.276156\pi\)
−0.983910 + 0.178663i \(0.942823\pi\)
\(84\) 0 0
\(85\) 2.21874 + 1.28099i 0.240656 + 0.138943i
\(86\) 1.11131 + 0.641614i 0.119835 + 0.0691870i
\(87\) 0 0
\(88\) 3.12313 + 1.11626i 0.332927 + 0.118994i
\(89\) 2.58819i 0.274348i 0.990547 + 0.137174i \(0.0438019\pi\)
−0.990547 + 0.137174i \(0.956198\pi\)
\(90\) 0 0
\(91\) 1.57163 0.164751
\(92\) −2.16508 + 1.25001i −0.225725 + 0.130322i
\(93\) 0 0
\(94\) −9.14162 − 5.27792i −0.942886 − 0.544376i
\(95\) 2.27424 − 3.93910i 0.233332 − 0.404144i
\(96\) 0 0
\(97\) −0.505626 − 0.875770i −0.0513385 − 0.0889209i 0.839214 − 0.543801i \(-0.183015\pi\)
−0.890553 + 0.454880i \(0.849682\pi\)
\(98\) −5.73205 −0.579025
\(99\) 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.1187.3 16
3.2 odd 2 1782.2.i.m.1187.6 16
9.2 odd 6 1782.2.i.m.593.3 16
9.4 even 3 1782.2.b.c.1781.6 yes 8
9.5 odd 6 1782.2.b.d.1781.3 yes 8
9.7 even 3 inner 1782.2.i.n.593.6 16
11.10 odd 2 1782.2.i.m.1187.3 16
33.32 even 2 inner 1782.2.i.n.1187.6 16
99.32 even 6 1782.2.b.c.1781.3 ✓ 8
99.43 odd 6 1782.2.i.m.593.6 16
99.65 even 6 inner 1782.2.i.n.593.3 16
99.76 odd 6 1782.2.b.d.1781.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1782.2.b.c.1781.3 ✓ 8 99.32 even 6
1782.2.b.c.1781.6 yes 8 9.4 even 3
1782.2.b.d.1781.3 yes 8 9.5 odd 6
1782.2.b.d.1781.6 yes 8 99.76 odd 6
1782.2.i.m.593.3 16 9.2 odd 6
1782.2.i.m.593.6 16 99.43 odd 6
1782.2.i.m.1187.3 16 11.10 odd 2
1782.2.i.m.1187.6 16 3.2 odd 2
1782.2.i.n.593.3 16 99.65 even 6 inner
1782.2.i.n.593.6 16 9.7 even 3 inner
1782.2.i.n.1187.3 16 1.1 even 1 trivial
1782.2.i.n.1187.6 16 33.32 even 2 inner