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Properties

Label 1782.2.b.c.1781.6
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 1781.6
Root \(-1.16225 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1781
Dual form 1782.2.b.c.1781.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.608394i q^{5} +1.12603i q^{7} -1.00000 q^{8} -0.608394i q^{10} +(2.52828 - 2.14658i) q^{11} -1.39572i q^{13} -1.12603i q^{14} +1.00000 q^{16} -4.21106 q^{17} +7.47621i q^{19} +0.608394i q^{20} +(-2.52828 + 2.14658i) q^{22} -2.50002i q^{23} +4.62986 q^{25} +1.39572i q^{26} +1.12603i q^{28} +1.93353 q^{29} +10.1184 q^{31} -1.00000 q^{32} +4.21106 q^{34} -0.685072 q^{35} -3.22063 q^{37} -7.47621i q^{38} -0.608394i q^{40} -3.08271 q^{41} +1.28323i q^{43} +(2.52828 - 2.14658i) q^{44} +2.50002i q^{46} -10.5558i q^{47} +5.73205 q^{49} -4.62986 q^{50} -1.39572i q^{52} +8.81321i q^{53} +(1.30597 + 1.53819i) q^{55} -1.12603i q^{56} -1.93353 q^{58} +8.48204i q^{59} -1.56369i q^{61} -10.1184 q^{62} +1.00000 q^{64} +0.849148 q^{65} -1.26262 q^{67} -4.21106 q^{68} +0.685072 q^{70} +1.17980i q^{71} +6.76439i q^{73} +3.22063 q^{74} +7.47621i q^{76} +(2.41712 + 2.84692i) q^{77} +11.0517i q^{79} +0.608394i q^{80} +3.08271 q^{82} +6.14459 q^{83} -2.56198i q^{85} -1.28323i q^{86} +(-2.52828 + 2.14658i) q^{88} +2.58819i q^{89} +1.57163 q^{91} -2.50002i q^{92} +10.5558i q^{94} -4.54849 q^{95} +1.01125 q^{97} -5.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{8} + 8 q^{16} - 24 q^{29} + 16 q^{31} - 8 q^{32} - 24 q^{35} + 16 q^{37} + 24 q^{41} + 32 q^{49} + 40 q^{55} + 24 q^{58} - 16 q^{62} + 8 q^{64} + 24 q^{65} - 8 q^{67} + 24 q^{70}+ \cdots - 32 q^{98}+O(q^{100}) \) 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
\(5\) 0.608394i 0.272082i 0.990703 + 0.136041i \(0.0434379\pi\)
−0.990703 + 0.136041i \(0.956562\pi\)
\(6\) 0 0
\(7\) 1.12603i 0.425600i 0.977096 + 0.212800i \(0.0682583\pi\)
−0.977096 + 0.212800i \(0.931742\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) − 0.608394i − 0.192391i
\(11\) 2.52828 − 2.14658i 0.762304 − 0.647219i
\(12\) 0 0
\(13\) − 1.39572i − 0.387103i −0.981090 − 0.193551i \(-0.937999\pi\)
0.981090 − 0.193551i \(-0.0620007\pi\)
\(14\) − 1.12603i − 0.300945i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −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 \(0.671967\pi\)
\(20\) 0.608394i 0.136041i
\(21\) 0 0
\(22\) −2.52828 + 2.14658i −0.539030 + 0.457653i
\(23\) − 2.50002i − 0.521289i −0.965435 − 0.260645i \(-0.916065\pi\)
0.965435 − 0.260645i \(-0.0839351\pi\)
\(24\) 0 0
\(25\) 4.62986 0.925971
\(26\) 1.39572i 0.273723i
\(27\) 0 0
\(28\) 1.12603i 0.212800i
\(29\) 1.93353 0.359048 0.179524 − 0.983754i \(-0.442544\pi\)
0.179524 + 0.983754i \(0.442544\pi\)
\(30\) 0 0
\(31\) 10.1184 1.81732 0.908662 − 0.417532i \(-0.137105\pi\)
0.908662 + 0.417532i \(0.137105\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.21106 0.722190
\(35\) −0.685072 −0.115798
\(36\) 0 0
\(37\) −3.22063 −0.529468 −0.264734 − 0.964321i \(-0.585284\pi\)
−0.264734 + 0.964321i \(0.585284\pi\)
\(38\) − 7.47621i − 1.21280i
\(39\) 0 0
\(40\) − 0.608394i − 0.0961956i
\(41\) −3.08271 −0.481438 −0.240719 − 0.970595i \(-0.577383\pi\)
−0.240719 + 0.970595i \(0.577383\pi\)
\(42\) 0 0
\(43\) 1.28323i 0.195690i 0.995202 + 0.0978451i \(0.0311950\pi\)
−0.995202 + 0.0978451i \(0.968805\pi\)
\(44\) 2.52828 − 2.14658i 0.381152 − 0.323610i
\(45\) 0 0
\(46\) 2.50002i 0.368607i
\(47\) − 10.5558i − 1.53973i −0.638209 − 0.769863i \(-0.720324\pi\)
0.638209 − 0.769863i \(-0.279676\pi\)
\(48\) 0 0
\(49\) 5.73205 0.818864
\(50\) −4.62986 −0.654761
\(51\) 0 0
\(52\) − 1.39572i − 0.193551i
\(53\) 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
\(56\) − 1.12603i − 0.150472i
\(57\) 0 0
\(58\) −1.93353 −0.253885
\(59\) 8.48204i 1.10427i 0.833756 + 0.552134i \(0.186186\pi\)
−0.833756 + 0.552134i \(0.813814\pi\)
\(60\) 0 0
\(61\) − 1.56369i − 0.200210i −0.994977 − 0.100105i \(-0.968082\pi\)
0.994977 − 0.100105i \(-0.0319179\pi\)
\(62\) −10.1184 −1.28504
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.849148 0.105324
\(66\) 0 0
\(67\) −1.26262 −0.154254 −0.0771268 − 0.997021i \(-0.524575\pi\)
−0.0771268 + 0.997021i \(0.524575\pi\)
\(68\) −4.21106 −0.510665
\(69\) 0 0
\(70\) 0.685072 0.0818818
\(71\) 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 \(0.870448\pi\)
\(74\) 3.22063 0.374390
\(75\) 0 0
\(76\) 7.47621i 0.857580i
\(77\) 2.41712 + 2.84692i 0.275457 + 0.324437i
\(78\) 0 0
\(79\) 11.0517i 1.24342i 0.783249 + 0.621709i \(0.213561\pi\)
−0.783249 + 0.621709i \(0.786439\pi\)
\(80\) 0.608394i 0.0680206i
\(81\) 0 0
\(82\) 3.08271 0.340428
\(83\) 6.14459 0.674456 0.337228 − 0.941423i \(-0.390511\pi\)
0.337228 + 0.941423i \(0.390511\pi\)
\(84\) 0 0
\(85\) − 2.56198i − 0.277886i
\(86\) − 1.28323i − 0.138374i
\(87\) 0 0
\(88\) −2.52828 + 2.14658i −0.269515 + 0.228827i
\(89\) 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
\(92\) − 2.50002i − 0.260645i
\(93\) 0 0
\(94\) 10.5558i 1.08875i
\(95\) −4.54849 −0.466665
\(96\) 0 0
\(97\) 1.01125 0.102677 0.0513385 − 0.998681i \(-0.483651\pi\)
0.0513385 + 0.998681i \(0.483651\pi\)
\(98\) −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.b.c.1781.6 yes 8
3.2 odd 2 1782.2.b.d.1781.3 yes 8
9.2 odd 6 1782.2.i.m.1187.6 16
9.4 even 3 1782.2.i.n.593.6 16
9.5 odd 6 1782.2.i.m.593.3 16
9.7 even 3 1782.2.i.n.1187.3 16
11.10 odd 2 1782.2.b.d.1781.6 yes 8
33.32 even 2 inner 1782.2.b.c.1781.3 ✓ 8
99.32 even 6 1782.2.i.n.593.3 16
99.43 odd 6 1782.2.i.m.1187.3 16
99.65 even 6 1782.2.i.n.1187.6 16
99.76 odd 6 1782.2.i.m.593.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1782.2.b.c.1781.3 ✓ 8 33.32 even 2 inner
1782.2.b.c.1781.6 yes 8 1.1 even 1 trivial
1782.2.b.d.1781.3 yes 8 3.2 odd 2
1782.2.b.d.1781.6 yes 8 11.10 odd 2
1782.2.i.m.593.3 16 9.5 odd 6
1782.2.i.m.593.6 16 99.76 odd 6
1782.2.i.m.1187.3 16 99.43 odd 6
1782.2.i.m.1187.6 16 9.2 odd 6
1782.2.i.n.593.3 16 99.32 even 6
1782.2.i.n.593.6 16 9.4 even 3
1782.2.i.n.1187.3 16 9.7 even 3
1782.2.i.n.1187.6 16 99.65 even 6