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Properties

Label 1782.2.i.l.1187.4
Level $1782$
Weight $2$
Character 1782.1187
Analytic conductor $14.229$
Analytic rank $0$
Dimension $8$
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 = [8,4,0,-4,0,0,0,-8,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,12] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1187.4
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1187
Dual form 1782.2.i.l.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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(3.34607 + 1.93185i) q^{5} +(4.24264 - 2.44949i) q^{7} -1.00000 q^{8} +3.86370i q^{10} +(-1.58346 + 2.91421i) q^{11} +(3.67423 + 2.12132i) q^{13} +(4.24264 + 2.44949i) q^{14} +(-0.500000 - 0.866025i) q^{16} -3.46410 q^{17} +3.34607i q^{19} +(-3.34607 + 1.93185i) q^{20} +(-3.31552 + 0.0857864i) q^{22} +(-7.46859 - 4.31199i) q^{23} +(4.96410 + 8.59808i) q^{25} +4.24264i q^{26} +4.89898i q^{28} +(0.866025 + 1.50000i) q^{29} +(1.13397 - 1.96410i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-1.73205 - 3.00000i) q^{34} +18.9282 q^{35} +2.92820 q^{37} +(-2.89778 + 1.67303i) q^{38} +(-3.34607 - 1.93185i) q^{40} +(1.26795 - 2.19615i) q^{41} +(-0.776457 + 0.448288i) q^{43} +(-1.73205 - 2.82843i) q^{44} -8.62398i q^{46} +(1.10463 - 0.637756i) q^{47} +(8.50000 - 14.7224i) q^{49} +(-4.96410 + 8.59808i) q^{50} +(-3.67423 + 2.12132i) q^{52} -5.65685i q^{53} +(-10.9282 + 6.69213i) q^{55} +(-4.24264 + 2.44949i) q^{56} +(-0.866025 + 1.50000i) q^{58} +(-8.24504 - 4.76028i) q^{59} +(6.57201 - 3.79435i) q^{61} +2.26795 q^{62} +1.00000 q^{64} +(8.19615 + 14.1962i) q^{65} +(2.46410 - 4.26795i) q^{67} +(1.73205 - 3.00000i) q^{68} +(9.46410 + 16.3923i) q^{70} -3.20736i q^{71} +4.89898i q^{73} +(1.46410 + 2.53590i) q^{74} +(-2.89778 - 1.67303i) q^{76} +(0.420266 + 16.2426i) q^{77} +(-8.90138 + 5.13922i) q^{79} -3.86370i q^{80} +2.53590 q^{82} +(-0.866025 - 1.50000i) q^{83} +(-11.5911 - 6.69213i) q^{85} +(-0.776457 - 0.448288i) q^{86} +(1.58346 - 2.91421i) q^{88} +15.9725i q^{89} +20.7846 q^{91} +(7.46859 - 4.31199i) q^{92} +(1.10463 + 0.637756i) q^{94} +(-6.46410 + 11.1962i) q^{95} +(-4.86603 - 8.42820i) q^{97} +17.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} - 4 q^{16} + 12 q^{25} + 16 q^{31} + 4 q^{32} + 96 q^{35} - 32 q^{37} + 24 q^{41} + 68 q^{49} - 12 q^{50} - 32 q^{55} + 32 q^{62} + 8 q^{64} + 24 q^{65} - 8 q^{67} + 48 q^{70}+ \cdots + 136 q^{98}+O(q^{100}) \) 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
\(5\) 3.34607 + 1.93185i 1.49641 + 0.863950i 0.999991 − 0.00413535i \(-0.00131633\pi\)
0.496414 + 0.868086i \(0.334650\pi\)
\(6\) 0 0
\(7\) 4.24264 − 2.44949i 1.60357 − 0.925820i 0.612801 − 0.790237i \(-0.290043\pi\)
0.990766 − 0.135583i \(-0.0432908\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.86370i 1.22181i
\(11\) −1.58346 + 2.91421i −0.477432 + 0.878668i
\(12\) 0 0
\(13\) 3.67423 + 2.12132i 1.01905 + 0.588348i 0.913828 − 0.406100i \(-0.133112\pi\)
0.105221 + 0.994449i \(0.466445\pi\)
\(14\) 4.24264 + 2.44949i 1.13389 + 0.654654i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −3.46410 −0.840168 −0.420084 − 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 3.34607i 0.767640i 0.923408 + 0.383820i \(0.125392\pi\)
−0.923408 + 0.383820i \(0.874608\pi\)
\(20\) −3.34607 + 1.93185i −0.748203 + 0.431975i
\(21\) 0 0
\(22\) −3.31552 + 0.0857864i −0.706870 + 0.0182897i
\(23\) −7.46859 − 4.31199i −1.55731 − 0.899112i −0.997513 − 0.0704812i \(-0.977547\pi\)
−0.559795 − 0.828631i \(-0.689120\pi\)
\(24\) 0 0
\(25\) 4.96410 + 8.59808i 0.992820 + 1.71962i
\(26\) 4.24264i 0.832050i
\(27\) 0 0
\(28\) 4.89898i 0.925820i
\(29\) 0.866025 + 1.50000i 0.160817 + 0.278543i 0.935162 − 0.354221i \(-0.115254\pi\)
−0.774345 + 0.632764i \(0.781920\pi\)
\(30\) 0 0
\(31\) 1.13397 − 1.96410i 0.203668 − 0.352763i −0.746040 − 0.665902i \(-0.768047\pi\)
0.949707 + 0.313138i \(0.101380\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −1.73205 − 3.00000i −0.297044 − 0.514496i
\(35\) 18.9282 3.19945
\(36\) 0 0
\(37\) 2.92820 0.481394 0.240697 − 0.970600i \(-0.422624\pi\)
0.240697 + 0.970600i \(0.422624\pi\)
\(38\) −2.89778 + 1.67303i −0.470082 + 0.271402i
\(39\) 0 0
\(40\) −3.34607 − 1.93185i −0.529059 − 0.305453i
\(41\) 1.26795 − 2.19615i 0.198020 − 0.342981i −0.749866 − 0.661590i \(-0.769882\pi\)
0.947886 + 0.318608i \(0.103215\pi\)
\(42\) 0 0
\(43\) −0.776457 + 0.448288i −0.118409 + 0.0683632i −0.558035 − 0.829818i \(-0.688444\pi\)
0.439626 + 0.898181i \(0.355111\pi\)
\(44\) −1.73205 − 2.82843i −0.261116 − 0.426401i
\(45\) 0 0
\(46\) − 8.62398i − 1.27154i
\(47\) 1.10463 − 0.637756i 0.161126 − 0.0930263i −0.417269 − 0.908783i \(-0.637013\pi\)
0.578395 + 0.815757i \(0.303679\pi\)
\(48\) 0 0
\(49\) 8.50000 − 14.7224i 1.21429 − 2.10320i
\(50\) −4.96410 + 8.59808i −0.702030 + 1.21595i
\(51\) 0 0
\(52\) −3.67423 + 2.12132i −0.509525 + 0.294174i
\(53\) − 5.65685i − 0.777029i −0.921443 − 0.388514i \(-0.872988\pi\)
0.921443 − 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) −10.9282 + 6.69213i −1.47356 + 0.902367i
\(56\) −4.24264 + 2.44949i −0.566947 + 0.327327i
\(57\) 0 0
\(58\) −0.866025 + 1.50000i −0.113715 + 0.196960i
\(59\) −8.24504 − 4.76028i −1.07341 − 0.619736i −0.144301 − 0.989534i \(-0.546093\pi\)
−0.929112 + 0.369798i \(0.879427\pi\)
\(60\) 0 0
\(61\) 6.57201 − 3.79435i 0.841460 − 0.485817i −0.0163003 − 0.999867i \(-0.505189\pi\)
0.857760 + 0.514050i \(0.171855\pi\)
\(62\) 2.26795 0.288030
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.19615 + 14.1962i 1.01661 + 1.76082i
\(66\) 0 0
\(67\) 2.46410 − 4.26795i 0.301038 − 0.521413i −0.675333 − 0.737513i \(-0.736000\pi\)
0.976371 + 0.216100i \(0.0693336\pi\)
\(68\) 1.73205 − 3.00000i 0.210042 − 0.363803i
\(69\) 0 0
\(70\) 9.46410 + 16.3923i 1.13118 + 1.95926i
\(71\) − 3.20736i − 0.380644i −0.981722 − 0.190322i \(-0.939047\pi\)
0.981722 − 0.190322i \(-0.0609532\pi\)
\(72\) 0 0
\(73\) 4.89898i 0.573382i 0.958023 + 0.286691i \(0.0925553\pi\)
−0.958023 + 0.286691i \(0.907445\pi\)
\(74\) 1.46410 + 2.53590i 0.170198 + 0.294792i
\(75\) 0 0
\(76\) −2.89778 − 1.67303i −0.332398 − 0.191910i
\(77\) 0.420266 + 16.2426i 0.0478938 + 1.85102i
\(78\) 0 0
\(79\) −8.90138 + 5.13922i −1.00148 + 0.578207i −0.908687 − 0.417478i \(-0.862914\pi\)
−0.0927969 + 0.995685i \(0.529581\pi\)
\(80\) − 3.86370i − 0.431975i
\(81\) 0 0
\(82\) 2.53590 0.280043
\(83\) −0.866025 − 1.50000i −0.0950586 − 0.164646i 0.814574 − 0.580059i \(-0.196971\pi\)
−0.909633 + 0.415413i \(0.863637\pi\)
\(84\) 0 0
\(85\) −11.5911 − 6.69213i −1.25723 − 0.725863i
\(86\) −0.776457 − 0.448288i −0.0837275 − 0.0483401i
\(87\) 0 0
\(88\) 1.58346 − 2.91421i 0.168798 − 0.310656i
\(89\) 15.9725i 1.69308i 0.532328 + 0.846538i \(0.321317\pi\)
−0.532328 + 0.846538i \(0.678683\pi\)
\(90\) 0 0
\(91\) 20.7846 2.17882
\(92\) 7.46859 − 4.31199i 0.778654 − 0.449556i
\(93\) 0 0
\(94\) 1.10463 + 0.637756i 0.113934 + 0.0657796i
\(95\) −6.46410 + 11.1962i −0.663203 + 1.14870i
\(96\) 0 0
\(97\) −4.86603 − 8.42820i −0.494070 − 0.855754i 0.505907 − 0.862588i \(-0.331158\pi\)
−0.999977 + 0.00683387i \(0.997825\pi\)
\(98\) 17.0000 1.71726
\(99\) 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.l.1187.4 8
3.2 odd 2 1782.2.i.j.1187.1 8
9.2 odd 6 1782.2.i.j.593.4 8
9.4 even 3 1782.2.b.a.1781.1 ✓ 4
9.5 odd 6 1782.2.b.b.1781.4 yes 4
9.7 even 3 inner 1782.2.i.l.593.1 8
11.10 odd 2 1782.2.i.j.1187.4 8
33.32 even 2 inner 1782.2.i.l.1187.1 8
99.32 even 6 1782.2.b.a.1781.4 yes 4
99.43 odd 6 1782.2.i.j.593.1 8
99.65 even 6 inner 1782.2.i.l.593.4 8
99.76 odd 6 1782.2.b.b.1781.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1782.2.b.a.1781.1 ✓ 4 9.4 even 3
1782.2.b.a.1781.4 yes 4 99.32 even 6
1782.2.b.b.1781.1 yes 4 99.76 odd 6
1782.2.b.b.1781.4 yes 4 9.5 odd 6
1782.2.i.j.593.1 8 99.43 odd 6
1782.2.i.j.593.4 8 9.2 odd 6
1782.2.i.j.1187.1 8 3.2 odd 2
1782.2.i.j.1187.4 8 11.10 odd 2
1782.2.i.l.593.1 8 9.7 even 3 inner
1782.2.i.l.593.4 8 99.65 even 6 inner
1782.2.i.l.1187.1 8 33.32 even 2 inner
1782.2.i.l.1187.4 8 1.1 even 1 trivial