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Properties

Label 1782.2.i.f.1187.1
Level $1782$
Weight $2$
Character 1782.1187
Analytic conductor $14.229$
Analytic rank $0$
Dimension $4$
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 = [4,2,0,-2,0,0,0,-4,0,0,-6,0,0,0,0,-2,0,0,0,0,0,6,0,0,-6] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1187.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1782.1187
Dual form 1782.2.i.f.593.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.22474 - 0.707107i) q^{5} +(3.67423 - 2.12132i) q^{7} -1.00000 q^{8} -1.41421i q^{10} +(-2.72474 - 1.89097i) q^{11} +(3.67423 + 2.12132i) q^{13} +(3.67423 + 2.12132i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.22474 - 0.707107i) q^{20} +(0.275255 - 3.30518i) q^{22} +(-1.22474 - 0.707107i) q^{23} +(-1.50000 - 2.59808i) q^{25} +4.24264i q^{26} +4.24264i q^{28} +(-3.00000 - 5.19615i) q^{29} +(2.00000 - 3.46410i) q^{31} +(0.500000 - 0.866025i) q^{32} -6.00000 q^{35} +2.00000 q^{37} +(1.22474 + 0.707107i) q^{40} +(3.00000 - 5.19615i) q^{41} +(-7.34847 + 4.24264i) q^{43} +(3.00000 - 1.41421i) q^{44} -1.41421i q^{46} +(8.57321 - 4.94975i) q^{47} +(5.50000 - 9.52628i) q^{49} +(1.50000 - 2.59808i) q^{50} +(-3.67423 + 2.12132i) q^{52} -7.07107i q^{53} +(2.00000 + 4.24264i) q^{55} +(-3.67423 + 2.12132i) q^{56} +(3.00000 - 5.19615i) q^{58} +(9.79796 + 5.65685i) q^{59} +(3.67423 - 2.12132i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-3.00000 - 5.19615i) q^{65} +(2.00000 - 3.46410i) q^{67} +(-3.00000 - 5.19615i) q^{70} -7.07107i q^{71} +(1.00000 + 1.73205i) q^{74} +(-14.0227 - 1.16781i) q^{77} +(3.67423 - 2.12132i) q^{79} +1.41421i q^{80} +6.00000 q^{82} +(6.00000 + 10.3923i) q^{83} +(-7.34847 - 4.24264i) q^{86} +(2.72474 + 1.89097i) q^{88} +5.65685i q^{89} +18.0000 q^{91} +(1.22474 - 0.707107i) q^{92} +(8.57321 + 4.94975i) q^{94} +(-4.00000 - 6.92820i) q^{97} +11.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 6 q^{11} - 2 q^{16} + 6 q^{22} - 6 q^{25} - 12 q^{29} + 8 q^{31} + 2 q^{32} - 24 q^{35} + 8 q^{37} + 12 q^{41} + 12 q^{44} + 22 q^{49} + 6 q^{50} + 8 q^{55} + 12 q^{58}+ \cdots + 44 q^{98}+O(q^{100}) \) 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
\(5\) −1.22474 − 0.707107i −0.547723 − 0.316228i 0.200480 − 0.979698i \(-0.435750\pi\)
−0.748203 + 0.663470i \(0.769083\pi\)
\(6\) 0 0
\(7\) 3.67423 − 2.12132i 1.38873 − 0.801784i 0.395558 − 0.918441i \(-0.370551\pi\)
0.993172 + 0.116657i \(0.0372179\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) − 1.41421i − 0.447214i
\(11\) −2.72474 − 1.89097i −0.821541 − 0.570149i
\(12\) 0 0
\(13\) 3.67423 + 2.12132i 1.01905 + 0.588348i 0.913828 − 0.406100i \(-0.133112\pi\)
0.105221 + 0.994449i \(0.466445\pi\)
\(14\) 3.67423 + 2.12132i 0.981981 + 0.566947i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.22474 − 0.707107i 0.273861 − 0.158114i
\(21\) 0 0
\(22\) 0.275255 − 3.30518i 0.0586846 − 0.704667i
\(23\) −1.22474 − 0.707107i −0.255377 − 0.147442i 0.366847 − 0.930281i \(-0.380437\pi\)
−0.622224 + 0.782839i \(0.713771\pi\)
\(24\) 0 0
\(25\) −1.50000 − 2.59808i −0.300000 − 0.519615i
\(26\) 4.24264i 0.832050i
\(27\) 0 0
\(28\) 4.24264i 0.801784i
\(29\) −3.00000 − 5.19615i −0.557086 − 0.964901i −0.997738 − 0.0672232i \(-0.978586\pi\)
0.440652 − 0.897678i \(-0.354747\pi\)
\(30\) 0 0
\(31\) 2.00000 − 3.46410i 0.359211 − 0.622171i −0.628619 − 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.22474 + 0.707107i 0.193649 + 0.111803i
\(41\) 3.00000 − 5.19615i 0.468521 − 0.811503i −0.530831 − 0.847477i \(-0.678120\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) −7.34847 + 4.24264i −1.12063 + 0.646997i −0.941562 − 0.336840i \(-0.890642\pi\)
−0.179069 + 0.983836i \(0.557309\pi\)
\(44\) 3.00000 − 1.41421i 0.452267 − 0.213201i
\(45\) 0 0
\(46\) − 1.41421i − 0.208514i
\(47\) 8.57321 − 4.94975i 1.25053 − 0.721995i 0.279317 − 0.960199i \(-0.409892\pi\)
0.971215 + 0.238204i \(0.0765587\pi\)
\(48\) 0 0
\(49\) 5.50000 − 9.52628i 0.785714 − 1.36090i
\(50\) 1.50000 − 2.59808i 0.212132 − 0.367423i
\(51\) 0 0
\(52\) −3.67423 + 2.12132i −0.509525 + 0.294174i
\(53\) − 7.07107i − 0.971286i −0.874157 − 0.485643i \(-0.838586\pi\)
0.874157 − 0.485643i \(-0.161414\pi\)
\(54\) 0 0
\(55\) 2.00000 + 4.24264i 0.269680 + 0.572078i
\(56\) −3.67423 + 2.12132i −0.490990 + 0.283473i
\(57\) 0 0
\(58\) 3.00000 − 5.19615i 0.393919 − 0.682288i
\(59\) 9.79796 + 5.65685i 1.27559 + 0.736460i 0.976034 − 0.217620i \(-0.0698294\pi\)
0.299552 + 0.954080i \(0.403163\pi\)
\(60\) 0 0
\(61\) 3.67423 − 2.12132i 0.470438 − 0.271607i −0.245985 − 0.969274i \(-0.579112\pi\)
0.716423 + 0.697666i \(0.245778\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 − 5.19615i −0.372104 − 0.644503i
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −3.00000 − 5.19615i −0.358569 − 0.621059i
\(71\) − 7.07107i − 0.839181i −0.907713 − 0.419591i \(-0.862174\pi\)
0.907713 − 0.419591i \(-0.137826\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 1.00000 + 1.73205i 0.116248 + 0.201347i
\(75\) 0 0
\(76\) 0 0
\(77\) −14.0227 − 1.16781i −1.59804 − 0.133084i
\(78\) 0 0
\(79\) 3.67423 − 2.12132i 0.413384 − 0.238667i −0.278859 − 0.960332i \(-0.589956\pi\)
0.692243 + 0.721665i \(0.256623\pi\)
\(80\) 1.41421i 0.158114i
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.34847 − 4.24264i −0.792406 − 0.457496i
\(87\) 0 0
\(88\) 2.72474 + 1.89097i 0.290459 + 0.201578i
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 18.0000 1.88691
\(92\) 1.22474 − 0.707107i 0.127688 − 0.0737210i
\(93\) 0 0
\(94\) 8.57321 + 4.94975i 0.884260 + 0.510527i
\(95\) 0 0
\(96\) 0 0
\(97\) −4.00000 − 6.92820i −0.406138 − 0.703452i 0.588315 − 0.808632i \(-0.299792\pi\)
−0.994453 + 0.105180i \(0.966458\pi\)
\(98\) 11.0000 1.11117
\(99\) 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.f.1187.1 4
3.2 odd 2 1782.2.i.c.1187.2 4
9.2 odd 6 1782.2.i.c.593.1 4
9.4 even 3 66.2.b.a.65.2 yes 2
9.5 odd 6 66.2.b.b.65.1 yes 2
9.7 even 3 inner 1782.2.i.f.593.2 4
11.10 odd 2 1782.2.i.c.1187.1 4
33.32 even 2 inner 1782.2.i.f.1187.2 4
36.23 even 6 528.2.b.c.65.2 2
36.31 odd 6 528.2.b.b.65.1 2
45.4 even 6 1650.2.d.b.1451.1 2
45.13 odd 12 1650.2.f.b.1649.3 4
45.14 odd 6 1650.2.d.a.1451.2 2
45.22 odd 12 1650.2.f.b.1649.2 4
45.23 even 12 1650.2.f.a.1649.2 4
45.32 even 12 1650.2.f.a.1649.3 4
72.5 odd 6 2112.2.b.i.65.2 2
72.13 even 6 2112.2.b.g.65.1 2
72.59 even 6 2112.2.b.b.65.1 2
72.67 odd 6 2112.2.b.d.65.2 2
99.4 even 15 726.2.h.i.215.2 8
99.5 odd 30 726.2.h.e.239.2 8
99.13 odd 30 726.2.h.e.161.2 8
99.14 odd 30 726.2.h.e.233.2 8
99.31 even 15 726.2.h.i.161.2 8
99.32 even 6 66.2.b.a.65.1 ✓ 2
99.40 odd 30 726.2.h.e.215.2 8
99.41 even 30 726.2.h.i.233.2 8
99.43 odd 6 1782.2.i.c.593.2 4
99.49 even 15 726.2.h.i.239.1 8
99.50 even 30 726.2.h.i.239.2 8
99.58 even 15 726.2.h.i.233.1 8
99.59 odd 30 726.2.h.e.215.1 8
99.65 even 6 inner 1782.2.i.f.593.1 4
99.68 even 30 726.2.h.i.161.1 8
99.76 odd 6 66.2.b.b.65.2 yes 2
99.85 odd 30 726.2.h.e.233.1 8
99.86 odd 30 726.2.h.e.161.1 8
99.94 odd 30 726.2.h.e.239.1 8
99.95 even 30 726.2.h.i.215.1 8
396.131 odd 6 528.2.b.b.65.2 2
396.175 even 6 528.2.b.c.65.1 2
495.32 odd 12 1650.2.f.b.1649.1 4
495.274 odd 6 1650.2.d.a.1451.1 2
495.329 even 6 1650.2.d.b.1451.2 2
495.373 even 12 1650.2.f.a.1649.1 4
495.428 odd 12 1650.2.f.b.1649.4 4
495.472 even 12 1650.2.f.a.1649.4 4
792.131 odd 6 2112.2.b.d.65.1 2
792.373 odd 6 2112.2.b.i.65.1 2
792.571 even 6 2112.2.b.b.65.2 2
792.725 even 6 2112.2.b.g.65.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.b.a.65.1 ✓ 2 99.32 even 6
66.2.b.a.65.2 yes 2 9.4 even 3
66.2.b.b.65.1 yes 2 9.5 odd 6
66.2.b.b.65.2 yes 2 99.76 odd 6
528.2.b.b.65.1 2 36.31 odd 6
528.2.b.b.65.2 2 396.131 odd 6
528.2.b.c.65.1 2 396.175 even 6
528.2.b.c.65.2 2 36.23 even 6
726.2.h.e.161.1 8 99.86 odd 30
726.2.h.e.161.2 8 99.13 odd 30
726.2.h.e.215.1 8 99.59 odd 30
726.2.h.e.215.2 8 99.40 odd 30
726.2.h.e.233.1 8 99.85 odd 30
726.2.h.e.233.2 8 99.14 odd 30
726.2.h.e.239.1 8 99.94 odd 30
726.2.h.e.239.2 8 99.5 odd 30
726.2.h.i.161.1 8 99.68 even 30
726.2.h.i.161.2 8 99.31 even 15
726.2.h.i.215.1 8 99.95 even 30
726.2.h.i.215.2 8 99.4 even 15
726.2.h.i.233.1 8 99.58 even 15
726.2.h.i.233.2 8 99.41 even 30
726.2.h.i.239.1 8 99.49 even 15
726.2.h.i.239.2 8 99.50 even 30
1650.2.d.a.1451.1 2 495.274 odd 6
1650.2.d.a.1451.2 2 45.14 odd 6
1650.2.d.b.1451.1 2 45.4 even 6
1650.2.d.b.1451.2 2 495.329 even 6
1650.2.f.a.1649.1 4 495.373 even 12
1650.2.f.a.1649.2 4 45.23 even 12
1650.2.f.a.1649.3 4 45.32 even 12
1650.2.f.a.1649.4 4 495.472 even 12
1650.2.f.b.1649.1 4 495.32 odd 12
1650.2.f.b.1649.2 4 45.22 odd 12
1650.2.f.b.1649.3 4 45.13 odd 12
1650.2.f.b.1649.4 4 495.428 odd 12
1782.2.i.c.593.1 4 9.2 odd 6
1782.2.i.c.593.2 4 99.43 odd 6
1782.2.i.c.1187.1 4 11.10 odd 2
1782.2.i.c.1187.2 4 3.2 odd 2
1782.2.i.f.593.1 4 99.65 even 6 inner
1782.2.i.f.593.2 4 9.7 even 3 inner
1782.2.i.f.1187.1 4 1.1 even 1 trivial
1782.2.i.f.1187.2 4 33.32 even 2 inner
2112.2.b.b.65.1 2 72.59 even 6
2112.2.b.b.65.2 2 792.571 even 6
2112.2.b.d.65.1 2 792.131 odd 6
2112.2.b.d.65.2 2 72.67 odd 6
2112.2.b.g.65.1 2 72.13 even 6
2112.2.b.g.65.2 2 792.725 even 6
2112.2.b.i.65.1 2 792.373 odd 6
2112.2.b.i.65.2 2 72.5 odd 6