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Properties

Label 1134.2.f.a.757.1
Level $1134$
Weight $2$
Character 1134.757
Analytic conductor $9.055$
Analytic rank $0$
Dimension $2$
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] = [1134,2,Mod(379,1134)] 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("1134.379"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,-4,0,1,2,0,8,4,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 757.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1134.757
Dual form 1134.2.f.a.379.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 + 3.46410i) q^{5} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +4.00000 q^{10} +(2.00000 + 3.46410i) q^{11} +(-1.50000 + 2.59808i) q^{13} +(0.500000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -7.00000 q^{17} +2.00000 q^{19} +(-2.00000 - 3.46410i) q^{20} +(2.00000 - 3.46410i) q^{22} +(0.500000 - 0.866025i) q^{23} +(-5.50000 - 9.52628i) q^{25} +3.00000 q^{26} -1.00000 q^{28} +(-0.500000 - 0.866025i) q^{29} +(4.50000 - 7.79423i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(3.50000 + 6.06218i) q^{34} -4.00000 q^{35} +2.00000 q^{37} +(-1.00000 - 1.73205i) q^{38} +(-2.00000 + 3.46410i) q^{40} +(-3.00000 + 5.19615i) q^{41} +(-5.50000 - 9.52628i) q^{43} -4.00000 q^{44} -1.00000 q^{46} +(3.00000 + 5.19615i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-5.50000 + 9.52628i) q^{50} +(-1.50000 - 2.59808i) q^{52} -9.00000 q^{53} -16.0000 q^{55} +(0.500000 + 0.866025i) q^{56} +(-0.500000 + 0.866025i) q^{58} +(2.50000 - 4.33013i) q^{59} +(3.00000 + 5.19615i) q^{61} -9.00000 q^{62} +1.00000 q^{64} +(-6.00000 - 10.3923i) q^{65} +(-3.50000 + 6.06218i) q^{67} +(3.50000 - 6.06218i) q^{68} +(2.00000 + 3.46410i) q^{70} -7.00000 q^{71} -14.0000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(-1.00000 + 1.73205i) q^{76} +(-2.00000 + 3.46410i) q^{77} +(3.00000 + 5.19615i) q^{79} +4.00000 q^{80} +6.00000 q^{82} +(2.00000 + 3.46410i) q^{83} +(14.0000 - 24.2487i) q^{85} +(-5.50000 + 9.52628i) q^{86} +(2.00000 + 3.46410i) q^{88} -3.00000 q^{89} -3.00000 q^{91} +(0.500000 + 0.866025i) q^{92} +(3.00000 - 5.19615i) q^{94} +(-4.00000 + 6.92820i) q^{95} +(4.00000 + 6.92820i) q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 4 q^{5} + q^{7} + 2 q^{8} + 8 q^{10} + 4 q^{11} - 3 q^{13} + q^{14} - q^{16} - 14 q^{17} + 4 q^{19} - 4 q^{20} + 4 q^{22} + q^{23} - 11 q^{25} + 6 q^{26} - 2 q^{28} - q^{29}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −2.00000 + 3.46410i −0.894427 + 1.54919i −0.0599153 + 0.998203i \(0.519083\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 4.00000 1.26491
\(11\) 2.00000 + 3.46410i 0.603023 + 1.04447i 0.992361 + 0.123371i \(0.0393705\pi\)
−0.389338 + 0.921095i \(0.627296\pi\)
\(12\) 0 0
\(13\) −1.50000 + 2.59808i −0.416025 + 0.720577i −0.995535 − 0.0943882i \(-0.969911\pi\)
0.579510 + 0.814965i \(0.303244\pi\)
\(14\) 0.500000 − 0.866025i 0.133631 − 0.231455i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −2.00000 − 3.46410i −0.447214 − 0.774597i
\(21\) 0 0
\(22\) 2.00000 − 3.46410i 0.426401 − 0.738549i
\(23\) 0.500000 − 0.866025i 0.104257 − 0.180579i −0.809177 − 0.587565i \(-0.800087\pi\)
0.913434 + 0.406986i \(0.133420\pi\)
\(24\) 0 0
\(25\) −5.50000 − 9.52628i −1.10000 − 1.90526i
\(26\) 3.00000 0.588348
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −0.500000 − 0.866025i −0.0928477 − 0.160817i 0.815861 − 0.578249i \(-0.196264\pi\)
−0.908708 + 0.417432i \(0.862930\pi\)
\(30\) 0 0
\(31\) 4.50000 − 7.79423i 0.808224 − 1.39988i −0.105869 − 0.994380i \(-0.533762\pi\)
0.914093 − 0.405505i \(-0.132904\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 3.50000 + 6.06218i 0.600245 + 1.03965i
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 − 1.73205i −0.162221 − 0.280976i
\(39\) 0 0
\(40\) −2.00000 + 3.46410i −0.316228 + 0.547723i
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 − 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) −5.50000 − 9.52628i −0.838742 − 1.45274i −0.890947 − 0.454108i \(-0.849958\pi\)
0.0522047 − 0.998636i \(-0.483375\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 − 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) −5.50000 + 9.52628i −0.777817 + 1.34722i
\(51\) 0 0
\(52\) −1.50000 − 2.59808i −0.208013 − 0.360288i
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −16.0000 −2.15744
\(56\) 0.500000 + 0.866025i 0.0668153 + 0.115728i
\(57\) 0 0
\(58\) −0.500000 + 0.866025i −0.0656532 + 0.113715i
\(59\) 2.50000 − 4.33013i 0.325472 − 0.563735i −0.656136 − 0.754643i \(-0.727810\pi\)
0.981608 + 0.190909i \(0.0611434\pi\)
\(60\) 0 0
\(61\) 3.00000 + 5.19615i 0.384111 + 0.665299i 0.991645 − 0.128994i \(-0.0411748\pi\)
−0.607535 + 0.794293i \(0.707841\pi\)
\(62\) −9.00000 −1.14300
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 − 10.3923i −0.744208 − 1.28901i
\(66\) 0 0
\(67\) −3.50000 + 6.06218i −0.427593 + 0.740613i −0.996659 − 0.0816792i \(-0.973972\pi\)
0.569066 + 0.822292i \(0.307305\pi\)
\(68\) 3.50000 − 6.06218i 0.424437 − 0.735147i
\(69\) 0 0
\(70\) 2.00000 + 3.46410i 0.239046 + 0.414039i
\(71\) −7.00000 −0.830747 −0.415374 − 0.909651i \(-0.636349\pi\)
−0.415374 + 0.909651i \(0.636349\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −1.00000 − 1.73205i −0.116248 − 0.201347i
\(75\) 0 0
\(76\) −1.00000 + 1.73205i −0.114708 + 0.198680i
\(77\) −2.00000 + 3.46410i −0.227921 + 0.394771i
\(78\) 0 0
\(79\) 3.00000 + 5.19615i 0.337526 + 0.584613i 0.983967 − 0.178352i \(-0.0570765\pi\)
−0.646440 + 0.762964i \(0.723743\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 2.00000 + 3.46410i 0.219529 + 0.380235i 0.954664 − 0.297686i \(-0.0962148\pi\)
−0.735135 + 0.677920i \(0.762881\pi\)
\(84\) 0 0
\(85\) 14.0000 − 24.2487i 1.51851 − 2.63014i
\(86\) −5.50000 + 9.52628i −0.593080 + 1.02725i
\(87\) 0 0
\(88\) 2.00000 + 3.46410i 0.213201 + 0.369274i
\(89\) −3.00000 −0.317999 −0.159000 − 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 0.500000 + 0.866025i 0.0521286 + 0.0902894i
\(93\) 0 0
\(94\) 3.00000 − 5.19615i 0.309426 − 0.535942i
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) 4.00000 + 6.92820i 0.406138 + 0.703452i 0.994453 − 0.105180i \(-0.0335417\pi\)
−0.588315 + 0.808632i \(0.700208\pi\)
\(98\) 1.00000 0.101015
\(99\) 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 1134.2.f.a.757.1 2
3.2 odd 2 1134.2.f.p.757.1 2
9.2 odd 6 1134.2.f.p.379.1 2
9.4 even 3 378.2.a.h.1.1 yes 1
9.5 odd 6 378.2.a.a.1.1 ✓ 1
9.7 even 3 inner 1134.2.f.a.379.1 2
36.23 even 6 3024.2.a.a.1.1 1
36.31 odd 6 3024.2.a.bd.1.1 1
45.4 even 6 9450.2.a.bc.1.1 1
45.14 odd 6 9450.2.a.dv.1.1 1
63.13 odd 6 2646.2.a.p.1.1 1
63.41 even 6 2646.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.a.1.1 ✓ 1 9.5 odd 6
378.2.a.h.1.1 yes 1 9.4 even 3
1134.2.f.a.379.1 2 9.7 even 3 inner
1134.2.f.a.757.1 2 1.1 even 1 trivial
1134.2.f.p.379.1 2 9.2 odd 6
1134.2.f.p.757.1 2 3.2 odd 2
2646.2.a.o.1.1 1 63.41 even 6
2646.2.a.p.1.1 1 63.13 odd 6
3024.2.a.a.1.1 1 36.23 even 6
3024.2.a.bd.1.1 1 36.31 odd 6
9450.2.a.bc.1.1 1 45.4 even 6
9450.2.a.dv.1.1 1 45.14 odd 6