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Properties

Label 1134.2.t.d.1025.1
Level $1134$
Weight $2$
Character 1134.1025
Analytic conductor $9.055$
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] = [1134,2,Mod(593,1134)] 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("1134.593"); 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([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,0,8,0,0,-6,0,0,-12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.05503558921\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1025.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1134.1025
Dual form 1134.2.t.d.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)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.73205 q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +(-1.50000 - 0.866025i) q^{10} -3.00000i q^{11} +(-3.00000 - 1.73205i) q^{13} +(-0.866025 - 2.50000i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(1.73205 - 3.00000i) q^{17} +(3.00000 - 1.73205i) q^{19} +(0.866025 + 1.50000i) q^{20} +(-1.50000 + 2.59808i) q^{22} -6.00000i q^{23} -2.00000 q^{25} +(1.73205 + 3.00000i) q^{26} +(-0.500000 + 2.59808i) q^{28} +(2.59808 - 1.50000i) q^{29} +(1.50000 - 0.866025i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-3.00000 + 1.73205i) q^{34} +(3.46410 + 3.00000i) q^{35} +(1.00000 + 1.73205i) q^{37} -3.46410 q^{38} -1.73205i q^{40} +(3.46410 - 6.00000i) q^{41} +(4.00000 + 6.92820i) q^{43} +(2.59808 - 1.50000i) q^{44} +(-3.00000 + 5.19615i) q^{46} +(-3.46410 + 6.00000i) q^{47} +(1.00000 + 6.92820i) q^{49} +(1.73205 + 1.00000i) q^{50} -3.46410i q^{52} +(7.79423 + 4.50000i) q^{53} -5.19615i q^{55} +(1.73205 - 2.00000i) q^{56} -3.00000 q^{58} +(0.866025 + 1.50000i) q^{59} -1.73205 q^{62} -1.00000 q^{64} +(-5.19615 - 3.00000i) q^{65} +(-1.00000 - 1.73205i) q^{67} +3.46410 q^{68} +(-1.50000 - 4.33013i) q^{70} -12.0000i q^{71} +(6.00000 + 3.46410i) q^{73} -2.00000i q^{74} +(3.00000 + 1.73205i) q^{76} +(5.19615 - 6.00000i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-0.866025 + 1.50000i) q^{80} +(-6.00000 + 3.46410i) q^{82} +(-4.33013 - 7.50000i) q^{83} +(3.00000 - 5.19615i) q^{85} -8.00000i q^{86} -3.00000 q^{88} +(5.19615 + 9.00000i) q^{89} +(-3.00000 - 8.66025i) q^{91} +(5.19615 - 3.00000i) q^{92} +(6.00000 - 3.46410i) q^{94} +(5.19615 - 3.00000i) q^{95} +(4.50000 - 2.59808i) q^{97} +(2.59808 - 6.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 8 q^{7} - 6 q^{10} - 12 q^{13} - 2 q^{16} + 12 q^{19} - 6 q^{22} - 8 q^{25} - 2 q^{28} + 6 q^{31} - 12 q^{34} + 4 q^{37} + 16 q^{43} - 12 q^{46} + 4 q^{49} - 12 q^{58} - 4 q^{64} - 4 q^{67}+ \cdots + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 − 0.500000i −0.612372 − 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 1.73205 0.774597 0.387298 − 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) −1.50000 − 0.866025i −0.474342 − 0.273861i
\(11\) − 3.00000i − 0.904534i −0.891883 − 0.452267i \(-0.850615\pi\)
0.891883 − 0.452267i \(-0.149385\pi\)
\(12\) 0 0
\(13\) −3.00000 − 1.73205i −0.832050 − 0.480384i 0.0225039 − 0.999747i \(-0.492836\pi\)
−0.854554 + 0.519362i \(0.826170\pi\)
\(14\) −0.866025 − 2.50000i −0.231455 − 0.668153i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.73205 − 3.00000i 0.420084 − 0.727607i −0.575863 − 0.817546i \(-0.695334\pi\)
0.995947 + 0.0899392i \(0.0286673\pi\)
\(18\) 0 0
\(19\) 3.00000 − 1.73205i 0.688247 − 0.397360i −0.114708 − 0.993399i \(-0.536593\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0.866025 + 1.50000i 0.193649 + 0.335410i
\(21\) 0 0
\(22\) −1.50000 + 2.59808i −0.319801 + 0.553912i
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 1.73205 + 3.00000i 0.339683 + 0.588348i
\(27\) 0 0
\(28\) −0.500000 + 2.59808i −0.0944911 + 0.490990i
\(29\) 2.59808 − 1.50000i 0.482451 − 0.278543i −0.238987 − 0.971023i \(-0.576815\pi\)
0.721437 + 0.692480i \(0.243482\pi\)
\(30\) 0 0
\(31\) 1.50000 − 0.866025i 0.269408 − 0.155543i −0.359211 − 0.933257i \(-0.616954\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0.866025 − 0.500000i 0.153093 − 0.0883883i
\(33\) 0 0
\(34\) −3.00000 + 1.73205i −0.514496 + 0.297044i
\(35\) 3.46410 + 3.00000i 0.585540 + 0.507093i
\(36\) 0 0
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 − 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) −3.46410 −0.561951
\(39\) 0 0
\(40\) − 1.73205i − 0.273861i
\(41\) 3.46410 − 6.00000i 0.541002 − 0.937043i −0.457845 − 0.889032i \(-0.651379\pi\)
0.998847 − 0.0480106i \(-0.0152881\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 2.59808 − 1.50000i 0.391675 − 0.226134i
\(45\) 0 0
\(46\) −3.00000 + 5.19615i −0.442326 + 0.766131i
\(47\) −3.46410 + 6.00000i −0.505291 + 0.875190i 0.494690 + 0.869069i \(0.335282\pi\)
−0.999981 + 0.00612051i \(0.998052\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 1.73205 + 1.00000i 0.244949 + 0.141421i
\(51\) 0 0
\(52\) − 3.46410i − 0.480384i
\(53\) 7.79423 + 4.50000i 1.07062 + 0.618123i 0.928351 − 0.371706i \(-0.121227\pi\)
0.142269 + 0.989828i \(0.454560\pi\)
\(54\) 0 0
\(55\) − 5.19615i − 0.700649i
\(56\) 1.73205 − 2.00000i 0.231455 − 0.267261i
\(57\) 0 0
\(58\) −3.00000 −0.393919
\(59\) 0.866025 + 1.50000i 0.112747 + 0.195283i 0.916877 − 0.399170i \(-0.130702\pi\)
−0.804130 + 0.594454i \(0.797368\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) −1.73205 −0.219971
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −5.19615 − 3.00000i −0.644503 − 0.372104i
\(66\) 0 0
\(67\) −1.00000 − 1.73205i −0.122169 − 0.211604i 0.798454 − 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 3.46410 0.420084
\(69\) 0 0
\(70\) −1.50000 − 4.33013i −0.179284 − 0.517549i
\(71\) − 12.0000i − 1.42414i −0.702109 − 0.712069i \(-0.747758\pi\)
0.702109 − 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 6.00000 + 3.46410i 0.702247 + 0.405442i 0.808184 − 0.588930i \(-0.200451\pi\)
−0.105937 + 0.994373i \(0.533784\pi\)
\(74\) − 2.00000i − 0.232495i
\(75\) 0 0
\(76\) 3.00000 + 1.73205i 0.344124 + 0.198680i
\(77\) 5.19615 − 6.00000i 0.592157 − 0.683763i
\(78\) 0 0
\(79\) 0.500000 − 0.866025i 0.0562544 − 0.0974355i −0.836527 − 0.547926i \(-0.815418\pi\)
0.892781 + 0.450490i \(0.148751\pi\)
\(80\) −0.866025 + 1.50000i −0.0968246 + 0.167705i
\(81\) 0 0
\(82\) −6.00000 + 3.46410i −0.662589 + 0.382546i
\(83\) −4.33013 − 7.50000i −0.475293 − 0.823232i 0.524306 − 0.851530i \(-0.324325\pi\)
−0.999600 + 0.0282978i \(0.990991\pi\)
\(84\) 0 0
\(85\) 3.00000 − 5.19615i 0.325396 − 0.563602i
\(86\) − 8.00000i − 0.862662i
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 5.19615 + 9.00000i 0.550791 + 0.953998i 0.998218 + 0.0596775i \(0.0190072\pi\)
−0.447427 + 0.894321i \(0.647659\pi\)
\(90\) 0 0
\(91\) −3.00000 − 8.66025i −0.314485 − 0.907841i
\(92\) 5.19615 − 3.00000i 0.541736 − 0.312772i
\(93\) 0 0
\(94\) 6.00000 − 3.46410i 0.618853 − 0.357295i
\(95\) 5.19615 − 3.00000i 0.533114 − 0.307794i
\(96\) 0 0
\(97\) 4.50000 − 2.59808i 0.456906 − 0.263795i −0.253837 − 0.967247i \(-0.581693\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 2.59808 − 6.50000i 0.262445 − 0.656599i
\(99\) 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.t.d.1025.1 4
3.2 odd 2 inner 1134.2.t.d.1025.2 4
7.5 odd 6 1134.2.l.c.215.2 4
9.2 odd 6 1134.2.l.c.269.1 4
9.4 even 3 42.2.f.a.17.2 yes 4
9.5 odd 6 42.2.f.a.17.1 yes 4
9.7 even 3 1134.2.l.c.269.2 4
21.5 even 6 1134.2.l.c.215.1 4
36.23 even 6 336.2.bc.e.17.1 4
36.31 odd 6 336.2.bc.e.17.2 4
45.4 even 6 1050.2.s.b.101.1 4
45.13 odd 12 1050.2.u.a.899.2 4
45.14 odd 6 1050.2.s.b.101.2 4
45.22 odd 12 1050.2.u.d.899.1 4
45.23 even 12 1050.2.u.d.899.2 4
45.32 even 12 1050.2.u.a.899.1 4
63.4 even 3 294.2.d.a.293.4 4
63.5 even 6 42.2.f.a.5.2 yes 4
63.13 odd 6 294.2.f.a.227.2 4
63.23 odd 6 294.2.f.a.215.2 4
63.31 odd 6 294.2.d.a.293.3 4
63.32 odd 6 294.2.d.a.293.1 4
63.40 odd 6 42.2.f.a.5.1 ✓ 4
63.41 even 6 294.2.f.a.227.1 4
63.47 even 6 inner 1134.2.t.d.593.1 4
63.58 even 3 294.2.f.a.215.1 4
63.59 even 6 294.2.d.a.293.2 4
63.61 odd 6 inner 1134.2.t.d.593.2 4
252.31 even 6 2352.2.k.e.881.4 4
252.59 odd 6 2352.2.k.e.881.1 4
252.67 odd 6 2352.2.k.e.881.2 4
252.95 even 6 2352.2.k.e.881.3 4
252.103 even 6 336.2.bc.e.257.1 4
252.131 odd 6 336.2.bc.e.257.2 4
315.68 odd 12 1050.2.u.d.299.1 4
315.103 even 12 1050.2.u.a.299.1 4
315.194 even 6 1050.2.s.b.551.1 4
315.229 odd 6 1050.2.s.b.551.2 4
315.257 odd 12 1050.2.u.a.299.2 4
315.292 even 12 1050.2.u.d.299.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.f.a.5.1 ✓ 4 63.40 odd 6
42.2.f.a.5.2 yes 4 63.5 even 6
42.2.f.a.17.1 yes 4 9.5 odd 6
42.2.f.a.17.2 yes 4 9.4 even 3
294.2.d.a.293.1 4 63.32 odd 6
294.2.d.a.293.2 4 63.59 even 6
294.2.d.a.293.3 4 63.31 odd 6
294.2.d.a.293.4 4 63.4 even 3
294.2.f.a.215.1 4 63.58 even 3
294.2.f.a.215.2 4 63.23 odd 6
294.2.f.a.227.1 4 63.41 even 6
294.2.f.a.227.2 4 63.13 odd 6
336.2.bc.e.17.1 4 36.23 even 6
336.2.bc.e.17.2 4 36.31 odd 6
336.2.bc.e.257.1 4 252.103 even 6
336.2.bc.e.257.2 4 252.131 odd 6
1050.2.s.b.101.1 4 45.4 even 6
1050.2.s.b.101.2 4 45.14 odd 6
1050.2.s.b.551.1 4 315.194 even 6
1050.2.s.b.551.2 4 315.229 odd 6
1050.2.u.a.299.1 4 315.103 even 12
1050.2.u.a.299.2 4 315.257 odd 12
1050.2.u.a.899.1 4 45.32 even 12
1050.2.u.a.899.2 4 45.13 odd 12
1050.2.u.d.299.1 4 315.68 odd 12
1050.2.u.d.299.2 4 315.292 even 12
1050.2.u.d.899.1 4 45.22 odd 12
1050.2.u.d.899.2 4 45.23 even 12
1134.2.l.c.215.1 4 21.5 even 6
1134.2.l.c.215.2 4 7.5 odd 6
1134.2.l.c.269.1 4 9.2 odd 6
1134.2.l.c.269.2 4 9.7 even 3
1134.2.t.d.593.1 4 63.47 even 6 inner
1134.2.t.d.593.2 4 63.61 odd 6 inner
1134.2.t.d.1025.1 4 1.1 even 1 trivial
1134.2.t.d.1025.2 4 3.2 odd 2 inner
2352.2.k.e.881.1 4 252.59 odd 6
2352.2.k.e.881.2 4 252.67 odd 6
2352.2.k.e.881.3 4 252.95 even 6
2352.2.k.e.881.4 4 252.31 even 6