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Properties

Label 1134.2.t.f.593.2
Level $1134$
Weight $2$
Character 1134.593
Analytic conductor $9.055$
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] = [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 = [8,0,0,4,0,0,4,0,0,0,0,0,-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: \(8\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.2
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1134.593
Dual form 1134.2.t.f.1025.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +2.44949 q^{5} +(-1.62132 - 2.09077i) q^{7} +1.00000i q^{8} +(-2.12132 + 1.22474i) q^{10} -4.24264i q^{11} +(-3.62132 + 2.09077i) q^{13} +(2.44949 + 1.00000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.22474 - 2.12132i) q^{17} +(4.24264 + 2.44949i) q^{19} +(1.22474 - 2.12132i) q^{20} +(2.12132 + 3.67423i) q^{22} -6.00000i q^{23} +1.00000 q^{25} +(2.09077 - 3.62132i) q^{26} +(-2.62132 + 0.358719i) q^{28} +(-8.87039 - 5.12132i) q^{29} +(-4.86396 - 2.80821i) q^{31} +(0.866025 + 0.500000i) q^{32} +(2.12132 + 1.22474i) q^{34} +(-3.97141 - 5.12132i) q^{35} +(1.62132 - 2.80821i) q^{37} -4.89898 q^{38} +2.44949i q^{40} +(1.22474 + 2.12132i) q^{41} +(-3.50000 + 6.06218i) q^{43} +(-3.67423 - 2.12132i) q^{44} +(3.00000 + 5.19615i) q^{46} +(3.97141 + 6.87868i) q^{47} +(-1.74264 + 6.77962i) q^{49} +(-0.866025 + 0.500000i) q^{50} +4.18154i q^{52} +(2.15232 - 1.24264i) q^{53} -10.3923i q^{55} +(2.09077 - 1.62132i) q^{56} +10.2426 q^{58} +(1.22474 - 2.12132i) q^{59} +(0.621320 - 0.358719i) q^{61} +5.61642 q^{62} -1.00000 q^{64} +(-8.87039 + 5.12132i) q^{65} +(1.74264 - 3.01834i) q^{67} -2.44949 q^{68} +(6.00000 + 2.44949i) q^{70} -12.7279i q^{71} +(-13.2426 + 7.64564i) q^{73} +3.24264i q^{74} +(4.24264 - 2.44949i) q^{76} +(-8.87039 + 6.87868i) q^{77} +(-4.62132 - 8.00436i) q^{79} +(-1.22474 - 2.12132i) q^{80} +(-2.12132 - 1.22474i) q^{82} +(2.74666 - 4.75736i) q^{83} +(-3.00000 - 5.19615i) q^{85} -7.00000i q^{86} +4.24264 q^{88} +(8.87039 - 15.3640i) q^{89} +(10.2426 + 4.18154i) q^{91} +(-5.19615 - 3.00000i) q^{92} +(-6.87868 - 3.97141i) q^{94} +(10.3923 + 6.00000i) q^{95} +(-5.74264 - 3.31552i) q^{97} +(-1.88064 - 6.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{7} - 12 q^{13} - 4 q^{16} + 8 q^{25} - 4 q^{28} + 12 q^{31} - 4 q^{37} - 28 q^{43} + 24 q^{46} + 20 q^{49} + 48 q^{58} - 12 q^{61} - 8 q^{64} - 20 q^{67} + 48 q^{70} - 72 q^{73} - 20 q^{79}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 2.44949 1.09545 0.547723 − 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) −1.62132 − 2.09077i −0.612801 − 0.790237i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −2.12132 + 1.22474i −0.670820 + 0.387298i
\(11\) − 4.24264i − 1.27920i −0.768706 − 0.639602i \(-0.779099\pi\)
0.768706 − 0.639602i \(-0.220901\pi\)
\(12\) 0 0
\(13\) −3.62132 + 2.09077i −1.00437 + 0.579875i −0.909539 − 0.415618i \(-0.863565\pi\)
−0.0948342 + 0.995493i \(0.530232\pi\)
\(14\) 2.44949 + 1.00000i 0.654654 + 0.267261i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.22474 − 2.12132i −0.297044 − 0.514496i 0.678414 − 0.734680i \(-0.262668\pi\)
−0.975458 + 0.220184i \(0.929334\pi\)
\(18\) 0 0
\(19\) 4.24264 + 2.44949i 0.973329 + 0.561951i 0.900249 − 0.435375i \(-0.143384\pi\)
0.0730792 + 0.997326i \(0.476717\pi\)
\(20\) 1.22474 − 2.12132i 0.273861 − 0.474342i
\(21\) 0 0
\(22\) 2.12132 + 3.67423i 0.452267 + 0.783349i
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.09077 − 3.62132i 0.410034 − 0.710199i
\(27\) 0 0
\(28\) −2.62132 + 0.358719i −0.495383 + 0.0677916i
\(29\) −8.87039 − 5.12132i −1.64719 − 0.951005i −0.978182 − 0.207750i \(-0.933386\pi\)
−0.669007 − 0.743256i \(-0.733281\pi\)
\(30\) 0 0
\(31\) −4.86396 − 2.80821i −0.873593 − 0.504369i −0.00505256 − 0.999987i \(-0.501608\pi\)
−0.868541 + 0.495618i \(0.834942\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 2.12132 + 1.22474i 0.363803 + 0.210042i
\(35\) −3.97141 − 5.12132i −0.671290 − 0.865661i
\(36\) 0 0
\(37\) 1.62132 − 2.80821i 0.266543 − 0.461667i −0.701423 − 0.712745i \(-0.747452\pi\)
0.967967 + 0.251078i \(0.0807851\pi\)
\(38\) −4.89898 −0.794719
\(39\) 0 0
\(40\) 2.44949i 0.387298i
\(41\) 1.22474 + 2.12132i 0.191273 + 0.331295i 0.945672 − 0.325121i \(-0.105405\pi\)
−0.754399 + 0.656416i \(0.772072\pi\)
\(42\) 0 0
\(43\) −3.50000 + 6.06218i −0.533745 + 0.924473i 0.465478 + 0.885059i \(0.345882\pi\)
−0.999223 + 0.0394140i \(0.987451\pi\)
\(44\) −3.67423 − 2.12132i −0.553912 − 0.319801i
\(45\) 0 0
\(46\) 3.00000 + 5.19615i 0.442326 + 0.766131i
\(47\) 3.97141 + 6.87868i 0.579289 + 1.00336i 0.995561 + 0.0941183i \(0.0300032\pi\)
−0.416272 + 0.909240i \(0.636663\pi\)
\(48\) 0 0
\(49\) −1.74264 + 6.77962i −0.248949 + 0.968517i
\(50\) −0.866025 + 0.500000i −0.122474 + 0.0707107i
\(51\) 0 0
\(52\) 4.18154i 0.579875i
\(53\) 2.15232 − 1.24264i 0.295643 − 0.170690i −0.344841 − 0.938661i \(-0.612067\pi\)
0.640484 + 0.767971i \(0.278734\pi\)
\(54\) 0 0
\(55\) − 10.3923i − 1.40130i
\(56\) 2.09077 − 1.62132i 0.279391 − 0.216658i
\(57\) 0 0
\(58\) 10.2426 1.34492
\(59\) 1.22474 − 2.12132i 0.159448 − 0.276172i −0.775222 − 0.631689i \(-0.782362\pi\)
0.934670 + 0.355517i \(0.115695\pi\)
\(60\) 0 0
\(61\) 0.621320 − 0.358719i 0.0795519 − 0.0459293i −0.459696 − 0.888076i \(-0.652042\pi\)
0.539248 + 0.842147i \(0.318708\pi\)
\(62\) 5.61642 0.713286
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −8.87039 + 5.12132i −1.10024 + 0.635222i
\(66\) 0 0
\(67\) 1.74264 − 3.01834i 0.212897 − 0.368749i −0.739723 − 0.672912i \(-0.765043\pi\)
0.952620 + 0.304163i \(0.0983766\pi\)
\(68\) −2.44949 −0.297044
\(69\) 0 0
\(70\) 6.00000 + 2.44949i 0.717137 + 0.292770i
\(71\) − 12.7279i − 1.51053i −0.655422 − 0.755263i \(-0.727509\pi\)
0.655422 − 0.755263i \(-0.272491\pi\)
\(72\) 0 0
\(73\) −13.2426 + 7.64564i −1.54993 + 0.894855i −0.551788 + 0.833984i \(0.686054\pi\)
−0.998146 + 0.0608704i \(0.980612\pi\)
\(74\) 3.24264i 0.376949i
\(75\) 0 0
\(76\) 4.24264 − 2.44949i 0.486664 − 0.280976i
\(77\) −8.87039 + 6.87868i −1.01087 + 0.783898i
\(78\) 0 0
\(79\) −4.62132 − 8.00436i −0.519939 − 0.900561i −0.999731 − 0.0231789i \(-0.992621\pi\)
0.479792 − 0.877382i \(-0.340712\pi\)
\(80\) −1.22474 − 2.12132i −0.136931 − 0.237171i
\(81\) 0 0
\(82\) −2.12132 − 1.22474i −0.234261 − 0.135250i
\(83\) 2.74666 − 4.75736i 0.301485 − 0.522188i −0.674987 − 0.737829i \(-0.735851\pi\)
0.976473 + 0.215641i \(0.0691842\pi\)
\(84\) 0 0
\(85\) −3.00000 − 5.19615i −0.325396 − 0.563602i
\(86\) − 7.00000i − 0.754829i
\(87\) 0 0
\(88\) 4.24264 0.452267
\(89\) 8.87039 − 15.3640i 0.940259 − 1.62858i 0.175283 − 0.984518i \(-0.443916\pi\)
0.764976 − 0.644059i \(-0.222751\pi\)
\(90\) 0 0
\(91\) 10.2426 + 4.18154i 1.07372 + 0.438345i
\(92\) −5.19615 − 3.00000i −0.541736 − 0.312772i
\(93\) 0 0
\(94\) −6.87868 − 3.97141i −0.709482 − 0.409619i
\(95\) 10.3923 + 6.00000i 1.06623 + 0.615587i
\(96\) 0 0
\(97\) −5.74264 − 3.31552i −0.583077 − 0.336640i 0.179278 − 0.983798i \(-0.442624\pi\)
−0.762355 + 0.647159i \(0.775957\pi\)
\(98\) −1.88064 − 6.74264i −0.189973 − 0.681110i
\(99\) 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.f.593.2 8
3.2 odd 2 inner 1134.2.t.f.593.3 8
7.3 odd 6 1134.2.l.e.269.2 8
9.2 odd 6 378.2.k.d.215.2 ✓ 8
9.4 even 3 1134.2.l.e.215.1 8
9.5 odd 6 1134.2.l.e.215.4 8
9.7 even 3 378.2.k.d.215.3 yes 8
21.17 even 6 1134.2.l.e.269.3 8
63.2 odd 6 2646.2.d.d.2645.6 8
63.16 even 3 2646.2.d.d.2645.3 8
63.31 odd 6 inner 1134.2.t.f.1025.3 8
63.38 even 6 378.2.k.d.269.3 yes 8
63.47 even 6 2646.2.d.d.2645.8 8
63.52 odd 6 378.2.k.d.269.2 yes 8
63.59 even 6 inner 1134.2.t.f.1025.2 8
63.61 odd 6 2646.2.d.d.2645.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.d.215.2 ✓ 8 9.2 odd 6
378.2.k.d.215.3 yes 8 9.7 even 3
378.2.k.d.269.2 yes 8 63.52 odd 6
378.2.k.d.269.3 yes 8 63.38 even 6
1134.2.l.e.215.1 8 9.4 even 3
1134.2.l.e.215.4 8 9.5 odd 6
1134.2.l.e.269.2 8 7.3 odd 6
1134.2.l.e.269.3 8 21.17 even 6
1134.2.t.f.593.2 8 1.1 even 1 trivial
1134.2.t.f.593.3 8 3.2 odd 2 inner
1134.2.t.f.1025.2 8 63.59 even 6 inner
1134.2.t.f.1025.3 8 63.31 odd 6 inner
2646.2.d.d.2645.1 8 63.61 odd 6
2646.2.d.d.2645.3 8 63.16 even 3
2646.2.d.d.2645.6 8 63.2 odd 6
2646.2.d.d.2645.8 8 63.47 even 6