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Properties

Label 1134.2.t.f.593.3
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 593.3
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1134.593
Dual form 1134.2.t.f.1025.3

$q$-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 \(-0.684505\pi\)
−0.547723 + 0.836660i \(0.684505\pi\)
\(6\) 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 \(0.220901\pi\)
−0.768706 + 0.639602i \(0.779099\pi\)
\(12\) 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 0.975458 − 0.220184i \(-0.0706658\pi\)
−0.678414 + 0.734680i \(0.737332\pi\)
\(18\) 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 \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 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 \(0.0666139\pi\)
0.669007 + 0.743256i \(0.266719\pi\)
\(30\) 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 0.754399 − 0.656416i \(-0.227928\pi\)
−0.945672 + 0.325121i \(0.894595\pi\)
\(42\) 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 \(-0.969997\pi\)
0.416272 − 0.909240i \(-0.363337\pi\)
\(48\) 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 −0.640484 − 0.767971i \(-0.721266\pi\)
0.344841 + 0.938661i \(0.387933\pi\)
\(54\) 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 −0.934670 − 0.355517i \(-0.884305\pi\)
0.775222 + 0.631689i \(0.217638\pi\)
\(60\) 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 \(0.272491\pi\)
−0.655422 + 0.755263i \(0.727509\pi\)
\(72\) 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 −0.976473 − 0.215641i \(-0.930816\pi\)
0.674987 + 0.737829i \(0.264149\pi\)
\(84\) 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 \(0.556084\pi\)
−0.764976 + 0.644059i \(0.777249\pi\)
\(90\) 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.593.3 8
3.2 odd 2 inner 1134.2.t.f.593.2 8
7.3 odd 6 1134.2.l.e.269.3 8
9.2 odd 6 378.2.k.d.215.3 yes 8
9.4 even 3 1134.2.l.e.215.4 8
9.5 odd 6 1134.2.l.e.215.1 8
9.7 even 3 378.2.k.d.215.2 ✓ 8
21.17 even 6 1134.2.l.e.269.2 8
63.2 odd 6 2646.2.d.d.2645.3 8
63.16 even 3 2646.2.d.d.2645.6 8
63.31 odd 6 inner 1134.2.t.f.1025.2 8
63.38 even 6 378.2.k.d.269.2 yes 8
63.47 even 6 2646.2.d.d.2645.1 8
63.52 odd 6 378.2.k.d.269.3 yes 8
63.59 even 6 inner 1134.2.t.f.1025.3 8
63.61 odd 6 2646.2.d.d.2645.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.d.215.2 ✓ 8 9.7 even 3
378.2.k.d.215.3 yes 8 9.2 odd 6
378.2.k.d.269.2 yes 8 63.38 even 6
378.2.k.d.269.3 yes 8 63.52 odd 6
1134.2.l.e.215.1 8 9.5 odd 6
1134.2.l.e.215.4 8 9.4 even 3
1134.2.l.e.269.2 8 21.17 even 6
1134.2.l.e.269.3 8 7.3 odd 6
1134.2.t.f.593.2 8 3.2 odd 2 inner
1134.2.t.f.593.3 8 1.1 even 1 trivial
1134.2.t.f.1025.2 8 63.31 odd 6 inner
1134.2.t.f.1025.3 8 63.59 even 6 inner
2646.2.d.d.2645.1 8 63.47 even 6
2646.2.d.d.2645.3 8 63.2 odd 6
2646.2.d.d.2645.6 8 63.16 even 3
2646.2.d.d.2645.8 8 63.61 odd 6