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Properties

Label 1134.2.t.b.593.2
Level $1134$
Weight $2$
Character 1134.593
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 = [4,0,0,2,0,0,2,0,0,-12,0,0,18,0] 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 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 593.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1134.593
Dual form 1134.2.t.b.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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} -3.46410 q^{5} +(0.500000 + 2.59808i) q^{7} -1.00000i q^{8} +(-3.00000 + 1.73205i) q^{10} +(4.50000 - 2.59808i) q^{13} +(1.73205 + 2.00000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.46410 - 6.00000i) q^{17} +(-3.00000 - 1.73205i) q^{19} +(-1.73205 + 3.00000i) q^{20} -6.00000i q^{23} +7.00000 q^{25} +(2.59808 - 4.50000i) q^{26} +(2.50000 + 0.866025i) q^{28} +(-7.50000 - 4.33013i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(-6.00000 - 3.46410i) q^{34} +(-1.73205 - 9.00000i) q^{35} +(2.50000 - 4.33013i) q^{37} -3.46410 q^{38} +3.46410i q^{40} +(3.46410 + 6.00000i) q^{41} +(-0.500000 + 0.866025i) q^{43} +(-3.00000 - 5.19615i) q^{46} +(-3.46410 - 6.00000i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(6.06218 - 3.50000i) q^{50} -5.19615i q^{52} +(-5.19615 + 3.00000i) q^{53} +(2.59808 - 0.500000i) q^{56} +(3.46410 - 6.00000i) q^{59} +(1.50000 - 0.866025i) q^{61} -8.66025 q^{62} -1.00000 q^{64} +(-15.5885 + 9.00000i) q^{65} +(6.50000 - 11.2583i) q^{67} -6.92820 q^{68} +(-6.00000 - 6.92820i) q^{70} +6.00000i q^{71} +(-6.00000 + 3.46410i) q^{73} -5.00000i q^{74} +(-3.00000 + 1.73205i) q^{76} +(3.50000 + 6.06218i) q^{79} +(1.73205 + 3.00000i) q^{80} +(6.00000 + 3.46410i) q^{82} +(-1.73205 + 3.00000i) q^{83} +(12.0000 + 20.7846i) q^{85} +1.00000i q^{86} +(5.19615 - 9.00000i) q^{89} +(9.00000 + 10.3923i) q^{91} +(-5.19615 - 3.00000i) q^{92} +(-6.00000 - 3.46410i) q^{94} +(10.3923 + 6.00000i) q^{95} +(1.50000 + 0.866025i) q^{97} +(-4.33013 + 5.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{7} - 12 q^{10} + 18 q^{13} - 2 q^{16} - 12 q^{19} + 28 q^{25} + 10 q^{28} - 30 q^{31} - 24 q^{34} + 10 q^{37} - 2 q^{43} - 12 q^{46} - 26 q^{49} + 6 q^{61} - 4 q^{64} + 26 q^{67} - 24 q^{70}+ \cdots + 6 q^{97}+O(q^{100}) \) 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
\(5\) −3.46410 −1.54919 −0.774597 − 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 0.500000 + 2.59808i 0.188982 + 0.981981i
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) −3.00000 + 1.73205i −0.948683 + 0.547723i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 4.50000 − 2.59808i 1.24808 − 0.720577i 0.277350 − 0.960769i \(-0.410544\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 1.73205 + 2.00000i 0.462910 + 0.534522i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −3.46410 − 6.00000i −0.840168 − 1.45521i −0.889752 − 0.456444i \(-0.849123\pi\)
0.0495842 − 0.998770i \(-0.484210\pi\)
\(18\) 0 0
\(19\) −3.00000 − 1.73205i −0.688247 − 0.397360i 0.114708 − 0.993399i \(-0.463407\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −1.73205 + 3.00000i −0.387298 + 0.670820i
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 2.59808 − 4.50000i 0.509525 − 0.882523i
\(27\) 0 0
\(28\) 2.50000 + 0.866025i 0.472456 + 0.163663i
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −7.50000 − 4.33013i −1.34704 − 0.777714i −0.359211 − 0.933257i \(-0.616954\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) −6.00000 − 3.46410i −1.02899 − 0.594089i
\(35\) −1.73205 − 9.00000i −0.292770 − 1.52128i
\(36\) 0 0
\(37\) 2.50000 − 4.33013i 0.410997 − 0.711868i −0.584002 − 0.811752i \(-0.698514\pi\)
0.994999 + 0.0998840i \(0.0318472\pi\)
\(38\) −3.46410 −0.561951
\(39\) 0 0
\(40\) 3.46410i 0.547723i
\(41\) 3.46410 + 6.00000i 0.541002 + 0.937043i 0.998847 + 0.0480106i \(0.0152881\pi\)
−0.457845 + 0.889032i \(0.651379\pi\)
\(42\) 0 0
\(43\) −0.500000 + 0.866025i −0.0762493 + 0.132068i −0.901629 − 0.432511i \(-0.857628\pi\)
0.825380 + 0.564578i \(0.190961\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 − 5.19615i −0.442326 − 0.766131i
\(47\) −3.46410 − 6.00000i −0.505291 − 0.875190i −0.999981 − 0.00612051i \(-0.998052\pi\)
0.494690 − 0.869069i \(-0.335282\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 6.06218 − 3.50000i 0.857321 − 0.494975i
\(51\) 0 0
\(52\) − 5.19615i − 0.720577i
\(53\) −5.19615 + 3.00000i −0.713746 + 0.412082i −0.812447 − 0.583036i \(-0.801865\pi\)
0.0987002 + 0.995117i \(0.468532\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.59808 − 0.500000i 0.347183 − 0.0668153i
\(57\) 0 0
\(58\) 0 0
\(59\) 3.46410 − 6.00000i 0.450988 − 0.781133i −0.547460 − 0.836832i \(-0.684405\pi\)
0.998448 + 0.0556984i \(0.0177385\pi\)
\(60\) 0 0
\(61\) 1.50000 − 0.866025i 0.192055 − 0.110883i −0.400889 − 0.916127i \(-0.631299\pi\)
0.592944 + 0.805243i \(0.297965\pi\)
\(62\) −8.66025 −1.09985
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −15.5885 + 9.00000i −1.93351 + 1.11631i
\(66\) 0 0
\(67\) 6.50000 − 11.2583i 0.794101 − 1.37542i −0.129307 − 0.991605i \(-0.541275\pi\)
0.923408 − 0.383819i \(-0.125391\pi\)
\(68\) −6.92820 −0.840168
\(69\) 0 0
\(70\) −6.00000 − 6.92820i −0.717137 − 0.828079i
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 0 0
\(73\) −6.00000 + 3.46410i −0.702247 + 0.405442i −0.808184 − 0.588930i \(-0.799549\pi\)
0.105937 + 0.994373i \(0.466216\pi\)
\(74\) − 5.00000i − 0.581238i
\(75\) 0 0
\(76\) −3.00000 + 1.73205i −0.344124 + 0.198680i
\(77\) 0 0
\(78\) 0 0
\(79\) 3.50000 + 6.06218i 0.393781 + 0.682048i 0.992945 − 0.118578i \(-0.0378336\pi\)
−0.599164 + 0.800626i \(0.704500\pi\)
\(80\) 1.73205 + 3.00000i 0.193649 + 0.335410i
\(81\) 0 0
\(82\) 6.00000 + 3.46410i 0.662589 + 0.382546i
\(83\) −1.73205 + 3.00000i −0.190117 + 0.329293i −0.945289 − 0.326234i \(-0.894220\pi\)
0.755172 + 0.655527i \(0.227553\pi\)
\(84\) 0 0
\(85\) 12.0000 + 20.7846i 1.30158 + 2.25441i
\(86\) 1.00000i 0.107833i
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 − 9.00000i 0.550791 − 0.953998i −0.447427 − 0.894321i \(-0.647659\pi\)
0.998218 − 0.0596775i \(-0.0190072\pi\)
\(90\) 0 0
\(91\) 9.00000 + 10.3923i 0.943456 + 1.08941i
\(92\) −5.19615 − 3.00000i −0.541736 − 0.312772i
\(93\) 0 0
\(94\) −6.00000 − 3.46410i −0.618853 − 0.357295i
\(95\) 10.3923 + 6.00000i 1.06623 + 0.615587i
\(96\) 0 0
\(97\) 1.50000 + 0.866025i 0.152302 + 0.0879316i 0.574214 − 0.818705i \(-0.305308\pi\)
−0.421912 + 0.906637i \(0.638641\pi\)
\(98\) −4.33013 + 5.50000i −0.437409 + 0.555584i
\(99\) 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.b.593.2 4
3.2 odd 2 inner 1134.2.t.b.593.1 4
7.3 odd 6 1134.2.l.a.269.2 4
9.2 odd 6 378.2.k.b.215.2 yes 4
9.4 even 3 1134.2.l.a.215.2 4
9.5 odd 6 1134.2.l.a.215.1 4
9.7 even 3 378.2.k.b.215.1 ✓ 4
21.17 even 6 1134.2.l.a.269.1 4
63.2 odd 6 2646.2.d.b.2645.2 4
63.16 even 3 2646.2.d.b.2645.3 4
63.31 odd 6 inner 1134.2.t.b.1025.1 4
63.38 even 6 378.2.k.b.269.1 yes 4
63.47 even 6 2646.2.d.b.2645.1 4
63.52 odd 6 378.2.k.b.269.2 yes 4
63.59 even 6 inner 1134.2.t.b.1025.2 4
63.61 odd 6 2646.2.d.b.2645.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.k.b.215.1 ✓ 4 9.7 even 3
378.2.k.b.215.2 yes 4 9.2 odd 6
378.2.k.b.269.1 yes 4 63.38 even 6
378.2.k.b.269.2 yes 4 63.52 odd 6
1134.2.l.a.215.1 4 9.5 odd 6
1134.2.l.a.215.2 4 9.4 even 3
1134.2.l.a.269.1 4 21.17 even 6
1134.2.l.a.269.2 4 7.3 odd 6
1134.2.t.b.593.1 4 3.2 odd 2 inner
1134.2.t.b.593.2 4 1.1 even 1 trivial
1134.2.t.b.1025.1 4 63.31 odd 6 inner
1134.2.t.b.1025.2 4 63.59 even 6 inner
2646.2.d.b.2645.1 4 63.47 even 6
2646.2.d.b.2645.2 4 63.2 odd 6
2646.2.d.b.2645.3 4 63.16 even 3
2646.2.d.b.2645.4 4 63.61 odd 6