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Properties

Label 912.2.q.l.577.3
Level $912$
Weight $2$
Character 912.577
Analytic conductor $7.282$
Analytic rank $0$
Dimension $6$
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] = [912,2,Mod(49,912)] 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("912.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,3,0,-2,0,2,0,-3,0,0,0,1] 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.3
Root \(-1.62241 + 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 912.577
Dual form 912.2.q.l.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(1.33641 + 2.31473i) q^{5} +3.67282 q^{7} +(-0.500000 + 0.866025i) q^{9} +3.81681 q^{11} +(-0.0719933 + 0.124696i) q^{13} +(-1.33641 + 2.31473i) q^{15} +(-4.24482 + 0.990721i) q^{19} +(1.83641 + 3.18076i) q^{21} +(3.76442 - 6.52016i) q^{23} +(-1.07199 + 1.85675i) q^{25} -1.00000 q^{27} +(2.67282 - 4.62947i) q^{29} -8.81681 q^{31} +(1.90841 + 3.30545i) q^{33} +(4.90841 + 8.50161i) q^{35} -1.00000 q^{37} -0.143987 q^{39} +(-2.67282 - 4.62947i) q^{41} +(1.40841 + 2.43943i) q^{43} -2.67282 q^{45} +(-3.00000 + 5.19615i) q^{47} +6.48963 q^{49} +(-4.00924 + 6.94420i) q^{53} +(5.10083 + 8.83490i) q^{55} +(-2.98040 - 3.18076i) q^{57} +(1.90841 + 3.30545i) q^{59} +(-5.74482 + 9.95031i) q^{61} +(-1.83641 + 3.18076i) q^{63} -0.384851 q^{65} +(2.69243 - 4.66342i) q^{67} +7.52884 q^{69} +(-6.81681 - 11.8071i) q^{71} +(0.172824 + 0.299339i) q^{73} -2.14399 q^{75} +14.0185 q^{77} +(3.26442 + 5.65414i) q^{79} +(-0.500000 - 0.866025i) q^{81} +2.28797 q^{83} +5.34565 q^{87} +(4.33641 - 7.51089i) q^{89} +(-0.264419 + 0.457986i) q^{91} +(-4.40841 - 7.63558i) q^{93} +(-7.96608 - 8.50161i) q^{95} +(2.95684 + 5.12140i) q^{97} +(-1.90841 + 3.30545i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 2 q^{5} + 2 q^{7} - 3 q^{9} + q^{13} + 2 q^{15} - 4 q^{19} + q^{21} + 14 q^{23} - 5 q^{25} - 6 q^{27} - 4 q^{29} - 30 q^{31} + 18 q^{35} - 6 q^{37} + 2 q^{39} + 4 q^{41} - 3 q^{43} + 4 q^{45}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 1.33641 + 2.31473i 0.597662 + 1.03518i 0.993165 + 0.116716i \(0.0372367\pi\)
−0.395504 + 0.918464i \(0.629430\pi\)
\(6\) 0 0
\(7\) 3.67282 1.38820 0.694098 − 0.719880i \(-0.255803\pi\)
0.694098 + 0.719880i \(0.255803\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 3.81681 1.15081 0.575406 − 0.817868i \(-0.304844\pi\)
0.575406 + 0.817868i \(0.304844\pi\)
\(12\) 0 0
\(13\) −0.0719933 + 0.124696i −0.0199673 + 0.0345844i −0.875836 − 0.482608i \(-0.839690\pi\)
0.855869 + 0.517193i \(0.173023\pi\)
\(14\) 0 0
\(15\) −1.33641 + 2.31473i −0.345060 + 0.597662i
\(16\) 0 0
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −4.24482 + 0.990721i −0.973828 + 0.227287i
\(20\) 0 0
\(21\) 1.83641 + 3.18076i 0.400738 + 0.694098i
\(22\) 0 0
\(23\) 3.76442 − 6.52016i 0.784936 − 1.35955i −0.144102 − 0.989563i \(-0.546029\pi\)
0.929038 − 0.369985i \(-0.120637\pi\)
\(24\) 0 0
\(25\) −1.07199 + 1.85675i −0.214399 + 0.371349i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.67282 − 4.62947i 0.496331 − 0.859670i −0.503660 − 0.863902i \(-0.668014\pi\)
0.999991 + 0.00423154i \(0.00134695\pi\)
\(30\) 0 0
\(31\) −8.81681 −1.58355 −0.791773 − 0.610816i \(-0.790842\pi\)
−0.791773 + 0.610816i \(0.790842\pi\)
\(32\) 0 0
\(33\) 1.90841 + 3.30545i 0.332211 + 0.575406i
\(34\) 0 0
\(35\) 4.90841 + 8.50161i 0.829672 + 1.43703i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) −0.143987 −0.0230563
\(40\) 0 0
\(41\) −2.67282 − 4.62947i −0.417425 − 0.723001i 0.578255 − 0.815856i \(-0.303734\pi\)
−0.995680 + 0.0928551i \(0.970401\pi\)
\(42\) 0 0
\(43\) 1.40841 + 2.43943i 0.214780 + 0.372009i 0.953204 − 0.302327i \(-0.0977633\pi\)
−0.738425 + 0.674336i \(0.764430\pi\)
\(44\) 0 0
\(45\) −2.67282 −0.398441
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 − 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) 6.48963 0.927091
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00924 + 6.94420i −0.550711 + 0.953859i 0.447513 + 0.894278i \(0.352310\pi\)
−0.998223 + 0.0595815i \(0.981023\pi\)
\(54\) 0 0
\(55\) 5.10083 + 8.83490i 0.687796 + 1.19130i
\(56\) 0 0
\(57\) −2.98040 − 3.18076i −0.394763 − 0.421302i
\(58\) 0 0
\(59\) 1.90841 + 3.30545i 0.248453 + 0.430334i 0.963097 − 0.269155i \(-0.0867445\pi\)
−0.714644 + 0.699489i \(0.753411\pi\)
\(60\) 0 0
\(61\) −5.74482 + 9.95031i −0.735548 + 1.27401i 0.218934 + 0.975740i \(0.429742\pi\)
−0.954482 + 0.298268i \(0.903591\pi\)
\(62\) 0 0
\(63\) −1.83641 + 3.18076i −0.231366 + 0.400738i
\(64\) 0 0
\(65\) −0.384851 −0.0477348
\(66\) 0 0
\(67\) 2.69243 − 4.66342i 0.328932 − 0.569727i −0.653368 − 0.757040i \(-0.726645\pi\)
0.982300 + 0.187313i \(0.0599779\pi\)
\(68\) 0 0
\(69\) 7.52884 0.906365
\(70\) 0 0
\(71\) −6.81681 − 11.8071i −0.809007 − 1.40124i −0.913553 − 0.406720i \(-0.866672\pi\)
0.104546 − 0.994520i \(-0.466661\pi\)
\(72\) 0 0
\(73\) 0.172824 + 0.299339i 0.0202275 + 0.0350350i 0.875962 − 0.482380i \(-0.160228\pi\)
−0.855734 + 0.517415i \(0.826894\pi\)
\(74\) 0 0
\(75\) −2.14399 −0.247566
\(76\) 0 0
\(77\) 14.0185 1.59755
\(78\) 0 0
\(79\) 3.26442 + 5.65414i 0.367276 + 0.636140i 0.989139 − 0.146986i \(-0.0469572\pi\)
−0.621863 + 0.783126i \(0.713624\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 2.28797 0.251138 0.125569 − 0.992085i \(-0.459924\pi\)
0.125569 + 0.992085i \(0.459924\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.34565 0.573114
\(88\) 0 0
\(89\) 4.33641 − 7.51089i 0.459659 − 0.796152i −0.539284 − 0.842124i \(-0.681305\pi\)
0.998943 + 0.0459717i \(0.0146384\pi\)
\(90\) 0 0
\(91\) −0.264419 + 0.457986i −0.0277186 + 0.0480100i
\(92\) 0 0
\(93\) −4.40841 − 7.63558i −0.457130 − 0.791773i
\(94\) 0 0
\(95\) −7.96608 − 8.50161i −0.817303 − 0.872246i
\(96\) 0 0
\(97\) 2.95684 + 5.12140i 0.300222 + 0.520000i 0.976186 − 0.216935i \(-0.0696059\pi\)
−0.675964 + 0.736935i \(0.736273\pi\)
\(98\) 0 0
\(99\) −1.90841 + 3.30545i −0.191802 + 0.332211i
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 912.2.q.l.577.3 6
3.2 odd 2 2736.2.s.z.577.1 6
4.3 odd 2 57.2.e.b.7.2 ✓ 6
12.11 even 2 171.2.f.b.64.2 6
19.11 even 3 inner 912.2.q.l.49.3 6
57.11 odd 6 2736.2.s.z.1873.1 6
76.7 odd 6 1083.2.a.l.1.2 3
76.11 odd 6 57.2.e.b.49.2 yes 6
76.31 even 6 1083.2.a.o.1.2 3
228.11 even 6 171.2.f.b.163.2 6
228.83 even 6 3249.2.a.y.1.2 3
228.107 odd 6 3249.2.a.t.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.2 ✓ 6 4.3 odd 2
57.2.e.b.49.2 yes 6 76.11 odd 6
171.2.f.b.64.2 6 12.11 even 2
171.2.f.b.163.2 6 228.11 even 6
912.2.q.l.49.3 6 19.11 even 3 inner
912.2.q.l.577.3 6 1.1 even 1 trivial
1083.2.a.l.1.2 3 76.7 odd 6
1083.2.a.o.1.2 3 76.31 even 6
2736.2.s.z.577.1 6 3.2 odd 2
2736.2.s.z.1873.1 6 57.11 odd 6
3249.2.a.t.1.2 3 228.107 odd 6
3249.2.a.y.1.2 3 228.83 even 6