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Properties

Label 3042.2.b.i.1351.1
Level $3042$
Weight $2$
Character 3042.1351
Analytic conductor $24.290$
Analytic rank $0$
Dimension $4$
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] = [3042,2,Mod(1351,3042)] 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("3042.1351"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,0,0,0,0,-8,0,0,0,4,0,4,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.2904922949\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3042.1351
Dual form 3042.2.b.i.1351.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -3.73205i q^{5} +2.73205i q^{7} +1.00000i q^{8} -3.73205 q^{10} +1.26795i q^{11} +2.73205 q^{14} +1.00000 q^{16} -5.73205 q^{17} +4.73205i q^{19} +3.73205i q^{20} +1.26795 q^{22} +4.19615 q^{23} -8.92820 q^{25} -2.73205i q^{28} +4.46410 q^{29} +1.46410i q^{31} -1.00000i q^{32} +5.73205i q^{34} +10.1962 q^{35} -3.53590i q^{37} +4.73205 q^{38} +3.73205 q^{40} +9.39230i q^{41} +9.66025 q^{43} -1.26795i q^{44} -4.19615i q^{46} +2.19615i q^{47} -0.464102 q^{49} +8.92820i q^{50} +6.46410 q^{53} +4.73205 q^{55} -2.73205 q^{56} -4.46410i q^{58} +8.00000i q^{59} -9.19615 q^{61} +1.46410 q^{62} -1.00000 q^{64} +13.1244i q^{67} +5.73205 q^{68} -10.1962i q^{70} -4.73205i q^{71} +6.26795i q^{73} -3.53590 q^{74} -4.73205i q^{76} -3.46410 q^{77} -2.53590 q^{79} -3.73205i q^{80} +9.39230 q^{82} +0.196152i q^{83} +21.3923i q^{85} -9.66025i q^{86} -1.26795 q^{88} -9.46410i q^{89} -4.19615 q^{92} +2.19615 q^{94} +17.6603 q^{95} -6.00000i q^{97} +0.464102i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 8 q^{10} + 4 q^{14} + 4 q^{16} - 16 q^{17} + 12 q^{22} - 4 q^{23} - 8 q^{25} + 4 q^{29} + 20 q^{35} + 12 q^{38} + 8 q^{40} + 4 q^{43} + 12 q^{49} + 12 q^{53} + 12 q^{55} - 4 q^{56} - 16 q^{61}+ \cdots + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) − 3.73205i − 1.66902i −0.550990 − 0.834512i \(-0.685750\pi\)
0.550990 − 0.834512i \(-0.314250\pi\)
\(6\) 0 0
\(7\) 2.73205i 1.03262i 0.856402 + 0.516309i \(0.172694\pi\)
−0.856402 + 0.516309i \(0.827306\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) −3.73205 −1.18018
\(11\) 1.26795i 0.382301i 0.981561 + 0.191151i \(0.0612219\pi\)
−0.981561 + 0.191151i \(0.938778\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 2.73205 0.730171
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.73205 −1.39023 −0.695113 − 0.718900i \(-0.744646\pi\)
−0.695113 + 0.718900i \(0.744646\pi\)
\(18\) 0 0
\(19\) 4.73205i 1.08561i 0.839860 + 0.542803i \(0.182637\pi\)
−0.839860 + 0.542803i \(0.817363\pi\)
\(20\) 3.73205i 0.834512i
\(21\) 0 0
\(22\) 1.26795 0.270328
\(23\) 4.19615 0.874958 0.437479 − 0.899229i \(-0.355871\pi\)
0.437479 + 0.899229i \(0.355871\pi\)
\(24\) 0 0
\(25\) −8.92820 −1.78564
\(26\) 0 0
\(27\) 0 0
\(28\) − 2.73205i − 0.516309i
\(29\) 4.46410 0.828963 0.414481 − 0.910058i \(-0.363963\pi\)
0.414481 + 0.910058i \(0.363963\pi\)
\(30\) 0 0
\(31\) 1.46410i 0.262960i 0.991319 + 0.131480i \(0.0419730\pi\)
−0.991319 + 0.131480i \(0.958027\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 5.73205i 0.983039i
\(35\) 10.1962 1.72346
\(36\) 0 0
\(37\) − 3.53590i − 0.581298i −0.956830 − 0.290649i \(-0.906129\pi\)
0.956830 − 0.290649i \(-0.0938712\pi\)
\(38\) 4.73205 0.767640
\(39\) 0 0
\(40\) 3.73205 0.590089
\(41\) 9.39230i 1.46683i 0.679780 + 0.733416i \(0.262075\pi\)
−0.679780 + 0.733416i \(0.737925\pi\)
\(42\) 0 0
\(43\) 9.66025 1.47317 0.736587 − 0.676342i \(-0.236436\pi\)
0.736587 + 0.676342i \(0.236436\pi\)
\(44\) − 1.26795i − 0.191151i
\(45\) 0 0
\(46\) − 4.19615i − 0.618689i
\(47\) 2.19615i 0.320342i 0.987089 + 0.160171i \(0.0512045\pi\)
−0.987089 + 0.160171i \(0.948795\pi\)
\(48\) 0 0
\(49\) −0.464102 −0.0663002
\(50\) 8.92820i 1.26264i
\(51\) 0 0
\(52\) 0 0
\(53\) 6.46410 0.887913 0.443956 − 0.896048i \(-0.353575\pi\)
0.443956 + 0.896048i \(0.353575\pi\)
\(54\) 0 0
\(55\) 4.73205 0.638070
\(56\) −2.73205 −0.365086
\(57\) 0 0
\(58\) − 4.46410i − 0.586165i
\(59\) 8.00000i 1.04151i 0.853706 + 0.520756i \(0.174350\pi\)
−0.853706 + 0.520756i \(0.825650\pi\)
\(60\) 0 0
\(61\) −9.19615 −1.17745 −0.588723 − 0.808335i \(-0.700369\pi\)
−0.588723 + 0.808335i \(0.700369\pi\)
\(62\) 1.46410 0.185941
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.1244i 1.60340i 0.597730 + 0.801698i \(0.296070\pi\)
−0.597730 + 0.801698i \(0.703930\pi\)
\(68\) 5.73205 0.695113
\(69\) 0 0
\(70\) − 10.1962i − 1.21867i
\(71\) − 4.73205i − 0.561591i −0.959768 − 0.280796i \(-0.909402\pi\)
0.959768 − 0.280796i \(-0.0905983\pi\)
\(72\) 0 0
\(73\) 6.26795i 0.733608i 0.930298 + 0.366804i \(0.119548\pi\)
−0.930298 + 0.366804i \(0.880452\pi\)
\(74\) −3.53590 −0.411040
\(75\) 0 0
\(76\) − 4.73205i − 0.542803i
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −2.53590 −0.285311 −0.142655 − 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) − 3.73205i − 0.417256i
\(81\) 0 0
\(82\) 9.39230 1.03721
\(83\) 0.196152i 0.0215305i 0.999942 + 0.0107653i \(0.00342676\pi\)
−0.999942 + 0.0107653i \(0.996573\pi\)
\(84\) 0 0
\(85\) 21.3923i 2.32032i
\(86\) − 9.66025i − 1.04169i
\(87\) 0 0
\(88\) −1.26795 −0.135164
\(89\) − 9.46410i − 1.00319i −0.865102 − 0.501596i \(-0.832746\pi\)
0.865102 − 0.501596i \(-0.167254\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.19615 −0.437479
\(93\) 0 0
\(94\) 2.19615 0.226516
\(95\) 17.6603 1.81190
\(96\) 0 0
\(97\) − 6.00000i − 0.609208i −0.952479 − 0.304604i \(-0.901476\pi\)
0.952479 − 0.304604i \(-0.0985241\pi\)
\(98\) 0.464102i 0.0468813i
\(99\) 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 3042.2.b.i.1351.1 4
3.2 odd 2 1014.2.b.e.337.4 4
13.3 even 3 234.2.l.c.199.2 4
13.4 even 6 234.2.l.c.127.2 4
13.5 odd 4 3042.2.a.p.1.1 2
13.8 odd 4 3042.2.a.y.1.2 2
13.12 even 2 inner 3042.2.b.i.1351.4 4
39.2 even 12 1014.2.e.g.529.2 4
39.5 even 4 1014.2.a.k.1.2 2
39.8 even 4 1014.2.a.i.1.1 2
39.11 even 12 1014.2.e.i.529.1 4
39.17 odd 6 78.2.i.a.49.1 yes 4
39.20 even 12 1014.2.e.i.991.1 4
39.23 odd 6 1014.2.i.a.823.2 4
39.29 odd 6 78.2.i.a.43.1 ✓ 4
39.32 even 12 1014.2.e.g.991.2 4
39.35 odd 6 1014.2.i.a.361.2 4
39.38 odd 2 1014.2.b.e.337.1 4
52.3 odd 6 1872.2.by.h.433.1 4
52.43 odd 6 1872.2.by.h.1297.2 4
156.47 odd 4 8112.2.a.bj.1.1 2
156.83 odd 4 8112.2.a.bp.1.2 2
156.95 even 6 624.2.bv.e.49.1 4
156.107 even 6 624.2.bv.e.433.2 4
195.17 even 12 1950.2.y.b.49.1 4
195.29 odd 6 1950.2.bc.d.901.2 4
195.68 even 12 1950.2.y.b.199.1 4
195.107 even 12 1950.2.y.g.199.2 4
195.134 odd 6 1950.2.bc.d.751.2 4
195.173 even 12 1950.2.y.g.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 ✓ 4 39.29 odd 6
78.2.i.a.49.1 yes 4 39.17 odd 6
234.2.l.c.127.2 4 13.4 even 6
234.2.l.c.199.2 4 13.3 even 3
624.2.bv.e.49.1 4 156.95 even 6
624.2.bv.e.433.2 4 156.107 even 6
1014.2.a.i.1.1 2 39.8 even 4
1014.2.a.k.1.2 2 39.5 even 4
1014.2.b.e.337.1 4 39.38 odd 2
1014.2.b.e.337.4 4 3.2 odd 2
1014.2.e.g.529.2 4 39.2 even 12
1014.2.e.g.991.2 4 39.32 even 12
1014.2.e.i.529.1 4 39.11 even 12
1014.2.e.i.991.1 4 39.20 even 12
1014.2.i.a.361.2 4 39.35 odd 6
1014.2.i.a.823.2 4 39.23 odd 6
1872.2.by.h.433.1 4 52.3 odd 6
1872.2.by.h.1297.2 4 52.43 odd 6
1950.2.y.b.49.1 4 195.17 even 12
1950.2.y.b.199.1 4 195.68 even 12
1950.2.y.g.49.2 4 195.173 even 12
1950.2.y.g.199.2 4 195.107 even 12
1950.2.bc.d.751.2 4 195.134 odd 6
1950.2.bc.d.901.2 4 195.29 odd 6
3042.2.a.p.1.1 2 13.5 odd 4
3042.2.a.y.1.2 2 13.8 odd 4
3042.2.b.i.1351.1 4 1.1 even 1 trivial
3042.2.b.i.1351.4 4 13.12 even 2 inner
8112.2.a.bj.1.1 2 156.47 odd 4
8112.2.a.bp.1.2 2 156.83 odd 4