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Properties

Label 2646.2.h.o.361.3
Level $2646$
Weight $2$
Character 2646.361
Analytic conductor $21.128$
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] = [2646,2,Mod(361,2646)] 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("2646.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2646, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3,0,-3,-2,0,0,6,0,1,-2,0,-8] 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: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.3
Root \(0.500000 + 0.224437i\) of defining polynomial
Character \(\chi\) \(=\) 2646.361
Dual form 2646.2.h.o.667.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +1.58836 q^{5} +1.00000 q^{8} +(-0.794182 + 1.37556i) q^{10} +1.58836 q^{11} +(-2.40545 + 4.16635i) q^{13} +(-0.500000 + 0.866025i) q^{16} +(-2.69963 + 4.67589i) q^{17} +(3.54944 + 6.14781i) q^{19} +(-0.794182 - 1.37556i) q^{20} +(-0.794182 + 1.37556i) q^{22} -0.300372 q^{23} -2.47710 q^{25} +(-2.40545 - 4.16635i) q^{26} +(-4.13781 - 7.16689i) q^{29} +(-1.35600 - 2.34867i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-2.69963 - 4.67589i) q^{34} +(0.500000 + 0.866025i) q^{37} -7.09888 q^{38} +1.58836 q^{40} +(2.93818 - 5.08907i) q^{41} +(-0.833104 - 1.44298i) q^{43} +(-0.794182 - 1.37556i) q^{44} +(0.150186 - 0.260130i) q^{46} +(-1.33310 + 2.30900i) q^{47} +(1.23855 - 2.14523i) q^{50} +4.81089 q^{52} +(-2.44437 + 4.23377i) q^{53} +2.52290 q^{55} +8.27561 q^{58} +(-3.23855 - 5.60933i) q^{59} +(-2.23855 + 3.87728i) q^{61} +2.71201 q^{62} +1.00000 q^{64} +(-3.82072 + 6.61769i) q^{65} +(5.02654 + 8.70623i) q^{67} +5.39926 q^{68} -12.7207 q^{71} +(-8.02654 + 13.9024i) q^{73} -1.00000 q^{74} +(3.54944 - 6.14781i) q^{76} +(-4.19344 + 7.26325i) q^{79} +(-0.794182 + 1.37556i) q^{80} +(2.93818 + 5.08907i) q^{82} +(1.18292 + 2.04887i) q^{83} +(-4.28799 + 7.42702i) q^{85} +1.66621 q^{86} +1.58836 q^{88} +(1.60507 + 2.78007i) q^{89} +(0.150186 + 0.260130i) q^{92} +(-1.33310 - 2.30900i) q^{94} +(5.63781 + 9.76497i) q^{95} +(-0.712008 - 1.23323i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - 2 q^{5} + 6 q^{8} + q^{10} - 2 q^{11} - 8 q^{13} - 3 q^{16} - 4 q^{17} + 3 q^{19} + q^{20} + q^{22} - 14 q^{23} - 4 q^{25} - 8 q^{26} + 5 q^{29} - 20 q^{31} - 3 q^{32} - 4 q^{34}+ \cdots - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 1.58836 0.710338 0.355169 − 0.934802i \(-0.384423\pi\)
0.355169 + 0.934802i \(0.384423\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −0.794182 + 1.37556i −0.251142 + 0.434991i
\(11\) 1.58836 0.478910 0.239455 − 0.970907i \(-0.423031\pi\)
0.239455 + 0.970907i \(0.423031\pi\)
\(12\) 0 0
\(13\) −2.40545 + 4.16635i −0.667151 + 1.15554i 0.311547 + 0.950231i \(0.399153\pi\)
−0.978697 + 0.205308i \(0.934180\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −2.69963 + 4.67589i −0.654756 + 1.13407i 0.327199 + 0.944955i \(0.393895\pi\)
−0.981955 + 0.189115i \(0.939438\pi\)
\(18\) 0 0
\(19\) 3.54944 + 6.14781i 0.814298 + 1.41041i 0.909831 + 0.414979i \(0.136211\pi\)
−0.0955331 + 0.995426i \(0.530456\pi\)
\(20\) −0.794182 − 1.37556i −0.177584 − 0.307585i
\(21\) 0 0
\(22\) −0.794182 + 1.37556i −0.169320 + 0.293271i
\(23\) −0.300372 −0.0626319 −0.0313159 − 0.999510i \(-0.509970\pi\)
−0.0313159 + 0.999510i \(0.509970\pi\)
\(24\) 0 0
\(25\) −2.47710 −0.495420
\(26\) −2.40545 − 4.16635i −0.471747 − 0.817089i
\(27\) 0 0
\(28\) 0 0
\(29\) −4.13781 − 7.16689i −0.768371 − 1.33086i −0.938446 − 0.345427i \(-0.887734\pi\)
0.170074 − 0.985431i \(-0.445599\pi\)
\(30\) 0 0
\(31\) −1.35600 − 2.34867i −0.243545 − 0.421833i 0.718176 − 0.695861i \(-0.244977\pi\)
−0.961722 + 0.274028i \(0.911644\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −2.69963 − 4.67589i −0.462982 − 0.801909i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −7.09888 −1.15159
\(39\) 0 0
\(40\) 1.58836 0.251142
\(41\) 2.93818 − 5.08907i 0.458866 − 0.794780i −0.540035 − 0.841643i \(-0.681589\pi\)
0.998901 + 0.0468628i \(0.0149223\pi\)
\(42\) 0 0
\(43\) −0.833104 − 1.44298i −0.127047 − 0.220052i 0.795484 − 0.605974i \(-0.207217\pi\)
−0.922531 + 0.385922i \(0.873883\pi\)
\(44\) −0.794182 − 1.37556i −0.119727 − 0.207374i
\(45\) 0 0
\(46\) 0.150186 − 0.260130i 0.0221437 − 0.0383540i
\(47\) −1.33310 + 2.30900i −0.194453 + 0.336803i −0.946721 − 0.322055i \(-0.895627\pi\)
0.752268 + 0.658857i \(0.228960\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.23855 − 2.14523i 0.175157 − 0.303382i
\(51\) 0 0
\(52\) 4.81089 0.667151
\(53\) −2.44437 + 4.23377i −0.335760 + 0.581553i −0.983630 − 0.180197i \(-0.942326\pi\)
0.647871 + 0.761750i \(0.275660\pi\)
\(54\) 0 0
\(55\) 2.52290 0.340188
\(56\) 0 0
\(57\) 0 0
\(58\) 8.27561 1.08664
\(59\) −3.23855 − 5.60933i −0.421623 − 0.730273i 0.574475 − 0.818522i \(-0.305206\pi\)
−0.996098 + 0.0882491i \(0.971873\pi\)
\(60\) 0 0
\(61\) −2.23855 + 3.87728i −0.286617 + 0.496435i −0.973000 − 0.230805i \(-0.925864\pi\)
0.686383 + 0.727240i \(0.259197\pi\)
\(62\) 2.71201 0.344425
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.82072 + 6.61769i −0.473902 + 0.820823i
\(66\) 0 0
\(67\) 5.02654 + 8.70623i 0.614090 + 1.06363i 0.990543 + 0.137199i \(0.0438101\pi\)
−0.376454 + 0.926435i \(0.622857\pi\)
\(68\) 5.39926 0.654756
\(69\) 0 0
\(70\) 0 0
\(71\) −12.7207 −1.50967 −0.754833 − 0.655917i \(-0.772282\pi\)
−0.754833 + 0.655917i \(0.772282\pi\)
\(72\) 0 0
\(73\) −8.02654 + 13.9024i −0.939436 + 1.62715i −0.172909 + 0.984938i \(0.555317\pi\)
−0.766527 + 0.642213i \(0.778017\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 3.54944 − 6.14781i 0.407149 − 0.705203i
\(77\) 0 0
\(78\) 0 0
\(79\) −4.19344 + 7.26325i −0.471799 + 0.817179i −0.999479 − 0.0322635i \(-0.989728\pi\)
0.527681 + 0.849443i \(0.323062\pi\)
\(80\) −0.794182 + 1.37556i −0.0887922 + 0.153793i
\(81\) 0 0
\(82\) 2.93818 + 5.08907i 0.324467 + 0.561994i
\(83\) 1.18292 + 2.04887i 0.129842 + 0.224893i 0.923615 − 0.383321i \(-0.125220\pi\)
−0.793773 + 0.608214i \(0.791886\pi\)
\(84\) 0 0
\(85\) −4.28799 + 7.42702i −0.465098 + 0.805573i
\(86\) 1.66621 0.179672
\(87\) 0 0
\(88\) 1.58836 0.169320
\(89\) 1.60507 + 2.78007i 0.170138 + 0.294687i 0.938468 − 0.345367i \(-0.112245\pi\)
−0.768330 + 0.640054i \(0.778912\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0.150186 + 0.260130i 0.0156580 + 0.0271204i
\(93\) 0 0
\(94\) −1.33310 − 2.30900i −0.137499 − 0.238156i
\(95\) 5.63781 + 9.76497i 0.578427 + 1.00186i
\(96\) 0 0
\(97\) −0.712008 − 1.23323i −0.0722934 − 0.125216i 0.827613 − 0.561300i \(-0.189698\pi\)
−0.899906 + 0.436084i \(0.856365\pi\)
\(98\) 0 0
\(99\) 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 2646.2.h.o.361.3 6
3.2 odd 2 882.2.h.p.67.1 6
7.2 even 3 2646.2.e.p.1549.1 6
7.3 odd 6 2646.2.f.l.1765.3 6
7.4 even 3 2646.2.f.m.1765.1 6
7.5 odd 6 378.2.e.d.37.3 6
7.6 odd 2 378.2.h.c.361.1 6
9.2 odd 6 882.2.e.o.655.3 6
9.7 even 3 2646.2.e.p.2125.1 6
21.2 odd 6 882.2.e.o.373.3 6
21.5 even 6 126.2.e.c.121.1 yes 6
21.11 odd 6 882.2.f.o.589.1 6
21.17 even 6 882.2.f.n.589.3 6
21.20 even 2 126.2.h.d.67.3 yes 6
28.19 even 6 3024.2.q.g.2305.3 6
28.27 even 2 3024.2.t.h.1873.1 6
63.2 odd 6 882.2.h.p.79.1 6
63.4 even 3 7938.2.a.bz.1.3 3
63.5 even 6 1134.2.g.m.163.1 6
63.11 odd 6 882.2.f.o.295.1 6
63.13 odd 6 1134.2.g.l.487.3 6
63.16 even 3 inner 2646.2.h.o.667.3 6
63.20 even 6 126.2.e.c.25.1 ✓ 6
63.25 even 3 2646.2.f.m.883.1 6
63.31 odd 6 7938.2.a.ca.1.1 3
63.32 odd 6 7938.2.a.bw.1.1 3
63.34 odd 6 378.2.e.d.235.3 6
63.38 even 6 882.2.f.n.295.3 6
63.40 odd 6 1134.2.g.l.163.3 6
63.41 even 6 1134.2.g.m.487.1 6
63.47 even 6 126.2.h.d.79.3 yes 6
63.52 odd 6 2646.2.f.l.883.3 6
63.59 even 6 7938.2.a.bv.1.3 3
63.61 odd 6 378.2.h.c.289.1 6
84.47 odd 6 1008.2.q.g.625.3 6
84.83 odd 2 1008.2.t.h.193.1 6
252.47 odd 6 1008.2.t.h.961.1 6
252.83 odd 6 1008.2.q.g.529.3 6
252.187 even 6 3024.2.t.h.289.1 6
252.223 even 6 3024.2.q.g.2881.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.2.e.c.25.1 ✓ 6 63.20 even 6
126.2.e.c.121.1 yes 6 21.5 even 6
126.2.h.d.67.3 yes 6 21.20 even 2
126.2.h.d.79.3 yes 6 63.47 even 6
378.2.e.d.37.3 6 7.5 odd 6
378.2.e.d.235.3 6 63.34 odd 6
378.2.h.c.289.1 6 63.61 odd 6
378.2.h.c.361.1 6 7.6 odd 2
882.2.e.o.373.3 6 21.2 odd 6
882.2.e.o.655.3 6 9.2 odd 6
882.2.f.n.295.3 6 63.38 even 6
882.2.f.n.589.3 6 21.17 even 6
882.2.f.o.295.1 6 63.11 odd 6
882.2.f.o.589.1 6 21.11 odd 6
882.2.h.p.67.1 6 3.2 odd 2
882.2.h.p.79.1 6 63.2 odd 6
1008.2.q.g.529.3 6 252.83 odd 6
1008.2.q.g.625.3 6 84.47 odd 6
1008.2.t.h.193.1 6 84.83 odd 2
1008.2.t.h.961.1 6 252.47 odd 6
1134.2.g.l.163.3 6 63.40 odd 6
1134.2.g.l.487.3 6 63.13 odd 6
1134.2.g.m.163.1 6 63.5 even 6
1134.2.g.m.487.1 6 63.41 even 6
2646.2.e.p.1549.1 6 7.2 even 3
2646.2.e.p.2125.1 6 9.7 even 3
2646.2.f.l.883.3 6 63.52 odd 6
2646.2.f.l.1765.3 6 7.3 odd 6
2646.2.f.m.883.1 6 63.25 even 3
2646.2.f.m.1765.1 6 7.4 even 3
2646.2.h.o.361.3 6 1.1 even 1 trivial
2646.2.h.o.667.3 6 63.16 even 3 inner
3024.2.q.g.2305.3 6 28.19 even 6
3024.2.q.g.2881.3 6 252.223 even 6
3024.2.t.h.289.1 6 252.187 even 6
3024.2.t.h.1873.1 6 28.27 even 2
7938.2.a.bv.1.3 3 63.59 even 6
7938.2.a.bw.1.1 3 63.32 odd 6
7938.2.a.bz.1.3 3 63.4 even 3
7938.2.a.ca.1.1 3 63.31 odd 6