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Properties

Label 336.2.bc.d.257.1
Level $336$
Weight $2$
Character 336.257
Analytic conductor $2.683$
Analytic rank $0$
Dimension $2$
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] = [336,2,Mod(17,336)] 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("336.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.68297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 257.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 336.257
Dual form 336.2.bc.d.17.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} +(4.50000 - 2.59808i) q^{11} +(4.50000 - 2.59808i) q^{15} +(1.50000 + 2.59808i) q^{17} +(-1.50000 - 0.866025i) q^{19} +(-4.50000 + 0.866025i) q^{21} +(-4.50000 - 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} +5.19615i q^{27} +(1.50000 - 0.866025i) q^{31} +9.00000 q^{33} +(1.50000 + 7.79423i) q^{35} +(-3.50000 + 6.06218i) q^{37} -6.00000 q^{41} -4.00000 q^{43} +9.00000 q^{45} +(-1.50000 + 2.59808i) q^{47} +(1.00000 - 6.92820i) q^{49} +5.19615i q^{51} +(4.50000 - 2.59808i) q^{53} -15.5885i q^{55} +(-1.50000 - 2.59808i) q^{57} +(1.50000 + 2.59808i) q^{59} +(-10.5000 - 6.06218i) q^{61} +(-7.50000 - 2.59808i) q^{63} +(2.50000 + 4.33013i) q^{67} +(-4.50000 - 7.79423i) q^{69} -10.3923i q^{71} +(-10.5000 + 6.06218i) q^{73} -6.92820i q^{75} +(-4.50000 + 12.9904i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(-4.50000 + 7.79423i) q^{81} -12.0000 q^{83} +9.00000 q^{85} +(-4.50000 + 7.79423i) q^{89} +3.00000 q^{93} +(-4.50000 + 2.59808i) q^{95} -6.92820i q^{97} +(13.5000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} + 9 q^{11} + 9 q^{15} + 3 q^{17} - 3 q^{19} - 9 q^{21} - 9 q^{23} - 4 q^{25} + 3 q^{31} + 18 q^{33} + 3 q^{35} - 7 q^{37} - 12 q^{41} - 8 q^{43} + 18 q^{45} - 3 q^{47}+ \cdots + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) 1.50000 − 2.59808i 0.670820 − 1.16190i −0.306851 − 0.951757i \(-0.599275\pi\)
0.977672 − 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) −2.00000 + 1.73205i −0.755929 + 0.654654i
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 4.50000 − 2.59808i 1.35680 − 0.783349i 0.367610 − 0.929980i \(-0.380176\pi\)
0.989191 + 0.146631i \(0.0468429\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 4.50000 − 2.59808i 1.16190 − 0.670820i
\(16\) 0 0
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) −1.50000 − 0.866025i −0.344124 − 0.198680i 0.317970 − 0.948101i \(-0.396999\pi\)
−0.662094 + 0.749421i \(0.730332\pi\)
\(20\) 0 0
\(21\) −4.50000 + 0.866025i −0.981981 + 0.188982i
\(22\) 0 0
\(23\) −4.50000 − 2.59808i −0.938315 − 0.541736i −0.0488832 − 0.998805i \(-0.515566\pi\)
−0.889432 + 0.457068i \(0.848900\pi\)
\(24\) 0 0
\(25\) −2.00000 − 3.46410i −0.400000 − 0.692820i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.50000 − 0.866025i 0.269408 − 0.155543i −0.359211 − 0.933257i \(-0.616954\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 9.00000 1.56670
\(34\) 0 0
\(35\) 1.50000 + 7.79423i 0.253546 + 1.31747i
\(36\) 0 0
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 9.00000 1.34164
\(46\) 0 0
\(47\) −1.50000 + 2.59808i −0.218797 + 0.378968i −0.954441 − 0.298401i \(-0.903547\pi\)
0.735643 + 0.677369i \(0.236880\pi\)
\(48\) 0 0
\(49\) 1.00000 − 6.92820i 0.142857 − 0.989743i
\(50\) 0 0
\(51\) 5.19615i 0.727607i
\(52\) 0 0
\(53\) 4.50000 − 2.59808i 0.618123 − 0.356873i −0.158015 − 0.987437i \(-0.550509\pi\)
0.776138 + 0.630563i \(0.217176\pi\)
\(54\) 0 0
\(55\) − 15.5885i − 2.10195i
\(56\) 0 0
\(57\) −1.50000 − 2.59808i −0.198680 − 0.344124i
\(58\) 0 0
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 − 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 0 0
\(61\) −10.5000 − 6.06218i −1.34439 − 0.776182i −0.356939 − 0.934128i \(-0.616180\pi\)
−0.987448 + 0.157945i \(0.949513\pi\)
\(62\) 0 0
\(63\) −7.50000 − 2.59808i −0.944911 − 0.327327i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.50000 + 4.33013i 0.305424 + 0.529009i 0.977356 − 0.211604i \(-0.0678686\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) −4.50000 − 7.79423i −0.541736 − 0.938315i
\(70\) 0 0
\(71\) − 10.3923i − 1.23334i −0.787222 − 0.616670i \(-0.788481\pi\)
0.787222 − 0.616670i \(-0.211519\pi\)
\(72\) 0 0
\(73\) −10.5000 + 6.06218i −1.22893 + 0.709524i −0.966807 − 0.255510i \(-0.917757\pi\)
−0.262126 + 0.965034i \(0.584423\pi\)
\(74\) 0 0
\(75\) − 6.92820i − 0.800000i
\(76\) 0 0
\(77\) −4.50000 + 12.9904i −0.512823 + 1.48039i
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.0562544 + 0.0974355i −0.892781 − 0.450490i \(-0.851249\pi\)
0.836527 + 0.547926i \(0.184582\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.50000 + 7.79423i −0.476999 + 0.826187i −0.999653 − 0.0263586i \(-0.991609\pi\)
0.522654 + 0.852545i \(0.324942\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −4.50000 + 2.59808i −0.461690 + 0.266557i
\(96\) 0 0
\(97\) − 6.92820i − 0.703452i −0.936103 − 0.351726i \(-0.885595\pi\)
0.936103 − 0.351726i \(-0.114405\pi\)
\(98\) 0 0
\(99\) 13.5000 + 7.79423i 1.35680 + 0.783349i
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 336.2.bc.d.257.1 2
3.2 odd 2 336.2.bc.b.257.1 2
4.3 odd 2 84.2.k.a.5.1 ✓ 2
7.2 even 3 2352.2.k.a.881.2 2
7.3 odd 6 336.2.bc.b.17.1 2
7.5 odd 6 2352.2.k.d.881.1 2
12.11 even 2 84.2.k.b.5.1 yes 2
20.3 even 4 2100.2.bo.a.1349.2 4
20.7 even 4 2100.2.bo.a.1349.1 4
20.19 odd 2 2100.2.bi.f.1601.1 2
21.2 odd 6 2352.2.k.d.881.2 2
21.5 even 6 2352.2.k.a.881.1 2
21.17 even 6 inner 336.2.bc.d.17.1 2
28.3 even 6 84.2.k.b.17.1 yes 2
28.11 odd 6 588.2.k.c.521.1 2
28.19 even 6 588.2.f.a.293.2 2
28.23 odd 6 588.2.f.c.293.1 2
28.27 even 2 588.2.k.d.509.1 2
36.7 odd 6 2268.2.w.f.1349.1 2
36.11 even 6 2268.2.w.a.1349.1 2
36.23 even 6 2268.2.bm.f.593.1 2
36.31 odd 6 2268.2.bm.a.593.1 2
60.23 odd 4 2100.2.bo.f.1349.1 4
60.47 odd 4 2100.2.bo.f.1349.2 4
60.59 even 2 2100.2.bi.e.1601.1 2
84.11 even 6 588.2.k.d.521.1 2
84.23 even 6 588.2.f.a.293.1 2
84.47 odd 6 588.2.f.c.293.2 2
84.59 odd 6 84.2.k.a.17.1 yes 2
84.83 odd 2 588.2.k.c.509.1 2
140.3 odd 12 2100.2.bo.f.1949.2 4
140.59 even 6 2100.2.bi.e.101.1 2
140.87 odd 12 2100.2.bo.f.1949.1 4
252.31 even 6 2268.2.w.a.269.1 2
252.59 odd 6 2268.2.w.f.269.1 2
252.115 even 6 2268.2.bm.f.1025.1 2
252.227 odd 6 2268.2.bm.a.1025.1 2
420.59 odd 6 2100.2.bi.f.101.1 2
420.143 even 12 2100.2.bo.a.1949.1 4
420.227 even 12 2100.2.bo.a.1949.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.k.a.5.1 ✓ 2 4.3 odd 2
84.2.k.a.17.1 yes 2 84.59 odd 6
84.2.k.b.5.1 yes 2 12.11 even 2
84.2.k.b.17.1 yes 2 28.3 even 6
336.2.bc.b.17.1 2 7.3 odd 6
336.2.bc.b.257.1 2 3.2 odd 2
336.2.bc.d.17.1 2 21.17 even 6 inner
336.2.bc.d.257.1 2 1.1 even 1 trivial
588.2.f.a.293.1 2 84.23 even 6
588.2.f.a.293.2 2 28.19 even 6
588.2.f.c.293.1 2 28.23 odd 6
588.2.f.c.293.2 2 84.47 odd 6
588.2.k.c.509.1 2 84.83 odd 2
588.2.k.c.521.1 2 28.11 odd 6
588.2.k.d.509.1 2 28.27 even 2
588.2.k.d.521.1 2 84.11 even 6
2100.2.bi.e.101.1 2 140.59 even 6
2100.2.bi.e.1601.1 2 60.59 even 2
2100.2.bi.f.101.1 2 420.59 odd 6
2100.2.bi.f.1601.1 2 20.19 odd 2
2100.2.bo.a.1349.1 4 20.7 even 4
2100.2.bo.a.1349.2 4 20.3 even 4
2100.2.bo.a.1949.1 4 420.143 even 12
2100.2.bo.a.1949.2 4 420.227 even 12
2100.2.bo.f.1349.1 4 60.23 odd 4
2100.2.bo.f.1349.2 4 60.47 odd 4
2100.2.bo.f.1949.1 4 140.87 odd 12
2100.2.bo.f.1949.2 4 140.3 odd 12
2268.2.w.a.269.1 2 252.31 even 6
2268.2.w.a.1349.1 2 36.11 even 6
2268.2.w.f.269.1 2 252.59 odd 6
2268.2.w.f.1349.1 2 36.7 odd 6
2268.2.bm.a.593.1 2 36.31 odd 6
2268.2.bm.a.1025.1 2 252.227 odd 6
2268.2.bm.f.593.1 2 36.23 even 6
2268.2.bm.f.1025.1 2 252.115 even 6
2352.2.k.a.881.1 2 21.5 even 6
2352.2.k.a.881.2 2 7.2 even 3
2352.2.k.d.881.1 2 7.5 odd 6
2352.2.k.d.881.2 2 21.2 odd 6