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Properties

Label 234.2.l.c.127.2
Level $234$
Weight $2$
Character 234.127
Analytic conductor $1.868$
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] = [234,2,Mod(127,234)] 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("234.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.86849940730\)
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 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 127.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 234.127
Dual form 234.2.l.c.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +3.73205i q^{5} +(2.36603 + 1.36603i) q^{7} -1.00000i q^{8} +(1.86603 + 3.23205i) q^{10} +(-1.09808 + 0.633975i) q^{11} +(-2.59808 - 2.50000i) q^{13} +2.73205 q^{14} +(-0.500000 - 0.866025i) q^{16} +(2.86603 - 4.96410i) q^{17} +(4.09808 + 2.36603i) q^{19} +(3.23205 + 1.86603i) q^{20} +(-0.633975 + 1.09808i) q^{22} +(-2.09808 - 3.63397i) q^{23} -8.92820 q^{25} +(-3.50000 - 0.866025i) q^{26} +(2.36603 - 1.36603i) q^{28} +(-2.23205 - 3.86603i) q^{29} -1.46410i q^{31} +(-0.866025 - 0.500000i) q^{32} -5.73205i q^{34} +(-5.09808 + 8.83013i) q^{35} +(3.06218 - 1.76795i) q^{37} +4.73205 q^{38} +3.73205 q^{40} +(-8.13397 + 4.69615i) q^{41} +(-4.83013 + 8.36603i) q^{43} +1.26795i q^{44} +(-3.63397 - 2.09808i) q^{46} -2.19615i q^{47} +(0.232051 + 0.401924i) q^{49} +(-7.73205 + 4.46410i) q^{50} +(-3.46410 + 1.00000i) q^{52} +6.46410 q^{53} +(-2.36603 - 4.09808i) q^{55} +(1.36603 - 2.36603i) q^{56} +(-3.86603 - 2.23205i) q^{58} +(6.92820 + 4.00000i) q^{59} +(4.59808 - 7.96410i) q^{61} +(-0.732051 - 1.26795i) q^{62} -1.00000 q^{64} +(9.33013 - 9.69615i) q^{65} +(-11.3660 + 6.56218i) q^{67} +(-2.86603 - 4.96410i) q^{68} +10.1962i q^{70} +(-4.09808 - 2.36603i) q^{71} -6.26795i q^{73} +(1.76795 - 3.06218i) q^{74} +(4.09808 - 2.36603i) q^{76} -3.46410 q^{77} -2.53590 q^{79} +(3.23205 - 1.86603i) q^{80} +(-4.69615 + 8.13397i) q^{82} -0.196152i q^{83} +(18.5263 + 10.6962i) q^{85} +9.66025i q^{86} +(0.633975 + 1.09808i) q^{88} +(8.19615 - 4.73205i) q^{89} +(-2.73205 - 9.46410i) q^{91} -4.19615 q^{92} +(-1.09808 - 1.90192i) q^{94} +(-8.83013 + 15.2942i) q^{95} +(-5.19615 - 3.00000i) q^{97} +(0.401924 + 0.232051i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 6 q^{7} + 4 q^{10} + 6 q^{11} + 4 q^{14} - 2 q^{16} + 8 q^{17} + 6 q^{19} + 6 q^{20} - 6 q^{22} + 2 q^{23} - 8 q^{25} - 14 q^{26} + 6 q^{28} - 2 q^{29} - 10 q^{35} - 12 q^{37} + 12 q^{38}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 3.73205i 1.66902i 0.550990 + 0.834512i \(0.314250\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) 2.36603 + 1.36603i 0.894274 + 0.516309i 0.875338 − 0.483512i \(-0.160639\pi\)
0.0189356 + 0.999821i \(0.493972\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 1.86603 + 3.23205i 0.590089 + 1.02206i
\(11\) −1.09808 + 0.633975i −0.331082 + 0.191151i −0.656322 − 0.754481i \(-0.727889\pi\)
0.325239 + 0.945632i \(0.394555\pi\)
\(12\) 0 0
\(13\) −2.59808 − 2.50000i −0.720577 − 0.693375i
\(14\) 2.73205 0.730171
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.86603 − 4.96410i 0.695113 − 1.20397i −0.275029 − 0.961436i \(-0.588688\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 4.09808 + 2.36603i 0.940163 + 0.542803i 0.890011 − 0.455938i \(-0.150696\pi\)
0.0501517 + 0.998742i \(0.484030\pi\)
\(20\) 3.23205 + 1.86603i 0.722709 + 0.417256i
\(21\) 0 0
\(22\) −0.633975 + 1.09808i −0.135164 + 0.234111i
\(23\) −2.09808 − 3.63397i −0.437479 − 0.757736i 0.560015 − 0.828482i \(-0.310795\pi\)
−0.997494 + 0.0707462i \(0.977462\pi\)
\(24\) 0 0
\(25\) −8.92820 −1.78564
\(26\) −3.50000 − 0.866025i −0.686406 − 0.169842i
\(27\) 0 0
\(28\) 2.36603 − 1.36603i 0.447137 − 0.258155i
\(29\) −2.23205 − 3.86603i −0.414481 − 0.717903i 0.580892 − 0.813980i \(-0.302704\pi\)
−0.995374 + 0.0960774i \(0.969370\pi\)
\(30\) 0 0
\(31\) − 1.46410i − 0.262960i −0.991319 − 0.131480i \(-0.958027\pi\)
0.991319 − 0.131480i \(-0.0419730\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) − 5.73205i − 0.983039i
\(35\) −5.09808 + 8.83013i −0.861732 + 1.49256i
\(36\) 0 0
\(37\) 3.06218 − 1.76795i 0.503419 − 0.290649i −0.226705 − 0.973963i \(-0.572795\pi\)
0.730124 + 0.683314i \(0.239462\pi\)
\(38\) 4.73205 0.767640
\(39\) 0 0
\(40\) 3.73205 0.590089
\(41\) −8.13397 + 4.69615i −1.27031 + 0.733416i −0.975047 − 0.221999i \(-0.928742\pi\)
−0.295267 + 0.955415i \(0.595408\pi\)
\(42\) 0 0
\(43\) −4.83013 + 8.36603i −0.736587 + 1.27581i 0.217436 + 0.976075i \(0.430231\pi\)
−0.954023 + 0.299732i \(0.903103\pi\)
\(44\) 1.26795i 0.191151i
\(45\) 0 0
\(46\) −3.63397 − 2.09808i −0.535800 − 0.309344i
\(47\) − 2.19615i − 0.320342i −0.987089 − 0.160171i \(-0.948795\pi\)
0.987089 − 0.160171i \(-0.0512045\pi\)
\(48\) 0 0
\(49\) 0.232051 + 0.401924i 0.0331501 + 0.0574177i
\(50\) −7.73205 + 4.46410i −1.09348 + 0.631319i
\(51\) 0 0
\(52\) −3.46410 + 1.00000i −0.480384 + 0.138675i
\(53\) 6.46410 0.887913 0.443956 − 0.896048i \(-0.353575\pi\)
0.443956 + 0.896048i \(0.353575\pi\)
\(54\) 0 0
\(55\) −2.36603 − 4.09808i −0.319035 − 0.552584i
\(56\) 1.36603 − 2.36603i 0.182543 − 0.316173i
\(57\) 0 0
\(58\) −3.86603 − 2.23205i −0.507634 − 0.293083i
\(59\) 6.92820 + 4.00000i 0.901975 + 0.520756i 0.877841 − 0.478953i \(-0.158984\pi\)
0.0241347 + 0.999709i \(0.492317\pi\)
\(60\) 0 0
\(61\) 4.59808 − 7.96410i 0.588723 − 1.01970i −0.405677 − 0.914017i \(-0.632964\pi\)
0.994400 − 0.105682i \(-0.0337026\pi\)
\(62\) −0.732051 − 1.26795i −0.0929705 − 0.161030i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 9.33013 − 9.69615i 1.15726 − 1.20266i
\(66\) 0 0
\(67\) −11.3660 + 6.56218i −1.38858 + 0.801698i −0.993155 − 0.116800i \(-0.962736\pi\)
−0.395426 + 0.918498i \(0.629403\pi\)
\(68\) −2.86603 − 4.96410i −0.347557 − 0.601986i
\(69\) 0 0
\(70\) 10.1962i 1.21867i
\(71\) −4.09808 − 2.36603i −0.486352 − 0.280796i 0.236708 − 0.971581i \(-0.423932\pi\)
−0.723060 + 0.690785i \(0.757265\pi\)
\(72\) 0 0
\(73\) − 6.26795i − 0.733608i −0.930298 − 0.366804i \(-0.880452\pi\)
0.930298 − 0.366804i \(-0.119548\pi\)
\(74\) 1.76795 − 3.06218i 0.205520 − 0.355971i
\(75\) 0 0
\(76\) 4.09808 − 2.36603i 0.470082 − 0.271402i
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −2.53590 −0.285311 −0.142655 − 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) 3.23205 − 1.86603i 0.361354 − 0.208628i
\(81\) 0 0
\(82\) −4.69615 + 8.13397i −0.518603 + 0.898247i
\(83\) − 0.196152i − 0.0215305i −0.999942 − 0.0107653i \(-0.996573\pi\)
0.999942 − 0.0107653i \(-0.00342676\pi\)
\(84\) 0 0
\(85\) 18.5263 + 10.6962i 2.00946 + 1.16016i
\(86\) 9.66025i 1.04169i
\(87\) 0 0
\(88\) 0.633975 + 1.09808i 0.0675819 + 0.117055i
\(89\) 8.19615 − 4.73205i 0.868790 − 0.501596i 0.00184433 − 0.999998i \(-0.499413\pi\)
0.866946 + 0.498402i \(0.166080\pi\)
\(90\) 0 0
\(91\) −2.73205 − 9.46410i −0.286397 − 0.992107i
\(92\) −4.19615 −0.437479
\(93\) 0 0
\(94\) −1.09808 − 1.90192i −0.113258 − 0.196168i
\(95\) −8.83013 + 15.2942i −0.905952 + 1.56915i
\(96\) 0 0
\(97\) −5.19615 − 3.00000i −0.527589 − 0.304604i 0.212445 − 0.977173i \(-0.431857\pi\)
−0.740034 + 0.672569i \(0.765191\pi\)
\(98\) 0.401924 + 0.232051i 0.0406004 + 0.0234407i
\(99\) 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 234.2.l.c.127.2 4
3.2 odd 2 78.2.i.a.49.1 yes 4
4.3 odd 2 1872.2.by.h.1297.2 4
12.11 even 2 624.2.bv.e.49.1 4
13.2 odd 12 3042.2.a.y.1.2 2
13.3 even 3 3042.2.b.i.1351.4 4
13.4 even 6 inner 234.2.l.c.199.2 4
13.10 even 6 3042.2.b.i.1351.1 4
13.11 odd 12 3042.2.a.p.1.1 2
15.2 even 4 1950.2.y.b.49.1 4
15.8 even 4 1950.2.y.g.49.2 4
15.14 odd 2 1950.2.bc.d.751.2 4
39.2 even 12 1014.2.a.i.1.1 2
39.5 even 4 1014.2.e.i.991.1 4
39.8 even 4 1014.2.e.g.991.2 4
39.11 even 12 1014.2.a.k.1.2 2
39.17 odd 6 78.2.i.a.43.1 ✓ 4
39.20 even 12 1014.2.e.g.529.2 4
39.23 odd 6 1014.2.b.e.337.4 4
39.29 odd 6 1014.2.b.e.337.1 4
39.32 even 12 1014.2.e.i.529.1 4
39.35 odd 6 1014.2.i.a.823.2 4
39.38 odd 2 1014.2.i.a.361.2 4
52.43 odd 6 1872.2.by.h.433.1 4
156.11 odd 12 8112.2.a.bp.1.2 2
156.95 even 6 624.2.bv.e.433.2 4
156.119 odd 12 8112.2.a.bj.1.1 2
195.17 even 12 1950.2.y.g.199.2 4
195.134 odd 6 1950.2.bc.d.901.2 4
195.173 even 12 1950.2.y.b.199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 ✓ 4 39.17 odd 6
78.2.i.a.49.1 yes 4 3.2 odd 2
234.2.l.c.127.2 4 1.1 even 1 trivial
234.2.l.c.199.2 4 13.4 even 6 inner
624.2.bv.e.49.1 4 12.11 even 2
624.2.bv.e.433.2 4 156.95 even 6
1014.2.a.i.1.1 2 39.2 even 12
1014.2.a.k.1.2 2 39.11 even 12
1014.2.b.e.337.1 4 39.29 odd 6
1014.2.b.e.337.4 4 39.23 odd 6
1014.2.e.g.529.2 4 39.20 even 12
1014.2.e.g.991.2 4 39.8 even 4
1014.2.e.i.529.1 4 39.32 even 12
1014.2.e.i.991.1 4 39.5 even 4
1014.2.i.a.361.2 4 39.38 odd 2
1014.2.i.a.823.2 4 39.35 odd 6
1872.2.by.h.433.1 4 52.43 odd 6
1872.2.by.h.1297.2 4 4.3 odd 2
1950.2.y.b.49.1 4 15.2 even 4
1950.2.y.b.199.1 4 195.173 even 12
1950.2.y.g.49.2 4 15.8 even 4
1950.2.y.g.199.2 4 195.17 even 12
1950.2.bc.d.751.2 4 15.14 odd 2
1950.2.bc.d.901.2 4 195.134 odd 6
3042.2.a.p.1.1 2 13.11 odd 12
3042.2.a.y.1.2 2 13.2 odd 12
3042.2.b.i.1351.1 4 13.10 even 6
3042.2.b.i.1351.4 4 13.3 even 3
8112.2.a.bj.1.1 2 156.119 odd 12
8112.2.a.bp.1.2 2 156.11 odd 12