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Properties

Label 2268.2.w.a.1349.1
Level $2268$
Weight $2$
Character 2268.1349
Analytic conductor $18.110$
Analytic rank $1$
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] = [2268,2,Mod(269,2268)] 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("2268.269"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-3,0,-5,0,0,0,-9,0,0] 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: \(18.1100711784\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1349.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2268.1349
Dual form 2268.2.w.a.269.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 2.59808i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(-4.50000 - 2.59808i) q^{11} +(-1.50000 - 2.59808i) q^{17} +(1.50000 + 0.866025i) q^{19} +(4.50000 - 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} -1.73205i q^{31} +(1.50000 + 7.79423i) q^{35} +(-3.50000 + 6.06218i) q^{37} +(-3.00000 + 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +3.00000 q^{47} +(5.50000 + 4.33013i) q^{49} +(-4.50000 + 2.59808i) q^{53} +15.5885i q^{55} -3.00000 q^{59} +12.1244i q^{61} +5.00000 q^{67} -10.3923i q^{71} +(-10.5000 + 6.06218i) q^{73} +(9.00000 + 10.3923i) q^{77} -1.00000 q^{79} +(6.00000 + 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(4.50000 - 7.79423i) q^{89} -5.19615i q^{95} +(-6.00000 + 3.46410i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - 5 q^{7} - 9 q^{11} - 3 q^{17} + 3 q^{19} + 9 q^{23} - 4 q^{25} + 3 q^{35} - 7 q^{37} - 6 q^{41} - 4 q^{43} + 6 q^{47} + 11 q^{49} - 9 q^{53} - 6 q^{59} + 10 q^{67} - 21 q^{73} + 18 q^{77}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.50000 − 2.59808i −0.670820 − 1.16190i −0.977672 − 0.210138i \(-0.932609\pi\)
0.306851 − 0.951757i \(-0.400725\pi\)
\(6\) 0 0
\(7\) −2.50000 − 0.866025i −0.944911 − 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.50000 − 2.59808i −1.35680 − 0.783349i −0.367610 − 0.929980i \(-0.619824\pi\)
−0.989191 + 0.146631i \(0.953157\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.50000 − 2.59808i −0.363803 − 0.630126i 0.624780 − 0.780801i \(-0.285189\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(18\) 0 0
\(19\) 1.50000 + 0.866025i 0.344124 + 0.198680i 0.662094 − 0.749421i \(-0.269668\pi\)
−0.317970 + 0.948101i \(0.603001\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.50000 − 2.59808i 0.938315 − 0.541736i 0.0488832 − 0.998805i \(-0.484434\pi\)
0.889432 + 0.457068i \(0.151100\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) − 1.73205i − 0.311086i −0.987829 − 0.155543i \(-0.950287\pi\)
0.987829 − 0.155543i \(-0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 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
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 − 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) −2.00000 − 3.46410i −0.304997 − 0.528271i 0.672264 − 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 + 2.59808i −0.618123 + 0.356873i −0.776138 − 0.630563i \(-0.782824\pi\)
0.158015 + 0.987437i \(0.449491\pi\)
\(54\) 0 0
\(55\) 15.5885i 2.10195i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 12.1244i 1.55236i 0.630509 + 0.776182i \(0.282846\pi\)
−0.630509 + 0.776182i \(0.717154\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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
\(75\) 0 0
\(76\) 0 0
\(77\) 9.00000 + 10.3923i 1.02565 + 1.18431i
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) −4.50000 + 7.79423i −0.488094 + 0.845403i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.50000 − 7.79423i 0.476999 − 0.826187i −0.522654 − 0.852545i \(-0.675058\pi\)
0.999653 + 0.0263586i \(0.00839118\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 5.19615i − 0.533114i
\(96\) 0 0
\(97\) −6.00000 + 3.46410i −0.609208 + 0.351726i −0.772655 − 0.634826i \(-0.781072\pi\)
0.163448 + 0.986552i \(0.447739\pi\)
\(98\) 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 2268.2.w.a.1349.1 2
3.2 odd 2 2268.2.w.f.1349.1 2
7.3 odd 6 2268.2.bm.a.1025.1 2
9.2 odd 6 2268.2.bm.a.593.1 2
9.4 even 3 84.2.k.b.5.1 yes 2
9.5 odd 6 84.2.k.a.5.1 ✓ 2
9.7 even 3 2268.2.bm.f.593.1 2
21.17 even 6 2268.2.bm.f.1025.1 2
36.23 even 6 336.2.bc.d.257.1 2
36.31 odd 6 336.2.bc.b.257.1 2
45.4 even 6 2100.2.bi.e.1601.1 2
45.13 odd 12 2100.2.bo.f.1349.1 4
45.14 odd 6 2100.2.bi.f.1601.1 2
45.22 odd 12 2100.2.bo.f.1349.2 4
45.23 even 12 2100.2.bo.a.1349.2 4
45.32 even 12 2100.2.bo.a.1349.1 4
63.4 even 3 588.2.k.d.521.1 2
63.5 even 6 588.2.f.a.293.2 2
63.13 odd 6 588.2.k.c.509.1 2
63.23 odd 6 588.2.f.c.293.1 2
63.31 odd 6 84.2.k.a.17.1 yes 2
63.32 odd 6 588.2.k.c.521.1 2
63.38 even 6 inner 2268.2.w.a.269.1 2
63.40 odd 6 588.2.f.c.293.2 2
63.41 even 6 588.2.k.d.509.1 2
63.52 odd 6 2268.2.w.f.269.1 2
63.58 even 3 588.2.f.a.293.1 2
63.59 even 6 84.2.k.b.17.1 yes 2
252.23 even 6 2352.2.k.a.881.2 2
252.31 even 6 336.2.bc.d.17.1 2
252.59 odd 6 336.2.bc.b.17.1 2
252.103 even 6 2352.2.k.a.881.1 2
252.131 odd 6 2352.2.k.d.881.1 2
252.247 odd 6 2352.2.k.d.881.2 2
315.59 even 6 2100.2.bi.e.101.1 2
315.94 odd 6 2100.2.bi.f.101.1 2
315.122 odd 12 2100.2.bo.f.1949.1 4
315.157 even 12 2100.2.bo.a.1949.2 4
315.248 odd 12 2100.2.bo.f.1949.2 4
315.283 even 12 2100.2.bo.a.1949.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.k.a.5.1 ✓ 2 9.5 odd 6
84.2.k.a.17.1 yes 2 63.31 odd 6
84.2.k.b.5.1 yes 2 9.4 even 3
84.2.k.b.17.1 yes 2 63.59 even 6
336.2.bc.b.17.1 2 252.59 odd 6
336.2.bc.b.257.1 2 36.31 odd 6
336.2.bc.d.17.1 2 252.31 even 6
336.2.bc.d.257.1 2 36.23 even 6
588.2.f.a.293.1 2 63.58 even 3
588.2.f.a.293.2 2 63.5 even 6
588.2.f.c.293.1 2 63.23 odd 6
588.2.f.c.293.2 2 63.40 odd 6
588.2.k.c.509.1 2 63.13 odd 6
588.2.k.c.521.1 2 63.32 odd 6
588.2.k.d.509.1 2 63.41 even 6
588.2.k.d.521.1 2 63.4 even 3
2100.2.bi.e.101.1 2 315.59 even 6
2100.2.bi.e.1601.1 2 45.4 even 6
2100.2.bi.f.101.1 2 315.94 odd 6
2100.2.bi.f.1601.1 2 45.14 odd 6
2100.2.bo.a.1349.1 4 45.32 even 12
2100.2.bo.a.1349.2 4 45.23 even 12
2100.2.bo.a.1949.1 4 315.283 even 12
2100.2.bo.a.1949.2 4 315.157 even 12
2100.2.bo.f.1349.1 4 45.13 odd 12
2100.2.bo.f.1349.2 4 45.22 odd 12
2100.2.bo.f.1949.1 4 315.122 odd 12
2100.2.bo.f.1949.2 4 315.248 odd 12
2268.2.w.a.269.1 2 63.38 even 6 inner
2268.2.w.a.1349.1 2 1.1 even 1 trivial
2268.2.w.f.269.1 2 63.52 odd 6
2268.2.w.f.1349.1 2 3.2 odd 2
2268.2.bm.a.593.1 2 9.2 odd 6
2268.2.bm.a.1025.1 2 7.3 odd 6
2268.2.bm.f.593.1 2 9.7 even 3
2268.2.bm.f.1025.1 2 21.17 even 6
2352.2.k.a.881.1 2 252.103 even 6
2352.2.k.a.881.2 2 252.23 even 6
2352.2.k.d.881.1 2 252.131 odd 6
2352.2.k.d.881.2 2 252.247 odd 6