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Properties

Label 1274.2.o.d.459.4
Level $1274$
Weight $2$
Character 1274.459
Analytic conductor $10.173$
Analytic rank $0$
Dimension $12$
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] = [1274,2,Mod(459,1274)] 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("1274.459"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1274, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-2,-12,0,6,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1729412175\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 459.4
Root \(0.500000 + 2.47866i\) of defining polynomial
Character \(\chi\) \(=\) 1274.459
Dual form 1274.2.o.d.569.1

$q$-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+1.00000i q^{2} +(-1.67234 - 2.89658i) q^{3} -1.00000 q^{4} +(1.35408 - 0.781779i) q^{5} +(2.89658 - 1.67234i) q^{6} -1.00000i q^{8} +(-4.09347 + 7.09010i) q^{9} +(0.781779 + 1.35408i) q^{10} +(2.48215 - 1.43307i) q^{11} +(1.67234 + 2.89658i) q^{12} +(2.99598 + 2.00602i) q^{13} +(-4.52898 - 2.61481i) q^{15} +1.00000 q^{16} +2.22961 q^{17} +(-7.09010 - 4.09347i) q^{18} +(6.26657 + 3.61801i) q^{19} +(-1.35408 + 0.781779i) q^{20} +(1.43307 + 2.48215i) q^{22} +1.66735 q^{23} +(-2.89658 + 1.67234i) q^{24} +(-1.27764 + 2.21294i) q^{25} +(-2.00602 + 2.99598i) q^{26} +17.3487 q^{27} +(-2.41379 + 4.18080i) q^{29} +(2.61481 - 4.52898i) q^{30} +(0.517851 + 0.298982i) q^{31} +1.00000i q^{32} +(-8.30201 - 4.79317i) q^{33} +2.22961i q^{34} +(4.09347 - 7.09010i) q^{36} -0.0385636i q^{37} +(-3.61801 + 6.26657i) q^{38} +(0.800299 - 12.0329i) q^{39} +(-0.781779 - 1.35408i) q^{40} +(-6.88896 - 3.97734i) q^{41} +(5.04571 + 8.73942i) q^{43} +(-2.48215 + 1.43307i) q^{44} +12.8007i q^{45} +1.66735i q^{46} +(6.08501 - 3.51318i) q^{47} +(-1.67234 - 2.89658i) q^{48} +(-2.21294 - 1.27764i) q^{50} +(-3.72868 - 6.45826i) q^{51} +(-2.99598 - 2.00602i) q^{52} +(2.99202 - 5.18233i) q^{53} +17.3487i q^{54} +(2.24069 - 3.88098i) q^{55} -24.2022i q^{57} +(-4.18080 - 2.41379i) q^{58} +0.896206i q^{59} +(4.52898 + 2.61481i) q^{60} +(7.12846 - 12.3469i) q^{61} +(-0.298982 + 0.517851i) q^{62} -1.00000 q^{64} +(5.62506 + 0.374120i) q^{65} +(4.79317 - 8.30201i) q^{66} +(1.42103 - 0.820432i) q^{67} -2.22961 q^{68} +(-2.78838 - 4.82962i) q^{69} +(-1.98724 + 1.14733i) q^{71} +(7.09010 + 4.09347i) q^{72} +(-9.72351 - 5.61387i) q^{73} +0.0385636 q^{74} +8.54664 q^{75} +(-6.26657 - 3.61801i) q^{76} +(12.0329 + 0.800299i) q^{78} +(-2.13049 - 3.69011i) q^{79} +(1.35408 - 0.781779i) q^{80} +(-16.7326 - 28.9816i) q^{81} +(3.97734 - 6.88896i) q^{82} -4.94829i q^{83} +(3.01907 - 1.74306i) q^{85} +(-8.73942 + 5.04571i) q^{86} +16.1467 q^{87} +(-1.43307 - 2.48215i) q^{88} -2.42120i q^{89} -12.8007 q^{90} -1.66735 q^{92} -2.00000i q^{93} +(3.51318 + 6.08501i) q^{94} +11.3139 q^{95} +(2.89658 - 1.67234i) q^{96} +(-4.23338 + 2.44414i) q^{97} +23.4649i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} - 12 q^{4} + 6 q^{6} - 6 q^{9} - 2 q^{10} + 18 q^{11} + 2 q^{12} - 8 q^{13} - 6 q^{15} + 12 q^{16} - 8 q^{17} - 12 q^{19} - 2 q^{22} + 12 q^{23} - 6 q^{24} + 12 q^{25} - 2 q^{26} + 40 q^{27}+ \cdots + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1274\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.67234 − 2.89658i −0.965528 − 1.67234i −0.708189 − 0.706023i \(-0.750488\pi\)
−0.257339 − 0.966321i \(-0.582846\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.35408 − 0.781779i 0.605563 − 0.349622i −0.165664 − 0.986182i \(-0.552977\pi\)
0.771227 + 0.636560i \(0.219643\pi\)
\(6\) 2.89658 − 1.67234i 1.18253 − 0.682732i
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) −4.09347 + 7.09010i −1.36449 + 2.36337i
\(10\) 0.781779 + 1.35408i 0.247220 + 0.428198i
\(11\) 2.48215 − 1.43307i 0.748396 − 0.432087i −0.0767180 − 0.997053i \(-0.524444\pi\)
0.825114 + 0.564966i \(0.191111\pi\)
\(12\) 1.67234 + 2.89658i 0.482764 + 0.836172i
\(13\) 2.99598 + 2.00602i 0.830935 + 0.556370i
\(14\) 0 0
\(15\) −4.52898 − 2.61481i −1.16938 − 0.675140i
\(16\) 1.00000 0.250000
\(17\) 2.22961 0.540760 0.270380 − 0.962754i \(-0.412851\pi\)
0.270380 + 0.962754i \(0.412851\pi\)
\(18\) −7.09010 − 4.09347i −1.67115 − 0.964840i
\(19\) 6.26657 + 3.61801i 1.43765 + 0.830028i 0.997686 − 0.0679872i \(-0.0216577\pi\)
0.439964 + 0.898015i \(0.354991\pi\)
\(20\) −1.35408 + 0.781779i −0.302782 + 0.174811i
\(21\) 0 0
\(22\) 1.43307 + 2.48215i 0.305531 + 0.529196i
\(23\) 1.66735 0.347667 0.173833 − 0.984775i \(-0.444385\pi\)
0.173833 + 0.984775i \(0.444385\pi\)
\(24\) −2.89658 + 1.67234i −0.591263 + 0.341366i
\(25\) −1.27764 + 2.21294i −0.255529 + 0.442589i
\(26\) −2.00602 + 2.99598i −0.393413 + 0.587560i
\(27\) 17.3487 3.33876
\(28\) 0 0
\(29\) −2.41379 + 4.18080i −0.448229 + 0.776356i −0.998271 − 0.0587816i \(-0.981278\pi\)
0.550042 + 0.835137i \(0.314612\pi\)
\(30\) 2.61481 − 4.52898i 0.477396 − 0.826874i
\(31\) 0.517851 + 0.298982i 0.0930088 + 0.0536987i 0.545783 − 0.837927i \(-0.316232\pi\)
−0.452774 + 0.891625i \(0.649566\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −8.30201 − 4.79317i −1.44519 − 0.834384i
\(34\) 2.22961i 0.382375i
\(35\) 0 0
\(36\) 4.09347 − 7.09010i 0.682245 − 1.18168i
\(37\) − 0.0385636i − 0.00633982i −0.999995 − 0.00316991i \(-0.998991\pi\)
0.999995 − 0.00316991i \(-0.00100902\pi\)
\(38\) −3.61801 + 6.26657i −0.586918 + 1.01657i
\(39\) 0.800299 − 12.0329i 0.128150 − 1.92680i
\(40\) −0.781779 − 1.35408i −0.123610 − 0.214099i
\(41\) −6.88896 − 3.97734i −1.07588 − 0.621157i −0.146095 − 0.989271i \(-0.546670\pi\)
−0.929781 + 0.368114i \(0.880004\pi\)
\(42\) 0 0
\(43\) 5.04571 + 8.73942i 0.769463 + 1.33275i 0.937854 + 0.347029i \(0.112809\pi\)
−0.168391 + 0.985720i \(0.553857\pi\)
\(44\) −2.48215 + 1.43307i −0.374198 + 0.216043i
\(45\) 12.8007i 1.90822i
\(46\) 1.66735i 0.245838i
\(47\) 6.08501 − 3.51318i 0.887590 − 0.512450i 0.0144363 − 0.999896i \(-0.495405\pi\)
0.873153 + 0.487446i \(0.162071\pi\)
\(48\) −1.67234 − 2.89658i −0.241382 − 0.418086i
\(49\) 0 0
\(50\) −2.21294 − 1.27764i −0.312958 − 0.180686i
\(51\) −3.72868 − 6.45826i −0.522119 − 0.904337i
\(52\) −2.99598 − 2.00602i −0.415467 − 0.278185i
\(53\) 2.99202 − 5.18233i 0.410985 − 0.711848i −0.584012 − 0.811745i \(-0.698518\pi\)
0.994998 + 0.0998972i \(0.0318514\pi\)
\(54\) 17.3487i 2.36086i
\(55\) 2.24069 − 3.88098i 0.302134 − 0.523312i
\(56\) 0 0
\(57\) − 24.2022i − 3.20566i
\(58\) −4.18080 − 2.41379i −0.548966 − 0.316946i
\(59\) 0.896206i 0.116676i 0.998297 + 0.0583381i \(0.0185801\pi\)
−0.998297 + 0.0583381i \(0.981420\pi\)
\(60\) 4.52898 + 2.61481i 0.584688 + 0.337570i
\(61\) 7.12846 − 12.3469i 0.912706 − 1.58085i 0.102481 − 0.994735i \(-0.467322\pi\)
0.810225 − 0.586119i \(-0.199345\pi\)
\(62\) −0.298982 + 0.517851i −0.0379707 + 0.0657672i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 5.62506 + 0.374120i 0.697703 + 0.0464038i
\(66\) 4.79317 − 8.30201i 0.589998 − 1.02191i
\(67\) 1.42103 − 0.820432i 0.173606 − 0.100232i −0.410679 − 0.911780i \(-0.634708\pi\)
0.584285 + 0.811548i \(0.301375\pi\)
\(68\) −2.22961 −0.270380
\(69\) −2.78838 − 4.82962i −0.335682 − 0.581418i
\(70\) 0 0
\(71\) −1.98724 + 1.14733i −0.235841 + 0.136163i −0.613264 − 0.789878i \(-0.710144\pi\)
0.377422 + 0.926041i \(0.376810\pi\)
\(72\) 7.09010 + 4.09347i 0.835576 + 0.482420i
\(73\) −9.72351 − 5.61387i −1.13805 − 0.657054i −0.192104 − 0.981375i \(-0.561531\pi\)
−0.945947 + 0.324321i \(0.894864\pi\)
\(74\) 0.0385636 0.00448293
\(75\) 8.54664 0.986881
\(76\) −6.26657 − 3.61801i −0.718825 − 0.415014i
\(77\) 0 0
\(78\) 12.0329 + 0.800299i 1.36245 + 0.0906160i
\(79\) −2.13049 − 3.69011i −0.239699 − 0.415170i 0.720929 − 0.693009i \(-0.243715\pi\)
−0.960628 + 0.277839i \(0.910382\pi\)
\(80\) 1.35408 − 0.781779i 0.151391 − 0.0874055i
\(81\) −16.7326 − 28.9816i −1.85917 − 3.22018i
\(82\) 3.97734 − 6.88896i 0.439224 − 0.760759i
\(83\) − 4.94829i − 0.543145i −0.962418 − 0.271572i \(-0.912456\pi\)
0.962418 − 0.271572i \(-0.0875437\pi\)
\(84\) 0 0
\(85\) 3.01907 − 1.74306i 0.327465 − 0.189062i
\(86\) −8.73942 + 5.04571i −0.942396 + 0.544092i
\(87\) 16.1467 1.73111
\(88\) −1.43307 − 2.48215i −0.152766 − 0.264598i
\(89\) − 2.42120i − 0.256647i −0.991732 − 0.128323i \(-0.959040\pi\)
0.991732 − 0.128323i \(-0.0409595\pi\)
\(90\) −12.8007 −1.34932
\(91\) 0 0
\(92\) −1.66735 −0.173833
\(93\) − 2.00000i − 0.207390i
\(94\) 3.51318 + 6.08501i 0.362357 + 0.627621i
\(95\) 11.3139 1.16078
\(96\) 2.89658 − 1.67234i 0.295631 − 0.170683i
\(97\) −4.23338 + 2.44414i −0.429835 + 0.248165i −0.699276 − 0.714852i \(-0.746494\pi\)
0.269442 + 0.963017i \(0.413161\pi\)
\(98\) 0 0
\(99\) 23.4649i 2.35831i
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 1274.2.o.d.459.4 12
7.2 even 3 1274.2.v.e.667.6 12
7.3 odd 6 1274.2.m.c.589.3 12
7.4 even 3 182.2.m.b.43.1 ✓ 12
7.5 odd 6 1274.2.v.d.667.4 12
7.6 odd 2 1274.2.o.e.459.6 12
13.10 even 6 1274.2.v.e.361.6 12
21.11 odd 6 1638.2.bj.g.1135.4 12
28.11 odd 6 1456.2.cc.d.225.6 12
91.4 even 6 2366.2.d.r.337.6 12
91.10 odd 6 1274.2.m.c.491.3 12
91.23 even 6 inner 1274.2.o.d.569.1 12
91.32 odd 12 2366.2.a.bh.1.6 6
91.46 odd 12 2366.2.a.bf.1.6 6
91.62 odd 6 1274.2.v.d.361.4 12
91.74 even 3 2366.2.d.r.337.12 12
91.75 odd 6 1274.2.o.e.569.3 12
91.88 even 6 182.2.m.b.127.1 yes 12
273.179 odd 6 1638.2.bj.g.127.6 12
364.179 odd 6 1456.2.cc.d.673.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.m.b.43.1 ✓ 12 7.4 even 3
182.2.m.b.127.1 yes 12 91.88 even 6
1274.2.m.c.491.3 12 91.10 odd 6
1274.2.m.c.589.3 12 7.3 odd 6
1274.2.o.d.459.4 12 1.1 even 1 trivial
1274.2.o.d.569.1 12 91.23 even 6 inner
1274.2.o.e.459.6 12 7.6 odd 2
1274.2.o.e.569.3 12 91.75 odd 6
1274.2.v.d.361.4 12 91.62 odd 6
1274.2.v.d.667.4 12 7.5 odd 6
1274.2.v.e.361.6 12 13.10 even 6
1274.2.v.e.667.6 12 7.2 even 3
1456.2.cc.d.225.6 12 28.11 odd 6
1456.2.cc.d.673.6 12 364.179 odd 6
1638.2.bj.g.127.6 12 273.179 odd 6
1638.2.bj.g.1135.4 12 21.11 odd 6
2366.2.a.bf.1.6 6 91.46 odd 12
2366.2.a.bh.1.6 6 91.32 odd 12
2366.2.d.r.337.6 12 91.4 even 6
2366.2.d.r.337.12 12 91.74 even 3