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Properties

Label 1274.2.f.i.79.1
Level $1274$
Weight $2$
Character 1274.79
Analytic conductor $10.173$
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] = [1274,2,Mod(79,1274)] 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("1274.79"); 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([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,1,-1,0] 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: \(10.1729412175\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1274.79
Dual form 1274.2.f.i.1145.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{6} +1.00000 q^{8} +(1.00000 - 1.73205i) q^{9} +(1.50000 + 2.59808i) q^{11} +(0.500000 - 0.866025i) q^{12} -1.00000 q^{13} +(-0.500000 + 0.866025i) q^{16} +(1.00000 + 1.73205i) q^{18} +(1.00000 - 1.73205i) q^{19} -3.00000 q^{22} +(1.50000 - 2.59808i) q^{23} +(0.500000 + 0.866025i) q^{24} +(2.50000 + 4.33013i) q^{25} +(0.500000 - 0.866025i) q^{26} +5.00000 q^{27} +(2.50000 + 4.33013i) q^{31} +(-0.500000 - 0.866025i) q^{32} +(-1.50000 + 2.59808i) q^{33} -2.00000 q^{36} +(3.50000 - 6.06218i) q^{37} +(1.00000 + 1.73205i) q^{38} +(-0.500000 - 0.866025i) q^{39} -3.00000 q^{41} +8.00000 q^{43} +(1.50000 - 2.59808i) q^{44} +(1.50000 + 2.59808i) q^{46} +(-1.50000 + 2.59808i) q^{47} -1.00000 q^{48} -5.00000 q^{50} +(0.500000 + 0.866025i) q^{52} +(6.00000 + 10.3923i) q^{53} +(-2.50000 + 4.33013i) q^{54} +2.00000 q^{57} +(3.00000 + 5.19615i) q^{59} +(-0.500000 + 0.866025i) q^{61} -5.00000 q^{62} +1.00000 q^{64} +(-1.50000 - 2.59808i) q^{66} +(-2.50000 - 4.33013i) q^{67} +3.00000 q^{69} +12.0000 q^{71} +(1.00000 - 1.73205i) q^{72} +(5.50000 + 9.52628i) q^{73} +(3.50000 + 6.06218i) q^{74} +(-2.50000 + 4.33013i) q^{75} -2.00000 q^{76} +1.00000 q^{78} +(0.500000 - 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(1.50000 - 2.59808i) q^{82} -12.0000 q^{83} +(-4.00000 + 6.92820i) q^{86} +(1.50000 + 2.59808i) q^{88} +(-9.00000 + 15.5885i) q^{89} -3.00000 q^{92} +(-2.50000 + 4.33013i) q^{93} +(-1.50000 - 2.59808i) q^{94} +(0.500000 - 0.866025i) q^{96} -17.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9} + 3 q^{11} + q^{12} - 2 q^{13} - q^{16} + 2 q^{18} + 2 q^{19} - 6 q^{22} + 3 q^{23} + q^{24} + 5 q^{25} + q^{26} + 10 q^{27} + 5 q^{31} - q^{32}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i 0.973494 − 0.228714i \(-0.0734519\pi\)
−0.684819 + 0.728714i \(0.740119\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 − 1.73205i 0.333333 − 0.577350i
\(10\) 0 0
\(11\) 1.50000 + 2.59808i 0.452267 + 0.783349i 0.998526 − 0.0542666i \(-0.0172821\pi\)
−0.546259 + 0.837616i \(0.683949\pi\)
\(12\) 0.500000 − 0.866025i 0.144338 − 0.250000i
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 1.00000 + 1.73205i 0.235702 + 0.408248i
\(19\) 1.00000 − 1.73205i 0.229416 − 0.397360i −0.728219 − 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 1.50000 − 2.59808i 0.312772 − 0.541736i −0.666190 − 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0.500000 + 0.866025i 0.102062 + 0.176777i
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0.500000 − 0.866025i 0.0980581 − 0.169842i
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.50000 + 4.33013i 0.449013 + 0.777714i 0.998322 − 0.0579057i \(-0.0184423\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) −1.50000 + 2.59808i −0.261116 + 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 3.50000 − 6.06218i 0.575396 − 0.996616i −0.420602 − 0.907245i \(-0.638181\pi\)
0.995998 − 0.0893706i \(-0.0284856\pi\)
\(38\) 1.00000 + 1.73205i 0.162221 + 0.280976i
\(39\) −0.500000 − 0.866025i −0.0800641 − 0.138675i
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 1.50000 − 2.59808i 0.226134 − 0.391675i
\(45\) 0 0
\(46\) 1.50000 + 2.59808i 0.221163 + 0.383065i
\(47\) −1.50000 + 2.59808i −0.218797 + 0.378968i −0.954441 − 0.298401i \(-0.903547\pi\)
0.735643 + 0.677369i \(0.236880\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) 0 0
\(52\) 0.500000 + 0.866025i 0.0693375 + 0.120096i
\(53\) 6.00000 + 10.3923i 0.824163 + 1.42749i 0.902557 + 0.430570i \(0.141688\pi\)
−0.0783936 + 0.996922i \(0.524979\pi\)
\(54\) −2.50000 + 4.33013i −0.340207 + 0.589256i
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 3.00000 + 5.19615i 0.390567 + 0.676481i 0.992524 − 0.122047i \(-0.0389457\pi\)
−0.601958 + 0.798528i \(0.705612\pi\)
\(60\) 0 0
\(61\) −0.500000 + 0.866025i −0.0640184 + 0.110883i −0.896258 − 0.443533i \(-0.853725\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.50000 − 2.59808i −0.184637 − 0.319801i
\(67\) −2.50000 − 4.33013i −0.305424 − 0.529009i 0.671932 − 0.740613i \(-0.265465\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 − 1.73205i 0.117851 − 0.204124i
\(73\) 5.50000 + 9.52628i 0.643726 + 1.11497i 0.984594 + 0.174855i \(0.0559458\pi\)
−0.340868 + 0.940111i \(0.610721\pi\)
\(74\) 3.50000 + 6.06218i 0.406867 + 0.704714i
\(75\) −2.50000 + 4.33013i −0.288675 + 0.500000i
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 0.500000 − 0.866025i 0.0562544 − 0.0974355i −0.836527 − 0.547926i \(-0.815418\pi\)
0.892781 + 0.450490i \(0.148751\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 1.50000 − 2.59808i 0.165647 − 0.286910i
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 + 6.92820i −0.431331 + 0.747087i
\(87\) 0 0
\(88\) 1.50000 + 2.59808i 0.159901 + 0.276956i
\(89\) −9.00000 + 15.5885i −0.953998 + 1.65237i −0.217354 + 0.976093i \(0.569742\pi\)
−0.736644 + 0.676280i \(0.763591\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) −2.50000 + 4.33013i −0.259238 + 0.449013i
\(94\) −1.50000 − 2.59808i −0.154713 − 0.267971i
\(95\) 0 0
\(96\) 0.500000 − 0.866025i 0.0510310 − 0.0883883i
\(97\) −17.0000 −1.72609 −0.863044 − 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
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.f.i.79.1 2
7.2 even 3 1274.2.a.j.1.1 1
7.3 odd 6 1274.2.f.d.1145.1 2
7.4 even 3 inner 1274.2.f.i.1145.1 2
7.5 odd 6 182.2.a.d.1.1 ✓ 1
7.6 odd 2 1274.2.f.d.79.1 2
21.5 even 6 1638.2.a.f.1.1 1
28.19 even 6 1456.2.a.d.1.1 1
35.19 odd 6 4550.2.a.c.1.1 1
56.5 odd 6 5824.2.a.k.1.1 1
56.19 even 6 5824.2.a.x.1.1 1
91.5 even 12 2366.2.d.e.337.1 2
91.12 odd 6 2366.2.a.e.1.1 1
91.47 even 12 2366.2.d.e.337.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.d.1.1 ✓ 1 7.5 odd 6
1274.2.a.j.1.1 1 7.2 even 3
1274.2.f.d.79.1 2 7.6 odd 2
1274.2.f.d.1145.1 2 7.3 odd 6
1274.2.f.i.79.1 2 1.1 even 1 trivial
1274.2.f.i.1145.1 2 7.4 even 3 inner
1456.2.a.d.1.1 1 28.19 even 6
1638.2.a.f.1.1 1 21.5 even 6
2366.2.a.e.1.1 1 91.12 odd 6
2366.2.d.e.337.1 2 91.5 even 12
2366.2.d.e.337.2 2 91.47 even 12
4550.2.a.c.1.1 1 35.19 odd 6
5824.2.a.k.1.1 1 56.5 odd 6
5824.2.a.x.1.1 1 56.19 even 6