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Properties

Label 1014.2.i.c.823.1
Level $1014$
Weight $2$
Character 1014.823
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [1014,2,Mod(361,1014)] 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("1014.361"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,2,0,0,0,0,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.09683076496\)
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 823.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.823
Dual form 1014.2.i.c.361.1

$q$-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+(-0.866025 - 0.500000i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{4} +2.00000i q^{5} +(0.866025 - 0.500000i) q^{6} +(-1.73205 + 1.00000i) q^{7} -1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +(1.00000 - 1.73205i) q^{10} -1.00000 q^{12} +2.00000 q^{14} +(-1.73205 - 1.00000i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(1.00000 + 1.73205i) q^{17} +1.00000i q^{18} +(-5.19615 + 3.00000i) q^{19} +(-1.73205 + 1.00000i) q^{20} -2.00000i q^{21} +(-2.00000 + 3.46410i) q^{23} +(0.866025 + 0.500000i) q^{24} +1.00000 q^{25} +1.00000 q^{27} +(-1.73205 - 1.00000i) q^{28} +(5.00000 - 8.66025i) q^{29} +(1.00000 + 1.73205i) q^{30} +10.0000i q^{31} +(0.866025 - 0.500000i) q^{32} -2.00000i q^{34} +(-2.00000 - 3.46410i) q^{35} +(0.500000 - 0.866025i) q^{36} +(-6.92820 - 4.00000i) q^{37} +6.00000 q^{38} +2.00000 q^{40} +(-8.66025 - 5.00000i) q^{41} +(-1.00000 + 1.73205i) q^{42} +(-2.00000 - 3.46410i) q^{43} +(1.73205 - 1.00000i) q^{45} +(3.46410 - 2.00000i) q^{46} -12.0000i q^{47} +(-0.500000 - 0.866025i) q^{48} +(-1.50000 + 2.59808i) q^{49} +(-0.866025 - 0.500000i) q^{50} -2.00000 q^{51} -6.00000 q^{53} +(-0.866025 - 0.500000i) q^{54} +(1.00000 + 1.73205i) q^{56} -6.00000i q^{57} +(-8.66025 + 5.00000i) q^{58} +(3.46410 - 2.00000i) q^{59} -2.00000i q^{60} +(-1.00000 - 1.73205i) q^{61} +(5.00000 - 8.66025i) q^{62} +(1.73205 + 1.00000i) q^{63} -1.00000 q^{64} +(1.73205 + 1.00000i) q^{67} +(-1.00000 + 1.73205i) q^{68} +(-2.00000 - 3.46410i) q^{69} +4.00000i q^{70} +(-0.866025 + 0.500000i) q^{72} -4.00000i q^{73} +(4.00000 + 6.92820i) q^{74} +(-0.500000 + 0.866025i) q^{75} +(-5.19615 - 3.00000i) q^{76} +(-1.73205 - 1.00000i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(5.00000 + 8.66025i) q^{82} -4.00000i q^{83} +(1.73205 - 1.00000i) q^{84} +(-3.46410 + 2.00000i) q^{85} +4.00000i q^{86} +(5.00000 + 8.66025i) q^{87} +(5.19615 + 3.00000i) q^{89} -2.00000 q^{90} -4.00000 q^{92} +(-8.66025 - 5.00000i) q^{93} +(-6.00000 + 10.3923i) q^{94} +(-6.00000 - 10.3923i) q^{95} +1.00000i q^{96} +(-10.3923 + 6.00000i) q^{97} +(2.59808 - 1.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} - 2 q^{9} + 4 q^{10} - 4 q^{12} + 8 q^{14} - 2 q^{16} + 4 q^{17} - 8 q^{23} + 4 q^{25} + 4 q^{27} + 20 q^{29} + 4 q^{30} - 8 q^{35} + 2 q^{36} + 24 q^{38} + 8 q^{40} - 4 q^{42} - 8 q^{43}+ \cdots - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 − 0.500000i −0.612372 − 0.353553i
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0.866025 − 0.500000i 0.353553 − 0.204124i
\(7\) −1.73205 + 1.00000i −0.654654 + 0.377964i −0.790237 − 0.612801i \(-0.790043\pi\)
0.135583 + 0.990766i \(0.456709\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 1.00000 − 1.73205i 0.316228 − 0.547723i
\(11\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) −1.73205 − 1.00000i −0.447214 − 0.258199i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.00000 + 1.73205i 0.242536 + 0.420084i 0.961436 − 0.275029i \(-0.0886875\pi\)
−0.718900 + 0.695113i \(0.755354\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −5.19615 + 3.00000i −1.19208 + 0.688247i −0.958778 − 0.284157i \(-0.908286\pi\)
−0.233301 + 0.972404i \(0.574953\pi\)
\(20\) −1.73205 + 1.00000i −0.387298 + 0.223607i
\(21\) − 2.00000i − 0.436436i
\(22\) 0 0
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 − 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) 0.866025 + 0.500000i 0.176777 + 0.102062i
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.73205 − 1.00000i −0.327327 − 0.188982i
\(29\) 5.00000 − 8.66025i 0.928477 − 1.60817i 0.142605 − 0.989780i \(-0.454452\pi\)
0.785872 − 0.618389i \(-0.212214\pi\)
\(30\) 1.00000 + 1.73205i 0.182574 + 0.316228i
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 0.866025 − 0.500000i 0.153093 − 0.0883883i
\(33\) 0 0
\(34\) − 2.00000i − 0.342997i
\(35\) −2.00000 − 3.46410i −0.338062 − 0.585540i
\(36\) 0.500000 − 0.866025i 0.0833333 − 0.144338i
\(37\) −6.92820 − 4.00000i −1.13899 − 0.657596i −0.192809 − 0.981236i \(-0.561760\pi\)
−0.946180 + 0.323640i \(0.895093\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 2.00000 0.316228
\(41\) −8.66025 − 5.00000i −1.35250 − 0.780869i −0.363905 − 0.931436i \(-0.618557\pi\)
−0.988600 + 0.150567i \(0.951890\pi\)
\(42\) −1.00000 + 1.73205i −0.154303 + 0.267261i
\(43\) −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
\(45\) 1.73205 − 1.00000i 0.258199 − 0.149071i
\(46\) 3.46410 − 2.00000i 0.510754 − 0.294884i
\(47\) − 12.0000i − 1.75038i −0.483779 − 0.875190i \(-0.660736\pi\)
0.483779 − 0.875190i \(-0.339264\pi\)
\(48\) −0.500000 − 0.866025i −0.0721688 − 0.125000i
\(49\) −1.50000 + 2.59808i −0.214286 + 0.371154i
\(50\) −0.866025 − 0.500000i −0.122474 − 0.0707107i
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −0.866025 − 0.500000i −0.117851 − 0.0680414i
\(55\) 0 0
\(56\) 1.00000 + 1.73205i 0.133631 + 0.231455i
\(57\) − 6.00000i − 0.794719i
\(58\) −8.66025 + 5.00000i −1.13715 + 0.656532i
\(59\) 3.46410 − 2.00000i 0.450988 − 0.260378i −0.257260 − 0.966342i \(-0.582820\pi\)
0.708247 + 0.705965i \(0.249486\pi\)
\(60\) − 2.00000i − 0.258199i
\(61\) −1.00000 − 1.73205i −0.128037 − 0.221766i 0.794879 − 0.606768i \(-0.207534\pi\)
−0.922916 + 0.385002i \(0.874201\pi\)
\(62\) 5.00000 − 8.66025i 0.635001 − 1.09985i
\(63\) 1.73205 + 1.00000i 0.218218 + 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.73205 + 1.00000i 0.211604 + 0.122169i 0.602056 − 0.798454i \(-0.294348\pi\)
−0.390453 + 0.920623i \(0.627682\pi\)
\(68\) −1.00000 + 1.73205i −0.121268 + 0.210042i
\(69\) −2.00000 − 3.46410i −0.240772 − 0.417029i
\(70\) 4.00000i 0.478091i
\(71\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) −0.866025 + 0.500000i −0.102062 + 0.0589256i
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) 4.00000 + 6.92820i 0.464991 + 0.805387i
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) −5.19615 − 3.00000i −0.596040 − 0.344124i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.73205 − 1.00000i −0.193649 − 0.111803i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 5.00000 + 8.66025i 0.552158 + 0.956365i
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 1.73205 − 1.00000i 0.188982 − 0.109109i
\(85\) −3.46410 + 2.00000i −0.375735 + 0.216930i
\(86\) 4.00000i 0.431331i
\(87\) 5.00000 + 8.66025i 0.536056 + 0.928477i
\(88\) 0 0
\(89\) 5.19615 + 3.00000i 0.550791 + 0.317999i 0.749441 − 0.662071i \(-0.230322\pi\)
−0.198650 + 0.980071i \(0.563656\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −8.66025 − 5.00000i −0.898027 − 0.518476i
\(94\) −6.00000 + 10.3923i −0.618853 + 1.07188i
\(95\) −6.00000 − 10.3923i −0.615587 − 1.06623i
\(96\) 1.00000i 0.102062i
\(97\) −10.3923 + 6.00000i −1.05518 + 0.609208i −0.924095 − 0.382164i \(-0.875179\pi\)
−0.131084 + 0.991371i \(0.541846\pi\)
\(98\) 2.59808 − 1.50000i 0.262445 − 0.151523i
\(99\) 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 1014.2.i.c.823.1 4
13.2 odd 12 1014.2.e.b.991.1 2
13.3 even 3 inner 1014.2.i.c.361.2 4
13.4 even 6 78.2.b.a.25.1 ✓ 2
13.5 odd 4 1014.2.e.b.529.1 2
13.6 odd 12 1014.2.a.g.1.1 1
13.7 odd 12 1014.2.a.b.1.1 1
13.8 odd 4 1014.2.e.e.529.1 2
13.9 even 3 78.2.b.a.25.2 yes 2
13.10 even 6 inner 1014.2.i.c.361.1 4
13.11 odd 12 1014.2.e.e.991.1 2
13.12 even 2 inner 1014.2.i.c.823.2 4
39.17 odd 6 234.2.b.a.181.2 2
39.20 even 12 3042.2.a.n.1.1 1
39.32 even 12 3042.2.a.c.1.1 1
39.35 odd 6 234.2.b.a.181.1 2
52.7 even 12 8112.2.a.g.1.1 1
52.19 even 12 8112.2.a.j.1.1 1
52.35 odd 6 624.2.c.a.337.2 2
52.43 odd 6 624.2.c.a.337.1 2
65.4 even 6 1950.2.b.c.1351.2 2
65.9 even 6 1950.2.b.c.1351.1 2
65.17 odd 12 1950.2.f.g.649.1 2
65.22 odd 12 1950.2.f.d.649.1 2
65.43 odd 12 1950.2.f.d.649.2 2
65.48 odd 12 1950.2.f.g.649.2 2
91.48 odd 6 3822.2.c.d.883.2 2
91.69 odd 6 3822.2.c.d.883.1 2
104.35 odd 6 2496.2.c.m.961.1 2
104.43 odd 6 2496.2.c.m.961.2 2
104.61 even 6 2496.2.c.f.961.1 2
104.69 even 6 2496.2.c.f.961.2 2
156.35 even 6 1872.2.c.b.1585.1 2
156.95 even 6 1872.2.c.b.1585.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 ✓ 2 13.4 even 6
78.2.b.a.25.2 yes 2 13.9 even 3
234.2.b.a.181.1 2 39.35 odd 6
234.2.b.a.181.2 2 39.17 odd 6
624.2.c.a.337.1 2 52.43 odd 6
624.2.c.a.337.2 2 52.35 odd 6
1014.2.a.b.1.1 1 13.7 odd 12
1014.2.a.g.1.1 1 13.6 odd 12
1014.2.e.b.529.1 2 13.5 odd 4
1014.2.e.b.991.1 2 13.2 odd 12
1014.2.e.e.529.1 2 13.8 odd 4
1014.2.e.e.991.1 2 13.11 odd 12
1014.2.i.c.361.1 4 13.10 even 6 inner
1014.2.i.c.361.2 4 13.3 even 3 inner
1014.2.i.c.823.1 4 1.1 even 1 trivial
1014.2.i.c.823.2 4 13.12 even 2 inner
1872.2.c.b.1585.1 2 156.35 even 6
1872.2.c.b.1585.2 2 156.95 even 6
1950.2.b.c.1351.1 2 65.9 even 6
1950.2.b.c.1351.2 2 65.4 even 6
1950.2.f.d.649.1 2 65.22 odd 12
1950.2.f.d.649.2 2 65.43 odd 12
1950.2.f.g.649.1 2 65.17 odd 12
1950.2.f.g.649.2 2 65.48 odd 12
2496.2.c.f.961.1 2 104.61 even 6
2496.2.c.f.961.2 2 104.69 even 6
2496.2.c.m.961.1 2 104.35 odd 6
2496.2.c.m.961.2 2 104.43 odd 6
3042.2.a.c.1.1 1 39.32 even 12
3042.2.a.n.1.1 1 39.20 even 12
3822.2.c.d.883.1 2 91.69 odd 6
3822.2.c.d.883.2 2 91.48 odd 6
8112.2.a.g.1.1 1 52.7 even 12
8112.2.a.j.1.1 1 52.19 even 12