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Properties

Label 1014.2.b.d.337.3
Level $1014$
Weight $2$
Character 1014.337
Analytic conductor $8.097$
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] = [1014,2,Mod(337,1014)] 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("1014.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,-4,0,0,0,0,4,0] 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: \(4\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.3
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.2.b.d.337.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.73205i q^{5} -1.00000i q^{6} +1.26795i q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.73205 q^{10} +1.26795i q^{11} +1.00000 q^{12} -1.26795 q^{14} +1.73205i q^{15} +1.00000 q^{16} -5.19615 q^{17} +1.00000i q^{18} -4.73205i q^{19} +1.73205i q^{20} -1.26795i q^{21} -1.26795 q^{22} +8.19615 q^{23} +1.00000i q^{24} +2.00000 q^{25} -1.00000 q^{27} -1.26795i q^{28} -3.00000 q^{29} -1.73205 q^{30} -9.46410i q^{31} +1.00000i q^{32} -1.26795i q^{33} -5.19615i q^{34} +2.19615 q^{35} -1.00000 q^{36} +3.00000i q^{37} +4.73205 q^{38} -1.73205 q^{40} -6.46410i q^{41} +1.26795 q^{42} +4.19615 q^{43} -1.26795i q^{44} -1.73205i q^{45} +8.19615i q^{46} -4.73205i q^{47} -1.00000 q^{48} +5.39230 q^{49} +2.00000i q^{50} +5.19615 q^{51} +3.00000 q^{53} -1.00000i q^{54} +2.19615 q^{55} +1.26795 q^{56} +4.73205i q^{57} -3.00000i q^{58} -13.8564i q^{59} -1.73205i q^{60} +15.1962 q^{61} +9.46410 q^{62} +1.26795i q^{63} -1.00000 q^{64} +1.26795 q^{66} -7.26795i q^{67} +5.19615 q^{68} -8.19615 q^{69} +2.19615i q^{70} +2.19615i q^{71} -1.00000i q^{72} +12.1244i q^{73} -3.00000 q^{74} -2.00000 q^{75} +4.73205i q^{76} -1.60770 q^{77} +8.39230 q^{79} -1.73205i q^{80} +1.00000 q^{81} +6.46410 q^{82} -5.66025i q^{83} +1.26795i q^{84} +9.00000i q^{85} +4.19615i q^{86} +3.00000 q^{87} +1.26795 q^{88} +9.46410i q^{89} +1.73205 q^{90} -8.19615 q^{92} +9.46410i q^{93} +4.73205 q^{94} -8.19615 q^{95} -1.00000i q^{96} -6.00000i q^{97} +5.39230i q^{98} +1.26795i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{12} - 12 q^{14} + 4 q^{16} - 12 q^{22} + 12 q^{23} + 8 q^{25} - 4 q^{27} - 12 q^{29} - 12 q^{35} - 4 q^{36} + 12 q^{38} + 12 q^{42} - 4 q^{43} - 4 q^{48} - 20 q^{49}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.73205i − 0.774597i −0.921954 − 0.387298i \(-0.873408\pi\)
0.921954 − 0.387298i \(-0.126592\pi\)
\(6\) − 1.00000i − 0.408248i
\(7\) 1.26795i 0.479240i 0.970867 + 0.239620i \(0.0770228\pi\)
−0.970867 + 0.239620i \(0.922977\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 1.73205 0.547723
\(11\) 1.26795i 0.382301i 0.981561 + 0.191151i \(0.0612219\pi\)
−0.981561 + 0.191151i \(0.938778\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.26795 −0.338874
\(15\) 1.73205i 0.447214i
\(16\) 1.00000 0.250000
\(17\) −5.19615 −1.26025 −0.630126 − 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 4.73205i − 1.08561i −0.839860 − 0.542803i \(-0.817363\pi\)
0.839860 − 0.542803i \(-0.182637\pi\)
\(20\) 1.73205i 0.387298i
\(21\) − 1.26795i − 0.276689i
\(22\) −1.26795 −0.270328
\(23\) 8.19615 1.70902 0.854508 − 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 1.26795i − 0.239620i
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −1.73205 −0.316228
\(31\) − 9.46410i − 1.69980i −0.526942 − 0.849901i \(-0.676661\pi\)
0.526942 − 0.849901i \(-0.323339\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.26795i − 0.220722i
\(34\) − 5.19615i − 0.891133i
\(35\) 2.19615 0.371218
\(36\) −1.00000 −0.166667
\(37\) 3.00000i 0.493197i 0.969118 + 0.246598i \(0.0793129\pi\)
−0.969118 + 0.246598i \(0.920687\pi\)
\(38\) 4.73205 0.767640
\(39\) 0 0
\(40\) −1.73205 −0.273861
\(41\) − 6.46410i − 1.00952i −0.863259 − 0.504762i \(-0.831580\pi\)
0.863259 − 0.504762i \(-0.168420\pi\)
\(42\) 1.26795 0.195649
\(43\) 4.19615 0.639907 0.319954 − 0.947433i \(-0.396333\pi\)
0.319954 + 0.947433i \(0.396333\pi\)
\(44\) − 1.26795i − 0.191151i
\(45\) − 1.73205i − 0.258199i
\(46\) 8.19615i 1.20846i
\(47\) − 4.73205i − 0.690241i −0.938558 − 0.345120i \(-0.887838\pi\)
0.938558 − 0.345120i \(-0.112162\pi\)
\(48\) −1.00000 −0.144338
\(49\) 5.39230 0.770329
\(50\) 2.00000i 0.282843i
\(51\) 5.19615 0.727607
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 2.19615 0.296129
\(56\) 1.26795 0.169437
\(57\) 4.73205i 0.626775i
\(58\) − 3.00000i − 0.393919i
\(59\) − 13.8564i − 1.80395i −0.431788 − 0.901975i \(-0.642117\pi\)
0.431788 − 0.901975i \(-0.357883\pi\)
\(60\) − 1.73205i − 0.223607i
\(61\) 15.1962 1.94567 0.972834 − 0.231504i \(-0.0743646\pi\)
0.972834 + 0.231504i \(0.0743646\pi\)
\(62\) 9.46410 1.20194
\(63\) 1.26795i 0.159747i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.26795 0.156074
\(67\) − 7.26795i − 0.887921i −0.896046 − 0.443961i \(-0.853573\pi\)
0.896046 − 0.443961i \(-0.146427\pi\)
\(68\) 5.19615 0.630126
\(69\) −8.19615 −0.986701
\(70\) 2.19615i 0.262490i
\(71\) 2.19615i 0.260635i 0.991472 + 0.130318i \(0.0415997\pi\)
−0.991472 + 0.130318i \(0.958400\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 12.1244i 1.41905i 0.704681 + 0.709524i \(0.251090\pi\)
−0.704681 + 0.709524i \(0.748910\pi\)
\(74\) −3.00000 −0.348743
\(75\) −2.00000 −0.230940
\(76\) 4.73205i 0.542803i
\(77\) −1.60770 −0.183214
\(78\) 0 0
\(79\) 8.39230 0.944208 0.472104 − 0.881543i \(-0.343495\pi\)
0.472104 + 0.881543i \(0.343495\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) 1.00000 0.111111
\(82\) 6.46410 0.713841
\(83\) − 5.66025i − 0.621294i −0.950525 − 0.310647i \(-0.899454\pi\)
0.950525 − 0.310647i \(-0.100546\pi\)
\(84\) 1.26795i 0.138345i
\(85\) 9.00000i 0.976187i
\(86\) 4.19615i 0.452483i
\(87\) 3.00000 0.321634
\(88\) 1.26795 0.135164
\(89\) 9.46410i 1.00319i 0.865102 + 0.501596i \(0.167254\pi\)
−0.865102 + 0.501596i \(0.832746\pi\)
\(90\) 1.73205 0.182574
\(91\) 0 0
\(92\) −8.19615 −0.854508
\(93\) 9.46410i 0.981382i
\(94\) 4.73205 0.488074
\(95\) −8.19615 −0.840907
\(96\) − 1.00000i − 0.102062i
\(97\) − 6.00000i − 0.609208i −0.952479 − 0.304604i \(-0.901476\pi\)
0.952479 − 0.304604i \(-0.0985241\pi\)
\(98\) 5.39230i 0.544705i
\(99\) 1.26795i 0.127434i
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.b.d.337.3 4
3.2 odd 2 3042.2.b.l.1351.2 4
13.2 odd 12 1014.2.e.h.529.1 4
13.3 even 3 78.2.i.b.43.1 ✓ 4
13.4 even 6 78.2.i.b.49.1 yes 4
13.5 odd 4 1014.2.a.j.1.1 2
13.6 odd 12 1014.2.e.h.991.1 4
13.7 odd 12 1014.2.e.j.991.2 4
13.8 odd 4 1014.2.a.h.1.2 2
13.9 even 3 1014.2.i.f.361.2 4
13.10 even 6 1014.2.i.f.823.2 4
13.11 odd 12 1014.2.e.j.529.2 4
13.12 even 2 inner 1014.2.b.d.337.2 4
39.5 even 4 3042.2.a.s.1.2 2
39.8 even 4 3042.2.a.v.1.1 2
39.17 odd 6 234.2.l.a.127.2 4
39.29 odd 6 234.2.l.a.199.2 4
39.38 odd 2 3042.2.b.l.1351.3 4
52.3 odd 6 624.2.bv.d.433.1 4
52.31 even 4 8112.2.a.bx.1.1 2
52.43 odd 6 624.2.bv.d.49.1 4
52.47 even 4 8112.2.a.bq.1.2 2
65.3 odd 12 1950.2.y.a.199.2 4
65.4 even 6 1950.2.bc.c.751.2 4
65.17 odd 12 1950.2.y.a.49.2 4
65.29 even 6 1950.2.bc.c.901.2 4
65.42 odd 12 1950.2.y.h.199.1 4
65.43 odd 12 1950.2.y.h.49.1 4
156.95 even 6 1872.2.by.k.1297.1 4
156.107 even 6 1872.2.by.k.433.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.b.43.1 ✓ 4 13.3 even 3
78.2.i.b.49.1 yes 4 13.4 even 6
234.2.l.a.127.2 4 39.17 odd 6
234.2.l.a.199.2 4 39.29 odd 6
624.2.bv.d.49.1 4 52.43 odd 6
624.2.bv.d.433.1 4 52.3 odd 6
1014.2.a.h.1.2 2 13.8 odd 4
1014.2.a.j.1.1 2 13.5 odd 4
1014.2.b.d.337.2 4 13.12 even 2 inner
1014.2.b.d.337.3 4 1.1 even 1 trivial
1014.2.e.h.529.1 4 13.2 odd 12
1014.2.e.h.991.1 4 13.6 odd 12
1014.2.e.j.529.2 4 13.11 odd 12
1014.2.e.j.991.2 4 13.7 odd 12
1014.2.i.f.361.2 4 13.9 even 3
1014.2.i.f.823.2 4 13.10 even 6
1872.2.by.k.433.1 4 156.107 even 6
1872.2.by.k.1297.1 4 156.95 even 6
1950.2.y.a.49.2 4 65.17 odd 12
1950.2.y.a.199.2 4 65.3 odd 12
1950.2.y.h.49.1 4 65.43 odd 12
1950.2.y.h.199.1 4 65.42 odd 12
1950.2.bc.c.751.2 4 65.4 even 6
1950.2.bc.c.901.2 4 65.29 even 6
3042.2.a.s.1.2 2 39.5 even 4
3042.2.a.v.1.1 2 39.8 even 4
3042.2.b.l.1351.2 4 3.2 odd 2
3042.2.b.l.1351.3 4 39.38 odd 2
8112.2.a.bq.1.2 2 52.47 even 4
8112.2.a.bx.1.1 2 52.31 even 4