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Properties

Label 624.2.c.a.337.2
Level $624$
Weight $2$
Character 624.337
Analytic conductor $4.983$
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] = [624,2,Mod(337,624)] 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("624.337"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,0,0,2,0,0,0,-6,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 624.337
Dual form 624.2.c.a.337.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.00000i q^{5} +2.00000i q^{7} +1.00000 q^{9} +(-3.00000 - 2.00000i) q^{13} -2.00000i q^{15} -2.00000 q^{17} +6.00000i q^{19} -2.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} -10.0000i q^{31} -4.00000 q^{35} +8.00000i q^{37} +(3.00000 + 2.00000i) q^{39} +10.0000i q^{41} -4.00000 q^{43} +2.00000i q^{45} +12.0000i q^{47} +3.00000 q^{49} +2.00000 q^{51} -6.00000 q^{53} -6.00000i q^{57} -4.00000i q^{59} +2.00000 q^{61} +2.00000i q^{63} +(4.00000 - 6.00000i) q^{65} +2.00000i q^{67} +4.00000 q^{69} -4.00000i q^{73} -1.00000 q^{75} +1.00000 q^{81} +4.00000i q^{83} -4.00000i q^{85} +10.0000 q^{87} -6.00000i q^{89} +(4.00000 - 6.00000i) q^{91} +10.0000i q^{93} -12.0000 q^{95} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{13} - 4 q^{17} - 8 q^{23} + 2 q^{25} - 2 q^{27} - 20 q^{29} - 8 q^{35} + 6 q^{39} - 8 q^{43} + 6 q^{49} + 4 q^{51} - 12 q^{53} + 4 q^{61} + 8 q^{65} + 8 q^{69} - 2 q^{75}+ \cdots - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −3.00000 − 2.00000i −0.832050 − 0.554700i
\(14\) 0 0
\(15\) − 2.00000i − 0.516398i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 0 0
\(21\) − 2.00000i − 0.436436i
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) − 10.0000i − 1.79605i −0.439941 − 0.898027i \(-0.645001\pi\)
0.439941 − 0.898027i \(-0.354999\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 3.00000 + 2.00000i 0.480384 + 0.320256i
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 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 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 − 0.260378i \(-0.916153\pi\)
0.965507 − 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 4.00000 − 6.00000i 0.496139 − 0.744208i
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 − 0.317999i \(-0.896989\pi\)
0.948091 − 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 4.00000 − 6.00000i 0.419314 − 0.628971i
\(92\) 0 0
\(93\) 10.0000i 1.03695i
\(94\) 0 0
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) − 12.0000i − 1.21842i −0.793011 − 0.609208i \(-0.791488\pi\)
0.793011 − 0.609208i \(-0.208512\pi\)
\(98\) 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 624.2.c.a.337.2 2
3.2 odd 2 1872.2.c.b.1585.1 2
4.3 odd 2 78.2.b.a.25.2 yes 2
8.3 odd 2 2496.2.c.f.961.1 2
8.5 even 2 2496.2.c.m.961.1 2
12.11 even 2 234.2.b.a.181.1 2
13.5 odd 4 8112.2.a.j.1.1 1
13.8 odd 4 8112.2.a.g.1.1 1
13.12 even 2 inner 624.2.c.a.337.1 2
20.3 even 4 1950.2.f.g.649.2 2
20.7 even 4 1950.2.f.d.649.1 2
20.19 odd 2 1950.2.b.c.1351.1 2
28.27 even 2 3822.2.c.d.883.2 2
39.38 odd 2 1872.2.c.b.1585.2 2
52.3 odd 6 1014.2.i.c.823.1 4
52.7 even 12 1014.2.e.e.991.1 2
52.11 even 12 1014.2.e.e.529.1 2
52.15 even 12 1014.2.e.b.529.1 2
52.19 even 12 1014.2.e.b.991.1 2
52.23 odd 6 1014.2.i.c.823.2 4
52.31 even 4 1014.2.a.g.1.1 1
52.35 odd 6 1014.2.i.c.361.2 4
52.43 odd 6 1014.2.i.c.361.1 4
52.47 even 4 1014.2.a.b.1.1 1
52.51 odd 2 78.2.b.a.25.1 ✓ 2
104.51 odd 2 2496.2.c.f.961.2 2
104.77 even 2 2496.2.c.m.961.2 2
156.47 odd 4 3042.2.a.n.1.1 1
156.83 odd 4 3042.2.a.c.1.1 1
156.155 even 2 234.2.b.a.181.2 2
260.103 even 4 1950.2.f.d.649.2 2
260.207 even 4 1950.2.f.g.649.1 2
260.259 odd 2 1950.2.b.c.1351.2 2
364.363 even 2 3822.2.c.d.883.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 ✓ 2 52.51 odd 2
78.2.b.a.25.2 yes 2 4.3 odd 2
234.2.b.a.181.1 2 12.11 even 2
234.2.b.a.181.2 2 156.155 even 2
624.2.c.a.337.1 2 13.12 even 2 inner
624.2.c.a.337.2 2 1.1 even 1 trivial
1014.2.a.b.1.1 1 52.47 even 4
1014.2.a.g.1.1 1 52.31 even 4
1014.2.e.b.529.1 2 52.15 even 12
1014.2.e.b.991.1 2 52.19 even 12
1014.2.e.e.529.1 2 52.11 even 12
1014.2.e.e.991.1 2 52.7 even 12
1014.2.i.c.361.1 4 52.43 odd 6
1014.2.i.c.361.2 4 52.35 odd 6
1014.2.i.c.823.1 4 52.3 odd 6
1014.2.i.c.823.2 4 52.23 odd 6
1872.2.c.b.1585.1 2 3.2 odd 2
1872.2.c.b.1585.2 2 39.38 odd 2
1950.2.b.c.1351.1 2 20.19 odd 2
1950.2.b.c.1351.2 2 260.259 odd 2
1950.2.f.d.649.1 2 20.7 even 4
1950.2.f.d.649.2 2 260.103 even 4
1950.2.f.g.649.1 2 260.207 even 4
1950.2.f.g.649.2 2 20.3 even 4
2496.2.c.f.961.1 2 8.3 odd 2
2496.2.c.f.961.2 2 104.51 odd 2
2496.2.c.m.961.1 2 8.5 even 2
2496.2.c.m.961.2 2 104.77 even 2
3042.2.a.c.1.1 1 156.83 odd 4
3042.2.a.n.1.1 1 156.47 odd 4
3822.2.c.d.883.1 2 364.363 even 2
3822.2.c.d.883.2 2 28.27 even 2
8112.2.a.g.1.1 1 13.8 odd 4
8112.2.a.j.1.1 1 13.5 odd 4