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Properties

Label 3822.2.c.d.883.2
Level $3822$
Weight $2$
Character 3822.883
Analytic conductor $30.519$
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] = [3822,2,Mod(883,3822)] 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("3822.883"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3822, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,-2,0,0,0,0,2,4,0,2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(30.5188236525\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 883.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3822.883
Dual form 3822.2.c.d.883.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000i q^{5} -1.00000i q^{6} -1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{12} +(3.00000 + 2.00000i) q^{13} +2.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +6.00000i q^{19} +2.00000i q^{20} +4.00000 q^{23} +1.00000i q^{24} +1.00000 q^{25} +(-2.00000 + 3.00000i) q^{26} -1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} -10.0000i q^{31} +1.00000i q^{32} +2.00000i q^{34} -1.00000 q^{36} +8.00000i q^{37} -6.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} -2.00000 q^{40} -10.0000i q^{41} +4.00000 q^{43} -2.00000i q^{45} +4.00000i q^{46} +12.0000i q^{47} -1.00000 q^{48} +1.00000i q^{50} -2.00000 q^{51} +(-3.00000 - 2.00000i) q^{52} -6.00000 q^{53} -1.00000i q^{54} -6.00000i q^{57} -10.0000i q^{58} -4.00000i q^{59} -2.00000i q^{60} -2.00000 q^{61} +10.0000 q^{62} -1.00000 q^{64} +(4.00000 - 6.00000i) q^{65} -2.00000i q^{67} -2.00000 q^{68} -4.00000 q^{69} -1.00000i q^{72} +4.00000i q^{73} -8.00000 q^{74} -1.00000 q^{75} -6.00000i q^{76} +(2.00000 - 3.00000i) q^{78} -2.00000i q^{80} +1.00000 q^{81} +10.0000 q^{82} +4.00000i q^{83} -4.00000i q^{85} +4.00000i q^{86} +10.0000 q^{87} +6.00000i q^{89} +2.00000 q^{90} -4.00000 q^{92} +10.0000i q^{93} -12.0000 q^{94} +12.0000 q^{95} -1.00000i q^{96} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} + 6 q^{13} + 2 q^{16} + 4 q^{17} + 8 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 20 q^{29} - 4 q^{30} - 2 q^{36} - 12 q^{38} - 6 q^{39} - 4 q^{40} + 8 q^{43}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times\).

\(n\) \(1471\) \(2549\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) − 2.00000i − 0.894427i −0.894427 − 0.447214i \(-0.852416\pi\)
0.894427 − 0.447214i \(-0.147584\pi\)
\(6\) − 1.00000i − 0.408248i
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.00000 + 2.00000i 0.832050 + 0.554700i
\(14\) 0 0
\(15\) 2.00000i 0.516398i
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.00000i 1.37649i 0.725476 + 0.688247i \(0.241620\pi\)
−0.725476 + 0.688247i \(0.758380\pi\)
\(20\) 2.00000i 0.447214i
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 1.00000 0.200000
\(26\) −2.00000 + 3.00000i −0.392232 + 0.588348i
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) − 10.0000i − 1.79605i −0.439941 − 0.898027i \(-0.645001\pi\)
0.439941 − 0.898027i \(-0.354999\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.00000i 0.342997i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) −6.00000 −0.973329
\(39\) −3.00000 − 2.00000i −0.480384 − 0.320256i
\(40\) −2.00000 −0.316228
\(41\) − 10.0000i − 1.56174i −0.624695 − 0.780869i \(-0.714777\pi\)
0.624695 − 0.780869i \(-0.285223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) − 2.00000i − 0.298142i
\(46\) 4.00000i 0.589768i
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000i 0.141421i
\(51\) −2.00000 −0.280056
\(52\) −3.00000 − 2.00000i −0.416025 − 0.277350i
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) − 10.0000i − 1.31306i
\(59\) − 4.00000i − 0.520756i −0.965507 − 0.260378i \(-0.916153\pi\)
0.965507 − 0.260378i \(-0.0838471\pi\)
\(60\) − 2.00000i − 0.258199i
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 4.00000 − 6.00000i 0.496139 − 0.744208i
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 − 0.122169i \(-0.961015\pi\)
0.992509 − 0.122169i \(-0.0389851\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) −8.00000 −0.929981
\(75\) −1.00000 −0.115470
\(76\) − 6.00000i − 0.688247i
\(77\) 0 0
\(78\) 2.00000 − 3.00000i 0.226455 − 0.339683i
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) − 2.00000i − 0.223607i
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 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
\(86\) 4.00000i 0.431331i
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 10.0000i 1.03695i
\(94\) −12.0000 −1.23771
\(95\) 12.0000 1.23117
\(96\) − 1.00000i − 0.102062i
\(97\) 12.0000i 1.21842i 0.793011 + 0.609208i \(0.208512\pi\)
−0.793011 + 0.609208i \(0.791488\pi\)
\(98\) 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 3822.2.c.d.883.2 2
7.6 odd 2 78.2.b.a.25.2 yes 2
13.12 even 2 inner 3822.2.c.d.883.1 2
21.20 even 2 234.2.b.a.181.1 2
28.27 even 2 624.2.c.a.337.2 2
35.13 even 4 1950.2.f.g.649.2 2
35.27 even 4 1950.2.f.d.649.1 2
35.34 odd 2 1950.2.b.c.1351.1 2
56.13 odd 2 2496.2.c.f.961.1 2
56.27 even 2 2496.2.c.m.961.1 2
84.83 odd 2 1872.2.c.b.1585.1 2
91.6 even 12 1014.2.e.b.991.1 2
91.20 even 12 1014.2.e.e.991.1 2
91.34 even 4 1014.2.a.b.1.1 1
91.41 even 12 1014.2.e.b.529.1 2
91.48 odd 6 1014.2.i.c.361.2 4
91.55 odd 6 1014.2.i.c.823.1 4
91.62 odd 6 1014.2.i.c.823.2 4
91.69 odd 6 1014.2.i.c.361.1 4
91.76 even 12 1014.2.e.e.529.1 2
91.83 even 4 1014.2.a.g.1.1 1
91.90 odd 2 78.2.b.a.25.1 ✓ 2
273.83 odd 4 3042.2.a.c.1.1 1
273.125 odd 4 3042.2.a.n.1.1 1
273.272 even 2 234.2.b.a.181.2 2
364.83 odd 4 8112.2.a.j.1.1 1
364.307 odd 4 8112.2.a.g.1.1 1
364.363 even 2 624.2.c.a.337.1 2
455.272 even 4 1950.2.f.g.649.1 2
455.363 even 4 1950.2.f.d.649.2 2
455.454 odd 2 1950.2.b.c.1351.2 2
728.181 odd 2 2496.2.c.f.961.2 2
728.363 even 2 2496.2.c.m.961.2 2
1092.1091 odd 2 1872.2.c.b.1585.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.b.a.25.1 ✓ 2 91.90 odd 2
78.2.b.a.25.2 yes 2 7.6 odd 2
234.2.b.a.181.1 2 21.20 even 2
234.2.b.a.181.2 2 273.272 even 2
624.2.c.a.337.1 2 364.363 even 2
624.2.c.a.337.2 2 28.27 even 2
1014.2.a.b.1.1 1 91.34 even 4
1014.2.a.g.1.1 1 91.83 even 4
1014.2.e.b.529.1 2 91.41 even 12
1014.2.e.b.991.1 2 91.6 even 12
1014.2.e.e.529.1 2 91.76 even 12
1014.2.e.e.991.1 2 91.20 even 12
1014.2.i.c.361.1 4 91.69 odd 6
1014.2.i.c.361.2 4 91.48 odd 6
1014.2.i.c.823.1 4 91.55 odd 6
1014.2.i.c.823.2 4 91.62 odd 6
1872.2.c.b.1585.1 2 84.83 odd 2
1872.2.c.b.1585.2 2 1092.1091 odd 2
1950.2.b.c.1351.1 2 35.34 odd 2
1950.2.b.c.1351.2 2 455.454 odd 2
1950.2.f.d.649.1 2 35.27 even 4
1950.2.f.d.649.2 2 455.363 even 4
1950.2.f.g.649.1 2 455.272 even 4
1950.2.f.g.649.2 2 35.13 even 4
2496.2.c.f.961.1 2 56.13 odd 2
2496.2.c.f.961.2 2 728.181 odd 2
2496.2.c.m.961.1 2 56.27 even 2
2496.2.c.m.961.2 2 728.363 even 2
3042.2.a.c.1.1 1 273.83 odd 4
3042.2.a.n.1.1 1 273.125 odd 4
3822.2.c.d.883.1 2 13.12 even 2 inner
3822.2.c.d.883.2 2 1.1 even 1 trivial
8112.2.a.g.1.1 1 364.307 odd 4
8112.2.a.j.1.1 1 364.83 odd 4