Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [242,2,Mod(1,242)] 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("242.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 242.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.93237972891\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 242.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.23607 q^{5} +2.61803 q^{6} -2.00000 q^{7} +1.00000 q^{8} +3.85410 q^{9} -1.23607 q^{10} +2.61803 q^{12} -3.23607 q^{13} -2.00000 q^{14} -3.23607 q^{15} +1.00000 q^{16} +1.61803 q^{17} +3.85410 q^{18} +0.854102 q^{19} -1.23607 q^{20} -5.23607 q^{21} -3.23607 q^{23} +2.61803 q^{24} -3.47214 q^{25} -3.23607 q^{26} +2.23607 q^{27} -2.00000 q^{28} -4.47214 q^{29} -3.23607 q^{30} +2.00000 q^{31} +1.00000 q^{32} +1.61803 q^{34} +2.47214 q^{35} +3.85410 q^{36} +9.70820 q^{37} +0.854102 q^{38} -8.47214 q^{39} -1.23607 q^{40} +3.38197 q^{41} -5.23607 q^{42} +11.5623 q^{43} -4.76393 q^{45} -3.23607 q^{46} +2.47214 q^{47} +2.61803 q^{48} -3.00000 q^{49} -3.47214 q^{50} +4.23607 q^{51} -3.23607 q^{52} -10.4721 q^{53} +2.23607 q^{54} -2.00000 q^{56} +2.23607 q^{57} -4.47214 q^{58} -6.38197 q^{59} -3.23607 q^{60} +6.47214 q^{61} +2.00000 q^{62} -7.70820 q^{63} +1.00000 q^{64} +4.00000 q^{65} -0.0901699 q^{67} +1.61803 q^{68} -8.47214 q^{69} +2.47214 q^{70} -0.763932 q^{71} +3.85410 q^{72} +12.6180 q^{73} +9.70820 q^{74} -9.09017 q^{75} +0.854102 q^{76} -8.47214 q^{78} -13.4164 q^{79} -1.23607 q^{80} -5.70820 q^{81} +3.38197 q^{82} -6.32624 q^{83} -5.23607 q^{84} -2.00000 q^{85} +11.5623 q^{86} -11.7082 q^{87} +3.09017 q^{89} -4.76393 q^{90} +6.47214 q^{91} -3.23607 q^{92} +5.23607 q^{93} +2.47214 q^{94} -1.05573 q^{95} +2.61803 q^{96} +13.8541 q^{97} -3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} + 3 q^{6} - 4 q^{7} + 2 q^{8} + q^{9} + 2 q^{10} + 3 q^{12} - 2 q^{13} - 4 q^{14} - 2 q^{15} + 2 q^{16} + q^{17} + q^{18} - 5 q^{19} + 2 q^{20} - 6 q^{21}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.61803 1.51152 0.755761 − 0.654847i \(-0.227267\pi\)
0.755761 + 0.654847i \(0.227267\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.23607 −0.552786 −0.276393 − 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 2.61803 1.06881
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.85410 1.28470
\(10\) −1.23607 −0.390879
\(11\) 0 0
\(12\) 2.61803 0.755761
\(13\) −3.23607 −0.897524 −0.448762 − 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) −2.00000 −0.534522
\(15\) −3.23607 −0.835549
\(16\) 1.00000 0.250000
\(17\) 1.61803 0.392431 0.196215 − 0.980561i \(-0.437135\pi\)
0.196215 + 0.980561i \(0.437135\pi\)
\(18\) 3.85410 0.908421
\(19\) 0.854102 0.195944 0.0979722 − 0.995189i \(-0.468764\pi\)
0.0979722 + 0.995189i \(0.468764\pi\)
\(20\) −1.23607 −0.276393
\(21\) −5.23607 −1.14260
\(22\) 0 0
\(23\) −3.23607 −0.674767 −0.337383 − 0.941367i \(-0.609542\pi\)
−0.337383 + 0.941367i \(0.609542\pi\)
\(24\) 2.61803 0.534404
\(25\) −3.47214 −0.694427
\(26\) −3.23607 −0.634645
\(27\) 2.23607 0.430331
\(28\) −2.00000 −0.377964
\(29\) −4.47214 −0.830455 −0.415227 − 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) −3.23607 −0.590822
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 1.61803 0.277491
\(35\) 2.47214 0.417867
\(36\) 3.85410 0.642350
\(37\) 9.70820 1.59602 0.798009 − 0.602645i \(-0.205886\pi\)
0.798009 + 0.602645i \(0.205886\pi\)
\(38\) 0.854102 0.138554
\(39\) −8.47214 −1.35663
\(40\) −1.23607 −0.195440
\(41\) 3.38197 0.528174 0.264087 − 0.964499i \(-0.414929\pi\)
0.264087 + 0.964499i \(0.414929\pi\)
\(42\) −5.23607 −0.807943
\(43\) 11.5623 1.76324 0.881618 − 0.471964i \(-0.156455\pi\)
0.881618 + 0.471964i \(0.156455\pi\)
\(44\) 0 0
\(45\) −4.76393 −0.710165
\(46\) −3.23607 −0.477132
\(47\) 2.47214 0.360598 0.180299 − 0.983612i \(-0.442293\pi\)
0.180299 + 0.983612i \(0.442293\pi\)
\(48\) 2.61803 0.377881
\(49\) −3.00000 −0.428571
\(50\) −3.47214 −0.491034
\(51\) 4.23607 0.593168
\(52\) −3.23607 −0.448762
\(53\) −10.4721 −1.43846 −0.719229 − 0.694773i \(-0.755505\pi\)
−0.719229 + 0.694773i \(0.755505\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 2.23607 0.296174
\(58\) −4.47214 −0.587220
\(59\) −6.38197 −0.830861 −0.415431 − 0.909625i \(-0.636369\pi\)
−0.415431 + 0.909625i \(0.636369\pi\)
\(60\) −3.23607 −0.417775
\(61\) 6.47214 0.828672 0.414336 − 0.910124i \(-0.364014\pi\)
0.414336 + 0.910124i \(0.364014\pi\)
\(62\) 2.00000 0.254000
\(63\) −7.70820 −0.971142
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −0.0901699 −0.0110160 −0.00550801 − 0.999985i \(-0.501753\pi\)
−0.00550801 + 0.999985i \(0.501753\pi\)
\(68\) 1.61803 0.196215
\(69\) −8.47214 −1.01993
\(70\) 2.47214 0.295477
\(71\) −0.763932 −0.0906621 −0.0453310 − 0.998972i \(-0.514434\pi\)
−0.0453310 + 0.998972i \(0.514434\pi\)
\(72\) 3.85410 0.454210
\(73\) 12.6180 1.47683 0.738415 − 0.674347i \(-0.235575\pi\)
0.738415 + 0.674347i \(0.235575\pi\)
\(74\) 9.70820 1.12856
\(75\) −9.09017 −1.04964
\(76\) 0.854102 0.0979722
\(77\) 0 0
\(78\) −8.47214 −0.959280
\(79\) −13.4164 −1.50946 −0.754732 − 0.656033i \(-0.772233\pi\)
−0.754732 + 0.656033i \(0.772233\pi\)
\(80\) −1.23607 −0.138197
\(81\) −5.70820 −0.634245
\(82\) 3.38197 0.373476
\(83\) −6.32624 −0.694395 −0.347197 − 0.937792i \(-0.612867\pi\)
−0.347197 + 0.937792i \(0.612867\pi\)
\(84\) −5.23607 −0.571302
\(85\) −2.00000 −0.216930
\(86\) 11.5623 1.24680
\(87\) −11.7082 −1.25525
\(88\) 0 0
\(89\) 3.09017 0.327557 0.163779 − 0.986497i \(-0.447632\pi\)
0.163779 + 0.986497i \(0.447632\pi\)
\(90\) −4.76393 −0.502163
\(91\) 6.47214 0.678464
\(92\) −3.23607 −0.337383
\(93\) 5.23607 0.542955
\(94\) 2.47214 0.254981
\(95\) −1.05573 −0.108315
\(96\) 2.61803 0.267202
\(97\) 13.8541 1.40667 0.703335 − 0.710858i \(-0.251693\pi\)
0.703335 + 0.710858i \(0.251693\pi\)
\(98\) −3.00000 −0.303046
\(99\) 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 242.2.a.f.1.2 2
3.2 odd 2 2178.2.a.p.1.2 2
4.3 odd 2 1936.2.a.o.1.1 2
5.4 even 2 6050.2.a.bs.1.1 2
8.3 odd 2 7744.2.a.cz.1.2 2
8.5 even 2 7744.2.a.bm.1.1 2
11.2 odd 10 242.2.c.d.81.1 4
11.3 even 5 22.2.c.a.9.1 yes 4
11.4 even 5 22.2.c.a.5.1 ✓ 4
11.5 even 5 242.2.c.a.3.1 4
11.6 odd 10 242.2.c.d.3.1 4
11.7 odd 10 242.2.c.c.27.1 4
11.8 odd 10 242.2.c.c.9.1 4
11.9 even 5 242.2.c.a.81.1 4
11.10 odd 2 242.2.a.d.1.2 2
33.14 odd 10 198.2.f.e.163.1 4
33.26 odd 10 198.2.f.e.181.1 4
33.32 even 2 2178.2.a.x.1.2 2
44.3 odd 10 176.2.m.c.97.1 4
44.15 odd 10 176.2.m.c.49.1 4
44.43 even 2 1936.2.a.n.1.1 2
55.3 odd 20 550.2.ba.c.449.1 8
55.4 even 10 550.2.h.h.401.1 4
55.14 even 10 550.2.h.h.251.1 4
55.37 odd 20 550.2.ba.c.49.1 8
55.47 odd 20 550.2.ba.c.449.2 8
55.48 odd 20 550.2.ba.c.49.2 8
55.54 odd 2 6050.2.a.ci.1.1 2
88.3 odd 10 704.2.m.a.449.1 4
88.21 odd 2 7744.2.a.bn.1.1 2
88.37 even 10 704.2.m.h.577.1 4
88.43 even 2 7744.2.a.cy.1.2 2
88.59 odd 10 704.2.m.a.577.1 4
88.69 even 10 704.2.m.h.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.2.c.a.5.1 ✓ 4 11.4 even 5
22.2.c.a.9.1 yes 4 11.3 even 5
176.2.m.c.49.1 4 44.15 odd 10
176.2.m.c.97.1 4 44.3 odd 10
198.2.f.e.163.1 4 33.14 odd 10
198.2.f.e.181.1 4 33.26 odd 10
242.2.a.d.1.2 2 11.10 odd 2
242.2.a.f.1.2 2 1.1 even 1 trivial
242.2.c.a.3.1 4 11.5 even 5
242.2.c.a.81.1 4 11.9 even 5
242.2.c.c.9.1 4 11.8 odd 10
242.2.c.c.27.1 4 11.7 odd 10
242.2.c.d.3.1 4 11.6 odd 10
242.2.c.d.81.1 4 11.2 odd 10
550.2.h.h.251.1 4 55.14 even 10
550.2.h.h.401.1 4 55.4 even 10
550.2.ba.c.49.1 8 55.37 odd 20
550.2.ba.c.49.2 8 55.48 odd 20
550.2.ba.c.449.1 8 55.3 odd 20
550.2.ba.c.449.2 8 55.47 odd 20
704.2.m.a.449.1 4 88.3 odd 10
704.2.m.a.577.1 4 88.59 odd 10
704.2.m.h.449.1 4 88.69 even 10
704.2.m.h.577.1 4 88.37 even 10
1936.2.a.n.1.1 2 44.43 even 2
1936.2.a.o.1.1 2 4.3 odd 2
2178.2.a.p.1.2 2 3.2 odd 2
2178.2.a.x.1.2 2 33.32 even 2
6050.2.a.bs.1.1 2 5.4 even 2
6050.2.a.ci.1.1 2 55.54 odd 2
7744.2.a.bm.1.1 2 8.5 even 2
7744.2.a.bn.1.1 2 88.21 odd 2
7744.2.a.cy.1.2 2 88.43 even 2
7744.2.a.cz.1.2 2 8.3 odd 2