Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [363,2,Mod(1,363)] 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("363.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,2] 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: \(2.89856959337\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 363.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -2.61803 q^{5} -0.618034 q^{6} +3.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.61803 q^{10} -1.61803 q^{12} +1.76393 q^{13} -1.85410 q^{14} -2.61803 q^{15} +1.85410 q^{16} +1.61803 q^{17} -0.618034 q^{18} +5.85410 q^{19} +4.23607 q^{20} +3.00000 q^{21} +3.47214 q^{23} +2.23607 q^{24} +1.85410 q^{25} -1.09017 q^{26} +1.00000 q^{27} -4.85410 q^{28} +4.47214 q^{29} +1.61803 q^{30} +2.85410 q^{31} -5.61803 q^{32} -1.00000 q^{34} -7.85410 q^{35} -1.61803 q^{36} +0.236068 q^{37} -3.61803 q^{38} +1.76393 q^{39} -5.85410 q^{40} -11.9443 q^{41} -1.85410 q^{42} +6.23607 q^{43} -2.61803 q^{45} -2.14590 q^{46} +1.61803 q^{47} +1.85410 q^{48} +2.00000 q^{49} -1.14590 q^{50} +1.61803 q^{51} -2.85410 q^{52} -9.61803 q^{53} -0.618034 q^{54} +6.70820 q^{56} +5.85410 q^{57} -2.76393 q^{58} +10.3262 q^{59} +4.23607 q^{60} +7.85410 q^{61} -1.76393 q^{62} +3.00000 q^{63} -0.236068 q^{64} -4.61803 q^{65} -9.56231 q^{67} -2.61803 q^{68} +3.47214 q^{69} +4.85410 q^{70} -5.56231 q^{71} +2.23607 q^{72} -3.23607 q^{73} -0.145898 q^{74} +1.85410 q^{75} -9.47214 q^{76} -1.09017 q^{78} +9.47214 q^{79} -4.85410 q^{80} +1.00000 q^{81} +7.38197 q^{82} +0.708204 q^{83} -4.85410 q^{84} -4.23607 q^{85} -3.85410 q^{86} +4.47214 q^{87} +0.527864 q^{89} +1.61803 q^{90} +5.29180 q^{91} -5.61803 q^{92} +2.85410 q^{93} -1.00000 q^{94} -15.3262 q^{95} -5.61803 q^{96} -14.0344 q^{97} -1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} - q^{4} - 3 q^{5} + q^{6} + 6 q^{7} + 2 q^{9} + q^{10} - q^{12} + 8 q^{13} + 3 q^{14} - 3 q^{15} - 3 q^{16} + q^{17} + q^{18} + 5 q^{19} + 4 q^{20} + 6 q^{21} - 2 q^{23} - 3 q^{25}+ \cdots + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 − 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.61803 −0.809017
\(5\) −2.61803 −1.17082 −0.585410 − 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) −0.618034 −0.252311
\(7\) 3.00000 1.13389 0.566947 − 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 1.61803 0.511667
\(11\) 0 0
\(12\) −1.61803 −0.467086
\(13\) 1.76393 0.489227 0.244613 − 0.969621i \(-0.421339\pi\)
0.244613 + 0.969621i \(0.421339\pi\)
\(14\) −1.85410 −0.495530
\(15\) −2.61803 −0.675973
\(16\) 1.85410 0.463525
\(17\) 1.61803 0.392431 0.196215 − 0.980561i \(-0.437135\pi\)
0.196215 + 0.980561i \(0.437135\pi\)
\(18\) −0.618034 −0.145672
\(19\) 5.85410 1.34302 0.671512 − 0.740994i \(-0.265645\pi\)
0.671512 + 0.740994i \(0.265645\pi\)
\(20\) 4.23607 0.947214
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) 3.47214 0.723990 0.361995 − 0.932180i \(-0.382096\pi\)
0.361995 + 0.932180i \(0.382096\pi\)
\(24\) 2.23607 0.456435
\(25\) 1.85410 0.370820
\(26\) −1.09017 −0.213800
\(27\) 1.00000 0.192450
\(28\) −4.85410 −0.917339
\(29\) 4.47214 0.830455 0.415227 − 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 1.61803 0.295411
\(31\) 2.85410 0.512612 0.256306 − 0.966596i \(-0.417495\pi\)
0.256306 + 0.966596i \(0.417495\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) −7.85410 −1.32759
\(36\) −1.61803 −0.269672
\(37\) 0.236068 0.0388093 0.0194047 − 0.999812i \(-0.493823\pi\)
0.0194047 + 0.999812i \(0.493823\pi\)
\(38\) −3.61803 −0.586923
\(39\) 1.76393 0.282455
\(40\) −5.85410 −0.925615
\(41\) −11.9443 −1.86538 −0.932691 − 0.360677i \(-0.882546\pi\)
−0.932691 + 0.360677i \(0.882546\pi\)
\(42\) −1.85410 −0.286094
\(43\) 6.23607 0.950991 0.475496 − 0.879718i \(-0.342269\pi\)
0.475496 + 0.879718i \(0.342269\pi\)
\(44\) 0 0
\(45\) −2.61803 −0.390273
\(46\) −2.14590 −0.316395
\(47\) 1.61803 0.236015 0.118007 − 0.993013i \(-0.462349\pi\)
0.118007 + 0.993013i \(0.462349\pi\)
\(48\) 1.85410 0.267617
\(49\) 2.00000 0.285714
\(50\) −1.14590 −0.162054
\(51\) 1.61803 0.226570
\(52\) −2.85410 −0.395793
\(53\) −9.61803 −1.32114 −0.660569 − 0.750765i \(-0.729685\pi\)
−0.660569 + 0.750765i \(0.729685\pi\)
\(54\) −0.618034 −0.0841038
\(55\) 0 0
\(56\) 6.70820 0.896421
\(57\) 5.85410 0.775395
\(58\) −2.76393 −0.362922
\(59\) 10.3262 1.34436 0.672181 − 0.740387i \(-0.265358\pi\)
0.672181 + 0.740387i \(0.265358\pi\)
\(60\) 4.23607 0.546874
\(61\) 7.85410 1.00561 0.502807 − 0.864398i \(-0.332301\pi\)
0.502807 + 0.864398i \(0.332301\pi\)
\(62\) −1.76393 −0.224020
\(63\) 3.00000 0.377964
\(64\) −0.236068 −0.0295085
\(65\) −4.61803 −0.572797
\(66\) 0 0
\(67\) −9.56231 −1.16822 −0.584111 − 0.811674i \(-0.698557\pi\)
−0.584111 + 0.811674i \(0.698557\pi\)
\(68\) −2.61803 −0.317483
\(69\) 3.47214 0.417996
\(70\) 4.85410 0.580176
\(71\) −5.56231 −0.660124 −0.330062 − 0.943959i \(-0.607070\pi\)
−0.330062 + 0.943959i \(0.607070\pi\)
\(72\) 2.23607 0.263523
\(73\) −3.23607 −0.378753 −0.189377 − 0.981905i \(-0.560647\pi\)
−0.189377 + 0.981905i \(0.560647\pi\)
\(74\) −0.145898 −0.0169603
\(75\) 1.85410 0.214093
\(76\) −9.47214 −1.08653
\(77\) 0 0
\(78\) −1.09017 −0.123437
\(79\) 9.47214 1.06570 0.532849 − 0.846210i \(-0.321121\pi\)
0.532849 + 0.846210i \(0.321121\pi\)
\(80\) −4.85410 −0.542705
\(81\) 1.00000 0.111111
\(82\) 7.38197 0.815202
\(83\) 0.708204 0.0777355 0.0388677 − 0.999244i \(-0.487625\pi\)
0.0388677 + 0.999244i \(0.487625\pi\)
\(84\) −4.85410 −0.529626
\(85\) −4.23607 −0.459466
\(86\) −3.85410 −0.415599
\(87\) 4.47214 0.479463
\(88\) 0 0
\(89\) 0.527864 0.0559535 0.0279767 − 0.999609i \(-0.491094\pi\)
0.0279767 + 0.999609i \(0.491094\pi\)
\(90\) 1.61803 0.170556
\(91\) 5.29180 0.554731
\(92\) −5.61803 −0.585721
\(93\) 2.85410 0.295957
\(94\) −1.00000 −0.103142
\(95\) −15.3262 −1.57244
\(96\) −5.61803 −0.573388
\(97\) −14.0344 −1.42498 −0.712491 − 0.701681i \(-0.752433\pi\)
−0.712491 + 0.701681i \(0.752433\pi\)
\(98\) −1.23607 −0.124862
\(99\) 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 363.2.a.h.1.1 2
3.2 odd 2 1089.2.a.m.1.2 2
4.3 odd 2 5808.2.a.bl.1.1 2
5.4 even 2 9075.2.a.x.1.2 2
11.2 odd 10 363.2.e.c.202.1 4
11.3 even 5 33.2.e.a.31.1 yes 4
11.4 even 5 33.2.e.a.16.1 ✓ 4
11.5 even 5 363.2.e.h.124.1 4
11.6 odd 10 363.2.e.c.124.1 4
11.7 odd 10 363.2.e.j.148.1 4
11.8 odd 10 363.2.e.j.130.1 4
11.9 even 5 363.2.e.h.202.1 4
11.10 odd 2 363.2.a.e.1.2 2
33.14 odd 10 99.2.f.b.64.1 4
33.26 odd 10 99.2.f.b.82.1 4
33.32 even 2 1089.2.a.s.1.1 2
44.3 odd 10 528.2.y.f.97.1 4
44.15 odd 10 528.2.y.f.49.1 4
44.43 even 2 5808.2.a.bm.1.1 2
55.3 odd 20 825.2.bx.b.724.2 8
55.4 even 10 825.2.n.f.676.1 4
55.14 even 10 825.2.n.f.526.1 4
55.37 odd 20 825.2.bx.b.49.2 8
55.47 odd 20 825.2.bx.b.724.1 8
55.48 odd 20 825.2.bx.b.49.1 8
55.54 odd 2 9075.2.a.bv.1.1 2
99.4 even 15 891.2.n.d.379.1 8
99.14 odd 30 891.2.n.a.460.1 8
99.25 even 15 891.2.n.d.757.1 8
99.47 odd 30 891.2.n.a.757.1 8
99.58 even 15 891.2.n.d.460.1 8
99.59 odd 30 891.2.n.a.379.1 8
99.70 even 15 891.2.n.d.676.1 8
99.92 odd 30 891.2.n.a.676.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.2.e.a.16.1 ✓ 4 11.4 even 5
33.2.e.a.31.1 yes 4 11.3 even 5
99.2.f.b.64.1 4 33.14 odd 10
99.2.f.b.82.1 4 33.26 odd 10
363.2.a.e.1.2 2 11.10 odd 2
363.2.a.h.1.1 2 1.1 even 1 trivial
363.2.e.c.124.1 4 11.6 odd 10
363.2.e.c.202.1 4 11.2 odd 10
363.2.e.h.124.1 4 11.5 even 5
363.2.e.h.202.1 4 11.9 even 5
363.2.e.j.130.1 4 11.8 odd 10
363.2.e.j.148.1 4 11.7 odd 10
528.2.y.f.49.1 4 44.15 odd 10
528.2.y.f.97.1 4 44.3 odd 10
825.2.n.f.526.1 4 55.14 even 10
825.2.n.f.676.1 4 55.4 even 10
825.2.bx.b.49.1 8 55.48 odd 20
825.2.bx.b.49.2 8 55.37 odd 20
825.2.bx.b.724.1 8 55.47 odd 20
825.2.bx.b.724.2 8 55.3 odd 20
891.2.n.a.379.1 8 99.59 odd 30
891.2.n.a.460.1 8 99.14 odd 30
891.2.n.a.676.1 8 99.92 odd 30
891.2.n.a.757.1 8 99.47 odd 30
891.2.n.d.379.1 8 99.4 even 15
891.2.n.d.460.1 8 99.58 even 15
891.2.n.d.676.1 8 99.70 even 15
891.2.n.d.757.1 8 99.25 even 15
1089.2.a.m.1.2 2 3.2 odd 2
1089.2.a.s.1.1 2 33.32 even 2
5808.2.a.bl.1.1 2 4.3 odd 2
5808.2.a.bm.1.1 2 44.43 even 2
9075.2.a.x.1.2 2 5.4 even 2
9075.2.a.bv.1.1 2 55.54 odd 2