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Properties

Label 1369.2.b.c.1368.1
Level $1369$
Weight $2$
Character 1369.1368
Analytic conductor $10.932$
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] = [1369,2,Mod(1368,1369)] 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("1369.1368"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1369, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1369 = 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1369.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,6] 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: no
Analytic conductor: \(10.9315200367\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1368.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1369.1368
Dual form 1369.2.b.c.1368.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} +3.00000 q^{3} -2.00000 q^{4} +2.00000i q^{5} -6.00000i q^{6} -1.00000 q^{7} +6.00000 q^{9} +4.00000 q^{10} +5.00000 q^{11} -6.00000 q^{12} +2.00000i q^{13} +2.00000i q^{14} +6.00000i q^{15} -4.00000 q^{16} -12.0000i q^{18} -4.00000i q^{20} -3.00000 q^{21} -10.0000i q^{22} -2.00000i q^{23} +1.00000 q^{25} +4.00000 q^{26} +9.00000 q^{27} +2.00000 q^{28} +6.00000i q^{29} +12.0000 q^{30} -4.00000i q^{31} +8.00000i q^{32} +15.0000 q^{33} -2.00000i q^{35} -12.0000 q^{36} +6.00000i q^{39} +9.00000 q^{41} +6.00000i q^{42} -2.00000i q^{43} -10.0000 q^{44} +12.0000i q^{45} -4.00000 q^{46} -9.00000 q^{47} -12.0000 q^{48} -6.00000 q^{49} -2.00000i q^{50} -4.00000i q^{52} +1.00000 q^{53} -18.0000i q^{54} +10.0000i q^{55} +12.0000 q^{58} -8.00000i q^{59} -12.0000i q^{60} -8.00000i q^{61} -8.00000 q^{62} -6.00000 q^{63} +8.00000 q^{64} -4.00000 q^{65} -30.0000i q^{66} -8.00000 q^{67} -6.00000i q^{69} -4.00000 q^{70} +9.00000 q^{71} +1.00000 q^{73} +3.00000 q^{75} -5.00000 q^{77} +12.0000 q^{78} -4.00000i q^{79} -8.00000i q^{80} +9.00000 q^{81} -18.0000i q^{82} -15.0000 q^{83} +6.00000 q^{84} -4.00000 q^{86} +18.0000i q^{87} +4.00000i q^{89} +24.0000 q^{90} -2.00000i q^{91} +4.00000i q^{92} -12.0000i q^{93} +18.0000i q^{94} +24.0000i q^{96} -4.00000i q^{97} +12.0000i q^{98} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} - 4 q^{4} - 2 q^{7} + 12 q^{9} + 8 q^{10} + 10 q^{11} - 12 q^{12} - 8 q^{16} - 6 q^{21} + 2 q^{25} + 8 q^{26} + 18 q^{27} + 4 q^{28} + 24 q^{30} + 30 q^{33} - 24 q^{36} + 18 q^{41} - 20 q^{44}+ \cdots + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 1.41421i −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 − 0.707107i \(-0.250000\pi\)
\(3\) 3.00000 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −2.00000 −1.00000
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) − 6.00000i − 2.44949i
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 4.00000 1.26491
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) −6.00000 −1.73205
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 6.00000i 1.54919i
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 12.0000i − 2.82843i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) − 4.00000i − 0.894427i
\(21\) −3.00000 −0.654654
\(22\) − 10.0000i − 2.13201i
\(23\) − 2.00000i − 0.417029i −0.978019 − 0.208514i \(-0.933137\pi\)
0.978019 − 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 9.00000 1.73205
\(28\) 2.00000 0.377964
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 12.0000 2.19089
\(31\) − 4.00000i − 0.718421i −0.933257 − 0.359211i \(-0.883046\pi\)
0.933257 − 0.359211i \(-0.116954\pi\)
\(32\) 8.00000i 1.41421i
\(33\) 15.0000 2.61116
\(34\) 0 0
\(35\) − 2.00000i − 0.338062i
\(36\) −12.0000 −2.00000
\(37\) 0 0
\(38\) 0 0
\(39\) 6.00000i 0.960769i
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 6.00000i 0.925820i
\(43\) − 2.00000i − 0.304997i −0.988304 − 0.152499i \(-0.951268\pi\)
0.988304 − 0.152499i \(-0.0487319\pi\)
\(44\) −10.0000 −1.50756
\(45\) 12.0000i 1.78885i
\(46\) −4.00000 −0.589768
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −12.0000 −1.73205
\(49\) −6.00000 −0.857143
\(50\) − 2.00000i − 0.282843i
\(51\) 0 0
\(52\) − 4.00000i − 0.554700i
\(53\) 1.00000 0.137361 0.0686803 − 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) − 18.0000i − 2.44949i
\(55\) 10.0000i 1.34840i
\(56\) 0 0
\(57\) 0 0
\(58\) 12.0000 1.57568
\(59\) − 8.00000i − 1.04151i −0.853706 − 0.520756i \(-0.825650\pi\)
0.853706 − 0.520756i \(-0.174350\pi\)
\(60\) − 12.0000i − 1.54919i
\(61\) − 8.00000i − 1.02430i −0.858898 − 0.512148i \(-0.828850\pi\)
0.858898 − 0.512148i \(-0.171150\pi\)
\(62\) −8.00000 −1.01600
\(63\) −6.00000 −0.755929
\(64\) 8.00000 1.00000
\(65\) −4.00000 −0.496139
\(66\) − 30.0000i − 3.69274i
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) − 6.00000i − 0.722315i
\(70\) −4.00000 −0.478091
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) 12.0000 1.35873
\(79\) − 4.00000i − 0.450035i −0.974355 − 0.225018i \(-0.927756\pi\)
0.974355 − 0.225018i \(-0.0722440\pi\)
\(80\) − 8.00000i − 0.894427i
\(81\) 9.00000 1.00000
\(82\) − 18.0000i − 1.98777i
\(83\) −15.0000 −1.64646 −0.823232 − 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 6.00000 0.654654
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 18.0000i 1.92980i
\(88\) 0 0
\(89\) 4.00000i 0.423999i 0.977270 + 0.212000i \(0.0679975\pi\)
−0.977270 + 0.212000i \(0.932002\pi\)
\(90\) 24.0000 2.52982
\(91\) − 2.00000i − 0.209657i
\(92\) 4.00000i 0.417029i
\(93\) − 12.0000i − 1.24434i
\(94\) 18.0000i 1.85656i
\(95\) 0 0
\(96\) 24.0000i 2.44949i
\(97\) − 4.00000i − 0.406138i −0.979164 − 0.203069i \(-0.934908\pi\)
0.979164 − 0.203069i \(-0.0650917\pi\)
\(98\) 12.0000i 1.21218i
\(99\) 30.0000 3.01511
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 1369.2.b.c.1368.1 2
37.6 odd 4 37.2.a.a.1.1 ✓ 1
37.31 odd 4 1369.2.a.e.1.1 1
37.36 even 2 inner 1369.2.b.c.1368.2 2
111.80 even 4 333.2.a.d.1.1 1
148.43 even 4 592.2.a.e.1.1 1
185.43 even 4 925.2.b.b.149.2 2
185.117 even 4 925.2.b.b.149.1 2
185.154 odd 4 925.2.a.e.1.1 1
259.6 even 4 1813.2.a.a.1.1 1
296.43 even 4 2368.2.a.b.1.1 1
296.117 odd 4 2368.2.a.q.1.1 1
407.43 even 4 4477.2.a.b.1.1 1
444.191 odd 4 5328.2.a.r.1.1 1
481.376 odd 4 6253.2.a.c.1.1 1
555.524 even 4 8325.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 ✓ 1 37.6 odd 4
333.2.a.d.1.1 1 111.80 even 4
592.2.a.e.1.1 1 148.43 even 4
925.2.a.e.1.1 1 185.154 odd 4
925.2.b.b.149.1 2 185.117 even 4
925.2.b.b.149.2 2 185.43 even 4
1369.2.a.e.1.1 1 37.31 odd 4
1369.2.b.c.1368.1 2 1.1 even 1 trivial
1369.2.b.c.1368.2 2 37.36 even 2 inner
1813.2.a.a.1.1 1 259.6 even 4
2368.2.a.b.1.1 1 296.43 even 4
2368.2.a.q.1.1 1 296.117 odd 4
4477.2.a.b.1.1 1 407.43 even 4
5328.2.a.r.1.1 1 444.191 odd 4
6253.2.a.c.1.1 1 481.376 odd 4
8325.2.a.e.1.1 1 555.524 even 4