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Properties

Label 2368.2.a.ba.1.2
Level $2368$
Weight $2$
Character 2368.1
Self dual yes
Analytic conductor $18.909$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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] = [2368,2,Mod(1,2368)] 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("2368.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,3,0,1,0,-2,0,5,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) 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 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 2368.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.30278 q^{3} +2.30278 q^{5} +2.60555 q^{7} +7.90833 q^{9} -2.30278 q^{11} -1.30278 q^{13} +7.60555 q^{15} -6.00000 q^{17} +2.00000 q^{19} +8.60555 q^{21} -3.90833 q^{23} +0.302776 q^{25} +16.2111 q^{27} +3.90833 q^{29} +0.302776 q^{31} -7.60555 q^{33} +6.00000 q^{35} -1.00000 q^{37} -4.30278 q^{39} +9.90833 q^{41} +0.605551 q^{43} +18.2111 q^{45} -4.60555 q^{47} -0.211103 q^{49} -19.8167 q^{51} +6.00000 q^{53} -5.30278 q^{55} +6.60555 q^{57} +10.6056 q^{59} -7.51388 q^{61} +20.6056 q^{63} -3.00000 q^{65} -3.51388 q^{67} -12.9083 q^{69} -6.00000 q^{71} -12.3028 q^{73} +1.00000 q^{75} -6.00000 q^{77} -9.11943 q^{79} +29.8167 q^{81} +2.78890 q^{83} -13.8167 q^{85} +12.9083 q^{87} -9.21110 q^{89} -3.39445 q^{91} +1.00000 q^{93} +4.60555 q^{95} -16.4222 q^{97} -18.2111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} + q^{5} - 2 q^{7} + 5 q^{9} - q^{11} + q^{13} + 8 q^{15} - 12 q^{17} + 4 q^{19} + 10 q^{21} + 3 q^{23} - 3 q^{25} + 18 q^{27} - 3 q^{29} - 3 q^{31} - 8 q^{33} + 12 q^{35} - 2 q^{37} - 5 q^{39}+ \cdots - 22 q^{99}+O(q^{100}) \) 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 0
\(3\) 3.30278 1.90686 0.953429 − 0.301617i \(-0.0975264\pi\)
0.953429 + 0.301617i \(0.0975264\pi\)
\(4\) 0 0
\(5\) 2.30278 1.02983 0.514916 − 0.857240i \(-0.327823\pi\)
0.514916 + 0.857240i \(0.327823\pi\)
\(6\) 0 0
\(7\) 2.60555 0.984806 0.492403 − 0.870367i \(-0.336119\pi\)
0.492403 + 0.870367i \(0.336119\pi\)
\(8\) 0 0
\(9\) 7.90833 2.63611
\(10\) 0 0
\(11\) −2.30278 −0.694313 −0.347156 − 0.937807i \(-0.612853\pi\)
−0.347156 + 0.937807i \(0.612853\pi\)
\(12\) 0 0
\(13\) −1.30278 −0.361325 −0.180662 − 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(14\) 0 0
\(15\) 7.60555 1.96374
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 8.60555 1.87789
\(22\) 0 0
\(23\) −3.90833 −0.814942 −0.407471 − 0.913218i \(-0.633589\pi\)
−0.407471 + 0.913218i \(0.633589\pi\)
\(24\) 0 0
\(25\) 0.302776 0.0605551
\(26\) 0 0
\(27\) 16.2111 3.11983
\(28\) 0 0
\(29\) 3.90833 0.725758 0.362879 − 0.931836i \(-0.381794\pi\)
0.362879 + 0.931836i \(0.381794\pi\)
\(30\) 0 0
\(31\) 0.302776 0.0543801 0.0271901 − 0.999630i \(-0.491344\pi\)
0.0271901 + 0.999630i \(0.491344\pi\)
\(32\) 0 0
\(33\) −7.60555 −1.32396
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −4.30278 −0.688996
\(40\) 0 0
\(41\) 9.90833 1.54742 0.773710 − 0.633540i \(-0.218399\pi\)
0.773710 + 0.633540i \(0.218399\pi\)
\(42\) 0 0
\(43\) 0.605551 0.0923457 0.0461729 − 0.998933i \(-0.485297\pi\)
0.0461729 + 0.998933i \(0.485297\pi\)
\(44\) 0 0
\(45\) 18.2111 2.71475
\(46\) 0 0
\(47\) −4.60555 −0.671789 −0.335894 − 0.941900i \(-0.609039\pi\)
−0.335894 + 0.941900i \(0.609039\pi\)
\(48\) 0 0
\(49\) −0.211103 −0.0301575
\(50\) 0 0
\(51\) −19.8167 −2.77489
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −5.30278 −0.715026
\(56\) 0 0
\(57\) 6.60555 0.874927
\(58\) 0 0
\(59\) 10.6056 1.38073 0.690363 − 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) 0 0
\(61\) −7.51388 −0.962054 −0.481027 − 0.876706i \(-0.659736\pi\)
−0.481027 + 0.876706i \(0.659736\pi\)
\(62\) 0 0
\(63\) 20.6056 2.59606
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −3.51388 −0.429289 −0.214644 − 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(68\) 0 0
\(69\) −12.9083 −1.55398
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −12.3028 −1.43993 −0.719965 − 0.694010i \(-0.755842\pi\)
−0.719965 + 0.694010i \(0.755842\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −9.11943 −1.02602 −0.513008 − 0.858384i \(-0.671469\pi\)
−0.513008 + 0.858384i \(0.671469\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) 0 0
\(83\) 2.78890 0.306121 0.153061 − 0.988217i \(-0.451087\pi\)
0.153061 + 0.988217i \(0.451087\pi\)
\(84\) 0 0
\(85\) −13.8167 −1.49863
\(86\) 0 0
\(87\) 12.9083 1.38392
\(88\) 0 0
\(89\) −9.21110 −0.976375 −0.488187 − 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(90\) 0 0
\(91\) −3.39445 −0.355835
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 4.60555 0.472520
\(96\) 0 0
\(97\) −16.4222 −1.66742 −0.833711 − 0.552201i \(-0.813788\pi\)
−0.833711 + 0.552201i \(0.813788\pi\)
\(98\) 0 0
\(99\) −18.2111 −1.83028
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 2368.2.a.ba.1.2 2
4.3 odd 2 2368.2.a.s.1.1 2
8.3 odd 2 74.2.a.a.1.2 ✓ 2
8.5 even 2 592.2.a.f.1.1 2
24.5 odd 2 5328.2.a.bf.1.2 2
24.11 even 2 666.2.a.j.1.2 2
40.3 even 4 1850.2.b.i.149.4 4
40.19 odd 2 1850.2.a.u.1.1 2
40.27 even 4 1850.2.b.i.149.1 4
56.27 even 2 3626.2.a.a.1.1 2
88.43 even 2 8954.2.a.p.1.2 2
296.147 odd 2 2738.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.2 ✓ 2 8.3 odd 2
592.2.a.f.1.1 2 8.5 even 2
666.2.a.j.1.2 2 24.11 even 2
1850.2.a.u.1.1 2 40.19 odd 2
1850.2.b.i.149.1 4 40.27 even 4
1850.2.b.i.149.4 4 40.3 even 4
2368.2.a.s.1.1 2 4.3 odd 2
2368.2.a.ba.1.2 2 1.1 even 1 trivial
2738.2.a.l.1.2 2 296.147 odd 2
3626.2.a.a.1.1 2 56.27 even 2
5328.2.a.bf.1.2 2 24.5 odd 2
8954.2.a.p.1.2 2 88.43 even 2