↑

Properties

Label 8954.2.a.p.1.2
Level $8954$
Weight $2$
Character 8954.1
Self dual yes
Analytic conductor $71.498$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8954,2,Mod(1,8954)] 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("8954.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8954, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8954 = 2 \cdot 11^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8954.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,3,2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.4980499699\)
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 \(\chi\) \(=\) 8954.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.30278 q^{3} +1.00000 q^{4} -2.30278 q^{5} +3.30278 q^{6} +2.60555 q^{7} +1.00000 q^{8} +7.90833 q^{9} -2.30278 q^{10} +3.30278 q^{12} -1.30278 q^{13} +2.60555 q^{14} -7.60555 q^{15} +1.00000 q^{16} +6.00000 q^{17} +7.90833 q^{18} -2.00000 q^{19} -2.30278 q^{20} +8.60555 q^{21} +3.90833 q^{23} +3.30278 q^{24} +0.302776 q^{25} -1.30278 q^{26} +16.2111 q^{27} +2.60555 q^{28} +3.90833 q^{29} -7.60555 q^{30} -0.302776 q^{31} +1.00000 q^{32} +6.00000 q^{34} -6.00000 q^{35} +7.90833 q^{36} +1.00000 q^{37} -2.00000 q^{38} -4.30278 q^{39} -2.30278 q^{40} -9.90833 q^{41} +8.60555 q^{42} -0.605551 q^{43} -18.2111 q^{45} +3.90833 q^{46} +4.60555 q^{47} +3.30278 q^{48} -0.211103 q^{49} +0.302776 q^{50} +19.8167 q^{51} -1.30278 q^{52} -6.00000 q^{53} +16.2111 q^{54} +2.60555 q^{56} -6.60555 q^{57} +3.90833 q^{58} +10.6056 q^{59} -7.60555 q^{60} -7.51388 q^{61} -0.302776 q^{62} +20.6056 q^{63} +1.00000 q^{64} +3.00000 q^{65} -3.51388 q^{67} +6.00000 q^{68} +12.9083 q^{69} -6.00000 q^{70} +6.00000 q^{71} +7.90833 q^{72} +12.3028 q^{73} +1.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} -4.30278 q^{78} -9.11943 q^{79} -2.30278 q^{80} +29.8167 q^{81} -9.90833 q^{82} -2.78890 q^{83} +8.60555 q^{84} -13.8167 q^{85} -0.605551 q^{86} +12.9083 q^{87} -9.21110 q^{89} -18.2111 q^{90} -3.39445 q^{91} +3.90833 q^{92} -1.00000 q^{93} +4.60555 q^{94} +4.60555 q^{95} +3.30278 q^{96} -16.4222 q^{97} -0.211103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - 2 q^{7} + 2 q^{8} + 5 q^{9} - q^{10} + 3 q^{12} + q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} + 12 q^{17} + 5 q^{18} - 4 q^{19} - q^{20} + 10 q^{21}+ \cdots + 14 q^{98}+O(q^{100}) \) 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
\(3\) 3.30278 1.90686 0.953429 − 0.301617i \(-0.0975264\pi\)
0.953429 + 0.301617i \(0.0975264\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.30278 −1.02983 −0.514916 − 0.857240i \(-0.672177\pi\)
−0.514916 + 0.857240i \(0.672177\pi\)
\(6\) 3.30278 1.34835
\(7\) 2.60555 0.984806 0.492403 − 0.870367i \(-0.336119\pi\)
0.492403 + 0.870367i \(0.336119\pi\)
\(8\) 1.00000 0.353553
\(9\) 7.90833 2.63611
\(10\) −2.30278 −0.728202
\(11\) 0 0
\(12\) 3.30278 0.953429
\(13\) −1.30278 −0.361325 −0.180662 − 0.983545i \(-0.557824\pi\)
−0.180662 + 0.983545i \(0.557824\pi\)
\(14\) 2.60555 0.696363
\(15\) −7.60555 −1.96374
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 7.90833 1.86401
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −2.30278 −0.514916
\(21\) 8.60555 1.87789
\(22\) 0 0
\(23\) 3.90833 0.814942 0.407471 − 0.913218i \(-0.366411\pi\)
0.407471 + 0.913218i \(0.366411\pi\)
\(24\) 3.30278 0.674176
\(25\) 0.302776 0.0605551
\(26\) −1.30278 −0.255495
\(27\) 16.2111 3.11983
\(28\) 2.60555 0.492403
\(29\) 3.90833 0.725758 0.362879 − 0.931836i \(-0.381794\pi\)
0.362879 + 0.931836i \(0.381794\pi\)
\(30\) −7.60555 −1.38858
\(31\) −0.302776 −0.0543801 −0.0271901 − 0.999630i \(-0.508656\pi\)
−0.0271901 + 0.999630i \(0.508656\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −6.00000 −1.01419
\(36\) 7.90833 1.31805
\(37\) 1.00000 0.164399
\(38\) −2.00000 −0.324443
\(39\) −4.30278 −0.688996
\(40\) −2.30278 −0.364101
\(41\) −9.90833 −1.54742 −0.773710 − 0.633540i \(-0.781601\pi\)
−0.773710 + 0.633540i \(0.781601\pi\)
\(42\) 8.60555 1.32787
\(43\) −0.605551 −0.0923457 −0.0461729 − 0.998933i \(-0.514703\pi\)
−0.0461729 + 0.998933i \(0.514703\pi\)
\(44\) 0 0
\(45\) −18.2111 −2.71475
\(46\) 3.90833 0.576251
\(47\) 4.60555 0.671789 0.335894 − 0.941900i \(-0.390961\pi\)
0.335894 + 0.941900i \(0.390961\pi\)
\(48\) 3.30278 0.476715
\(49\) −0.211103 −0.0301575
\(50\) 0.302776 0.0428189
\(51\) 19.8167 2.77489
\(52\) −1.30278 −0.180662
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 16.2111 2.20605
\(55\) 0 0
\(56\) 2.60555 0.348181
\(57\) −6.60555 −0.874927
\(58\) 3.90833 0.513188
\(59\) 10.6056 1.38073 0.690363 − 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) −7.60555 −0.981872
\(61\) −7.51388 −0.962054 −0.481027 − 0.876706i \(-0.659736\pi\)
−0.481027 + 0.876706i \(0.659736\pi\)
\(62\) −0.302776 −0.0384525
\(63\) 20.6056 2.59606
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −3.51388 −0.429289 −0.214644 − 0.976692i \(-0.568859\pi\)
−0.214644 + 0.976692i \(0.568859\pi\)
\(68\) 6.00000 0.727607
\(69\) 12.9083 1.55398
\(70\) −6.00000 −0.717137
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 7.90833 0.932005
\(73\) 12.3028 1.43993 0.719965 − 0.694010i \(-0.244158\pi\)
0.719965 + 0.694010i \(0.244158\pi\)
\(74\) 1.00000 0.116248
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −4.30278 −0.487193
\(79\) −9.11943 −1.02602 −0.513008 − 0.858384i \(-0.671469\pi\)
−0.513008 + 0.858384i \(0.671469\pi\)
\(80\) −2.30278 −0.257458
\(81\) 29.8167 3.31296
\(82\) −9.90833 −1.09419
\(83\) −2.78890 −0.306121 −0.153061 − 0.988217i \(-0.548913\pi\)
−0.153061 + 0.988217i \(0.548913\pi\)
\(84\) 8.60555 0.938943
\(85\) −13.8167 −1.49863
\(86\) −0.605551 −0.0652983
\(87\) 12.9083 1.38392
\(88\) 0 0
\(89\) −9.21110 −0.976375 −0.488187 − 0.872739i \(-0.662342\pi\)
−0.488187 + 0.872739i \(0.662342\pi\)
\(90\) −18.2111 −1.91962
\(91\) −3.39445 −0.355835
\(92\) 3.90833 0.407471
\(93\) −1.00000 −0.103695
\(94\) 4.60555 0.475026
\(95\) 4.60555 0.472520
\(96\) 3.30278 0.337088
\(97\) −16.4222 −1.66742 −0.833711 − 0.552201i \(-0.813788\pi\)
−0.833711 + 0.552201i \(0.813788\pi\)
\(98\) −0.211103 −0.0213246
\(99\) 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 8954.2.a.p.1.2 2
11.10 odd 2 74.2.a.a.1.2 ✓ 2
33.32 even 2 666.2.a.j.1.2 2
44.43 even 2 592.2.a.f.1.1 2
55.32 even 4 1850.2.b.i.149.1 4
55.43 even 4 1850.2.b.i.149.4 4
55.54 odd 2 1850.2.a.u.1.1 2
77.76 even 2 3626.2.a.a.1.1 2
88.21 odd 2 2368.2.a.s.1.1 2
88.43 even 2 2368.2.a.ba.1.2 2
132.131 odd 2 5328.2.a.bf.1.2 2
407.406 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 11.10 odd 2
592.2.a.f.1.1 2 44.43 even 2
666.2.a.j.1.2 2 33.32 even 2
1850.2.a.u.1.1 2 55.54 odd 2
1850.2.b.i.149.1 4 55.32 even 4
1850.2.b.i.149.4 4 55.43 even 4
2368.2.a.s.1.1 2 88.21 odd 2
2368.2.a.ba.1.2 2 88.43 even 2
2738.2.a.l.1.2 2 407.406 odd 2
3626.2.a.a.1.1 2 77.76 even 2
5328.2.a.bf.1.2 2 132.131 odd 2
8954.2.a.p.1.2 2 1.1 even 1 trivial