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Properties

Label 4050.2.c.t.649.1
Level $4050$
Weight $2$
Character 4050.649
Analytic conductor $32.339$
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] = [4050,2,Mod(649,4050)] 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("4050.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4050.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2,0,0,0,0,0,0,12,0,0,2,0,2,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.3394128186\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4050.649
Dual form 4050.2.c.t.649.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +6.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{19} -6.00000i q^{22} +9.00000i q^{23} +2.00000 q^{26} -1.00000i q^{28} -3.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -8.00000i q^{37} -4.00000i q^{38} -3.00000 q^{41} +8.00000i q^{43} -6.00000 q^{44} +9.00000 q^{46} +3.00000i q^{47} +6.00000 q^{49} -2.00000i q^{52} +6.00000i q^{53} -1.00000 q^{56} +3.00000i q^{58} -6.00000 q^{59} -13.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +13.0000i q^{67} -6.00000 q^{71} -4.00000i q^{73} -8.00000 q^{74} -4.00000 q^{76} +6.00000i q^{77} +10.0000 q^{79} +3.00000i q^{82} -9.00000i q^{83} +8.00000 q^{86} +6.00000i q^{88} -9.00000 q^{89} -2.00000 q^{91} -9.00000i q^{92} +3.00000 q^{94} -2.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 12 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{19} + 4 q^{26} - 6 q^{29} - 8 q^{31} - 6 q^{41} - 12 q^{44} + 18 q^{46} + 12 q^{49} - 2 q^{56} - 12 q^{59} - 26 q^{61} - 2 q^{64} - 12 q^{71} - 16 q^{74}+ \cdots + 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times\).

\(n\) \(2351\) \(3727\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 6.00000i − 1.27920i
\(23\) 9.00000i 1.87663i 0.345782 + 0.938315i \(0.387614\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) − 1.00000i − 0.188982i
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 3.00000i 0.393919i
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 − 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000i 1.58820i 0.607785 + 0.794101i \(0.292058\pi\)
−0.607785 + 0.794101i \(0.707942\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 6.00000i 0.683763i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.00000i 0.331295i
\(83\) − 9.00000i − 0.987878i −0.869496 − 0.493939i \(-0.835557\pi\)
0.869496 − 0.493939i \(-0.164443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) 6.00000i 0.639602i
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) − 9.00000i − 0.938315i
\(93\) 0 0
\(94\) 3.00000 0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) 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 4050.2.c.t.649.1 2
3.2 odd 2 4050.2.c.a.649.2 2
5.2 odd 4 810.2.a.g.1.1 1
5.3 odd 4 4050.2.a.n.1.1 1
5.4 even 2 inner 4050.2.c.t.649.2 2
9.2 odd 6 1350.2.j.e.199.2 4
9.4 even 3 450.2.j.c.349.2 4
9.5 odd 6 1350.2.j.e.1099.1 4
9.7 even 3 450.2.j.c.49.1 4
15.2 even 4 810.2.a.b.1.1 1
15.8 even 4 4050.2.a.ba.1.1 1
15.14 odd 2 4050.2.c.a.649.1 2
20.7 even 4 6480.2.a.g.1.1 1
45.2 even 12 270.2.e.b.91.1 2
45.4 even 6 450.2.j.c.349.1 4
45.7 odd 12 90.2.e.a.31.1 ✓ 2
45.13 odd 12 450.2.e.e.151.1 2
45.14 odd 6 1350.2.j.e.1099.2 4
45.22 odd 12 90.2.e.a.61.1 yes 2
45.23 even 12 1350.2.e.b.451.1 2
45.29 odd 6 1350.2.j.e.199.1 4
45.32 even 12 270.2.e.b.181.1 2
45.34 even 6 450.2.j.c.49.2 4
45.38 even 12 1350.2.e.b.901.1 2
45.43 odd 12 450.2.e.e.301.1 2
60.47 odd 4 6480.2.a.v.1.1 1
180.7 even 12 720.2.q.b.481.1 2
180.47 odd 12 2160.2.q.b.1441.1 2
180.67 even 12 720.2.q.b.241.1 2
180.167 odd 12 2160.2.q.b.721.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.e.a.31.1 ✓ 2 45.7 odd 12
90.2.e.a.61.1 yes 2 45.22 odd 12
270.2.e.b.91.1 2 45.2 even 12
270.2.e.b.181.1 2 45.32 even 12
450.2.e.e.151.1 2 45.13 odd 12
450.2.e.e.301.1 2 45.43 odd 12
450.2.j.c.49.1 4 9.7 even 3
450.2.j.c.49.2 4 45.34 even 6
450.2.j.c.349.1 4 45.4 even 6
450.2.j.c.349.2 4 9.4 even 3
720.2.q.b.241.1 2 180.67 even 12
720.2.q.b.481.1 2 180.7 even 12
810.2.a.b.1.1 1 15.2 even 4
810.2.a.g.1.1 1 5.2 odd 4
1350.2.e.b.451.1 2 45.23 even 12
1350.2.e.b.901.1 2 45.38 even 12
1350.2.j.e.199.1 4 45.29 odd 6
1350.2.j.e.199.2 4 9.2 odd 6
1350.2.j.e.1099.1 4 9.5 odd 6
1350.2.j.e.1099.2 4 45.14 odd 6
2160.2.q.b.721.1 2 180.167 odd 12
2160.2.q.b.1441.1 2 180.47 odd 12
4050.2.a.n.1.1 1 5.3 odd 4
4050.2.a.ba.1.1 1 15.8 even 4
4050.2.c.a.649.1 2 15.14 odd 2
4050.2.c.a.649.2 2 3.2 odd 2
4050.2.c.t.649.1 2 1.1 even 1 trivial
4050.2.c.t.649.2 2 5.4 even 2 inner
6480.2.a.g.1.1 1 20.7 even 4
6480.2.a.v.1.1 1 60.47 odd 4