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Properties

Label 4800.2.f.p.3649.1
Level $4800$
Weight $2$
Character 4800.3649
Analytic conductor $38.328$
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] = [4800,2,Mod(3649,4800)] 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("4800.3649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,-8,0,8,0,0,0,0,0,0,0,-12, 0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.3281929702\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4800.3649
Dual form 4800.2.f.p.3649.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -2.00000i q^{13} -6.00000i q^{17} -4.00000 q^{19} +4.00000 q^{21} +1.00000i q^{27} -6.00000 q^{29} +8.00000 q^{31} +2.00000i q^{37} -2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} -9.00000 q^{49} -6.00000 q^{51} +6.00000i q^{53} +4.00000i q^{57} +10.0000 q^{61} -4.00000i q^{63} -4.00000i q^{67} +2.00000i q^{73} -8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +6.00000i q^{87} -18.0000 q^{89} +8.00000 q^{91} -8.00000i q^{93} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} - 8 q^{19} + 8 q^{21} - 12 q^{29} + 16 q^{31} - 4 q^{39} - 12 q^{41} - 18 q^{49} - 12 q^{51} + 20 q^{61} - 16 q^{79} + 2 q^{81} - 36 q^{89} + 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 − 0.727607i \(-0.740633\pi\)
0.685994 − 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 − 0.658586i \(-0.771155\pi\)
0.752506 − 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000i 0.643268i
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 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 4800.2.f.p.3649.1 2
4.3 odd 2 4800.2.f.w.3649.2 2
5.2 odd 4 960.2.a.e.1.1 1
5.3 odd 4 4800.2.a.cq.1.1 1
5.4 even 2 inner 4800.2.f.p.3649.2 2
8.3 odd 2 1200.2.f.e.49.1 2
8.5 even 2 150.2.c.a.49.2 2
15.2 even 4 2880.2.a.a.1.1 1
20.3 even 4 4800.2.a.d.1.1 1
20.7 even 4 960.2.a.p.1.1 1
20.19 odd 2 4800.2.f.w.3649.1 2
24.5 odd 2 450.2.c.b.199.1 2
24.11 even 2 3600.2.f.i.2449.1 2
40.3 even 4 1200.2.a.k.1.1 1
40.13 odd 4 150.2.a.b.1.1 1
40.19 odd 2 1200.2.f.e.49.2 2
40.27 even 4 240.2.a.b.1.1 1
40.29 even 2 150.2.c.a.49.1 2
40.37 odd 4 30.2.a.a.1.1 ✓ 1
60.47 odd 4 2880.2.a.q.1.1 1
80.27 even 4 3840.2.k.f.1921.1 2
80.37 odd 4 3840.2.k.y.1921.2 2
80.67 even 4 3840.2.k.f.1921.2 2
80.77 odd 4 3840.2.k.y.1921.1 2
120.29 odd 2 450.2.c.b.199.2 2
120.53 even 4 450.2.a.d.1.1 1
120.59 even 2 3600.2.f.i.2449.2 2
120.77 even 4 90.2.a.c.1.1 1
120.83 odd 4 3600.2.a.f.1.1 1
120.107 odd 4 720.2.a.j.1.1 1
280.13 even 4 7350.2.a.ct.1.1 1
280.37 odd 12 1470.2.i.o.361.1 2
280.117 even 12 1470.2.i.q.361.1 2
280.157 even 12 1470.2.i.q.961.1 2
280.237 even 4 1470.2.a.d.1.1 1
280.277 odd 12 1470.2.i.o.961.1 2
360.77 even 12 810.2.e.b.541.1 2
360.157 odd 12 810.2.e.l.541.1 2
360.277 odd 12 810.2.e.l.271.1 2
360.317 even 12 810.2.e.b.271.1 2
440.197 even 4 3630.2.a.w.1.1 1
520.77 odd 4 5070.2.a.w.1.1 1
520.317 even 4 5070.2.b.k.1351.2 2
520.437 even 4 5070.2.b.k.1351.1 2
680.237 odd 4 8670.2.a.g.1.1 1
840.797 odd 4 4410.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 ✓ 1 40.37 odd 4
90.2.a.c.1.1 1 120.77 even 4
150.2.a.b.1.1 1 40.13 odd 4
150.2.c.a.49.1 2 40.29 even 2
150.2.c.a.49.2 2 8.5 even 2
240.2.a.b.1.1 1 40.27 even 4
450.2.a.d.1.1 1 120.53 even 4
450.2.c.b.199.1 2 24.5 odd 2
450.2.c.b.199.2 2 120.29 odd 2
720.2.a.j.1.1 1 120.107 odd 4
810.2.e.b.271.1 2 360.317 even 12
810.2.e.b.541.1 2 360.77 even 12
810.2.e.l.271.1 2 360.277 odd 12
810.2.e.l.541.1 2 360.157 odd 12
960.2.a.e.1.1 1 5.2 odd 4
960.2.a.p.1.1 1 20.7 even 4
1200.2.a.k.1.1 1 40.3 even 4
1200.2.f.e.49.1 2 8.3 odd 2
1200.2.f.e.49.2 2 40.19 odd 2
1470.2.a.d.1.1 1 280.237 even 4
1470.2.i.o.361.1 2 280.37 odd 12
1470.2.i.o.961.1 2 280.277 odd 12
1470.2.i.q.361.1 2 280.117 even 12
1470.2.i.q.961.1 2 280.157 even 12
2880.2.a.a.1.1 1 15.2 even 4
2880.2.a.q.1.1 1 60.47 odd 4
3600.2.a.f.1.1 1 120.83 odd 4
3600.2.f.i.2449.1 2 24.11 even 2
3600.2.f.i.2449.2 2 120.59 even 2
3630.2.a.w.1.1 1 440.197 even 4
3840.2.k.f.1921.1 2 80.27 even 4
3840.2.k.f.1921.2 2 80.67 even 4
3840.2.k.y.1921.1 2 80.77 odd 4
3840.2.k.y.1921.2 2 80.37 odd 4
4410.2.a.z.1.1 1 840.797 odd 4
4800.2.a.d.1.1 1 20.3 even 4
4800.2.a.cq.1.1 1 5.3 odd 4
4800.2.f.p.3649.1 2 1.1 even 1 trivial
4800.2.f.p.3649.2 2 5.4 even 2 inner
4800.2.f.w.3649.1 2 20.19 odd 2
4800.2.f.w.3649.2 2 4.3 odd 2
5070.2.a.w.1.1 1 520.77 odd 4
5070.2.b.k.1351.1 2 520.437 even 4
5070.2.b.k.1351.2 2 520.317 even 4
7350.2.a.ct.1.1 1 280.13 even 4
8670.2.a.g.1.1 1 680.237 odd 4