Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3600,2,Mod(2449,3600)] 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("3600.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.7461447277\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3600.2449
Dual form 3600.2.f.i.2449.2

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 − 0.755929i \(-0.727186\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 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 \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 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 \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 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
\(51\) 0 0
\(52\) 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
\(57\) 0 0
\(58\) 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 \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 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 \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 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 3600.2.f.i.2449.1 2
3.2 odd 2 1200.2.f.e.49.1 2
4.3 odd 2 450.2.c.b.199.1 2
5.2 odd 4 720.2.a.j.1.1 1
5.3 odd 4 3600.2.a.f.1.1 1
5.4 even 2 inner 3600.2.f.i.2449.2 2
12.11 even 2 150.2.c.a.49.2 2
15.2 even 4 240.2.a.b.1.1 1
15.8 even 4 1200.2.a.k.1.1 1
15.14 odd 2 1200.2.f.e.49.2 2
20.3 even 4 450.2.a.d.1.1 1
20.7 even 4 90.2.a.c.1.1 1
20.19 odd 2 450.2.c.b.199.2 2
24.5 odd 2 4800.2.f.w.3649.2 2
24.11 even 2 4800.2.f.p.3649.1 2
40.27 even 4 2880.2.a.a.1.1 1
40.37 odd 4 2880.2.a.q.1.1 1
60.23 odd 4 150.2.a.b.1.1 1
60.47 odd 4 30.2.a.a.1.1 ✓ 1
60.59 even 2 150.2.c.a.49.1 2
120.29 odd 2 4800.2.f.w.3649.1 2
120.53 even 4 4800.2.a.d.1.1 1
120.59 even 2 4800.2.f.p.3649.2 2
120.77 even 4 960.2.a.p.1.1 1
120.83 odd 4 4800.2.a.cq.1.1 1
120.107 odd 4 960.2.a.e.1.1 1
140.27 odd 4 4410.2.a.z.1.1 1
180.7 even 12 810.2.e.b.271.1 2
180.47 odd 12 810.2.e.l.271.1 2
180.67 even 12 810.2.e.b.541.1 2
180.167 odd 12 810.2.e.l.541.1 2
240.77 even 4 3840.2.k.f.1921.1 2
240.107 odd 4 3840.2.k.y.1921.1 2
240.197 even 4 3840.2.k.f.1921.2 2
240.227 odd 4 3840.2.k.y.1921.2 2
420.47 even 12 1470.2.i.q.361.1 2
420.83 even 4 7350.2.a.ct.1.1 1
420.107 odd 12 1470.2.i.o.361.1 2
420.167 even 4 1470.2.a.d.1.1 1
420.227 even 12 1470.2.i.q.961.1 2
420.347 odd 12 1470.2.i.o.961.1 2
660.527 even 4 3630.2.a.w.1.1 1
780.47 even 4 5070.2.b.k.1351.1 2
780.467 odd 4 5070.2.a.w.1.1 1
780.707 even 4 5070.2.b.k.1351.2 2
1020.407 odd 4 8670.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 ✓ 1 60.47 odd 4
90.2.a.c.1.1 1 20.7 even 4
150.2.a.b.1.1 1 60.23 odd 4
150.2.c.a.49.1 2 60.59 even 2
150.2.c.a.49.2 2 12.11 even 2
240.2.a.b.1.1 1 15.2 even 4
450.2.a.d.1.1 1 20.3 even 4
450.2.c.b.199.1 2 4.3 odd 2
450.2.c.b.199.2 2 20.19 odd 2
720.2.a.j.1.1 1 5.2 odd 4
810.2.e.b.271.1 2 180.7 even 12
810.2.e.b.541.1 2 180.67 even 12
810.2.e.l.271.1 2 180.47 odd 12
810.2.e.l.541.1 2 180.167 odd 12
960.2.a.e.1.1 1 120.107 odd 4
960.2.a.p.1.1 1 120.77 even 4
1200.2.a.k.1.1 1 15.8 even 4
1200.2.f.e.49.1 2 3.2 odd 2
1200.2.f.e.49.2 2 15.14 odd 2
1470.2.a.d.1.1 1 420.167 even 4
1470.2.i.o.361.1 2 420.107 odd 12
1470.2.i.o.961.1 2 420.347 odd 12
1470.2.i.q.361.1 2 420.47 even 12
1470.2.i.q.961.1 2 420.227 even 12
2880.2.a.a.1.1 1 40.27 even 4
2880.2.a.q.1.1 1 40.37 odd 4
3600.2.a.f.1.1 1 5.3 odd 4
3600.2.f.i.2449.1 2 1.1 even 1 trivial
3600.2.f.i.2449.2 2 5.4 even 2 inner
3630.2.a.w.1.1 1 660.527 even 4
3840.2.k.f.1921.1 2 240.77 even 4
3840.2.k.f.1921.2 2 240.197 even 4
3840.2.k.y.1921.1 2 240.107 odd 4
3840.2.k.y.1921.2 2 240.227 odd 4
4410.2.a.z.1.1 1 140.27 odd 4
4800.2.a.d.1.1 1 120.53 even 4
4800.2.a.cq.1.1 1 120.83 odd 4
4800.2.f.p.3649.1 2 24.11 even 2
4800.2.f.p.3649.2 2 120.59 even 2
4800.2.f.w.3649.1 2 120.29 odd 2
4800.2.f.w.3649.2 2 24.5 odd 2
5070.2.a.w.1.1 1 780.467 odd 4
5070.2.b.k.1351.1 2 780.47 even 4
5070.2.b.k.1351.2 2 780.707 even 4
7350.2.a.ct.1.1 1 420.83 even 4
8670.2.a.g.1.1 1 1020.407 odd 4