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Properties

Label 1800.4.a.w.1.1
Level $1800$
Weight $4$
Character 1800.1
Self dual yes
Analytic conductor $106.203$
Analytic rank $1$
Dimension $1$
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] = [1800,4,Mod(1,1800)] 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("1800.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,6,0,0,0,19,0,-12,0,0,0,-75] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(106.203438010\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1800.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.00000 q^{7} +19.0000 q^{11} -12.0000 q^{13} -75.0000 q^{17} -91.0000 q^{19} +174.000 q^{23} +272.000 q^{29} -230.000 q^{31} +182.000 q^{37} -117.000 q^{41} -372.000 q^{43} -52.0000 q^{47} -307.000 q^{49} -402.000 q^{53} -312.000 q^{59} +170.000 q^{61} -763.000 q^{67} +52.0000 q^{71} +981.000 q^{73} +114.000 q^{77} +1054.00 q^{79} +351.000 q^{83} -799.000 q^{89} -72.0000 q^{91} -962.000 q^{97} +O(q^{100})\)

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
\(7\) 6.00000 0.323970 0.161985 − 0.986793i \(-0.448210\pi\)
0.161985 + 0.986793i \(0.448210\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 19.0000 0.520792 0.260396 − 0.965502i \(-0.416147\pi\)
0.260396 + 0.965502i \(0.416147\pi\)
\(12\) 0 0
\(13\) −12.0000 −0.256015 −0.128008 − 0.991773i \(-0.540858\pi\)
−0.128008 + 0.991773i \(0.540858\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −75.0000 −1.07001 −0.535005 − 0.844849i \(-0.679690\pi\)
−0.535005 + 0.844849i \(0.679690\pi\)
\(18\) 0 0
\(19\) −91.0000 −1.09878 −0.549390 − 0.835566i \(-0.685140\pi\)
−0.549390 + 0.835566i \(0.685140\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 174.000 1.57746 0.788728 − 0.614742i \(-0.210740\pi\)
0.788728 + 0.614742i \(0.210740\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 272.000 1.74169 0.870847 − 0.491554i \(-0.163571\pi\)
0.870847 + 0.491554i \(0.163571\pi\)
\(30\) 0 0
\(31\) −230.000 −1.33256 −0.666278 − 0.745704i \(-0.732113\pi\)
−0.666278 + 0.745704i \(0.732113\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 182.000 0.808665 0.404333 − 0.914612i \(-0.367504\pi\)
0.404333 + 0.914612i \(0.367504\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −117.000 −0.445667 −0.222833 − 0.974857i \(-0.571531\pi\)
−0.222833 + 0.974857i \(0.571531\pi\)
\(42\) 0 0
\(43\) −372.000 −1.31929 −0.659645 − 0.751577i \(-0.729293\pi\)
−0.659645 + 0.751577i \(0.729293\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −52.0000 −0.161383 −0.0806913 − 0.996739i \(-0.525713\pi\)
−0.0806913 + 0.996739i \(0.525713\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −402.000 −1.04187 −0.520933 − 0.853597i \(-0.674416\pi\)
−0.520933 + 0.853597i \(0.674416\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −312.000 −0.688457 −0.344228 − 0.938886i \(-0.611859\pi\)
−0.344228 + 0.938886i \(0.611859\pi\)
\(60\) 0 0
\(61\) 170.000 0.356824 0.178412 − 0.983956i \(-0.442904\pi\)
0.178412 + 0.983956i \(0.442904\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −763.000 −1.39127 −0.695636 − 0.718394i \(-0.744878\pi\)
−0.695636 + 0.718394i \(0.744878\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 52.0000 0.0869192 0.0434596 − 0.999055i \(-0.486162\pi\)
0.0434596 + 0.999055i \(0.486162\pi\)
\(72\) 0 0
\(73\) 981.000 1.57284 0.786420 − 0.617692i \(-0.211932\pi\)
0.786420 + 0.617692i \(0.211932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 114.000 0.168721
\(78\) 0 0
\(79\) 1054.00 1.50107 0.750533 − 0.660833i \(-0.229797\pi\)
0.750533 + 0.660833i \(0.229797\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 351.000 0.464184 0.232092 − 0.972694i \(-0.425443\pi\)
0.232092 + 0.972694i \(0.425443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −799.000 −0.951616 −0.475808 − 0.879549i \(-0.657844\pi\)
−0.475808 + 0.879549i \(0.657844\pi\)
\(90\) 0 0
\(91\) −72.0000 −0.0829412
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −962.000 −1.00697 −0.503486 − 0.864003i \(-0.667949\pi\)
−0.503486 + 0.864003i \(0.667949\pi\)
\(98\) 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 1800.4.a.w.1.1 1
3.2 odd 2 200.4.a.e.1.1 ✓ 1
5.2 odd 4 1800.4.f.p.649.2 2
5.3 odd 4 1800.4.f.p.649.1 2
5.4 even 2 1800.4.a.l.1.1 1
12.11 even 2 400.4.a.k.1.1 1
15.2 even 4 200.4.c.g.49.2 2
15.8 even 4 200.4.c.g.49.1 2
15.14 odd 2 200.4.a.f.1.1 yes 1
24.5 odd 2 1600.4.a.bf.1.1 1
24.11 even 2 1600.4.a.v.1.1 1
60.23 odd 4 400.4.c.m.49.2 2
60.47 odd 4 400.4.c.m.49.1 2
60.59 even 2 400.4.a.j.1.1 1
120.29 odd 2 1600.4.a.w.1.1 1
120.59 even 2 1600.4.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 ✓ 1 3.2 odd 2
200.4.a.f.1.1 yes 1 15.14 odd 2
200.4.c.g.49.1 2 15.8 even 4
200.4.c.g.49.2 2 15.2 even 4
400.4.a.j.1.1 1 60.59 even 2
400.4.a.k.1.1 1 12.11 even 2
400.4.c.m.49.1 2 60.47 odd 4
400.4.c.m.49.2 2 60.23 odd 4
1600.4.a.v.1.1 1 24.11 even 2
1600.4.a.w.1.1 1 120.29 odd 2
1600.4.a.be.1.1 1 120.59 even 2
1600.4.a.bf.1.1 1 24.5 odd 2
1800.4.a.l.1.1 1 5.4 even 2
1800.4.a.w.1.1 1 1.1 even 1 trivial
1800.4.f.p.649.1 2 5.3 odd 4
1800.4.f.p.649.2 2 5.2 odd 4