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Properties

Label 1800.4.f.p.649.1
Level $1800$
Weight $4$
Character 1800.649
Analytic conductor $106.203$
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] = [1800,4,Mod(649,1800)] 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("1800.649"); 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,182,0,0,0,0,0,0,0,0,0,-544] 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: \(106.203438010\)
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: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.649
Dual form 1800.4.f.p.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)
 
\(f(q)\) \(=\) \(q-6.00000i q^{7} +19.0000 q^{11} -12.0000i q^{13} +75.0000i q^{17} +91.0000 q^{19} +174.000i q^{23} -272.000 q^{29} -230.000 q^{31} -182.000i q^{37} -117.000 q^{41} -372.000i q^{43} +52.0000i q^{47} +307.000 q^{49} -402.000i q^{53} +312.000 q^{59} +170.000 q^{61} +763.000i q^{67} +52.0000 q^{71} +981.000i q^{73} -114.000i q^{77} -1054.00 q^{79} +351.000i q^{83} +799.000 q^{89} -72.0000 q^{91} +962.000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 38 q^{11} + 182 q^{19} - 544 q^{29} - 460 q^{31} - 234 q^{41} + 614 q^{49} + 624 q^{59} + 340 q^{61} + 104 q^{71} - 2108 q^{79} + 1598 q^{89} - 144 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(7\) − 6.00000i − 0.323970i −0.986793 − 0.161985i \(-0.948210\pi\)
0.986793 − 0.161985i \(-0.0517895\pi\)
\(8\) 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
\(13\) − 12.0000i − 0.256015i −0.991773 − 0.128008i \(-0.959142\pi\)
0.991773 − 0.128008i \(-0.0408582\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 75.0000i 1.07001i 0.844849 + 0.535005i \(0.179690\pi\)
−0.844849 + 0.535005i \(0.820310\pi\)
\(18\) 0 0
\(19\) 91.0000 1.09878 0.549390 − 0.835566i \(-0.314860\pi\)
0.549390 + 0.835566i \(0.314860\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 174.000i 1.57746i 0.614742 + 0.788728i \(0.289260\pi\)
−0.614742 + 0.788728i \(0.710740\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −272.000 −1.74169 −0.870847 − 0.491554i \(-0.836429\pi\)
−0.870847 + 0.491554i \(0.836429\pi\)
\(30\) 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
\(37\) − 182.000i − 0.808665i −0.914612 − 0.404333i \(-0.867504\pi\)
0.914612 − 0.404333i \(-0.132496\pi\)
\(38\) 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
\(43\) − 372.000i − 1.31929i −0.751577 − 0.659645i \(-0.770707\pi\)
0.751577 − 0.659645i \(-0.229293\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 52.0000i 0.161383i 0.996739 + 0.0806913i \(0.0257128\pi\)
−0.996739 + 0.0806913i \(0.974287\pi\)
\(48\) 0 0
\(49\) 307.000 0.895044
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 402.000i − 1.04187i −0.853597 − 0.520933i \(-0.825584\pi\)
0.853597 − 0.520933i \(-0.174416\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 312.000 0.688457 0.344228 − 0.938886i \(-0.388141\pi\)
0.344228 + 0.938886i \(0.388141\pi\)
\(60\) 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
\(67\) 763.000i 1.39127i 0.718394 + 0.695636i \(0.244878\pi\)
−0.718394 + 0.695636i \(0.755122\pi\)
\(68\) 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
\(73\) 981.000i 1.57284i 0.617692 + 0.786420i \(0.288068\pi\)
−0.617692 + 0.786420i \(0.711932\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 114.000i − 0.168721i
\(78\) 0 0
\(79\) −1054.00 −1.50107 −0.750533 − 0.660833i \(-0.770203\pi\)
−0.750533 + 0.660833i \(0.770203\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 351.000i 0.464184i 0.972694 + 0.232092i \(0.0745570\pi\)
−0.972694 + 0.232092i \(0.925443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 799.000 0.951616 0.475808 − 0.879549i \(-0.342156\pi\)
0.475808 + 0.879549i \(0.342156\pi\)
\(90\) 0 0
\(91\) −72.0000 −0.0829412
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 962.000i 1.00697i 0.864003 + 0.503486i \(0.167949\pi\)
−0.864003 + 0.503486i \(0.832051\pi\)
\(98\) 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.f.p.649.1 2
3.2 odd 2 200.4.c.g.49.1 2
5.2 odd 4 1800.4.a.w.1.1 1
5.3 odd 4 1800.4.a.l.1.1 1
5.4 even 2 inner 1800.4.f.p.649.2 2
12.11 even 2 400.4.c.m.49.2 2
15.2 even 4 200.4.a.e.1.1 ✓ 1
15.8 even 4 200.4.a.f.1.1 yes 1
15.14 odd 2 200.4.c.g.49.2 2
60.23 odd 4 400.4.a.j.1.1 1
60.47 odd 4 400.4.a.k.1.1 1
60.59 even 2 400.4.c.m.49.1 2
120.53 even 4 1600.4.a.w.1.1 1
120.77 even 4 1600.4.a.bf.1.1 1
120.83 odd 4 1600.4.a.be.1.1 1
120.107 odd 4 1600.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.e.1.1 ✓ 1 15.2 even 4
200.4.a.f.1.1 yes 1 15.8 even 4
200.4.c.g.49.1 2 3.2 odd 2
200.4.c.g.49.2 2 15.14 odd 2
400.4.a.j.1.1 1 60.23 odd 4
400.4.a.k.1.1 1 60.47 odd 4
400.4.c.m.49.1 2 60.59 even 2
400.4.c.m.49.2 2 12.11 even 2
1600.4.a.v.1.1 1 120.107 odd 4
1600.4.a.w.1.1 1 120.53 even 4
1600.4.a.be.1.1 1 120.83 odd 4
1600.4.a.bf.1.1 1 120.77 even 4
1800.4.a.l.1.1 1 5.3 odd 4
1800.4.a.w.1.1 1 5.2 odd 4
1800.4.f.p.649.1 2 1.1 even 1 trivial
1800.4.f.p.649.2 2 5.4 even 2 inner