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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,5,Mod(701,900)] 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("900.701"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 900.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(93.0329667755\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 701.3
Root \(-1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 900.701
Dual form 900.5.g.b.701.4

$q$-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)
 
\(f(q)\) \(=\) \(q+37.4868 q^{7} -174.525i q^{11} -130.355 q^{13} -220.617i q^{17} +588.394 q^{19} +475.306i q^{23} -610.847i q^{29} +413.762 q^{31} -2638.09 q^{37} -1756.15i q^{41} -1848.13 q^{43} +1602.19i q^{47} -995.737 q^{49} -265.798i q^{53} -1647.32i q^{59} +2289.97 q^{61} -308.000 q^{67} +3509.52i q^{71} +5587.39 q^{73} -6542.39i q^{77} +399.183 q^{79} +8933.29i q^{83} -6024.96i q^{89} -4886.60 q^{91} +8842.01 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 112 q^{7} - 104 q^{13} + 608 q^{19} - 1912 q^{31} - 3608 q^{37} + 880 q^{43} - 6108 q^{49} + 6200 q^{61} - 1232 q^{67} + 6184 q^{73} - 3640 q^{79} - 6872 q^{91} - 19352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(7\) 37.4868 0.765037 0.382519 − 0.923948i \(-0.375057\pi\)
0.382519 + 0.923948i \(0.375057\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 174.525i − 1.44236i −0.692750 − 0.721178i \(-0.743601\pi\)
0.692750 − 0.721178i \(-0.256399\pi\)
\(12\) 0 0
\(13\) −130.355 −0.771332 −0.385666 − 0.922638i \(-0.626028\pi\)
−0.385666 + 0.922638i \(0.626028\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 220.617i − 0.763382i −0.924290 − 0.381691i \(-0.875342\pi\)
0.924290 − 0.381691i \(-0.124658\pi\)
\(18\) 0 0
\(19\) 588.394 1.62990 0.814951 − 0.579530i \(-0.196764\pi\)
0.814951 + 0.579530i \(0.196764\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 475.306i 0.898498i 0.893407 + 0.449249i \(0.148308\pi\)
−0.893407 + 0.449249i \(0.851692\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 610.847i − 0.726334i −0.931724 − 0.363167i \(-0.881695\pi\)
0.931724 − 0.363167i \(-0.118305\pi\)
\(30\) 0 0
\(31\) 413.762 0.430554 0.215277 − 0.976553i \(-0.430935\pi\)
0.215277 + 0.976553i \(0.430935\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2638.09 −1.92702 −0.963510 − 0.267672i \(-0.913746\pi\)
−0.963510 + 0.267672i \(0.913746\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1756.15i − 1.04471i −0.852729 − 0.522354i \(-0.825054\pi\)
0.852729 − 0.522354i \(-0.174946\pi\)
\(42\) 0 0
\(43\) −1848.13 −0.999529 −0.499765 − 0.866161i \(-0.666580\pi\)
−0.499765 + 0.866161i \(0.666580\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1602.19i 0.725301i 0.931925 + 0.362651i \(0.118128\pi\)
−0.931925 + 0.362651i \(0.881872\pi\)
\(48\) 0 0
\(49\) −995.737 −0.414718
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 265.798i − 0.0946236i −0.998880 − 0.0473118i \(-0.984935\pi\)
0.998880 − 0.0473118i \(-0.0150654\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1647.32i − 0.473231i −0.971603 − 0.236615i \(-0.923962\pi\)
0.971603 − 0.236615i \(-0.0760381\pi\)
\(60\) 0 0
\(61\) 2289.97 0.615419 0.307709 − 0.951480i \(-0.400438\pi\)
0.307709 + 0.951480i \(0.400438\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −308.000 −0.0686122 −0.0343061 − 0.999411i \(-0.510922\pi\)
−0.0343061 + 0.999411i \(0.510922\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3509.52i 0.696195i 0.937458 + 0.348097i \(0.113172\pi\)
−0.937458 + 0.348097i \(0.886828\pi\)
\(72\) 0 0
\(73\) 5587.39 1.04849 0.524244 − 0.851568i \(-0.324348\pi\)
0.524244 + 0.851568i \(0.324348\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6542.39i − 1.10346i
\(78\) 0 0
\(79\) 399.183 0.0639614 0.0319807 − 0.999488i \(-0.489818\pi\)
0.0319807 + 0.999488i \(0.489818\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8933.29i 1.29675i 0.761323 + 0.648373i \(0.224550\pi\)
−0.761323 + 0.648373i \(0.775450\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 6024.96i − 0.760632i −0.924857 − 0.380316i \(-0.875815\pi\)
0.924857 − 0.380316i \(-0.124185\pi\)
\(90\) 0 0
\(91\) −4886.60 −0.590098
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8842.01 0.939740 0.469870 − 0.882736i \(-0.344301\pi\)
0.469870 + 0.882736i \(0.344301\pi\)
\(98\) 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 900.5.g.b.701.3 4
3.2 odd 2 inner 900.5.g.b.701.4 4
5.2 odd 4 900.5.b.b.449.7 8
5.3 odd 4 900.5.b.b.449.1 8
5.4 even 2 180.5.g.a.161.3 yes 4
15.2 even 4 900.5.b.b.449.8 8
15.8 even 4 900.5.b.b.449.2 8
15.14 odd 2 180.5.g.a.161.1 ✓ 4
20.19 odd 2 720.5.l.d.161.4 4
60.59 even 2 720.5.l.d.161.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.5.g.a.161.1 ✓ 4 15.14 odd 2
180.5.g.a.161.3 yes 4 5.4 even 2
720.5.l.d.161.2 4 60.59 even 2
720.5.l.d.161.4 4 20.19 odd 2
900.5.b.b.449.1 8 5.3 odd 4
900.5.b.b.449.2 8 15.8 even 4
900.5.b.b.449.7 8 5.2 odd 4
900.5.b.b.449.8 8 15.2 even 4
900.5.g.b.701.3 4 1.1 even 1 trivial
900.5.g.b.701.4 4 3.2 odd 2 inner