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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,5,Mod(449,450)] 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("450.449"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 450.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,32,0,0,0,0,0,0,0,0,0,0,0,256,0,0,-404] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.5164833877\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 450.449
Dual form 450.5.b.a.449.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843 q^{2} +8.00000 q^{4} -13.0000i q^{7} -22.6274 q^{8} +80.6102i q^{11} -89.0000i q^{13} +36.7696i q^{14} +64.0000 q^{16} -38.1838 q^{17} -101.000 q^{19} -228.000i q^{22} +521.845 q^{23} +251.730i q^{26} -104.000i q^{28} +683.065i q^{29} -91.0000 q^{31} -181.019 q^{32} +108.000 q^{34} -2248.00i q^{37} +285.671 q^{38} +492.146i q^{41} -125.000i q^{43} +644.881i q^{44} -1476.00 q^{46} -1590.99 q^{47} +2232.00 q^{49} -712.000i q^{52} +2257.08 q^{53} +294.156i q^{56} -1932.00i q^{58} +5697.87i q^{59} -4975.00 q^{61} +257.387 q^{62} +512.000 q^{64} -3829.00i q^{67} -305.470 q^{68} -2935.91i q^{71} -2072.00i q^{73} +6358.30i q^{74} -808.000 q^{76} +1047.93 q^{77} +12304.0 q^{79} -1392.00i q^{82} +8862.88 q^{83} +353.553i q^{86} -1824.00i q^{88} +1391.59i q^{89} -1157.00 q^{91} +4174.76 q^{92} +4500.00 q^{94} -10783.0i q^{97} -6313.05 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4} + 256 q^{16} - 404 q^{19} - 364 q^{31} + 432 q^{34} - 5904 q^{46} + 8928 q^{49} - 19900 q^{61} + 2048 q^{64} - 3232 q^{76} + 49216 q^{79} - 4628 q^{91} + 18000 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82843 −0.707107
\(3\) 0 0
\(4\) 8.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 13.0000i − 0.265306i −0.991163 − 0.132653i \(-0.957650\pi\)
0.991163 − 0.132653i \(-0.0423496\pi\)
\(8\) −22.6274 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 80.6102i 0.666200i 0.942892 + 0.333100i \(0.108095\pi\)
−0.942892 + 0.333100i \(0.891905\pi\)
\(12\) 0 0
\(13\) − 89.0000i − 0.526627i −0.964710 − 0.263314i \(-0.915185\pi\)
0.964710 − 0.263314i \(-0.0848154\pi\)
\(14\) 36.7696i 0.187600i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) −38.1838 −0.132124 −0.0660619 − 0.997816i \(-0.521043\pi\)
−0.0660619 + 0.997816i \(0.521043\pi\)
\(18\) 0 0
\(19\) −101.000 −0.279778 −0.139889 − 0.990167i \(-0.544675\pi\)
−0.139889 + 0.990167i \(0.544675\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 228.000i − 0.471074i
\(23\) 521.845 0.986474 0.493237 − 0.869895i \(-0.335814\pi\)
0.493237 + 0.869895i \(0.335814\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 251.730i 0.372382i
\(27\) 0 0
\(28\) − 104.000i − 0.132653i
\(29\) 683.065i 0.812206i 0.913827 + 0.406103i \(0.133113\pi\)
−0.913827 + 0.406103i \(0.866887\pi\)
\(30\) 0 0
\(31\) −91.0000 −0.0946930 −0.0473465 − 0.998879i \(-0.515076\pi\)
−0.0473465 + 0.998879i \(0.515076\pi\)
\(32\) −181.019 −0.176777
\(33\) 0 0
\(34\) 108.000 0.0934256
\(35\) 0 0
\(36\) 0 0
\(37\) − 2248.00i − 1.64207i −0.570875 − 0.821037i \(-0.693396\pi\)
0.570875 − 0.821037i \(-0.306604\pi\)
\(38\) 285.671 0.197833
\(39\) 0 0
\(40\) 0 0
\(41\) 492.146i 0.292770i 0.989228 + 0.146385i \(0.0467638\pi\)
−0.989228 + 0.146385i \(0.953236\pi\)
\(42\) 0 0
\(43\) − 125.000i − 0.0676041i −0.999429 − 0.0338021i \(-0.989238\pi\)
0.999429 − 0.0338021i \(-0.0107616\pi\)
\(44\) 644.881i 0.333100i
\(45\) 0 0
\(46\) −1476.00 −0.697543
\(47\) −1590.99 −0.720231 −0.360115 − 0.932908i \(-0.617263\pi\)
−0.360115 + 0.932908i \(0.617263\pi\)
\(48\) 0 0
\(49\) 2232.00 0.929613
\(50\) 0 0
\(51\) 0 0
\(52\) − 712.000i − 0.263314i
\(53\) 2257.08 0.803519 0.401759 − 0.915745i \(-0.368399\pi\)
0.401759 + 0.915745i \(0.368399\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 294.156i 0.0937999i
\(57\) 0 0
\(58\) − 1932.00i − 0.574316i
\(59\) 5697.87i 1.63685i 0.574615 + 0.818424i \(0.305152\pi\)
−0.574615 + 0.818424i \(0.694848\pi\)
\(60\) 0 0
\(61\) −4975.00 −1.33701 −0.668503 − 0.743709i \(-0.733065\pi\)
−0.668503 + 0.743709i \(0.733065\pi\)
\(62\) 257.387 0.0669581
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 3829.00i − 0.852974i −0.904494 − 0.426487i \(-0.859751\pi\)
0.904494 − 0.426487i \(-0.140249\pi\)
\(68\) −305.470 −0.0660619
\(69\) 0 0
\(70\) 0 0
\(71\) − 2935.91i − 0.582406i −0.956661 − 0.291203i \(-0.905945\pi\)
0.956661 − 0.291203i \(-0.0940555\pi\)
\(72\) 0 0
\(73\) − 2072.00i − 0.388816i −0.980921 − 0.194408i \(-0.937721\pi\)
0.980921 − 0.194408i \(-0.0622786\pi\)
\(74\) 6358.30i 1.16112i
\(75\) 0 0
\(76\) −808.000 −0.139889
\(77\) 1047.93 0.176747
\(78\) 0 0
\(79\) 12304.0 1.97148 0.985739 − 0.168279i \(-0.0538208\pi\)
0.985739 + 0.168279i \(0.0538208\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 1392.00i − 0.207020i
\(83\) 8862.88 1.28653 0.643263 − 0.765645i \(-0.277580\pi\)
0.643263 + 0.765645i \(0.277580\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 353.553i 0.0478033i
\(87\) 0 0
\(88\) − 1824.00i − 0.235537i
\(89\) 1391.59i 0.175683i 0.996134 + 0.0878416i \(0.0279969\pi\)
−0.996134 + 0.0878416i \(0.972003\pi\)
\(90\) 0 0
\(91\) −1157.00 −0.139717
\(92\) 4174.76 0.493237
\(93\) 0 0
\(94\) 4500.00 0.509280
\(95\) 0 0
\(96\) 0 0
\(97\) − 10783.0i − 1.14603i −0.819545 − 0.573015i \(-0.805774\pi\)
0.819545 − 0.573015i \(-0.194226\pi\)
\(98\) −6313.05 −0.657335
\(99\) 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 450.5.b.a.449.1 4
3.2 odd 2 inner 450.5.b.a.449.3 4
5.2 odd 4 450.5.d.c.251.1 yes 2
5.3 odd 4 450.5.d.b.251.2 yes 2
5.4 even 2 inner 450.5.b.a.449.4 4
15.2 even 4 450.5.d.c.251.2 yes 2
15.8 even 4 450.5.d.b.251.1 ✓ 2
15.14 odd 2 inner 450.5.b.a.449.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.5.b.a.449.1 4 1.1 even 1 trivial
450.5.b.a.449.2 4 15.14 odd 2 inner
450.5.b.a.449.3 4 3.2 odd 2 inner
450.5.b.a.449.4 4 5.4 even 2 inner
450.5.d.b.251.1 ✓ 2 15.8 even 4
450.5.d.b.251.2 yes 2 5.3 odd 4
450.5.d.c.251.1 yes 2 5.2 odd 4
450.5.d.c.251.2 yes 2 15.2 even 4