Show commands: Magma / Pari/GP / SageMath

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 = [4,0,0,32,0,0,0,0,0,0,0,0,0,0,0,256,0,0,556] 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 449.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 450.449
Dual form 450.5.b.b.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843 q^{2} +8.00000 q^{4} +83.0000i q^{7} -22.6274 q^{8} -80.6102i q^{11} -41.0000i q^{13} -234.759i q^{14} +64.0000 q^{16} +513.360 q^{17} +139.000 q^{19} +228.000i q^{22} +224.860 q^{23} +115.966i q^{26} +664.000i q^{28} +674.580i q^{29} -1051.00 q^{31} -181.019 q^{32} -1452.00 q^{34} -1672.00i q^{37} -393.151 q^{38} -831.558i q^{41} +2515.00i q^{43} -644.881i q^{44} -636.000 q^{46} +2948.64 q^{47} -4488.00 q^{49} -328.000i q^{52} +390.323 q^{53} -1878.08i q^{56} -1908.00i q^{58} +750.947i q^{59} +5825.00 q^{61} +2972.68 q^{62} +512.000 q^{64} +7259.00i q^{67} +4106.88 q^{68} -6907.02i q^{71} +4552.00i q^{73} +4729.13i q^{74} +1112.00 q^{76} +6690.64 q^{77} -9296.00 q^{79} +2352.00i q^{82} -7980.41 q^{83} -7113.49i q^{86} +1824.00i q^{88} +6075.46i q^{89} +3403.00 q^{91} +1798.88 q^{92} -8340.00 q^{94} +1793.00i q^{97} +12694.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4} + 256 q^{16} + 556 q^{19} - 4204 q^{31} - 5808 q^{34} - 2544 q^{46} - 17952 q^{49} + 23300 q^{61} + 2048 q^{64} + 4448 q^{76} - 37184 q^{79} + 13612 q^{91} - 33360 q^{94}+O(q^{100}) \) 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
\(7\) 83.0000i 1.69388i 0.531690 + 0.846939i \(0.321557\pi\)
−0.531690 + 0.846939i \(0.678443\pi\)
\(8\) −22.6274 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) − 80.6102i − 0.666200i −0.942892 − 0.333100i \(-0.891905\pi\)
0.942892 − 0.333100i \(-0.108095\pi\)
\(12\) 0 0
\(13\) − 41.0000i − 0.242604i −0.992616 − 0.121302i \(-0.961293\pi\)
0.992616 − 0.121302i \(-0.0387069\pi\)
\(14\) − 234.759i − 1.19775i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 513.360 1.77633 0.888165 − 0.459524i \(-0.151980\pi\)
0.888165 + 0.459524i \(0.151980\pi\)
\(18\) 0 0
\(19\) 139.000 0.385042 0.192521 − 0.981293i \(-0.438334\pi\)
0.192521 + 0.981293i \(0.438334\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 228.000i 0.471074i
\(23\) 224.860 0.425066 0.212533 − 0.977154i \(-0.431829\pi\)
0.212533 + 0.977154i \(0.431829\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 115.966i 0.171547i
\(27\) 0 0
\(28\) 664.000i 0.846939i
\(29\) 674.580i 0.802116i 0.916053 + 0.401058i \(0.131357\pi\)
−0.916053 + 0.401058i \(0.868643\pi\)
\(30\) 0 0
\(31\) −1051.00 −1.09365 −0.546826 − 0.837246i \(-0.684164\pi\)
−0.546826 + 0.837246i \(0.684164\pi\)
\(32\) −181.019 −0.176777
\(33\) 0 0
\(34\) −1452.00 −1.25606
\(35\) 0 0
\(36\) 0 0
\(37\) − 1672.00i − 1.22133i −0.791889 − 0.610665i \(-0.790902\pi\)
0.791889 − 0.610665i \(-0.209098\pi\)
\(38\) −393.151 −0.272265
\(39\) 0 0
\(40\) 0 0
\(41\) − 831.558i − 0.494680i −0.968929 − 0.247340i \(-0.920443\pi\)
0.968929 − 0.247340i \(-0.0795565\pi\)
\(42\) 0 0
\(43\) 2515.00i 1.36019i 0.733122 + 0.680097i \(0.238063\pi\)
−0.733122 + 0.680097i \(0.761937\pi\)
\(44\) − 644.881i − 0.333100i
\(45\) 0 0
\(46\) −636.000 −0.300567
\(47\) 2948.64 1.33483 0.667414 − 0.744687i \(-0.267401\pi\)
0.667414 + 0.744687i \(0.267401\pi\)
\(48\) 0 0
\(49\) −4488.00 −1.86922
\(50\) 0 0
\(51\) 0 0
\(52\) − 328.000i − 0.121302i
\(53\) 390.323 0.138954 0.0694772 − 0.997584i \(-0.477867\pi\)
0.0694772 + 0.997584i \(0.477867\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) − 1878.08i − 0.598876i
\(57\) 0 0
\(58\) − 1908.00i − 0.567182i
\(59\) 750.947i 0.215727i 0.994166 + 0.107864i \(0.0344010\pi\)
−0.994166 + 0.107864i \(0.965599\pi\)
\(60\) 0 0
\(61\) 5825.00 1.56544 0.782720 − 0.622374i \(-0.213832\pi\)
0.782720 + 0.622374i \(0.213832\pi\)
\(62\) 2972.68 0.773329
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 7259.00i 1.61706i 0.588452 + 0.808532i \(0.299737\pi\)
−0.588452 + 0.808532i \(0.700263\pi\)
\(68\) 4106.88 0.888165
\(69\) 0 0
\(70\) 0 0
\(71\) − 6907.02i − 1.37017i −0.728464 − 0.685084i \(-0.759765\pi\)
0.728464 − 0.685084i \(-0.240235\pi\)
\(72\) 0 0
\(73\) 4552.00i 0.854194i 0.904206 + 0.427097i \(0.140464\pi\)
−0.904206 + 0.427097i \(0.859536\pi\)
\(74\) 4729.13i 0.863610i
\(75\) 0 0
\(76\) 1112.00 0.192521
\(77\) 6690.64 1.12846
\(78\) 0 0
\(79\) −9296.00 −1.48950 −0.744752 − 0.667341i \(-0.767432\pi\)
−0.744752 + 0.667341i \(0.767432\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2352.00i 0.349792i
\(83\) −7980.41 −1.15843 −0.579214 − 0.815176i \(-0.696640\pi\)
−0.579214 + 0.815176i \(0.696640\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 7113.49i − 0.961803i
\(87\) 0 0
\(88\) 1824.00i 0.235537i
\(89\) 6075.46i 0.767007i 0.923539 + 0.383503i \(0.125283\pi\)
−0.923539 + 0.383503i \(0.874717\pi\)
\(90\) 0 0
\(91\) 3403.00 0.410941
\(92\) 1798.88 0.212533
\(93\) 0 0
\(94\) −8340.00 −0.943866
\(95\) 0 0
\(96\) 0 0
\(97\) 1793.00i 0.190562i 0.995450 + 0.0952811i \(0.0303750\pi\)
−0.995450 + 0.0952811i \(0.969625\pi\)
\(98\) 12694.0 1.32174
\(99\) 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.b.449.2 4
3.2 odd 2 inner 450.5.b.b.449.4 4
5.2 odd 4 450.5.d.a.251.1 ✓ 2
5.3 odd 4 450.5.d.d.251.2 yes 2
5.4 even 2 inner 450.5.b.b.449.3 4
15.2 even 4 450.5.d.a.251.2 yes 2
15.8 even 4 450.5.d.d.251.1 yes 2
15.14 odd 2 inner 450.5.b.b.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.5.b.b.449.1 4 15.14 odd 2 inner
450.5.b.b.449.2 4 1.1 even 1 trivial
450.5.b.b.449.3 4 5.4 even 2 inner
450.5.b.b.449.4 4 3.2 odd 2 inner
450.5.d.a.251.1 ✓ 2 5.2 odd 4
450.5.d.a.251.2 yes 2 15.2 even 4
450.5.d.d.251.1 yes 2 15.8 even 4
450.5.d.d.251.2 yes 2 5.3 odd 4