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 = [8,0,0,64,0,0,0,0,0,0,0,0,0,0,0,512,0,0,-2368] 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: \(8\)
Coefficient field: \(\Q(i, \sqrt{2}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.8
Root \(0.437016 + 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 450.449
Dual form 450.5.b.c.449.5

$q$-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)
 
\(f(q)\) \(=\) \(q+2.82843 q^{2} +8.00000 q^{4} +79.4342i q^{7} +22.6274 q^{8} -197.003i q^{11} +308.302i q^{13} +224.674i q^{14} +64.0000 q^{16} +200.446 q^{17} -201.132 q^{19} -557.210i q^{22} -496.631 q^{23} +872.011i q^{26} +635.473i q^{28} +437.234i q^{29} -810.605 q^{31} +181.019 q^{32} +566.947 q^{34} +1828.38i q^{37} -568.886 q^{38} +1962.26i q^{41} -2101.44i q^{43} -1576.03i q^{44} -1404.68 q^{46} -1192.59 q^{47} -3908.79 q^{49} +2466.42i q^{52} -2781.09 q^{53} +1797.39i q^{56} +1236.68i q^{58} -475.417i q^{59} +3458.02 q^{61} -2292.74 q^{62} +512.000 q^{64} +3897.58i q^{67} +1603.57 q^{68} +9881.37i q^{71} +6375.23i q^{73} +5171.44i q^{74} -1609.05 q^{76} +15648.8 q^{77} +4278.86 q^{79} +5550.11i q^{82} +2371.56 q^{83} -5943.78i q^{86} -4457.68i q^{88} -3539.89i q^{89} -24489.8 q^{91} -3973.04 q^{92} -3373.15 q^{94} +2936.34i q^{97} -11055.7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{4} + 512 q^{16} - 2368 q^{19} - 4208 q^{31} - 1536 q^{34} - 3648 q^{46} - 6984 q^{49} - 12560 q^{61} + 4096 q^{64} - 18944 q^{76} - 24208 q^{79} - 96496 q^{91} + 12480 q^{94}+O(q^{100}) \) 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
\(7\) 79.4342i 1.62111i 0.585666 + 0.810553i \(0.300833\pi\)
−0.585666 + 0.810553i \(0.699167\pi\)
\(8\) 22.6274 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) − 197.003i − 1.62813i −0.580775 − 0.814064i \(-0.697250\pi\)
0.580775 − 0.814064i \(-0.302750\pi\)
\(12\) 0 0
\(13\) 308.302i 1.82428i 0.409884 + 0.912138i \(0.365569\pi\)
−0.409884 + 0.912138i \(0.634431\pi\)
\(14\) 224.674i 1.14629i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 200.446 0.693584 0.346792 − 0.937942i \(-0.387271\pi\)
0.346792 + 0.937942i \(0.387271\pi\)
\(18\) 0 0
\(19\) −201.132 −0.557151 −0.278576 − 0.960414i \(-0.589862\pi\)
−0.278576 + 0.960414i \(0.589862\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 557.210i − 1.15126i
\(23\) −496.631 −0.938810 −0.469405 − 0.882983i \(-0.655532\pi\)
−0.469405 + 0.882983i \(0.655532\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 872.011i 1.28996i
\(27\) 0 0
\(28\) 635.473i 0.810553i
\(29\) 437.234i 0.519897i 0.965622 + 0.259949i \(0.0837056\pi\)
−0.965622 + 0.259949i \(0.916294\pi\)
\(30\) 0 0
\(31\) −810.605 −0.843502 −0.421751 − 0.906712i \(-0.638584\pi\)
−0.421751 + 0.906712i \(0.638584\pi\)
\(32\) 181.019 0.176777
\(33\) 0 0
\(34\) 566.947 0.490438
\(35\) 0 0
\(36\) 0 0
\(37\) 1828.38i 1.33556i 0.744359 + 0.667780i \(0.232755\pi\)
−0.744359 + 0.667780i \(0.767245\pi\)
\(38\) −568.886 −0.393966
\(39\) 0 0
\(40\) 0 0
\(41\) 1962.26i 1.16732i 0.811999 + 0.583658i \(0.198379\pi\)
−0.811999 + 0.583658i \(0.801621\pi\)
\(42\) 0 0
\(43\) − 2101.44i − 1.13653i −0.822845 − 0.568265i \(-0.807615\pi\)
0.822845 − 0.568265i \(-0.192385\pi\)
\(44\) − 1576.03i − 0.814064i
\(45\) 0 0
\(46\) −1404.68 −0.663839
\(47\) −1192.59 −0.539878 −0.269939 − 0.962877i \(-0.587003\pi\)
−0.269939 + 0.962877i \(0.587003\pi\)
\(48\) 0 0
\(49\) −3908.79 −1.62798
\(50\) 0 0
\(51\) 0 0
\(52\) 2466.42i 0.912138i
\(53\) −2781.09 −0.990064 −0.495032 − 0.868875i \(-0.664844\pi\)
−0.495032 + 0.868875i \(0.664844\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1797.39i 0.573147i
\(57\) 0 0
\(58\) 1236.68i 0.367623i
\(59\) − 475.417i − 0.136575i −0.997666 − 0.0682875i \(-0.978246\pi\)
0.997666 − 0.0682875i \(-0.0217535\pi\)
\(60\) 0 0
\(61\) 3458.02 0.929326 0.464663 − 0.885488i \(-0.346176\pi\)
0.464663 + 0.885488i \(0.346176\pi\)
\(62\) −2292.74 −0.596446
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3897.58i 0.868251i 0.900852 + 0.434126i \(0.142943\pi\)
−0.900852 + 0.434126i \(0.857057\pi\)
\(68\) 1603.57 0.346792
\(69\) 0 0
\(70\) 0 0
\(71\) 9881.37i 1.96020i 0.198505 + 0.980100i \(0.436391\pi\)
−0.198505 + 0.980100i \(0.563609\pi\)
\(72\) 0 0
\(73\) 6375.23i 1.19633i 0.801374 + 0.598164i \(0.204103\pi\)
−0.801374 + 0.598164i \(0.795897\pi\)
\(74\) 5171.44i 0.944383i
\(75\) 0 0
\(76\) −1609.05 −0.278576
\(77\) 15648.8 2.63937
\(78\) 0 0
\(79\) 4278.86 0.685605 0.342803 − 0.939407i \(-0.388624\pi\)
0.342803 + 0.939407i \(0.388624\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 5550.11i 0.825417i
\(83\) 2371.56 0.344253 0.172127 − 0.985075i \(-0.444936\pi\)
0.172127 + 0.985075i \(0.444936\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 5943.78i − 0.803648i
\(87\) 0 0
\(88\) − 4457.68i − 0.575630i
\(89\) − 3539.89i − 0.446899i −0.974716 − 0.223450i \(-0.928268\pi\)
0.974716 − 0.223450i \(-0.0717318\pi\)
\(90\) 0 0
\(91\) −24489.8 −2.95734
\(92\) −3973.04 −0.469405
\(93\) 0 0
\(94\) −3373.15 −0.381751
\(95\) 0 0
\(96\) 0 0
\(97\) 2936.34i 0.312078i 0.987751 + 0.156039i \(0.0498726\pi\)
−0.987751 + 0.156039i \(0.950127\pi\)
\(98\) −11055.7 −1.15116
\(99\) 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.c.449.8 8
3.2 odd 2 inner 450.5.b.c.449.4 8
5.2 odd 4 450.5.d.e.251.3 4
5.3 odd 4 90.5.d.b.71.1 ✓ 4
5.4 even 2 inner 450.5.b.c.449.1 8
15.2 even 4 450.5.d.e.251.1 4
15.8 even 4 90.5.d.b.71.4 yes 4
15.14 odd 2 inner 450.5.b.c.449.5 8
20.3 even 4 720.5.l.a.161.1 4
60.23 odd 4 720.5.l.a.161.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.5.d.b.71.1 ✓ 4 5.3 odd 4
90.5.d.b.71.4 yes 4 15.8 even 4
450.5.b.c.449.1 8 5.4 even 2 inner
450.5.b.c.449.4 8 3.2 odd 2 inner
450.5.b.c.449.5 8 15.14 odd 2 inner
450.5.b.c.449.8 8 1.1 even 1 trivial
450.5.d.e.251.1 4 15.2 even 4
450.5.d.e.251.3 4 5.2 odd 4
720.5.l.a.161.1 4 20.3 even 4
720.5.l.a.161.3 4 60.23 odd 4