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Properties

Label 450.8.c.g.199.2
Level $450$
Weight $8$
Character 450.199
Analytic conductor $140.573$
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] = [450,8,Mod(199,450)] 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("450.199"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-128,0,0,0,0,0,0,-2184] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(140.573261468\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.8.c.g.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.00000i q^{2} -64.0000 q^{4} +1016.00i q^{7} -512.000i q^{8} -1092.00 q^{11} -1382.00i q^{13} -8128.00 q^{14} +4096.00 q^{16} -14706.0i q^{17} +39940.0 q^{19} -8736.00i q^{22} +68712.0i q^{23} +11056.0 q^{26} -65024.0i q^{28} -102570. q^{29} +227552. q^{31} +32768.0i q^{32} +117648. q^{34} +160526. i q^{37} +319520. i q^{38} -10842.0 q^{41} +630748. i q^{43} +69888.0 q^{44} -549696. q^{46} -472656. i q^{47} -208713. q^{49} +88448.0i q^{52} -1.49402e6i q^{53} +520192. q^{56} -820560. i q^{58} +2.64066e6 q^{59} +827702. q^{61} +1.82042e6i q^{62} -262144. q^{64} -126004. i q^{67} +941184. i q^{68} +1.41473e6 q^{71} -980282. i q^{73} -1.28421e6 q^{74} -2.55616e6 q^{76} -1.10947e6i q^{77} +3.56680e6 q^{79} -86736.0i q^{82} +5.67289e6i q^{83} -5.04598e6 q^{86} +559104. i q^{88} -1.19512e7 q^{89} +1.40411e6 q^{91} -4.39757e6i q^{92} +3.78125e6 q^{94} +8.68215e6i q^{97} -1.66970e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} - 2184 q^{11} - 16256 q^{14} + 8192 q^{16} + 79880 q^{19} + 22112 q^{26} - 205140 q^{29} + 455104 q^{31} + 235296 q^{34} - 21684 q^{41} + 139776 q^{44} - 1099392 q^{46} - 417426 q^{49} + 1040384 q^{56}+ \cdots + 7562496 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000i 0.707107i
\(3\) 0 0
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1016.00i 1.11957i 0.828638 + 0.559784i \(0.189116\pi\)
−0.828638 + 0.559784i \(0.810884\pi\)
\(8\) − 512.000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −1092.00 −0.247371 −0.123685 − 0.992321i \(-0.539471\pi\)
−0.123685 + 0.992321i \(0.539471\pi\)
\(12\) 0 0
\(13\) − 1382.00i − 0.174464i −0.996188 − 0.0872321i \(-0.972198\pi\)
0.996188 − 0.0872321i \(-0.0278022\pi\)
\(14\) −8128.00 −0.791654
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) − 14706.0i − 0.725978i −0.931793 − 0.362989i \(-0.881756\pi\)
0.931793 − 0.362989i \(-0.118244\pi\)
\(18\) 0 0
\(19\) 39940.0 1.33589 0.667945 − 0.744211i \(-0.267174\pi\)
0.667945 + 0.744211i \(0.267174\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 8736.00i − 0.174917i
\(23\) 68712.0i 1.17757i 0.808291 + 0.588783i \(0.200393\pi\)
−0.808291 + 0.588783i \(0.799607\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 11056.0 0.123365
\(27\) 0 0
\(28\) − 65024.0i − 0.559784i
\(29\) −102570. −0.780957 −0.390479 − 0.920612i \(-0.627690\pi\)
−0.390479 + 0.920612i \(0.627690\pi\)
\(30\) 0 0
\(31\) 227552. 1.37188 0.685938 − 0.727660i \(-0.259392\pi\)
0.685938 + 0.727660i \(0.259392\pi\)
\(32\) 32768.0i 0.176777i
\(33\) 0 0
\(34\) 117648. 0.513344
\(35\) 0 0
\(36\) 0 0
\(37\) 160526.i 0.521002i 0.965474 + 0.260501i \(0.0838877\pi\)
−0.965474 + 0.260501i \(0.916112\pi\)
\(38\) 319520.i 0.944616i
\(39\) 0 0
\(40\) 0 0
\(41\) −10842.0 −0.0245678 −0.0122839 − 0.999925i \(-0.503910\pi\)
−0.0122839 + 0.999925i \(0.503910\pi\)
\(42\) 0 0
\(43\) 630748.i 1.20981i 0.796299 + 0.604904i \(0.206788\pi\)
−0.796299 + 0.604904i \(0.793212\pi\)
\(44\) 69888.0 0.123685
\(45\) 0 0
\(46\) −549696. −0.832665
\(47\) − 472656.i − 0.664053i −0.943270 − 0.332026i \(-0.892268\pi\)
0.943270 − 0.332026i \(-0.107732\pi\)
\(48\) 0 0
\(49\) −208713. −0.253433
\(50\) 0 0
\(51\) 0 0
\(52\) 88448.0i 0.0872321i
\(53\) − 1.49402e6i − 1.37845i −0.724548 − 0.689224i \(-0.757952\pi\)
0.724548 − 0.689224i \(-0.242048\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 520192. 0.395827
\(57\) 0 0
\(58\) − 820560.i − 0.552220i
\(59\) 2.64066e6 1.67390 0.836952 − 0.547277i \(-0.184335\pi\)
0.836952 + 0.547277i \(0.184335\pi\)
\(60\) 0 0
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) 1.82042e6i 0.970063i
\(63\) 0 0
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 126004.i − 0.0511826i −0.999672 − 0.0255913i \(-0.991853\pi\)
0.999672 − 0.0255913i \(-0.00814686\pi\)
\(68\) 941184.i 0.362989i
\(69\) 0 0
\(70\) 0 0
\(71\) 1.41473e6 0.469104 0.234552 − 0.972104i \(-0.424638\pi\)
0.234552 + 0.972104i \(0.424638\pi\)
\(72\) 0 0
\(73\) − 980282.i − 0.294931i −0.989067 − 0.147466i \(-0.952888\pi\)
0.989067 − 0.147466i \(-0.0471116\pi\)
\(74\) −1.28421e6 −0.368404
\(75\) 0 0
\(76\) −2.55616e6 −0.667945
\(77\) − 1.10947e6i − 0.276948i
\(78\) 0 0
\(79\) 3.56680e6 0.813924 0.406962 − 0.913445i \(-0.366588\pi\)
0.406962 + 0.913445i \(0.366588\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 86736.0i − 0.0173720i
\(83\) 5.67289e6i 1.08901i 0.838758 + 0.544504i \(0.183282\pi\)
−0.838758 + 0.544504i \(0.816718\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.04598e6 −0.855463
\(87\) 0 0
\(88\) 559104.i 0.0874587i
\(89\) −1.19512e7 −1.79699 −0.898496 − 0.438982i \(-0.855339\pi\)
−0.898496 + 0.438982i \(0.855339\pi\)
\(90\) 0 0
\(91\) 1.40411e6 0.195325
\(92\) − 4.39757e6i − 0.588783i
\(93\) 0 0
\(94\) 3.78125e6 0.469556
\(95\) 0 0
\(96\) 0 0
\(97\) 8.68215e6i 0.965886i 0.875652 + 0.482943i \(0.160432\pi\)
−0.875652 + 0.482943i \(0.839568\pi\)
\(98\) − 1.66970e6i − 0.179204i
\(99\) 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 450.8.c.g.199.2 2
3.2 odd 2 50.8.b.c.49.1 2
5.2 odd 4 450.8.a.c.1.1 1
5.3 odd 4 18.8.a.b.1.1 1
5.4 even 2 inner 450.8.c.g.199.1 2
12.11 even 2 400.8.c.j.49.2 2
15.2 even 4 50.8.a.g.1.1 1
15.8 even 4 2.8.a.a.1.1 ✓ 1
15.14 odd 2 50.8.b.c.49.2 2
20.3 even 4 144.8.a.i.1.1 1
40.3 even 4 576.8.a.f.1.1 1
40.13 odd 4 576.8.a.g.1.1 1
45.13 odd 12 162.8.c.a.55.1 2
45.23 even 12 162.8.c.l.55.1 2
45.38 even 12 162.8.c.l.109.1 2
45.43 odd 12 162.8.c.a.109.1 2
60.23 odd 4 16.8.a.b.1.1 1
60.47 odd 4 400.8.a.l.1.1 1
60.59 even 2 400.8.c.j.49.1 2
105.23 even 12 98.8.c.d.67.1 2
105.38 odd 12 98.8.c.e.79.1 2
105.53 even 12 98.8.c.d.79.1 2
105.68 odd 12 98.8.c.e.67.1 2
105.83 odd 4 98.8.a.a.1.1 1
120.53 even 4 64.8.a.c.1.1 1
120.83 odd 4 64.8.a.e.1.1 1
165.98 odd 4 242.8.a.e.1.1 1
195.8 odd 4 338.8.b.d.337.1 2
195.38 even 4 338.8.a.d.1.1 1
195.83 odd 4 338.8.b.d.337.2 2
240.53 even 4 256.8.b.b.129.1 2
240.83 odd 4 256.8.b.f.129.1 2
240.173 even 4 256.8.b.b.129.2 2
240.203 odd 4 256.8.b.f.129.2 2
255.203 even 4 578.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 15.8 even 4
16.8.a.b.1.1 1 60.23 odd 4
18.8.a.b.1.1 1 5.3 odd 4
50.8.a.g.1.1 1 15.2 even 4
50.8.b.c.49.1 2 3.2 odd 2
50.8.b.c.49.2 2 15.14 odd 2
64.8.a.c.1.1 1 120.53 even 4
64.8.a.e.1.1 1 120.83 odd 4
98.8.a.a.1.1 1 105.83 odd 4
98.8.c.d.67.1 2 105.23 even 12
98.8.c.d.79.1 2 105.53 even 12
98.8.c.e.67.1 2 105.68 odd 12
98.8.c.e.79.1 2 105.38 odd 12
144.8.a.i.1.1 1 20.3 even 4
162.8.c.a.55.1 2 45.13 odd 12
162.8.c.a.109.1 2 45.43 odd 12
162.8.c.l.55.1 2 45.23 even 12
162.8.c.l.109.1 2 45.38 even 12
242.8.a.e.1.1 1 165.98 odd 4
256.8.b.b.129.1 2 240.53 even 4
256.8.b.b.129.2 2 240.173 even 4
256.8.b.f.129.1 2 240.83 odd 4
256.8.b.f.129.2 2 240.203 odd 4
338.8.a.d.1.1 1 195.38 even 4
338.8.b.d.337.1 2 195.8 odd 4
338.8.b.d.337.2 2 195.83 odd 4
400.8.a.l.1.1 1 60.47 odd 4
400.8.c.j.49.1 2 60.59 even 2
400.8.c.j.49.2 2 12.11 even 2
450.8.a.c.1.1 1 5.2 odd 4
450.8.c.g.199.1 2 5.4 even 2 inner
450.8.c.g.199.2 2 1.1 even 1 trivial
576.8.a.f.1.1 1 40.3 even 4
576.8.a.g.1.1 1 40.13 odd 4
578.8.a.b.1.1 1 255.203 even 4