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Properties

Label 450.8.a.c.1.1
Level $450$
Weight $8$
Character 450.1
Self dual yes
Analytic conductor $140.573$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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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(1,450)] 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("450.1"); 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, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-8,0,64,0,0,-1016] 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: yes
Analytic conductor: \(140.573261468\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 450.1

$q$-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)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +64.0000 q^{4} -1016.00 q^{7} -512.000 q^{8} -1092.00 q^{11} -1382.00 q^{13} +8128.00 q^{14} +4096.00 q^{16} +14706.0 q^{17} -39940.0 q^{19} +8736.00 q^{22} +68712.0 q^{23} +11056.0 q^{26} -65024.0 q^{28} +102570. q^{29} +227552. q^{31} -32768.0 q^{32} -117648. q^{34} -160526. q^{37} +319520. q^{38} -10842.0 q^{41} +630748. q^{43} -69888.0 q^{44} -549696. q^{46} +472656. q^{47} +208713. q^{49} -88448.0 q^{52} -1.49402e6 q^{53} +520192. q^{56} -820560. q^{58} -2.64066e6 q^{59} +827702. q^{61} -1.82042e6 q^{62} +262144. q^{64} +126004. q^{67} +941184. q^{68} +1.41473e6 q^{71} -980282. q^{73} +1.28421e6 q^{74} -2.55616e6 q^{76} +1.10947e6 q^{77} -3.56680e6 q^{79} +86736.0 q^{82} +5.67289e6 q^{83} -5.04598e6 q^{86} +559104. q^{88} +1.19512e7 q^{89} +1.40411e6 q^{91} +4.39757e6 q^{92} -3.78125e6 q^{94} -8.68215e6 q^{97} -1.66970e6 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.00000 −0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1016.00 −1.11957 −0.559784 − 0.828638i \(-0.689116\pi\)
−0.559784 + 0.828638i \(0.689116\pi\)
\(8\) −512.000 −0.353553
\(9\) 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
\(13\) −1382.00 −0.174464 −0.0872321 − 0.996188i \(-0.527802\pi\)
−0.0872321 + 0.996188i \(0.527802\pi\)
\(14\) 8128.00 0.791654
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) 14706.0 0.725978 0.362989 − 0.931793i \(-0.381756\pi\)
0.362989 + 0.931793i \(0.381756\pi\)
\(18\) 0 0
\(19\) −39940.0 −1.33589 −0.667945 − 0.744211i \(-0.732826\pi\)
−0.667945 + 0.744211i \(0.732826\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8736.00 0.174917
\(23\) 68712.0 1.17757 0.588783 − 0.808291i \(-0.299607\pi\)
0.588783 + 0.808291i \(0.299607\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 11056.0 0.123365
\(27\) 0 0
\(28\) −65024.0 −0.559784
\(29\) 102570. 0.780957 0.390479 − 0.920612i \(-0.372310\pi\)
0.390479 + 0.920612i \(0.372310\pi\)
\(30\) 0 0
\(31\) 227552. 1.37188 0.685938 − 0.727660i \(-0.259392\pi\)
0.685938 + 0.727660i \(0.259392\pi\)
\(32\) −32768.0 −0.176777
\(33\) 0 0
\(34\) −117648. −0.513344
\(35\) 0 0
\(36\) 0 0
\(37\) −160526. −0.521002 −0.260501 − 0.965474i \(-0.583888\pi\)
−0.260501 + 0.965474i \(0.583888\pi\)
\(38\) 319520. 0.944616
\(39\) 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. 1.20981 0.604904 − 0.796299i \(-0.293212\pi\)
0.604904 + 0.796299i \(0.293212\pi\)
\(44\) −69888.0 −0.123685
\(45\) 0 0
\(46\) −549696. −0.832665
\(47\) 472656. 0.664053 0.332026 − 0.943270i \(-0.392268\pi\)
0.332026 + 0.943270i \(0.392268\pi\)
\(48\) 0 0
\(49\) 208713. 0.253433
\(50\) 0 0
\(51\) 0 0
\(52\) −88448.0 −0.0872321
\(53\) −1.49402e6 −1.37845 −0.689224 − 0.724548i \(-0.742048\pi\)
−0.689224 + 0.724548i \(0.742048\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 520192. 0.395827
\(57\) 0 0
\(58\) −820560. −0.552220
\(59\) −2.64066e6 −1.67390 −0.836952 − 0.547277i \(-0.815665\pi\)
−0.836952 + 0.547277i \(0.815665\pi\)
\(60\) 0 0
\(61\) 827702. 0.466895 0.233448 − 0.972369i \(-0.424999\pi\)
0.233448 + 0.972369i \(0.424999\pi\)
\(62\) −1.82042e6 −0.970063
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 126004. 0.0511826 0.0255913 − 0.999672i \(-0.491853\pi\)
0.0255913 + 0.999672i \(0.491853\pi\)
\(68\) 941184. 0.362989
\(69\) 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. −0.294931 −0.147466 − 0.989067i \(-0.547112\pi\)
−0.147466 + 0.989067i \(0.547112\pi\)
\(74\) 1.28421e6 0.368404
\(75\) 0 0
\(76\) −2.55616e6 −0.667945
\(77\) 1.10947e6 0.276948
\(78\) 0 0
\(79\) −3.56680e6 −0.813924 −0.406962 − 0.913445i \(-0.633412\pi\)
−0.406962 + 0.913445i \(0.633412\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 86736.0 0.0173720
\(83\) 5.67289e6 1.08901 0.544504 − 0.838758i \(-0.316718\pi\)
0.544504 + 0.838758i \(0.316718\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.04598e6 −0.855463
\(87\) 0 0
\(88\) 559104. 0.0874587
\(89\) 1.19512e7 1.79699 0.898496 − 0.438982i \(-0.144661\pi\)
0.898496 + 0.438982i \(0.144661\pi\)
\(90\) 0 0
\(91\) 1.40411e6 0.195325
\(92\) 4.39757e6 0.588783
\(93\) 0 0
\(94\) −3.78125e6 −0.469556
\(95\) 0 0
\(96\) 0 0
\(97\) −8.68215e6 −0.965886 −0.482943 − 0.875652i \(-0.660432\pi\)
−0.482943 + 0.875652i \(0.660432\pi\)
\(98\) −1.66970e6 −0.179204
\(99\) 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.a.c.1.1 1
3.2 odd 2 50.8.a.g.1.1 1
5.2 odd 4 450.8.c.g.199.1 2
5.3 odd 4 450.8.c.g.199.2 2
5.4 even 2 18.8.a.b.1.1 1
12.11 even 2 400.8.a.l.1.1 1
15.2 even 4 50.8.b.c.49.2 2
15.8 even 4 50.8.b.c.49.1 2
15.14 odd 2 2.8.a.a.1.1 ✓ 1
20.19 odd 2 144.8.a.i.1.1 1
40.19 odd 2 576.8.a.f.1.1 1
40.29 even 2 576.8.a.g.1.1 1
45.4 even 6 162.8.c.a.55.1 2
45.14 odd 6 162.8.c.l.55.1 2
45.29 odd 6 162.8.c.l.109.1 2
45.34 even 6 162.8.c.a.109.1 2
60.23 odd 4 400.8.c.j.49.2 2
60.47 odd 4 400.8.c.j.49.1 2
60.59 even 2 16.8.a.b.1.1 1
105.44 odd 6 98.8.c.d.67.1 2
105.59 even 6 98.8.c.e.79.1 2
105.74 odd 6 98.8.c.d.79.1 2
105.89 even 6 98.8.c.e.67.1 2
105.104 even 2 98.8.a.a.1.1 1
120.29 odd 2 64.8.a.c.1.1 1
120.59 even 2 64.8.a.e.1.1 1
165.164 even 2 242.8.a.e.1.1 1
195.44 even 4 338.8.b.d.337.2 2
195.164 even 4 338.8.b.d.337.1 2
195.194 odd 2 338.8.a.d.1.1 1
240.29 odd 4 256.8.b.b.129.2 2
240.59 even 4 256.8.b.f.129.2 2
240.149 odd 4 256.8.b.b.129.1 2
240.179 even 4 256.8.b.f.129.1 2
255.254 odd 2 578.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.8.a.a.1.1 ✓ 1 15.14 odd 2
16.8.a.b.1.1 1 60.59 even 2
18.8.a.b.1.1 1 5.4 even 2
50.8.a.g.1.1 1 3.2 odd 2
50.8.b.c.49.1 2 15.8 even 4
50.8.b.c.49.2 2 15.2 even 4
64.8.a.c.1.1 1 120.29 odd 2
64.8.a.e.1.1 1 120.59 even 2
98.8.a.a.1.1 1 105.104 even 2
98.8.c.d.67.1 2 105.44 odd 6
98.8.c.d.79.1 2 105.74 odd 6
98.8.c.e.67.1 2 105.89 even 6
98.8.c.e.79.1 2 105.59 even 6
144.8.a.i.1.1 1 20.19 odd 2
162.8.c.a.55.1 2 45.4 even 6
162.8.c.a.109.1 2 45.34 even 6
162.8.c.l.55.1 2 45.14 odd 6
162.8.c.l.109.1 2 45.29 odd 6
242.8.a.e.1.1 1 165.164 even 2
256.8.b.b.129.1 2 240.149 odd 4
256.8.b.b.129.2 2 240.29 odd 4
256.8.b.f.129.1 2 240.179 even 4
256.8.b.f.129.2 2 240.59 even 4
338.8.a.d.1.1 1 195.194 odd 2
338.8.b.d.337.1 2 195.164 even 4
338.8.b.d.337.2 2 195.44 even 4
400.8.a.l.1.1 1 12.11 even 2
400.8.c.j.49.1 2 60.47 odd 4
400.8.c.j.49.2 2 60.23 odd 4
450.8.a.c.1.1 1 1.1 even 1 trivial
450.8.c.g.199.1 2 5.2 odd 4
450.8.c.g.199.2 2 5.3 odd 4
576.8.a.f.1.1 1 40.19 odd 2
576.8.a.g.1.1 1 40.29 even 2
578.8.a.b.1.1 1 255.254 odd 2