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Properties

Label 450.4.a.a.1.1
Level $450$
Weight $4$
Character 450.1
Self dual yes
Analytic conductor $26.551$
Analytic rank $0$
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] = [450,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,0,4,0,0,-34,-8,0,0,-27] 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: yes
Analytic conductor: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} -34.0000 q^{7} -8.00000 q^{8} -27.0000 q^{11} -28.0000 q^{13} +68.0000 q^{14} +16.0000 q^{16} -21.0000 q^{17} +35.0000 q^{19} +54.0000 q^{22} +78.0000 q^{23} +56.0000 q^{26} -136.000 q^{28} +120.000 q^{29} +182.000 q^{31} -32.0000 q^{32} +42.0000 q^{34} +146.000 q^{37} -70.0000 q^{38} -357.000 q^{41} -148.000 q^{43} -108.000 q^{44} -156.000 q^{46} +84.0000 q^{47} +813.000 q^{49} -112.000 q^{52} -702.000 q^{53} +272.000 q^{56} -240.000 q^{58} +840.000 q^{59} -238.000 q^{61} -364.000 q^{62} +64.0000 q^{64} +461.000 q^{67} -84.0000 q^{68} +708.000 q^{71} -133.000 q^{73} -292.000 q^{74} +140.000 q^{76} +918.000 q^{77} +650.000 q^{79} +714.000 q^{82} +903.000 q^{83} +296.000 q^{86} +216.000 q^{88} -735.000 q^{89} +952.000 q^{91} +312.000 q^{92} -168.000 q^{94} +1106.00 q^{97} -1626.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −34.0000 −1.83583 −0.917914 − 0.396780i \(-0.870128\pi\)
−0.917914 + 0.396780i \(0.870128\pi\)
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −27.0000 −0.740073 −0.370037 − 0.929017i \(-0.620655\pi\)
−0.370037 + 0.929017i \(0.620655\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 − 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) 68.0000 1.29813
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −21.0000 −0.299603 −0.149801 − 0.988716i \(-0.547863\pi\)
−0.149801 + 0.988716i \(0.547863\pi\)
\(18\) 0 0
\(19\) 35.0000 0.422608 0.211304 − 0.977420i \(-0.432229\pi\)
0.211304 + 0.977420i \(0.432229\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 54.0000 0.523311
\(23\) 78.0000 0.707136 0.353568 − 0.935409i \(-0.384968\pi\)
0.353568 + 0.935409i \(0.384968\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 56.0000 0.422404
\(27\) 0 0
\(28\) −136.000 −0.917914
\(29\) 120.000 0.768395 0.384197 − 0.923251i \(-0.374478\pi\)
0.384197 + 0.923251i \(0.374478\pi\)
\(30\) 0 0
\(31\) 182.000 1.05446 0.527228 − 0.849724i \(-0.323231\pi\)
0.527228 + 0.849724i \(0.323231\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) 42.0000 0.211851
\(35\) 0 0
\(36\) 0 0
\(37\) 146.000 0.648710 0.324355 − 0.945936i \(-0.394853\pi\)
0.324355 + 0.945936i \(0.394853\pi\)
\(38\) −70.0000 −0.298829
\(39\) 0 0
\(40\) 0 0
\(41\) −357.000 −1.35985 −0.679927 − 0.733280i \(-0.737989\pi\)
−0.679927 + 0.733280i \(0.737989\pi\)
\(42\) 0 0
\(43\) −148.000 −0.524879 −0.262439 − 0.964948i \(-0.584527\pi\)
−0.262439 + 0.964948i \(0.584527\pi\)
\(44\) −108.000 −0.370037
\(45\) 0 0
\(46\) −156.000 −0.500021
\(47\) 84.0000 0.260695 0.130347 − 0.991468i \(-0.458391\pi\)
0.130347 + 0.991468i \(0.458391\pi\)
\(48\) 0 0
\(49\) 813.000 2.37026
\(50\) 0 0
\(51\) 0 0
\(52\) −112.000 −0.298685
\(53\) −702.000 −1.81938 −0.909690 − 0.415288i \(-0.863681\pi\)
−0.909690 + 0.415288i \(0.863681\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 272.000 0.649063
\(57\) 0 0
\(58\) −240.000 −0.543337
\(59\) 840.000 1.85354 0.926769 − 0.375633i \(-0.122575\pi\)
0.926769 + 0.375633i \(0.122575\pi\)
\(60\) 0 0
\(61\) −238.000 −0.499554 −0.249777 − 0.968303i \(-0.580357\pi\)
−0.249777 + 0.968303i \(0.580357\pi\)
\(62\) −364.000 −0.745614
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 461.000 0.840599 0.420299 − 0.907386i \(-0.361925\pi\)
0.420299 + 0.907386i \(0.361925\pi\)
\(68\) −84.0000 −0.149801
\(69\) 0 0
\(70\) 0 0
\(71\) 708.000 1.18344 0.591719 − 0.806144i \(-0.298449\pi\)
0.591719 + 0.806144i \(0.298449\pi\)
\(72\) 0 0
\(73\) −133.000 −0.213239 −0.106620 − 0.994300i \(-0.534003\pi\)
−0.106620 + 0.994300i \(0.534003\pi\)
\(74\) −292.000 −0.458707
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) 918.000 1.35865
\(78\) 0 0
\(79\) 650.000 0.925705 0.462853 − 0.886435i \(-0.346826\pi\)
0.462853 + 0.886435i \(0.346826\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 714.000 0.961562
\(83\) 903.000 1.19418 0.597091 − 0.802173i \(-0.296323\pi\)
0.597091 + 0.802173i \(0.296323\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 296.000 0.371145
\(87\) 0 0
\(88\) 216.000 0.261655
\(89\) −735.000 −0.875392 −0.437696 − 0.899123i \(-0.644205\pi\)
−0.437696 + 0.899123i \(0.644205\pi\)
\(90\) 0 0
\(91\) 952.000 1.09667
\(92\) 312.000 0.353568
\(93\) 0 0
\(94\) −168.000 −0.184339
\(95\) 0 0
\(96\) 0 0
\(97\) 1106.00 1.15770 0.578852 − 0.815433i \(-0.303501\pi\)
0.578852 + 0.815433i \(0.303501\pi\)
\(98\) −1626.00 −1.67603
\(99\) 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 450.4.a.a.1.1 1
3.2 odd 2 50.4.a.e.1.1 yes 1
5.2 odd 4 450.4.c.c.199.1 2
5.3 odd 4 450.4.c.c.199.2 2
5.4 even 2 450.4.a.t.1.1 1
12.11 even 2 400.4.a.d.1.1 1
15.2 even 4 50.4.b.b.49.2 2
15.8 even 4 50.4.b.b.49.1 2
15.14 odd 2 50.4.a.a.1.1 ✓ 1
21.20 even 2 2450.4.a.y.1.1 1
24.5 odd 2 1600.4.a.f.1.1 1
24.11 even 2 1600.4.a.bv.1.1 1
60.23 odd 4 400.4.c.d.49.1 2
60.47 odd 4 400.4.c.d.49.2 2
60.59 even 2 400.4.a.r.1.1 1
105.104 even 2 2450.4.a.t.1.1 1
120.29 odd 2 1600.4.a.bu.1.1 1
120.59 even 2 1600.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.4.a.a.1.1 ✓ 1 15.14 odd 2
50.4.a.e.1.1 yes 1 3.2 odd 2
50.4.b.b.49.1 2 15.8 even 4
50.4.b.b.49.2 2 15.2 even 4
400.4.a.d.1.1 1 12.11 even 2
400.4.a.r.1.1 1 60.59 even 2
400.4.c.d.49.1 2 60.23 odd 4
400.4.c.d.49.2 2 60.47 odd 4
450.4.a.a.1.1 1 1.1 even 1 trivial
450.4.a.t.1.1 1 5.4 even 2
450.4.c.c.199.1 2 5.2 odd 4
450.4.c.c.199.2 2 5.3 odd 4
1600.4.a.f.1.1 1 24.5 odd 2
1600.4.a.g.1.1 1 120.59 even 2
1600.4.a.bu.1.1 1 120.29 odd 2
1600.4.a.bv.1.1 1 24.11 even 2
2450.4.a.t.1.1 1 105.104 even 2
2450.4.a.y.1.1 1 21.20 even 2