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Properties

Label 450.4.c.c.199.1
Level $450$
Weight $4$
Character 450.199
Analytic conductor $26.551$
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] = [450,4,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, 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, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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,-8,0,0,0,0,0,0,-54,0,0,-136] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.5508595026\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.4.c.c.199.2

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000i q^{2} -4.00000 q^{4} -34.0000i q^{7} +8.00000i q^{8} -27.0000 q^{11} +28.0000i q^{13} -68.0000 q^{14} +16.0000 q^{16} -21.0000i q^{17} -35.0000 q^{19} +54.0000i q^{22} -78.0000i q^{23} +56.0000 q^{26} +136.000i q^{28} -120.000 q^{29} +182.000 q^{31} -32.0000i q^{32} -42.0000 q^{34} +146.000i q^{37} +70.0000i q^{38} -357.000 q^{41} +148.000i q^{43} +108.000 q^{44} -156.000 q^{46} +84.0000i q^{47} -813.000 q^{49} -112.000i q^{52} +702.000i q^{53} +272.000 q^{56} +240.000i q^{58} -840.000 q^{59} -238.000 q^{61} -364.000i q^{62} -64.0000 q^{64} +461.000i q^{67} +84.0000i q^{68} +708.000 q^{71} +133.000i q^{73} +292.000 q^{74} +140.000 q^{76} +918.000i q^{77} -650.000 q^{79} +714.000i q^{82} -903.000i q^{83} +296.000 q^{86} -216.000i q^{88} +735.000 q^{89} +952.000 q^{91} +312.000i q^{92} +168.000 q^{94} +1106.00i q^{97} +1626.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 54 q^{11} - 136 q^{14} + 32 q^{16} - 70 q^{19} + 112 q^{26} - 240 q^{29} + 364 q^{31} - 84 q^{34} - 714 q^{41} + 216 q^{44} - 312 q^{46} - 1626 q^{49} + 544 q^{56} - 1680 q^{59} - 476 q^{61}+ \cdots + 336 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.00000i − 0.707107i
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 34.0000i − 1.83583i −0.396780 − 0.917914i \(-0.629872\pi\)
0.396780 − 0.917914i \(-0.370128\pi\)
\(8\) 8.00000i 0.353553i
\(9\) 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
\(13\) 28.0000i 0.597369i 0.954352 + 0.298685i \(0.0965479\pi\)
−0.954352 + 0.298685i \(0.903452\pi\)
\(14\) −68.0000 −1.29813
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) − 21.0000i − 0.299603i −0.988716 − 0.149801i \(-0.952137\pi\)
0.988716 − 0.149801i \(-0.0478634\pi\)
\(18\) 0 0
\(19\) −35.0000 −0.422608 −0.211304 − 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 54.0000i 0.523311i
\(23\) − 78.0000i − 0.707136i −0.935409 − 0.353568i \(-0.884968\pi\)
0.935409 − 0.353568i \(-0.115032\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 56.0000 0.422404
\(27\) 0 0
\(28\) 136.000i 0.917914i
\(29\) −120.000 −0.768395 −0.384197 − 0.923251i \(-0.625522\pi\)
−0.384197 + 0.923251i \(0.625522\pi\)
\(30\) 0 0
\(31\) 182.000 1.05446 0.527228 − 0.849724i \(-0.323231\pi\)
0.527228 + 0.849724i \(0.323231\pi\)
\(32\) − 32.0000i − 0.176777i
\(33\) 0 0
\(34\) −42.0000 −0.211851
\(35\) 0 0
\(36\) 0 0
\(37\) 146.000i 0.648710i 0.945936 + 0.324355i \(0.105147\pi\)
−0.945936 + 0.324355i \(0.894853\pi\)
\(38\) 70.0000i 0.298829i
\(39\) 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
\(43\) 148.000i 0.524879i 0.964948 + 0.262439i \(0.0845270\pi\)
−0.964948 + 0.262439i \(0.915473\pi\)
\(44\) 108.000 0.370037
\(45\) 0 0
\(46\) −156.000 −0.500021
\(47\) 84.0000i 0.260695i 0.991468 + 0.130347i \(0.0416093\pi\)
−0.991468 + 0.130347i \(0.958391\pi\)
\(48\) 0 0
\(49\) −813.000 −2.37026
\(50\) 0 0
\(51\) 0 0
\(52\) − 112.000i − 0.298685i
\(53\) 702.000i 1.81938i 0.415288 + 0.909690i \(0.363681\pi\)
−0.415288 + 0.909690i \(0.636319\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 272.000 0.649063
\(57\) 0 0
\(58\) 240.000i 0.543337i
\(59\) −840.000 −1.85354 −0.926769 − 0.375633i \(-0.877425\pi\)
−0.926769 + 0.375633i \(0.877425\pi\)
\(60\) 0 0
\(61\) −238.000 −0.499554 −0.249777 − 0.968303i \(-0.580357\pi\)
−0.249777 + 0.968303i \(0.580357\pi\)
\(62\) − 364.000i − 0.745614i
\(63\) 0 0
\(64\) −64.0000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 461.000i 0.840599i 0.907386 + 0.420299i \(0.138075\pi\)
−0.907386 + 0.420299i \(0.861925\pi\)
\(68\) 84.0000i 0.149801i
\(69\) 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
\(73\) 133.000i 0.213239i 0.994300 + 0.106620i \(0.0340027\pi\)
−0.994300 + 0.106620i \(0.965997\pi\)
\(74\) 292.000 0.458707
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) 918.000i 1.35865i
\(78\) 0 0
\(79\) −650.000 −0.925705 −0.462853 − 0.886435i \(-0.653174\pi\)
−0.462853 + 0.886435i \(0.653174\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 714.000i 0.961562i
\(83\) − 903.000i − 1.19418i −0.802173 − 0.597091i \(-0.796323\pi\)
0.802173 − 0.597091i \(-0.203677\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 296.000 0.371145
\(87\) 0 0
\(88\) − 216.000i − 0.261655i
\(89\) 735.000 0.875392 0.437696 − 0.899123i \(-0.355795\pi\)
0.437696 + 0.899123i \(0.355795\pi\)
\(90\) 0 0
\(91\) 952.000 1.09667
\(92\) 312.000i 0.353568i
\(93\) 0 0
\(94\) 168.000 0.184339
\(95\) 0 0
\(96\) 0 0
\(97\) 1106.00i 1.15770i 0.815433 + 0.578852i \(0.196499\pi\)
−0.815433 + 0.578852i \(0.803501\pi\)
\(98\) 1626.00i 1.67603i
\(99\) 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.c.c.199.1 2
3.2 odd 2 50.4.b.b.49.2 2
5.2 odd 4 450.4.a.t.1.1 1
5.3 odd 4 450.4.a.a.1.1 1
5.4 even 2 inner 450.4.c.c.199.2 2
12.11 even 2 400.4.c.d.49.2 2
15.2 even 4 50.4.a.a.1.1 ✓ 1
15.8 even 4 50.4.a.e.1.1 yes 1
15.14 odd 2 50.4.b.b.49.1 2
60.23 odd 4 400.4.a.d.1.1 1
60.47 odd 4 400.4.a.r.1.1 1
60.59 even 2 400.4.c.d.49.1 2
105.62 odd 4 2450.4.a.t.1.1 1
105.83 odd 4 2450.4.a.y.1.1 1
120.53 even 4 1600.4.a.f.1.1 1
120.77 even 4 1600.4.a.bu.1.1 1
120.83 odd 4 1600.4.a.bv.1.1 1
120.107 odd 4 1600.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.4.a.a.1.1 ✓ 1 15.2 even 4
50.4.a.e.1.1 yes 1 15.8 even 4
50.4.b.b.49.1 2 15.14 odd 2
50.4.b.b.49.2 2 3.2 odd 2
400.4.a.d.1.1 1 60.23 odd 4
400.4.a.r.1.1 1 60.47 odd 4
400.4.c.d.49.1 2 60.59 even 2
400.4.c.d.49.2 2 12.11 even 2
450.4.a.a.1.1 1 5.3 odd 4
450.4.a.t.1.1 1 5.2 odd 4
450.4.c.c.199.1 2 1.1 even 1 trivial
450.4.c.c.199.2 2 5.4 even 2 inner
1600.4.a.f.1.1 1 120.53 even 4
1600.4.a.g.1.1 1 120.107 odd 4
1600.4.a.bu.1.1 1 120.77 even 4
1600.4.a.bv.1.1 1 120.83 odd 4
2450.4.a.t.1.1 1 105.62 odd 4
2450.4.a.y.1.1 1 105.83 odd 4