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Properties

Label 450.2.c.d.199.2
Level $450$
Weight $2$
Character 450.199
Analytic conductor $3.593$
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] = [450,2,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, 2); 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, 2, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
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,-2,0,0,0,0,0,0,12] 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: \(3.59326809096\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
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 450.2.c.d.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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.00000i q^{8} +6.00000 q^{11} -4.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} +6.00000i q^{22} +4.00000 q^{26} +2.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} -6.00000 q^{34} -8.00000i q^{37} +4.00000i q^{38} +8.00000i q^{43} -6.00000 q^{44} +3.00000 q^{49} +4.00000i q^{52} -6.00000i q^{53} -2.00000 q^{56} +6.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +4.00000i q^{67} -6.00000i q^{68} -12.0000 q^{71} -10.0000i q^{73} +8.00000 q^{74} -4.00000 q^{76} -12.0000i q^{77} +4.00000 q^{79} +12.0000i q^{83} -8.00000 q^{86} -6.00000i q^{88} -12.0000 q^{89} -8.00000 q^{91} -2.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 12 q^{11} + 4 q^{14} + 2 q^{16} + 8 q^{19} + 8 q^{26} + 12 q^{29} - 8 q^{31} - 12 q^{34} - 12 q^{44} + 6 q^{49} - 4 q^{56} - 12 q^{59} + 4 q^{61} - 2 q^{64} - 24 q^{71} + 16 q^{74} - 8 q^{76}+ \cdots - 16 q^{91}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 − 0.377964i \(-0.876624\pi\)
0.925820 − 0.377964i \(-0.123376\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) − 4.00000i − 1.10940i −0.832050 − 0.554700i \(-0.812833\pi\)
0.832050 − 0.554700i \(-0.187167\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000i 1.27920i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 4.00000i 0.648886i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000i 0.554700i
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 4.00000i − 0.508001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 − 0.585206i \(-0.801014\pi\)
0.810885 − 0.585206i \(-0.198986\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) − 6.00000i − 0.639602i
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 − 0.101535i \(-0.967625\pi\)
0.994832 − 0.101535i \(-0.0323753\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 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 450.2.c.d.199.2 2
3.2 odd 2 450.2.c.a.199.1 2
4.3 odd 2 3600.2.f.a.2449.2 2
5.2 odd 4 90.2.a.a.1.1 ✓ 1
5.3 odd 4 450.2.a.e.1.1 1
5.4 even 2 inner 450.2.c.d.199.1 2
12.11 even 2 3600.2.f.u.2449.2 2
15.2 even 4 90.2.a.b.1.1 yes 1
15.8 even 4 450.2.a.a.1.1 1
15.14 odd 2 450.2.c.a.199.2 2
20.3 even 4 3600.2.a.ba.1.1 1
20.7 even 4 720.2.a.g.1.1 1
20.19 odd 2 3600.2.f.a.2449.1 2
35.27 even 4 4410.2.a.k.1.1 1
40.27 even 4 2880.2.a.h.1.1 1
40.37 odd 4 2880.2.a.k.1.1 1
45.2 even 12 810.2.e.e.271.1 2
45.7 odd 12 810.2.e.h.271.1 2
45.22 odd 12 810.2.e.h.541.1 2
45.32 even 12 810.2.e.e.541.1 2
60.23 odd 4 3600.2.a.bj.1.1 1
60.47 odd 4 720.2.a.b.1.1 1
60.59 even 2 3600.2.f.u.2449.1 2
105.62 odd 4 4410.2.a.bf.1.1 1
120.77 even 4 2880.2.a.bf.1.1 1
120.107 odd 4 2880.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.a.a.1.1 ✓ 1 5.2 odd 4
90.2.a.b.1.1 yes 1 15.2 even 4
450.2.a.a.1.1 1 15.8 even 4
450.2.a.e.1.1 1 5.3 odd 4
450.2.c.a.199.1 2 3.2 odd 2
450.2.c.a.199.2 2 15.14 odd 2
450.2.c.d.199.1 2 5.4 even 2 inner
450.2.c.d.199.2 2 1.1 even 1 trivial
720.2.a.b.1.1 1 60.47 odd 4
720.2.a.g.1.1 1 20.7 even 4
810.2.e.e.271.1 2 45.2 even 12
810.2.e.e.541.1 2 45.32 even 12
810.2.e.h.271.1 2 45.7 odd 12
810.2.e.h.541.1 2 45.22 odd 12
2880.2.a.h.1.1 1 40.27 even 4
2880.2.a.k.1.1 1 40.37 odd 4
2880.2.a.u.1.1 1 120.107 odd 4
2880.2.a.bf.1.1 1 120.77 even 4
3600.2.a.ba.1.1 1 20.3 even 4
3600.2.a.bj.1.1 1 60.23 odd 4
3600.2.f.a.2449.1 2 20.19 odd 2
3600.2.f.a.2449.2 2 4.3 odd 2
3600.2.f.u.2449.1 2 60.59 even 2
3600.2.f.u.2449.2 2 12.11 even 2
4410.2.a.k.1.1 1 35.27 even 4
4410.2.a.bf.1.1 1 105.62 odd 4