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Properties

Label 450.3.d.c.251.2
Level $450$
Weight $3$
Character 450.251
Analytic conductor $12.262$
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] = [450,3,Mod(251,450)] 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("450.251"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4,0,0,-2,0,0,0,0,0,34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2616118962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 450.251
Dual form 450.3.d.c.251.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} -1.00000 q^{7} -2.82843i q^{8} +4.24264i q^{11} +17.0000 q^{13} -1.41421i q^{14} +4.00000 q^{16} +29.6985i q^{17} -31.0000 q^{19} -6.00000 q^{22} +4.24264i q^{23} +24.0416i q^{26} +2.00000 q^{28} +38.1838i q^{29} -31.0000 q^{31} +5.65685i q^{32} -42.0000 q^{34} -16.0000 q^{37} -43.8406i q^{38} +16.9706i q^{41} +65.0000 q^{43} -8.48528i q^{44} -6.00000 q^{46} +21.2132i q^{47} -48.0000 q^{49} -34.0000 q^{52} +16.9706i q^{53} +2.82843i q^{56} -54.0000 q^{58} +114.551i q^{59} -55.0000 q^{61} -43.8406i q^{62} -8.00000 q^{64} -73.0000 q^{67} -59.3970i q^{68} -118.794i q^{71} +56.0000 q^{73} -22.6274i q^{74} +62.0000 q^{76} -4.24264i q^{77} +104.000 q^{79} -24.0000 q^{82} +12.7279i q^{83} +91.9239i q^{86} +12.0000 q^{88} -33.9411i q^{89} -17.0000 q^{91} -8.48528i q^{92} -30.0000 q^{94} +89.0000 q^{97} -67.8823i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 2 q^{7} + 34 q^{13} + 8 q^{16} - 62 q^{19} - 12 q^{22} + 4 q^{28} - 62 q^{31} - 84 q^{34} - 32 q^{37} + 130 q^{43} - 12 q^{46} - 96 q^{49} - 68 q^{52} - 108 q^{58} - 110 q^{61} - 16 q^{64}+ \cdots + 178 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.142857 −0.0714286 − 0.997446i \(-0.522756\pi\)
−0.0714286 + 0.997446i \(0.522756\pi\)
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 4.24264i 0.385695i 0.981229 + 0.192847i \(0.0617722\pi\)
−0.981229 + 0.192847i \(0.938228\pi\)
\(12\) 0 0
\(13\) 17.0000 1.30769 0.653846 − 0.756628i \(-0.273154\pi\)
0.653846 + 0.756628i \(0.273154\pi\)
\(14\) − 1.41421i − 0.101015i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 29.6985i 1.74697i 0.486851 + 0.873485i \(0.338145\pi\)
−0.486851 + 0.873485i \(0.661855\pi\)
\(18\) 0 0
\(19\) −31.0000 −1.63158 −0.815789 − 0.578349i \(-0.803697\pi\)
−0.815789 + 0.578349i \(0.803697\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −0.272727
\(23\) 4.24264i 0.184463i 0.995738 + 0.0922313i \(0.0293999\pi\)
−0.995738 + 0.0922313i \(0.970600\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 24.0416i 0.924678i
\(27\) 0 0
\(28\) 2.00000 0.0714286
\(29\) 38.1838i 1.31668i 0.752720 + 0.658341i \(0.228741\pi\)
−0.752720 + 0.658341i \(0.771259\pi\)
\(30\) 0 0
\(31\) −31.0000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) −42.0000 −1.23529
\(35\) 0 0
\(36\) 0 0
\(37\) −16.0000 −0.432432 −0.216216 − 0.976346i \(-0.569372\pi\)
−0.216216 + 0.976346i \(0.569372\pi\)
\(38\) − 43.8406i − 1.15370i
\(39\) 0 0
\(40\) 0 0
\(41\) 16.9706i 0.413916i 0.978350 + 0.206958i \(0.0663564\pi\)
−0.978350 + 0.206958i \(0.933644\pi\)
\(42\) 0 0
\(43\) 65.0000 1.51163 0.755814 − 0.654786i \(-0.227241\pi\)
0.755814 + 0.654786i \(0.227241\pi\)
\(44\) − 8.48528i − 0.192847i
\(45\) 0 0
\(46\) −6.00000 −0.130435
\(47\) 21.2132i 0.451345i 0.974203 + 0.225672i \(0.0724579\pi\)
−0.974203 + 0.225672i \(0.927542\pi\)
\(48\) 0 0
\(49\) −48.0000 −0.979592
\(50\) 0 0
\(51\) 0 0
\(52\) −34.0000 −0.653846
\(53\) 16.9706i 0.320199i 0.987101 + 0.160100i \(0.0511816\pi\)
−0.987101 + 0.160100i \(0.948818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.82843i 0.0505076i
\(57\) 0 0
\(58\) −54.0000 −0.931034
\(59\) 114.551i 1.94155i 0.239997 + 0.970774i \(0.422854\pi\)
−0.239997 + 0.970774i \(0.577146\pi\)
\(60\) 0 0
\(61\) −55.0000 −0.901639 −0.450820 − 0.892615i \(-0.648868\pi\)
−0.450820 + 0.892615i \(0.648868\pi\)
\(62\) − 43.8406i − 0.707107i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −73.0000 −1.08955 −0.544776 − 0.838582i \(-0.683385\pi\)
−0.544776 + 0.838582i \(0.683385\pi\)
\(68\) − 59.3970i − 0.873485i
\(69\) 0 0
\(70\) 0 0
\(71\) − 118.794i − 1.67315i −0.547849 − 0.836577i \(-0.684553\pi\)
0.547849 − 0.836577i \(-0.315447\pi\)
\(72\) 0 0
\(73\) 56.0000 0.767123 0.383562 − 0.923515i \(-0.374697\pi\)
0.383562 + 0.923515i \(0.374697\pi\)
\(74\) − 22.6274i − 0.305776i
\(75\) 0 0
\(76\) 62.0000 0.815789
\(77\) − 4.24264i − 0.0550992i
\(78\) 0 0
\(79\) 104.000 1.31646 0.658228 − 0.752819i \(-0.271306\pi\)
0.658228 + 0.752819i \(0.271306\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −24.0000 −0.292683
\(83\) 12.7279i 0.153348i 0.997056 + 0.0766742i \(0.0244301\pi\)
−0.997056 + 0.0766742i \(0.975570\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 91.9239i 1.06888i
\(87\) 0 0
\(88\) 12.0000 0.136364
\(89\) − 33.9411i − 0.381361i −0.981652 − 0.190680i \(-0.938931\pi\)
0.981652 − 0.190680i \(-0.0610694\pi\)
\(90\) 0 0
\(91\) −17.0000 −0.186813
\(92\) − 8.48528i − 0.0922313i
\(93\) 0 0
\(94\) −30.0000 −0.319149
\(95\) 0 0
\(96\) 0 0
\(97\) 89.0000 0.917526 0.458763 − 0.888559i \(-0.348293\pi\)
0.458763 + 0.888559i \(0.348293\pi\)
\(98\) − 67.8823i − 0.692676i
\(99\) 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 450.3.d.c.251.2 yes 2
3.2 odd 2 inner 450.3.d.c.251.1 ✓ 2
4.3 odd 2 3600.3.l.h.1601.1 2
5.2 odd 4 450.3.b.c.449.1 4
5.3 odd 4 450.3.b.c.449.4 4
5.4 even 2 450.3.d.d.251.1 yes 2
12.11 even 2 3600.3.l.h.1601.2 2
15.2 even 4 450.3.b.c.449.3 4
15.8 even 4 450.3.b.c.449.2 4
15.14 odd 2 450.3.d.d.251.2 yes 2
20.3 even 4 3600.3.c.a.449.1 4
20.7 even 4 3600.3.c.a.449.3 4
20.19 odd 2 3600.3.l.e.1601.1 2
60.23 odd 4 3600.3.c.a.449.2 4
60.47 odd 4 3600.3.c.a.449.4 4
60.59 even 2 3600.3.l.e.1601.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.3.b.c.449.1 4 5.2 odd 4
450.3.b.c.449.2 4 15.8 even 4
450.3.b.c.449.3 4 15.2 even 4
450.3.b.c.449.4 4 5.3 odd 4
450.3.d.c.251.1 ✓ 2 3.2 odd 2 inner
450.3.d.c.251.2 yes 2 1.1 even 1 trivial
450.3.d.d.251.1 yes 2 5.4 even 2
450.3.d.d.251.2 yes 2 15.14 odd 2
3600.3.c.a.449.1 4 20.3 even 4
3600.3.c.a.449.2 4 60.23 odd 4
3600.3.c.a.449.3 4 20.7 even 4
3600.3.c.a.449.4 4 60.47 odd 4
3600.3.l.e.1601.1 2 20.19 odd 2
3600.3.l.e.1601.2 2 60.59 even 2
3600.3.l.h.1601.1 2 4.3 odd 2
3600.3.l.h.1601.2 2 12.11 even 2