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Properties

Label 1450.2.d.a.1449.1
Level $1450$
Weight $2$
Character 1450.1449
Analytic conductor $11.578$
Analytic rank $1$
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] = [1450,2,Mod(1449,1450)] 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("1450.1449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5783082931\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 290)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1449.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1450.1449
Dual form 1450.2.d.a.1449.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.00000i q^{7} -1.00000 q^{8} -3.00000 q^{9} -2.00000i q^{11} -2.00000i q^{13} +4.00000i q^{14} +1.00000 q^{16} -6.00000 q^{17} +3.00000 q^{18} +2.00000i q^{19} +2.00000i q^{22} +2.00000i q^{26} -4.00000i q^{28} +(-5.00000 + 2.00000i) q^{29} +10.0000i q^{31} -1.00000 q^{32} +6.00000 q^{34} -3.00000 q^{36} +6.00000 q^{37} -2.00000i q^{38} +12.0000i q^{41} +4.00000 q^{43} -2.00000i q^{44} -8.00000 q^{47} -9.00000 q^{49} -2.00000i q^{52} -2.00000i q^{53} +4.00000i q^{56} +(5.00000 - 2.00000i) q^{58} +12.0000 q^{59} +4.00000i q^{61} -10.0000i q^{62} +12.0000i q^{63} +1.00000 q^{64} +8.00000i q^{67} -6.00000 q^{68} -8.00000 q^{71} +3.00000 q^{72} -2.00000 q^{73} -6.00000 q^{74} +2.00000i q^{76} -8.00000 q^{77} -10.0000i q^{79} +9.00000 q^{81} -12.0000i q^{82} -4.00000i q^{83} -4.00000 q^{86} +2.00000i q^{88} -12.0000i q^{89} -8.00000 q^{91} +8.00000 q^{94} -2.00000 q^{97} +9.00000 q^{98} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9} + 2 q^{16} - 12 q^{17} + 6 q^{18} - 10 q^{29} - 2 q^{32} + 12 q^{34} - 6 q^{36} + 12 q^{37} + 8 q^{43} - 16 q^{47} - 18 q^{49} + 10 q^{58} + 24 q^{59} + 2 q^{64}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 − 0.755929i \(-0.727186\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) 4.00000i 1.06904i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 3.00000 0.707107
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000i 0.392232i
\(27\) 0 0
\(28\) − 4.00000i − 0.755929i
\(29\) −5.00000 + 2.00000i −0.928477 + 0.371391i
\(30\) 0 0
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) − 2.00000i − 0.324443i
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000i 1.87409i 0.349215 + 0.937043i \(0.386448\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) − 2.00000i − 0.301511i
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) − 2.00000i − 0.274721i −0.990521 − 0.137361i \(-0.956138\pi\)
0.990521 − 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.00000i 0.534522i
\(57\) 0 0
\(58\) 5.00000 − 2.00000i 0.656532 − 0.262613i
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 4.00000i 0.512148i 0.966657 + 0.256074i \(0.0824290\pi\)
−0.966657 + 0.256074i \(0.917571\pi\)
\(62\) − 10.0000i − 1.27000i
\(63\) 12.0000i 1.51186i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 3.00000 0.353553
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 2.00000i 0.229416i
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) − 10.0000i − 1.12509i −0.826767 − 0.562544i \(-0.809823\pi\)
0.826767 − 0.562544i \(-0.190177\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 12.0000i − 1.32518i
\(83\) − 4.00000i − 0.439057i −0.975606 − 0.219529i \(-0.929548\pi\)
0.975606 − 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 2.00000i 0.213201i
\(89\) − 12.0000i − 1.27200i −0.771690 − 0.635999i \(-0.780588\pi\)
0.771690 − 0.635999i \(-0.219412\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 9.00000 0.909137
\(99\) 6.00000i 0.603023i
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 1450.2.d.a.1449.1 2
5.2 odd 4 290.2.c.a.231.1 ✓ 2
5.3 odd 4 1450.2.c.b.1101.2 2
5.4 even 2 1450.2.d.d.1449.2 2
15.2 even 4 2610.2.f.c.811.2 2
20.7 even 4 2320.2.g.a.1681.2 2
29.28 even 2 1450.2.d.d.1449.1 2
145.12 even 4 8410.2.a.l.1.1 1
145.17 even 4 8410.2.a.e.1.1 1
145.28 odd 4 1450.2.c.b.1101.1 2
145.57 odd 4 290.2.c.a.231.2 yes 2
145.144 even 2 inner 1450.2.d.a.1449.2 2
435.347 even 4 2610.2.f.c.811.1 2
580.347 even 4 2320.2.g.a.1681.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
290.2.c.a.231.1 ✓ 2 5.2 odd 4
290.2.c.a.231.2 yes 2 145.57 odd 4
1450.2.c.b.1101.1 2 145.28 odd 4
1450.2.c.b.1101.2 2 5.3 odd 4
1450.2.d.a.1449.1 2 1.1 even 1 trivial
1450.2.d.a.1449.2 2 145.144 even 2 inner
1450.2.d.d.1449.1 2 29.28 even 2
1450.2.d.d.1449.2 2 5.4 even 2
2320.2.g.a.1681.1 2 580.347 even 4
2320.2.g.a.1681.2 2 20.7 even 4
2610.2.f.c.811.1 2 435.347 even 4
2610.2.f.c.811.2 2 15.2 even 4
8410.2.a.e.1.1 1 145.17 even 4
8410.2.a.l.1.1 1 145.12 even 4