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Properties

Label 6000.2.f.k.1249.1
Level $6000$
Weight $2$
Character 6000.1249
Analytic conductor $47.910$
Analytic rank $0$
Dimension $4$
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] = [6000,2,Mod(1249,6000)] 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("6000.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6000.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,10,0,0,0,0,0,0,0,-6,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.9102412128\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 750)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 6000.1249
Dual form 6000.2.f.k.1249.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.38197i q^{7} -1.00000 q^{9} +5.85410 q^{11} +3.38197i q^{13} -7.70820i q^{17} +4.09017 q^{19} +2.38197 q^{21} +3.09017i q^{23} +1.00000i q^{27} -5.70820 q^{29} +6.47214 q^{31} -5.85410i q^{33} +1.61803i q^{37} +3.38197 q^{39} +0.381966 q^{41} -7.70820i q^{43} +8.61803i q^{47} +1.32624 q^{49} -7.70820 q^{51} +0.381966i q^{53} -4.09017i q^{57} -10.8541 q^{59} +11.7082 q^{61} -2.38197i q^{63} +3.23607i q^{67} +3.09017 q^{69} +4.47214 q^{71} -8.00000i q^{73} +13.9443i q^{77} +12.4721 q^{79} +1.00000 q^{81} -2.00000i q^{83} +5.70820i q^{87} -15.5623 q^{89} -8.05573 q^{91} -6.47214i q^{93} +14.1803i q^{97} -5.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 10 q^{11} - 6 q^{19} + 14 q^{21} + 4 q^{29} + 8 q^{31} + 18 q^{39} + 6 q^{41} - 26 q^{49} - 4 q^{51} - 30 q^{59} + 20 q^{61} - 10 q^{69} + 32 q^{79} + 4 q^{81} - 22 q^{89} - 68 q^{91} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6000\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(4001\) \(4501\) \(5377\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.38197i 0.900299i 0.892953 + 0.450149i \(0.148629\pi\)
−0.892953 + 0.450149i \(0.851371\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.85410 1.76508 0.882539 − 0.470239i \(-0.155832\pi\)
0.882539 + 0.470239i \(0.155832\pi\)
\(12\) 0 0
\(13\) 3.38197i 0.937989i 0.883201 + 0.468994i \(0.155384\pi\)
−0.883201 + 0.468994i \(0.844616\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.70820i − 1.86951i −0.355288 − 0.934757i \(-0.615617\pi\)
0.355288 − 0.934757i \(-0.384383\pi\)
\(18\) 0 0
\(19\) 4.09017 0.938349 0.469175 − 0.883105i \(-0.344551\pi\)
0.469175 + 0.883105i \(0.344551\pi\)
\(20\) 0 0
\(21\) 2.38197 0.519788
\(22\) 0 0
\(23\) 3.09017i 0.644345i 0.946681 + 0.322172i \(0.104413\pi\)
−0.946681 + 0.322172i \(0.895587\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −5.70820 −1.05999 −0.529993 − 0.848002i \(-0.677806\pi\)
−0.529993 + 0.848002i \(0.677806\pi\)
\(30\) 0 0
\(31\) 6.47214 1.16243 0.581215 − 0.813750i \(-0.302578\pi\)
0.581215 + 0.813750i \(0.302578\pi\)
\(32\) 0 0
\(33\) − 5.85410i − 1.01907i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.61803i 0.266003i 0.991116 + 0.133002i \(0.0424615\pi\)
−0.991116 + 0.133002i \(0.957538\pi\)
\(38\) 0 0
\(39\) 3.38197 0.541548
\(40\) 0 0
\(41\) 0.381966 0.0596531 0.0298265 − 0.999555i \(-0.490505\pi\)
0.0298265 + 0.999555i \(0.490505\pi\)
\(42\) 0 0
\(43\) − 7.70820i − 1.17549i −0.809046 − 0.587745i \(-0.800016\pi\)
0.809046 − 0.587745i \(-0.199984\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.61803i 1.25707i 0.777782 + 0.628535i \(0.216345\pi\)
−0.777782 + 0.628535i \(0.783655\pi\)
\(48\) 0 0
\(49\) 1.32624 0.189463
\(50\) 0 0
\(51\) −7.70820 −1.07936
\(52\) 0 0
\(53\) 0.381966i 0.0524671i 0.999656 + 0.0262335i \(0.00835135\pi\)
−0.999656 + 0.0262335i \(0.991649\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.09017i − 0.541756i
\(58\) 0 0
\(59\) −10.8541 −1.41308 −0.706542 − 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(60\) 0 0
\(61\) 11.7082 1.49908 0.749541 − 0.661958i \(-0.230274\pi\)
0.749541 + 0.661958i \(0.230274\pi\)
\(62\) 0 0
\(63\) − 2.38197i − 0.300100i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.23607i 0.395349i 0.980268 + 0.197674i \(0.0633388\pi\)
−0.980268 + 0.197674i \(0.936661\pi\)
\(68\) 0 0
\(69\) 3.09017 0.372013
\(70\) 0 0
\(71\) 4.47214 0.530745 0.265372 − 0.964146i \(-0.414505\pi\)
0.265372 + 0.964146i \(0.414505\pi\)
\(72\) 0 0
\(73\) − 8.00000i − 0.936329i −0.883641 − 0.468165i \(-0.844915\pi\)
0.883641 − 0.468165i \(-0.155085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.9443i 1.58910i
\(78\) 0 0
\(79\) 12.4721 1.40322 0.701612 − 0.712559i \(-0.252464\pi\)
0.701612 + 0.712559i \(0.252464\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 2.00000i − 0.219529i −0.993958 − 0.109764i \(-0.964990\pi\)
0.993958 − 0.109764i \(-0.0350096\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.70820i 0.611984i
\(88\) 0 0
\(89\) −15.5623 −1.64960 −0.824801 − 0.565424i \(-0.808713\pi\)
−0.824801 + 0.565424i \(0.808713\pi\)
\(90\) 0 0
\(91\) −8.05573 −0.844470
\(92\) 0 0
\(93\) − 6.47214i − 0.671129i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.1803i 1.43980i 0.694080 + 0.719898i \(0.255811\pi\)
−0.694080 + 0.719898i \(0.744189\pi\)
\(98\) 0 0
\(99\) −5.85410 −0.588359
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 6000.2.f.k.1249.1 4
4.3 odd 2 750.2.c.a.499.4 4
5.2 odd 4 6000.2.a.a.1.2 2
5.3 odd 4 6000.2.a.bb.1.1 2
5.4 even 2 inner 6000.2.f.k.1249.4 4
12.11 even 2 2250.2.c.g.1999.2 4
20.3 even 4 750.2.a.e.1.2 yes 2
20.7 even 4 750.2.a.d.1.1 ✓ 2
20.19 odd 2 750.2.c.a.499.1 4
60.23 odd 4 2250.2.a.a.1.2 2
60.47 odd 4 2250.2.a.p.1.1 2
60.59 even 2 2250.2.c.g.1999.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
750.2.a.d.1.1 ✓ 2 20.7 even 4
750.2.a.e.1.2 yes 2 20.3 even 4
750.2.c.a.499.1 4 20.19 odd 2
750.2.c.a.499.4 4 4.3 odd 2
2250.2.a.a.1.2 2 60.23 odd 4
2250.2.a.p.1.1 2 60.47 odd 4
2250.2.c.g.1999.2 4 12.11 even 2
2250.2.c.g.1999.3 4 60.59 even 2
6000.2.a.a.1.2 2 5.2 odd 4
6000.2.a.bb.1.1 2 5.3 odd 4
6000.2.f.k.1249.1 4 1.1 even 1 trivial
6000.2.f.k.1249.4 4 5.4 even 2 inner