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Properties

Label 6300.2.a.bk.1.3
Level $6300$
Weight $2$
Character 6300.1
Self dual yes
Analytic conductor $50.306$
Analytic rank $1$
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] = [6300,2,Mod(1,6300)] 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("6300.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(50.3057532734\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1260)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.456850\) of defining polynomial
Character \(\chi\) \(=\) 6300.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{7} +3.46410 q^{11} -5.58258 q^{13} -7.84190 q^{17} +5.58258 q^{19} +4.37780 q^{23} +1.82740 q^{29} +5.58258 q^{31} -4.00000 q^{37} -6.20520 q^{41} +7.16515 q^{43} -8.75560 q^{47} +1.00000 q^{49} +6.20520 q^{53} +8.75560 q^{59} -9.16515 q^{61} -8.00000 q^{67} -3.46410 q^{71} +5.58258 q^{73} -3.46410 q^{77} -7.16515 q^{79} +6.92820 q^{83} -18.2342 q^{89} +5.58258 q^{91} -13.5826 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7} - 4 q^{13} + 4 q^{19} + 4 q^{31} - 16 q^{37} - 8 q^{43} + 4 q^{49} - 32 q^{67} + 4 q^{73} + 8 q^{79} + 4 q^{91} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −5.58258 −1.54833 −0.774164 − 0.632985i \(-0.781829\pi\)
−0.774164 + 0.632985i \(0.781829\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.84190 −1.90194 −0.950971 − 0.309282i \(-0.899911\pi\)
−0.950971 + 0.309282i \(0.899911\pi\)
\(18\) 0 0
\(19\) 5.58258 1.28073 0.640365 − 0.768070i \(-0.278783\pi\)
0.640365 + 0.768070i \(0.278783\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.37780 0.912835 0.456417 − 0.889766i \(-0.349132\pi\)
0.456417 + 0.889766i \(0.349132\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.82740 0.339340 0.169670 − 0.985501i \(-0.445730\pi\)
0.169670 + 0.985501i \(0.445730\pi\)
\(30\) 0 0
\(31\) 5.58258 1.00266 0.501330 − 0.865256i \(-0.332844\pi\)
0.501330 + 0.865256i \(0.332844\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.20520 −0.969090 −0.484545 − 0.874766i \(-0.661015\pi\)
−0.484545 + 0.874766i \(0.661015\pi\)
\(42\) 0 0
\(43\) 7.16515 1.09268 0.546338 − 0.837565i \(-0.316022\pi\)
0.546338 + 0.837565i \(0.316022\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.75560 −1.27714 −0.638568 − 0.769565i \(-0.720473\pi\)
−0.638568 + 0.769565i \(0.720473\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.20520 0.852350 0.426175 − 0.904641i \(-0.359861\pi\)
0.426175 + 0.904641i \(0.359861\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.75560 1.13988 0.569941 − 0.821685i \(-0.306966\pi\)
0.569941 + 0.821685i \(0.306966\pi\)
\(60\) 0 0
\(61\) −9.16515 −1.17348 −0.586739 − 0.809776i \(-0.699588\pi\)
−0.586739 + 0.809776i \(0.699588\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 − 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 5.58258 0.653391 0.326696 − 0.945130i \(-0.394065\pi\)
0.326696 + 0.945130i \(0.394065\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −7.16515 −0.806143 −0.403071 − 0.915169i \(-0.632057\pi\)
−0.403071 + 0.915169i \(0.632057\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.92820 0.760469 0.380235 − 0.924890i \(-0.375843\pi\)
0.380235 + 0.924890i \(0.375843\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −18.2342 −1.93282 −0.966411 − 0.257001i \(-0.917266\pi\)
−0.966411 + 0.257001i \(0.917266\pi\)
\(90\) 0 0
\(91\) 5.58258 0.585213
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.5826 −1.37910 −0.689551 − 0.724237i \(-0.742192\pi\)
−0.689551 + 0.724237i \(0.742192\pi\)
\(98\) 0 0
\(99\) 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 6300.2.a.bk.1.3 4
3.2 odd 2 inner 6300.2.a.bk.1.1 4
5.2 odd 4 1260.2.k.e.1009.1 ✓ 8
5.3 odd 4 1260.2.k.e.1009.2 yes 8
5.4 even 2 6300.2.a.bl.1.4 4
15.2 even 4 1260.2.k.e.1009.8 yes 8
15.8 even 4 1260.2.k.e.1009.7 yes 8
15.14 odd 2 6300.2.a.bl.1.2 4
20.3 even 4 5040.2.t.bb.1009.2 8
20.7 even 4 5040.2.t.bb.1009.1 8
60.23 odd 4 5040.2.t.bb.1009.7 8
60.47 odd 4 5040.2.t.bb.1009.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.k.e.1009.1 ✓ 8 5.2 odd 4
1260.2.k.e.1009.2 yes 8 5.3 odd 4
1260.2.k.e.1009.7 yes 8 15.8 even 4
1260.2.k.e.1009.8 yes 8 15.2 even 4
5040.2.t.bb.1009.1 8 20.7 even 4
5040.2.t.bb.1009.2 8 20.3 even 4
5040.2.t.bb.1009.7 8 60.23 odd 4
5040.2.t.bb.1009.8 8 60.47 odd 4
6300.2.a.bk.1.1 4 3.2 odd 2 inner
6300.2.a.bk.1.3 4 1.1 even 1 trivial
6300.2.a.bl.1.2 4 15.14 odd 2
6300.2.a.bl.1.4 4 5.4 even 2