Properties

Label 504.6.a.k.1.1
Level $504$
Weight $6$
Character 504.1
Self dual yes
Analytic conductor $80.833$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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] = [504,6,Mod(1,504)] 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("504.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-76,0,98] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(80.8334451857\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 106 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-10.2956\) of defining polynomial
Character \(\chi\) \(=\) 504.1

$q$-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)
 
\(f(q)\) \(=\) \(q-99.7738 q^{5} +49.0000 q^{7} +26.8689 q^{11} +10.9049 q^{13} +593.964 q^{17} +1427.10 q^{19} -1514.73 q^{23} +6829.81 q^{25} -2924.52 q^{29} +2049.00 q^{31} -4888.92 q^{35} +2476.43 q^{37} +20188.3 q^{41} +9587.24 q^{43} -20069.8 q^{47} +2401.00 q^{49} +6554.12 q^{53} -2680.81 q^{55} -18212.8 q^{59} -35155.6 q^{61} -1088.02 q^{65} -28813.0 q^{67} -17116.0 q^{71} +21872.2 q^{73} +1316.58 q^{77} -87602.0 q^{79} -71085.9 q^{83} -59262.0 q^{85} -3299.03 q^{89} +534.339 q^{91} -142387. q^{95} -117193. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 76 q^{5} + 98 q^{7} - 564 q^{11} + 516 q^{13} + 76 q^{17} + 2360 q^{19} + 2036 q^{23} + 4270 q^{25} - 8320 q^{29} - 6280 q^{31} - 3724 q^{35} - 2460 q^{37} + 14308 q^{41} + 23128 q^{43} - 12712 q^{47}+ \cdots - 272932 q^{97}+O(q^{100}) \) 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
\(5\) −99.7738 −1.78481 −0.892404 − 0.451238i \(-0.850983\pi\)
−0.892404 + 0.451238i \(0.850983\pi\)
\(6\) 0 0
\(7\) 49.0000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 26.8689 0.0669527 0.0334764 − 0.999440i \(-0.489342\pi\)
0.0334764 + 0.999440i \(0.489342\pi\)
\(12\) 0 0
\(13\) 10.9049 0.0178963 0.00894813 − 0.999960i \(-0.497152\pi\)
0.00894813 + 0.999960i \(0.497152\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 593.964 0.498469 0.249234 − 0.968443i \(-0.419821\pi\)
0.249234 + 0.968443i \(0.419821\pi\)
\(18\) 0 0
\(19\) 1427.10 0.906920 0.453460 − 0.891277i \(-0.350190\pi\)
0.453460 + 0.891277i \(0.350190\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1514.73 −0.597055 −0.298527 − 0.954401i \(-0.596495\pi\)
−0.298527 + 0.954401i \(0.596495\pi\)
\(24\) 0 0
\(25\) 6829.81 2.18554
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2924.52 −0.645744 −0.322872 − 0.946443i \(-0.604648\pi\)
−0.322872 + 0.946443i \(0.604648\pi\)
\(30\) 0 0
\(31\) 2049.00 0.382946 0.191473 − 0.981498i \(-0.438674\pi\)
0.191473 + 0.981498i \(0.438674\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4888.92 −0.674594
\(36\) 0 0
\(37\) 2476.43 0.297386 0.148693 − 0.988883i \(-0.452493\pi\)
0.148693 + 0.988883i \(0.452493\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 20188.3 1.87560 0.937798 − 0.347181i \(-0.112861\pi\)
0.937798 + 0.347181i \(0.112861\pi\)
\(42\) 0 0
\(43\) 9587.24 0.790719 0.395360 − 0.918526i \(-0.370620\pi\)
0.395360 + 0.918526i \(0.370620\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −20069.8 −1.32525 −0.662625 − 0.748951i \(-0.730558\pi\)
−0.662625 + 0.748951i \(0.730558\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6554.12 0.320497 0.160249 − 0.987077i \(-0.448770\pi\)
0.160249 + 0.987077i \(0.448770\pi\)
\(54\) 0 0
\(55\) −2680.81 −0.119498
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −18212.8 −0.681156 −0.340578 − 0.940216i \(-0.610623\pi\)
−0.340578 + 0.940216i \(0.610623\pi\)
\(60\) 0 0
\(61\) −35155.6 −1.20968 −0.604840 − 0.796347i \(-0.706763\pi\)
−0.604840 + 0.796347i \(0.706763\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1088.02 −0.0319414
\(66\) 0 0
\(67\) −28813.0 −0.784156 −0.392078 − 0.919932i \(-0.628244\pi\)
−0.392078 + 0.919932i \(0.628244\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −17116.0 −0.402954 −0.201477 − 0.979493i \(-0.564574\pi\)
−0.201477 + 0.979493i \(0.564574\pi\)
\(72\) 0 0
\(73\) 21872.2 0.480380 0.240190 − 0.970726i \(-0.422790\pi\)
0.240190 + 0.970726i \(0.422790\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1316.58 0.0253058
\(78\) 0 0
\(79\) −87602.0 −1.57923 −0.789616 − 0.613601i \(-0.789720\pi\)
−0.789616 + 0.613601i \(0.789720\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −71085.9 −1.13263 −0.566315 − 0.824189i \(-0.691632\pi\)
−0.566315 + 0.824189i \(0.691632\pi\)
\(84\) 0 0
\(85\) −59262.0 −0.889671
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3299.03 −0.0441480 −0.0220740 − 0.999756i \(-0.507027\pi\)
−0.0220740 + 0.999756i \(0.507027\pi\)
\(90\) 0 0
\(91\) 534.339 0.00676415
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −142387. −1.61868
\(96\) 0 0
\(97\) −117193. −1.26465 −0.632326 − 0.774703i \(-0.717900\pi\)
−0.632326 + 0.774703i \(0.717900\pi\)
\(98\) 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 504.6.a.k.1.1 ✓ 2
3.2 odd 2 504.6.a.u.1.2 yes 2
4.3 odd 2 1008.6.a.bg.1.1 2
12.11 even 2 1008.6.a.bv.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.6.a.k.1.1 ✓ 2 1.1 even 1 trivial
504.6.a.u.1.2 yes 2 3.2 odd 2
1008.6.a.bg.1.1 2 4.3 odd 2
1008.6.a.bv.1.2 2 12.11 even 2