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Properties

Label 504.4.s.h.361.2
Level $504$
Weight $4$
Character 504.361
Analytic conductor $29.737$
Analytic rank $0$
Dimension $6$
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] = [504,4,Mod(289,504)] 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("504.289"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.7369626429\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11163123.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 49x^{2} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.2
Root \(-0.821510i\) of defining polynomial
Character \(\chi\) \(=\) 504.361
Dual form 504.4.s.h.289.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-3.34580 - 5.79509i) q^{5} +(-8.65024 + 16.3760i) q^{7} +(15.7441 - 27.2695i) q^{11} +18.6837 q^{13} +(-43.9769 + 76.1703i) q^{17} +(-6.54850 - 11.3423i) q^{19} +(-4.68840 - 8.12055i) q^{23} +(40.1113 - 69.4748i) q^{25} +5.11923 q^{29} +(128.706 - 222.925i) q^{31} +(123.842 - 4.66183i) q^{35} +(190.107 + 329.276i) q^{37} +217.959 q^{41} +377.049 q^{43} +(-178.855 - 309.786i) q^{47} +(-193.347 - 283.313i) q^{49} +(382.195 - 661.981i) q^{53} -210.706 q^{55} +(-225.336 + 390.293i) q^{59} +(87.0388 + 150.756i) q^{61} +(-62.5117 - 108.273i) q^{65} +(248.617 - 430.617i) q^{67} -350.238 q^{71} +(531.343 - 920.312i) q^{73} +(310.375 + 493.712i) q^{77} +(-280.224 - 485.363i) q^{79} +1105.27 q^{83} +588.551 q^{85} +(603.357 + 1045.04i) q^{89} +(-161.618 + 305.964i) q^{91} +(-43.8199 + 75.8983i) q^{95} +1442.99 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 4 q^{7} - 3 q^{11} - 52 q^{13} - 31 q^{17} + 89 q^{19} - 201 q^{23} - 300 q^{25} - 380 q^{29} + 339 q^{31} + 473 q^{35} + 535 q^{37} - 116 q^{41} + 536 q^{43} - 205 q^{47} - 1530 q^{49}+ \cdots - 684 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.34580 − 5.79509i −0.299257 − 0.518328i 0.676709 − 0.736250i \(-0.263405\pi\)
−0.975966 + 0.217922i \(0.930072\pi\)
\(6\) 0 0
\(7\) −8.65024 + 16.3760i −0.467069 + 0.884221i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 15.7441 − 27.2695i 0.431546 − 0.747460i −0.565460 − 0.824776i \(-0.691301\pi\)
0.997007 + 0.0773151i \(0.0246348\pi\)
\(12\) 0 0
\(13\) 18.6837 0.398609 0.199304 − 0.979938i \(-0.436132\pi\)
0.199304 + 0.979938i \(0.436132\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −43.9769 + 76.1703i −0.627410 + 1.08671i 0.360660 + 0.932698i \(0.382552\pi\)
−0.988070 + 0.154008i \(0.950782\pi\)
\(18\) 0 0
\(19\) −6.54850 − 11.3423i −0.0790700 − 0.136953i 0.823779 − 0.566911i \(-0.191862\pi\)
−0.902849 + 0.429958i \(0.858528\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.68840 − 8.12055i −0.0425043 − 0.0736197i 0.843991 − 0.536358i \(-0.180200\pi\)
−0.886495 + 0.462738i \(0.846867\pi\)
\(24\) 0 0
\(25\) 40.1113 − 69.4748i 0.320890 − 0.555799i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.11923 0.0327799 0.0163900 − 0.999866i \(-0.494783\pi\)
0.0163900 + 0.999866i \(0.494783\pi\)
\(30\) 0 0
\(31\) 128.706 − 222.925i 0.745685 − 1.29156i −0.204189 − 0.978932i \(-0.565456\pi\)
0.949874 − 0.312633i \(-0.101211\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 123.842 − 4.66183i 0.598090 − 0.0225141i
\(36\) 0 0
\(37\) 190.107 + 329.276i 0.844688 + 1.46304i 0.885892 + 0.463892i \(0.153547\pi\)
−0.0412040 + 0.999151i \(0.513119\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 217.959 0.830230 0.415115 − 0.909769i \(-0.363741\pi\)
0.415115 + 0.909769i \(0.363741\pi\)
\(42\) 0 0
\(43\) 377.049 1.33720 0.668598 − 0.743624i \(-0.266895\pi\)
0.668598 + 0.743624i \(0.266895\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −178.855 − 309.786i −0.555079 − 0.961425i −0.997897 − 0.0648137i \(-0.979355\pi\)
0.442818 − 0.896611i \(-0.353979\pi\)
\(48\) 0 0
\(49\) −193.347 − 283.313i −0.563693 − 0.825984i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 382.195 − 661.981i 0.990538 − 1.71566i 0.376415 − 0.926451i \(-0.377157\pi\)
0.614122 − 0.789211i \(-0.289510\pi\)
\(54\) 0 0
\(55\) −210.706 −0.516573
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −225.336 + 390.293i −0.497224 + 0.861216i −0.999995 − 0.00320303i \(-0.998980\pi\)
0.502771 + 0.864419i \(0.332314\pi\)
\(60\) 0 0
\(61\) 87.0388 + 150.756i 0.182691 + 0.316431i 0.942796 − 0.333370i \(-0.108186\pi\)
−0.760105 + 0.649801i \(0.774852\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −62.5117 − 108.273i −0.119287 − 0.206610i
\(66\) 0 0
\(67\) 248.617 − 430.617i 0.453334 − 0.785198i −0.545256 − 0.838269i \(-0.683568\pi\)
0.998591 + 0.0530711i \(0.0169010\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −350.238 −0.585432 −0.292716 − 0.956199i \(-0.594559\pi\)
−0.292716 + 0.956199i \(0.594559\pi\)
\(72\) 0 0
\(73\) 531.343 − 920.312i 0.851903 − 1.47554i −0.0275851 − 0.999619i \(-0.508782\pi\)
0.879488 − 0.475920i \(-0.157885\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 310.375 + 493.712i 0.459358 + 0.730698i
\(78\) 0 0
\(79\) −280.224 − 485.363i −0.399085 − 0.691235i 0.594528 − 0.804075i \(-0.297339\pi\)
−0.993613 + 0.112839i \(0.964005\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1105.27 1.46168 0.730840 − 0.682549i \(-0.239129\pi\)
0.730840 + 0.682549i \(0.239129\pi\)
\(84\) 0 0
\(85\) 588.551 0.751027
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 603.357 + 1045.04i 0.718604 + 1.24466i 0.961553 + 0.274619i \(0.0885517\pi\)
−0.242950 + 0.970039i \(0.578115\pi\)
\(90\) 0 0
\(91\) −161.618 + 305.964i −0.186178 + 0.352458i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −43.8199 + 75.8983i −0.0473245 + 0.0819684i
\(96\) 0 0
\(97\) 1442.99 1.51045 0.755226 − 0.655465i \(-0.227527\pi\)
0.755226 + 0.655465i \(0.227527\pi\)
\(98\) 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 504.4.s.h.361.2 6
3.2 odd 2 56.4.i.b.25.1 yes 6
7.2 even 3 inner 504.4.s.h.289.2 6
12.11 even 2 112.4.i.e.81.3 6
21.2 odd 6 56.4.i.b.9.1 ✓ 6
21.5 even 6 392.4.i.m.177.3 6
21.11 odd 6 392.4.a.i.1.3 3
21.17 even 6 392.4.a.l.1.1 3
21.20 even 2 392.4.i.m.361.3 6
24.5 odd 2 448.4.i.j.193.3 6
24.11 even 2 448.4.i.m.193.1 6
84.11 even 6 784.4.a.be.1.1 3
84.23 even 6 112.4.i.e.65.3 6
84.59 odd 6 784.4.a.bb.1.3 3
168.107 even 6 448.4.i.m.65.1 6
168.149 odd 6 448.4.i.j.65.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.4.i.b.9.1 ✓ 6 21.2 odd 6
56.4.i.b.25.1 yes 6 3.2 odd 2
112.4.i.e.65.3 6 84.23 even 6
112.4.i.e.81.3 6 12.11 even 2
392.4.a.i.1.3 3 21.11 odd 6
392.4.a.l.1.1 3 21.17 even 6
392.4.i.m.177.3 6 21.5 even 6
392.4.i.m.361.3 6 21.20 even 2
448.4.i.j.65.3 6 168.149 odd 6
448.4.i.j.193.3 6 24.5 odd 2
448.4.i.m.65.1 6 168.107 even 6
448.4.i.m.193.1 6 24.11 even 2
504.4.s.h.289.2 6 7.2 even 3 inner
504.4.s.h.361.2 6 1.1 even 1 trivial
784.4.a.bb.1.3 3 84.59 odd 6
784.4.a.be.1.1 3 84.11 even 6