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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(169,504)] 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("504.169"); S:= CuspForms(chi, 2); 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, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0] 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 337.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 504.337
Dual form 504.2.r.a.169.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{7} -3.00000 q^{9} +(3.00000 + 5.19615i) q^{11} +(-3.00000 + 5.19615i) q^{13} +(1.50000 + 0.866025i) q^{15} -2.00000 q^{17} +7.00000 q^{19} +(1.50000 - 0.866025i) q^{21} +(0.500000 - 0.866025i) q^{23} +(2.00000 + 3.46410i) q^{25} -5.19615i q^{27} +(-1.00000 - 1.73205i) q^{29} +(-5.00000 + 8.66025i) q^{31} +(-9.00000 + 5.19615i) q^{33} -1.00000 q^{35} -6.00000 q^{37} +(-9.00000 - 5.19615i) q^{39} +(4.00000 - 6.92820i) q^{41} +(5.00000 + 8.66025i) q^{43} +(-1.50000 + 2.59808i) q^{45} +(-4.00000 - 6.92820i) q^{47} +(-0.500000 + 0.866025i) q^{49} -3.46410i q^{51} +2.00000 q^{53} +6.00000 q^{55} +12.1244i q^{57} +(-3.50000 - 6.06218i) q^{61} +(1.50000 + 2.59808i) q^{63} +(3.00000 + 5.19615i) q^{65} +(6.00000 - 10.3923i) q^{67} +(1.50000 + 0.866025i) q^{69} +15.0000 q^{71} -2.00000 q^{73} +(-6.00000 + 3.46410i) q^{75} +(3.00000 - 5.19615i) q^{77} +(-0.500000 - 0.866025i) q^{79} +9.00000 q^{81} +(-6.00000 - 10.3923i) q^{83} +(-1.00000 + 1.73205i) q^{85} +(3.00000 - 1.73205i) q^{87} +4.00000 q^{89} +6.00000 q^{91} +(-15.0000 - 8.66025i) q^{93} +(3.50000 - 6.06218i) q^{95} +(1.00000 + 1.73205i) q^{97} +(-9.00000 - 15.5885i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - q^{7} - 6 q^{9} + 6 q^{11} - 6 q^{13} + 3 q^{15} - 4 q^{17} + 14 q^{19} + 3 q^{21} + q^{23} + 4 q^{25} - 2 q^{29} - 10 q^{31} - 18 q^{33} - 2 q^{35} - 12 q^{37} - 18 q^{39} + 8 q^{41} + 10 q^{43}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i −0.732294 − 0.680989i \(-0.761550\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 3.00000 + 5.19615i 0.904534 + 1.56670i 0.821541 + 0.570149i \(0.193114\pi\)
0.0829925 + 0.996550i \(0.473552\pi\)
\(12\) 0 0
\(13\) −3.00000 + 5.19615i −0.832050 + 1.44115i 0.0643593 + 0.997927i \(0.479500\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) 0 0
\(15\) 1.50000 + 0.866025i 0.387298 + 0.223607i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 1.50000 − 0.866025i 0.327327 − 0.188982i
\(22\) 0 0
\(23\) 0.500000 − 0.866025i 0.104257 − 0.180579i −0.809177 − 0.587565i \(-0.800087\pi\)
0.913434 + 0.406986i \(0.133420\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) −1.00000 − 1.73205i −0.185695 − 0.321634i 0.758115 − 0.652121i \(-0.226120\pi\)
−0.943811 + 0.330487i \(0.892787\pi\)
\(30\) 0 0
\(31\) −5.00000 + 8.66025i −0.898027 + 1.55543i −0.0680129 + 0.997684i \(0.521666\pi\)
−0.830014 + 0.557743i \(0.811667\pi\)
\(32\) 0 0
\(33\) −9.00000 + 5.19615i −1.56670 + 0.904534i
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −9.00000 − 5.19615i −1.44115 − 0.832050i
\(40\) 0 0
\(41\) 4.00000 − 6.92820i 0.624695 − 1.08200i −0.363905 − 0.931436i \(-0.618557\pi\)
0.988600 − 0.150567i \(-0.0481100\pi\)
\(42\) 0 0
\(43\) 5.00000 + 8.66025i 0.762493 + 1.32068i 0.941562 + 0.336840i \(0.109358\pi\)
−0.179069 + 0.983836i \(0.557309\pi\)
\(44\) 0 0
\(45\) −1.50000 + 2.59808i −0.223607 + 0.387298i
\(46\) 0 0
\(47\) −4.00000 − 6.92820i −0.583460 − 1.01058i −0.995066 − 0.0992202i \(-0.968365\pi\)
0.411606 − 0.911362i \(-0.364968\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) − 3.46410i − 0.485071i
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 12.1244i 1.60591i
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −3.50000 − 6.06218i −0.448129 − 0.776182i 0.550135 − 0.835076i \(-0.314576\pi\)
−0.998264 + 0.0588933i \(0.981243\pi\)
\(62\) 0 0
\(63\) 1.50000 + 2.59808i 0.188982 + 0.327327i
\(64\) 0 0
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) 6.00000 − 10.3923i 0.733017 − 1.26962i −0.222571 − 0.974916i \(-0.571445\pi\)
0.955588 − 0.294706i \(-0.0952216\pi\)
\(68\) 0 0
\(69\) 1.50000 + 0.866025i 0.180579 + 0.104257i
\(70\) 0 0
\(71\) 15.0000 1.78017 0.890086 − 0.455792i \(-0.150644\pi\)
0.890086 + 0.455792i \(0.150644\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) −6.00000 + 3.46410i −0.692820 + 0.400000i
\(76\) 0 0
\(77\) 3.00000 − 5.19615i 0.341882 − 0.592157i
\(78\) 0 0
\(79\) −0.500000 − 0.866025i −0.0562544 − 0.0974355i 0.836527 − 0.547926i \(-0.184582\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 − 10.3923i −0.658586 − 1.14070i −0.980982 − 0.194099i \(-0.937822\pi\)
0.322396 − 0.946605i \(-0.395512\pi\)
\(84\) 0 0
\(85\) −1.00000 + 1.73205i −0.108465 + 0.187867i
\(86\) 0 0
\(87\) 3.00000 − 1.73205i 0.321634 − 0.185695i
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) −15.0000 − 8.66025i −1.55543 − 0.898027i
\(94\) 0 0
\(95\) 3.50000 − 6.06218i 0.359092 − 0.621966i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 − 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 0 0
\(99\) −9.00000 − 15.5885i −0.904534 − 1.56670i
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 504.2.r.a.337.1 yes 2
3.2 odd 2 1512.2.r.a.1009.1 2
4.3 odd 2 1008.2.r.c.337.1 2
9.2 odd 6 1512.2.r.a.505.1 2
9.4 even 3 4536.2.a.d.1.1 1
9.5 odd 6 4536.2.a.g.1.1 1
9.7 even 3 inner 504.2.r.a.169.1 ✓ 2
12.11 even 2 3024.2.r.b.1009.1 2
36.7 odd 6 1008.2.r.c.673.1 2
36.11 even 6 3024.2.r.b.2017.1 2
36.23 even 6 9072.2.a.n.1.1 1
36.31 odd 6 9072.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.a.169.1 ✓ 2 9.7 even 3 inner
504.2.r.a.337.1 yes 2 1.1 even 1 trivial
1008.2.r.c.337.1 2 4.3 odd 2
1008.2.r.c.673.1 2 36.7 odd 6
1512.2.r.a.505.1 2 9.2 odd 6
1512.2.r.a.1009.1 2 3.2 odd 2
3024.2.r.b.1009.1 2 12.11 even 2
3024.2.r.b.2017.1 2 36.11 even 6
4536.2.a.d.1.1 1 9.4 even 3
4536.2.a.g.1.1 1 9.5 odd 6
9072.2.a.i.1.1 1 36.31 odd 6
9072.2.a.n.1.1 1 36.23 even 6