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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,2,Mod(25,504)] 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("504.25"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,-2,0,1] 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.7
Character \(\chi\) \(=\) 504.121
Dual form 504.2.q.c.25.7

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.748111 - 1.56216i) q^{3} +(2.11148 + 3.65719i) q^{5} +(2.19338 - 1.47956i) q^{7} +(-1.88066 - 2.33733i) q^{9} +(-0.964575 + 1.67069i) q^{11} +(-0.291529 + 0.504943i) q^{13} +(7.29273 - 0.562477i) q^{15} +(3.61082 + 6.25412i) q^{17} +(2.10268 - 3.64194i) q^{19} +(-0.670409 - 4.53327i) q^{21} +(-0.639939 - 1.10841i) q^{23} +(-6.41671 + 11.1141i) q^{25} +(-5.05822 + 1.18930i) q^{27} +(-4.20305 - 7.27990i) q^{29} -0.952121 q^{31} +(1.88827 + 2.75668i) q^{33} +(10.0423 + 4.89755i) q^{35} +(3.03329 - 5.25381i) q^{37} +(0.570704 + 0.833168i) q^{39} +(1.31299 - 2.27416i) q^{41} +(0.442349 + 0.766171i) q^{43} +(4.57709 - 11.8132i) q^{45} +5.76401 q^{47} +(2.62182 - 6.49046i) q^{49} +(12.4712 - 0.961885i) q^{51} +(-0.962456 - 1.66702i) q^{53} -8.14673 q^{55} +(-4.11625 - 6.00929i) q^{57} -4.55229 q^{59} -10.5802 q^{61} +(-7.58322 - 2.34411i) q^{63} -2.46223 q^{65} -4.86383 q^{67} +(-2.21025 + 0.170473i) q^{69} +11.5443 q^{71} +(0.446138 + 0.772734i) q^{73} +(12.5615 + 18.3384i) q^{75} +(0.356209 + 5.09160i) q^{77} -11.8704 q^{79} +(-1.92623 + 8.79145i) q^{81} +(-5.24250 - 9.08028i) q^{83} +(-15.2484 + 26.4109i) q^{85} +(-14.5167 + 1.11965i) q^{87} +(3.87906 - 6.71874i) q^{89} +(0.107659 + 1.53887i) q^{91} +(-0.712292 + 1.48736i) q^{93} +17.7591 q^{95} +(-1.98651 - 3.44073i) q^{97} +(5.71900 - 0.887474i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q - 2 q^{3} + q^{5} + 5 q^{7} + 6 q^{9} + 3 q^{11} + 7 q^{13} - q^{15} - q^{17} + 13 q^{19} - 22 q^{25} - 2 q^{27} - 7 q^{29} - 12 q^{31} - 3 q^{33} + 2 q^{35} + 6 q^{37} - 4 q^{39} + 4 q^{41} + 2 q^{43}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) 0.748111 − 1.56216i 0.431922 − 0.901911i
\(4\) 0 0
\(5\) 2.11148 + 3.65719i 0.944283 + 1.63555i 0.757180 + 0.653206i \(0.226577\pi\)
0.187103 + 0.982340i \(0.440090\pi\)
\(6\) 0 0
\(7\) 2.19338 − 1.47956i 0.829019 − 0.559220i
\(8\) 0 0
\(9\) −1.88066 − 2.33733i −0.626887 − 0.779110i
\(10\) 0 0
\(11\) −0.964575 + 1.67069i −0.290830 + 0.503733i −0.974006 − 0.226521i \(-0.927265\pi\)
0.683176 + 0.730254i \(0.260598\pi\)
\(12\) 0 0
\(13\) −0.291529 + 0.504943i −0.0808557 + 0.140046i −0.903618 − 0.428340i \(-0.859099\pi\)
0.822762 + 0.568386i \(0.192432\pi\)
\(14\) 0 0
\(15\) 7.29273 − 0.562477i 1.88297 − 0.145231i
\(16\) 0 0
\(17\) 3.61082 + 6.25412i 0.875753 + 1.51685i 0.855959 + 0.517044i \(0.172968\pi\)
0.0197936 + 0.999804i \(0.493699\pi\)
\(18\) 0 0
\(19\) 2.10268 − 3.64194i 0.482387 − 0.835519i −0.517408 − 0.855739i \(-0.673103\pi\)
0.999796 + 0.0202194i \(0.00643646\pi\)
\(20\) 0 0
\(21\) −0.670409 − 4.53327i −0.146295 − 0.989241i
\(22\) 0 0
\(23\) −0.639939 − 1.10841i −0.133437 − 0.231119i 0.791563 − 0.611088i \(-0.209268\pi\)
−0.924999 + 0.379969i \(0.875935\pi\)
\(24\) 0 0
\(25\) −6.41671 + 11.1141i −1.28334 + 2.22281i
\(26\) 0 0
\(27\) −5.05822 + 1.18930i −0.973454 + 0.228881i
\(28\) 0 0
\(29\) −4.20305 − 7.27990i −0.780487 − 1.35184i −0.931658 − 0.363335i \(-0.881638\pi\)
0.151171 − 0.988508i \(-0.451695\pi\)
\(30\) 0 0
\(31\) −0.952121 −0.171006 −0.0855030 − 0.996338i \(-0.527250\pi\)
−0.0855030 + 0.996338i \(0.527250\pi\)
\(32\) 0 0
\(33\) 1.88827 + 2.75668i 0.328706 + 0.479876i
\(34\) 0 0
\(35\) 10.0423 + 4.89755i 1.69746 + 0.827837i
\(36\) 0 0
\(37\) 3.03329 − 5.25381i 0.498669 − 0.863721i −0.501330 − 0.865256i \(-0.667156\pi\)
0.999999 + 0.00153588i \(0.000488885\pi\)
\(38\) 0 0
\(39\) 0.570704 + 0.833168i 0.0913858 + 0.133414i
\(40\) 0 0
\(41\) 1.31299 − 2.27416i 0.205054 − 0.355164i −0.745096 − 0.666957i \(-0.767596\pi\)
0.950150 + 0.311794i \(0.100930\pi\)
\(42\) 0 0
\(43\) 0.442349 + 0.766171i 0.0674576 + 0.116840i 0.897782 − 0.440441i \(-0.145178\pi\)
−0.830324 + 0.557281i \(0.811845\pi\)
\(44\) 0 0
\(45\) 4.57709 − 11.8132i 0.682313 − 1.76100i
\(46\) 0 0
\(47\) 5.76401 0.840767 0.420384 − 0.907346i \(-0.361895\pi\)
0.420384 + 0.907346i \(0.361895\pi\)
\(48\) 0 0
\(49\) 2.62182 − 6.49046i 0.374545 − 0.927209i
\(50\) 0 0
\(51\) 12.4712 − 0.961885i 1.74632 − 0.134691i
\(52\) 0 0
\(53\) −0.962456 − 1.66702i −0.132204 − 0.228983i 0.792322 − 0.610103i \(-0.208872\pi\)
−0.924526 + 0.381120i \(0.875539\pi\)
\(54\) 0 0
\(55\) −8.14673 −1.09850
\(56\) 0 0
\(57\) −4.11625 − 6.00929i −0.545210 − 0.795950i
\(58\) 0 0
\(59\) −4.55229 −0.592657 −0.296329 − 0.955086i \(-0.595762\pi\)
−0.296329 + 0.955086i \(0.595762\pi\)
\(60\) 0 0
\(61\) −10.5802 −1.35465 −0.677325 − 0.735684i \(-0.736861\pi\)
−0.677325 + 0.735684i \(0.736861\pi\)
\(62\) 0 0
\(63\) −7.58322 − 2.34411i −0.955395 − 0.295330i
\(64\) 0 0
\(65\) −2.46223 −0.305403
\(66\) 0 0
\(67\) −4.86383 −0.594211 −0.297106 − 0.954845i \(-0.596021\pi\)
−0.297106 + 0.954845i \(0.596021\pi\)
\(68\) 0 0
\(69\) −2.21025 + 0.170473i −0.266083 + 0.0205226i
\(70\) 0 0
\(71\) 11.5443 1.37005 0.685027 − 0.728518i \(-0.259791\pi\)
0.685027 + 0.728518i \(0.259791\pi\)
\(72\) 0 0
\(73\) 0.446138 + 0.772734i 0.0522165 + 0.0904417i 0.890952 − 0.454097i \(-0.150038\pi\)
−0.838736 + 0.544539i \(0.816705\pi\)
\(74\) 0 0
\(75\) 12.5615 + 18.3384i 1.45048 + 2.11754i
\(76\) 0 0
\(77\) 0.356209 + 5.09160i 0.0405937 + 0.580242i
\(78\) 0 0
\(79\) −11.8704 −1.33553 −0.667763 − 0.744374i \(-0.732748\pi\)
−0.667763 + 0.744374i \(0.732748\pi\)
\(80\) 0 0
\(81\) −1.92623 + 8.79145i −0.214026 + 0.976828i
\(82\) 0 0
\(83\) −5.24250 − 9.08028i −0.575439 − 0.996690i −0.995994 − 0.0894227i \(-0.971498\pi\)
0.420555 − 0.907267i \(-0.361836\pi\)
\(84\) 0 0
\(85\) −15.2484 + 26.4109i −1.65392 + 2.86467i
\(86\) 0 0
\(87\) −14.5167 + 1.11965i −1.55635 + 0.120039i
\(88\) 0 0
\(89\) 3.87906 − 6.71874i 0.411180 − 0.712185i −0.583839 − 0.811869i \(-0.698450\pi\)
0.995019 + 0.0996849i \(0.0317835\pi\)
\(90\) 0 0
\(91\) 0.107659 + 1.53887i 0.0112857 + 0.161317i
\(92\) 0 0
\(93\) −0.712292 + 1.48736i −0.0738613 + 0.154232i
\(94\) 0 0
\(95\) 17.7591 1.82204
\(96\) 0 0
\(97\) −1.98651 − 3.44073i −0.201699 − 0.349353i 0.747377 − 0.664400i \(-0.231313\pi\)
−0.949076 + 0.315047i \(0.897980\pi\)
\(98\) 0 0
\(99\) 5.71900 − 0.887474i 0.574781 − 0.0891945i
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.q.c.121.7 yes 22
3.2 odd 2 1512.2.q.d.793.1 22
4.3 odd 2 1008.2.q.l.625.5 22
7.4 even 3 504.2.t.c.193.1 yes 22
9.2 odd 6 1512.2.t.c.289.11 22
9.7 even 3 504.2.t.c.457.1 yes 22
12.11 even 2 3024.2.q.l.2305.1 22
21.11 odd 6 1512.2.t.c.361.11 22
28.11 odd 6 1008.2.t.l.193.11 22
36.7 odd 6 1008.2.t.l.961.11 22
36.11 even 6 3024.2.t.k.289.11 22
63.11 odd 6 1512.2.q.d.1369.1 22
63.25 even 3 inner 504.2.q.c.25.7 ✓ 22
84.11 even 6 3024.2.t.k.1873.11 22
252.11 even 6 3024.2.q.l.2881.1 22
252.151 odd 6 1008.2.q.l.529.5 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.c.25.7 ✓ 22 63.25 even 3 inner
504.2.q.c.121.7 yes 22 1.1 even 1 trivial
504.2.t.c.193.1 yes 22 7.4 even 3
504.2.t.c.457.1 yes 22 9.7 even 3
1008.2.q.l.529.5 22 252.151 odd 6
1008.2.q.l.625.5 22 4.3 odd 2
1008.2.t.l.193.11 22 28.11 odd 6
1008.2.t.l.961.11 22 36.7 odd 6
1512.2.q.d.793.1 22 3.2 odd 2
1512.2.q.d.1369.1 22 63.11 odd 6
1512.2.t.c.289.11 22 9.2 odd 6
1512.2.t.c.361.11 22 21.11 odd 6
3024.2.q.l.2305.1 22 12.11 even 2
3024.2.q.l.2881.1 22 252.11 even 6
3024.2.t.k.289.11 22 36.11 even 6
3024.2.t.k.1873.11 22 84.11 even 6