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Properties

Label 252.4.k.f.37.1
Level $252$
Weight $4$
Character 252.37
Analytic conductor $14.868$
Analytic rank $0$
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] = [252,4,Mod(37,252)] 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("252.37"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,11,0,6] 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: no
Analytic conductor: \(14.8684813214\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.1
Root \(3.72311 - 6.44862i\) of defining polynomial
Character \(\chi\) \(=\) 252.37
Dual form 252.4.k.f.109.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.723111 + 1.25246i) q^{5} +(-12.3924 - 13.7633i) q^{7} +(23.0618 + 39.9442i) q^{11} -32.2311 q^{13} +(38.8924 + 67.3637i) q^{17} +(-6.33067 + 10.9650i) q^{19} +(-50.4622 + 87.4031i) q^{23} +(61.4542 + 106.442i) q^{25} -213.908 q^{29} +(-21.0378 - 36.4385i) q^{31} +(26.1991 - 5.56874i) q^{35} +(-155.040 + 268.537i) q^{37} -44.0320 q^{41} +381.339 q^{43} +(-179.032 + 310.093i) q^{47} +(-35.8547 + 341.121i) q^{49} +(92.4920 + 160.201i) q^{53} -66.7049 q^{55} +(227.325 + 393.738i) q^{59} +(-5.92444 + 10.2614i) q^{61} +(23.3067 - 40.3683i) q^{65} +(-295.180 - 511.266i) q^{67} -494.366 q^{71} +(-487.825 - 844.937i) q^{73} +(263.970 - 812.411i) q^{77} +(-149.667 + 259.231i) q^{79} +1406.07 q^{83} -112.494 q^{85} +(-347.629 + 602.112i) q^{89} +(399.422 + 443.605i) q^{91} +(-9.15555 - 15.8579i) q^{95} +481.940 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 11 q^{5} + 6 q^{7} - 5 q^{11} + 10 q^{13} + 100 q^{17} - 67 q^{19} + 76 q^{23} + 93 q^{25} - 550 q^{29} - 362 q^{31} + 466 q^{35} + 5 q^{37} + 324 q^{41} + 1442 q^{43} - 216 q^{47} + 190 q^{49} + 495 q^{53}+ \cdots + 1122 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(5\) −0.723111 + 1.25246i −0.0646770 + 0.112024i −0.896551 − 0.442941i \(-0.853935\pi\)
0.831874 + 0.554965i \(0.187268\pi\)
\(6\) 0 0
\(7\) −12.3924 − 13.7633i −0.669129 − 0.743146i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 23.0618 + 39.9442i 0.632126 + 1.09487i 0.987116 + 0.160005i \(0.0511509\pi\)
−0.354990 + 0.934870i \(0.615516\pi\)
\(12\) 0 0
\(13\) −32.2311 −0.687639 −0.343819 − 0.939036i \(-0.611721\pi\)
−0.343819 + 0.939036i \(0.611721\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 38.8924 + 67.3637i 0.554871 + 0.961064i 0.997914 + 0.0645639i \(0.0205656\pi\)
−0.443043 + 0.896500i \(0.646101\pi\)
\(18\) 0 0
\(19\) −6.33067 + 10.9650i −0.0764397 + 0.132397i −0.901711 − 0.432338i \(-0.857689\pi\)
0.825272 + 0.564736i \(0.191022\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −50.4622 + 87.4031i −0.457483 + 0.792383i −0.998827 − 0.0484177i \(-0.984582\pi\)
0.541345 + 0.840801i \(0.317915\pi\)
\(24\) 0 0
\(25\) 61.4542 + 106.442i 0.491634 + 0.851535i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −213.908 −1.36972 −0.684859 − 0.728676i \(-0.740136\pi\)
−0.684859 + 0.728676i \(0.740136\pi\)
\(30\) 0 0
\(31\) −21.0378 − 36.4385i −0.121887 − 0.211114i 0.798625 − 0.601829i \(-0.205561\pi\)
−0.920512 + 0.390715i \(0.872228\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 26.1991 − 5.56874i 0.126527 − 0.0268940i
\(36\) 0 0
\(37\) −155.040 + 268.537i −0.688876 + 1.19317i 0.283326 + 0.959024i \(0.408562\pi\)
−0.972202 + 0.234145i \(0.924771\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −44.0320 −0.167723 −0.0838615 − 0.996477i \(-0.526725\pi\)
−0.0838615 + 0.996477i \(0.526725\pi\)
\(42\) 0 0
\(43\) 381.339 1.35241 0.676205 − 0.736714i \(-0.263624\pi\)
0.676205 + 0.736714i \(0.263624\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −179.032 + 310.093i −0.555628 + 0.962375i 0.442227 + 0.896903i \(0.354189\pi\)
−0.997854 + 0.0654721i \(0.979145\pi\)
\(48\) 0 0
\(49\) −35.8547 + 341.121i −0.104533 + 0.994521i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 92.4920 + 160.201i 0.239712 + 0.415194i 0.960632 − 0.277825i \(-0.0896135\pi\)
−0.720919 + 0.693019i \(0.756280\pi\)
\(54\) 0 0
\(55\) −66.7049 −0.163536
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 227.325 + 393.738i 0.501613 + 0.868820i 0.999998 + 0.00186377i \(0.000593256\pi\)
−0.498385 + 0.866956i \(0.666073\pi\)
\(60\) 0 0
\(61\) −5.92444 + 10.2614i −0.0124352 + 0.0215384i −0.872176 − 0.489192i \(-0.837292\pi\)
0.859741 + 0.510731i \(0.170625\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 23.3067 − 40.3683i 0.0444744 − 0.0770319i
\(66\) 0 0
\(67\) −295.180 − 511.266i −0.538238 − 0.932255i −0.998999 − 0.0447309i \(-0.985757\pi\)
0.460761 − 0.887524i \(-0.347576\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −494.366 −0.826345 −0.413172 − 0.910653i \(-0.635579\pi\)
−0.413172 + 0.910653i \(0.635579\pi\)
\(72\) 0 0
\(73\) −487.825 − 844.937i −0.782131 − 1.35469i −0.930698 − 0.365789i \(-0.880799\pi\)
0.148567 − 0.988902i \(-0.452534\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 263.970 − 812.411i 0.390678 − 1.20237i
\(78\) 0 0
\(79\) −149.667 + 259.231i −0.213150 + 0.369187i −0.952699 − 0.303916i \(-0.901706\pi\)
0.739549 + 0.673103i \(0.235039\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1406.07 1.85947 0.929735 − 0.368229i \(-0.120036\pi\)
0.929735 + 0.368229i \(0.120036\pi\)
\(84\) 0 0
\(85\) −112.494 −0.143550
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −347.629 + 602.112i −0.414030 + 0.717120i −0.995326 − 0.0965715i \(-0.969212\pi\)
0.581296 + 0.813692i \(0.302546\pi\)
\(90\) 0 0
\(91\) 399.422 + 443.605i 0.460119 + 0.511016i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −9.15555 − 15.8579i −0.00988779 − 0.0171261i
\(96\) 0 0
\(97\) 481.940 0.504470 0.252235 − 0.967666i \(-0.418834\pi\)
0.252235 + 0.967666i \(0.418834\pi\)
\(98\) 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 252.4.k.f.37.1 4
3.2 odd 2 84.4.i.a.37.2 yes 4
7.2 even 3 1764.4.a.o.1.2 2
7.3 odd 6 1764.4.k.q.361.2 4
7.4 even 3 inner 252.4.k.f.109.1 4
7.5 odd 6 1764.4.a.y.1.1 2
7.6 odd 2 1764.4.k.q.1549.2 4
12.11 even 2 336.4.q.i.289.2 4
21.2 odd 6 588.4.a.i.1.1 2
21.5 even 6 588.4.a.f.1.2 2
21.11 odd 6 84.4.i.a.25.2 ✓ 4
21.17 even 6 588.4.i.j.361.1 4
21.20 even 2 588.4.i.j.373.1 4
84.11 even 6 336.4.q.i.193.2 4
84.23 even 6 2352.4.a.bt.1.1 2
84.47 odd 6 2352.4.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.a.25.2 ✓ 4 21.11 odd 6
84.4.i.a.37.2 yes 4 3.2 odd 2
252.4.k.f.37.1 4 1.1 even 1 trivial
252.4.k.f.109.1 4 7.4 even 3 inner
336.4.q.i.193.2 4 84.11 even 6
336.4.q.i.289.2 4 12.11 even 2
588.4.a.f.1.2 2 21.5 even 6
588.4.a.i.1.1 2 21.2 odd 6
588.4.i.j.361.1 4 21.17 even 6
588.4.i.j.373.1 4 21.20 even 2
1764.4.a.o.1.2 2 7.2 even 3
1764.4.a.y.1.1 2 7.5 odd 6
1764.4.k.q.361.2 4 7.3 odd 6
1764.4.k.q.1549.2 4 7.6 odd 2
2352.4.a.bt.1.1 2 84.23 even 6
2352.4.a.bx.1.2 2 84.47 odd 6