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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [252,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,196] 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: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(-0.00737575 + 0.0127752i\) of defining polynomial
Character \(\chi\) \(=\) 252.109
Dual form 252.8.k.b.37.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-80.0854 - 138.712i) q^{5} +(-254.682 - 871.022i) q^{7} +(1581.93 - 2739.98i) q^{11} -4771.53 q^{13} +(-6504.73 + 11266.5i) q^{17} +(-22058.1 - 38205.8i) q^{19} +(-32533.0 - 56348.8i) q^{23} +(26235.2 - 45440.6i) q^{25} +246115. q^{29} +(-150328. + 260376. i) q^{31} +(-100425. + 105084. i) q^{35} +(-258479. - 447698. i) q^{37} +377844. q^{41} -71420.3 q^{43} +(558378. + 967140. i) q^{47} +(-693817. + 443667. i) q^{49} +(-184365. + 319330. i) q^{53} -506758. q^{55} +(515977. - 893698. i) q^{59} +(161341. + 279450. i) q^{61} +(382130. + 661869. i) q^{65} +(11763.8 - 20375.5i) q^{67} -2.84199e6 q^{71} +(-336523. + 582875. i) q^{73} +(-2.78947e6 - 680072. i) q^{77} +(429448. + 743826. i) q^{79} -7.32276e6 q^{83} +2.08373e6 q^{85} +(2.31400e6 + 4.00797e6i) q^{89} +(1.21522e6 + 4.15611e6i) q^{91} +(-3.53307e6 + 6.11946e6i) q^{95} +951821. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 196 q^{5} - 434 q^{7} - 406 q^{11} + 3948 q^{13} + 7436 q^{17} + 15874 q^{19} + 6788 q^{23} + 69898 q^{25} + 189088 q^{29} - 55890 q^{31} + 750596 q^{35} + 93742 q^{37} - 13944 q^{41} - 487844 q^{43}+ \cdots + 23722580 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) −80.0854 − 138.712i −0.286522 − 0.496271i 0.686455 − 0.727172i \(-0.259166\pi\)
−0.972977 + 0.230901i \(0.925833\pi\)
\(6\) 0 0
\(7\) −254.682 − 871.022i −0.280644 − 0.959812i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1581.93 − 2739.98i 0.358354 − 0.620688i −0.629332 − 0.777137i \(-0.716671\pi\)
0.987686 + 0.156449i \(0.0500046\pi\)
\(12\) 0 0
\(13\) −4771.53 −0.602360 −0.301180 − 0.953567i \(-0.597380\pi\)
−0.301180 + 0.953567i \(0.597380\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6504.73 + 11266.5i −0.321113 + 0.556184i −0.980718 − 0.195429i \(-0.937390\pi\)
0.659605 + 0.751612i \(0.270724\pi\)
\(18\) 0 0
\(19\) −22058.1 − 38205.8i −0.737787 − 1.27789i −0.953490 − 0.301426i \(-0.902537\pi\)
0.215702 − 0.976459i \(-0.430796\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −32533.0 − 56348.8i −0.557541 − 0.965690i −0.997701 − 0.0677701i \(-0.978412\pi\)
0.440160 − 0.897919i \(-0.354922\pi\)
\(24\) 0 0
\(25\) 26235.2 − 45440.6i 0.335810 − 0.581640i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 246115. 1.87390 0.936949 − 0.349466i \(-0.113637\pi\)
0.936949 + 0.349466i \(0.113637\pi\)
\(30\) 0 0
\(31\) −150328. + 260376.i −0.906306 + 1.56977i −0.0871514 + 0.996195i \(0.527776\pi\)
−0.819155 + 0.573573i \(0.805557\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −100425. + 105084.i −0.395916 + 0.414283i
\(36\) 0 0
\(37\) −258479. − 447698.i −0.838916 − 1.45305i −0.890801 − 0.454393i \(-0.849856\pi\)
0.0518851 − 0.998653i \(-0.483477\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 377844. 0.856188 0.428094 − 0.903734i \(-0.359185\pi\)
0.428094 + 0.903734i \(0.359185\pi\)
\(42\) 0 0
\(43\) −71420.3 −0.136988 −0.0684940 − 0.997652i \(-0.521819\pi\)
−0.0684940 + 0.997652i \(0.521819\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 558378. + 967140.i 0.784488 + 1.35877i 0.929305 + 0.369314i \(0.120407\pi\)
−0.144817 + 0.989458i \(0.546259\pi\)
\(48\) 0 0
\(49\) −693817. + 443667.i −0.842478 + 0.538730i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −184365. + 319330.i −0.170104 + 0.294628i −0.938456 − 0.345399i \(-0.887744\pi\)
0.768352 + 0.640027i \(0.221077\pi\)
\(54\) 0 0
\(55\) −506758. −0.410706
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 515977. − 893698.i 0.327076 − 0.566512i −0.654855 − 0.755755i \(-0.727270\pi\)
0.981930 + 0.189243i \(0.0606035\pi\)
\(60\) 0 0
\(61\) 161341. + 279450.i 0.0910101 + 0.157634i 0.907936 − 0.419108i \(-0.137657\pi\)
−0.816926 + 0.576742i \(0.804324\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 382130. + 661869.i 0.172590 + 0.298934i
\(66\) 0 0
\(67\) 11763.8 − 20375.5i 0.00477844 − 0.00827650i −0.863626 − 0.504133i \(-0.831812\pi\)
0.868405 + 0.495856i \(0.165146\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.84199e6 −0.942363 −0.471181 − 0.882036i \(-0.656172\pi\)
−0.471181 + 0.882036i \(0.656172\pi\)
\(72\) 0 0
\(73\) −336523. + 582875.i −0.101248 + 0.175366i −0.912199 − 0.409748i \(-0.865617\pi\)
0.810951 + 0.585114i \(0.198950\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.78947e6 − 680072.i −0.696314 − 0.169761i
\(78\) 0 0
\(79\) 429448. + 743826.i 0.0979977 + 0.169737i 0.910856 − 0.412725i \(-0.135423\pi\)
−0.812858 + 0.582462i \(0.802090\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.32276e6 −1.40573 −0.702864 − 0.711324i \(-0.748096\pi\)
−0.702864 + 0.711324i \(0.748096\pi\)
\(84\) 0 0
\(85\) 2.08373e6 0.368024
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.31400e6 + 4.00797e6i 0.347935 + 0.602641i 0.985883 − 0.167439i \(-0.0535496\pi\)
−0.637947 + 0.770080i \(0.720216\pi\)
\(90\) 0 0
\(91\) 1.21522e6 + 4.15611e6i 0.169048 + 0.578152i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.53307e6 + 6.11946e6i −0.422785 + 0.732285i
\(96\) 0 0
\(97\) 951821. 0.105890 0.0529449 − 0.998597i \(-0.483139\pi\)
0.0529449 + 0.998597i \(0.483139\pi\)
\(98\) 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 252.8.k.b.109.1 8
3.2 odd 2 84.8.i.a.25.4 ✓ 8
7.2 even 3 inner 252.8.k.b.37.1 8
21.2 odd 6 84.8.i.a.37.4 yes 8
21.5 even 6 588.8.i.o.373.1 8
21.11 odd 6 588.8.a.i.1.1 4
21.17 even 6 588.8.a.j.1.4 4
21.20 even 2 588.8.i.o.361.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.i.a.25.4 ✓ 8 3.2 odd 2
84.8.i.a.37.4 yes 8 21.2 odd 6
252.8.k.b.37.1 8 7.2 even 3 inner
252.8.k.b.109.1 8 1.1 even 1 trivial
588.8.a.i.1.1 4 21.11 odd 6
588.8.a.j.1.4 4 21.17 even 6
588.8.i.o.361.1 8 21.20 even 2
588.8.i.o.373.1 8 21.5 even 6