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Properties

Label 252.6.k.f.109.1
Level $252$
Weight $6$
Character 252.109
Analytic conductor $40.417$
Analytic rank $0$
Dimension $8$
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] = [252,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0] 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: \(40.4167225929\)
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: \( x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3}\cdot 7 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 109.1
Root \(-11.2416 + 19.4709i\) of defining polynomial
Character \(\chi\) \(=\) 252.109
Dual form 252.6.k.f.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)
 
\(f(q)\) \(=\) \(q+(-46.4128 - 80.3893i) q^{5} +(-118.369 + 52.8745i) q^{7} +(70.3812 - 121.904i) q^{11} -1111.24 q^{13} +(27.4435 - 47.5335i) q^{17} +(855.929 + 1482.51i) q^{19} +(-1643.95 - 2847.41i) q^{23} +(-2745.79 + 4755.85i) q^{25} +3790.72 q^{29} +(2423.66 - 4197.90i) q^{31} +(9744.39 + 7061.57i) q^{35} +(5683.35 + 9843.86i) q^{37} +10385.6 q^{41} +7137.16 q^{43} +(-8207.53 - 14215.9i) q^{47} +(11215.6 - 12517.4i) q^{49} +(-10487.2 + 18164.3i) q^{53} -13066.3 q^{55} +(-18106.0 + 31360.5i) q^{59} +(-2474.35 - 4285.70i) q^{61} +(51575.9 + 89332.0i) q^{65} +(11482.8 - 19888.8i) q^{67} +26341.8 q^{71} +(-27693.7 + 47966.9i) q^{73} +(-1885.36 + 18151.0i) q^{77} +(24978.3 + 43263.7i) q^{79} -44858.9 q^{83} -5094.91 q^{85} +(63972.5 + 110804. i) q^{89} +(131537. - 58756.4i) q^{91} +(79452.1 - 137615. i) q^{95} -65685.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) \) 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
\(5\) −46.4128 − 80.3893i −0.830257 − 1.43805i −0.897834 − 0.440333i \(-0.854860\pi\)
0.0675775 − 0.997714i \(-0.478473\pi\)
\(6\) 0 0
\(7\) −118.369 + 52.8745i −0.913049 + 0.407851i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 70.3812 − 121.904i 0.175378 − 0.303763i −0.764914 − 0.644132i \(-0.777219\pi\)
0.940292 + 0.340369i \(0.110552\pi\)
\(12\) 0 0
\(13\) −1111.24 −1.82369 −0.911844 − 0.410537i \(-0.865341\pi\)
−0.911844 + 0.410537i \(0.865341\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 27.4435 − 47.5335i 0.0230312 − 0.0398912i −0.854280 − 0.519813i \(-0.826002\pi\)
0.877311 + 0.479922i \(0.159335\pi\)
\(18\) 0 0
\(19\) 855.929 + 1482.51i 0.543943 + 0.942138i 0.998673 + 0.0515079i \(0.0164027\pi\)
−0.454729 + 0.890630i \(0.650264\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1643.95 − 2847.41i −0.647993 − 1.12236i −0.983602 − 0.180355i \(-0.942275\pi\)
0.335609 − 0.942001i \(-0.391058\pi\)
\(24\) 0 0
\(25\) −2745.79 + 4755.85i −0.878653 + 1.52187i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3790.72 0.837003 0.418501 − 0.908216i \(-0.362555\pi\)
0.418501 + 0.908216i \(0.362555\pi\)
\(30\) 0 0
\(31\) 2423.66 − 4197.90i 0.452968 − 0.784563i −0.545601 − 0.838045i \(-0.683699\pi\)
0.998569 + 0.0534817i \(0.0170319\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9744.39 + 7061.57i 1.34457 + 0.974386i
\(36\) 0 0
\(37\) 5683.35 + 9843.86i 0.682496 + 1.18212i 0.974217 + 0.225615i \(0.0724391\pi\)
−0.291720 + 0.956504i \(0.594228\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10385.6 0.964881 0.482440 − 0.875929i \(-0.339751\pi\)
0.482440 + 0.875929i \(0.339751\pi\)
\(42\) 0 0
\(43\) 7137.16 0.588646 0.294323 − 0.955706i \(-0.404906\pi\)
0.294323 + 0.955706i \(0.404906\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8207.53 − 14215.9i −0.541961 − 0.938704i −0.998791 − 0.0491499i \(-0.984349\pi\)
0.456831 − 0.889554i \(-0.348985\pi\)
\(48\) 0 0
\(49\) 11215.6 − 12517.4i 0.667315 − 0.744775i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10487.2 + 18164.3i −0.512824 + 0.888238i 0.487065 + 0.873366i \(0.338067\pi\)
−0.999889 + 0.0148720i \(0.995266\pi\)
\(54\) 0 0
\(55\) −13066.3 −0.582435
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −18106.0 + 31360.5i −0.677161 + 1.17288i 0.298672 + 0.954356i \(0.403456\pi\)
−0.975832 + 0.218521i \(0.929877\pi\)
\(60\) 0 0
\(61\) −2474.35 − 4285.70i −0.0851405 − 0.147468i 0.820311 − 0.571918i \(-0.193801\pi\)
−0.905451 + 0.424451i \(0.860467\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 51575.9 + 89332.0i 1.51413 + 2.62255i
\(66\) 0 0
\(67\) 11482.8 − 19888.8i 0.312507 − 0.541279i −0.666397 − 0.745597i \(-0.732164\pi\)
0.978905 + 0.204318i \(0.0654978\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 26341.8 0.620154 0.310077 − 0.950711i \(-0.399645\pi\)
0.310077 + 0.950711i \(0.399645\pi\)
\(72\) 0 0
\(73\) −27693.7 + 47966.9i −0.608238 + 1.05350i 0.383293 + 0.923627i \(0.374790\pi\)
−0.991531 + 0.129872i \(0.958543\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1885.36 + 18151.0i −0.0362383 + 0.348879i
\(78\) 0 0
\(79\) 24978.3 + 43263.7i 0.450293 + 0.779930i 0.998404 − 0.0564752i \(-0.0179862\pi\)
−0.548111 + 0.836406i \(0.684653\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −44858.9 −0.714749 −0.357374 − 0.933961i \(-0.616328\pi\)
−0.357374 + 0.933961i \(0.616328\pi\)
\(84\) 0 0
\(85\) −5094.91 −0.0764873
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 63972.5 + 110804.i 0.856087 + 1.48279i 0.875633 + 0.482978i \(0.160445\pi\)
−0.0195454 + 0.999809i \(0.506222\pi\)
\(90\) 0 0
\(91\) 131537. − 58756.4i 1.66512 − 0.743793i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 79452.1 − 137615.i 0.903226 − 1.56443i
\(96\) 0 0
\(97\) −65685.9 −0.708831 −0.354415 − 0.935088i \(-0.615320\pi\)
−0.354415 + 0.935088i \(0.615320\pi\)
\(98\) 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 252.6.k.f.109.1 8
3.2 odd 2 84.6.i.c.25.4 ✓ 8
7.2 even 3 inner 252.6.k.f.37.1 8
12.11 even 2 336.6.q.i.193.4 8
21.2 odd 6 84.6.i.c.37.4 yes 8
21.5 even 6 588.6.i.o.373.1 8
21.11 odd 6 588.6.a.n.1.1 4
21.17 even 6 588.6.a.p.1.4 4
21.20 even 2 588.6.i.o.361.1 8
84.23 even 6 336.6.q.i.289.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.4 ✓ 8 3.2 odd 2
84.6.i.c.37.4 yes 8 21.2 odd 6
252.6.k.f.37.1 8 7.2 even 3 inner
252.6.k.f.109.1 8 1.1 even 1 trivial
336.6.q.i.193.4 8 12.11 even 2
336.6.q.i.289.4 8 84.23 even 6
588.6.a.n.1.1 4 21.11 odd 6
588.6.a.p.1.4 4 21.17 even 6
588.6.i.o.361.1 8 21.20 even 2
588.6.i.o.373.1 8 21.5 even 6