↑

Properties

Label 336.6.q.i.193.1
Level $336$
Weight $6$
Character 336.193
Analytic conductor $53.889$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,6,Mod(193,336)] 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("336.193"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-36,0,0] 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: \(53.8889634572\)
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 193.1
Root \(4.59067 - 7.95128i\) of defining polynomial
Character \(\chi\) \(=\) 336.193
Dual form 336.6.q.i.289.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 + 7.79423i) q^{3} +(-39.3359 - 68.1317i) q^{5} +(100.606 - 81.7641i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(345.759 - 598.873i) q^{11} +818.732 q^{13} +708.046 q^{15} +(-554.638 + 960.661i) q^{17} +(-286.523 - 496.273i) q^{19} +(184.559 + 1152.09i) q^{21} +(1258.80 + 2180.30i) q^{23} +(-1532.12 + 2653.71i) q^{25} +729.000 q^{27} -3258.19 q^{29} +(5059.55 - 8763.40i) q^{31} +(3111.84 + 5389.86i) q^{33} +(-9528.17 - 3638.23i) q^{35} +(-2434.81 - 4217.21i) q^{37} +(-3684.30 + 6381.39i) q^{39} -13094.3 q^{41} +9303.64 q^{43} +(-3186.21 + 5518.67i) q^{45} +(6452.90 + 11176.8i) q^{47} +(3436.28 - 16452.0i) q^{49} +(-4991.74 - 8645.95i) q^{51} +(9770.12 - 16922.3i) q^{53} -54403.0 q^{55} +5157.42 q^{57} +(-12560.2 + 21755.0i) q^{59} +(15681.1 + 27160.5i) q^{61} +(-9810.15 - 3745.90i) q^{63} +(-32205.6 - 55781.7i) q^{65} +(27971.9 - 48448.8i) q^{67} -22658.4 q^{69} -20501.0 q^{71} +(33825.0 - 58586.6i) q^{73} +(-13789.1 - 23883.4i) q^{75} +(-14180.7 - 88521.1i) q^{77} +(7039.95 + 12193.5i) q^{79} +(-3280.50 + 5681.99i) q^{81} +77129.1 q^{83} +87268.6 q^{85} +(14661.9 - 25395.1i) q^{87} +(-160.396 - 277.815i) q^{89} +(82369.7 - 66942.9i) q^{91} +(45536.0 + 78870.6i) q^{93} +(-22541.3 + 39042.6i) q^{95} -112009. q^{97} -56013.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 36 q^{3} + 42 q^{7} - 324 q^{9} + 462 q^{11} - 1204 q^{13} + 228 q^{17} - 358 q^{19} + 1404 q^{21} + 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} + 4158 q^{33} - 7692 q^{35}+ \cdots - 74844 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −4.50000 + 7.79423i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) −39.3359 − 68.1317i −0.703662 − 1.21878i −0.967172 − 0.254121i \(-0.918214\pi\)
0.263511 − 0.964656i \(-0.415120\pi\)
\(6\) 0 0
\(7\) 100.606 − 81.7641i 0.776033 − 0.630692i
\(8\) 0 0
\(9\) −40.5000 − 70.1481i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 345.759 − 598.873i 0.861574 − 1.49229i −0.00883597 − 0.999961i \(-0.502813\pi\)
0.870410 − 0.492328i \(-0.163854\pi\)
\(12\) 0 0
\(13\) 818.732 1.34364 0.671821 − 0.740714i \(-0.265512\pi\)
0.671821 + 0.740714i \(0.265512\pi\)
\(14\) 0 0
\(15\) 708.046 0.812518
\(16\) 0 0
\(17\) −554.638 + 960.661i −0.465465 + 0.806209i −0.999222 − 0.0394286i \(-0.987446\pi\)
0.533757 + 0.845638i \(0.320780\pi\)
\(18\) 0 0
\(19\) −286.523 − 496.273i −0.182086 − 0.315382i 0.760505 − 0.649332i \(-0.224951\pi\)
−0.942591 + 0.333951i \(0.891618\pi\)
\(20\) 0 0
\(21\) 184.559 + 1152.09i 0.0913245 + 0.570082i
\(22\) 0 0
\(23\) 1258.80 + 2180.30i 0.496177 + 0.859403i 0.999990 − 0.00440926i \(-0.00140352\pi\)
−0.503814 + 0.863812i \(0.668070\pi\)
\(24\) 0 0
\(25\) −1532.12 + 2653.71i −0.490279 + 0.849189i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −3258.19 −0.719419 −0.359710 − 0.933064i \(-0.617124\pi\)
−0.359710 + 0.933064i \(0.617124\pi\)
\(30\) 0 0
\(31\) 5059.55 − 8763.40i 0.945601 − 1.63783i 0.191056 − 0.981579i \(-0.438809\pi\)
0.754544 − 0.656249i \(-0.227858\pi\)
\(32\) 0 0
\(33\) 3111.84 + 5389.86i 0.497430 + 0.861574i
\(34\) 0 0
\(35\) −9528.17 − 3638.23i −1.31474 − 0.502018i
\(36\) 0 0
\(37\) −2434.81 − 4217.21i −0.292389 − 0.506432i 0.681985 − 0.731366i \(-0.261117\pi\)
−0.974374 + 0.224934i \(0.927783\pi\)
\(38\) 0 0
\(39\) −3684.30 + 6381.39i −0.387876 + 0.671821i
\(40\) 0 0
\(41\) −13094.3 −1.21653 −0.608265 − 0.793734i \(-0.708134\pi\)
−0.608265 + 0.793734i \(0.708134\pi\)
\(42\) 0 0
\(43\) 9303.64 0.767329 0.383664 − 0.923473i \(-0.374662\pi\)
0.383664 + 0.923473i \(0.374662\pi\)
\(44\) 0 0
\(45\) −3186.21 + 5518.67i −0.234554 + 0.406259i
\(46\) 0 0
\(47\) 6452.90 + 11176.8i 0.426099 + 0.738025i 0.996522 − 0.0833256i \(-0.0265542\pi\)
−0.570423 + 0.821351i \(0.693221\pi\)
\(48\) 0 0
\(49\) 3436.28 − 16452.0i 0.204455 − 0.978876i
\(50\) 0 0
\(51\) −4991.74 − 8645.95i −0.268736 − 0.465465i
\(52\) 0 0
\(53\) 9770.12 − 16922.3i 0.477760 − 0.827505i −0.521915 − 0.852998i \(-0.674782\pi\)
0.999675 + 0.0254926i \(0.00811542\pi\)
\(54\) 0 0
\(55\) −54403.0 −2.42503
\(56\) 0 0
\(57\) 5157.42 0.210254
\(58\) 0 0
\(59\) −12560.2 + 21755.0i −0.469751 + 0.813632i −0.999402 − 0.0345835i \(-0.988990\pi\)
0.529651 + 0.848216i \(0.322323\pi\)
\(60\) 0 0
\(61\) 15681.1 + 27160.5i 0.539575 + 0.934571i 0.998927 + 0.0463168i \(0.0147484\pi\)
−0.459352 + 0.888254i \(0.651918\pi\)
\(62\) 0 0
\(63\) −9810.15 − 3745.90i −0.311404 − 0.118906i
\(64\) 0 0
\(65\) −32205.6 − 55781.7i −0.945469 − 1.63760i
\(66\) 0 0
\(67\) 27971.9 − 48448.8i 0.761264 − 1.31855i −0.180936 − 0.983495i \(-0.557913\pi\)
0.942200 − 0.335052i \(-0.108754\pi\)
\(68\) 0 0
\(69\) −22658.4 −0.572935
\(70\) 0 0
\(71\) −20501.0 −0.482647 −0.241323 − 0.970445i \(-0.577581\pi\)
−0.241323 + 0.970445i \(0.577581\pi\)
\(72\) 0 0
\(73\) 33825.0 − 58586.6i 0.742900 − 1.28674i −0.208270 − 0.978071i \(-0.566783\pi\)
0.951170 − 0.308669i \(-0.0998835\pi\)
\(74\) 0 0
\(75\) −13789.1 − 23883.4i −0.283063 − 0.490279i
\(76\) 0 0
\(77\) −14180.7 − 88521.1i −0.272565 − 1.70145i
\(78\) 0 0
\(79\) 7039.95 + 12193.5i 0.126912 + 0.219818i 0.922479 − 0.386048i \(-0.126160\pi\)
−0.795567 + 0.605866i \(0.792827\pi\)
\(80\) 0 0
\(81\) −3280.50 + 5681.99i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 77129.1 1.22892 0.614459 − 0.788949i \(-0.289374\pi\)
0.614459 + 0.788949i \(0.289374\pi\)
\(84\) 0 0
\(85\) 87268.6 1.31012
\(86\) 0 0
\(87\) 14661.9 − 25395.1i 0.207678 − 0.359710i
\(88\) 0 0
\(89\) −160.396 − 277.815i −0.00214644 − 0.00371775i 0.864950 − 0.501858i \(-0.167350\pi\)
−0.867097 + 0.498140i \(0.834017\pi\)
\(90\) 0 0
\(91\) 82369.7 − 66942.9i 1.04271 − 0.847424i
\(92\) 0 0
\(93\) 45536.0 + 78870.6i 0.545943 + 0.945601i
\(94\) 0 0
\(95\) −22541.3 + 39042.6i −0.256253 + 0.443844i
\(96\) 0 0
\(97\) −112009. −1.20871 −0.604355 − 0.796715i \(-0.706569\pi\)
−0.604355 + 0.796715i \(0.706569\pi\)
\(98\) 0 0
\(99\) −56013.0 −0.574382
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 336.6.q.i.193.1 8
4.3 odd 2 84.6.i.c.25.1 ✓ 8
7.2 even 3 inner 336.6.q.i.289.1 8
12.11 even 2 252.6.k.f.109.4 8
28.3 even 6 588.6.a.p.1.1 4
28.11 odd 6 588.6.a.n.1.4 4
28.19 even 6 588.6.i.o.373.4 8
28.23 odd 6 84.6.i.c.37.1 yes 8
28.27 even 2 588.6.i.o.361.4 8
84.23 even 6 252.6.k.f.37.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.i.c.25.1 ✓ 8 4.3 odd 2
84.6.i.c.37.1 yes 8 28.23 odd 6
252.6.k.f.37.4 8 84.23 even 6
252.6.k.f.109.4 8 12.11 even 2
336.6.q.i.193.1 8 1.1 even 1 trivial
336.6.q.i.289.1 8 7.2 even 3 inner
588.6.a.n.1.4 4 28.11 odd 6
588.6.a.p.1.1 4 28.3 even 6
588.6.i.o.361.4 8 28.27 even 2
588.6.i.o.373.4 8 28.19 even 6