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Properties

Label 560.3.bx.c.241.6
Level $560$
Weight $3$
Character 560.241
Analytic conductor $15.259$
Analytic rank $0$
Dimension $12$
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] = [560,3,Mod(241,560)] 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("560.241"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 560.bx (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,6,0,0,0,2] 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: \(15.2588948042\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 241.6
Root \(0.242987 - 0.420865i\) of defining polynomial
Character \(\chi\) \(=\) 560.241
Dual form 560.3.bx.c.481.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.74894 + 2.74180i) q^{3} +(-1.93649 + 1.11803i) q^{5} +(-5.87843 + 3.80053i) q^{7} +(10.5350 + 18.2471i) q^{9} +(-3.29893 + 5.71391i) q^{11} -3.29523i q^{13} -12.2617 q^{15} +(9.44949 + 5.45566i) q^{17} +(-8.27840 + 4.77954i) q^{19} +(-38.3366 + 1.93099i) q^{21} +(-5.80641 - 10.0570i) q^{23} +(2.50000 - 4.33013i) q^{25} +66.1866i q^{27} -23.6149 q^{29} +(-9.22362 - 5.32526i) q^{31} +(-31.3328 + 18.0900i) q^{33} +(7.13441 - 13.9320i) q^{35} +(31.3288 + 54.2630i) q^{37} +(9.03486 - 15.6488i) q^{39} +49.7889i q^{41} +7.82740 q^{43} +(-40.8017 - 23.5569i) q^{45} +(-26.4161 + 15.2513i) q^{47} +(20.1119 - 44.6823i) q^{49} +(29.9167 + 51.8172i) q^{51} +(31.0976 - 53.8627i) q^{53} -14.7532i q^{55} -52.4182 q^{57} +(39.0701 + 22.5571i) q^{59} +(35.0955 - 20.2624i) q^{61} +(-131.278 - 67.2258i) q^{63} +(3.68418 + 6.38118i) q^{65} +(26.3967 - 45.7204i) q^{67} -63.6802i q^{69} +48.2295 q^{71} +(-25.2610 - 14.5844i) q^{73} +(23.7447 - 13.7090i) q^{75} +(-2.32337 - 46.1265i) q^{77} +(-5.47308 - 9.47965i) q^{79} +(-86.6559 + 150.092i) q^{81} -14.0223i q^{83} -24.3985 q^{85} +(-112.146 - 64.7474i) q^{87} +(91.7289 - 52.9597i) q^{89} +(12.5236 + 19.3708i) q^{91} +(-29.2016 - 50.5787i) q^{93} +(10.6874 - 18.5111i) q^{95} +118.638i q^{97} -139.016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 2 q^{7} + 14 q^{9} + 14 q^{11} + 20 q^{15} + 48 q^{17} + 30 q^{19} - 84 q^{21} + 14 q^{23} + 30 q^{25} + 64 q^{29} - 132 q^{31} - 192 q^{33} - 30 q^{35} + 44 q^{37} + 24 q^{39} + 4 q^{43}+ \cdots + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\) \(1\)

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
\(3\) 4.74894 + 2.74180i 1.58298 + 0.913934i 0.994422 + 0.105479i \(0.0336376\pi\)
0.588558 + 0.808455i \(0.299696\pi\)
\(4\) 0 0
\(5\) −1.93649 + 1.11803i −0.387298 + 0.223607i
\(6\) 0 0
\(7\) −5.87843 + 3.80053i −0.839776 + 0.542933i
\(8\) 0 0
\(9\) 10.5350 + 18.2471i 1.17055 + 2.02745i
\(10\) 0 0
\(11\) −3.29893 + 5.71391i −0.299902 + 0.519446i −0.976113 − 0.217262i \(-0.930287\pi\)
0.676211 + 0.736708i \(0.263621\pi\)
\(12\) 0 0
\(13\) − 3.29523i − 0.253479i −0.991936 − 0.126740i \(-0.959549\pi\)
0.991936 − 0.126740i \(-0.0404513\pi\)
\(14\) 0 0
\(15\) −12.2617 −0.817447
\(16\) 0 0
\(17\) 9.44949 + 5.45566i 0.555852 + 0.320921i 0.751479 − 0.659757i \(-0.229341\pi\)
−0.195627 + 0.980678i \(0.562674\pi\)
\(18\) 0 0
\(19\) −8.27840 + 4.77954i −0.435705 + 0.251555i −0.701774 − 0.712399i \(-0.747608\pi\)
0.266069 + 0.963954i \(0.414275\pi\)
\(20\) 0 0
\(21\) −38.3366 + 1.93099i −1.82555 + 0.0919521i
\(22\) 0 0
\(23\) −5.80641 − 10.0570i −0.252453 − 0.437261i 0.711748 − 0.702435i \(-0.247904\pi\)
−0.964201 + 0.265174i \(0.914571\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.100000 − 0.173205i
\(26\) 0 0
\(27\) 66.1866i 2.45135i
\(28\) 0 0
\(29\) −23.6149 −0.814308 −0.407154 − 0.913360i \(-0.633479\pi\)
−0.407154 + 0.913360i \(0.633479\pi\)
\(30\) 0 0
\(31\) −9.22362 − 5.32526i −0.297536 − 0.171783i 0.343799 − 0.939043i \(-0.388286\pi\)
−0.641336 + 0.767261i \(0.721619\pi\)
\(32\) 0 0
\(33\) −31.3328 + 18.0900i −0.949479 + 0.548182i
\(34\) 0 0
\(35\) 7.13441 − 13.9320i 0.203840 − 0.398057i
\(36\) 0 0
\(37\) 31.3288 + 54.2630i 0.846724 + 1.46657i 0.884116 + 0.467268i \(0.154762\pi\)
−0.0373921 + 0.999301i \(0.511905\pi\)
\(38\) 0 0
\(39\) 9.03486 − 15.6488i 0.231663 − 0.401252i
\(40\) 0 0
\(41\) 49.7889i 1.21436i 0.794563 + 0.607182i \(0.207700\pi\)
−0.794563 + 0.607182i \(0.792300\pi\)
\(42\) 0 0
\(43\) 7.82740 0.182033 0.0910163 − 0.995849i \(-0.470988\pi\)
0.0910163 + 0.995849i \(0.470988\pi\)
\(44\) 0 0
\(45\) −40.8017 − 23.5569i −0.906704 − 0.523486i
\(46\) 0 0
\(47\) −26.4161 + 15.2513i −0.562044 + 0.324496i −0.753966 − 0.656914i \(-0.771862\pi\)
0.191921 + 0.981410i \(0.438528\pi\)
\(48\) 0 0
\(49\) 20.1119 − 44.6823i 0.410448 − 0.911884i
\(50\) 0 0
\(51\) 29.9167 + 51.8172i 0.586602 + 1.01602i
\(52\) 0 0
\(53\) 31.0976 − 53.8627i 0.586748 − 1.01628i −0.407907 − 0.913023i \(-0.633741\pi\)
0.994655 − 0.103254i \(-0.0329254\pi\)
\(54\) 0 0
\(55\) − 14.7532i − 0.268241i
\(56\) 0 0
\(57\) −52.4182 −0.919617
\(58\) 0 0
\(59\) 39.0701 + 22.5571i 0.662205 + 0.382324i 0.793117 − 0.609070i \(-0.208457\pi\)
−0.130911 + 0.991394i \(0.541790\pi\)
\(60\) 0 0
\(61\) 35.0955 − 20.2624i 0.575336 − 0.332170i −0.183942 − 0.982937i \(-0.558886\pi\)
0.759278 + 0.650767i \(0.225552\pi\)
\(62\) 0 0
\(63\) −131.278 − 67.2258i −2.08377 − 1.06708i
\(64\) 0 0
\(65\) 3.68418 + 6.38118i 0.0566797 + 0.0981721i
\(66\) 0 0
\(67\) 26.3967 − 45.7204i 0.393981 − 0.682395i −0.598990 − 0.800757i \(-0.704431\pi\)
0.992970 + 0.118362i \(0.0377644\pi\)
\(68\) 0 0
\(69\) − 63.6802i − 0.922901i
\(70\) 0 0
\(71\) 48.2295 0.679289 0.339644 − 0.940554i \(-0.389693\pi\)
0.339644 + 0.940554i \(0.389693\pi\)
\(72\) 0 0
\(73\) −25.2610 − 14.5844i −0.346040 − 0.199787i 0.316900 − 0.948459i \(-0.397358\pi\)
−0.662940 + 0.748673i \(0.730692\pi\)
\(74\) 0 0
\(75\) 23.7447 − 13.7090i 0.316596 − 0.182787i
\(76\) 0 0
\(77\) −2.32337 − 46.1265i −0.0301736 − 0.599045i
\(78\) 0 0
\(79\) −5.47308 − 9.47965i −0.0692795 − 0.119996i 0.829305 − 0.558796i \(-0.188737\pi\)
−0.898584 + 0.438801i \(0.855403\pi\)
\(80\) 0 0
\(81\) −86.6559 + 150.092i −1.06983 + 1.85299i
\(82\) 0 0
\(83\) − 14.0223i − 0.168943i −0.996426 − 0.0844717i \(-0.973080\pi\)
0.996426 − 0.0844717i \(-0.0269203\pi\)
\(84\) 0 0
\(85\) −24.3985 −0.287041
\(86\) 0 0
\(87\) −112.146 − 64.7474i −1.28903 − 0.744223i
\(88\) 0 0
\(89\) 91.7289 − 52.9597i 1.03066 − 0.595053i 0.113487 − 0.993539i \(-0.463798\pi\)
0.917174 + 0.398487i \(0.130465\pi\)
\(90\) 0 0
\(91\) 12.5236 + 19.3708i 0.137622 + 0.212866i
\(92\) 0 0
\(93\) −29.2016 − 50.5787i −0.313996 − 0.543857i
\(94\) 0 0
\(95\) 10.6874 − 18.5111i 0.112499 − 0.194853i
\(96\) 0 0
\(97\) 118.638i 1.22307i 0.791218 + 0.611534i \(0.209447\pi\)
−0.791218 + 0.611534i \(0.790553\pi\)
\(98\) 0 0
\(99\) −139.016 −1.40420
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 560.3.bx.c.241.6 12
4.3 odd 2 35.3.h.a.31.4 yes 12
7.5 odd 6 inner 560.3.bx.c.481.6 12
12.11 even 2 315.3.w.c.136.3 12
20.3 even 4 175.3.j.b.24.6 24
20.7 even 4 175.3.j.b.24.7 24
20.19 odd 2 175.3.i.d.101.3 12
28.3 even 6 245.3.d.a.146.5 12
28.11 odd 6 245.3.d.a.146.6 12
28.19 even 6 35.3.h.a.26.4 ✓ 12
28.23 odd 6 245.3.h.c.166.4 12
28.27 even 2 245.3.h.c.31.4 12
84.47 odd 6 315.3.w.c.271.3 12
140.19 even 6 175.3.i.d.26.3 12
140.47 odd 12 175.3.j.b.124.6 24
140.103 odd 12 175.3.j.b.124.7 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.3.h.a.26.4 ✓ 12 28.19 even 6
35.3.h.a.31.4 yes 12 4.3 odd 2
175.3.i.d.26.3 12 140.19 even 6
175.3.i.d.101.3 12 20.19 odd 2
175.3.j.b.24.6 24 20.3 even 4
175.3.j.b.24.7 24 20.7 even 4
175.3.j.b.124.6 24 140.47 odd 12
175.3.j.b.124.7 24 140.103 odd 12
245.3.d.a.146.5 12 28.3 even 6
245.3.d.a.146.6 12 28.11 odd 6
245.3.h.c.31.4 12 28.27 even 2
245.3.h.c.166.4 12 28.23 odd 6
315.3.w.c.136.3 12 12.11 even 2
315.3.w.c.271.3 12 84.47 odd 6
560.3.bx.c.241.6 12 1.1 even 1 trivial
560.3.bx.c.481.6 12 7.5 odd 6 inner