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Properties

Label 56.4.i.b.25.1
Level $56$
Weight $4$
Character 56.25
Analytic conductor $3.304$
Analytic rank $0$
Dimension $6$
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] = [56,4,Mod(9,56)] 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("56.9"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 56.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.30410696032\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11163123.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 14x^{4} + 49x^{2} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 25.1
Root \(0.821510i\) of defining polynomial
Character \(\chi\) \(=\) 56.25
Dual form 56.4.i.b.9.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.40222 + 2.42872i) q^{3} +(3.34580 + 5.79509i) q^{5} +(-8.65024 + 16.3760i) q^{7} +(9.56754 + 16.5715i) q^{9} +(-15.7441 + 27.2695i) q^{11} +18.6837 q^{13} -18.7662 q^{15} +(43.9769 - 76.1703i) q^{17} +(-6.54850 - 11.3423i) q^{19} +(-27.6432 - 43.9718i) q^{21} +(4.68840 + 8.12055i) q^{23} +(40.1113 - 69.4748i) q^{25} -129.383 q^{27} -5.11923 q^{29} +(128.706 - 222.925i) q^{31} +(-44.1533 - 76.4758i) q^{33} +(-123.842 + 4.66183i) q^{35} +(190.107 + 329.276i) q^{37} +(-26.1987 + 45.3774i) q^{39} -217.959 q^{41} +377.049 q^{43} +(-64.0221 + 110.889i) q^{45} +(178.855 + 309.786i) q^{47} +(-193.347 - 283.313i) q^{49} +(123.331 + 213.615i) q^{51} +(-382.195 + 661.981i) q^{53} -210.706 q^{55} +36.7298 q^{57} +(225.336 - 390.293i) q^{59} +(87.0388 + 150.756i) q^{61} +(-354.136 + 13.3308i) q^{63} +(62.5117 + 108.273i) q^{65} +(248.617 - 430.617i) q^{67} -26.2967 q^{69} +350.238 q^{71} +(531.343 - 920.312i) q^{73} +(112.490 + 194.838i) q^{75} +(-310.375 - 493.712i) q^{77} +(-280.224 - 485.363i) q^{79} +(-76.8993 + 133.194i) q^{81} -1105.27 q^{83} +588.551 q^{85} +(7.17830 - 12.4332i) q^{87} +(-603.357 - 1045.04i) q^{89} +(-161.618 + 305.964i) q^{91} +(360.948 + 625.181i) q^{93} +(43.8199 - 75.8983i) q^{95} +1442.99 q^{97} -602.528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 7 q^{3} + 3 q^{5} - 4 q^{7} - 18 q^{9} + 3 q^{11} - 52 q^{13} + 254 q^{15} + 31 q^{17} + 89 q^{19} - 375 q^{21} + 201 q^{23} - 300 q^{25} - 938 q^{27} + 380 q^{29} + 339 q^{31} + 105 q^{33} - 473 q^{35}+ \cdots + 4620 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) −1.40222 + 2.42872i −0.269858 + 0.467408i −0.968825 − 0.247746i \(-0.920310\pi\)
0.698967 + 0.715154i \(0.253643\pi\)
\(4\) 0 0
\(5\) 3.34580 + 5.79509i 0.299257 + 0.518328i 0.975966 − 0.217922i \(-0.0699279\pi\)
−0.676709 + 0.736250i \(0.736595\pi\)
\(6\) 0 0
\(7\) −8.65024 + 16.3760i −0.467069 + 0.884221i
\(8\) 0 0
\(9\) 9.56754 + 16.5715i 0.354353 + 0.613758i
\(10\) 0 0
\(11\) −15.7441 + 27.2695i −0.431546 + 0.747460i −0.997007 − 0.0773151i \(-0.975365\pi\)
0.565460 + 0.824776i \(0.308699\pi\)
\(12\) 0 0
\(13\) 18.6837 0.398609 0.199304 − 0.979938i \(-0.436132\pi\)
0.199304 + 0.979938i \(0.436132\pi\)
\(14\) 0 0
\(15\) −18.7662 −0.323028
\(16\) 0 0
\(17\) 43.9769 − 76.1703i 0.627410 − 1.08671i −0.360660 − 0.932698i \(-0.617448\pi\)
0.988070 − 0.154008i \(-0.0492183\pi\)
\(18\) 0 0
\(19\) −6.54850 − 11.3423i −0.0790700 − 0.136953i 0.823779 − 0.566911i \(-0.191862\pi\)
−0.902849 + 0.429958i \(0.858528\pi\)
\(20\) 0 0
\(21\) −27.6432 − 43.9718i −0.287249 − 0.456926i
\(22\) 0 0
\(23\) 4.68840 + 8.12055i 0.0425043 + 0.0736197i 0.886495 − 0.462738i \(-0.153133\pi\)
−0.843991 + 0.536358i \(0.819800\pi\)
\(24\) 0 0
\(25\) 40.1113 − 69.4748i 0.320890 − 0.555799i
\(26\) 0 0
\(27\) −129.383 −0.922216
\(28\) 0 0
\(29\) −5.11923 −0.0327799 −0.0163900 − 0.999866i \(-0.505217\pi\)
−0.0163900 + 0.999866i \(0.505217\pi\)
\(30\) 0 0
\(31\) 128.706 − 222.925i 0.745685 − 1.29156i −0.204189 − 0.978932i \(-0.565456\pi\)
0.949874 − 0.312633i \(-0.101211\pi\)
\(32\) 0 0
\(33\) −44.1533 − 76.4758i −0.232912 − 0.403416i
\(34\) 0 0
\(35\) −123.842 + 4.66183i −0.598090 + 0.0225141i
\(36\) 0 0
\(37\) 190.107 + 329.276i 0.844688 + 1.46304i 0.885892 + 0.463892i \(0.153547\pi\)
−0.0412040 + 0.999151i \(0.513119\pi\)
\(38\) 0 0
\(39\) −26.1987 + 45.3774i −0.107568 + 0.186313i
\(40\) 0 0
\(41\) −217.959 −0.830230 −0.415115 − 0.909769i \(-0.636259\pi\)
−0.415115 + 0.909769i \(0.636259\pi\)
\(42\) 0 0
\(43\) 377.049 1.33720 0.668598 − 0.743624i \(-0.266895\pi\)
0.668598 + 0.743624i \(0.266895\pi\)
\(44\) 0 0
\(45\) −64.0221 + 110.889i −0.212086 + 0.367343i
\(46\) 0 0
\(47\) 178.855 + 309.786i 0.555079 + 0.961425i 0.997897 + 0.0648137i \(0.0206453\pi\)
−0.442818 + 0.896611i \(0.646021\pi\)
\(48\) 0 0
\(49\) −193.347 − 283.313i −0.563693 − 0.825984i
\(50\) 0 0
\(51\) 123.331 + 213.615i 0.338623 + 0.586512i
\(52\) 0 0
\(53\) −382.195 + 661.981i −0.990538 + 1.71566i −0.376415 + 0.926451i \(0.622843\pi\)
−0.614122 + 0.789211i \(0.710490\pi\)
\(54\) 0 0
\(55\) −210.706 −0.516573
\(56\) 0 0
\(57\) 36.7298 0.0853506
\(58\) 0 0
\(59\) 225.336 − 390.293i 0.497224 − 0.861216i −0.502771 − 0.864419i \(-0.667686\pi\)
0.999995 + 0.00320303i \(0.00101956\pi\)
\(60\) 0 0
\(61\) 87.0388 + 150.756i 0.182691 + 0.316431i 0.942796 − 0.333370i \(-0.108186\pi\)
−0.760105 + 0.649801i \(0.774852\pi\)
\(62\) 0 0
\(63\) −354.136 + 13.3308i −0.708205 + 0.0266592i
\(64\) 0 0
\(65\) 62.5117 + 108.273i 0.119287 + 0.206610i
\(66\) 0 0
\(67\) 248.617 − 430.617i 0.453334 − 0.785198i −0.545256 − 0.838269i \(-0.683568\pi\)
0.998591 + 0.0530711i \(0.0169010\pi\)
\(68\) 0 0
\(69\) −26.2967 −0.0458805
\(70\) 0 0
\(71\) 350.238 0.585432 0.292716 − 0.956199i \(-0.405441\pi\)
0.292716 + 0.956199i \(0.405441\pi\)
\(72\) 0 0
\(73\) 531.343 − 920.312i 0.851903 − 1.47554i −0.0275851 − 0.999619i \(-0.508782\pi\)
0.879488 − 0.475920i \(-0.157885\pi\)
\(74\) 0 0
\(75\) 112.490 + 194.838i 0.173190 + 0.299973i
\(76\) 0 0
\(77\) −310.375 − 493.712i −0.459358 − 0.730698i
\(78\) 0 0
\(79\) −280.224 − 485.363i −0.399085 − 0.691235i 0.594528 − 0.804075i \(-0.297339\pi\)
−0.993613 + 0.112839i \(0.964005\pi\)
\(80\) 0 0
\(81\) −76.8993 + 133.194i −0.105486 + 0.182707i
\(82\) 0 0
\(83\) −1105.27 −1.46168 −0.730840 − 0.682549i \(-0.760871\pi\)
−0.730840 + 0.682549i \(0.760871\pi\)
\(84\) 0 0
\(85\) 588.551 0.751027
\(86\) 0 0
\(87\) 7.17830 − 12.4332i 0.00884592 − 0.0153216i
\(88\) 0 0
\(89\) −603.357 − 1045.04i −0.718604 − 1.24466i −0.961553 − 0.274619i \(-0.911448\pi\)
0.242950 − 0.970039i \(-0.421885\pi\)
\(90\) 0 0
\(91\) −161.618 + 305.964i −0.186178 + 0.352458i
\(92\) 0 0
\(93\) 360.948 + 625.181i 0.402458 + 0.697078i
\(94\) 0 0
\(95\) 43.8199 − 75.8983i 0.0473245 − 0.0819684i
\(96\) 0 0
\(97\) 1442.99 1.51045 0.755226 − 0.655465i \(-0.227527\pi\)
0.755226 + 0.655465i \(0.227527\pi\)
\(98\) 0 0
\(99\) −602.528 −0.611680
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 56.4.i.b.25.1 yes 6
3.2 odd 2 504.4.s.h.361.2 6
4.3 odd 2 112.4.i.e.81.3 6
7.2 even 3 inner 56.4.i.b.9.1 ✓ 6
7.3 odd 6 392.4.a.l.1.1 3
7.4 even 3 392.4.a.i.1.3 3
7.5 odd 6 392.4.i.m.177.3 6
7.6 odd 2 392.4.i.m.361.3 6
8.3 odd 2 448.4.i.m.193.1 6
8.5 even 2 448.4.i.j.193.3 6
21.2 odd 6 504.4.s.h.289.2 6
28.3 even 6 784.4.a.bb.1.3 3
28.11 odd 6 784.4.a.be.1.1 3
28.23 odd 6 112.4.i.e.65.3 6
56.37 even 6 448.4.i.j.65.3 6
56.51 odd 6 448.4.i.m.65.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.4.i.b.9.1 ✓ 6 7.2 even 3 inner
56.4.i.b.25.1 yes 6 1.1 even 1 trivial
112.4.i.e.65.3 6 28.23 odd 6
112.4.i.e.81.3 6 4.3 odd 2
392.4.a.i.1.3 3 7.4 even 3
392.4.a.l.1.1 3 7.3 odd 6
392.4.i.m.177.3 6 7.5 odd 6
392.4.i.m.361.3 6 7.6 odd 2
448.4.i.j.65.3 6 56.37 even 6
448.4.i.j.193.3 6 8.5 even 2
448.4.i.m.65.1 6 56.51 odd 6
448.4.i.m.193.1 6 8.3 odd 2
504.4.s.h.289.2 6 21.2 odd 6
504.4.s.h.361.2 6 3.2 odd 2
784.4.a.bb.1.3 3 28.3 even 6
784.4.a.be.1.1 3 28.11 odd 6