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Properties

Label 448.4.i.j.193.3
Level $448$
Weight $4$
Character 448.193
Analytic conductor $26.433$
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] = [448,4,Mod(65,448)] 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("448.65"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-7,0,-3,0,-4] 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: \(26.4328556826\)
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: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.3
Root \(-0.821510i\) of defining polynomial
Character \(\chi\) \(=\) 448.193
Dual form 448.4.i.j.65.3

$q$-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 \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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 −0.698967 − 0.715154i \(-0.746357\pi\)
0.968825 + 0.247746i \(0.0796899\pi\)
\(4\) 0 0
\(5\) −3.34580 − 5.79509i −0.299257 − 0.518328i 0.676709 − 0.736250i \(-0.263405\pi\)
−0.975966 + 0.217922i \(0.930072\pi\)
\(6\) 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 −0.565460 − 0.824776i \(-0.691301\pi\)
0.997007 + 0.0773151i \(0.0246348\pi\)
\(12\) 0 0
\(13\) −18.6837 −0.398609 −0.199304 − 0.979938i \(-0.563868\pi\)
−0.199304 + 0.979938i \(0.563868\pi\)
\(14\) 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 0.902849 − 0.429958i \(-0.141472\pi\)
−0.823779 + 0.566911i \(0.808138\pi\)
\(20\) 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 \(-0.494783\pi\)
0.0163900 + 0.999866i \(0.494783\pi\)
\(30\) 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 \(-0.846453\pi\)
0.0412040 − 0.999151i \(-0.486881\pi\)
\(38\) 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 \(-0.733105\pi\)
−0.668598 + 0.743624i \(0.733105\pi\)
\(44\) 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 \(-0.377157\pi\)
0.614122 − 0.789211i \(-0.289510\pi\)
\(54\) 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 −0.999995 − 0.00320303i \(-0.998980\pi\)
0.502771 + 0.864419i \(0.332314\pi\)
\(60\) 0 0
\(61\) −87.0388 − 150.756i −0.182691 − 0.316431i 0.760105 − 0.649801i \(-0.225148\pi\)
−0.942796 + 0.333370i \(0.891814\pi\)
\(62\) 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 −0.998591 − 0.0530711i \(-0.983099\pi\)
0.545256 + 0.838269i \(0.316432\pi\)
\(68\) 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 \(-0.239129\pi\)
0.730840 + 0.682549i \(0.239129\pi\)
\(84\) 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 448.4.i.j.193.3 6
4.3 odd 2 448.4.i.m.193.1 6
7.2 even 3 inner 448.4.i.j.65.3 6
8.3 odd 2 112.4.i.e.81.3 6
8.5 even 2 56.4.i.b.25.1 yes 6
24.5 odd 2 504.4.s.h.361.2 6
28.23 odd 6 448.4.i.m.65.1 6
56.3 even 6 784.4.a.bb.1.3 3
56.5 odd 6 392.4.i.m.177.3 6
56.11 odd 6 784.4.a.be.1.1 3
56.13 odd 2 392.4.i.m.361.3 6
56.37 even 6 56.4.i.b.9.1 ✓ 6
56.45 odd 6 392.4.a.l.1.1 3
56.51 odd 6 112.4.i.e.65.3 6
56.53 even 6 392.4.a.i.1.3 3
168.149 odd 6 504.4.s.h.289.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.4.i.b.9.1 ✓ 6 56.37 even 6
56.4.i.b.25.1 yes 6 8.5 even 2
112.4.i.e.65.3 6 56.51 odd 6
112.4.i.e.81.3 6 8.3 odd 2
392.4.a.i.1.3 3 56.53 even 6
392.4.a.l.1.1 3 56.45 odd 6
392.4.i.m.177.3 6 56.5 odd 6
392.4.i.m.361.3 6 56.13 odd 2
448.4.i.j.65.3 6 7.2 even 3 inner
448.4.i.j.193.3 6 1.1 even 1 trivial
448.4.i.m.65.1 6 28.23 odd 6
448.4.i.m.193.1 6 4.3 odd 2
504.4.s.h.289.2 6 168.149 odd 6
504.4.s.h.361.2 6 24.5 odd 2
784.4.a.bb.1.3 3 56.3 even 6
784.4.a.be.1.1 3 56.11 odd 6