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Properties

Label 8464.2.a.cj.1.3
Level $8464$
Weight $2$
Character 8464.1
Self dual yes
Analytic conductor $67.585$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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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] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8464 = 2^{4} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8464.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,0,1,0,10,0,-10,0,16,0,9,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.5853802708\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 30 x^{13} + 26 x^{12} + 338 x^{11} - 238 x^{10} - 1773 x^{9} + 894 x^{8} + 4319 x^{7} + \cdots + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.34707\) of defining polynomial
Character \(\chi\) \(=\) 8464.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.34707 q^{3} +1.00622 q^{5} -0.336699 q^{7} +2.50876 q^{9} -1.49175 q^{11} +2.73389 q^{13} -2.36166 q^{15} +1.85305 q^{17} -7.68716 q^{19} +0.790257 q^{21} -3.98753 q^{25} +1.15298 q^{27} -4.19474 q^{29} -4.98507 q^{31} +3.50124 q^{33} -0.338791 q^{35} +10.7555 q^{37} -6.41664 q^{39} -2.63631 q^{41} +3.44181 q^{43} +2.52435 q^{45} +4.41079 q^{47} -6.88663 q^{49} -4.34925 q^{51} -6.26898 q^{53} -1.50102 q^{55} +18.0423 q^{57} -4.19975 q^{59} +4.96167 q^{61} -0.844696 q^{63} +2.75088 q^{65} +4.20911 q^{67} +14.8113 q^{71} +15.3401 q^{73} +9.35903 q^{75} +0.502269 q^{77} -12.6126 q^{79} -10.2324 q^{81} +7.46472 q^{83} +1.86457 q^{85} +9.84536 q^{87} -10.2282 q^{89} -0.920496 q^{91} +11.7003 q^{93} -7.73494 q^{95} -13.5307 q^{97} -3.74244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{3} + 10 q^{5} - 10 q^{7} + 16 q^{9} + 9 q^{11} - 12 q^{15} + 20 q^{17} + 3 q^{19} + 21 q^{21} + 11 q^{25} + 7 q^{27} - 4 q^{29} + 14 q^{31} + 34 q^{33} - 18 q^{35} + 26 q^{37} - 33 q^{39} - 11 q^{41}+ \cdots + 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.34707 −1.35508 −0.677542 − 0.735484i \(-0.736955\pi\)
−0.677542 + 0.735484i \(0.736955\pi\)
\(4\) 0 0
\(5\) 1.00622 0.449993 0.224997 − 0.974360i \(-0.427763\pi\)
0.224997 + 0.974360i \(0.427763\pi\)
\(6\) 0 0
\(7\) −0.336699 −0.127260 −0.0636300 − 0.997974i \(-0.520268\pi\)
−0.0636300 + 0.997974i \(0.520268\pi\)
\(8\) 0 0
\(9\) 2.50876 0.836254
\(10\) 0 0
\(11\) −1.49175 −0.449779 −0.224889 − 0.974384i \(-0.572202\pi\)
−0.224889 + 0.974384i \(0.572202\pi\)
\(12\) 0 0
\(13\) 2.73389 0.758244 0.379122 − 0.925347i \(-0.376226\pi\)
0.379122 + 0.925347i \(0.376226\pi\)
\(14\) 0 0
\(15\) −2.36166 −0.609779
\(16\) 0 0
\(17\) 1.85305 0.449431 0.224716 − 0.974424i \(-0.427855\pi\)
0.224716 + 0.974424i \(0.427855\pi\)
\(18\) 0 0
\(19\) −7.68716 −1.76355 −0.881777 − 0.471666i \(-0.843653\pi\)
−0.881777 + 0.471666i \(0.843653\pi\)
\(20\) 0 0
\(21\) 0.790257 0.172448
\(22\) 0 0
\(23\) 0 0
\(24\) 0 0
\(25\) −3.98753 −0.797506
\(26\) 0 0
\(27\) 1.15298 0.221890
\(28\) 0 0
\(29\) −4.19474 −0.778943 −0.389472 − 0.921038i \(-0.627342\pi\)
−0.389472 + 0.921038i \(0.627342\pi\)
\(30\) 0 0
\(31\) −4.98507 −0.895344 −0.447672 − 0.894198i \(-0.647747\pi\)
−0.447672 + 0.894198i \(0.647747\pi\)
\(32\) 0 0
\(33\) 3.50124 0.609488
\(34\) 0 0
\(35\) −0.338791 −0.0572662
\(36\) 0 0
\(37\) 10.7555 1.76819 0.884096 − 0.467305i \(-0.154775\pi\)
0.884096 + 0.467305i \(0.154775\pi\)
\(38\) 0 0
\(39\) −6.41664 −1.02748
\(40\) 0 0
\(41\) −2.63631 −0.411723 −0.205861 − 0.978581i \(-0.566000\pi\)
−0.205861 + 0.978581i \(0.566000\pi\)
\(42\) 0 0
\(43\) 3.44181 0.524871 0.262435 − 0.964950i \(-0.415474\pi\)
0.262435 + 0.964950i \(0.415474\pi\)
\(44\) 0 0
\(45\) 2.52435 0.376308
\(46\) 0 0
\(47\) 4.41079 0.643380 0.321690 − 0.946845i \(-0.395749\pi\)
0.321690 + 0.946845i \(0.395749\pi\)
\(48\) 0 0
\(49\) −6.88663 −0.983805
\(50\) 0 0
\(51\) −4.34925 −0.609017
\(52\) 0 0
\(53\) −6.26898 −0.861111 −0.430555 − 0.902564i \(-0.641682\pi\)
−0.430555 + 0.902564i \(0.641682\pi\)
\(54\) 0 0
\(55\) −1.50102 −0.202397
\(56\) 0 0
\(57\) 18.0423 2.38977
\(58\) 0 0
\(59\) −4.19975 −0.546761 −0.273381 − 0.961906i \(-0.588142\pi\)
−0.273381 + 0.961906i \(0.588142\pi\)
\(60\) 0 0
\(61\) 4.96167 0.635276 0.317638 − 0.948212i \(-0.397110\pi\)
0.317638 + 0.948212i \(0.397110\pi\)
\(62\) 0 0
\(63\) −0.844696 −0.106422
\(64\) 0 0
\(65\) 2.75088 0.341205
\(66\) 0 0
\(67\) 4.20911 0.514225 0.257113 − 0.966381i \(-0.417229\pi\)
0.257113 + 0.966381i \(0.417229\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.8113 1.75778 0.878889 − 0.477027i \(-0.158285\pi\)
0.878889 + 0.477027i \(0.158285\pi\)
\(72\) 0 0
\(73\) 15.3401 1.79542 0.897709 − 0.440589i \(-0.145230\pi\)
0.897709 + 0.440589i \(0.145230\pi\)
\(74\) 0 0
\(75\) 9.35903 1.08069
\(76\) 0 0
\(77\) 0.502269 0.0572389
\(78\) 0 0
\(79\) −12.6126 −1.41903 −0.709514 − 0.704692i \(-0.751085\pi\)
−0.709514 + 0.704692i \(0.751085\pi\)
\(80\) 0 0
\(81\) −10.2324 −1.13693
\(82\) 0 0
\(83\) 7.46472 0.819359 0.409680 − 0.912229i \(-0.365640\pi\)
0.409680 + 0.912229i \(0.365640\pi\)
\(84\) 0 0
\(85\) 1.86457 0.202241
\(86\) 0 0
\(87\) 9.84536 1.05553
\(88\) 0 0
\(89\) −10.2282 −1.08418 −0.542092 − 0.840319i \(-0.682367\pi\)
−0.542092 + 0.840319i \(0.682367\pi\)
\(90\) 0 0
\(91\) −0.920496 −0.0964942
\(92\) 0 0
\(93\) 11.7003 1.21327
\(94\) 0 0
\(95\) −7.73494 −0.793588
\(96\) 0 0
\(97\) −13.5307 −1.37384 −0.686918 − 0.726735i \(-0.741037\pi\)
−0.686918 + 0.726735i \(0.741037\pi\)
\(98\) 0 0
\(99\) −3.74244 −0.376129
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 8464.2.a.cj.1.3 15
4.3 odd 2 4232.2.a.z.1.13 15
23.17 odd 22 368.2.m.f.289.1 30
23.19 odd 22 368.2.m.f.177.1 30
23.22 odd 2 8464.2.a.ci.1.3 15
92.19 even 22 184.2.i.a.177.3 yes 30
92.63 even 22 184.2.i.a.105.3 ✓ 30
92.91 even 2 4232.2.a.y.1.13 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.i.a.105.3 ✓ 30 92.63 even 22
184.2.i.a.177.3 yes 30 92.19 even 22
368.2.m.f.177.1 30 23.19 odd 22
368.2.m.f.289.1 30 23.17 odd 22
4232.2.a.y.1.13 15 92.91 even 2
4232.2.a.z.1.13 15 4.3 odd 2
8464.2.a.ci.1.3 15 23.22 odd 2
8464.2.a.cj.1.3 15 1.1 even 1 trivial