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Properties

Label 1008.5.f.h.433.4
Level $1008$
Weight $5$
Character 1008.433
Analytic conductor $104.197$
Analytic rank $0$
Dimension $4$
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] = [1008,5,Mod(433,1008)] 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("1008.433"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,76,0,0,0,360] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(104.196922789\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-36 +3 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 72x^{2} + 1278 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 433.4
Root \(5.63537i\) of defining polynomial
Character \(\chi\) \(=\) 1008.433
Dual form 1008.5.f.h.433.1

$q$-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)
 
\(f(q)\) \(=\) \(q+43.1492i q^{5} +(44.4558 - 20.6077i) q^{7} -11.8234 q^{11} -20.6077i q^{13} -289.172i q^{17} -104.641i q^{19} +73.5290 q^{23} -1236.85 q^{25} -950.881 q^{29} +1385.30i q^{31} +(889.206 + 1918.23i) q^{35} -1279.47 q^{37} +1303.54i q^{41} +96.2338 q^{43} -186.190i q^{47} +(1551.64 - 1832.27i) q^{49} -4376.94 q^{53} -510.169i q^{55} -1650.28i q^{59} +5200.50i q^{61} +889.206 q^{65} -552.587 q^{67} -8487.61 q^{71} -317.344i q^{73} +(-525.618 + 243.653i) q^{77} +624.377 q^{79} +7662.33i q^{83} +12477.5 q^{85} -4190.72i q^{89} +(-424.678 - 916.133i) q^{91} +4515.15 q^{95} -12994.4i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 76 q^{7} + 360 q^{11} - 792 q^{23} - 2300 q^{25} - 1224 q^{29} + 4032 q^{35} - 3896 q^{37} - 3688 q^{43} - 1532 q^{49} - 5832 q^{53} + 4032 q^{65} + 1048 q^{67} - 21528 q^{71} - 3528 q^{77} - 12776 q^{79}+ \cdots + 36864 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 43.1492i 1.72597i 0.505232 + 0.862984i \(0.331407\pi\)
−0.505232 + 0.862984i \(0.668593\pi\)
\(6\) 0 0
\(7\) 44.4558 − 20.6077i 0.907262 − 0.420566i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −11.8234 −0.0977139 −0.0488569 − 0.998806i \(-0.515558\pi\)
−0.0488569 + 0.998806i \(0.515558\pi\)
\(12\) 0 0
\(13\) − 20.6077i − 0.121939i −0.998140 − 0.0609696i \(-0.980581\pi\)
0.998140 − 0.0609696i \(-0.0194193\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 289.172i − 1.00059i −0.865854 − 0.500297i \(-0.833224\pi\)
0.865854 − 0.500297i \(-0.166776\pi\)
\(18\) 0 0
\(19\) − 104.641i − 0.289863i −0.989442 − 0.144932i \(-0.953704\pi\)
0.989442 − 0.144932i \(-0.0462962\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 73.5290 0.138996 0.0694981 − 0.997582i \(-0.477860\pi\)
0.0694981 + 0.997582i \(0.477860\pi\)
\(24\) 0 0
\(25\) −1236.85 −1.97896
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −950.881 −1.13066 −0.565328 − 0.824866i \(-0.691250\pi\)
−0.565328 + 0.824866i \(0.691250\pi\)
\(30\) 0 0
\(31\) 1385.30i 1.44152i 0.693182 + 0.720762i \(0.256208\pi\)
−0.693182 + 0.720762i \(0.743792\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 889.206 + 1918.23i 0.725882 + 1.56590i
\(36\) 0 0
\(37\) −1279.47 −0.934602 −0.467301 − 0.884098i \(-0.654774\pi\)
−0.467301 + 0.884098i \(0.654774\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1303.54i 0.775454i 0.921774 + 0.387727i \(0.126740\pi\)
−0.921774 + 0.387727i \(0.873260\pi\)
\(42\) 0 0
\(43\) 96.2338 0.0520464 0.0260232 − 0.999661i \(-0.491716\pi\)
0.0260232 + 0.999661i \(0.491716\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 186.190i − 0.0842870i −0.999112 − 0.0421435i \(-0.986581\pi\)
0.999112 − 0.0421435i \(-0.0134187\pi\)
\(48\) 0 0
\(49\) 1551.64 − 1832.27i 0.646249 − 0.763127i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4376.94 −1.55818 −0.779092 − 0.626910i \(-0.784319\pi\)
−0.779092 + 0.626910i \(0.784319\pi\)
\(54\) 0 0
\(55\) − 510.169i − 0.168651i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1650.28i − 0.474081i −0.971500 − 0.237041i \(-0.923823\pi\)
0.971500 − 0.237041i \(-0.0761774\pi\)
\(60\) 0 0
\(61\) 5200.50i 1.39761i 0.715313 + 0.698804i \(0.246284\pi\)
−0.715313 + 0.698804i \(0.753716\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 889.206 0.210463
\(66\) 0 0
\(67\) −552.587 −0.123098 −0.0615490 − 0.998104i \(-0.519604\pi\)
−0.0615490 + 0.998104i \(0.519604\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8487.61 −1.68372 −0.841858 − 0.539699i \(-0.818538\pi\)
−0.841858 + 0.539699i \(0.818538\pi\)
\(72\) 0 0
\(73\) − 317.344i − 0.0595503i −0.999557 − 0.0297751i \(-0.990521\pi\)
0.999557 − 0.0297751i \(-0.00947912\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −525.618 + 243.653i −0.0886521 + 0.0410951i
\(78\) 0 0
\(79\) 624.377 0.100044 0.0500222 − 0.998748i \(-0.484071\pi\)
0.0500222 + 0.998748i \(0.484071\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7662.33i 1.11226i 0.831097 + 0.556128i \(0.187713\pi\)
−0.831097 + 0.556128i \(0.812287\pi\)
\(84\) 0 0
\(85\) 12477.5 1.72699
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 4190.72i − 0.529065i −0.964377 − 0.264532i \(-0.914782\pi\)
0.964377 − 0.264532i \(-0.0852176\pi\)
\(90\) 0 0
\(91\) −424.678 − 916.133i −0.0512834 − 0.110631i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4515.15 0.500294
\(96\) 0 0
\(97\) − 12994.4i − 1.38106i −0.723305 − 0.690529i \(-0.757378\pi\)
0.723305 − 0.690529i \(-0.242622\pi\)
\(98\) 0 0
\(99\) 0 0
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 1008.5.f.h.433.4 4
3.2 odd 2 112.5.c.c.97.2 4
4.3 odd 2 126.5.c.a.55.2 4
7.6 odd 2 inner 1008.5.f.h.433.1 4
12.11 even 2 14.5.b.a.13.4 yes 4
21.20 even 2 112.5.c.c.97.3 4
24.5 odd 2 448.5.c.f.321.3 4
24.11 even 2 448.5.c.e.321.2 4
28.27 even 2 126.5.c.a.55.1 4
60.23 odd 4 350.5.d.a.349.2 8
60.47 odd 4 350.5.d.a.349.7 8
60.59 even 2 350.5.b.a.251.1 4
84.11 even 6 98.5.d.d.19.2 8
84.23 even 6 98.5.d.d.31.1 8
84.47 odd 6 98.5.d.d.31.2 8
84.59 odd 6 98.5.d.d.19.1 8
84.83 odd 2 14.5.b.a.13.3 ✓ 4
168.83 odd 2 448.5.c.e.321.3 4
168.125 even 2 448.5.c.f.321.2 4
420.83 even 4 350.5.d.a.349.3 8
420.167 even 4 350.5.d.a.349.6 8
420.419 odd 2 350.5.b.a.251.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.5.b.a.13.3 ✓ 4 84.83 odd 2
14.5.b.a.13.4 yes 4 12.11 even 2
98.5.d.d.19.1 8 84.59 odd 6
98.5.d.d.19.2 8 84.11 even 6
98.5.d.d.31.1 8 84.23 even 6
98.5.d.d.31.2 8 84.47 odd 6
112.5.c.c.97.2 4 3.2 odd 2
112.5.c.c.97.3 4 21.20 even 2
126.5.c.a.55.1 4 28.27 even 2
126.5.c.a.55.2 4 4.3 odd 2
350.5.b.a.251.1 4 60.59 even 2
350.5.b.a.251.2 4 420.419 odd 2
350.5.d.a.349.2 8 60.23 odd 4
350.5.d.a.349.3 8 420.83 even 4
350.5.d.a.349.6 8 420.167 even 4
350.5.d.a.349.7 8 60.47 odd 4
448.5.c.e.321.2 4 24.11 even 2
448.5.c.e.321.3 4 168.83 odd 2
448.5.c.f.321.2 4 168.125 even 2
448.5.c.f.321.3 4 24.5 odd 2
1008.5.f.h.433.1 4 7.6 odd 2 inner
1008.5.f.h.433.4 4 1.1 even 1 trivial