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Properties

Label 126.5.c.a.55.2
Level $126$
Weight $5$
Character 126.55
Analytic conductor $13.025$
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] = [126,5,Mod(55,126)] 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("126.55"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 126.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,32,0,0,-76] 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: \(13.0246153486\)
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: \( 2^{4} \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 55.2
Root \(5.63537i\) of defining polynomial
Character \(\chi\) \(=\) 126.55
Dual form 126.5.c.a.55.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.82843 q^{2} +8.00000 q^{4} +43.1492i q^{5} +(-44.4558 + 20.6077i) q^{7} -22.6274 q^{8} -122.044i q^{10} +11.8234 q^{11} -20.6077i q^{13} +(125.740 - 58.2874i) q^{14} +64.0000 q^{16} -289.172i q^{17} +104.641i q^{19} +345.193i q^{20} -33.4416 q^{22} -73.5290 q^{23} -1236.85 q^{25} +58.2874i q^{26} +(-355.647 + 164.862i) q^{28} -950.881 q^{29} -1385.30i q^{31} -181.019 q^{32} +817.901i q^{34} +(-889.206 - 1918.23i) q^{35} -1279.47 q^{37} -295.968i q^{38} -976.355i q^{40} +1303.54i q^{41} -96.2338 q^{43} +94.5870 q^{44} +207.971 q^{46} +186.190i q^{47} +(1551.64 - 1832.27i) q^{49} +3498.35 q^{50} -164.862i q^{52} -4376.94 q^{53} +510.169i q^{55} +(1005.92 - 466.299i) q^{56} +2689.50 q^{58} +1650.28i q^{59} +5200.50i q^{61} +3918.23i q^{62} +512.000 q^{64} +889.206 q^{65} +552.587 q^{67} -2313.37i q^{68} +(2515.05 + 5425.58i) q^{70} +8487.61 q^{71} -317.344i q^{73} +3618.89 q^{74} +837.124i q^{76} +(-525.618 + 243.653i) q^{77} -624.377 q^{79} +2761.55i q^{80} -3686.96i q^{82} -7662.33i q^{83} +12477.5 q^{85} +272.190 q^{86} -267.532 q^{88} -4190.72i q^{89} +(424.678 + 916.133i) q^{91} -588.232 q^{92} -526.625i q^{94} -4515.15 q^{95} -12994.4i q^{97} +(-4388.71 + 5182.43i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{4} - 76 q^{7} - 360 q^{11} + 288 q^{14} + 256 q^{16} - 1152 q^{22} + 792 q^{23} - 2300 q^{25} - 608 q^{28} - 1224 q^{29} - 4032 q^{35} - 3896 q^{37} + 3688 q^{43} - 2880 q^{44} + 3072 q^{46}+ \cdots - 21888 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.82843 −0.707107
\(3\) 0 0
\(4\) 8.00000 0.500000
\(5\) 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
\(8\) −22.6274 −0.353553
\(9\) 0 0
\(10\) − 122.044i − 1.22044i
\(11\) 11.8234 0.0977139 0.0488569 − 0.998806i \(-0.484442\pi\)
0.0488569 + 0.998806i \(0.484442\pi\)
\(12\) 0 0
\(13\) − 20.6077i − 0.121939i −0.998140 − 0.0609696i \(-0.980581\pi\)
0.998140 − 0.0609696i \(-0.0194193\pi\)
\(14\) 125.740 − 58.2874i 0.641531 − 0.297385i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 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 \(0.0462962\pi\)
−0.989442 + 0.144932i \(0.953704\pi\)
\(20\) 345.193i 0.862984i
\(21\) 0 0
\(22\) −33.4416 −0.0690941
\(23\) −73.5290 −0.138996 −0.0694981 − 0.997582i \(-0.522140\pi\)
−0.0694981 + 0.997582i \(0.522140\pi\)
\(24\) 0 0
\(25\) −1236.85 −1.97896
\(26\) 58.2874i 0.0862240i
\(27\) 0 0
\(28\) −355.647 + 164.862i −0.453631 + 0.210283i
\(29\) −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 \(-0.743792\pi\)
0.693182 − 0.720762i \(-0.256208\pi\)
\(32\) −181.019 −0.176777
\(33\) 0 0
\(34\) 817.901i 0.707527i
\(35\) −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 \(0.654774\pi\)
\(38\) − 295.968i − 0.204964i
\(39\) 0 0
\(40\) − 976.355i − 0.610222i
\(41\) 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 \(-0.508284\pi\)
−0.0260232 + 0.999661i \(0.508284\pi\)
\(44\) 94.5870 0.0488569
\(45\) 0 0
\(46\) 207.971 0.0982852
\(47\) 186.190i 0.0842870i 0.999112 + 0.0421435i \(0.0134187\pi\)
−0.999112 + 0.0421435i \(0.986581\pi\)
\(48\) 0 0
\(49\) 1551.64 − 1832.27i 0.646249 − 0.763127i
\(50\) 3498.35 1.39934
\(51\) 0 0
\(52\) − 164.862i − 0.0609696i
\(53\) −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
\(56\) 1005.92 − 466.299i 0.320766 − 0.148692i
\(57\) 0 0
\(58\) 2689.50 0.799494
\(59\) 1650.28i 0.474081i 0.971500 + 0.237041i \(0.0761774\pi\)
−0.971500 + 0.237041i \(0.923823\pi\)
\(60\) 0 0
\(61\) 5200.50i 1.39761i 0.715313 + 0.698804i \(0.246284\pi\)
−0.715313 + 0.698804i \(0.753716\pi\)
\(62\) 3918.23i 1.01931i
\(63\) 0 0
\(64\) 512.000 0.125000
\(65\) 889.206 0.210463
\(66\) 0 0
\(67\) 552.587 0.123098 0.0615490 − 0.998104i \(-0.480396\pi\)
0.0615490 + 0.998104i \(0.480396\pi\)
\(68\) − 2313.37i − 0.500297i
\(69\) 0 0
\(70\) 2515.05 + 5425.58i 0.513276 + 1.10726i
\(71\) 8487.61 1.68372 0.841858 − 0.539699i \(-0.181462\pi\)
0.841858 + 0.539699i \(0.181462\pi\)
\(72\) 0 0
\(73\) − 317.344i − 0.0595503i −0.999557 − 0.0297751i \(-0.990521\pi\)
0.999557 − 0.0297751i \(-0.00947912\pi\)
\(74\) 3618.89 0.660863
\(75\) 0 0
\(76\) 837.124i 0.144932i
\(77\) −525.618 + 243.653i −0.0886521 + 0.0410951i
\(78\) 0 0
\(79\) −624.377 −0.100044 −0.0500222 − 0.998748i \(-0.515929\pi\)
−0.0500222 + 0.998748i \(0.515929\pi\)
\(80\) 2761.55i 0.431492i
\(81\) 0 0
\(82\) − 3686.96i − 0.548329i
\(83\) − 7662.33i − 1.11226i −0.831097 − 0.556128i \(-0.812287\pi\)
0.831097 − 0.556128i \(-0.187713\pi\)
\(84\) 0 0
\(85\) 12477.5 1.72699
\(86\) 272.190 0.0368024
\(87\) 0 0
\(88\) −267.532 −0.0345471
\(89\) − 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
\(92\) −588.232 −0.0694981
\(93\) 0 0
\(94\) − 526.625i − 0.0595999i
\(95\) −4515.15 −0.500294
\(96\) 0 0
\(97\) − 12994.4i − 1.38106i −0.723305 − 0.690529i \(-0.757378\pi\)
0.723305 − 0.690529i \(-0.242622\pi\)
\(98\) −4388.71 + 5182.43i −0.456967 + 0.539612i
\(99\) 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 126.5.c.a.55.2 4
3.2 odd 2 14.5.b.a.13.4 yes 4
4.3 odd 2 1008.5.f.h.433.4 4
7.6 odd 2 inner 126.5.c.a.55.1 4
12.11 even 2 112.5.c.c.97.2 4
15.2 even 4 350.5.d.a.349.7 8
15.8 even 4 350.5.d.a.349.2 8
15.14 odd 2 350.5.b.a.251.1 4
21.2 odd 6 98.5.d.d.31.1 8
21.5 even 6 98.5.d.d.31.2 8
21.11 odd 6 98.5.d.d.19.2 8
21.17 even 6 98.5.d.d.19.1 8
21.20 even 2 14.5.b.a.13.3 ✓ 4
24.5 odd 2 448.5.c.e.321.2 4
24.11 even 2 448.5.c.f.321.3 4
28.27 even 2 1008.5.f.h.433.1 4
84.83 odd 2 112.5.c.c.97.3 4
105.62 odd 4 350.5.d.a.349.6 8
105.83 odd 4 350.5.d.a.349.3 8
105.104 even 2 350.5.b.a.251.2 4
168.83 odd 2 448.5.c.f.321.2 4
168.125 even 2 448.5.c.e.321.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.5.b.a.13.3 ✓ 4 21.20 even 2
14.5.b.a.13.4 yes 4 3.2 odd 2
98.5.d.d.19.1 8 21.17 even 6
98.5.d.d.19.2 8 21.11 odd 6
98.5.d.d.31.1 8 21.2 odd 6
98.5.d.d.31.2 8 21.5 even 6
112.5.c.c.97.2 4 12.11 even 2
112.5.c.c.97.3 4 84.83 odd 2
126.5.c.a.55.1 4 7.6 odd 2 inner
126.5.c.a.55.2 4 1.1 even 1 trivial
350.5.b.a.251.1 4 15.14 odd 2
350.5.b.a.251.2 4 105.104 even 2
350.5.d.a.349.2 8 15.8 even 4
350.5.d.a.349.3 8 105.83 odd 4
350.5.d.a.349.6 8 105.62 odd 4
350.5.d.a.349.7 8 15.2 even 4
448.5.c.e.321.2 4 24.5 odd 2
448.5.c.e.321.3 4 168.125 even 2
448.5.c.f.321.2 4 168.83 odd 2
448.5.c.f.321.3 4 24.11 even 2
1008.5.f.h.433.1 4 28.27 even 2
1008.5.f.h.433.4 4 4.3 odd 2