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Properties

Label 1125.4.a.q.1.8
Level $1125$
Weight $4$
Character 1125.1
Self dual yes
Analytic conductor $66.377$
Analytic rank $1$
Dimension $12$
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] = [1125,4,Mod(1,1125)] 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("1125.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1125 = 3^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1125.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,54,0,0,0,0,0,0,-170] 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: yes
Analytic conductor: \(66.3771487565\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 75x^{10} + 2040x^{8} - 24475x^{6} + 126325x^{4} - 223000x^{2} + 122000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.24842\) of defining polynomial
Character \(\chi\) \(=\) 1125.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.24842 q^{2} -6.44144 q^{4} -22.2415 q^{7} -18.0290 q^{8} +32.0660 q^{11} +32.8850 q^{13} -27.7668 q^{14} +29.0236 q^{16} +100.891 q^{17} -61.5890 q^{19} +40.0320 q^{22} +39.8195 q^{23} +41.0544 q^{26} +143.267 q^{28} +29.0478 q^{29} -110.843 q^{31} +180.466 q^{32} +125.955 q^{34} -76.2508 q^{37} -76.8892 q^{38} +338.998 q^{41} -318.789 q^{43} -206.551 q^{44} +49.7116 q^{46} -322.319 q^{47} +151.684 q^{49} -211.826 q^{52} +460.753 q^{53} +400.993 q^{56} +36.2640 q^{58} -686.251 q^{59} +45.3181 q^{61} -138.380 q^{62} -6.89028 q^{64} -1072.88 q^{67} -649.885 q^{68} -343.658 q^{71} -261.220 q^{73} -95.1933 q^{74} +396.721 q^{76} -713.195 q^{77} -349.061 q^{79} +423.214 q^{82} -399.964 q^{83} -397.984 q^{86} -578.119 q^{88} -1372.06 q^{89} -731.411 q^{91} -256.495 q^{92} -402.390 q^{94} +747.241 q^{97} +189.366 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 54 q^{4} - 170 q^{11} - 290 q^{14} + 258 q^{16} + 152 q^{19} - 360 q^{26} - 630 q^{29} - 454 q^{31} - 780 q^{34} - 1260 q^{41} - 3340 q^{44} - 640 q^{46} - 16 q^{49} - 4410 q^{56} - 2970 q^{59} - 256 q^{61}+ \cdots + 2200 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.24842 0.441385 0.220692 − 0.975343i \(-0.429168\pi\)
0.220692 + 0.975343i \(0.429168\pi\)
\(3\) 0 0
\(4\) −6.44144 −0.805179
\(5\) 0 0
\(6\) 0 0
\(7\) −22.2415 −1.20093 −0.600464 − 0.799652i \(-0.705017\pi\)
−0.600464 + 0.799652i \(0.705017\pi\)
\(8\) −18.0290 −0.796779
\(9\) 0 0
\(10\) 0 0
\(11\) 32.0660 0.878932 0.439466 − 0.898259i \(-0.355168\pi\)
0.439466 + 0.898259i \(0.355168\pi\)
\(12\) 0 0
\(13\) 32.8850 0.701588 0.350794 − 0.936453i \(-0.385912\pi\)
0.350794 + 0.936453i \(0.385912\pi\)
\(14\) −27.7668 −0.530071
\(15\) 0 0
\(16\) 29.0236 0.453493
\(17\) 100.891 1.43940 0.719699 − 0.694286i \(-0.244280\pi\)
0.719699 + 0.694286i \(0.244280\pi\)
\(18\) 0 0
\(19\) −61.5890 −0.743657 −0.371828 − 0.928301i \(-0.621269\pi\)
−0.371828 + 0.928301i \(0.621269\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 40.0320 0.387947
\(23\) 39.8195 0.360997 0.180499 − 0.983575i \(-0.442229\pi\)
0.180499 + 0.983575i \(0.442229\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 41.0544 0.309670
\(27\) 0 0
\(28\) 143.267 0.966962
\(29\) 29.0478 0.186002 0.0930008 − 0.995666i \(-0.470354\pi\)
0.0930008 + 0.995666i \(0.470354\pi\)
\(30\) 0 0
\(31\) −110.843 −0.642196 −0.321098 − 0.947046i \(-0.604052\pi\)
−0.321098 + 0.947046i \(0.604052\pi\)
\(32\) 180.466 0.996944
\(33\) 0 0
\(34\) 125.955 0.635328
\(35\) 0 0
\(36\) 0 0
\(37\) −76.2508 −0.338799 −0.169399 − 0.985548i \(-0.554183\pi\)
−0.169399 + 0.985548i \(0.554183\pi\)
\(38\) −76.8892 −0.328239
\(39\) 0 0
\(40\) 0 0
\(41\) 338.998 1.29128 0.645642 − 0.763641i \(-0.276590\pi\)
0.645642 + 0.763641i \(0.276590\pi\)
\(42\) 0 0
\(43\) −318.789 −1.13058 −0.565289 − 0.824893i \(-0.691235\pi\)
−0.565289 + 0.824893i \(0.691235\pi\)
\(44\) −206.551 −0.707698
\(45\) 0 0
\(46\) 49.7116 0.159339
\(47\) −322.319 −1.00032 −0.500159 − 0.865933i \(-0.666725\pi\)
−0.500159 + 0.865933i \(0.666725\pi\)
\(48\) 0 0
\(49\) 151.684 0.442227
\(50\) 0 0
\(51\) 0 0
\(52\) −211.826 −0.564904
\(53\) 460.753 1.19414 0.597069 − 0.802190i \(-0.296332\pi\)
0.597069 + 0.802190i \(0.296332\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 400.993 0.956874
\(57\) 0 0
\(58\) 36.2640 0.0820983
\(59\) −686.251 −1.51428 −0.757138 − 0.653255i \(-0.773403\pi\)
−0.757138 + 0.653255i \(0.773403\pi\)
\(60\) 0 0
\(61\) 45.3181 0.0951211 0.0475605 − 0.998868i \(-0.484855\pi\)
0.0475605 + 0.998868i \(0.484855\pi\)
\(62\) −138.380 −0.283455
\(63\) 0 0
\(64\) −6.89028 −0.0134576
\(65\) 0 0
\(66\) 0 0
\(67\) −1072.88 −1.95632 −0.978159 − 0.207856i \(-0.933351\pi\)
−0.978159 + 0.207856i \(0.933351\pi\)
\(68\) −649.885 −1.15897
\(69\) 0 0
\(70\) 0 0
\(71\) −343.658 −0.574433 −0.287216 − 0.957866i \(-0.592730\pi\)
−0.287216 + 0.957866i \(0.592730\pi\)
\(72\) 0 0
\(73\) −261.220 −0.418814 −0.209407 − 0.977829i \(-0.567153\pi\)
−0.209407 + 0.977829i \(0.567153\pi\)
\(74\) −95.1933 −0.149541
\(75\) 0 0
\(76\) 396.721 0.598777
\(77\) −713.195 −1.05553
\(78\) 0 0
\(79\) −349.061 −0.497119 −0.248560 − 0.968617i \(-0.579957\pi\)
−0.248560 + 0.968617i \(0.579957\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 423.214 0.569953
\(83\) −399.964 −0.528936 −0.264468 − 0.964394i \(-0.585196\pi\)
−0.264468 + 0.964394i \(0.585196\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −397.984 −0.499020
\(87\) 0 0
\(88\) −578.119 −0.700315
\(89\) −1372.06 −1.63414 −0.817071 − 0.576537i \(-0.804404\pi\)
−0.817071 + 0.576537i \(0.804404\pi\)
\(90\) 0 0
\(91\) −731.411 −0.842557
\(92\) −256.495 −0.290667
\(93\) 0 0
\(94\) −402.390 −0.441526
\(95\) 0 0
\(96\) 0 0
\(97\) 747.241 0.782173 0.391087 − 0.920354i \(-0.372099\pi\)
0.391087 + 0.920354i \(0.372099\pi\)
\(98\) 189.366 0.195192
\(99\) 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 1125.4.a.q.1.8 yes 12
3.2 odd 2 1125.4.a.r.1.5 yes 12
5.4 even 2 inner 1125.4.a.q.1.5 ✓ 12
15.14 odd 2 1125.4.a.r.1.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1125.4.a.q.1.5 ✓ 12 5.4 even 2 inner
1125.4.a.q.1.8 yes 12 1.1 even 1 trivial
1125.4.a.r.1.5 yes 12 3.2 odd 2
1125.4.a.r.1.8 yes 12 15.14 odd 2