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Properties

Label 1125.4.a.p.1.2
Level $1125$
Weight $4$
Character 1125.1
Self dual yes
Analytic conductor $66.377$
Analytic rank $0$
Dimension $8$
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] = [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 = [8,7,0,41,0,0,-19,99,0,0,-27] 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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 49x^{6} + 19x^{5} + 711x^{4} - 70x^{3} - 3215x^{2} + 4400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: no (minimal twist has level 375)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.91962\) 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)
 
\(f(q)\) \(=\) \(q-2.91962 q^{2} +0.524200 q^{4} -12.0754 q^{7} +21.8265 q^{8} -45.6229 q^{11} -11.8301 q^{13} +35.2557 q^{14} -67.9188 q^{16} -136.463 q^{17} +51.4445 q^{19} +133.202 q^{22} +23.7841 q^{23} +34.5395 q^{26} -6.32994 q^{28} -216.013 q^{29} -14.3619 q^{31} +23.6852 q^{32} +398.419 q^{34} +342.589 q^{37} -150.199 q^{38} -286.502 q^{41} -159.491 q^{43} -23.9155 q^{44} -69.4406 q^{46} +314.063 q^{47} -197.184 q^{49} -6.20135 q^{52} +138.174 q^{53} -263.565 q^{56} +630.677 q^{58} -760.013 q^{59} -372.123 q^{61} +41.9312 q^{62} +474.199 q^{64} -389.961 q^{67} -71.5336 q^{68} -1085.49 q^{71} +257.362 q^{73} -1000.23 q^{74} +26.9672 q^{76} +550.916 q^{77} +680.342 q^{79} +836.478 q^{82} -395.985 q^{83} +465.652 q^{86} -995.788 q^{88} +797.491 q^{89} +142.854 q^{91} +12.4676 q^{92} -916.945 q^{94} -200.021 q^{97} +575.702 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 7 q^{2} + 41 q^{4} - 19 q^{7} + 99 q^{8} - 27 q^{11} + 13 q^{13} - 119 q^{14} + 305 q^{16} + 223 q^{17} + 304 q^{19} - 31 q^{22} + 168 q^{23} - 221 q^{26} - 97 q^{28} - 168 q^{29} + 383 q^{31} + 774 q^{32}+ \cdots + 1378 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.91962 −1.03224 −0.516121 − 0.856515i \(-0.672625\pi\)
−0.516121 + 0.856515i \(0.672625\pi\)
\(3\) 0 0
\(4\) 0.524200 0.0655249
\(5\) 0 0
\(6\) 0 0
\(7\) −12.0754 −0.652013 −0.326006 − 0.945368i \(-0.605703\pi\)
−0.326006 + 0.945368i \(0.605703\pi\)
\(8\) 21.8265 0.964605
\(9\) 0 0
\(10\) 0 0
\(11\) −45.6229 −1.25053 −0.625264 − 0.780413i \(-0.715009\pi\)
−0.625264 + 0.780413i \(0.715009\pi\)
\(12\) 0 0
\(13\) −11.8301 −0.252391 −0.126196 − 0.992005i \(-0.540277\pi\)
−0.126196 + 0.992005i \(0.540277\pi\)
\(14\) 35.2557 0.673035
\(15\) 0 0
\(16\) −67.9188 −1.06123
\(17\) −136.463 −1.94688 −0.973442 − 0.228932i \(-0.926477\pi\)
−0.973442 + 0.228932i \(0.926477\pi\)
\(18\) 0 0
\(19\) 51.4445 0.621167 0.310584 − 0.950546i \(-0.399476\pi\)
0.310584 + 0.950546i \(0.399476\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 133.202 1.29085
\(23\) 23.7841 0.215623 0.107812 − 0.994171i \(-0.465616\pi\)
0.107812 + 0.994171i \(0.465616\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 34.5395 0.260529
\(27\) 0 0
\(28\) −6.32994 −0.0427231
\(29\) −216.013 −1.38320 −0.691598 − 0.722283i \(-0.743093\pi\)
−0.691598 + 0.722283i \(0.743093\pi\)
\(30\) 0 0
\(31\) −14.3619 −0.0832086 −0.0416043 − 0.999134i \(-0.513247\pi\)
−0.0416043 + 0.999134i \(0.513247\pi\)
\(32\) 23.6852 0.130843
\(33\) 0 0
\(34\) 398.419 2.00966
\(35\) 0 0
\(36\) 0 0
\(37\) 342.589 1.52220 0.761099 − 0.648636i \(-0.224660\pi\)
0.761099 + 0.648636i \(0.224660\pi\)
\(38\) −150.199 −0.641195
\(39\) 0 0
\(40\) 0 0
\(41\) −286.502 −1.09132 −0.545660 − 0.838007i \(-0.683721\pi\)
−0.545660 + 0.838007i \(0.683721\pi\)
\(42\) 0 0
\(43\) −159.491 −0.565630 −0.282815 − 0.959175i \(-0.591268\pi\)
−0.282815 + 0.959175i \(0.591268\pi\)
\(44\) −23.9155 −0.0819408
\(45\) 0 0
\(46\) −69.4406 −0.222575
\(47\) 314.063 0.974697 0.487349 − 0.873207i \(-0.337964\pi\)
0.487349 + 0.873207i \(0.337964\pi\)
\(48\) 0 0
\(49\) −197.184 −0.574880
\(50\) 0 0
\(51\) 0 0
\(52\) −6.20135 −0.0165379
\(53\) 138.174 0.358106 0.179053 − 0.983839i \(-0.442697\pi\)
0.179053 + 0.983839i \(0.442697\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −263.565 −0.628935
\(57\) 0 0
\(58\) 630.677 1.42779
\(59\) −760.013 −1.67704 −0.838519 − 0.544873i \(-0.816578\pi\)
−0.838519 + 0.544873i \(0.816578\pi\)
\(60\) 0 0
\(61\) −372.123 −0.781074 −0.390537 − 0.920587i \(-0.627711\pi\)
−0.390537 + 0.920587i \(0.627711\pi\)
\(62\) 41.9312 0.0858915
\(63\) 0 0
\(64\) 474.199 0.926169
\(65\) 0 0
\(66\) 0 0
\(67\) −389.961 −0.711064 −0.355532 − 0.934664i \(-0.615700\pi\)
−0.355532 + 0.934664i \(0.615700\pi\)
\(68\) −71.5336 −0.127570
\(69\) 0 0
\(70\) 0 0
\(71\) −1085.49 −1.81443 −0.907213 − 0.420672i \(-0.861794\pi\)
−0.907213 + 0.420672i \(0.861794\pi\)
\(72\) 0 0
\(73\) 257.362 0.412629 0.206314 − 0.978486i \(-0.433853\pi\)
0.206314 + 0.978486i \(0.433853\pi\)
\(74\) −1000.23 −1.57128
\(75\) 0 0
\(76\) 26.9672 0.0407019
\(77\) 550.916 0.815360
\(78\) 0 0
\(79\) 680.342 0.968917 0.484459 − 0.874814i \(-0.339017\pi\)
0.484459 + 0.874814i \(0.339017\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 836.478 1.12651
\(83\) −395.985 −0.523675 −0.261837 − 0.965112i \(-0.584328\pi\)
−0.261837 + 0.965112i \(0.584328\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 465.652 0.583867
\(87\) 0 0
\(88\) −995.788 −1.20627
\(89\) 797.491 0.949818 0.474909 − 0.880035i \(-0.342481\pi\)
0.474909 + 0.880035i \(0.342481\pi\)
\(90\) 0 0
\(91\) 142.854 0.164562
\(92\) 12.4676 0.0141287
\(93\) 0 0
\(94\) −916.945 −1.00612
\(95\) 0 0
\(96\) 0 0
\(97\) −200.021 −0.209372 −0.104686 − 0.994505i \(-0.533384\pi\)
−0.104686 + 0.994505i \(0.533384\pi\)
\(98\) 575.702 0.593415
\(99\) 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.p.1.2 8
3.2 odd 2 375.4.a.g.1.7 ✓ 8
5.4 even 2 1125.4.a.l.1.7 8
15.2 even 4 375.4.b.d.124.13 16
15.8 even 4 375.4.b.d.124.4 16
15.14 odd 2 375.4.a.h.1.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
375.4.a.g.1.7 ✓ 8 3.2 odd 2
375.4.a.h.1.2 yes 8 15.14 odd 2
375.4.b.d.124.4 16 15.8 even 4
375.4.b.d.124.13 16 15.2 even 4
1125.4.a.l.1.7 8 5.4 even 2
1125.4.a.p.1.2 8 1.1 even 1 trivial