Properties

Label 1035.4.a.s.1.9
Level $1035$
Weight $4$
Character 1035.1
Self dual yes
Analytic conductor $61.067$
Analytic rank $1$
Dimension $9$
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] = [1035,4,Mod(1,1035)] 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("1035.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1035.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,0,24,-45,0,-12] 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: yes
Analytic conductor: \(61.0669768559\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 48x^{7} + 713x^{5} - 8x^{4} - 3438x^{3} + 488x^{2} + 3732x + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 13 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(4.83690\) of defining polynomial
Character \(\chi\) \(=\) 1035.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.83690 q^{2} +15.3956 q^{4} -5.00000 q^{5} +15.1704 q^{7} +35.7717 q^{8} -24.1845 q^{10} -51.7468 q^{11} -67.2319 q^{13} +73.3779 q^{14} +49.8593 q^{16} -135.882 q^{17} -59.8420 q^{19} -76.9779 q^{20} -250.294 q^{22} +23.0000 q^{23} +25.0000 q^{25} -325.194 q^{26} +233.558 q^{28} +268.616 q^{29} -143.293 q^{31} -45.0090 q^{32} -657.246 q^{34} -75.8522 q^{35} -283.364 q^{37} -289.450 q^{38} -178.858 q^{40} +398.811 q^{41} -24.2431 q^{43} -796.672 q^{44} +111.249 q^{46} +320.113 q^{47} -112.857 q^{49} +120.922 q^{50} -1035.07 q^{52} -269.696 q^{53} +258.734 q^{55} +542.672 q^{56} +1299.27 q^{58} +434.625 q^{59} +441.172 q^{61} -693.093 q^{62} -616.579 q^{64} +336.160 q^{65} -723.365 q^{67} -2091.98 q^{68} -366.890 q^{70} +279.075 q^{71} -250.273 q^{73} -1370.60 q^{74} -921.302 q^{76} -785.022 q^{77} +371.741 q^{79} -249.297 q^{80} +1929.01 q^{82} +713.345 q^{83} +679.409 q^{85} -117.262 q^{86} -1851.07 q^{88} +1299.08 q^{89} -1019.94 q^{91} +354.098 q^{92} +1548.35 q^{94} +299.210 q^{95} +929.717 q^{97} -545.880 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 24 q^{4} - 45 q^{5} - 12 q^{7} + 2 q^{11} - 118 q^{13} + 206 q^{14} + 28 q^{16} - 50 q^{19} - 120 q^{20} - 314 q^{22} + 207 q^{23} + 225 q^{25} + 212 q^{26} - 170 q^{28} + 532 q^{29} - 292 q^{31}+ \cdots - 1804 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.83690 1.71010 0.855051 − 0.518544i \(-0.173526\pi\)
0.855051 + 0.518544i \(0.173526\pi\)
\(3\) 0 0
\(4\) 15.3956 1.92445
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 15.1704 0.819127 0.409564 − 0.912282i \(-0.365681\pi\)
0.409564 + 0.912282i \(0.365681\pi\)
\(8\) 35.7717 1.58090
\(9\) 0 0
\(10\) −24.1845 −0.764781
\(11\) −51.7468 −1.41839 −0.709193 − 0.705014i \(-0.750941\pi\)
−0.709193 + 0.705014i \(0.750941\pi\)
\(12\) 0 0
\(13\) −67.2319 −1.43437 −0.717184 − 0.696884i \(-0.754569\pi\)
−0.717184 + 0.696884i \(0.754569\pi\)
\(14\) 73.3779 1.40079
\(15\) 0 0
\(16\) 49.8593 0.779052
\(17\) −135.882 −1.93860 −0.969299 − 0.245884i \(-0.920922\pi\)
−0.969299 + 0.245884i \(0.920922\pi\)
\(18\) 0 0
\(19\) −59.8420 −0.722563 −0.361281 − 0.932457i \(-0.617661\pi\)
−0.361281 + 0.932457i \(0.617661\pi\)
\(20\) −76.9779 −0.860639
\(21\) 0 0
\(22\) −250.294 −2.42559
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −325.194 −2.45291
\(27\) 0 0
\(28\) 233.558 1.57637
\(29\) 268.616 1.72003 0.860013 − 0.510271i \(-0.170455\pi\)
0.860013 + 0.510271i \(0.170455\pi\)
\(30\) 0 0
\(31\) −143.293 −0.830199 −0.415099 − 0.909776i \(-0.636253\pi\)
−0.415099 + 0.909776i \(0.636253\pi\)
\(32\) −45.0090 −0.248642
\(33\) 0 0
\(34\) −657.246 −3.31520
\(35\) −75.8522 −0.366325
\(36\) 0 0
\(37\) −283.364 −1.25905 −0.629523 − 0.776982i \(-0.716750\pi\)
−0.629523 + 0.776982i \(0.716750\pi\)
\(38\) −289.450 −1.23566
\(39\) 0 0
\(40\) −178.858 −0.707000
\(41\) 398.811 1.51912 0.759558 − 0.650440i \(-0.225415\pi\)
0.759558 + 0.650440i \(0.225415\pi\)
\(42\) 0 0
\(43\) −24.2431 −0.0859777 −0.0429889 − 0.999076i \(-0.513688\pi\)
−0.0429889 + 0.999076i \(0.513688\pi\)
\(44\) −796.672 −2.72961
\(45\) 0 0
\(46\) 111.249 0.356581
\(47\) 320.113 0.993474 0.496737 − 0.867901i \(-0.334531\pi\)
0.496737 + 0.867901i \(0.334531\pi\)
\(48\) 0 0
\(49\) −112.857 −0.329031
\(50\) 120.922 0.342020
\(51\) 0 0
\(52\) −1035.07 −2.76037
\(53\) −269.696 −0.698973 −0.349486 − 0.936941i \(-0.613644\pi\)
−0.349486 + 0.936941i \(0.613644\pi\)
\(54\) 0 0
\(55\) 258.734 0.634322
\(56\) 542.672 1.29496
\(57\) 0 0
\(58\) 1299.27 2.94142
\(59\) 434.625 0.959039 0.479519 − 0.877531i \(-0.340811\pi\)
0.479519 + 0.877531i \(0.340811\pi\)
\(60\) 0 0
\(61\) 441.172 0.926004 0.463002 − 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(62\) −693.093 −1.41972
\(63\) 0 0
\(64\) −616.579 −1.20426
\(65\) 336.160 0.641469
\(66\) 0 0
\(67\) −723.365 −1.31900 −0.659500 − 0.751704i \(-0.729232\pi\)
−0.659500 + 0.751704i \(0.729232\pi\)
\(68\) −2091.98 −3.73073
\(69\) 0 0
\(70\) −366.890 −0.626453
\(71\) 279.075 0.466480 0.233240 − 0.972419i \(-0.425067\pi\)
0.233240 + 0.972419i \(0.425067\pi\)
\(72\) 0 0
\(73\) −250.273 −0.401264 −0.200632 − 0.979667i \(-0.564299\pi\)
−0.200632 + 0.979667i \(0.564299\pi\)
\(74\) −1370.60 −2.15310
\(75\) 0 0
\(76\) −921.302 −1.39053
\(77\) −785.022 −1.16184
\(78\) 0 0
\(79\) 371.741 0.529419 0.264710 − 0.964328i \(-0.414724\pi\)
0.264710 + 0.964328i \(0.414724\pi\)
\(80\) −249.297 −0.348403
\(81\) 0 0
\(82\) 1929.01 2.59784
\(83\) 713.345 0.943371 0.471685 − 0.881767i \(-0.343646\pi\)
0.471685 + 0.881767i \(0.343646\pi\)
\(84\) 0 0
\(85\) 679.409 0.866968
\(86\) −117.262 −0.147031
\(87\) 0 0
\(88\) −1851.07 −2.24233
\(89\) 1299.08 1.54722 0.773610 − 0.633662i \(-0.218449\pi\)
0.773610 + 0.633662i \(0.218449\pi\)
\(90\) 0 0
\(91\) −1019.94 −1.17493
\(92\) 354.098 0.401275
\(93\) 0 0
\(94\) 1548.35 1.69894
\(95\) 299.210 0.323140
\(96\) 0 0
\(97\) 929.717 0.973180 0.486590 − 0.873631i \(-0.338241\pi\)
0.486590 + 0.873631i \(0.338241\pi\)
\(98\) −545.880 −0.562676
\(99\) 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 1035.4.a.s.1.9 ✓ 9
3.2 odd 2 1035.4.a.t.1.1 yes 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1035.4.a.s.1.9 ✓ 9 1.1 even 1 trivial
1035.4.a.t.1.1 yes 9 3.2 odd 2