Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [735,4,Mod(1,735)] 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("735.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 735.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-6,12,18,-20,-18,0,-30,36,30,-24] 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: \(43.3664038542\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{33 +12 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 15x^{2} + 16x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.03449\) of defining polynomial
Character \(\chi\) \(=\) 735.1

$q$-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)
 
\(f(q)\) \(=\) \(q-3.62028 q^{2} +3.00000 q^{3} +5.10642 q^{4} -5.00000 q^{5} -10.8608 q^{6} +10.4756 q^{8} +9.00000 q^{9} +18.1014 q^{10} -7.56472 q^{11} +15.3193 q^{12} +0.301010 q^{13} -15.0000 q^{15} -78.7758 q^{16} +47.8330 q^{17} -32.5825 q^{18} -79.2076 q^{19} -25.5321 q^{20} +27.3864 q^{22} -22.1884 q^{23} +31.4267 q^{24} +25.0000 q^{25} -1.08974 q^{26} +27.0000 q^{27} -138.150 q^{29} +54.3042 q^{30} +288.891 q^{31} +201.386 q^{32} -22.6942 q^{33} -173.169 q^{34} +45.9578 q^{36} +177.022 q^{37} +286.754 q^{38} +0.903030 q^{39} -52.3778 q^{40} -256.075 q^{41} +143.601 q^{43} -38.6287 q^{44} -45.0000 q^{45} +80.3282 q^{46} -575.163 q^{47} -236.327 q^{48} -90.5070 q^{50} +143.499 q^{51} +1.53708 q^{52} +34.5430 q^{53} -97.7475 q^{54} +37.8236 q^{55} -237.623 q^{57} +500.140 q^{58} +136.831 q^{59} -76.5963 q^{60} +251.136 q^{61} -1045.87 q^{62} -98.8672 q^{64} -1.50505 q^{65} +82.1592 q^{66} -435.802 q^{67} +244.256 q^{68} -66.5652 q^{69} +121.343 q^{71} +94.2800 q^{72} +544.257 q^{73} -640.868 q^{74} +75.0000 q^{75} -404.467 q^{76} -3.26922 q^{78} +92.9247 q^{79} +393.879 q^{80} +81.0000 q^{81} +927.062 q^{82} -908.302 q^{83} -239.165 q^{85} -519.876 q^{86} -414.449 q^{87} -79.2446 q^{88} -562.936 q^{89} +162.913 q^{90} -113.303 q^{92} +866.674 q^{93} +2082.25 q^{94} +396.038 q^{95} +604.158 q^{96} -310.119 q^{97} -68.0825 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} + 12 q^{3} + 18 q^{4} - 20 q^{5} - 18 q^{6} - 30 q^{8} + 36 q^{9} + 30 q^{10} - 24 q^{11} + 54 q^{12} - 60 q^{15} - 46 q^{16} - 88 q^{17} - 54 q^{18} + 72 q^{19} - 90 q^{20} - 28 q^{22} - 90 q^{24}+ \cdots - 216 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.62028 −1.27996 −0.639981 − 0.768391i \(-0.721058\pi\)
−0.639981 + 0.768391i \(0.721058\pi\)
\(3\) 3.00000 0.577350
\(4\) 5.10642 0.638303
\(5\) −5.00000 −0.447214
\(6\) −10.8608 −0.738986
\(7\) 0 0
\(8\) 10.4756 0.462959
\(9\) 9.00000 0.333333
\(10\) 18.1014 0.572416
\(11\) −7.56472 −0.207350 −0.103675 − 0.994611i \(-0.533060\pi\)
−0.103675 + 0.994611i \(0.533060\pi\)
\(12\) 15.3193 0.368524
\(13\) 0.301010 0.00642194 0.00321097 − 0.999995i \(-0.498978\pi\)
0.00321097 + 0.999995i \(0.498978\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) −78.7758 −1.23087
\(17\) 47.8330 0.682425 0.341212 − 0.939986i \(-0.389162\pi\)
0.341212 + 0.939986i \(0.389162\pi\)
\(18\) −32.5825 −0.426654
\(19\) −79.2076 −0.956393 −0.478196 − 0.878253i \(-0.658709\pi\)
−0.478196 + 0.878253i \(0.658709\pi\)
\(20\) −25.5321 −0.285458
\(21\) 0 0
\(22\) 27.3864 0.265400
\(23\) −22.1884 −0.201157 −0.100578 − 0.994929i \(-0.532069\pi\)
−0.100578 + 0.994929i \(0.532069\pi\)
\(24\) 31.4267 0.267289
\(25\) 25.0000 0.200000
\(26\) −1.08974 −0.00821983
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −138.150 −0.884611 −0.442306 − 0.896864i \(-0.645839\pi\)
−0.442306 + 0.896864i \(0.645839\pi\)
\(30\) 54.3042 0.330485
\(31\) 288.891 1.67376 0.836878 − 0.547389i \(-0.184378\pi\)
0.836878 + 0.547389i \(0.184378\pi\)
\(32\) 201.386 1.11251
\(33\) −22.6942 −0.119713
\(34\) −173.169 −0.873478
\(35\) 0 0
\(36\) 45.9578 0.212768
\(37\) 177.022 0.786545 0.393273 − 0.919422i \(-0.371343\pi\)
0.393273 + 0.919422i \(0.371343\pi\)
\(38\) 286.754 1.22415
\(39\) 0.903030 0.00370771
\(40\) −52.3778 −0.207041
\(41\) −256.075 −0.975419 −0.487709 − 0.873006i \(-0.662167\pi\)
−0.487709 + 0.873006i \(0.662167\pi\)
\(42\) 0 0
\(43\) 143.601 0.509279 0.254639 − 0.967036i \(-0.418043\pi\)
0.254639 + 0.967036i \(0.418043\pi\)
\(44\) −38.6287 −0.132352
\(45\) −45.0000 −0.149071
\(46\) 80.3282 0.257473
\(47\) −575.163 −1.78502 −0.892512 − 0.451024i \(-0.851059\pi\)
−0.892512 + 0.451024i \(0.851059\pi\)
\(48\) −236.327 −0.710644
\(49\) 0 0
\(50\) −90.5070 −0.255992
\(51\) 143.499 0.393998
\(52\) 1.53708 0.00409914
\(53\) 34.5430 0.0895254 0.0447627 − 0.998998i \(-0.485747\pi\)
0.0447627 + 0.998998i \(0.485747\pi\)
\(54\) −97.7475 −0.246329
\(55\) 37.8236 0.0927297
\(56\) 0 0
\(57\) −237.623 −0.552174
\(58\) 500.140 1.13227
\(59\) 136.831 0.301930 0.150965 − 0.988539i \(-0.451762\pi\)
0.150965 + 0.988539i \(0.451762\pi\)
\(60\) −76.5963 −0.164809
\(61\) 251.136 0.527126 0.263563 − 0.964642i \(-0.415102\pi\)
0.263563 + 0.964642i \(0.415102\pi\)
\(62\) −1045.87 −2.14234
\(63\) 0 0
\(64\) −98.8672 −0.193100
\(65\) −1.50505 −0.00287198
\(66\) 82.1592 0.153229
\(67\) −435.802 −0.794653 −0.397326 − 0.917677i \(-0.630062\pi\)
−0.397326 + 0.917677i \(0.630062\pi\)
\(68\) 244.256 0.435594
\(69\) −66.5652 −0.116138
\(70\) 0 0
\(71\) 121.343 0.202828 0.101414 − 0.994844i \(-0.467663\pi\)
0.101414 + 0.994844i \(0.467663\pi\)
\(72\) 94.2800 0.154320
\(73\) 544.257 0.872608 0.436304 − 0.899799i \(-0.356287\pi\)
0.436304 + 0.899799i \(0.356287\pi\)
\(74\) −640.868 −1.00675
\(75\) 75.0000 0.115470
\(76\) −404.467 −0.610468
\(77\) 0 0
\(78\) −3.26922 −0.00474572
\(79\) 92.9247 0.132340 0.0661699 − 0.997808i \(-0.478922\pi\)
0.0661699 + 0.997808i \(0.478922\pi\)
\(80\) 393.879 0.550463
\(81\) 81.0000 0.111111
\(82\) 927.062 1.24850
\(83\) −908.302 −1.20119 −0.600597 − 0.799552i \(-0.705070\pi\)
−0.600597 + 0.799552i \(0.705070\pi\)
\(84\) 0 0
\(85\) −239.165 −0.305190
\(86\) −519.876 −0.651857
\(87\) −414.449 −0.510731
\(88\) −79.2446 −0.0959944
\(89\) −562.936 −0.670462 −0.335231 − 0.942136i \(-0.608814\pi\)
−0.335231 + 0.942136i \(0.608814\pi\)
\(90\) 162.913 0.190805
\(91\) 0 0
\(92\) −113.303 −0.128399
\(93\) 866.674 0.966344
\(94\) 2082.25 2.28476
\(95\) 396.038 0.427712
\(96\) 604.158 0.642309
\(97\) −310.119 −0.324617 −0.162308 − 0.986740i \(-0.551894\pi\)
−0.162308 + 0.986740i \(0.551894\pi\)
\(98\) 0 0
\(99\) −68.0825 −0.0691166
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 735.4.a.u.1.2 yes 4
3.2 odd 2 2205.4.a.bq.1.3 4
7.6 odd 2 735.4.a.t.1.2 ✓ 4
21.20 even 2 2205.4.a.bp.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
735.4.a.t.1.2 ✓ 4 7.6 odd 2
735.4.a.u.1.2 yes 4 1.1 even 1 trivial
2205.4.a.bp.1.3 4 21.20 even 2
2205.4.a.bq.1.3 4 3.2 odd 2