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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [153,4,Mod(118,153)] 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("153.118"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(153, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-37 +3 \sqrt{33}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 118.2
Root \(-4.44593i\) of defining polynomial
Character \(\chi\) \(=\) 153.118
Dual form 153.4.d.b.118.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.37228 q^{2} -2.37228 q^{4} +19.4389i q^{5} +14.9929i q^{7} +24.6060 q^{8} -46.1145i q^{10} -31.1215i q^{11} -5.21194 q^{13} -35.5675i q^{14} -39.3940 q^{16} +(-68.2119 + 16.1286i) q^{17} -28.0000 q^{19} -46.1145i q^{20} +73.8290i q^{22} +167.194i q^{23} -252.870 q^{25} +12.3642 q^{26} -35.5675i q^{28} -136.072i q^{29} -50.5604i q^{31} -103.394 q^{32} +(161.818 - 38.2616i) q^{34} -291.446 q^{35} -260.558i q^{37} +66.4239 q^{38} +478.312i q^{40} -183.225i q^{41} -348.293 q^{43} +73.8290i q^{44} -396.630i q^{46} -318.424 q^{47} +118.212 q^{49} +599.878 q^{50} +12.3642 q^{52} +408.250 q^{53} +604.967 q^{55} +368.916i q^{56} +322.801i q^{58} -108.119 q^{59} -123.677i q^{61} +119.943i q^{62} +560.432 q^{64} -101.314i q^{65} -243.826 q^{67} +(161.818 - 38.2616i) q^{68} +691.391 q^{70} -42.7075i q^{71} +875.172i q^{73} +618.117i q^{74} +66.4239 q^{76} +466.603 q^{77} +750.553i q^{79} -765.775i q^{80} +434.662i q^{82} +472.728 q^{83} +(-313.522 - 1325.96i) q^{85} +826.250 q^{86} -765.775i q^{88} -376.364 q^{89} -78.1422i q^{91} -396.630i q^{92} +755.391 q^{94} -544.288i q^{95} -303.169i q^{97} -280.432 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 18 q^{8} + 140 q^{13} - 238 q^{16} - 112 q^{17} - 112 q^{19} - 460 q^{25} + 532 q^{26} - 494 q^{32} + 406 q^{34} - 936 q^{35} - 56 q^{38} - 520 q^{43} - 952 q^{47} + 312 q^{49}+ \cdots - 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(137\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37228 −0.838728 −0.419364 − 0.907818i \(-0.637747\pi\)
−0.419364 + 0.907818i \(0.637747\pi\)
\(3\) 0 0
\(4\) −2.37228 −0.296535
\(5\) 19.4389i 1.73867i 0.494228 + 0.869333i \(0.335451\pi\)
−0.494228 + 0.869333i \(0.664549\pi\)
\(6\) 0 0
\(7\) 14.9929i 0.809542i 0.914418 + 0.404771i \(0.132649\pi\)
−0.914418 + 0.404771i \(0.867351\pi\)
\(8\) 24.6060 1.08744
\(9\) 0 0
\(10\) − 46.1145i − 1.45827i
\(11\) − 31.1215i − 0.853045i −0.904477 − 0.426522i \(-0.859739\pi\)
0.904477 − 0.426522i \(-0.140261\pi\)
\(12\) 0 0
\(13\) −5.21194 −0.111195 −0.0555974 − 0.998453i \(-0.517706\pi\)
−0.0555974 + 0.998453i \(0.517706\pi\)
\(14\) − 35.5675i − 0.678986i
\(15\) 0 0
\(16\) −39.3940 −0.615532
\(17\) −68.2119 + 16.1286i −0.973166 + 0.230103i
\(18\) 0 0
\(19\) −28.0000 −0.338086 −0.169043 − 0.985609i \(-0.554068\pi\)
−0.169043 + 0.985609i \(0.554068\pi\)
\(20\) − 46.1145i − 0.515575i
\(21\) 0 0
\(22\) 73.8290i 0.715473i
\(23\) 167.194i 1.51575i 0.652399 + 0.757875i \(0.273763\pi\)
−0.652399 + 0.757875i \(0.726237\pi\)
\(24\) 0 0
\(25\) −252.870 −2.02296
\(26\) 12.3642 0.0932622
\(27\) 0 0
\(28\) − 35.5675i − 0.240058i
\(29\) − 136.072i − 0.871309i −0.900114 − 0.435654i \(-0.856517\pi\)
0.900114 − 0.435654i \(-0.143483\pi\)
\(30\) 0 0
\(31\) − 50.5604i − 0.292933i −0.989216 − 0.146466i \(-0.953210\pi\)
0.989216 − 0.146466i \(-0.0467900\pi\)
\(32\) −103.394 −0.571177
\(33\) 0 0
\(34\) 161.818 − 38.2616i 0.816222 − 0.192994i
\(35\) −291.446 −1.40752
\(36\) 0 0
\(37\) − 260.558i − 1.15772i −0.815428 − 0.578858i \(-0.803499\pi\)
0.815428 − 0.578858i \(-0.196501\pi\)
\(38\) 66.4239 0.283563
\(39\) 0 0
\(40\) 478.312i 1.89069i
\(41\) − 183.225i − 0.697927i −0.937136 − 0.348964i \(-0.886534\pi\)
0.937136 − 0.348964i \(-0.113466\pi\)
\(42\) 0 0
\(43\) −348.293 −1.23521 −0.617607 − 0.786486i \(-0.711898\pi\)
−0.617607 + 0.786486i \(0.711898\pi\)
\(44\) 73.8290i 0.252958i
\(45\) 0 0
\(46\) − 396.630i − 1.27130i
\(47\) −318.424 −0.988232 −0.494116 − 0.869396i \(-0.664508\pi\)
−0.494116 + 0.869396i \(0.664508\pi\)
\(48\) 0 0
\(49\) 118.212 0.344641
\(50\) 599.878 1.69671
\(51\) 0 0
\(52\) 12.3642 0.0329732
\(53\) 408.250 1.05806 0.529032 − 0.848602i \(-0.322555\pi\)
0.529032 + 0.848602i \(0.322555\pi\)
\(54\) 0 0
\(55\) 604.967 1.48316
\(56\) 368.916i 0.880329i
\(57\) 0 0
\(58\) 322.801i 0.730791i
\(59\) −108.119 −0.238575 −0.119288 − 0.992860i \(-0.538061\pi\)
−0.119288 + 0.992860i \(0.538061\pi\)
\(60\) 0 0
\(61\) − 123.677i − 0.259593i −0.991541 − 0.129796i \(-0.958568\pi\)
0.991541 − 0.129796i \(-0.0414324\pi\)
\(62\) 119.943i 0.245691i
\(63\) 0 0
\(64\) 560.432 1.09459
\(65\) − 101.314i − 0.193330i
\(66\) 0 0
\(67\) −243.826 −0.444598 −0.222299 − 0.974979i \(-0.571356\pi\)
−0.222299 + 0.974979i \(0.571356\pi\)
\(68\) 161.818 − 38.2616i 0.288578 − 0.0682338i
\(69\) 0 0
\(70\) 691.391 1.18053
\(71\) − 42.7075i − 0.0713866i −0.999363 − 0.0356933i \(-0.988636\pi\)
0.999363 − 0.0356933i \(-0.0113639\pi\)
\(72\) 0 0
\(73\) 875.172i 1.40317i 0.712588 + 0.701583i \(0.247523\pi\)
−0.712588 + 0.701583i \(0.752477\pi\)
\(74\) 618.117i 0.971009i
\(75\) 0 0
\(76\) 66.4239 0.100254
\(77\) 466.603 0.690576
\(78\) 0 0
\(79\) 750.553i 1.06891i 0.845197 + 0.534454i \(0.179483\pi\)
−0.845197 + 0.534454i \(0.820517\pi\)
\(80\) − 765.775i − 1.07020i
\(81\) 0 0
\(82\) 434.662i 0.585371i
\(83\) 472.728 0.625165 0.312582 − 0.949891i \(-0.398806\pi\)
0.312582 + 0.949891i \(0.398806\pi\)
\(84\) 0 0
\(85\) −313.522 − 1325.96i −0.400073 − 1.69201i
\(86\) 826.250 1.03601
\(87\) 0 0
\(88\) − 765.775i − 0.927635i
\(89\) −376.364 −0.448253 −0.224127 − 0.974560i \(-0.571953\pi\)
−0.224127 + 0.974560i \(0.571953\pi\)
\(90\) 0 0
\(91\) − 78.1422i − 0.0900169i
\(92\) − 396.630i − 0.449473i
\(93\) 0 0
\(94\) 755.391 0.828858
\(95\) − 544.288i − 0.587819i
\(96\) 0 0
\(97\) − 303.169i − 0.317342i −0.987332 − 0.158671i \(-0.949279\pi\)
0.987332 − 0.158671i \(-0.0507209\pi\)
\(98\) −280.432 −0.289060
\(99\) 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 153.4.d.b.118.2 4
3.2 odd 2 17.4.b.a.16.4 yes 4
12.11 even 2 272.4.b.d.33.2 4
15.2 even 4 425.4.c.c.424.6 8
15.8 even 4 425.4.c.c.424.3 8
15.14 odd 2 425.4.d.c.101.1 4
17.16 even 2 inner 153.4.d.b.118.1 4
51.38 odd 4 289.4.a.e.1.2 4
51.47 odd 4 289.4.a.e.1.1 4
51.50 odd 2 17.4.b.a.16.3 ✓ 4
204.203 even 2 272.4.b.d.33.3 4
255.152 even 4 425.4.c.c.424.5 8
255.203 even 4 425.4.c.c.424.4 8
255.254 odd 2 425.4.d.c.101.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.3 ✓ 4 51.50 odd 2
17.4.b.a.16.4 yes 4 3.2 odd 2
153.4.d.b.118.1 4 17.16 even 2 inner
153.4.d.b.118.2 4 1.1 even 1 trivial
272.4.b.d.33.2 4 12.11 even 2
272.4.b.d.33.3 4 204.203 even 2
289.4.a.e.1.1 4 51.47 odd 4
289.4.a.e.1.2 4 51.38 odd 4
425.4.c.c.424.3 8 15.8 even 4
425.4.c.c.424.4 8 255.203 even 4
425.4.c.c.424.5 8 255.152 even 4
425.4.c.c.424.6 8 15.2 even 4
425.4.d.c.101.1 4 15.14 odd 2
425.4.d.c.101.2 4 255.254 odd 2