Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [17,4,Mod(16,17)] 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("17.16"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(17, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 17.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.00303247010\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 16.3
Root \(-4.44593i\) of defining polynomial
Character \(\chi\) \(=\) 17.16
Dual form 17.4.b.a.16.4

$q$-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)
 
\(f(q)\) \(=\) \(q+2.37228 q^{2} -4.44593i q^{3} -2.37228 q^{4} +19.4389i q^{5} -10.5470i q^{6} -14.9929i q^{7} -24.6060 q^{8} +7.23369 q^{9} +46.1145i q^{10} -31.1215i q^{11} +10.5470i q^{12} -5.21194 q^{13} -35.5675i q^{14} +86.4239 q^{15} -39.3940 q^{16} +(68.2119 + 16.1286i) q^{17} +17.1603 q^{18} -28.0000 q^{19} -46.1145i q^{20} -66.6576 q^{21} -73.8290i q^{22} +167.194i q^{23} +109.396i q^{24} -252.870 q^{25} -12.3642 q^{26} -152.201i q^{27} +35.5675i q^{28} -136.072i q^{29} +205.022 q^{30} +50.5604i q^{31} +103.394 q^{32} -138.364 q^{33} +(161.818 + 38.2616i) q^{34} +291.446 q^{35} -17.1603 q^{36} +260.558i q^{37} -66.4239 q^{38} +23.1719i q^{39} -478.312i q^{40} -183.225i q^{41} -158.130 q^{42} -348.293 q^{43} +73.8290i q^{44} +140.615i q^{45} +396.630i q^{46} +318.424 q^{47} +175.143i q^{48} +118.212 q^{49} -599.878 q^{50} +(71.7066 - 303.266i) q^{51} +12.3642 q^{52} -408.250 q^{53} -361.063i q^{54} +604.967 q^{55} +368.916i q^{56} +124.486i q^{57} -322.801i q^{58} +108.119 q^{59} -205.022 q^{60} +123.677i q^{61} +119.943i q^{62} -108.454i q^{63} +560.432 q^{64} -101.314i q^{65} -328.239 q^{66} -243.826 q^{67} +(-161.818 - 38.2616i) q^{68} +743.331 q^{69} +691.391 q^{70} -42.7075i q^{71} -177.992 q^{72} -875.172i q^{73} +618.117i q^{74} +1124.24i q^{75} +66.4239 q^{76} -466.603 q^{77} +54.9703i q^{78} -750.553i q^{79} -765.775i q^{80} -481.364 q^{81} -434.662i q^{82} -472.728 q^{83} +158.130 q^{84} +(-313.522 + 1325.96i) q^{85} -826.250 q^{86} -604.967 q^{87} +765.775i q^{88} +376.364 q^{89} +333.578i q^{90} +78.1422i q^{91} -396.630i q^{92} +224.788 q^{93} +755.391 q^{94} -544.288i q^{95} -459.683i q^{96} +303.169i q^{97} +280.432 q^{98} -225.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 2 q^{4} - 18 q^{8} - 40 q^{9} + 140 q^{13} + 24 q^{15} - 238 q^{16} + 112 q^{17} + 218 q^{18} - 112 q^{19} + 124 q^{21} - 460 q^{25} - 532 q^{26} + 912 q^{30} + 494 q^{32} - 1036 q^{33}+ \cdots + 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(-0.362253\pi\)
0.419364 + 0.907818i \(0.362253\pi\)
\(3\) − 4.44593i − 0.855620i −0.903869 − 0.427810i \(-0.859285\pi\)
0.903869 − 0.427810i \(-0.140715\pi\)
\(4\) −2.37228 −0.296535
\(5\) 19.4389i 1.73867i 0.494228 + 0.869333i \(0.335451\pi\)
−0.494228 + 0.869333i \(0.664549\pi\)
\(6\) − 10.5470i − 0.717633i
\(7\) − 14.9929i − 0.809542i −0.914418 − 0.404771i \(-0.867351\pi\)
0.914418 − 0.404771i \(-0.132649\pi\)
\(8\) −24.6060 −1.08744
\(9\) 7.23369 0.267914
\(10\) 46.1145i 1.45827i
\(11\) − 31.1215i − 0.853045i −0.904477 − 0.426522i \(-0.859739\pi\)
0.904477 − 0.426522i \(-0.140261\pi\)
\(12\) 10.5470i 0.253721i
\(13\) −5.21194 −0.111195 −0.0555974 − 0.998453i \(-0.517706\pi\)
−0.0555974 + 0.998453i \(0.517706\pi\)
\(14\) − 35.5675i − 0.678986i
\(15\) 86.4239 1.48764
\(16\) −39.3940 −0.615532
\(17\) 68.2119 + 16.1286i 0.973166 + 0.230103i
\(18\) 17.1603 0.224707
\(19\) −28.0000 −0.338086 −0.169043 − 0.985609i \(-0.554068\pi\)
−0.169043 + 0.985609i \(0.554068\pi\)
\(20\) − 46.1145i − 0.515575i
\(21\) −66.6576 −0.692661
\(22\) − 73.8290i − 0.715473i
\(23\) 167.194i 1.51575i 0.652399 + 0.757875i \(0.273763\pi\)
−0.652399 + 0.757875i \(0.726237\pi\)
\(24\) 109.396i 0.930436i
\(25\) −252.870 −2.02296
\(26\) −12.3642 −0.0932622
\(27\) − 152.201i − 1.08485i
\(28\) 35.5675i 0.240058i
\(29\) − 136.072i − 0.871309i −0.900114 − 0.435654i \(-0.856517\pi\)
0.900114 − 0.435654i \(-0.143483\pi\)
\(30\) 205.022 1.24772
\(31\) 50.5604i 0.292933i 0.989216 + 0.146466i \(0.0467900\pi\)
−0.989216 + 0.146466i \(0.953210\pi\)
\(32\) 103.394 0.571177
\(33\) −138.364 −0.729882
\(34\) 161.818 + 38.2616i 0.816222 + 0.192994i
\(35\) 291.446 1.40752
\(36\) −17.1603 −0.0794460
\(37\) 260.558i 1.15772i 0.815428 + 0.578858i \(0.196501\pi\)
−0.815428 + 0.578858i \(0.803499\pi\)
\(38\) −66.4239 −0.283563
\(39\) 23.1719i 0.0951404i
\(40\) − 478.312i − 1.89069i
\(41\) − 183.225i − 0.697927i −0.937136 − 0.348964i \(-0.886534\pi\)
0.937136 − 0.348964i \(-0.113466\pi\)
\(42\) −158.130 −0.580954
\(43\) −348.293 −1.23521 −0.617607 − 0.786486i \(-0.711898\pi\)
−0.617607 + 0.786486i \(0.711898\pi\)
\(44\) 73.8290i 0.252958i
\(45\) 140.615i 0.465813i
\(46\) 396.630i 1.27130i
\(47\) 318.424 0.988232 0.494116 − 0.869396i \(-0.335492\pi\)
0.494116 + 0.869396i \(0.335492\pi\)
\(48\) 175.143i 0.526661i
\(49\) 118.212 0.344641
\(50\) −599.878 −1.69671
\(51\) 71.7066 − 303.266i 0.196881 − 0.832660i
\(52\) 12.3642 0.0329732
\(53\) −408.250 −1.05806 −0.529032 − 0.848602i \(-0.677445\pi\)
−0.529032 + 0.848602i \(0.677445\pi\)
\(54\) − 361.063i − 0.909897i
\(55\) 604.967 1.48316
\(56\) 368.916i 0.880329i
\(57\) 124.486i 0.289273i
\(58\) − 322.801i − 0.730791i
\(59\) 108.119 0.238575 0.119288 − 0.992860i \(-0.461939\pi\)
0.119288 + 0.992860i \(0.461939\pi\)
\(60\) −205.022 −0.441137
\(61\) 123.677i 0.259593i 0.991541 + 0.129796i \(0.0414324\pi\)
−0.991541 + 0.129796i \(0.958568\pi\)
\(62\) 119.943i 0.245691i
\(63\) − 108.454i − 0.216888i
\(64\) 560.432 1.09459
\(65\) − 101.314i − 0.193330i
\(66\) −328.239 −0.612173
\(67\) −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
\(69\) 743.331 1.29691
\(70\) 691.391 1.18053
\(71\) − 42.7075i − 0.0713866i −0.999363 − 0.0356933i \(-0.988636\pi\)
0.999363 − 0.0356933i \(-0.0113639\pi\)
\(72\) −177.992 −0.291341
\(73\) − 875.172i − 1.40317i −0.712588 − 0.701583i \(-0.752477\pi\)
0.712588 − 0.701583i \(-0.247523\pi\)
\(74\) 618.117i 0.971009i
\(75\) 1124.24i 1.73088i
\(76\) 66.4239 0.100254
\(77\) −466.603 −0.690576
\(78\) 54.9703i 0.0797970i
\(79\) − 750.553i − 1.06891i −0.845197 − 0.534454i \(-0.820517\pi\)
0.845197 − 0.534454i \(-0.179483\pi\)
\(80\) − 765.775i − 1.07020i
\(81\) −481.364 −0.660308
\(82\) − 434.662i − 0.585371i
\(83\) −472.728 −0.625165 −0.312582 − 0.949891i \(-0.601194\pi\)
−0.312582 + 0.949891i \(0.601194\pi\)
\(84\) 158.130 0.205398
\(85\) −313.522 + 1325.96i −0.400073 + 1.69201i
\(86\) −826.250 −1.03601
\(87\) −604.967 −0.745509
\(88\) 765.775i 0.927635i
\(89\) 376.364 0.448253 0.224127 − 0.974560i \(-0.428047\pi\)
0.224127 + 0.974560i \(0.428047\pi\)
\(90\) 333.578i 0.390691i
\(91\) 78.1422i 0.0900169i
\(92\) − 396.630i − 0.449473i
\(93\) 224.788 0.250639
\(94\) 755.391 0.828858
\(95\) − 544.288i − 0.587819i
\(96\) − 459.683i − 0.488710i
\(97\) 303.169i 0.317342i 0.987332 + 0.158671i \(0.0507209\pi\)
−0.987332 + 0.158671i \(0.949279\pi\)
\(98\) 280.432 0.289060
\(99\) − 225.123i − 0.228543i
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 17.4.b.a.16.3 ✓ 4
3.2 odd 2 153.4.d.b.118.1 4
4.3 odd 2 272.4.b.d.33.3 4
5.2 odd 4 425.4.c.c.424.5 8
5.3 odd 4 425.4.c.c.424.4 8
5.4 even 2 425.4.d.c.101.2 4
17.4 even 4 289.4.a.e.1.1 4
17.13 even 4 289.4.a.e.1.2 4
17.16 even 2 inner 17.4.b.a.16.4 yes 4
51.50 odd 2 153.4.d.b.118.2 4
68.67 odd 2 272.4.b.d.33.2 4
85.33 odd 4 425.4.c.c.424.3 8
85.67 odd 4 425.4.c.c.424.6 8
85.84 even 2 425.4.d.c.101.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.3 ✓ 4 1.1 even 1 trivial
17.4.b.a.16.4 yes 4 17.16 even 2 inner
153.4.d.b.118.1 4 3.2 odd 2
153.4.d.b.118.2 4 51.50 odd 2
272.4.b.d.33.2 4 68.67 odd 2
272.4.b.d.33.3 4 4.3 odd 2
289.4.a.e.1.1 4 17.4 even 4
289.4.a.e.1.2 4 17.13 even 4
425.4.c.c.424.3 8 85.33 odd 4
425.4.c.c.424.4 8 5.3 odd 4
425.4.c.c.424.5 8 5.2 odd 4
425.4.c.c.424.6 8 85.67 odd 4
425.4.d.c.101.1 4 85.84 even 2
425.4.d.c.101.2 4 5.4 even 2