Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,4,Mod(424,425)] 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("425.424"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
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: \( x^{8} + 833x^{4} + 71824 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 424.3
Root \(2.22297 - 2.22297i\) of defining polynomial
Character \(\chi\) \(=\) 425.424
Dual form 425.4.c.c.424.5

$q$-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)
 
\(f(q)\) \(=\) \(q-2.37228i q^{2} -4.44593 q^{3} +2.37228 q^{4} +10.5470i q^{6} +14.9929 q^{7} -24.6060i q^{8} -7.23369 q^{9} +31.1215i q^{11} -10.5470 q^{12} -5.21194i q^{13} -35.5675i q^{14} -39.3940 q^{16} +(-16.1286 - 68.2119i) q^{17} +17.1603i q^{18} +28.0000 q^{19} -66.6576 q^{21} +73.8290 q^{22} +167.194 q^{23} +109.396i q^{24} -12.3642 q^{26} +152.201 q^{27} +35.5675 q^{28} -136.072i q^{29} -50.5604i q^{31} -103.394i q^{32} -138.364i q^{33} +(-161.818 + 38.2616i) q^{34} -17.1603 q^{36} -260.558 q^{37} -66.4239i q^{38} +23.1719i q^{39} +183.225i q^{41} +158.130i q^{42} -348.293i q^{43} +73.8290i q^{44} -396.630i q^{46} -318.424i q^{47} +175.143 q^{48} -118.212 q^{49} +(71.7066 + 303.266i) q^{51} -12.3642i q^{52} -408.250i q^{53} -361.063i q^{54} -368.916i q^{56} -124.486 q^{57} -322.801 q^{58} -108.119 q^{59} -123.677i q^{61} -119.943 q^{62} -108.454 q^{63} -560.432 q^{64} -328.239 q^{66} +243.826i q^{67} +(-38.2616 - 161.818i) q^{68} -743.331 q^{69} +42.7075i q^{71} +177.992i q^{72} -875.172 q^{73} +618.117i q^{74} +66.4239 q^{76} +466.603i q^{77} +54.9703 q^{78} -750.553i q^{79} -481.364 q^{81} +434.662 q^{82} -472.728i q^{83} -158.130 q^{84} -826.250 q^{86} +604.967i q^{87} +765.775 q^{88} -376.364 q^{89} -78.1422i q^{91} +396.630 q^{92} +224.788i q^{93} -755.391 q^{94} +459.683i q^{96} -303.169 q^{97} +280.432i q^{98} -225.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{4} + 80 q^{9} - 476 q^{16} + 224 q^{19} + 248 q^{21} - 1064 q^{26} - 812 q^{34} - 436 q^{36} - 624 q^{49} + 2320 q^{51} + 2352 q^{59} - 2852 q^{64} + 3808 q^{66} - 2408 q^{69} - 112 q^{76} - 4816 q^{81}+ \cdots - 896 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.37228i − 0.838728i −0.907818 − 0.419364i \(-0.862253\pi\)
0.907818 − 0.419364i \(-0.137747\pi\)
\(3\) −4.44593 −0.855620 −0.427810 − 0.903869i \(-0.640715\pi\)
−0.427810 + 0.903869i \(0.640715\pi\)
\(4\) 2.37228 0.296535
\(5\) 0 0
\(6\) 10.5470i 0.717633i
\(7\) 14.9929 0.809542 0.404771 − 0.914418i \(-0.367351\pi\)
0.404771 + 0.914418i \(0.367351\pi\)
\(8\) − 24.6060i − 1.08744i
\(9\) −7.23369 −0.267914
\(10\) 0 0
\(11\) 31.1215i 0.853045i 0.904477 + 0.426522i \(0.140261\pi\)
−0.904477 + 0.426522i \(0.859739\pi\)
\(12\) −10.5470 −0.253721
\(13\) − 5.21194i − 0.111195i −0.998453 − 0.0555974i \(-0.982294\pi\)
0.998453 − 0.0555974i \(-0.0177063\pi\)
\(14\) − 35.5675i − 0.678986i
\(15\) 0 0
\(16\) −39.3940 −0.615532
\(17\) −16.1286 − 68.2119i −0.230103 − 0.973166i
\(18\) 17.1603i 0.224707i
\(19\) 28.0000 0.338086 0.169043 − 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 0 0
\(21\) −66.6576 −0.692661
\(22\) 73.8290 0.715473
\(23\) 167.194 1.51575 0.757875 − 0.652399i \(-0.226237\pi\)
0.757875 + 0.652399i \(0.226237\pi\)
\(24\) 109.396i 0.930436i
\(25\) 0 0
\(26\) −12.3642 −0.0932622
\(27\) 152.201 1.08485
\(28\) 35.5675 0.240058
\(29\) − 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 \(-0.0467900\pi\)
\(32\) − 103.394i − 0.571177i
\(33\) − 138.364i − 0.729882i
\(34\) −161.818 + 38.2616i −0.816222 + 0.192994i
\(35\) 0 0
\(36\) −17.1603 −0.0794460
\(37\) −260.558 −1.15772 −0.578858 − 0.815428i \(-0.696501\pi\)
−0.578858 + 0.815428i \(0.696501\pi\)
\(38\) − 66.4239i − 0.283563i
\(39\) 23.1719i 0.0951404i
\(40\) 0 0
\(41\) 183.225i 0.697927i 0.937136 + 0.348964i \(0.113466\pi\)
−0.937136 + 0.348964i \(0.886534\pi\)
\(42\) 158.130i 0.580954i
\(43\) − 348.293i − 1.23521i −0.786486 − 0.617607i \(-0.788102\pi\)
0.786486 − 0.617607i \(-0.211898\pi\)
\(44\) 73.8290i 0.252958i
\(45\) 0 0
\(46\) − 396.630i − 1.27130i
\(47\) − 318.424i − 0.988232i −0.869396 − 0.494116i \(-0.835492\pi\)
0.869396 − 0.494116i \(-0.164508\pi\)
\(48\) 175.143 0.526661
\(49\) −118.212 −0.344641
\(50\) 0 0
\(51\) 71.7066 + 303.266i 0.196881 + 0.832660i
\(52\) − 12.3642i − 0.0329732i
\(53\) − 408.250i − 1.05806i −0.848602 − 0.529032i \(-0.822555\pi\)
0.848602 − 0.529032i \(-0.177445\pi\)
\(54\) − 361.063i − 0.909897i
\(55\) 0 0
\(56\) − 368.916i − 0.880329i
\(57\) −124.486 −0.289273
\(58\) −322.801 −0.730791
\(59\) −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 \(-0.0414324\pi\)
\(62\) −119.943 −0.245691
\(63\) −108.454 −0.216888
\(64\) −560.432 −1.09459
\(65\) 0 0
\(66\) −328.239 −0.612173
\(67\) 243.826i 0.444598i 0.974979 + 0.222299i \(0.0713562\pi\)
−0.974979 + 0.222299i \(0.928644\pi\)
\(68\) −38.2616 − 161.818i −0.0682338 − 0.288578i
\(69\) −743.331 −1.29691
\(70\) 0 0
\(71\) 42.7075i 0.0713866i 0.999363 + 0.0356933i \(0.0113639\pi\)
−0.999363 + 0.0356933i \(0.988636\pi\)
\(72\) 177.992i 0.291341i
\(73\) −875.172 −1.40317 −0.701583 − 0.712588i \(-0.747523\pi\)
−0.701583 + 0.712588i \(0.747523\pi\)
\(74\) 618.117i 0.971009i
\(75\) 0 0
\(76\) 66.4239 0.100254
\(77\) 466.603i 0.690576i
\(78\) 54.9703 0.0797970
\(79\) − 750.553i − 1.06891i −0.845197 − 0.534454i \(-0.820517\pi\)
0.845197 − 0.534454i \(-0.179483\pi\)
\(80\) 0 0
\(81\) −481.364 −0.660308
\(82\) 434.662 0.585371
\(83\) − 472.728i − 0.625165i −0.949891 − 0.312582i \(-0.898806\pi\)
0.949891 − 0.312582i \(-0.101194\pi\)
\(84\) −158.130 −0.205398
\(85\) 0 0
\(86\) −826.250 −1.03601
\(87\) 604.967i 0.745509i
\(88\) 765.775 0.927635
\(89\) −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
\(92\) 396.630 0.449473
\(93\) 224.788i 0.250639i
\(94\) −755.391 −0.828858
\(95\) 0 0
\(96\) 459.683i 0.488710i
\(97\) −303.169 −0.317342 −0.158671 − 0.987332i \(-0.550721\pi\)
−0.158671 + 0.987332i \(0.550721\pi\)
\(98\) 280.432i 0.289060i
\(99\) − 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 425.4.c.c.424.3 8
5.2 odd 4 17.4.b.a.16.4 yes 4
5.3 odd 4 425.4.d.c.101.1 4
5.4 even 2 inner 425.4.c.c.424.6 8
15.2 even 4 153.4.d.b.118.2 4
17.16 even 2 inner 425.4.c.c.424.4 8
20.7 even 4 272.4.b.d.33.2 4
85.33 odd 4 425.4.d.c.101.2 4
85.47 odd 4 289.4.a.e.1.1 4
85.67 odd 4 17.4.b.a.16.3 ✓ 4
85.72 odd 4 289.4.a.e.1.2 4
85.84 even 2 inner 425.4.c.c.424.5 8
255.152 even 4 153.4.d.b.118.1 4
340.67 even 4 272.4.b.d.33.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.3 ✓ 4 85.67 odd 4
17.4.b.a.16.4 yes 4 5.2 odd 4
153.4.d.b.118.1 4 255.152 even 4
153.4.d.b.118.2 4 15.2 even 4
272.4.b.d.33.2 4 20.7 even 4
272.4.b.d.33.3 4 340.67 even 4
289.4.a.e.1.1 4 85.47 odd 4
289.4.a.e.1.2 4 85.72 odd 4
425.4.c.c.424.3 8 1.1 even 1 trivial
425.4.c.c.424.4 8 17.16 even 2 inner
425.4.c.c.424.5 8 85.84 even 2 inner
425.4.c.c.424.6 8 5.4 even 2 inner
425.4.d.c.101.1 4 5.3 odd 4
425.4.d.c.101.2 4 85.33 odd 4