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(101,425)] 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("425.101"); 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([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,2,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
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: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.2
Root \(4.44593i\) of defining polynomial
Character \(\chi\) \(=\) 425.101
Dual form 425.4.d.c.101.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.37228 q^{2} +4.44593i q^{3} -2.37228 q^{4} -10.5470i q^{6} +14.9929i q^{7} +24.6060 q^{8} +7.23369 q^{9} -31.1215i q^{11} -10.5470i q^{12} +5.21194 q^{13} -35.5675i q^{14} -39.3940 q^{16} +(-68.2119 - 16.1286i) q^{17} -17.1603 q^{18} -28.0000 q^{19} -66.6576 q^{21} +73.8290i q^{22} -167.194i q^{23} +109.396i q^{24} -12.3642 q^{26} +152.201i q^{27} -35.5675i q^{28} -136.072i q^{29} +50.5604i q^{31} -103.394 q^{32} +138.364 q^{33} +(161.818 + 38.2616i) q^{34} -17.1603 q^{36} -260.558i q^{37} +66.4239 q^{38} +23.1719i q^{39} -183.225i q^{41} +158.130 q^{42} +348.293 q^{43} +73.8290i q^{44} +396.630i q^{46} -318.424 q^{47} -175.143i q^{48} +118.212 q^{49} +(71.7066 - 303.266i) q^{51} -12.3642 q^{52} +408.250 q^{53} -361.063i q^{54} +368.916i q^{56} -124.486i q^{57} +322.801i q^{58} +108.119 q^{59} +123.677i q^{61} -119.943i q^{62} +108.454i q^{63} +560.432 q^{64} -328.239 q^{66} +243.826 q^{67} +(161.818 + 38.2616i) q^{68} +743.331 q^{69} -42.7075i q^{71} +177.992 q^{72} +875.172i q^{73} +618.117i q^{74} +66.4239 q^{76} +466.603 q^{77} -54.9703i q^{78} -750.553i q^{79} -481.364 q^{81} +434.662i q^{82} +472.728 q^{83} +158.130 q^{84} -826.250 q^{86} +604.967 q^{87} -765.775i q^{88} +376.364 q^{89} +78.1422i q^{91} +396.630i q^{92} -224.788 q^{93} +755.391 q^{94} -459.683i q^{96} -303.169i q^{97} -280.432 q^{98} -225.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{4} + 18 q^{8} - 40 q^{9} - 140 q^{13} - 238 q^{16} - 112 q^{17} - 218 q^{18} - 112 q^{19} + 124 q^{21} - 532 q^{26} - 494 q^{32} + 1036 q^{33} + 406 q^{34} - 218 q^{36} - 56 q^{38}+ \cdots - 306 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.37228 −0.838728 −0.419364 − 0.907818i \(-0.637747\pi\)
−0.419364 + 0.907818i \(0.637747\pi\)
\(3\) 4.44593i 0.855620i 0.903869 + 0.427810i \(0.140715\pi\)
−0.903869 + 0.427810i \(0.859285\pi\)
\(4\) −2.37228 −0.296535
\(5\) 0 0
\(6\) − 10.5470i − 0.717633i
\(7\) 14.9929i 0.809542i 0.914418 + 0.404771i \(0.132649\pi\)
−0.914418 + 0.404771i \(0.867351\pi\)
\(8\) 24.6060 1.08744
\(9\) 7.23369 0.267914
\(10\) 0 0
\(11\) − 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 \(-0.482294\pi\)
0.0555974 + 0.998453i \(0.482294\pi\)
\(14\) − 35.5675i − 0.678986i
\(15\) 0 0
\(16\) −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 \(0.554068\pi\)
\(20\) 0 0
\(21\) −66.6576 −0.692661
\(22\) 73.8290i 0.715473i
\(23\) − 167.194i − 1.51575i −0.652399 − 0.757875i \(-0.726237\pi\)
0.652399 − 0.757875i \(-0.273763\pi\)
\(24\) 109.396i 0.930436i
\(25\) 0 0
\(26\) −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 \(-0.143483\pi\)
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) −17.1603 −0.0794460
\(37\) − 260.558i − 1.15772i −0.815428 − 0.578858i \(-0.803499\pi\)
0.815428 − 0.578858i \(-0.196501\pi\)
\(38\) 66.4239 0.283563
\(39\) 23.1719i 0.0951404i
\(40\) 0 0
\(41\) − 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 \(-0.288102\pi\)
0.617607 + 0.786486i \(0.288102\pi\)
\(44\) 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 \(0.664508\pi\)
\(48\) − 175.143i − 0.526661i
\(49\) 118.212 0.344641
\(50\) 0 0
\(51\) 71.7066 − 303.266i 0.196881 − 0.832660i
\(52\) −12.3642 −0.0329732
\(53\) 408.250 1.05806 0.529032 − 0.848602i \(-0.322555\pi\)
0.529032 + 0.848602i \(0.322555\pi\)
\(54\) − 361.063i − 0.909897i
\(55\) 0 0
\(56\) 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 \(0.461939\pi\)
\(60\) 0 0
\(61\) 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
\(65\) 0 0
\(66\) −328.239 −0.612173
\(67\) 243.826 0.444598 0.222299 − 0.974979i \(-0.428644\pi\)
0.222299 + 0.974979i \(0.428644\pi\)
\(68\) 161.818 + 38.2616i 0.288578 + 0.0682338i
\(69\) 743.331 1.29691
\(70\) 0 0
\(71\) − 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 \(0.247523\pi\)
−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
\(78\) − 54.9703i − 0.0797970i
\(79\) − 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
\(82\) 434.662i 0.585371i
\(83\) 472.728 0.625165 0.312582 − 0.949891i \(-0.398806\pi\)
0.312582 + 0.949891i \(0.398806\pi\)
\(84\) 158.130 0.205398
\(85\) 0 0
\(86\) −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 \(0.428047\pi\)
\(90\) 0 0
\(91\) 78.1422i 0.0900169i
\(92\) 396.630i 0.449473i
\(93\) −224.788 −0.250639
\(94\) 755.391 0.828858
\(95\) 0 0
\(96\) − 459.683i − 0.488710i
\(97\) − 303.169i − 0.317342i −0.987332 − 0.158671i \(-0.949279\pi\)
0.987332 − 0.158671i \(-0.0507209\pi\)
\(98\) −280.432 −0.289060
\(99\) − 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.d.c.101.2 4
5.2 odd 4 425.4.c.c.424.4 8
5.3 odd 4 425.4.c.c.424.5 8
5.4 even 2 17.4.b.a.16.3 ✓ 4
15.14 odd 2 153.4.d.b.118.1 4
17.16 even 2 inner 425.4.d.c.101.1 4
20.19 odd 2 272.4.b.d.33.3 4
85.4 even 4 289.4.a.e.1.1 4
85.33 odd 4 425.4.c.c.424.6 8
85.64 even 4 289.4.a.e.1.2 4
85.67 odd 4 425.4.c.c.424.3 8
85.84 even 2 17.4.b.a.16.4 yes 4
255.254 odd 2 153.4.d.b.118.2 4
340.339 odd 2 272.4.b.d.33.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.3 ✓ 4 5.4 even 2
17.4.b.a.16.4 yes 4 85.84 even 2
153.4.d.b.118.1 4 15.14 odd 2
153.4.d.b.118.2 4 255.254 odd 2
272.4.b.d.33.2 4 340.339 odd 2
272.4.b.d.33.3 4 20.19 odd 2
289.4.a.e.1.1 4 85.4 even 4
289.4.a.e.1.2 4 85.64 even 4
425.4.c.c.424.3 8 85.67 odd 4
425.4.c.c.424.4 8 5.2 odd 4
425.4.c.c.424.5 8 5.3 odd 4
425.4.c.c.424.6 8 85.33 odd 4
425.4.d.c.101.1 4 17.16 even 2 inner
425.4.d.c.101.2 4 1.1 even 1 trivial