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(324,425)] 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("425.324"); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-20,0,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27206656.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 4x^{3} + 25x^{2} - 30x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.4
Root \(1.90022 - 1.90022i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.g.324.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.22168i q^{2} +1.15753i q^{3} +6.50750 q^{4} -1.41413 q^{6} +14.6743i q^{7} +17.7235i q^{8} +25.6601 q^{9} -71.3743 q^{11} +7.53262i q^{12} +28.4035i q^{13} -17.9273 q^{14} +30.4076 q^{16} +17.0000i q^{17} +31.3484i q^{18} -85.9592 q^{19} -16.9860 q^{21} -87.1964i q^{22} +7.17872i q^{23} -20.5154 q^{24} -34.7000 q^{26} +60.9556i q^{27} +95.4934i q^{28} +27.1610 q^{29} -15.6170 q^{31} +178.936i q^{32} -82.6178i q^{33} -20.7685 q^{34} +166.983 q^{36} -67.0036i q^{37} -105.014i q^{38} -32.8779 q^{39} -38.5701 q^{41} -20.7514i q^{42} +251.495i q^{43} -464.469 q^{44} -8.77008 q^{46} +57.9121i q^{47} +35.1977i q^{48} +127.663 q^{49} -19.6780 q^{51} +184.836i q^{52} +677.807i q^{53} -74.4680 q^{54} -260.081 q^{56} -99.5002i q^{57} +33.1819i q^{58} -598.645 q^{59} +346.063 q^{61} -19.0789i q^{62} +376.546i q^{63} +24.6588 q^{64} +100.932 q^{66} -849.923i q^{67} +110.628i q^{68} -8.30957 q^{69} -911.836 q^{71} +454.787i q^{72} -704.352i q^{73} +81.8568 q^{74} -559.380 q^{76} -1047.37i q^{77} -40.1662i q^{78} -55.8414 q^{79} +622.266 q^{81} -47.1202i q^{82} +1235.19i q^{83} -110.536 q^{84} -307.245 q^{86} +31.4396i q^{87} -1265.00i q^{88} +72.8069 q^{89} -416.803 q^{91} +46.7155i q^{92} -18.0771i q^{93} -70.7499 q^{94} -207.124 q^{96} +972.939i q^{97} +155.964i q^{98} -1831.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 20 q^{4} + 72 q^{6} - 78 q^{9} - 236 q^{11} + 136 q^{14} + 484 q^{16} + 320 q^{19} - 944 q^{21} - 1608 q^{24} - 88 q^{26} - 748 q^{29} + 140 q^{31} + 204 q^{34} + 2516 q^{36} - 496 q^{39} - 708 q^{41}+ \cdots - 1540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22168i 0.431928i 0.976401 + 0.215964i \(0.0692894\pi\)
−0.976401 + 0.215964i \(0.930711\pi\)
\(3\) 1.15753i 0.222766i 0.993778 + 0.111383i \(0.0355281\pi\)
−0.993778 + 0.111383i \(0.964472\pi\)
\(4\) 6.50750 0.813438
\(5\) 0 0
\(6\) −1.41413 −0.0962191
\(7\) 14.6743i 0.792340i 0.918177 + 0.396170i \(0.129661\pi\)
−0.918177 + 0.396170i \(0.870339\pi\)
\(8\) 17.7235i 0.783275i
\(9\) 25.6601 0.950375
\(10\) 0 0
\(11\) −71.3743 −1.95638 −0.978189 − 0.207716i \(-0.933397\pi\)
−0.978189 + 0.207716i \(0.933397\pi\)
\(12\) 7.53262i 0.181207i
\(13\) 28.4035i 0.605979i 0.952994 + 0.302989i \(0.0979847\pi\)
−0.952994 + 0.302989i \(0.902015\pi\)
\(14\) −17.9273 −0.342234
\(15\) 0 0
\(16\) 30.4076 0.475119
\(17\) 17.0000i 0.242536i
\(18\) 31.3484i 0.410494i
\(19\) −85.9592 −1.03792 −0.518958 − 0.854800i \(-0.673680\pi\)
−0.518958 + 0.854800i \(0.673680\pi\)
\(20\) 0 0
\(21\) −16.9860 −0.176507
\(22\) − 87.1964i − 0.845015i
\(23\) 7.17872i 0.0650811i 0.999470 + 0.0325406i \(0.0103598\pi\)
−0.999470 + 0.0325406i \(0.989640\pi\)
\(24\) −20.5154 −0.174487
\(25\) 0 0
\(26\) −34.7000 −0.261739
\(27\) 60.9556i 0.434478i
\(28\) 95.4934i 0.644520i
\(29\) 27.1610 0.173919 0.0869597 − 0.996212i \(-0.472285\pi\)
0.0869597 + 0.996212i \(0.472285\pi\)
\(30\) 0 0
\(31\) −15.6170 −0.0904803 −0.0452401 − 0.998976i \(-0.514405\pi\)
−0.0452401 + 0.998976i \(0.514405\pi\)
\(32\) 178.936i 0.988493i
\(33\) − 82.6178i − 0.435815i
\(34\) −20.7685 −0.104758
\(35\) 0 0
\(36\) 166.983 0.773071
\(37\) − 67.0036i − 0.297712i −0.988859 − 0.148856i \(-0.952441\pi\)
0.988859 − 0.148856i \(-0.0475590\pi\)
\(38\) − 105.014i − 0.448305i
\(39\) −32.8779 −0.134992
\(40\) 0 0
\(41\) −38.5701 −0.146918 −0.0734590 − 0.997298i \(-0.523404\pi\)
−0.0734590 + 0.997298i \(0.523404\pi\)
\(42\) − 20.7514i − 0.0762383i
\(43\) 251.495i 0.891920i 0.895053 + 0.445960i \(0.147138\pi\)
−0.895053 + 0.445960i \(0.852862\pi\)
\(44\) −464.469 −1.59139
\(45\) 0 0
\(46\) −8.77008 −0.0281104
\(47\) 57.9121i 0.179731i 0.995954 + 0.0898654i \(0.0286437\pi\)
−0.995954 + 0.0898654i \(0.971356\pi\)
\(48\) 35.1977i 0.105841i
\(49\) 127.663 0.372197
\(50\) 0 0
\(51\) −19.6780 −0.0540288
\(52\) 184.836i 0.492926i
\(53\) 677.807i 1.75668i 0.478039 + 0.878339i \(0.341348\pi\)
−0.478039 + 0.878339i \(0.658652\pi\)
\(54\) −74.4680 −0.187663
\(55\) 0 0
\(56\) −260.081 −0.620620
\(57\) − 99.5002i − 0.231213i
\(58\) 33.1819i 0.0751207i
\(59\) −598.645 −1.32097 −0.660483 − 0.750841i \(-0.729648\pi\)
−0.660483 + 0.750841i \(0.729648\pi\)
\(60\) 0 0
\(61\) 346.063 0.726375 0.363187 − 0.931716i \(-0.381689\pi\)
0.363187 + 0.931716i \(0.381689\pi\)
\(62\) − 19.0789i − 0.0390810i
\(63\) 376.546i 0.753021i
\(64\) 24.6588 0.0481617
\(65\) 0 0
\(66\) 100.932 0.188241
\(67\) − 849.923i − 1.54977i −0.632102 − 0.774885i \(-0.717808\pi\)
0.632102 − 0.774885i \(-0.282192\pi\)
\(68\) 110.628i 0.197288i
\(69\) −8.30957 −0.0144979
\(70\) 0 0
\(71\) −911.836 −1.52415 −0.762077 − 0.647486i \(-0.775820\pi\)
−0.762077 + 0.647486i \(0.775820\pi\)
\(72\) 454.787i 0.744405i
\(73\) − 704.352i − 1.12929i −0.825334 − 0.564645i \(-0.809013\pi\)
0.825334 − 0.564645i \(-0.190987\pi\)
\(74\) 81.8568 0.128590
\(75\) 0 0
\(76\) −559.380 −0.844280
\(77\) − 1047.37i − 1.55012i
\(78\) − 40.1662i − 0.0583067i
\(79\) −55.8414 −0.0795272 −0.0397636 − 0.999209i \(-0.512660\pi\)
−0.0397636 + 0.999209i \(0.512660\pi\)
\(80\) 0 0
\(81\) 622.266 0.853588
\(82\) − 47.1202i − 0.0634580i
\(83\) 1235.19i 1.63350i 0.576995 + 0.816748i \(0.304225\pi\)
−0.576995 + 0.816748i \(0.695775\pi\)
\(84\) −110.536 −0.143577
\(85\) 0 0
\(86\) −307.245 −0.385246
\(87\) 31.4396i 0.0387434i
\(88\) − 1265.00i − 1.53238i
\(89\) 72.8069 0.0867136 0.0433568 − 0.999060i \(-0.486195\pi\)
0.0433568 + 0.999060i \(0.486195\pi\)
\(90\) 0 0
\(91\) −416.803 −0.480141
\(92\) 46.7155i 0.0529395i
\(93\) − 18.0771i − 0.0201560i
\(94\) −70.7499 −0.0776308
\(95\) 0 0
\(96\) −207.124 −0.220203
\(97\) 972.939i 1.01842i 0.860642 + 0.509211i \(0.170063\pi\)
−0.860642 + 0.509211i \(0.829937\pi\)
\(98\) 155.964i 0.160762i
\(99\) −1831.47 −1.85929
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.b.g.324.4 6
5.2 odd 4 425.4.a.h.1.1 3
5.3 odd 4 85.4.a.e.1.3 ✓ 3
5.4 even 2 inner 425.4.b.g.324.3 6
15.8 even 4 765.4.a.l.1.1 3
20.3 even 4 1360.4.a.s.1.2 3
85.33 odd 4 1445.4.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.3 ✓ 3 5.3 odd 4
425.4.a.h.1.1 3 5.2 odd 4
425.4.b.g.324.3 6 5.4 even 2 inner
425.4.b.g.324.4 6 1.1 even 1 trivial
765.4.a.l.1.1 3 15.8 even 4
1360.4.a.s.1.2 3 20.3 even 4
1445.4.a.j.1.3 3 85.33 odd 4