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(1,425)] 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("425.1"); 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, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,6,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1304.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 11x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.22168\) of defining polynomial
Character \(\chi\) \(=\) 425.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.22168 q^{2} +1.15753 q^{3} -6.50750 q^{4} -1.41413 q^{6} -14.6743 q^{7} +17.7235 q^{8} -25.6601 q^{9} -71.3743 q^{11} -7.53262 q^{12} +28.4035 q^{13} +17.9273 q^{14} +30.4076 q^{16} -17.0000 q^{17} +31.3484 q^{18} +85.9592 q^{19} -16.9860 q^{21} +87.1964 q^{22} +7.17872 q^{23} +20.5154 q^{24} -34.7000 q^{26} -60.9556 q^{27} +95.4934 q^{28} -27.1610 q^{29} -15.6170 q^{31} -178.936 q^{32} -82.6178 q^{33} +20.7685 q^{34} +166.983 q^{36} +67.0036 q^{37} -105.014 q^{38} +32.8779 q^{39} -38.5701 q^{41} +20.7514 q^{42} +251.495 q^{43} +464.469 q^{44} -8.77008 q^{46} -57.9121 q^{47} +35.1977 q^{48} -127.663 q^{49} -19.6780 q^{51} -184.836 q^{52} +677.807 q^{53} +74.4680 q^{54} -260.081 q^{56} +99.5002 q^{57} +33.1819 q^{58} +598.645 q^{59} +346.063 q^{61} +19.0789 q^{62} +376.546 q^{63} -24.6588 q^{64} +100.932 q^{66} +849.923 q^{67} +110.628 q^{68} +8.30957 q^{69} -911.836 q^{71} -454.787 q^{72} -704.352 q^{73} -81.8568 q^{74} -559.380 q^{76} +1047.37 q^{77} -40.1662 q^{78} +55.8414 q^{79} +622.266 q^{81} +47.1202 q^{82} +1235.19 q^{83} +110.536 q^{84} -307.245 q^{86} -31.4396 q^{87} -1265.00 q^{88} -72.8069 q^{89} -416.803 q^{91} -46.7155 q^{92} -18.0771 q^{93} +70.7499 q^{94} -207.124 q^{96} -972.939 q^{97} +155.964 q^{98} +1831.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} + 4 q^{3} + 10 q^{4} + 36 q^{6} - 8 q^{7} + 66 q^{8} + 39 q^{9} - 118 q^{11} + 212 q^{12} + 10 q^{13} - 68 q^{14} + 242 q^{16} - 51 q^{17} + 342 q^{18} - 160 q^{19} - 472 q^{21} - 44 q^{22}+ \cdots + 770 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22168 −0.431928 −0.215964 − 0.976401i \(-0.569289\pi\)
−0.215964 + 0.976401i \(0.569289\pi\)
\(3\) 1.15753 0.222766 0.111383 − 0.993778i \(-0.464472\pi\)
0.111383 + 0.993778i \(0.464472\pi\)
\(4\) −6.50750 −0.813438
\(5\) 0 0
\(6\) −1.41413 −0.0962191
\(7\) −14.6743 −0.792340 −0.396170 − 0.918177i \(-0.629661\pi\)
−0.396170 + 0.918177i \(0.629661\pi\)
\(8\) 17.7235 0.783275
\(9\) −25.6601 −0.950375
\(10\) 0 0
\(11\) −71.3743 −1.95638 −0.978189 − 0.207716i \(-0.933397\pi\)
−0.978189 + 0.207716i \(0.933397\pi\)
\(12\) −7.53262 −0.181207
\(13\) 28.4035 0.605979 0.302989 − 0.952994i \(-0.402015\pi\)
0.302989 + 0.952994i \(0.402015\pi\)
\(14\) 17.9273 0.342234
\(15\) 0 0
\(16\) 30.4076 0.475119
\(17\) −17.0000 −0.242536
\(18\) 31.3484 0.410494
\(19\) 85.9592 1.03792 0.518958 − 0.854800i \(-0.326320\pi\)
0.518958 + 0.854800i \(0.326320\pi\)
\(20\) 0 0
\(21\) −16.9860 −0.176507
\(22\) 87.1964 0.845015
\(23\) 7.17872 0.0650811 0.0325406 − 0.999470i \(-0.489640\pi\)
0.0325406 + 0.999470i \(0.489640\pi\)
\(24\) 20.5154 0.174487
\(25\) 0 0
\(26\) −34.7000 −0.261739
\(27\) −60.9556 −0.434478
\(28\) 95.4934 0.644520
\(29\) −27.1610 −0.173919 −0.0869597 − 0.996212i \(-0.527715\pi\)
−0.0869597 + 0.996212i \(0.527715\pi\)
\(30\) 0 0
\(31\) −15.6170 −0.0904803 −0.0452401 − 0.998976i \(-0.514405\pi\)
−0.0452401 + 0.998976i \(0.514405\pi\)
\(32\) −178.936 −0.988493
\(33\) −82.6178 −0.435815
\(34\) 20.7685 0.104758
\(35\) 0 0
\(36\) 166.983 0.773071
\(37\) 67.0036 0.297712 0.148856 − 0.988859i \(-0.452441\pi\)
0.148856 + 0.988859i \(0.452441\pi\)
\(38\) −105.014 −0.448305
\(39\) 32.8779 0.134992
\(40\) 0 0
\(41\) −38.5701 −0.146918 −0.0734590 − 0.997298i \(-0.523404\pi\)
−0.0734590 + 0.997298i \(0.523404\pi\)
\(42\) 20.7514 0.0762383
\(43\) 251.495 0.891920 0.445960 − 0.895053i \(-0.352862\pi\)
0.445960 + 0.895053i \(0.352862\pi\)
\(44\) 464.469 1.59139
\(45\) 0 0
\(46\) −8.77008 −0.0281104
\(47\) −57.9121 −0.179731 −0.0898654 − 0.995954i \(-0.528644\pi\)
−0.0898654 + 0.995954i \(0.528644\pi\)
\(48\) 35.1977 0.105841
\(49\) −127.663 −0.372197
\(50\) 0 0
\(51\) −19.6780 −0.0540288
\(52\) −184.836 −0.492926
\(53\) 677.807 1.75668 0.878339 − 0.478039i \(-0.158652\pi\)
0.878339 + 0.478039i \(0.158652\pi\)
\(54\) 74.4680 0.187663
\(55\) 0 0
\(56\) −260.081 −0.620620
\(57\) 99.5002 0.231213
\(58\) 33.1819 0.0751207
\(59\) 598.645 1.32097 0.660483 − 0.750841i \(-0.270352\pi\)
0.660483 + 0.750841i \(0.270352\pi\)
\(60\) 0 0
\(61\) 346.063 0.726375 0.363187 − 0.931716i \(-0.381689\pi\)
0.363187 + 0.931716i \(0.381689\pi\)
\(62\) 19.0789 0.0390810
\(63\) 376.546 0.753021
\(64\) −24.6588 −0.0481617
\(65\) 0 0
\(66\) 100.932 0.188241
\(67\) 849.923 1.54977 0.774885 − 0.632102i \(-0.217808\pi\)
0.774885 + 0.632102i \(0.217808\pi\)
\(68\) 110.628 0.197288
\(69\) 8.30957 0.0144979
\(70\) 0 0
\(71\) −911.836 −1.52415 −0.762077 − 0.647486i \(-0.775820\pi\)
−0.762077 + 0.647486i \(0.775820\pi\)
\(72\) −454.787 −0.744405
\(73\) −704.352 −1.12929 −0.564645 − 0.825334i \(-0.690987\pi\)
−0.564645 + 0.825334i \(0.690987\pi\)
\(74\) −81.8568 −0.128590
\(75\) 0 0
\(76\) −559.380 −0.844280
\(77\) 1047.37 1.55012
\(78\) −40.1662 −0.0583067
\(79\) 55.8414 0.0795272 0.0397636 − 0.999209i \(-0.487340\pi\)
0.0397636 + 0.999209i \(0.487340\pi\)
\(80\) 0 0
\(81\) 622.266 0.853588
\(82\) 47.1202 0.0634580
\(83\) 1235.19 1.63350 0.816748 − 0.576995i \(-0.195775\pi\)
0.816748 + 0.576995i \(0.195775\pi\)
\(84\) 110.536 0.143577
\(85\) 0 0
\(86\) −307.245 −0.385246
\(87\) −31.4396 −0.0387434
\(88\) −1265.00 −1.53238
\(89\) −72.8069 −0.0867136 −0.0433568 − 0.999060i \(-0.513805\pi\)
−0.0433568 + 0.999060i \(0.513805\pi\)
\(90\) 0 0
\(91\) −416.803 −0.480141
\(92\) −46.7155 −0.0529395
\(93\) −18.0771 −0.0201560
\(94\) 70.7499 0.0776308
\(95\) 0 0
\(96\) −207.124 −0.220203
\(97\) −972.939 −1.01842 −0.509211 − 0.860642i \(-0.670063\pi\)
−0.509211 + 0.860642i \(0.670063\pi\)
\(98\) 155.964 0.160762
\(99\) 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.a.h.1.1 3
5.2 odd 4 425.4.b.g.324.3 6
5.3 odd 4 425.4.b.g.324.4 6
5.4 even 2 85.4.a.e.1.3 ✓ 3
15.14 odd 2 765.4.a.l.1.1 3
20.19 odd 2 1360.4.a.s.1.2 3
85.84 even 2 1445.4.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.3 ✓ 3 5.4 even 2
425.4.a.h.1.1 3 1.1 even 1 trivial
425.4.b.g.324.3 6 5.2 odd 4
425.4.b.g.324.4 6 5.3 odd 4
765.4.a.l.1.1 3 15.14 odd 2
1360.4.a.s.1.2 3 20.19 odd 2
1445.4.a.j.1.3 3 85.84 even 2