Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-6,4,10,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(1\)
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 1.3
Root \(-3.22168\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.22168 q^{2} +1.15753 q^{3} -6.50750 q^{4} +5.00000 q^{5} +1.41413 q^{6} -14.6743 q^{7} -17.7235 q^{8} -25.6601 q^{9} +6.10839 q^{10} +71.3743 q^{11} -7.53262 q^{12} -28.4035 q^{13} -17.9273 q^{14} +5.78764 q^{15} +30.4076 q^{16} -31.3484 q^{18} +85.9592 q^{19} -32.5375 q^{20} -16.9860 q^{21} +87.1964 q^{22} +7.17872 q^{23} -20.5154 q^{24} +25.0000 q^{25} -34.7000 q^{26} -60.9556 q^{27} +95.4934 q^{28} +27.1610 q^{29} +7.07063 q^{30} +15.6170 q^{31} +178.936 q^{32} +82.6178 q^{33} -73.3717 q^{35} +166.983 q^{36} +67.0036 q^{37} +105.014 q^{38} -32.8779 q^{39} -88.6175 q^{40} +38.5701 q^{41} -20.7514 q^{42} -251.495 q^{43} -464.469 q^{44} -128.301 q^{45} +8.77008 q^{46} +57.9121 q^{47} +35.1977 q^{48} -127.663 q^{49} +30.5419 q^{50} +184.836 q^{52} -677.807 q^{53} -74.4680 q^{54} +356.872 q^{55} +260.081 q^{56} +99.5002 q^{57} +33.1819 q^{58} +598.645 q^{59} -37.6631 q^{60} -346.063 q^{61} +19.0789 q^{62} +376.546 q^{63} -24.6588 q^{64} -142.018 q^{65} +100.932 q^{66} -849.923 q^{67} +8.30957 q^{69} -89.6366 q^{70} +911.836 q^{71} +454.787 q^{72} -704.352 q^{73} +81.8568 q^{74} +28.9382 q^{75} -559.380 q^{76} -1047.37 q^{77} -40.1662 q^{78} -55.8414 q^{79} +152.038 q^{80} +622.266 q^{81} +47.1202 q^{82} -1235.19 q^{83} +110.536 q^{84} -307.245 q^{86} +31.4396 q^{87} -1265.00 q^{88} -72.8069 q^{89} -156.742 q^{90} +416.803 q^{91} -46.7155 q^{92} +18.0771 q^{93} +70.7499 q^{94} +429.796 q^{95} +207.124 q^{96} -972.939 q^{97} -155.964 q^{98} -1831.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} + 4 q^{3} + 10 q^{4} + 15 q^{5} - 36 q^{6} - 8 q^{7} - 66 q^{8} + 39 q^{9} - 30 q^{10} + 118 q^{11} + 212 q^{12} - 10 q^{13} + 68 q^{14} + 20 q^{15} + 242 q^{16} - 342 q^{18} - 160 q^{19}+ \cdots - 770 q^{99}+O(q^{100}) \) 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 \(-0.430711\pi\)
0.215964 + 0.976401i \(0.430711\pi\)
\(3\) 1.15753 0.222766 0.111383 − 0.993778i \(-0.464472\pi\)
0.111383 + 0.993778i \(0.464472\pi\)
\(4\) −6.50750 −0.813438
\(5\) 5.00000 0.447214
\(6\) 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
\(10\) 6.10839 0.193164
\(11\) 71.3743 1.95638 0.978189 − 0.207716i \(-0.0666030\pi\)
0.978189 + 0.207716i \(0.0666030\pi\)
\(12\) −7.53262 −0.181207
\(13\) −28.4035 −0.605979 −0.302989 − 0.952994i \(-0.597985\pi\)
−0.302989 + 0.952994i \(0.597985\pi\)
\(14\) −17.9273 −0.342234
\(15\) 5.78764 0.0996241
\(16\) 30.4076 0.475119
\(17\) 0 0
\(18\) −31.3484 −0.410494
\(19\) 85.9592 1.03792 0.518958 − 0.854800i \(-0.326320\pi\)
0.518958 + 0.854800i \(0.326320\pi\)
\(20\) −32.5375 −0.363781
\(21\) −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
\(25\) 25.0000 0.200000
\(26\) −34.7000 −0.261739
\(27\) −60.9556 −0.434478
\(28\) 95.4934 0.644520
\(29\) 27.1610 0.173919 0.0869597 − 0.996212i \(-0.472285\pi\)
0.0869597 + 0.996212i \(0.472285\pi\)
\(30\) 7.07063 0.0430305
\(31\) 15.6170 0.0904803 0.0452401 − 0.998976i \(-0.485595\pi\)
0.0452401 + 0.998976i \(0.485595\pi\)
\(32\) 178.936 0.988493
\(33\) 82.6178 0.435815
\(34\) 0 0
\(35\) −73.3717 −0.354345
\(36\) 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
\(40\) −88.6175 −0.350291
\(41\) 38.5701 0.146918 0.0734590 − 0.997298i \(-0.476596\pi\)
0.0734590 + 0.997298i \(0.476596\pi\)
\(42\) −20.7514 −0.0762383
\(43\) −251.495 −0.891920 −0.445960 − 0.895053i \(-0.647138\pi\)
−0.445960 + 0.895053i \(0.647138\pi\)
\(44\) −464.469 −1.59139
\(45\) −128.301 −0.425021
\(46\) 8.77008 0.0281104
\(47\) 57.9121 0.179731 0.0898654 − 0.995954i \(-0.471356\pi\)
0.0898654 + 0.995954i \(0.471356\pi\)
\(48\) 35.1977 0.105841
\(49\) −127.663 −0.372197
\(50\) 30.5419 0.0863856
\(51\) 0 0
\(52\) 184.836 0.492926
\(53\) −677.807 −1.75668 −0.878339 − 0.478039i \(-0.841348\pi\)
−0.878339 + 0.478039i \(0.841348\pi\)
\(54\) −74.4680 −0.187663
\(55\) 356.872 0.874919
\(56\) 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 \(0.270352\pi\)
\(60\) −37.6631 −0.0810381
\(61\) −346.063 −0.726375 −0.363187 − 0.931716i \(-0.618311\pi\)
−0.363187 + 0.931716i \(0.618311\pi\)
\(62\) 19.0789 0.0390810
\(63\) 376.546 0.753021
\(64\) −24.6588 −0.0481617
\(65\) −142.018 −0.271002
\(66\) 100.932 0.188241
\(67\) −849.923 −1.54977 −0.774885 − 0.632102i \(-0.782192\pi\)
−0.774885 + 0.632102i \(0.782192\pi\)
\(68\) 0 0
\(69\) 8.30957 0.0144979
\(70\) −89.6366 −0.153052
\(71\) 911.836 1.52415 0.762077 − 0.647486i \(-0.224180\pi\)
0.762077 + 0.647486i \(0.224180\pi\)
\(72\) 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
\(75\) 28.9382 0.0445533
\(76\) −559.380 −0.844280
\(77\) −1047.37 −1.55012
\(78\) −40.1662 −0.0583067
\(79\) −55.8414 −0.0795272 −0.0397636 − 0.999209i \(-0.512660\pi\)
−0.0397636 + 0.999209i \(0.512660\pi\)
\(80\) 152.038 0.212480
\(81\) 622.266 0.853588
\(82\) 47.1202 0.0634580
\(83\) −1235.19 −1.63350 −0.816748 − 0.576995i \(-0.804225\pi\)
−0.816748 + 0.576995i \(0.804225\pi\)
\(84\) 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 \(0.513805\pi\)
\(90\) −156.742 −0.183578
\(91\) 416.803 0.480141
\(92\) −46.7155 −0.0529395
\(93\) 18.0771 0.0201560
\(94\) 70.7499 0.0776308
\(95\) 429.796 0.464170
\(96\) 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 1445.4.a.j.1.3 3
17.16 even 2 85.4.a.e.1.3 ✓ 3
51.50 odd 2 765.4.a.l.1.1 3
68.67 odd 2 1360.4.a.s.1.2 3
85.33 odd 4 425.4.b.g.324.3 6
85.67 odd 4 425.4.b.g.324.4 6
85.84 even 2 425.4.a.h.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.3 ✓ 3 17.16 even 2
425.4.a.h.1.1 3 85.84 even 2
425.4.b.g.324.3 6 85.33 odd 4
425.4.b.g.324.4 6 85.67 odd 4
765.4.a.l.1.1 3 51.50 odd 2
1360.4.a.s.1.2 3 68.67 odd 2
1445.4.a.j.1.3 3 1.1 even 1 trivial