Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [765,4,Mod(1,765)] 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("765.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(765, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 765 = 3^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 765.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,6,0,10,15,0,8,66,0,30,118] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(45.1364611544\)
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 \(\chi\) \(=\) 765.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.22168 q^{2} -6.50750 q^{4} +5.00000 q^{5} +14.6743 q^{7} +17.7235 q^{8} -6.10839 q^{10} +71.3743 q^{11} -28.4035 q^{13} -17.9273 q^{14} +30.4076 q^{16} -17.0000 q^{17} +85.9592 q^{19} -32.5375 q^{20} -87.1964 q^{22} +7.17872 q^{23} +25.0000 q^{25} +34.7000 q^{26} -95.4934 q^{28} +27.1610 q^{29} -15.6170 q^{31} -178.936 q^{32} +20.7685 q^{34} +73.3717 q^{35} -67.0036 q^{37} -105.014 q^{38} +88.6175 q^{40} +38.5701 q^{41} -251.495 q^{43} -464.469 q^{44} -8.77008 q^{46} -57.9121 q^{47} -127.663 q^{49} -30.5419 q^{50} +184.836 q^{52} +677.807 q^{53} +356.872 q^{55} +260.081 q^{56} -33.1819 q^{58} -598.645 q^{59} +346.063 q^{61} +19.0789 q^{62} -24.6588 q^{64} -142.018 q^{65} -849.923 q^{67} +110.628 q^{68} -89.6366 q^{70} +911.836 q^{71} +704.352 q^{73} +81.8568 q^{74} -559.380 q^{76} +1047.37 q^{77} +55.8414 q^{79} +152.038 q^{80} -47.1202 q^{82} +1235.19 q^{83} -85.0000 q^{85} +307.245 q^{86} +1265.00 q^{88} +72.8069 q^{89} -416.803 q^{91} -46.7155 q^{92} +70.7499 q^{94} +429.796 q^{95} +972.939 q^{97} +155.964 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} + 10 q^{4} + 15 q^{5} + 8 q^{7} + 66 q^{8} + 30 q^{10} + 118 q^{11} - 10 q^{13} + 68 q^{14} + 242 q^{16} - 51 q^{17} - 160 q^{19} + 50 q^{20} + 44 q^{22} + 92 q^{23} + 75 q^{25} + 44 q^{26}+ \cdots + 3822 q^{98}+O(q^{100}) \) 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 \(0.569289\pi\)
\(3\) 0 0
\(4\) −6.50750 −0.813438
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 14.6743 0.792340 0.396170 − 0.918177i \(-0.370339\pi\)
0.396170 + 0.918177i \(0.370339\pi\)
\(8\) 17.7235 0.783275
\(9\) 0 0
\(10\) −6.10839 −0.193164
\(11\) 71.3743 1.95638 0.978189 − 0.207716i \(-0.0666030\pi\)
0.978189 + 0.207716i \(0.0666030\pi\)
\(12\) 0 0
\(13\) −28.4035 −0.605979 −0.302989 − 0.952994i \(-0.597985\pi\)
−0.302989 + 0.952994i \(0.597985\pi\)
\(14\) −17.9273 −0.342234
\(15\) 0 0
\(16\) 30.4076 0.475119
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) 85.9592 1.03792 0.518958 − 0.854800i \(-0.326320\pi\)
0.518958 + 0.854800i \(0.326320\pi\)
\(20\) −32.5375 −0.363781
\(21\) 0 0
\(22\) −87.1964 −0.845015
\(23\) 7.17872 0.0650811 0.0325406 − 0.999470i \(-0.489640\pi\)
0.0325406 + 0.999470i \(0.489640\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 34.7000 0.261739
\(27\) 0 0
\(28\) −95.4934 −0.644520
\(29\) 27.1610 0.173919 0.0869597 − 0.996212i \(-0.472285\pi\)
0.0869597 + 0.996212i \(0.472285\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 20.7685 0.104758
\(35\) 73.3717 0.354345
\(36\) 0 0
\(37\) −67.0036 −0.297712 −0.148856 − 0.988859i \(-0.547559\pi\)
−0.148856 + 0.988859i \(0.547559\pi\)
\(38\) −105.014 −0.448305
\(39\) 0 0
\(40\) 88.6175 0.350291
\(41\) 38.5701 0.146918 0.0734590 − 0.997298i \(-0.476596\pi\)
0.0734590 + 0.997298i \(0.476596\pi\)
\(42\) 0 0
\(43\) −251.495 −0.891920 −0.445960 − 0.895053i \(-0.647138\pi\)
−0.445960 + 0.895053i \(0.647138\pi\)
\(44\) −464.469 −1.59139
\(45\) 0 0
\(46\) −8.77008 −0.0281104
\(47\) −57.9121 −0.179731 −0.0898654 − 0.995954i \(-0.528644\pi\)
−0.0898654 + 0.995954i \(0.528644\pi\)
\(48\) 0 0
\(49\) −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 \(-0.158652\pi\)
0.878339 + 0.478039i \(0.158652\pi\)
\(54\) 0 0
\(55\) 356.872 0.874919
\(56\) 260.081 0.620620
\(57\) 0 0
\(58\) −33.1819 −0.0751207
\(59\) −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 \(0.381689\pi\)
\(62\) 19.0789 0.0390810
\(63\) 0 0
\(64\) −24.6588 −0.0481617
\(65\) −142.018 −0.271002
\(66\) 0 0
\(67\) −849.923 −1.54977 −0.774885 − 0.632102i \(-0.782192\pi\)
−0.774885 + 0.632102i \(0.782192\pi\)
\(68\) 110.628 0.197288
\(69\) 0 0
\(70\) −89.6366 −0.153052
\(71\) 911.836 1.52415 0.762077 − 0.647486i \(-0.224180\pi\)
0.762077 + 0.647486i \(0.224180\pi\)
\(72\) 0 0
\(73\) 704.352 1.12929 0.564645 − 0.825334i \(-0.309013\pi\)
0.564645 + 0.825334i \(0.309013\pi\)
\(74\) 81.8568 0.128590
\(75\) 0 0
\(76\) −559.380 −0.844280
\(77\) 1047.37 1.55012
\(78\) 0 0
\(79\) 55.8414 0.0795272 0.0397636 − 0.999209i \(-0.487340\pi\)
0.0397636 + 0.999209i \(0.487340\pi\)
\(80\) 152.038 0.212480
\(81\) 0 0
\(82\) −47.1202 −0.0634580
\(83\) 1235.19 1.63350 0.816748 − 0.576995i \(-0.195775\pi\)
0.816748 + 0.576995i \(0.195775\pi\)
\(84\) 0 0
\(85\) −85.0000 −0.108465
\(86\) 307.245 0.385246
\(87\) 0 0
\(88\) 1265.00 1.53238
\(89\) 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
\(92\) −46.7155 −0.0529395
\(93\) 0 0
\(94\) 70.7499 0.0776308
\(95\) 429.796 0.464170
\(96\) 0 0
\(97\) 972.939 1.01842 0.509211 − 0.860642i \(-0.329937\pi\)
0.509211 + 0.860642i \(0.329937\pi\)
\(98\) 155.964 0.160762
\(99\) 0 0
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 765.4.a.l.1.1 3
3.2 odd 2 85.4.a.e.1.3 ✓ 3
12.11 even 2 1360.4.a.s.1.2 3
15.2 even 4 425.4.b.g.324.4 6
15.8 even 4 425.4.b.g.324.3 6
15.14 odd 2 425.4.a.h.1.1 3
51.50 odd 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 3.2 odd 2
425.4.a.h.1.1 3 15.14 odd 2
425.4.b.g.324.3 6 15.8 even 4
425.4.b.g.324.4 6 15.2 even 4
765.4.a.l.1.1 3 1.1 even 1 trivial
1360.4.a.s.1.2 3 12.11 even 2
1445.4.a.j.1.3 3 51.50 odd 2