Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [465,2,Mod(94,465)] 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("465.94"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.1016580161536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 94.7
Root \(-0.815403i\) of defining polynomial
Character \(\chi\) \(=\) 465.94
Dual form 465.2.c.a.94.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.697747i q^{2} -1.00000i q^{3} +1.51315 q^{4} +(0.815403 - 2.08209i) q^{5} +0.697747 q^{6} -0.302253i q^{7} +2.45129i q^{8} -1.00000 q^{9} +(1.45277 + 0.568945i) q^{10} +2.71947 q^{11} -1.51315i q^{12} +3.96444i q^{13} +0.210896 q^{14} +(-2.08209 - 0.815403i) q^{15} +1.31592 q^{16} -6.05458i q^{17} -0.697747i q^{18} -0.452774 q^{19} +(1.23383 - 3.15052i) q^{20} -0.302253 q^{21} +1.89750i q^{22} -4.37050i q^{23} +2.45129 q^{24} +(-3.67024 - 3.39549i) q^{25} -2.76617 q^{26} +1.00000i q^{27} -0.457355i q^{28} +3.48564 q^{29} +(0.568945 - 1.45277i) q^{30} -1.00000 q^{31} +5.82076i q^{32} -2.71947i q^{33} +4.22456 q^{34} +(-0.629320 - 0.246458i) q^{35} -1.51315 q^{36} +1.56089i q^{37} -0.315922i q^{38} +3.96444 q^{39} +(5.10381 + 1.99879i) q^{40} -9.00441 q^{41} -0.210896i q^{42} +8.93693i q^{43} +4.11496 q^{44} +(-0.815403 + 2.08209i) q^{45} +3.04950 q^{46} -6.64299i q^{47} -1.31592i q^{48} +6.90864 q^{49} +(2.36919 - 2.56089i) q^{50} -6.05458 q^{51} +5.99879i q^{52} +8.49423i q^{53} -0.697747 q^{54} +(2.21746 - 5.66218i) q^{55} +0.740910 q^{56} +0.452774i q^{57} +2.43209i q^{58} +3.25033 q^{59} +(-3.15052 - 1.23383i) q^{60} -5.35535 q^{61} -0.697747i q^{62} +0.302253i q^{63} -1.42957 q^{64} +(8.25433 + 3.23262i) q^{65} +1.89750 q^{66} +4.26548i q^{67} -9.16149i q^{68} -4.37050 q^{69} +(0.171966 - 0.439106i) q^{70} +8.83190 q^{71} -2.45129i q^{72} +10.4968i q^{73} -1.08911 q^{74} +(-3.39549 + 3.67024i) q^{75} -0.685115 q^{76} -0.821968i q^{77} +2.76617i q^{78} -14.6348 q^{79} +(1.07301 - 2.73987i) q^{80} +1.00000 q^{81} -6.28279i q^{82} +10.2063i q^{83} -0.457355 q^{84} +(-12.6062 - 4.93693i) q^{85} -6.23571 q^{86} -3.48564i q^{87} +6.66619i q^{88} -15.6279 q^{89} +(-1.45277 - 0.568945i) q^{90} +1.19827 q^{91} -6.61323i q^{92} +1.00000i q^{93} +4.63512 q^{94} +(-0.369194 + 0.942719i) q^{95} +5.82076 q^{96} +8.85267i q^{97} +4.82048i q^{98} -2.71947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6 q^{4} - 6 q^{6} - 10 q^{9} + 2 q^{10} - 32 q^{14} + 2 q^{15} + 6 q^{16} + 8 q^{19} + 28 q^{20} - 16 q^{21} + 18 q^{24} - 2 q^{25} - 12 q^{26} - 8 q^{29} + 4 q^{30} - 10 q^{31} - 12 q^{34} + 4 q^{35}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(406\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.697747i 0.493381i 0.969094 + 0.246691i \(0.0793432\pi\)
−0.969094 + 0.246691i \(0.920657\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.51315 0.756575
\(5\) 0.815403 − 2.08209i 0.364659 − 0.931141i
\(6\) 0.697747 0.284854
\(7\) − 0.302253i − 0.114241i −0.998367 − 0.0571205i \(-0.981808\pi\)
0.998367 − 0.0571205i \(-0.0181919\pi\)
\(8\) 2.45129i 0.866661i
\(9\) −1.00000 −0.333333
\(10\) 1.45277 + 0.568945i 0.459408 + 0.179916i
\(11\) 2.71947 0.819950 0.409975 − 0.912097i \(-0.365537\pi\)
0.409975 + 0.912097i \(0.365537\pi\)
\(12\) − 1.51315i − 0.436809i
\(13\) 3.96444i 1.09954i 0.835317 + 0.549769i \(0.185284\pi\)
−0.835317 + 0.549769i \(0.814716\pi\)
\(14\) 0.210896 0.0563644
\(15\) −2.08209 − 0.815403i −0.537594 − 0.210536i
\(16\) 1.31592 0.328980
\(17\) − 6.05458i − 1.46845i −0.678905 − 0.734226i \(-0.737545\pi\)
0.678905 − 0.734226i \(-0.262455\pi\)
\(18\) − 0.697747i − 0.164460i
\(19\) −0.452774 −0.103874 −0.0519368 − 0.998650i \(-0.516539\pi\)
−0.0519368 + 0.998650i \(0.516539\pi\)
\(20\) 1.23383 − 3.15052i 0.275892 − 0.704478i
\(21\) −0.302253 −0.0659571
\(22\) 1.89750i 0.404548i
\(23\) − 4.37050i − 0.911313i −0.890156 − 0.455657i \(-0.849404\pi\)
0.890156 − 0.455657i \(-0.150596\pi\)
\(24\) 2.45129 0.500367
\(25\) −3.67024 − 3.39549i −0.734047 − 0.679099i
\(26\) −2.76617 −0.542491
\(27\) 1.00000i 0.192450i
\(28\) − 0.457355i − 0.0864319i
\(29\) 3.48564 0.647267 0.323633 − 0.946183i \(-0.395096\pi\)
0.323633 + 0.946183i \(0.395096\pi\)
\(30\) 0.568945 − 1.45277i 0.103875 − 0.265239i
\(31\) −1.00000 −0.179605
\(32\) 5.82076i 1.02897i
\(33\) − 2.71947i − 0.473398i
\(34\) 4.22456 0.724507
\(35\) −0.629320 − 0.246458i −0.106375 − 0.0416591i
\(36\) −1.51315 −0.252192
\(37\) 1.56089i 0.256609i 0.991735 + 0.128305i \(0.0409535\pi\)
−0.991735 + 0.128305i \(0.959046\pi\)
\(38\) − 0.315922i − 0.0512493i
\(39\) 3.96444 0.634818
\(40\) 5.10381 + 1.99879i 0.806984 + 0.316036i
\(41\) −9.00441 −1.40625 −0.703126 − 0.711065i \(-0.748213\pi\)
−0.703126 + 0.711065i \(0.748213\pi\)
\(42\) − 0.210896i − 0.0325420i
\(43\) 8.93693i 1.36287i 0.731879 + 0.681434i \(0.238643\pi\)
−0.731879 + 0.681434i \(0.761357\pi\)
\(44\) 4.11496 0.620353
\(45\) −0.815403 + 2.08209i −0.121553 + 0.310380i
\(46\) 3.04950 0.449625
\(47\) − 6.64299i − 0.968979i −0.874797 − 0.484490i \(-0.839005\pi\)
0.874797 − 0.484490i \(-0.160995\pi\)
\(48\) − 1.31592i − 0.189937i
\(49\) 6.90864 0.986949
\(50\) 2.36919 − 2.56089i 0.335055 − 0.362165i
\(51\) −6.05458 −0.847811
\(52\) 5.99879i 0.831882i
\(53\) 8.49423i 1.16677i 0.812195 + 0.583386i \(0.198272\pi\)
−0.812195 + 0.583386i \(0.801728\pi\)
\(54\) −0.697747 −0.0949513
\(55\) 2.21746 − 5.66218i 0.299002 − 0.763489i
\(56\) 0.740910 0.0990083
\(57\) 0.452774i 0.0599714i
\(58\) 2.43209i 0.319349i
\(59\) 3.25033 0.423156 0.211578 − 0.977361i \(-0.432140\pi\)
0.211578 + 0.977361i \(0.432140\pi\)
\(60\) −3.15052 − 1.23383i −0.406730 − 0.159286i
\(61\) −5.35535 −0.685682 −0.342841 − 0.939393i \(-0.611389\pi\)
−0.342841 + 0.939393i \(0.611389\pi\)
\(62\) − 0.697747i − 0.0886139i
\(63\) 0.302253i 0.0380804i
\(64\) −1.42957 −0.178696
\(65\) 8.25433 + 3.23262i 1.02382 + 0.400957i
\(66\) 1.89750 0.233566
\(67\) 4.26548i 0.521111i 0.965459 + 0.260556i \(0.0839057\pi\)
−0.965459 + 0.260556i \(0.916094\pi\)
\(68\) − 9.16149i − 1.11099i
\(69\) −4.37050 −0.526147
\(70\) 0.171966 − 0.439106i 0.0205538 − 0.0524832i
\(71\) 8.83190 1.04815 0.524077 − 0.851671i \(-0.324410\pi\)
0.524077 + 0.851671i \(0.324410\pi\)
\(72\) − 2.45129i − 0.288887i
\(73\) 10.4968i 1.22855i 0.789091 + 0.614276i \(0.210552\pi\)
−0.789091 + 0.614276i \(0.789448\pi\)
\(74\) −1.08911 −0.126606
\(75\) −3.39549 + 3.67024i −0.392078 + 0.423802i
\(76\) −0.685115 −0.0785881
\(77\) − 0.821968i − 0.0936719i
\(78\) 2.76617i 0.313207i
\(79\) −14.6348 −1.64654 −0.823270 − 0.567650i \(-0.807853\pi\)
−0.823270 + 0.567650i \(0.807853\pi\)
\(80\) 1.07301 − 2.73987i 0.119966 − 0.306327i
\(81\) 1.00000 0.111111
\(82\) − 6.28279i − 0.693818i
\(83\) 10.2063i 1.12029i 0.828395 + 0.560144i \(0.189254\pi\)
−0.828395 + 0.560144i \(0.810746\pi\)
\(84\) −0.457355 −0.0499015
\(85\) −12.6062 − 4.93693i −1.36734 − 0.535485i
\(86\) −6.23571 −0.672414
\(87\) − 3.48564i − 0.373700i
\(88\) 6.66619i 0.710619i
\(89\) −15.6279 −1.65656 −0.828279 − 0.560316i \(-0.810680\pi\)
−0.828279 + 0.560316i \(0.810680\pi\)
\(90\) −1.45277 − 0.568945i −0.153136 − 0.0599720i
\(91\) 1.19827 0.125612
\(92\) − 6.61323i − 0.689477i
\(93\) 1.00000i 0.103695i
\(94\) 4.63512 0.478076
\(95\) −0.369194 + 0.942719i −0.0378785 + 0.0967209i
\(96\) 5.82076 0.594078
\(97\) 8.85267i 0.898853i 0.893317 + 0.449426i \(0.148372\pi\)
−0.893317 + 0.449426i \(0.851628\pi\)
\(98\) 4.82048i 0.486942i
\(99\) −2.71947 −0.273317
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 465.2.c.a.94.7 yes 10
3.2 odd 2 1395.2.c.f.559.4 10
5.2 odd 4 2325.2.a.x.1.2 5
5.3 odd 4 2325.2.a.w.1.4 5
5.4 even 2 inner 465.2.c.a.94.4 ✓ 10
15.2 even 4 6975.2.a.bs.1.4 5
15.8 even 4 6975.2.a.bv.1.2 5
15.14 odd 2 1395.2.c.f.559.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.c.a.94.4 ✓ 10 5.4 even 2 inner
465.2.c.a.94.7 yes 10 1.1 even 1 trivial
1395.2.c.f.559.4 10 3.2 odd 2
1395.2.c.f.559.7 10 15.14 odd 2
2325.2.a.w.1.4 5 5.3 odd 4
2325.2.a.x.1.2 5 5.2 odd 4
6975.2.a.bs.1.4 5 15.2 even 4
6975.2.a.bv.1.2 5 15.8 even 4