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 = [5,-2,0,34,-25,0,10,-30,0,10,-126] 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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 20x^{3} + 38x^{2} + 69x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(4.05155\) 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)
 
\(f(q)\) \(=\) \(q-0.290753 q^{2} -7.91546 q^{4} -5.00000 q^{5} +22.4661 q^{7} +4.62746 q^{8} +1.45376 q^{10} -39.6109 q^{11} -6.70527 q^{13} -6.53208 q^{14} +61.9783 q^{16} -17.0000 q^{17} +42.9490 q^{19} +39.5773 q^{20} +11.5170 q^{22} +115.726 q^{23} +25.0000 q^{25} +1.94958 q^{26} -177.830 q^{28} -184.009 q^{29} +201.848 q^{31} -55.0401 q^{32} +4.94280 q^{34} -112.330 q^{35} -189.885 q^{37} -12.4876 q^{38} -23.1373 q^{40} -297.987 q^{41} +428.676 q^{43} +313.539 q^{44} -33.6476 q^{46} +311.134 q^{47} +161.725 q^{49} -7.26882 q^{50} +53.0753 q^{52} +307.329 q^{53} +198.055 q^{55} +103.961 q^{56} +53.5011 q^{58} -704.985 q^{59} -929.140 q^{61} -58.6879 q^{62} -479.823 q^{64} +33.5263 q^{65} +587.978 q^{67} +134.563 q^{68} +32.6604 q^{70} -507.339 q^{71} -13.3237 q^{73} +55.2094 q^{74} -339.962 q^{76} -889.903 q^{77} -143.573 q^{79} -309.891 q^{80} +86.6407 q^{82} -1017.25 q^{83} +85.0000 q^{85} -124.639 q^{86} -183.298 q^{88} -772.337 q^{89} -150.641 q^{91} -916.023 q^{92} -90.4630 q^{94} -214.745 q^{95} -1700.56 q^{97} -47.0221 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 34 q^{4} - 25 q^{5} + 10 q^{7} - 30 q^{8} + 10 q^{10} - 126 q^{11} + 83 q^{13} - 90 q^{14} + 322 q^{16} - 85 q^{17} + 55 q^{19} - 170 q^{20} - 240 q^{22} + 2 q^{23} + 125 q^{25} + 395 q^{26}+ \cdots - 502 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.290753 −0.102797 −0.0513983 − 0.998678i \(-0.516368\pi\)
−0.0513983 + 0.998678i \(0.516368\pi\)
\(3\) 0 0
\(4\) −7.91546 −0.989433
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 22.4661 1.21306 0.606528 − 0.795062i \(-0.292562\pi\)
0.606528 + 0.795062i \(0.292562\pi\)
\(8\) 4.62746 0.204507
\(9\) 0 0
\(10\) 1.45376 0.0459720
\(11\) −39.6109 −1.08574 −0.542870 − 0.839817i \(-0.682662\pi\)
−0.542870 + 0.839817i \(0.682662\pi\)
\(12\) 0 0
\(13\) −6.70527 −0.143054 −0.0715272 − 0.997439i \(-0.522787\pi\)
−0.0715272 + 0.997439i \(0.522787\pi\)
\(14\) −6.53208 −0.124698
\(15\) 0 0
\(16\) 61.9783 0.968410
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) 42.9490 0.518589 0.259294 − 0.965798i \(-0.416510\pi\)
0.259294 + 0.965798i \(0.416510\pi\)
\(20\) 39.5773 0.442488
\(21\) 0 0
\(22\) 11.5170 0.111610
\(23\) 115.726 1.04915 0.524576 − 0.851364i \(-0.324224\pi\)
0.524576 + 0.851364i \(0.324224\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 1.94958 0.0147055
\(27\) 0 0
\(28\) −177.830 −1.20024
\(29\) −184.009 −1.17826 −0.589131 − 0.808037i \(-0.700530\pi\)
−0.589131 + 0.808037i \(0.700530\pi\)
\(30\) 0 0
\(31\) 201.848 1.16945 0.584726 − 0.811231i \(-0.301202\pi\)
0.584726 + 0.811231i \(0.301202\pi\)
\(32\) −55.0401 −0.304056
\(33\) 0 0
\(34\) 4.94280 0.0249318
\(35\) −112.330 −0.542495
\(36\) 0 0
\(37\) −189.885 −0.843698 −0.421849 − 0.906666i \(-0.638619\pi\)
−0.421849 + 0.906666i \(0.638619\pi\)
\(38\) −12.4876 −0.0533092
\(39\) 0 0
\(40\) −23.1373 −0.0914583
\(41\) −297.987 −1.13507 −0.567534 − 0.823350i \(-0.692103\pi\)
−0.567534 + 0.823350i \(0.692103\pi\)
\(42\) 0 0
\(43\) 428.676 1.52029 0.760144 − 0.649754i \(-0.225128\pi\)
0.760144 + 0.649754i \(0.225128\pi\)
\(44\) 313.539 1.07427
\(45\) 0 0
\(46\) −33.6476 −0.107849
\(47\) 311.134 0.965607 0.482803 − 0.875729i \(-0.339619\pi\)
0.482803 + 0.875729i \(0.339619\pi\)
\(48\) 0 0
\(49\) 161.725 0.471503
\(50\) −7.26882 −0.0205593
\(51\) 0 0
\(52\) 53.0753 0.141543
\(53\) 307.329 0.796507 0.398253 − 0.917275i \(-0.369617\pi\)
0.398253 + 0.917275i \(0.369617\pi\)
\(54\) 0 0
\(55\) 198.055 0.485558
\(56\) 103.961 0.248078
\(57\) 0 0
\(58\) 53.5011 0.121121
\(59\) −704.985 −1.55561 −0.777807 − 0.628503i \(-0.783668\pi\)
−0.777807 + 0.628503i \(0.783668\pi\)
\(60\) 0 0
\(61\) −929.140 −1.95023 −0.975117 − 0.221693i \(-0.928842\pi\)
−0.975117 + 0.221693i \(0.928842\pi\)
\(62\) −58.6879 −0.120216
\(63\) 0 0
\(64\) −479.823 −0.937154
\(65\) 33.5263 0.0639759
\(66\) 0 0
\(67\) 587.978 1.07213 0.536067 − 0.844175i \(-0.319909\pi\)
0.536067 + 0.844175i \(0.319909\pi\)
\(68\) 134.563 0.239973
\(69\) 0 0
\(70\) 32.6604 0.0557666
\(71\) −507.339 −0.848030 −0.424015 − 0.905655i \(-0.639380\pi\)
−0.424015 + 0.905655i \(0.639380\pi\)
\(72\) 0 0
\(73\) −13.3237 −0.0213619 −0.0106810 − 0.999943i \(-0.503400\pi\)
−0.0106810 + 0.999943i \(0.503400\pi\)
\(74\) 55.2094 0.0867293
\(75\) 0 0
\(76\) −339.962 −0.513109
\(77\) −889.903 −1.31706
\(78\) 0 0
\(79\) −143.573 −0.204472 −0.102236 − 0.994760i \(-0.532600\pi\)
−0.102236 + 0.994760i \(0.532600\pi\)
\(80\) −309.891 −0.433086
\(81\) 0 0
\(82\) 86.6407 0.116681
\(83\) −1017.25 −1.34528 −0.672638 − 0.739972i \(-0.734839\pi\)
−0.672638 + 0.739972i \(0.734839\pi\)
\(84\) 0 0
\(85\) 85.0000 0.108465
\(86\) −124.639 −0.156280
\(87\) 0 0
\(88\) −183.298 −0.222041
\(89\) −772.337 −0.919860 −0.459930 − 0.887955i \(-0.652126\pi\)
−0.459930 + 0.887955i \(0.652126\pi\)
\(90\) 0 0
\(91\) −150.641 −0.173533
\(92\) −916.023 −1.03806
\(93\) 0 0
\(94\) −90.4630 −0.0992611
\(95\) −214.745 −0.231920
\(96\) 0 0
\(97\) −1700.56 −1.78006 −0.890031 − 0.455899i \(-0.849318\pi\)
−0.890031 + 0.455899i \(0.849318\pi\)
\(98\) −47.0221 −0.0484689
\(99\) 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.m.1.3 5
3.2 odd 2 85.4.a.g.1.3 ✓ 5
12.11 even 2 1360.4.a.w.1.5 5
15.2 even 4 425.4.b.i.324.6 10
15.8 even 4 425.4.b.i.324.5 10
15.14 odd 2 425.4.a.i.1.3 5
51.50 odd 2 1445.4.a.l.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.g.1.3 ✓ 5 3.2 odd 2
425.4.a.i.1.3 5 15.14 odd 2
425.4.b.i.324.5 10 15.8 even 4
425.4.b.i.324.6 10 15.2 even 4
765.4.a.m.1.3 5 1.1 even 1 trivial
1360.4.a.w.1.5 5 12.11 even 2
1445.4.a.l.1.3 5 51.50 odd 2