Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,2,Mod(866,1445)] 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("1445.866"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-6,0,42,0,0,0,-24,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5383830921\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 866.13
Character \(\chi\) \(=\) 1445.866
Dual form 1445.2.d.i.866.14

$q$-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)
 
\(f(q)\) \(=\) \(q+0.0540929 q^{2} +0.605946i q^{3} -1.99707 q^{4} +1.00000i q^{5} +0.0327774i q^{6} +4.08179i q^{7} -0.216213 q^{8} +2.63283 q^{9} +0.0540929i q^{10} -1.50083i q^{11} -1.21012i q^{12} +3.30340 q^{13} +0.220796i q^{14} -0.605946 q^{15} +3.98245 q^{16} +0.142417 q^{18} +4.02443 q^{19} -1.99707i q^{20} -2.47334 q^{21} -0.0811844i q^{22} -2.09556i q^{23} -0.131014i q^{24} -1.00000 q^{25} +0.178691 q^{26} +3.41319i q^{27} -8.15163i q^{28} +10.4424i q^{29} -0.0327774 q^{30} +2.12355i q^{31} +0.647849 q^{32} +0.909424 q^{33} -4.08179 q^{35} -5.25796 q^{36} -5.75990i q^{37} +0.217693 q^{38} +2.00168i q^{39} -0.216213i q^{40} +4.09160i q^{41} -0.133790 q^{42} -11.4355 q^{43} +2.99728i q^{44} +2.63283i q^{45} -0.113355i q^{46} +11.0877 q^{47} +2.41315i q^{48} -9.66099 q^{49} -0.0540929 q^{50} -6.59714 q^{52} -6.78569 q^{53} +0.184629i q^{54} +1.50083 q^{55} -0.882536i q^{56} +2.43859i q^{57} +0.564859i q^{58} -9.05391 q^{59} +1.21012 q^{60} +5.81828i q^{61} +0.114869i q^{62} +10.7466i q^{63} -7.92986 q^{64} +3.30340i q^{65} +0.0491934 q^{66} +6.85741 q^{67} +1.26980 q^{69} -0.220796 q^{70} +12.5305i q^{71} -0.569252 q^{72} -11.5717i q^{73} -0.311569i q^{74} -0.605946i q^{75} -8.03709 q^{76} +6.12608 q^{77} +0.108277i q^{78} -0.181191i q^{79} +3.98245i q^{80} +5.83028 q^{81} +0.221327i q^{82} -1.27674 q^{83} +4.93945 q^{84} -0.618579 q^{86} -6.32753 q^{87} +0.324500i q^{88} -11.3962 q^{89} +0.142417i q^{90} +13.4838i q^{91} +4.18499i q^{92} -1.28676 q^{93} +0.599763 q^{94} +4.02443i q^{95} +0.392561i q^{96} -8.27401i q^{97} -0.522590 q^{98} -3.95144i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{2} + 42 q^{4} - 24 q^{8} - 42 q^{9} + 18 q^{13} - 6 q^{15} + 78 q^{16} - 18 q^{18} - 54 q^{19} + 12 q^{21} - 24 q^{25} - 12 q^{26} - 18 q^{30} - 24 q^{32} + 12 q^{35} - 96 q^{36} - 6 q^{38}+ \cdots - 84 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times\).

\(n\) \(581\) \(1157\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0540929 0.0382494 0.0191247 − 0.999817i \(-0.493912\pi\)
0.0191247 + 0.999817i \(0.493912\pi\)
\(3\) 0.605946i 0.349843i 0.984582 + 0.174922i \(0.0559672\pi\)
−0.984582 + 0.174922i \(0.944033\pi\)
\(4\) −1.99707 −0.998537
\(5\) 1.00000i 0.447214i
\(6\) 0.0327774i 0.0133813i
\(7\) 4.08179i 1.54277i 0.636368 + 0.771385i \(0.280436\pi\)
−0.636368 + 0.771385i \(0.719564\pi\)
\(8\) −0.216213 −0.0764429
\(9\) 2.63283 0.877610
\(10\) 0.0540929i 0.0171057i
\(11\) − 1.50083i − 0.452518i −0.974067 − 0.226259i \(-0.927350\pi\)
0.974067 − 0.226259i \(-0.0726496\pi\)
\(12\) − 1.21012i − 0.349331i
\(13\) 3.30340 0.916199 0.458100 − 0.888901i \(-0.348530\pi\)
0.458100 + 0.888901i \(0.348530\pi\)
\(14\) 0.220796i 0.0590101i
\(15\) −0.605946 −0.156455
\(16\) 3.98245 0.995613
\(17\) 0 0
\(18\) 0.142417 0.0335681
\(19\) 4.02443 0.923268 0.461634 − 0.887071i \(-0.347263\pi\)
0.461634 + 0.887071i \(0.347263\pi\)
\(20\) − 1.99707i − 0.446559i
\(21\) −2.47334 −0.539728
\(22\) − 0.0811844i − 0.0173086i
\(23\) − 2.09556i − 0.436955i −0.975842 − 0.218477i \(-0.929891\pi\)
0.975842 − 0.218477i \(-0.0701090\pi\)
\(24\) − 0.131014i − 0.0267430i
\(25\) −1.00000 −0.200000
\(26\) 0.178691 0.0350441
\(27\) 3.41319i 0.656869i
\(28\) − 8.15163i − 1.54051i
\(29\) 10.4424i 1.93910i 0.244887 + 0.969552i \(0.421249\pi\)
−0.244887 + 0.969552i \(0.578751\pi\)
\(30\) −0.0327774 −0.00598430
\(31\) 2.12355i 0.381401i 0.981648 + 0.190701i \(0.0610760\pi\)
−0.981648 + 0.190701i \(0.938924\pi\)
\(32\) 0.647849 0.114525
\(33\) 0.909424 0.158310
\(34\) 0 0
\(35\) −4.08179 −0.689948
\(36\) −5.25796 −0.876326
\(37\) − 5.75990i − 0.946921i −0.880815 − 0.473461i \(-0.843005\pi\)
0.880815 − 0.473461i \(-0.156995\pi\)
\(38\) 0.217693 0.0353145
\(39\) 2.00168i 0.320526i
\(40\) − 0.216213i − 0.0341863i
\(41\) 4.09160i 0.639001i 0.947586 + 0.319501i \(0.103515\pi\)
−0.947586 + 0.319501i \(0.896485\pi\)
\(42\) −0.133790 −0.0206443
\(43\) −11.4355 −1.74390 −0.871949 − 0.489597i \(-0.837144\pi\)
−0.871949 + 0.489597i \(0.837144\pi\)
\(44\) 2.99728i 0.451856i
\(45\) 2.63283i 0.392479i
\(46\) − 0.113355i − 0.0167133i
\(47\) 11.0877 1.61730 0.808650 − 0.588290i \(-0.200199\pi\)
0.808650 + 0.588290i \(0.200199\pi\)
\(48\) 2.41315i 0.348308i
\(49\) −9.66099 −1.38014
\(50\) −0.0540929 −0.00764989
\(51\) 0 0
\(52\) −6.59714 −0.914859
\(53\) −6.78569 −0.932086 −0.466043 − 0.884762i \(-0.654321\pi\)
−0.466043 + 0.884762i \(0.654321\pi\)
\(54\) 0.184629i 0.0251249i
\(55\) 1.50083 0.202372
\(56\) − 0.882536i − 0.117934i
\(57\) 2.43859i 0.322999i
\(58\) 0.564859i 0.0741696i
\(59\) −9.05391 −1.17872 −0.589359 − 0.807871i \(-0.700620\pi\)
−0.589359 + 0.807871i \(0.700620\pi\)
\(60\) 1.21012 0.156226
\(61\) 5.81828i 0.744954i 0.928041 + 0.372477i \(0.121491\pi\)
−0.928041 + 0.372477i \(0.878509\pi\)
\(62\) 0.114869i 0.0145884i
\(63\) 10.7466i 1.35395i
\(64\) −7.92986 −0.991233
\(65\) 3.30340i 0.409737i
\(66\) 0.0491934 0.00605528
\(67\) 6.85741 0.837766 0.418883 − 0.908040i \(-0.362422\pi\)
0.418883 + 0.908040i \(0.362422\pi\)
\(68\) 0 0
\(69\) 1.26980 0.152866
\(70\) −0.220796 −0.0263901
\(71\) 12.5305i 1.48710i 0.668683 + 0.743548i \(0.266859\pi\)
−0.668683 + 0.743548i \(0.733141\pi\)
\(72\) −0.569252 −0.0670870
\(73\) − 11.5717i − 1.35437i −0.735815 − 0.677183i \(-0.763201\pi\)
0.735815 − 0.677183i \(-0.236799\pi\)
\(74\) − 0.311569i − 0.0362192i
\(75\) − 0.605946i − 0.0699686i
\(76\) −8.03709 −0.921917
\(77\) 6.12608 0.698132
\(78\) 0.108277i 0.0122599i
\(79\) − 0.181191i − 0.0203856i −0.999948 − 0.0101928i \(-0.996755\pi\)
0.999948 − 0.0101928i \(-0.00324452\pi\)
\(80\) 3.98245i 0.445252i
\(81\) 5.83028 0.647809
\(82\) 0.221327i 0.0244414i
\(83\) −1.27674 −0.140140 −0.0700701 − 0.997542i \(-0.522322\pi\)
−0.0700701 + 0.997542i \(0.522322\pi\)
\(84\) 4.93945 0.538938
\(85\) 0 0
\(86\) −0.618579 −0.0667031
\(87\) −6.32753 −0.678382
\(88\) 0.324500i 0.0345918i
\(89\) −11.3962 −1.20799 −0.603996 − 0.796987i \(-0.706426\pi\)
−0.603996 + 0.796987i \(0.706426\pi\)
\(90\) 0.142417i 0.0150121i
\(91\) 13.4838i 1.41348i
\(92\) 4.18499i 0.436316i
\(93\) −1.28676 −0.133431
\(94\) 0.599763 0.0618608
\(95\) 4.02443i 0.412898i
\(96\) 0.392561i 0.0400656i
\(97\) − 8.27401i − 0.840098i −0.907501 − 0.420049i \(-0.862013\pi\)
0.907501 − 0.420049i \(-0.137987\pi\)
\(98\) −0.522590 −0.0527896
\(99\) − 3.95144i − 0.397135i
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 1445.2.d.i.866.13 24
17.4 even 4 1445.2.a.s.1.6 yes 12
17.13 even 4 1445.2.a.r.1.6 ✓ 12
17.16 even 2 inner 1445.2.d.i.866.14 24
85.4 even 4 7225.2.a.bn.1.7 12
85.64 even 4 7225.2.a.bo.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1445.2.a.r.1.6 ✓ 12 17.13 even 4
1445.2.a.s.1.6 yes 12 17.4 even 4
1445.2.d.i.866.13 24 1.1 even 1 trivial
1445.2.d.i.866.14 24 17.16 even 2 inner
7225.2.a.bn.1.7 12 85.4 even 4
7225.2.a.bo.1.7 12 85.64 even 4