Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9300,2,Mod(1,9300)] 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("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9300.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,0,0,-4,0,3,0,-2] 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: \(74.2608738798\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1860)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.571993\) of defining polynomial
Character \(\chi\) \(=\) 9300.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.24482 q^{7} +1.00000 q^{9} -1.14399 q^{11} +2.24482 q^{13} -2.67282 q^{17} -5.34565 q^{19} +2.24482 q^{21} -2.67282 q^{23} +1.00000 q^{27} +0.715980 q^{29} -1.00000 q^{31} -1.14399 q^{33} -6.53279 q^{37} +2.24482 q^{39} -3.14399 q^{41} -12.2017 q^{43} +10.8745 q^{47} -1.96080 q^{49} -2.67282 q^{51} -11.5288 q^{53} -5.34565 q^{57} -2.71598 q^{59} +6.00000 q^{61} +2.24482 q^{63} -13.5905 q^{67} -2.67282 q^{69} -5.48568 q^{71} -3.75518 q^{73} -2.56804 q^{77} +10.0185 q^{79} +1.00000 q^{81} +0.384851 q^{83} +0.715980 q^{87} -2.62967 q^{89} +5.03920 q^{91} -1.00000 q^{93} -16.1233 q^{97} -1.14399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 4 q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{17} + 4 q^{19} - 4 q^{21} + 2 q^{23} + 3 q^{27} - 3 q^{31} - 2 q^{33} - 6 q^{37} - 4 q^{39} - 8 q^{41} - 18 q^{43} + 4 q^{47} + 7 q^{49} + 2 q^{51}+ \cdots - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.24482 0.848461 0.424231 − 0.905554i \(-0.360545\pi\)
0.424231 + 0.905554i \(0.360545\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.14399 −0.344925 −0.172462 − 0.985016i \(-0.555172\pi\)
−0.172462 + 0.985016i \(0.555172\pi\)
\(12\) 0 0
\(13\) 2.24482 0.622600 0.311300 − 0.950312i \(-0.399236\pi\)
0.311300 + 0.950312i \(0.399236\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.67282 −0.648255 −0.324127 − 0.946013i \(-0.605071\pi\)
−0.324127 + 0.946013i \(0.605071\pi\)
\(18\) 0 0
\(19\) −5.34565 −1.22638 −0.613188 − 0.789937i \(-0.710113\pi\)
−0.613188 + 0.789937i \(0.710113\pi\)
\(20\) 0 0
\(21\) 2.24482 0.489859
\(22\) 0 0
\(23\) −2.67282 −0.557322 −0.278661 − 0.960389i \(-0.589891\pi\)
−0.278661 + 0.960389i \(0.589891\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.715980 0.132954 0.0664771 − 0.997788i \(-0.478824\pi\)
0.0664771 + 0.997788i \(0.478824\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −1.14399 −0.199142
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.53279 −1.07398 −0.536992 − 0.843587i \(-0.680439\pi\)
−0.536992 + 0.843587i \(0.680439\pi\)
\(38\) 0 0
\(39\) 2.24482 0.359458
\(40\) 0 0
\(41\) −3.14399 −0.491008 −0.245504 − 0.969396i \(-0.578953\pi\)
−0.245504 + 0.969396i \(0.578953\pi\)
\(42\) 0 0
\(43\) −12.2017 −1.86074 −0.930368 − 0.366627i \(-0.880512\pi\)
−0.930368 + 0.366627i \(0.880512\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.8745 1.58621 0.793103 − 0.609087i \(-0.208464\pi\)
0.793103 + 0.609087i \(0.208464\pi\)
\(48\) 0 0
\(49\) −1.96080 −0.280114
\(50\) 0 0
\(51\) −2.67282 −0.374270
\(52\) 0 0
\(53\) −11.5288 −1.58361 −0.791804 − 0.610776i \(-0.790858\pi\)
−0.791804 + 0.610776i \(0.790858\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.34565 −0.708048
\(58\) 0 0
\(59\) −2.71598 −0.353590 −0.176795 − 0.984248i \(-0.556573\pi\)
−0.176795 + 0.984248i \(0.556573\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 2.24482 0.282820
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.5905 −1.66034 −0.830170 − 0.557511i \(-0.811757\pi\)
−0.830170 + 0.557511i \(0.811757\pi\)
\(68\) 0 0
\(69\) −2.67282 −0.321770
\(70\) 0 0
\(71\) −5.48568 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(72\) 0 0
\(73\) −3.75518 −0.439511 −0.219755 − 0.975555i \(-0.570526\pi\)
−0.219755 + 0.975555i \(0.570526\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.56804 −0.292655
\(78\) 0 0
\(79\) 10.0185 1.12717 0.563583 − 0.826059i \(-0.309422\pi\)
0.563583 + 0.826059i \(0.309422\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.384851 0.0422428 0.0211214 − 0.999777i \(-0.493276\pi\)
0.0211214 + 0.999777i \(0.493276\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.715980 0.0767611
\(88\) 0 0
\(89\) −2.62967 −0.278744 −0.139372 − 0.990240i \(-0.544508\pi\)
−0.139372 + 0.990240i \(0.544508\pi\)
\(90\) 0 0
\(91\) 5.03920 0.528252
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.1233 −1.63707 −0.818534 − 0.574458i \(-0.805213\pi\)
−0.818534 + 0.574458i \(0.805213\pi\)
\(98\) 0 0
\(99\) −1.14399 −0.114975
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 9300.2.a.v.1.3 3
5.2 odd 4 9300.2.g.p.3349.3 6
5.3 odd 4 9300.2.g.p.3349.4 6
5.4 even 2 1860.2.a.f.1.1 ✓ 3
15.14 odd 2 5580.2.a.l.1.1 3
20.19 odd 2 7440.2.a.bt.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.f.1.1 ✓ 3 5.4 even 2
5580.2.a.l.1.1 3 15.14 odd 2
7440.2.a.bt.1.3 3 20.19 odd 2
9300.2.a.v.1.3 3 1.1 even 1 trivial
9300.2.g.p.3349.3 6 5.2 odd 4
9300.2.g.p.3349.4 6 5.3 odd 4