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(3349,9300)] 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("9300.3349"); 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, 1, 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.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,-6,0,-4,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(74.2608738798\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5089536.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1860)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.4
Root \(-1.33641 - 1.33641i\) of defining polynomial
Character \(\chi\) \(=\) 9300.3349
Dual form 9300.2.g.p.3349.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.24482i q^{7} -1.00000 q^{9} -1.14399 q^{11} +2.24482i q^{13} +2.67282i q^{17} +5.34565 q^{19} +2.24482 q^{21} -2.67282i q^{23} -1.00000i q^{27} -0.715980 q^{29} -1.00000 q^{31} -1.14399i q^{33} +6.53279i q^{37} -2.24482 q^{39} -3.14399 q^{41} -12.2017i q^{43} -10.8745i q^{47} +1.96080 q^{49} -2.67282 q^{51} -11.5288i q^{53} +5.34565i q^{57} +2.71598 q^{59} +6.00000 q^{61} +2.24482i q^{63} +13.5905i q^{67} +2.67282 q^{69} -5.48568 q^{71} -3.75518i q^{73} +2.56804i q^{77} -10.0185 q^{79} +1.00000 q^{81} +0.384851i q^{83} -0.715980i q^{87} +2.62967 q^{89} +5.03920 q^{91} -1.00000i q^{93} +16.1233i q^{97} +1.14399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9} - 4 q^{11} - 8 q^{19} - 8 q^{21} - 6 q^{31} + 8 q^{39} - 16 q^{41} - 14 q^{49} + 4 q^{51} + 12 q^{59} + 36 q^{61} - 4 q^{69} + 6 q^{81} - 20 q^{89} + 56 q^{91} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\) \(4651\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.24482i − 0.848461i −0.905554 − 0.424231i \(-0.860545\pi\)
0.905554 − 0.424231i \(-0.139455\pi\)
\(8\) 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
\(13\) 2.24482i 0.622600i 0.950312 + 0.311300i \(0.100764\pi\)
−0.950312 + 0.311300i \(0.899236\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.67282i 0.648255i 0.946013 + 0.324127i \(0.105071\pi\)
−0.946013 + 0.324127i \(0.894929\pi\)
\(18\) 0 0
\(19\) 5.34565 1.22638 0.613188 − 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(20\) 0 0
\(21\) 2.24482 0.489859
\(22\) 0 0
\(23\) − 2.67282i − 0.557322i −0.960389 − 0.278661i \(-0.910109\pi\)
0.960389 − 0.278661i \(-0.0898906\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) −0.715980 −0.132954 −0.0664771 − 0.997788i \(-0.521176\pi\)
−0.0664771 + 0.997788i \(0.521176\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) − 1.14399i − 0.199142i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.53279i 1.07398i 0.843587 + 0.536992i \(0.180439\pi\)
−0.843587 + 0.536992i \(0.819561\pi\)
\(38\) 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
\(43\) − 12.2017i − 1.86074i −0.366627 − 0.930368i \(-0.619488\pi\)
0.366627 − 0.930368i \(-0.380512\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 10.8745i − 1.58621i −0.609087 − 0.793103i \(-0.708464\pi\)
0.609087 − 0.793103i \(-0.291536\pi\)
\(48\) 0 0
\(49\) 1.96080 0.280114
\(50\) 0 0
\(51\) −2.67282 −0.374270
\(52\) 0 0
\(53\) − 11.5288i − 1.58361i −0.610776 − 0.791804i \(-0.709142\pi\)
0.610776 − 0.791804i \(-0.290858\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.34565i 0.708048i
\(58\) 0 0
\(59\) 2.71598 0.353590 0.176795 − 0.984248i \(-0.443427\pi\)
0.176795 + 0.984248i \(0.443427\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 2.24482i 0.282820i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.5905i 1.66034i 0.557511 + 0.830170i \(0.311757\pi\)
−0.557511 + 0.830170i \(0.688243\pi\)
\(68\) 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
\(73\) − 3.75518i − 0.439511i −0.975555 − 0.219755i \(-0.929474\pi\)
0.975555 − 0.219755i \(-0.0705259\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.56804i 0.292655i
\(78\) 0 0
\(79\) −10.0185 −1.12717 −0.563583 − 0.826059i \(-0.690578\pi\)
−0.563583 + 0.826059i \(0.690578\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.384851i 0.0422428i 0.999777 + 0.0211214i \(0.00672366\pi\)
−0.999777 + 0.0211214i \(0.993276\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.715980i − 0.0767611i
\(88\) 0 0
\(89\) 2.62967 0.278744 0.139372 − 0.990240i \(-0.455492\pi\)
0.139372 + 0.990240i \(0.455492\pi\)
\(90\) 0 0
\(91\) 5.03920 0.528252
\(92\) 0 0
\(93\) − 1.00000i − 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.1233i 1.63707i 0.574458 + 0.818534i \(0.305213\pi\)
−0.574458 + 0.818534i \(0.694787\pi\)
\(98\) 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.g.p.3349.4 6
5.2 odd 4 9300.2.a.v.1.3 3
5.3 odd 4 1860.2.a.f.1.1 ✓ 3
5.4 even 2 inner 9300.2.g.p.3349.3 6
15.8 even 4 5580.2.a.l.1.1 3
20.3 even 4 7440.2.a.bt.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.f.1.1 ✓ 3 5.3 odd 4
5580.2.a.l.1.1 3 15.8 even 4
7440.2.a.bt.1.3 3 20.3 even 4
9300.2.a.v.1.3 3 5.2 odd 4
9300.2.g.p.3349.3 6 5.4 even 2 inner
9300.2.g.p.3349.4 6 1.1 even 1 trivial