Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9680,2,Mod(1,9680)] 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("9680.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-2,0,-6,0,6,0,10,0,0,0,6,0,2,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45753625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.36422\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.843136 q^{3} -1.00000 q^{5} +4.75693 q^{7} -2.28912 q^{9} -3.78308 q^{13} +0.843136 q^{15} -4.98203 q^{17} +3.12115 q^{19} -4.01074 q^{21} -9.39001 q^{23} +1.00000 q^{25} +4.45945 q^{27} +2.60693 q^{29} +3.68869 q^{31} -4.75693 q^{35} -3.44757 q^{37} +3.18965 q^{39} -9.32837 q^{41} -11.7051 q^{43} +2.28912 q^{45} +1.66156 q^{47} +15.6284 q^{49} +4.20053 q^{51} -1.93994 q^{53} -2.63156 q^{57} -0.103915 q^{59} +10.1513 q^{61} -10.8892 q^{63} +3.78308 q^{65} +4.99277 q^{67} +7.91706 q^{69} +6.22929 q^{71} +7.30650 q^{73} -0.843136 q^{75} +10.5445 q^{79} +3.10744 q^{81} +5.69010 q^{83} +4.98203 q^{85} -2.19800 q^{87} +13.6371 q^{89} -17.9958 q^{91} -3.11007 q^{93} -3.12115 q^{95} +13.6967 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} + 6 q^{13} + 2 q^{15} - 11 q^{17} - 11 q^{19} - 2 q^{21} - 18 q^{23} + 6 q^{25} + q^{27} + 6 q^{29} - q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} + 4 q^{41} + 3 q^{43}+ \cdots + 20 q^{97}+O(q^{100}) \) 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
\(3\) −0.843136 −0.486785 −0.243392 − 0.969928i \(-0.578260\pi\)
−0.243392 + 0.969928i \(0.578260\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.75693 1.79795 0.898975 − 0.438000i \(-0.144313\pi\)
0.898975 + 0.438000i \(0.144313\pi\)
\(8\) 0 0
\(9\) −2.28912 −0.763040
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −3.78308 −1.04924 −0.524619 − 0.851337i \(-0.675792\pi\)
−0.524619 + 0.851337i \(0.675792\pi\)
\(14\) 0 0
\(15\) 0.843136 0.217697
\(16\) 0 0
\(17\) −4.98203 −1.20832 −0.604160 − 0.796863i \(-0.706491\pi\)
−0.604160 + 0.796863i \(0.706491\pi\)
\(18\) 0 0
\(19\) 3.12115 0.716041 0.358021 − 0.933714i \(-0.383452\pi\)
0.358021 + 0.933714i \(0.383452\pi\)
\(20\) 0 0
\(21\) −4.01074 −0.875215
\(22\) 0 0
\(23\) −9.39001 −1.95795 −0.978976 − 0.203975i \(-0.934614\pi\)
−0.978976 + 0.203975i \(0.934614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.45945 0.858222
\(28\) 0 0
\(29\) 2.60693 0.484095 0.242047 − 0.970264i \(-0.422181\pi\)
0.242047 + 0.970264i \(0.422181\pi\)
\(30\) 0 0
\(31\) 3.68869 0.662508 0.331254 − 0.943542i \(-0.392528\pi\)
0.331254 + 0.943542i \(0.392528\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.75693 −0.804068
\(36\) 0 0
\(37\) −3.44757 −0.566776 −0.283388 − 0.959005i \(-0.591458\pi\)
−0.283388 + 0.959005i \(0.591458\pi\)
\(38\) 0 0
\(39\) 3.18965 0.510753
\(40\) 0 0
\(41\) −9.32837 −1.45685 −0.728423 − 0.685127i \(-0.759747\pi\)
−0.728423 + 0.685127i \(0.759747\pi\)
\(42\) 0 0
\(43\) −11.7051 −1.78500 −0.892502 − 0.451043i \(-0.851052\pi\)
−0.892502 + 0.451043i \(0.851052\pi\)
\(44\) 0 0
\(45\) 2.28912 0.341242
\(46\) 0 0
\(47\) 1.66156 0.242364 0.121182 − 0.992630i \(-0.461332\pi\)
0.121182 + 0.992630i \(0.461332\pi\)
\(48\) 0 0
\(49\) 15.6284 2.23262
\(50\) 0 0
\(51\) 4.20053 0.588192
\(52\) 0 0
\(53\) −1.93994 −0.266472 −0.133236 − 0.991084i \(-0.542537\pi\)
−0.133236 + 0.991084i \(0.542537\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.63156 −0.348558
\(58\) 0 0
\(59\) −0.103915 −0.0135285 −0.00676427 − 0.999977i \(-0.502153\pi\)
−0.00676427 + 0.999977i \(0.502153\pi\)
\(60\) 0 0
\(61\) 10.1513 1.29974 0.649870 − 0.760045i \(-0.274823\pi\)
0.649870 + 0.760045i \(0.274823\pi\)
\(62\) 0 0
\(63\) −10.8892 −1.37191
\(64\) 0 0
\(65\) 3.78308 0.469233
\(66\) 0 0
\(67\) 4.99277 0.609964 0.304982 − 0.952358i \(-0.401350\pi\)
0.304982 + 0.952358i \(0.401350\pi\)
\(68\) 0 0
\(69\) 7.91706 0.953102
\(70\) 0 0
\(71\) 6.22929 0.739281 0.369640 − 0.929175i \(-0.379481\pi\)
0.369640 + 0.929175i \(0.379481\pi\)
\(72\) 0 0
\(73\) 7.30650 0.855161 0.427580 − 0.903977i \(-0.359366\pi\)
0.427580 + 0.903977i \(0.359366\pi\)
\(74\) 0 0
\(75\) −0.843136 −0.0973570
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 10.5445 1.18635 0.593175 − 0.805073i \(-0.297874\pi\)
0.593175 + 0.805073i \(0.297874\pi\)
\(80\) 0 0
\(81\) 3.10744 0.345271
\(82\) 0 0
\(83\) 5.69010 0.624569 0.312285 − 0.949989i \(-0.398906\pi\)
0.312285 + 0.949989i \(0.398906\pi\)
\(84\) 0 0
\(85\) 4.98203 0.540377
\(86\) 0 0
\(87\) −2.19800 −0.235650
\(88\) 0 0
\(89\) 13.6371 1.44553 0.722767 − 0.691091i \(-0.242870\pi\)
0.722767 + 0.691091i \(0.242870\pi\)
\(90\) 0 0
\(91\) −17.9958 −1.88648
\(92\) 0 0
\(93\) −3.11007 −0.322499
\(94\) 0 0
\(95\) −3.12115 −0.320223
\(96\) 0 0
\(97\) 13.6967 1.39069 0.695345 − 0.718676i \(-0.255252\pi\)
0.695345 + 0.718676i \(0.255252\pi\)
\(98\) 0 0
\(99\) 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 9680.2.a.cy.1.3 6
4.3 odd 2 4840.2.a.be.1.4 6
11.2 odd 10 880.2.bo.j.81.2 12
11.6 odd 10 880.2.bo.j.641.2 12
11.10 odd 2 9680.2.a.cx.1.3 6
44.35 even 10 440.2.y.b.81.2 ✓ 12
44.39 even 10 440.2.y.b.201.2 yes 12
44.43 even 2 4840.2.a.bf.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.2 ✓ 12 44.35 even 10
440.2.y.b.201.2 yes 12 44.39 even 10
880.2.bo.j.81.2 12 11.2 odd 10
880.2.bo.j.641.2 12 11.6 odd 10
4840.2.a.be.1.4 6 4.3 odd 2
4840.2.a.bf.1.4 6 44.43 even 2
9680.2.a.cx.1.3 6 11.10 odd 2
9680.2.a.cy.1.3 6 1.1 even 1 trivial