Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3344,2,Mod(1,3344)] 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("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,-2,0,2,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.61330\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.19599 q^{3} +4.07680 q^{5} -3.61829 q^{7} -1.56960 q^{9} +1.00000 q^{11} -1.47857 q^{13} -4.87582 q^{15} -3.27003 q^{17} -1.00000 q^{19} +4.32745 q^{21} +7.45793 q^{23} +11.6203 q^{25} +5.46521 q^{27} +1.02535 q^{29} -1.64921 q^{31} -1.19599 q^{33} -14.7511 q^{35} -6.71293 q^{37} +1.76836 q^{39} -3.92451 q^{41} -5.38113 q^{43} -6.39896 q^{45} +3.71597 q^{47} +6.09205 q^{49} +3.91093 q^{51} -0.102902 q^{53} +4.07680 q^{55} +1.19599 q^{57} -13.2986 q^{59} -6.49664 q^{61} +5.67929 q^{63} -6.02783 q^{65} +3.70989 q^{67} -8.91962 q^{69} -6.32968 q^{71} -1.37759 q^{73} -13.8978 q^{75} -3.61829 q^{77} -13.6725 q^{79} -1.82753 q^{81} -5.44061 q^{83} -13.3313 q^{85} -1.22631 q^{87} +12.1357 q^{89} +5.34990 q^{91} +1.97244 q^{93} -4.07680 q^{95} -13.7910 q^{97} -1.56960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 2 q^{3} + 2 q^{5} - 10 q^{7} + 11 q^{9} + 7 q^{11} - 4 q^{13} - 12 q^{15} + 2 q^{17} - 7 q^{19} - 14 q^{21} - 10 q^{23} + 9 q^{25} + 4 q^{27} - 18 q^{29} - 24 q^{31} - 2 q^{33} - 8 q^{35} - 24 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 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
\(3\) −1.19599 −0.690506 −0.345253 − 0.938510i \(-0.612207\pi\)
−0.345253 + 0.938510i \(0.612207\pi\)
\(4\) 0 0
\(5\) 4.07680 1.82320 0.911600 − 0.411078i \(-0.134847\pi\)
0.911600 + 0.411078i \(0.134847\pi\)
\(6\) 0 0
\(7\) −3.61829 −1.36759 −0.683793 − 0.729676i \(-0.739671\pi\)
−0.683793 + 0.729676i \(0.739671\pi\)
\(8\) 0 0
\(9\) −1.56960 −0.523201
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.47857 −0.410081 −0.205041 − 0.978753i \(-0.565733\pi\)
−0.205041 + 0.978753i \(0.565733\pi\)
\(14\) 0 0
\(15\) −4.87582 −1.25893
\(16\) 0 0
\(17\) −3.27003 −0.793099 −0.396549 − 0.918013i \(-0.629792\pi\)
−0.396549 + 0.918013i \(0.629792\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 4.32745 0.944327
\(22\) 0 0
\(23\) 7.45793 1.55509 0.777543 − 0.628830i \(-0.216466\pi\)
0.777543 + 0.628830i \(0.216466\pi\)
\(24\) 0 0
\(25\) 11.6203 2.32406
\(26\) 0 0
\(27\) 5.46521 1.05178
\(28\) 0 0
\(29\) 1.02535 0.190403 0.0952013 − 0.995458i \(-0.469651\pi\)
0.0952013 + 0.995458i \(0.469651\pi\)
\(30\) 0 0
\(31\) −1.64921 −0.296207 −0.148104 − 0.988972i \(-0.547317\pi\)
−0.148104 + 0.988972i \(0.547317\pi\)
\(32\) 0 0
\(33\) −1.19599 −0.208195
\(34\) 0 0
\(35\) −14.7511 −2.49338
\(36\) 0 0
\(37\) −6.71293 −1.10360 −0.551799 − 0.833977i \(-0.686059\pi\)
−0.551799 + 0.833977i \(0.686059\pi\)
\(38\) 0 0
\(39\) 1.76836 0.283164
\(40\) 0 0
\(41\) −3.92451 −0.612905 −0.306453 − 0.951886i \(-0.599142\pi\)
−0.306453 + 0.951886i \(0.599142\pi\)
\(42\) 0 0
\(43\) −5.38113 −0.820614 −0.410307 − 0.911947i \(-0.634578\pi\)
−0.410307 + 0.911947i \(0.634578\pi\)
\(44\) 0 0
\(45\) −6.39896 −0.953901
\(46\) 0 0
\(47\) 3.71597 0.542030 0.271015 − 0.962575i \(-0.412641\pi\)
0.271015 + 0.962575i \(0.412641\pi\)
\(48\) 0 0
\(49\) 6.09205 0.870292
\(50\) 0 0
\(51\) 3.91093 0.547640
\(52\) 0 0
\(53\) −0.102902 −0.0141347 −0.00706733 − 0.999975i \(-0.502250\pi\)
−0.00706733 + 0.999975i \(0.502250\pi\)
\(54\) 0 0
\(55\) 4.07680 0.549716
\(56\) 0 0
\(57\) 1.19599 0.158413
\(58\) 0 0
\(59\) −13.2986 −1.73134 −0.865668 − 0.500619i \(-0.833106\pi\)
−0.865668 + 0.500619i \(0.833106\pi\)
\(60\) 0 0
\(61\) −6.49664 −0.831809 −0.415905 − 0.909408i \(-0.636535\pi\)
−0.415905 + 0.909408i \(0.636535\pi\)
\(62\) 0 0
\(63\) 5.67929 0.715523
\(64\) 0 0
\(65\) −6.02783 −0.747661
\(66\) 0 0
\(67\) 3.70989 0.453235 0.226618 − 0.973984i \(-0.427233\pi\)
0.226618 + 0.973984i \(0.427233\pi\)
\(68\) 0 0
\(69\) −8.91962 −1.07380
\(70\) 0 0
\(71\) −6.32968 −0.751194 −0.375597 − 0.926783i \(-0.622562\pi\)
−0.375597 + 0.926783i \(0.622562\pi\)
\(72\) 0 0
\(73\) −1.37759 −0.161235 −0.0806173 − 0.996745i \(-0.525689\pi\)
−0.0806173 + 0.996745i \(0.525689\pi\)
\(74\) 0 0
\(75\) −13.8978 −1.60478
\(76\) 0 0
\(77\) −3.61829 −0.412343
\(78\) 0 0
\(79\) −13.6725 −1.53828 −0.769141 − 0.639079i \(-0.779316\pi\)
−0.769141 + 0.639079i \(0.779316\pi\)
\(80\) 0 0
\(81\) −1.82753 −0.203059
\(82\) 0 0
\(83\) −5.44061 −0.597184 −0.298592 − 0.954381i \(-0.596517\pi\)
−0.298592 + 0.954381i \(0.596517\pi\)
\(84\) 0 0
\(85\) −13.3313 −1.44598
\(86\) 0 0
\(87\) −1.22631 −0.131474
\(88\) 0 0
\(89\) 12.1357 1.28638 0.643191 − 0.765706i \(-0.277610\pi\)
0.643191 + 0.765706i \(0.277610\pi\)
\(90\) 0 0
\(91\) 5.34990 0.560822
\(92\) 0 0
\(93\) 1.97244 0.204533
\(94\) 0 0
\(95\) −4.07680 −0.418271
\(96\) 0 0
\(97\) −13.7910 −1.40026 −0.700131 − 0.714014i \(-0.746875\pi\)
−0.700131 + 0.714014i \(0.746875\pi\)
\(98\) 0 0
\(99\) −1.56960 −0.157751
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 3344.2.a.ba.1.4 7
4.3 odd 2 209.2.a.d.1.2 ✓ 7
12.11 even 2 1881.2.a.p.1.6 7
20.19 odd 2 5225.2.a.n.1.6 7
44.43 even 2 2299.2.a.q.1.6 7
76.75 even 2 3971.2.a.i.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.2 ✓ 7 4.3 odd 2
1881.2.a.p.1.6 7 12.11 even 2
2299.2.a.q.1.6 7 44.43 even 2
3344.2.a.ba.1.4 7 1.1 even 1 trivial
3971.2.a.i.1.6 7 76.75 even 2
5225.2.a.n.1.6 7 20.19 odd 2