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,-6,0,0,0,20,0,0,0,0,0,-2,0,0] 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: 6.6.507624192.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 20x^{4} + 100x^{2} - 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2420)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.22147\) 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)
 
\(f(q)\) \(=\) \(q+3.39234 q^{3} -1.00000 q^{5} +2.95353 q^{7} +8.50800 q^{9} -5.36514 q^{13} -3.39234 q^{15} -2.44295 q^{17} +6.38629 q^{19} +10.0194 q^{21} +0.276689 q^{23} +1.00000 q^{25} +18.6850 q^{27} +2.44295 q^{29} +0.508000 q^{31} -2.95353 q^{35} -0.784689 q^{37} -18.2004 q^{39} -10.5613 q^{41} +6.41763 q^{43} -8.50800 q^{45} +10.1770 q^{47} +1.72331 q^{49} -8.28732 q^{51} +0.276689 q^{53} +21.6645 q^{57} +9.29269 q^{59} -4.65424 q^{61} +25.1286 q^{63} +5.36514 q^{65} +11.6237 q^{67} +0.938623 q^{69} -9.29269 q^{71} +6.92820 q^{73} +3.39234 q^{75} +16.6373 q^{79} +37.8621 q^{81} -13.3145 q^{83} +2.44295 q^{85} +8.28732 q^{87} +7.78469 q^{89} -15.8461 q^{91} +1.72331 q^{93} -6.38629 q^{95} +8.27669 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 6 q^{5} + 20 q^{9} - 2 q^{15} - 4 q^{23} + 6 q^{25} + 26 q^{27} - 28 q^{31} + 32 q^{37} - 20 q^{45} + 6 q^{47} + 16 q^{49} - 4 q^{53} - 12 q^{59} + 26 q^{67} + 48 q^{69} + 12 q^{71} + 2 q^{75}+ \cdots + 44 q^{97}+O(q^{100}) \) 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
\(3\) 3.39234 1.95857 0.979285 − 0.202485i \(-0.0649016\pi\)
0.979285 + 0.202485i \(0.0649016\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.95353 1.11633 0.558164 − 0.829731i \(-0.311506\pi\)
0.558164 + 0.829731i \(0.311506\pi\)
\(8\) 0 0
\(9\) 8.50800 2.83600
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.36514 −1.48802 −0.744011 − 0.668168i \(-0.767079\pi\)
−0.744011 + 0.668168i \(0.767079\pi\)
\(14\) 0 0
\(15\) −3.39234 −0.875900
\(16\) 0 0
\(17\) −2.44295 −0.592502 −0.296251 − 0.955110i \(-0.595737\pi\)
−0.296251 + 0.955110i \(0.595737\pi\)
\(18\) 0 0
\(19\) 6.38629 1.46512 0.732558 − 0.680705i \(-0.238326\pi\)
0.732558 + 0.680705i \(0.238326\pi\)
\(20\) 0 0
\(21\) 10.0194 2.18641
\(22\) 0 0
\(23\) 0.276689 0.0576935 0.0288468 − 0.999584i \(-0.490817\pi\)
0.0288468 + 0.999584i \(0.490817\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 18.6850 3.59594
\(28\) 0 0
\(29\) 2.44295 0.453644 0.226822 − 0.973936i \(-0.427166\pi\)
0.226822 + 0.973936i \(0.427166\pi\)
\(30\) 0 0
\(31\) 0.508000 0.0912395 0.0456198 − 0.998959i \(-0.485474\pi\)
0.0456198 + 0.998959i \(0.485474\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.95353 −0.499237
\(36\) 0 0
\(37\) −0.784689 −0.129002 −0.0645010 − 0.997918i \(-0.520546\pi\)
−0.0645010 + 0.997918i \(0.520546\pi\)
\(38\) 0 0
\(39\) −18.2004 −2.91439
\(40\) 0 0
\(41\) −10.5613 −1.64940 −0.824698 − 0.565573i \(-0.808655\pi\)
−0.824698 + 0.565573i \(0.808655\pi\)
\(42\) 0 0
\(43\) 6.41763 0.978679 0.489339 − 0.872093i \(-0.337238\pi\)
0.489339 + 0.872093i \(0.337238\pi\)
\(44\) 0 0
\(45\) −8.50800 −1.26830
\(46\) 0 0
\(47\) 10.1770 1.48447 0.742236 − 0.670138i \(-0.233765\pi\)
0.742236 + 0.670138i \(0.233765\pi\)
\(48\) 0 0
\(49\) 1.72331 0.246187
\(50\) 0 0
\(51\) −8.28732 −1.16046
\(52\) 0 0
\(53\) 0.276689 0.0380061 0.0190030 − 0.999819i \(-0.493951\pi\)
0.0190030 + 0.999819i \(0.493951\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 21.6645 2.86953
\(58\) 0 0
\(59\) 9.29269 1.20981 0.604903 − 0.796300i \(-0.293212\pi\)
0.604903 + 0.796300i \(0.293212\pi\)
\(60\) 0 0
\(61\) −4.65424 −0.595914 −0.297957 − 0.954579i \(-0.596305\pi\)
−0.297957 + 0.954579i \(0.596305\pi\)
\(62\) 0 0
\(63\) 25.1286 3.16591
\(64\) 0 0
\(65\) 5.36514 0.665463
\(66\) 0 0
\(67\) 11.6237 1.42006 0.710028 − 0.704174i \(-0.248682\pi\)
0.710028 + 0.704174i \(0.248682\pi\)
\(68\) 0 0
\(69\) 0.938623 0.112997
\(70\) 0 0
\(71\) −9.29269 −1.10284 −0.551420 − 0.834228i \(-0.685914\pi\)
−0.551420 + 0.834228i \(0.685914\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 − 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 0 0
\(75\) 3.39234 0.391714
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.6373 1.87185 0.935923 − 0.352206i \(-0.114568\pi\)
0.935923 + 0.352206i \(0.114568\pi\)
\(80\) 0 0
\(81\) 37.8621 4.20690
\(82\) 0 0
\(83\) −13.3145 −1.46146 −0.730728 − 0.682669i \(-0.760819\pi\)
−0.730728 + 0.682669i \(0.760819\pi\)
\(84\) 0 0
\(85\) 2.44295 0.264975
\(86\) 0 0
\(87\) 8.28732 0.888494
\(88\) 0 0
\(89\) 7.78469 0.825175 0.412588 − 0.910918i \(-0.364625\pi\)
0.412588 + 0.910918i \(0.364625\pi\)
\(90\) 0 0
\(91\) −15.8461 −1.66112
\(92\) 0 0
\(93\) 1.72331 0.178699
\(94\) 0 0
\(95\) −6.38629 −0.655219
\(96\) 0 0
\(97\) 8.27669 0.840370 0.420185 − 0.907438i \(-0.361965\pi\)
0.420185 + 0.907438i \(0.361965\pi\)
\(98\) 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.cz.1.6 6
4.3 odd 2 2420.2.a.o.1.1 ✓ 6
11.10 odd 2 inner 9680.2.a.cz.1.5 6
44.43 even 2 2420.2.a.o.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2420.2.a.o.1.1 ✓ 6 4.3 odd 2
2420.2.a.o.1.2 yes 6 44.43 even 2
9680.2.a.cz.1.5 6 11.10 odd 2 inner
9680.2.a.cz.1.6 6 1.1 even 1 trivial