Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7728,2,Mod(1,7728)] 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("7728.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-5,0,-2,0,2,0,2,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.38197 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.47214 q^{11} -2.38197 q^{13} +1.38197 q^{15} -1.00000 q^{17} +3.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.09017 q^{25} -1.00000 q^{27} -7.47214 q^{29} +3.76393 q^{31} -5.47214 q^{33} +1.38197 q^{35} +1.47214 q^{37} +2.38197 q^{39} -4.70820 q^{41} -8.09017 q^{43} -1.38197 q^{45} -1.70820 q^{47} +1.00000 q^{49} +1.00000 q^{51} -3.38197 q^{53} -7.56231 q^{55} -3.00000 q^{57} +6.14590 q^{59} +13.7984 q^{61} -1.00000 q^{63} +3.29180 q^{65} -4.14590 q^{67} -1.00000 q^{69} +3.90983 q^{71} +2.70820 q^{73} +3.09017 q^{75} -5.47214 q^{77} -0.527864 q^{79} +1.00000 q^{81} +3.00000 q^{83} +1.38197 q^{85} +7.47214 q^{87} +3.14590 q^{89} +2.38197 q^{91} -3.76393 q^{93} -4.14590 q^{95} +5.00000 q^{97} +5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 7 q^{13} + 5 q^{15} - 2 q^{17} + 6 q^{19} + 2 q^{21} + 2 q^{23} + 5 q^{25} - 2 q^{27} - 6 q^{29} + 12 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + 7 q^{39}+ \cdots + 2 q^{99}+O(q^{100}) \) 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
\(5\) −1.38197 −0.618034 −0.309017 − 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.47214 1.64991 0.824956 − 0.565198i \(-0.191200\pi\)
0.824956 + 0.565198i \(0.191200\pi\)
\(12\) 0 0
\(13\) −2.38197 −0.660639 −0.330319 − 0.943869i \(-0.607156\pi\)
−0.330319 + 0.943869i \(0.607156\pi\)
\(14\) 0 0
\(15\) 1.38197 0.356822
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.09017 −0.618034
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −7.47214 −1.38754 −0.693770 − 0.720196i \(-0.744052\pi\)
−0.693770 + 0.720196i \(0.744052\pi\)
\(30\) 0 0
\(31\) 3.76393 0.676022 0.338011 − 0.941142i \(-0.390246\pi\)
0.338011 + 0.941142i \(0.390246\pi\)
\(32\) 0 0
\(33\) −5.47214 −0.952577
\(34\) 0 0
\(35\) 1.38197 0.233595
\(36\) 0 0
\(37\) 1.47214 0.242018 0.121009 − 0.992651i \(-0.461387\pi\)
0.121009 + 0.992651i \(0.461387\pi\)
\(38\) 0 0
\(39\) 2.38197 0.381420
\(40\) 0 0
\(41\) −4.70820 −0.735298 −0.367649 − 0.929965i \(-0.619837\pi\)
−0.367649 + 0.929965i \(0.619837\pi\)
\(42\) 0 0
\(43\) −8.09017 −1.23374 −0.616870 − 0.787065i \(-0.711599\pi\)
−0.616870 + 0.787065i \(0.711599\pi\)
\(44\) 0 0
\(45\) −1.38197 −0.206011
\(46\) 0 0
\(47\) −1.70820 −0.249167 −0.124584 − 0.992209i \(-0.539759\pi\)
−0.124584 + 0.992209i \(0.539759\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −3.38197 −0.464549 −0.232274 − 0.972650i \(-0.574617\pi\)
−0.232274 + 0.972650i \(0.574617\pi\)
\(54\) 0 0
\(55\) −7.56231 −1.01970
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 6.14590 0.800128 0.400064 − 0.916487i \(-0.368988\pi\)
0.400064 + 0.916487i \(0.368988\pi\)
\(60\) 0 0
\(61\) 13.7984 1.76670 0.883350 − 0.468713i \(-0.155282\pi\)
0.883350 + 0.468713i \(0.155282\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 3.29180 0.408297
\(66\) 0 0
\(67\) −4.14590 −0.506502 −0.253251 − 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.90983 0.464011 0.232006 − 0.972714i \(-0.425471\pi\)
0.232006 + 0.972714i \(0.425471\pi\)
\(72\) 0 0
\(73\) 2.70820 0.316971 0.158486 − 0.987361i \(-0.449339\pi\)
0.158486 + 0.987361i \(0.449339\pi\)
\(74\) 0 0
\(75\) 3.09017 0.356822
\(76\) 0 0
\(77\) −5.47214 −0.623608
\(78\) 0 0
\(79\) −0.527864 −0.0593893 −0.0296947 − 0.999559i \(-0.509453\pi\)
−0.0296947 + 0.999559i \(0.509453\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 − 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 1.38197 0.149895
\(86\) 0 0
\(87\) 7.47214 0.801097
\(88\) 0 0
\(89\) 3.14590 0.333465 0.166732 − 0.986002i \(-0.446678\pi\)
0.166732 + 0.986002i \(0.446678\pi\)
\(90\) 0 0
\(91\) 2.38197 0.249698
\(92\) 0 0
\(93\) −3.76393 −0.390302
\(94\) 0 0
\(95\) −4.14590 −0.425360
\(96\) 0 0
\(97\) 5.00000 0.507673 0.253837 − 0.967247i \(-0.418307\pi\)
0.253837 + 0.967247i \(0.418307\pi\)
\(98\) 0 0
\(99\) 5.47214 0.549970
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 7728.2.a.v.1.2 2
4.3 odd 2 483.2.a.c.1.2 ✓ 2
12.11 even 2 1449.2.a.k.1.1 2
28.27 even 2 3381.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.c.1.2 ✓ 2 4.3 odd 2
1449.2.a.k.1.1 2 12.11 even 2
3381.2.a.n.1.2 2 28.27 even 2
7728.2.a.v.1.2 2 1.1 even 1 trivial