↑

Properties

Label 104.2.b.c.53.3
Level $104$
Weight $2$
Character 104.53
Analytic conductor $0.830$
Analytic rank $0$
Dimension $6$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [104,2,Mod(53,104)] 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("104.53"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 104.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.830444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 53.3
Root \(0.264658 + 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 104.53
Dual form 104.2.b.c.53.4

$q$-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+(-0.264658 - 1.38923i) q^{2} -3.24914i q^{3} +(-1.85991 + 0.735342i) q^{4} +1.00000i q^{5} +(-4.51380 + 0.859912i) q^{6} +3.24914 q^{7} +(1.51380 + 2.38923i) q^{8} -7.55691 q^{9} +(1.38923 - 0.264658i) q^{10} +1.05863i q^{11} +(2.38923 + 6.04312i) q^{12} -1.00000i q^{13} +(-0.859912 - 4.51380i) q^{14} +3.24914 q^{15} +(2.91855 - 2.73534i) q^{16} +1.00000 q^{17} +(2.00000 + 10.4983i) q^{18} -4.00000i q^{19} +(-0.735342 - 1.85991i) q^{20} -10.5569i q^{21} +(1.47068 - 0.280176i) q^{22} -2.94137 q^{23} +(7.76294 - 4.91855i) q^{24} +4.00000 q^{25} +(-1.38923 + 0.264658i) q^{26} +14.8061i q^{27} +(-6.04312 + 2.38923i) q^{28} +7.43965i q^{29} +(-0.859912 - 4.51380i) q^{30} +5.05863 q^{31} +(-4.57243 - 3.33060i) q^{32} +3.43965 q^{33} +(-0.264658 - 1.38923i) q^{34} +3.24914i q^{35} +(14.0552 - 5.55691i) q^{36} +6.55691i q^{37} +(-5.55691 + 1.05863i) q^{38} -3.24914 q^{39} +(-2.38923 + 1.51380i) q^{40} -9.43965 q^{41} +(-14.6660 + 2.79397i) q^{42} -0.307774i q^{43} +(-0.778457 - 1.96896i) q^{44} -7.55691i q^{45} +(0.778457 + 4.08623i) q^{46} +6.80605 q^{47} +(-8.88751 - 9.48276i) q^{48} +3.55691 q^{49} +(-1.05863 - 5.55691i) q^{50} -3.24914i q^{51} +(0.735342 + 1.85991i) q^{52} -1.55691i q^{53} +(20.5690 - 3.91855i) q^{54} -1.05863 q^{55} +(4.91855 + 7.76294i) q^{56} -12.9966 q^{57} +(10.3354 - 1.96896i) q^{58} -5.67418i q^{59} +(-6.04312 + 2.38923i) q^{60} +9.67418i q^{61} +(-1.33881 - 7.02760i) q^{62} -24.5535 q^{63} +(-3.41683 + 7.23362i) q^{64} +1.00000 q^{65} +(-0.910331 - 4.77846i) q^{66} -1.50172i q^{67} +(-1.85991 + 0.735342i) q^{68} +9.55691i q^{69} +(4.51380 - 0.859912i) q^{70} -5.36641 q^{71} +(-11.4396 - 18.0552i) q^{72} +3.55691 q^{73} +(9.10905 - 1.73534i) q^{74} -12.9966i q^{75} +(2.94137 + 7.43965i) q^{76} +3.43965i q^{77} +(0.859912 + 4.51380i) q^{78} -6.73281 q^{79} +(2.73534 + 2.91855i) q^{80} +25.4362 q^{81} +(2.49828 + 13.1138i) q^{82} +2.49828i q^{83} +(7.76294 + 19.6349i) q^{84} +1.00000i q^{85} +(-0.427568 + 0.0814549i) q^{86} +24.1725 q^{87} +(-2.52932 + 1.60256i) q^{88} +2.11727 q^{89} +(-10.4983 + 2.00000i) q^{90} -3.24914i q^{91} +(5.47068 - 2.16291i) q^{92} -16.4362i q^{93} +(-1.80128 - 9.45517i) q^{94} +4.00000 q^{95} +(-10.8216 + 14.8565i) q^{96} +7.67418 q^{97} +(-0.941367 - 4.94137i) q^{98} -8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 2 q^{4} - 10 q^{6} + 2 q^{7} - 8 q^{8} - 12 q^{9} + 6 q^{12} + 4 q^{14} + 2 q^{15} + 10 q^{16} + 6 q^{17} + 12 q^{18} - 4 q^{20} + 8 q^{22} - 16 q^{23} + 12 q^{24} + 24 q^{25} - 20 q^{28}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.264658 − 1.38923i −0.187142 − 0.982333i
\(3\) − 3.24914i − 1.87589i −0.346781 − 0.937946i \(-0.612725\pi\)
0.346781 − 0.937946i \(-0.387275\pi\)
\(4\) −1.85991 + 0.735342i −0.929956 + 0.367671i
\(5\) 1.00000i 0.447214i 0.974679 + 0.223607i \(0.0717831\pi\)
−0.974679 + 0.223607i \(0.928217\pi\)
\(6\) −4.51380 + 0.859912i −1.84275 + 0.351058i
\(7\) 3.24914 1.22806 0.614030 − 0.789283i \(-0.289547\pi\)
0.614030 + 0.789283i \(0.289547\pi\)
\(8\) 1.51380 + 2.38923i 0.535209 + 0.844720i
\(9\) −7.55691 −2.51897
\(10\) 1.38923 − 0.264658i 0.439313 − 0.0836923i
\(11\) 1.05863i 0.319190i 0.987183 + 0.159595i \(0.0510188\pi\)
−0.987183 + 0.159595i \(0.948981\pi\)
\(12\) 2.38923 + 6.04312i 0.689711 + 1.74450i
\(13\) − 1.00000i − 0.277350i
\(14\) −0.859912 − 4.51380i −0.229821 − 1.20636i
\(15\) 3.24914 0.838924
\(16\) 2.91855 − 2.73534i 0.729636 − 0.683835i
\(17\) 1.00000 0.242536 0.121268 − 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 2.00000 + 10.4983i 0.471405 + 2.47447i
\(19\) − 4.00000i − 0.917663i −0.888523 − 0.458831i \(-0.848268\pi\)
0.888523 − 0.458831i \(-0.151732\pi\)
\(20\) −0.735342 − 1.85991i −0.164427 − 0.415889i
\(21\) − 10.5569i − 2.30371i
\(22\) 1.47068 − 0.280176i 0.313551 − 0.0597337i
\(23\) −2.94137 −0.613317 −0.306659 − 0.951820i \(-0.599211\pi\)
−0.306659 + 0.951820i \(0.599211\pi\)
\(24\) 7.76294 − 4.91855i 1.58460 − 1.00399i
\(25\) 4.00000 0.800000
\(26\) −1.38923 + 0.264658i −0.272450 + 0.0519038i
\(27\) 14.8061i 2.84943i
\(28\) −6.04312 + 2.38923i −1.14204 + 0.451522i
\(29\) 7.43965i 1.38151i 0.723090 + 0.690754i \(0.242721\pi\)
−0.723090 + 0.690754i \(0.757279\pi\)
\(30\) −0.859912 − 4.51380i −0.156998 − 0.824103i
\(31\) 5.05863 0.908557 0.454279 − 0.890860i \(-0.349897\pi\)
0.454279 + 0.890860i \(0.349897\pi\)
\(32\) −4.57243 − 3.33060i −0.808299 − 0.588772i
\(33\) 3.43965 0.598766
\(34\) −0.264658 − 1.38923i −0.0453885 − 0.238251i
\(35\) 3.24914i 0.549205i
\(36\) 14.0552 − 5.55691i 2.34253 − 0.926152i
\(37\) 6.55691i 1.07795i 0.842322 + 0.538975i \(0.181188\pi\)
−0.842322 + 0.538975i \(0.818812\pi\)
\(38\) −5.55691 + 1.05863i −0.901451 + 0.171733i
\(39\) −3.24914 −0.520279
\(40\) −2.38923 + 1.51380i −0.377770 + 0.239353i
\(41\) −9.43965 −1.47423 −0.737113 − 0.675770i \(-0.763811\pi\)
−0.737113 + 0.675770i \(0.763811\pi\)
\(42\) −14.6660 + 2.79397i −2.26301 + 0.431120i
\(43\) − 0.307774i − 0.0469350i −0.999725 − 0.0234675i \(-0.992529\pi\)
0.999725 − 0.0234675i \(-0.00747063\pi\)
\(44\) −0.778457 − 1.96896i −0.117357 − 0.296833i
\(45\) − 7.55691i − 1.12652i
\(46\) 0.778457 + 4.08623i 0.114777 + 0.602482i
\(47\) 6.80605 0.992765 0.496383 − 0.868104i \(-0.334661\pi\)
0.496383 + 0.868104i \(0.334661\pi\)
\(48\) −8.88751 − 9.48276i −1.28280 − 1.36872i
\(49\) 3.55691 0.508131
\(50\) −1.05863 − 5.55691i −0.149713 − 0.785866i
\(51\) − 3.24914i − 0.454971i
\(52\) 0.735342 + 1.85991i 0.101974 + 0.257923i
\(53\) − 1.55691i − 0.213859i −0.994267 − 0.106929i \(-0.965898\pi\)
0.994267 − 0.106929i \(-0.0341019\pi\)
\(54\) 20.5690 − 3.91855i 2.79909 − 0.533246i
\(55\) −1.05863 −0.142746
\(56\) 4.91855 + 7.76294i 0.657268 + 1.03737i
\(57\) −12.9966 −1.72144
\(58\) 10.3354 − 1.96896i 1.35710 − 0.258538i
\(59\) − 5.67418i − 0.738715i −0.929287 − 0.369358i \(-0.879578\pi\)
0.929287 − 0.369358i \(-0.120422\pi\)
\(60\) −6.04312 + 2.38923i −0.780163 + 0.308448i
\(61\) 9.67418i 1.23865i 0.785134 + 0.619326i \(0.212594\pi\)
−0.785134 + 0.619326i \(0.787406\pi\)
\(62\) −1.33881 − 7.02760i −0.170029 − 0.892506i
\(63\) −24.5535 −3.09345
\(64\) −3.41683 + 7.23362i −0.427103 + 0.904203i
\(65\) 1.00000 0.124035
\(66\) −0.910331 − 4.77846i −0.112054 − 0.588187i
\(67\) − 1.50172i − 0.183464i −0.995784 − 0.0917321i \(-0.970760\pi\)
0.995784 − 0.0917321i \(-0.0292403\pi\)
\(68\) −1.85991 + 0.735342i −0.225547 + 0.0891733i
\(69\) 9.55691i 1.15052i
\(70\) 4.51380 − 0.859912i 0.539502 − 0.102779i
\(71\) −5.36641 −0.636875 −0.318438 − 0.947944i \(-0.603158\pi\)
−0.318438 + 0.947944i \(0.603158\pi\)
\(72\) −11.4396 − 18.0552i −1.34818 − 2.12783i
\(73\) 3.55691 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(74\) 9.10905 − 1.73534i 1.05891 − 0.201729i
\(75\) − 12.9966i − 1.50071i
\(76\) 2.94137 + 7.43965i 0.337398 + 0.853386i
\(77\) 3.43965i 0.391984i
\(78\) 0.859912 + 4.51380i 0.0973659 + 0.511087i
\(79\) −6.73281 −0.757501 −0.378750 − 0.925499i \(-0.623646\pi\)
−0.378750 + 0.925499i \(0.623646\pi\)
\(80\) 2.73534 + 2.91855i 0.305821 + 0.326303i
\(81\) 25.4362 2.82625
\(82\) 2.49828 + 13.1138i 0.275889 + 1.44818i
\(83\) 2.49828i 0.274222i 0.990556 + 0.137111i \(0.0437817\pi\)
−0.990556 + 0.137111i \(0.956218\pi\)
\(84\) 7.76294 + 19.6349i 0.847006 + 2.14235i
\(85\) 1.00000i 0.108465i
\(86\) −0.427568 + 0.0814549i −0.0461058 + 0.00878350i
\(87\) 24.1725 2.59156
\(88\) −2.52932 + 1.60256i −0.269626 + 0.170833i
\(89\) 2.11727 0.224430 0.112215 − 0.993684i \(-0.464205\pi\)
0.112215 + 0.993684i \(0.464205\pi\)
\(90\) −10.4983 + 2.00000i −1.10662 + 0.210819i
\(91\) − 3.24914i − 0.340602i
\(92\) 5.47068 − 2.16291i 0.570358 − 0.225499i
\(93\) − 16.4362i − 1.70436i
\(94\) −1.80128 − 9.45517i −0.185788 − 0.975226i
\(95\) 4.00000 0.410391
\(96\) −10.8216 + 14.8565i −1.10447 + 1.51628i
\(97\) 7.67418 0.779195 0.389597 − 0.920985i \(-0.372614\pi\)
0.389597 + 0.920985i \(0.372614\pi\)
\(98\) −0.941367 − 4.94137i −0.0950924 − 0.499153i
\(99\) − 8.00000i − 0.804030i
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 104.2.b.c.53.3 ✓ 6
3.2 odd 2 936.2.g.c.469.4 6
4.3 odd 2 416.2.b.c.209.6 6
8.3 odd 2 416.2.b.c.209.1 6
8.5 even 2 inner 104.2.b.c.53.4 yes 6
12.11 even 2 3744.2.g.c.1873.1 6
16.3 odd 4 3328.2.a.bf.1.1 3
16.5 even 4 3328.2.a.be.1.1 3
16.11 odd 4 3328.2.a.bg.1.3 3
16.13 even 4 3328.2.a.bh.1.3 3
24.5 odd 2 936.2.g.c.469.3 6
24.11 even 2 3744.2.g.c.1873.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.b.c.53.3 ✓ 6 1.1 even 1 trivial
104.2.b.c.53.4 yes 6 8.5 even 2 inner
416.2.b.c.209.1 6 8.3 odd 2
416.2.b.c.209.6 6 4.3 odd 2
936.2.g.c.469.3 6 24.5 odd 2
936.2.g.c.469.4 6 3.2 odd 2
3328.2.a.be.1.1 3 16.5 even 4
3328.2.a.bf.1.1 3 16.3 odd 4
3328.2.a.bg.1.3 3 16.11 odd 4
3328.2.a.bh.1.3 3 16.13 even 4
3744.2.g.c.1873.1 6 12.11 even 2
3744.2.g.c.1873.4 6 24.11 even 2