Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [120,2,Mod(59,120)] 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("120.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 59.13
Root \(-0.357857i\) of defining polynomial
Character \(\chi\) \(=\) 120.59
Dual form 120.2.m.b.59.16

$q$-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+(1.30656 - 0.541196i) q^{2} +(-0.541196 - 1.64533i) q^{3} +(1.41421 - 1.41421i) q^{4} +(1.25928 + 1.84776i) q^{5} +(-1.59755 - 1.85683i) q^{6} -3.29066 q^{7} +(1.08239 - 2.61313i) q^{8} +(-2.41421 + 1.78089i) q^{9} +(2.64533 + 1.73270i) q^{10} +2.51856i q^{11} +(-3.09221 - 1.56148i) q^{12} +4.65369 q^{13} +(-4.29945 + 1.78089i) q^{14} +(2.35865 - 3.07193i) q^{15} -4.00000i q^{16} -3.69552 q^{17} +(-2.19051 + 3.63341i) q^{18} +0.828427 q^{19} +(4.39402 + 0.832235i) q^{20} +(1.78089 + 5.41421i) q^{21} +(1.36303 + 3.29066i) q^{22} +2.61313i q^{23} +(-4.88524 - 0.366677i) q^{24} +(-1.82843 + 4.65369i) q^{25} +(6.08034 - 2.51856i) q^{26} +(4.23671 + 3.00836i) q^{27} +(-4.65369 + 4.65369i) q^{28} -6.08034 q^{29} +(1.41921 - 5.29016i) q^{30} -1.17157i q^{31} +(-2.16478 - 5.22625i) q^{32} +(4.14386 - 1.36303i) q^{33} +(-4.82843 + 2.00000i) q^{34} +(-4.14386 - 6.08034i) q^{35} +(-0.895653 + 5.93277i) q^{36} +1.92762 q^{37} +(1.08239 - 0.448342i) q^{38} +(-2.51856 - 7.65685i) q^{39} +(6.19146 - 1.29066i) q^{40} -8.59890i q^{41} +(5.25700 + 6.11020i) q^{42} -6.01673i q^{43} +(3.56178 + 3.56178i) q^{44} +(-6.33083 - 2.21824i) q^{45} +(1.41421 + 3.41421i) q^{46} -2.61313i q^{47} +(-6.58132 + 2.16478i) q^{48} +3.82843 q^{49} +(0.129605 + 7.06988i) q^{50} +(2.00000 + 6.08034i) q^{51} +(6.58132 - 6.58132i) q^{52} +4.59220i q^{53} +(7.16365 + 1.63772i) q^{54} +(-4.65369 + 3.17157i) q^{55} +(-3.56178 + 8.59890i) q^{56} +(-0.448342 - 1.36303i) q^{57} +(-7.94435 + 3.29066i) q^{58} -2.51856i q^{59} +(-1.00872 - 7.68000i) q^{60} -8.48528i q^{61} +(-0.634051 - 1.53073i) q^{62} +(7.94435 - 5.86030i) q^{63} +(-5.65685 - 5.65685i) q^{64} +(5.86030 + 8.59890i) q^{65} +(4.67654 - 4.02353i) q^{66} +3.29066i q^{67} +(-5.22625 + 5.22625i) q^{68} +(4.29945 - 1.41421i) q^{69} +(-8.70487 - 5.70171i) q^{70} -7.12356 q^{71} +(2.04057 + 8.23627i) q^{72} -6.58132i q^{73} +(2.51856 - 1.04322i) q^{74} +(8.64639 + 0.489804i) q^{75} +(1.17157 - 1.17157i) q^{76} -8.28772i q^{77} +(-7.43452 - 8.64113i) q^{78} +16.4853i q^{79} +(7.39104 - 5.03712i) q^{80} +(2.65685 - 8.59890i) q^{81} +(-4.65369 - 11.2350i) q^{82} +9.37011 q^{83} +(10.1754 + 5.13829i) q^{84} +(-4.65369 - 6.82843i) q^{85} +(-3.25623 - 7.86123i) q^{86} +(3.29066 + 10.0042i) q^{87} +(6.58132 + 2.72607i) q^{88} +5.03712i q^{89} +(-9.47213 + 0.527944i) q^{90} -15.3137 q^{91} +(3.69552 + 3.69552i) q^{92} +(-1.92762 + 0.634051i) q^{93} +(-1.41421 - 3.41421i) q^{94} +(1.04322 + 1.53073i) q^{95} +(-7.42733 + 6.39021i) q^{96} +2.72607i q^{97} +(5.00208 - 2.07193i) q^{98} +(-4.48528 - 6.08034i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9} + 16 q^{10} - 32 q^{19} - 32 q^{24} + 16 q^{25} + 16 q^{30} - 32 q^{34} - 32 q^{36} + 32 q^{40} + 16 q^{49} + 32 q^{51} + 32 q^{54} + 64 q^{66} - 64 q^{70} + 32 q^{75} + 64 q^{76} - 48 q^{81}+ \cdots + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30656 − 0.541196i 0.923880 − 0.382683i
\(3\) −0.541196 − 1.64533i −0.312460 − 0.949931i
\(4\) 1.41421 − 1.41421i 0.707107 − 0.707107i
\(5\) 1.25928 + 1.84776i 0.563167 + 0.826343i
\(6\) −1.59755 − 1.85683i −0.652198 − 0.758049i
\(7\) −3.29066 −1.24375 −0.621876 − 0.783116i \(-0.713629\pi\)
−0.621876 + 0.783116i \(0.713629\pi\)
\(8\) 1.08239 − 2.61313i 0.382683 − 0.923880i
\(9\) −2.41421 + 1.78089i −0.804738 + 0.593630i
\(10\) 2.64533 + 1.73270i 0.836526 + 0.547927i
\(11\) 2.51856i 0.759374i 0.925115 + 0.379687i \(0.123968\pi\)
−0.925115 + 0.379687i \(0.876032\pi\)
\(12\) −3.09221 − 1.56148i −0.892645 − 0.450760i
\(13\) 4.65369 1.29070 0.645351 − 0.763886i \(-0.276711\pi\)
0.645351 + 0.763886i \(0.276711\pi\)
\(14\) −4.29945 + 1.78089i −1.14908 + 0.475963i
\(15\) 2.35865 − 3.07193i 0.609002 − 0.793169i
\(16\) − 4.00000i − 1.00000i
\(17\) −3.69552 −0.896295 −0.448147 − 0.893960i \(-0.647916\pi\)
−0.448147 + 0.893960i \(0.647916\pi\)
\(18\) −2.19051 + 3.63341i −0.516308 + 0.856403i
\(19\) 0.828427 0.190054 0.0950271 − 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) 4.39402 + 0.832235i 0.982532 + 0.186093i
\(21\) 1.78089 + 5.41421i 0.388622 + 1.18148i
\(22\) 1.36303 + 3.29066i 0.290600 + 0.701571i
\(23\) 2.61313i 0.544874i 0.962174 + 0.272437i \(0.0878297\pi\)
−0.962174 + 0.272437i \(0.912170\pi\)
\(24\) −4.88524 − 0.366677i −0.997195 − 0.0748477i
\(25\) −1.82843 + 4.65369i −0.365685 + 0.930739i
\(26\) 6.08034 − 2.51856i 1.19245 − 0.493930i
\(27\) 4.23671 + 3.00836i 0.815356 + 0.578960i
\(28\) −4.65369 + 4.65369i −0.879465 + 0.879465i
\(29\) −6.08034 −1.12909 −0.564546 − 0.825402i \(-0.690949\pi\)
−0.564546 + 0.825402i \(0.690949\pi\)
\(30\) 1.41921 − 5.29016i 0.259112 − 0.965847i
\(31\) − 1.17157i − 0.210421i −0.994450 − 0.105210i \(-0.966448\pi\)
0.994450 − 0.105210i \(-0.0335516\pi\)
\(32\) −2.16478 − 5.22625i −0.382683 − 0.923880i
\(33\) 4.14386 − 1.36303i 0.721353 − 0.237274i
\(34\) −4.82843 + 2.00000i −0.828068 + 0.342997i
\(35\) −4.14386 − 6.08034i −0.700440 − 1.02777i
\(36\) −0.895653 + 5.93277i −0.149276 + 0.988796i
\(37\) 1.92762 0.316899 0.158450 − 0.987367i \(-0.449350\pi\)
0.158450 + 0.987367i \(0.449350\pi\)
\(38\) 1.08239 − 0.448342i 0.175587 − 0.0727306i
\(39\) −2.51856 − 7.65685i −0.403292 − 1.22608i
\(40\) 6.19146 − 1.29066i 0.978956 − 0.204071i
\(41\) − 8.59890i − 1.34292i −0.741039 − 0.671461i \(-0.765667\pi\)
0.741039 − 0.671461i \(-0.234333\pi\)
\(42\) 5.25700 + 6.11020i 0.811172 + 0.942824i
\(43\) − 6.01673i − 0.917542i −0.888554 − 0.458771i \(-0.848290\pi\)
0.888554 − 0.458771i \(-0.151710\pi\)
\(44\) 3.56178 + 3.56178i 0.536959 + 0.536959i
\(45\) −6.33083 − 2.21824i −0.943744 − 0.330676i
\(46\) 1.41421 + 3.41421i 0.208514 + 0.503398i
\(47\) − 2.61313i − 0.381164i −0.981671 − 0.190582i \(-0.938963\pi\)
0.981671 − 0.190582i \(-0.0610374\pi\)
\(48\) −6.58132 + 2.16478i −0.949931 + 0.312460i
\(49\) 3.82843 0.546918
\(50\) 0.129605 + 7.06988i 0.0183289 + 0.999832i
\(51\) 2.00000 + 6.08034i 0.280056 + 0.851418i
\(52\) 6.58132 − 6.58132i 0.912664 − 0.912664i
\(53\) 4.59220i 0.630787i 0.948961 + 0.315394i \(0.102137\pi\)
−0.948961 + 0.315394i \(0.897863\pi\)
\(54\) 7.16365 + 1.63772i 0.974849 + 0.222866i
\(55\) −4.65369 + 3.17157i −0.627504 + 0.427655i
\(56\) −3.56178 + 8.59890i −0.475963 + 1.14908i
\(57\) −0.448342 − 1.36303i −0.0593843 − 0.180538i
\(58\) −7.94435 + 3.29066i −1.04314 + 0.432085i
\(59\) − 2.51856i − 0.327889i −0.986470 − 0.163944i \(-0.947578\pi\)
0.986470 − 0.163944i \(-0.0524217\pi\)
\(60\) −1.00872 − 7.68000i −0.130226 − 0.991484i
\(61\) − 8.48528i − 1.08643i −0.839594 − 0.543214i \(-0.817207\pi\)
0.839594 − 0.543214i \(-0.182793\pi\)
\(62\) −0.634051 − 1.53073i −0.0805245 − 0.194403i
\(63\) 7.94435 − 5.86030i 1.00089 − 0.738329i
\(64\) −5.65685 − 5.65685i −0.707107 − 0.707107i
\(65\) 5.86030 + 8.59890i 0.726881 + 1.06656i
\(66\) 4.67654 − 4.02353i 0.575643 − 0.495263i
\(67\) 3.29066i 0.402018i 0.979589 + 0.201009i \(0.0644220\pi\)
−0.979589 + 0.201009i \(0.935578\pi\)
\(68\) −5.22625 + 5.22625i −0.633776 + 0.633776i
\(69\) 4.29945 − 1.41421i 0.517593 − 0.170251i
\(70\) −8.70487 − 5.70171i −1.04043 − 0.681485i
\(71\) −7.12356 −0.845412 −0.422706 − 0.906267i \(-0.638920\pi\)
−0.422706 + 0.906267i \(0.638920\pi\)
\(72\) 2.04057 + 8.23627i 0.240483 + 0.970653i
\(73\) − 6.58132i − 0.770285i −0.922857 − 0.385142i \(-0.874152\pi\)
0.922857 − 0.385142i \(-0.125848\pi\)
\(74\) 2.51856 − 1.04322i 0.292777 − 0.121272i
\(75\) 8.64639 + 0.489804i 0.998399 + 0.0565576i
\(76\) 1.17157 − 1.17157i 0.134389 − 0.134389i
\(77\) − 8.28772i − 0.944473i
\(78\) −7.43452 − 8.64113i −0.841793 − 0.978415i
\(79\) 16.4853i 1.85474i 0.374147 + 0.927370i \(0.377936\pi\)
−0.374147 + 0.927370i \(0.622064\pi\)
\(80\) 7.39104 − 5.03712i 0.826343 − 0.563167i
\(81\) 2.65685 − 8.59890i 0.295206 − 0.955434i
\(82\) −4.65369 − 11.2350i −0.513914 − 1.24070i
\(83\) 9.37011 1.02850 0.514252 − 0.857639i \(-0.328070\pi\)
0.514252 + 0.857639i \(0.328070\pi\)
\(84\) 10.1754 + 5.13829i 1.11023 + 0.560634i
\(85\) −4.65369 − 6.82843i −0.504764 − 0.740647i
\(86\) −3.25623 − 7.86123i −0.351128 − 0.847699i
\(87\) 3.29066 + 10.0042i 0.352796 + 1.07256i
\(88\) 6.58132 + 2.72607i 0.701571 + 0.290600i
\(89\) 5.03712i 0.533934i 0.963706 + 0.266967i \(0.0860214\pi\)
−0.963706 + 0.266967i \(0.913979\pi\)
\(90\) −9.47213 + 0.527944i −0.998450 + 0.0556502i
\(91\) −15.3137 −1.60531
\(92\) 3.69552 + 3.69552i 0.385284 + 0.385284i
\(93\) −1.92762 + 0.634051i −0.199885 + 0.0657480i
\(94\) −1.41421 − 3.41421i −0.145865 − 0.352149i
\(95\) 1.04322 + 1.53073i 0.107032 + 0.157050i
\(96\) −7.42733 + 6.39021i −0.758049 + 0.652198i
\(97\) 2.72607i 0.276790i 0.990377 + 0.138395i \(0.0441944\pi\)
−0.990377 + 0.138395i \(0.955806\pi\)
\(98\) 5.00208 − 2.07193i 0.505286 − 0.209297i
\(99\) −4.48528 − 6.08034i −0.450788 − 0.611097i
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 120.2.m.b.59.13 yes 16
3.2 odd 2 inner 120.2.m.b.59.3 yes 16
4.3 odd 2 480.2.m.b.239.12 16
5.2 odd 4 600.2.b.i.251.11 16
5.3 odd 4 600.2.b.i.251.6 16
5.4 even 2 inner 120.2.m.b.59.4 yes 16
8.3 odd 2 inner 120.2.m.b.59.15 yes 16
8.5 even 2 480.2.m.b.239.11 16
12.11 even 2 480.2.m.b.239.7 16
15.2 even 4 600.2.b.i.251.5 16
15.8 even 4 600.2.b.i.251.12 16
15.14 odd 2 inner 120.2.m.b.59.14 yes 16
20.3 even 4 2400.2.b.i.2351.3 16
20.7 even 4 2400.2.b.i.2351.14 16
20.19 odd 2 480.2.m.b.239.6 16
24.5 odd 2 480.2.m.b.239.8 16
24.11 even 2 inner 120.2.m.b.59.1 ✓ 16
40.3 even 4 600.2.b.i.251.10 16
40.13 odd 4 2400.2.b.i.2351.4 16
40.19 odd 2 inner 120.2.m.b.59.2 yes 16
40.27 even 4 600.2.b.i.251.7 16
40.29 even 2 480.2.m.b.239.5 16
40.37 odd 4 2400.2.b.i.2351.13 16
60.23 odd 4 2400.2.b.i.2351.1 16
60.47 odd 4 2400.2.b.i.2351.16 16
60.59 even 2 480.2.m.b.239.9 16
120.29 odd 2 480.2.m.b.239.10 16
120.53 even 4 2400.2.b.i.2351.2 16
120.59 even 2 inner 120.2.m.b.59.16 yes 16
120.77 even 4 2400.2.b.i.2351.15 16
120.83 odd 4 600.2.b.i.251.8 16
120.107 odd 4 600.2.b.i.251.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.m.b.59.1 ✓ 16 24.11 even 2 inner
120.2.m.b.59.2 yes 16 40.19 odd 2 inner
120.2.m.b.59.3 yes 16 3.2 odd 2 inner
120.2.m.b.59.4 yes 16 5.4 even 2 inner
120.2.m.b.59.13 yes 16 1.1 even 1 trivial
120.2.m.b.59.14 yes 16 15.14 odd 2 inner
120.2.m.b.59.15 yes 16 8.3 odd 2 inner
120.2.m.b.59.16 yes 16 120.59 even 2 inner
480.2.m.b.239.5 16 40.29 even 2
480.2.m.b.239.6 16 20.19 odd 2
480.2.m.b.239.7 16 12.11 even 2
480.2.m.b.239.8 16 24.5 odd 2
480.2.m.b.239.9 16 60.59 even 2
480.2.m.b.239.10 16 120.29 odd 2
480.2.m.b.239.11 16 8.5 even 2
480.2.m.b.239.12 16 4.3 odd 2
600.2.b.i.251.5 16 15.2 even 4
600.2.b.i.251.6 16 5.3 odd 4
600.2.b.i.251.7 16 40.27 even 4
600.2.b.i.251.8 16 120.83 odd 4
600.2.b.i.251.9 16 120.107 odd 4
600.2.b.i.251.10 16 40.3 even 4
600.2.b.i.251.11 16 5.2 odd 4
600.2.b.i.251.12 16 15.8 even 4
2400.2.b.i.2351.1 16 60.23 odd 4
2400.2.b.i.2351.2 16 120.53 even 4
2400.2.b.i.2351.3 16 20.3 even 4
2400.2.b.i.2351.4 16 40.13 odd 4
2400.2.b.i.2351.13 16 40.37 odd 4
2400.2.b.i.2351.14 16 20.7 even 4
2400.2.b.i.2351.15 16 120.77 even 4
2400.2.b.i.2351.16 16 60.47 odd 4