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 59.9
Root \(-3.49930i\) of defining polynomial
Character \(\chi\) \(=\) 120.59
Dual form 120.2.m.b.59.12

$q$-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+(0.541196 - 1.30656i) q^{2} +(1.30656 - 1.13705i) q^{3} +(-1.41421 - 1.41421i) q^{4} +(-2.10100 + 0.765367i) q^{5} +(-0.778527 - 2.32248i) q^{6} +2.27411 q^{7} +(-2.61313 + 1.08239i) q^{8} +(0.414214 - 2.97127i) q^{9} +(-0.137055 + 3.15931i) q^{10} +4.20201i q^{11} +(-3.45580 - 0.239721i) q^{12} +3.21608 q^{13} +(1.23074 - 2.97127i) q^{14} +(-1.87483 + 3.38896i) q^{15} +4.00000i q^{16} +1.53073 q^{17} +(-3.65798 - 2.14923i) q^{18} -4.82843 q^{19} +(4.05366 + 1.88887i) q^{20} +(2.97127 - 2.58579i) q^{21} +(5.49019 + 2.27411i) q^{22} -1.08239i q^{23} +(-2.18347 + 4.38548i) q^{24} +(3.82843 - 3.21608i) q^{25} +(1.74053 - 4.20201i) q^{26} +(-2.83730 - 4.35313i) q^{27} +(-3.21608 - 3.21608i) q^{28} -1.74053 q^{29} +(3.41323 + 4.28367i) q^{30} +6.82843i q^{31} +(5.22625 + 2.16478i) q^{32} +(4.77791 + 5.49019i) q^{33} +(0.828427 - 2.00000i) q^{34} +(-4.77791 + 1.74053i) q^{35} +(-4.78779 + 3.61622i) q^{36} -7.76429 q^{37} +(-2.61313 + 6.30864i) q^{38} +(4.20201 - 3.65685i) q^{39} +(4.66176 - 4.27411i) q^{40} -2.46148i q^{41} +(-1.77045 - 5.28156i) q^{42} +8.70626i q^{43} +(5.94253 - 5.94253i) q^{44} +(1.40385 + 6.55967i) q^{45} +(-1.41421 - 0.585786i) q^{46} +1.08239i q^{47} +(4.54822 + 5.22625i) q^{48} -1.82843 q^{49} +(-2.13008 - 6.74261i) q^{50} +(2.00000 - 1.74053i) q^{51} +(-4.54822 - 4.54822i) q^{52} -11.0866i q^{53} +(-7.22317 + 1.35121i) q^{54} +(-3.21608 - 8.82843i) q^{55} +(-5.94253 + 2.46148i) q^{56} +(-6.30864 + 5.49019i) q^{57} +(-0.941967 + 2.27411i) q^{58} -4.20201i q^{59} +(7.44411 - 2.14130i) q^{60} -8.48528i q^{61} +(8.92177 + 3.69552i) q^{62} +(0.941967 - 6.75699i) q^{63} +(5.65685 - 5.65685i) q^{64} +(-6.75699 + 2.46148i) q^{65} +(9.75906 - 3.27137i) q^{66} +2.27411i q^{67} +(-2.16478 - 2.16478i) q^{68} +(-1.23074 - 1.41421i) q^{69} +(-0.311677 + 7.18461i) q^{70} -11.8851 q^{71} +(2.13368 + 8.21264i) q^{72} -4.54822i q^{73} +(-4.20201 + 10.1445i) q^{74} +(1.34523 - 8.55514i) q^{75} +(6.82843 + 6.82843i) q^{76} +9.55582i q^{77} +(-2.50380 - 7.46926i) q^{78} +0.485281i q^{79} +(-3.06147 - 8.40401i) q^{80} +(-8.65685 - 2.46148i) q^{81} +(-3.21608 - 1.33214i) q^{82} +6.94269 q^{83} +(-7.85886 - 0.545152i) q^{84} +(-3.21608 + 1.17157i) q^{85} +(11.3753 + 4.71179i) q^{86} +(-2.27411 + 1.97908i) q^{87} +(-4.54822 - 10.9804i) q^{88} +8.40401i q^{89} +(9.33037 + 1.71585i) q^{90} +7.31371 q^{91} +(-1.53073 + 1.53073i) q^{92} +(7.76429 + 8.92177i) q^{93} +(1.41421 + 0.585786i) q^{94} +(10.1445 - 3.69552i) q^{95} +(9.28991 - 3.11411i) q^{96} -10.9804i q^{97} +(-0.989538 + 2.38896i) q^{98} +(12.4853 + 1.74053i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.541196 − 1.30656i 0.382683 − 0.923880i
\(3\) 1.30656 − 1.13705i 0.754344 − 0.656479i
\(4\) −1.41421 − 1.41421i −0.707107 − 0.707107i
\(5\) −2.10100 + 0.765367i −0.939597 + 0.342282i
\(6\) −0.778527 − 2.32248i −0.317832 − 0.948147i
\(7\) 2.27411 0.859533 0.429766 − 0.902940i \(-0.358596\pi\)
0.429766 + 0.902940i \(0.358596\pi\)
\(8\) −2.61313 + 1.08239i −0.923880 + 0.382683i
\(9\) 0.414214 − 2.97127i 0.138071 − 0.990422i
\(10\) −0.137055 + 3.15931i −0.0433405 + 0.999060i
\(11\) 4.20201i 1.26695i 0.773762 + 0.633476i \(0.218373\pi\)
−0.773762 + 0.633476i \(0.781627\pi\)
\(12\) −3.45580 − 0.239721i −0.997603 − 0.0692015i
\(13\) 3.21608 0.891979 0.445990 − 0.895038i \(-0.352852\pi\)
0.445990 + 0.895038i \(0.352852\pi\)
\(14\) 1.23074 − 2.97127i 0.328929 − 0.794104i
\(15\) −1.87483 + 3.38896i −0.484079 + 0.875024i
\(16\) 4.00000i 1.00000i
\(17\) 1.53073 0.371257 0.185629 − 0.982620i \(-0.440568\pi\)
0.185629 + 0.982620i \(0.440568\pi\)
\(18\) −3.65798 − 2.14923i −0.862193 − 0.506579i
\(19\) −4.82843 −1.10772 −0.553859 − 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(20\) 4.05366 + 1.88887i 0.906426 + 0.422365i
\(21\) 2.97127 − 2.58579i 0.648384 − 0.564265i
\(22\) 5.49019 + 2.27411i 1.17051 + 0.484842i
\(23\) − 1.08239i − 0.225694i −0.993612 − 0.112847i \(-0.964003\pi\)
0.993612 − 0.112847i \(-0.0359971\pi\)
\(24\) −2.18347 + 4.38548i −0.445700 + 0.895182i
\(25\) 3.82843 − 3.21608i 0.765685 − 0.643215i
\(26\) 1.74053 − 4.20201i 0.341346 − 0.824081i
\(27\) −2.83730 − 4.35313i −0.546038 − 0.837760i
\(28\) −3.21608 − 3.21608i −0.607781 − 0.607781i
\(29\) −1.74053 −0.323208 −0.161604 − 0.986856i \(-0.551667\pi\)
−0.161604 + 0.986856i \(0.551667\pi\)
\(30\) 3.41323 + 4.28367i 0.623168 + 0.782088i
\(31\) 6.82843i 1.22642i 0.789919 + 0.613211i \(0.210122\pi\)
−0.789919 + 0.613211i \(0.789878\pi\)
\(32\) 5.22625 + 2.16478i 0.923880 + 0.382683i
\(33\) 4.77791 + 5.49019i 0.831727 + 0.955719i
\(34\) 0.828427 − 2.00000i 0.142074 − 0.342997i
\(35\) −4.77791 + 1.74053i −0.807614 + 0.294203i
\(36\) −4.78779 + 3.61622i −0.797965 + 0.602703i
\(37\) −7.76429 −1.27644 −0.638221 − 0.769853i \(-0.720329\pi\)
−0.638221 + 0.769853i \(0.720329\pi\)
\(38\) −2.61313 + 6.30864i −0.423905 + 1.02340i
\(39\) 4.20201 − 3.65685i 0.672859 − 0.585565i
\(40\) 4.66176 − 4.27411i 0.737089 − 0.675796i
\(41\) − 2.46148i − 0.384418i −0.981354 − 0.192209i \(-0.938435\pi\)
0.981354 − 0.192209i \(-0.0615652\pi\)
\(42\) −1.77045 − 5.28156i −0.273187 − 0.814963i
\(43\) 8.70626i 1.32769i 0.747869 + 0.663846i \(0.231077\pi\)
−0.747869 + 0.663846i \(0.768923\pi\)
\(44\) 5.94253 − 5.94253i 0.895871 − 0.895871i
\(45\) 1.40385 + 6.55967i 0.209273 + 0.977857i
\(46\) −1.41421 − 0.585786i −0.208514 − 0.0863695i
\(47\) 1.08239i 0.157883i 0.996879 + 0.0789416i \(0.0251541\pi\)
−0.996879 + 0.0789416i \(0.974846\pi\)
\(48\) 4.54822 + 5.22625i 0.656479 + 0.754344i
\(49\) −1.82843 −0.261204
\(50\) −2.13008 − 6.74261i −0.301238 − 0.953549i
\(51\) 2.00000 − 1.74053i 0.280056 − 0.243723i
\(52\) −4.54822 − 4.54822i −0.630724 − 0.630724i
\(53\) − 11.0866i − 1.52286i −0.648250 − 0.761428i \(-0.724499\pi\)
0.648250 − 0.761428i \(-0.275501\pi\)
\(54\) −7.22317 + 1.35121i −0.982949 + 0.183876i
\(55\) −3.21608 − 8.82843i −0.433656 − 1.19042i
\(56\) −5.94253 + 2.46148i −0.794104 + 0.328929i
\(57\) −6.30864 + 5.49019i −0.835600 + 0.727193i
\(58\) −0.941967 + 2.27411i −0.123686 + 0.298605i
\(59\) − 4.20201i − 0.547055i −0.961864 − 0.273527i \(-0.911810\pi\)
0.961864 − 0.273527i \(-0.0881904\pi\)
\(60\) 7.44411 − 2.14130i 0.961031 − 0.276440i
\(61\) − 8.48528i − 1.08643i −0.839594 − 0.543214i \(-0.817207\pi\)
0.839594 − 0.543214i \(-0.182793\pi\)
\(62\) 8.92177 + 3.69552i 1.13307 + 0.469331i
\(63\) 0.941967 − 6.75699i 0.118677 − 0.851300i
\(64\) 5.65685 − 5.65685i 0.707107 − 0.707107i
\(65\) −6.75699 + 2.46148i −0.838101 + 0.305309i
\(66\) 9.75906 − 3.27137i 1.20126 − 0.402678i
\(67\) 2.27411i 0.277827i 0.990305 + 0.138913i \(0.0443609\pi\)
−0.990305 + 0.138913i \(0.955639\pi\)
\(68\) −2.16478 − 2.16478i −0.262519 − 0.262519i
\(69\) −1.23074 − 1.41421i −0.148164 − 0.170251i
\(70\) −0.311677 + 7.18461i −0.0372525 + 0.858725i
\(71\) −11.8851 −1.41050 −0.705249 − 0.708960i \(-0.749165\pi\)
−0.705249 + 0.708960i \(0.749165\pi\)
\(72\) 2.13368 + 8.21264i 0.251457 + 0.967868i
\(73\) − 4.54822i − 0.532329i −0.963928 − 0.266164i \(-0.914244\pi\)
0.963928 − 0.266164i \(-0.0857564\pi\)
\(74\) −4.20201 + 10.1445i −0.488473 + 1.17928i
\(75\) 1.34523 − 8.55514i 0.155333 − 0.987862i
\(76\) 6.82843 + 6.82843i 0.783274 + 0.783274i
\(77\) 9.55582i 1.08899i
\(78\) −2.50380 − 7.46926i −0.283500 − 0.845727i
\(79\) 0.485281i 0.0545984i 0.999627 + 0.0272992i \(0.00869069\pi\)
−0.999627 + 0.0272992i \(0.991309\pi\)
\(80\) −3.06147 − 8.40401i −0.342282 − 0.939597i
\(81\) −8.65685 − 2.46148i −0.961873 − 0.273498i
\(82\) −3.21608 − 1.33214i −0.355156 − 0.147111i
\(83\) 6.94269 0.762060 0.381030 − 0.924563i \(-0.375569\pi\)
0.381030 + 0.924563i \(0.375569\pi\)
\(84\) −7.85886 − 0.545152i −0.857472 − 0.0594809i
\(85\) −3.21608 + 1.17157i −0.348832 + 0.127075i
\(86\) 11.3753 + 4.71179i 1.22663 + 0.508086i
\(87\) −2.27411 + 1.97908i −0.243810 + 0.212179i
\(88\) −4.54822 − 10.9804i −0.484842 − 1.17051i
\(89\) 8.40401i 0.890823i 0.895326 + 0.445412i \(0.146943\pi\)
−0.895326 + 0.445412i \(0.853057\pi\)
\(90\) 9.33037 + 1.71585i 0.983508 + 0.180867i
\(91\) 7.31371 0.766685
\(92\) −1.53073 + 1.53073i −0.159590 + 0.159590i
\(93\) 7.76429 + 8.92177i 0.805120 + 0.925144i
\(94\) 1.41421 + 0.585786i 0.145865 + 0.0604193i
\(95\) 10.1445 − 3.69552i 1.04081 − 0.379152i
\(96\) 9.28991 − 3.11411i 0.948147 − 0.317832i
\(97\) − 10.9804i − 1.11489i −0.830215 − 0.557444i \(-0.811782\pi\)
0.830215 − 0.557444i \(-0.188218\pi\)
\(98\) −0.989538 + 2.38896i −0.0999584 + 0.241321i
\(99\) 12.4853 + 1.74053i 1.25482 + 0.174930i
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.59.9 yes 16
3.2 odd 2 inner 120.2.m.b.59.7 yes 16
4.3 odd 2 480.2.m.b.239.3 16
5.2 odd 4 600.2.b.i.251.15 16
5.3 odd 4 600.2.b.i.251.2 16
5.4 even 2 inner 120.2.m.b.59.8 yes 16
8.3 odd 2 inner 120.2.m.b.59.11 yes 16
8.5 even 2 480.2.m.b.239.4 16
12.11 even 2 480.2.m.b.239.16 16
15.2 even 4 600.2.b.i.251.1 16
15.8 even 4 600.2.b.i.251.16 16
15.14 odd 2 inner 120.2.m.b.59.10 yes 16
20.3 even 4 2400.2.b.i.2351.6 16
20.7 even 4 2400.2.b.i.2351.11 16
20.19 odd 2 480.2.m.b.239.13 16
24.5 odd 2 480.2.m.b.239.15 16
24.11 even 2 inner 120.2.m.b.59.5 ✓ 16
40.3 even 4 600.2.b.i.251.14 16
40.13 odd 4 2400.2.b.i.2351.5 16
40.19 odd 2 inner 120.2.m.b.59.6 yes 16
40.27 even 4 600.2.b.i.251.3 16
40.29 even 2 480.2.m.b.239.14 16
40.37 odd 4 2400.2.b.i.2351.12 16
60.23 odd 4 2400.2.b.i.2351.8 16
60.47 odd 4 2400.2.b.i.2351.9 16
60.59 even 2 480.2.m.b.239.2 16
120.29 odd 2 480.2.m.b.239.1 16
120.53 even 4 2400.2.b.i.2351.7 16
120.59 even 2 inner 120.2.m.b.59.12 yes 16
120.77 even 4 2400.2.b.i.2351.10 16
120.83 odd 4 600.2.b.i.251.4 16
120.107 odd 4 600.2.b.i.251.13 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.m.b.59.5 ✓ 16 24.11 even 2 inner
120.2.m.b.59.6 yes 16 40.19 odd 2 inner
120.2.m.b.59.7 yes 16 3.2 odd 2 inner
120.2.m.b.59.8 yes 16 5.4 even 2 inner
120.2.m.b.59.9 yes 16 1.1 even 1 trivial
120.2.m.b.59.10 yes 16 15.14 odd 2 inner
120.2.m.b.59.11 yes 16 8.3 odd 2 inner
120.2.m.b.59.12 yes 16 120.59 even 2 inner
480.2.m.b.239.1 16 120.29 odd 2
480.2.m.b.239.2 16 60.59 even 2
480.2.m.b.239.3 16 4.3 odd 2
480.2.m.b.239.4 16 8.5 even 2
480.2.m.b.239.13 16 20.19 odd 2
480.2.m.b.239.14 16 40.29 even 2
480.2.m.b.239.15 16 24.5 odd 2
480.2.m.b.239.16 16 12.11 even 2
600.2.b.i.251.1 16 15.2 even 4
600.2.b.i.251.2 16 5.3 odd 4
600.2.b.i.251.3 16 40.27 even 4
600.2.b.i.251.4 16 120.83 odd 4
600.2.b.i.251.13 16 120.107 odd 4
600.2.b.i.251.14 16 40.3 even 4
600.2.b.i.251.15 16 5.2 odd 4
600.2.b.i.251.16 16 15.8 even 4
2400.2.b.i.2351.5 16 40.13 odd 4
2400.2.b.i.2351.6 16 20.3 even 4
2400.2.b.i.2351.7 16 120.53 even 4
2400.2.b.i.2351.8 16 60.23 odd 4
2400.2.b.i.2351.9 16 60.47 odd 4
2400.2.b.i.2351.10 16 120.77 even 4
2400.2.b.i.2351.11 16 20.7 even 4
2400.2.b.i.2351.12 16 40.37 odd 4