Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [74,2,Mod(7,74)] 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("74.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] 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: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 71.1
Root \(-2.20976 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 74.71
Dual form 74.2.f.b.49.1

$q$-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+(0.766044 + 0.642788i) q^{2} +(-2.41262 + 2.02443i) q^{3} +(0.173648 + 0.984808i) q^{4} +(-1.43969 - 0.524005i) q^{5} -3.14945 q^{6} +(4.53424 + 1.65033i) q^{7} +(-0.500000 + 0.866025i) q^{8} +(1.20148 - 6.81391i) q^{9} +(-0.766044 - 1.32683i) q^{10} +(0.546896 - 0.947252i) q^{11} +(-2.41262 - 2.02443i) q^{12} +(0.307284 + 1.74269i) q^{13} +(2.41262 + 4.17878i) q^{14} +(4.53424 - 1.65033i) q^{15} +(-0.939693 + 0.342020i) q^{16} +(0.511542 - 2.90110i) q^{17} +(5.30028 - 4.44747i) q^{18} +(-0.632167 + 0.530451i) q^{19} +(0.266044 - 1.50881i) q^{20} +(-14.2804 + 5.19762i) q^{21} +(1.02783 - 0.374099i) q^{22} +(0.121620 + 0.210653i) q^{23} +(-0.546896 - 3.10160i) q^{24} +(-2.03209 - 1.70513i) q^{25} +(-0.884789 + 1.53250i) q^{26} +(6.17139 + 10.6892i) q^{27} +(-0.837893 + 4.75193i) q^{28} +(2.78587 - 4.82526i) q^{29} +(4.53424 + 1.65033i) q^{30} -5.73495 q^{31} +(-0.939693 - 0.342020i) q^{32} +(0.598191 + 3.39251i) q^{33} +(2.25665 - 1.89356i) q^{34} +(-5.66313 - 4.75193i) q^{35} +6.91903 q^{36} +(5.84038 - 1.69999i) q^{37} -0.825235 q^{38} +(-4.26932 - 3.58238i) q^{39} +(1.17365 - 0.984808i) q^{40} +(0.505653 + 2.86770i) q^{41} +(-14.2804 - 5.19762i) q^{42} +4.37987 q^{43} +(1.02783 + 0.374099i) q^{44} +(-5.30028 + 9.18036i) q^{45} +(-0.0422383 + 0.239545i) q^{46} +(-1.13249 - 1.96153i) q^{47} +(1.57472 - 2.72750i) q^{48} +(12.4734 + 10.4664i) q^{49} +(-0.460637 - 2.61240i) q^{50} +(4.63890 + 8.03481i) q^{51} +(-1.66286 + 0.605232i) q^{52} +(-5.77859 + 2.10324i) q^{53} +(-2.14330 + 12.1553i) q^{54} +(-1.28373 + 1.07717i) q^{55} +(-3.69634 + 3.10160i) q^{56} +(0.451318 - 2.55955i) q^{57} +(5.23571 - 1.90564i) q^{58} +(-8.29301 + 3.01841i) q^{59} +(2.41262 + 4.17878i) q^{60} +(-2.04075 - 11.5737i) q^{61} +(-4.39322 - 3.68635i) q^{62} +(16.6930 - 28.9131i) q^{63} +(-0.500000 - 0.866025i) q^{64} +(0.470787 - 2.66996i) q^{65} +(-1.72242 + 2.98332i) q^{66} +(-8.44220 - 3.07271i) q^{67} +2.94585 q^{68} +(-0.719875 - 0.262013i) q^{69} +(-1.28373 - 7.28038i) q^{70} +(-0.933535 + 0.783329i) q^{71} +(5.30028 + 4.44747i) q^{72} +8.79418 q^{73} +(5.56672 + 2.45185i) q^{74} +8.35455 q^{75} +(-0.632167 - 0.530451i) q^{76} +(4.04303 - 3.39251i) q^{77} +(-0.967776 - 5.48853i) q^{78} +(-2.02379 - 0.736599i) q^{79} +1.53209 q^{80} +(-17.0233 - 6.19599i) q^{81} +(-1.45597 + 2.52181i) q^{82} +(0.884528 - 5.01641i) q^{83} +(-7.59842 - 13.1608i) q^{84} +(-2.25665 + 3.90864i) q^{85} +(3.35518 + 2.81533i) q^{86} +(3.04716 + 17.2813i) q^{87} +(0.546896 + 0.947252i) q^{88} +(-8.43486 + 3.07004i) q^{89} +(-9.96127 + 3.62561i) q^{90} +(-1.48272 + 8.40891i) q^{91} +(-0.186333 + 0.156352i) q^{92} +(13.8362 - 11.6100i) q^{93} +(0.393310 - 2.23057i) q^{94} +(1.18809 - 0.432428i) q^{95} +(2.95951 - 1.07717i) q^{96} +(-2.17954 - 3.77507i) q^{97} +(2.82750 + 16.0355i) q^{98} +(-5.79741 - 4.86460i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} - 3 q^{17} + 6 q^{18} - 3 q^{19} - 6 q^{20} - 33 q^{21} - 3 q^{22} - 21 q^{23} - 3 q^{24} - 6 q^{25}+ \cdots - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.766044 + 0.642788i 0.541675 + 0.454519i
\(3\) −2.41262 + 2.02443i −1.39293 + 1.16880i −0.428786 + 0.903406i \(0.641059\pi\)
−0.964139 + 0.265397i \(0.914497\pi\)
\(4\) 0.173648 + 0.984808i 0.0868241 + 0.492404i
\(5\) −1.43969 − 0.524005i −0.643850 − 0.234342i −0.000601695 − 1.00000i \(-0.500192\pi\)
−0.643248 + 0.765658i \(0.722414\pi\)
\(6\) −3.14945 −1.28576
\(7\) 4.53424 + 1.65033i 1.71378 + 0.623765i 0.997272 − 0.0738128i \(-0.0235167\pi\)
0.716509 + 0.697578i \(0.245739\pi\)
\(8\) −0.500000 + 0.866025i −0.176777 + 0.306186i
\(9\) 1.20148 − 6.81391i 0.400492 − 2.27130i
\(10\) −0.766044 − 1.32683i −0.242245 − 0.419580i
\(11\) 0.546896 − 0.947252i 0.164895 − 0.285607i −0.771723 − 0.635959i \(-0.780605\pi\)
0.936618 + 0.350352i \(0.113938\pi\)
\(12\) −2.41262 − 2.02443i −0.696463 − 0.584402i
\(13\) 0.307284 + 1.74269i 0.0852253 + 0.483337i 0.997308 + 0.0733298i \(0.0233626\pi\)
−0.912082 + 0.410007i \(0.865526\pi\)
\(14\) 2.41262 + 4.17878i 0.644799 + 1.11682i
\(15\) 4.53424 − 1.65033i 1.17074 − 0.426113i
\(16\) −0.939693 + 0.342020i −0.234923 + 0.0855050i
\(17\) 0.511542 − 2.90110i 0.124067 − 0.703619i −0.857791 − 0.513999i \(-0.828163\pi\)
0.981858 − 0.189620i \(-0.0607255\pi\)
\(18\) 5.30028 − 4.44747i 1.24929 − 1.04828i
\(19\) −0.632167 + 0.530451i −0.145029 + 0.121694i −0.712416 − 0.701758i \(-0.752399\pi\)
0.567387 + 0.823451i \(0.307954\pi\)
\(20\) 0.266044 − 1.50881i 0.0594893 − 0.337381i
\(21\) −14.2804 + 5.19762i −3.11623 + 1.13421i
\(22\) 1.02783 − 0.374099i 0.219134 − 0.0797582i
\(23\) 0.121620 + 0.210653i 0.0253596 + 0.0439241i 0.878427 − 0.477877i \(-0.158594\pi\)
−0.853067 + 0.521801i \(0.825260\pi\)
\(24\) −0.546896 − 3.10160i −0.111635 − 0.633112i
\(25\) −2.03209 − 1.70513i −0.406418 − 0.341025i
\(26\) −0.884789 + 1.53250i −0.173521 + 0.300548i
\(27\) 6.17139 + 10.6892i 1.18768 + 2.05713i
\(28\) −0.837893 + 4.75193i −0.158347 + 0.898030i
\(29\) 2.78587 − 4.82526i 0.517322 − 0.896028i −0.482475 − 0.875910i \(-0.660262\pi\)
0.999798 − 0.0201189i \(-0.00640447\pi\)
\(30\) 4.53424 + 1.65033i 0.827835 + 0.301307i
\(31\) −5.73495 −1.03003 −0.515013 − 0.857182i \(-0.672213\pi\)
−0.515013 + 0.857182i \(0.672213\pi\)
\(32\) −0.939693 − 0.342020i −0.166116 − 0.0604612i
\(33\) 0.598191 + 3.39251i 0.104132 + 0.590560i
\(34\) 2.25665 − 1.89356i 0.387013 − 0.324742i
\(35\) −5.66313 − 4.75193i −0.957243 − 0.803223i
\(36\) 6.91903 1.15317
\(37\) 5.84038 − 1.69999i 0.960152 − 0.279477i
\(38\) −0.825235 −0.133871
\(39\) −4.26932 − 3.58238i −0.683638 − 0.573640i
\(40\) 1.17365 − 0.984808i 0.185570 − 0.155712i
\(41\) 0.505653 + 2.86770i 0.0789697 + 0.447859i 0.998496 + 0.0548301i \(0.0174617\pi\)
−0.919526 + 0.393029i \(0.871427\pi\)
\(42\) −14.2804 − 5.19762i −2.20351 − 0.802011i
\(43\) 4.37987 0.667924 0.333962 − 0.942587i \(-0.391614\pi\)
0.333962 + 0.942587i \(0.391614\pi\)
\(44\) 1.02783 + 0.374099i 0.154951 + 0.0563975i
\(45\) −5.30028 + 9.18036i −0.790119 + 1.36853i
\(46\) −0.0422383 + 0.239545i −0.00622770 + 0.0353191i
\(47\) −1.13249 − 1.96153i −0.165191 − 0.286119i 0.771532 − 0.636190i \(-0.219491\pi\)
−0.936723 + 0.350071i \(0.886157\pi\)
\(48\) 1.57472 − 2.72750i 0.227292 − 0.393681i
\(49\) 12.4734 + 10.4664i 1.78192 + 1.49521i
\(50\) −0.460637 − 2.61240i −0.0651439 − 0.369450i
\(51\) 4.63890 + 8.03481i 0.649576 + 1.12510i
\(52\) −1.66286 + 0.605232i −0.230597 + 0.0839305i
\(53\) −5.77859 + 2.10324i −0.793751 + 0.288902i −0.706894 − 0.707319i \(-0.749904\pi\)
−0.0868564 + 0.996221i \(0.527682\pi\)
\(54\) −2.14330 + 12.1553i −0.291666 + 1.65412i
\(55\) −1.28373 + 1.07717i −0.173098 + 0.145246i
\(56\) −3.69634 + 3.10160i −0.493945 + 0.414469i
\(57\) 0.451318 − 2.55955i 0.0597785 − 0.339021i
\(58\) 5.23571 − 1.90564i 0.687483 − 0.250223i
\(59\) −8.29301 + 3.01841i −1.07966 + 0.392963i −0.819780 − 0.572678i \(-0.805904\pi\)
−0.259878 + 0.965642i \(0.583682\pi\)
\(60\) 2.41262 + 4.17878i 0.311468 + 0.539478i
\(61\) −2.04075 − 11.5737i −0.261292 − 1.48186i −0.779390 − 0.626539i \(-0.784471\pi\)
0.518098 − 0.855321i \(-0.326640\pi\)
\(62\) −4.39322 − 3.68635i −0.557940 − 0.468167i
\(63\) 16.6930 − 28.9131i 2.10312 − 3.64270i
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) 0.470787 − 2.66996i 0.0583939 − 0.331168i
\(66\) −1.72242 + 2.98332i −0.212015 + 0.367221i
\(67\) −8.44220 − 3.07271i −1.03138 − 0.375391i −0.229773 − 0.973244i \(-0.573798\pi\)
−0.801606 + 0.597853i \(0.796021\pi\)
\(68\) 2.94585 0.357237
\(69\) −0.719875 − 0.262013i −0.0866627 − 0.0315427i
\(70\) −1.28373 − 7.28038i −0.153435 − 0.870172i
\(71\) −0.933535 + 0.783329i −0.110790 + 0.0929640i −0.696500 − 0.717557i \(-0.745260\pi\)
0.585710 + 0.810521i \(0.300816\pi\)
\(72\) 5.30028 + 4.44747i 0.624644 + 0.524139i
\(73\) 8.79418 1.02928 0.514640 − 0.857406i \(-0.327925\pi\)
0.514640 + 0.857406i \(0.327925\pi\)
\(74\) 5.56672 + 2.45185i 0.647119 + 0.285022i
\(75\) 8.35455 0.964701
\(76\) −0.632167 − 0.530451i −0.0725145 − 0.0608469i
\(77\) 4.04303 − 3.39251i 0.460746 − 0.386612i
\(78\) −0.967776 − 5.48853i −0.109579 − 0.621453i
\(79\) −2.02379 − 0.736599i −0.227694 − 0.0828739i 0.225654 − 0.974208i \(-0.427548\pi\)
−0.453348 + 0.891334i \(0.649770\pi\)
\(80\) 1.53209 0.171293
\(81\) −17.0233 − 6.19599i −1.89148 − 0.688443i
\(82\) −1.45597 + 2.52181i −0.160785 + 0.278488i
\(83\) 0.884528 − 5.01641i 0.0970895 − 0.550622i −0.896997 − 0.442036i \(-0.854256\pi\)
0.994087 − 0.108586i \(-0.0346324\pi\)
\(84\) −7.59842 − 13.1608i −0.829055 − 1.43597i
\(85\) −2.25665 + 3.90864i −0.244768 + 0.423951i
\(86\) 3.35518 + 2.81533i 0.361798 + 0.303584i
\(87\) 3.04716 + 17.2813i 0.326690 + 1.85275i
\(88\) 0.546896 + 0.947252i 0.0582993 + 0.100977i
\(89\) −8.43486 + 3.07004i −0.894093 + 0.325423i −0.747883 − 0.663830i \(-0.768930\pi\)
−0.146210 + 0.989254i \(0.546708\pi\)
\(90\) −9.96127 + 3.62561i −1.05001 + 0.382173i
\(91\) −1.48272 + 8.40891i −0.155431 + 0.881494i
\(92\) −0.186333 + 0.156352i −0.0194266 + 0.0163008i
\(93\) 13.8362 − 11.6100i 1.43475 − 1.20390i
\(94\) 0.393310 − 2.23057i 0.0405669 − 0.230066i
\(95\) 1.18809 − 0.432428i 0.121895 − 0.0443661i
\(96\) 2.95951 − 1.07717i 0.302054 − 0.109939i
\(97\) −2.17954 − 3.77507i −0.221298 − 0.383300i 0.733904 − 0.679253i \(-0.237696\pi\)
−0.955203 + 0.295953i \(0.904363\pi\)
\(98\) 2.82750 + 16.0355i 0.285620 + 1.61983i
\(99\) −5.79741 − 4.86460i −0.582661 − 0.488911i
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 74.2.f.b.71.1 yes 12
3.2 odd 2 666.2.x.g.145.2 12
4.3 odd 2 592.2.bc.d.145.2 12
37.7 even 9 2738.2.a.t.1.1 6
37.12 even 9 inner 74.2.f.b.49.1 ✓ 12
37.30 even 18 2738.2.a.q.1.1 6
111.86 odd 18 666.2.x.g.271.2 12
148.123 odd 18 592.2.bc.d.49.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.1 ✓ 12 37.12 even 9 inner
74.2.f.b.71.1 yes 12 1.1 even 1 trivial
592.2.bc.d.49.2 12 148.123 odd 18
592.2.bc.d.145.2 12 4.3 odd 2
666.2.x.g.145.2 12 3.2 odd 2
666.2.x.g.271.2 12 111.86 odd 18
2738.2.a.q.1.1 6 37.30 even 18
2738.2.a.t.1.1 6 37.7 even 9