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 49.1
Root \(-2.20976 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 74.49
Dual form 74.2.f.b.71.1

$q$-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 \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{7}{9}\right)\)

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 −0.964139 − 0.265397i \(-0.914497\pi\)
−0.428786 − 0.903406i \(-0.641059\pi\)
\(4\) 0.173648 − 0.984808i 0.0868241 − 0.492404i
\(5\) −1.43969 + 0.524005i −0.643850 + 0.234342i −0.643248 − 0.765658i \(-0.722414\pi\)
−0.000601695 1.00000i \(0.500192\pi\)
\(6\) −3.14945 −1.28576
\(7\) 4.53424 − 1.65033i 1.71378 − 0.623765i 0.716509 − 0.697578i \(-0.245739\pi\)
0.997272 + 0.0738128i \(0.0235167\pi\)
\(8\) −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 0.936618 − 0.350352i \(-0.113938\pi\)
−0.771723 + 0.635959i \(0.780605\pi\)
\(12\) −2.41262 + 2.02443i −0.696463 + 0.584402i
\(13\) 0.307284 − 1.74269i 0.0852253 − 0.483337i −0.912082 − 0.410007i \(-0.865526\pi\)
0.997308 − 0.0733298i \(-0.0233626\pi\)
\(14\) 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 0.981858 + 0.189620i \(0.0607255\pi\)
−0.857791 + 0.513999i \(0.828163\pi\)
\(18\) 5.30028 + 4.44747i 1.24929 + 1.04828i
\(19\) −0.632167 − 0.530451i −0.145029 − 0.121694i 0.567387 − 0.823451i \(-0.307954\pi\)
−0.712416 + 0.701758i \(0.752399\pi\)
\(20\) 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 −0.853067 − 0.521801i \(-0.825260\pi\)
0.878427 + 0.477877i \(0.158594\pi\)
\(24\) −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 0.999798 + 0.0201189i \(0.00640447\pi\)
−0.482475 + 0.875910i \(0.660262\pi\)
\(30\) 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 −0.919526 − 0.393029i \(-0.871427\pi\)
0.998496 − 0.0548301i \(-0.0174617\pi\)
\(42\) −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 −0.936723 − 0.350071i \(-0.886157\pi\)
0.771532 + 0.636190i \(0.219491\pi\)
\(48\) 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 −0.0868564 − 0.996221i \(-0.527682\pi\)
−0.706894 + 0.707319i \(0.749904\pi\)
\(54\) −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 −0.259878 − 0.965642i \(-0.583682\pi\)
−0.819780 + 0.572678i \(0.805904\pi\)
\(60\) 2.41262 − 4.17878i 0.311468 − 0.539478i
\(61\) −2.04075 + 11.5737i −0.261292 + 1.48186i 0.518098 + 0.855321i \(0.326640\pi\)
−0.779390 + 0.626539i \(0.784471\pi\)
\(62\) −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 −0.801606 − 0.597853i \(-0.796021\pi\)
−0.229773 + 0.973244i \(0.573798\pi\)
\(68\) 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 0.585710 − 0.810521i \(-0.300816\pi\)
−0.696500 + 0.717557i \(0.745260\pi\)
\(72\) 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 −0.453348 − 0.891334i \(-0.649770\pi\)
0.225654 + 0.974208i \(0.427548\pi\)
\(80\) 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 0.994087 + 0.108586i \(0.0346324\pi\)
−0.896997 + 0.442036i \(0.854256\pi\)
\(84\) −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 −0.146210 − 0.989254i \(-0.546708\pi\)
−0.747883 + 0.663830i \(0.768930\pi\)
\(90\) −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 −0.955203 − 0.295953i \(-0.904363\pi\)
0.733904 + 0.679253i \(0.237696\pi\)
\(98\) 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.49.1 ✓ 12
3.2 odd 2 666.2.x.g.271.2 12
4.3 odd 2 592.2.bc.d.49.2 12
37.16 even 9 2738.2.a.t.1.1 6
37.21 even 18 2738.2.a.q.1.1 6
37.34 even 9 inner 74.2.f.b.71.1 yes 12
111.71 odd 18 666.2.x.g.145.2 12
148.71 odd 18 592.2.bc.d.145.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.1 ✓ 12 1.1 even 1 trivial
74.2.f.b.71.1 yes 12 37.34 even 9 inner
592.2.bc.d.49.2 12 4.3 odd 2
592.2.bc.d.145.2 12 148.71 odd 18
666.2.x.g.145.2 12 111.71 odd 18
666.2.x.g.271.2 12 3.2 odd 2
2738.2.a.q.1.1 6 37.21 even 18
2738.2.a.t.1.1 6 37.16 even 9