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(3,74)] 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("74.3"); 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([13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == 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 \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 41.1
Root \(-0.984808 - 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 74.41
Dual form 74.2.h.a.65.1

$q$-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+(-0.984808 - 0.173648i) q^{2} +(-0.266044 - 1.50881i) q^{3} +(0.939693 + 0.342020i) q^{4} +(-1.97937 - 2.35892i) q^{5} +1.53209i q^{6} +(0.153180 - 0.128533i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(0.613341 - 0.223238i) q^{9} +(1.53967 + 2.66679i) q^{10} +(2.17174 - 3.76157i) q^{11} +(0.266044 - 1.50881i) q^{12} +(-1.59735 + 4.38867i) q^{13} +(-0.173172 + 0.0999810i) q^{14} +(-3.03256 + 3.61407i) q^{15} +(0.766044 + 0.642788i) q^{16} +(2.32445 + 6.38637i) q^{17} +(-0.642788 + 0.113341i) q^{18} +(4.07964 - 0.719350i) q^{19} +(-1.05320 - 2.89364i) q^{20} +(-0.234685 - 0.196924i) q^{21} +(-2.79194 + 3.32730i) q^{22} +(-0.896915 + 0.517834i) q^{23} +(-0.524005 + 1.43969i) q^{24} +(-0.778357 + 4.41428i) q^{25} +(2.33516 - 4.04462i) q^{26} +(-2.79813 - 4.84651i) q^{27} +(0.187903 - 0.0683910i) q^{28} +(1.25937 + 0.727100i) q^{29} +(3.61407 - 3.03256i) q^{30} -5.10852i q^{31} +(-0.642788 - 0.766044i) q^{32} +(-6.25329 - 2.27601i) q^{33} +(-1.18015 - 6.69298i) q^{34} +(-0.606398 - 0.106924i) q^{35} +0.652704 q^{36} +(5.64160 - 2.27429i) q^{37} -4.14257 q^{38} +(7.04665 + 1.24252i) q^{39} +(0.534723 + 3.03256i) q^{40} +(-4.46505 - 1.62515i) q^{41} +(0.196924 + 0.234685i) q^{42} +0.399970i q^{43} +(3.32730 - 2.79194i) q^{44} +(-1.74063 - 1.00495i) q^{45} +(0.973210 - 0.354220i) q^{46} +(4.10475 + 7.10963i) q^{47} +(0.766044 - 1.32683i) q^{48} +(-1.20859 + 6.85428i) q^{49} +(1.53306 - 4.21206i) q^{50} +(9.01743 - 5.20621i) q^{51} +(-3.00203 + 3.57768i) q^{52} +(-8.65606 - 7.26330i) q^{53} +(1.91404 + 5.25877i) q^{54} +(-13.1719 + 2.32256i) q^{55} +(-0.196924 + 0.0347230i) q^{56} +(-2.17073 - 5.96403i) q^{57} +(-1.11398 - 0.934742i) q^{58} +(-2.69714 + 3.21433i) q^{59} +(-4.08576 + 2.35892i) q^{60} +(-3.60153 + 9.89514i) q^{61} +(-0.887086 + 5.03091i) q^{62} +(0.0652579 - 0.113030i) q^{63} +(0.500000 + 0.866025i) q^{64} +(13.5143 - 4.91879i) q^{65} +(5.76306 + 3.32730i) q^{66} +(6.67299 - 5.59930i) q^{67} +6.79623i q^{68} +(1.01993 + 1.21551i) q^{69} +(0.578618 + 0.210600i) q^{70} +(2.45953 + 13.9487i) q^{71} +(-0.642788 - 0.113341i) q^{72} +7.27588 q^{73} +(-5.95081 + 1.26008i) q^{74} +6.86740 q^{75} +(4.07964 + 0.719350i) q^{76} +(-0.150819 - 0.855337i) q^{77} +(-6.72384 - 2.44728i) q^{78} +(-4.04665 - 4.82261i) q^{79} -3.07935i q^{80} +(-5.06805 + 4.25260i) q^{81} +(4.11502 + 2.37581i) q^{82} +(-14.4861 + 5.27251i) q^{83} +(-0.153180 - 0.265315i) q^{84} +(10.4640 - 18.1241i) q^{85} +(0.0694540 - 0.393893i) q^{86} +(0.762009 - 2.09360i) q^{87} +(-3.76157 + 2.17174i) q^{88} +(2.06611 - 2.46229i) q^{89} +(1.53967 + 1.29194i) q^{90} +(0.319409 + 0.877568i) q^{91} +(-1.01993 + 0.179842i) q^{92} +(-7.70781 + 1.35909i) q^{93} +(-2.80781 - 7.71440i) q^{94} +(-9.77198 - 8.19967i) q^{95} +(-0.984808 + 1.17365i) q^{96} +(2.07350 - 1.19713i) q^{97} +(2.38047 - 6.54027i) q^{98} +(0.492294 - 2.79194i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 12 q^{7} - 6 q^{9} + 6 q^{10} - 6 q^{11} - 6 q^{12} + 6 q^{13} - 18 q^{14} - 18 q^{19} - 6 q^{21} - 18 q^{25} + 12 q^{26} - 6 q^{27} - 6 q^{28} + 18 q^{29} + 24 q^{30} - 6 q^{33} + 12 q^{34}+ \cdots + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.984808 − 0.173648i −0.696364 − 0.122788i
\(3\) −0.266044 − 1.50881i −0.153601 − 0.871114i −0.960054 − 0.279815i \(-0.909727\pi\)
0.806453 − 0.591298i \(-0.201384\pi\)
\(4\) 0.939693 + 0.342020i 0.469846 + 0.171010i
\(5\) −1.97937 − 2.35892i −0.885199 − 1.05494i −0.998117 − 0.0613353i \(-0.980464\pi\)
0.112918 − 0.993604i \(-0.463980\pi\)
\(6\) 1.53209i 0.625473i
\(7\) 0.153180 − 0.128533i 0.0578965 − 0.0485809i −0.613380 − 0.789788i \(-0.710191\pi\)
0.671277 + 0.741207i \(0.265746\pi\)
\(8\) −0.866025 − 0.500000i −0.306186 − 0.176777i
\(9\) 0.613341 − 0.223238i 0.204447 − 0.0744126i
\(10\) 1.53967 + 2.66679i 0.486888 + 0.843314i
\(11\) 2.17174 − 3.76157i 0.654805 − 1.13416i −0.327137 − 0.944977i \(-0.606084\pi\)
0.981943 − 0.189179i \(-0.0605827\pi\)
\(12\) 0.266044 − 1.50881i 0.0768004 − 0.435557i
\(13\) −1.59735 + 4.38867i −0.443024 + 1.21720i 0.494469 + 0.869195i \(0.335363\pi\)
−0.937493 + 0.348004i \(0.886860\pi\)
\(14\) −0.173172 + 0.0999810i −0.0462822 + 0.0267210i
\(15\) −3.03256 + 3.61407i −0.783005 + 0.933149i
\(16\) 0.766044 + 0.642788i 0.191511 + 0.160697i
\(17\) 2.32445 + 6.38637i 0.563761 + 1.54892i 0.814077 + 0.580757i \(0.197243\pi\)
−0.250316 + 0.968164i \(0.580534\pi\)
\(18\) −0.642788 + 0.113341i −0.151506 + 0.0267147i
\(19\) 4.07964 − 0.719350i 0.935933 − 0.165030i 0.315176 − 0.949033i \(-0.397936\pi\)
0.620757 + 0.784003i \(0.286825\pi\)
\(20\) −1.05320 − 2.89364i −0.235502 − 0.647038i
\(21\) −0.234685 − 0.196924i −0.0512125 − 0.0429724i
\(22\) −2.79194 + 3.32730i −0.595244 + 0.709384i
\(23\) −0.896915 + 0.517834i −0.187020 + 0.107976i −0.590587 − 0.806974i \(-0.701104\pi\)
0.403567 + 0.914950i \(0.367770\pi\)
\(24\) −0.524005 + 1.43969i −0.106962 + 0.293876i
\(25\) −0.778357 + 4.41428i −0.155671 + 0.882856i
\(26\) 2.33516 − 4.04462i 0.457964 − 0.793216i
\(27\) −2.79813 − 4.84651i −0.538501 − 0.932711i
\(28\) 0.187903 − 0.0683910i 0.0355103 − 0.0129247i
\(29\) 1.25937 + 0.727100i 0.233860 + 0.135019i 0.612351 − 0.790586i \(-0.290224\pi\)
−0.378492 + 0.925605i \(0.623557\pi\)
\(30\) 3.61407 − 3.03256i 0.659836 − 0.553668i
\(31\) − 5.10852i − 0.917518i −0.888561 − 0.458759i \(-0.848294\pi\)
0.888561 − 0.458759i \(-0.151706\pi\)
\(32\) −0.642788 − 0.766044i −0.113630 − 0.135419i
\(33\) −6.25329 − 2.27601i −1.08856 − 0.396202i
\(34\) −1.18015 − 6.69298i −0.202395 − 1.14784i
\(35\) −0.606398 − 0.106924i −0.102500 − 0.0180735i
\(36\) 0.652704 0.108784
\(37\) 5.64160 − 2.27429i 0.927473 − 0.373891i
\(38\) −4.14257 −0.672014
\(39\) 7.04665 + 1.24252i 1.12837 + 0.198962i
\(40\) 0.534723 + 3.03256i 0.0845471 + 0.479491i
\(41\) −4.46505 − 1.62515i −0.697324 − 0.253805i −0.0310562 − 0.999518i \(-0.509887\pi\)
−0.666268 + 0.745712i \(0.732109\pi\)
\(42\) 0.196924 + 0.234685i 0.0303860 + 0.0362127i
\(43\) 0.399970i 0.0609948i 0.999535 + 0.0304974i \(0.00970913\pi\)
−0.999535 + 0.0304974i \(0.990291\pi\)
\(44\) 3.32730 − 2.79194i 0.501610 − 0.420901i
\(45\) −1.74063 − 1.00495i −0.259477 − 0.149809i
\(46\) 0.973210 − 0.354220i 0.143492 − 0.0522268i
\(47\) 4.10475 + 7.10963i 0.598739 + 1.03705i 0.993008 + 0.118051i \(0.0376646\pi\)
−0.394269 + 0.918995i \(0.629002\pi\)
\(48\) 0.766044 − 1.32683i 0.110569 − 0.191511i
\(49\) −1.20859 + 6.85428i −0.172656 + 0.979182i
\(50\) 1.53306 − 4.21206i 0.216808 − 0.595675i
\(51\) 9.01743 − 5.20621i 1.26269 − 0.729016i
\(52\) −3.00203 + 3.57768i −0.416307 + 0.496135i
\(53\) −8.65606 − 7.26330i −1.18900 − 0.997690i −0.999876 − 0.0157372i \(-0.994990\pi\)
−0.189125 − 0.981953i \(-0.560565\pi\)
\(54\) 1.91404 + 5.25877i 0.260467 + 0.715628i
\(55\) −13.1719 + 2.32256i −1.77610 + 0.313174i
\(56\) −0.196924 + 0.0347230i −0.0263151 + 0.00464006i
\(57\) −2.17073 − 5.96403i −0.287520 − 0.789955i
\(58\) −1.11398 − 0.934742i −0.146273 − 0.122738i
\(59\) −2.69714 + 3.21433i −0.351138 + 0.418470i −0.912485 − 0.409111i \(-0.865839\pi\)
0.561347 + 0.827581i \(0.310283\pi\)
\(60\) −4.08576 + 2.35892i −0.527470 + 0.304535i
\(61\) −3.60153 + 9.89514i −0.461129 + 1.26694i 0.463508 + 0.886093i \(0.346591\pi\)
−0.924637 + 0.380849i \(0.875632\pi\)
\(62\) −0.887086 + 5.03091i −0.112660 + 0.638927i
\(63\) 0.0652579 − 0.113030i 0.00822173 − 0.0142404i
\(64\) 0.500000 + 0.866025i 0.0625000 + 0.108253i
\(65\) 13.5143 − 4.91879i 1.67624 − 0.610100i
\(66\) 5.76306 + 3.32730i 0.709384 + 0.409563i
\(67\) 6.67299 − 5.59930i 0.815235 − 0.684063i −0.136616 − 0.990624i \(-0.543623\pi\)
0.951851 + 0.306561i \(0.0991782\pi\)
\(68\) 6.79623i 0.824164i
\(69\) 1.01993 + 1.21551i 0.122786 + 0.146330i
\(70\) 0.578618 + 0.210600i 0.0691581 + 0.0251715i
\(71\) 2.45953 + 13.9487i 0.291893 + 1.65541i 0.679569 + 0.733612i \(0.262167\pi\)
−0.387675 + 0.921796i \(0.626722\pi\)
\(72\) −0.642788 − 0.113341i −0.0757532 − 0.0133573i
\(73\) 7.27588 0.851577 0.425789 − 0.904823i \(-0.359997\pi\)
0.425789 + 0.904823i \(0.359997\pi\)
\(74\) −5.95081 + 1.26008i −0.691768 + 0.146482i
\(75\) 6.86740 0.792979
\(76\) 4.07964 + 0.719350i 0.467967 + 0.0825151i
\(77\) −0.150819 − 0.855337i −0.0171874 − 0.0974747i
\(78\) −6.72384 − 2.44728i −0.761325 − 0.277100i
\(79\) −4.04665 − 4.82261i −0.455284 − 0.542587i 0.488754 − 0.872421i \(-0.337451\pi\)
−0.944039 + 0.329835i \(0.893007\pi\)
\(80\) − 3.07935i − 0.344281i
\(81\) −5.06805 + 4.25260i −0.563116 + 0.472511i
\(82\) 4.11502 + 2.37581i 0.454427 + 0.262364i
\(83\) −14.4861 + 5.27251i −1.59006 + 0.578733i −0.977360 − 0.211585i \(-0.932137\pi\)
−0.612696 + 0.790318i \(0.709915\pi\)
\(84\) −0.153180 − 0.265315i −0.0167133 − 0.0289482i
\(85\) 10.4640 − 18.1241i 1.13498 − 1.96584i
\(86\) 0.0694540 − 0.393893i 0.00748942 − 0.0424746i
\(87\) 0.762009 − 2.09360i 0.0816959 − 0.224458i
\(88\) −3.76157 + 2.17174i −0.400985 + 0.231509i
\(89\) 2.06611 − 2.46229i 0.219007 − 0.261002i −0.645343 − 0.763893i \(-0.723286\pi\)
0.864350 + 0.502890i \(0.167730\pi\)
\(90\) 1.53967 + 1.29194i 0.162296 + 0.136182i
\(91\) 0.319409 + 0.877568i 0.0334831 + 0.0919941i
\(92\) −1.01993 + 0.179842i −0.106336 + 0.0187498i
\(93\) −7.70781 + 1.35909i −0.799262 + 0.140932i
\(94\) −2.80781 − 7.71440i −0.289604 − 0.795680i
\(95\) −9.77198 − 8.19967i −1.00258 − 0.841268i
\(96\) −0.984808 + 1.17365i −0.100512 + 0.119785i
\(97\) 2.07350 − 1.19713i 0.210532 − 0.121551i −0.391027 − 0.920379i \(-0.627880\pi\)
0.601559 + 0.798829i \(0.294547\pi\)
\(98\) 2.38047 − 6.54027i 0.240463 − 0.660668i
\(99\) 0.492294 − 2.79194i 0.0494774 − 0.280600i
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.h.a.41.1 ✓ 12
3.2 odd 2 666.2.bj.c.559.2 12
4.3 odd 2 592.2.bq.b.337.1 12
37.18 odd 36 2738.2.a.s.1.5 6
37.19 odd 36 2738.2.a.r.1.6 6
37.28 even 18 inner 74.2.h.a.65.1 yes 12
111.65 odd 18 666.2.bj.c.361.2 12
148.139 odd 18 592.2.bq.b.65.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.1 ✓ 12 1.1 even 1 trivial
74.2.h.a.65.1 yes 12 37.28 even 18 inner
592.2.bq.b.65.1 12 148.139 odd 18
592.2.bq.b.337.1 12 4.3 odd 2
666.2.bj.c.361.2 12 111.65 odd 18
666.2.bj.c.559.2 12 3.2 odd 2
2738.2.a.r.1.6 6 37.19 odd 36
2738.2.a.s.1.5 6 37.18 odd 36