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 25.2
Root \(0.642788 + 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 74.25
Dual form 74.2.h.a.3.2

$q$-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+(0.642788 + 0.766044i) q^{2} +(1.43969 + 1.20805i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(-1.45842 - 4.00698i) q^{5} +1.87939i q^{6} +(-3.39364 + 1.23518i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(0.0923963 + 0.524005i) q^{9} +(2.13207 - 3.69285i) q^{10} +(1.05840 + 1.83321i) q^{11} +(-1.43969 + 1.20805i) q^{12} +(2.84019 + 0.500802i) q^{13} +(-3.12760 - 1.80572i) q^{14} +(2.74094 - 7.53066i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(0.0263137 - 0.00463982i) q^{17} +(-0.342020 + 0.407604i) q^{18} +(-2.07522 + 2.47315i) q^{19} +(4.19936 - 0.740460i) q^{20} +(-6.37796 - 2.32139i) q^{21} +(-0.723990 + 1.98915i) q^{22} +(2.57421 + 1.48622i) q^{23} +(-1.85083 - 0.326352i) q^{24} +(-10.0987 + 8.47380i) q^{25} +(1.44200 + 2.49762i) q^{26} +(2.31908 - 4.01676i) q^{27} +(-0.627119 - 3.55657i) q^{28} +(4.96493 - 2.86650i) q^{29} +(7.53066 - 2.74094i) q^{30} -6.76932i q^{31} +(-0.342020 - 0.939693i) q^{32} +(-0.690823 + 3.91785i) q^{33} +(0.0204685 + 0.0171751i) q^{34} +(9.89872 + 11.7968i) q^{35} -0.532089 q^{36} +(-4.49375 + 4.09954i) q^{37} -3.22847 q^{38} +(3.48401 + 4.15208i) q^{39} +(3.26652 + 2.74094i) q^{40} +(0.259000 - 1.46886i) q^{41} +(-2.32139 - 6.37796i) q^{42} +5.53737i q^{43} +(-1.98915 + 0.723990i) q^{44} +(1.96493 - 1.13445i) q^{45} +(0.516159 + 2.92728i) q^{46} +(-1.30654 + 2.26300i) q^{47} +(-0.939693 - 1.62760i) q^{48} +(4.62880 - 3.88403i) q^{49} +(-12.9826 - 2.28918i) q^{50} +(0.0434888 + 0.0251083i) q^{51} +(-0.986387 + 2.71008i) q^{52} +(-1.79389 - 0.652924i) q^{53} +(4.56769 - 0.805407i) q^{54} +(5.80203 - 6.91459i) q^{55} +(2.32139 - 2.76652i) q^{56} +(-5.97536 + 1.05362i) q^{57} +(5.38726 + 1.96080i) q^{58} +(1.92380 - 5.28560i) q^{59} +(6.94029 + 4.00698i) q^{60} +(-5.65366 - 0.996892i) q^{61} +(5.18560 - 4.35124i) q^{62} +(-0.960802 - 1.66416i) q^{63} +(0.500000 - 0.866025i) q^{64} +(-2.13549 - 12.1110i) q^{65} +(-3.44530 + 1.98915i) q^{66} +(-6.50406 + 2.36728i) q^{67} +0.0267197i q^{68} +(1.91065 + 5.24947i) q^{69} +(-2.67412 + 15.1657i) q^{70} +(-7.10830 - 5.96457i) q^{71} +(-0.342020 - 0.407604i) q^{72} +16.2707 q^{73} +(-6.02896 - 0.807274i) q^{74} -24.7757 q^{75} +(-2.07522 - 2.47315i) q^{76} +(-5.85619 - 4.91392i) q^{77} +(-0.941199 + 5.33781i) q^{78} +(-0.484006 - 1.32980i) q^{79} +4.26414i q^{80} +(9.69119 - 3.52730i) q^{81} +(1.29170 - 0.745761i) q^{82} +(-0.294580 - 1.67065i) q^{83} +(3.39364 - 5.87796i) q^{84} +(-0.0569682 - 0.0986718i) q^{85} +(-4.24187 + 3.55935i) q^{86} +(10.6108 + 1.87098i) q^{87} +(-1.83321 - 1.05840i) q^{88} +(2.82882 - 7.77213i) q^{89} +(2.13207 + 0.776010i) q^{90} +(-10.2572 + 1.80861i) q^{91} +(-1.91065 + 2.27702i) q^{92} +(8.17765 - 9.74574i) q^{93} +(-2.57339 + 0.453757i) q^{94} +(12.9364 + 4.70847i) q^{95} +(0.642788 - 1.76604i) q^{96} +(5.65105 + 3.26264i) q^{97} +(5.95067 + 1.04926i) q^{98} +(-0.862818 + 0.723990i) 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 \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.642788 + 0.766044i 0.454519 + 0.541675i
\(3\) 1.43969 + 1.20805i 0.831207 + 0.697465i 0.955568 − 0.294772i \(-0.0952436\pi\)
−0.124361 + 0.992237i \(0.539688\pi\)
\(4\) −0.173648 + 0.984808i −0.0868241 + 0.492404i
\(5\) −1.45842 − 4.00698i −0.652226 − 1.79198i −0.609358 − 0.792895i \(-0.708573\pi\)
−0.0428683 − 0.999081i \(-0.513650\pi\)
\(6\) 1.87939i 0.767256i
\(7\) −3.39364 + 1.23518i −1.28268 + 0.466856i −0.891316 − 0.453382i \(-0.850217\pi\)
−0.391359 + 0.920238i \(0.627995\pi\)
\(8\) −0.866025 + 0.500000i −0.306186 + 0.176777i
\(9\) 0.0923963 + 0.524005i 0.0307988 + 0.174668i
\(10\) 2.13207 − 3.69285i 0.674220 − 1.16778i
\(11\) 1.05840 + 1.83321i 0.319120 + 0.552733i 0.980305 − 0.197491i \(-0.0632792\pi\)
−0.661184 + 0.750223i \(0.729946\pi\)
\(12\) −1.43969 + 1.20805i −0.415603 + 0.348733i
\(13\) 2.84019 + 0.500802i 0.787726 + 0.138897i 0.553019 − 0.833169i \(-0.313476\pi\)
0.234707 + 0.972066i \(0.424587\pi\)
\(14\) −3.12760 − 1.80572i −0.835885 − 0.482598i
\(15\) 2.74094 − 7.53066i 0.707707 − 1.94441i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) 0.0263137 − 0.00463982i 0.00638202 − 0.00112532i −0.170456 − 0.985365i \(-0.554524\pi\)
0.176838 + 0.984240i \(0.443413\pi\)
\(18\) −0.342020 + 0.407604i −0.0806149 + 0.0960731i
\(19\) −2.07522 + 2.47315i −0.476088 + 0.567380i −0.949623 − 0.313395i \(-0.898534\pi\)
0.473534 + 0.880775i \(0.342978\pi\)
\(20\) 4.19936 − 0.740460i 0.939005 − 0.165572i
\(21\) −6.37796 − 2.32139i −1.39178 − 0.506568i
\(22\) −0.723990 + 1.98915i −0.154355 + 0.424087i
\(23\) 2.57421 + 1.48622i 0.536760 + 0.309899i 0.743765 − 0.668441i \(-0.233038\pi\)
−0.207005 + 0.978340i \(0.566372\pi\)
\(24\) −1.85083 − 0.326352i −0.377800 − 0.0666163i
\(25\) −10.0987 + 8.47380i −2.01974 + 1.69476i
\(26\) 1.44200 + 2.49762i 0.282800 + 0.489823i
\(27\) 2.31908 − 4.01676i 0.446307 − 0.773026i
\(28\) −0.627119 − 3.55657i −0.118514 − 0.672129i
\(29\) 4.96493 − 2.86650i 0.921964 − 0.532296i 0.0377027 − 0.999289i \(-0.487996\pi\)
0.884261 + 0.466993i \(0.154663\pi\)
\(30\) 7.53066 − 2.74094i 1.37490 − 0.500424i
\(31\) − 6.76932i − 1.21581i −0.794011 − 0.607903i \(-0.792011\pi\)
0.794011 − 0.607903i \(-0.207989\pi\)
\(32\) −0.342020 − 0.939693i −0.0604612 − 0.166116i
\(33\) −0.690823 + 3.91785i −0.120257 + 0.682011i
\(34\) 0.0204685 + 0.0171751i 0.00351031 + 0.00294550i
\(35\) 9.89872 + 11.7968i 1.67319 + 1.99403i
\(36\) −0.532089 −0.0886815
\(37\) −4.49375 + 4.09954i −0.738767 + 0.673961i
\(38\) −3.22847 −0.523727
\(39\) 3.48401 + 4.15208i 0.557887 + 0.664864i
\(40\) 3.26652 + 2.74094i 0.516482 + 0.433380i
\(41\) 0.259000 − 1.46886i 0.0404490 − 0.229398i −0.957881 − 0.287165i \(-0.907287\pi\)
0.998330 + 0.0577674i \(0.0183982\pi\)
\(42\) −2.32139 − 6.37796i −0.358198 − 0.984140i
\(43\) 5.53737i 0.844441i 0.906493 + 0.422221i \(0.138749\pi\)
−0.906493 + 0.422221i \(0.861251\pi\)
\(44\) −1.98915 + 0.723990i −0.299875 + 0.109146i
\(45\) 1.96493 − 1.13445i 0.292914 − 0.169114i
\(46\) 0.516159 + 2.92728i 0.0761035 + 0.431605i
\(47\) −1.30654 + 2.26300i −0.190579 + 0.330092i −0.945442 − 0.325790i \(-0.894370\pi\)
0.754863 + 0.655882i \(0.227703\pi\)
\(48\) −0.939693 − 1.62760i −0.135633 − 0.234923i
\(49\) 4.62880 − 3.88403i 0.661257 − 0.554861i
\(50\) −12.9826 − 2.28918i −1.83602 − 0.323740i
\(51\) 0.0434888 + 0.0251083i 0.00608965 + 0.00351586i
\(52\) −0.986387 + 2.71008i −0.136787 + 0.375820i
\(53\) −1.79389 − 0.652924i −0.246410 − 0.0896860i 0.215862 − 0.976424i \(-0.430744\pi\)
−0.462273 + 0.886738i \(0.652966\pi\)
\(54\) 4.56769 − 0.805407i 0.621584 − 0.109602i
\(55\) 5.80203 − 6.91459i 0.782345 − 0.932363i
\(56\) 2.32139 − 2.76652i 0.310208 − 0.369692i
\(57\) −5.97536 + 1.05362i −0.791456 + 0.139555i
\(58\) 5.38726 + 1.96080i 0.707382 + 0.257466i
\(59\) 1.92380 − 5.28560i 0.250457 − 0.688126i −0.749210 − 0.662333i \(-0.769567\pi\)
0.999667 − 0.0257935i \(-0.00821124\pi\)
\(60\) 6.94029 + 4.00698i 0.895988 + 0.517299i
\(61\) −5.65366 − 0.996892i −0.723876 − 0.127639i −0.200443 − 0.979705i \(-0.564238\pi\)
−0.523434 + 0.852066i \(0.675349\pi\)
\(62\) 5.18560 − 4.35124i 0.658572 − 0.552608i
\(63\) −0.960802 − 1.66416i −0.121050 − 0.209664i
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) −2.13549 − 12.1110i −0.264875 − 1.50218i
\(66\) −3.44530 + 1.98915i −0.424087 + 0.244847i
\(67\) −6.50406 + 2.36728i −0.794597 + 0.289210i −0.707246 − 0.706968i \(-0.750063\pi\)
−0.0873516 + 0.996178i \(0.527840\pi\)
\(68\) 0.0267197i 0.00324024i
\(69\) 1.91065 + 5.24947i 0.230015 + 0.631962i
\(70\) −2.67412 + 15.1657i −0.319619 + 1.81265i
\(71\) −7.10830 − 5.96457i −0.843600 − 0.707865i 0.114770 − 0.993392i \(-0.463387\pi\)
−0.958371 + 0.285527i \(0.907831\pi\)
\(72\) −0.342020 − 0.407604i −0.0403075 − 0.0480366i
\(73\) 16.2707 1.90434 0.952169 − 0.305571i \(-0.0988473\pi\)
0.952169 + 0.305571i \(0.0988473\pi\)
\(74\) −6.02896 − 0.807274i −0.700852 − 0.0938437i
\(75\) −24.7757 −2.86085
\(76\) −2.07522 − 2.47315i −0.238044 − 0.283690i
\(77\) −5.85619 − 4.91392i −0.667374 − 0.559993i
\(78\) −0.941199 + 5.33781i −0.106570 + 0.604388i
\(79\) −0.484006 − 1.32980i −0.0544549 − 0.149614i 0.909483 − 0.415742i \(-0.136478\pi\)
−0.963938 + 0.266128i \(0.914256\pi\)
\(80\) 4.26414i 0.476745i
\(81\) 9.69119 − 3.52730i 1.07680 − 0.391923i
\(82\) 1.29170 − 0.745761i 0.142644 − 0.0823556i
\(83\) −0.294580 − 1.67065i −0.0323343 − 0.183377i 0.964363 − 0.264583i \(-0.0852342\pi\)
−0.996697 + 0.0812054i \(0.974123\pi\)
\(84\) 3.39364 − 5.87796i 0.370276 − 0.641338i
\(85\) −0.0569682 − 0.0986718i −0.00617907 − 0.0107025i
\(86\) −4.24187 + 3.55935i −0.457413 + 0.383815i
\(87\) 10.6108 + 1.87098i 1.13760 + 0.200590i
\(88\) −1.83321 − 1.05840i −0.195421 − 0.112826i
\(89\) 2.82882 − 7.77213i 0.299855 − 0.823844i −0.694669 − 0.719330i \(-0.744449\pi\)
0.994523 − 0.104514i \(-0.0333288\pi\)
\(90\) 2.13207 + 0.776010i 0.224740 + 0.0817986i
\(91\) −10.2572 + 1.80861i −1.07524 + 0.189594i
\(92\) −1.91065 + 2.27702i −0.199199 + 0.237396i
\(93\) 8.17765 − 9.74574i 0.847983 − 1.01059i
\(94\) −2.57339 + 0.453757i −0.265425 + 0.0468015i
\(95\) 12.9364 + 4.70847i 1.32725 + 0.483079i
\(96\) 0.642788 − 1.76604i 0.0656042 − 0.180246i
\(97\) 5.65105 + 3.26264i 0.573777 + 0.331270i 0.758657 − 0.651491i \(-0.225856\pi\)
−0.184879 + 0.982761i \(0.559189\pi\)
\(98\) 5.95067 + 1.04926i 0.601109 + 0.105992i
\(99\) −0.862818 + 0.723990i −0.0867164 + 0.0727637i
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.25.2 yes 12
3.2 odd 2 666.2.bj.c.469.1 12
4.3 odd 2 592.2.bq.b.321.1 12
37.3 even 18 inner 74.2.h.a.3.2 ✓ 12
37.15 odd 36 2738.2.a.r.1.2 6
37.22 odd 36 2738.2.a.s.1.1 6
111.77 odd 18 666.2.bj.c.595.1 12
148.3 odd 18 592.2.bq.b.225.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.2 ✓ 12 37.3 even 18 inner
74.2.h.a.25.2 yes 12 1.1 even 1 trivial
592.2.bq.b.225.1 12 148.3 odd 18
592.2.bq.b.321.1 12 4.3 odd 2
666.2.bj.c.469.1 12 3.2 odd 2
666.2.bj.c.595.1 12 111.77 odd 18
2738.2.a.r.1.2 6 37.15 odd 36
2738.2.a.s.1.1 6 37.22 odd 36