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 3.2
Root \(0.642788 - 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 74.3
Dual form 74.2.h.a.25.2

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\)

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 −0.124361 − 0.992237i \(-0.539688\pi\)
0.955568 + 0.294772i \(0.0952436\pi\)
\(4\) −0.173648 − 0.984808i −0.0868241 − 0.492404i
\(5\) −1.45842 + 4.00698i −0.652226 + 1.79198i −0.0428683 + 0.999081i \(0.513650\pi\)
−0.609358 + 0.792895i \(0.708573\pi\)
\(6\) − 1.87939i − 0.767256i
\(7\) −3.39364 − 1.23518i −1.28268 − 0.466856i −0.391359 − 0.920238i \(-0.627995\pi\)
−0.891316 + 0.453382i \(0.850217\pi\)
\(8\) −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 −0.661184 − 0.750223i \(-0.729946\pi\)
0.980305 + 0.197491i \(0.0632792\pi\)
\(12\) −1.43969 − 1.20805i −0.415603 − 0.348733i
\(13\) 2.84019 − 0.500802i 0.787726 − 0.138897i 0.234707 − 0.972066i \(-0.424587\pi\)
0.553019 + 0.833169i \(0.313476\pi\)
\(14\) −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 0.176838 − 0.984240i \(-0.443413\pi\)
−0.170456 + 0.985365i \(0.554524\pi\)
\(18\) −0.342020 − 0.407604i −0.0806149 − 0.0960731i
\(19\) −2.07522 − 2.47315i −0.476088 − 0.567380i 0.473534 − 0.880775i \(-0.342978\pi\)
−0.949623 + 0.313395i \(0.898534\pi\)
\(20\) 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 −0.207005 − 0.978340i \(-0.566372\pi\)
0.743765 + 0.668441i \(0.233038\pi\)
\(24\) −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 0.884261 − 0.466993i \(-0.154663\pi\)
0.0377027 + 0.999289i \(0.487996\pi\)
\(30\) 7.53066 + 2.74094i 1.37490 + 0.500424i
\(31\) 6.76932i 1.21581i 0.794011 + 0.607903i \(0.207989\pi\)
−0.794011 + 0.607903i \(0.792011\pi\)
\(32\) −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 0.998330 − 0.0577674i \(-0.0183982\pi\)
−0.957881 + 0.287165i \(0.907287\pi\)
\(42\) −2.32139 + 6.37796i −0.358198 + 0.984140i
\(43\) − 5.53737i − 0.844441i −0.906493 − 0.422221i \(-0.861251\pi\)
0.906493 − 0.422221i \(-0.138749\pi\)
\(44\) −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 0.754863 − 0.655882i \(-0.227703\pi\)
−0.945442 + 0.325790i \(0.894370\pi\)
\(48\) −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 −0.462273 − 0.886738i \(-0.652966\pi\)
0.215862 + 0.976424i \(0.430744\pi\)
\(54\) 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 0.999667 + 0.0257935i \(0.00821124\pi\)
−0.749210 + 0.662333i \(0.769567\pi\)
\(60\) 6.94029 − 4.00698i 0.895988 − 0.517299i
\(61\) −5.65366 + 0.996892i −0.723876 + 0.127639i −0.523434 − 0.852066i \(-0.675349\pi\)
−0.200443 + 0.979705i \(0.564238\pi\)
\(62\) 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 −0.0873516 − 0.996178i \(-0.527840\pi\)
−0.707246 + 0.706968i \(0.750063\pi\)
\(68\) − 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 −0.958371 − 0.285527i \(-0.907831\pi\)
0.114770 + 0.993392i \(0.463387\pi\)
\(72\) −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 −0.963938 − 0.266128i \(-0.914256\pi\)
0.909483 + 0.415742i \(0.136478\pi\)
\(80\) − 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 −0.996697 − 0.0812054i \(-0.974123\pi\)
0.964363 + 0.264583i \(0.0852342\pi\)
\(84\) 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 0.994523 + 0.104514i \(0.0333288\pi\)
−0.694669 + 0.719330i \(0.744449\pi\)
\(90\) 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 −0.184879 − 0.982761i \(-0.559189\pi\)
0.758657 + 0.651491i \(0.225856\pi\)
\(98\) 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.3.2 ✓ 12
3.2 odd 2 666.2.bj.c.595.1 12
4.3 odd 2 592.2.bq.b.225.1 12
37.5 odd 36 2738.2.a.r.1.2 6
37.25 even 18 inner 74.2.h.a.25.2 yes 12
37.32 odd 36 2738.2.a.s.1.1 6
111.62 odd 18 666.2.bj.c.469.1 12
148.99 odd 18 592.2.bq.b.321.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.2 ✓ 12 1.1 even 1 trivial
74.2.h.a.25.2 yes 12 37.25 even 18 inner
592.2.bq.b.225.1 12 4.3 odd 2
592.2.bq.b.321.1 12 148.99 odd 18
666.2.bj.c.469.1 12 111.62 odd 18
666.2.bj.c.595.1 12 3.2 odd 2
2738.2.a.r.1.2 6 37.5 odd 36
2738.2.a.s.1.1 6 37.32 odd 36