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 21.1
Root \(-0.342020 + 0.939693i\) of defining polynomial
Character \(\chi\) \(=\) 74.21
Dual form 74.2.h.a.67.1

$q$-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+(-0.342020 + 0.939693i) q^{2} +(0.326352 - 0.118782i) q^{3} +(-0.766044 - 0.642788i) q^{4} +(2.57176 + 0.453471i) q^{5} +0.347296i q^{6} +(-0.361075 + 2.04776i) q^{7} +(0.866025 - 0.500000i) q^{8} +(-2.20574 + 1.85083i) q^{9} +(-1.30572 + 2.26157i) q^{10} +(-2.99810 - 5.19285i) q^{11} +(-0.326352 - 0.118782i) q^{12} +(2.64632 - 3.15377i) q^{13} +(-1.80077 - 1.03967i) q^{14} +(0.893164 - 0.157489i) q^{15} +(0.173648 + 0.984808i) q^{16} +(-0.618710 - 0.737350i) q^{17} +(-0.984808 - 2.70574i) q^{18} +(0.534946 + 1.46975i) q^{19} +(-1.67860 - 2.00048i) q^{20} +(0.125400 + 0.711179i) q^{21} +(5.90509 - 1.04123i) q^{22} +(-5.51705 - 3.18527i) q^{23} +(0.223238 - 0.266044i) q^{24} +(1.70986 + 0.622339i) q^{25} +(2.05847 + 3.56538i) q^{26} +(-1.02094 + 1.76833i) q^{27} +(1.59287 - 1.33658i) q^{28} +(-3.51193 + 2.02761i) q^{29} +(-0.157489 + 0.893164i) q^{30} +3.39997i q^{31} +(-0.984808 - 0.173648i) q^{32} +(-1.59525 - 1.33858i) q^{33} +(0.904494 - 0.329209i) q^{34} +(-1.85720 + 5.10261i) q^{35} +2.87939 q^{36} +(6.07068 + 0.383130i) q^{37} -1.56408 q^{38} +(0.489021 - 1.34357i) q^{39} +(2.45395 - 0.893164i) q^{40} +(7.94502 + 6.66666i) q^{41} +(-0.711179 - 0.125400i) q^{42} +3.76932i q^{43} +(-1.04123 + 5.90509i) q^{44} +(-6.51193 + 3.75967i) q^{45} +(4.88011 - 4.09490i) q^{46} +(3.08750 - 5.34771i) q^{47} +(0.173648 + 0.300767i) q^{48} +(2.51491 + 0.915354i) q^{49} +(-1.16962 + 1.39389i) q^{50} +(-0.289501 - 0.167144i) q^{51} +(-4.05440 + 0.714901i) q^{52} +(-1.39401 - 7.90585i) q^{53} +(-1.31250 - 1.56418i) q^{54} +(-5.35558 - 14.7143i) q^{55} +(0.711179 + 1.95395i) q^{56} +(0.349161 + 0.416114i) q^{57} +(-0.704183 - 3.99362i) q^{58} +(5.02269 - 0.885636i) q^{59} +(-0.785435 - 0.453471i) q^{60} +(-6.25519 + 7.45465i) q^{61} +(-3.19493 - 1.16286i) q^{62} +(-2.99362 - 5.18510i) q^{63} +(0.500000 - 0.866025i) q^{64} +(8.23586 - 6.91071i) q^{65} +(1.80346 - 1.04123i) q^{66} +(-1.83263 + 10.3934i) q^{67} +0.962542i q^{68} +(-2.17885 - 0.384190i) q^{69} +(-4.15968 - 3.49039i) q^{70} +(-10.1503 + 3.69442i) q^{71} +(-0.984808 + 2.70574i) q^{72} +3.55293 q^{73} +(-2.43632 + 5.57354i) q^{74} +0.631940 q^{75} +(0.534946 - 1.46975i) q^{76} +(11.7162 - 4.26436i) q^{77} +(1.09529 + 0.919059i) q^{78} +(2.51098 + 0.442753i) q^{79} +2.61144i q^{80} +(1.37686 - 7.80856i) q^{81} +(-8.98197 + 5.18574i) q^{82} +(5.29798 - 4.44553i) q^{83} +(0.361075 - 0.625400i) q^{84} +(-1.25681 - 2.17686i) q^{85} +(-3.54200 - 1.28918i) q^{86} +(-0.905280 + 1.07887i) q^{87} +(-5.19285 - 2.99810i) q^{88} +(-16.0165 + 2.82414i) q^{89} +(-1.30572 - 7.40509i) q^{90} +(5.50263 + 6.55778i) q^{91} +(2.17885 + 5.98635i) q^{92} +(0.403856 + 1.10959i) q^{93} +(3.96922 + 4.73033i) q^{94} +(0.709264 + 4.02243i) q^{95} +(-0.342020 + 0.0603074i) q^{96} +(-14.1175 - 8.15074i) q^{97} +(-1.72030 + 2.05018i) q^{98} +(16.2241 + 5.90509i) 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 \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.342020 + 0.939693i −0.241845 + 0.664463i
\(3\) 0.326352 − 0.118782i 0.188419 − 0.0685790i −0.246087 − 0.969248i \(-0.579145\pi\)
0.434507 + 0.900669i \(0.356923\pi\)
\(4\) −0.766044 − 0.642788i −0.383022 − 0.321394i
\(5\) 2.57176 + 0.453471i 1.15013 + 0.202798i 0.716031 − 0.698068i \(-0.245957\pi\)
0.434096 + 0.900867i \(0.357068\pi\)
\(6\) 0.347296i 0.141783i
\(7\) −0.361075 + 2.04776i −0.136473 + 0.773979i 0.837349 + 0.546669i \(0.184104\pi\)
−0.973822 + 0.227311i \(0.927007\pi\)
\(8\) 0.866025 − 0.500000i 0.306186 − 0.176777i
\(9\) −2.20574 + 1.85083i −0.735246 + 0.616944i
\(10\) −1.30572 + 2.26157i −0.412904 + 0.715171i
\(11\) −2.99810 − 5.19285i −0.903960 − 1.56570i −0.822308 − 0.569043i \(-0.807314\pi\)
−0.0816522 − 0.996661i \(-0.526020\pi\)
\(12\) −0.326352 − 0.118782i −0.0942097 − 0.0342895i
\(13\) 2.64632 − 3.15377i 0.733958 − 0.874697i −0.261949 − 0.965082i \(-0.584365\pi\)
0.995907 + 0.0903843i \(0.0288095\pi\)
\(14\) −1.80077 − 1.03967i −0.481275 − 0.277864i
\(15\) 0.893164 − 0.157489i 0.230614 − 0.0406634i
\(16\) 0.173648 + 0.984808i 0.0434120 + 0.246202i
\(17\) −0.618710 − 0.737350i −0.150059 − 0.178834i 0.685778 − 0.727810i \(-0.259462\pi\)
−0.835838 + 0.548977i \(0.815018\pi\)
\(18\) −0.984808 − 2.70574i −0.232121 − 0.637748i
\(19\) 0.534946 + 1.46975i 0.122725 + 0.337184i 0.985808 − 0.167878i \(-0.0536916\pi\)
−0.863083 + 0.505063i \(0.831469\pi\)
\(20\) −1.67860 − 2.00048i −0.375346 − 0.447320i
\(21\) 0.125400 + 0.711179i 0.0273645 + 0.155192i
\(22\) 5.90509 − 1.04123i 1.25897 − 0.221990i
\(23\) −5.51705 − 3.18527i −1.15038 − 0.664174i −0.201403 − 0.979508i \(-0.564550\pi\)
−0.948981 + 0.315334i \(0.897883\pi\)
\(24\) 0.223238 − 0.266044i 0.0455682 − 0.0543061i
\(25\) 1.70986 + 0.622339i 0.341973 + 0.124468i
\(26\) 2.05847 + 3.56538i 0.403700 + 0.699229i
\(27\) −1.02094 + 1.76833i −0.196481 + 0.340315i
\(28\) 1.59287 − 1.33658i 0.301025 − 0.252590i
\(29\) −3.51193 + 2.02761i −0.652149 + 0.376519i −0.789279 − 0.614035i \(-0.789546\pi\)
0.137130 + 0.990553i \(0.456212\pi\)
\(30\) −0.157489 + 0.893164i −0.0287534 + 0.163069i
\(31\) 3.39997i 0.610653i 0.952248 + 0.305326i \(0.0987655\pi\)
−0.952248 + 0.305326i \(0.901234\pi\)
\(32\) −0.984808 − 0.173648i −0.174091 − 0.0306970i
\(33\) −1.59525 − 1.33858i −0.277698 − 0.233016i
\(34\) 0.904494 − 0.329209i 0.155119 − 0.0564588i
\(35\) −1.85720 + 5.10261i −0.313924 + 0.862498i
\(36\) 2.87939 0.479898
\(37\) 6.07068 + 0.383130i 0.998014 + 0.0629862i
\(38\) −1.56408 −0.253727
\(39\) 0.489021 − 1.34357i 0.0783060 − 0.215144i
\(40\) 2.45395 − 0.893164i 0.388003 − 0.141222i
\(41\) 7.94502 + 6.66666i 1.24080 + 1.04116i 0.997461 + 0.0712179i \(0.0226886\pi\)
0.243343 + 0.969940i \(0.421756\pi\)
\(42\) −0.711179 − 0.125400i −0.109737 − 0.0193496i
\(43\) 3.76932i 0.574816i 0.957808 + 0.287408i \(0.0927936\pi\)
−0.957808 + 0.287408i \(0.907206\pi\)
\(44\) −1.04123 + 5.90509i −0.156971 + 0.890227i
\(45\) −6.51193 + 3.75967i −0.970741 + 0.560458i
\(46\) 4.88011 − 4.09490i 0.719534 − 0.603760i
\(47\) 3.08750 − 5.34771i 0.450359 − 0.780044i −0.548050 − 0.836446i \(-0.684629\pi\)
0.998408 + 0.0564019i \(0.0179628\pi\)
\(48\) 0.173648 + 0.300767i 0.0250640 + 0.0434120i
\(49\) 2.51491 + 0.915354i 0.359273 + 0.130765i
\(50\) −1.16962 + 1.39389i −0.165409 + 0.197126i
\(51\) −0.289501 − 0.167144i −0.0405383 − 0.0234048i
\(52\) −4.05440 + 0.714901i −0.562245 + 0.0991389i
\(53\) −1.39401 − 7.90585i −0.191483 − 1.08595i −0.917339 − 0.398106i \(-0.869668\pi\)
0.725857 − 0.687846i \(-0.241444\pi\)
\(54\) −1.31250 − 1.56418i −0.178609 − 0.212858i
\(55\) −5.35558 − 14.7143i −0.722146 − 1.98408i
\(56\) 0.711179 + 1.95395i 0.0950352 + 0.261107i
\(57\) 0.349161 + 0.416114i 0.0462475 + 0.0551156i
\(58\) −0.704183 − 3.99362i −0.0924638 − 0.524388i
\(59\) 5.02269 − 0.885636i 0.653899 − 0.115300i 0.163150 − 0.986601i \(-0.447834\pi\)
0.490749 + 0.871301i \(0.336723\pi\)
\(60\) −0.785435 − 0.453471i −0.101399 − 0.0585429i
\(61\) −6.25519 + 7.45465i −0.800895 + 0.954470i −0.999673 − 0.0255750i \(-0.991858\pi\)
0.198778 + 0.980045i \(0.436303\pi\)
\(62\) −3.19493 − 1.16286i −0.405756 − 0.147683i
\(63\) −2.99362 − 5.18510i −0.377161 − 0.653262i
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) 8.23586 − 6.91071i 1.02153 − 0.857168i
\(66\) 1.80346 − 1.04123i 0.221990 − 0.128166i
\(67\) −1.83263 + 10.3934i −0.223892 + 1.26975i 0.640901 + 0.767624i \(0.278561\pi\)
−0.864792 + 0.502130i \(0.832550\pi\)
\(68\) 0.962542i 0.116725i
\(69\) −2.17885 − 0.384190i −0.262303 − 0.0462511i
\(70\) −4.15968 − 3.49039i −0.497177 − 0.417181i
\(71\) −10.1503 + 3.69442i −1.20462 + 0.438447i −0.864835 − 0.502056i \(-0.832577\pi\)
−0.339787 + 0.940502i \(0.610355\pi\)
\(72\) −0.984808 + 2.70574i −0.116061 + 0.318874i
\(73\) 3.55293 0.415839 0.207920 − 0.978146i \(-0.433331\pi\)
0.207920 + 0.978146i \(0.433331\pi\)
\(74\) −2.43632 + 5.57354i −0.283217 + 0.647911i
\(75\) 0.631940 0.0729701
\(76\) 0.534946 − 1.46975i 0.0613625 − 0.168592i
\(77\) 11.7162 − 4.26436i 1.33519 − 0.485969i
\(78\) 1.09529 + 0.919059i 0.124017 + 0.104063i
\(79\) 2.51098 + 0.442753i 0.282507 + 0.0498136i 0.313106 − 0.949718i \(-0.398630\pi\)
−0.0305991 + 0.999532i \(0.509742\pi\)
\(80\) 2.61144i 0.291967i
\(81\) 1.37686 − 7.80856i 0.152984 − 0.867617i
\(82\) −8.98197 + 5.18574i −0.991893 + 0.572670i
\(83\) 5.29798 − 4.44553i 0.581529 − 0.487960i −0.303920 − 0.952698i \(-0.598296\pi\)
0.885449 + 0.464737i \(0.153851\pi\)
\(84\) 0.361075 − 0.625400i 0.0393965 − 0.0682367i
\(85\) −1.25681 − 2.17686i −0.136320 − 0.236113i
\(86\) −3.54200 − 1.28918i −0.381944 − 0.139016i
\(87\) −0.905280 + 1.07887i −0.0970562 + 0.115667i
\(88\) −5.19285 − 2.99810i −0.553560 − 0.319598i
\(89\) −16.0165 + 2.82414i −1.69774 + 0.299358i −0.936905 − 0.349584i \(-0.886323\pi\)
−0.760838 + 0.648942i \(0.775212\pi\)
\(90\) −1.30572 − 7.40509i −0.137635 − 0.780566i
\(91\) 5.50263 + 6.55778i 0.576832 + 0.687442i
\(92\) 2.17885 + 5.98635i 0.227161 + 0.624120i
\(93\) 0.403856 + 1.10959i 0.0418780 + 0.115059i
\(94\) 3.96922 + 4.73033i 0.409394 + 0.487896i
\(95\) 0.709264 + 4.02243i 0.0727689 + 0.412693i
\(96\) −0.342020 + 0.0603074i −0.0349073 + 0.00615510i
\(97\) −14.1175 − 8.15074i −1.43342 − 0.827583i −0.436036 − 0.899929i \(-0.643618\pi\)
−0.997380 + 0.0723469i \(0.976951\pi\)
\(98\) −1.72030 + 2.05018i −0.173777 + 0.207099i
\(99\) 16.2241 + 5.90509i 1.63058 + 0.593484i
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.21.1 ✓ 12
3.2 odd 2 666.2.bj.c.613.2 12
4.3 odd 2 592.2.bq.b.465.2 12
37.17 odd 36 2738.2.a.r.1.3 6
37.20 odd 36 2738.2.a.s.1.4 6
37.30 even 18 inner 74.2.h.a.67.1 yes 12
111.104 odd 18 666.2.bj.c.289.2 12
148.67 odd 18 592.2.bq.b.289.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.21.1 ✓ 12 1.1 even 1 trivial
74.2.h.a.67.1 yes 12 37.30 even 18 inner
592.2.bq.b.289.2 12 148.67 odd 18
592.2.bq.b.465.2 12 4.3 odd 2
666.2.bj.c.289.2 12 111.104 odd 18
666.2.bj.c.613.2 12 3.2 odd 2
2738.2.a.r.1.3 6 37.17 odd 36
2738.2.a.s.1.4 6 37.20 odd 36