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 67.1
Root \(-0.342020 - 0.939693i\) of defining polynomial
Character \(\chi\) \(=\) 74.67
Dual form 74.2.h.a.21.1

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{7}{18}\right)\)

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 0.434507 − 0.900669i \(-0.356923\pi\)
−0.246087 + 0.969248i \(0.579145\pi\)
\(4\) −0.766044 + 0.642788i −0.383022 + 0.321394i
\(5\) 2.57176 − 0.453471i 1.15013 − 0.202798i 0.434096 − 0.900867i \(-0.357068\pi\)
0.716031 + 0.698068i \(0.245957\pi\)
\(6\) − 0.347296i − 0.141783i
\(7\) −0.361075 − 2.04776i −0.136473 − 0.773979i −0.973822 − 0.227311i \(-0.927007\pi\)
0.837349 − 0.546669i \(-0.184104\pi\)
\(8\) 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 −0.0816522 + 0.996661i \(0.526020\pi\)
−0.822308 + 0.569043i \(0.807314\pi\)
\(12\) −0.326352 + 0.118782i −0.0942097 + 0.0342895i
\(13\) 2.64632 + 3.15377i 0.733958 + 0.874697i 0.995907 − 0.0903843i \(-0.0288095\pi\)
−0.261949 + 0.965082i \(0.584365\pi\)
\(14\) −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 −0.835838 − 0.548977i \(-0.815018\pi\)
0.685778 + 0.727810i \(0.259462\pi\)
\(18\) −0.984808 + 2.70574i −0.232121 + 0.637748i
\(19\) 0.534946 − 1.46975i 0.122725 − 0.337184i −0.863083 − 0.505063i \(-0.831469\pi\)
0.985808 + 0.167878i \(0.0536916\pi\)
\(20\) −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 −0.948981 − 0.315334i \(-0.897883\pi\)
−0.201403 + 0.979508i \(0.564550\pi\)
\(24\) 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 0.137130 − 0.990553i \(-0.456212\pi\)
−0.789279 + 0.614035i \(0.789546\pi\)
\(30\) −0.157489 − 0.893164i −0.0287534 − 0.163069i
\(31\) − 3.39997i − 0.610653i −0.952248 − 0.305326i \(-0.901234\pi\)
0.952248 − 0.305326i \(-0.0987655\pi\)
\(32\) −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 0.243343 − 0.969940i \(-0.421756\pi\)
0.997461 − 0.0712179i \(-0.0226886\pi\)
\(42\) −0.711179 + 0.125400i −0.109737 + 0.0193496i
\(43\) − 3.76932i − 0.574816i −0.957808 − 0.287408i \(-0.907206\pi\)
0.957808 − 0.287408i \(-0.0927936\pi\)
\(44\) −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 0.998408 − 0.0564019i \(-0.0179628\pi\)
−0.548050 + 0.836446i \(0.684629\pi\)
\(48\) 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 0.725857 + 0.687846i \(0.241444\pi\)
−0.917339 + 0.398106i \(0.869668\pi\)
\(54\) −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 0.490749 − 0.871301i \(-0.336723\pi\)
0.163150 + 0.986601i \(0.447834\pi\)
\(60\) −0.785435 + 0.453471i −0.101399 + 0.0585429i
\(61\) −6.25519 − 7.45465i −0.800895 − 0.954470i 0.198778 − 0.980045i \(-0.436303\pi\)
−0.999673 + 0.0255750i \(0.991858\pi\)
\(62\) −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 −0.864792 − 0.502130i \(-0.832550\pi\)
0.640901 − 0.767624i \(-0.278561\pi\)
\(68\) − 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 −0.339787 − 0.940502i \(-0.610355\pi\)
−0.864835 + 0.502056i \(0.832577\pi\)
\(72\) −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 −0.0305991 − 0.999532i \(-0.509742\pi\)
0.313106 + 0.949718i \(0.398630\pi\)
\(80\) − 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 0.885449 − 0.464737i \(-0.153851\pi\)
−0.303920 + 0.952698i \(0.598296\pi\)
\(84\) 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 −0.760838 − 0.648942i \(-0.775212\pi\)
−0.936905 + 0.349584i \(0.886323\pi\)
\(90\) −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 −0.997380 − 0.0723469i \(-0.976951\pi\)
−0.436036 + 0.899929i \(0.643618\pi\)
\(98\) −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.67.1 yes 12
3.2 odd 2 666.2.bj.c.289.2 12
4.3 odd 2 592.2.bq.b.289.2 12
37.13 odd 36 2738.2.a.s.1.4 6
37.21 even 18 inner 74.2.h.a.21.1 ✓ 12
37.24 odd 36 2738.2.a.r.1.3 6
111.95 odd 18 666.2.bj.c.613.2 12
148.95 odd 18 592.2.bq.b.465.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.21.1 ✓ 12 37.21 even 18 inner
74.2.h.a.67.1 yes 12 1.1 even 1 trivial
592.2.bq.b.289.2 12 4.3 odd 2
592.2.bq.b.465.2 12 148.95 odd 18
666.2.bj.c.289.2 12 3.2 odd 2
666.2.bj.c.613.2 12 111.95 odd 18
2738.2.a.r.1.3 6 37.24 odd 36
2738.2.a.s.1.4 6 37.13 odd 36