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 21.2
Root \(0.342020 - 0.939693i\) of defining polynomial
Character \(\chi\) \(=\) 74.21
Dual form 74.2.h.a.67.2

$q$-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+(0.342020 - 0.939693i) q^{2} +(0.326352 - 0.118782i) q^{3} +(-0.766044 - 0.642788i) q^{4} +(0.839712 + 0.148064i) q^{5} -0.347296i q^{6} +(0.240460 - 1.36372i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(-2.20574 + 1.85083i) q^{9} +(0.426333 - 0.738430i) q^{10} +(0.466006 + 0.807147i) q^{11} +(-0.326352 - 0.118782i) q^{12} +(-2.34092 + 2.78980i) q^{13} +(-1.19923 - 0.692377i) q^{14} +(0.291629 - 0.0514220i) q^{15} +(0.173648 + 0.984808i) q^{16} +(2.84539 + 3.39101i) q^{17} +(0.984808 + 2.70574i) q^{18} +(-1.30826 - 3.59443i) q^{19} +(-0.548083 - 0.653180i) q^{20} +(-0.0835109 - 0.473614i) q^{21} +(0.917853 - 0.161842i) q^{22} +(0.920780 + 0.531613i) q^{23} +(-0.223238 + 0.266044i) q^{24} +(-4.01527 - 1.46144i) q^{25} +(1.82091 + 3.15391i) q^{26} +(-1.02094 + 1.76833i) q^{27} +(-1.06078 + 0.890103i) q^{28} +(0.873775 - 0.504474i) q^{29} +(0.0514220 - 0.291629i) q^{30} -7.33920i q^{31} +(0.984808 + 0.173648i) q^{32} +(0.247957 + 0.208060i) q^{33} +(4.15968 - 1.51400i) q^{34} +(0.403834 - 1.10953i) q^{35} +2.87939 q^{36} +(1.15600 - 5.97191i) q^{37} -3.82511 q^{38} +(-0.432584 + 1.18851i) q^{39} +(-0.801244 + 0.291629i) q^{40} +(-0.186251 - 0.156283i) q^{41} +(-0.473614 - 0.0835109i) q^{42} -5.13740i q^{43} +(0.161842 - 0.917853i) q^{44} +(-2.12622 + 1.22758i) q^{45} +(0.814478 - 0.683428i) q^{46} +(3.89795 - 6.75145i) q^{47} +(0.173648 + 0.300767i) q^{48} +(4.77595 + 1.73830i) q^{49} +(-2.74661 + 3.27328i) q^{50} +(1.33139 + 0.768679i) q^{51} +(3.58649 - 0.632396i) q^{52} +(2.25380 + 12.7819i) q^{53} +(1.31250 + 1.56418i) q^{54} +(0.271802 + 0.746769i) q^{55} +(0.473614 + 1.30124i) q^{56} +(-0.853909 - 1.01765i) q^{57} +(-0.175202 - 0.993621i) q^{58} +(-9.61896 + 1.69608i) q^{59} +(-0.256454 - 0.148064i) q^{60} +(0.255191 - 0.304124i) q^{61} +(-6.89659 - 2.51015i) q^{62} +(1.99362 + 3.45305i) q^{63} +(0.500000 - 0.866025i) q^{64} +(-2.37876 + 1.99602i) q^{65} +(0.280319 - 0.161842i) q^{66} +(-2.47277 + 14.0238i) q^{67} -4.42664i q^{68} +(0.363645 + 0.0641204i) q^{69} +(-0.904494 - 0.758960i) q^{70} +(12.8449 - 4.67517i) q^{71} +(0.984808 - 2.70574i) q^{72} -13.1543 q^{73} +(-5.21638 - 3.12879i) q^{74} -1.48398 q^{75} +(-1.30826 + 3.59443i) q^{76} +(1.21278 - 0.441414i) q^{77} +(0.968886 + 0.812992i) q^{78} +(3.43258 + 0.605257i) q^{79} +0.852666i q^{80} +(1.37686 - 7.80856i) q^{81} +(-0.210560 + 0.121567i) q^{82} +(-12.8078 + 10.7470i) q^{83} +(-0.240460 + 0.416489i) q^{84} +(1.88722 + 3.26877i) q^{85} +(-4.82758 - 1.75710i) q^{86} +(0.225236 - 0.268425i) q^{87} +(-0.807147 - 0.466006i) q^{88} +(6.19352 - 1.09209i) q^{89} +(0.426333 + 2.41785i) q^{90} +(3.24160 + 3.86318i) q^{91} +(-0.363645 - 0.999105i) q^{92} +(-0.871767 - 2.39516i) q^{93} +(-5.01111 - 5.97200i) q^{94} +(-0.566360 - 3.21199i) q^{95} +(0.342020 - 0.0603074i) q^{96} +(6.47160 + 3.73638i) q^{97} +(3.26694 - 3.89339i) q^{98} +(-2.52178 - 0.917853i) 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
\(5\) 0.839712 + 0.148064i 0.375530 + 0.0662162i 0.358228 − 0.933634i \(-0.383381\pi\)
0.0173025 + 0.999850i \(0.494492\pi\)
\(6\) − 0.347296i − 0.141783i
\(7\) 0.240460 − 1.36372i 0.0908854 − 0.515437i −0.905045 − 0.425315i \(-0.860163\pi\)
0.995931 − 0.0901216i \(-0.0287256\pi\)
\(8\) −0.866025 + 0.500000i −0.306186 + 0.176777i
\(9\) −2.20574 + 1.85083i −0.735246 + 0.616944i
\(10\) 0.426333 − 0.738430i 0.134818 − 0.233512i
\(11\) 0.466006 + 0.807147i 0.140506 + 0.243364i 0.927687 − 0.373358i \(-0.121794\pi\)
−0.787181 + 0.616722i \(0.788460\pi\)
\(12\) −0.326352 − 0.118782i −0.0942097 − 0.0342895i
\(13\) −2.34092 + 2.78980i −0.649254 + 0.773750i −0.985801 − 0.167916i \(-0.946296\pi\)
0.336548 + 0.941666i \(0.390741\pi\)
\(14\) −1.19923 − 0.692377i −0.320508 − 0.185046i
\(15\) 0.291629 − 0.0514220i 0.0752982 − 0.0132771i
\(16\) 0.173648 + 0.984808i 0.0434120 + 0.246202i
\(17\) 2.84539 + 3.39101i 0.690109 + 0.822440i 0.991369 − 0.131102i \(-0.0418517\pi\)
−0.301260 + 0.953542i \(0.597407\pi\)
\(18\) 0.984808 + 2.70574i 0.232121 + 0.637748i
\(19\) −1.30826 − 3.59443i −0.300136 − 0.824618i −0.994475 − 0.104970i \(-0.966525\pi\)
0.694339 − 0.719648i \(-0.255697\pi\)
\(20\) −0.548083 − 0.653180i −0.122555 − 0.146055i
\(21\) −0.0835109 − 0.473614i −0.0182236 − 0.103351i
\(22\) 0.917853 − 0.161842i 0.195687 − 0.0345049i
\(23\) 0.920780 + 0.531613i 0.191996 + 0.110849i 0.592917 − 0.805264i \(-0.297976\pi\)
−0.400921 + 0.916113i \(0.631310\pi\)
\(24\) −0.223238 + 0.266044i −0.0455682 + 0.0543061i
\(25\) −4.01527 − 1.46144i −0.803054 − 0.292288i
\(26\) 1.82091 + 3.15391i 0.357110 + 0.618532i
\(27\) −1.02094 + 1.76833i −0.196481 + 0.340315i
\(28\) −1.06078 + 0.890103i −0.200469 + 0.168214i
\(29\) 0.873775 − 0.504474i 0.162256 − 0.0936786i −0.416673 − 0.909056i \(-0.636804\pi\)
0.578929 + 0.815378i \(0.303471\pi\)
\(30\) 0.0514220 − 0.291629i 0.00938833 − 0.0532439i
\(31\) − 7.33920i − 1.31816i −0.752073 − 0.659080i \(-0.770946\pi\)
0.752073 − 0.659080i \(-0.229054\pi\)
\(32\) 0.984808 + 0.173648i 0.174091 + 0.0306970i
\(33\) 0.247957 + 0.208060i 0.0431637 + 0.0362187i
\(34\) 4.15968 − 1.51400i 0.713380 − 0.259649i
\(35\) 0.403834 − 1.10953i 0.0682605 − 0.187544i
\(36\) 2.87939 0.479898
\(37\) 1.15600 − 5.97191i 0.190045 − 0.981775i
\(38\) −3.82511 −0.620515
\(39\) −0.432584 + 1.18851i −0.0692689 + 0.190315i
\(40\) −0.801244 + 0.291629i −0.126688 + 0.0461106i
\(41\) −0.186251 − 0.156283i −0.0290876 − 0.0244074i 0.628128 − 0.778110i \(-0.283821\pi\)
−0.657216 + 0.753703i \(0.728266\pi\)
\(42\) −0.473614 − 0.0835109i −0.0730802 − 0.0128860i
\(43\) − 5.13740i − 0.783447i −0.920083 − 0.391723i \(-0.871879\pi\)
0.920083 − 0.391723i \(-0.128121\pi\)
\(44\) 0.161842 − 0.917853i 0.0243986 − 0.138372i
\(45\) −2.12622 + 1.22758i −0.316959 + 0.182996i
\(46\) 0.814478 − 0.683428i 0.120088 − 0.100766i
\(47\) 3.89795 − 6.75145i 0.568574 − 0.984800i −0.428133 − 0.903716i \(-0.640829\pi\)
0.996707 − 0.0810838i \(-0.0258381\pi\)
\(48\) 0.173648 + 0.300767i 0.0250640 + 0.0434120i
\(49\) 4.77595 + 1.73830i 0.682278 + 0.248329i
\(50\) −2.74661 + 3.27328i −0.388429 + 0.462911i
\(51\) 1.33139 + 0.768679i 0.186432 + 0.107637i
\(52\) 3.58649 − 0.632396i 0.497357 − 0.0876975i
\(53\) 2.25380 + 12.7819i 0.309583 + 1.75573i 0.601106 + 0.799170i \(0.294727\pi\)
−0.291522 + 0.956564i \(0.594162\pi\)
\(54\) 1.31250 + 1.56418i 0.178609 + 0.212858i
\(55\) 0.271802 + 0.746769i 0.0366497 + 0.100694i
\(56\) 0.473614 + 1.30124i 0.0632893 + 0.173886i
\(57\) −0.853909 − 1.01765i −0.113103 − 0.134791i
\(58\) −0.175202 − 0.993621i −0.0230052 − 0.130469i
\(59\) −9.61896 + 1.69608i −1.25228 + 0.220811i −0.760172 − 0.649722i \(-0.774885\pi\)
−0.492110 + 0.870533i \(0.663774\pi\)
\(60\) −0.256454 − 0.148064i −0.0331081 − 0.0191150i
\(61\) 0.255191 − 0.304124i 0.0326738 − 0.0389391i −0.749460 − 0.662050i \(-0.769687\pi\)
0.782134 + 0.623110i \(0.214131\pi\)
\(62\) −6.89659 − 2.51015i −0.875868 − 0.318790i
\(63\) 1.99362 + 3.45305i 0.251173 + 0.435044i
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) −2.37876 + 1.99602i −0.295049 + 0.247576i
\(66\) 0.280319 − 0.161842i 0.0345049 − 0.0199214i
\(67\) −2.47277 + 14.0238i −0.302097 + 1.71328i 0.334765 + 0.942302i \(0.391343\pi\)
−0.636862 + 0.770978i \(0.719768\pi\)
\(68\) − 4.42664i − 0.536809i
\(69\) 0.363645 + 0.0641204i 0.0437777 + 0.00771918i
\(70\) −0.904494 − 0.758960i −0.108108 − 0.0907131i
\(71\) 12.8449 − 4.67517i 1.52441 − 0.554840i 0.562166 − 0.827024i \(-0.309968\pi\)
0.962245 + 0.272184i \(0.0877459\pi\)
\(72\) 0.984808 − 2.70574i 0.116061 − 0.318874i
\(73\) −13.1543 −1.53959 −0.769794 − 0.638292i \(-0.779641\pi\)
−0.769794 + 0.638292i \(0.779641\pi\)
\(74\) −5.21638 − 3.12879i −0.606392 − 0.363715i
\(75\) −1.48398 −0.171356
\(76\) −1.30826 + 3.59443i −0.150068 + 0.412309i
\(77\) 1.21278 − 0.441414i 0.138209 − 0.0503038i
\(78\) 0.968886 + 0.812992i 0.109705 + 0.0920532i
\(79\) 3.43258 + 0.605257i 0.386196 + 0.0680968i 0.363375 − 0.931643i \(-0.381624\pi\)
0.0228205 + 0.999740i \(0.492735\pi\)
\(80\) 0.852666i 0.0953309i
\(81\) 1.37686 − 7.80856i 0.152984 − 0.867617i
\(82\) −0.210560 + 0.121567i −0.0232525 + 0.0134248i
\(83\) −12.8078 + 10.7470i −1.40584 + 1.17964i −0.447399 + 0.894335i \(0.647649\pi\)
−0.958438 + 0.285302i \(0.907906\pi\)
\(84\) −0.240460 + 0.416489i −0.0262363 + 0.0454427i
\(85\) 1.88722 + 3.26877i 0.204698 + 0.354548i
\(86\) −4.82758 − 1.75710i −0.520571 − 0.189472i
\(87\) 0.225236 − 0.268425i 0.0241478 − 0.0287782i
\(88\) −0.807147 − 0.466006i −0.0860421 − 0.0496764i
\(89\) 6.19352 − 1.09209i 0.656512 − 0.115761i 0.164538 − 0.986371i \(-0.447387\pi\)
0.491974 + 0.870610i \(0.336275\pi\)
\(90\) 0.426333 + 2.41785i 0.0449394 + 0.254864i
\(91\) 3.24160 + 3.86318i 0.339812 + 0.404972i
\(92\) −0.363645 − 0.999105i −0.0379126 − 0.104164i
\(93\) −0.871767 − 2.39516i −0.0903981 − 0.248367i
\(94\) −5.01111 − 5.97200i −0.516856 − 0.615965i
\(95\) −0.566360 − 3.21199i −0.0581073 − 0.329543i
\(96\) 0.342020 − 0.0603074i 0.0349073 − 0.00615510i
\(97\) 6.47160 + 3.73638i 0.657092 + 0.379372i 0.791168 − 0.611599i \(-0.209473\pi\)
−0.134076 + 0.990971i \(0.542807\pi\)
\(98\) 3.26694 − 3.89339i 0.330011 − 0.393291i
\(99\) −2.52178 − 0.917853i −0.253449 − 0.0922477i
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.21.2 ✓ 12
3.2 odd 2 666.2.bj.c.613.1 12
4.3 odd 2 592.2.bq.b.465.1 12
37.17 odd 36 2738.2.a.s.1.3 6
37.20 odd 36 2738.2.a.r.1.4 6
37.30 even 18 inner 74.2.h.a.67.2 yes 12
111.104 odd 18 666.2.bj.c.289.1 12
148.67 odd 18 592.2.bq.b.289.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.21.2 ✓ 12 1.1 even 1 trivial
74.2.h.a.67.2 yes 12 37.30 even 18 inner
592.2.bq.b.289.1 12 148.67 odd 18
592.2.bq.b.465.1 12 4.3 odd 2
666.2.bj.c.289.1 12 111.104 odd 18
666.2.bj.c.613.1 12 3.2 odd 2
2738.2.a.r.1.4 6 37.20 odd 36
2738.2.a.s.1.3 6 37.17 odd 36