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 65.2
Root \(0.984808 - 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 74.65
Dual form 74.2.h.a.41.2

$q$-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+(0.984808 - 0.173648i) q^{2} +(-0.266044 + 1.50881i) q^{3} +(0.939693 - 0.342020i) q^{4} +(-0.247315 + 0.294739i) q^{5} +1.53209i q^{6} +(-2.50048 - 2.09815i) q^{7} +(0.866025 - 0.500000i) q^{8} +(0.613341 + 0.223238i) q^{9} +(-0.192377 + 0.333207i) q^{10} +(-1.29236 - 2.23843i) q^{11} +(0.266044 + 1.50881i) q^{12} +(-0.466831 - 1.28261i) q^{13} +(-2.82683 - 1.63207i) q^{14} +(-0.378909 - 0.451566i) q^{15} +(0.766044 - 0.642788i) q^{16} +(-1.13965 + 3.13118i) q^{17} +(0.642788 + 0.113341i) q^{18} +(-5.89485 - 1.03942i) q^{19} +(-0.131594 + 0.361551i) q^{20} +(3.83095 - 3.21455i) q^{21} +(-1.66142 - 1.98001i) q^{22} +(6.53507 + 3.77303i) q^{23} +(0.524005 + 1.43969i) q^{24} +(0.842535 + 4.77825i) q^{25} +(-0.682461 - 1.18206i) q^{26} +(-2.79813 + 4.84651i) q^{27} +(-3.06729 - 1.11640i) q^{28} +(2.78251 - 1.60649i) q^{29} +(-0.451566 - 0.378909i) q^{30} -2.53737i q^{31} +(0.642788 - 0.766044i) q^{32} +(3.72120 - 1.35440i) q^{33} +(-0.578618 + 3.28150i) q^{34} +(1.23681 - 0.218083i) q^{35} +0.652704 q^{36} +(0.543196 - 6.05846i) q^{37} -5.98578 q^{38} +(2.05941 - 0.363130i) q^{39} +(-0.0668119 + 0.378909i) q^{40} +(7.77046 - 2.82822i) q^{41} +(3.21455 - 3.83095i) q^{42} +4.33920i q^{43} +(-1.98001 - 1.66142i) q^{44} +(-0.217486 + 0.125565i) q^{45} +(7.09097 + 2.58090i) q^{46} +(-2.61455 + 4.52853i) q^{47} +(0.766044 + 1.32683i) q^{48} +(0.634616 + 3.59909i) q^{49} +(1.65947 + 4.55935i) q^{50} +(-4.42116 - 2.55256i) q^{51} +(-0.877355 - 1.04559i) q^{52} +(-6.64254 + 5.57375i) q^{53} +(-1.91404 + 5.25877i) q^{54} +(0.979373 + 0.172690i) q^{55} +(-3.21455 - 0.566812i) q^{56} +(3.13658 - 8.61769i) q^{57} +(2.46128 - 2.06526i) q^{58} +(8.33530 + 9.93362i) q^{59} +(-0.510503 - 0.294739i) q^{60} +(-2.39847 - 6.58973i) q^{61} +(-0.440610 - 2.49882i) q^{62} +(-1.06526 - 1.84508i) q^{63} +(0.500000 - 0.866025i) q^{64} +(0.493489 + 0.179615i) q^{65} +(3.42947 - 1.98001i) q^{66} +(-8.60881 - 7.22365i) q^{67} +3.33213i q^{68} +(-7.43141 + 8.85641i) q^{69} +(1.18015 - 0.429540i) q^{70} +(2.60464 - 14.7717i) q^{71} +(0.642788 - 0.113341i) q^{72} -15.0792 q^{73} +(-0.517097 - 6.06074i) q^{74} -7.43364 q^{75} +(-5.89485 + 1.03942i) q^{76} +(-1.46505 + 8.30870i) q^{77} +(1.96507 - 0.715227i) q^{78} +(0.940587 - 1.12095i) q^{79} +0.384754i q^{80} +(-5.06805 - 4.25260i) q^{81} +(7.16130 - 4.13458i) q^{82} +(0.0104473 + 0.00380252i) q^{83} +(2.50048 - 4.33095i) q^{84} +(-0.641025 - 1.11029i) q^{85} +(0.753494 + 4.27328i) q^{86} +(1.68361 + 4.62569i) q^{87} +(-2.23843 - 1.29236i) q^{88} +(-0.612745 - 0.730241i) q^{89} +(-0.192377 + 0.161424i) q^{90} +(-1.52380 + 4.18661i) q^{91} +(7.43141 + 1.31036i) q^{92} +(3.82842 + 0.675054i) q^{93} +(-1.78845 + 4.91374i) q^{94} +(1.76424 - 1.48038i) q^{95} +(0.984808 + 1.17365i) q^{96} +(12.8332 + 7.40927i) q^{97} +(1.24995 + 3.43421i) q^{98} +(-0.292954 - 1.66142i) 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 \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.984808 − 0.173648i 0.696364 − 0.122788i
\(3\) −0.266044 + 1.50881i −0.153601 + 0.871114i 0.806453 + 0.591298i \(0.201384\pi\)
−0.960054 + 0.279815i \(0.909727\pi\)
\(4\) 0.939693 − 0.342020i 0.469846 − 0.171010i
\(5\) −0.247315 + 0.294739i −0.110603 + 0.131811i −0.818506 − 0.574499i \(-0.805197\pi\)
0.707903 + 0.706310i \(0.249641\pi\)
\(6\) 1.53209i 0.625473i
\(7\) −2.50048 − 2.09815i −0.945091 − 0.793026i 0.0333729 − 0.999443i \(-0.489375\pi\)
−0.978464 + 0.206417i \(0.933820\pi\)
\(8\) 0.866025 − 0.500000i 0.306186 − 0.176777i
\(9\) 0.613341 + 0.223238i 0.204447 + 0.0744126i
\(10\) −0.192377 + 0.333207i −0.0608350 + 0.105369i
\(11\) −1.29236 − 2.23843i −0.389661 − 0.674912i 0.602743 − 0.797935i \(-0.294074\pi\)
−0.992404 + 0.123023i \(0.960741\pi\)
\(12\) 0.266044 + 1.50881i 0.0768004 + 0.435557i
\(13\) −0.466831 − 1.28261i −0.129476 − 0.355731i 0.857968 − 0.513703i \(-0.171727\pi\)
−0.987444 + 0.157972i \(0.949504\pi\)
\(14\) −2.82683 − 1.63207i −0.755502 − 0.436189i
\(15\) −0.378909 − 0.451566i −0.0978339 − 0.116594i
\(16\) 0.766044 − 0.642788i 0.191511 − 0.160697i
\(17\) −1.13965 + 3.13118i −0.276407 + 0.759422i 0.721356 + 0.692565i \(0.243519\pi\)
−0.997763 + 0.0668568i \(0.978703\pi\)
\(18\) 0.642788 + 0.113341i 0.151506 + 0.0267147i
\(19\) −5.89485 − 1.03942i −1.35237 − 0.238459i −0.549939 − 0.835205i \(-0.685349\pi\)
−0.802431 + 0.596745i \(0.796460\pi\)
\(20\) −0.131594 + 0.361551i −0.0294253 + 0.0808452i
\(21\) 3.83095 − 3.21455i 0.835982 − 0.701472i
\(22\) −1.66142 − 1.98001i −0.354217 − 0.422139i
\(23\) 6.53507 + 3.77303i 1.36266 + 0.786730i 0.989977 − 0.141230i \(-0.0451057\pi\)
0.372680 + 0.927960i \(0.378439\pi\)
\(24\) 0.524005 + 1.43969i 0.106962 + 0.293876i
\(25\) 0.842535 + 4.77825i 0.168507 + 0.955650i
\(26\) −0.682461 − 1.18206i −0.133842 − 0.231821i
\(27\) −2.79813 + 4.84651i −0.538501 + 0.932711i
\(28\) −3.06729 − 1.11640i −0.579663 − 0.210980i
\(29\) 2.78251 − 1.60649i 0.516700 − 0.298317i −0.218883 − 0.975751i \(-0.570241\pi\)
0.735583 + 0.677434i \(0.236908\pi\)
\(30\) −0.451566 − 0.378909i −0.0824444 − 0.0691790i
\(31\) − 2.53737i − 0.455726i −0.973693 − 0.227863i \(-0.926826\pi\)
0.973693 − 0.227863i \(-0.0731738\pi\)
\(32\) 0.642788 − 0.766044i 0.113630 − 0.135419i
\(33\) 3.72120 − 1.35440i 0.647777 − 0.235772i
\(34\) −0.578618 + 3.28150i −0.0992321 + 0.562773i
\(35\) 1.23681 − 0.218083i 0.209059 − 0.0368628i
\(36\) 0.652704 0.108784
\(37\) 0.543196 − 6.05846i 0.0893009 − 0.996005i
\(38\) −5.98578 −0.971022
\(39\) 2.05941 − 0.363130i 0.329770 − 0.0581473i
\(40\) −0.0668119 + 0.378909i −0.0105639 + 0.0599108i
\(41\) 7.77046 − 2.82822i 1.21354 − 0.441693i 0.345611 − 0.938378i \(-0.387672\pi\)
0.867931 + 0.496685i \(0.165449\pi\)
\(42\) 3.21455 − 3.83095i 0.496016 − 0.591129i
\(43\) 4.33920i 0.661722i 0.943680 + 0.330861i \(0.107339\pi\)
−0.943680 + 0.330861i \(0.892661\pi\)
\(44\) −1.98001 − 1.66142i −0.298497 − 0.250469i
\(45\) −0.217486 + 0.125565i −0.0324208 + 0.0187182i
\(46\) 7.09097 + 2.58090i 1.04551 + 0.380533i
\(47\) −2.61455 + 4.52853i −0.381371 + 0.660554i −0.991258 − 0.131934i \(-0.957881\pi\)
0.609888 + 0.792488i \(0.291215\pi\)
\(48\) 0.766044 + 1.32683i 0.110569 + 0.191511i
\(49\) 0.634616 + 3.59909i 0.0906594 + 0.514155i
\(50\) 1.65947 + 4.55935i 0.234684 + 0.644790i
\(51\) −4.42116 − 2.55256i −0.619086 − 0.357430i
\(52\) −0.877355 − 1.04559i −0.121667 − 0.144997i
\(53\) −6.64254 + 5.57375i −0.912423 + 0.765614i −0.972578 − 0.232575i \(-0.925285\pi\)
0.0601551 + 0.998189i \(0.480840\pi\)
\(54\) −1.91404 + 5.25877i −0.260467 + 0.715628i
\(55\) 0.979373 + 0.172690i 0.132059 + 0.0232855i
\(56\) −3.21455 − 0.566812i −0.429562 − 0.0757434i
\(57\) 3.13658 − 8.61769i 0.415450 − 1.14144i
\(58\) 2.46128 − 2.06526i 0.323182 − 0.271182i
\(59\) 8.33530 + 9.93362i 1.08516 + 1.29325i 0.953315 + 0.301978i \(0.0976469\pi\)
0.131849 + 0.991270i \(0.457909\pi\)
\(60\) −0.510503 − 0.294739i −0.0659056 − 0.0380506i
\(61\) −2.39847 − 6.58973i −0.307092 − 0.843728i −0.993220 − 0.116248i \(-0.962913\pi\)
0.686128 − 0.727481i \(-0.259309\pi\)
\(62\) −0.440610 − 2.49882i −0.0559576 − 0.317351i
\(63\) −1.06526 − 1.84508i −0.134210 − 0.232458i
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) 0.493489 + 0.179615i 0.0612098 + 0.0222785i
\(66\) 3.42947 − 1.98001i 0.422139 − 0.243722i
\(67\) −8.60881 − 7.22365i −1.05173 − 0.882509i −0.0584586 − 0.998290i \(-0.518619\pi\)
−0.993275 + 0.115781i \(0.963063\pi\)
\(68\) 3.33213i 0.404080i
\(69\) −7.43141 + 8.85641i −0.894636 + 1.06619i
\(70\) 1.18015 − 0.429540i 0.141055 − 0.0513399i
\(71\) 2.60464 − 14.7717i 0.309114 − 1.75307i −0.294363 − 0.955694i \(-0.595107\pi\)
0.603477 − 0.797380i \(-0.293781\pi\)
\(72\) 0.642788 − 0.113341i 0.0757532 − 0.0133573i
\(73\) −15.0792 −1.76489 −0.882445 − 0.470416i \(-0.844104\pi\)
−0.882445 + 0.470416i \(0.844104\pi\)
\(74\) −0.517097 − 6.06074i −0.0601113 − 0.704547i
\(75\) −7.43364 −0.858363
\(76\) −5.89485 + 1.03942i −0.676185 + 0.119230i
\(77\) −1.46505 + 8.30870i −0.166958 + 0.946864i
\(78\) 1.96507 − 0.715227i 0.222500 − 0.0809835i
\(79\) 0.940587 − 1.12095i 0.105824 − 0.126117i −0.710532 − 0.703664i \(-0.751546\pi\)
0.816357 + 0.577548i \(0.195990\pi\)
\(80\) 0.384754i 0.0430169i
\(81\) −5.06805 − 4.25260i −0.563116 − 0.472511i
\(82\) 7.16130 − 4.13458i 0.790833 − 0.456588i
\(83\) 0.0104473 + 0.00380252i 0.00114674 + 0.000417381i 0.342593 − 0.939484i \(-0.388695\pi\)
−0.341447 + 0.939901i \(0.610917\pi\)
\(84\) 2.50048 − 4.33095i 0.272824 − 0.472546i
\(85\) −0.641025 − 1.11029i −0.0695290 − 0.120428i
\(86\) 0.753494 + 4.27328i 0.0812514 + 0.460799i
\(87\) 1.68361 + 4.62569i 0.180502 + 0.495926i
\(88\) −2.23843 − 1.29236i −0.238617 − 0.137766i
\(89\) −0.612745 − 0.730241i −0.0649509 − 0.0774054i 0.732591 − 0.680669i \(-0.238311\pi\)
−0.797542 + 0.603264i \(0.793867\pi\)
\(90\) −0.192377 + 0.161424i −0.0202783 + 0.0170155i
\(91\) −1.52380 + 4.18661i −0.159738 + 0.438876i
\(92\) 7.43141 + 1.31036i 0.774778 + 0.136614i
\(93\) 3.82842 + 0.675054i 0.396989 + 0.0699998i
\(94\) −1.78845 + 4.91374i −0.184465 + 0.506814i
\(95\) 1.76424 − 1.48038i 0.181008 − 0.151883i
\(96\) 0.984808 + 1.17365i 0.100512 + 0.119785i
\(97\) 12.8332 + 7.40927i 1.30302 + 0.752297i 0.980920 − 0.194410i \(-0.0622791\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(98\) 1.24995 + 3.43421i 0.126264 + 0.346907i
\(99\) −0.292954 − 1.66142i −0.0294430 − 0.166979i
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.65.2 yes 12
3.2 odd 2 666.2.bj.c.361.1 12
4.3 odd 2 592.2.bq.b.65.2 12
37.2 odd 36 2738.2.a.s.1.6 6
37.4 even 18 inner 74.2.h.a.41.2 ✓ 12
37.35 odd 36 2738.2.a.r.1.5 6
111.41 odd 18 666.2.bj.c.559.1 12
148.115 odd 18 592.2.bq.b.337.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.2 ✓ 12 37.4 even 18 inner
74.2.h.a.65.2 yes 12 1.1 even 1 trivial
592.2.bq.b.65.2 12 4.3 odd 2
592.2.bq.b.337.2 12 148.115 odd 18
666.2.bj.c.361.1 12 3.2 odd 2
666.2.bj.c.559.1 12 111.41 odd 18
2738.2.a.r.1.5 6 37.35 odd 36
2738.2.a.s.1.6 6 37.2 odd 36