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 41.2
Root \(0.984808 + 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 74.41
Dual form 74.2.h.a.65.2

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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 −0.960054 − 0.279815i \(-0.909727\pi\)
0.806453 − 0.591298i \(-0.201384\pi\)
\(4\) 0.939693 + 0.342020i 0.469846 + 0.171010i
\(5\) −0.247315 − 0.294739i −0.110603 − 0.131811i 0.707903 − 0.706310i \(-0.249641\pi\)
−0.818506 + 0.574499i \(0.805197\pi\)
\(6\) − 1.53209i − 0.625473i
\(7\) −2.50048 + 2.09815i −0.945091 + 0.793026i −0.978464 − 0.206417i \(-0.933820\pi\)
0.0333729 + 0.999443i \(0.489375\pi\)
\(8\) 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 −0.992404 − 0.123023i \(-0.960741\pi\)
0.602743 + 0.797935i \(0.294074\pi\)
\(12\) 0.266044 − 1.50881i 0.0768004 − 0.435557i
\(13\) −0.466831 + 1.28261i −0.129476 + 0.355731i −0.987444 − 0.157972i \(-0.949504\pi\)
0.857968 + 0.513703i \(0.171727\pi\)
\(14\) −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 −0.997763 − 0.0668568i \(-0.978703\pi\)
0.721356 − 0.692565i \(-0.243519\pi\)
\(18\) 0.642788 − 0.113341i 0.151506 − 0.0267147i
\(19\) −5.89485 + 1.03942i −1.35237 + 0.238459i −0.802431 − 0.596745i \(-0.796460\pi\)
−0.549939 + 0.835205i \(0.685349\pi\)
\(20\) −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 0.372680 − 0.927960i \(-0.378439\pi\)
0.989977 + 0.141230i \(0.0451057\pi\)
\(24\) 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 0.735583 − 0.677434i \(-0.236908\pi\)
−0.218883 + 0.975751i \(0.570241\pi\)
\(30\) −0.451566 + 0.378909i −0.0824444 + 0.0691790i
\(31\) 2.53737i 0.455726i 0.973693 + 0.227863i \(0.0731738\pi\)
−0.973693 + 0.227863i \(0.926826\pi\)
\(32\) 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 0.867931 − 0.496685i \(-0.165449\pi\)
0.345611 + 0.938378i \(0.387672\pi\)
\(42\) 3.21455 + 3.83095i 0.496016 + 0.591129i
\(43\) − 4.33920i − 0.661722i −0.943680 − 0.330861i \(-0.892661\pi\)
0.943680 − 0.330861i \(-0.107339\pi\)
\(44\) −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 0.609888 − 0.792488i \(-0.291215\pi\)
−0.991258 + 0.131934i \(0.957881\pi\)
\(48\) 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 0.0601551 − 0.998189i \(-0.480840\pi\)
−0.972578 + 0.232575i \(0.925285\pi\)
\(54\) −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 0.131849 − 0.991270i \(-0.457909\pi\)
0.953315 − 0.301978i \(-0.0976469\pi\)
\(60\) −0.510503 + 0.294739i −0.0659056 + 0.0380506i
\(61\) −2.39847 + 6.58973i −0.307092 + 0.843728i 0.686128 + 0.727481i \(0.259309\pi\)
−0.993220 + 0.116248i \(0.962913\pi\)
\(62\) −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 −0.993275 − 0.115781i \(-0.963063\pi\)
−0.0584586 + 0.998290i \(0.518619\pi\)
\(68\) − 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 0.603477 + 0.797380i \(0.293781\pi\)
−0.294363 + 0.955694i \(0.595107\pi\)
\(72\) 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 0.816357 − 0.577548i \(-0.195990\pi\)
−0.710532 + 0.703664i \(0.751546\pi\)
\(80\) − 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 −0.341447 − 0.939901i \(-0.610917\pi\)
0.342593 + 0.939484i \(0.388695\pi\)
\(84\) 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 −0.797542 − 0.603264i \(-0.793867\pi\)
0.732591 + 0.680669i \(0.238311\pi\)
\(90\) −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 0.322097 − 0.946707i \(-0.395612\pi\)
0.980920 + 0.194410i \(0.0622791\pi\)
\(98\) 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.41.2 ✓ 12
3.2 odd 2 666.2.bj.c.559.1 12
4.3 odd 2 592.2.bq.b.337.2 12
37.18 odd 36 2738.2.a.r.1.5 6
37.19 odd 36 2738.2.a.s.1.6 6
37.28 even 18 inner 74.2.h.a.65.2 yes 12
111.65 odd 18 666.2.bj.c.361.1 12
148.139 odd 18 592.2.bq.b.65.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.2 ✓ 12 1.1 even 1 trivial
74.2.h.a.65.2 yes 12 37.28 even 18 inner
592.2.bq.b.65.2 12 148.139 odd 18
592.2.bq.b.337.2 12 4.3 odd 2
666.2.bj.c.361.1 12 111.65 odd 18
666.2.bj.c.559.1 12 3.2 odd 2
2738.2.a.r.1.5 6 37.18 odd 36
2738.2.a.s.1.6 6 37.19 odd 36