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(7,74)] 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("74.7"); 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([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == 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 \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 53.1
Root \(2.00752 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 74.53
Dual form 74.2.f.b.7.1

$q$-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+(-0.939693 - 0.342020i) q^{2} +(-2.04963 + 0.746005i) q^{3} +(0.766044 + 0.642788i) q^{4} +(-0.326352 + 1.85083i) q^{5} +2.18117 q^{6} +(-0.711830 + 4.03699i) q^{7} +(-0.500000 - 0.866025i) q^{8} +(1.34633 - 1.12971i) q^{9} +(0.939693 - 1.62760i) q^{10} +(-1.67088 - 2.89404i) q^{11} +(-2.04963 - 0.746005i) q^{12} +(1.48512 + 1.24616i) q^{13} +(2.04963 - 3.55007i) q^{14} +(-0.711830 - 4.03699i) q^{15} +(0.173648 + 0.984808i) q^{16} +(3.40626 - 2.85819i) q^{17} +(-1.65152 + 0.601105i) q^{18} +(0.0932770 - 0.0339500i) q^{19} +(-1.43969 + 1.20805i) q^{20} +(-1.55262 - 8.80536i) q^{21} +(0.580289 + 3.29098i) q^{22} +(-4.76146 + 8.24709i) q^{23} +(1.67088 + 1.40203i) q^{24} +(1.37939 + 0.502055i) q^{25} +(-0.969343 - 1.67895i) q^{26} +(1.35504 - 2.34700i) q^{27} +(-3.14022 + 2.63496i) q^{28} +(-0.387288 - 0.670802i) q^{29} +(-0.711830 + 4.03699i) q^{30} +10.3914 q^{31} +(0.173648 - 0.984808i) q^{32} +(5.58365 + 4.68524i) q^{33} +(-4.17840 + 1.52081i) q^{34} +(-7.23948 - 2.63496i) q^{35} +1.75751 q^{36} +(-3.50302 + 4.97281i) q^{37} -0.0992633 q^{38} +(-3.97359 - 1.44627i) q^{39} +(1.76604 - 0.642788i) q^{40} +(4.36218 + 3.66031i) q^{41} +(-1.55262 + 8.80536i) q^{42} -1.11986 q^{43} +(0.580289 - 3.29098i) q^{44} +(1.65152 + 2.86052i) q^{45} +(7.29498 - 6.12122i) q^{46} +(3.55724 - 6.16132i) q^{47} +(-1.09059 - 1.88895i) q^{48} +(-9.21271 - 3.35315i) q^{49} +(-1.12449 - 0.943555i) q^{50} +(-4.84936 + 8.39933i) q^{51} +(0.336649 + 1.90923i) q^{52} +(0.142188 + 0.806388i) q^{53} +(-2.07604 + 1.74200i) q^{54} +(5.90168 - 2.14804i) q^{55} +(3.85205 - 1.40203i) q^{56} +(-0.165857 + 0.139170i) q^{57} +(0.134504 + 0.762808i) q^{58} +(1.40642 + 7.97622i) q^{59} +(2.04963 - 3.55007i) q^{60} +(-8.19737 - 6.87841i) q^{61} +(-9.76476 - 3.55408i) q^{62} +(3.60225 + 6.23929i) q^{63} +(-0.500000 + 0.866025i) q^{64} +(-2.79111 + 2.34202i) q^{65} +(-3.64447 - 6.31240i) q^{66} +(2.28630 - 12.9663i) q^{67} +4.44656 q^{68} +(3.60687 - 20.4556i) q^{69} +(5.90168 + 4.95210i) q^{70} +(2.79386 - 1.01688i) q^{71} +(-1.65152 - 0.601105i) q^{72} -2.55740 q^{73} +(4.99257 - 3.47481i) q^{74} -3.20177 q^{75} +(0.0932770 + 0.0339500i) q^{76} +(12.8726 - 4.68524i) q^{77} +(3.23930 + 2.71810i) q^{78} +(-0.943888 + 5.35305i) q^{79} -1.87939 q^{80} +(-1.94203 + 11.0138i) q^{81} +(-2.84721 - 4.93152i) q^{82} +(0.504249 - 0.423115i) q^{83} +(4.47060 - 7.74331i) q^{84} +(4.17840 + 7.23720i) q^{85} +(1.05232 + 0.383014i) q^{86} +(1.29422 + 1.08598i) q^{87} +(-1.67088 + 2.89404i) q^{88} +(0.612149 + 3.47167i) q^{89} +(-0.573568 - 3.25286i) q^{90} +(-6.08790 + 5.10835i) q^{91} +(-8.94862 + 3.25703i) q^{92} +(-21.2986 + 7.75206i) q^{93} +(-5.45001 + 4.57310i) q^{94} +(0.0323948 + 0.183720i) q^{95} +(0.378757 + 2.14804i) q^{96} +(-1.81012 + 3.13522i) q^{97} +(7.51027 + 6.30186i) q^{98} +(-5.51897 - 2.00874i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} - 3 q^{17} + 6 q^{18} - 3 q^{19} - 6 q^{20} - 33 q^{21} - 3 q^{22} - 21 q^{23} - 3 q^{24} - 6 q^{25}+ \cdots - 33 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.939693 − 0.342020i −0.664463 − 0.241845i
\(3\) −2.04963 + 0.746005i −1.18336 + 0.430706i −0.857386 − 0.514674i \(-0.827913\pi\)
−0.325970 + 0.945380i \(0.605691\pi\)
\(4\) 0.766044 + 0.642788i 0.383022 + 0.321394i
\(5\) −0.326352 + 1.85083i −0.145949 + 0.827718i 0.820651 + 0.571429i \(0.193611\pi\)
−0.966600 + 0.256289i \(0.917500\pi\)
\(6\) 2.18117 0.890460
\(7\) −0.711830 + 4.03699i −0.269046 + 1.52584i 0.488214 + 0.872724i \(0.337649\pi\)
−0.757260 + 0.653113i \(0.773463\pi\)
\(8\) −0.500000 − 0.866025i −0.176777 − 0.306186i
\(9\) 1.34633 − 1.12971i 0.448778 − 0.376569i
\(10\) 0.939693 − 1.62760i 0.297157 − 0.514691i
\(11\) −1.67088 − 2.89404i −0.503788 − 0.872586i −0.999990 − 0.00437929i \(-0.998606\pi\)
0.496203 − 0.868207i \(-0.334727\pi\)
\(12\) −2.04963 − 0.746005i −0.591678 − 0.215353i
\(13\) 1.48512 + 1.24616i 0.411898 + 0.345623i 0.825071 − 0.565029i \(-0.191135\pi\)
−0.413173 + 0.910653i \(0.635580\pi\)
\(14\) 2.04963 − 3.55007i 0.547787 − 0.948795i
\(15\) −0.711830 − 4.03699i −0.183794 − 1.04235i
\(16\) 0.173648 + 0.984808i 0.0434120 + 0.246202i
\(17\) 3.40626 − 2.85819i 0.826140 − 0.693214i −0.128261 − 0.991740i \(-0.540940\pi\)
0.954401 + 0.298527i \(0.0964952\pi\)
\(18\) −1.65152 + 0.601105i −0.389268 + 0.141682i
\(19\) 0.0932770 − 0.0339500i 0.0213992 − 0.00778868i −0.331298 − 0.943526i \(-0.607487\pi\)
0.352698 + 0.935737i \(0.385264\pi\)
\(20\) −1.43969 + 1.20805i −0.321925 + 0.270127i
\(21\) −1.55262 − 8.80536i −0.338810 − 1.92149i
\(22\) 0.580289 + 3.29098i 0.123718 + 0.701640i
\(23\) −4.76146 + 8.24709i −0.992833 + 1.71964i −0.392920 + 0.919573i \(0.628535\pi\)
−0.599913 + 0.800065i \(0.704798\pi\)
\(24\) 1.67088 + 1.40203i 0.341066 + 0.286188i
\(25\) 1.37939 + 0.502055i 0.275877 + 0.100411i
\(26\) −0.969343 − 1.67895i −0.190104 − 0.329269i
\(27\) 1.35504 − 2.34700i 0.260777 − 0.451680i
\(28\) −3.14022 + 2.63496i −0.593445 + 0.497960i
\(29\) −0.387288 − 0.670802i −0.0719175 − 0.124565i 0.827824 − 0.560988i \(-0.189579\pi\)
−0.899742 + 0.436423i \(0.856245\pi\)
\(30\) −0.711830 + 4.03699i −0.129962 + 0.737049i
\(31\) 10.3914 1.86636 0.933179 − 0.359413i \(-0.117023\pi\)
0.933179 + 0.359413i \(0.117023\pi\)
\(32\) 0.173648 − 0.984808i 0.0306970 − 0.174091i
\(33\) 5.58365 + 4.68524i 0.971988 + 0.815595i
\(34\) −4.17840 + 1.52081i −0.716590 + 0.260817i
\(35\) −7.23948 − 2.63496i −1.22370 − 0.445389i
\(36\) 1.75751 0.292919
\(37\) −3.50302 + 4.97281i −0.575894 + 0.817525i
\(38\) −0.0992633 −0.0161026
\(39\) −3.97359 − 1.44627i −0.636284 − 0.231588i
\(40\) 1.76604 − 0.642788i 0.279236 − 0.101634i
\(41\) 4.36218 + 3.66031i 0.681259 + 0.571644i 0.916374 − 0.400324i \(-0.131102\pi\)
−0.235115 + 0.971968i \(0.575547\pi\)
\(42\) −1.55262 + 8.80536i −0.239575 + 1.35870i
\(43\) −1.11986 −0.170777 −0.0853884 − 0.996348i \(-0.527213\pi\)
−0.0853884 + 0.996348i \(0.527213\pi\)
\(44\) 0.580289 − 3.29098i 0.0874818 − 0.496134i
\(45\) 1.65152 + 2.86052i 0.246194 + 0.426421i
\(46\) 7.29498 − 6.12122i 1.07559 − 0.902524i
\(47\) 3.55724 − 6.16132i 0.518877 − 0.898721i −0.480883 − 0.876785i \(-0.659684\pi\)
0.999759 − 0.0219357i \(-0.00698290\pi\)
\(48\) −1.09059 − 1.88895i −0.157413 − 0.272647i
\(49\) −9.21271 − 3.35315i −1.31610 − 0.479022i
\(50\) −1.12449 − 0.943555i −0.159026 − 0.133439i
\(51\) −4.84936 + 8.39933i −0.679046 + 1.17614i
\(52\) 0.336649 + 1.90923i 0.0466848 + 0.264763i
\(53\) 0.142188 + 0.806388i 0.0195310 + 0.110766i 0.993015 − 0.117990i \(-0.0376452\pi\)
−0.973484 + 0.228756i \(0.926534\pi\)
\(54\) −2.07604 + 1.74200i −0.282513 + 0.237057i
\(55\) 5.90168 − 2.14804i 0.795782 − 0.289641i
\(56\) 3.85205 − 1.40203i 0.514751 − 0.187354i
\(57\) −0.165857 + 0.139170i −0.0219682 + 0.0184335i
\(58\) 0.134504 + 0.762808i 0.0176612 + 0.100162i
\(59\) 1.40642 + 7.97622i 0.183101 + 1.03842i 0.928372 + 0.371653i \(0.121209\pi\)
−0.745271 + 0.666762i \(0.767680\pi\)
\(60\) 2.04963 − 3.55007i 0.264606 − 0.458312i
\(61\) −8.19737 − 6.87841i −1.04957 − 0.880691i −0.0565183 − 0.998402i \(-0.518000\pi\)
−0.993048 + 0.117711i \(0.962444\pi\)
\(62\) −9.76476 − 3.55408i −1.24013 − 0.451369i
\(63\) 3.60225 + 6.23929i 0.453841 + 0.786076i
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) −2.79111 + 2.34202i −0.346195 + 0.290492i
\(66\) −3.64447 − 6.31240i −0.448603 − 0.777003i
\(67\) 2.28630 − 12.9663i 0.279317 − 1.58408i −0.445591 − 0.895237i \(-0.647006\pi\)
0.724907 − 0.688846i \(-0.241883\pi\)
\(68\) 4.44656 0.539225
\(69\) 3.60687 − 20.4556i 0.434216 − 2.46256i
\(70\) 5.90168 + 4.95210i 0.705386 + 0.591889i
\(71\) 2.79386 − 1.01688i 0.331570 − 0.120682i −0.170870 − 0.985294i \(-0.554658\pi\)
0.502440 + 0.864612i \(0.332436\pi\)
\(72\) −1.65152 − 0.601105i −0.194634 − 0.0708409i
\(73\) −2.55740 −0.299321 −0.149660 − 0.988737i \(-0.547818\pi\)
−0.149660 + 0.988737i \(0.547818\pi\)
\(74\) 4.99257 − 3.47481i 0.580374 − 0.403938i
\(75\) −3.20177 −0.369708
\(76\) 0.0932770 + 0.0339500i 0.0106996 + 0.00389434i
\(77\) 12.8726 − 4.68524i 1.46697 − 0.533932i
\(78\) 3.23930 + 2.71810i 0.366779 + 0.307764i
\(79\) −0.943888 + 5.35305i −0.106196 + 0.602265i 0.884540 + 0.466464i \(0.154472\pi\)
−0.990736 + 0.135802i \(0.956639\pi\)
\(80\) −1.87939 −0.210122
\(81\) −1.94203 + 11.0138i −0.215781 + 1.22375i
\(82\) −2.84721 − 4.93152i −0.314422 − 0.544595i
\(83\) 0.504249 − 0.423115i 0.0553485 − 0.0464429i −0.614694 − 0.788766i \(-0.710720\pi\)
0.670042 + 0.742323i \(0.266276\pi\)
\(84\) 4.47060 − 7.74331i 0.487782 − 0.844864i
\(85\) 4.17840 + 7.23720i 0.453211 + 0.784985i
\(86\) 1.05232 + 0.383014i 0.113475 + 0.0413015i
\(87\) 1.29422 + 1.08598i 0.138755 + 0.116429i
\(88\) −1.67088 + 2.89404i −0.178116 + 0.308506i
\(89\) 0.612149 + 3.47167i 0.0648877 + 0.367996i 0.999910 + 0.0134078i \(0.00426798\pi\)
−0.935022 + 0.354589i \(0.884621\pi\)
\(90\) −0.573568 − 3.25286i −0.0604593 − 0.342882i
\(91\) −6.08790 + 5.10835i −0.638185 + 0.535501i
\(92\) −8.94862 + 3.25703i −0.932958 + 0.339569i
\(93\) −21.2986 + 7.75206i −2.20856 + 0.803852i
\(94\) −5.45001 + 4.57310i −0.562125 + 0.471679i
\(95\) 0.0323948 + 0.183720i 0.00332363 + 0.0188493i
\(96\) 0.378757 + 2.14804i 0.0386567 + 0.219233i
\(97\) −1.81012 + 3.13522i −0.183790 + 0.318334i −0.943168 − 0.332316i \(-0.892170\pi\)
0.759378 + 0.650650i \(0.225503\pi\)
\(98\) 7.51027 + 6.30186i 0.758652 + 0.636584i
\(99\) −5.51897 − 2.00874i −0.554678 − 0.201886i
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.f.b.53.1 yes 12
3.2 odd 2 666.2.x.g.127.1 12
4.3 odd 2 592.2.bc.d.497.2 12
37.7 even 9 inner 74.2.f.b.7.1 ✓ 12
37.9 even 9 2738.2.a.t.1.5 6
37.28 even 18 2738.2.a.q.1.5 6
111.44 odd 18 666.2.x.g.451.1 12
148.7 odd 18 592.2.bc.d.81.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.7.1 ✓ 12 37.7 even 9 inner
74.2.f.b.53.1 yes 12 1.1 even 1 trivial
592.2.bc.d.81.2 12 148.7 odd 18
592.2.bc.d.497.2 12 4.3 odd 2
666.2.x.g.127.1 12 3.2 odd 2
666.2.x.g.451.1 12 111.44 odd 18
2738.2.a.q.1.5 6 37.28 even 18
2738.2.a.t.1.5 6 37.9 even 9