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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [74,2,Mod(47,74)] 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("74.47"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(74, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.4406832.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 47.3
Root \(-0.827721 - 1.43366i\) of defining polynomial
Character \(\chi\) \(=\) 74.47
Dual form 74.2.c.c.63.3

$q$-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+(-0.500000 + 0.866025i) q^{2} +(1.52569 + 2.64257i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.629755 - 1.09077i) q^{5} -3.05137 q^{6} +(-1.52569 - 2.64257i) q^{7} +1.00000 q^{8} +(-3.15544 + 5.46539i) q^{9} +1.25951 q^{10} +5.31088 q^{11} +(1.52569 - 2.64257i) q^{12} +(-1.00000 - 1.73205i) q^{13} +3.05137 q^{14} +(1.92162 - 3.32834i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(-2.55137 + 4.41911i) q^{17} +(-3.15544 - 5.46539i) q^{18} +(-2.39593 - 4.14988i) q^{19} +(-0.629755 + 1.09077i) q^{20} +(4.65544 - 8.06346i) q^{21} +(-2.65544 + 4.59936i) q^{22} +1.74049 q^{23} +(1.52569 + 2.64257i) q^{24} +(1.70682 - 2.95629i) q^{25} +2.00000 q^{26} -10.1027 q^{27} +(-1.52569 + 2.64257i) q^{28} -4.05137 q^{29} +(1.92162 + 3.32834i) q^{30} -0.791864 q^{31} +(-0.500000 - 0.866025i) q^{32} +(8.10275 + 14.0344i) q^{33} +(-2.55137 - 4.41911i) q^{34} +(-1.92162 + 3.32834i) q^{35} +6.31088 q^{36} +(-1.37024 + 5.92642i) q^{37} +4.79186 q^{38} +(3.05137 - 5.28514i) q^{39} +(-0.629755 - 1.09077i) q^{40} +(0.104068 + 0.180251i) q^{41} +(4.65544 + 8.06346i) q^{42} -4.36226 q^{43} +(-2.65544 - 4.59936i) q^{44} +7.94863 q^{45} +(-0.870245 + 1.50731i) q^{46} -3.20814 q^{47} -3.05137 q^{48} +(-1.15544 + 2.00128i) q^{49} +(1.70682 + 2.95629i) q^{50} -15.5704 q^{51} +(-1.00000 + 1.73205i) q^{52} +(-5.65544 + 9.79551i) q^{53} +(5.05137 - 8.74924i) q^{54} +(-3.34456 - 5.79294i) q^{55} +(-1.52569 - 2.64257i) q^{56} +(7.31088 - 12.6628i) q^{57} +(2.02569 - 3.50859i) q^{58} +(6.36226 - 11.0198i) q^{59} -3.84324 q^{60} +(-3.02569 - 5.24064i) q^{61} +(0.395932 - 0.685774i) q^{62} +19.2569 q^{63} +1.00000 q^{64} +(-1.25951 + 2.18154i) q^{65} -16.2055 q^{66} +(4.65544 + 8.06346i) q^{67} +5.10275 q^{68} +(2.65544 + 4.59936i) q^{69} +(-1.92162 - 3.32834i) q^{70} +(3.52569 + 6.10667i) q^{71} +(-3.15544 + 5.46539i) q^{72} -3.31088 q^{73} +(-4.44731 - 4.14988i) q^{74} +10.4163 q^{75} +(-2.39593 + 4.14988i) q^{76} +(-8.10275 - 14.0344i) q^{77} +(3.05137 + 5.28514i) q^{78} +(5.70682 + 9.88450i) q^{79} +1.25951 q^{80} +(-5.94731 - 10.3010i) q^{81} -0.208136 q^{82} +(1.78520 - 3.09205i) q^{83} -9.31088 q^{84} +6.42697 q^{85} +(2.18113 - 3.77783i) q^{86} +(-6.18113 - 10.7060i) q^{87} +5.31088 q^{88} +(-1.63642 + 2.83436i) q^{89} +(-3.97431 + 6.88371i) q^{90} +(-3.05137 + 5.28514i) q^{91} +(-0.870245 - 1.50731i) q^{92} +(-1.20814 - 2.09255i) q^{93} +(1.60407 - 2.77833i) q^{94} +(-3.01770 + 5.22681i) q^{95} +(1.52569 - 2.64257i) q^{96} +2.20814 q^{97} +(-1.15544 - 2.00128i) q^{98} +(-16.7582 + 29.0260i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} - 3 q^{4} - q^{5} + 6 q^{8} - 7 q^{9} + 2 q^{10} + 8 q^{11} - 6 q^{13} - 4 q^{15} - 3 q^{16} + 3 q^{17} - 7 q^{18} - 8 q^{19} - q^{20} + 16 q^{21} - 4 q^{22} + 16 q^{23} - 20 q^{25} + 12 q^{26}+ \cdots - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 + 0.866025i −0.353553 + 0.612372i
\(3\) 1.52569 + 2.64257i 0.880856 + 1.52569i 0.850391 + 0.526151i \(0.176365\pi\)
0.0304649 + 0.999536i \(0.490301\pi\)
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −0.629755 − 1.09077i −0.281635 − 0.487806i 0.690153 − 0.723664i \(-0.257543\pi\)
−0.971788 + 0.235858i \(0.924210\pi\)
\(6\) −3.05137 −1.24572
\(7\) −1.52569 − 2.64257i −0.576656 − 0.998797i −0.995860 − 0.0909046i \(-0.971024\pi\)
0.419204 − 0.907892i \(-0.362309\pi\)
\(8\) 1.00000 0.353553
\(9\) −3.15544 + 5.46539i −1.05181 + 1.82180i
\(10\) 1.25951 0.398292
\(11\) 5.31088 1.60129 0.800646 − 0.599138i \(-0.204490\pi\)
0.800646 + 0.599138i \(0.204490\pi\)
\(12\) 1.52569 − 2.64257i 0.440428 − 0.762844i
\(13\) −1.00000 − 1.73205i −0.277350 − 0.480384i 0.693375 − 0.720577i \(-0.256123\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 3.05137 0.815514
\(15\) 1.92162 − 3.32834i 0.496160 − 0.859374i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −2.55137 + 4.41911i −0.618799 + 1.07179i 0.370906 + 0.928670i \(0.379047\pi\)
−0.989705 + 0.143121i \(0.954286\pi\)
\(18\) −3.15544 − 5.46539i −0.743745 − 1.28820i
\(19\) −2.39593 − 4.14988i −0.549664 − 0.952047i −0.998297 − 0.0583306i \(-0.981422\pi\)
0.448633 − 0.893716i \(-0.351911\pi\)
\(20\) −0.629755 + 1.09077i −0.140818 + 0.243903i
\(21\) 4.65544 − 8.06346i 1.01590 − 1.75959i
\(22\) −2.65544 + 4.59936i −0.566142 + 0.980587i
\(23\) 1.74049 0.362917 0.181459 − 0.983399i \(-0.441918\pi\)
0.181459 + 0.983399i \(0.441918\pi\)
\(24\) 1.52569 + 2.64257i 0.311430 + 0.539412i
\(25\) 1.70682 − 2.95629i 0.341363 − 0.591259i
\(26\) 2.00000 0.392232
\(27\) −10.1027 −1.94427
\(28\) −1.52569 + 2.64257i −0.288328 + 0.499398i
\(29\) −4.05137 −0.752321 −0.376161 − 0.926554i \(-0.622756\pi\)
−0.376161 + 0.926554i \(0.622756\pi\)
\(30\) 1.92162 + 3.32834i 0.350838 + 0.607669i
\(31\) −0.791864 −0.142223 −0.0711115 − 0.997468i \(-0.522655\pi\)
−0.0711115 + 0.997468i \(0.522655\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 8.10275 + 14.0344i 1.41051 + 2.44307i
\(34\) −2.55137 − 4.41911i −0.437557 − 0.757871i
\(35\) −1.92162 + 3.32834i −0.324813 + 0.562592i
\(36\) 6.31088 1.05181
\(37\) −1.37024 + 5.92642i −0.225267 + 0.974297i
\(38\) 4.79186 0.777343
\(39\) 3.05137 − 5.28514i 0.488611 − 0.846299i
\(40\) −0.629755 − 1.09077i −0.0995730 − 0.172466i
\(41\) 0.104068 + 0.180251i 0.0162527 + 0.0281505i 0.874037 − 0.485859i \(-0.161493\pi\)
−0.857785 + 0.514009i \(0.828160\pi\)
\(42\) 4.65544 + 8.06346i 0.718350 + 1.24422i
\(43\) −4.36226 −0.665238 −0.332619 − 0.943061i \(-0.607932\pi\)
−0.332619 + 0.943061i \(0.607932\pi\)
\(44\) −2.65544 − 4.59936i −0.400323 − 0.693380i
\(45\) 7.94863 1.18491
\(46\) −0.870245 + 1.50731i −0.128311 + 0.222240i
\(47\) −3.20814 −0.467955 −0.233977 − 0.972242i \(-0.575174\pi\)
−0.233977 + 0.972242i \(0.575174\pi\)
\(48\) −3.05137 −0.440428
\(49\) −1.15544 + 2.00128i −0.165063 + 0.285898i
\(50\) 1.70682 + 2.95629i 0.241380 + 0.418083i
\(51\) −15.5704 −2.18029
\(52\) −1.00000 + 1.73205i −0.138675 + 0.240192i
\(53\) −5.65544 + 9.79551i −0.776835 + 1.34552i 0.156923 + 0.987611i \(0.449843\pi\)
−0.933757 + 0.357906i \(0.883491\pi\)
\(54\) 5.05137 − 8.74924i 0.687405 − 1.19062i
\(55\) −3.34456 − 5.79294i −0.450980 − 0.781120i
\(56\) −1.52569 − 2.64257i −0.203879 − 0.353128i
\(57\) 7.31088 − 12.6628i 0.968350 − 1.67723i
\(58\) 2.02569 − 3.50859i 0.265986 − 0.460701i
\(59\) 6.36226 − 11.0198i 0.828296 − 1.43465i −0.0710788 − 0.997471i \(-0.522644\pi\)
0.899374 − 0.437179i \(-0.144022\pi\)
\(60\) −3.84324 −0.496160
\(61\) −3.02569 − 5.24064i −0.387400 − 0.670996i 0.604699 − 0.796454i \(-0.293293\pi\)
−0.992099 + 0.125458i \(0.959960\pi\)
\(62\) 0.395932 − 0.685774i 0.0502834 − 0.0870934i
\(63\) 19.2569 2.42614
\(64\) 1.00000 0.125000
\(65\) −1.25951 + 2.18154i −0.156223 + 0.270586i
\(66\) −16.2055 −1.99476
\(67\) 4.65544 + 8.06346i 0.568753 + 0.985109i 0.996690 + 0.0813001i \(0.0259072\pi\)
−0.427937 + 0.903809i \(0.640759\pi\)
\(68\) 5.10275 0.618799
\(69\) 2.65544 + 4.59936i 0.319678 + 0.553698i
\(70\) −1.92162 − 3.32834i −0.229677 − 0.397813i
\(71\) 3.52569 + 6.10667i 0.418422 + 0.724728i 0.995781 − 0.0917622i \(-0.0292500\pi\)
−0.577359 + 0.816491i \(0.695917\pi\)
\(72\) −3.15544 + 5.46539i −0.371872 + 0.644102i
\(73\) −3.31088 −0.387510 −0.193755 − 0.981050i \(-0.562067\pi\)
−0.193755 + 0.981050i \(0.562067\pi\)
\(74\) −4.44731 − 4.14988i −0.516989 − 0.482413i
\(75\) 10.4163 1.20277
\(76\) −2.39593 + 4.14988i −0.274832 + 0.476023i
\(77\) −8.10275 − 14.0344i −0.923394 − 1.59937i
\(78\) 3.05137 + 5.28514i 0.345500 + 0.598424i
\(79\) 5.70682 + 9.88450i 0.642067 + 1.11209i 0.984971 + 0.172722i \(0.0552563\pi\)
−0.342904 + 0.939371i \(0.611410\pi\)
\(80\) 1.25951 0.140818
\(81\) −5.94731 − 10.3010i −0.660812 − 1.14456i
\(82\) −0.208136 −0.0229848
\(83\) 1.78520 − 3.09205i 0.195951 − 0.339397i −0.751261 − 0.660005i \(-0.770554\pi\)
0.947212 + 0.320608i \(0.103887\pi\)
\(84\) −9.31088 −1.01590
\(85\) 6.42697 0.697102
\(86\) 2.18113 − 3.77783i 0.235197 − 0.407374i
\(87\) −6.18113 − 10.7060i −0.662687 − 1.14781i
\(88\) 5.31088 0.566142
\(89\) −1.63642 + 2.83436i −0.173460 + 0.300442i −0.939627 − 0.342199i \(-0.888828\pi\)
0.766167 + 0.642641i \(0.222162\pi\)
\(90\) −3.97431 + 6.88371i −0.418929 + 0.725607i
\(91\) −3.05137 + 5.28514i −0.319871 + 0.554033i
\(92\) −0.870245 − 1.50731i −0.0907293 − 0.157148i
\(93\) −1.20814 − 2.09255i −0.125278 − 0.216988i
\(94\) 1.60407 − 2.77833i 0.165447 − 0.286563i
\(95\) −3.01770 + 5.22681i −0.309610 + 0.536260i
\(96\) 1.52569 − 2.64257i 0.155715 − 0.269706i
\(97\) 2.20814 0.224202 0.112101 − 0.993697i \(-0.464242\pi\)
0.112101 + 0.993697i \(0.464242\pi\)
\(98\) −1.15544 − 2.00128i −0.116717 − 0.202160i
\(99\) −16.7582 + 29.0260i −1.68426 + 2.91723i
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.c.c.47.3 ✓ 6
3.2 odd 2 666.2.f.j.343.2 6
4.3 odd 2 592.2.i.e.417.1 6
37.10 even 3 2738.2.a.o.1.1 3
37.26 even 3 inner 74.2.c.c.63.3 yes 6
37.27 even 6 2738.2.a.n.1.1 3
111.26 odd 6 666.2.f.j.433.2 6
148.63 odd 6 592.2.i.e.433.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.c.47.3 ✓ 6 1.1 even 1 trivial
74.2.c.c.63.3 yes 6 37.26 even 3 inner
592.2.i.e.417.1 6 4.3 odd 2
592.2.i.e.433.1 6 148.63 odd 6
666.2.f.j.343.2 6 3.2 odd 2
666.2.f.j.433.2 6 111.26 odd 6
2738.2.a.n.1.1 3 37.27 even 6
2738.2.a.o.1.1 3 37.10 even 3