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(11,74)] 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("74.11"); 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([5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,2,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 11.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 74.11
Dual form 74.2.e.a.27.1

$q$-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+(-0.866025 + 0.500000i) q^{2} +(-1.36603 + 2.36603i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-1.50000 - 0.866025i) q^{5} -2.73205i q^{6} +(-2.00000 + 3.46410i) q^{7} +1.00000i q^{8} +(-2.23205 - 3.86603i) q^{9} +1.73205 q^{10} +1.26795 q^{11} +(1.36603 + 2.36603i) q^{12} +(5.19615 + 3.00000i) q^{13} -4.00000i q^{14} +(4.09808 - 2.36603i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 + 0.866025i) q^{17} +(3.86603 + 2.23205i) q^{18} +(4.09808 + 2.36603i) q^{19} +(-1.50000 + 0.866025i) q^{20} +(-5.46410 - 9.46410i) q^{21} +(-1.09808 + 0.633975i) q^{22} -4.73205i q^{23} +(-2.36603 - 1.36603i) q^{24} +(-1.00000 - 1.73205i) q^{25} -6.00000 q^{26} +4.00000 q^{27} +(2.00000 + 3.46410i) q^{28} +7.73205i q^{29} +(-2.36603 + 4.09808i) q^{30} +4.73205i q^{31} +(0.866025 + 0.500000i) q^{32} +(-1.73205 + 3.00000i) q^{33} +(0.866025 - 1.50000i) q^{34} +(6.00000 - 3.46410i) q^{35} -4.46410 q^{36} +(-0.500000 - 6.06218i) q^{37} -4.73205 q^{38} +(-14.1962 + 8.19615i) q^{39} +(0.866025 - 1.50000i) q^{40} +(3.23205 - 5.59808i) q^{41} +(9.46410 + 5.46410i) q^{42} -2.53590i q^{43} +(0.633975 - 1.09808i) q^{44} +7.73205i q^{45} +(2.36603 + 4.09808i) q^{46} -5.66025 q^{47} +2.73205 q^{48} +(-4.50000 - 7.79423i) q^{49} +(1.73205 + 1.00000i) q^{50} -4.73205i q^{51} +(5.19615 - 3.00000i) q^{52} +(4.73205 + 8.19615i) q^{53} +(-3.46410 + 2.00000i) q^{54} +(-1.90192 - 1.09808i) q^{55} +(-3.46410 - 2.00000i) q^{56} +(-11.1962 + 6.46410i) q^{57} +(-3.86603 - 6.69615i) q^{58} +(8.19615 - 4.73205i) q^{59} -4.73205i q^{60} +(2.30385 + 1.33013i) q^{61} +(-2.36603 - 4.09808i) q^{62} +17.8564 q^{63} -1.00000 q^{64} +(-5.19615 - 9.00000i) q^{65} -3.46410i q^{66} +(2.09808 - 3.63397i) q^{67} +1.73205i q^{68} +(11.1962 + 6.46410i) q^{69} +(-3.46410 + 6.00000i) q^{70} +(-4.73205 + 8.19615i) q^{71} +(3.86603 - 2.23205i) q^{72} +8.39230 q^{73} +(3.46410 + 5.00000i) q^{74} +5.46410 q^{75} +(4.09808 - 2.36603i) q^{76} +(-2.53590 + 4.39230i) q^{77} +(8.19615 - 14.1962i) q^{78} +(-1.90192 - 1.09808i) q^{79} +1.73205i q^{80} +(1.23205 - 2.13397i) q^{81} +6.46410i q^{82} +(2.83013 + 4.90192i) q^{83} -10.9282 q^{84} +3.00000 q^{85} +(1.26795 + 2.19615i) q^{86} +(-18.2942 - 10.5622i) q^{87} +1.26795i q^{88} +(4.50000 - 2.59808i) q^{89} +(-3.86603 - 6.69615i) q^{90} +(-20.7846 + 12.0000i) q^{91} +(-4.09808 - 2.36603i) q^{92} +(-11.1962 - 6.46410i) q^{93} +(4.90192 - 2.83013i) q^{94} +(-4.09808 - 7.09808i) q^{95} +(-2.36603 + 1.36603i) q^{96} +5.19615i q^{97} +(7.79423 + 4.50000i) q^{98} +(-2.83013 - 4.90192i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} - 6 q^{5} - 8 q^{7} - 2 q^{9} + 12 q^{11} + 2 q^{12} + 6 q^{15} - 2 q^{16} - 6 q^{17} + 12 q^{18} + 6 q^{19} - 6 q^{20} - 8 q^{21} + 6 q^{22} - 6 q^{24} - 4 q^{25} - 24 q^{26} + 16 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) −1.36603 + 2.36603i −0.788675 + 1.36603i 0.138104 + 0.990418i \(0.455899\pi\)
−0.926779 + 0.375608i \(0.877434\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −1.50000 − 0.866025i −0.670820 − 0.387298i 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) − 2.73205i − 1.11536i
\(7\) −2.00000 + 3.46410i −0.755929 + 1.30931i 0.188982 + 0.981981i \(0.439481\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −2.23205 − 3.86603i −0.744017 − 1.28868i
\(10\) 1.73205 0.547723
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 1.36603 + 2.36603i 0.394338 + 0.683013i
\(13\) 5.19615 + 3.00000i 1.44115 + 0.832050i 0.997927 − 0.0643593i \(-0.0205004\pi\)
0.443227 + 0.896410i \(0.353834\pi\)
\(14\) − 4.00000i − 1.06904i
\(15\) 4.09808 − 2.36603i 1.05812 − 0.610905i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.50000 + 0.866025i −0.363803 + 0.210042i −0.670748 − 0.741685i \(-0.734027\pi\)
0.306944 + 0.951727i \(0.400693\pi\)
\(18\) 3.86603 + 2.23205i 0.911231 + 0.526099i
\(19\) 4.09808 + 2.36603i 0.940163 + 0.542803i 0.890011 − 0.455938i \(-0.150696\pi\)
0.0501517 + 0.998742i \(0.484030\pi\)
\(20\) −1.50000 + 0.866025i −0.335410 + 0.193649i
\(21\) −5.46410 − 9.46410i −1.19236 − 2.06524i
\(22\) −1.09808 + 0.633975i −0.234111 + 0.135164i
\(23\) − 4.73205i − 0.986701i −0.869831 − 0.493350i \(-0.835772\pi\)
0.869831 − 0.493350i \(-0.164228\pi\)
\(24\) −2.36603 − 1.36603i −0.482963 − 0.278839i
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) −6.00000 −1.17670
\(27\) 4.00000 0.769800
\(28\) 2.00000 + 3.46410i 0.377964 + 0.654654i
\(29\) 7.73205i 1.43581i 0.696143 + 0.717903i \(0.254898\pi\)
−0.696143 + 0.717903i \(0.745102\pi\)
\(30\) −2.36603 + 4.09808i −0.431975 + 0.748203i
\(31\) 4.73205i 0.849901i 0.905216 + 0.424951i \(0.139709\pi\)
−0.905216 + 0.424951i \(0.860291\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) −1.73205 + 3.00000i −0.301511 + 0.522233i
\(34\) 0.866025 − 1.50000i 0.148522 − 0.257248i
\(35\) 6.00000 − 3.46410i 1.01419 − 0.585540i
\(36\) −4.46410 −0.744017
\(37\) −0.500000 − 6.06218i −0.0821995 − 0.996616i
\(38\) −4.73205 −0.767640
\(39\) −14.1962 + 8.19615i −2.27320 + 1.31243i
\(40\) 0.866025 − 1.50000i 0.136931 − 0.237171i
\(41\) 3.23205 − 5.59808i 0.504762 − 0.874273i −0.495223 − 0.868766i \(-0.664914\pi\)
0.999985 − 0.00550690i \(-0.00175291\pi\)
\(42\) 9.46410 + 5.46410i 1.46034 + 0.843129i
\(43\) − 2.53590i − 0.386721i −0.981128 − 0.193360i \(-0.938061\pi\)
0.981128 − 0.193360i \(-0.0619387\pi\)
\(44\) 0.633975 − 1.09808i 0.0955753 − 0.165541i
\(45\) 7.73205i 1.15263i
\(46\) 2.36603 + 4.09808i 0.348851 + 0.604228i
\(47\) −5.66025 −0.825633 −0.412816 − 0.910814i \(-0.635455\pi\)
−0.412816 + 0.910814i \(0.635455\pi\)
\(48\) 2.73205 0.394338
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) 1.73205 + 1.00000i 0.244949 + 0.141421i
\(51\) − 4.73205i − 0.662620i
\(52\) 5.19615 − 3.00000i 0.720577 − 0.416025i
\(53\) 4.73205 + 8.19615i 0.649997 + 1.12583i 0.983123 + 0.182946i \(0.0585633\pi\)
−0.333126 + 0.942882i \(0.608103\pi\)
\(54\) −3.46410 + 2.00000i −0.471405 + 0.272166i
\(55\) −1.90192 − 1.09808i −0.256455 − 0.148065i
\(56\) −3.46410 − 2.00000i −0.462910 − 0.267261i
\(57\) −11.1962 + 6.46410i −1.48297 + 0.856191i
\(58\) −3.86603 − 6.69615i −0.507634 − 0.879248i
\(59\) 8.19615 − 4.73205i 1.06705 − 0.616061i 0.139675 − 0.990197i \(-0.455394\pi\)
0.927373 + 0.374137i \(0.122061\pi\)
\(60\) − 4.73205i − 0.610905i
\(61\) 2.30385 + 1.33013i 0.294977 + 0.170305i 0.640184 − 0.768221i \(-0.278858\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) −2.36603 − 4.09808i −0.300486 − 0.520456i
\(63\) 17.8564 2.24970
\(64\) −1.00000 −0.125000
\(65\) −5.19615 − 9.00000i −0.644503 − 1.11631i
\(66\) − 3.46410i − 0.426401i
\(67\) 2.09808 − 3.63397i 0.256321 − 0.443961i −0.708933 − 0.705276i \(-0.750823\pi\)
0.965253 + 0.261316i \(0.0841563\pi\)
\(68\) 1.73205i 0.210042i
\(69\) 11.1962 + 6.46410i 1.34786 + 0.778186i
\(70\) −3.46410 + 6.00000i −0.414039 + 0.717137i
\(71\) −4.73205 + 8.19615i −0.561591 + 0.972704i 0.435767 + 0.900060i \(0.356477\pi\)
−0.997358 + 0.0726447i \(0.976856\pi\)
\(72\) 3.86603 − 2.23205i 0.455615 − 0.263050i
\(73\) 8.39230 0.982245 0.491122 − 0.871091i \(-0.336587\pi\)
0.491122 + 0.871091i \(0.336587\pi\)
\(74\) 3.46410 + 5.00000i 0.402694 + 0.581238i
\(75\) 5.46410 0.630940
\(76\) 4.09808 − 2.36603i 0.470082 − 0.271402i
\(77\) −2.53590 + 4.39230i −0.288992 + 0.500550i
\(78\) 8.19615 − 14.1962i 0.928032 − 1.60740i
\(79\) −1.90192 − 1.09808i −0.213983 − 0.123543i 0.389178 − 0.921163i \(-0.372759\pi\)
−0.603161 + 0.797619i \(0.706092\pi\)
\(80\) 1.73205i 0.193649i
\(81\) 1.23205 − 2.13397i 0.136895 − 0.237108i
\(82\) 6.46410i 0.713841i
\(83\) 2.83013 + 4.90192i 0.310647 + 0.538056i 0.978503 − 0.206235i \(-0.0661210\pi\)
−0.667856 + 0.744291i \(0.732788\pi\)
\(84\) −10.9282 −1.19236
\(85\) 3.00000 0.325396
\(86\) 1.26795 + 2.19615i 0.136726 + 0.236817i
\(87\) −18.2942 − 10.5622i −1.96135 − 1.13238i
\(88\) 1.26795i 0.135164i
\(89\) 4.50000 − 2.59808i 0.476999 − 0.275396i −0.242166 − 0.970235i \(-0.577858\pi\)
0.719165 + 0.694839i \(0.244525\pi\)
\(90\) −3.86603 − 6.69615i −0.407515 − 0.705836i
\(91\) −20.7846 + 12.0000i −2.17882 + 1.25794i
\(92\) −4.09808 − 2.36603i −0.427254 − 0.246675i
\(93\) −11.1962 − 6.46410i −1.16099 − 0.670296i
\(94\) 4.90192 − 2.83013i 0.505595 − 0.291905i
\(95\) −4.09808 − 7.09808i −0.420454 − 0.728247i
\(96\) −2.36603 + 1.36603i −0.241481 + 0.139419i
\(97\) 5.19615i 0.527589i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 7.79423 + 4.50000i 0.787336 + 0.454569i
\(99\) −2.83013 − 4.90192i −0.284438 − 0.492662i
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.e.a.11.1 ✓ 4
3.2 odd 2 666.2.s.e.307.2 4
4.3 odd 2 592.2.w.d.529.2 4
37.8 odd 12 2738.2.a.f.1.1 2
37.27 even 6 inner 74.2.e.a.27.1 yes 4
37.29 odd 12 2738.2.a.j.1.1 2
111.101 odd 6 666.2.s.e.397.2 4
148.27 odd 6 592.2.w.d.545.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.1 ✓ 4 1.1 even 1 trivial
74.2.e.a.27.1 yes 4 37.27 even 6 inner
592.2.w.d.529.2 4 4.3 odd 2
592.2.w.d.545.2 4 148.27 odd 6
666.2.s.e.307.2 4 3.2 odd 2
666.2.s.e.397.2 4 111.101 odd 6
2738.2.a.f.1.1 2 37.8 odd 12
2738.2.a.j.1.1 2 37.29 odd 12