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Properties

Label 74.2.e.b.11.1
Level $74$
Weight $2$
Character 74.11
Analytic conductor $0.591$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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.b.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} +(0.366025 - 0.633975i) q^{3} +(0.500000 - 0.866025i) q^{4} +(1.50000 + 0.866025i) q^{5} +0.732051i q^{6} +(1.00000 - 1.73205i) q^{7} +1.00000i q^{8} +(1.23205 + 2.13397i) q^{9} -1.73205 q^{10} -1.26795 q^{11} +(-0.366025 - 0.633975i) q^{12} +(-3.00000 - 1.73205i) q^{13} +2.00000i q^{14} +(1.09808 - 0.633975i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-3.69615 + 2.13397i) q^{17} +(-2.13397 - 1.23205i) q^{18} +(-4.09808 - 2.36603i) q^{19} +(1.50000 - 0.866025i) q^{20} +(-0.732051 - 1.26795i) q^{21} +(1.09808 - 0.633975i) q^{22} -1.26795i q^{23} +(0.633975 + 0.366025i) q^{24} +(-1.00000 - 1.73205i) q^{25} +3.46410 q^{26} +4.00000 q^{27} +(-1.00000 - 1.73205i) q^{28} +8.66025i q^{29} +(-0.633975 + 1.09808i) q^{30} +4.73205i q^{31} +(0.866025 + 0.500000i) q^{32} +(-0.464102 + 0.803848i) q^{33} +(2.13397 - 3.69615i) q^{34} +(3.00000 - 1.73205i) q^{35} +2.46410 q^{36} +(4.69615 + 3.86603i) q^{37} +4.73205 q^{38} +(-2.19615 + 1.26795i) q^{39} +(-0.866025 + 1.50000i) q^{40} +(1.96410 - 3.40192i) q^{41} +(1.26795 + 0.732051i) q^{42} -12.9282i q^{43} +(-0.633975 + 1.09808i) q^{44} +4.26795i q^{45} +(0.633975 + 1.09808i) q^{46} +1.26795 q^{47} -0.732051 q^{48} +(1.50000 + 2.59808i) q^{49} +(1.73205 + 1.00000i) q^{50} +3.12436i q^{51} +(-3.00000 + 1.73205i) q^{52} +(-4.73205 - 8.19615i) q^{53} +(-3.46410 + 2.00000i) q^{54} +(-1.90192 - 1.09808i) q^{55} +(1.73205 + 1.00000i) q^{56} +(-3.00000 + 1.73205i) q^{57} +(-4.33013 - 7.50000i) q^{58} +(-8.19615 + 4.73205i) q^{59} -1.26795i q^{60} +(1.50000 + 0.866025i) q^{61} +(-2.36603 - 4.09808i) q^{62} +4.92820 q^{63} -1.00000 q^{64} +(-3.00000 - 5.19615i) q^{65} -0.928203i q^{66} +(-0.0980762 + 0.169873i) q^{67} +4.26795i q^{68} +(-0.803848 - 0.464102i) q^{69} +(-1.73205 + 3.00000i) q^{70} +(1.73205 - 3.00000i) q^{71} +(-2.13397 + 1.23205i) q^{72} +4.00000 q^{73} +(-6.00000 - 1.00000i) q^{74} -1.46410 q^{75} +(-4.09808 + 2.36603i) q^{76} +(-1.26795 + 2.19615i) q^{77} +(1.26795 - 2.19615i) q^{78} +(14.4904 + 8.36603i) q^{79} -1.73205i q^{80} +(-2.23205 + 3.86603i) q^{81} +3.92820i q^{82} +(-5.83013 - 10.0981i) q^{83} -1.46410 q^{84} -7.39230 q^{85} +(6.46410 + 11.1962i) q^{86} +(5.49038 + 3.16987i) q^{87} -1.26795i q^{88} +(14.8923 - 8.59808i) q^{89} +(-2.13397 - 3.69615i) q^{90} +(-6.00000 + 3.46410i) q^{91} +(-1.09808 - 0.633975i) q^{92} +(3.00000 + 1.73205i) q^{93} +(-1.09808 + 0.633975i) q^{94} +(-4.09808 - 7.09808i) q^{95} +(0.633975 - 0.366025i) q^{96} -4.26795i q^{97} +(-2.59808 - 1.50000i) q^{98} +(-1.56218 - 2.70577i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{4} + 6 q^{5} + 4 q^{7} - 2 q^{9} - 12 q^{11} + 2 q^{12} - 12 q^{13} - 6 q^{15} - 2 q^{16} + 6 q^{17} - 12 q^{18} - 6 q^{19} + 6 q^{20} + 4 q^{21} - 6 q^{22} + 6 q^{24} - 4 q^{25} + 16 q^{27}+ \cdots + 18 q^{99}+O(q^{100}) \) 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
\(3\) 0.366025 − 0.633975i 0.211325 − 0.366025i −0.740805 − 0.671721i \(-0.765556\pi\)
0.952129 + 0.305695i \(0.0988889\pi\)
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 1.50000 + 0.866025i 0.670820 + 0.387298i 0.796387 − 0.604787i \(-0.206742\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0.732051i 0.298858i
\(7\) 1.00000 − 1.73205i 0.377964 − 0.654654i −0.612801 − 0.790237i \(-0.709957\pi\)
0.990766 + 0.135583i \(0.0432908\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.23205 + 2.13397i 0.410684 + 0.711325i
\(10\) −1.73205 −0.547723
\(11\) −1.26795 −0.382301 −0.191151 − 0.981561i \(-0.561222\pi\)
−0.191151 + 0.981561i \(0.561222\pi\)
\(12\) −0.366025 − 0.633975i −0.105662 − 0.183013i
\(13\) −3.00000 − 1.73205i −0.832050 − 0.480384i 0.0225039 − 0.999747i \(-0.492836\pi\)
−0.854554 + 0.519362i \(0.826170\pi\)
\(14\) 2.00000i 0.534522i
\(15\) 1.09808 − 0.633975i 0.283522 − 0.163692i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −3.69615 + 2.13397i −0.896449 + 0.517565i −0.876046 − 0.482227i \(-0.839828\pi\)
−0.0204023 + 0.999792i \(0.506495\pi\)
\(18\) −2.13397 − 1.23205i −0.502983 − 0.290397i
\(19\) −4.09808 − 2.36603i −0.940163 − 0.542803i −0.0501517 − 0.998742i \(-0.515970\pi\)
−0.890011 + 0.455938i \(0.849304\pi\)
\(20\) 1.50000 − 0.866025i 0.335410 − 0.193649i
\(21\) −0.732051 − 1.26795i −0.159747 − 0.276689i
\(22\) 1.09808 − 0.633975i 0.234111 − 0.135164i
\(23\) − 1.26795i − 0.264386i −0.991224 − 0.132193i \(-0.957798\pi\)
0.991224 − 0.132193i \(-0.0422018\pi\)
\(24\) 0.633975 + 0.366025i 0.129410 + 0.0747146i
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 3.46410 0.679366
\(27\) 4.00000 0.769800
\(28\) −1.00000 − 1.73205i −0.188982 − 0.327327i
\(29\) 8.66025i 1.60817i 0.594515 + 0.804084i \(0.297344\pi\)
−0.594515 + 0.804084i \(0.702656\pi\)
\(30\) −0.633975 + 1.09808i −0.115747 + 0.200480i
\(31\) 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
\(33\) −0.464102 + 0.803848i −0.0807897 + 0.139932i
\(34\) 2.13397 − 3.69615i 0.365974 − 0.633885i
\(35\) 3.00000 − 1.73205i 0.507093 − 0.292770i
\(36\) 2.46410 0.410684
\(37\) 4.69615 + 3.86603i 0.772043 + 0.635571i
\(38\) 4.73205 0.767640
\(39\) −2.19615 + 1.26795i −0.351666 + 0.203034i
\(40\) −0.866025 + 1.50000i −0.136931 + 0.237171i
\(41\) 1.96410 − 3.40192i 0.306741 − 0.531291i −0.670906 − 0.741542i \(-0.734095\pi\)
0.977647 + 0.210251i \(0.0674281\pi\)
\(42\) 1.26795 + 0.732051i 0.195649 + 0.112958i
\(43\) − 12.9282i − 1.97153i −0.168122 − 0.985766i \(-0.553770\pi\)
0.168122 − 0.985766i \(-0.446230\pi\)
\(44\) −0.633975 + 1.09808i −0.0955753 + 0.165541i
\(45\) 4.26795i 0.636228i
\(46\) 0.633975 + 1.09808i 0.0934745 + 0.161903i
\(47\) 1.26795 0.184949 0.0924747 − 0.995715i \(-0.470522\pi\)
0.0924747 + 0.995715i \(0.470522\pi\)
\(48\) −0.732051 −0.105662
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) 1.73205 + 1.00000i 0.244949 + 0.141421i
\(51\) 3.12436i 0.437497i
\(52\) −3.00000 + 1.73205i −0.416025 + 0.240192i
\(53\) −4.73205 − 8.19615i −0.649997 − 1.12583i −0.983123 − 0.182946i \(-0.941437\pi\)
0.333126 − 0.942882i \(-0.391897\pi\)
\(54\) −3.46410 + 2.00000i −0.471405 + 0.272166i
\(55\) −1.90192 − 1.09808i −0.256455 − 0.148065i
\(56\) 1.73205 + 1.00000i 0.231455 + 0.133631i
\(57\) −3.00000 + 1.73205i −0.397360 + 0.229416i
\(58\) −4.33013 − 7.50000i −0.568574 − 0.984798i
\(59\) −8.19615 + 4.73205i −1.06705 + 0.616061i −0.927373 − 0.374137i \(-0.877939\pi\)
−0.139675 + 0.990197i \(0.544606\pi\)
\(60\) − 1.26795i − 0.163692i
\(61\) 1.50000 + 0.866025i 0.192055 + 0.110883i 0.592944 − 0.805243i \(-0.297965\pi\)
−0.400889 + 0.916127i \(0.631299\pi\)
\(62\) −2.36603 − 4.09808i −0.300486 − 0.520456i
\(63\) 4.92820 0.620895
\(64\) −1.00000 −0.125000
\(65\) −3.00000 − 5.19615i −0.372104 − 0.644503i
\(66\) − 0.928203i − 0.114254i
\(67\) −0.0980762 + 0.169873i −0.0119819 + 0.0207533i −0.871954 − 0.489587i \(-0.837147\pi\)
0.859972 + 0.510341i \(0.170481\pi\)
\(68\) 4.26795i 0.517565i
\(69\) −0.803848 − 0.464102i −0.0967719 − 0.0558713i
\(70\) −1.73205 + 3.00000i −0.207020 + 0.358569i
\(71\) 1.73205 − 3.00000i 0.205557 − 0.356034i −0.744753 − 0.667340i \(-0.767433\pi\)
0.950310 + 0.311305i \(0.100766\pi\)
\(72\) −2.13397 + 1.23205i −0.251491 + 0.145199i
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −6.00000 − 1.00000i −0.697486 − 0.116248i
\(75\) −1.46410 −0.169060
\(76\) −4.09808 + 2.36603i −0.470082 + 0.271402i
\(77\) −1.26795 + 2.19615i −0.144496 + 0.250275i
\(78\) 1.26795 − 2.19615i 0.143567 − 0.248665i
\(79\) 14.4904 + 8.36603i 1.63030 + 0.941251i 0.984001 + 0.178163i \(0.0570155\pi\)
0.646294 + 0.763088i \(0.276318\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) −2.23205 + 3.86603i −0.248006 + 0.429558i
\(82\) 3.92820i 0.433797i
\(83\) −5.83013 − 10.0981i −0.639940 − 1.10841i −0.985446 − 0.169991i \(-0.945626\pi\)
0.345506 − 0.938417i \(-0.387707\pi\)
\(84\) −1.46410 −0.159747
\(85\) −7.39230 −0.801808
\(86\) 6.46410 + 11.1962i 0.697042 + 1.20731i
\(87\) 5.49038 + 3.16987i 0.588631 + 0.339846i
\(88\) − 1.26795i − 0.135164i
\(89\) 14.8923 − 8.59808i 1.57858 − 0.911394i 0.583523 − 0.812096i \(-0.301674\pi\)
0.995058 − 0.0992979i \(-0.0316597\pi\)
\(90\) −2.13397 − 3.69615i −0.224941 − 0.389609i
\(91\) −6.00000 + 3.46410i −0.628971 + 0.363137i
\(92\) −1.09808 − 0.633975i −0.114482 − 0.0660964i
\(93\) 3.00000 + 1.73205i 0.311086 + 0.179605i
\(94\) −1.09808 + 0.633975i −0.113258 + 0.0653895i
\(95\) −4.09808 − 7.09808i −0.420454 − 0.728247i
\(96\) 0.633975 − 0.366025i 0.0647048 − 0.0373573i
\(97\) − 4.26795i − 0.433345i −0.976244 − 0.216672i \(-0.930480\pi\)
0.976244 − 0.216672i \(-0.0695203\pi\)
\(98\) −2.59808 − 1.50000i −0.262445 − 0.151523i
\(99\) −1.56218 − 2.70577i −0.157005 − 0.271940i
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.b.11.1 ✓ 4
3.2 odd 2 666.2.s.a.307.2 4
4.3 odd 2 592.2.w.e.529.1 4
37.8 odd 12 2738.2.a.e.1.2 2
37.27 even 6 inner 74.2.e.b.27.1 yes 4
37.29 odd 12 2738.2.a.i.1.2 2
111.101 odd 6 666.2.s.a.397.2 4
148.27 odd 6 592.2.w.e.545.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.1 ✓ 4 1.1 even 1 trivial
74.2.e.b.27.1 yes 4 37.27 even 6 inner
592.2.w.e.529.1 4 4.3 odd 2
592.2.w.e.545.1 4 148.27 odd 6
666.2.s.a.307.2 4 3.2 odd 2
666.2.s.a.397.2 4 111.101 odd 6
2738.2.a.e.1.2 2 37.8 odd 12
2738.2.a.i.1.2 2 37.29 odd 12