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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 11.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 74.11
Dual form 74.2.e.a.27.2

$q$-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+(0.866025 - 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} +(-2.00000 + 3.46410i) q^{7} -1.00000i q^{8} +(1.23205 + 2.13397i) q^{9} -1.73205 q^{10} +4.73205 q^{11} +(-0.366025 - 0.633975i) q^{12} +(-5.19615 - 3.00000i) q^{13} +4.00000i q^{14} +(-1.09808 + 0.633975i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 + 0.866025i) q^{17} +(2.13397 + 1.23205i) q^{18} +(-1.09808 - 0.633975i) q^{19} +(-1.50000 + 0.866025i) q^{20} +(1.46410 + 2.53590i) q^{21} +(4.09808 - 2.36603i) q^{22} +1.26795i q^{23} +(-0.633975 - 0.366025i) q^{24} +(-1.00000 - 1.73205i) q^{25} -6.00000 q^{26} +4.00000 q^{27} +(2.00000 + 3.46410i) q^{28} -4.26795i q^{29} +(-0.633975 + 1.09808i) q^{30} -1.26795i 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} +2.46410 q^{36} +(-0.500000 - 6.06218i) q^{37} -1.26795 q^{38} +(-3.80385 + 2.19615i) q^{39} +(-0.866025 + 1.50000i) q^{40} +(-0.232051 + 0.401924i) q^{41} +(2.53590 + 1.46410i) q^{42} +9.46410i q^{43} +(2.36603 - 4.09808i) q^{44} -4.26795i q^{45} +(0.633975 + 1.09808i) q^{46} +11.6603 q^{47} -0.732051 q^{48} +(-4.50000 - 7.79423i) q^{49} +(-1.73205 - 1.00000i) q^{50} +1.26795i q^{51} +(-5.19615 + 3.00000i) q^{52} +(1.26795 + 2.19615i) q^{53} +(3.46410 - 2.00000i) q^{54} +(-7.09808 - 4.09808i) q^{55} +(3.46410 + 2.00000i) q^{56} +(-0.803848 + 0.464102i) q^{57} +(-2.13397 - 3.69615i) q^{58} +(-2.19615 + 1.26795i) q^{59} +1.26795i q^{60} +(12.6962 + 7.33013i) q^{61} +(-0.633975 - 1.09808i) q^{62} -9.85641 q^{63} -1.00000 q^{64} +(5.19615 + 9.00000i) q^{65} -3.46410i q^{66} +(-3.09808 + 5.36603i) q^{67} +1.73205i q^{68} +(0.803848 + 0.464102i) q^{69} +(3.46410 - 6.00000i) q^{70} +(-1.26795 + 2.19615i) q^{71} +(2.13397 - 1.23205i) q^{72} -12.3923 q^{73} +(-3.46410 - 5.00000i) q^{74} -1.46410 q^{75} +(-1.09808 + 0.633975i) q^{76} +(-9.46410 + 16.3923i) q^{77} +(-2.19615 + 3.80385i) q^{78} +(-7.09808 - 4.09808i) q^{79} +1.73205i q^{80} +(-2.23205 + 3.86603i) q^{81} +0.464102i q^{82} +(-5.83013 - 10.0981i) q^{83} +2.92820 q^{84} +3.00000 q^{85} +(4.73205 + 8.19615i) q^{86} +(-2.70577 - 1.56218i) q^{87} -4.73205i q^{88} +(4.50000 - 2.59808i) q^{89} +(-2.13397 - 3.69615i) q^{90} +(20.7846 - 12.0000i) q^{91} +(1.09808 + 0.633975i) q^{92} +(-0.803848 - 0.464102i) q^{93} +(10.0981 - 5.83013i) q^{94} +(1.09808 + 1.90192i) q^{95} +(-0.633975 + 0.366025i) q^{96} +5.19615i q^{97} +(-7.79423 - 4.50000i) q^{98} +(5.83013 + 10.0981i) 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
\(3\) 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 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) − 0.732051i − 0.298858i
\(7\) −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
\(9\) 1.23205 + 2.13397i 0.410684 + 0.711325i
\(10\) −1.73205 −0.547723
\(11\) 4.73205 1.42677 0.713384 − 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) −0.366025 − 0.633975i −0.105662 − 0.183013i
\(13\) −5.19615 − 3.00000i −1.44115 − 0.832050i −0.443227 − 0.896410i \(-0.646166\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 4.00000i 1.06904i
\(15\) −1.09808 + 0.633975i −0.283522 + 0.163692i
\(16\) −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 \(0.400693\pi\)
\(18\) 2.13397 + 1.23205i 0.502983 + 0.290397i
\(19\) −1.09808 − 0.633975i −0.251916 − 0.145444i 0.368725 − 0.929538i \(-0.379794\pi\)
−0.620641 + 0.784095i \(0.713128\pi\)
\(20\) −1.50000 + 0.866025i −0.335410 + 0.193649i
\(21\) 1.46410 + 2.53590i 0.319493 + 0.553378i
\(22\) 4.09808 − 2.36603i 0.873713 − 0.504438i
\(23\) 1.26795i 0.264386i 0.991224 + 0.132193i \(0.0422018\pi\)
−0.991224 + 0.132193i \(0.957798\pi\)
\(24\) −0.633975 − 0.366025i −0.129410 − 0.0747146i
\(25\) −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
\(29\) − 4.26795i − 0.792538i −0.918134 − 0.396269i \(-0.870305\pi\)
0.918134 − 0.396269i \(-0.129695\pi\)
\(30\) −0.633975 + 1.09808i −0.115747 + 0.200480i
\(31\) − 1.26795i − 0.227730i −0.993496 − 0.113865i \(-0.963677\pi\)
0.993496 − 0.113865i \(-0.0363232\pi\)
\(32\) −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
\(36\) 2.46410 0.410684
\(37\) −0.500000 − 6.06218i −0.0821995 − 0.996616i
\(38\) −1.26795 −0.205689
\(39\) −3.80385 + 2.19615i −0.609103 + 0.351666i
\(40\) −0.866025 + 1.50000i −0.136931 + 0.237171i
\(41\) −0.232051 + 0.401924i −0.0362402 + 0.0627700i −0.883577 − 0.468287i \(-0.844871\pi\)
0.847336 + 0.531057i \(0.178205\pi\)
\(42\) 2.53590 + 1.46410i 0.391298 + 0.225916i
\(43\) 9.46410i 1.44326i 0.692278 + 0.721631i \(0.256607\pi\)
−0.692278 + 0.721631i \(0.743393\pi\)
\(44\) 2.36603 − 4.09808i 0.356692 − 0.617808i
\(45\) − 4.26795i − 0.636228i
\(46\) 0.633975 + 1.09808i 0.0934745 + 0.161903i
\(47\) 11.6603 1.70082 0.850411 − 0.526118i \(-0.176353\pi\)
0.850411 + 0.526118i \(0.176353\pi\)
\(48\) −0.732051 −0.105662
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) −1.73205 − 1.00000i −0.244949 − 0.141421i
\(51\) 1.26795i 0.177548i
\(52\) −5.19615 + 3.00000i −0.720577 + 0.416025i
\(53\) 1.26795 + 2.19615i 0.174166 + 0.301665i 0.939872 − 0.341526i \(-0.110944\pi\)
−0.765706 + 0.643191i \(0.777610\pi\)
\(54\) 3.46410 − 2.00000i 0.471405 − 0.272166i
\(55\) −7.09808 − 4.09808i −0.957104 − 0.552584i
\(56\) 3.46410 + 2.00000i 0.462910 + 0.267261i
\(57\) −0.803848 + 0.464102i −0.106472 + 0.0614718i
\(58\) −2.13397 − 3.69615i −0.280205 − 0.485329i
\(59\) −2.19615 + 1.26795i −0.285915 + 0.165073i −0.636098 − 0.771608i \(-0.719453\pi\)
0.350183 + 0.936681i \(0.386119\pi\)
\(60\) 1.26795i 0.163692i
\(61\) 12.6962 + 7.33013i 1.62558 + 0.938527i 0.985391 + 0.170305i \(0.0544754\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −0.633975 − 1.09808i −0.0805149 − 0.139456i
\(63\) −9.85641 −1.24179
\(64\) −1.00000 −0.125000
\(65\) 5.19615 + 9.00000i 0.644503 + 1.11631i
\(66\) − 3.46410i − 0.426401i
\(67\) −3.09808 + 5.36603i −0.378490 + 0.655564i −0.990843 − 0.135020i \(-0.956890\pi\)
0.612353 + 0.790585i \(0.290223\pi\)
\(68\) 1.73205i 0.210042i
\(69\) 0.803848 + 0.464102i 0.0967719 + 0.0558713i
\(70\) 3.46410 − 6.00000i 0.414039 − 0.717137i
\(71\) −1.26795 + 2.19615i −0.150478 + 0.260635i −0.931403 − 0.363989i \(-0.881415\pi\)
0.780925 + 0.624624i \(0.214748\pi\)
\(72\) 2.13397 − 1.23205i 0.251491 − 0.145199i
\(73\) −12.3923 −1.45041 −0.725205 − 0.688533i \(-0.758255\pi\)
−0.725205 + 0.688533i \(0.758255\pi\)
\(74\) −3.46410 − 5.00000i −0.402694 − 0.581238i
\(75\) −1.46410 −0.169060
\(76\) −1.09808 + 0.633975i −0.125958 + 0.0727219i
\(77\) −9.46410 + 16.3923i −1.07853 + 1.86808i
\(78\) −2.19615 + 3.80385i −0.248665 + 0.430701i
\(79\) −7.09808 − 4.09808i −0.798596 − 0.461070i 0.0443840 − 0.999015i \(-0.485867\pi\)
−0.842980 + 0.537945i \(0.819201\pi\)
\(80\) 1.73205i 0.193649i
\(81\) −2.23205 + 3.86603i −0.248006 + 0.429558i
\(82\) 0.464102i 0.0512514i
\(83\) −5.83013 − 10.0981i −0.639940 − 1.10841i −0.985446 − 0.169991i \(-0.945626\pi\)
0.345506 − 0.938417i \(-0.387707\pi\)
\(84\) 2.92820 0.319493
\(85\) 3.00000 0.325396
\(86\) 4.73205 + 8.19615i 0.510270 + 0.883814i
\(87\) −2.70577 − 1.56218i −0.290089 − 0.167483i
\(88\) − 4.73205i − 0.504438i
\(89\) 4.50000 − 2.59808i 0.476999 − 0.275396i −0.242166 − 0.970235i \(-0.577858\pi\)
0.719165 + 0.694839i \(0.244525\pi\)
\(90\) −2.13397 − 3.69615i −0.224941 − 0.389609i
\(91\) 20.7846 − 12.0000i 2.17882 − 1.25794i
\(92\) 1.09808 + 0.633975i 0.114482 + 0.0660964i
\(93\) −0.803848 − 0.464102i −0.0833551 − 0.0481251i
\(94\) 10.0981 − 5.83013i 1.04154 − 0.601332i
\(95\) 1.09808 + 1.90192i 0.112660 + 0.195133i
\(96\) −0.633975 + 0.366025i −0.0647048 + 0.0373573i
\(97\) 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
\(99\) 5.83013 + 10.0981i 0.585950 + 1.01489i
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.11.2 ✓ 4
3.2 odd 2 666.2.s.e.307.1 4
4.3 odd 2 592.2.w.d.529.1 4
37.8 odd 12 2738.2.a.j.1.2 2
37.27 even 6 inner 74.2.e.a.27.2 yes 4
37.29 odd 12 2738.2.a.f.1.2 2
111.101 odd 6 666.2.s.e.397.1 4
148.27 odd 6 592.2.w.d.545.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.2 ✓ 4 1.1 even 1 trivial
74.2.e.a.27.2 yes 4 37.27 even 6 inner
592.2.w.d.529.1 4 4.3 odd 2
592.2.w.d.545.1 4 148.27 odd 6
666.2.s.e.307.1 4 3.2 odd 2
666.2.s.e.397.1 4 111.101 odd 6
2738.2.a.f.1.2 2 37.29 odd 12
2738.2.a.j.1.2 2 37.8 odd 12